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Metamath Proof Explorer |
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| Ref | Description |
| idi 1 | (_Note_: This inference r... |
| a1ii 2 | (_Note_: This inference r... |
| mp2 9 | A double modus ponens infe... |
| mp2b 10 | A double modus ponens infe... |
| a1i 11 | Inference introducing an a... |
| 2a1i 12 | Inference introducing two ... |
| mp1i 13 | Inference detaching an ant... |
| a2i 14 | Inference distributing an ... |
| mpd 15 | A modus ponens deduction. ... |
| imim2i 16 | Inference adding common an... |
| syl 17 | An inference version of th... |
| 3syl 18 | Inference chaining two syl... |
| 4syl 19 | Inference chaining three s... |
| mpi 20 | A nested modus ponens infe... |
| mpisyl 21 | A syllogism combined with ... |
| id 22 | Principle of identity. Th... |
| idALT 23 | Alternate proof of ~ id . ... |
| idd 24 | Principle of identity ~ id... |
| a1d 25 | Deduction introducing an e... |
| 2a1d 26 | Deduction introducing two ... |
| a1i13 27 | Add two antecedents to a w... |
| 2a1 28 | A double form of ~ ax-1 . ... |
| a2d 29 | Deduction distributing an ... |
| sylcom 30 | Syllogism inference with c... |
| syl5com 31 | Syllogism inference with c... |
| com12 32 | Inference that swaps (comm... |
| syl11 33 | A syllogism inference. Co... |
| syl5 34 | A syllogism rule of infere... |
| syl6 35 | A syllogism rule of infere... |
| syl56 36 | Combine ~ syl5 and ~ syl6 ... |
| syl6com 37 | Syllogism inference with c... |
| mpcom 38 | Modus ponens inference wit... |
| syli 39 | Syllogism inference with c... |
| syl2im 40 | Replace two antecedents. ... |
| syl2imc 41 | A commuted version of ~ sy... |
| pm2.27 42 | This theorem, sometimes ca... |
| mpdd 43 | A nested modus ponens dedu... |
| mpid 44 | A nested modus ponens dedu... |
| mpdi 45 | A nested modus ponens dedu... |
| mpii 46 | A doubly nested modus pone... |
| syld 47 | Syllogism deduction. Dedu... |
| syldc 48 | Syllogism deduction. Comm... |
| mp2d 49 | A double modus ponens dedu... |
| a1dd 50 | Double deduction introduci... |
| 2a1dd 51 | Double deduction introduci... |
| pm2.43i 52 | Inference absorbing redund... |
| pm2.43d 53 | Deduction absorbing redund... |
| pm2.43a 54 | Inference absorbing redund... |
| pm2.43b 55 | Inference absorbing redund... |
| pm2.43 56 | Absorption of redundant an... |
| imim2d 57 | Deduction adding nested an... |
| imim2 58 | A closed form of syllogism... |
| embantd 59 | Deduction embedding an ant... |
| 3syld 60 | Triple syllogism deduction... |
| sylsyld 61 | A double syllogism inferen... |
| imim12i 62 | Inference joining two impl... |
| imim1i 63 | Inference adding common co... |
| imim3i 64 | Inference adding three nes... |
| sylc 65 | A syllogism inference comb... |
| syl3c 66 | A syllogism inference comb... |
| syl6mpi 67 | A syllogism inference. (C... |
| mpsyl 68 | Modus ponens combined with... |
| mpsylsyld 69 | Modus ponens combined with... |
| syl6c 70 | Inference combining ~ syl6... |
| syl6ci 71 | A syllogism inference comb... |
| syldd 72 | Nested syllogism deduction... |
| syl5d 73 | A nested syllogism deducti... |
| syl7 74 | A syllogism rule of infere... |
| syl6d 75 | A nested syllogism deducti... |
| syl8 76 | A syllogism rule of infere... |
| syl9 77 | A nested syllogism inferen... |
| syl9r 78 | A nested syllogism inferen... |
| syl10 79 | A nested syllogism inferen... |
| a1ddd 80 | Triple deduction introduci... |
| imim12d 81 | Deduction combining antece... |
| imim1d 82 | Deduction adding nested co... |
| imim1 83 | A closed form of syllogism... |
| pm2.83 84 | Theorem *2.83 of [Whitehea... |
| peirceroll 85 | Over minimal implicational... |
| com23 86 | Commutation of antecedents... |
| com3r 87 | Commutation of antecedents... |
| com13 88 | Commutation of antecedents... |
| com3l 89 | Commutation of antecedents... |
| pm2.04 90 | Swap antecedents. Theorem... |
| com34 91 | Commutation of antecedents... |
| com4l 92 | Commutation of antecedents... |
| com4t 93 | Commutation of antecedents... |
| com4r 94 | Commutation of antecedents... |
| com24 95 | Commutation of antecedents... |
| com14 96 | Commutation of antecedents... |
| com45 97 | Commutation of antecedents... |
| com35 98 | Commutation of antecedents... |
| com25 99 | Commutation of antecedents... |
| com5l 100 | Commutation of antecedents... |
| com15 101 | Commutation of antecedents... |
| com52l 102 | Commutation of antecedents... |
| com52r 103 | Commutation of antecedents... |
| com5r 104 | Commutation of antecedents... |
| imim12 105 | Closed form of ~ imim12i a... |
| jarr 106 | Elimination of a nested an... |
| jarri 107 | Inference associated with ... |
| pm2.86d 108 | Deduction associated with ... |
| pm2.86 109 | Converse of Axiom ~ ax-2 .... |
| pm2.86i 110 | Inference associated with ... |
| loolin 111 | The Linearity Axiom of the... |
| loowoz 112 | An alternate for the Linea... |
| con4 113 | Alias for ~ ax-3 to be use... |
| con4i 114 | Inference associated with ... |
| con4d 115 | Deduction associated with ... |
| mt4 116 | The rule of modus tollens.... |
| mt4d 117 | Modus tollens deduction. ... |
| mt4i 118 | Modus tollens inference. ... |
| pm2.21i 119 | A contradiction implies an... |
| pm2.24ii 120 | A contradiction implies an... |
| pm2.21d 121 | A contradiction implies an... |
| pm2.21ddALT 122 | Alternate proof of ~ pm2.2... |
| pm2.21 123 | From a wff and its negatio... |
| pm2.24 124 | Theorem *2.24 of [Whitehea... |
| jarl 125 | Elimination of a nested an... |
| jarli 126 | Inference associated with ... |
| pm2.18d 127 | Deduction form of the Clav... |
| pm2.18 128 | Clavius law, or "consequen... |
| pm2.18i 129 | Inference associated with ... |
| notnotr 130 | Double negation eliminatio... |
| notnotri 131 | Inference associated with ... |
| notnotriALT 132 | Alternate proof of ~ notno... |
| notnotrd 133 | Deduction associated with ... |
| con2d 134 | A contraposition deduction... |
| con2 135 | Contraposition. Theorem *... |
| mt2d 136 | Modus tollens deduction. ... |
| mt2i 137 | Modus tollens inference. ... |
| nsyl3 138 | A negated syllogism infere... |
| con2i 139 | A contraposition inference... |
| nsyl 140 | A negated syllogism infere... |
| nsyl2 141 | A negated syllogism infere... |
| notnot 142 | Double negation introducti... |
| notnoti 143 | Inference associated with ... |
| notnotd 144 | Deduction associated with ... |
| con1d 145 | A contraposition deduction... |
| con1 146 | Contraposition. Theorem *... |
| con1i 147 | A contraposition inference... |
| mt3d 148 | Modus tollens deduction. ... |
| mt3i 149 | Modus tollens inference. ... |
| pm2.24i 150 | Inference associated with ... |
| pm2.24d 151 | Deduction form of ~ pm2.24... |
| con3d 152 | A contraposition deduction... |
| con3 153 | Contraposition. Theorem *... |
| con3i 154 | A contraposition inference... |
| con3rr3 155 | Rotate through consequent ... |
| nsyld 156 | A negated syllogism deduct... |
| nsyli 157 | A negated syllogism infere... |
| nsyl4 158 | A negated syllogism infere... |
| nsyl5 159 | A negated syllogism infere... |
| pm3.2im 160 | Theorem *3.2 of [Whitehead... |
| jc 161 | Deduction joining the cons... |
| jcn 162 | Theorem joining the conseq... |
| jcnd 163 | Deduction joining the cons... |
| impi 164 | An importation inference. ... |
| expi 165 | An exportation inference. ... |
| simprim 166 | Simplification. Similar t... |
| simplim 167 | Simplification. Similar t... |
| pm2.5g 168 | General instance of Theore... |
| pm2.5 169 | Theorem *2.5 of [Whitehead... |
| conax1 170 | Contrapositive of ~ ax-1 .... |
| conax1k 171 | Weakening of ~ conax1 . G... |
| pm2.51 172 | Theorem *2.51 of [Whitehea... |
| pm2.52 173 | Theorem *2.52 of [Whitehea... |
| pm2.521g 174 | A general instance of Theo... |
| pm2.521g2 175 | A general instance of Theo... |
| pm2.521 176 | Theorem *2.521 of [Whitehe... |
| expt 177 | Exportation theorem ~ pm3.... |
| exptOLD 178 | Obsolete version of ~ expt... |
| impt 179 | Importation theorem ~ pm3.... |
| pm2.61d 180 | Deduction eliminating an a... |
| pm2.61d1 181 | Inference eliminating an a... |
| pm2.61d2 182 | Inference eliminating an a... |
| pm2.61i 183 | Inference eliminating an a... |
| pm2.61ii 184 | Inference eliminating two ... |
| pm2.61nii 185 | Inference eliminating two ... |
| pm2.61iii 186 | Inference eliminating thre... |
| ja 187 | Inference joining the ante... |
| jad 188 | Deduction form of ~ ja . ... |
| pm2.01 189 | Weak Clavius law. If a fo... |
| pm2.01i 190 | Inference associated with ... |
| pm2.01d 191 | Deduction based on reducti... |
| pm2.6 192 | Theorem *2.6 of [Whitehead... |
| pm2.61 193 | Theorem *2.61 of [Whitehea... |
| pm2.65 194 | Theorem *2.65 of [Whitehea... |
| pm2.65i 195 | Inference for proof by con... |
| pm2.65iOLD 196 | Obsolete version of ~ pm2.... |
| pm2.21dd 197 | A contradiction implies an... |
| pm2.65d 198 | Deduction for proof by con... |
| mto 199 | The rule of modus tollens.... |
| mtod 200 | Modus tollens deduction. ... |
| mtoi 201 | Modus tollens inference. ... |
| mt2 202 | A rule similar to modus to... |
| mt3 203 | A rule similar to modus to... |
| peirce 204 | Peirce's axiom. A non-int... |
| looinv 205 | The Inversion Axiom of the... |
| bijust0 206 | A self-implication (see ~ ... |
| bijust 207 | Theorem used to justify th... |
| impbi 210 | Property of the biconditio... |
| impbii 211 | Infer an equivalence from ... |
| impbidd 212 | Deduce an equivalence from... |
| impbid21d 213 | Deduce an equivalence from... |
| impbid 214 | Deduce an equivalence from... |
| dfbi1 215 | Relate the biconditional c... |
| dfbi1ALT 216 | Alternate proof of ~ dfbi1... |
| biimp 217 | Property of the biconditio... |
| biimpi 218 | Infer an implication from ... |
| sylbi 219 | A mixed syllogism inferenc... |
| sylib 220 | A mixed syllogism inferenc... |
| sylbb 221 | A mixed syllogism inferenc... |
| biimpr 222 | Property of the biconditio... |
| bicom1 223 | Commutative law for the bi... |
| bicom 224 | Commutative law for the bi... |
| bicomd 225 | Commute two sides of a bic... |
| bicomi 226 | Inference from commutative... |
| impbid1 227 | Infer an equivalence from ... |
| impbid2 228 | Infer an equivalence from ... |
| impcon4bid 229 | A variation on ~ impbid wi... |
| biimpri 230 | Infer a converse implicati... |
| biimpd 231 | Deduce an implication from... |
| mpbi 232 | An inference from a bicond... |
| mpbir 233 | An inference from a bicond... |
| mpbid 234 | A deduction from a bicondi... |
| mpbii 235 | An inference from a nested... |
| sylibr 236 | A mixed syllogism inferenc... |
| sylbir 237 | A mixed syllogism inferenc... |
| sylbbr 238 | A mixed syllogism inferenc... |
| sylbb1 239 | A mixed syllogism inferenc... |
| sylbb2 240 | A mixed syllogism inferenc... |
| sylibd 241 | A syllogism deduction. (C... |
| sylbid 242 | A syllogism deduction. (C... |
| mpbidi 243 | A deduction from a bicondi... |
| biimtrid 244 | A mixed syllogism inferenc... |
| biimtrrid 245 | A mixed syllogism inferenc... |
| imbitrid 246 | A mixed syllogism inferenc... |
| syl5ibcom 247 | A mixed syllogism inferenc... |
| imbitrrid 248 | A mixed syllogism inferenc... |
| syl5ibrcom 249 | A mixed syllogism inferenc... |
| biimprd 250 | Deduce a converse implicat... |
| biimpcd 251 | Deduce a commuted implicat... |
| biimprcd 252 | Deduce a converse commuted... |
| imbitrdi 253 | A mixed syllogism inferenc... |
| imbitrrdi 254 | A mixed syllogism inferenc... |
| biimtrdi 255 | A mixed syllogism inferenc... |
| biimtrrdi 256 | A mixed syllogism inferenc... |
| syl7bi 257 | A mixed syllogism inferenc... |
| syl8ib 258 | A syllogism rule of infere... |
| mpbird 259 | A deduction from a bicondi... |
| mpbiri 260 | An inference from a nested... |
| sylibrd 261 | A syllogism deduction. (C... |
| sylbird 262 | A syllogism deduction. (C... |
| biid 263 | Principle of identity for ... |
| biidd 264 | Principle of identity with... |
| pm5.1im 265 | Two propositions are equiv... |
| 2th 266 | Two truths are equivalent.... |
| 2thd 267 | Two truths are equivalent.... |
| monothetic 268 | Two self-implications (see... |
| ibi 269 | Inference that converts a ... |
| ibir 270 | Inference that converts a ... |
| ibd 271 | Deduction that converts a ... |
| pm5.74 272 | Distribution of implicatio... |
| pm5.74i 273 | Distribution of implicatio... |
| pm5.74ri 274 | Distribution of implicatio... |
| pm5.74d 275 | Distribution of implicatio... |
| pm5.74rd 276 | Distribution of implicatio... |
| bitri 277 | An inference from transiti... |
| bitr2i 278 | An inference from transiti... |
| bitr3i 279 | An inference from transiti... |
| bitr4i 280 | An inference from transiti... |
| bitrd 281 | Deduction form of ~ bitri ... |
| bitr2d 282 | Deduction form of ~ bitr2i... |
| bitr3d 283 | Deduction form of ~ bitr3i... |
| bitr4d 284 | Deduction form of ~ bitr4i... |
| bitrid 285 | A syllogism inference from... |
| bitr2id 286 | A syllogism inference from... |
| bitr3id 287 | A syllogism inference from... |
| bitr3di 288 | A syllogism inference from... |
| bitrdi 289 | A syllogism inference from... |
| bitr2di 290 | A syllogism inference from... |
| bitr4di 291 | A syllogism inference from... |
| bitr4id 292 | A syllogism inference from... |
| 3imtr3i 293 | A mixed syllogism inferenc... |
| 3imtr4i 294 | A mixed syllogism inferenc... |
| 3imtr3d 295 | More general version of ~ ... |
| 3imtr4d 296 | More general version of ~ ... |
| 3imtr3g 297 | More general version of ~ ... |
| 3imtr4g 298 | More general version of ~ ... |
| 3bitri 299 | A chained inference from t... |
| 3bitrri 300 | A chained inference from t... |
| 3bitr2i 301 | A chained inference from t... |
| 3bitr2ri 302 | A chained inference from t... |
| 3bitr3i 303 | A chained inference from t... |
| 3bitr3ri 304 | A chained inference from t... |
| 3bitr4i 305 | A chained inference from t... |
| 3bitr4ri 306 | A chained inference from t... |
| 3bitrd 307 | Deduction from transitivit... |
| 3bitrrd 308 | Deduction from transitivit... |
| 3bitr2d 309 | Deduction from transitivit... |
| 3bitr2rd 310 | Deduction from transitivit... |
| 3bitr3d 311 | Deduction from transitivit... |
| 3bitr3rd 312 | Deduction from transitivit... |
| 3bitr4d 313 | Deduction from transitivit... |
| 3bitr4rd 314 | Deduction from transitivit... |
| 3bitr3g 315 | More general version of ~ ... |
| 3bitr4g 316 | More general version of ~ ... |
| notnotb 317 | Double negation. Theorem ... |
| con34b 318 | A biconditional form of co... |
| con4bid 319 | A contraposition deduction... |
| notbid 320 | Deduction negating both si... |
| notbi 321 | Contraposition. Theorem *... |
| notbii 322 | Negate both sides of a log... |
| con4bii 323 | A contraposition inference... |
| mtbi 324 | An inference from a bicond... |
| mtbir 325 | An inference from a bicond... |
| mtbid 326 | A deduction from a bicondi... |
| mtbird 327 | A deduction from a bicondi... |
| mtbii 328 | An inference from a bicond... |
| mtbiri 329 | An inference from a bicond... |
| sylnib 330 | A mixed syllogism inferenc... |
| sylnibr 331 | A mixed syllogism inferenc... |
| sylnbi 332 | A mixed syllogism inferenc... |
| sylnbir 333 | A mixed syllogism inferenc... |
| xchnxbi 334 | Replacement of a subexpres... |
| xchnxbir 335 | Replacement of a subexpres... |
| xchbinx 336 | Replacement of a subexpres... |
| xchbinxr 337 | Replacement of a subexpres... |
| imbi2i 338 | Introduce an antecedent to... |
| bibi2i 339 | Inference adding a bicondi... |
| bibi1i 340 | Inference adding a bicondi... |
| bibi12i 341 | The equivalence of two equ... |
| imbi2d 342 | Deduction adding an antece... |
| imbi1d 343 | Deduction adding a consequ... |
| bibi2d 344 | Deduction adding a bicondi... |
| bibi1d 345 | Deduction adding a bicondi... |
| imbi12d 346 | Deduction joining two equi... |
| bibi12d 347 | Deduction joining two equi... |
| imbi12 348 | Closed form of ~ imbi12i .... |
| imbi1 349 | Theorem *4.84 of [Whitehea... |
| imbi2 350 | Theorem *4.85 of [Whitehea... |
| imbi1i 351 | Introduce a consequent to ... |
| imbi12i 352 | Join two logical equivalen... |
| bibi1 353 | Theorem *4.86 of [Whitehea... |
| bitr3 354 | Closed nested implication ... |
| con2bi 355 | Contraposition. Theorem *... |
| con2bid 356 | A contraposition deduction... |
| con1bid 357 | A contraposition deduction... |
| con1bii 358 | A contraposition inference... |
| con2bii 359 | A contraposition inference... |
| con1b 360 | Contraposition. Bidirecti... |
| con2b 361 | Contraposition. Bidirecti... |
| biimt 362 | A wff is equivalent to its... |
| pm5.5 363 | Theorem *5.5 of [Whitehead... |
| a1bi 364 | Inference introducing a th... |
| mt2bi 365 | A false consequent falsifi... |
| mtt 366 | Modus-tollens-like theorem... |
| imnot 367 | If a proposition is false,... |
| pm5.501 368 | Theorem *5.501 of [Whitehe... |
| ibib 369 | Implication in terms of im... |
| ibibr 370 | Implication in terms of im... |
| tbt 371 | A wff is equivalent to its... |
| nbn2 372 | The negation of a wff is e... |
| bibif 373 | Transfer negation via an e... |
| nbn 374 | The negation of a wff is e... |
| nbn3 375 | Transfer falsehood via equ... |
| pm5.21im 376 | Two propositions are equiv... |
| 2false 377 | Two falsehoods are equival... |
| 2falsed 378 | Two falsehoods are equival... |
| pm5.21ni 379 | Two propositions implying ... |
| pm5.21nii 380 | Eliminate an antecedent im... |
| pm5.21ndd 381 | Eliminate an antecedent im... |
| bija 382 | Combine antecedents into a... |
| pm5.18 383 | Theorem *5.18 of [Whitehea... |
| xor3 384 | Two ways to express "exclu... |
| nbbn 385 | Move negation outside of b... |
| nbbnOLD 386 | Obsolete version of ~ nbbn... |
| biass 387 | Associative law for the bi... |
| biluk 388 | Lukasiewicz's shortest axi... |
| pm5.19 389 | Theorem *5.19 of [Whitehea... |
| bi2.04 390 | Logical equivalence of com... |
| pm5.4 391 | Antecedent absorption impl... |
| imdi 392 | Distributive law for impli... |
| pm5.41 393 | Theorem *5.41 of [Whitehea... |
| imbibi 394 | The antecedent of one side... |
| imbibiOLD 395 | Obsolete version of ~ imbi... |
| pm4.8 396 | Theorem *4.8 of [Whitehead... |
| pm4.81 397 | A formula is equivalent to... |
| imim21b 398 | Simplify an implication be... |
| pm4.63 401 | Theorem *4.63 of [Whitehea... |
| pm4.67 402 | Theorem *4.67 of [Whitehea... |
| imnan 403 | Express an implication in ... |
| imnani 404 | Infer an implication from ... |
| iman 405 | Implication in terms of co... |
| pm3.24 406 | Law of noncontradiction. ... |
| annim 407 | Express a conjunction in t... |
| pm4.61 408 | Theorem *4.61 of [Whitehea... |
| pm4.65 409 | Theorem *4.65 of [Whitehea... |
| imp 410 | Importation inference. (C... |
| impcom 411 | Importation inference with... |
| con3dimp 412 | Variant of ~ con3d with im... |
| mpnanrd 413 | Eliminate the right side o... |
| impd 414 | Importation deduction. (C... |
| impcomd 415 | Importation deduction with... |
| ex 416 | Exportation inference. (T... |
| expcom 417 | Exportation inference with... |
| expdcom 418 | Commuted form of ~ expd . ... |
| expd 419 | Exportation deduction. (C... |
| expcomd 420 | Deduction form of ~ expcom... |
| imp31 421 | An importation inference. ... |
| imp32 422 | An importation inference. ... |
| exp31 423 | An exportation inference. ... |
| exp32 424 | An exportation inference. ... |
| imp4b 425 | An importation inference. ... |
| imp4a 426 | An importation inference. ... |
| imp4c 427 | An importation inference. ... |
| imp4d 428 | An importation inference. ... |
| imp41 429 | An importation inference. ... |
| imp42 430 | An importation inference. ... |
| imp43 431 | An importation inference. ... |
| imp44 432 | An importation inference. ... |
| imp45 433 | An importation inference. ... |
| exp4b 434 | An exportation inference. ... |
| exp4a 435 | An exportation inference. ... |
| exp4c 436 | An exportation inference. ... |
| exp4d 437 | An exportation inference. ... |
| exp41 438 | An exportation inference. ... |
| exp42 439 | An exportation inference. ... |
| exp43 440 | An exportation inference. ... |
| exp44 441 | An exportation inference. ... |
| exp45 442 | An exportation inference. ... |
| imp5d 443 | An importation inference. ... |
| imp5a 444 | An importation inference. ... |
| imp5g 445 | An importation inference. ... |
| imp55 446 | An importation inference. ... |
| imp511 447 | An importation inference. ... |
| exp5c 448 | An exportation inference. ... |
| exp5j 449 | An exportation inference. ... |
| exp5l 450 | An exportation inference. ... |
| exp53 451 | An exportation inference. ... |
| pm3.3 452 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 453 | Theorem *3.31 (Imp) of [Wh... |
| impexp 454 | Import-export theorem. Pa... |
| impancom 455 | Mixed importation/commutat... |
| expdimp 456 | A deduction version of exp... |
| expimpd 457 | Exportation followed by a ... |
| impr 458 | Import a wff into a right ... |
| impl 459 | Export a wff from a left c... |
| expr 460 | Export a wff from a right ... |
| expl 461 | Export a wff from a left c... |
| ancoms 462 | Inference commuting conjun... |
| pm3.22 463 | Theorem *3.22 of [Whitehea... |
| ancom 464 | Commutative law for conjun... |
| ancomd 465 | Commutation of conjuncts i... |
| biancomi 466 | Commuting conjunction in a... |
| biancomd 467 | Commuting conjunction in a... |
| ancomst 468 | Closed form of ~ ancoms . ... |
| ancomsd 469 | Deduction commuting conjun... |
| anasss 470 | Associative law for conjun... |
| anassrs 471 | Associative law for conjun... |
| anass 472 | Associative law for conjun... |
| pm3.2 473 | Join antecedents with conj... |
| pm3.2i 474 | Infer conjunction of premi... |
| pm3.21 475 | Join antecedents with conj... |
| pm3.43i 476 | Nested conjunction of ante... |
| pm3.43 477 | Theorem *3.43 (Comp) of [W... |
| dfbi2 478 | A theorem similar to the s... |
| dfbi 479 | Definition ~ df-bi rewritt... |
| biimpa 480 | Importation inference from... |
| biimpar 481 | Importation inference from... |
| biimpac 482 | Importation inference from... |
| biimparc 483 | Importation inference from... |
| adantr 484 | Inference adding a conjunc... |
| adantl 485 | Inference adding a conjunc... |
| simpl 486 | Elimination of a conjunct.... |
| simpli 487 | Inference eliminating a co... |
| simpr 488 | Elimination of a conjunct.... |
| simpri 489 | Inference eliminating a co... |
| intnan 490 | Introduction of conjunct i... |
| intnanr 491 | Introduction of conjunct i... |
| intnand 492 | Introduction of conjunct i... |
| intnanrd 493 | Introduction of conjunct i... |
| adantld 494 | Deduction adding a conjunc... |
| adantrd 495 | Deduction adding a conjunc... |
| pm3.41 496 | Theorem *3.41 of [Whitehea... |
| pm3.42 497 | Theorem *3.42 of [Whitehea... |
| simpld 498 | Deduction eliminating a co... |
| simprd 499 | Deduction eliminating a co... |
| simplbi 500 | Deduction eliminating a co... |
| simprbi 501 | Deduction eliminating a co... |
| simprbda 502 | Deduction eliminating a co... |
| simplbda 503 | Deduction eliminating a co... |
| simplbi2 504 | Deduction eliminating a co... |
| simplbi2comt 505 | Closed form of ~ simplbi2c... |
| simplbi2com 506 | A deduction eliminating a ... |
| birani 507 | Inference adding a conjunc... |
| bilani 508 | Inference adding a conjunc... |
| biranri 509 | Inference adding a conjunc... |
| bilanri 510 | Inference adding a conjunc... |
| simpl2im 511 | Implication from an elimin... |
| simplbiim 512 | Implication from an elimin... |
| impel 513 | An inference for implicati... |
| mpan9 514 | Modus ponens conjoining di... |
| sylan9 515 | Nested syllogism inference... |
| sylan9r 516 | Nested syllogism inference... |
| sylan9bb 517 | Nested syllogism inference... |
| sylan9bbr 518 | Nested syllogism inference... |
| jca 519 | Deduce conjunction of the ... |
| jcad 520 | Deduction conjoining the c... |
| jca2 521 | Inference conjoining the c... |
| jca31 522 | Join three consequents. (... |
| jca32 523 | Join three consequents. (... |
| jcai 524 | Deduction replacing implic... |
| jcab 525 | Distributive law for impli... |
| pm4.76 526 | Theorem *4.76 of [Whitehea... |
| jctil 527 | Inference conjoining a the... |
| jctir 528 | Inference conjoining a the... |
| jccir 529 | Inference conjoining a con... |
| jccil 530 | Inference conjoining a con... |
| jctl 531 | Inference conjoining a the... |
| jctr 532 | Inference conjoining a the... |
| jctild 533 | Deduction conjoining a the... |
| jctird 534 | Deduction conjoining a the... |
| iba 535 | Introduction of antecedent... |
| ibar 536 | Introduction of antecedent... |
| biantru 537 | A wff is equivalent to its... |
| biantrur 538 | A wff is equivalent to its... |
| biantrud 539 | A wff is equivalent to its... |
| biantrurd 540 | A wff is equivalent to its... |
| bianfi 541 | A wff conjoined with false... |
| bianfd 542 | A wff conjoined with false... |
| baib 543 | Move conjunction outside o... |
| baibr 544 | Move conjunction outside o... |
| rbaibr 545 | Move conjunction outside o... |
| rbaib 546 | Move conjunction outside o... |
| baibd 547 | Move conjunction outside o... |
| rbaibd 548 | Move conjunction outside o... |
| bianabs 549 | Absorb a hypothesis into t... |
| pm5.44 550 | Theorem *5.44 of [Whitehea... |
| pm5.42 551 | Theorem *5.42 of [Whitehea... |
| ancl 552 | Conjoin antecedent to left... |
| anclb 553 | Conjoin antecedent to left... |
| ancr 554 | Conjoin antecedent to righ... |
| ancrb 555 | Conjoin antecedent to righ... |
| ancli 556 | Deduction conjoining antec... |
| ancri 557 | Deduction conjoining antec... |
| ancld 558 | Deduction conjoining antec... |
| ancrd 559 | Deduction conjoining antec... |
| impac 560 | Importation with conjuncti... |
| anc2l 561 | Conjoin antecedent to left... |
| anc2r 562 | Conjoin antecedent to righ... |
| anc2li 563 | Deduction conjoining antec... |
| anc2ri 564 | Deduction conjoining antec... |
| pm4.71 565 | Implication in terms of bi... |
| pm4.71r 566 | Implication in terms of bi... |
| pm4.71i 567 | Inference converting an im... |
| pm4.71ri 568 | Inference converting an im... |
| pm4.71d 569 | Deduction converting an im... |
| pm4.71rd 570 | Deduction converting an im... |
| pm4.24 571 | Theorem *4.24 of [Whitehea... |
| anidm 572 | Idempotent law for conjunc... |
| anidmdbi 573 | Conjunction idempotence wi... |
| anidms 574 | Inference from idempotent ... |
| imdistan 575 | Distribution of implicatio... |
| imdistani 576 | Distribution of implicatio... |
| imdistanri 577 | Distribution of implicatio... |
| imdistand 578 | Distribution of implicatio... |
| imdistanda 579 | Distribution of implicatio... |
| pm5.3 580 | Theorem *5.3 of [Whitehead... |
| pm5.32 581 | Distribution of implicatio... |
| pm5.32i 582 | Distribution of implicatio... |
| pm5.32ri 583 | Distribution of implicatio... |
| bianim 584 | Exchanging conjunction in ... |
| pm5.32d 585 | Distribution of implicatio... |
| pm5.32rd 586 | Distribution of implicatio... |
| pm5.32da 587 | Distribution of implicatio... |
| bian1d 588 | Adding a superfluous conju... |
| sylan 589 | A syllogism inference. (C... |
| sylanb 590 | A syllogism inference. (C... |
| sylanbr 591 | A syllogism inference. (C... |
| sylanbrc 592 | Syllogism inference. (Con... |
| syl2anc 593 | Syllogism inference combin... |
| syl2anc2 594 | Double syllogism inference... |
| sylancl 595 | Syllogism inference combin... |
| sylancr 596 | Syllogism inference combin... |
| sylancom 597 | Syllogism inference with c... |
| sylanblc 598 | Syllogism inference combin... |
| sylanblrc 599 | Syllogism inference combin... |
| syldan 600 | A syllogism deduction with... |
| sylbida 601 | A syllogism deduction. (C... |
| sylan2 602 | A syllogism inference. (C... |
| sylan2b 603 | A syllogism inference. (C... |
| sylan2br 604 | A syllogism inference. (C... |
| syl2an 605 | A double syllogism inferen... |
| syl2anr 606 | A double syllogism inferen... |
| syl2anb 607 | A double syllogism inferen... |
| syl2anbr 608 | A double syllogism inferen... |
| sylancb 609 | A syllogism inference comb... |
| sylancbr 610 | A syllogism inference comb... |
| syldanl 611 | A syllogism deduction with... |
| syland 612 | A syllogism deduction. (C... |
| sylani 613 | A syllogism inference. (C... |
| sylan2d 614 | A syllogism deduction. (C... |
| sylan2i 615 | A syllogism inference. (C... |
| syl2ani 616 | A syllogism inference. (C... |
| syl2and 617 | A syllogism deduction. (C... |
| anim12d 618 | Conjoin antecedents and co... |
| anim12d1 619 | Variant of ~ anim12d where... |
| anim1d 620 | Add a conjunct to right of... |
| anim2d 621 | Add a conjunct to left of ... |
| anim12i 622 | Conjoin antecedents and co... |
| anim12ci 623 | Variant of ~ anim12i with ... |
| anim1i 624 | Introduce conjunct to both... |
| anim1ci 625 | Introduce conjunct to both... |
| anim2i 626 | Introduce conjunct to both... |
| anim12ii 627 | Conjoin antecedents and co... |
| anim12dan 628 | Conjoin antecedents and co... |
| im2anan9 629 | Deduction joining nested i... |
| im2anan9r 630 | Deduction joining nested i... |
| pm3.45 631 | Theorem *3.45 (Fact) of [W... |
| anbi2i 632 | Introduce a left conjunct ... |
| anbi1i 633 | Introduce a right conjunct... |
| anbi2ci 634 | Variant of ~ anbi2i with c... |
| anbi1ci 635 | Variant of ~ anbi1i with c... |
| bianbi 636 | Exchanging conjunction in ... |
| anbi12i 637 | Conjoin both sides of two ... |
| anbi12ci 638 | Variant of ~ anbi12i with ... |
| anbi2d 639 | Deduction adding a left co... |
| anbi1d 640 | Deduction adding a right c... |
| anbi12d 641 | Deduction joining two equi... |
| anbi1 642 | Introduce a right conjunct... |
| anbi2 643 | Introduce a left conjunct ... |
| anbi1cd 644 | Introduce a proposition as... |
| an2anr 645 | Double commutation in conj... |
| pm4.38 646 | Theorem *4.38 of [Whitehea... |
| bi2anan9 647 | Deduction joining two equi... |
| bi2anan9r 648 | Deduction joining two equi... |
| bi2bian9 649 | Deduction joining two bico... |
| anbiim 650 | Adding biconditional when ... |
| anbiimOLD 651 | Obsolete version of ~ anbi... |
| bianass 652 | An inference to merge two ... |
| bianassc 653 | An inference to merge two ... |
| an21 654 | Swap two conjuncts. (Cont... |
| an12 655 | Swap two conjuncts. Note ... |
| an32 656 | A rearrangement of conjunc... |
| an13 657 | A rearrangement of conjunc... |
| an31 658 | A rearrangement of conjunc... |
| an12s 659 | Swap two conjuncts in ante... |
| ancom2s 660 | Inference commuting a nest... |
| an13s 661 | Swap two conjuncts in ante... |
| an32s 662 | Swap two conjuncts in ante... |
| ancom1s 663 | Inference commuting a nest... |
| an31s 664 | Swap two conjuncts in ante... |
| anass1rs 665 | Commutative-associative la... |
| an4 666 | Rearrangement of 4 conjunc... |
| an42 667 | Rearrangement of 4 conjunc... |
| an43 668 | Rearrangement of 4 conjunc... |
| an3 669 | A rearrangement of conjunc... |
| an4s 670 | Inference rearranging 4 co... |
| an42s 671 | Inference rearranging 4 co... |
| anabs1 672 | Absorption into embedded c... |
| anabs5 673 | Absorption into embedded c... |
| anabs7 674 | Absorption into embedded c... |
| anabsan 675 | Absorption of antecedent w... |
| anabss1 676 | Absorption of antecedent i... |
| anabss4 677 | Absorption of antecedent i... |
| anabss5 678 | Absorption of antecedent i... |
| anabsi5 679 | Absorption of antecedent i... |
| anabsi6 680 | Absorption of antecedent i... |
| anabsi7 681 | Absorption of antecedent i... |
| anabsi8 682 | Absorption of antecedent i... |
| anabss7 683 | Absorption of antecedent i... |
| anabsan2 684 | Absorption of antecedent w... |
| anabss3 685 | Absorption of antecedent i... |
| anandi 686 | Distribution of conjunctio... |
| anandir 687 | Distribution of conjunctio... |
| anandis 688 | Inference that undistribut... |
| anandirs 689 | Inference that undistribut... |
| sylanl1 690 | A syllogism inference. (C... |
| sylanl2 691 | A syllogism inference. (C... |
| sylanr1 692 | A syllogism inference. (C... |
| sylanr2 693 | A syllogism inference. (C... |
| syl6an 694 | A syllogism deduction comb... |
| syl2an2r 695 | ~ syl2anr with antecedents... |
| syl2an2 696 | ~ syl2an with antecedents ... |
| mpdan 697 | An inference based on modu... |
| mpancom 698 | An inference based on modu... |
| mpidan 699 | A deduction which "stacks"... |
| mpan 700 | An inference based on modu... |
| mpan2 701 | An inference based on modu... |
| mp2an 702 | An inference based on modu... |
| mp4an 703 | An inference based on modu... |
| mpan2d 704 | A deduction based on modus... |
| mpand 705 | A deduction based on modus... |
| mpani 706 | An inference based on modu... |
| mpan2i 707 | An inference based on modu... |
| mp2ani 708 | An inference based on modu... |
| mp2and 709 | A deduction based on modus... |
| mpanl1 710 | An inference based on modu... |
| mpanl2 711 | An inference based on modu... |
| mpanl12 712 | An inference based on modu... |
| mpanr1 713 | An inference based on modu... |
| mpanr2 714 | An inference based on modu... |
| mpanr12 715 | An inference based on modu... |
| mpanlr1 716 | An inference based on modu... |
| mpbirand 717 | Detach truth from conjunct... |
| mpbiran2d 718 | Detach truth from conjunct... |
| mpbiran 719 | Detach truth from conjunct... |
| mpbiran2 720 | Detach truth from conjunct... |
| mpbir2an 721 | Detach a conjunction of tr... |
| mpbi2and 722 | Detach a conjunction of tr... |
| mpbir2and 723 | Detach a conjunction of tr... |
| adantll 724 | Deduction adding a conjunc... |
| adantlr 725 | Deduction adding a conjunc... |
| adantrl 726 | Deduction adding a conjunc... |
| adantrr 727 | Deduction adding a conjunc... |
| adantlll 728 | Deduction adding a conjunc... |
| adantllr 729 | Deduction adding a conjunc... |
| adantlrl 730 | Deduction adding a conjunc... |
| adantlrr 731 | Deduction adding a conjunc... |
| adantrll 732 | Deduction adding a conjunc... |
| adantrlr 733 | Deduction adding a conjunc... |
| adantrrl 734 | Deduction adding a conjunc... |
| adantrrr 735 | Deduction adding a conjunc... |
| ad2antrr 736 | Deduction adding two conju... |
| ad2antlr 737 | Deduction adding two conju... |
| ad2antrl 738 | Deduction adding two conju... |
| ad2antll 739 | Deduction adding conjuncts... |
| ad3antrrr 740 | Deduction adding three con... |
| ad3antlr 741 | Deduction adding three con... |
| ad4antr 742 | Deduction adding 4 conjunc... |
| ad4antlr 743 | Deduction adding 4 conjunc... |
| ad5antr 744 | Deduction adding 5 conjunc... |
| ad5antlr 745 | Deduction adding 5 conjunc... |
| ad6antr 746 | Deduction adding 6 conjunc... |
| ad6antlr 747 | Deduction adding 6 conjunc... |
| ad7antr 748 | Deduction adding 7 conjunc... |
| ad7antlr 749 | Deduction adding 7 conjunc... |
| ad8antr 750 | Deduction adding 8 conjunc... |
| ad8antlr 751 | Deduction adding 8 conjunc... |
| ad9antr 752 | Deduction adding 9 conjunc... |
| ad9antlr 753 | Deduction adding 9 conjunc... |
| ad10antr 754 | Deduction adding 10 conjun... |
| ad10antlr 755 | Deduction adding 10 conjun... |
| ad2ant2l 756 | Deduction adding two conju... |
| ad2ant2r 757 | Deduction adding two conju... |
| ad2ant2lr 758 | Deduction adding two conju... |
| ad2ant2rl 759 | Deduction adding two conju... |
| adantl3r 760 | Deduction adding 1 conjunc... |
| ad4ant13 761 | Deduction adding conjuncts... |
| ad4ant14 762 | Deduction adding conjuncts... |
| ad4ant23 763 | Deduction adding conjuncts... |
| ad4ant24 764 | Deduction adding conjuncts... |
| adantl4r 765 | Deduction adding 1 conjunc... |
| ad5ant13 766 | Deduction adding conjuncts... |
| ad5ant14 767 | Deduction adding conjuncts... |
| ad5ant15 768 | Deduction adding conjuncts... |
| ad5ant23 769 | Deduction adding conjuncts... |
| ad5ant24 770 | Deduction adding conjuncts... |
| ad5ant25 771 | Deduction adding conjuncts... |
| adantl5r 772 | Deduction adding 1 conjunc... |
| adantl6r 773 | Deduction adding 1 conjunc... |
| pm3.33 774 | Theorem *3.33 (Syll) of [W... |
| pm3.34 775 | Theorem *3.34 (Syll) of [W... |
| simpll 776 | Simplification of a conjun... |
| simplld 777 | Deduction form of ~ simpll... |
| simplr 778 | Simplification of a conjun... |
| simplrd 779 | Deduction eliminating a do... |
| simprl 780 | Simplification of a conjun... |
| simprld 781 | Deduction eliminating a do... |
| simprr 782 | Simplification of a conjun... |
| simprrd 783 | Deduction form of ~ simprr... |
| simplll 784 | Simplification of a conjun... |
| simpllr 785 | Simplification of a conjun... |
| simplrl 786 | Simplification of a conjun... |
| simplrr 787 | Simplification of a conjun... |
| simprll 788 | Simplification of a conjun... |
| simprlr 789 | Simplification of a conjun... |
| simprrl 790 | Simplification of a conjun... |
| simprrr 791 | Simplification of a conjun... |
| simp-4l 792 | Simplification of a conjun... |
| simp-4r 793 | Simplification of a conjun... |
| simp-5l 794 | Simplification of a conjun... |
| simp-5r 795 | Simplification of a conjun... |
| simp-6l 796 | Simplification of a conjun... |
| simp-6r 797 | Simplification of a conjun... |
| simp-7l 798 | Simplification of a conjun... |
| simp-7r 799 | Simplification of a conjun... |
| simp-8l 800 | Simplification of a conjun... |
| simp-8r 801 | Simplification of a conjun... |
| simp-9l 802 | Simplification of a conjun... |
| simp-9r 803 | Simplification of a conjun... |
| simp-10l 804 | Simplification of a conjun... |
| simp-10r 805 | Simplification of a conjun... |
| simp-11l 806 | Simplification of a conjun... |
| simp-11r 807 | Simplification of a conjun... |
| pm2.01da 808 | Deduction based on reducti... |
| pm2.18da 809 | Deduction based on reducti... |
| impbida 810 | Deduce an equivalence from... |
| pm5.21nd 811 | Eliminate an antecedent im... |
| pm3.35 812 | Conjunctive detachment. T... |
| pm5.74da 813 | Distribution of implicatio... |
| bitr 814 | Theorem *4.22 of [Whitehea... |
| biantr 815 | A transitive law of equiva... |
| pm4.14 816 | Theorem *4.14 of [Whitehea... |
| pm3.37 817 | Theorem *3.37 (Transp) of ... |
| anim12 818 | Conjoin antecedents and co... |
| pm3.4 819 | Conjunction implies implic... |
| exbiri 820 | Inference form of ~ exbir ... |
| pm2.61ian 821 | Elimination of an antecede... |
| pm2.61dan 822 | Elimination of an antecede... |
| pm2.61ddan 823 | Elimination of two anteced... |
| pm2.61dda 824 | Elimination of two anteced... |
| mtand 825 | A modus tollens deduction.... |
| pm2.65da 826 | Deduction for proof by con... |
| condan 827 | Proof by contradiction. (... |
| biadan 828 | An implication is equivale... |
| biadani 829 | Inference associated with ... |
| biadaniALT 830 | Alternate proof of ~ biada... |
| biadanii 831 | Inference associated with ... |
| biadanid 832 | Deduction associated with ... |
| pm5.1 833 | Two propositions are equiv... |
| pm5.21 834 | Two propositions are equiv... |
| pm5.35 835 | Theorem *5.35 of [Whitehea... |
| abai 836 | Introduce one conjunct as ... |
| abab 837 | Introduce one conjunct as ... |
| pm4.45im 838 | Conjunction with implicati... |
| impimprbi 839 | An implication and its rev... |
| nan 840 | Theorem to move a conjunct... |
| pm5.31 841 | Theorem *5.31 of [Whitehea... |
| pm5.31r 842 | Variant of ~ pm5.31 . (Co... |
| pm4.15 843 | Theorem *4.15 of [Whitehea... |
| pm5.36 844 | Theorem *5.36 of [Whitehea... |
| annotanannot 845 | A conjunction with a negat... |
| pm5.33 846 | Theorem *5.33 of [Whitehea... |
| syl12anc 847 | Syllogism combined with co... |
| syl21anc 848 | Syllogism combined with co... |
| syl22anc 849 | Syllogism combined with co... |
| bibiad 850 | Eliminate an hypothesis ` ... |
| syl1111anc 851 | Four-hypothesis eliminatio... |
| syldbl2 852 | Stacked hypotheseis implie... |
| mpsyl4anc 853 | An elimination deduction. ... |
| pm4.87 854 | Theorem *4.87 of [Whitehea... |
| bimsc1 855 | Removal of conjunct from o... |
| a2and 856 | Deduction distributing a c... |
| animpimp2impd 857 | Deduction deriving nested ... |
| pm4.64 860 | Theorem *4.64 of [Whitehea... |
| pm4.66 861 | Theorem *4.66 of [Whitehea... |
| pm2.53 862 | Theorem *2.53 of [Whitehea... |
| pm2.54 863 | Theorem *2.54 of [Whitehea... |
| imor 864 | Implication in terms of di... |
| imori 865 | Infer disjunction from imp... |
| imorri 866 | Infer implication from dis... |
| pm4.62 867 | Theorem *4.62 of [Whitehea... |
| jaoi 868 | Inference disjoining the a... |
| jao1i 869 | Add a disjunct in the ante... |
| jaod 870 | Deduction disjoining the a... |
| mpjaod 871 | Eliminate a disjunction in... |
| ori 872 | Infer implication from dis... |
| orri 873 | Infer disjunction from imp... |
| orrd 874 | Deduce disjunction from im... |
| ord 875 | Deduce implication from di... |
| orci 876 | Deduction introducing a di... |
| olci 877 | Deduction introducing a di... |
| orc 878 | Introduction of a disjunct... |
| olc 879 | Introduction of a disjunct... |
| pm1.4 880 | Axiom *1.4 of [WhiteheadRu... |
| orcom 881 | Commutative law for disjun... |
| orcomd 882 | Commutation of disjuncts i... |
| orcoms 883 | Commutation of disjuncts i... |
| orcd 884 | Deduction introducing a di... |
| olcd 885 | Deduction introducing a di... |
| orcs 886 | Deduction eliminating disj... |
| olcs 887 | Deduction eliminating disj... |
| olcnd 888 | A lemma for Conjunctive No... |
| orcnd 889 | A lemma for Conjunctive No... |
| mtord 890 | A modus tollens deduction ... |
| pm3.2ni 891 | Infer negated disjunction ... |
| pm2.45 892 | Theorem *2.45 of [Whitehea... |
| pm2.46 893 | Theorem *2.46 of [Whitehea... |
| pm2.47 894 | Theorem *2.47 of [Whitehea... |
| pm2.48 895 | Theorem *2.48 of [Whitehea... |
| pm2.49 896 | Theorem *2.49 of [Whitehea... |
| norbi 897 | If neither of two proposit... |
| nbior 898 | If two propositions are no... |
| orel1 899 | Elimination of disjunction... |
| pm2.25 900 | Theorem *2.25 of [Whitehea... |
| orel2 901 | Elimination of disjunction... |
| pm2.67-2 902 | Slight generalization of T... |
| pm2.67 903 | Theorem *2.67 of [Whitehea... |
| curryax 904 | A non-intuitionistic posit... |
| exmid 905 | Law of excluded middle, al... |
| exmidd 906 | Law of excluded middle in ... |
| pm2.1 907 | Theorem *2.1 of [Whitehead... |
| pm2.13 908 | Theorem *2.13 of [Whitehea... |
| pm2.621 909 | Theorem *2.621 of [Whitehe... |
| pm2.62 910 | Theorem *2.62 of [Whitehea... |
| pm2.68 911 | Theorem *2.68 of [Whitehea... |
| dfor2 912 | Logical 'or' expressed in ... |
| pm2.07 913 | Theorem *2.07 of [Whitehea... |
| pm1.2 914 | Axiom *1.2 of [WhiteheadRu... |
| oridm 915 | Idempotent law for disjunc... |
| pm4.25 916 | Theorem *4.25 of [Whitehea... |
| pm2.4 917 | Theorem *2.4 of [Whitehead... |
| pm2.41 918 | Theorem *2.41 of [Whitehea... |
| orim12i 919 | Disjoin antecedents and co... |
| orim1i 920 | Introduce disjunct to both... |
| orim2i 921 | Introduce disjunct to both... |
| orim12dALT 922 | Alternate proof of ~ orim1... |
| orbi2i 923 | Inference adding a left di... |
| orbi1i 924 | Inference adding a right d... |
| orbi12i 925 | Infer the disjunction of t... |
| orbi2d 926 | Deduction adding a left di... |
| orbi1d 927 | Deduction adding a right d... |
| orbi1 928 | Theorem *4.37 of [Whitehea... |
| orbi12d 929 | Deduction joining two equi... |
| pm1.5 930 | Axiom *1.5 (Assoc) of [Whi... |
| or12 931 | Swap two disjuncts. (Cont... |
| orass 932 | Associative law for disjun... |
| pm2.31 933 | Theorem *2.31 of [Whitehea... |
| pm2.32 934 | Theorem *2.32 of [Whitehea... |
| pm2.3 935 | Theorem *2.3 of [Whitehead... |
| or32 936 | A rearrangement of disjunc... |
| or4 937 | Rearrangement of 4 disjunc... |
| or42 938 | Rearrangement of 4 disjunc... |
| orordi 939 | Distribution of disjunctio... |
| orordir 940 | Distribution of disjunctio... |
| orimdi 941 | Disjunction distributes ov... |
| pm2.76 942 | Theorem *2.76 of [Whitehea... |
| pm2.85 943 | Theorem *2.85 of [Whitehea... |
| pm2.75 944 | Theorem *2.75 of [Whitehea... |
| pm4.78 945 | Implication distributes ov... |
| biort 946 | A disjunction with a true ... |
| biorf 947 | A wff is equivalent to its... |
| biortn 948 | A wff is equivalent to its... |
| biorfi 949 | The dual of ~ biorf is not... |
| biorfri 950 | A wff is equivalent to its... |
| biorfriOLD 951 | Obsolete version of ~ bior... |
| pm2.26 952 | Theorem *2.26 of [Whitehea... |
| pm2.63 953 | Theorem *2.63 of [Whitehea... |
| pm2.64 954 | Theorem *2.64 of [Whitehea... |
| pm2.42 955 | Theorem *2.42 of [Whitehea... |
| pm5.11g 956 | A general instance of Theo... |
| pm5.11 957 | Theorem *5.11 of [Whitehea... |
| pm5.12 958 | Theorem *5.12 of [Whitehea... |
| pm5.14 959 | Theorem *5.14 of [Whitehea... |
| pm5.13 960 | Theorem *5.13 of [Whitehea... |
| pm5.55 961 | Theorem *5.55 of [Whitehea... |
| pm4.72 962 | Implication in terms of bi... |
| imimorb 963 | Simplify an implication be... |
| oibabs 964 | Absorption of disjunction ... |
| orbidi 965 | Disjunction distributes ov... |
| pm5.7 966 | Disjunction distributes ov... |
| jaao 967 | Inference conjoining and d... |
| jaoa 968 | Inference disjoining and c... |
| jaoian 969 | Inference disjoining the a... |
| jaodan 970 | Deduction disjoining the a... |
| mpjaodan 971 | Eliminate a disjunction in... |
| pm3.44 972 | Theorem *3.44 of [Whitehea... |
| jao 973 | Disjunction of antecedents... |
| jaob 974 | Disjunction of antecedents... |
| pm4.77 975 | Theorem *4.77 of [Whitehea... |
| pm3.48 976 | Theorem *3.48 of [Whitehea... |
| orim12d 977 | Disjoin antecedents and co... |
| orim12da 978 | Deduce a disjunction from ... |
| orim1d 979 | Disjoin antecedents and co... |
| orim2d 980 | Disjoin antecedents and co... |
| orim2 981 | Axiom *1.6 (Sum) of [White... |
| pm2.38 982 | Theorem *2.38 of [Whitehea... |
| pm2.36 983 | Theorem *2.36 of [Whitehea... |
| pm2.37 984 | Theorem *2.37 of [Whitehea... |
| pm2.81 985 | Theorem *2.81 of [Whitehea... |
| pm2.8 986 | Theorem *2.8 of [Whitehead... |
| pm2.73 987 | Theorem *2.73 of [Whitehea... |
| pm2.74 988 | Theorem *2.74 of [Whitehea... |
| pm2.82 989 | Theorem *2.82 of [Whitehea... |
| pm4.39 990 | Theorem *4.39 of [Whitehea... |
| animorl 991 | Conjunction implies disjun... |
| animorr 992 | Conjunction implies disjun... |
| animorlr 993 | Conjunction implies disjun... |
| animorrl 994 | Conjunction implies disjun... |
| ianor 995 | Negated conjunction in ter... |
| anor 996 | Conjunction in terms of di... |
| ioran 997 | Negated disjunction in ter... |
| pm4.52 998 | Theorem *4.52 of [Whitehea... |
| pm4.53 999 | Theorem *4.53 of [Whitehea... |
| pm4.54 1000 | Theorem *4.54 of [Whitehea... |
| pm4.55 1001 | Theorem *4.55 of [Whitehea... |
| pm4.56 1002 | Theorem *4.56 of [Whitehea... |
| oran 1003 | Disjunction in terms of co... |
| pm4.57 1004 | Theorem *4.57 of [Whitehea... |
| pm3.1 1005 | Theorem *3.1 of [Whitehead... |
| pm3.11 1006 | Theorem *3.11 of [Whitehea... |
| pm3.12 1007 | Theorem *3.12 of [Whitehea... |
| pm3.13 1008 | Theorem *3.13 of [Whitehea... |
| pm3.14 1009 | Theorem *3.14 of [Whitehea... |
| pm4.44 1010 | Theorem *4.44 of [Whitehea... |
| pm4.45 1011 | Theorem *4.45 of [Whitehea... |
| orabs 1012 | Absorption of redundant in... |
| oranabs 1013 | Absorb a disjunct into a c... |
| pm5.61 1014 | Theorem *5.61 of [Whitehea... |
| pm5.6 1015 | Conjunction in antecedent ... |
| orcanai 1016 | Change disjunction in cons... |
| pm4.79 1017 | Theorem *4.79 of [Whitehea... |
| pm5.53 1018 | Theorem *5.53 of [Whitehea... |
| ordi 1019 | Distributive law for disju... |
| ordir 1020 | Distributive law for disju... |
| andi 1021 | Distributive law for conju... |
| andir 1022 | Distributive law for conju... |
| orddi 1023 | Double distributive law fo... |
| anddi 1024 | Double distributive law fo... |
| pm5.17 1025 | Theorem *5.17 of [Whitehea... |
| pm5.15 1026 | Theorem *5.15 of [Whitehea... |
| pm5.16 1027 | Theorem *5.16 of [Whitehea... |
| xor 1028 | Two ways to express exclus... |
| nbi2 1029 | Two ways to express "exclu... |
| xordi 1030 | Conjunction distributes ov... |
| pm5.54 1031 | Theorem *5.54 of [Whitehea... |
| pm5.62 1032 | Theorem *5.62 of [Whitehea... |
| pm5.63 1033 | Theorem *5.63 of [Whitehea... |
| niabn 1034 | Miscellaneous inference re... |
| ninba 1035 | Miscellaneous inference re... |
| pm4.43 1036 | Theorem *4.43 of [Whitehea... |
| pm4.82 1037 | Theorem *4.82 of [Whitehea... |
| pm4.83 1038 | Theorem *4.83 of [Whitehea... |
| pclem6 1039 | Negation inferred from emb... |
| bigolden 1040 | Dijkstra-Scholten's Golden... |
| pm5.71 1041 | Theorem *5.71 of [Whitehea... |
| pm5.75 1042 | Theorem *5.75 of [Whitehea... |
| ecase2d 1043 | Deduction for elimination ... |
| ecase3 1044 | Inference for elimination ... |
| ecase 1045 | Inference for elimination ... |
| ecase3d 1046 | Deduction for elimination ... |
| ecased 1047 | Deduction for elimination ... |
| ecase3ad 1048 | Deduction for elimination ... |
| ccase 1049 | Inference for combining ca... |
| ccased 1050 | Deduction for combining ca... |
| ccase2 1051 | Inference for combining ca... |
| 4cases 1052 | Inference eliminating two ... |
| 4casesdan 1053 | Deduction eliminating two ... |
| cases 1054 | Case disjunction according... |
| dedlem0a 1055 | Lemma for an alternate ver... |
| dedlem0b 1056 | Lemma for an alternate ver... |
| dedlema 1057 | Lemma for weak deduction t... |
| dedlemb 1058 | Lemma for weak deduction t... |
| cases2 1059 | Case disjunction according... |
| cases2ALT 1060 | Alternate proof of ~ cases... |
| dfbi3 1061 | An alternate definition of... |
| pm5.24 1062 | Theorem *5.24 of [Whitehea... |
| 4exmid 1063 | The disjunction of the fou... |
| consensus 1064 | The consensus theorem. Th... |
| pm4.42 1065 | Theorem *4.42 of [Whitehea... |
| prlem1 1066 | A specialized lemma for se... |
| prlem2 1067 | A specialized lemma for se... |
| oplem1 1068 | A specialized lemma for se... |
| dn1 1069 | A single axiom for Boolean... |
| bianir 1070 | A closed form of ~ mpbir ,... |
| jaoi2 1071 | Inference removing a negat... |
| jaoi3 1072 | Inference separating a dis... |
| ornld 1073 | Selecting one statement fr... |
| dfifp2 1076 | Alternate definition of th... |
| dfifp3 1077 | Alternate definition of th... |
| dfifp4 1078 | Alternate definition of th... |
| dfifp5 1079 | Alternate definition of th... |
| dfifp6 1080 | Alternate definition of th... |
| dfifp7 1081 | Alternate definition of th... |
| ifpdfbi 1082 | Define the biconditional a... |
| ifpdfbiOLD 1083 | Obsolete version of ~ ifpd... |
| anifp 1084 | The conditional operator i... |
| ifpor 1085 | The conditional operator i... |
| ifpn 1086 | Conditional operator for t... |
| ifptru 1087 | Value of the conditional o... |
| ifpfal 1088 | Value of the conditional o... |
| ifpid 1089 | Value of the conditional o... |
| casesifp 1090 | Version of ~ cases express... |
| ifpbi123d 1091 | Equivalence deduction for ... |
| ifpbi23d 1092 | Equivalence deduction for ... |
| ifpimpda 1093 | Separation of the values o... |
| 1fpid3 1094 | The value of the condition... |
| elimh 1095 | Hypothesis builder for the... |
| dedt 1096 | The weak deduction theorem... |
| con3ALT 1097 | Proof of ~ con3 from its a... |
| 3orass 1102 | Associative law for triple... |
| 3orel1 1103 | Partial elimination of a t... |
| 3orrot 1104 | Rotation law for triple di... |
| 3orcoma 1105 | Commutation law for triple... |
| 3orcomb 1106 | Commutation law for triple... |
| 3anass 1107 | Associative law for triple... |
| 3anan12 1108 | Convert triple conjunction... |
| 3anan32 1109 | Convert triple conjunction... |
| 3anan32OLD 1110 | Obsolete version of ~ 3ana... |
| 3ancoma 1111 | Commutation law for triple... |
| 3ancomb 1112 | Commutation law for triple... |
| 3anrot 1113 | Rotation law for triple co... |
| 3anrev 1114 | Reversal law for triple co... |
| anandi3 1115 | Distribution of triple con... |
| anandi3r 1116 | Distribution of triple con... |
| 3anidm 1117 | Idempotent law for conjunc... |
| 3an4anass 1118 | Associative law for four c... |
| 3ioran 1119 | Negated triple disjunction... |
| 3ianor 1120 | Negated triple conjunction... |
| 3anor 1121 | Triple conjunction express... |
| 3oran 1122 | Triple disjunction in term... |
| 3impa 1123 | Importation from double to... |
| 3imp 1124 | Importation inference. (C... |
| 3imp31 1125 | The importation inference ... |
| 3imp231 1126 | Importation inference. (C... |
| 3imp21 1127 | The importation inference ... |
| 3impb 1128 | Importation from double to... |
| bi23imp13 1129 | ~ 3imp with middle implica... |
| 3impib 1130 | Importation to triple conj... |
| 3impia 1131 | Importation to triple conj... |
| 3expa 1132 | Exportation from triple to... |
| 3exp 1133 | Exportation inference. (C... |
| 3expb 1134 | Exportation from triple to... |
| 3expia 1135 | Exportation from triple co... |
| 3expib 1136 | Exportation from triple co... |
| 3com12 1137 | Commutation in antecedent.... |
| 3com13 1138 | Commutation in antecedent.... |
| 3comr 1139 | Commutation in antecedent.... |
| 3com23 1140 | Commutation in antecedent.... |
| 3coml 1141 | Commutation in antecedent.... |
| 3jca 1142 | Join consequents with conj... |
| 3jcad 1143 | Deduction conjoining the c... |
| 3adant1 1144 | Deduction adding a conjunc... |
| 3adant2 1145 | Deduction adding a conjunc... |
| 3adant3 1146 | Deduction adding a conjunc... |
| 3ad2ant1 1147 | Deduction adding conjuncts... |
| 3ad2ant2 1148 | Deduction adding conjuncts... |
| 3ad2ant3 1149 | Deduction adding conjuncts... |
| simp1 1150 | Simplification of triple c... |
| simp2 1151 | Simplification of triple c... |
| simp3 1152 | Simplification of triple c... |
| simp1i 1153 | Infer a conjunct from a tr... |
| simp2i 1154 | Infer a conjunct from a tr... |
| simp3i 1155 | Infer a conjunct from a tr... |
| simp1d 1156 | Deduce a conjunct from a t... |
| simp2d 1157 | Deduce a conjunct from a t... |
| simp3d 1158 | Deduce a conjunct from a t... |
| simp1bi 1159 | Deduce a conjunct from a t... |
| simp2bi 1160 | Deduce a conjunct from a t... |
| simp3bi 1161 | Deduce a conjunct from a t... |
| 3simpa 1162 | Simplification of triple c... |
| 3simpb 1163 | Simplification of triple c... |
| 3simpc 1164 | Simplification of triple c... |
| 3anim123i 1165 | Join antecedents and conse... |
| 3anim1i 1166 | Add two conjuncts to antec... |
| 3anim2i 1167 | Add two conjuncts to antec... |
| 3anim3i 1168 | Add two conjuncts to antec... |
| 3anbi123i 1169 | Join 3 biconditionals with... |
| 3orbi123i 1170 | Join 3 biconditionals with... |
| 3anbi1i 1171 | Inference adding two conju... |
| 3anbi2i 1172 | Inference adding two conju... |
| 3anbi3i 1173 | Inference adding two conju... |
| syl3an 1174 | A triple syllogism inferen... |
| syl3anb 1175 | A triple syllogism inferen... |
| syl3anbr 1176 | A triple syllogism inferen... |
| syl3an1 1177 | A syllogism inference. (C... |
| syl3an2 1178 | A syllogism inference. (C... |
| syl3an3 1179 | A syllogism inference. (C... |
| syl3an132 1180 | ~ syl2an with antecedents ... |
| 3adantl1 1181 | Deduction adding a conjunc... |
| 3adantl2 1182 | Deduction adding a conjunc... |
| 3adantl3 1183 | Deduction adding a conjunc... |
| 3adantr1 1184 | Deduction adding a conjunc... |
| 3adantr2 1185 | Deduction adding a conjunc... |
| 3adantr3 1186 | Deduction adding a conjunc... |
| ad4ant123 1187 | Deduction adding conjuncts... |
| ad4ant124 1188 | Deduction adding conjuncts... |
| ad4ant134 1189 | Deduction adding conjuncts... |
| ad4ant234 1190 | Deduction adding conjuncts... |
| 3adant1l 1191 | Deduction adding a conjunc... |
| 3adant1r 1192 | Deduction adding a conjunc... |
| 3adant2l 1193 | Deduction adding a conjunc... |
| 3adant2r 1194 | Deduction adding a conjunc... |
| 3adant3l 1195 | Deduction adding a conjunc... |
| 3adant3r 1196 | Deduction adding a conjunc... |
| 3adant3r1 1197 | Deduction adding a conjunc... |
| 3adant3r2 1198 | Deduction adding a conjunc... |
| 3adant3r3 1199 | Deduction adding a conjunc... |
| 3ad2antl1 1200 | Deduction adding conjuncts... |
| 3ad2antl2 1201 | Deduction adding conjuncts... |
| 3ad2antl3 1202 | Deduction adding conjuncts... |
| 3ad2antr1 1203 | Deduction adding conjuncts... |
| 3ad2antr2 1204 | Deduction adding conjuncts... |
| 3ad2antr3 1205 | Deduction adding conjuncts... |
| simpl1 1206 | Simplification of conjunct... |
| simpl2 1207 | Simplification of conjunct... |
| simpl3 1208 | Simplification of conjunct... |
| simpr1 1209 | Simplification of conjunct... |
| simpr2 1210 | Simplification of conjunct... |
| simpr3 1211 | Simplification of conjunct... |
| simp1l 1212 | Simplification of triple c... |
| simp1r 1213 | Simplification of triple c... |
| simp2l 1214 | Simplification of triple c... |
| simp2r 1215 | Simplification of triple c... |
| simp3l 1216 | Simplification of triple c... |
| simp3r 1217 | Simplification of triple c... |
| simp11 1218 | Simplification of doubly t... |
| simp12 1219 | Simplification of doubly t... |
| simp13 1220 | Simplification of doubly t... |
| simp21 1221 | Simplification of doubly t... |
| simp22 1222 | Simplification of doubly t... |
| simp23 1223 | Simplification of doubly t... |
| simp31 1224 | Simplification of doubly t... |
| simp32 1225 | Simplification of doubly t... |
| simp33 1226 | Simplification of doubly t... |
| simpll1 1227 | Simplification of conjunct... |
| simpll2 1228 | Simplification of conjunct... |
| simpll3 1229 | Simplification of conjunct... |
| simplr1 1230 | Simplification of conjunct... |
| simplr2 1231 | Simplification of conjunct... |
| simplr3 1232 | Simplification of conjunct... |
| simprl1 1233 | Simplification of conjunct... |
| simprl2 1234 | Simplification of conjunct... |
| simprl3 1235 | Simplification of conjunct... |
| simprr1 1236 | Simplification of conjunct... |
| simprr2 1237 | Simplification of conjunct... |
| simprr3 1238 | Simplification of conjunct... |
| simpl1l 1239 | Simplification of conjunct... |
| simpl1r 1240 | Simplification of conjunct... |
| simpl2l 1241 | Simplification of conjunct... |
| simpl2r 1242 | Simplification of conjunct... |
| simpl3l 1243 | Simplification of conjunct... |
| simpl3r 1244 | Simplification of conjunct... |
| simpr1l 1245 | Simplification of conjunct... |
| simpr1r 1246 | Simplification of conjunct... |
| simpr2l 1247 | Simplification of conjunct... |
| simpr2r 1248 | Simplification of conjunct... |
| simpr3l 1249 | Simplification of conjunct... |
| simpr3r 1250 | Simplification of conjunct... |
| simp1ll 1251 | Simplification of conjunct... |
| simp1lr 1252 | Simplification of conjunct... |
| simp1rl 1253 | Simplification of conjunct... |
| simp1rr 1254 | Simplification of conjunct... |
| simp2ll 1255 | Simplification of conjunct... |
| simp2lr 1256 | Simplification of conjunct... |
| simp2rl 1257 | Simplification of conjunct... |
| simp2rr 1258 | Simplification of conjunct... |
| simp3ll 1259 | Simplification of conjunct... |
| simp3lr 1260 | Simplification of conjunct... |
| simp3rl 1261 | Simplification of conjunct... |
| simp3rr 1262 | Simplification of conjunct... |
| simpl11 1263 | Simplification of conjunct... |
| simpl12 1264 | Simplification of conjunct... |
| simpl13 1265 | Simplification of conjunct... |
| simpl21 1266 | Simplification of conjunct... |
| simpl22 1267 | Simplification of conjunct... |
| simpl23 1268 | Simplification of conjunct... |
| simpl31 1269 | Simplification of conjunct... |
| simpl32 1270 | Simplification of conjunct... |
| simpl33 1271 | Simplification of conjunct... |
| simpr11 1272 | Simplification of conjunct... |
| simpr12 1273 | Simplification of conjunct... |
| simpr13 1274 | Simplification of conjunct... |
| simpr21 1275 | Simplification of conjunct... |
| simpr22 1276 | Simplification of conjunct... |
| simpr23 1277 | Simplification of conjunct... |
| simpr31 1278 | Simplification of conjunct... |
| simpr32 1279 | Simplification of conjunct... |
| simpr33 1280 | Simplification of conjunct... |
| simp1l1 1281 | Simplification of conjunct... |
| simp1l2 1282 | Simplification of conjunct... |
| simp1l3 1283 | Simplification of conjunct... |
| simp1r1 1284 | Simplification of conjunct... |
| simp1r2 1285 | Simplification of conjunct... |
| simp1r3 1286 | Simplification of conjunct... |
| simp2l1 1287 | Simplification of conjunct... |
| simp2l2 1288 | Simplification of conjunct... |
| simp2l3 1289 | Simplification of conjunct... |
| simp2r1 1290 | Simplification of conjunct... |
| simp2r2 1291 | Simplification of conjunct... |
| simp2r3 1292 | Simplification of conjunct... |
| simp3l1 1293 | Simplification of conjunct... |
| simp3l2 1294 | Simplification of conjunct... |
| simp3l3 1295 | Simplification of conjunct... |
| simp3r1 1296 | Simplification of conjunct... |
| simp3r2 1297 | Simplification of conjunct... |
| simp3r3 1298 | Simplification of conjunct... |
| simp11l 1299 | Simplification of conjunct... |
| simp11r 1300 | Simplification of conjunct... |
| simp12l 1301 | Simplification of conjunct... |
| simp12r 1302 | Simplification of conjunct... |
| simp13l 1303 | Simplification of conjunct... |
| simp13r 1304 | Simplification of conjunct... |
| simp21l 1305 | Simplification of conjunct... |
| simp21r 1306 | Simplification of conjunct... |
| simp22l 1307 | Simplification of conjunct... |
| simp22r 1308 | Simplification of conjunct... |
| simp23l 1309 | Simplification of conjunct... |
| simp23r 1310 | Simplification of conjunct... |
| simp31l 1311 | Simplification of conjunct... |
| simp31r 1312 | Simplification of conjunct... |
| simp32l 1313 | Simplification of conjunct... |
| simp32r 1314 | Simplification of conjunct... |
| simp33l 1315 | Simplification of conjunct... |
| simp33r 1316 | Simplification of conjunct... |
| simp111 1317 | Simplification of conjunct... |
| simp112 1318 | Simplification of conjunct... |
| simp113 1319 | Simplification of conjunct... |
| simp121 1320 | Simplification of conjunct... |
| simp122 1321 | Simplification of conjunct... |
| simp123 1322 | Simplification of conjunct... |
| simp131 1323 | Simplification of conjunct... |
| simp132 1324 | Simplification of conjunct... |
| simp133 1325 | Simplification of conjunct... |
| simp211 1326 | Simplification of conjunct... |
| simp212 1327 | Simplification of conjunct... |
| simp213 1328 | Simplification of conjunct... |
| simp221 1329 | Simplification of conjunct... |
| simp222 1330 | Simplification of conjunct... |
| simp223 1331 | Simplification of conjunct... |
| simp231 1332 | Simplification of conjunct... |
| simp232 1333 | Simplification of conjunct... |
| simp233 1334 | Simplification of conjunct... |
| simp311 1335 | Simplification of conjunct... |
| simp312 1336 | Simplification of conjunct... |
| simp313 1337 | Simplification of conjunct... |
| simp321 1338 | Simplification of conjunct... |
| simp322 1339 | Simplification of conjunct... |
| simp323 1340 | Simplification of conjunct... |
| simp331 1341 | Simplification of conjunct... |
| simp332 1342 | Simplification of conjunct... |
| simp333 1343 | Simplification of conjunct... |
| 3anibar 1344 | Remove a hypothesis from t... |
| 3mix1 1345 | Introduction in triple dis... |
| 3mix2 1346 | Introduction in triple dis... |
| 3mix3 1347 | Introduction in triple dis... |
| 3mix1i 1348 | Introduction in triple dis... |
| 3mix2i 1349 | Introduction in triple dis... |
| 3mix3i 1350 | Introduction in triple dis... |
| 3mix1d 1351 | Deduction introducing trip... |
| 3mix2d 1352 | Deduction introducing trip... |
| 3mix3d 1353 | Deduction introducing trip... |
| 3pm3.2i 1354 | Infer conjunction of premi... |
| pm3.2an3 1355 | Version of ~ pm3.2 for a t... |
| mpbir3an 1356 | Detach a conjunction of tr... |
| mpbir3and 1357 | Detach a conjunction of tr... |
| syl3anbrc 1358 | Syllogism inference. (Con... |
| syl21anbrc 1359 | Syllogism inference. (Con... |
| 3imp3i2an 1360 | An elimination deduction. ... |
| ex3 1361 | Apply ~ ex to a hypothesis... |
| 3imp1 1362 | Importation to left triple... |
| 3impd 1363 | Importation deduction for ... |
| 3imp2 1364 | Importation to right tripl... |
| 3impdi 1365 | Importation inference (und... |
| 3impdir 1366 | Importation inference (und... |
| 3exp1 1367 | Exportation from left trip... |
| 3expd 1368 | Exportation deduction for ... |
| 3exp2 1369 | Exportation from right tri... |
| exp5o 1370 | A triple exportation infer... |
| exp516 1371 | A triple exportation infer... |
| exp520 1372 | A triple exportation infer... |
| 3impexp 1373 | Version of ~ impexp for a ... |
| 3an1rs 1374 | Swap conjuncts. (Contribu... |
| 3anassrs 1375 | Associative law for conjun... |
| 4anpull2 1376 | An equivalence of two four... |
| 4anpull2OLD 1377 | Obsolete version of ~ 4anp... |
| ad5ant245 1378 | Deduction adding conjuncts... |
| ad5ant234 1379 | Deduction adding conjuncts... |
| ad5ant235 1380 | Deduction adding conjuncts... |
| ad5ant123 1381 | Deduction adding conjuncts... |
| ad5ant124 1382 | Deduction adding conjuncts... |
| ad5ant124OLD 1383 | Obsolete version of ~ ad5a... |
| ad5ant125 1384 | Deduction adding conjuncts... |
| ad5ant125OLD 1385 | Obsolete version of ~ ad5a... |
| ad5ant134 1386 | Deduction adding conjuncts... |
| ad5ant134OLD 1387 | Obsolete version of ~ ad5a... |
| ad5ant135 1388 | Deduction adding conjuncts... |
| ad5ant135OLD 1389 | Obsolete version of ~ ad5a... |
| ad5ant145 1390 | Deduction adding conjuncts... |
| ad5ant2345 1391 | Deduction adding conjuncts... |
| syl3anc 1392 | Syllogism combined with co... |
| syl13anc 1393 | Syllogism combined with co... |
| syl31anc 1394 | Syllogism combined with co... |
| syl112anc 1395 | Syllogism combined with co... |
| syl121anc 1396 | Syllogism combined with co... |
| syl211anc 1397 | Syllogism combined with co... |
| syl23anc 1398 | Syllogism combined with co... |
| syl32anc 1399 | Syllogism combined with co... |
| syl122anc 1400 | Syllogism combined with co... |
| syl212anc 1401 | Syllogism combined with co... |
| syl221anc 1402 | Syllogism combined with co... |
| syl113anc 1403 | Syllogism combined with co... |
| syl131anc 1404 | Syllogism combined with co... |
| syl311anc 1405 | Syllogism combined with co... |
| syl33anc 1406 | Syllogism combined with co... |
| syl222anc 1407 | Syllogism combined with co... |
| syl123anc 1408 | Syllogism combined with co... |
| syl132anc 1409 | Syllogism combined with co... |
| syl213anc 1410 | Syllogism combined with co... |
| syl231anc 1411 | Syllogism combined with co... |
| syl312anc 1412 | Syllogism combined with co... |
| syl321anc 1413 | Syllogism combined with co... |
| syl133anc 1414 | Syllogism combined with co... |
| syl313anc 1415 | Syllogism combined with co... |
| syl331anc 1416 | Syllogism combined with co... |
| syl223anc 1417 | Syllogism combined with co... |
| syl232anc 1418 | Syllogism combined with co... |
| syl322anc 1419 | Syllogism combined with co... |
| syl233anc 1420 | Syllogism combined with co... |
| syl323anc 1421 | Syllogism combined with co... |
| syl332anc 1422 | Syllogism combined with co... |
| syl333anc 1423 | A syllogism inference comb... |
| syl3an1b 1424 | A syllogism inference. (C... |
| syl3an2b 1425 | A syllogism inference. (C... |
| syl3an3b 1426 | A syllogism inference. (C... |
| syl3an1br 1427 | A syllogism inference. (C... |
| syl3an2br 1428 | A syllogism inference. (C... |
| syl3an3br 1429 | A syllogism inference. (C... |
| syld3an3 1430 | A syllogism inference. (C... |
| syld3an1 1431 | A syllogism inference. (C... |
| syld3an2 1432 | A syllogism inference. (C... |
| syl3anl1 1433 | A syllogism inference. (C... |
| syl3anl2 1434 | A syllogism inference. (C... |
| syl3anl3 1435 | A syllogism inference. (C... |
| syl3anl 1436 | A triple syllogism inferen... |
| syl3anr1 1437 | A syllogism inference. (C... |
| syl3anr2 1438 | A syllogism inference. (C... |
| syl3anr3 1439 | A syllogism inference. (C... |
| 3anidm12 1440 | Inference from idempotent ... |
| 3anidm13 1441 | Inference from idempotent ... |
| 3anidm23 1442 | Inference from idempotent ... |
| syl2an3an 1443 | ~ syl3an with antecedents ... |
| syl2an23an 1444 | Deduction related to ~ syl... |
| 3ori 1445 | Infer implication from tri... |
| 3jao 1446 | Disjunction of three antec... |
| 3jaob 1447 | Disjunction of three antec... |
| 3jaobOLD 1448 | Obsolete version of ~ 3jao... |
| 3jaoi 1449 | Disjunction of three antec... |
| 3jaoiOLD 1450 | Obsolete version of ~ 3jao... |
| 3jaod 1451 | Disjunction of three antec... |
| 3jaoian 1452 | Disjunction of three antec... |
| 3jaodan 1453 | Disjunction of three antec... |
| mpjao3dan 1454 | Eliminate a three-way disj... |
| 3jaao 1455 | Inference conjoining and d... |
| 3jaaoOLD 1456 | Obsolete version of ~ 3jaa... |
| syl3an9b 1457 | Nested syllogism inference... |
| 3orbi123d 1458 | Deduction joining 3 equiva... |
| 3anbi123d 1459 | Deduction joining 3 equiva... |
| 3anbi12d 1460 | Deduction conjoining and a... |
| 3anbi13d 1461 | Deduction conjoining and a... |
| 3anbi23d 1462 | Deduction conjoining and a... |
| 3anbi1d 1463 | Deduction adding conjuncts... |
| 3anbi2d 1464 | Deduction adding conjuncts... |
| 3anbi3d 1465 | Deduction adding conjuncts... |
| 3anim123d 1466 | Deduction joining 3 implic... |
| 3orim123d 1467 | Deduction joining 3 implic... |
| an6 1468 | Rearrangement of 6 conjunc... |
| 3an6 1469 | Analogue of ~ an4 for trip... |
| 3or6 1470 | Analogue of ~ or4 for trip... |
| mp3an1 1471 | An inference based on modu... |
| mp3an2 1472 | An inference based on modu... |
| mp3an3 1473 | An inference based on modu... |
| mp3an12 1474 | An inference based on modu... |
| mp3an13 1475 | An inference based on modu... |
| mp3an23 1476 | An inference based on modu... |
| mp3an1i 1477 | An inference based on modu... |
| mp3anl1 1478 | An inference based on modu... |
| mp3anl2 1479 | An inference based on modu... |
| mp3anl3 1480 | An inference based on modu... |
| mp3anr1 1481 | An inference based on modu... |
| mp3anr2 1482 | An inference based on modu... |
| mp3anr3 1483 | An inference based on modu... |
| mp3an 1484 | An inference based on modu... |
| mpd3an3 1485 | An inference based on modu... |
| mpd3an23 1486 | An inference based on modu... |
| mp3and 1487 | A deduction based on modus... |
| mp3an12i 1488 | ~ mp3an with antecedents i... |
| mp3an2i 1489 | ~ mp3an with antecedents i... |
| mp3an3an 1490 | ~ mp3an with antecedents i... |
| mp3an2ani 1491 | An elimination deduction. ... |
| biimp3a 1492 | Infer implication from a l... |
| biimp3ar 1493 | Infer implication from a l... |
| 3anandis 1494 | Inference that undistribut... |
| 3anandirs 1495 | Inference that undistribut... |
| ecase23d 1496 | Deduction for elimination ... |
| 3ecase 1497 | Inference for elimination ... |
| 3bior1fd 1498 | A disjunction is equivalen... |
| 3bior1fand 1499 | A disjunction is equivalen... |
| 3bior2fd 1500 | A wff is equivalent to its... |
| 3biant1d 1501 | A conjunction is equivalen... |
| intn3an1d 1502 | Introduction of a triple c... |
| intn3an2d 1503 | Introduction of a triple c... |
| intn3an3d 1504 | Introduction of a triple c... |
| an3andi 1505 | Distribution of conjunctio... |
| an33rean 1506 | Rearrange a 9-fold conjunc... |
| 3orel2 1507 | Partial elimination of a t... |
| 3orel2OLD 1508 | Obsolete version of ~ 3ore... |
| 3orel3 1509 | Partial elimination of a t... |
| 3orel13 1510 | Elimination of two disjunc... |
| 3pm3.2ni 1511 | Triple negated disjunction... |
| an42ds 1512 | Inference exchanging the l... |
| nanan 1515 | Conjunction in terms of al... |
| dfnan2 1516 | Alternative denial in term... |
| nanor 1517 | Alternative denial in term... |
| nancom 1518 | Alternative denial is comm... |
| nannan 1519 | Nested alternative denials... |
| nanim 1520 | Implication in terms of al... |
| nannot 1521 | Negation in terms of alter... |
| nanbi 1522 | Biconditional in terms of ... |
| nanbi1 1523 | Introduce a right anti-con... |
| nanbi2 1524 | Introduce a left anti-conj... |
| nanbi12 1525 | Join two logical equivalen... |
| nanbi1i 1526 | Introduce a right anti-con... |
| nanbi2i 1527 | Introduce a left anti-conj... |
| nanbi12i 1528 | Join two logical equivalen... |
| nanbi1d 1529 | Introduce a right anti-con... |
| nanbi2d 1530 | Introduce a left anti-conj... |
| nanbi12d 1531 | Join two logical equivalen... |
| nanass 1532 | A characterization of when... |
| xnor 1535 | Two ways to write XNOR (ex... |
| xorcom 1536 | The connector ` \/_ ` is c... |
| xorass 1537 | The connector ` \/_ ` is a... |
| excxor 1538 | This tautology shows that ... |
| xor2 1539 | Two ways to express "exclu... |
| xoror 1540 | Exclusive disjunction impl... |
| xornan 1541 | Exclusive disjunction impl... |
| xornan2 1542 | XOR implies NAND (written ... |
| xorneg2 1543 | The connector ` \/_ ` is n... |
| xorneg1 1544 | The connector ` \/_ ` is n... |
| xorneg 1545 | The connector ` \/_ ` is u... |
| xorbi12i 1546 | Equality property for excl... |
| xorbi12d 1547 | Equality property for excl... |
| anxordi 1548 | Conjunction distributes ov... |
| xorexmid 1549 | Exclusive-or variant of th... |
| norcom 1552 | The connector ` -\/ ` is c... |
| nornot 1553 | ` -. ` is expressible via ... |
| noran 1554 | ` /\ ` is expressible via ... |
| noror 1555 | ` \/ ` is expressible via ... |
| norasslem1 1556 | This lemma shows the equiv... |
| norasslem2 1557 | This lemma specializes ~ b... |
| norasslem3 1558 | This lemma specializes ~ b... |
| norass 1559 | A characterization of when... |
| trujust 1564 | Soundness justification th... |
| tru 1566 | The truth value ` T. ` is ... |
| dftru2 1567 | An alternate definition of... |
| trut 1568 | A proposition is equivalen... |
| mptru 1569 | Eliminate ` T. ` as an ant... |
| tbtru 1570 | A proposition is equivalen... |
| bitru 1571 | A theorem is equivalent to... |
| trud 1572 | Anything implies ` T. ` . ... |
| truan 1573 | True can be removed from a... |
| fal 1576 | The truth value ` F. ` is ... |
| nbfal 1577 | The negation of a proposit... |
| bifal 1578 | A contradiction is equival... |
| falim 1579 | The truth value ` F. ` imp... |
| falimd 1580 | The truth value ` F. ` imp... |
| dfnot 1581 | Given falsum ` F. ` , we c... |
| inegd 1582 | Negation introduction rule... |
| efald 1583 | Deduction based on reducti... |
| pm2.21fal 1584 | If a wff and its negation ... |
| truimtru 1585 | A ` -> ` identity. (Contr... |
| truimfal 1586 | A ` -> ` identity. (Contr... |
| falimtru 1587 | A ` -> ` identity. (Contr... |
| falimfal 1588 | A ` -> ` identity. (Contr... |
| nottru 1589 | A ` -. ` identity. (Contr... |
| notfal 1590 | A ` -. ` identity. (Contr... |
| trubitru 1591 | A ` <-> ` identity. (Cont... |
| falbitru 1592 | A ` <-> ` identity. (Cont... |
| trubifal 1593 | A ` <-> ` identity. (Cont... |
| falbifal 1594 | A ` <-> ` identity. (Cont... |
| truantru 1595 | A ` /\ ` identity. (Contr... |
| truanfal 1596 | A ` /\ ` identity. (Contr... |
| falantru 1597 | A ` /\ ` identity. (Contr... |
| falanfal 1598 | A ` /\ ` identity. (Contr... |
| truortru 1599 | A ` \/ ` identity. (Contr... |
| truorfal 1600 | A ` \/ ` identity. (Contr... |
| falortru 1601 | A ` \/ ` identity. (Contr... |
| falorfal 1602 | A ` \/ ` identity. (Contr... |
| trunantru 1603 | A ` -/\ ` identity. (Cont... |
| trunanfal 1604 | A ` -/\ ` identity. (Cont... |
| falnantru 1605 | A ` -/\ ` identity. (Cont... |
| falnanfal 1606 | A ` -/\ ` identity. (Cont... |
| truxortru 1607 | A ` \/_ ` identity. (Cont... |
| truxorfal 1608 | A ` \/_ ` identity. (Cont... |
| falxortru 1609 | A ` \/_ ` identity. (Cont... |
| falxorfal 1610 | A ` \/_ ` identity. (Cont... |
| trunortru 1611 | A ` -\/ ` identity. (Cont... |
| trunorfal 1612 | A ` -\/ ` identity. (Cont... |
| falnortru 1613 | A ` -\/ ` identity. (Cont... |
| falnorfal 1614 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1617 | Equality theorem for the a... |
| hadbi123i 1618 | Equality theorem for the a... |
| hadass 1619 | Associative law for the ad... |
| hadbi 1620 | The adder sum is the same ... |
| hadcoma 1621 | Commutative law for the ad... |
| hadcomb 1622 | Commutative law for the ad... |
| hadrot 1623 | Rotation law for the adder... |
| hadnot 1624 | The adder sum distributes ... |
| had1 1625 | If the first input is true... |
| had0 1626 | If the first input is fals... |
| hadifp 1627 | The value of the adder sum... |
| cador 1630 | The adder carry in disjunc... |
| cadan 1631 | The adder carry in conjunc... |
| cadbi123d 1632 | Equality theorem for the a... |
| cadbi123i 1633 | Equality theorem for the a... |
| cadcoma 1634 | Commutative law for the ad... |
| cadcomb 1635 | Commutative law for the ad... |
| cadrot 1636 | Rotation law for the adder... |
| cadnot 1637 | The adder carry distribute... |
| cad11 1638 | If (at least) two inputs a... |
| cad1 1639 | If one input is true, then... |
| cad0 1640 | If one input is false, the... |
| cadifp 1641 | The value of the carry is,... |
| cadtru 1642 | The adder carry is true as... |
| minimp 1643 | A single axiom for minimal... |
| minimp-syllsimp 1644 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1645 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1646 | Derivation of a commuted f... |
| minimp-ax2 1647 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1648 | Derivation of ~ pm2.43 (al... |
| impsingle 1649 | The shortest single axiom ... |
| impsingle-step4 1650 | Derivation of impsingle-st... |
| impsingle-step8 1651 | Derivation of impsingle-st... |
| impsingle-ax1 1652 | Derivation of impsingle-ax... |
| impsingle-step15 1653 | Derivation of impsingle-st... |
| impsingle-step18 1654 | Derivation of impsingle-st... |
| impsingle-step19 1655 | Derivation of impsingle-st... |
| impsingle-step20 1656 | Derivation of impsingle-st... |
| impsingle-step21 1657 | Derivation of impsingle-st... |
| impsingle-step22 1658 | Derivation of impsingle-st... |
| impsingle-step25 1659 | Derivation of impsingle-st... |
| impsingle-imim1 1660 | Derivation of impsingle-im... |
| impsingle-peirce 1661 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1662 | Derivation of ~ ax-2 from ... |
| meredith 1663 | Carew Meredith's sole axio... |
| merlem1 1664 | Step 3 of Meredith's proof... |
| merlem2 1665 | Step 4 of Meredith's proof... |
| merlem3 1666 | Step 7 of Meredith's proof... |
| merlem4 1667 | Step 8 of Meredith's proof... |
| merlem5 1668 | Step 11 of Meredith's proo... |
| merlem6 1669 | Step 12 of Meredith's proo... |
| merlem7 1670 | Between steps 14 and 15 of... |
| merlem8 1671 | Step 15 of Meredith's proo... |
| merlem9 1672 | Step 18 of Meredith's proo... |
| merlem10 1673 | Step 19 of Meredith's proo... |
| merlem11 1674 | Step 20 of Meredith's proo... |
| merlem12 1675 | Step 28 of Meredith's proo... |
| merlem13 1676 | Step 35 of Meredith's proo... |
| luk-1 1677 | 1 of 3 axioms for proposit... |
| luk-2 1678 | 2 of 3 axioms for proposit... |
| luk-3 1679 | 3 of 3 axioms for proposit... |
| luklem1 1680 | Used to rederive standard ... |
| luklem2 1681 | Used to rederive standard ... |
| luklem3 1682 | Used to rederive standard ... |
| luklem4 1683 | Used to rederive standard ... |
| luklem5 1684 | Used to rederive standard ... |
| luklem6 1685 | Used to rederive standard ... |
| luklem7 1686 | Used to rederive standard ... |
| luklem8 1687 | Used to rederive standard ... |
| ax1 1688 | Standard propositional axi... |
| ax2 1689 | Standard propositional axi... |
| ax3 1690 | Standard propositional axi... |
| nic-dfim 1691 | This theorem "defines" imp... |
| nic-dfneg 1692 | This theorem "defines" neg... |
| nic-mp 1693 | Derive Nicod's rule of mod... |
| nic-mpALT 1694 | A direct proof of ~ nic-mp... |
| nic-ax 1695 | Nicod's axiom derived from... |
| nic-axALT 1696 | A direct proof of ~ nic-ax... |
| nic-imp 1697 | Inference for ~ nic-mp usi... |
| nic-idlem1 1698 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1699 | Lemma for ~ nic-id . Infe... |
| nic-id 1700 | Theorem ~ id expressed wit... |
| nic-swap 1701 | The connector ` -/\ ` is s... |
| nic-isw1 1702 | Inference version of ~ nic... |
| nic-isw2 1703 | Inference for swapping nes... |
| nic-iimp1 1704 | Inference version of ~ nic... |
| nic-iimp2 1705 | Inference version of ~ nic... |
| nic-idel 1706 | Inference to remove the tr... |
| nic-ich 1707 | Chained inference. (Contr... |
| nic-idbl 1708 | Double the terms. Since d... |
| nic-bijust 1709 | Biconditional justificatio... |
| nic-bi1 1710 | Inference to extract one s... |
| nic-bi2 1711 | Inference to extract the o... |
| nic-stdmp 1712 | Derive the standard modus ... |
| nic-luk1 1713 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1714 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1715 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1716 | This alternative axiom for... |
| lukshefth1 1717 | Lemma for ~ renicax . (Co... |
| lukshefth2 1718 | Lemma for ~ renicax . (Co... |
| renicax 1719 | A rederivation of ~ nic-ax... |
| tbw-bijust 1720 | Justification for ~ tbw-ne... |
| tbw-negdf 1721 | The definition of negation... |
| tbw-ax1 1722 | The first of four axioms i... |
| tbw-ax2 1723 | The second of four axioms ... |
| tbw-ax3 1724 | The third of four axioms i... |
| tbw-ax4 1725 | The fourth of four axioms ... |
| tbwsyl 1726 | Used to rederive the Lukas... |
| tbwlem1 1727 | Used to rederive the Lukas... |
| tbwlem2 1728 | Used to rederive the Lukas... |
| tbwlem3 1729 | Used to rederive the Lukas... |
| tbwlem4 1730 | Used to rederive the Lukas... |
| tbwlem5 1731 | Used to rederive the Lukas... |
| re1luk1 1732 | ~ luk-1 derived from the T... |
| re1luk2 1733 | ~ luk-2 derived from the T... |
| re1luk3 1734 | ~ luk-3 derived from the T... |
| merco1 1735 | A single axiom for proposi... |
| merco1lem1 1736 | Used to rederive the Tarsk... |
| retbwax4 1737 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1738 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1739 | Used to rederive the Tarsk... |
| merco1lem3 1740 | Used to rederive the Tarsk... |
| merco1lem4 1741 | Used to rederive the Tarsk... |
| merco1lem5 1742 | Used to rederive the Tarsk... |
| merco1lem6 1743 | Used to rederive the Tarsk... |
| merco1lem7 1744 | Used to rederive the Tarsk... |
| retbwax3 1745 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1746 | Used to rederive the Tarsk... |
| merco1lem9 1747 | Used to rederive the Tarsk... |
| merco1lem10 1748 | Used to rederive the Tarsk... |
| merco1lem11 1749 | Used to rederive the Tarsk... |
| merco1lem12 1750 | Used to rederive the Tarsk... |
| merco1lem13 1751 | Used to rederive the Tarsk... |
| merco1lem14 1752 | Used to rederive the Tarsk... |
| merco1lem15 1753 | Used to rederive the Tarsk... |
| merco1lem16 1754 | Used to rederive the Tarsk... |
| merco1lem17 1755 | Used to rederive the Tarsk... |
| merco1lem18 1756 | Used to rederive the Tarsk... |
| retbwax1 1757 | ~ tbw-ax1 rederived from ~... |
| merco2 1758 | A single axiom for proposi... |
| mercolem1 1759 | Used to rederive the Tarsk... |
| mercolem2 1760 | Used to rederive the Tarsk... |
| mercolem3 1761 | Used to rederive the Tarsk... |
| mercolem4 1762 | Used to rederive the Tarsk... |
| mercolem5 1763 | Used to rederive the Tarsk... |
| mercolem6 1764 | Used to rederive the Tarsk... |
| mercolem7 1765 | Used to rederive the Tarsk... |
| mercolem8 1766 | Used to rederive the Tarsk... |
| re1tbw1 1767 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1768 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1769 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1770 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1771 | Justification for ~ rb-imd... |
| rb-imdf 1772 | The definition of implicat... |
| anmp 1773 | Modus ponens for ` { \/ , ... |
| rb-ax1 1774 | The first of four axioms i... |
| rb-ax2 1775 | The second of four axioms ... |
| rb-ax3 1776 | The third of four axioms i... |
| rb-ax4 1777 | The fourth of four axioms ... |
| rbsyl 1778 | Used to rederive the Lukas... |
| rblem1 1779 | Used to rederive the Lukas... |
| rblem2 1780 | Used to rederive the Lukas... |
| rblem3 1781 | Used to rederive the Lukas... |
| rblem4 1782 | Used to rederive the Lukas... |
| rblem5 1783 | Used to rederive the Lukas... |
| rblem6 1784 | Used to rederive the Lukas... |
| rblem7 1785 | Used to rederive the Lukas... |
| re1axmp 1786 | ~ ax-mp derived from Russe... |
| re2luk1 1787 | ~ luk-1 derived from Russe... |
| re2luk2 1788 | ~ luk-2 derived from Russe... |
| re2luk3 1789 | ~ luk-3 derived from Russe... |
| mptnan 1790 | Modus ponendo tollens 1, o... |
| mptxor 1791 | Modus ponendo tollens 2, o... |
| mtpor 1792 | Modus tollendo ponens (inc... |
| mtpxor 1793 | Modus tollendo ponens (ori... |
| stoic1a 1794 | Stoic logic Thema 1 (part ... |
| stoic1b 1795 | Stoic logic Thema 1 (part ... |
| stoic2a 1796 | Stoic logic Thema 2 versio... |
| stoic2b 1797 | Stoic logic Thema 2 versio... |
| stoic3 1798 | Stoic logic Thema 3. Stat... |
| stoic4a 1799 | Stoic logic Thema 4 versio... |
| stoic4b 1800 | Stoic logic Thema 4 versio... |
| alnex 1803 | Universal quantification o... |
| eximal 1804 | An equivalence between an ... |
| nf2 1807 | Alternate definition of no... |
| nf3 1808 | Alternate definition of no... |
| nf4 1809 | Alternate definition of no... |
| nfi 1810 | Deduce that ` x ` is not f... |
| nfri 1811 | Consequence of the definit... |
| nfd 1812 | Deduce that ` x ` is not f... |
| nfrd 1813 | Consequence of the definit... |
| nftht 1814 | Closed form of ~ nfth . (... |
| nfntht 1815 | Closed form of ~ nfnth . ... |
| nfntht2 1816 | Closed form of ~ nfnth . ... |
| gen2 1818 | Generalization applied twi... |
| mpg 1819 | Modus ponens combined with... |
| mpgbi 1820 | Modus ponens on biconditio... |
| mpgbir 1821 | Modus ponens on biconditio... |
| nex 1822 | Generalization rule for ne... |
| nfth 1823 | No variable is (effectivel... |
| nfnth 1824 | No variable is (effectivel... |
| hbth 1825 | No variable is (effectivel... |
| nftru 1826 | The true constant has no f... |
| nffal 1827 | The false constant has no ... |
| sptruw 1828 | Version of ~ sp when ` ph ... |
| altru 1829 | For all sets, ` T. ` is tr... |
| alfal 1830 | For all sets, ` -. F. ` is... |
| alim 1832 | Restatement of Axiom ~ ax-... |
| alimi 1833 | Inference quantifying both... |
| 2alimi 1834 | Inference doubly quantifyi... |
| ala1 1835 | Add an antecedent in a uni... |
| al2im 1836 | Closed form of ~ al2imi . ... |
| al2imi 1837 | Inference quantifying ante... |
| alanimi 1838 | Variant of ~ al2imi with c... |
| alimdh 1839 | Deduction form of Theorem ... |
| albi 1840 | Theorem 19.15 of [Margaris... |
| albii 1841 | Inference adding universal... |
| 2albii 1842 | Inference adding two unive... |
| 3albii 1843 | Inference adding three uni... |
| sylgt 1844 | Closed form of ~ sylg . (... |
| sylg 1845 | A syllogism combined with ... |
| alrimih 1846 | Inference form of Theorem ... |
| hbxfrbi 1847 | A utility lemma to transfe... |
| alex 1848 | Universal quantifier in te... |
| exnal 1849 | Existential quantification... |
| 2nalexn 1850 | Part of theorem *11.5 in [... |
| 2exnaln 1851 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1852 | Theorem *11.25 in [Whitehe... |
| alimex 1853 | An equivalence between an ... |
| aleximi 1854 | A variant of ~ al2imi : in... |
| alexbii 1855 | Biconditional form of ~ al... |
| exim 1856 | Theorem 19.22 of [Margaris... |
| eximi 1857 | Inference adding existenti... |
| 2eximi 1858 | Inference adding two exist... |
| eximii 1859 | Inference associated with ... |
| exa1 1860 | Add an antecedent in an ex... |
| 19.38 1861 | Theorem 19.38 of [Margaris... |
| 19.38a 1862 | Under a nonfreeness hypoth... |
| 19.38b 1863 | Under a nonfreeness hypoth... |
| imnang 1864 | Quantified implication in ... |
| alinexa 1865 | A transformation of quanti... |
| exnalimn 1866 | Existential quantification... |
| alexn 1867 | A relationship between two... |
| 2exnexn 1868 | Theorem *11.51 in [Whitehe... |
| exbi 1869 | Theorem 19.18 of [Margaris... |
| exbii 1870 | Inference adding existenti... |
| 2exbii 1871 | Inference adding two exist... |
| 3exbii 1872 | Inference adding three exi... |
| nfbiit 1873 | Equivalence theorem for th... |
| nfbii 1874 | Equality theorem for the n... |
| nfxfr 1875 | A utility lemma to transfe... |
| nfxfrd 1876 | A utility lemma to transfe... |
| nfnbi 1877 | A variable is nonfree in a... |
| nfnt 1878 | If a variable is nonfree i... |
| nfn 1879 | Inference associated with ... |
| nfnd 1880 | Deduction associated with ... |
| exanali 1881 | A transformation of quanti... |
| 2exanali 1882 | Theorem *11.521 in [Whiteh... |
| exancom 1883 | Commutation of conjunction... |
| exan 1884 | Place a conjunct in the sc... |
| alrimdh 1885 | Deduction form of Theorem ... |
| eximdh 1886 | Deduction from Theorem 19.... |
| nexdh 1887 | Deduction for generalizati... |
| albidh 1888 | Formula-building rule for ... |
| exbidh 1889 | Formula-building rule for ... |
| exsimpl 1890 | Simplification of an exist... |
| exsimpr 1891 | Simplification of an exist... |
| 19.26 1892 | Theorem 19.26 of [Margaris... |
| 19.26-2 1893 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1894 | Theorem ~ 19.26 with tripl... |
| 19.29 1895 | Theorem 19.29 of [Margaris... |
| 19.29r 1896 | Variation of ~ 19.29 . (C... |
| 19.29r2 1897 | Variation of ~ 19.29r with... |
| 19.29x 1898 | Variation of ~ 19.29 with ... |
| 19.35 1899 | Theorem 19.35 of [Margaris... |
| 19.35i 1900 | Inference associated with ... |
| 19.35ri 1901 | Inference associated with ... |
| 19.25 1902 | Theorem 19.25 of [Margaris... |
| 19.30 1903 | Theorem 19.30 of [Margaris... |
| 19.43 1904 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1905 | Obsolete proof of ~ 19.43 ... |
| 19.33 1906 | Theorem 19.33 of [Margaris... |
| 19.33b 1907 | The antecedent provides a ... |
| 19.40 1908 | Theorem 19.40 of [Margaris... |
| 19.40-2 1909 | Theorem *11.42 in [Whitehe... |
| 19.40b 1910 | The antecedent provides a ... |
| albiim 1911 | Split a biconditional and ... |
| 2albiim 1912 | Split a biconditional and ... |
| exintrbi 1913 | Add/remove a conjunct in t... |
| exintr 1914 | Introduce a conjunct in th... |
| alsyl 1915 | Universally quantified and... |
| nfimd 1916 | If in a context ` x ` is n... |
| nfimt 1917 | Closed form of ~ nfim and ... |
| nfim 1918 | If ` x ` is not free in ` ... |
| nfand 1919 | If in a context ` x ` is n... |
| nf3and 1920 | Deduction form of bound-va... |
| nfan 1921 | If ` x ` is not free in ` ... |
| nfnan 1922 | If ` x ` is not free in ` ... |
| nf3an 1923 | If ` x ` is not free in ` ... |
| nfbid 1924 | If in a context ` x ` is n... |
| nfbi 1925 | If ` x ` is not free in ` ... |
| nfor 1926 | If ` x ` is not free in ` ... |
| nf3or 1927 | If ` x ` is not free in ` ... |
| empty 1928 | Two characterizations of t... |
| emptyex 1929 | On the empty domain, any e... |
| emptyal 1930 | On the empty domain, any u... |
| emptynf 1931 | On the empty domain, any v... |
| ax5d 1933 | Version of ~ ax-5 with ant... |
| ax5e 1934 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1935 | If a formula holds for som... |
| nfv 1936 | If ` x ` is not present in... |
| nfvd 1937 | ~ nfv with antecedent. Us... |
| alimdv 1938 | Deduction form of Theorem ... |
| eximdv 1939 | Deduction form of Theorem ... |
| 2alimdv 1940 | Deduction form of Theorem ... |
| 2eximdv 1941 | Deduction form of Theorem ... |
| albidv 1942 | Formula-building rule for ... |
| exbidv 1943 | Formula-building rule for ... |
| nfbidv 1944 | An equality theorem for no... |
| 2albidv 1945 | Formula-building rule for ... |
| 2exbidv 1946 | Formula-building rule for ... |
| 3exbidv 1947 | Formula-building rule for ... |
| 4exbidv 1948 | Formula-building rule for ... |
| alrimiv 1949 | Inference form of Theorem ... |
| alrimivv 1950 | Inference form of Theorem ... |
| alrimdv 1951 | Deduction form of Theorem ... |
| exlimiv 1952 | Inference form of Theorem ... |
| exlimiiv 1953 | Inference (Rule C) associa... |
| exlimivv 1954 | Inference form of Theorem ... |
| exlimdv 1955 | Deduction form of Theorem ... |
| exlimdvv 1956 | Deduction form of Theorem ... |
| exlimddv 1957 | Existential elimination ru... |
| nexdv 1958 | Deduction for generalizati... |
| 2ax5 1959 | Quantification of two vari... |
| stdpc5v 1960 | Version of ~ stdpc5 with a... |
| 19.21v 1961 | Version of ~ 19.21 with a ... |
| 19.32v 1962 | Version of ~ 19.32 with a ... |
| 19.31v 1963 | Version of ~ 19.31 with a ... |
| 19.23v 1964 | Version of ~ 19.23 with a ... |
| 19.23vv 1965 | Theorem ~ 19.23v extended ... |
| pm11.53v 1966 | Version of ~ pm11.53 with ... |
| 19.36imv 1967 | One direction of ~ 19.36v ... |
| 19.36iv 1968 | Inference associated with ... |
| 19.37imv 1969 | One direction of ~ 19.37v ... |
| 19.37iv 1970 | Inference associated with ... |
| 19.41v 1971 | Version of ~ 19.41 with a ... |
| 19.41vv 1972 | Version of ~ 19.41 with tw... |
| 19.41vvv 1973 | Version of ~ 19.41 with th... |
| 19.41vvvv 1974 | Version of ~ 19.41 with fo... |
| 19.42v 1975 | Version of ~ 19.42 with a ... |
| exdistr 1976 | Distribution of existentia... |
| exdistrv 1977 | Distribute a pair of exist... |
| 4exdistrv 1978 | Distribute two pairs of ex... |
| 19.42vv 1979 | Version of ~ 19.42 with tw... |
| exdistr2 1980 | Distribution of existentia... |
| 19.42vvv 1981 | Version of ~ 19.42 with th... |
| 3exdistr 1982 | Distribution of existentia... |
| 4exdistr 1983 | Distribution of existentia... |
| weq 1984 | Extend wff definition to i... |
| speimfw 1985 | Specialization, with addit... |
| speimfwALT 1986 | Alternate proof of ~ speim... |
| spimfw 1987 | Specialization, with addit... |
| ax12i 1988 | Inference that has ~ ax-12... |
| ax6v 1990 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1991 | At least one individual ex... |
| spimw 1992 | Specialization. Lemma 8 o... |
| spimew 1993 | Existential introduction, ... |
| speiv 1994 | Inference from existential... |
| speivw 1995 | Version of ~ spei with a d... |
| exgen 1996 | Rule of existential genera... |
| extru 1997 | There exists a variable su... |
| 19.2 1998 | Theorem 19.2 of [Margaris]... |
| 19.2d 1999 | Deduction associated with ... |
| 19.8w 2000 | Weak version of ~ 19.8a an... |
| spnfw 2001 | Weak version of ~ sp . Us... |
| spfalw 2002 | Version of ~ sp when ` ph ... |
| spvw 2003 | Version of ~ sp when ` x `... |
| 19.3v 2004 | Version of ~ 19.3 with a d... |
| 19.8v 2005 | Version of ~ 19.8a with a ... |
| 19.9v 2006 | Version of ~ 19.9 with a d... |
| spimevw 2007 | Existential introduction, ... |
| spimvw 2008 | A weak form of specializat... |
| spsv 2009 | Generalization of antecede... |
| spvv 2010 | Specialization, using impl... |
| chvarvv 2011 | Implicit substitution of `... |
| 19.39 2012 | Theorem 19.39 of [Margaris... |
| 19.24 2013 | Theorem 19.24 of [Margaris... |
| 19.34 2014 | Theorem 19.34 of [Margaris... |
| 19.36v 2015 | Version of ~ 19.36 with a ... |
| 19.12vvv 2016 | Version of ~ 19.12vv with ... |
| 19.27v 2017 | Version of ~ 19.27 with a ... |
| 19.28v 2018 | Version of ~ 19.28 with a ... |
| 19.37v 2019 | Version of ~ 19.37 with a ... |
| 19.44v 2020 | Version of ~ 19.44 with a ... |
| 19.45v 2021 | Version of ~ 19.45 with a ... |
| equs4v 2022 | Version of ~ equs4 with a ... |
| alequexv 2023 | Version of ~ equs4v with i... |
| exsbim 2024 | One direction of the equiv... |
| equsv 2025 | If a formula does not cont... |
| equsalvw 2026 | Version of ~ equsalv with ... |
| equsexvw 2027 | Version of ~ equsexv with ... |
| cbvaliw 2028 | Change bound variable. Us... |
| cbvalivw 2029 | Change bound variable. Us... |
| ax7v 2031 | Weakened version of ~ ax-7... |
| ax7v1 2032 | First of two weakened vers... |
| ax7v2 2033 | Second of two weakened ver... |
| equid 2034 | Identity law for equality.... |
| nfequid 2035 | Bound-variable hypothesis ... |
| equcomiv 2036 | Weaker form of ~ equcomi w... |
| ax6evr 2037 | A commuted form of ~ ax6ev... |
| ax7 2038 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2039 | Commutative law for equali... |
| equcom 2040 | Commutative law for equali... |
| equcomd 2041 | Deduction form of ~ equcom... |
| equcoms 2042 | An inference commuting equ... |
| equtr 2043 | A transitive law for equal... |
| equtrr 2044 | A transitive law for equal... |
| equeuclr 2045 | Commuted version of ~ eque... |
| equeucl 2046 | Equality is a left-Euclide... |
| equequ1 2047 | An equivalence law for equ... |
| equequ2 2048 | An equivalence law for equ... |
| equtr2 2049 | Equality is a left-Euclide... |
| stdpc6 2050 | One of the two equality ax... |
| equvinv 2051 | A variable introduction la... |
| equvinva 2052 | A modified version of the ... |
| equvelv 2053 | A biconditional form of ~ ... |
| ax13b 2054 | An equivalence between two... |
| spfw 2055 | Weak version of ~ sp . Us... |
| spw 2056 | Weak version of the specia... |
| cbvalw 2057 | Change bound variable. Us... |
| cbvalvw 2058 | Change bound variable. Us... |
| cbvexvw 2059 | Change bound variable. Us... |
| cbvaldvaw 2060 | Rule used to change the bo... |
| cbvexdvaw 2061 | Rule used to change the bo... |
| cbval2vw 2062 | Rule used to change bound ... |
| cbvex2vw 2063 | Rule used to change bound ... |
| cbvex4vw 2064 | Rule used to change bound ... |
| alcomimw 2065 | Weak version of ~ ax-11 . ... |
| excomimw 2066 | Weak version of ~ excomim ... |
| alcomw 2067 | Weak version of ~ alcom an... |
| excomw 2068 | Weak version of ~ excom an... |
| hbn1fw 2069 | Weak version of ~ ax-10 fr... |
| hbn1w 2070 | Weak version of ~ hbn1 . ... |
| hba1w 2071 | Weak version of ~ hba1 . ... |
| hbe1w 2072 | Weak version of ~ hbe1 . ... |
| hbalw 2073 | Weak version of ~ hbal . ... |
| 19.8aw 2074 | If a formula is true, then... |
| exexw 2075 | Existential quantification... |
| spaev 2076 | A special instance of ~ sp... |
| cbvaev 2077 | Change bound variable in a... |
| aevlem0 2078 | Lemma for ~ aevlem . Inst... |
| aevlem 2079 | Lemma for ~ aev and ~ axc1... |
| aeveq 2080 | The antecedent ` A. x x = ... |
| aev 2081 | A "distinctor elimination"... |
| aev2 2082 | A version of ~ aev with tw... |
| hbaev 2083 | All variables are effectiv... |
| naev 2084 | If some set variables can ... |
| naev2 2085 | Generalization of ~ hbnaev... |
| hbnaev 2086 | Any variable is free in ` ... |
| justify-df 2087 | Metamath handles substitut... |
| just1-df 2088 | First justification theore... |
| just2-df 2089 | Second justification theor... |
| just3-df 2090 | Third justification theore... |
| rename-sb 2091 | The equivalence needed for... |
| dfsbimp 2094 | A simple consequence of ~ ... |
| dfsb 2095 | Simplify definition ~ df-s... |
| sbtlem 2096 | In the case of ~ sbt , ~ r... |
| sbt 2097 | A substitution into a theo... |
| sbtru 2098 | The result of substituting... |
| stdpc4lem 2099 | In the case of ~ stdpc4 , ... |
| stdpc4 2100 | The specialization axiom o... |
| stdpc4ALT 2101 | Alternate proof of ~ stdpc... |
| sbtALT 2102 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2103 | A double specialization us... |
| sbi1lem 2104 | Lemma for ~ sbi1 . The co... |
| sbi1 2105 | Distribute substitution ov... |
| sbi1ALT 2106 | Alternate proof of ~ sbt ,... |
| spsbim 2107 | Distribute substitution ov... |
| spsbbi 2108 | Biconditional property for... |
| sbimi 2109 | Distribute substitution ov... |
| sb2imi 2110 | Distribute substitution ov... |
| sbbii 2111 | Infer substitution into bo... |
| 2sbbii 2112 | Infer double substitution ... |
| sbimdv 2113 | Deduction substituting bot... |
| sbbidv 2114 | Deduction substituting bot... |
| sban 2115 | Conjunction inside and out... |
| sb3an 2116 | Threefold conjunction insi... |
| spsbe 2117 | Existential generalization... |
| sbequ 2118 | Equality property for subs... |
| sbequi 2119 | An equality theorem for su... |
| sb6 2120 | Alternate definition of su... |
| 2sb6 2121 | Equivalence for double sub... |
| sb1v 2122 | One direction of ~ sb5 , p... |
| sbv 2123 | Substitution for a variabl... |
| sbcom4 2124 | Commutativity law for subs... |
| pm11.07 2125 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2126 | Substitution in an implica... |
| sbrimvwOLD 2127 | Obsolete version of ~ sbri... |
| sbbiiev 2128 | An equivalence of substitu... |
| sbievw 2129 | Conversion of implicit sub... |
| sbievwOLD 2130 | Obsolete version of ~ sbie... |
| sbiedvw 2131 | Conversion of implicit sub... |
| 2sbievw 2132 | Conversion of double impli... |
| sbcom3vv 2133 | Substituting ` y ` for ` x... |
| sbievw2 2134 | ~ sbievw applied twice, av... |
| sbco2vv 2135 | A composition law for subs... |
| cbvsbv 2136 | Change the bound variable ... |
| sbco4lem 2137 | Lemma for ~ sbco4 . It re... |
| sbco4 2138 | Two ways of exchanging two... |
| equsb3 2139 | Substitution in an equalit... |
| equsb3r 2140 | Substitution applied to th... |
| equsb1v 2141 | Substitution applied to an... |
| nsb 2142 | Any substitution in an alw... |
| sbn1 2143 | One direction of ~ sbn , u... |
| wel 2145 | Extend wff definition to i... |
| ax8v 2147 | Weakened version of ~ ax-8... |
| ax8v1 2148 | First of two weakened vers... |
| ax8v2 2149 | Second of two weakened ver... |
| ax8 2150 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2151 | An identity law for the no... |
| elsb1 2152 | Substitution for the first... |
| cleljust 2153 | When the class variables i... |
| ax9v 2155 | Weakened version of ~ ax-9... |
| ax9v1 2156 | First of two weakened vers... |
| ax9v2 2157 | Second of two weakened ver... |
| ax9 2158 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2159 | An identity law for the no... |
| elequ2g 2160 | A form of ~ elequ2 with a ... |
| elsb2 2161 | Substitution for the secon... |
| elequ12 2162 | An identity law for the no... |
| ru0 2163 | The FOL statement used in ... |
| ax6dgen 2164 | Tarski's system uses the w... |
| ax10w 2165 | Weak version of ~ ax-10 fr... |
| ax11w 2166 | Weak version of ~ ax-11 fr... |
| ax11dgen 2167 | Degenerate instance of ~ a... |
| ax12wlem 2168 | Lemma for weak version of ... |
| ax12w 2169 | Weak version of ~ ax-12 fr... |
| ax12dgen 2170 | Degenerate instance of ~ a... |
| ax12wdemo 2171 | Example of an application ... |
| ax13w 2172 | Weak version (principal in... |
| ax13dgen1 2173 | Degenerate instance of ~ a... |
| ax13dgen2 2174 | Degenerate instance of ~ a... |
| ax13dgen3 2175 | Degenerate instance of ~ a... |
| ax13dgen4 2176 | Degenerate instance of ~ a... |
| hbn1 2178 | Alias for ~ ax-10 to be us... |
| hbe1 2179 | The setvar ` x ` is not fr... |
| hbe1a 2180 | Dual statement of ~ hbe1 .... |
| nf5-1 2181 | One direction of ~ nf5 can... |
| nf5i 2182 | Deduce that ` x ` is not f... |
| nf5dh 2183 | Deduce that ` x ` is not f... |
| nf5dv 2184 | Apply the definition of no... |
| nfnaew 2185 | All variables are effectiv... |
| nfe1 2186 | The setvar ` x ` is not fr... |
| nfa1 2187 | The setvar ` x ` is not fr... |
| nfna1 2188 | A convenience theorem part... |
| nfia1 2189 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2190 | The setvar ` x ` is not fr... |
| modal5 2191 | The analogue in our predic... |
| nfs1v 2192 | The setvar ` x ` is not fr... |
| alcoms 2194 | Swap quantifiers in an ant... |
| alcom 2195 | Theorem 19.5 of [Margaris]... |
| alrot3 2196 | Theorem *11.21 in [Whitehe... |
| alrot4 2197 | Rotate four universal quan... |
| excom 2198 | Theorem 19.11 of [Margaris... |
| excomim 2199 | One direction of Theorem 1... |
| excom13 2200 | Swap 1st and 3rd existenti... |
| exrot3 2201 | Rotate existential quantif... |
| exrot4 2202 | Rotate existential quantif... |
| hbal 2203 | If ` x ` is not free in ` ... |
| hbald 2204 | Deduction form of bound-va... |
| sbal 2205 | Move universal quantifier ... |
| sbalv 2206 | Quantify with new variable... |
| hbsbw 2207 | If ` z ` is not free in ` ... |
| sbcom2 2208 | Commutativity law for subs... |
| sbco4lemOLD 2209 | Obsolete version of ~ sbco... |
| sbco4OLD 2210 | Obsolete version of ~ sbco... |
| nfa2 2211 | Lemma 24 of [Monk2] p. 114... |
| nfexhe 2212 | Version of ~ nfex with the... |
| nfexa2 2213 | An inner universal quantif... |
| ax12v 2215 | This is essentially Axiom ... |
| ax12v2 2216 | It is possible to remove a... |
| ax12ev2 2217 | Version of ~ ax12v2 rewrit... |
| 19.8a 2218 | If a wff is true, it is tr... |
| 19.8ad 2219 | If a wff is true, it is tr... |
| sp 2220 | Specialization. A univers... |
| spi 2221 | Inference rule of universa... |
| sps 2222 | Generalization of antecede... |
| 2sp 2223 | A double specialization (s... |
| spsd 2224 | Deduction generalizing ant... |
| 19.2g 2225 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2226 | Inference form of ~ 19.21 ... |
| 19.21bbi 2227 | Inference removing two uni... |
| 19.23bi 2228 | Inference form of Theorem ... |
| nexr 2229 | Inference associated with ... |
| qexmid 2230 | Quantified excluded middle... |
| nf5r 2231 | Consequence of the definit... |
| nf5ri 2232 | Consequence of the definit... |
| nf5rd 2233 | Consequence of the definit... |
| spimedv 2234 | Deduction version of ~ spi... |
| spimefv 2235 | Version of ~ spime with a ... |
| nfim1 2236 | A closed form of ~ nfim . ... |
| nfan1 2237 | A closed form of ~ nfan . ... |
| 19.3t 2238 | Closed form of ~ 19.3 and ... |
| 19.3 2239 | A wff may be quantified wi... |
| 19.9d 2240 | A deduction version of one... |
| 19.9t 2241 | Closed form of ~ 19.9 and ... |
| 19.9 2242 | A wff may be existentially... |
| 19.21t 2243 | Closed form of Theorem 19.... |
| 19.21 2244 | Theorem 19.21 of [Margaris... |
| stdpc5 2245 | An axiom scheme of standar... |
| 19.21-2 2246 | Version of ~ 19.21 with tw... |
| 19.23t 2247 | Closed form of Theorem 19.... |
| 19.23 2248 | Theorem 19.23 of [Margaris... |
| alimd 2249 | Deduction form of Theorem ... |
| alrimi 2250 | Inference form of Theorem ... |
| alrimdd 2251 | Deduction form of Theorem ... |
| alrimd 2252 | Deduction form of Theorem ... |
| eximd 2253 | Deduction form of Theorem ... |
| exlimi 2254 | Inference associated with ... |
| exlimd 2255 | Deduction form of Theorem ... |
| exlimimdd 2256 | Existential elimination ru... |
| exlimdd 2257 | Existential elimination ru... |
| nexd 2258 | Deduction for generalizati... |
| albid 2259 | Formula-building rule for ... |
| exbid 2260 | Formula-building rule for ... |
| nfbidf 2261 | An equality theorem for ef... |
| 19.16 2262 | Theorem 19.16 of [Margaris... |
| 19.17 2263 | Theorem 19.17 of [Margaris... |
| 19.27 2264 | Theorem 19.27 of [Margaris... |
| 19.28 2265 | Theorem 19.28 of [Margaris... |
| 19.19 2266 | Theorem 19.19 of [Margaris... |
| 19.36 2267 | Theorem 19.36 of [Margaris... |
| 19.36i 2268 | Inference associated with ... |
| 19.37 2269 | Theorem 19.37 of [Margaris... |
| 19.32 2270 | Theorem 19.32 of [Margaris... |
| 19.31 2271 | Theorem 19.31 of [Margaris... |
| 19.41 2272 | Theorem 19.41 of [Margaris... |
| 19.42 2273 | Theorem 19.42 of [Margaris... |
| 19.44 2274 | Theorem 19.44 of [Margaris... |
| 19.45 2275 | Theorem 19.45 of [Margaris... |
| spimfv 2276 | Specialization, using impl... |
| chvarfv 2277 | Implicit substitution of `... |
| cbv3v2 2278 | Version of ~ cbv3 with two... |
| sbalex 2279 | Equivalence of two ways to... |
| sbalexOLD 2280 | Obsolete version of ~ sbal... |
| sb4av 2281 | Version of ~ sb4a with a d... |
| sbimd 2282 | Deduction substituting bot... |
| sbbid 2283 | Deduction substituting bot... |
| 2sbbid 2284 | Deduction doubly substitut... |
| sbequ1 2285 | An equality theorem for su... |
| sbequ2 2286 | An equality theorem for su... |
| stdpc7 2287 | One of the two equality ax... |
| sbequ12 2288 | An equality theorem for su... |
| sbequ12r 2289 | An equality theorem for su... |
| sbelx 2290 | Elimination of substitutio... |
| sbequ12a 2291 | An equality theorem for su... |
| sbid 2292 | An identity theorem for su... |
| sbcov 2293 | A composition law for subs... |
| sbcovOLD 2294 | Obsolete version of ~ sbco... |
| sb6a 2295 | Equivalence for substituti... |
| sbid2vw 2296 | Reverting substitution yie... |
| axc16g 2297 | Generalization of ~ axc16 ... |
| axc16 2298 | Proof of older axiom ~ ax-... |
| axc16gb 2299 | Biconditional strengthenin... |
| axc16nf 2300 | If ~ dtru is false, then t... |
| axc11v 2301 | Version of ~ axc11 with a ... |
| axc11rv 2302 | Version of ~ axc11r with a... |
| drsb2 2303 | Formula-building lemma for... |
| equsalv 2304 | An equivalence related to ... |
| equsexv 2305 | An equivalence related to ... |
| sbft 2306 | Substitution has no effect... |
| sbf 2307 | Substitution for a variabl... |
| sbf2 2308 | Substitution has no effect... |
| sbh 2309 | Substitution for a variabl... |
| hbs1 2310 | The setvar ` x ` is not fr... |
| nfs1f 2311 | If ` x ` is not free in ` ... |
| sb5 2312 | Alternate definition of su... |
| equs5av 2313 | A property related to subs... |
| 2sb5 2314 | Equivalence for double sub... |
| dfsb7 2315 | An alternate definition of... |
| sbn 2316 | Negation inside and outsid... |
| sbex 2317 | Move existential quantifie... |
| nf5 2318 | Alternate definition of ~ ... |
| nf6 2319 | An alternate definition of... |
| nf5d 2320 | Deduce that ` x ` is not f... |
| nf5di 2321 | Since the converse holds b... |
| 19.9h 2322 | A wff may be existentially... |
| 19.21h 2323 | Theorem 19.21 of [Margaris... |
| 19.23h 2324 | Theorem 19.23 of [Margaris... |
| exlimih 2325 | Inference associated with ... |
| exlimdh 2326 | Deduction form of Theorem ... |
| equsalhw 2327 | Version of ~ equsalh with ... |
| equsexhv 2328 | An equivalence related to ... |
| hba1 2329 | The setvar ` x ` is not fr... |
| hbnt 2330 | Closed theorem version of ... |
| hbn 2331 | If ` x ` is not free in ` ... |
| hbnd 2332 | Deduction form of bound-va... |
| hbim1 2333 | A closed form of ~ hbim . ... |
| hbimd 2334 | Deduction form of bound-va... |
| hbim 2335 | If ` x ` is not free in ` ... |
| hban 2336 | If ` x ` is not free in ` ... |
| hb3an 2337 | If ` x ` is not free in ` ... |
| sbi2 2338 | Introduction of implicatio... |
| sbim 2339 | Implication inside and out... |
| sbrim 2340 | Substitution in an implica... |
| sblim 2341 | Substitution in an implica... |
| sbor 2342 | Disjunction inside and out... |
| sbbi 2343 | Equivalence inside and out... |
| sblbis 2344 | Introduce left bicondition... |
| sbrbis 2345 | Introduce right biconditio... |
| sbrbif 2346 | Introduce right biconditio... |
| sbnf 2347 | Move nonfree predicate in ... |
| sbiev 2348 | Conversion of implicit sub... |
| sbievOLD 2349 | Obsolete version of ~ sbie... |
| sbiedw 2350 | Conversion of implicit sub... |
| axc7 2351 | Show that the original axi... |
| axc7e 2352 | Abbreviated version of ~ a... |
| modal-b 2353 | The analogue in our predic... |
| 19.9ht 2354 | A closed version of ~ 19.9... |
| axc4 2355 | Show that the original axi... |
| axc4i 2356 | Inference version of ~ axc... |
| nfal 2357 | If ` x ` is not free in ` ... |
| nfex 2358 | If ` x ` is not free in ` ... |
| hbex 2359 | If ` x ` is not free in ` ... |
| nfnf 2360 | If ` x ` is not free in ` ... |
| 19.12 2361 | Theorem 19.12 of [Margaris... |
| nfald 2362 | Deduction form of ~ nfal .... |
| nfexd 2363 | If ` x ` is not free in ` ... |
| nfsbv 2364 | If ` z ` is not free in ` ... |
| sbco2v 2365 | A composition law for subs... |
| aaan 2366 | Distribute universal quant... |
| eeor 2367 | Distribute existential qua... |
| cbv3v 2368 | Rule used to change bound ... |
| cbv1v 2369 | Rule used to change bound ... |
| cbv2w 2370 | Rule used to change bound ... |
| cbvaldw 2371 | Deduction used to change b... |
| cbvexdw 2372 | Deduction used to change b... |
| cbv3hv 2373 | Rule used to change bound ... |
| cbvalv1 2374 | Rule used to change bound ... |
| cbvexv1 2375 | Rule used to change bound ... |
| cbval2v 2376 | Rule used to change bound ... |
| cbvex2v 2377 | Rule used to change bound ... |
| dvelimhw 2378 | Proof of ~ dvelimh without... |
| pm11.53 2379 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2380 | Special case of ~ 19.12 wh... |
| eean 2381 | Distribute existential qua... |
| eeanv 2382 | Distribute a pair of exist... |
| eeeanv 2383 | Distribute three existenti... |
| ee4anv 2384 | Distribute two pairs of ex... |
| ee4anvOLD 2385 | Obsolete version of ~ ee4a... |
| sb8v 2386 | Substitution of variable i... |
| sb8f 2387 | Substitution of variable i... |
| sb8ef 2388 | Substitution of variable i... |
| 2sb8ef 2389 | An equivalent expression f... |
| sb6rfv 2390 | Reversed substitution. Ve... |
| sbnf2 2391 | Two ways of expressing " `... |
| exsb 2392 | An equivalent expression f... |
| 2exsb 2393 | An equivalent expression f... |
| sbbib 2394 | Reversal of substitution. ... |
| sbbibvv 2395 | Reversal of substitution. ... |
| cbvsbvf 2396 | Change the bound variable ... |
| cleljustALT 2397 | Alternate proof of ~ clelj... |
| cleljustALT2 2398 | Alternate proof of ~ clelj... |
| equs5aALT 2399 | Alternate proof of ~ equs5... |
| equs5eALT 2400 | Alternate proof of ~ equs5... |
| axc11r 2401 | Same as ~ axc11 but with r... |
| dral1v 2402 | Formula-building lemma for... |
| drex1v 2403 | Formula-building lemma for... |
| drnf1v 2404 | Formula-building lemma for... |
| ax13v 2406 | A weaker version of ~ ax-1... |
| ax13lem1 2407 | A version of ~ ax13v with ... |
| ax13 2408 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2409 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2410 | An equation between setvar... |
| dveeq2 2411 | Quantifier introduction wh... |
| nfeqf1 2412 | An equation between setvar... |
| dveeq1 2413 | Quantifier introduction wh... |
| nfeqf 2414 | A variable is effectively ... |
| axc9 2415 | Derive set.mm's original ~... |
| ax6e 2416 | At least one individual ex... |
| ax6 2417 | Theorem showing that ~ ax-... |
| axc10 2418 | Show that the original axi... |
| spimt 2419 | Closed theorem form of ~ s... |
| spim 2420 | Specialization, using impl... |
| spimed 2421 | Deduction version of ~ spi... |
| spime 2422 | Existential introduction, ... |
| spimv 2423 | A version of ~ spim with a... |
| spimvALT 2424 | Alternate proof of ~ spimv... |
| spimev 2425 | Distinct-variable version ... |
| spv 2426 | Specialization, using impl... |
| spei 2427 | Inference from existential... |
| chvar 2428 | Implicit substitution of `... |
| chvarv 2429 | Implicit substitution of `... |
| cbv3 2430 | Rule used to change bound ... |
| cbval 2431 | Rule used to change bound ... |
| cbvex 2432 | Rule used to change bound ... |
| cbvalv 2433 | Rule used to change bound ... |
| cbvexv 2434 | Rule used to change bound ... |
| cbv1 2435 | Rule used to change bound ... |
| cbv2 2436 | Rule used to change bound ... |
| cbv3h 2437 | Rule used to change bound ... |
| cbv1h 2438 | Rule used to change bound ... |
| cbv2h 2439 | Rule used to change bound ... |
| cbvald 2440 | Deduction used to change b... |
| cbvexd 2441 | Deduction used to change b... |
| cbvaldva 2442 | Rule used to change the bo... |
| cbvexdva 2443 | Rule used to change the bo... |
| cbval2 2444 | Rule used to change bound ... |
| cbvex2 2445 | Rule used to change bound ... |
| cbval2vv 2446 | Rule used to change bound ... |
| cbvex2vv 2447 | Rule used to change bound ... |
| cbvex4v 2448 | Rule used to change bound ... |
| equs4 2449 | Lemma used in proofs of im... |
| equsal 2450 | An equivalence related to ... |
| equsex 2451 | An equivalence related to ... |
| equsexALT 2452 | Alternate proof of ~ equse... |
| equsalh 2453 | An equivalence related to ... |
| equsexh 2454 | An equivalence related to ... |
| axc15 2455 | Derivation of set.mm's ori... |
| ax12 2456 | Rederivation of Axiom ~ ax... |
| ax12b 2457 | A bidirectional version of... |
| ax13ALT 2458 | Alternate proof of ~ ax13 ... |
| axc11n 2459 | Derive set.mm's original ~... |
| aecom 2460 | Commutation law for identi... |
| aecoms 2461 | A commutation rule for ide... |
| naecoms 2462 | A commutation rule for dis... |
| axc11 2463 | Show that ~ ax-c11 can be ... |
| hbae 2464 | All variables are effectiv... |
| hbnae 2465 | All variables are effectiv... |
| nfae 2466 | All variables are effectiv... |
| nfnae 2467 | All variables are effectiv... |
| hbnaes 2468 | Rule that applies ~ hbnae ... |
| axc16i 2469 | Inference with ~ axc16 as ... |
| axc16nfALT 2470 | Alternate proof of ~ axc16... |
| dral2 2471 | Formula-building lemma for... |
| dral1 2472 | Formula-building lemma for... |
| dral1ALT 2473 | Alternate proof of ~ dral1... |
| drex1 2474 | Formula-building lemma for... |
| drex2 2475 | Formula-building lemma for... |
| drnf1 2476 | Formula-building lemma for... |
| drnf2 2477 | Formula-building lemma for... |
| nfald2 2478 | Variation on ~ nfald which... |
| nfexd2 2479 | Variation on ~ nfexd which... |
| exdistrf 2480 | Distribution of existentia... |
| dvelimf 2481 | Version of ~ dvelimv witho... |
| dvelimdf 2482 | Deduction form of ~ dvelim... |
| dvelimh 2483 | Version of ~ dvelim withou... |
| dvelim 2484 | This theorem can be used t... |
| dvelimv 2485 | Similar to ~ dvelim with f... |
| dvelimnf 2486 | Version of ~ dvelim using ... |
| dveeq2ALT 2487 | Alternate proof of ~ dveeq... |
| equvini 2488 | A variable introduction la... |
| equvel 2489 | A variable elimination law... |
| equs5a 2490 | A property related to subs... |
| equs5e 2491 | A property related to subs... |
| equs45f 2492 | Two ways of expressing sub... |
| equs5 2493 | Lemma used in proofs of su... |
| dveel1 2494 | Quantifier introduction wh... |
| dveel2 2495 | Quantifier introduction wh... |
| axc14 2496 | Axiom ~ ax-c14 is redundan... |
| sb6x 2497 | Equivalence involving subs... |
| sbequ5 2498 | Substitution does not chan... |
| sbequ6 2499 | Substitution does not chan... |
| sb5rf 2500 | Reversed substitution. Us... |
| sb6rf 2501 | Reversed substitution. Fo... |
| ax12vALT 2502 | Alternate proof of ~ ax12v... |
| 2ax6elem 2503 | We can always find values ... |
| 2ax6e 2504 | We can always find values ... |
| 2sb5rf 2505 | Reversed double substituti... |
| 2sb6rf 2506 | Reversed double substituti... |
| sbel2x 2507 | Elimination of double subs... |
| sb4b 2508 | Simplified definition of s... |
| sb3b 2509 | Simplified definition of s... |
| sb3 2510 | One direction of a simplif... |
| sb1 2511 | One direction of a simplif... |
| sb2 2512 | One direction of a simplif... |
| sb4a 2513 | A version of one implicati... |
| dfsb1 2514 | Alternate definition of su... |
| hbsb2 2515 | Bound-variable hypothesis ... |
| nfsb2 2516 | Bound-variable hypothesis ... |
| hbsb2a 2517 | Special case of a bound-va... |
| sb4e 2518 | One direction of a simplif... |
| hbsb2e 2519 | Special case of a bound-va... |
| hbsb3 2520 | If ` y ` is not free in ` ... |
| nfs1 2521 | If ` y ` is not free in ` ... |
| axc16ALT 2522 | Alternate proof of ~ axc16... |
| axc16gALT 2523 | Alternate proof of ~ axc16... |
| equsb1 2524 | Substitution applied to an... |
| equsb2 2525 | Substitution applied to an... |
| dfsb2 2526 | An alternate definition of... |
| dfsb3 2527 | An alternate definition of... |
| drsb1 2528 | Formula-building lemma for... |
| sb2ae 2529 | In the case of two success... |
| sb6f 2530 | Equivalence for substituti... |
| sb5f 2531 | Equivalence for substituti... |
| nfsb4t 2532 | A variable not free in a p... |
| nfsb4 2533 | A variable not free in a p... |
| sbequ8 2534 | Elimination of equality fr... |
| sbie 2535 | Conversion of implicit sub... |
| sbied 2536 | Conversion of implicit sub... |
| sbiedv 2537 | Conversion of implicit sub... |
| 2sbiev 2538 | Conversion of double impli... |
| sbcom3 2539 | Substituting ` y ` for ` x... |
| sbco 2540 | A composition law for subs... |
| sbid2 2541 | An identity law for substi... |
| sbid2v 2542 | An identity law for substi... |
| sbidm 2543 | An idempotent law for subs... |
| sbco2 2544 | A composition law for subs... |
| sbco2d 2545 | A composition law for subs... |
| sbco3 2546 | A composition law for subs... |
| sbcom 2547 | A commutativity law for su... |
| sbtrt 2548 | Partially closed form of ~... |
| sbtr 2549 | A partial converse to ~ sb... |
| sb8 2550 | Substitution of variable i... |
| sb8e 2551 | Substitution of variable i... |
| sb9 2552 | Commutation of quantificat... |
| sb9i 2553 | Commutation of quantificat... |
| sbhb 2554 | Two ways of expressing " `... |
| nfsbd 2555 | Deduction version of ~ nfs... |
| nfsb 2556 | If ` z ` is not free in ` ... |
| hbsb 2557 | If ` z ` is not free in ` ... |
| sb7f 2558 | This version of ~ dfsb7 do... |
| sb7h 2559 | This version of ~ dfsb7 do... |
| sb10f 2560 | Hao Wang's identity axiom ... |
| sbal1 2561 | Check out ~ sbal for a ver... |
| sbal2 2562 | Move quantifier in and out... |
| 2sb8e 2563 | An equivalent expression f... |
| dfmoeu 2564 | An elementary proof of ~ m... |
| dfeumo 2565 | An elementary proof showin... |
| mojust 2567 | Soundness justification th... |
| dfmo 2569 | Simplify definition ~ df-m... |
| nexmo 2570 | Nonexistence implies uniqu... |
| exmo 2571 | Any proposition holds for ... |
| moabs 2572 | Absorption of existence co... |
| moim 2573 | The at-most-one quantifier... |
| moimi 2574 | The at-most-one quantifier... |
| moimdv 2575 | The at-most-one quantifier... |
| mobi 2576 | Equivalence theorem for th... |
| mobii 2577 | Formula-building rule for ... |
| mobidv 2578 | Formula-building rule for ... |
| mobid 2579 | Formula-building rule for ... |
| moa1 2580 | If an implication holds fo... |
| moan 2581 | "At most one" is still the... |
| moani 2582 | "At most one" is still tru... |
| moor 2583 | "At most one" is still the... |
| mooran1 2584 | "At most one" imports disj... |
| mooran2 2585 | "At most one" exports disj... |
| nfmo1 2586 | Bound-variable hypothesis ... |
| nfmod2 2587 | Bound-variable hypothesis ... |
| nfmodv 2588 | Bound-variable hypothesis ... |
| nfmov 2589 | Bound-variable hypothesis ... |
| nfmod 2590 | Bound-variable hypothesis ... |
| nfmo 2591 | Bound-variable hypothesis ... |
| mof 2592 | Version of ~ df-mo with di... |
| mo3 2593 | Alternate definition of th... |
| mo 2594 | Equivalent definitions of ... |
| mo4 2595 | At-most-one quantifier exp... |
| mo4f 2596 | At-most-one quantifier exp... |
| eu3v 2599 | An alternate way to expres... |
| eujust 2600 | Soundness justification th... |
| eujustALT 2601 | Alternate proof of ~ eujus... |
| eu6lem 2602 | Lemma of ~ eu6im . A diss... |
| eu6 2603 | Alternate definition of th... |
| eu6im 2604 | One direction of ~ eu6 nee... |
| euf 2605 | Version of ~ eu6 with disj... |
| euex 2606 | Existential uniqueness imp... |
| eumo 2607 | Existential uniqueness imp... |
| eumoi 2608 | Uniqueness inferred from e... |
| exmoeub 2609 | Existence implies that uni... |
| exmoeu 2610 | Existence is equivalent to... |
| moeuex 2611 | Uniqueness implies that ex... |
| moeu 2612 | Uniqueness is equivalent t... |
| eubi 2613 | Equivalence theorem for th... |
| eubii 2614 | Introduce unique existenti... |
| eubidv 2615 | Formula-building rule for ... |
| eubid 2616 | Formula-building rule for ... |
| nfeu1ALT 2617 | Alternate version of ~ nfe... |
| nfeu1 2618 | Bound-variable hypothesis ... |
| nfeud2 2619 | Bound-variable hypothesis ... |
| nfeudw 2620 | Bound-variable hypothesis ... |
| nfeud 2621 | Bound-variable hypothesis ... |
| nfeuw 2622 | Bound-variable hypothesis ... |
| nfeu 2623 | Bound-variable hypothesis ... |
| dfeu 2624 | Rederive ~ df-eu from the ... |
| dfmo2 2625 | Rederive ~ df-mo from the ... |
| euequ 2626 | There exists a unique set ... |
| sb8eulem 2627 | Lemma. Factor out the com... |
| sb8euv 2628 | Variable substitution in u... |
| sb8eu 2629 | Variable substitution in u... |
| sb8mo 2630 | Variable substitution for ... |
| cbvmovw 2631 | Change bound variable. Us... |
| cbvmow 2632 | Rule used to change bound ... |
| cbvmo 2633 | Rule used to change bound ... |
| cbveuvw 2634 | Change bound variable. Us... |
| cbveuw 2635 | Version of ~ cbveu with a ... |
| cbveu 2636 | Rule used to change bound ... |
| cbveuALT 2637 | Alternative proof of ~ cbv... |
| eu2 2638 | An alternate way of defini... |
| eu1 2639 | An alternate way to expres... |
| euor 2640 | Introduce a disjunct into ... |
| euorv 2641 | Introduce a disjunct into ... |
| euor2 2642 | Introduce or eliminate a d... |
| sbmo 2643 | Substitution into an at-mo... |
| eu4 2644 | Uniqueness using implicit ... |
| euimmo 2645 | Existential uniqueness imp... |
| euim 2646 | Add unique existential qua... |
| moanimlem 2647 | Factor out the common proo... |
| moanimv 2648 | Introduction of a conjunct... |
| moanim 2649 | Introduction of a conjunct... |
| euan 2650 | Introduction of a conjunct... |
| moanmo 2651 | Nested at-most-one quantif... |
| moaneu 2652 | Nested at-most-one and uni... |
| euanv 2653 | Introduction of a conjunct... |
| mopick 2654 | "At most one" picks a vari... |
| moexexlem 2655 | Factor out the proof skele... |
| 2moexv 2656 | Double quantification with... |
| moexexvw 2657 | "At most one" double quant... |
| 2moswapv 2658 | A condition allowing to sw... |
| 2euswapv 2659 | A condition allowing to sw... |
| 2euexv 2660 | Double quantification with... |
| 2exeuv 2661 | Double existential uniquen... |
| eupick 2662 | Existential uniqueness "pi... |
| eupicka 2663 | Version of ~ eupick with c... |
| eupickb 2664 | Existential uniqueness "pi... |
| eupickbi 2665 | Theorem *14.26 in [Whitehe... |
| mopick2 2666 | "At most one" can show the... |
| moexex 2667 | "At most one" double quant... |
| moexexv 2668 | "At most one" double quant... |
| 2moex 2669 | Double quantification with... |
| 2euex 2670 | Double quantification with... |
| 2eumo 2671 | Nested unique existential ... |
| 2eu2ex 2672 | Double existential uniquen... |
| 2moswap 2673 | A condition allowing to sw... |
| 2euswap 2674 | A condition allowing to sw... |
| 2exeu 2675 | Double existential uniquen... |
| 2mo2 2676 | Two ways of expressing "th... |
| 2mo 2677 | Two ways of expressing "th... |
| 2mos 2678 | Double "there exists at mo... |
| 2eu1 2679 | Double existential uniquen... |
| 2eu1v 2680 | Double existential uniquen... |
| 2eu2 2681 | Double existential uniquen... |
| 2eu3 2682 | Double existential uniquen... |
| 2eu4 2683 | This theorem provides us w... |
| 2eu5 2684 | An alternate definition of... |
| 2eu6 2685 | Two equivalent expressions... |
| 2eu7 2686 | Two equivalent expressions... |
| 2eu8 2687 | Two equivalent expressions... |
| euae 2688 | Two ways to express "exact... |
| exists1 2689 | Two ways to express "exact... |
| exists2 2690 | A condition implying that ... |
| barbara 2691 | "Barbara", one of the fund... |
| celarent 2692 | "Celarent", one of the syl... |
| darii 2693 | "Darii", one of the syllog... |
| dariiALT 2694 | Alternate proof of ~ darii... |
| ferio 2695 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2696 | Lemma for ~ barbari and th... |
| barbari 2697 | "Barbari", one of the syll... |
| barbariALT 2698 | Alternate proof of ~ barba... |
| celaront 2699 | "Celaront", one of the syl... |
| cesare 2700 | "Cesare", one of the syllo... |
| camestres 2701 | "Camestres", one of the sy... |
| festino 2702 | "Festino", one of the syll... |
| festinoALT 2703 | Alternate proof of ~ festi... |
| baroco 2704 | "Baroco", one of the syllo... |
| barocoALT 2705 | Alternate proof of ~ festi... |
| cesaro 2706 | "Cesaro", one of the syllo... |
| camestros 2707 | "Camestros", one of the sy... |
| datisi 2708 | "Datisi", one of the syllo... |
| disamis 2709 | "Disamis", one of the syll... |
| ferison 2710 | "Ferison", one of the syll... |
| bocardo 2711 | "Bocardo", one of the syll... |
| darapti 2712 | "Darapti", one of the syll... |
| daraptiALT 2713 | Alternate proof of ~ darap... |
| felapton 2714 | "Felapton", one of the syl... |
| calemes 2715 | "Calemes", one of the syll... |
| dimatis 2716 | "Dimatis", one of the syll... |
| fresison 2717 | "Fresison", one of the syl... |
| calemos 2718 | "Calemos", one of the syll... |
| fesapo 2719 | "Fesapo", one of the syllo... |
| bamalip 2720 | "Bamalip", one of the syll... |
| axia1 2721 | Left 'and' elimination (in... |
| axia2 2722 | Right 'and' elimination (i... |
| axia3 2723 | 'And' introduction (intuit... |
| axin1 2724 | 'Not' introduction (intuit... |
| axin2 2725 | 'Not' elimination (intuiti... |
| axio 2726 | Definition of 'or' (intuit... |
| axi4 2727 | Specialization (intuitioni... |
| axi5r 2728 | Converse of ~ axc4 (intuit... |
| axial 2729 | The setvar ` x ` is not fr... |
| axie1 2730 | The setvar ` x ` is not fr... |
| axie2 2731 | A key property of existent... |
| axi9 2732 | Axiom of existence (intuit... |
| axi10 2733 | Axiom of Quantifier Substi... |
| axi12 2734 | Axiom of Quantifier Introd... |
| axbnd 2735 | Axiom of Bundling (intuiti... |
| axexte 2737 | The axiom of extensionalit... |
| axextg 2738 | A generalization of the ax... |
| axextb 2739 | A bidirectional version of... |
| axextmo 2740 | There exists at most one s... |
| nulmo 2741 | There exists at most one e... |
| eleq1ab 2744 | Extension (in the sense of... |
| cleljustab 2745 | Extension of ~ cleljust fr... |
| abid 2746 | Simplification of class ab... |
| vexwt 2747 | A standard theorem of pred... |
| vexw 2748 | If ` ph ` is a theorem, th... |
| vextru 2749 | Every setvar is a member o... |
| nfsab1 2750 | Bound-variable hypothesis ... |
| hbab1 2751 | Bound-variable hypothesis ... |
| hbab 2752 | Bound-variable hypothesis ... |
| hbabg 2753 | Bound-variable hypothesis ... |
| nfsab 2754 | Bound-variable hypothesis ... |
| nfsabg 2755 | Bound-variable hypothesis ... |
| dfcleq 2757 | The defining characterizat... |
| cvjust 2758 | Every set is a class. Pro... |
| ax9ALT 2759 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2760 | A weaker version of ~ eleq... |
| eqriv 2761 | Infer equality of classes ... |
| eqrdv 2762 | Deduce equality of classes... |
| eqrdav 2763 | Deduce equality of classes... |
| eqid 2764 | Law of identity (reflexivi... |
| eqidd 2765 | Class identity law with an... |
| eqeq1d 2766 | Deduction from equality to... |
| eqeq1dALT 2767 | Alternate proof of ~ eqeq1... |
| eqeq1 2768 | Equality implies equivalen... |
| eqeq1i 2769 | Inference from equality to... |
| eqcomd 2770 | Deduction from commutative... |
| eqcom 2771 | Commutative law for class ... |
| eqcoms 2772 | Inference applying commuta... |
| eqcomi 2773 | Inference from commutative... |
| neqcomd 2774 | Commute an inequality. (C... |
| eqeq2d 2775 | Deduction from equality to... |
| eqeq2 2776 | Equality implies equivalen... |
| eqeq2i 2777 | Inference from equality to... |
| eqeqan12d 2778 | A useful inference for sub... |
| eqeqan12rd 2779 | A useful inference for sub... |
| eqeq12d 2780 | A useful inference for sub... |
| eqeq12 2781 | Equality relationship amon... |
| eqeq12i 2782 | A useful inference for sub... |
| eqeqan12dALT 2783 | Alternate proof of ~ eqeqa... |
| eqtr 2784 | Transitive law for class e... |
| eqtr2 2785 | A transitive law for class... |
| eqtr3 2786 | A transitive law for class... |
| eqtri 2787 | An equality transitivity i... |
| eqtr2i 2788 | An equality transitivity i... |
| eqtr3i 2789 | An equality transitivity i... |
| eqtr4i 2790 | An equality transitivity i... |
| 3eqtri 2791 | An inference from three ch... |
| 3eqtrri 2792 | An inference from three ch... |
| 3eqtr2i 2793 | An inference from three ch... |
| 3eqtr2ri 2794 | An inference from three ch... |
| 3eqtr3i 2795 | An inference from three ch... |
| 3eqtr3ri 2796 | An inference from three ch... |
| 3eqtr4i 2797 | An inference from three ch... |
| 3eqtr4ri 2798 | An inference from three ch... |
| eqtrd 2799 | An equality transitivity d... |
| eqtr2d 2800 | An equality transitivity d... |
| eqtr3d 2801 | An equality transitivity e... |
| eqtr4d 2802 | An equality transitivity e... |
| 3eqtrd 2803 | A deduction from three cha... |
| 3eqtrrd 2804 | A deduction from three cha... |
| 3eqtr2d 2805 | A deduction from three cha... |
| 3eqtr2rd 2806 | A deduction from three cha... |
| 3eqtr3d 2807 | A deduction from three cha... |
| 3eqtr3rd 2808 | A deduction from three cha... |
| 3eqtr4d 2809 | A deduction from three cha... |
| 3eqtr4rd 2810 | A deduction from three cha... |
| eqtrid 2811 | An equality transitivity d... |
| eqtr2id 2812 | An equality transitivity d... |
| eqtr3id 2813 | An equality transitivity d... |
| eqtr3di 2814 | An equality transitivity d... |
| eqtrdi 2815 | An equality transitivity d... |
| eqtr2di 2816 | An equality transitivity d... |
| eqtr4di 2817 | An equality transitivity d... |
| eqtr4id 2818 | An equality transitivity d... |
| sylan9eq 2819 | An equality transitivity d... |
| sylan9req 2820 | An equality transitivity d... |
| sylan9eqr 2821 | An equality transitivity d... |
| 3eqtr3g 2822 | A chained equality inferen... |
| 3eqtr3a 2823 | A chained equality inferen... |
| 3eqtr4g 2824 | A chained equality inferen... |
| 3eqtr4a 2825 | A chained equality inferen... |
| eq2tri 2826 | A compound transitive infe... |
| iseqsetvlem 2827 | Lemma for ~ iseqsetv-cleq ... |
| iseqsetv-cleq 2828 | Alternate proof of ~ iseqs... |
| abbi 2829 | Equivalent formulas yield ... |
| abbidv 2830 | Equivalent wff's yield equ... |
| abbii 2831 | Equivalent wff's yield equ... |
| abbid 2832 | Equivalent wff's yield equ... |
| abbib 2833 | Equal class abstractions r... |
| cbvabv 2834 | Rule used to change bound ... |
| cbvabw 2835 | Rule used to change bound ... |
| cbvab 2836 | Rule used to change bound ... |
| eqabbw 2837 | Version of ~ eqabb using i... |
| eqabcbw 2838 | Version of ~ eqabcb using ... |
| dfclel 2840 | Characterization of the el... |
| elex2 2841 | If a class contains anothe... |
| issettru 2842 | Weak version of ~ isset . ... |
| iseqsetv-clel 2843 | Alternate proof of ~ iseqs... |
| issetlem 2844 | Lemma for ~ elisset and ~ ... |
| elissetv 2845 | An element of a class exis... |
| elisset 2846 | An element of a class exis... |
| eleq1w 2847 | Weaker version of ~ eleq1 ... |
| eleq2w 2848 | Weaker version of ~ eleq2 ... |
| eleq1d 2849 | Deduction from equality to... |
| eleq2d 2850 | Deduction from equality to... |
| eleq2dALT 2851 | Alternate proof of ~ eleq2... |
| eleq1 2852 | Equality implies equivalen... |
| eleq2 2853 | Equality implies equivalen... |
| eleq12 2854 | Equality implies equivalen... |
| eleq1i 2855 | Inference from equality to... |
| eleq2i 2856 | Inference from equality to... |
| eleq12i 2857 | Inference from equality to... |
| eleq12d 2858 | Deduction from equality to... |
| eleq1a 2859 | A transitive-type law rela... |
| eqeltri 2860 | Substitution of equal clas... |
| eqeltrri 2861 | Substitution of equal clas... |
| eleqtri 2862 | Substitution of equal clas... |
| eleqtrri 2863 | Substitution of equal clas... |
| eqeltrd 2864 | Substitution of equal clas... |
| eqeltrrd 2865 | Deduction that substitutes... |
| eleqtrd 2866 | Deduction that substitutes... |
| eleqtrrd 2867 | Deduction that substitutes... |
| eqeltrid 2868 | A membership and equality ... |
| eqeltrrid 2869 | A membership and equality ... |
| eleqtrid 2870 | A membership and equality ... |
| eleqtrrid 2871 | A membership and equality ... |
| eqeltrdi 2872 | A membership and equality ... |
| eqeltrrdi 2873 | A membership and equality ... |
| eleqtrdi 2874 | A membership and equality ... |
| eleqtrrdi 2875 | A membership and equality ... |
| 3eltr3i 2876 | Substitution of equal clas... |
| 3eltr4i 2877 | Substitution of equal clas... |
| 3eltr3d 2878 | Substitution of equal clas... |
| 3eltr4d 2879 | Substitution of equal clas... |
| 3eltr3g 2880 | Substitution of equal clas... |
| 3eltr4g 2881 | Substitution of equal clas... |
| eleq2s 2882 | Substitution of equal clas... |
| eqneltri 2883 | If a class is not an eleme... |
| eqneltrd 2884 | If a class is not an eleme... |
| eqneltrrd 2885 | If a class is not an eleme... |
| neleqtrd 2886 | If a class is not an eleme... |
| neleqtrrd 2887 | If a class is not an eleme... |
| nelneq 2888 | A way of showing two class... |
| nelneq2 2889 | A way of showing two class... |
| eqsb1 2890 | Substitution for the left-... |
| clelsb1 2891 | Substitution for the first... |
| clelsb2 2892 | Substitution for the secon... |
| cleqh 2893 | Establish equality between... |
| hbxfreq 2894 | A utility lemma to transfe... |
| hblem 2895 | Change the free variable o... |
| hblemg 2896 | Change the free variable o... |
| eqabdv 2897 | Deduction from a wff to a ... |
| eqabcdv 2898 | Deduction from a wff to a ... |
| eqabi 2899 | Equality of a class variab... |
| abid1 2900 | Every class is equal to a ... |
| abid2 2901 | A simplification of class ... |
| eqab 2902 | One direction of ~ eqabb i... |
| eqabb 2903 | Equality of a class variab... |
| eqabcb 2904 | Equality of a class variab... |
| eqabrd 2905 | Equality of a class variab... |
| eqabri 2906 | Equality of a class variab... |
| eqabcri 2907 | Equality of a class variab... |
| clelab 2908 | Membership of a class vari... |
| clabel 2909 | Membership of a class abst... |
| sbab 2910 | The right-hand side of the... |
| nfcjust 2912 | Justification theorem for ... |
| nfci 2914 | Deduce that a class ` A ` ... |
| nfcii 2915 | Deduce that a class ` A ` ... |
| nfcr 2916 | Consequence of the not-fre... |
| nfcrALT 2917 | Alternate version of ~ nfc... |
| nfcri 2918 | Consequence of the not-fre... |
| nfcd 2919 | Deduce that a class ` A ` ... |
| nfcrd 2920 | Consequence of the not-fre... |
| nfcrii 2921 | Consequence of the not-fre... |
| nfceqdf 2922 | An equality theorem for ef... |
| nfceqi 2923 | Equality theorem for class... |
| nfcxfr 2924 | A utility lemma to transfe... |
| nfcxfrd 2925 | A utility lemma to transfe... |
| nfcv 2926 | If ` x ` is disjoint from ... |
| nfcvd 2927 | If ` x ` is disjoint from ... |
| nfab1 2928 | Bound-variable hypothesis ... |
| nfnfc1 2929 | The setvar ` x ` is bound ... |
| clelsb1fw 2930 | Substitution for the first... |
| clelsb1f 2931 | Substitution for the first... |
| nfab 2932 | Bound-variable hypothesis ... |
| nfabg 2933 | Bound-variable hypothesis ... |
| nfaba1 2934 | Bound-variable hypothesis ... |
| nfaba1g 2935 | Bound-variable hypothesis ... |
| nfeqd 2936 | Hypothesis builder for equ... |
| nfeld 2937 | Hypothesis builder for ele... |
| nfnfc 2938 | Hypothesis builder for ` F... |
| nfeq 2939 | Hypothesis builder for equ... |
| nfel 2940 | Hypothesis builder for ele... |
| nfeq1 2941 | Hypothesis builder for equ... |
| nfel1 2942 | Hypothesis builder for ele... |
| nfeq2 2943 | Hypothesis builder for equ... |
| nfel2 2944 | Hypothesis builder for ele... |
| drnfc1 2945 | Formula-building lemma for... |
| drnfc2 2946 | Formula-building lemma for... |
| nfabdw 2947 | Bound-variable hypothesis ... |
| nfabd 2948 | Bound-variable hypothesis ... |
| nfabd2 2949 | Bound-variable hypothesis ... |
| dvelimdc 2950 | Deduction form of ~ dvelim... |
| dvelimc 2951 | Version of ~ dvelim for cl... |
| nfcvf 2952 | If ` x ` and ` y ` are dis... |
| nfcvf2 2953 | If ` x ` and ` y ` are dis... |
| cleqf 2954 | Establish equality between... |
| eqabf 2955 | Equality of a class variab... |
| abid2f 2956 | A simplification of class ... |
| abid2fOLD 2957 | Obsolete version of ~ abid... |
| sbabel 2958 | Theorem to move a substitu... |
| neii 2961 | Inference associated with ... |
| neir 2962 | Inference associated with ... |
| nne 2963 | Negation of inequality. (... |
| neneqd 2964 | Deduction eliminating ineq... |
| neneq 2965 | From inequality to non-equ... |
| neqned 2966 | If it is not the case that... |
| neqne 2967 | From non-equality to inequ... |
| neirr 2968 | No class is unequal to its... |
| exmidne 2969 | Excluded middle with equal... |
| eqneqall 2970 | A contradiction concerning... |
| nonconne 2971 | Law of noncontradiction wi... |
| necon3ad 2972 | Contrapositive law deducti... |
| necon3bd 2973 | Contrapositive law deducti... |
| necon2ad 2974 | Contrapositive inference f... |
| necon2bd 2975 | Contrapositive inference f... |
| necon1ad 2976 | Contrapositive deduction f... |
| necon1bd 2977 | Contrapositive deduction f... |
| necon4ad 2978 | Contrapositive inference f... |
| necon4bd 2979 | Contrapositive inference f... |
| necon3d 2980 | Contrapositive law deducti... |
| necon1d 2981 | Contrapositive law deducti... |
| necon2d 2982 | Contrapositive inference f... |
| necon4d 2983 | Contrapositive inference f... |
| necon3ai 2984 | Contrapositive inference f... |
| necon3bi 2985 | Contrapositive inference f... |
| necon1ai 2986 | Contrapositive inference f... |
| necon1bi 2987 | Contrapositive inference f... |
| necon2ai 2988 | Contrapositive inference f... |
| necon2bi 2989 | Contrapositive inference f... |
| necon4ai 2990 | Contrapositive inference f... |
| necon3i 2991 | Contrapositive inference f... |
| necon1i 2992 | Contrapositive inference f... |
| necon2i 2993 | Contrapositive inference f... |
| necon4i 2994 | Contrapositive inference f... |
| necon3abid 2995 | Deduction from equality to... |
| necon3bbid 2996 | Deduction from equality to... |
| necon1abid 2997 | Contrapositive deduction f... |
| necon1bbid 2998 | Contrapositive inference f... |
| necon4abid 2999 | Contrapositive law deducti... |
| necon4bbid 3000 | Contrapositive law deducti... |
| necon2abid 3001 | Contrapositive deduction f... |
| necon2bbid 3002 | Contrapositive deduction f... |
| necon3bid 3003 | Deduction from equality to... |
| necon4bid 3004 | Contrapositive law deducti... |
| necon3abii 3005 | Deduction from equality to... |
| necon3bbii 3006 | Deduction from equality to... |
| necon1abii 3007 | Contrapositive inference f... |
| necon1bbii 3008 | Contrapositive inference f... |
| necon2abii 3009 | Contrapositive inference f... |
| necon2bbii 3010 | Contrapositive inference f... |
| necon3bii 3011 | Inference from equality to... |
| necom 3012 | Commutation of inequality.... |
| necomi 3013 | Inference from commutative... |
| necomd 3014 | Deduction from commutative... |
| nesym 3015 | Characterization of inequa... |
| nesymi 3016 | Inference associated with ... |
| nesymir 3017 | Inference associated with ... |
| neeq1d 3018 | Deduction for inequality. ... |
| neeq2d 3019 | Deduction for inequality. ... |
| neeq12d 3020 | Deduction for inequality. ... |
| neeq1 3021 | Equality theorem for inequ... |
| neeq2 3022 | Equality theorem for inequ... |
| neeq1i 3023 | Inference for inequality. ... |
| neeq2i 3024 | Inference for inequality. ... |
| neeq12i 3025 | Inference for inequality. ... |
| eqnetrd 3026 | Substitution of equal clas... |
| eqnetrrd 3027 | Substitution of equal clas... |
| neeqtrd 3028 | Substitution of equal clas... |
| eqnetri 3029 | Substitution of equal clas... |
| eqnetrri 3030 | Substitution of equal clas... |
| neeqtri 3031 | Substitution of equal clas... |
| neeqtrri 3032 | Substitution of equal clas... |
| neeqtrrd 3033 | Substitution of equal clas... |
| eqnetrrid 3034 | A chained equality inferen... |
| 3netr3d 3035 | Substitution of equality i... |
| 3netr4d 3036 | Substitution of equality i... |
| 3netr3g 3037 | Substitution of equality i... |
| 3netr4g 3038 | Substitution of equality i... |
| nebi 3039 | Contraposition law for ine... |
| pm13.18 3040 | Theorem *13.18 in [Whitehe... |
| pm13.181 3041 | Theorem *13.181 in [Whiteh... |
| pm2.61ine 3042 | Inference eliminating an i... |
| pm2.21ddne 3043 | A contradiction implies an... |
| pm2.61ne 3044 | Deduction eliminating an i... |
| pm2.61dne 3045 | Deduction eliminating an i... |
| pm2.61dane 3046 | Deduction eliminating an i... |
| pm2.61da2ne 3047 | Deduction eliminating two ... |
| pm2.61da3ne 3048 | Deduction eliminating thre... |
| pm2.61iine 3049 | Equality version of ~ pm2.... |
| mteqand 3050 | A modus tollens deduction ... |
| neor 3051 | Logical OR with an equalit... |
| neanior 3052 | A De Morgan's law for ineq... |
| ne3anior 3053 | A De Morgan's law for ineq... |
| neorian 3054 | A De Morgan's law for ineq... |
| nemtbir 3055 | An inference from an inequ... |
| nelne1 3056 | Two classes are different ... |
| nelne2 3057 | Two classes are different ... |
| nelelne 3058 | Two classes are different ... |
| neneor 3059 | If two classes are differe... |
| nfne 3060 | Bound-variable hypothesis ... |
| nfned 3061 | Bound-variable hypothesis ... |
| nabbib 3062 | Not equivalent wff's corre... |
| neli 3065 | Inference associated with ... |
| nelir 3066 | Inference associated with ... |
| nelcon3d 3067 | Contrapositive law deducti... |
| neleq12d 3068 | Equality theorem for negat... |
| neleq1 3069 | Equality theorem for negat... |
| neleq2 3070 | Equality theorem for negat... |
| nfnel 3071 | Bound-variable hypothesis ... |
| nfneld 3072 | Bound-variable hypothesis ... |
| nnel 3073 | Negation of negated member... |
| elnelne1 3074 | Two classes are different ... |
| elnelne2 3075 | Two classes are different ... |
| pm2.24nel 3076 | A contradiction concerning... |
| pm2.61danel 3077 | Deduction eliminating an e... |
| rgen 3080 | Generalization rule for re... |
| ralel 3081 | All elements of a class ar... |
| rgenw 3082 | Generalization rule for re... |
| rgen2w 3083 | Generalization rule for re... |
| mprg 3084 | Modus ponens combined with... |
| mprgbir 3085 | Modus ponens on biconditio... |
| ralrid 3086 | Sufficient condition for t... |
| raln 3087 | Restricted universally qua... |
| ralnex 3090 | Relationship between restr... |
| dfrex2 3091 | Relationship between restr... |
| nrex 3092 | Inference adding restricte... |
| alral 3093 | Universal quantification i... |
| rexex 3094 | Restricted existence impli... |
| rextru 3095 | Two ways of expressing tha... |
| ralimi2 3096 | Inference quantifying both... |
| reximi2 3097 | Inference quantifying both... |
| ralimia 3098 | Inference quantifying both... |
| reximia 3099 | Inference quantifying both... |
| ralimiaa 3100 | Inference quantifying both... |
| ralimi 3101 | Inference quantifying both... |
| reximi 3102 | Inference quantifying both... |
| ral2imi 3103 | Inference quantifying ante... |
| ralim 3104 | Distribution of restricted... |
| rexim 3105 | Theorem 19.22 of [Margaris... |
| ralbii2 3106 | Inference adding different... |
| rexbii2 3107 | Inference adding different... |
| ralbiia 3108 | Inference adding restricte... |
| rexbiia 3109 | Inference adding restricte... |
| ralbii 3110 | Inference adding restricte... |
| rexbii 3111 | Inference adding restricte... |
| ralanid 3112 | Cancellation law for restr... |
| rexanid 3113 | Cancellation law for restr... |
| ralcom3 3114 | A commutation law for rest... |
| dfral2 3115 | Relationship between restr... |
| rexnal 3116 | Relationship between restr... |
| ralinexa 3117 | A transformation of restri... |
| rexanali 3118 | A transformation of restri... |
| ralbi 3119 | Distribute a restricted un... |
| rexbi 3120 | Distribute restricted quan... |
| ralrexbid 3121 | Formula-building rule for ... |
| r19.35 3122 | Restricted quantifier vers... |
| r19.26m 3123 | Version of ~ 19.26 and ~ r... |
| r19.26 3124 | Restricted quantifier vers... |
| r19.26-3 3125 | Version of ~ r19.26 with t... |
| ralbiim 3126 | Split a biconditional and ... |
| r19.29 3127 | Restricted quantifier vers... |
| r19.29r 3128 | Restricted quantifier vers... |
| r19.29imd 3129 | Theorem 19.29 of [Margaris... |
| r19.40 3130 | Restricted quantifier vers... |
| r19.30 3131 | Restricted quantifier vers... |
| r19.43 3132 | Restricted quantifier vers... |
| 3r19.43 3133 | Restricted quantifier vers... |
| 2ralimi 3134 | Inference quantifying both... |
| 3ralimi 3135 | Inference quantifying both... |
| 4ralimi 3136 | Inference quantifying both... |
| 5ralimi 3137 | Inference quantifying both... |
| 6ralimi 3138 | Inference quantifying both... |
| 2ralbii 3139 | Inference adding two restr... |
| 2rexbii 3140 | Inference adding two restr... |
| 3ralbii 3141 | Inference adding three res... |
| 4ralbii 3142 | Inference adding four rest... |
| 2ralbiim 3143 | Split a biconditional and ... |
| ralnex2 3144 | Relationship between two r... |
| ralnex3 3145 | Relationship between three... |
| rexnal2 3146 | Relationship between two r... |
| rexnal3 3147 | Relationship between three... |
| nrexralim 3148 | Negation of a complex pred... |
| r19.26-2 3149 | Restricted quantifier vers... |
| 2r19.29 3150 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3151 | Theorem 19.29 of [Margaris... |
| r2allem 3152 | Lemma factoring out common... |
| r2exlem 3153 | Lemma factoring out common... |
| hbralrimi 3154 | Inference from Theorem 19.... |
| ralrimiv 3155 | Inference from Theorem 19.... |
| ralrimiva 3156 | Inference from Theorem 19.... |
| rexlimiva 3157 | Inference from Theorem 19.... |
| rexlimiv 3158 | Inference from Theorem 19.... |
| nrexdv 3159 | Deduction adding restricte... |
| ralrimivw 3160 | Inference from Theorem 19.... |
| rexlimivw 3161 | Weaker version of ~ rexlim... |
| ralrimdv 3162 | Inference from Theorem 19.... |
| rexlimdv 3163 | Inference from Theorem 19.... |
| ralrimdva 3164 | Inference from Theorem 19.... |
| rexlimdva 3165 | Inference from Theorem 19.... |
| rexlimdvaa 3166 | Inference from Theorem 19.... |
| rexlimdva2 3167 | Inference from Theorem 19.... |
| r19.29an 3168 | A commonly used pattern in... |
| rexlimdv3a 3169 | Inference from Theorem 19.... |
| rexlimdvw 3170 | Inference from Theorem 19.... |
| rexlimddv 3171 | Restricted existential eli... |
| r19.29a 3172 | A commonly used pattern in... |
| ralimdv2 3173 | Inference quantifying both... |
| reximdv2 3174 | Deduction quantifying both... |
| reximdvai 3175 | Deduction quantifying both... |
| ralimdva 3176 | Deduction quantifying both... |
| reximdva 3177 | Deduction quantifying both... |
| ralimdv 3178 | Deduction quantifying both... |
| reximdv 3179 | Deduction from Theorem 19.... |
| reximddv 3180 | Deduction from Theorem 19.... |
| reximddv3 3181 | Deduction from Theorem 19.... |
| reximssdv 3182 | Derivation of a restricted... |
| ralbidv2 3183 | Formula-building rule for ... |
| rexbidv2 3184 | Formula-building rule for ... |
| ralbidva 3185 | Formula-building rule for ... |
| rexbidva 3186 | Formula-building rule for ... |
| ralbidv 3187 | Formula-building rule for ... |
| rexbidv 3188 | Formula-building rule for ... |
| r19.21v 3189 | Restricted quantifier vers... |
| r19.37v 3190 | Restricted quantifier vers... |
| r19.23v 3191 | Restricted quantifier vers... |
| r19.36v 3192 | Restricted quantifier vers... |
| r19.27v 3193 | Restricted quantitifer ver... |
| r19.41v 3194 | Restricted quantifier vers... |
| r19.28v 3195 | Restricted quantifier vers... |
| r19.42v 3196 | Restricted quantifier vers... |
| r19.32v 3197 | Restricted quantifier vers... |
| r19.45v 3198 | Restricted quantifier vers... |
| r19.44v 3199 | One direction of a restric... |
| r2al 3200 | Double restricted universa... |
| r2ex 3201 | Double restricted existent... |
| r3al 3202 | Triple restricted universa... |
| r3ex 3203 | Triple existential quantif... |
| rgen2 3204 | Generalization rule for re... |
| ralrimivv 3205 | Inference from Theorem 19.... |
| rexlimivv 3206 | Inference from Theorem 19.... |
| ralrimivva 3207 | Inference from Theorem 19.... |
| ralrimdvv 3208 | Inference from Theorem 19.... |
| rgen3 3209 | Generalization rule for re... |
| ralrimivvva 3210 | Inference from Theorem 19.... |
| ralimdvva 3211 | Deduction doubly quantifyi... |
| reximdvva 3212 | Deduction doubly quantifyi... |
| ralimdvv 3213 | Deduction doubly quantifyi... |
| ralimdvvOLD 3214 | Obsolete version of ~ rali... |
| ralimd4v 3215 | Deduction quadrupally quan... |
| ralimd4vOLD 3216 | Obsolete version of ~ rali... |
| ralimd6v 3217 | Deduction sextupally quant... |
| ralimd6vOLD 3218 | Obsolete version of ~ rali... |
| ralrimdvva 3219 | Inference from Theorem 19.... |
| rexlimdvv 3220 | Inference from Theorem 19.... |
| rexlimdvva 3221 | Inference from Theorem 19.... |
| rexlimdvvva 3222 | Inference from Theorem 19.... |
| reximddv2 3223 | Double deduction from Theo... |
| r19.29vva 3224 | A commonly used pattern ba... |
| 2rexbiia 3225 | Inference adding two restr... |
| 2ralbidva 3226 | Formula-building rule for ... |
| 2rexbidva 3227 | Formula-building rule for ... |
| 2ralbidv 3228 | Formula-building rule for ... |
| 2rexbidv 3229 | Formula-building rule for ... |
| rexralbidv 3230 | Formula-building rule for ... |
| 3ralbidv 3231 | Formula-building rule for ... |
| 4ralbidv 3232 | Formula-building rule for ... |
| 6ralbidv 3233 | Formula-building rule for ... |
| r19.41vv 3234 | Version of ~ r19.41v with ... |
| reeanlem 3235 | Lemma factoring out common... |
| reeanv 3236 | Rearrange restricted exist... |
| 3reeanv 3237 | Rearrange three restricted... |
| 2ralor 3238 | Distribute restricted univ... |
| risset 3239 | Two ways to say " ` A ` be... |
| nelb 3240 | A definition of ` -. A e. ... |
| rspw 3241 | Restricted specialization.... |
| cbvralvw 3242 | Change the bound variable ... |
| cbvrexvw 3243 | Change the bound variable ... |
| cbvraldva 3244 | Rule used to change the bo... |
| cbvrexdva 3245 | Rule used to change the bo... |
| cbvral2vw 3246 | Change bound variables of ... |
| cbvrex2vw 3247 | Change bound variables of ... |
| cbvral3vw 3248 | Change bound variables of ... |
| cbvral4vw 3249 | Change bound variables of ... |
| cbvral6vw 3250 | Change bound variables of ... |
| cbvral8vw 3251 | Change bound variables of ... |
| rsp 3252 | Restricted specialization.... |
| rspa 3253 | Restricted specialization.... |
| rspe 3254 | Restricted specialization.... |
| rspec 3255 | Specialization rule for re... |
| r19.21bi 3256 | Inference from Theorem 19.... |
| r19.21be 3257 | Inference from Theorem 19.... |
| r19.21t 3258 | Restricted quantifier vers... |
| r19.21 3259 | Restricted quantifier vers... |
| r19.23t 3260 | Closed theorem form of ~ r... |
| r19.23 3261 | Restricted quantifier vers... |
| ralrimi 3262 | Inference from Theorem 19.... |
| ralrimia 3263 | Inference from Theorem 19.... |
| rexlimi 3264 | Restricted quantifier vers... |
| ralimdaa 3265 | Deduction quantifying both... |
| reximdai 3266 | Deduction from Theorem 19.... |
| r19.37 3267 | Restricted quantifier vers... |
| r19.41 3268 | Restricted quantifier vers... |
| ralrimd 3269 | Inference from Theorem 19.... |
| rexlimd2 3270 | Version of ~ rexlimd with ... |
| rexlimd 3271 | Deduction form of ~ rexlim... |
| r19.29af2 3272 | A commonly used pattern ba... |
| r19.29af 3273 | A commonly used pattern ba... |
| reximd2a 3274 | Deduction quantifying both... |
| ralbida 3275 | Formula-building rule for ... |
| rexbida 3276 | Formula-building rule for ... |
| ralbid 3277 | Formula-building rule for ... |
| rexbid 3278 | Formula-building rule for ... |
| rexbidvALT 3279 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3280 | Alternate proof of ~ rexbi... |
| rsp2 3281 | Restricted specialization,... |
| rsp2e 3282 | Restricted specialization.... |
| rspec2 3283 | Specialization rule for re... |
| rspec3 3284 | Specialization rule for re... |
| r2alf 3285 | Double restricted universa... |
| r2exf 3286 | Double restricted existent... |
| 2ralbida 3287 | Formula-building rule for ... |
| nfra1 3288 | The setvar ` x ` is not fr... |
| nfre1 3289 | The setvar ` x ` is not fr... |
| ralcom4 3290 | Commutation of restricted ... |
| rexcom4 3291 | Commutation of restricted ... |
| ralcom 3292 | Commutation of restricted ... |
| rexcom 3293 | Commutation of restricted ... |
| rexcom4a 3294 | Specialized existential co... |
| ralrot3 3295 | Rotate three restricted un... |
| ralcom13 3296 | Swap first and third restr... |
| rexcom13 3297 | Swap first and third restr... |
| rexrot4 3298 | Rotate four restricted exi... |
| 2ex2rexrot 3299 | Rotate two existential qua... |
| nfra2w 3300 | Similar to Lemma 24 of [Mo... |
| hbra1 3301 | The setvar ` x ` is not fr... |
| ralcomf 3302 | Commutation of restricted ... |
| rexcomf 3303 | Commutation of restricted ... |
| cbvralfw 3304 | Rule used to change bound ... |
| cbvrexfw 3305 | Rule used to change bound ... |
| cbvralw 3306 | Rule used to change bound ... |
| cbvrexw 3307 | Rule used to change bound ... |
| hbral 3308 | Bound-variable hypothesis ... |
| nfraldw 3309 | Deduction version of ~ nfr... |
| nfrexdw 3310 | Deduction version of ~ nfr... |
| nfralw 3311 | Bound-variable hypothesis ... |
| nfrexw 3312 | Bound-variable hypothesis ... |
| r19.12 3313 | Restricted quantifier vers... |
| reean 3314 | Rearrange restricted exist... |
| cbvralsvw 3315 | Change bound variable by u... |
| cbvrexsvw 3316 | Change bound variable by u... |
| cbvralsvwOLD 3317 | Obsolete version of ~ cbvr... |
| rexeq 3318 | Equality theorem for restr... |
| raleq 3319 | Equality theorem for restr... |
| raleqi 3320 | Equality inference for res... |
| rexeqi 3321 | Equality inference for res... |
| raleqdv 3322 | Equality deduction for res... |
| rexeqdv 3323 | Equality deduction for res... |
| raleqtrdv 3324 | Substitution of equal clas... |
| rexeqtrdv 3325 | Substitution of equal clas... |
| raleqtrrdv 3326 | Substitution of equal clas... |
| rexeqtrrdv 3327 | Substitution of equal clas... |
| raleqbidva 3328 | Equality deduction for res... |
| rexeqbidva 3329 | Equality deduction for res... |
| raleqbidvv 3330 | Version of ~ raleqbidv wit... |
| rexeqbidvv 3331 | Version of ~ rexeqbidv wit... |
| raleqbi1dv 3332 | Equality deduction for res... |
| rexeqbi1dv 3333 | Equality deduction for res... |
| raleleq 3334 | All elements of a class ar... |
| raleleqOLD 3335 | Obsolete version of ~ rale... |
| raleqbii 3336 | Equality deduction for res... |
| rexeqbii 3337 | Equality deduction for res... |
| raleqbidv 3338 | Equality deduction for res... |
| rexeqbidv 3339 | Equality deduction for res... |
| cbvraldva2 3340 | Rule used to change the bo... |
| cbvrexdva2 3341 | Rule used to change the bo... |
| sbralie 3342 | Implicit to explicit subst... |
| sbralieALT 3343 | Alternative shorter proof ... |
| sbralieOLD 3344 | Obsolete version of ~ sbra... |
| raleqf 3345 | Equality theorem for restr... |
| rexeqf 3346 | Equality theorem for restr... |
| raleqbid 3347 | Equality deduction for res... |
| rexeqbid 3348 | Equality deduction for res... |
| cbvralf 3349 | Rule used to change bound ... |
| cbvrexf 3350 | Rule used to change bound ... |
| cbvral 3351 | Rule used to change bound ... |
| cbvrex 3352 | Rule used to change bound ... |
| cbvralv 3353 | Change the bound variable ... |
| cbvrexv 3354 | Change the bound variable ... |
| cbvralsv 3355 | Change bound variable by u... |
| cbvrexsv 3356 | Change bound variable by u... |
| cbvral2v 3357 | Change bound variables of ... |
| cbvrex2v 3358 | Change bound variables of ... |
| cbvral3v 3359 | Change bound variables of ... |
| rgen2a 3360 | Generalization rule for re... |
| nfrald 3361 | Deduction version of ~ nfr... |
| nfrexd 3362 | Deduction version of ~ nfr... |
| nfral 3363 | Bound-variable hypothesis ... |
| nfrex 3364 | Bound-variable hypothesis ... |
| nfra2 3365 | Similar to Lemma 24 of [Mo... |
| ralcom2 3366 | Commutation of restricted ... |
| reu5 3371 | Restricted uniqueness in t... |
| reurmo 3372 | Restricted existential uni... |
| reurex 3373 | Restricted unique existenc... |
| mormo 3374 | Unrestricted "at most one"... |
| rmobiia 3375 | Formula-building rule for ... |
| reubiia 3376 | Formula-building rule for ... |
| rmobii 3377 | Formula-building rule for ... |
| reubii 3378 | Formula-building rule for ... |
| rmoanid 3379 | Cancellation law for restr... |
| reuanid 3380 | Cancellation law for restr... |
| 2reu2rex 3381 | Double restricted existent... |
| rmobidva 3382 | Formula-building rule for ... |
| reubidva 3383 | Formula-building rule for ... |
| rmobidv 3384 | Formula-building rule for ... |
| reubidv 3385 | Formula-building rule for ... |
| reueubd 3386 | Restricted existential uni... |
| rmo5 3387 | Restricted "at most one" i... |
| nrexrmo 3388 | Nonexistence implies restr... |
| moel 3389 | "At most one" element in a... |
| cbvrmovw 3390 | Change the bound variable ... |
| cbvreuvw 3391 | Change the bound variable ... |
| rmobida 3392 | Formula-building rule for ... |
| reubida 3393 | Formula-building rule for ... |
| cbvrmow 3394 | Change the bound variable ... |
| cbvreuw 3395 | Change the bound variable ... |
| nfrmo1 3396 | The setvar ` x ` is not fr... |
| nfreu1 3397 | The setvar ` x ` is not fr... |
| nfrmow 3398 | Bound-variable hypothesis ... |
| nfreuw 3399 | Bound-variable hypothesis ... |
| rmoeq1 3400 | Equality theorem for restr... |
| reueq1 3401 | Equality theorem for restr... |
| rmoeqd 3402 | Equality deduction for res... |
| reueqd 3403 | Equality deduction for res... |
| reueqdv 3404 | Formula-building rule for ... |
| reueqbidv 3405 | Formula-building rule for ... |
| rmoeq1f 3406 | Equality theorem for restr... |
| reueq1f 3407 | Equality theorem for restr... |
| cbvreu 3408 | Change the bound variable ... |
| cbvrmo 3409 | Change the bound variable ... |
| cbvrmov 3410 | Change the bound variable ... |
| cbvreuv 3411 | Change the bound variable ... |
| nfrmod 3412 | Deduction version of ~ nfr... |
| nfreud 3413 | Deduction version of ~ nfr... |
| nfrmo 3414 | Bound-variable hypothesis ... |
| nfreu 3415 | Bound-variable hypothesis ... |
| rabbidva2 3418 | Equivalent wff's yield equ... |
| rabbia2 3419 | Equivalent wff's yield equ... |
| rabbiia 3420 | Equivalent formulas yield ... |
| rabbii 3421 | Equivalent wff's correspon... |
| rabbidva 3422 | Equivalent wff's yield equ... |
| rabbidv 3423 | Equivalent wff's yield equ... |
| rabbieq 3424 | Equivalent wff's correspon... |
| rabswap 3425 | Swap with a membership rel... |
| cbvrabv 3426 | Rule to change the bound v... |
| rabeqcda 3427 | When ` ps ` is always true... |
| rabeqc 3428 | A restricted class abstrac... |
| rabeqi 3429 | Equality theorem for restr... |
| rabeq 3430 | Equality theorem for restr... |
| rabeqdv 3431 | Equality of restricted cla... |
| rabeqbidva 3432 | Equality of restricted cla... |
| rabeqbidvaOLD 3433 | Obsolete version of ~ rabe... |
| rabeqbidv 3434 | Equality of restricted cla... |
| rabrabi 3435 | Abstract builder restricte... |
| nfrab1 3436 | The abstraction variable i... |
| rabid 3437 | An "identity" law of concr... |
| rabidim1 3438 | Membership in a restricted... |
| reqabi 3439 | Inference from equality of... |
| rabrab 3440 | Abstract builder restricte... |
| rabbida4 3441 | Version of ~ rabbidva2 wit... |
| rabbida 3442 | Equivalent wff's yield equ... |
| rabbid 3443 | Version of ~ rabbidv with ... |
| rabeqd 3444 | Deduction form of ~ rabeq ... |
| rabeqbida 3445 | Version of ~ rabeqbidva wi... |
| rabbi 3446 | Equivalent wff's correspon... |
| rabid2f 3447 | An "identity" law for rest... |
| rabid2im 3448 | One direction of ~ rabid2 ... |
| rabid2 3449 | An "identity" law for rest... |
| rabeqf 3450 | Equality theorem for restr... |
| cbvrabw 3451 | Rule to change the bound v... |
| cbvrabwOLD 3452 | Obsolete version of ~ cbvr... |
| nfrabw 3453 | A variable not free in a w... |
| nfrab 3454 | A variable not free in a w... |
| cbvrab 3455 | Rule to change the bound v... |
| vjust 3457 | Justification theorem for ... |
| dfv2 3459 | Alternate definition of th... |
| vex 3460 | All setvar variables are s... |
| elv 3461 | If a proposition is implie... |
| elvd 3462 | If a proposition is implie... |
| el2v 3463 | If a proposition is implie... |
| el3v 3464 | If a proposition is implie... |
| el3v3 3465 | If a proposition is implie... |
| eqv 3466 | The universe contains ever... |
| eqvf 3467 | The universe contains ever... |
| abv 3468 | The class of sets verifyin... |
| abvALT 3469 | Alternate proof of ~ abv ,... |
| isset 3470 | Two ways to express that "... |
| cbvexeqsetf 3471 | The expression ` E. x x = ... |
| issetft 3472 | Closed theorem form of ~ i... |
| issetf 3473 | A version of ~ isset that ... |
| isseti 3474 | A way to say " ` A ` is a ... |
| issetri 3475 | A way to say " ` A ` is a ... |
| eqvisset 3476 | A class equal to a variabl... |
| elex 3477 | If a class is a member of ... |
| elexi 3478 | If a class is a member of ... |
| elexd 3479 | If a class is a member of ... |
| elex22 3480 | If two classes each contai... |
| prcnel 3481 | A proper class doesn't bel... |
| ralv 3482 | A universal quantifier res... |
| rexv 3483 | An existential quantifier ... |
| reuv 3484 | A unique existential quant... |
| rmov 3485 | An at-most-one quantifier ... |
| rabab 3486 | A class abstraction restri... |
| rexcom4b 3487 | Specialized existential co... |
| ceqsal1t 3488 | One direction of ~ ceqsalt... |
| ceqsalt 3489 | Closed theorem version of ... |
| ceqsralt 3490 | Restricted quantifier vers... |
| ceqsalg 3491 | A representation of explic... |
| ceqsalgALT 3492 | Alternate proof of ~ ceqsa... |
| ceqsal 3493 | A representation of explic... |
| ceqsalALT 3494 | A representation of explic... |
| ceqsalv 3495 | A representation of explic... |
| ceqsralv 3496 | Restricted quantifier vers... |
| gencl 3497 | Implicit substitution for ... |
| 2gencl 3498 | Implicit substitution for ... |
| 3gencl 3499 | Implicit substitution for ... |
| cgsexg 3500 | Implicit substitution infe... |
| cgsex2g 3501 | Implicit substitution infe... |
| cgsex4g 3502 | An implicit substitution i... |
| ceqsex 3503 | Elimination of an existent... |
| ceqsexv 3504 | Elimination of an existent... |
| ceqsexv2d 3505 | Elimination of an existent... |
| ceqsex2 3506 | Elimination of two existen... |
| ceqsex2v 3507 | Elimination of two existen... |
| ceqsex3v 3508 | Elimination of three exist... |
| ceqsex4v 3509 | Elimination of four existe... |
| ceqsex6v 3510 | Elimination of six existen... |
| ceqsex8v 3511 | Elimination of eight exist... |
| gencbvex 3512 | Change of bound variable u... |
| gencbvex2 3513 | Restatement of ~ gencbvex ... |
| gencbval 3514 | Change of bound variable u... |
| sbhypf 3515 | Introduce an explicit subs... |
| spcimgft 3516 | Closed theorem form of ~ s... |
| spcimgfi1 3517 | A closed version of ~ spci... |
| spcimgfi1OLD 3518 | Obsolete version of ~ spci... |
| spcgft 3519 | A closed version of ~ spcg... |
| spcimgf 3520 | Rule of specialization, us... |
| spcimegf 3521 | Existential specialization... |
| vtoclgft 3522 | Closed theorem form of ~ v... |
| vtocleg 3523 | Implicit substitution of a... |
| vtoclg 3524 | Implicit substitution of a... |
| vtocle 3525 | Implicit substitution of a... |
| vtoclbg 3526 | Implicit substitution of a... |
| vtocl 3527 | Implicit substitution of a... |
| vtocldf 3528 | Implicit substitution of a... |
| vtocld 3529 | Implicit substitution of a... |
| vtocl2d 3530 | Implicit substitution of t... |
| vtoclef 3531 | Implicit substitution of a... |
| vtoclf 3532 | Implicit substitution of a... |
| vtocl2 3533 | Implicit substitution of c... |
| vtocl3 3534 | Implicit substitution of c... |
| vtoclb 3535 | Implicit substitution of a... |
| vtoclgf 3536 | Implicit substitution of a... |
| vtoclg1f 3537 | Version of ~ vtoclgf with ... |
| vtocl2gf 3538 | Implicit substitution of a... |
| vtocl3gf 3539 | Implicit substitution of a... |
| vtocl2g 3540 | Implicit substitution of 2... |
| vtocl3g 3541 | Implicit substitution of a... |
| vtoclgaf 3542 | Implicit substitution of a... |
| vtoclga 3543 | Implicit substitution of a... |
| vtocl2ga 3544 | Implicit substitution of 2... |
| vtocl2gaf 3545 | Implicit substitution of 2... |
| vtocl3gaf 3546 | Implicit substitution of 3... |
| vtocl3ga 3547 | Implicit substitution of 3... |
| vtocl4g 3548 | Implicit substitution of 4... |
| vtocl4ga 3549 | Implicit substitution of 4... |
| vtoclegft 3550 | Implicit substitution of a... |
| vtoclri 3551 | Implicit substitution of a... |
| spcgf 3552 | Rule of specialization, us... |
| spcegf 3553 | Existential specialization... |
| spcimdv 3554 | Restricted specialization,... |
| spcdv 3555 | Rule of specialization, us... |
| spcimedv 3556 | Restricted existential spe... |
| spcgv 3557 | Rule of specialization, us... |
| spcegv 3558 | Existential specialization... |
| spcedv 3559 | Existential specialization... |
| spc2egv 3560 | Existential specialization... |
| spc2gv 3561 | Specialization with two qu... |
| spc2ed 3562 | Existential specialization... |
| spc2d 3563 | Specialization with 2 quan... |
| spc3egv 3564 | Existential specialization... |
| spc3gv 3565 | Specialization with three ... |
| spcv 3566 | Rule of specialization, us... |
| spcev 3567 | Existential specialization... |
| spc2ev 3568 | Existential specialization... |
| rspct 3569 | A closed version of ~ rspc... |
| rspcdf 3570 | Restricted specialization,... |
| rspc 3571 | Restricted specialization,... |
| rspce 3572 | Restricted existential spe... |
| rspcimdv 3573 | Restricted specialization,... |
| rspcimedv 3574 | Restricted existential spe... |
| rspcdv 3575 | Restricted specialization,... |
| rspcedv 3576 | Restricted existential spe... |
| rspcebdv 3577 | Restricted existential spe... |
| rspcdv2 3578 | Restricted specialization,... |
| rspcv 3579 | Restricted specialization,... |
| rspccv 3580 | Restricted specialization,... |
| rspcva 3581 | Restricted specialization,... |
| rspccva 3582 | Restricted specialization,... |
| rspcev 3583 | Restricted existential spe... |
| rspcdva 3584 | Restricted specialization,... |
| rspcedvd 3585 | Restricted existential spe... |
| rspcedvdw 3586 | Version of ~ rspcedvd wher... |
| rspceb2dv 3587 | Restricted existential spe... |
| rspcime 3588 | Prove a restricted existen... |
| rspceaimv 3589 | Restricted existential spe... |
| rspcedeq1vd 3590 | Restricted existential spe... |
| rspcedeq2vd 3591 | Restricted existential spe... |
| rspc2 3592 | Restricted specialization ... |
| rspc2gv 3593 | Restricted specialization ... |
| rspc2v 3594 | 2-variable restricted spec... |
| rspc2va 3595 | 2-variable restricted spec... |
| rspc2ev 3596 | 2-variable restricted exis... |
| 2rspcedvdw 3597 | Double application of ~ rs... |
| rspc2dv 3598 | 2-variable restricted spec... |
| rspc3v 3599 | 3-variable restricted spec... |
| rspc3ev 3600 | 3-variable restricted exis... |
| 3rspcedvdw 3601 | Triple application of ~ rs... |
| rspc3dv 3602 | 3-variable restricted spec... |
| rspc4v 3603 | 4-variable restricted spec... |
| rspc6v 3604 | 6-variable restricted spec... |
| rspc8v 3605 | 8-variable restricted spec... |
| rspceeqv 3606 | Restricted existential spe... |
| ralxpxfr2d 3607 | Transfer a universal quant... |
| rexraleqim 3608 | Statement following from e... |
| eqvincg 3609 | A variable introduction la... |
| eqvinc 3610 | A variable introduction la... |
| eqvincf 3611 | A variable introduction la... |
| alexeqg 3612 | Two ways to express substi... |
| ceqex 3613 | Equality implies equivalen... |
| ceqsexg 3614 | A representation of explic... |
| ceqsexgv 3615 | Elimination of an existent... |
| ceqsrexv 3616 | Elimination of a restricte... |
| ceqsrexbv 3617 | Elimination of a restricte... |
| ceqsralbv 3618 | Elimination of a restricte... |
| ceqsrex2v 3619 | Elimination of a restricte... |
| clel2g 3620 | Alternate definition of me... |
| clel2 3621 | Alternate definition of me... |
| clel3g 3622 | Alternate definition of me... |
| clel3 3623 | Alternate definition of me... |
| clel4g 3624 | Alternate definition of me... |
| clel4 3625 | Alternate definition of me... |
| clel5 3626 | Alternate definition of cl... |
| pm13.183 3627 | Compare theorem *13.183 in... |
| rr19.3v 3628 | Restricted quantifier vers... |
| rr19.28v 3629 | Restricted quantifier vers... |
| elab6g 3630 | Membership in a class abst... |
| elabd2 3631 | Membership in a class abst... |
| elabd3 3632 | Membership in a class abst... |
| elabgt 3633 | Membership in a class abst... |
| elabgtOLD 3634 | Obsolete version of ~ elab... |
| elabgf 3635 | Membership in a class abst... |
| elabf 3636 | Membership in a class abst... |
| elabg 3637 | Membership in a class abst... |
| elabgw 3638 | Membership in a class abst... |
| elab2gw 3639 | Membership in a class abst... |
| elab 3640 | Membership in a class abst... |
| elab2g 3641 | Membership in a class abst... |
| elabd 3642 | Explicit demonstration the... |
| elab2 3643 | Membership in a class abst... |
| elab4g 3644 | Membership in a class abst... |
| elab3gf 3645 | Membership in a class abst... |
| elab3g 3646 | Membership in a class abst... |
| elab3 3647 | Membership in a class abst... |
| elrabi 3648 | Implication for the member... |
| elrabf 3649 | Membership in a restricted... |
| rabtru 3650 | Abstract builder using the... |
| elrab3t 3651 | Membership in a restricted... |
| elrab 3652 | Membership in a restricted... |
| elrab3 3653 | Membership in a restricted... |
| elrabd 3654 | Membership in a restricted... |
| elrabrd 3655 | Deduction version of ~ elr... |
| elrab2 3656 | Membership in a restricted... |
| elrab2w 3657 | Membership in a restricted... |
| ralab 3658 | Universal quantification o... |
| ralrab 3659 | Universal quantification o... |
| rexab 3660 | Existential quantification... |
| rexrab 3661 | Existential quantification... |
| ralab2 3662 | Universal quantification o... |
| ralrab2 3663 | Universal quantification o... |
| rexab2 3664 | Existential quantification... |
| rexrab2 3665 | Existential quantification... |
| reurab 3666 | Restricted existential uni... |
| abidnf 3667 | Identity used to create cl... |
| dedhb 3668 | A deduction theorem for co... |
| class2seteq 3669 | Writing a set as a class a... |
| nelrdva 3670 | Deduce negative membership... |
| eqeu 3671 | A condition which implies ... |
| moeq 3672 | There exists at most one s... |
| eueq 3673 | A class is a set if and on... |
| eueqi 3674 | There exists a unique set ... |
| eueq2 3675 | Equality has existential u... |
| eueq3 3676 | Equality has existential u... |
| moeq3 3677 | "At most one" property of ... |
| mosub 3678 | "At most one" remains true... |
| mo2icl 3679 | Theorem for inferring "at ... |
| mob2 3680 | Consequence of "at most on... |
| moi2 3681 | Consequence of "at most on... |
| mob 3682 | Equality implied by "at mo... |
| moi 3683 | Equality implied by "at mo... |
| morex 3684 | Derive membership from uni... |
| euxfr2w 3685 | Transfer existential uniqu... |
| euxfrw 3686 | Transfer existential uniqu... |
| euxfr2 3687 | Transfer existential uniqu... |
| euxfr 3688 | Transfer existential uniqu... |
| euind 3689 | Existential uniqueness via... |
| reu2 3690 | A way to express restricte... |
| reu6 3691 | A way to express restricte... |
| reu3 3692 | A way to express restricte... |
| reu6i 3693 | A condition which implies ... |
| eqreu 3694 | A condition which implies ... |
| rmo4 3695 | Restricted "at most one" u... |
| reu4 3696 | Restricted uniqueness usin... |
| reu7 3697 | Restricted uniqueness usin... |
| reu8 3698 | Restricted uniqueness usin... |
| rmo3f 3699 | Restricted "at most one" u... |
| rmo4f 3700 | Restricted "at most one" u... |
| reu2eqd 3701 | Deduce equality from restr... |
| reueq 3702 | Equality has existential u... |
| rmoeq 3703 | Equality's restricted exis... |
| rmoan 3704 | Restricted "at most one" s... |
| rmoim 3705 | Restricted "at most one" i... |
| rmoimia 3706 | Restricted "at most one" i... |
| rmoimi 3707 | Restricted "at most one" i... |
| rmoimi2 3708 | Restricted "at most one" i... |
| 2reu5a 3709 | Double restricted existent... |
| reuimrmo 3710 | Restricted uniqueness impl... |
| 2reuswap 3711 | A condition allowing swap ... |
| 2reuswap2 3712 | A condition allowing swap ... |
| reuxfrd 3713 | Transfer existential uniqu... |
| reuxfr 3714 | Transfer existential uniqu... |
| reuxfr1d 3715 | Transfer existential uniqu... |
| reuxfr1ds 3716 | Transfer existential uniqu... |
| reuxfr1 3717 | Transfer existential uniqu... |
| reuind 3718 | Existential uniqueness via... |
| 2rmorex 3719 | Double restricted quantifi... |
| 2reu5lem1 3720 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3721 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3722 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3723 | Double restricted existent... |
| 2reurmo 3724 | Double restricted quantifi... |
| 2reurex 3725 | Double restricted quantifi... |
| 2rmoswap 3726 | A condition allowing to sw... |
| 2rexreu 3727 | Double restricted existent... |
| cdeqi 3730 | Deduce conditional equalit... |
| cdeqri 3731 | Property of conditional eq... |
| cdeqth 3732 | Deduce conditional equalit... |
| cdeqnot 3733 | Distribute conditional equ... |
| cdeqal 3734 | Distribute conditional equ... |
| cdeqab 3735 | Distribute conditional equ... |
| cdeqal1 3736 | Distribute conditional equ... |
| cdeqab1 3737 | Distribute conditional equ... |
| cdeqim 3738 | Distribute conditional equ... |
| cdeqcv 3739 | Conditional equality for s... |
| cdeqeq 3740 | Distribute conditional equ... |
| cdeqel 3741 | Distribute conditional equ... |
| nfcdeq 3742 | If we have a conditional e... |
| nfccdeq 3743 | Variation of ~ nfcdeq for ... |
| rru 3744 | Relative version of Russel... |
| ru 3745 | Russell's Paradox. Propos... |
| dfsbcq 3748 | Proper substitution of a c... |
| dfsbcq2 3749 | This theorem, which is sim... |
| sbsbc 3750 | Show that ~ df-sb and ~ df... |
| sbceq1d 3751 | Equality theorem for class... |
| sbceq1dd 3752 | Equality theorem for class... |
| sbceqbid 3753 | Equality theorem for class... |
| sbc8g 3754 | This is the closest we can... |
| sbc2or 3755 | The disjunction of two equ... |
| sbcex 3756 | By our definition of prope... |
| sbceq1a 3757 | Equality theorem for class... |
| sbceq2a 3758 | Equality theorem for class... |
| spsbc 3759 | Specialization: if a formu... |
| spsbcd 3760 | Specialization: if a formu... |
| sbcth 3761 | A substitution into a theo... |
| sbcthdv 3762 | Deduction version of ~ sbc... |
| sbcid 3763 | An identity theorem for su... |
| nfsbc1d 3764 | Deduction version of ~ nfs... |
| nfsbc1 3765 | Bound-variable hypothesis ... |
| nfsbc1v 3766 | Bound-variable hypothesis ... |
| nfsbcdw 3767 | Deduction version of ~ nfs... |
| nfsbcw 3768 | Bound-variable hypothesis ... |
| sbccow 3769 | A composition law for clas... |
| nfsbcd 3770 | Deduction version of ~ nfs... |
| nfsbc 3771 | Bound-variable hypothesis ... |
| sbcco 3772 | A composition law for clas... |
| sbcco2 3773 | A composition law for clas... |
| sbc5 3774 | An equivalence for class s... |
| sbc5ALT 3775 | Alternate proof of ~ sbc5 ... |
| sbc6g 3776 | An equivalence for class s... |
| sbc6 3777 | An equivalence for class s... |
| sbc7 3778 | An equivalence for class s... |
| cbvsbcw 3779 | Change bound variables in ... |
| cbvsbcvw 3780 | Change the bound variable ... |
| cbvsbc 3781 | Change bound variables in ... |
| cbvsbcv 3782 | Change the bound variable ... |
| sbciegft 3783 | Conversion of implicit sub... |
| sbciegf 3784 | Conversion of implicit sub... |
| sbcieg 3785 | Conversion of implicit sub... |
| sbcie2g 3786 | Conversion of implicit sub... |
| sbcie 3787 | Conversion of implicit sub... |
| sbciedf 3788 | Conversion of implicit sub... |
| sbcied 3789 | Conversion of implicit sub... |
| sbcied2 3790 | Conversion of implicit sub... |
| elrabsf 3791 | Membership in a restricted... |
| eqsbc1 3792 | Substitution for the left-... |
| sbcng 3793 | Move negation in and out o... |
| sbcimg 3794 | Distribution of class subs... |
| sbcan 3795 | Distribution of class subs... |
| sbcor 3796 | Distribution of class subs... |
| sbcbig 3797 | Distribution of class subs... |
| sbcn1 3798 | Move negation in and out o... |
| sbcim1 3799 | Distribution of class subs... |
| sbcbid 3800 | Formula-building deduction... |
| sbcbidv 3801 | Formula-building deduction... |
| sbcbii 3802 | Formula-building inference... |
| sbcbi1 3803 | Distribution of class subs... |
| sbcbi2 3804 | Substituting into equivale... |
| sbcal 3805 | Move universal quantifier ... |
| sbcex2 3806 | Move existential quantifie... |
| sbceqal 3807 | Class version of one impli... |
| sbeqalb 3808 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3809 | Substitution for the right... |
| sbc3an 3810 | Distribution of class subs... |
| sbcel1v 3811 | Class substitution into a ... |
| sbcel2gv 3812 | Class substitution into a ... |
| sbcel21v 3813 | Class substitution into a ... |
| sbcimdv 3814 | Substitution analogue of T... |
| sbctt 3815 | Substitution for a variabl... |
| sbcgf 3816 | Substitution for a variabl... |
| sbc19.21g 3817 | Substitution for a variabl... |
| sbcg 3818 | Substitution for a variabl... |
| sbcgfi 3819 | Substitution for a variabl... |
| sbc2iegf 3820 | Conversion of implicit sub... |
| sbc2ie 3821 | Conversion of implicit sub... |
| sbc2iedv 3822 | Conversion of implicit sub... |
| sbc3ie 3823 | Conversion of implicit sub... |
| sbccomlem 3824 | Lemma for ~ sbccom . (Con... |
| sbccomlemOLD 3825 | Obsolete version of ~ sbcc... |
| sbccom 3826 | Commutative law for double... |
| sbcralt 3827 | Interchange class substitu... |
| sbcrext 3828 | Interchange class substitu... |
| sbcralg 3829 | Interchange class substitu... |
| sbcrex 3830 | Interchange class substitu... |
| sbcreu 3831 | Interchange class substitu... |
| reu8nf 3832 | Restricted uniqueness usin... |
| sbcabel 3833 | Interchange class substitu... |
| rspsbc 3834 | Restricted quantifier vers... |
| rspsbca 3835 | Restricted quantifier vers... |
| rspesbca 3836 | Existence form of ~ rspsbc... |
| spesbc 3837 | Existence form of ~ spsbc ... |
| spesbcd 3838 | form of ~ spsbc . (Contri... |
| sbcth2 3839 | A substitution into a theo... |
| ra4v 3840 | Version of ~ ra4 with a di... |
| ra4 3841 | Restricted quantifier vers... |
| rmo2 3842 | Alternate definition of re... |
| rmo2i 3843 | Condition implying restric... |
| rmo3 3844 | Restricted "at most one" u... |
| rmob 3845 | Consequence of "at most on... |
| rmoi 3846 | Consequence of "at most on... |
| rmob2 3847 | Consequence of "restricted... |
| rmoi2 3848 | Consequence of "restricted... |
| rmoanim 3849 | Introduction of a conjunct... |
| rmoanimALT 3850 | Alternate proof of ~ rmoan... |
| reuan 3851 | Introduction of a conjunct... |
| 2reu1 3852 | Double restricted existent... |
| 2reu2 3853 | Double restricted existent... |
| csb2 3856 | Alternate expression for t... |
| csbeq1 3857 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3858 | Equality deduction for pro... |
| csbeq2 3859 | Substituting into equivale... |
| csbeq2d 3860 | Formula-building deduction... |
| csbeq2dv 3861 | Formula-building deduction... |
| csbeq2i 3862 | Formula-building inference... |
| csbeq12dv 3863 | Formula-building inference... |
| cbvcsbw 3864 | Change bound variables in ... |
| cbvcsb 3865 | Change bound variables in ... |
| cbvcsbv 3866 | Change the bound variable ... |
| csbid 3867 | Analogue of ~ sbid for pro... |
| csbeq1a 3868 | Equality theorem for prope... |
| csbcow 3869 | Composition law for chaine... |
| csbco 3870 | Composition law for chaine... |
| csbtt 3871 | Substitution doesn't affec... |
| csbconstgf 3872 | Substitution doesn't affec... |
| csbconstg 3873 | Substitution doesn't affec... |
| csbgfi 3874 | Substitution for a variabl... |
| csbconstgi 3875 | The proper substitution of... |
| nfcsb1d 3876 | Bound-variable hypothesis ... |
| nfcsb1 3877 | Bound-variable hypothesis ... |
| nfcsb1v 3878 | Bound-variable hypothesis ... |
| nfcsbd 3879 | Deduction version of ~ nfc... |
| nfcsbw 3880 | Bound-variable hypothesis ... |
| nfcsb 3881 | Bound-variable hypothesis ... |
| csbhypf 3882 | Introduce an explicit subs... |
| csbiebt 3883 | Conversion of implicit sub... |
| csbiedf 3884 | Conversion of implicit sub... |
| csbieb 3885 | Bidirectional conversion b... |
| csbiebg 3886 | Bidirectional conversion b... |
| csbiegf 3887 | Conversion of implicit sub... |
| csbief 3888 | Conversion of implicit sub... |
| csbie 3889 | Conversion of implicit sub... |
| csbied 3890 | Conversion of implicit sub... |
| csbied2 3891 | Conversion of implicit sub... |
| csbie2t 3892 | Conversion of implicit sub... |
| csbie2 3893 | Conversion of implicit sub... |
| csbie2g 3894 | Conversion of implicit sub... |
| cbvrabcsfw 3895 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3896 | A more general version of ... |
| cbvrexcsf 3897 | A more general version of ... |
| cbvreucsf 3898 | A more general version of ... |
| cbvrabcsf 3899 | A more general version of ... |
| cbvralv2 3900 | Rule used to change the bo... |
| cbvrexv2 3901 | Rule used to change the bo... |
| rspc2vd 3902 | Deduction version of 2-var... |
| difjust 3908 | Soundness justification th... |
| unjust 3910 | Soundness justification th... |
| injust 3912 | Soundness justification th... |
| dfin5 3914 | Alternate definition for t... |
| dfdif2 3915 | Alternate definition of cl... |
| eldif 3916 | Expansion of membership in... |
| eldifd 3917 | If a class is in one class... |
| eldifad 3918 | If a class is in the diffe... |
| eldifbd 3919 | If a class is in the diffe... |
| elneeldif 3920 | The elements of a set diff... |
| velcomp 3921 | Characterization of setvar... |
| elin 3922 | Expansion of membership in... |
| dfss2 3924 | Alternate definition of th... |
| dfss 3925 | Variant of subclass defini... |
| dfss3 3927 | Alternate definition of su... |
| dfss6 3928 | Alternate definition of su... |
| dfssf 3929 | Equivalence for subclass r... |
| dfss3f 3930 | Equivalence for subclass r... |
| nfss 3931 | If ` x ` is not free in ` ... |
| ssel 3932 | Membership relationships f... |
| ssel2 3933 | Membership relationships f... |
| sseli 3934 | Membership implication fro... |
| sselii 3935 | Membership inference from ... |
| sselid 3936 | Membership inference from ... |
| sseld 3937 | Membership deduction from ... |
| sselda 3938 | Membership deduction from ... |
| sseldd 3939 | Membership inference from ... |
| ssneld 3940 | If a class is not in anoth... |
| ssneldd 3941 | If an element is not in a ... |
| ssriv 3942 | Inference based on subclas... |
| ssrd 3943 | Deduction based on subclas... |
| ssrdv 3944 | Deduction based on subclas... |
| sstr2 3945 | Transitivity of subclass r... |
| sstr 3946 | Transitivity of subclass r... |
| sstri 3947 | Subclass transitivity infe... |
| sstrd 3948 | Subclass transitivity dedu... |
| sstrid 3949 | Subclass transitivity dedu... |
| sstrdi 3950 | Subclass transitivity dedu... |
| sylan9ss 3951 | A subclass transitivity de... |
| sylan9ssr 3952 | A subclass transitivity de... |
| eqss 3953 | The subclass relationship ... |
| eqssi 3954 | Infer equality from two su... |
| eqssd 3955 | Equality deduction from tw... |
| sssseq 3956 | If a class is a subclass o... |
| eqrd 3957 | Deduce equality of classes... |
| eqri 3958 | Infer equality of classes ... |
| eqelssd 3959 | Equality deduction from su... |
| ssid 3960 | Any class is a subclass of... |
| ssidd 3961 | Weakening of ~ ssid . (Co... |
| ssv 3962 | Any class is a subclass of... |
| sseq1 3963 | Equality theorem for subcl... |
| sseq2 3964 | Equality theorem for the s... |
| sseq12 3965 | Equality theorem for the s... |
| sseq1i 3966 | An equality inference for ... |
| sseq2i 3967 | An equality inference for ... |
| sseq12i 3968 | An equality inference for ... |
| sseq1d 3969 | An equality deduction for ... |
| sseq2d 3970 | An equality deduction for ... |
| sseq12d 3971 | An equality deduction for ... |
| eqsstrd 3972 | Substitution of equality i... |
| eqsstrrd 3973 | Substitution of equality i... |
| sseqtrd 3974 | Substitution of equality i... |
| sseqtrrd 3975 | Substitution of equality i... |
| eqsstrid 3976 | A chained subclass and equ... |
| eqsstrrid 3977 | A chained subclass and equ... |
| sseqtrdi 3978 | A chained subclass and equ... |
| sseqtrrdi 3979 | A chained subclass and equ... |
| sseqtrid 3980 | Subclass transitivity dedu... |
| sseqtrrid 3981 | Subclass transitivity dedu... |
| eqsstrdi 3982 | A chained subclass and equ... |
| eqsstrrdi 3983 | A chained subclass and equ... |
| eqsstri 3984 | Substitution of equality i... |
| eqsstrri 3985 | Substitution of equality i... |
| sseqtri 3986 | Substitution of equality i... |
| sseqtrri 3987 | Substitution of equality i... |
| 3sstr3i 3988 | Substitution of equality i... |
| 3sstr4i 3989 | Substitution of equality i... |
| 3sstr3g 3990 | Substitution of equality i... |
| 3sstr4g 3991 | Substitution of equality i... |
| 3sstr3d 3992 | Substitution of equality i... |
| 3sstr4d 3993 | Substitution of equality i... |
| eqimssd 3994 | Equality implies inclusion... |
| eqimsscd 3995 | Equality implies inclusion... |
| eqimss 3996 | Equality implies inclusion... |
| eqimss2 3997 | Equality implies inclusion... |
| eqimssi 3998 | Infer subclass relationshi... |
| eqimss2i 3999 | Infer subclass relationshi... |
| nssne1 4000 | Two classes are different ... |
| nssne2 4001 | Two classes are different ... |
| nss 4002 | Negation of subclass relat... |
| nssrex 4003 | Negation of subclass relat... |
| nelss 4004 | Demonstrate by witnesses t... |
| ssrexf 4005 | Restricted existential qua... |
| ssrmof 4006 | "At most one" existential ... |
| ssralv 4007 | Quantification restricted ... |
| ssrexv 4008 | Existential quantification... |
| ss2ralv 4009 | Two quantifications restri... |
| ss2rexv 4010 | Two existential quantifica... |
| ralss 4011 | Restricted universal quant... |
| rexss 4012 | Restricted existential qua... |
| ralssOLD 4013 | Obsolete version of ~ rals... |
| rexssOLD 4014 | Obsolete version of ~ rexs... |
| ss2abim 4015 | Class abstractions in a su... |
| ss2ab 4016 | Class abstractions in a su... |
| abss 4017 | Class abstraction in a sub... |
| ssab 4018 | Subclass of a class abstra... |
| ssabral 4019 | The relation for a subclas... |
| ss2abdv 4020 | Deduction of abstraction s... |
| ss2abi 4021 | Inference of abstraction s... |
| abssdv 4022 | Deduction of abstraction s... |
| abssi 4023 | Inference of abstraction s... |
| ss2rab 4024 | Restricted abstraction cla... |
| rabss 4025 | Restricted class abstracti... |
| ssrab 4026 | Subclass of a restricted c... |
| ss2rabd 4027 | Subclass of a restricted c... |
| ssrabdv 4028 | Subclass of a restricted c... |
| rabssdv 4029 | Subclass of a restricted c... |
| ss2rabdv 4030 | Deduction of restricted ab... |
| ss2rabi 4031 | Inference of restricted ab... |
| rabss2 4032 | Subclass law for restricte... |
| rabss2OLD 4033 | Obsolete version of ~ rabs... |
| ssab2 4034 | Subclass relation for the ... |
| ssrab2 4035 | Subclass relation for a re... |
| rabss3d 4036 | Subclass law for restricte... |
| ssrab3 4037 | Subclass relation for a re... |
| rabssrabd 4038 | Subclass of a restricted c... |
| ssrabeq 4039 | If the restricting class o... |
| rabssab 4040 | A restricted class is a su... |
| eqrrabd 4041 | Deduce equality with a res... |
| uniiunlem 4042 | A subset relationship usef... |
| dfpss2 4043 | Alternate definition of pr... |
| dfpss3 4044 | Alternate definition of pr... |
| psseq1 4045 | Equality theorem for prope... |
| psseq2 4046 | Equality theorem for prope... |
| psseq1i 4047 | An equality inference for ... |
| psseq2i 4048 | An equality inference for ... |
| psseq12i 4049 | An equality inference for ... |
| psseq1d 4050 | An equality deduction for ... |
| psseq2d 4051 | An equality deduction for ... |
| psseq12d 4052 | An equality deduction for ... |
| pssss 4053 | A proper subclass is a sub... |
| pssne 4054 | Two classes in a proper su... |
| pssssd 4055 | Deduce subclass from prope... |
| pssned 4056 | Proper subclasses are uneq... |
| sspss 4057 | Subclass in terms of prope... |
| pssirr 4058 | Proper subclass is irrefle... |
| pssirrOLD 4059 | Obsolete version of ~ pssi... |
| pssn2lp 4060 | Proper subclass has no 2-c... |
| sspsstri 4061 | Two ways of stating tricho... |
| ssnpss 4062 | Partial trichotomy law for... |
| psstr 4063 | Transitive law for proper ... |
| sspsstr 4064 | Transitive law for subclas... |
| psssstr 4065 | Transitive law for subclas... |
| psstrd 4066 | Proper subclass inclusion ... |
| sspsstrd 4067 | Transitivity involving sub... |
| psssstrd 4068 | Transitivity involving sub... |
| npss 4069 | A class is not a proper su... |
| ssnelpss 4070 | A subclass missing a membe... |
| ssnelpssd 4071 | Subclass inclusion with on... |
| ssexnelpss 4072 | If there is an element of ... |
| dfdif3 4073 | Alternate definition of cl... |
| dfdif3OLD 4074 | Obsolete version of ~ dfdi... |
| difeq1 4075 | Equality theorem for class... |
| difeq2 4076 | Equality theorem for class... |
| difeq12 4077 | Equality theorem for class... |
| difeq1i 4078 | Inference adding differenc... |
| difeq2i 4079 | Inference adding differenc... |
| difeq12i 4080 | Equality inference for cla... |
| difeq1d 4081 | Deduction adding differenc... |
| difeq2d 4082 | Deduction adding differenc... |
| difeq12d 4083 | Equality deduction for cla... |
| difeqri 4084 | Inference from membership ... |
| nfdif 4085 | Bound-variable hypothesis ... |
| eldifi 4086 | Implication of membership ... |
| eldifn 4087 | Implication of membership ... |
| elndif 4088 | A set does not belong to a... |
| neldif 4089 | Implication of membership ... |
| difdif 4090 | Double class difference. ... |
| difss 4091 | Subclass relationship for ... |
| difssd 4092 | A difference of two classe... |
| difss2 4093 | If a class is contained in... |
| difss2d 4094 | If a class is contained in... |
| ssdifss 4095 | Preservation of a subclass... |
| ddif 4096 | Double complement under un... |
| ssconb 4097 | Contraposition law for sub... |
| sscon 4098 | Contraposition law for sub... |
| ssdif 4099 | Difference law for subsets... |
| ssdifd 4100 | If ` A ` is contained in `... |
| sscond 4101 | If ` A ` is contained in `... |
| ssdifssd 4102 | If ` A ` is contained in `... |
| ssdif2d 4103 | If ` A ` is contained in `... |
| raldifb 4104 | Restricted universal quant... |
| rexdifi 4105 | Restricted existential qua... |
| complss 4106 | Complementation reverses i... |
| compleq 4107 | Two classes are equal if a... |
| elun 4108 | Expansion of membership in... |
| elunnel1 4109 | A member of a union that i... |
| elunnel2 4110 | A member of a union that i... |
| uneqri 4111 | Inference from membership ... |
| unidm 4112 | Idempotent law for union o... |
| uncom 4113 | Commutative law for union ... |
| equncom 4114 | If a class equals the unio... |
| equncomi 4115 | Inference form of ~ equnco... |
| uneq1 4116 | Equality theorem for the u... |
| uneq2 4117 | Equality theorem for the u... |
| uneq12 4118 | Equality theorem for the u... |
| uneq1i 4119 | Inference adding union to ... |
| uneq2i 4120 | Inference adding union to ... |
| uneq12i 4121 | Equality inference for the... |
| uneq1d 4122 | Deduction adding union to ... |
| uneq2d 4123 | Deduction adding union to ... |
| uneq12d 4124 | Equality deduction for the... |
| nfun 4125 | Bound-variable hypothesis ... |
| unass 4126 | Associative law for union ... |
| un12 4127 | A rearrangement of union. ... |
| un23 4128 | A rearrangement of union. ... |
| un4 4129 | A rearrangement of the uni... |
| unundi 4130 | Union distributes over its... |
| unundir 4131 | Union distributes over its... |
| ssun1 4132 | Subclass relationship for ... |
| ssun2 4133 | Subclass relationship for ... |
| ssun3 4134 | Subclass law for union of ... |
| ssun4 4135 | Subclass law for union of ... |
| elun1 4136 | Membership law for union o... |
| elun2 4137 | Membership law for union o... |
| elunant 4138 | A statement is true for ev... |
| unss1 4139 | Subclass law for union of ... |
| ssequn1 4140 | A relationship between sub... |
| unss2 4141 | Subclass law for union of ... |
| unss12 4142 | Subclass law for union of ... |
| ssequn2 4143 | A relationship between sub... |
| unss 4144 | The union of two subclasse... |
| unssi 4145 | An inference showing the u... |
| unssd 4146 | A deduction showing the un... |
| unssad 4147 | If ` ( A u. B ) ` is conta... |
| unssbd 4148 | If ` ( A u. B ) ` is conta... |
| ssun 4149 | A condition that implies i... |
| rexun 4150 | Restricted existential qua... |
| ralunb 4151 | Restricted quantification ... |
| ralun 4152 | Restricted quantification ... |
| elini 4153 | Membership in an intersect... |
| elind 4154 | Deduce membership in an in... |
| elinel1 4155 | Membership in an intersect... |
| elinel2 4156 | Membership in an intersect... |
| elin2 4157 | Membership in a class defi... |
| elin1d 4158 | Elementhood in the first s... |
| elin2d 4159 | Elementhood in the first s... |
| elin3 4160 | Membership in a class defi... |
| nel1nelin 4161 | Membership in an intersect... |
| nel2nelin 4162 | Membership in an intersect... |
| incom 4163 | Commutative law for inters... |
| ineqcom 4164 | Two ways of expressing tha... |
| ineqcomi 4165 | Two ways of expressing tha... |
| ineqri 4166 | Inference from membership ... |
| ineq1 4167 | Equality theorem for inter... |
| ineq2 4168 | Equality theorem for inter... |
| ineq12 4169 | Equality theorem for inter... |
| ineq1i 4170 | Equality inference for int... |
| ineq2i 4171 | Equality inference for int... |
| ineq12i 4172 | Equality inference for int... |
| ineq1d 4173 | Equality deduction for int... |
| ineq2d 4174 | Equality deduction for int... |
| ineq12d 4175 | Equality deduction for int... |
| ineqan12d 4176 | Equality deduction for int... |
| sseqin2 4177 | A relationship between sub... |
| nfin 4178 | Bound-variable hypothesis ... |
| rabbi2dva 4179 | Deduction from a wff to a ... |
| inidm 4180 | Idempotent law for interse... |
| inass 4181 | Associative law for inters... |
| in12 4182 | A rearrangement of interse... |
| in32 4183 | A rearrangement of interse... |
| in13 4184 | A rearrangement of interse... |
| in31 4185 | A rearrangement of interse... |
| inrot 4186 | Rotate the intersection of... |
| in4 4187 | Rearrangement of intersect... |
| inindi 4188 | Intersection distributes o... |
| inindir 4189 | Intersection distributes o... |
| inss1 4190 | The intersection of two cl... |
| inss2 4191 | The intersection of two cl... |
| ssin 4192 | Subclass of intersection. ... |
| ssini 4193 | An inference showing that ... |
| ssind 4194 | A deduction showing that a... |
| ssrin 4195 | Add right intersection to ... |
| sslin 4196 | Add left intersection to s... |
| ssrind 4197 | Add right intersection to ... |
| ss2in 4198 | Intersection of subclasses... |
| ssinss1 4199 | Intersection preserves sub... |
| ssinss1OLD 4200 | Obsolete version of ~ ssin... |
| ssinss1d 4201 | Intersection preserves sub... |
| inss 4202 | Inclusion of an intersecti... |
| ralin 4203 | Restricted universal quant... |
| rexin 4204 | Restricted existential qua... |
| dfss7 4205 | Alternate definition of su... |
| symdifcom 4208 | Symmetric difference commu... |
| symdifeq1 4209 | Equality theorem for symme... |
| symdifeq2 4210 | Equality theorem for symme... |
| nfsymdif 4211 | Hypothesis builder for sym... |
| elsymdif 4212 | Membership in a symmetric ... |
| dfsymdif4 4213 | Alternate definition of th... |
| elsymdifxor 4214 | Membership in a symmetric ... |
| dfsymdif2 4215 | Alternate definition of th... |
| symdifass 4216 | Symmetric difference is as... |
| difsssymdif 4217 | The symmetric difference c... |
| difsymssdifssd 4218 | If the symmetric differenc... |
| unabs 4219 | Absorption law for union. ... |
| inabs 4220 | Absorption law for interse... |
| nssinpss 4221 | Negation of subclass expre... |
| nsspssun 4222 | Negation of subclass expre... |
| dfss4 4223 | Subclass defined in terms ... |
| dfun2 4224 | An alternate definition of... |
| dfin2 4225 | An alternate definition of... |
| difin 4226 | Difference with intersecti... |
| ssdifim 4227 | Implication of a class dif... |
| ssdifsym 4228 | Symmetric class difference... |
| dfss5 4229 | Alternate definition of su... |
| dfun3 4230 | Union defined in terms of ... |
| dfin3 4231 | Intersection defined in te... |
| dfin4 4232 | Alternate definition of th... |
| invdif 4233 | Intersection with universa... |
| indif 4234 | Intersection with class di... |
| indif2 4235 | Bring an intersection in a... |
| indif1 4236 | Bring an intersection in a... |
| indifcom 4237 | Commutation law for inters... |
| indi 4238 | Distributive law for inter... |
| undi 4239 | Distributive law for union... |
| indir 4240 | Distributive law for inter... |
| undir 4241 | Distributive law for union... |
| unineq 4242 | Infer equality from equali... |
| uneqin 4243 | Equality of union and inte... |
| difundi 4244 | Distributive law for class... |
| difundir 4245 | Distributive law for class... |
| difindi 4246 | Distributive law for class... |
| difindir 4247 | Distributive law for class... |
| indifdi 4248 | Distribute intersection ov... |
| indifdir 4249 | Distribute intersection ov... |
| difdif2 4250 | Class difference by a clas... |
| undm 4251 | De Morgan's law for union.... |
| indm 4252 | De Morgan's law for inters... |
| difun1 4253 | A relationship involving d... |
| undif3 4254 | An equality involving clas... |
| difin2 4255 | Represent a class differen... |
| dif32 4256 | Swap second and third argu... |
| difabs 4257 | Absorption-like law for cl... |
| sscon34b 4258 | Relative complementation r... |
| rcompleq 4259 | Two subclasses are equal i... |
| dfsymdif3 4260 | Alternate definition of th... |
| unabw 4261 | Union of two class abstrac... |
| unab 4262 | Union of two class abstrac... |
| inab 4263 | Intersection of two class ... |
| difab 4264 | Difference of two class ab... |
| abanssl 4265 | A class abstraction with a... |
| abanssr 4266 | A class abstraction with a... |
| notabw 4267 | A class abstraction define... |
| notab 4268 | A class abstraction define... |
| unrab 4269 | Union of two restricted cl... |
| inrab 4270 | Intersection of two restri... |
| inrab2 4271 | Intersection with a restri... |
| difrab 4272 | Difference of two restrict... |
| dfrab3 4273 | Alternate definition of re... |
| dfrab2 4274 | Alternate definition of re... |
| rabdif 4275 | Move difference in and out... |
| notrab 4276 | Complementation of restric... |
| dfrab3ss 4277 | Restricted class abstracti... |
| rabun2 4278 | Abstraction restricted to ... |
| reuun2 4279 | Transfer uniqueness to a s... |
| reuss2 4280 | Transfer uniqueness to a s... |
| reuss 4281 | Transfer uniqueness to a s... |
| reuun1 4282 | Transfer uniqueness to a s... |
| reupick 4283 | Restricted uniqueness "pic... |
| reupick3 4284 | Restricted uniqueness "pic... |
| reupick2 4285 | Restricted uniqueness "pic... |
| euelss 4286 | Transfer uniqueness of an ... |
| dfnul4 4289 | Alternate definition of th... |
| dfnul2 4290 | Alternate definition of th... |
| dfnul3 4291 | Alternate definition of th... |
| noel 4292 | The empty set has no eleme... |
| nel02 4293 | The empty set has no eleme... |
| n0i 4294 | If a class has elements, t... |
| ne0i 4295 | If a class has elements, t... |
| ne0d 4296 | Deduction form of ~ ne0i .... |
| n0ii 4297 | If a class has elements, t... |
| ne0ii 4298 | If a class has elements, t... |
| vn0 4299 | The universal class is not... |
| vn0ALT 4300 | Alternate proof of ~ vn0 .... |
| eq0f 4301 | A class is equal to the em... |
| neq0f 4302 | A class is not empty if an... |
| n0f 4303 | A class is nonempty if and... |
| eq0 4304 | A class is equal to the em... |
| eq0ALT 4305 | Alternate proof of ~ eq0 .... |
| neq0 4306 | A class is not empty if an... |
| n0 4307 | A class is nonempty if and... |
| n0limd 4308 | Deduction rule for nonempt... |
| nel0 4309 | From the general negation ... |
| reximdva0 4310 | Restricted existence deduc... |
| rspn0 4311 | Specialization for restric... |
| n0rex 4312 | There is an element in a n... |
| ssn0rex 4313 | There is an element in a c... |
| n0moeu 4314 | A case of equivalence of "... |
| rex0 4315 | Vacuous restricted existen... |
| reu0 4316 | Vacuous restricted uniquen... |
| rmo0 4317 | Vacuous restricted at-most... |
| 0el 4318 | Membership of the empty se... |
| n0el 4319 | Negated membership of the ... |
| eqeuel 4320 | A condition which implies ... |
| ssdif0 4321 | Subclass expressed in term... |
| difn0 4322 | If the difference of two s... |
| pssdifn0 4323 | A proper subclass has a no... |
| pssdif 4324 | A proper subclass has a no... |
| ndisj 4325 | Express that an intersecti... |
| inn0f 4326 | A nonempty intersection. ... |
| inn0 4327 | A nonempty intersection. ... |
| difin0ss 4328 | Difference, intersection, ... |
| inssdif0 4329 | Intersection, subclass, an... |
| inindif 4330 | The intersection and class... |
| difid 4331 | The difference between a c... |
| difidALT 4332 | Alternate proof of ~ difid... |
| dif0 4333 | The difference between a c... |
| ab0w 4334 | The class of sets verifyin... |
| ab0 4335 | The class of sets verifyin... |
| ab0ALT 4336 | Alternate proof of ~ ab0 ,... |
| dfnf5 4337 | Characterization of nonfre... |
| ab0orv 4338 | The class abstraction defi... |
| ab0orvALT 4339 | Alternate proof of ~ ab0or... |
| abn0 4340 | Nonempty class abstraction... |
| rab0 4341 | Any restricted class abstr... |
| rab0OLD 4342 | Obsolete version of ~ rab0... |
| rabeq0w 4343 | Condition for a restricted... |
| rabeq0 4344 | Condition for a restricted... |
| rabn0 4345 | Nonempty restricted class ... |
| rabxm 4346 | Law of excluded middle, in... |
| rabnc 4347 | Law of noncontradiction, i... |
| elneldisj 4348 | The set of elements ` s ` ... |
| elnelun 4349 | The union of the set of el... |
| un0 4350 | The union of a class with ... |
| in0 4351 | The intersection of a clas... |
| 0un 4352 | The union of the empty set... |
| 0in 4353 | The intersection of the em... |
| inv1 4354 | The intersection of a clas... |
| unv 4355 | The union of a class with ... |
| 0ss 4356 | The null set is a subset o... |
| ss0b 4357 | Any subset of the empty se... |
| ss0 4358 | Any subset of the empty se... |
| sseq0 4359 | A subclass of an empty cla... |
| ssn0 4360 | A class with a nonempty su... |
| 0dif 4361 | The difference between the... |
| abf 4362 | A class abstraction determ... |
| eq0rdv 4363 | Deduction for equality to ... |
| eq0rdvALT 4364 | Alternate proof of ~ eq0rd... |
| csbprc 4365 | The proper substitution of... |
| csb0 4366 | The proper substitution of... |
| sbcel12 4367 | Distribute proper substitu... |
| sbceqg 4368 | Distribute proper substitu... |
| sbceqi 4369 | Distribution of class subs... |
| sbcnel12g 4370 | Distribute proper substitu... |
| sbcne12 4371 | Distribute proper substitu... |
| sbcel1g 4372 | Move proper substitution i... |
| sbceq1g 4373 | Move proper substitution t... |
| sbcel2 4374 | Move proper substitution i... |
| sbceq2g 4375 | Move proper substitution t... |
| csbcom 4376 | Commutative law for double... |
| sbcnestgfw 4377 | Nest the composition of tw... |
| csbnestgfw 4378 | Nest the composition of tw... |
| sbcnestgw 4379 | Nest the composition of tw... |
| csbnestgw 4380 | Nest the composition of tw... |
| sbcco3gw 4381 | Composition of two substit... |
| sbcnestgf 4382 | Nest the composition of tw... |
| csbnestgf 4383 | Nest the composition of tw... |
| sbcnestg 4384 | Nest the composition of tw... |
| csbnestg 4385 | Nest the composition of tw... |
| sbcco3g 4386 | Composition of two substit... |
| csbco3g 4387 | Composition of two class s... |
| csbnest1g 4388 | Nest the composition of tw... |
| csbidm 4389 | Idempotent law for class s... |
| csbvarg 4390 | The proper substitution of... |
| csbvargi 4391 | The proper substitution of... |
| sbccsb 4392 | Substitution into a wff ex... |
| sbccsb2 4393 | Substitution into a wff ex... |
| rspcsbela 4394 | Special case related to ~ ... |
| sbnfc2 4395 | Two ways of expressing " `... |
| csbab 4396 | Move substitution into a c... |
| csbun 4397 | Distribution of class subs... |
| csbin 4398 | Distribute proper substitu... |
| csbie2df 4399 | Conversion of implicit sub... |
| 2nreu 4400 | If there are two different... |
| un00 4401 | Two classes are empty iff ... |
| vss 4402 | Only the universal class h... |
| 0pss 4403 | The null set is a proper s... |
| npss0 4404 | No set is a proper subset ... |
| pssv 4405 | Any non-universal class is... |
| disj 4406 | Two ways of saying that tw... |
| disjr 4407 | Two ways of saying that tw... |
| disj1 4408 | Two ways of saying that tw... |
| reldisj 4409 | Two ways of saying that tw... |
| disj3 4410 | Two ways of saying that tw... |
| disjne 4411 | Members of disjoint sets a... |
| disjeq0 4412 | Two disjoint sets are equa... |
| disjel 4413 | A set can't belong to both... |
| disj2 4414 | Two ways of saying that tw... |
| disj4 4415 | Two ways of saying that tw... |
| ssdisj 4416 | Intersection with a subcla... |
| disjpss 4417 | A class is a proper subset... |
| undisj1 4418 | The union of disjoint clas... |
| undisj2 4419 | The union of disjoint clas... |
| ssindif0 4420 | Subclass expressed in term... |
| inelcm 4421 | The intersection of classe... |
| minel 4422 | A minimum element of a cla... |
| undif4 4423 | Distribute union over diff... |
| disjssun 4424 | Subset relation for disjoi... |
| vdif0 4425 | Universal class equality i... |
| difrab0eq 4426 | If the difference between ... |
| pssnel 4427 | A proper subclass has a me... |
| disjdif 4428 | A class and its relative c... |
| disjdifr 4429 | A class and its relative c... |
| difin0 4430 | The difference of a class ... |
| unvdif 4431 | The union of a class and i... |
| undif1 4432 | Absorption of difference b... |
| undif2 4433 | Absorption of difference b... |
| undifabs 4434 | Absorption of difference b... |
| inundif 4435 | The intersection and class... |
| disjdif2 4436 | The difference of a class ... |
| difun2 4437 | Absorption of union by dif... |
| undif 4438 | Union of complementary par... |
| undifr 4439 | Union of complementary par... |
| undif5 4440 | An equality involving clas... |
| ssdifin0 4441 | A subset of a difference d... |
| ssdifeq0 4442 | A class is a subclass of i... |
| ssundif 4443 | A condition equivalent to ... |
| difcom 4444 | Swap the arguments of a cl... |
| pssdifcom1 4445 | Two ways to express overla... |
| pssdifcom2 4446 | Two ways to express non-co... |
| difdifdir 4447 | Distributive law for class... |
| uneqdifeq 4448 | Two ways to say that ` A `... |
| raldifeq 4449 | Equality theorem for restr... |
| rzal 4450 | Vacuous quantification is ... |
| rzalALT 4451 | Alternate proof of ~ rzal ... |
| rexn0 4452 | Restricted existential qua... |
| ralf0 4453 | The quantification of a fa... |
| ral0 4454 | Vacuous universal quantifi... |
| r19.2z 4455 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4456 | A response to the notion t... |
| r19.3rz 4457 | Restricted quantification ... |
| r19.28z 4458 | Restricted quantifier vers... |
| r19.3rzv 4459 | Restricted quantification ... |
| r19.3rzvOLD 4460 | Obsolete version of ~ r19.... |
| r19.9rzv 4461 | Restricted quantification ... |
| r19.28zv 4462 | Restricted quantifier vers... |
| r19.37zv 4463 | Restricted quantifier vers... |
| r19.45zv 4464 | Restricted version of Theo... |
| r19.44zv 4465 | Restricted version of Theo... |
| r19.27z 4466 | Restricted quantifier vers... |
| r19.27zv 4467 | Restricted quantifier vers... |
| r19.36zv 4468 | Restricted quantifier vers... |
| ralnralall 4469 | A contradiction concerning... |
| falseral0 4470 | A false statement can only... |
| falseral0OLD 4471 | Obsolete version of ~ fals... |
| ralidmw 4472 | Idempotent law for restric... |
| ralidm 4473 | Idempotent law for restric... |
| raaan 4474 | Rearrange restricted quant... |
| raaanv 4475 | Rearrange restricted quant... |
| sbss 4476 | Set substitution into the ... |
| sbcssg 4477 | Distribute proper substitu... |
| raaan2 4478 | Rearrange restricted quant... |
| 2reu4lem 4479 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4480 | Definition of double restr... |
| csbdif 4481 | Distribution of class subs... |
| dfif2 4484 | An alternate definition of... |
| dfif6 4485 | An alternate definition of... |
| ifeq1 4486 | Equality theorem for condi... |
| ifeq2 4487 | Equality theorem for condi... |
| iftrue 4488 | Value of the conditional o... |
| iftruei 4489 | Inference associated with ... |
| iftrued 4490 | Value of the conditional o... |
| iffalse 4491 | Value of the conditional o... |
| iffalsei 4492 | Inference associated with ... |
| iffalsed 4493 | Value of the conditional o... |
| ifnefalse 4494 | When values are unequal, b... |
| iftrueb 4495 | When the branches are not ... |
| ifsb 4496 | Distribute a function over... |
| dfif3 4497 | Alternate definition of th... |
| dfif4 4498 | Alternate definition of th... |
| dfif5 4499 | Alternate definition of th... |
| ifssun 4500 | A conditional class is inc... |
| ifeq12 4501 | Equality theorem for condi... |
| ifeq1d 4502 | Equality deduction for con... |
| ifeq2d 4503 | Equality deduction for con... |
| ifeq12d 4504 | Equality deduction for con... |
| ifbi 4505 | Equivalence theorem for co... |
| ifbid 4506 | Equivalence deduction for ... |
| ifbieq1d 4507 | Equivalence/equality deduc... |
| ifbieq2i 4508 | Equivalence/equality infer... |
| ifbieq2d 4509 | Equivalence/equality deduc... |
| ifbieq12i 4510 | Equivalence deduction for ... |
| ifbieq12d 4511 | Equivalence deduction for ... |
| nfifd 4512 | Deduction form of ~ nfif .... |
| nfif 4513 | Bound-variable hypothesis ... |
| ifeq1da 4514 | Conditional equality. (Co... |
| ifeq2da 4515 | Conditional equality. (Co... |
| ifeq12da 4516 | Equivalence deduction for ... |
| ifbieq12d2 4517 | Equivalence deduction for ... |
| ifclda 4518 | Conditional closure. (Con... |
| ifeqda 4519 | Separation of the values o... |
| elimif 4520 | Elimination of a condition... |
| ifbothda 4521 | A wff ` th ` containing a ... |
| ifboth 4522 | A wff ` th ` containing a ... |
| ifid 4523 | Identical true and false a... |
| eqif 4524 | Expansion of an equality w... |
| ifval 4525 | Another expression of the ... |
| elif 4526 | Membership in a conditiona... |
| ifel 4527 | Membership of a conditiona... |
| ifcl 4528 | Membership (closure) of a ... |
| ifcld 4529 | Membership (closure) of a ... |
| ifcli 4530 | Inference associated with ... |
| ifexd 4531 | Existence of the condition... |
| ifexg 4532 | Existence of the condition... |
| ifex 4533 | Existence of the condition... |
| ifeqor 4534 | The possible values of a c... |
| ifnot 4535 | Negating the first argumen... |
| ifan 4536 | Rewrite a conjunction in a... |
| ifor 4537 | Rewrite a disjunction in a... |
| 2if2 4538 | Resolve two nested conditi... |
| ifcomnan 4539 | Commute the conditions in ... |
| csbif 4540 | Distribute proper substitu... |
| dedth 4541 | Weak deduction theorem tha... |
| dedth2h 4542 | Weak deduction theorem eli... |
| dedth3h 4543 | Weak deduction theorem eli... |
| dedth4h 4544 | Weak deduction theorem eli... |
| dedth2v 4545 | Weak deduction theorem for... |
| dedth3v 4546 | Weak deduction theorem for... |
| dedth4v 4547 | Weak deduction theorem for... |
| elimhyp 4548 | Eliminate a hypothesis con... |
| elimhyp2v 4549 | Eliminate a hypothesis con... |
| elimhyp3v 4550 | Eliminate a hypothesis con... |
| elimhyp4v 4551 | Eliminate a hypothesis con... |
| elimel 4552 | Eliminate a membership hyp... |
| elimdhyp 4553 | Version of ~ elimhyp where... |
| keephyp 4554 | Transform a hypothesis ` p... |
| keephyp2v 4555 | Keep a hypothesis containi... |
| keephyp3v 4556 | Keep a hypothesis containi... |
| pwjust 4558 | Soundness justification th... |
| elpwg 4560 | Membership in a power clas... |
| elpw 4561 | Membership in a power clas... |
| velpw 4562 | Setvar variable membership... |
| elpwd 4563 | Membership in a power clas... |
| elpwi 4564 | Subset relation implied by... |
| elpwb 4565 | Characterization of the el... |
| elpwid 4566 | An element of a power clas... |
| elelpwi 4567 | If ` A ` belongs to a part... |
| sspw 4568 | The powerclass preserves i... |
| sspwi 4569 | The powerclass preserves i... |
| sspwd 4570 | The powerclass preserves i... |
| pweq 4571 | Equality theorem for power... |
| pweqALT 4572 | Alternate proof of ~ pweq ... |
| pweqi 4573 | Equality inference for pow... |
| pweqd 4574 | Equality deduction for pow... |
| pwunss 4575 | The power class of the uni... |
| nfpw 4576 | Bound-variable hypothesis ... |
| pwidg 4577 | A set is an element of its... |
| pwidgOLD 4578 | Obsolete version of ~ pwid... |
| pwidb 4579 | A class is an element of i... |
| pwid 4580 | A set is a member of its p... |
| pwss 4581 | Subclass relationship for ... |
| pwundif 4582 | Break up the power class o... |
| snjust 4583 | Soundness justification th... |
| sneq 4594 | Equality theorem for singl... |
| sneqi 4595 | Equality inference for sin... |
| sneqd 4596 | Equality deduction for sin... |
| dfsn2 4597 | Alternate definition of si... |
| elsng 4598 | There is exactly one eleme... |
| elsn 4599 | There is exactly one eleme... |
| velsn 4600 | There is only one element ... |
| elsni 4601 | There is at most one eleme... |
| elsnd 4602 | There is at most one eleme... |
| rabsneq 4603 | Equality of class abstract... |
| absn 4604 | Condition for a class abst... |
| dfpr2 4605 | Alternate definition of a ... |
| dfsn2ALT 4606 | Alternate definition of si... |
| elprg 4607 | A member of a pair of clas... |
| elpri 4608 | If a class is an element o... |
| elpr 4609 | A member of a pair of clas... |
| elpr2g 4610 | A member of a pair of sets... |
| elpr2 4611 | A member of a pair of sets... |
| elprn1 4612 | A member of an unordered p... |
| elprn2 4613 | A member of an unordered p... |
| nelpr2 4614 | If a class is not an eleme... |
| nelpr1 4615 | If a class is not an eleme... |
| nelpri 4616 | If an element doesn't matc... |
| prneli 4617 | If an element doesn't matc... |
| nelprd 4618 | If an element doesn't matc... |
| eldifpr 4619 | Membership in a set with t... |
| rexdifpr 4620 | Restricted existential qua... |
| snidg 4621 | A set is a member of its s... |
| snidb 4622 | A class is a set iff it is... |
| snid 4623 | A set is a member of its s... |
| vsnid 4624 | A setvar variable is a mem... |
| elsn2g 4625 | There is exactly one eleme... |
| elsn2 4626 | There is exactly one eleme... |
| nelsn 4627 | If a class is not equal to... |
| rabeqsn 4628 | Conditions for a restricte... |
| rabsssn 4629 | Conditions for a restricte... |
| rabeqsnd 4630 | Conditions for a restricte... |
| ralsnsg 4631 | Substitution expressed in ... |
| rexsns 4632 | Restricted existential qua... |
| rexsngf 4633 | Restricted existential qua... |
| ralsngf 4634 | Restricted universal quant... |
| reusngf 4635 | Restricted existential uni... |
| ralsng 4636 | Substitution expressed in ... |
| rexsng 4637 | Restricted existential qua... |
| reusng 4638 | Restricted existential uni... |
| 2ralsng 4639 | Substitution expressed in ... |
| rexreusng 4640 | Restricted existential uni... |
| exsnrex 4641 | There is a set being the e... |
| ralsn 4642 | Convert a universal quanti... |
| rexsn 4643 | Convert an existential qua... |
| elunsn 4644 | Elementhood in a union wit... |
| elpwunsn 4645 | Membership in an extension... |
| eqoreldif 4646 | An element of a set is eit... |
| eltpg 4647 | Members of an unordered tr... |
| eldiftp 4648 | Membership in a set with t... |
| eltpi 4649 | A member of an unordered t... |
| eltp 4650 | A member of an unordered t... |
| el7g 4651 | Members of a set with seve... |
| dftp2 4652 | Alternate definition of un... |
| nfpr 4653 | Bound-variable hypothesis ... |
| ifpr 4654 | Membership of a conditiona... |
| ralprgf 4655 | Convert a restricted unive... |
| rexprgf 4656 | Convert a restricted exist... |
| ralprg 4657 | Convert a restricted unive... |
| rexprg 4658 | Convert a restricted exist... |
| raltpg 4659 | Convert a restricted unive... |
| rextpg 4660 | Convert a restricted exist... |
| ralpr 4661 | Convert a restricted unive... |
| rexpr 4662 | Convert a restricted exist... |
| reuprg0 4663 | Convert a restricted exist... |
| reuprg 4664 | Convert a restricted exist... |
| reurexprg 4665 | Convert a restricted exist... |
| raltp 4666 | Convert a universal quanti... |
| rextp 4667 | Convert an existential qua... |
| nfsn 4668 | Bound-variable hypothesis ... |
| csbsng 4669 | Distribute proper substitu... |
| csbprg 4670 | Distribute proper substitu... |
| elinsn 4671 | If the intersection of two... |
| disjsn 4672 | Intersection with the sing... |
| disjsn2 4673 | Two distinct singletons ar... |
| disjpr2 4674 | Two completely distinct un... |
| disjprsn 4675 | The disjoint intersection ... |
| disjtpsn 4676 | The disjoint intersection ... |
| disjtp2 4677 | Two completely distinct un... |
| snprc 4678 | The singleton of a proper ... |
| snnzb 4679 | A singleton is nonempty if... |
| rmosn 4680 | A restricted at-most-one q... |
| r19.12sn 4681 | Special case of ~ r19.12 w... |
| rabsn 4682 | Condition where a restrict... |
| rabsnifsb 4683 | A restricted class abstrac... |
| rabsnif 4684 | A restricted class abstrac... |
| rabrsn 4685 | A restricted class abstrac... |
| euabsn2 4686 | Another way to express exi... |
| euabsn 4687 | Another way to express exi... |
| reusn 4688 | A way to express restricte... |
| absneu 4689 | Restricted existential uni... |
| rabsneu 4690 | Restricted existential uni... |
| eusn 4691 | Two ways to express " ` A ... |
| rabsnt 4692 | Truth implied by equality ... |
| prcom 4693 | Commutative law for unorde... |
| preq1 4694 | Equality theorem for unord... |
| preq2 4695 | Equality theorem for unord... |
| preq12 4696 | Equality theorem for unord... |
| preq1i 4697 | Equality inference for uno... |
| preq2i 4698 | Equality inference for uno... |
| preq12i 4699 | Equality inference for uno... |
| preq1d 4700 | Equality deduction for uno... |
| preq2d 4701 | Equality deduction for uno... |
| preq12d 4702 | Equality deduction for uno... |
| tpeq1 4703 | Equality theorem for unord... |
| tpeq2 4704 | Equality theorem for unord... |
| tpeq3 4705 | Equality theorem for unord... |
| tpeq1d 4706 | Equality theorem for unord... |
| tpeq2d 4707 | Equality theorem for unord... |
| tpeq3d 4708 | Equality theorem for unord... |
| tpeq123d 4709 | Equality theorem for unord... |
| tprot 4710 | Rotation of the elements o... |
| tpcoma 4711 | Swap 1st and 2nd members o... |
| tpcomb 4712 | Swap 2nd and 3rd members o... |
| tpass 4713 | Split off the first elemen... |
| qdass 4714 | Two ways to write an unord... |
| qdassr 4715 | Two ways to write an unord... |
| tpidm12 4716 | Unordered triple ` { A , A... |
| tpidm13 4717 | Unordered triple ` { A , B... |
| tpidm23 4718 | Unordered triple ` { A , B... |
| tpidm 4719 | Unordered triple ` { A , A... |
| tppreq3 4720 | An unordered triple is an ... |
| prid1g 4721 | An unordered pair contains... |
| prid2g 4722 | An unordered pair contains... |
| prid1 4723 | An unordered pair contains... |
| prid2 4724 | An unordered pair contains... |
| ifpprsnss 4725 | An unordered pair is a sin... |
| prprc1 4726 | A proper class vanishes in... |
| prprc2 4727 | A proper class vanishes in... |
| prprc 4728 | An unordered pair containi... |
| tpid1 4729 | One of the three elements ... |
| tpid1g 4730 | Closed theorem form of ~ t... |
| tpid2 4731 | One of the three elements ... |
| tpid2g 4732 | Closed theorem form of ~ t... |
| tpid3g 4733 | Closed theorem form of ~ t... |
| tpid3 4734 | One of the three elements ... |
| snnzg 4735 | The singleton of a set is ... |
| snn0d 4736 | The singleton of a set is ... |
| snnz 4737 | The singleton of a set is ... |
| prnz 4738 | A pair containing a set is... |
| prnzg 4739 | A pair containing a set is... |
| tpnz 4740 | An unordered triple contai... |
| tpnzd 4741 | An unordered triple contai... |
| raltpd 4742 | Convert a universal quanti... |
| snssb 4743 | Characterization of the in... |
| snssg 4744 | The singleton formed on a ... |
| snss 4745 | The singleton of an elemen... |
| snssi 4746 | The singleton of an elemen... |
| snssd 4747 | The singleton of an elemen... |
| eldifsn 4748 | Membership in a set with a... |
| eldifsnd 4749 | Membership in a set with a... |
| ssdifsn 4750 | Subset of a set with an el... |
| elpwdifsn 4751 | A subset of a set is an el... |
| eldifsni 4752 | Membership in a set with a... |
| eldifsnneq 4753 | An element of a difference... |
| neldifsn 4754 | The class ` A ` is not in ... |
| neldifsnd 4755 | The class ` A ` is not in ... |
| rexdifsn 4756 | Restricted existential qua... |
| raldifsni 4757 | Rearrangement of a propert... |
| raldifsnb 4758 | Restricted universal quant... |
| eldifvsn 4759 | A set is an element of the... |
| difsn 4760 | An element not in a set ca... |
| difprsnss 4761 | Removal of a singleton fro... |
| difprsn1 4762 | Removal of a singleton fro... |
| difprsn2 4763 | Removal of a singleton fro... |
| diftpsn3 4764 | Removal of a singleton fro... |
| difpr 4765 | Removing two elements as p... |
| tpprceq3 4766 | An unordered triple is an ... |
| tppreqb 4767 | An unordered triple is an ... |
| difsnb 4768 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4769 | ` ( B \ { A } ) ` is a pro... |
| difsnid 4770 | If we remove a single elem... |
| eldifeldifsn 4771 | An element of a difference... |
| pw0 4772 | Compute the power set of t... |
| pwpw0 4773 | Compute the power set of t... |
| snsspr1 4774 | A singleton is a subset of... |
| snsspr2 4775 | A singleton is a subset of... |
| snsstp1 4776 | A singleton is a subset of... |
| snsstp2 4777 | A singleton is a subset of... |
| snsstp3 4778 | A singleton is a subset of... |
| prssg 4779 | A pair of elements of a cl... |
| prss 4780 | A pair of elements of a cl... |
| prssi 4781 | A pair of elements of a cl... |
| prssd 4782 | Deduction version of ~ prs... |
| prsspwg 4783 | An unordered pair belongs ... |
| ssprss 4784 | A pair as subset of a pair... |
| ssprsseq 4785 | A proper pair is a subset ... |
| sssn 4786 | The subsets of a singleton... |
| ssunsn2 4787 | The property of being sand... |
| ssunsn 4788 | Possible values for a set ... |
| eqsn 4789 | Two ways to express that a... |
| eqsnd 4790 | Deduce that a set is a sin... |
| eqsndOLD 4791 | Obsolete version of ~ eqsn... |
| issn 4792 | A sufficient condition for... |
| n0snor2el 4793 | A nonempty set is either a... |
| ssunpr 4794 | Possible values for a set ... |
| sspr 4795 | The subsets of a pair. (C... |
| sstp 4796 | The subsets of an unordere... |
| tpss 4797 | An unordered triple of ele... |
| tpssi 4798 | An unordered triple of ele... |
| sneqrg 4799 | Closed form of ~ sneqr . ... |
| sneqr 4800 | If the singletons of two s... |
| snsssn 4801 | If a singleton is a subset... |
| mosneq 4802 | There exists at most one s... |
| sneqbg 4803 | Two singletons of sets are... |
| snsspw 4804 | The singleton of a class i... |
| prsspw 4805 | An unordered pair belongs ... |
| preq1b 4806 | Biconditional equality lem... |
| preq2b 4807 | Biconditional equality lem... |
| preqr1 4808 | Reverse equality lemma for... |
| preqr2 4809 | Reverse equality lemma for... |
| preq12b 4810 | Equality relationship for ... |
| opthpr 4811 | An unordered pair has the ... |
| preqr1g 4812 | Reverse equality lemma for... |
| preq12bg 4813 | Closed form of ~ preq12b .... |
| prneimg 4814 | Two pairs are not equal if... |
| prneimg2 4815 | Two pairs are not equal if... |
| prnebg 4816 | A (proper) pair is not equ... |
| pr1eqbg 4817 | A (proper) pair is equal t... |
| pr1nebg 4818 | A (proper) pair is not equ... |
| preqsnd 4819 | Equivalence for a pair equ... |
| prnesn 4820 | A proper unordered pair is... |
| prneprprc 4821 | A proper unordered pair is... |
| preqsn 4822 | Equivalence for a pair equ... |
| preq12nebg 4823 | Equality relationship for ... |
| prel12g 4824 | Equality of two unordered ... |
| opthprneg 4825 | An unordered pair has the ... |
| elpreqprlem 4826 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4827 | Equality and membership ru... |
| elpreqprb 4828 | A set is an element of an ... |
| elpr2elpr 4829 | For an element ` A ` of an... |
| dfopif 4830 | Rewrite ~ df-op using ` if... |
| dfopg 4831 | Value of the ordered pair ... |
| dfop 4832 | Value of an ordered pair w... |
| opeq1 4833 | Equality theorem for order... |
| opeq2 4834 | Equality theorem for order... |
| opeq12 4835 | Equality theorem for order... |
| opeq1i 4836 | Equality inference for ord... |
| opeq2i 4837 | Equality inference for ord... |
| opeq12i 4838 | Equality inference for ord... |
| opeq1d 4839 | Equality deduction for ord... |
| opeq2d 4840 | Equality deduction for ord... |
| opeq12d 4841 | Equality deduction for ord... |
| oteq1 4842 | Equality theorem for order... |
| oteq2 4843 | Equality theorem for order... |
| oteq3 4844 | Equality theorem for order... |
| oteq1d 4845 | Equality deduction for ord... |
| oteq2d 4846 | Equality deduction for ord... |
| oteq3d 4847 | Equality deduction for ord... |
| oteq123d 4848 | Equality deduction for ord... |
| nfop 4849 | Bound-variable hypothesis ... |
| nfopd 4850 | Deduction version of bound... |
| csbopg 4851 | Distribution of class subs... |
| opidg 4852 | The ordered pair ` <. A , ... |
| opid 4853 | The ordered pair ` <. A , ... |
| ralunsn 4854 | Restricted quantification ... |
| 2ralunsn 4855 | Double restricted quantifi... |
| opprc 4856 | Expansion of an ordered pa... |
| opprc1 4857 | Expansion of an ordered pa... |
| opprc2 4858 | Expansion of an ordered pa... |
| oprcl 4859 | If an ordered pair has an ... |
| pwsn 4860 | The power set of a singlet... |
| pwpr 4861 | The power set of an unorde... |
| pwtp 4862 | The power set of an unorde... |
| pwpwpw0 4863 | Compute the power set of t... |
| pwv 4864 | The power class of the uni... |
| prproe 4865 | For an element of a proper... |
| 3elpr2eq 4866 | If there are three element... |
| dfuni2 4869 | Alternate definition of cl... |
| eluni 4870 | Membership in class union.... |
| eluni2 4871 | Membership in class union.... |
| elunii 4872 | Membership in class union.... |
| nfunid 4873 | Deduction version of ~ nfu... |
| nfuni 4874 | Bound-variable hypothesis ... |
| uniss 4875 | Subclass relationship for ... |
| unissi 4876 | Subclass relationship for ... |
| unissd 4877 | Subclass relationship for ... |
| unieq 4878 | Equality theorem for class... |
| unieqi 4879 | Inference of equality of t... |
| unieqd 4880 | Deduction of equality of t... |
| eluniab 4881 | Membership in union of a c... |
| elunirab 4882 | Membership in union of a c... |
| uniprg 4883 | The union of a pair is the... |
| unipr 4884 | The union of a pair is the... |
| unisng 4885 | A set equals the union of ... |
| unisn 4886 | A set equals the union of ... |
| unisnv 4887 | A set equals the union of ... |
| unisn3 4888 | Union of a singleton in th... |
| dfnfc2 4889 | An alternative statement o... |
| uniun 4890 | The class union of the uni... |
| uniin 4891 | The class union of the int... |
| uniinOLD 4892 | Obsolete version of ~ unii... |
| ssuni 4893 | Subclass relationship for ... |
| uni0b 4894 | The union of a set is empt... |
| uni0c 4895 | The union of a set is empt... |
| uni0 4896 | The union of the empty set... |
| uni0OLD 4897 | Obsolete version of ~ uni0... |
| csbuni 4898 | Distribute proper substitu... |
| elssuni 4899 | An element of a class is a... |
| unissel 4900 | Condition turning a subcla... |
| unissb 4901 | Relationship involving mem... |
| uniss2 4902 | A subclass condition on th... |
| unidif 4903 | If the difference ` A \ B ... |
| ssunieq 4904 | Relationship implying unio... |
| unimax 4905 | Any member of a class is t... |
| pwuni 4906 | A class is a subclass of t... |
| dfint2 4909 | Alternate definition of cl... |
| inteq 4910 | Equality law for intersect... |
| inteqi 4911 | Equality inference for cla... |
| inteqd 4912 | Equality deduction for cla... |
| elint 4913 | Membership in class inters... |
| elint2 4914 | Membership in class inters... |
| elintg 4915 | Membership in class inters... |
| elinti 4916 | Membership in class inters... |
| nfint 4917 | Bound-variable hypothesis ... |
| elintabg 4918 | Two ways of saying a set i... |
| elintab 4919 | Membership in the intersec... |
| elintrab 4920 | Membership in the intersec... |
| elintrabg 4921 | Membership in the intersec... |
| int0 4922 | The intersection of the em... |
| intss1 4923 | An element of a class incl... |
| ssint 4924 | Subclass of a class inters... |
| ssintab 4925 | Subclass of the intersecti... |
| ssintub 4926 | Subclass of the least uppe... |
| ssmin 4927 | Subclass of the minimum va... |
| intmin 4928 | Any member of a class is t... |
| intss 4929 | Intersection of subclasses... |
| intssuni 4930 | The intersection of a none... |
| ssintrab 4931 | Subclass of the intersecti... |
| unissint 4932 | If the union of a class is... |
| intssuni2 4933 | Subclass relationship for ... |
| intminss 4934 | Under subset ordering, the... |
| intmin2 4935 | Any set is the smallest of... |
| intmin3 4936 | Under subset ordering, the... |
| intmin4 4937 | Elimination of a conjunct ... |
| intab 4938 | The intersection of a spec... |
| int0el 4939 | The intersection of a clas... |
| intun 4940 | The class intersection of ... |
| intprg 4941 | The intersection of a pair... |
| intpr 4942 | The intersection of a pair... |
| intsng 4943 | Intersection of a singleto... |
| intsn 4944 | The intersection of a sing... |
| uniintsn 4945 | Two ways to express " ` A ... |
| uniintab 4946 | The union and the intersec... |
| intunsn 4947 | Theorem joining a singleto... |
| rint0 4948 | Relative intersection of a... |
| elrint 4949 | Membership in a restricted... |
| elrint2 4950 | Membership in a restricted... |
| eliun 4955 | Membership in indexed unio... |
| eliin 4956 | Membership in indexed inte... |
| eliuni 4957 | Membership in an indexed u... |
| eliund 4958 | Membership in indexed unio... |
| iuncom 4959 | Commutation of indexed uni... |
| iuncom4 4960 | Commutation of union with ... |
| iunconst 4961 | Indexed union of a constan... |
| iinconst 4962 | Indexed intersection of a ... |
| iuneqconst 4963 | Indexed union of identical... |
| iuniin 4964 | Law combining indexed unio... |
| iinssiun 4965 | An indexed intersection is... |
| iunss1 4966 | Subclass theorem for index... |
| iinss1 4967 | Subclass theorem for index... |
| iuneq1 4968 | Equality theorem for index... |
| iineq1 4969 | Equality theorem for index... |
| ss2iun 4970 | Subclass theorem for index... |
| iuneq2 4971 | Equality theorem for index... |
| iineq2 4972 | Equality theorem for index... |
| iuneq2i 4973 | Equality inference for ind... |
| iineq2i 4974 | Equality inference for ind... |
| iineq2d 4975 | Equality deduction for ind... |
| iuneq2dv 4976 | Equality deduction for ind... |
| iineq2dv 4977 | Equality deduction for ind... |
| iuneq12df 4978 | Equality deduction for ind... |
| iuneq1d 4979 | Equality theorem for index... |
| iuneq12dOLD 4980 | Obsolete version of ~ iune... |
| iuneq12d 4981 | Equality deduction for ind... |
| iuneq2d 4982 | Equality deduction for ind... |
| nfiun 4983 | Bound-variable hypothesis ... |
| nfiin 4984 | Bound-variable hypothesis ... |
| nfiung 4985 | Bound-variable hypothesis ... |
| nfiing 4986 | Bound-variable hypothesis ... |
| nfiu1 4987 | Bound-variable hypothesis ... |
| nfii1 4988 | Bound-variable hypothesis ... |
| dfiun2g 4989 | Alternate definition of in... |
| dfiin2g 4990 | Alternate definition of in... |
| dfiun2 4991 | Alternate definition of in... |
| dfiin2 4992 | Alternate definition of in... |
| dfiunv2 4993 | Define double indexed unio... |
| cbviun 4994 | Rule used to change the bo... |
| cbviin 4995 | Change bound variables in ... |
| cbviung 4996 | Rule used to change the bo... |
| cbviing 4997 | Change bound variables in ... |
| cbviunv 4998 | Rule used to change the bo... |
| cbviinv 4999 | Change bound variables in ... |
| cbviunvg 5000 | Rule used to change the bo... |
| cbviinvg 5001 | Change bound variables in ... |
| iunssf 5002 | Subset theorem for an inde... |
| iunssfOLD 5003 | Obsolete version of ~ iuns... |
| iunss 5004 | Subset theorem for an inde... |
| iunssOLD 5005 | Obsolete version of ~ iuns... |
| ssiun 5006 | Subset implication for an ... |
| ssiun2 5007 | Identity law for subset of... |
| ssiun2s 5008 | Subset relationship for an... |
| iunss2 5009 | A subclass condition on th... |
| iunssd 5010 | Subset theorem for an inde... |
| iunab 5011 | The indexed union of a cla... |
| iunrab 5012 | The indexed union of a res... |
| iunxdif2 5013 | Indexed union with a class... |
| ssiinf 5014 | Subset theorem for an inde... |
| ssiin 5015 | Subset theorem for an inde... |
| iinss 5016 | Subset implication for an ... |
| iinss2 5017 | An indexed intersection is... |
| uniiun 5018 | Class union in terms of in... |
| intiin 5019 | Class intersection in term... |
| iunid 5020 | An indexed union of single... |
| iun0 5021 | An indexed union of the em... |
| 0iun 5022 | An empty indexed union is ... |
| 0iin 5023 | An empty indexed intersect... |
| viin 5024 | Indexed intersection with ... |
| iunsn 5025 | Indexed union of a singlet... |
| iunn0 5026 | There is a nonempty class ... |
| iinab 5027 | Indexed intersection of a ... |
| iinrab 5028 | Indexed intersection of a ... |
| iinrab2 5029 | Indexed intersection of a ... |
| iunin2 5030 | Indexed union of intersect... |
| iunin1 5031 | Indexed union of intersect... |
| iinun2 5032 | Indexed intersection of un... |
| iundif2 5033 | Indexed union of class dif... |
| iindif1 5034 | Indexed intersection of cl... |
| 2iunin 5035 | Rearrange indexed unions o... |
| iindif2 5036 | Indexed intersection of cl... |
| iinin2 5037 | Indexed intersection of in... |
| iinin1 5038 | Indexed intersection of in... |
| iinvdif 5039 | The indexed intersection o... |
| elriin 5040 | Elementhood in a relative ... |
| riin0 5041 | Relative intersection of a... |
| riinn0 5042 | Relative intersection of a... |
| riinrab 5043 | Relative intersection of a... |
| symdif0 5044 | Symmetric difference with ... |
| symdifv 5045 | The symmetric difference w... |
| symdifid 5046 | The symmetric difference o... |
| iinxsng 5047 | A singleton index picks ou... |
| iinxprg 5048 | Indexed intersection with ... |
| iunxsng 5049 | A singleton index picks ou... |
| iunxsn 5050 | A singleton index picks ou... |
| iunxsngf 5051 | A singleton index picks ou... |
| iunun 5052 | Separate a union in an ind... |
| iunxun 5053 | Separate a union in the in... |
| iunxdif3 5054 | An indexed union where som... |
| iunxprg 5055 | A pair index picks out two... |
| iunxiun 5056 | Separate an indexed union ... |
| iinuni 5057 | A relationship involving u... |
| iununi 5058 | A relationship involving u... |
| sspwuni 5059 | Subclass relationship for ... |
| pwssb 5060 | Two ways to express a coll... |
| elpwpw 5061 | Characterization of the el... |
| pwpwab 5062 | The double power class wri... |
| pwpwssunieq 5063 | The class of sets whose un... |
| elpwuni 5064 | Relationship for power cla... |
| iinpw 5065 | The power class of an inte... |
| iunpwss 5066 | Inclusion of an indexed un... |
| intss2 5067 | A nonempty intersection of... |
| rintn0 5068 | Relative intersection of a... |
| dfdisj2 5071 | Alternate definition for d... |
| disjss2 5072 | If each element of a colle... |
| disjeq2 5073 | Equality theorem for disjo... |
| disjeq2dv 5074 | Equality deduction for dis... |
| disjss1 5075 | A subset of a disjoint col... |
| disjeq1 5076 | Equality theorem for disjo... |
| disjeq1d 5077 | Equality theorem for disjo... |
| disjeq12d 5078 | Equality theorem for disjo... |
| cbvdisj 5079 | Change bound variables in ... |
| cbvdisjv 5080 | Change bound variables in ... |
| nfdisjw 5081 | Bound-variable hypothesis ... |
| nfdisj 5082 | Bound-variable hypothesis ... |
| nfdisj1 5083 | Bound-variable hypothesis ... |
| disjor 5084 | Two ways to say that a col... |
| disjors 5085 | Two ways to say that a col... |
| disji2 5086 | Property of a disjoint col... |
| disji 5087 | Property of a disjoint col... |
| invdisj 5088 | If there is a function ` C... |
| invdisjrab 5089 | The restricted class abstr... |
| disjiun 5090 | A disjoint collection yiel... |
| disjord 5091 | Conditions for a collectio... |
| disjiunb 5092 | Two ways to say that a col... |
| disjiund 5093 | Conditions for a collectio... |
| sndisj 5094 | Any collection of singleto... |
| 0disj 5095 | Any collection of empty se... |
| disjxsn 5096 | A singleton collection is ... |
| disjx0 5097 | An empty collection is dis... |
| disjprg 5098 | A pair collection is disjo... |
| disjxiun 5099 | An indexed union of a disj... |
| disjxun 5100 | The union of two disjoint ... |
| disjss3 5101 | Expand a disjoint collecti... |
| breq 5104 | Equality theorem for binar... |
| breq1 5105 | Equality theorem for a bin... |
| breq2 5106 | Equality theorem for a bin... |
| breq12 5107 | Equality theorem for a bin... |
| breqi 5108 | Equality inference for bin... |
| breq1i 5109 | Equality inference for a b... |
| breq2i 5110 | Equality inference for a b... |
| breq12i 5111 | Equality inference for a b... |
| breq1d 5112 | Equality deduction for a b... |
| breqd 5113 | Equality deduction for a b... |
| breq2d 5114 | Equality deduction for a b... |
| breq12d 5115 | Equality deduction for a b... |
| breq123d 5116 | Equality deduction for a b... |
| breqdi 5117 | Equality deduction for a b... |
| breqan12d 5118 | Equality deduction for a b... |
| breqan12rd 5119 | Equality deduction for a b... |
| eqnbrtrd 5120 | Substitution of equal clas... |
| nbrne1 5121 | Two classes are different ... |
| nbrne2 5122 | Two classes are different ... |
| eqbrtri 5123 | Substitution of equal clas... |
| eqbrtrd 5124 | Substitution of equal clas... |
| eqbrtrri 5125 | Substitution of equal clas... |
| eqbrtrrd 5126 | Substitution of equal clas... |
| breqtri 5127 | Substitution of equal clas... |
| breqtrd 5128 | Substitution of equal clas... |
| breqtrri 5129 | Substitution of equal clas... |
| breqtrrd 5130 | Substitution of equal clas... |
| 3brtr3i 5131 | Substitution of equality i... |
| 3brtr4i 5132 | Substitution of equality i... |
| 3brtr3d 5133 | Substitution of equality i... |
| 3brtr4d 5134 | Substitution of equality i... |
| 3brtr3g 5135 | Substitution of equality i... |
| 3brtr4g 5136 | Substitution of equality i... |
| eqbrtrid 5137 | A chained equality inferen... |
| eqbrtrrid 5138 | A chained equality inferen... |
| breqtrid 5139 | A chained equality inferen... |
| breqtrrid 5140 | A chained equality inferen... |
| eqbrtrdi 5141 | A chained equality inferen... |
| eqbrtrrdi 5142 | A chained equality inferen... |
| breqtrdi 5143 | A chained equality inferen... |
| breqtrrdi 5144 | A chained equality inferen... |
| ssbrd 5145 | Deduction from a subclass ... |
| ssbr 5146 | Implication from a subclas... |
| ssbri 5147 | Inference from a subclass ... |
| nfbrd 5148 | Deduction version of bound... |
| nfbr 5149 | Bound-variable hypothesis ... |
| brab1 5150 | Relationship between a bin... |
| br0 5151 | The empty binary relation ... |
| brne0 5152 | If two sets are in a binar... |
| brun 5153 | The union of two binary re... |
| brin 5154 | The intersection of two re... |
| brdif 5155 | The difference of two bina... |
| sbcbr123 5156 | Move substitution in and o... |
| sbcbr 5157 | Move substitution in and o... |
| sbcbr12g 5158 | Move substitution in and o... |
| sbcbr1g 5159 | Move substitution in and o... |
| sbcbr2g 5160 | Move substitution in and o... |
| brsymdif 5161 | Characterization of the sy... |
| brralrspcev 5162 | Restricted existential spe... |
| brimralrspcev 5163 | Restricted existential spe... |
| opabss 5166 | The collection of ordered ... |
| opabbid 5167 | Equivalent wff's yield equ... |
| opabbidv 5168 | Equivalent wff's yield equ... |
| opabbii 5169 | Equivalent wff's yield equ... |
| nfopabd 5170 | Bound-variable hypothesis ... |
| nfopab 5171 | Bound-variable hypothesis ... |
| nfopab1 5172 | The first abstraction vari... |
| nfopab2 5173 | The second abstraction var... |
| cbvopab 5174 | Rule used to change bound ... |
| cbvopabv 5175 | Rule used to change bound ... |
| cbvopab1 5176 | Change first bound variabl... |
| cbvopab1g 5177 | Change first bound variabl... |
| cbvopab2 5178 | Change second bound variab... |
| cbvopab1s 5179 | Change first bound variabl... |
| cbvopab1v 5180 | Rule used to change the fi... |
| cbvopab2v 5181 | Rule used to change the se... |
| unopab 5182 | Union of two ordered pair ... |
| mpteq12da 5185 | An equality inference for ... |
| mpteq12df 5186 | An equality inference for ... |
| mpteq12f 5187 | An equality theorem for th... |
| mpteq12dva 5188 | An equality inference for ... |
| mpteq12dv 5189 | An equality inference for ... |
| mpteq12 5190 | An equality theorem for th... |
| mpteq1 5191 | An equality theorem for th... |
| mpteq1d 5192 | An equality theorem for th... |
| mpteq1i 5193 | An equality theorem for th... |
| mpteq2da 5194 | Slightly more general equa... |
| mpteq2dva 5195 | Slightly more general equa... |
| mpteq2dv 5196 | An equality inference for ... |
| mpteq2ia 5197 | An equality inference for ... |
| mpteq2i 5198 | An equality inference for ... |
| mpteq12i 5199 | An equality inference for ... |
| nfmpt 5200 | Bound-variable hypothesis ... |
| nfmpt1 5201 | Bound-variable hypothesis ... |
| cbvmptf 5202 | Rule to change the bound v... |
| cbvmptfg 5203 | Rule to change the bound v... |
| cbvmpt 5204 | Rule to change the bound v... |
| cbvmptg 5205 | Rule to change the bound v... |
| cbvmptv 5206 | Rule to change the bound v... |
| cbvmptvg 5207 | Rule to change the bound v... |
| mptv 5208 | Function with universal do... |
| dftr2 5211 | An alternate way of defini... |
| dftr2c 5212 | Variant of ~ dftr2 with co... |
| dftr5 5213 | An alternate way of defini... |
| dftr3 5214 | An alternate way of defini... |
| dftr4 5215 | An alternate way of defini... |
| treq 5216 | Equality theorem for the t... |
| trel 5217 | In a transitive class, the... |
| trel3 5218 | In a transitive class, the... |
| trss 5219 | An element of a transitive... |
| trun 5220 | The union of transitive cl... |
| trin 5221 | The intersection of transi... |
| tr0 5222 | The empty set is transitiv... |
| trv 5223 | The universe is transitive... |
| triun 5224 | An indexed union of a clas... |
| truni 5225 | The union of a class of tr... |
| triin 5226 | An indexed intersection of... |
| trint 5227 | The intersection of a clas... |
| trintss 5228 | Any nonempty transitive cl... |
| axrep1 5230 | The version of the Axiom o... |
| axreplem 5231 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5232 | Axiom of Replacement expre... |
| axrep3 5233 | Axiom of Replacement sligh... |
| axrep4v 5234 | Version of ~ axrep4 with a... |
| axrep4 5235 | A more traditional version... |
| axrep4OLD 5236 | Obsolete version of ~ axre... |
| axrep5 5237 | Axiom of Replacement (simi... |
| axrep6 5238 | A condensed form of ~ ax-r... |
| axrep6OLD 5239 | Obsolete version of ~ axre... |
| replem 5240 | A lemma for variants of th... |
| zfrep6 5241 | A version of the Axiom of ... |
| axrep6g 5242 | ~ axrep6 in class notation... |
| zfrepclf 5243 | An inference based on the ... |
| zfrep3cl 5244 | An inference based on the ... |
| zfrep4 5245 | A version of Replacement u... |
| axsepgfromrep 5246 | A more general version ~ a... |
| axsep 5247 | Axiom scheme of separation... |
| axsepg 5249 | A more general version of ... |
| zfauscl 5250 | Separation Scheme (Aussond... |
| sepexlem 5251 | Lemma for ~ sepex . Use ~... |
| sepex 5252 | Convert implication to equ... |
| sepexi 5253 | Convert implication to equ... |
| bm1.3iiOLD 5254 | Obsolete version of ~ sepe... |
| ax6vsep 5255 | Derive ~ ax6v (a weakened ... |
| axnulALT 5256 | Alternate proof of ~ axnul... |
| axnul 5257 | The Null Set Axiom of ZF s... |
| 0ex 5259 | The Null Set Axiom of ZF s... |
| al0ssb 5260 | The empty set is the uniqu... |
| sseliALT 5261 | Alternate proof of ~ sseli... |
| csbexg 5262 | The existence of proper su... |
| csbex 5263 | The existence of proper su... |
| unisn2 5264 | A version of ~ unisn witho... |
| exnelv 5265 | For any set ` x ` , there ... |
| nalset 5266 | No set contains all sets. ... |
| nalsetOLD 5267 | Obsolete version of ~ nals... |
| vneqv 5268 | The universal class is not... |
| vnex 5269 | The universal class does n... |
| vnexOLD 5270 | Obsolete proof of ~ vnex a... |
| nvel 5271 | The universal class does n... |
| vprc 5272 | The universal class is not... |
| vprcOLD 5273 | Obsolete proof of ~ vprc ,... |
| nvelOLD 5274 | Obsolete proof of ~ nvel ,... |
| inex1 5275 | Separation Scheme (Aussond... |
| inex2 5276 | Separation Scheme (Aussond... |
| inex1g 5277 | Closed-form, generalized S... |
| inex2g 5278 | Sufficient condition for a... |
| ssex 5279 | The subset of a set is als... |
| ssexi 5280 | The subset of a set is als... |
| ssexg 5281 | The subset of a set is als... |
| ssexd 5282 | A subclass of a set is a s... |
| abexd 5283 | Conditions for a class abs... |
| abex 5284 | Conditions for a class abs... |
| prcssprc 5285 | The superclass of a proper... |
| sselpwd 5286 | Elementhood to a power set... |
| difexg 5287 | Existence of a difference.... |
| difexi 5288 | Existence of a difference,... |
| difexd 5289 | Existence of a difference.... |
| zfausab 5290 | Separation Scheme (Aussond... |
| elpw2g 5291 | Membership in a power clas... |
| elpw2 5292 | Membership in a power clas... |
| elpwi2 5293 | Membership in a power clas... |
| rabelpw 5294 | A restricted class abstrac... |
| rabexg 5295 | Separation Scheme in terms... |
| rabexgOLD 5296 | Obsolete version of ~ rabe... |
| rabex 5297 | Separation Scheme in terms... |
| rabexd 5298 | Separation Scheme in terms... |
| rabex2 5299 | Separation Scheme in terms... |
| rab2ex 5300 | A class abstraction based ... |
| elssabg 5301 | Membership in a class abst... |
| intex 5302 | The intersection of a none... |
| intnex 5303 | If a class intersection is... |
| intexab 5304 | The intersection of a none... |
| intexrab 5305 | The intersection of a none... |
| iinexg 5306 | The existence of a class i... |
| intabs 5307 | Absorption of a redundant ... |
| inuni 5308 | The intersection of a unio... |
| axpweq 5309 | Two equivalent ways to exp... |
| pwnss 5310 | The power set of a set is ... |
| pwne 5311 | No set equals its power se... |
| difelpw 5312 | A difference is an element... |
| class2set 5313 | The class of elements of `... |
| 0elpw 5314 | Every power class contains... |
| pwne0 5315 | A power class is never emp... |
| 0nep0 5316 | The empty set and its powe... |
| 0inp0 5317 | Something cannot be equal ... |
| unidif0 5318 | The removal of the empty s... |
| unidif0OLD 5319 | Obsolete version of ~ unid... |
| eqsnuniex 5320 | If a class is equal to the... |
| iin0 5321 | An indexed intersection of... |
| notzfaus 5322 | In the Separation Scheme ~... |
| intv 5323 | The intersection of the un... |
| zfpow 5325 | Axiom of Power Sets expres... |
| axpow2 5326 | A variant of the Axiom of ... |
| axpow3 5327 | A variant of the Axiom of ... |
| elALT2 5328 | Alternate proof of ~ el us... |
| dtruALT2 5329 | Alternate proof of ~ dtru ... |
| dtrucor 5330 | Corollary of ~ dtru . Thi... |
| dtrucor2 5331 | The theorem form of the de... |
| dvdemo1 5332 | Demonstration of a theorem... |
| dvdemo2 5333 | Demonstration of a theorem... |
| nfnid 5334 | A setvar variable is not f... |
| nfcvb 5335 | The "distinctor" expressio... |
| vpwex 5336 | Power set axiom: the power... |
| pwexg 5337 | Power set axiom expressed ... |
| pwexd 5338 | Deduction version of the p... |
| pwex 5339 | Power set axiom expressed ... |
| pwel 5340 | Quantitative version of ~ ... |
| abssexg 5341 | Existence of a class of su... |
| snexALT 5342 | Alternate proof of ~ snex ... |
| p0ex 5343 | The power set of the empty... |
| p0exALT 5344 | Alternate proof of ~ p0ex ... |
| pp0ex 5345 | The power set of the power... |
| ord3ex 5346 | The ordinal number 3 is a ... |
| dtruALT 5347 | Alternate proof of ~ dtru ... |
| axc16b 5348 | This theorem shows that Ax... |
| eunex 5349 | Existential uniqueness imp... |
| eusv1 5350 | Two ways to express single... |
| eusvnf 5351 | Even if ` x ` is free in `... |
| eusvnfb 5352 | Two ways to say that ` A (... |
| eusv2i 5353 | Two ways to express single... |
| eusv2nf 5354 | Two ways to express single... |
| eusv2 5355 | Two ways to express single... |
| reusv1 5356 | Two ways to express single... |
| reusv2lem1 5357 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5358 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5359 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5360 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5361 | Lemma for ~ reusv2 . (Con... |
| reusv2 5362 | Two ways to express single... |
| reusv3i 5363 | Two ways of expressing exi... |
| reusv3 5364 | Two ways to express single... |
| eusv4 5365 | Two ways to express single... |
| alxfr 5366 | Transfer universal quantif... |
| ralxfrd 5367 | Transfer universal quantif... |
| rexxfrd 5368 | Transfer existential quant... |
| ralxfr2d 5369 | Transfer universal quantif... |
| rexxfr2d 5370 | Transfer existential quant... |
| ralxfrd2 5371 | Transfer universal quantif... |
| rexxfrd2 5372 | Transfer existence from a ... |
| ralxfr 5373 | Transfer universal quantif... |
| ralxfrALT 5374 | Alternate proof of ~ ralxf... |
| rexxfr 5375 | Transfer existence from a ... |
| rabxfrd 5376 | Membership in a restricted... |
| rabxfr 5377 | Membership in a restricted... |
| reuhypd 5378 | A theorem useful for elimi... |
| reuhyp 5379 | A theorem useful for elimi... |
| zfpair 5380 | The Axiom of Pairing of Ze... |
| axprALT 5381 | Alternate proof of ~ axpr ... |
| axprlem1 5382 | Lemma for ~ axpr . There ... |
| axprlem2 5383 | Lemma for ~ axpr . There ... |
| axprlem3 5384 | Lemma for ~ axpr . Elimin... |
| axprlem4 5385 | Lemma for ~ axpr . If an ... |
| axpr 5386 | Unabbreviated version of t... |
| axprlem1OLD 5387 | Obsolete version of ~ axpr... |
| axprlem3OLD 5388 | Obsolete version of ~ axpr... |
| axprlem4OLD 5389 | Obsolete version of ~ axpr... |
| axprlem5OLD 5390 | Obsolete version of ~ axpr... |
| axprOLD 5391 | Obsolete version of ~ axpr... |
| zfpair2 5393 | Derive the abbreviated ver... |
| vsnex 5394 | A singleton built on a set... |
| axprglem 5395 | Lemma for ~ axprg . (Cont... |
| axprg 5396 | Derive The Axiom of Pairin... |
| prex 5397 | The Axiom of Pairing using... |
| snex 5398 | A singleton is a set. The... |
| snexg 5399 | A singleton built on a set... |
| snexgALT 5400 | Alternate proof of ~ snexg... |
| snexOLD 5401 | Obsolete version of ~ snex... |
| prexOLD 5402 | Obsolete version of ~ prex... |
| exel 5403 | There exist two sets, one ... |
| exexneq 5404 | There exist two different ... |
| exneq 5405 | Given any set (the " ` y `... |
| dtru 5406 | Given any set (the " ` y `... |
| el 5407 | Any set is an element of s... |
| elOLD 5408 | Obsolete version of ~ el a... |
| sels 5409 | If a class is a set, then ... |
| selsALT 5410 | Alternate proof of ~ sels ... |
| elALT 5411 | Alternate proof of ~ el , ... |
| snelpwg 5412 | A singleton of a set is a ... |
| snelpwi 5413 | If a set is a member of a ... |
| snelpw 5414 | A singleton of a set is a ... |
| prelpw 5415 | An unordered pair of two s... |
| prelpwi 5416 | If two sets are members of... |
| rext 5417 | A theorem similar to exten... |
| sspwb 5418 | The powerclass constructio... |
| unipw 5419 | A class equals the union o... |
| univ 5420 | The union of the universe ... |
| pwtr 5421 | A class is transitive iff ... |
| ssextss 5422 | An extensionality-like pri... |
| ssext 5423 | An extensionality-like pri... |
| nssss 5424 | Negation of subclass relat... |
| pweqb 5425 | Classes are equal if and o... |
| intidg 5426 | The intersection of all se... |
| moabex 5427 | "At most one" existence im... |
| moabexOLD 5428 | Obsolete version of ~ moab... |
| rmorabex 5429 | Restricted "at most one" e... |
| euabex 5430 | The abstraction of a wff w... |
| nnullss 5431 | A nonempty class (even if ... |
| exss 5432 | Restricted existence in a ... |
| opex 5433 | An ordered pair of classes... |
| opexOLD 5434 | Obsolete version of ~ opex... |
| otex 5435 | An ordered triple of class... |
| elopg 5436 | Characterization of the el... |
| elop 5437 | Characterization of the el... |
| opi1 5438 | One of the two elements in... |
| opi2 5439 | One of the two elements of... |
| opeluu 5440 | Each member of an ordered ... |
| op1stb 5441 | Extract the first member o... |
| brv 5442 | Two classes are always in ... |
| opnz 5443 | An ordered pair is nonempt... |
| opnzi 5444 | An ordered pair is nonempt... |
| opth1 5445 | Equality of the first memb... |
| opth 5446 | The ordered pair theorem. ... |
| opthg 5447 | Ordered pair theorem. ` C ... |
| opth1g 5448 | Equality of the first memb... |
| opthg2 5449 | Ordered pair theorem. (Co... |
| opth2 5450 | Ordered pair theorem. (Co... |
| opthneg 5451 | Two ordered pairs are not ... |
| opthne 5452 | Two ordered pairs are not ... |
| otth2 5453 | Ordered triple theorem, wi... |
| otth 5454 | Ordered triple theorem. (... |
| otthg 5455 | Ordered triple theorem, cl... |
| otthne 5456 | Contrapositive of the orde... |
| eqvinop 5457 | A variable introduction la... |
| sbcop1 5458 | The proper substitution of... |
| sbcop 5459 | The proper substitution of... |
| copsexgw 5460 | Version of ~ copsexg with ... |
| copsexgwOLD 5461 | Obsolete version of ~ cops... |
| copsexg 5462 | Substitution of class ` A ... |
| copsex2t 5463 | Closed theorem form of ~ c... |
| copsex2g 5464 | Implicit substitution infe... |
| copsex2dv 5465 | Implicit substitution dedu... |
| copsex4g 5466 | An implicit substitution i... |
| 0nelop 5467 | A property of ordered pair... |
| opwo0id 5468 | An ordered pair is equal t... |
| opeqex 5469 | Equivalence of existence i... |
| oteqex2 5470 | Equivalence of existence i... |
| oteqex 5471 | Equivalence of existence i... |
| opcom 5472 | An ordered pair commutes i... |
| moop2 5473 | "At most one" property of ... |
| opeqsng 5474 | Equivalence for an ordered... |
| opeqsn 5475 | Equivalence for an ordered... |
| opeqpr 5476 | Equivalence for an ordered... |
| snopeqop 5477 | Equivalence for an ordered... |
| propeqop 5478 | Equivalence for an ordered... |
| propssopi 5479 | If a pair of ordered pairs... |
| snopeqopsnid 5480 | Equivalence for an ordered... |
| mosubopt 5481 | "At most one" remains true... |
| mosubop 5482 | "At most one" remains true... |
| euop2 5483 | Transfer existential uniqu... |
| euotd 5484 | Prove existential uniquene... |
| opthwiener 5485 | Justification theorem for ... |
| uniop 5486 | The union of an ordered pa... |
| uniopel 5487 | Ordered pair membership is... |
| opthhausdorff 5488 | Justification theorem for ... |
| opthhausdorff0 5489 | Justification theorem for ... |
| otsndisj 5490 | The singletons consisting ... |
| otiunsndisj 5491 | The union of singletons co... |
| iunopeqop 5492 | Implication of an ordered ... |
| iunopeqopOLD 5493 | Obsolete version of ~ iuno... |
| brsnop 5494 | Binary relation for an ord... |
| brtp 5495 | A necessary and sufficient... |
| opabidw 5496 | The law of concretion. Sp... |
| opabid 5497 | The law of concretion. Sp... |
| elopabw 5498 | Membership in a class abst... |
| elopab 5499 | Membership in a class abst... |
| rexopabb 5500 | Restricted existential qua... |
| vopelopabsb 5501 | The law of concretion in t... |
| opelopabsb 5502 | The law of concretion in t... |
| brabsb 5503 | The law of concretion in t... |
| opelopabt 5504 | Closed theorem form of ~ o... |
| opelopabga 5505 | The law of concretion. Th... |
| brabga 5506 | The law of concretion for ... |
| opelopab2a 5507 | Ordered pair membership in... |
| opelopaba 5508 | The law of concretion. Th... |
| braba 5509 | The law of concretion for ... |
| brab2d 5510 | Expressing that two sets a... |
| opelopabg 5511 | The law of concretion. Th... |
| brabg 5512 | The law of concretion for ... |
| opelopabgf 5513 | The law of concretion. Th... |
| opelopab2 5514 | Ordered pair membership in... |
| opelopab 5515 | The law of concretion. Th... |
| brab 5516 | The law of concretion for ... |
| opelopabaf 5517 | The law of concretion. Th... |
| opelopabf 5518 | The law of concretion. Th... |
| ssopab2 5519 | Equivalence of ordered pai... |
| ssopab2bw 5520 | Equivalence of ordered pai... |
| eqopab2bw 5521 | Equivalence of ordered pai... |
| ssopab2b 5522 | Equivalence of ordered pai... |
| ssopab2i 5523 | Inference of ordered pair ... |
| ssopab2dv 5524 | Inference of ordered pair ... |
| eqopab2b 5525 | Equivalence of ordered pai... |
| opabn0 5526 | Nonempty ordered pair clas... |
| opab0 5527 | Empty ordered pair class a... |
| csbopab 5528 | Move substitution into a c... |
| csbopabw 5529 | Move substitution into a c... |
| csbmpt12 5530 | Move substitution into a m... |
| csbmpt2 5531 | Move substitution into the... |
| iunopab 5532 | Move indexed union inside ... |
| elopabr 5533 | Membership in an ordered-p... |
| elopabran 5534 | Membership in an ordered-p... |
| rbropapd 5535 | Properties of a pair in an... |
| rbropap 5536 | Properties of a pair in a ... |
| 2rbropap 5537 | Properties of a pair in a ... |
| 0nelopab 5538 | The empty set is never an ... |
| brabv 5539 | If two classes are in a re... |
| pwin 5540 | The power class of the int... |
| pwssun 5541 | The power class of the uni... |
| pwun 5542 | The power class of the uni... |
| dfid4 5545 | The identity function expr... |
| dfid2 5546 | Alternate definition of th... |
| dfid3 5547 | A stronger version of ~ df... |
| epelg 5550 | The membership relation an... |
| epeli 5551 | The membership relation an... |
| epel 5552 | The membership relation an... |
| 0sn0ep 5553 | An example for the members... |
| epn0 5554 | The membership relation is... |
| poss 5559 | Subset theorem for the par... |
| poeq1 5560 | Equality theorem for parti... |
| poeq2 5561 | Equality theorem for parti... |
| poeq12d 5562 | Equality deduction for par... |
| nfpo 5563 | Bound-variable hypothesis ... |
| nfso 5564 | Bound-variable hypothesis ... |
| pocl 5565 | Characteristic properties ... |
| ispod 5566 | Sufficient conditions for ... |
| swopolem 5567 | Perform the substitutions ... |
| swopo 5568 | A strict weak order is a p... |
| poirr 5569 | A partial order is irrefle... |
| potr 5570 | A partial order is a trans... |
| po2nr 5571 | A partial order has no 2-c... |
| po3nr 5572 | A partial order has no 3-c... |
| po2ne 5573 | Two sets related by a part... |
| po0 5574 | Any relation is a partial ... |
| pofun 5575 | The inverse image of a par... |
| sopo 5576 | A strict linear order is a... |
| soss 5577 | Subset theorem for the str... |
| soeq1 5578 | Equality theorem for the s... |
| soeq2 5579 | Equality theorem for the s... |
| soeq12d 5580 | Equality deduction for tot... |
| sonr 5581 | A strict order relation is... |
| sotr 5582 | A strict order relation is... |
| sotrd 5583 | Transitivity law for stric... |
| solin 5584 | A strict order relation is... |
| so2nr 5585 | A strict order relation ha... |
| so3nr 5586 | A strict order relation ha... |
| sotric 5587 | A strict order relation sa... |
| sotrieq 5588 | Trichotomy law for strict ... |
| sotrieq2 5589 | Trichotomy law for strict ... |
| soasym 5590 | Asymmetry law for strict o... |
| sotr2 5591 | A transitivity relation. ... |
| issod 5592 | An irreflexive, transitive... |
| issoi 5593 | An irreflexive, transitive... |
| isso2i 5594 | Deduce strict ordering fro... |
| so0 5595 | Any relation is a strict o... |
| somo 5596 | A totally ordered set has ... |
| sotrine 5597 | Trichotomy law for strict ... |
| sotr3 5598 | Transitivity law for stric... |
| dffr6 5605 | Alternate definition of ~ ... |
| frd 5606 | A nonempty subset of an ` ... |
| fri 5607 | A nonempty subset of an ` ... |
| seex 5608 | The ` R ` -preimage of an ... |
| exse 5609 | Any relation on a set is s... |
| dffr2 5610 | Alternate definition of we... |
| dffr2ALT 5611 | Alternate proof of ~ dffr2... |
| frc 5612 | Property of well-founded r... |
| frss 5613 | Subset theorem for the wel... |
| sess1 5614 | Subset theorem for the set... |
| sess2 5615 | Subset theorem for the set... |
| freq1 5616 | Equality theorem for the w... |
| freq2 5617 | Equality theorem for the w... |
| freq12d 5618 | Equality deduction for wel... |
| seeq1 5619 | Equality theorem for the s... |
| seeq2 5620 | Equality theorem for the s... |
| seeq12d 5621 | Equality deduction for the... |
| nffr 5622 | Bound-variable hypothesis ... |
| nfse 5623 | Bound-variable hypothesis ... |
| nfwe 5624 | Bound-variable hypothesis ... |
| frirr 5625 | A well-founded relation is... |
| fr2nr 5626 | A well-founded relation ha... |
| fr0 5627 | Any relation is well-found... |
| frminex 5628 | If an element of a well-fo... |
| efrirr 5629 | A well-founded class does ... |
| efrn2lp 5630 | A well-founded class conta... |
| epse 5631 | The membership relation is... |
| tz7.2 5632 | Similar to Theorem 7.2 of ... |
| dfepfr 5633 | An alternate way of saying... |
| epfrc 5634 | A subset of a well-founded... |
| wess 5635 | Subset theorem for the wel... |
| weeq1 5636 | Equality theorem for the w... |
| weeq2 5637 | Equality theorem for the w... |
| weeq12d 5638 | Equality deduction for wel... |
| wefr 5639 | A well-ordering is well-fo... |
| weso 5640 | A well-ordering is a stric... |
| wecmpep 5641 | The elements of a class we... |
| wetrep 5642 | On a class well-ordered by... |
| wefrc 5643 | A nonempty subclass of a c... |
| we0 5644 | Any relation is a well-ord... |
| wereu 5645 | A nonempty subset of an ` ... |
| wereu2 5646 | A nonempty subclass of an ... |
| xpeq1 5663 | Equality theorem for Carte... |
| xpss12 5664 | Subset theorem for Cartesi... |
| xpss 5665 | A Cartesian product is inc... |
| inxpssres 5666 | Intersection with a Cartes... |
| relxp 5667 | A Cartesian product is a r... |
| xpss1 5668 | Subset relation for Cartes... |
| xpss2 5669 | Subset relation for Cartes... |
| xpeq2 5670 | Equality theorem for Carte... |
| elxpi 5671 | Membership in a Cartesian ... |
| elxp 5672 | Membership in a Cartesian ... |
| elxp2 5673 | Membership in a Cartesian ... |
| xpeq12 5674 | Equality theorem for Carte... |
| xpeq1i 5675 | Equality inference for Car... |
| xpeq2i 5676 | Equality inference for Car... |
| xpeq12i 5677 | Equality inference for Car... |
| xpeq1d 5678 | Equality deduction for Car... |
| xpeq2d 5679 | Equality deduction for Car... |
| xpeq12d 5680 | Equality deduction for Car... |
| sqxpeqd 5681 | Equality deduction for a C... |
| nfxp 5682 | Bound-variable hypothesis ... |
| 0nelxp 5683 | The empty set is not a mem... |
| 0nelelxp 5684 | A member of a Cartesian pr... |
| opelxp 5685 | Ordered pair membership in... |
| opelxpi 5686 | Ordered pair membership in... |
| opelxpii 5687 | Ordered pair membership in... |
| opelxpd 5688 | Ordered pair membership in... |
| opelvv 5689 | Ordered pair membership in... |
| opelvvg 5690 | Ordered pair membership in... |
| opelxp1 5691 | The first member of an ord... |
| opelxp2 5692 | The second member of an or... |
| otelxp 5693 | Ordered triple membership ... |
| otelxp1 5694 | The first member of an ord... |
| otel3xp 5695 | An ordered triple is an el... |
| opabssxpd 5696 | An ordered-pair class abst... |
| rabxp 5697 | Class abstraction restrict... |
| brxp 5698 | Binary relation on a Carte... |
| pwvrel 5699 | A set is a binary relation... |
| pwvabrel 5700 | The powerclass of the cart... |
| brrelex12 5701 | Two classes related by a b... |
| brrelex1 5702 | If two classes are related... |
| brrelex2 5703 | If two classes are related... |
| brrelex12i 5704 | Two classes that are relat... |
| brrelex1i 5705 | The first argument of a bi... |
| brrelex2i 5706 | The second argument of a b... |
| nprrel12 5707 | Proper classes are not rel... |
| nprrel 5708 | No proper class is related... |
| 0nelrel0 5709 | A binary relation does not... |
| 0nelrel 5710 | A binary relation does not... |
| fconstmpt 5711 | Representation of a consta... |
| vtoclr 5712 | Variable to class conversi... |
| opthprc 5713 | Justification theorem for ... |
| brel 5714 | Two things in a binary rel... |
| elxp3 5715 | Membership in a Cartesian ... |
| opeliunxp 5716 | Membership in a union of C... |
| opeliun2xp 5717 | Membership of an ordered p... |
| xpundi 5718 | Distributive law for Carte... |
| xpundir 5719 | Distributive law for Carte... |
| xpiundi 5720 | Distributive law for Carte... |
| xpiundir 5721 | Distributive law for Carte... |
| iunxpconst 5722 | Membership in a union of C... |
| xpun 5723 | The Cartesian product of t... |
| elvv 5724 | Membership in universal cl... |
| elvvv 5725 | Membership in universal cl... |
| elvvuni 5726 | An ordered pair contains i... |
| brinxp2 5727 | Intersection of binary rel... |
| brinxp 5728 | Intersection of binary rel... |
| opelinxp 5729 | Ordered pair element in an... |
| poinxp 5730 | Intersection of partial or... |
| soinxp 5731 | Intersection of total orde... |
| frinxp 5732 | Intersection of well-found... |
| seinxp 5733 | Intersection of set-like r... |
| weinxp 5734 | Intersection of well-order... |
| posn 5735 | Partial ordering of a sing... |
| sosn 5736 | Strict ordering on a singl... |
| frsn 5737 | Founded relation on a sing... |
| wesn 5738 | Well-ordering of a singlet... |
| elopaelxp 5739 | Membership in an ordered-p... |
| bropaex12 5740 | Two classes related by an ... |
| opabssxp 5741 | An abstraction relation is... |
| brab2a 5742 | The law of concretion for ... |
| optocl 5743 | Implicit substitution of c... |
| optoclOLD 5744 | Obsolete version of ~ opto... |
| 2optocl 5745 | Implicit substitution of c... |
| 3optocl 5746 | Implicit substitution of c... |
| opbrop 5747 | Ordered pair membership in... |
| 0xp 5748 | The Cartesian product with... |
| xp0 5749 | The Cartesian product with... |
| csbxp 5750 | Distribute proper substitu... |
| releq 5751 | Equality theorem for the r... |
| releqi 5752 | Equality inference for the... |
| releqd 5753 | Equality deduction for the... |
| nfrel 5754 | Bound-variable hypothesis ... |
| sbcrel 5755 | Distribute proper substitu... |
| relss 5756 | Subclass theorem for relat... |
| ssrel 5757 | A subclass relationship de... |
| eqrel 5758 | Extensionality principle f... |
| ssrel2 5759 | A subclass relationship de... |
| ssrel3 5760 | Subclass relation in anoth... |
| relssi 5761 | Inference from subclass pr... |
| relssdv 5762 | Deduction from subclass pr... |
| eqrelriv 5763 | Inference from extensional... |
| eqrelriiv 5764 | Inference from extensional... |
| eqbrriv 5765 | Inference from extensional... |
| eqrelrdv 5766 | Deduce equality of relatio... |
| eqbrrdv 5767 | Deduction from extensional... |
| eqbrrdiv 5768 | Deduction from extensional... |
| eqrelrdv2 5769 | A version of ~ eqrelrdv . ... |
| ssrelrel 5770 | A subclass relationship de... |
| eqrelrel 5771 | Extensionality principle f... |
| elrel 5772 | A member of a relation is ... |
| rel0 5773 | The empty set is a relatio... |
| nrelv 5774 | The universal class is not... |
| nrelvOLD 5775 | Obsolete version of ~ nrel... |
| relsng 5776 | A singleton is a relation ... |
| relsnb 5777 | An at-most-singleton is a ... |
| relsnopg 5778 | A singleton of an ordered ... |
| relsn 5779 | A singleton is a relation ... |
| relsnop 5780 | A singleton of an ordered ... |
| copsex2gb 5781 | Implicit substitution infe... |
| copsex2ga 5782 | Implicit substitution infe... |
| elopaba 5783 | Membership in an ordered-p... |
| xpsspw 5784 | A Cartesian product is inc... |
| unixpss 5785 | The double class union of ... |
| relun 5786 | The union of two relations... |
| relin1 5787 | The intersection with a re... |
| relin2 5788 | The intersection with a re... |
| relinxp 5789 | Intersection with a Cartes... |
| reldif 5790 | A difference cutting down ... |
| reliun 5791 | An indexed union is a rela... |
| reliin 5792 | An indexed intersection is... |
| reluni 5793 | The union of a class is a ... |
| relint 5794 | The intersection of a clas... |
| relopabiv 5795 | A class of ordered pairs i... |
| relopabv 5796 | A class of ordered pairs i... |
| relopabi 5797 | A class of ordered pairs i... |
| relopabiALT 5798 | Alternate proof of ~ relop... |
| relopab 5799 | A class of ordered pairs i... |
| mptrel 5800 | The maps-to notation alway... |
| reli 5801 | The identity relation is a... |
| rele 5802 | The membership relation is... |
| opabid2 5803 | A relation expressed as an... |
| inopab 5804 | Intersection of two ordere... |
| difopab 5805 | Difference of two ordered-... |
| inxp 5806 | Intersection of two Cartes... |
| xpindi 5807 | Distributive law for Carte... |
| xpindir 5808 | Distributive law for Carte... |
| xpiindi 5809 | Distributive law for Carte... |
| xpriindi 5810 | Distributive law for Carte... |
| eliunxp 5811 | Membership in a union of C... |
| opeliunxp2 5812 | Membership in a union of C... |
| raliunxp 5813 | Write a double restricted ... |
| rexiunxp 5814 | Write a double restricted ... |
| ralxp 5815 | Universal quantification r... |
| rexxp 5816 | Existential quantification... |
| exopxfr 5817 | Transfer ordered-pair exis... |
| exopxfr2 5818 | Transfer ordered-pair exis... |
| djussxp 5819 | Disjoint union is a subset... |
| ralxpf 5820 | Version of ~ ralxp with bo... |
| rexxpf 5821 | Version of ~ rexxp with bo... |
| iunxpf 5822 | Indexed union on a Cartesi... |
| opabbi2dv 5823 | Deduce equality of a relat... |
| relop 5824 | A necessary and sufficient... |
| ideqg 5825 | For sets, the identity rel... |
| ideq 5826 | For sets, the identity rel... |
| ididg 5827 | A set is identical to itse... |
| issetid 5828 | Two ways of expressing set... |
| coss1 5829 | Subclass theorem for compo... |
| coss2 5830 | Subclass theorem for compo... |
| coeq1 5831 | Equality theorem for compo... |
| coeq2 5832 | Equality theorem for compo... |
| coeq1i 5833 | Equality inference for com... |
| coeq2i 5834 | Equality inference for com... |
| coeq1d 5835 | Equality deduction for com... |
| coeq2d 5836 | Equality deduction for com... |
| coeq12i 5837 | Equality inference for com... |
| coeq12d 5838 | Equality deduction for com... |
| nfco 5839 | Bound-variable hypothesis ... |
| brcog 5840 | Ordered pair membership in... |
| opelco2g 5841 | Ordered pair membership in... |
| brcogw 5842 | Ordered pair membership in... |
| eqbrrdva 5843 | Deduction from extensional... |
| brco 5844 | Binary relation on a compo... |
| opelco 5845 | Ordered pair membership in... |
| cnvss 5846 | Subset theorem for convers... |
| cnveq 5847 | Equality theorem for conve... |
| cnveqi 5848 | Equality inference for con... |
| cnveqd 5849 | Equality deduction for con... |
| elcnv 5850 | Membership in a converse r... |
| elcnv2 5851 | Membership in a converse r... |
| nfcnv 5852 | Bound-variable hypothesis ... |
| brcnvg 5853 | The converse of a binary r... |
| opelcnvg 5854 | Ordered-pair membership in... |
| opelcnv 5855 | Ordered-pair membership in... |
| brcnv 5856 | The converse of a binary r... |
| cnv0 5857 | The converse of the empty ... |
| cnv0OLD 5858 | Obsolete version of ~ cnv0... |
| cnvi 5859 | The converse of the identi... |
| csbcnv 5860 | Move class substitution in... |
| csbcnvOLD 5861 | Obsolete version of ~ csbc... |
| csbcnvgALTOLD 5862 | Obsolete version of ~ csbc... |
| cnvco 5863 | Distributive law of conver... |
| cnvuni 5864 | The converse of a class un... |
| dfdm3 5865 | Alternate definition of do... |
| dfrn2 5866 | Alternate definition of ra... |
| dfrn3 5867 | Alternate definition of ra... |
| elrn2g 5868 | Membership in a range. (C... |
| elrng 5869 | Membership in a range. (C... |
| elrn2 5870 | Membership in a range. (C... |
| elrn 5871 | Membership in a range. (C... |
| ssrelrn 5872 | If a relation is a subset ... |
| dfdm4 5873 | Alternate definition of do... |
| dfdmf 5874 | Definition of domain, usin... |
| csbdm 5875 | Distribute proper substitu... |
| eldmg 5876 | Domain membership. Theore... |
| eldm2g 5877 | Domain membership. Theore... |
| eldm 5878 | Membership in a domain. T... |
| eldm2 5879 | Membership in a domain. T... |
| dmss 5880 | Subset theorem for domain.... |
| dmeq 5881 | Equality theorem for domai... |
| dmeqi 5882 | Equality inference for dom... |
| dmeqd 5883 | Equality deduction for dom... |
| opeldmd 5884 | Membership of first of an ... |
| opeldm 5885 | Membership of first of an ... |
| breldm 5886 | Membership of first of a b... |
| breldmg 5887 | Membership of first of a b... |
| dmun 5888 | The domain of a union is t... |
| dmin 5889 | The domain of an intersect... |
| breldmd 5890 | Membership of first of a b... |
| dmiun 5891 | The domain of an indexed u... |
| dmuni 5892 | The domain of a union. Pa... |
| dmopab 5893 | The domain of a class of o... |
| dmopabelb 5894 | A set is an element of the... |
| dmopab2rex 5895 | The domain of an ordered p... |
| dmopabss 5896 | Upper bound for the domain... |
| dmopab3 5897 | The domain of a restricted... |
| dm0 5898 | The domain of the empty se... |
| dmi 5899 | The domain of the identity... |
| dmv 5900 | The domain of the universe... |
| dmep 5901 | The domain of the membersh... |
| dm0rn0 5902 | An empty domain is equival... |
| dm0rn0OLD 5903 | Obsolete version of ~ dm0r... |
| rn0 5904 | The range of the empty set... |
| rnep 5905 | The range of the membershi... |
| reldm0 5906 | A relation is empty iff it... |
| dmxp 5907 | The domain of a Cartesian ... |
| dmxpid 5908 | The domain of a Cartesian ... |
| dmxpin 5909 | The domain of the intersec... |
| xpid11 5910 | The Cartesian square is a ... |
| dmcnvcnv 5911 | The domain of the double c... |
| rncnvcnv 5912 | The range of the double co... |
| elreldm 5913 | The first member of an ord... |
| rneq 5914 | Equality theorem for range... |
| rneqi 5915 | Equality inference for ran... |
| rneqd 5916 | Equality deduction for ran... |
| rnss 5917 | Subset theorem for range. ... |
| rnssi 5918 | Subclass inference for ran... |
| brelrng 5919 | The second argument of a b... |
| brelrn 5920 | The second argument of a b... |
| opelrn 5921 | Membership of second membe... |
| releldm 5922 | The first argument of a bi... |
| relelrn 5923 | The second argument of a b... |
| releldmb 5924 | Membership in a domain. (... |
| relelrnb 5925 | Membership in a range. (C... |
| releldmi 5926 | The first argument of a bi... |
| relelrni 5927 | The second argument of a b... |
| dfrnf 5928 | Definition of range, using... |
| nfdm 5929 | Bound-variable hypothesis ... |
| nfrn 5930 | Bound-variable hypothesis ... |
| dmiin 5931 | Domain of an intersection.... |
| rnopab 5932 | The range of a class of or... |
| rnopabss 5933 | Upper bound for the range ... |
| rnopab3 5934 | The range of a restricted ... |
| rnmpt 5935 | The range of a function in... |
| elrnmpt 5936 | The range of a function in... |
| elrnmpt1s 5937 | Elementhood in an image se... |
| elrnmpt1 5938 | Elementhood in an image se... |
| elrnmptg 5939 | Membership in the range of... |
| elrnmpti 5940 | Membership in the range of... |
| elrnmptd 5941 | The range of a function in... |
| elrnmpt1d 5942 | Elementhood in an image se... |
| elrnmptdv 5943 | Elementhood in the range o... |
| elrnmpt2d 5944 | Elementhood in the range o... |
| nelrnmpt 5945 | Non-membership in the rang... |
| dfiun3g 5946 | Alternate definition of in... |
| dfiin3g 5947 | Alternate definition of in... |
| dfiun3 5948 | Alternate definition of in... |
| dfiin3 5949 | Alternate definition of in... |
| riinint 5950 | Express a relative indexed... |
| relrn0 5951 | A relation is empty iff it... |
| dmrnssfld 5952 | The domain and range of a ... |
| dmcoss 5953 | Domain of a composition. ... |
| dmcossOLD 5954 | Obsolete version of ~ dmco... |
| rncoss 5955 | Range of a composition. (... |
| dmcosseq 5956 | Domain of a composition. ... |
| dmcosseqOLD 5957 | Obsolete version of ~ dmco... |
| dmcosseqOLDOLD 5958 | Obsolete version of ~ dmco... |
| dmcoeq 5959 | Domain of a composition. ... |
| rncoeq 5960 | Range of a composition. (... |
| reseq1 5961 | Equality theorem for restr... |
| reseq2 5962 | Equality theorem for restr... |
| reseq1i 5963 | Equality inference for res... |
| reseq2i 5964 | Equality inference for res... |
| reseq12i 5965 | Equality inference for res... |
| reseq1d 5966 | Equality deduction for res... |
| reseq2d 5967 | Equality deduction for res... |
| reseq12d 5968 | Equality deduction for res... |
| nfres 5969 | Bound-variable hypothesis ... |
| csbres 5970 | Distribute proper substitu... |
| res0 5971 | A restriction to the empty... |
| dfres3 5972 | Alternate definition of re... |
| opelres 5973 | Ordered pair elementhood i... |
| brres 5974 | Binary relation on a restr... |
| opelresi 5975 | Ordered pair membership in... |
| brresi 5976 | Binary relation on a restr... |
| opres 5977 | Ordered pair membership in... |
| resieq 5978 | A restricted identity rela... |
| opelidres 5979 | ` <. A , A >. ` belongs to... |
| resres 5980 | The restriction of a restr... |
| resundi 5981 | Distributive law for restr... |
| resundir 5982 | Distributive law for restr... |
| resindi 5983 | Class restriction distribu... |
| resindir 5984 | Class restriction distribu... |
| inres 5985 | Move intersection into cla... |
| resdifcom 5986 | Commutative law for restri... |
| resiun1 5987 | Distribution of restrictio... |
| resiun2 5988 | Distribution of restrictio... |
| resss 5989 | A class includes its restr... |
| rescom 5990 | Commutative law for restri... |
| ssres 5991 | Subclass theorem for restr... |
| ssres2 5992 | Subclass theorem for restr... |
| relres 5993 | A restriction is a relatio... |
| resabs1 5994 | Absorption law for restric... |
| resabs1i 5995 | Absorption law for restric... |
| resabs1d 5996 | Absorption law for restric... |
| resabs2 5997 | Absorption law for restric... |
| residm 5998 | Idempotent law for restric... |
| dmresss 5999 | The domain of a restrictio... |
| dmres 6000 | The domain of a restrictio... |
| ssdmres 6001 | A domain restricted to a s... |
| dmresexg 6002 | The domain of a restrictio... |
| resima 6003 | A restriction to an image.... |
| resima2 6004 | Image under a restricted c... |
| rnresss 6005 | The range of a restriction... |
| xpssres 6006 | Restriction of a constant ... |
| elinxp 6007 | Membership in an intersect... |
| elres 6008 | Membership in a restrictio... |
| elsnres 6009 | Membership in restriction ... |
| relssres 6010 | Simplification law for res... |
| dmressnsn 6011 | The domain of a restrictio... |
| eldmressnsn 6012 | The element of the domain ... |
| eldmeldmressn 6013 | An element of the domain (... |
| resdm 6014 | A relation restricted to i... |
| resexg 6015 | The restriction of a set i... |
| resexd 6016 | The restriction of a set i... |
| resex 6017 | The restriction of a set i... |
| resindm 6018 | When restricting a class, ... |
| resindmOLD 6019 | Obsolete version of ~ resi... |
| resdmdfsn 6020 | Restricting a class to its... |
| resdmdfsnOLD 6021 | Obsolete version of ~ resd... |
| reldmun 6022 | Split a relation into two ... |
| reldisjunOLD 6023 | Obsolete version of ~ reld... |
| relresdm1 6024 | Restriction of a disjoint ... |
| resopab 6025 | Restriction of a class abs... |
| iss 6026 | A subclass of the identity... |
| resopab2 6027 | Restriction of a class abs... |
| resmpt 6028 | Restriction of the mapping... |
| resmpt3 6029 | Unconditional restriction ... |
| resmptf 6030 | Restriction of the mapping... |
| resmptd 6031 | Restriction of the mapping... |
| dfres2 6032 | Alternate definition of th... |
| mptss 6033 | Sufficient condition for i... |
| elimampt 6034 | Membership in the image of... |
| elidinxp 6035 | Characterization of the el... |
| elidinxpid 6036 | Characterization of the el... |
| elrid 6037 | Characterization of the el... |
| idinxpres 6038 | The intersection of the id... |
| idinxpresid 6039 | The intersection of the id... |
| idssxp 6040 | A diagonal set as a subset... |
| opabresid 6041 | The restricted identity re... |
| mptresid 6042 | The restricted identity re... |
| dmresi 6043 | The domain of a restricted... |
| restidsing 6044 | Restriction of the identit... |
| iresn0n0 6045 | The identity function rest... |
| imaeq1 6046 | Equality theorem for image... |
| imaeq2 6047 | Equality theorem for image... |
| imaeq1i 6048 | Equality theorem for image... |
| imaeq2i 6049 | Equality theorem for image... |
| imaeq1d 6050 | Equality theorem for image... |
| imaeq2d 6051 | Equality theorem for image... |
| imaeq12d 6052 | Equality theorem for image... |
| dfima2 6053 | Alternate definition of im... |
| dfima3 6054 | Alternate definition of im... |
| elimag 6055 | Membership in an image. T... |
| elima 6056 | Membership in an image. T... |
| elima2 6057 | Membership in an image. T... |
| elima3 6058 | Membership in an image. T... |
| nfima 6059 | Bound-variable hypothesis ... |
| nfimad 6060 | Deduction version of bound... |
| imadmrn 6061 | The image of the domain of... |
| imassrn 6062 | The image of a class is a ... |
| mptima 6063 | Image of a function in map... |
| mptimass 6064 | Image of a function in map... |
| imai 6065 | Image under the identity r... |
| rnresi 6066 | The range of the restricte... |
| resiima 6067 | The image of a restriction... |
| ima0 6068 | Image of the empty set. T... |
| 0ima 6069 | Image under the empty rela... |
| csbima12 6070 | Move class substitution in... |
| imadisj 6071 | A class whose image under ... |
| imadisjlnd 6072 | Deduction form of one nega... |
| cnvimass 6073 | A preimage under any class... |
| cnvimarndm 6074 | The preimage of the range ... |
| imasng 6075 | The image of a singleton. ... |
| relimasn 6076 | The image of a singleton. ... |
| elrelimasn 6077 | Elementhood in the image o... |
| elimasng1 6078 | Membership in an image of ... |
| elimasn1 6079 | Membership in an image of ... |
| elimasng 6080 | Membership in an image of ... |
| elimasn 6081 | Membership in an image of ... |
| elimasni 6082 | Membership in an image of ... |
| args 6083 | Two ways to express the cl... |
| elinisegg 6084 | Membership in the inverse ... |
| eliniseg 6085 | Membership in the inverse ... |
| epin 6086 | Any set is equal to its pr... |
| epini 6087 | Any set is equal to its pr... |
| iniseg 6088 | An idiom that signifies an... |
| inisegn0 6089 | Nonemptiness of an initial... |
| dffr3 6090 | Alternate definition of we... |
| dfse2 6091 | Alternate definition of se... |
| imass1 6092 | Subset theorem for image. ... |
| imass2 6093 | Subset theorem for image. ... |
| ndmima 6094 | The image of a singleton o... |
| relcnv 6095 | A converse is a relation. ... |
| relbrcnvg 6096 | When ` R ` is a relation, ... |
| eliniseg2 6097 | Eliminate the class existe... |
| relbrcnv 6098 | When ` R ` is a relation, ... |
| relco 6099 | A composition is a relatio... |
| cotrg 6100 | Two ways of saying that th... |
| cotr 6101 | Two ways of saying a relat... |
| idrefALT 6102 | Alternate proof of ~ idref... |
| cnvsym 6103 | Two ways of saying a relat... |
| intasym 6104 | Two ways of saying a relat... |
| asymref 6105 | Two ways of saying a relat... |
| asymref2 6106 | Two ways of saying a relat... |
| intirr 6107 | Two ways of saying a relat... |
| brcodir 6108 | Two ways of saying that tw... |
| codir 6109 | Two ways of saying a relat... |
| qfto 6110 | A quantifier-free way of e... |
| xpidtr 6111 | A Cartesian square is a tr... |
| trin2 6112 | The intersection of two tr... |
| poirr2 6113 | A partial order is irrefle... |
| trinxp 6114 | The relation induced by a ... |
| soirri 6115 | A strict order relation is... |
| sotri 6116 | A strict order relation is... |
| son2lpi 6117 | A strict order relation ha... |
| sotri2 6118 | A transitivity relation. ... |
| sotri3 6119 | A transitivity relation. ... |
| poleloe 6120 | Express "less than or equa... |
| poltletr 6121 | Transitive law for general... |
| somin1 6122 | Property of a minimum in a... |
| somincom 6123 | Commutativity of minimum i... |
| somin2 6124 | Property of a minimum in a... |
| soltmin 6125 | Being less than a minimum,... |
| cnvopab 6126 | The converse of a class ab... |
| mptcnv 6127 | The converse of a mapping ... |
| cnvun 6128 | The converse of a union is... |
| cnvdif 6129 | Distributive law for conve... |
| cnvin 6130 | Distributive law for conve... |
| rnun 6131 | Distributive law for range... |
| rnin 6132 | The range of an intersecti... |
| rninOLD 6133 | Obsolete version of ~ rnin... |
| rniun 6134 | The range of an indexed un... |
| rnuni 6135 | The range of a union. Par... |
| imaundi 6136 | Distributive law for image... |
| imaundir 6137 | The image of a union. (Co... |
| imadifssran 6138 | Condition for the range of... |
| cnvimassrndm 6139 | The preimage of a superset... |
| dminss 6140 | An upper bound for interse... |
| imainss 6141 | An upper bound for interse... |
| inimass 6142 | The image of an intersecti... |
| inimasn 6143 | The intersection of the im... |
| cnvxp 6144 | The converse of a Cartesia... |
| xp0OLD 6145 | Obsolete version of ~ xp0 ... |
| xpnz 6146 | The Cartesian product of n... |
| xpeq0 6147 | At least one member of an ... |
| xpdisj1 6148 | Cartesian products with di... |
| xpdisj2 6149 | Cartesian products with di... |
| xpsndisj 6150 | Cartesian products with tw... |
| difxp 6151 | Difference of Cartesian pr... |
| difxp1 6152 | Difference law for Cartesi... |
| difxp2 6153 | Difference law for Cartesi... |
| djudisj 6154 | Disjoint unions with disjo... |
| xpdifid 6155 | The set of distinct couple... |
| xpdifcnvepel 6156 | The set of couples in a Ca... |
| resdisj 6157 | A double restriction to di... |
| rnxp 6158 | The range of a Cartesian p... |
| dmxpss 6159 | The domain of a Cartesian ... |
| rnxpss 6160 | The range of a Cartesian p... |
| rnxpid 6161 | The range of a Cartesian s... |
| ssxpb 6162 | A Cartesian product subcla... |
| xp11 6163 | The Cartesian product of n... |
| xpcan 6164 | Cancellation law for Carte... |
| xpcan2 6165 | Cancellation law for Carte... |
| ssrnres 6166 | Two ways to express surjec... |
| rninxp 6167 | Two ways to express surjec... |
| dminxp 6168 | Two ways to express totali... |
| imainrect 6169 | Image by a restricted and ... |
| xpima 6170 | Direct image by a Cartesia... |
| xpima1 6171 | Direct image by a Cartesia... |
| xpima2 6172 | Direct image by a Cartesia... |
| xpimasn 6173 | Direct image of a singleto... |
| sossfld 6174 | The base set of a strict o... |
| sofld 6175 | The base set of a nonempty... |
| cnvcnv3 6176 | The set of all ordered pai... |
| dfrel2 6177 | Alternate definition of re... |
| dfrel4v 6178 | A relation can be expresse... |
| dfrel4 6179 | A relation can be expresse... |
| cnvcnv 6180 | The double converse of a c... |
| cnvcnv2 6181 | The double converse of a c... |
| cnvcnvss 6182 | The double converse of a c... |
| cnvcnvssOLD 6183 | Obsolete version of ~ cnvc... |
| cnvrescnv 6184 | Two ways to express the co... |
| cnveqb 6185 | Equality theorem for conve... |
| cnveq0 6186 | A relation empty iff its c... |
| dfrel3 6187 | Alternate definition of re... |
| elid 6188 | Characterization of the el... |
| dmresv 6189 | The domain of a universal ... |
| rnresv 6190 | The range of a universal r... |
| dfrn4 6191 | Range defined in terms of ... |
| csbrn 6192 | Distribute proper substitu... |
| rescnvcnv 6193 | The restriction of the dou... |
| cnvcnvres 6194 | The double converse of the... |
| imacnvcnv 6195 | The image of the double co... |
| dmsnn0 6196 | The domain of a singleton ... |
| rnsnn0 6197 | The range of a singleton i... |
| dmsn0 6198 | The domain of the singleto... |
| cnvsn0 6199 | The converse of the single... |
| dmsn0el 6200 | The domain of a singleton ... |
| relsn2 6201 | A singleton is a relation ... |
| dmsnopg 6202 | The domain of a singleton ... |
| dmsnopss 6203 | The domain of a singleton ... |
| dmpropg 6204 | The domain of an unordered... |
| dmsnop 6205 | The domain of a singleton ... |
| dmprop 6206 | The domain of an unordered... |
| dmtpop 6207 | The domain of an unordered... |
| cnvcnvsn 6208 | Double converse of a singl... |
| dmsnsnsn 6209 | The domain of the singleto... |
| rnsnopg 6210 | The range of a singleton o... |
| rnpropg 6211 | The range of a pair of ord... |
| cnvsng 6212 | Converse of a singleton of... |
| rnsnop 6213 | The range of a singleton o... |
| op1sta 6214 | Extract the first member o... |
| cnvsn 6215 | Converse of a singleton of... |
| op2ndb 6216 | Extract the second member ... |
| op2nda 6217 | Extract the second member ... |
| opswap 6218 | Swap the members of an ord... |
| cnvresima 6219 | An image under the convers... |
| resdm2 6220 | A class restricted to its ... |
| resdmres 6221 | Restriction to the domain ... |
| resresdm 6222 | A restriction by an arbitr... |
| imadmres 6223 | The image of the domain of... |
| resdmss 6224 | Subset relationship for th... |
| resdifdi 6225 | Distributive law for restr... |
| resdifdir 6226 | Distributive law for restr... |
| mptpreima 6227 | The preimage of a function... |
| mptiniseg 6228 | Converse singleton image o... |
| dmmpt 6229 | The domain of the mapping ... |
| dmmptss 6230 | The domain of a mapping is... |
| dmmptg 6231 | The domain of the mapping ... |
| rnmpt0f 6232 | The range of a function in... |
| rnmptn0 6233 | The range of a function in... |
| dfco2 6234 | Alternate definition of a ... |
| dfco2a 6235 | Generalization of ~ dfco2 ... |
| coundi 6236 | Class composition distribu... |
| coundir 6237 | Class composition distribu... |
| cores 6238 | Restricted first member of... |
| resco 6239 | Associative law for the re... |
| imaco 6240 | Image of the composition o... |
| rnco 6241 | The range of the compositi... |
| rncoOLD 6242 | Obsolete version of ~ rnco... |
| rnco2 6243 | The range of the compositi... |
| dmco 6244 | The domain of a compositio... |
| coeq0 6245 | A composition of two relat... |
| coiun 6246 | Composition with an indexe... |
| cocnvcnv1 6247 | A composition is not affec... |
| cocnvcnv2 6248 | A composition is not affec... |
| cores2 6249 | Absorption of a reverse (p... |
| co02 6250 | Composition with the empty... |
| co01 6251 | Composition with the empty... |
| coi1 6252 | Composition with the ident... |
| coi2 6253 | Composition with the ident... |
| coires1 6254 | Composition with a restric... |
| coass 6255 | Associative law for class ... |
| relcnvtrg 6256 | General form of ~ relcnvtr... |
| relcnvtr 6257 | A relation is transitive i... |
| relssdmrn 6258 | A relation is included in ... |
| resssxp 6259 | If the ` R ` -image of a c... |
| cnvssrndm 6260 | The converse is a subset o... |
| cossxp 6261 | Composition as a subset of... |
| relrelss 6262 | Two ways to describe the s... |
| unielrel 6263 | The membership relation fo... |
| relfld 6264 | The double union of a rela... |
| relresfld 6265 | Restriction of a relation ... |
| relcoi2 6266 | Composition with the ident... |
| relcoi1 6267 | Composition with the ident... |
| unidmrn 6268 | The double union of the co... |
| relcnvfld 6269 | if ` R ` is a relation, it... |
| dfdm2 6270 | Alternate definition of do... |
| unixp 6271 | The double class union of ... |
| unixp0 6272 | A Cartesian product is emp... |
| unixpid 6273 | Field of a Cartesian squar... |
| ressn 6274 | Restriction of a class to ... |
| cnviin 6275 | The converse of an interse... |
| cnvpo 6276 | The converse of a partial ... |
| cnvso 6277 | The converse of a strict o... |
| xpco 6278 | Composition of two Cartesi... |
| xpcoid 6279 | Composition of two Cartesi... |
| elsnxp 6280 | Membership in a Cartesian ... |
| reu3op 6281 | There is a unique ordered ... |
| reuop 6282 | There is a unique ordered ... |
| opreu2reurex 6283 | There is a unique ordered ... |
| opreu2reu 6284 | If there is a unique order... |
| dfpo2 6285 | Quantifier-free definition... |
| csbcog 6286 | Distribute proper substitu... |
| snres0 6287 | Condition for restriction ... |
| imaindm 6288 | The image is unaffected by... |
| predeq123 6291 | Equality theorem for the p... |
| predeq1 6292 | Equality theorem for the p... |
| predeq2 6293 | Equality theorem for the p... |
| predeq3 6294 | Equality theorem for the p... |
| nfpred 6295 | Bound-variable hypothesis ... |
| csbpredg 6296 | Move class substitution in... |
| predpredss 6297 | If ` A ` is a subset of ` ... |
| predss 6298 | The predecessor class of `... |
| sspred 6299 | Another subset/predecessor... |
| dfpred2 6300 | An alternate definition of... |
| dfpred3 6301 | An alternate definition of... |
| dfpred3g 6302 | An alternate definition of... |
| elpredgg 6303 | Membership in a predecesso... |
| elpredg 6304 | Membership in a predecesso... |
| elpredimg 6305 | Membership in a predecesso... |
| elpredim 6306 | Membership in a predecesso... |
| elpred 6307 | Membership in a predecesso... |
| predexg 6308 | The predecessor class exis... |
| dffr4 6309 | Alternate definition of we... |
| predel 6310 | Membership in the predeces... |
| predtrss 6311 | If ` R ` is transitive ove... |
| predpo 6312 | Property of the predecesso... |
| predso 6313 | Property of the predecesso... |
| setlikespec 6314 | If ` R ` is set-like in ` ... |
| predidm 6315 | Idempotent law for the pre... |
| predin 6316 | Intersection law for prede... |
| predun 6317 | Union law for predecessor ... |
| preddif 6318 | Difference law for predece... |
| predep 6319 | The predecessor under the ... |
| trpred 6320 | The class of predecessors ... |
| preddowncl 6321 | A property of classes that... |
| predpoirr 6322 | Given a partial ordering, ... |
| predfrirr 6323 | Given a well-founded relat... |
| pred0 6324 | The predecessor class over... |
| dfse3 6325 | Alternate definition of se... |
| predrelss 6326 | Subset carries from relati... |
| predprc 6327 | The predecessor of a prope... |
| predres 6328 | Predecessor class is unaff... |
| frpomin 6329 | Every nonempty (possibly p... |
| frpomin2 6330 | Every nonempty (possibly p... |
| frpoind 6331 | The principle of well-foun... |
| frpoinsg 6332 | Well-Founded Induction Sch... |
| frpoins2fg 6333 | Well-Founded Induction sch... |
| frpoins2g 6334 | Well-Founded Induction sch... |
| frpoins3g 6335 | Well-Founded Induction sch... |
| tz6.26 6336 | All nonempty subclasses of... |
| tz6.26i 6337 | All nonempty subclasses of... |
| wfi 6338 | The Principle of Well-Orde... |
| wfii 6339 | The Principle of Well-Orde... |
| wfisg 6340 | Well-Ordered Induction Sch... |
| wfis 6341 | Well-Ordered Induction Sch... |
| wfis2fg 6342 | Well-Ordered Induction Sch... |
| wfis2f 6343 | Well-Ordered Induction sch... |
| wfis2g 6344 | Well-Ordered Induction Sch... |
| wfis2 6345 | Well-Ordered Induction sch... |
| wfis3 6346 | Well-Ordered Induction sch... |
| ordeq 6355 | Equality theorem for the o... |
| elong 6356 | An ordinal number is an or... |
| elon 6357 | An ordinal number is an or... |
| eloni 6358 | An ordinal number has the ... |
| elon2 6359 | An ordinal number is an or... |
| limeq 6360 | Equality theorem for the l... |
| ordwe 6361 | Membership well-orders eve... |
| ordtr 6362 | An ordinal class is transi... |
| ordfr 6363 | Membership is well-founded... |
| ordelss 6364 | An element of an ordinal c... |
| trssord 6365 | A transitive subclass of a... |
| ordirr 6366 | No ordinal class is a memb... |
| nordeq 6367 | A member of an ordinal cla... |
| ordn2lp 6368 | An ordinal class cannot be... |
| tz7.5 6369 | A nonempty subclass of an ... |
| ordelord 6370 | An element of an ordinal c... |
| tron 6371 | The class of all ordinal n... |
| ordelon 6372 | An element of an ordinal c... |
| onelon 6373 | An element of an ordinal n... |
| tz7.7 6374 | A transitive class belongs... |
| ordelssne 6375 | For ordinal classes, membe... |
| ordelpss 6376 | For ordinal classes, membe... |
| ordsseleq 6377 | For ordinal classes, inclu... |
| ordin 6378 | The intersection of two or... |
| onin 6379 | The intersection of two or... |
| ordtri3or 6380 | A trichotomy law for ordin... |
| ordtri1 6381 | A trichotomy law for ordin... |
| ontri1 6382 | A trichotomy law for ordin... |
| ordtri2 6383 | A trichotomy law for ordin... |
| ordtri3 6384 | A trichotomy law for ordin... |
| ordtri4 6385 | A trichotomy law for ordin... |
| orddisj 6386 | An ordinal class and its s... |
| onfr 6387 | The ordinal class is well-... |
| onelpss 6388 | Relationship between membe... |
| onsseleq 6389 | Relationship between subse... |
| onelss 6390 | An element of an ordinal n... |
| oneltri 6391 | The elementhood relation o... |
| ordtr1 6392 | Transitive law for ordinal... |
| ordtr2 6393 | Transitive law for ordinal... |
| ordtr3 6394 | Transitive law for ordinal... |
| ontr1 6395 | Transitive law for ordinal... |
| ontr2 6396 | Transitive law for ordinal... |
| onelssex 6397 | Ordinal less than is equiv... |
| ordunidif 6398 | The union of an ordinal st... |
| ordintdif 6399 | If ` B ` is smaller than `... |
| onintss 6400 | If a property is true for ... |
| oneqmini 6401 | A way to show that an ordi... |
| ord0 6402 | The empty set is an ordina... |
| 0elon 6403 | The empty set is an ordina... |
| ord0eln0 6404 | A nonempty ordinal contain... |
| on0eln0 6405 | An ordinal number contains... |
| dflim2 6406 | An alternate definition of... |
| inton 6407 | The intersection of the cl... |
| nlim0 6408 | The empty set is not a lim... |
| limord 6409 | A limit ordinal is ordinal... |
| limuni 6410 | A limit ordinal is its own... |
| limuni2 6411 | The union of a limit ordin... |
| 0ellim 6412 | A limit ordinal contains t... |
| limelon 6413 | A limit ordinal class that... |
| onn0 6414 | The class of all ordinal n... |
| suceqd 6415 | Deduction associated with ... |
| suceq 6416 | Equality of successors. (... |
| elsuci 6417 | Membership in a successor.... |
| elsucg 6418 | Membership in a successor.... |
| elsuc2g 6419 | Variant of membership in a... |
| elsuc 6420 | Membership in a successor.... |
| elsuc2 6421 | Membership in a successor.... |
| nfsuc 6422 | Bound-variable hypothesis ... |
| elelsuc 6423 | Membership in a successor.... |
| sucel 6424 | Membership of a successor ... |
| suc0 6425 | The successor of the empty... |
| sucprc 6426 | A proper class is its own ... |
| unisucs 6427 | The union of the successor... |
| unisucg 6428 | A transitive class is equa... |
| unisuc 6429 | A transitive class is equa... |
| sssucid 6430 | A class is included in its... |
| sucidg 6431 | Part of Proposition 7.23 o... |
| sucid 6432 | A set belongs to its succe... |
| nsuceq0 6433 | No successor is empty. (C... |
| eqelsuc 6434 | A set belongs to the succe... |
| iunsuc 6435 | Inductive definition for t... |
| suctr 6436 | The successor of a transit... |
| trsuc 6437 | A set whose successor belo... |
| trsucss 6438 | A member of the successor ... |
| ordsssuc 6439 | An ordinal is a subset of ... |
| onsssuc 6440 | A subset of an ordinal num... |
| ordsssuc2 6441 | An ordinal subset of an or... |
| onmindif 6442 | When its successor is subt... |
| ordnbtwn 6443 | There is no set between an... |
| onnbtwn 6444 | There is no set between an... |
| sucssel 6445 | A set whose successor is a... |
| orddif 6446 | Ordinal derived from its s... |
| orduniss 6447 | An ordinal class includes ... |
| ordtri2or 6448 | A trichotomy law for ordin... |
| ordtri2or2 6449 | A trichotomy law for ordin... |
| ordtri2or3 6450 | A consequence of total ord... |
| ordelinel 6451 | The intersection of two or... |
| ordssun 6452 | Property of a subclass of ... |
| ordequn 6453 | The maximum (i.e. union) o... |
| ordun 6454 | The maximum (i.e., union) ... |
| onunel 6455 | The union of two ordinals ... |
| ordunisssuc 6456 | A subclass relationship fo... |
| suc11 6457 | The successor operation be... |
| onun2 6458 | The union of two ordinals ... |
| ontr 6459 | An ordinal number is a tra... |
| onunisuc 6460 | An ordinal number is equal... |
| onordi 6461 | An ordinal number is an or... |
| onirri 6462 | An ordinal number is not a... |
| oneli 6463 | A member of an ordinal num... |
| onelssi 6464 | A member of an ordinal num... |
| onssneli 6465 | An ordering law for ordina... |
| onssnel2i 6466 | An ordering law for ordina... |
| onelini 6467 | An element of an ordinal n... |
| oneluni 6468 | An ordinal number equals i... |
| onunisuci 6469 | An ordinal number is equal... |
| onsseli 6470 | Subset is equivalent to me... |
| onun2i 6471 | The union of two ordinal n... |
| unizlim 6472 | An ordinal equal to its ow... |
| on0eqel 6473 | An ordinal number either e... |
| snsn0non 6474 | The singleton of the singl... |
| onxpdisj 6475 | Ordinal numbers and ordere... |
| onnev 6476 | The class of ordinal numbe... |
| iotajust 6478 | Soundness justification th... |
| dfiota2 6480 | Alternate definition for d... |
| nfiota1 6481 | Bound-variable hypothesis ... |
| nfiotadw 6482 | Deduction version of ~ nfi... |
| nfiotaw 6483 | Bound-variable hypothesis ... |
| nfiotad 6484 | Deduction version of ~ nfi... |
| nfiota 6485 | Bound-variable hypothesis ... |
| cbviotaw 6486 | Change bound variables in ... |
| cbviotavw 6487 | Change bound variables in ... |
| cbviota 6488 | Change bound variables in ... |
| cbviotav 6489 | Change bound variables in ... |
| sb8iota 6490 | Variable substitution in d... |
| iotaeq 6491 | Equality theorem for descr... |
| iotabi 6492 | Equivalence theorem for de... |
| uniabio 6493 | Part of Theorem 8.17 in [Q... |
| iotaval2 6494 | Version of ~ iotaval using... |
| iotauni2 6495 | Version of ~ iotauni using... |
| iotanul2 6496 | Version of ~ iotanul using... |
| iotaval 6497 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6498 | The ` iota ` class is a su... |
| iotaex 6499 | Theorem 8.23 in [Quine] p.... |
| iotauni 6500 | Equivalence between two di... |
| iotaint 6501 | Equivalence between two di... |
| iota1 6502 | Property of iota. (Contri... |
| iotanul 6503 | Theorem 8.22 in [Quine] p.... |
| iota4 6504 | Theorem *14.22 in [Whitehe... |
| iota4an 6505 | Theorem *14.23 in [Whitehe... |
| iota5 6506 | A method for computing iot... |
| iotabidv 6507 | Formula-building deduction... |
| iotabii 6508 | Formula-building deduction... |
| iotacl 6509 | Membership law for descrip... |
| iota2df 6510 | A condition that allows to... |
| iota2d 6511 | A condition that allows to... |
| iota2 6512 | The unique element such th... |
| iotan0 6513 | Representation of "the uni... |
| sniota 6514 | A class abstraction with a... |
| dfiota4 6515 | The ` iota ` operation usi... |
| csbiota 6516 | Class substitution within ... |
| dffun2 6533 | Alternate definition of a ... |
| dffun6 6534 | Alternate definition of a ... |
| dffun3 6535 | Alternate definition of fu... |
| dffun4 6536 | Alternate definition of a ... |
| dffun5 6537 | Alternate definition of fu... |
| dffun6f 6538 | Definition of function, us... |
| funmo 6539 | A function has at most one... |
| funrel 6540 | A function is a relation. ... |
| 0nelfun 6541 | A function does not contai... |
| funss 6542 | Subclass theorem for funct... |
| funeq 6543 | Equality theorem for funct... |
| funeqi 6544 | Equality inference for the... |
| funeqd 6545 | Equality deduction for the... |
| nffun 6546 | Bound-variable hypothesis ... |
| sbcfung 6547 | Distribute proper substitu... |
| funeu 6548 | There is exactly one value... |
| funeu2 6549 | There is exactly one value... |
| dffun7 6550 | Alternate definition of a ... |
| dffun8 6551 | Alternate definition of a ... |
| dffun9 6552 | Alternate definition of a ... |
| funfn 6553 | A class is a function if a... |
| funfnd 6554 | A function is a function o... |
| funi 6555 | The identity relation is a... |
| nfunv 6556 | The universal class is not... |
| funopg 6557 | A Kuratowski ordered pair ... |
| funopab 6558 | A class of ordered pairs i... |
| funopabeq 6559 | A class of ordered pairs o... |
| funopab4 6560 | A class of ordered pairs o... |
| funmpt 6561 | A function in maps-to nota... |
| funmpt2 6562 | Functionality of a class g... |
| funco 6563 | The composition of two fun... |
| funresfunco 6564 | Composition of two functio... |
| funres 6565 | A restriction of a functio... |
| funresd 6566 | A restriction of a functio... |
| funssres 6567 | The restriction of a funct... |
| fun2ssres 6568 | Equality of restrictions o... |
| funun 6569 | The union of functions wit... |
| fununmo 6570 | If the union of classes is... |
| fununfun 6571 | If the union of classes is... |
| fundif 6572 | A function with removed el... |
| funcnvsn 6573 | The converse singleton of ... |
| funsng 6574 | A singleton of an ordered ... |
| fnsng 6575 | Functionality and domain o... |
| funsn 6576 | A singleton of an ordered ... |
| funprg 6577 | A set of two pairs is a fu... |
| funtpg 6578 | A set of three pairs is a ... |
| funpr 6579 | A function with a domain o... |
| funtp 6580 | A function with a domain o... |
| fnsn 6581 | Functionality and domain o... |
| fnprg 6582 | Function with a domain of ... |
| fntpg 6583 | Function with a domain of ... |
| fntp 6584 | A function with a domain o... |
| funcnvpr 6585 | The converse pair of order... |
| funcnvtp 6586 | The converse triple of ord... |
| funcnvqp 6587 | The converse quadruple of ... |
| fun0 6588 | The empty set is a functio... |
| funcnv0 6589 | The converse of the empty ... |
| funcnvcnv 6590 | The double converse of a f... |
| funcnv2 6591 | A simpler equivalence for ... |
| funcnv 6592 | The converse of a class is... |
| funcnv3 6593 | A condition showing a clas... |
| fun2cnv 6594 | The double converse of a c... |
| svrelfun 6595 | A single-valued relation i... |
| fncnv 6596 | Single-rootedness (see ~ f... |
| fun11 6597 | Two ways of stating that `... |
| fununi 6598 | The union of a chain (with... |
| funin 6599 | The intersection with a fu... |
| funres11 6600 | The restriction of a one-t... |
| funcnvres 6601 | The converse of a restrict... |
| cnvresid 6602 | Converse of a restricted i... |
| funcnvres2 6603 | The converse of a restrict... |
| funimacnv 6604 | The image of the preimage ... |
| funimass1 6605 | A kind of contraposition l... |
| funimass2 6606 | A kind of contraposition l... |
| imadif 6607 | The image of a difference ... |
| imain 6608 | The image of an intersecti... |
| f1imadifssran 6609 | Condition for the range of... |
| funimaexg 6610 | Axiom of Replacement using... |
| funimaex 6611 | The image of a set under a... |
| isarep1 6612 | Part of a study of the Axi... |
| isarep2 6613 | Part of a study of the Axi... |
| fneq1 6614 | Equality theorem for funct... |
| fneq2 6615 | Equality theorem for funct... |
| fneq1d 6616 | Equality deduction for fun... |
| fneq2d 6617 | Equality deduction for fun... |
| fneq12d 6618 | Equality deduction for fun... |
| fneq12 6619 | Equality theorem for funct... |
| fneq1i 6620 | Equality inference for fun... |
| fneq2i 6621 | Equality inference for fun... |
| nffn 6622 | Bound-variable hypothesis ... |
| fnfun 6623 | A function with domain is ... |
| fnfund 6624 | A function with domain is ... |
| fnrel 6625 | A function with domain is ... |
| fndm 6626 | The domain of a function. ... |
| fndmi 6627 | The domain of a function. ... |
| fndmd 6628 | The domain of a function. ... |
| funfni 6629 | Inference to convert a fun... |
| fndmu 6630 | A function has a unique do... |
| fnbr 6631 | The first argument of bina... |
| fnop 6632 | The first argument of an o... |
| fneu 6633 | There is exactly one value... |
| fneu2 6634 | There is exactly one value... |
| fnunres1 6635 | Restriction of a disjoint ... |
| fnunres2 6636 | Restriction of a disjoint ... |
| fnun 6637 | The union of two functions... |
| fnund 6638 | The union of two functions... |
| fnunop 6639 | Extension of a function wi... |
| fncofn 6640 | Composition of a function ... |
| fnco 6641 | Composition of two functio... |
| fnresdm 6642 | A function does not change... |
| fnresdisj 6643 | A function restricted to a... |
| 2elresin 6644 | Membership in two function... |
| fnssresb 6645 | Restriction of a function ... |
| fnssres 6646 | Restriction of a function ... |
| fnssresd 6647 | Restriction of a function ... |
| fnresin1 6648 | Restriction of a function'... |
| fnresin2 6649 | Restriction of a function'... |
| fnres 6650 | An equivalence for functio... |
| idfn 6651 | The identity relation is a... |
| fnresi 6652 | The restricted identity re... |
| fnima 6653 | The image of a function's ... |
| fn0 6654 | A function with empty doma... |
| fnimadisj 6655 | A class that is disjoint w... |
| fnimaeq0 6656 | Images under a function ne... |
| dfmpt3 6657 | Alternate definition for t... |
| mptfnf 6658 | The maps-to notation defin... |
| fnmptf 6659 | The maps-to notation defin... |
| fnopabg 6660 | Functionality and domain o... |
| fnopab 6661 | Functionality and domain o... |
| mptfng 6662 | The maps-to notation defin... |
| fnmpt 6663 | The maps-to notation defin... |
| fnmptd 6664 | The maps-to notation defin... |
| mpt0 6665 | A mapping operation with e... |
| fnmpti 6666 | Functionality and domain o... |
| dmmpti 6667 | Domain of the mapping oper... |
| dmmptd 6668 | The domain of the mapping ... |
| mptun 6669 | Union of mappings which ar... |
| partfun 6670 | Rewrite a function defined... |
| feq1 6671 | Equality theorem for funct... |
| feq2 6672 | Equality theorem for funct... |
| feq3 6673 | Equality theorem for funct... |
| feq23 6674 | Equality theorem for funct... |
| feq1d 6675 | Equality deduction for fun... |
| feq1dd 6676 | Equality deduction for fun... |
| feq2d 6677 | Equality deduction for fun... |
| feq3d 6678 | Equality deduction for fun... |
| feq2dd 6679 | Equality deduction for fun... |
| feq3dd 6680 | Equality deduction for fun... |
| feq12d 6681 | Equality deduction for fun... |
| feq123d 6682 | Equality deduction for fun... |
| feq123 6683 | Equality theorem for funct... |
| feq1i 6684 | Equality inference for fun... |
| feq2i 6685 | Equality inference for fun... |
| feq12i 6686 | Equality inference for fun... |
| feq23i 6687 | Equality inference for fun... |
| feq23d 6688 | Equality deduction for fun... |
| nff 6689 | Bound-variable hypothesis ... |
| sbcfng 6690 | Distribute proper substitu... |
| sbcfg 6691 | Distribute proper substitu... |
| elimf 6692 | Eliminate a mapping hypoth... |
| ffn 6693 | A mapping is a function wi... |
| ffnd 6694 | A mapping is a function wi... |
| dffn2 6695 | Any function is a mapping ... |
| ffun 6696 | A mapping is a function. ... |
| ffunOLD 6697 | Obsolete version of ~ ffun... |
| ffund 6698 | A mapping is a function, d... |
| frel 6699 | A mapping is a relation. ... |
| freld 6700 | A mapping is a relation. ... |
| frn 6701 | The range of a mapping. (... |
| frnd 6702 | Deduction form of ~ frn . ... |
| fdm 6703 | The domain of a mapping. ... |
| fdmd 6704 | Deduction form of ~ fdm . ... |
| fdmi 6705 | Inference associated with ... |
| dffn3 6706 | A function maps to its ran... |
| ffrn 6707 | A function maps to its ran... |
| ffrnb 6708 | Characterization of a func... |
| ffrnbd 6709 | A function maps to its ran... |
| fss 6710 | Expanding the codomain of ... |
| fssd 6711 | Expanding the codomain of ... |
| fssdmd 6712 | Expressing that a class is... |
| fssdm 6713 | Expressing that a class is... |
| fimass 6714 | The image of a class under... |
| fimassd 6715 | The image of a class is a ... |
| fimacnv 6716 | The preimage of the codoma... |
| fcof 6717 | Composition of a function ... |
| fco 6718 | Composition of two functio... |
| fcod 6719 | Composition of two mapping... |
| fco2 6720 | Functionality of a composi... |
| fssxp 6721 | A mapping is a class of or... |
| funssxp 6722 | Two ways of specifying a p... |
| ffdm 6723 | A mapping is a partial fun... |
| ffdmd 6724 | The domain of a function. ... |
| fdmrn 6725 | A different way to write `... |
| funcofd 6726 | Composition of two functio... |
| opelf 6727 | The members of an ordered ... |
| fun 6728 | The union of two functions... |
| fun2 6729 | The union of two functions... |
| fun2d 6730 | The union of functions wit... |
| fnfco 6731 | Composition of two functio... |
| fssres 6732 | Restriction of a function ... |
| fssresd 6733 | Restriction of a function ... |
| fssres2 6734 | Restriction of a restricte... |
| fresin 6735 | An identity for the mappin... |
| resasplit 6736 | If two functions agree on ... |
| fresaun 6737 | The union of two functions... |
| fresaunres2 6738 | From the union of two func... |
| fresaunres1 6739 | From the union of two func... |
| fcoi1 6740 | Composition of a mapping a... |
| fcoi2 6741 | Composition of restricted ... |
| feu 6742 | There is exactly one value... |
| fcnvres 6743 | The converse of a restrict... |
| fimacnvdisj 6744 | The preimage of a class di... |
| fint 6745 | Function into an intersect... |
| fin 6746 | Mapping into an intersecti... |
| f0 6747 | The empty function. (Cont... |
| f00 6748 | A class is a function with... |
| f0bi 6749 | A function with empty doma... |
| f0dom0 6750 | A function is empty iff it... |
| f0rn0 6751 | If there is no element in ... |
| fconst 6752 | A Cartesian product with a... |
| fconstg 6753 | A Cartesian product with a... |
| fnconstg 6754 | A Cartesian product with a... |
| fconst6g 6755 | Constant function with loo... |
| fconst6 6756 | A constant function as a m... |
| f1eq1 6757 | Equality theorem for one-t... |
| f1eq2 6758 | Equality theorem for one-t... |
| f1eq3 6759 | Equality theorem for one-t... |
| nff1 6760 | Bound-variable hypothesis ... |
| dff12 6761 | Alternate definition of a ... |
| f1f 6762 | A one-to-one mapping is a ... |
| f1fn 6763 | A one-to-one mapping is a ... |
| f1fun 6764 | A one-to-one mapping is a ... |
| f1funOLD 6765 | Obsolete version of ~ f1fu... |
| f1rel 6766 | A one-to-one onto mapping ... |
| f1relOLD 6767 | Obsolete version of ~ f1re... |
| f1dm 6768 | The domain of a one-to-one... |
| f1ss 6769 | A function that is one-to-... |
| f1ssr 6770 | A function that is one-to-... |
| f1ssres 6771 | A function that is one-to-... |
| f1resf1 6772 | The restriction of an inje... |
| f1cnvcnv 6773 | Two ways to express that a... |
| f1cof1 6774 | Composition of two one-to-... |
| f1co 6775 | Composition of one-to-one ... |
| foeq1 6776 | Equality theorem for onto ... |
| foeq2 6777 | Equality theorem for onto ... |
| foeq3 6778 | Equality theorem for onto ... |
| nffo 6779 | Bound-variable hypothesis ... |
| fof 6780 | An onto mapping is a mappi... |
| fofun 6781 | An onto mapping is a funct... |
| fofn 6782 | An onto mapping is a funct... |
| forn 6783 | The codomain of an onto fu... |
| dffo2 6784 | Alternate definition of an... |
| foima 6785 | The image of the domain of... |
| dffn4 6786 | A function maps onto its r... |
| funforn 6787 | A function maps its domain... |
| fodmrnu 6788 | An onto function has uniqu... |
| fimadmfo 6789 | A function is a function o... |
| fores 6790 | Restriction of an onto fun... |
| fimadmfoALT 6791 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6792 | The preimage of the codoma... |
| focofo 6793 | Composition of onto functi... |
| foco 6794 | Composition of onto functi... |
| foconst 6795 | A nonzero constant functio... |
| f1oeq1 6796 | Equality theorem for one-t... |
| f1oeq2 6797 | Equality theorem for one-t... |
| f1oeq3 6798 | Equality theorem for one-t... |
| f1oeq23 6799 | Equality theorem for one-t... |
| f1eq123d 6800 | Equality deduction for one... |
| foeq123d 6801 | Equality deduction for ont... |
| f1oeq123d 6802 | Equality deduction for one... |
| f1oeq1d 6803 | Equality deduction for one... |
| f1oeq2d 6804 | Equality deduction for one... |
| f1oeq3d 6805 | Equality deduction for one... |
| nff1o 6806 | Bound-variable hypothesis ... |
| f1of1 6807 | A one-to-one onto mapping ... |
| f1of 6808 | A one-to-one onto mapping ... |
| f1ofn 6809 | A one-to-one onto mapping ... |
| f1ofun 6810 | A one-to-one onto mapping ... |
| f1orel 6811 | A one-to-one onto mapping ... |
| f1odm 6812 | The domain of a one-to-one... |
| f1odmOLD 6813 | Obsolete version of ~ f1od... |
| dff1o2 6814 | Alternate definition of on... |
| dff1o3 6815 | Alternate definition of on... |
| f1ofo 6816 | A one-to-one onto function... |
| dff1o4 6817 | Alternate definition of on... |
| dff1o5 6818 | Alternate definition of on... |
| f1orn 6819 | A one-to-one function maps... |
| f1f1orn 6820 | A one-to-one function maps... |
| f1ocnv 6821 | The converse of a one-to-o... |
| f1ocnvb 6822 | A relation is a one-to-one... |
| f1ores 6823 | The restriction of a one-t... |
| f1orescnv 6824 | The converse of a one-to-o... |
| f1imacnv 6825 | Preimage of an image. (Co... |
| foimacnv 6826 | A reverse version of ~ f1i... |
| foun 6827 | The union of two onto func... |
| f1oun 6828 | The union of two one-to-on... |
| f1un 6829 | The union of two one-to-on... |
| resdif 6830 | The restriction of a one-t... |
| resin 6831 | The restriction of a one-t... |
| f1oco 6832 | Composition of one-to-one ... |
| f1cnv 6833 | The converse of an injecti... |
| funcocnv2 6834 | Composition with the conve... |
| fococnv2 6835 | The composition of an onto... |
| f1ococnv2 6836 | The composition of a one-t... |
| f1cocnv2 6837 | Composition of an injectiv... |
| f1ococnv1 6838 | The composition of a one-t... |
| f1cocnv1 6839 | Composition of an injectiv... |
| funcoeqres 6840 | Express a constraint on a ... |
| f1ssf1 6841 | A subset of an injective f... |
| f10 6842 | The empty set maps one-to-... |
| f10d 6843 | The empty set maps one-to-... |
| f1o00 6844 | One-to-one onto mapping of... |
| fo00 6845 | Onto mapping of the empty ... |
| f1o0 6846 | One-to-one onto mapping of... |
| f1oi 6847 | A restriction of the ident... |
| f1oiOLD 6848 | Obsolete version of ~ f1oi... |
| f1ovi 6849 | The identity relation is a... |
| f1osn 6850 | A singleton of an ordered ... |
| f1osng 6851 | A singleton of an ordered ... |
| f1sng 6852 | A singleton of an ordered ... |
| fsnd 6853 | A singleton of an ordered ... |
| f1oprswap 6854 | A two-element swap is a bi... |
| f1oprg 6855 | An unordered pair of order... |
| tz6.12-2 6856 | Function value when ` F ` ... |
| tz6.12-2OLD 6857 | Obsolete version of ~ tz6.... |
| fveu 6858 | The value of a function at... |
| brprcneu 6859 | If ` A ` is a proper class... |
| brprcneuALT 6860 | Alternate proof of ~ brprc... |
| fvprc 6861 | A function's value at a pr... |
| fvprcALT 6862 | Alternate proof of ~ fvprc... |
| rnfvprc 6863 | The range of a function va... |
| fv2 6864 | Alternate definition of fu... |
| dffv3 6865 | A definition of function v... |
| dffv4 6866 | The previous definition of... |
| elfv 6867 | Membership in a function v... |
| fveq1 6868 | Equality theorem for funct... |
| fveq2 6869 | Equality theorem for funct... |
| fveq1i 6870 | Equality inference for fun... |
| fveq1d 6871 | Equality deduction for fun... |
| fveq2i 6872 | Equality inference for fun... |
| fveq2d 6873 | Equality deduction for fun... |
| 2fveq3 6874 | Equality theorem for neste... |
| fveq12i 6875 | Equality deduction for fun... |
| fveq12d 6876 | Equality deduction for fun... |
| fveqeq2d 6877 | Equality deduction for fun... |
| fveqeq2 6878 | Equality deduction for fun... |
| nffv 6879 | Bound-variable hypothesis ... |
| nffvmpt1 6880 | Bound-variable hypothesis ... |
| nffvd 6881 | Deduction version of bound... |
| fvex 6882 | The value of a class exist... |
| fvexi 6883 | The value of a class exist... |
| fvexd 6884 | The value of a class exist... |
| fvif 6885 | Move a conditional outside... |
| iffv 6886 | Move a conditional outside... |
| fv3 6887 | Alternate definition of th... |
| fvres 6888 | The value of a restricted ... |
| fvresd 6889 | The value of a restricted ... |
| funssfv 6890 | The value of a member of t... |
| tz6.12c 6891 | Corollary of Theorem 6.12(... |
| tz6.12-1 6892 | Function value. Theorem 6... |
| tz6.12 6893 | Function value. Theorem 6... |
| tz6.12f 6894 | Function value, using boun... |
| tz6.12i 6895 | Corollary of Theorem 6.12(... |
| fvbr0 6896 | Two possibilities for the ... |
| fvrn0 6897 | A function value is a memb... |
| fvn0fvelrn 6898 | If the value of a function... |
| elfvunirn 6899 | A function value is a subs... |
| fvssunirn 6900 | The result of a function v... |
| ndmfv 6901 | The value of a class outsi... |
| ndmfvrcl 6902 | Reverse closure law for fu... |
| elfvdm 6903 | If a function value has a ... |
| elfvex 6904 | If a function value has a ... |
| elfvexd 6905 | If a function value has a ... |
| eliman0 6906 | A nonempty function value ... |
| nfvres 6907 | The value of a non-member ... |
| nfunsn 6908 | If the restriction of a cl... |
| fvfundmfvn0 6909 | If the "value of a class" ... |
| 0fv 6910 | Function value of the empt... |
| fv2prc 6911 | A function value of a func... |
| elfv2ex 6912 | If a function value of a f... |
| fveqres 6913 | Equal values imply equal v... |
| csbfv12 6914 | Move class substitution in... |
| csbfv2g 6915 | Move class substitution in... |
| csbfv 6916 | Substitution for a functio... |
| funbrfv 6917 | The second argument of a b... |
| funopfv 6918 | The second element in an o... |
| fnbrfvb 6919 | Equivalence of function va... |
| fnopfvb 6920 | Equivalence of function va... |
| fvelima2 6921 | Function value in an image... |
| funbrfvb 6922 | Equivalence of function va... |
| funopfvb 6923 | Equivalence of function va... |
| fnbrfvb2 6924 | Version of ~ fnbrfvb for f... |
| fdmeu 6925 | There is exactly one codom... |
| funbrfv2b 6926 | Function value in terms of... |
| dffn5 6927 | Representation of a functi... |
| fnrnfv 6928 | The range of a function ex... |
| fvelrnb 6929 | A member of a function's r... |
| foelcdmi 6930 | A member of a surjective f... |
| dfimafn 6931 | Alternate definition of th... |
| dfimafn2 6932 | Alternate definition of th... |
| funimass4 6933 | Membership relation for th... |
| fvelima 6934 | Function value in an image... |
| funimassd 6935 | Sufficient condition for t... |
| fvelimad 6936 | Function value in an image... |
| feqmptd 6937 | Deduction form of ~ dffn5 ... |
| feqresmpt 6938 | Express a restricted funct... |
| feqmptdf 6939 | Deduction form of ~ dffn5f... |
| dffn5f 6940 | Representation of a functi... |
| fvelimab 6941 | Function value in an image... |
| fvelimabd 6942 | Deduction form of ~ fvelim... |
| fimarab 6943 | Expressing the image of a ... |
| unima 6944 | Image of a union. (Contri... |
| fvi 6945 | The value of the identity ... |
| fviss 6946 | The value of the identity ... |
| fniinfv 6947 | The indexed intersection o... |
| fnsnfv 6948 | Singleton of function valu... |
| opabiotafun 6949 | Define a function whose va... |
| opabiotadm 6950 | Define a function whose va... |
| opabiota 6951 | Define a function whose va... |
| fnimapr 6952 | The image of a pair under ... |
| fnimatpd 6953 | The image of an unordered ... |
| ssimaex 6954 | The existence of a subimag... |
| ssimaexg 6955 | The existence of a subimag... |
| funfv 6956 | A simplified expression fo... |
| funfv2 6957 | The value of a function. ... |
| funfv2f 6958 | The value of a function. ... |
| fvun 6959 | Value of the union of two ... |
| fvun1 6960 | The value of a union when ... |
| fvun2 6961 | The value of a union when ... |
| fvun1d 6962 | The value of a union when ... |
| fvun2d 6963 | The value of a union when ... |
| dffv2 6964 | Alternate definition of fu... |
| dmfco 6965 | Domains of a function comp... |
| fvco2 6966 | Value of a function compos... |
| fvco 6967 | Value of a function compos... |
| fvcod 6968 | Value of a function compos... |
| fvco3 6969 | Value of a function compos... |
| fvco3d 6970 | Value of a function compos... |
| fvco4i 6971 | Conditions for a compositi... |
| fvopab3g 6972 | Value of a function given ... |
| fvopab3ig 6973 | Value of a function given ... |
| brfvopabrbr 6974 | The binary relation of a f... |
| fvmptg 6975 | Value of a function given ... |
| fvmpti 6976 | Value of a function given ... |
| fvmpt 6977 | Value of a function given ... |
| fvmpt2f 6978 | Value of a function given ... |
| funcnvmpt 6979 | Condition for a function i... |
| fvtresfn 6980 | Functionality of a tuple-r... |
| fvmpts 6981 | Value of a function given ... |
| fvmpt3 6982 | Value of a function given ... |
| fvmpt3i 6983 | Value of a function given ... |
| fvmptdf 6984 | Deduction version of ~ fvm... |
| fvmptd 6985 | Deduction version of ~ fvm... |
| fvmptd2 6986 | Deduction version of ~ fvm... |
| mptrcl 6987 | Reverse closure for a mapp... |
| fvmpt2i 6988 | Value of a function given ... |
| fvmpt2 6989 | Value of a function given ... |
| fvmptss 6990 | If all the values of the m... |
| fvmpt2d 6991 | Deduction version of ~ fvm... |
| fvmptex 6992 | Express a function ` F ` w... |
| fvmptd3f 6993 | Alternate deduction versio... |
| fvmptd2f 6994 | Alternate deduction versio... |
| fvmptdv 6995 | Alternate deduction versio... |
| fvmptdv2 6996 | Alternate deduction versio... |
| mpteqb 6997 | Bidirectional equality the... |
| fvmptt 6998 | Closed theorem form of ~ f... |
| fvmptf 6999 | Value of a function given ... |
| fvmptnf 7000 | The value of a function gi... |
| fvmptd3 7001 | Deduction version of ~ fvm... |
| fvmptd4 7002 | Deduction version of ~ fvm... |
| fvmptn 7003 | This somewhat non-intuitiv... |
| fvmptss2 7004 | A mapping always evaluates... |
| elfvmptrab1w 7005 | Implications for the value... |
| elfvmptrab1 7006 | Implications for the value... |
| elfvmptrab 7007 | Implications for the value... |
| fvopab4ndm 7008 | Value of a function given ... |
| fvmptndm 7009 | Value of a function given ... |
| fvmptrabfv 7010 | Value of a function mappin... |
| fvopab5 7011 | The value of a function th... |
| fvopab6 7012 | Value of a function given ... |
| eqfnfv 7013 | Equality of functions is d... |
| eqfnfv2 7014 | Equality of functions is d... |
| eqfnfv3 7015 | Derive equality of functio... |
| eqfnfvd 7016 | Deduction for equality of ... |
| eqfnfv2f 7017 | Equality of functions is d... |
| fsneq 7018 | Equality condition for two... |
| eqfunfv 7019 | Equality of functions is d... |
| eqfnun 7020 | Two functions on ` A u. B ... |
| fvreseq0 7021 | Equality of restricted fun... |
| fvreseq1 7022 | Equality of a function res... |
| fvreseq 7023 | Equality of restricted fun... |
| fnmptfvd 7024 | A function with a given do... |
| fndmdif 7025 | Two ways to express the lo... |
| fndmdifcom 7026 | The difference set between... |
| fndmdifeq0 7027 | The difference set of two ... |
| fndmin 7028 | Two ways to express the lo... |
| fneqeql 7029 | Two functions are equal if... |
| fneqeql2 7030 | Two functions are equal if... |
| fnreseql 7031 | Two functions are equal on... |
| chfnrn 7032 | The range of a choice func... |
| funfvop 7033 | Ordered pair with function... |
| funfvbrb 7034 | Two ways to say that ` A `... |
| fvimacnvi 7035 | A member of a preimage is ... |
| fvimacnv 7036 | The argument of a function... |
| funimass3 7037 | A kind of contraposition l... |
| funimass5 7038 | A subclass of a preimage i... |
| funconstss 7039 | Two ways of specifying tha... |
| fvimacnvALT 7040 | Alternate proof of ~ fvima... |
| elpreima 7041 | Membership in the preimage... |
| elpreimad 7042 | Membership in the preimage... |
| fniniseg 7043 | Membership in the preimage... |
| fncnvima2 7044 | Inverse images under funct... |
| fniniseg2 7045 | Inverse point images under... |
| unpreima 7046 | Preimage of a union. (Con... |
| inpreima 7047 | Preimage of an intersectio... |
| difpreima 7048 | Preimage of a difference. ... |
| respreima 7049 | The preimage of a restrict... |
| cnvimainrn 7050 | The preimage of the inters... |
| sspreima 7051 | The preimage of a subset i... |
| iinpreima 7052 | Preimage of an intersectio... |
| intpreima 7053 | Preimage of an intersectio... |
| fimacnvinrn 7054 | Taking the converse image ... |
| fimacnvinrn2 7055 | Taking the converse image ... |
| rescnvimafod 7056 | The restriction of a funct... |
| fvn0ssdmfun 7057 | If a class' function value... |
| fnopfv 7058 | Ordered pair with function... |
| fvelrn 7059 | A function's value belongs... |
| nelrnfvne 7060 | A function value cannot be... |
| fveqdmss 7061 | If the empty set is not co... |
| fveqressseq 7062 | If the empty set is not co... |
| fnfvelrn 7063 | A function's value belongs... |
| ffvelcdm 7064 | A function's value belongs... |
| fnfvelrnd 7065 | A function's value belongs... |
| ffvelcdmi 7066 | A function's value belongs... |
| ffvelcdmda 7067 | A function's value belongs... |
| ffvelcdmd 7068 | A function's value belongs... |
| feldmfvelcdm 7069 | A class is an element of t... |
| rexrn 7070 | Restricted existential qua... |
| ralrn 7071 | Restricted universal quant... |
| elrnrexdm 7072 | For any element in the ran... |
| elrnrexdmb 7073 | For any element in the ran... |
| eldmrexrn 7074 | For any element in the dom... |
| eldmrexrnb 7075 | For any element in the dom... |
| fvcofneq 7076 | The values of two function... |
| ralrnmptw 7077 | A restricted quantifier ov... |
| rexrnmptw 7078 | A restricted quantifier ov... |
| ralrnmpt 7079 | A restricted quantifier ov... |
| rexrnmpt 7080 | A restricted quantifier ov... |
| f0cli 7081 | Unconditional closure of a... |
| dff2 7082 | Alternate definition of a ... |
| dff3 7083 | Alternate definition of a ... |
| dff4 7084 | Alternate definition of a ... |
| dffo3 7085 | An onto mapping expressed ... |
| dffo4 7086 | Alternate definition of an... |
| dffo5 7087 | Alternate definition of an... |
| exfo 7088 | A relation equivalent to t... |
| dffo3f 7089 | An onto mapping expressed ... |
| foelrn 7090 | Property of a surjective f... |
| foelrnf 7091 | Property of a surjective f... |
| foco2 7092 | If a composition of two fu... |
| fmpt 7093 | Functionality of the mappi... |
| f1ompt 7094 | Express bijection for a ma... |
| fmpti 7095 | Functionality of the mappi... |
| fvmptelcdm 7096 | The value of a function at... |
| fmptd 7097 | Domain and codomain of the... |
| fmpttd 7098 | Version of ~ fmptd with in... |
| fmpt3d 7099 | Domain and codomain of the... |
| fmptdf 7100 | A version of ~ fmptd using... |
| fompt 7101 | Express being onto for a m... |
| ffnfv 7102 | A function maps to a class... |
| ffnfvf 7103 | A function maps to a class... |
| fnfvrnss 7104 | An upper bound for range d... |
| fcdmssb 7105 | A function is a function i... |
| rnmptss 7106 | The range of an operation ... |
| rnmptssd 7107 | The range of a function gi... |
| fmpt2d 7108 | Domain and codomain of the... |
| ffvresb 7109 | A necessary and sufficient... |
| fssrescdmd 7110 | Restriction of a function ... |
| f1oresrab 7111 | Build a bijection between ... |
| f1ossf1o 7112 | Restricting a bijection, w... |
| fmptco 7113 | Composition of two functio... |
| fmptcof 7114 | Version of ~ fmptco where ... |
| fmptcos 7115 | Composition of two functio... |
| cofmpt 7116 | Express composition of a m... |
| fcompt 7117 | Express composition of two... |
| fcoconst 7118 | Composition with a constan... |
| fsn 7119 | A function maps a singleto... |
| fsn2 7120 | A function that maps a sin... |
| fsng 7121 | A function maps a singleto... |
| fsn2g 7122 | A function that maps a sin... |
| xpsng 7123 | The Cartesian product of t... |
| xpprsng 7124 | The Cartesian product of a... |
| xpsn 7125 | The Cartesian product of t... |
| f1o2sn 7126 | A singleton consisting in ... |
| residpr 7127 | Restriction of the identit... |
| dfmpt 7128 | Alternate definition for t... |
| fnasrn 7129 | A function expressed as th... |
| idref 7130 | Two ways to state that a r... |
| funiun 7131 | A function is a union of s... |
| funopsn 7132 | If a function is an ordere... |
| funopsnOLD 7133 | Obsolete version of ~ funo... |
| funop 7134 | An ordered pair is a funct... |
| funopdmsn 7135 | The domain of a function w... |
| funsndifnop 7136 | A singleton of an ordered ... |
| funsneqopb 7137 | A singleton of an ordered ... |
| ressnop0 7138 | If ` A ` is not in ` C ` ,... |
| fpr 7139 | A function with a domain o... |
| fprg 7140 | A function with a domain o... |
| ftpg 7141 | A function with a domain o... |
| ftp 7142 | A function with a domain o... |
| fnressn 7143 | A function restricted to a... |
| funressn 7144 | A function restricted to a... |
| fressnfv 7145 | The value of a function re... |
| fvrnressn 7146 | If the value of a function... |
| fvressn 7147 | The value of a function re... |
| fvconst 7148 | The value of a constant fu... |
| fnsnr 7149 | If a class belongs to a fu... |
| fnsnbg 7150 | A function's domain is a s... |
| fnsnb 7151 | A function whose domain is... |
| fnsnbOLD 7152 | Obsolete version of ~ fnsn... |
| fmptsn 7153 | Express a singleton functi... |
| fmptsng 7154 | Express a singleton functi... |
| fmptsnd 7155 | Express a singleton functi... |
| fmptap 7156 | Append an additional value... |
| fmptapd 7157 | Append an additional value... |
| fmptpr 7158 | Express a pair function in... |
| fvresi 7159 | The value of a restricted ... |
| fninfp 7160 | Express the class of fixed... |
| fnelfp 7161 | Property of a fixed point ... |
| fndifnfp 7162 | Express the class of non-f... |
| fnelnfp 7163 | Property of a non-fixed po... |
| fnnfpeq0 7164 | A function is the identity... |
| fvunsn 7165 | Remove an ordered pair not... |
| fvsng 7166 | The value of a singleton o... |
| fvsn 7167 | The value of a singleton o... |
| fvsnun1 7168 | The value of a function wi... |
| fvsnun2 7169 | The value of a function wi... |
| fnsnsplit 7170 | Split a function into a si... |
| fsnunf 7171 | Adjoining a point to a fun... |
| fsnunf2 7172 | Adjoining a point to a pun... |
| fsnunfv 7173 | Recover the added point fr... |
| fsnunres 7174 | Recover the original funct... |
| funresdfunsn 7175 | Restricting a function to ... |
| fvpr1g 7176 | The value of a function wi... |
| fvpr2g 7177 | The value of a function wi... |
| fvpr1 7178 | The value of a function wi... |
| fvpr2 7179 | The value of a function wi... |
| fprb 7180 | A condition for functionho... |
| fvtp1 7181 | The first value of a funct... |
| fvtp2 7182 | The second value of a func... |
| fvtp3 7183 | The third value of a funct... |
| fvtp1g 7184 | The value of a function wi... |
| fvtp2g 7185 | The value of a function wi... |
| fvtp3g 7186 | The value of a function wi... |
| tpres 7187 | An unordered triple of ord... |
| fvconst2g 7188 | The value of a constant fu... |
| fconst2g 7189 | A constant function expres... |
| fvconst2 7190 | The value of a constant fu... |
| fconst2 7191 | A constant function expres... |
| fconst5 7192 | Two ways to express that a... |
| rnmptc 7193 | Range of a constant functi... |
| fnprb 7194 | A function whose domain ha... |
| fntpb 7195 | A function whose domain ha... |
| fnpr2g 7196 | A function whose domain ha... |
| fpr2g 7197 | A function that maps a pai... |
| fconstfv 7198 | A constant function expres... |
| fconst3 7199 | Two ways to express a cons... |
| fconst4 7200 | Two ways to express a cons... |
| resfunexg 7201 | The restriction of a funct... |
| resiexd 7202 | The restriction of the ide... |
| fnex 7203 | If the domain of a functio... |
| fnexd 7204 | If the domain of a functio... |
| funex 7205 | If the domain of a functio... |
| opabex 7206 | Existence of a function ex... |
| mptexg 7207 | If the domain of a functio... |
| mptexgf 7208 | If the domain of a functio... |
| mptex 7209 | If the domain of a functio... |
| mptexd 7210 | If the domain of a functio... |
| mptrabex 7211 | If the domain of a functio... |
| fex 7212 | If the domain of a mapping... |
| fexd 7213 | If the domain of a mapping... |
| mptfvmpt 7214 | A function in maps-to nota... |
| eufnfv 7215 | A function is uniquely det... |
| funfvima 7216 | A function's value in a pr... |
| funfvima2 7217 | A function's value in an i... |
| funfvima2d 7218 | A function's value in a pr... |
| fnfvima 7219 | The function value of an o... |
| fnfvimad 7220 | A function's value belongs... |
| resfvresima 7221 | The value of the function ... |
| funfvima3 7222 | A class including a functi... |
| ralima 7223 | Universal quantification u... |
| rexima 7224 | Existential quantification... |
| reximaOLD 7225 | Obsolete version of ~ rexi... |
| ralimaOLD 7226 | Obsolete version of ~ rali... |
| fvclss 7227 | Upper bound for the class ... |
| elabrex 7228 | Elementhood in an image se... |
| elabrexg 7229 | Elementhood in an image se... |
| abrexco 7230 | Composition of two image m... |
| imaiun 7231 | The image of an indexed un... |
| imauni 7232 | The image of a union is th... |
| fniunfv 7233 | The indexed union of a fun... |
| funiunfv 7234 | The indexed union of a fun... |
| funiunfvf 7235 | The indexed union of a fun... |
| eluniima 7236 | Membership in the union of... |
| elunirn 7237 | Membership in the union of... |
| elunirnALT 7238 | Alternate proof of ~ eluni... |
| fnunirn 7239 | Membership in a union of s... |
| dff13 7240 | A one-to-one function in t... |
| dff13f 7241 | A one-to-one function in t... |
| f1veqaeq 7242 | If the values of a one-to-... |
| f1cofveqaeq 7243 | If the values of a composi... |
| f1cofveqaeqALT 7244 | Alternate proof of ~ f1cof... |
| dff14i 7245 | A one-to-one function maps... |
| 2f1fvneq 7246 | If two one-to-one function... |
| f1mpt 7247 | Express injection for a ma... |
| f1fveq 7248 | Equality of function value... |
| f1elima 7249 | Membership in the image of... |
| f1imass 7250 | Taking images under a one-... |
| f1imaeq 7251 | Taking images under a one-... |
| f1imapss 7252 | Taking images under a one-... |
| fpropnf1 7253 | A function, given by an un... |
| f1dom3fv3dif 7254 | The function values for a ... |
| f1dom3el3dif 7255 | The codomain of a 1-1 func... |
| dff14a 7256 | A one-to-one function in t... |
| dff14b 7257 | A one-to-one function in t... |
| f1ounsn 7258 | Extension of a bijection b... |
| f12dfv 7259 | A one-to-one function with... |
| f13dfv 7260 | A one-to-one function with... |
| dff1o6 7261 | A one-to-one onto function... |
| f1ocnvfv1 7262 | The converse value of the ... |
| f1ocnvfv2 7263 | The value of the converse ... |
| f1ocnvfv 7264 | Relationship between the v... |
| f1ocnvfvb 7265 | Relationship between the v... |
| nvof1o 7266 | An involution is a bijecti... |
| nvocnv 7267 | The converse of an involut... |
| f1cdmsn 7268 | If a one-to-one function w... |
| fsnex 7269 | Relate a function with a s... |
| f1prex 7270 | Relate a one-to-one functi... |
| f1ocnvdm 7271 | The value of the converse ... |
| f1ocnvfvrneq 7272 | If the values of a one-to-... |
| fcof1 7273 | An application is injectiv... |
| fcofo 7274 | An application is surjecti... |
| cbvfo 7275 | Change bound variable betw... |
| cbvexfo 7276 | Change bound variable betw... |
| cocan1 7277 | An injection is left-cance... |
| cocan2 7278 | A surjection is right-canc... |
| fcof1oinvd 7279 | Show that a function is th... |
| fcof1od 7280 | A function is bijective if... |
| 2fcoidinvd 7281 | Show that a function is th... |
| fcof1o 7282 | Show that two functions ar... |
| 2fvcoidd 7283 | Show that the composition ... |
| 2fvidf1od 7284 | A function is bijective if... |
| 2fvidinvd 7285 | Show that two functions ar... |
| foeqcnvco 7286 | Condition for function equ... |
| f1eqcocnv 7287 | Condition for function equ... |
| fveqf1o 7288 | Given a bijection ` F ` , ... |
| f1ocoima 7289 | The composition of two bij... |
| nf1const 7290 | A constant function from a... |
| nf1oconst 7291 | A constant function from a... |
| f1ofvswap 7292 | Swapping two values in a b... |
| fvf1pr 7293 | Values of a one-to-one fun... |
| fliftrel 7294 | ` F ` , a function lift, i... |
| fliftel 7295 | Elementhood in the relatio... |
| fliftel1 7296 | Elementhood in the relatio... |
| fliftcnv 7297 | Converse of the relation `... |
| fliftfun 7298 | The function ` F ` is the ... |
| fliftfund 7299 | The function ` F ` is the ... |
| fliftfuns 7300 | The function ` F ` is the ... |
| fliftf 7301 | The domain and range of th... |
| fliftval 7302 | The value of the function ... |
| isoeq1 7303 | Equality theorem for isomo... |
| isoeq2 7304 | Equality theorem for isomo... |
| isoeq3 7305 | Equality theorem for isomo... |
| isoeq4 7306 | Equality theorem for isomo... |
| isoeq5 7307 | Equality theorem for isomo... |
| nfiso 7308 | Bound-variable hypothesis ... |
| isof1o 7309 | An isomorphism is a one-to... |
| isof1oidb 7310 | A function is a bijection ... |
| isof1oopb 7311 | A function is a bijection ... |
| isorel 7312 | An isomorphism connects bi... |
| soisores 7313 | Express the condition of i... |
| soisoi 7314 | Infer isomorphism from one... |
| isoid 7315 | Identity law for isomorphi... |
| isocnv 7316 | Converse law for isomorphi... |
| isocnv2 7317 | Converse law for isomorphi... |
| isocnv3 7318 | Complementation law for is... |
| isores2 7319 | An isomorphism from one we... |
| isores1 7320 | An isomorphism from one we... |
| isores3 7321 | Induced isomorphism on a s... |
| isotr 7322 | Composition (transitive) l... |
| isomin 7323 | Isomorphisms preserve mini... |
| isoini 7324 | Isomorphisms preserve init... |
| isoini2 7325 | Isomorphisms are isomorphi... |
| isofrlem 7326 | Lemma for ~ isofr . (Cont... |
| isoselem 7327 | Lemma for ~ isose . (Cont... |
| isofr 7328 | An isomorphism preserves w... |
| isose 7329 | An isomorphism preserves s... |
| isofr2 7330 | A weak form of ~ isofr tha... |
| isopolem 7331 | Lemma for ~ isopo . (Cont... |
| isopo 7332 | An isomorphism preserves t... |
| isosolem 7333 | Lemma for ~ isoso . (Cont... |
| isoso 7334 | An isomorphism preserves t... |
| isowe 7335 | An isomorphism preserves t... |
| isowe2 7336 | A weak form of ~ isowe tha... |
| f1oiso 7337 | Any one-to-one onto functi... |
| f1oiso2 7338 | Any one-to-one onto functi... |
| f1owe 7339 | Well-ordering of isomorphi... |
| weniso 7340 | A set-like well-ordering h... |
| weisoeq 7341 | Thus, there is at most one... |
| weisoeq2 7342 | Thus, there is at most one... |
| knatar 7343 | The Knaster-Tarski theorem... |
| fvresval 7344 | The value of a restricted ... |
| funeldmb 7345 | If ` (/) ` is not part of ... |
| eqfunresadj 7346 | Law for adjoining an eleme... |
| eqfunressuc 7347 | Law for equality of restri... |
| fnssintima 7348 | Condition for subset of an... |
| imaeqsexvOLD 7349 | Obsolete version of ~ rexi... |
| imaeqsalvOLD 7350 | Obsolete version of ~ rali... |
| fnimasnd 7351 | The image of a function by... |
| canth 7352 | No set ` A ` is equinumero... |
| ncanth 7353 | Cantor's theorem fails for... |
| riotaeqdv 7356 | Formula-building deduction... |
| riotabidv 7357 | Formula-building deduction... |
| riotaeqbidv 7358 | Equality deduction for res... |
| riotaex 7359 | Restricted iota is a set. ... |
| riotav 7360 | An iota restricted to the ... |
| riotauni 7361 | Restricted iota in terms o... |
| nfriota1 7362 | The abstraction variable i... |
| nfriotadw 7363 | Deduction version of ~ nfr... |
| cbvriotaw 7364 | Change bound variable in a... |
| cbvriotavw 7365 | Change bound variable in a... |
| nfriotad 7366 | Deduction version of ~ nfr... |
| nfriota 7367 | A variable not free in a w... |
| cbvriota 7368 | Change bound variable in a... |
| cbvriotav 7369 | Change bound variable in a... |
| csbriota 7370 | Interchange class substitu... |
| riotacl2 7371 | Membership law for "the un... |
| riotacl 7372 | Closure of restricted iota... |
| riotasbc 7373 | Substitution law for descr... |
| riotabidva 7374 | Equivalent wff's yield equ... |
| riotabiia 7375 | Equivalent wff's yield equ... |
| riota1 7376 | Property of restricted iot... |
| riota1a 7377 | Property of iota. (Contri... |
| riota2df 7378 | A deduction version of ~ r... |
| riota2f 7379 | This theorem shows a condi... |
| riota2 7380 | This theorem shows a condi... |
| riotaeqimp 7381 | If two restricted iota des... |
| riotaprop 7382 | Properties of a restricted... |
| riota5f 7383 | A method for computing res... |
| riota5 7384 | A method for computing res... |
| riotass2 7385 | Restriction of a unique el... |
| riotass 7386 | Restriction of a unique el... |
| moriotass 7387 | Restriction of a unique el... |
| snriota 7388 | A restricted class abstrac... |
| riotaxfrd 7389 | Change the variable ` x ` ... |
| eusvobj2 7390 | Specify the same property ... |
| eusvobj1 7391 | Specify the same object in... |
| f1ofveu 7392 | There is one domain elemen... |
| f1ocnvfv3 7393 | Value of the converse of a... |
| riotaund 7394 | Restricted iota equals the... |
| riotassuni 7395 | The restricted iota class ... |
| riotaclb 7396 | Bidirectional closure of r... |
| riotarab 7397 | Restricted iota of a restr... |
| oveq 7404 | Equality theorem for opera... |
| oveq1 7405 | Equality theorem for opera... |
| oveq2 7406 | Equality theorem for opera... |
| oveq12 7407 | Equality theorem for opera... |
| oveq1i 7408 | Equality inference for ope... |
| oveq2i 7409 | Equality inference for ope... |
| oveq12i 7410 | Equality inference for ope... |
| oveqi 7411 | Equality inference for ope... |
| oveq123i 7412 | Equality inference for ope... |
| oveq1d 7413 | Equality deduction for ope... |
| oveq2d 7414 | Equality deduction for ope... |
| oveqd 7415 | Equality deduction for ope... |
| oveq12d 7416 | Equality deduction for ope... |
| oveqan12d 7417 | Equality deduction for ope... |
| oveqan12rd 7418 | Equality deduction for ope... |
| oveq123d 7419 | Equality deduction for ope... |
| fvoveq1d 7420 | Equality deduction for nes... |
| fvoveq1 7421 | Equality theorem for neste... |
| ovanraleqv 7422 | Equality theorem for a con... |
| imbrov2fvoveq 7423 | Equality theorem for neste... |
| ovrspc2v 7424 | If an operation value is a... |
| oveqrspc2v 7425 | Restricted specialization ... |
| oveqdr 7426 | Equality of two operations... |
| nfovd 7427 | Deduction version of bound... |
| nfov 7428 | Bound-variable hypothesis ... |
| oprabidw 7429 | The law of concretion. Sp... |
| oprabid 7430 | The law of concretion. Sp... |
| ovex 7431 | The result of an operation... |
| ovexi 7432 | The result of an operation... |
| ovexd 7433 | The result of an operation... |
| ovssunirn 7434 | The result of an operation... |
| 0ov 7435 | Operation value of the emp... |
| ovprc 7436 | The value of an operation ... |
| ovprc1 7437 | The value of an operation ... |
| ovprc2 7438 | The value of an operation ... |
| ovrcl 7439 | Reverse closure for an ope... |
| elfvov1 7440 | Utility theorem: reverse c... |
| elfvov2 7441 | Utility theorem: reverse c... |
| csbov123 7442 | Move class substitution in... |
| csbov 7443 | Move class substitution in... |
| csbov12g 7444 | Move class substitution in... |
| csbov1g 7445 | Move class substitution in... |
| csbov2g 7446 | Move class substitution in... |
| rspceov 7447 | A frequently used special ... |
| elovimad 7448 | Elementhood of the image s... |
| fnbrovb 7449 | Value of a binary operatio... |
| fnotovb 7450 | Equivalence of operation v... |
| opabbrex 7451 | A collection of ordered pa... |
| opabresex2 7452 | Restrictions of a collecti... |
| fvmptopab 7453 | The function value of a ma... |
| f1opr 7454 | Condition for an operation... |
| brfvopab 7455 | The classes involved in a ... |
| dfoprab2 7456 | Class abstraction for oper... |
| reloprab 7457 | An operation class abstrac... |
| oprabv 7458 | If a pair and a class are ... |
| nfoprab1 7459 | The abstraction variables ... |
| nfoprab2 7460 | The abstraction variables ... |
| nfoprab3 7461 | The abstraction variables ... |
| nfoprab 7462 | Bound-variable hypothesis ... |
| oprabbid 7463 | Equivalent wff's yield equ... |
| oprabbidv 7464 | Equivalent wff's yield equ... |
| oprabbii 7465 | Equivalent wff's yield equ... |
| ssoprab2 7466 | Equivalence of ordered pai... |
| ssoprab2b 7467 | Equivalence of ordered pai... |
| eqoprab2bw 7468 | Equivalence of ordered pai... |
| eqoprab2b 7469 | Equivalence of ordered pai... |
| mpoeq123 7470 | An equality theorem for th... |
| mpoeq12 7471 | An equality theorem for th... |
| mpoeq123dva 7472 | An equality deduction for ... |
| mpoeq123dv 7473 | An equality deduction for ... |
| mpoeq123i 7474 | An equality inference for ... |
| mpoeq3dva 7475 | Slightly more general equa... |
| mpoeq3ia 7476 | An equality inference for ... |
| mpoeq3dv 7477 | An equality deduction for ... |
| nfmpo1 7478 | Bound-variable hypothesis ... |
| nfmpo2 7479 | Bound-variable hypothesis ... |
| nfmpo 7480 | Bound-variable hypothesis ... |
| 0mpo0 7481 | A mapping operation with e... |
| mpo0v 7482 | A mapping operation with e... |
| mpo0 7483 | A mapping operation with e... |
| oprab4 7484 | Two ways to state the doma... |
| cbvoprab1 7485 | Rule used to change first ... |
| cbvoprab2 7486 | Change the second bound va... |
| cbvoprab12 7487 | Rule used to change first ... |
| cbvoprab12v 7488 | Rule used to change first ... |
| cbvoprab3 7489 | Rule used to change the th... |
| cbvoprab3v 7490 | Rule used to change the th... |
| cbvmpox 7491 | Rule to change the bound v... |
| cbvmpo 7492 | Rule to change the bound v... |
| cbvmpov 7493 | Rule to change the bound v... |
| elimdelov 7494 | Eliminate a hypothesis whi... |
| brif1 7495 | Move a relation inside and... |
| ovif 7496 | Move a conditional outside... |
| ovif2 7497 | Move a conditional outside... |
| ovif12 7498 | Move a conditional outside... |
| ifov 7499 | Move a conditional outside... |
| ifmpt2v 7500 | Move a conditional inside ... |
| dmoprab 7501 | The domain of an operation... |
| dmoprabss 7502 | The domain of an operation... |
| rnoprab 7503 | The range of an operation ... |
| rnoprab2 7504 | The range of a restricted ... |
| reldmoprab 7505 | The domain of an operation... |
| oprabss 7506 | Structure of an operation ... |
| eloprabga 7507 | The law of concretion for ... |
| eloprabg 7508 | The law of concretion for ... |
| ssoprab2i 7509 | Inference of operation cla... |
| mpov 7510 | Operation with universal d... |
| mpomptx 7511 | Express a two-argument fun... |
| mpompt 7512 | Express a two-argument fun... |
| mpodifsnif 7513 | A mapping with two argumen... |
| mposnif 7514 | A mapping with two argumen... |
| fconstmpo 7515 | Representation of a consta... |
| resoprab 7516 | Restriction of an operatio... |
| resoprab2 7517 | Restriction of an operator... |
| resmpo 7518 | Restriction of the mapping... |
| funoprabg 7519 | "At most one" is a suffici... |
| funoprab 7520 | "At most one" is a suffici... |
| fnoprabg 7521 | Functionality and domain o... |
| mpofun 7522 | The maps-to notation for a... |
| fnoprab 7523 | Functionality and domain o... |
| ffnov 7524 | An operation maps to a cla... |
| fovcld 7525 | Closure law for an operati... |
| fovcl 7526 | Closure law for an operati... |
| eqfnov 7527 | Equality of two operations... |
| eqfnov2 7528 | Two operators with the sam... |
| fnov 7529 | Representation of a functi... |
| mpo2eqb 7530 | Bidirectional equality the... |
| rnmpo 7531 | The range of an operation ... |
| reldmmpo 7532 | The domain of an operation... |
| elrnmpog 7533 | Membership in the range of... |
| elrnmpo 7534 | Membership in the range of... |
| elimampo 7535 | Membership in the image of... |
| elrnmpores 7536 | Membership in the range of... |
| ralrnmpo 7537 | A restricted quantifier ov... |
| rexrnmpo 7538 | A restricted quantifier ov... |
| ovid 7539 | The value of an operation ... |
| ovidig 7540 | The value of an operation ... |
| ovidi 7541 | The value of an operation ... |
| ov 7542 | The value of an operation ... |
| ovigg 7543 | The value of an operation ... |
| ovig 7544 | The value of an operation ... |
| ovmpt4g 7545 | Value of a function given ... |
| ovmpos 7546 | Value of a function given ... |
| ov2gf 7547 | The value of an operation ... |
| ovmpodxf 7548 | Value of an operation give... |
| ovmpodx 7549 | Value of an operation give... |
| ovmpod 7550 | Value of an operation give... |
| ovmpox 7551 | The value of an operation ... |
| ovmpoga 7552 | Value of an operation give... |
| ovmpoa 7553 | Value of an operation give... |
| ovmpodf 7554 | Alternate deduction versio... |
| ovmpodv 7555 | Alternate deduction versio... |
| ovmpodv2 7556 | Alternate deduction versio... |
| ovmpog 7557 | Value of an operation give... |
| ovmpo 7558 | Value of an operation give... |
| ovmpot 7559 | The value of an operation ... |
| fvmpopr2d 7560 | Value of an operation give... |
| ov3 7561 | The value of an operation ... |
| ov6g 7562 | The value of an operation ... |
| ovg 7563 | The value of an operation ... |
| ovres 7564 | The value of a restricted ... |
| ovresd 7565 | Lemma for converting metri... |
| oprres 7566 | The restriction of an oper... |
| oprssov 7567 | The value of a member of t... |
| fovcdm 7568 | An operation's value belon... |
| fovcdmda 7569 | An operation's value belon... |
| fovcdmd 7570 | An operation's value belon... |
| fnrnov 7571 | The range of an operation ... |
| foov 7572 | An onto mapping of an oper... |
| fnovrn 7573 | An operation's value belon... |
| ovelrn 7574 | A member of an operation's... |
| funimassov 7575 | Membership relation for th... |
| ovelimab 7576 | Operation value in an imag... |
| ovima0 7577 | An operation value is a me... |
| ovconst2 7578 | The value of a constant op... |
| oprssdm 7579 | Domain of closure of an op... |
| nssdmovg 7580 | The value of an operation ... |
| ndmovg 7581 | The value of an operation ... |
| ndmov 7582 | The value of an operation ... |
| ndmovcl 7583 | The closure of an operatio... |
| ndmovrcl 7584 | Reverse closure law, when ... |
| ndmovcom 7585 | Any operation is commutati... |
| ndmovass 7586 | Any operation is associati... |
| ndmovdistr 7587 | Any operation is distribut... |
| ndmovord 7588 | Elimination of redundant a... |
| ndmovordi 7589 | Elimination of redundant a... |
| caovclg 7590 | Convert an operation closu... |
| caovcld 7591 | Convert an operation closu... |
| caovcl 7592 | Convert an operation closu... |
| caovcomg 7593 | Convert an operation commu... |
| caovcomd 7594 | Convert an operation commu... |
| caovcom 7595 | Convert an operation commu... |
| caovassg 7596 | Convert an operation assoc... |
| caovassd 7597 | Convert an operation assoc... |
| caovass 7598 | Convert an operation assoc... |
| caovcang 7599 | Convert an operation cance... |
| caovcand 7600 | Convert an operation cance... |
| caovcanrd 7601 | Commute the arguments of a... |
| caovcan 7602 | Convert an operation cance... |
| caovordig 7603 | Convert an operation order... |
| caovordid 7604 | Convert an operation order... |
| caovordg 7605 | Convert an operation order... |
| caovordd 7606 | Convert an operation order... |
| caovord2d 7607 | Operation ordering law wit... |
| caovord3d 7608 | Ordering law. (Contribute... |
| caovord 7609 | Convert an operation order... |
| caovord2 7610 | Operation ordering law wit... |
| caovord3 7611 | Ordering law. (Contribute... |
| caovdig 7612 | Convert an operation distr... |
| caovdid 7613 | Convert an operation distr... |
| caovdir2d 7614 | Convert an operation distr... |
| caovdirg 7615 | Convert an operation rever... |
| caovdird 7616 | Convert an operation distr... |
| caovdi 7617 | Convert an operation distr... |
| caov32d 7618 | Rearrange arguments in a c... |
| caov12d 7619 | Rearrange arguments in a c... |
| caov31d 7620 | Rearrange arguments in a c... |
| caov13d 7621 | Rearrange arguments in a c... |
| caov4d 7622 | Rearrange arguments in a c... |
| caov411d 7623 | Rearrange arguments in a c... |
| caov42d 7624 | Rearrange arguments in a c... |
| caov32 7625 | Rearrange arguments in a c... |
| caov12 7626 | Rearrange arguments in a c... |
| caov31 7627 | Rearrange arguments in a c... |
| caov13 7628 | Rearrange arguments in a c... |
| caov4 7629 | Rearrange arguments in a c... |
| caov411 7630 | Rearrange arguments in a c... |
| caov42 7631 | Rearrange arguments in a c... |
| caovdir 7632 | Reverse distributive law. ... |
| caovdilem 7633 | Lemma used by real number ... |
| caovlem2 7634 | Lemma used in real number ... |
| caovmo 7635 | Uniqueness of inverse elem... |
| imaeqexov 7636 | Substitute an operation va... |
| imaeqalov 7637 | Substitute an operation va... |
| mpondm0 7638 | The value of an operation ... |
| elmpocl 7639 | If a two-parameter class i... |
| elmpocl1 7640 | If a two-parameter class i... |
| elmpocl2 7641 | If a two-parameter class i... |
| elovmpod 7642 | Utility lemma for two-para... |
| elovmpo 7643 | Utility lemma for two-para... |
| elovmporab 7644 | Implications for the value... |
| elovmporab1w 7645 | Implications for the value... |
| elovmporab1 7646 | Implications for the value... |
| 2mpo0 7647 | If the operation value of ... |
| relmptopab 7648 | Any function to sets of or... |
| f1ocnvd 7649 | Describe an implicit one-t... |
| f1od 7650 | Describe an implicit one-t... |
| f1ocnv2d 7651 | Describe an implicit one-t... |
| f1o2d 7652 | Describe an implicit one-t... |
| f1opw2 7653 | A one-to-one mapping induc... |
| f1opw 7654 | A one-to-one mapping induc... |
| elovmpt3imp 7655 | If the value of a function... |
| ovmpt3rab1 7656 | The value of an operation ... |
| ovmpt3rabdm 7657 | If the value of a function... |
| elovmpt3rab1 7658 | Implications for the value... |
| elovmpt3rab 7659 | Implications for the value... |
| ofeqd 7664 | Equality theorem for funct... |
| ofeq 7665 | Equality theorem for funct... |
| ofreq 7666 | Equality theorem for funct... |
| ofexg 7667 | A function operation restr... |
| nfof 7668 | Hypothesis builder for fun... |
| nfofr 7669 | Hypothesis builder for fun... |
| ofrfvalg 7670 | Value of a relation applie... |
| offval 7671 | Value of an operation appl... |
| ofrfval 7672 | Value of a relation applie... |
| ofval 7673 | Evaluate a function operat... |
| ofrval 7674 | Exhibit a function relatio... |
| offn 7675 | The function operation pro... |
| offun 7676 | The function operation pro... |
| offval2f 7677 | The function operation exp... |
| ofmresval 7678 | Value of a restriction of ... |
| fnfvof 7679 | Function value of a pointw... |
| off 7680 | The function operation pro... |
| ofres 7681 | Restrict the operands of a... |
| offval2 7682 | The function operation exp... |
| ofrfval2 7683 | The function relation acti... |
| offvalfv 7684 | The function operation exp... |
| ofmpteq 7685 | Value of a pointwise opera... |
| coof 7686 | The composition of a _homo... |
| ofco 7687 | The composition of a funct... |
| offveq 7688 | Convert an identity of the... |
| offveqb 7689 | Equivalent expressions for... |
| ofc1 7690 | Left operation by a consta... |
| ofc2 7691 | Right operation by a const... |
| ofc12 7692 | Function operation on two ... |
| caofref 7693 | Transfer a reflexive law t... |
| caofinvl 7694 | Transfer a left inverse la... |
| caofid0l 7695 | Transfer a left identity l... |
| caofid0r 7696 | Transfer a right identity ... |
| caofid1 7697 | Transfer a right absorptio... |
| caofid2 7698 | Transfer a right absorptio... |
| caofcom 7699 | Transfer a commutative law... |
| caofidlcan 7700 | Transfer a cancellation/id... |
| caofrss 7701 | Transfer a relation subset... |
| caofass 7702 | Transfer an associative la... |
| caoftrn 7703 | Transfer a transitivity la... |
| caofdi 7704 | Transfer a distributive la... |
| caofdir 7705 | Transfer a reverse distrib... |
| caonncan 7706 | Transfer ~ nncan -shaped l... |
| relrpss 7709 | The proper subset relation... |
| brrpssg 7710 | The proper subset relation... |
| brrpss 7711 | The proper subset relation... |
| porpss 7712 | Every class is partially o... |
| sorpss 7713 | Express strict ordering un... |
| sorpssi 7714 | Property of a chain of set... |
| sorpssun 7715 | A chain of sets is closed ... |
| sorpssin 7716 | A chain of sets is closed ... |
| sorpssuni 7717 | In a chain of sets, a maxi... |
| sorpssint 7718 | In a chain of sets, a mini... |
| sorpsscmpl 7719 | The componentwise compleme... |
| zfun 7721 | Axiom of Union expressed w... |
| axun2 7722 | A variant of the Axiom of ... |
| uniex2 7723 | The Axiom of Union using t... |
| vuniex 7724 | The union of a setvar is a... |
| uniexg 7725 | The ZF Axiom of Union in c... |
| uniex 7726 | The Axiom of Union in clas... |
| uniexd 7727 | Deduction version of the Z... |
| unexg 7728 | The union of two sets is a... |
| unex 7729 | The union of two sets is a... |
| unexOLD 7730 | Obsolete version of ~ unex... |
| tpex 7731 | An unordered triple of cla... |
| unexb 7732 | Existence of union is equi... |
| unexbOLD 7733 | Obsolete version of ~ unex... |
| unexgOLD 7734 | Obsolete version of ~ unex... |
| xpexg 7735 | The Cartesian product of t... |
| xpexd 7736 | The Cartesian product of t... |
| 3xpexg 7737 | The Cartesian product of t... |
| xpex 7738 | The Cartesian product of t... |
| unexd 7739 | The union of two sets is a... |
| sqxpexg 7740 | The Cartesian square of a ... |
| abnexg 7741 | Sufficient condition for a... |
| abnex 7742 | Sufficient condition for a... |
| snnex 7743 | The class of all singleton... |
| pwnex 7744 | The class of all power set... |
| difex2 7745 | If the subtrahend of a cla... |
| difsnexi 7746 | If the difference of a cla... |
| uniuni 7747 | Expression for double unio... |
| uniexr 7748 | Converse of the Axiom of U... |
| uniexb 7749 | The Axiom of Union and its... |
| pwexr 7750 | Converse of the Axiom of P... |
| pwexb 7751 | The Axiom of Power Sets an... |
| elpwpwel 7752 | A class belongs to a doubl... |
| eldifpw 7753 | Membership in a power clas... |
| elpwun 7754 | Membership in the power cl... |
| pwuncl 7755 | Power classes are closed u... |
| iunpw 7756 | An indexed union of a powe... |
| fr3nr 7757 | A well-founded relation ha... |
| epne3 7758 | A well-founded class conta... |
| dfwe2 7759 | Alternate definition of we... |
| epweon 7760 | The membership relation we... |
| epweonALT 7761 | Alternate proof of ~ epweo... |
| ordon 7762 | The class of all ordinal n... |
| onprc 7763 | No set contains all ordina... |
| ssorduni 7764 | The union of a class of or... |
| ssonuni 7765 | The union of a set of ordi... |
| ssonunii 7766 | The union of a set of ordi... |
| ordeleqon 7767 | A way to express the ordin... |
| ordsson 7768 | Any ordinal class is a sub... |
| dford5 7769 | A class is ordinal iff it ... |
| onss 7770 | An ordinal number is a sub... |
| predon 7771 | The predecessor of an ordi... |
| ssonprc 7772 | Two ways of saying a class... |
| onuni 7773 | The union of an ordinal nu... |
| orduni 7774 | The union of an ordinal cl... |
| onint 7775 | The intersection (infimum)... |
| onint0 7776 | The intersection of a clas... |
| onssmin 7777 | A nonempty class of ordina... |
| onminesb 7778 | If a property is true for ... |
| onminsb 7779 | If a property is true for ... |
| oninton 7780 | The intersection of a none... |
| onintrab 7781 | The intersection of a clas... |
| onintrab2 7782 | An existence condition equ... |
| onnmin 7783 | No member of a set of ordi... |
| onnminsb 7784 | An ordinal number smaller ... |
| oneqmin 7785 | A way to show that an ordi... |
| uniordint 7786 | The union of a set of ordi... |
| onminex 7787 | If a wff is true for an or... |
| sucon 7788 | The class of all ordinal n... |
| sucexb 7789 | A successor exists iff its... |
| sucexg 7790 | The successor of a set is ... |
| sucex 7791 | The successor of a set is ... |
| onmindif2 7792 | The minimum of a class of ... |
| ordsuci 7793 | The successor of an ordina... |
| sucexeloni 7794 | If the successor of an ord... |
| onsuc 7795 | The successor of an ordina... |
| ordsuc 7796 | A class is ordinal if and ... |
| ordpwsuc 7797 | The collection of ordinals... |
| onpwsuc 7798 | The collection of ordinal ... |
| onsucb 7799 | A class is an ordinal numb... |
| ordsucss 7800 | The successor of an elemen... |
| onpsssuc 7801 | An ordinal number is a pro... |
| ordelsuc 7802 | A set belongs to an ordina... |
| onsucmin 7803 | The successor of an ordina... |
| ordsucelsuc 7804 | Membership is inherited by... |
| ordsucsssuc 7805 | The subclass relationship ... |
| ordsucuniel 7806 | Given an element ` A ` of ... |
| ordsucun 7807 | The successor of the maxim... |
| ordunpr 7808 | The maximum of two ordinal... |
| ordunel 7809 | The maximum of two ordinal... |
| onsucuni 7810 | A class of ordinal numbers... |
| ordsucuni 7811 | An ordinal class is a subc... |
| orduniorsuc 7812 | An ordinal class is either... |
| unon 7813 | The class of all ordinal n... |
| ordunisuc 7814 | An ordinal class is equal ... |
| orduniss2 7815 | The union of the ordinal s... |
| onsucuni2 7816 | A successor ordinal is the... |
| 0elsuc 7817 | The successor of an ordina... |
| limon 7818 | The class of ordinal numbe... |
| onuniorsuc 7819 | An ordinal number is eithe... |
| onssi 7820 | An ordinal number is a sub... |
| onsuci 7821 | The successor of an ordina... |
| onuninsuci 7822 | An ordinal is equal to its... |
| onsucssi 7823 | A set belongs to an ordina... |
| nlimsucg 7824 | A successor is not a limit... |
| orduninsuc 7825 | An ordinal class is equal ... |
| ordunisuc2 7826 | An ordinal equal to its un... |
| ordzsl 7827 | An ordinal is zero, a succ... |
| onzsl 7828 | An ordinal number is zero,... |
| dflim3 7829 | An alternate definition of... |
| dflim4 7830 | An alternate definition of... |
| limsuc 7831 | The successor of a member ... |
| limsssuc 7832 | A class includes a limit o... |
| nlimon 7833 | Two ways to express the cl... |
| limuni3 7834 | The union of a nonempty cl... |
| tfi 7835 | The Principle of Transfini... |
| tfisg 7836 | A closed form of ~ tfis . ... |
| tfis 7837 | Transfinite Induction Sche... |
| tfis2f 7838 | Transfinite Induction Sche... |
| tfis2 7839 | Transfinite Induction Sche... |
| tfis3 7840 | Transfinite Induction Sche... |
| tfisi 7841 | A transfinite induction sc... |
| tfinds 7842 | Principle of Transfinite I... |
| tfindsg 7843 | Transfinite Induction (inf... |
| tfindsg2 7844 | Transfinite Induction (inf... |
| tfindes 7845 | Transfinite Induction with... |
| tfinds2 7846 | Transfinite Induction (inf... |
| tfinds3 7847 | Principle of Transfinite I... |
| dfom2 7850 | An alternate definition of... |
| elom 7851 | Membership in omega. The ... |
| omsson 7852 | Omega is a subset of ` On ... |
| limomss 7853 | The class of natural numbe... |
| nnon 7854 | A natural number is an ord... |
| nnoni 7855 | A natural number is an ord... |
| nnord 7856 | A natural number is ordina... |
| trom 7857 | The class of finite ordina... |
| ordom 7858 | The class of finite ordina... |
| elnn 7859 | A member of a natural numb... |
| omon 7860 | The class of natural numbe... |
| omelon2 7861 | Omega is an ordinal number... |
| nnlim 7862 | A natural number is not a ... |
| omssnlim 7863 | The class of natural numbe... |
| limom 7864 | Omega is a limit ordinal. ... |
| peano2b 7865 | A class belongs to omega i... |
| nnsuc 7866 | A nonzero natural number i... |
| omsucne 7867 | A natural number is not th... |
| ssnlim 7868 | An ordinal subclass of non... |
| omsinds 7869 | Strong (or "total") induct... |
| omun 7870 | The union of two finite or... |
| peano1 7871 | Zero is a natural number. ... |
| peano2 7872 | The successor of any natur... |
| peano3 7873 | The successor of any natur... |
| peano3OLD 7874 | Obsolete version of ~ pean... |
| peano4 7875 | Two natural numbers are eq... |
| peano5 7876 | The induction postulate: a... |
| nn0suc 7877 | A natural number is either... |
| find 7878 | The Principle of Finite In... |
| finds 7879 | Principle of Finite Induct... |
| findsg 7880 | Principle of Finite Induct... |
| finds2 7881 | Principle of Finite Induct... |
| finds1 7882 | Principle of Finite Induct... |
| findes 7883 | Finite induction with expl... |
| dmexg 7884 | The domain of a set is a s... |
| rnexg 7885 | The range of a set is a se... |
| dmexd 7886 | The domain of a set is a s... |
| fndmexd 7887 | If a function is a set, it... |
| dmfex 7888 | If a mapping is a set, its... |
| fndmexb 7889 | The domain of a function i... |
| fdmexb 7890 | The domain of a function i... |
| dmfexALT 7891 | Alternate proof of ~ dmfex... |
| dmex 7892 | The domain of a set is a s... |
| rnex 7893 | The range of a set is a se... |
| iprc 7894 | The identity function is a... |
| resiexg 7895 | The existence of a restric... |
| imaexg 7896 | The image of a set is a se... |
| imaex 7897 | The image of a set is a se... |
| rnexd 7898 | The range of a set is a se... |
| imaexd 7899 | The image of a set is a se... |
| exse2 7900 | Any set relation is set-li... |
| xpexr 7901 | If a Cartesian product is ... |
| xpexr2 7902 | If a nonempty Cartesian pr... |
| xpexcnv 7903 | A condition where the conv... |
| soex 7904 | If the relation in a stric... |
| elxp4 7905 | Membership in a Cartesian ... |
| elxp5 7906 | Membership in a Cartesian ... |
| cnvexg 7907 | The converse of a set is a... |
| cnvex 7908 | The converse of a set is a... |
| relcnvexb 7909 | A relation is a set iff it... |
| f1oexrnex 7910 | If the range of a 1-1 onto... |
| f1oexbi 7911 | There is a one-to-one onto... |
| coexg 7912 | The composition of two set... |
| coex 7913 | The composition of two set... |
| coexd 7914 | The composition of two set... |
| funcnvuni 7915 | The union of a chain (with... |
| fun11uni 7916 | The union of a chain (with... |
| resf1extb 7917 | Extension of an injection ... |
| resf1ext2b 7918 | Extension of an injection ... |
| fex2 7919 | A function with bounded do... |
| fabexd 7920 | Existence of a set of func... |
| fabexg 7921 | Existence of a set of func... |
| fabex 7922 | Existence of a set of func... |
| mapex 7923 | The class of all functions... |
| f1oabexg 7924 | The class of all 1-1-onto ... |
| fiunlem 7925 | Lemma for ~ fiun and ~ f1i... |
| fiun 7926 | The union of a chain (with... |
| f1iun 7927 | The union of a chain (with... |
| fviunfun 7928 | The function value of an i... |
| ffoss 7929 | Relationship between a map... |
| f11o 7930 | Relationship between one-t... |
| resfunexgALT 7931 | Alternate proof of ~ resfu... |
| cofunexg 7932 | Existence of a composition... |
| cofunex2g 7933 | Existence of a composition... |
| fnexALT 7934 | Alternate proof of ~ fnex ... |
| funexw 7935 | Weak version of ~ funex th... |
| mptexw 7936 | Weak version of ~ mptex th... |
| funrnex 7937 | If the domain of a functio... |
| zfrep6OLD 7938 | Obsolete proof of ~ zfrep6... |
| focdmex 7939 | If the domain of an onto f... |
| f1dmex 7940 | If the codomain of a one-t... |
| f1ovv 7941 | The codomain/range of a 1-... |
| fvclex 7942 | Existence of the class of ... |
| fvresex 7943 | Existence of the class of ... |
| abrexexg 7944 | Existence of a class abstr... |
| abrexex 7945 | Existence of a class abstr... |
| iunexg 7946 | The existence of an indexe... |
| abrexex2g 7947 | Existence of an existentia... |
| opabex3d 7948 | Existence of an ordered pa... |
| opabex3rd 7949 | Existence of an ordered pa... |
| opabex3 7950 | Existence of an ordered pa... |
| iunex 7951 | The existence of an indexe... |
| abrexex2 7952 | Existence of an existentia... |
| abexssex 7953 | Existence of a class abstr... |
| abexex 7954 | A condition where a class ... |
| f1oweALT 7955 | Alternate proof of ~ f1owe... |
| wemoiso 7956 | Thus, there is at most one... |
| wemoiso2 7957 | Thus, there is at most one... |
| oprabexd 7958 | Existence of an operator a... |
| oprabex 7959 | Existence of an operation ... |
| oprabex3 7960 | Existence of an operation ... |
| oprabrexex2 7961 | Existence of an existentia... |
| ab2rexex 7962 | Existence of a class abstr... |
| ab2rexex2 7963 | Existence of an existentia... |
| xpexgALT 7964 | Alternate proof of ~ xpexg... |
| offval3 7965 | General value of ` ( F oF ... |
| offres 7966 | Pointwise combination comm... |
| ofmres 7967 | Equivalent expressions for... |
| ofmresex 7968 | Existence of a restriction... |
| mptcnfimad 7969 | The converse of a mapping ... |
| 1stval 7974 | The value of the function ... |
| 2ndval 7975 | The value of the function ... |
| 1stnpr 7976 | Value of the first-member ... |
| 2ndnpr 7977 | Value of the second-member... |
| 1st0 7978 | The value of the first-mem... |
| 2nd0 7979 | The value of the second-me... |
| op1st 7980 | Extract the first member o... |
| op2nd 7981 | Extract the second member ... |
| op1std 7982 | Extract the first member o... |
| op2ndd 7983 | Extract the second member ... |
| op1stg 7984 | Extract the first member o... |
| op2ndg 7985 | Extract the second member ... |
| ot1stg 7986 | Extract the first member o... |
| ot2ndg 7987 | Extract the second member ... |
| ot3rdg 7988 | Extract the third member o... |
| 1stval2 7989 | Alternate value of the fun... |
| 2ndval2 7990 | Alternate value of the fun... |
| oteqimp 7991 | The components of an order... |
| fo1st 7992 | The ` 1st ` function maps ... |
| fo2nd 7993 | The ` 2nd ` function maps ... |
| br1steqg 7994 | Uniqueness condition for t... |
| br2ndeqg 7995 | Uniqueness condition for t... |
| f1stres 7996 | Mapping of a restriction o... |
| f2ndres 7997 | Mapping of a restriction o... |
| fo1stres 7998 | Onto mapping of a restrict... |
| fo2ndres 7999 | Onto mapping of a restrict... |
| 1st2val 8000 | Value of an alternate defi... |
| 2nd2val 8001 | Value of an alternate defi... |
| 1stcof 8002 | Composition of the first m... |
| 2ndcof 8003 | Composition of the second ... |
| xp1st 8004 | Location of the first elem... |
| xp2nd 8005 | Location of the second ele... |
| elxp6 8006 | Membership in a Cartesian ... |
| elxp7 8007 | Membership in a Cartesian ... |
| eqopi 8008 | Equality with an ordered p... |
| xp2 8009 | Representation of Cartesia... |
| unielxp 8010 | The membership relation fo... |
| 1st2nd2 8011 | Reconstruction of a member... |
| 1st2ndb 8012 | Reconstruction of an order... |
| xpopth 8013 | An ordered pair theorem fo... |
| eqop 8014 | Two ways to express equali... |
| eqop2 8015 | Two ways to express equali... |
| op1steq 8016 | Two ways of expressing tha... |
| opreuopreu 8017 | There is a unique ordered ... |
| el2xptp 8018 | A member of a nested Carte... |
| el2xptp0 8019 | A member of a nested Carte... |
| el2xpss 8020 | Version of ~ elrel for tri... |
| 2nd1st 8021 | Swap the members of an ord... |
| 1st2nd 8022 | Reconstruction of a member... |
| 1stdm 8023 | The first ordered pair com... |
| 2ndrn 8024 | The second ordered pair co... |
| 1st2ndbr 8025 | Express an element of a re... |
| releldm2 8026 | Two ways of expressing mem... |
| reldm 8027 | An expression for the doma... |
| releldmdifi 8028 | One way of expressing memb... |
| funfv1st2nd 8029 | The function value for the... |
| funelss 8030 | If the first component of ... |
| funeldmdif 8031 | Two ways of expressing mem... |
| sbcopeq1a 8032 | Equality theorem for subst... |
| csbopeq1a 8033 | Equality theorem for subst... |
| sbcoteq1a 8034 | Equality theorem for subst... |
| dfopab2 8035 | A way to define an ordered... |
| dfoprab3s 8036 | A way to define an operati... |
| dfoprab3 8037 | Operation class abstractio... |
| dfoprab4 8038 | Operation class abstractio... |
| dfoprab4f 8039 | Operation class abstractio... |
| opabex2 8040 | Condition for an operation... |
| opabn1stprc 8041 | An ordered-pair class abst... |
| opiota 8042 | The property of a uniquely... |
| cnvoprab 8043 | The converse of a class ab... |
| dfxp3 8044 | Define the Cartesian produ... |
| elopabi 8045 | A consequence of membershi... |
| eloprabi 8046 | A consequence of membershi... |
| mpomptsx 8047 | Express a two-argument fun... |
| mpompts 8048 | Express a two-argument fun... |
| dmmpossx 8049 | The domain of a mapping is... |
| fmpox 8050 | Functionality, domain and ... |
| fmpo 8051 | Functionality, domain and ... |
| fnmpo 8052 | Functionality and domain o... |
| fnmpoi 8053 | Functionality and domain o... |
| dmmpo 8054 | Domain of a class given by... |
| ovmpoelrn 8055 | An operation's value belon... |
| dmmpoga 8056 | Domain of an operation giv... |
| dmmpog 8057 | Domain of an operation giv... |
| mpoexxg 8058 | Existence of an operation ... |
| mpoexg 8059 | Existence of an operation ... |
| mpoexga 8060 | If the domain of an operat... |
| mpoexw 8061 | Weak version of ~ mpoex th... |
| mpoex 8062 | If the domain of an operat... |
| mpoexd 8063 | Existence of an operation ... |
| mptmpoopabbrd 8064 | The operation value of a f... |
| mptmpoopabovd 8065 | The operation value of a f... |
| el2mpocsbcl 8066 | If the operation value of ... |
| el2mpocl 8067 | If the operation value of ... |
| fnmpoovd 8068 | A function with a Cartesia... |
| offval22 8069 | The function operation exp... |
| brovpreldm 8070 | If a binary relation holds... |
| bropopvvv 8071 | If a binary relation holds... |
| bropfvvvvlem 8072 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8073 | If a binary relation holds... |
| ovmptss 8074 | If all the values of the m... |
| relmpoopab 8075 | Any function to sets of or... |
| fmpoco 8076 | Composition of two functio... |
| oprabco 8077 | Composition of a function ... |
| oprab2co 8078 | Composition of operator ab... |
| df1st2 8079 | An alternate possible defi... |
| df2nd2 8080 | An alternate possible defi... |
| 1stconst 8081 | The mapping of a restricti... |
| 2ndconst 8082 | The mapping of a restricti... |
| dfmpo 8083 | Alternate definition for t... |
| mposn 8084 | An operation (in maps-to n... |
| curry1 8085 | Composition with ` ``' ( 2... |
| curry1val 8086 | The value of a curried fun... |
| curry1f 8087 | Functionality of a curried... |
| curry2 8088 | Composition with ` ``' ( 1... |
| curry2f 8089 | Functionality of a curried... |
| curry2val 8090 | The value of a curried fun... |
| cnvf1olem 8091 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8092 | Describe a function that m... |
| fparlem1 8093 | Lemma for ~ fpar . (Contr... |
| fparlem2 8094 | Lemma for ~ fpar . (Contr... |
| fparlem3 8095 | Lemma for ~ fpar . (Contr... |
| fparlem4 8096 | Lemma for ~ fpar . (Contr... |
| fpar 8097 | Merge two functions in par... |
| fsplit 8098 | A function that can be use... |
| fsplitfpar 8099 | Merge two functions with a... |
| offsplitfpar 8100 | Express the function opera... |
| f2ndf 8101 | The ` 2nd ` (second compon... |
| fo2ndf 8102 | The ` 2nd ` (second compon... |
| f1o2ndf1 8103 | The ` 2nd ` (second compon... |
| opco1 8104 | Value of an operation prec... |
| opco2 8105 | Value of an operation prec... |
| opco1i 8106 | Inference form of ~ opco1 ... |
| mpof1o2d 8107 | Sufficient condition for a... |
| frxp 8108 | A lexicographical ordering... |
| xporderlem 8109 | Lemma for lexicographical ... |
| poxp 8110 | A lexicographical ordering... |
| soxp 8111 | A lexicographical ordering... |
| wexp 8112 | A lexicographical ordering... |
| fnwelem 8113 | Lemma for ~ fnwe . (Contr... |
| fnwe 8114 | A variant on lexicographic... |
| fnse 8115 | Condition for the well-ord... |
| fvproj 8116 | Value of a function on ord... |
| fimaproj 8117 | Image of a cartesian produ... |
| ralxpes 8118 | A version of ~ ralxp with ... |
| ralxp3f 8119 | Restricted for all over a ... |
| ralxp3 8120 | Restricted for all over a ... |
| ralxp3es 8121 | Restricted for-all over a ... |
| frpoins3xpg 8122 | Special case of founded pa... |
| frpoins3xp3g 8123 | Special case of founded pa... |
| xpord2lem 8124 | Lemma for Cartesian produc... |
| poxp2 8125 | Another way of partially o... |
| frxp2 8126 | Another way of giving a we... |
| xpord2pred 8127 | Calculate the predecessor ... |
| sexp2 8128 | Condition for the relation... |
| xpord2indlem 8129 | Induction over the Cartesi... |
| xpord2ind 8130 | Induction over the Cartesi... |
| xpord3lem 8131 | Lemma for triple ordering.... |
| poxp3 8132 | Triple Cartesian product p... |
| frxp3 8133 | Give well-foundedness over... |
| xpord3pred 8134 | Calculate the predecsessor... |
| sexp3 8135 | Show that the triple order... |
| xpord3inddlem 8136 | Induction over the triple ... |
| xpord3indd 8137 | Induction over the triple ... |
| xpord3ind 8138 | Induction over the triple ... |
| orderseqlem 8139 | Lemma for ~ poseq and ~ so... |
| poseq 8140 | A partial ordering of ordi... |
| soseq 8141 | A linear ordering of ordin... |
| suppval 8144 | The value of the operation... |
| supp0prc 8145 | The support of a class is ... |
| suppvalbr 8146 | The value of the operation... |
| supp0 8147 | The support of the empty s... |
| suppval1 8148 | The value of the operation... |
| suppvalfng 8149 | The value of the operation... |
| suppvalfn 8150 | The value of the operation... |
| elsuppfng 8151 | An element of the support ... |
| elsuppfn 8152 | An element of the support ... |
| fvdifsupp 8153 | Function value is zero out... |
| cnvimadfsn 8154 | The support of functions "... |
| suppimacnvss 8155 | The support of functions "... |
| suppimacnv 8156 | Support sets of functions ... |
| fsuppeq 8157 | Two ways of writing the su... |
| fsuppeqg 8158 | Version of ~ fsuppeq avoid... |
| suppssdm 8159 | The support of a function ... |
| suppsnop 8160 | The support of a singleton... |
| snopsuppss 8161 | The support of a singleton... |
| fvn0elsupp 8162 | If the function value for ... |
| fvn0elsuppb 8163 | The function value for a g... |
| rexsupp 8164 | Existential quantification... |
| ressuppss 8165 | The support of the restric... |
| suppun 8166 | The support of a class/fun... |
| ressuppssdif 8167 | The support of the restric... |
| mptsuppdifd 8168 | The support of a function ... |
| mptsuppd 8169 | The support of a function ... |
| extmptsuppeq 8170 | The support of an extended... |
| suppfnss 8171 | The support of a function ... |
| funsssuppss 8172 | The support of a function ... |
| fnsuppres 8173 | Two ways to express restri... |
| fnsuppeq0 8174 | The support of a function ... |
| fczsupp0 8175 | The support of a constant ... |
| suppss 8176 | Show that the support of a... |
| suppssr 8177 | A function is zero outside... |
| suppssrg 8178 | A function is zero outside... |
| suppssov1 8179 | Formula building theorem f... |
| suppssov2 8180 | Formula building theorem f... |
| suppssof1 8181 | Formula building theorem f... |
| suppss2 8182 | Show that the support of a... |
| suppsssn 8183 | Show that the support of a... |
| suppssfv 8184 | Formula building theorem f... |
| suppofssd 8185 | Condition for the support ... |
| suppofss1d 8186 | Condition for the support ... |
| suppofss2d 8187 | Condition for the support ... |
| suppco 8188 | The support of the composi... |
| suppcoss 8189 | The support of the composi... |
| supp0cosupp0 8190 | The support of the composi... |
| imacosupp 8191 | The image of the support o... |
| opeliunxp2f 8192 | Membership in a union of C... |
| mpoxeldm 8193 | If there is an element of ... |
| mpoxneldm 8194 | If the first argument of a... |
| mpoxopn0yelv 8195 | If there is an element of ... |
| mpoxopynvov0g 8196 | If the second argument of ... |
| mpoxopxnop0 8197 | If the first argument of a... |
| mpoxopx0ov0 8198 | If the first argument of a... |
| mpoxopxprcov0 8199 | If the components of the f... |
| mpoxopynvov0 8200 | If the second argument of ... |
| mpoxopoveq 8201 | Value of an operation give... |
| mpoxopovel 8202 | Element of the value of an... |
| mpoxopoveqd 8203 | Value of an operation give... |
| brovex 8204 | A binary relation of the v... |
| brovmpoex 8205 | A binary relation of the v... |
| sprmpod 8206 | The extension of a binary ... |
| tposss 8209 | Subset theorem for transpo... |
| tposeq 8210 | Equality theorem for trans... |
| tposeqd 8211 | Equality theorem for trans... |
| tposssxp 8212 | The transposition is a sub... |
| reltpos 8213 | The transposition is a rel... |
| brtpos2 8214 | Value of the transposition... |
| brtpos0 8215 | The behavior of ` tpos ` w... |
| reldmtpos 8216 | Necessary and sufficient c... |
| brtpos 8217 | The transposition swaps ar... |
| ottpos 8218 | The transposition swaps th... |
| relbrtpos 8219 | The transposition swaps ar... |
| dmtpos 8220 | The domain of ` tpos F ` w... |
| rntpos 8221 | The range of ` tpos F ` wh... |
| tposexg 8222 | The transposition of a set... |
| ovtpos 8223 | The transposition swaps th... |
| tposfun 8224 | The transposition of a fun... |
| dftpos2 8225 | Alternate definition of ` ... |
| dftpos3 8226 | Alternate definition of ` ... |
| dftpos4 8227 | Alternate definition of ` ... |
| tpostpos 8228 | Value of the double transp... |
| tpostpos2 8229 | Value of the double transp... |
| tposfn2 8230 | The domain of a transposit... |
| tposfo2 8231 | Condition for a surjective... |
| tposf2 8232 | The domain and codomain of... |
| tposf12 8233 | Condition for an injective... |
| tposf1o2 8234 | Condition of a bijective t... |
| tposfo 8235 | The domain and codomain/ra... |
| tposf 8236 | The domain and codomain of... |
| tposfn 8237 | Functionality of a transpo... |
| tpos0 8238 | Transposition of the empty... |
| tposco 8239 | Transposition of a composi... |
| tpossym 8240 | Two ways to say a function... |
| tposeqi 8241 | Equality theorem for trans... |
| tposex 8242 | A transposition is a set. ... |
| nftpos 8243 | Hypothesis builder for tra... |
| tposoprab 8244 | Transposition of a class o... |
| tposmpo 8245 | Transposition of a two-arg... |
| tposconst 8246 | The transposition of a con... |
| mpocurryd 8251 | The currying of an operati... |
| mpocurryvald 8252 | The value of a curried ope... |
| fvmpocurryd 8253 | The value of the value of ... |
| pwuninel2 8256 | Proof of ~ pwuninel under ... |
| pwuninel 8257 | The powerclass of the unio... |
| pwuninelOLD 8258 | Obsolete version of ~ pwun... |
| undefval 8259 | Value of the undefined val... |
| undefnel2 8260 | The undefined value genera... |
| undefnel 8261 | The undefined value genera... |
| undefne0 8262 | The undefined value genera... |
| frecseq123 8265 | Equality theorem for the w... |
| nffrecs 8266 | Bound-variable hypothesis ... |
| csbfrecsg 8267 | Move class substitution in... |
| fpr3g 8268 | Functions defined by well-... |
| frrlem1 8269 | Lemma for well-founded rec... |
| frrlem2 8270 | Lemma for well-founded rec... |
| frrlem3 8271 | Lemma for well-founded rec... |
| frrlem4 8272 | Lemma for well-founded rec... |
| frrlem5 8273 | Lemma for well-founded rec... |
| frrlem6 8274 | Lemma for well-founded rec... |
| frrlem7 8275 | Lemma for well-founded rec... |
| frrlem8 8276 | Lemma for well-founded rec... |
| frrlem9 8277 | Lemma for well-founded rec... |
| frrlem10 8278 | Lemma for well-founded rec... |
| frrlem11 8279 | Lemma for well-founded rec... |
| frrlem12 8280 | Lemma for well-founded rec... |
| frrlem13 8281 | Lemma for well-founded rec... |
| frrlem14 8282 | Lemma for well-founded rec... |
| fprlem1 8283 | Lemma for well-founded rec... |
| fprlem2 8284 | Lemma for well-founded rec... |
| fpr2a 8285 | Weak version of ~ fpr2 whi... |
| fpr1 8286 | Law of well-founded recurs... |
| fpr2 8287 | Law of well-founded recurs... |
| fpr3 8288 | Law of well-founded recurs... |
| frrrel 8289 | Show without using the axi... |
| frrdmss 8290 | Show without using the axi... |
| frrdmcl 8291 | Show without using the axi... |
| fprfung 8292 | A "function" defined by we... |
| fprresex 8293 | The restriction of a funct... |
| wrecseq123 8296 | General equality theorem f... |
| nfwrecs 8297 | Bound-variable hypothesis ... |
| wrecseq1 8298 | Equality theorem for the w... |
| wrecseq2 8299 | Equality theorem for the w... |
| wrecseq3 8300 | Equality theorem for the w... |
| csbwrecsg 8301 | Move class substitution in... |
| wfr3g 8302 | Functions defined by well-... |
| wfrrel 8303 | The well-ordered recursion... |
| wfrdmss 8304 | The domain of the well-ord... |
| wfrdmcl 8305 | The predecessor class of a... |
| wfrfun 8306 | The "function" generated b... |
| wfrresex 8307 | Show without using the axi... |
| wfr2a 8308 | A weak version of ~ wfr2 w... |
| wfr1 8309 | The Principle of Well-Orde... |
| wfr2 8310 | The Principle of Well-Orde... |
| wfr3 8311 | The principle of Well-Orde... |
| iunon 8312 | The indexed union of a set... |
| iinon 8313 | The nonempty indexed inter... |
| onfununi 8314 | A property of functions on... |
| onovuni 8315 | A variant of ~ onfununi fo... |
| onoviun 8316 | A variant of ~ onovuni wit... |
| onnseq 8317 | There are no length ` _om ... |
| dfsmo2 8320 | Alternate definition of a ... |
| issmo 8321 | Conditions for which ` A `... |
| issmo2 8322 | Alternate definition of a ... |
| smoeq 8323 | Equality theorem for stric... |
| smodm 8324 | The domain of a strictly m... |
| smores 8325 | A strictly monotone functi... |
| smores3 8326 | A strictly monotone functi... |
| smores2 8327 | A strictly monotone ordina... |
| smodm2 8328 | The domain of a strictly m... |
| smofvon2 8329 | The function values of a s... |
| iordsmo 8330 | The identity relation rest... |
| smo0 8331 | The null set is a strictly... |
| smofvon 8332 | If ` B ` is a strictly mon... |
| smoel 8333 | If ` x ` is less than ` y ... |
| smoiun 8334 | The value of a strictly mo... |
| smoiso 8335 | If ` F ` is an isomorphism... |
| smoel2 8336 | A strictly monotone ordina... |
| smo11 8337 | A strictly monotone ordina... |
| smoord 8338 | A strictly monotone ordina... |
| smoword 8339 | A strictly monotone ordina... |
| smogt 8340 | A strictly monotone ordina... |
| smocdmdom 8341 | The codomain of a strictly... |
| smoiso2 8342 | The strictly monotone ordi... |
| dfrecs3 8345 | The old definition of tran... |
| recseq 8346 | Equality theorem for ` rec... |
| nfrecs 8347 | Bound-variable hypothesis ... |
| tfrlem1 8348 | A technical lemma for tran... |
| tfrlem3a 8349 | Lemma for transfinite recu... |
| tfrlem3 8350 | Lemma for transfinite recu... |
| tfrlem4 8351 | Lemma for transfinite recu... |
| tfrlem5 8352 | Lemma for transfinite recu... |
| recsfval 8353 | Lemma for transfinite recu... |
| tfrlem6 8354 | Lemma for transfinite recu... |
| tfrlem6OLD 8355 | Obsolete version of ~ tfrl... |
| tfrlem7 8356 | Lemma for transfinite recu... |
| tfrlem8 8357 | Lemma for transfinite recu... |
| tfrlem9 8358 | Lemma for transfinite recu... |
| tfrlem9a 8359 | Lemma for transfinite recu... |
| tfrlem10 8360 | Lemma for transfinite recu... |
| tfrlem11 8361 | Lemma for transfinite recu... |
| tfrlem12 8362 | Lemma for transfinite recu... |
| tfrlem13 8363 | Lemma for transfinite recu... |
| tfrlem14 8364 | Lemma for transfinite recu... |
| tfrlem15 8365 | Lemma for transfinite recu... |
| tfrlem16 8366 | Lemma for finite recursion... |
| tfr1a 8367 | A weak version of ~ tfr1 w... |
| tfr2a 8368 | A weak version of ~ tfr2 w... |
| tfr2b 8369 | Without assuming ~ ax-rep ... |
| tfr1 8370 | Principle of Transfinite R... |
| tfr2 8371 | Principle of Transfinite R... |
| tfr3 8372 | Principle of Transfinite R... |
| tfr1ALT 8373 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8374 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8375 | Alternate proof of ~ tfr3 ... |
| recsfnon 8376 | Strong transfinite recursi... |
| recsval 8377 | Strong transfinite recursi... |
| tz7.44lem1 8378 | The ordered pair abstracti... |
| tz7.44-1 8379 | The value of ` F ` at ` (/... |
| tz7.44-2 8380 | The value of ` F ` at a su... |
| tz7.44-3 8381 | The value of ` F ` at a li... |
| rdgeq1 8384 | Equality theorem for the r... |
| rdgeq2 8385 | Equality theorem for the r... |
| rdgeq12 8386 | Equality theorem for the r... |
| nfrdg 8387 | Bound-variable hypothesis ... |
| rdglem1 8388 | Lemma used with the recurs... |
| rdgfun 8389 | The recursive definition g... |
| rdgdmlim 8390 | The domain of the recursiv... |
| rdgfnon 8391 | The recursive definition g... |
| rdgvalg 8392 | Value of the recursive def... |
| rdgval 8393 | Value of the recursive def... |
| rdg0 8394 | The initial value of the r... |
| rdgseg 8395 | The initial segments of th... |
| rdgsucg 8396 | The value of the recursive... |
| rdgsuc 8397 | The value of the recursive... |
| rdglimg 8398 | The value of the recursive... |
| rdglim 8399 | The value of the recursive... |
| rdg0g 8400 | The initial value of the r... |
| rdgsucmptf 8401 | The value of the recursive... |
| rdgsucmptnf 8402 | The value of the recursive... |
| rdgsucmpt2 8403 | This version of ~ rdgsucmp... |
| rdgsucmpt 8404 | The value of the recursive... |
| rdglim2 8405 | The value of the recursive... |
| rdglim2a 8406 | The value of the recursive... |
| rdg0n 8407 | If ` A ` is a proper class... |
| frfnom 8408 | The function generated by ... |
| fr0g 8409 | The initial value resultin... |
| frsuc 8410 | The successor value result... |
| frsucmpt 8411 | The successor value result... |
| frsucmptn 8412 | The value of the finite re... |
| frsucmpt2 8413 | The successor value result... |
| tz7.48lem 8414 | A way of showing an ordina... |
| tz7.48-2 8415 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8416 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8417 | Proposition 7.48(3) of [Ta... |
| tz7.49 8418 | Proposition 7.49 of [Takeu... |
| tz7.49c 8419 | Corollary of Proposition 7... |
| seqomlem0 8422 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8423 | Lemma for ` seqom ` . The... |
| seqomlem2 8424 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8425 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8426 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8427 | Equality theorem for ` seq... |
| fnseqom 8428 | An index-aware recursive d... |
| seqom0g 8429 | Value of an index-aware re... |
| seqomsuc 8430 | Value of an index-aware re... |
| omsucelsucb 8431 | Membership is inherited by... |
| df1o2 8446 | Expanded value of the ordi... |
| df2o3 8447 | Expanded value of the ordi... |
| df2o2 8448 | Expanded value of the ordi... |
| 1oex 8449 | Ordinal 1 is a set. (Cont... |
| 1oelpr 8450 | ` 1o ` is an element of ` ... |
| 2oex 8451 | ` 2o ` is a set. (Contrib... |
| 1on 8452 | Ordinal 1 is an ordinal nu... |
| 2on 8453 | Ordinal 2 is an ordinal nu... |
| 2on0 8454 | Ordinal two is not zero. ... |
| ord3 8455 | Ordinal 3 is an ordinal cl... |
| 3on 8456 | Ordinal 3 is an ordinal nu... |
| 4on 8457 | Ordinal 4 is an ordinal nu... |
| 1n0 8458 | Ordinal one is not equal t... |
| 1n0OLD 8459 | Obsolete version of ~ 1n0 ... |
| nlim1 8460 | 1 is not a limit ordinal. ... |
| nlim2 8461 | 2 is not a limit ordinal. ... |
| xp01disj 8462 | Cartesian products with th... |
| xp01disjl 8463 | Cartesian products with th... |
| ordgt0ge1 8464 | Two ways to express that a... |
| ordge1n0 8465 | An ordinal greater than or... |
| el1o 8466 | Membership in ordinal one.... |
| ord1eln01 8467 | An ordinal that is not 0 o... |
| ord2eln012 8468 | An ordinal that is not 0, ... |
| 1ellim 8469 | A limit ordinal contains 1... |
| 2ellim 8470 | A limit ordinal contains 2... |
| dif1o 8471 | Two ways to say that ` A `... |
| ondif1 8472 | Two ways to say that ` A `... |
| ondif2 8473 | Two ways to say that ` A `... |
| 2oconcl 8474 | Closure of the pair swappi... |
| 0lt1o 8475 | Ordinal zero is less than ... |
| dif20el 8476 | An ordinal greater than on... |
| 0we1 8477 | The empty set is a well-or... |
| brwitnlem 8478 | Lemma for relations which ... |
| fnoa 8479 | Functionality and domain o... |
| fnom 8480 | Functionality and domain o... |
| fnoe 8481 | Functionality and domain o... |
| oav 8482 | Value of ordinal addition.... |
| omv 8483 | Value of ordinal multiplic... |
| oe0lem 8484 | A helper lemma for ~ oe0 a... |
| oev 8485 | Value of ordinal exponenti... |
| oevn0 8486 | Value of ordinal exponenti... |
| oa0 8487 | Addition with zero. Propo... |
| om0 8488 | Ordinal multiplication wit... |
| oe0m 8489 | Value of zero raised to an... |
| om0x 8490 | Ordinal multiplication wit... |
| oe0m0 8491 | Ordinal exponentiation wit... |
| oe0m1 8492 | Ordinal exponentiation wit... |
| oe0 8493 | Ordinal exponentiation wit... |
| oev2 8494 | Alternate value of ordinal... |
| oasuc 8495 | Addition with successor. ... |
| oesuclem 8496 | Lemma for ~ oesuc . (Cont... |
| omsuc 8497 | Multiplication with succes... |
| oesuc 8498 | Ordinal exponentiation wit... |
| onasuc 8499 | Addition with successor. ... |
| onmsuc 8500 | Multiplication with succes... |
| onesuc 8501 | Exponentiation with a succ... |
| oa1suc 8502 | Addition with 1 is same as... |
| oalim 8503 | Ordinal addition with a li... |
| omlim 8504 | Ordinal multiplication wit... |
| oelim 8505 | Ordinal exponentiation wit... |
| oacl 8506 | Closure law for ordinal ad... |
| omcl 8507 | Closure law for ordinal mu... |
| oecl 8508 | Closure law for ordinal ex... |
| oa0r 8509 | Ordinal addition with zero... |
| om0r 8510 | Ordinal multiplication wit... |
| o1p1e2 8511 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8512 | 2 + 2 = 4 for ordinal numb... |
| om1 8513 | Ordinal multiplication wit... |
| om1r 8514 | Ordinal multiplication wit... |
| oe1 8515 | Ordinal exponentiation wit... |
| oe1m 8516 | Ordinal exponentiation wit... |
| oaordi 8517 | Ordering property of ordin... |
| oaord 8518 | Ordering property of ordin... |
| oacan 8519 | Left cancellation law for ... |
| oaword 8520 | Weak ordering property of ... |
| oawordri 8521 | Weak ordering property of ... |
| oaord1 8522 | An ordinal is less than it... |
| oaword1 8523 | An ordinal is less than or... |
| oaword2 8524 | An ordinal is less than or... |
| oawordeulem 8525 | Lemma for ~ oawordex . (C... |
| oawordeu 8526 | Existence theorem for weak... |
| oawordexr 8527 | Existence theorem for weak... |
| oawordex 8528 | Existence theorem for weak... |
| oaordex 8529 | Existence theorem for orde... |
| oa00 8530 | An ordinal sum is zero iff... |
| oalimcl 8531 | The ordinal sum with a lim... |
| oaass 8532 | Ordinal addition is associ... |
| oarec 8533 | Recursive definition of or... |
| oaf1o 8534 | Left addition by a constan... |
| oacomf1olem 8535 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8536 | Define a bijection from ` ... |
| omordi 8537 | Ordering property of ordin... |
| omord2 8538 | Ordering property of ordin... |
| omord 8539 | Ordering property of ordin... |
| omcan 8540 | Left cancellation law for ... |
| omword 8541 | Weak ordering property of ... |
| omwordi 8542 | Weak ordering property of ... |
| omwordri 8543 | Weak ordering property of ... |
| omword1 8544 | An ordinal is less than or... |
| omword2 8545 | An ordinal is less than or... |
| om00 8546 | The product of two ordinal... |
| om00el 8547 | The product of two nonzero... |
| omordlim 8548 | Ordering involving the pro... |
| omlimcl 8549 | The product of any nonzero... |
| odi 8550 | Distributive law for ordin... |
| omass 8551 | Multiplication of ordinal ... |
| oneo 8552 | If an ordinal number is ev... |
| omeulem1 8553 | Lemma for ~ omeu : existen... |
| omeulem2 8554 | Lemma for ~ omeu : uniquen... |
| omopth2 8555 | An ordered pair-like theor... |
| omeu 8556 | The division algorithm for... |
| om2 8557 | Two ways to double an ordi... |
| oen0 8558 | Ordinal exponentiation wit... |
| oeordi 8559 | Ordering law for ordinal e... |
| oeord 8560 | Ordering property of ordin... |
| oecan 8561 | Left cancellation law for ... |
| oeword 8562 | Weak ordering property of ... |
| oewordi 8563 | Weak ordering property of ... |
| oewordri 8564 | Weak ordering property of ... |
| oeworde 8565 | Ordinal exponentiation com... |
| oeordsuc 8566 | Ordering property of ordin... |
| oelim2 8567 | Ordinal exponentiation wit... |
| oeoalem 8568 | Lemma for ~ oeoa . (Contr... |
| oeoa 8569 | Sum of exponents law for o... |
| oeoelem 8570 | Lemma for ~ oeoe . (Contr... |
| oeoe 8571 | Product of exponents law f... |
| oelimcl 8572 | The ordinal exponential wi... |
| oeeulem 8573 | Lemma for ~ oeeu . (Contr... |
| oeeui 8574 | The division algorithm for... |
| oeeu 8575 | The division algorithm for... |
| nna0 8576 | Addition with zero. Theor... |
| nnm0 8577 | Multiplication with zero. ... |
| nnasuc 8578 | Addition with successor. ... |
| nnmsuc 8579 | Multiplication with succes... |
| nnesuc 8580 | Exponentiation with a succ... |
| nna0r 8581 | Addition to zero. Remark ... |
| nnm0r 8582 | Multiplication with zero. ... |
| nnacl 8583 | Closure of addition of nat... |
| nnmcl 8584 | Closure of multiplication ... |
| nnecl 8585 | Closure of exponentiation ... |
| nnacli 8586 | ` _om ` is closed under ad... |
| nnmcli 8587 | ` _om ` is closed under mu... |
| nnarcl 8588 | Reverse closure law for ad... |
| nnacom 8589 | Addition of natural number... |
| nnaordi 8590 | Ordering property of addit... |
| nnaord 8591 | Ordering property of addit... |
| nnaordr 8592 | Ordering property of addit... |
| nnawordi 8593 | Adding to both sides of an... |
| nnaass 8594 | Addition of natural number... |
| nndi 8595 | Distributive law for natur... |
| nnmass 8596 | Multiplication of natural ... |
| nnmsucr 8597 | Multiplication with succes... |
| nnmcom 8598 | Multiplication of natural ... |
| nnaword 8599 | Weak ordering property of ... |
| nnacan 8600 | Cancellation law for addit... |
| nnaword1 8601 | Weak ordering property of ... |
| nnaword2 8602 | Weak ordering property of ... |
| nnmordi 8603 | Ordering property of multi... |
| nnmord 8604 | Ordering property of multi... |
| nnmword 8605 | Weak ordering property of ... |
| nnmcan 8606 | Cancellation law for multi... |
| nnmwordi 8607 | Weak ordering property of ... |
| nnmwordri 8608 | Weak ordering property of ... |
| nnawordex 8609 | Equivalence for weak order... |
| nnaordex 8610 | Equivalence for ordering. ... |
| nnaordex2 8611 | Equivalence for ordering. ... |
| 1onn 8612 | The ordinal 1 is a natural... |
| 1onnALT 8613 | Shorter proof of ~ 1onn us... |
| 2onn 8614 | The ordinal 2 is a natural... |
| 2onnALT 8615 | Shorter proof of ~ 2onn us... |
| 3onn 8616 | The ordinal 3 is a natural... |
| 4onn 8617 | The ordinal 4 is a natural... |
| 1one2o 8618 | Ordinal one is not ordinal... |
| oaabslem 8619 | Lemma for ~ oaabs . (Cont... |
| oaabs 8620 | Ordinal addition absorbs a... |
| oaabs2 8621 | The absorption law ~ oaabs... |
| omabslem 8622 | Lemma for ~ omabs . (Cont... |
| omabs 8623 | Ordinal multiplication is ... |
| nnm1 8624 | Multiply an element of ` _... |
| nnm2 8625 | Multiply an element of ` _... |
| nn2m 8626 | Multiply an element of ` _... |
| nnneo 8627 | If a natural number is eve... |
| nneob 8628 | A natural number is even i... |
| omsmolem 8629 | Lemma for ~ omsmo . (Cont... |
| omsmo 8630 | A strictly monotonic ordin... |
| omopthlem1 8631 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8632 | Lemma for ~ omopthi . (Co... |
| omopthi 8633 | An ordered pair theorem fo... |
| omopth 8634 | An ordered pair theorem fo... |
| nnasmo 8635 | There is at most one left ... |
| eldifsucnn 8636 | Condition for membership i... |
| on2recsfn 8639 | Show that double recursion... |
| on2recsov 8640 | Calculate the value of the... |
| on2ind 8641 | Double induction over ordi... |
| on3ind 8642 | Triple induction over ordi... |
| coflton 8643 | Cofinality theorem for ord... |
| cofon1 8644 | Cofinality theorem for ord... |
| cofon2 8645 | Cofinality theorem for ord... |
| cofonr 8646 | Inverse cofinality law for... |
| naddfn 8647 | Natural addition is a func... |
| naddcllem 8648 | Lemma for ordinal addition... |
| naddcl 8649 | Closure law for natural ad... |
| naddov 8650 | The value of natural addit... |
| naddov2 8651 | Alternate expression for n... |
| naddcld 8652 | Closure law for natural ad... |
| naddov3 8653 | Alternate expression for n... |
| naddf 8654 | Function statement for nat... |
| naddcom 8655 | Natural addition commutes.... |
| naddrid 8656 | Ordinal zero is the additi... |
| naddlid 8657 | Ordinal zero is the additi... |
| naddssim 8658 | Ordinal less-than-or-equal... |
| naddelim 8659 | Ordinal less-than is prese... |
| naddel1 8660 | Ordinal less-than is not a... |
| naddel2 8661 | Ordinal less-than is not a... |
| naddss1 8662 | Ordinal less-than-or-equal... |
| naddss2 8663 | Ordinal less-than-or-equal... |
| naddword1 8664 | Weak-ordering principle fo... |
| naddword2 8665 | Weak-ordering principle fo... |
| naddunif 8666 | Uniformity theorem for nat... |
| naddasslem1 8667 | Lemma for ~ naddass . Exp... |
| naddasslem2 8668 | Lemma for ~ naddass . Exp... |
| naddass 8669 | Natural ordinal addition i... |
| nadd32 8670 | Commutative/associative la... |
| nadd4 8671 | Rearragement of terms in a... |
| nadd42 8672 | Rearragement of terms in a... |
| naddel12 8673 | Natural addition to both s... |
| naddsuc2 8674 | Natural addition with succ... |
| naddoa 8675 | Natural addition of a natu... |
| omnaddcl 8676 | The naturals are closed un... |
| dfer2 8681 | Alternate definition of eq... |
| dfec2 8683 | Alternate definition of ` ... |
| ecexg 8684 | An equivalence class modul... |
| ecexr 8685 | A nonempty equivalence cla... |
| dfqs2 8687 | Alternate definition of qu... |
| ereq1 8688 | Equality theorem for equiv... |
| ereq2 8689 | Equality theorem for equiv... |
| errel 8690 | An equivalence relation is... |
| erdm 8691 | The domain of an equivalen... |
| ercl 8692 | Elementhood in the field o... |
| ersym 8693 | An equivalence relation is... |
| ercl2 8694 | Elementhood in the field o... |
| ersymb 8695 | An equivalence relation is... |
| ertr 8696 | An equivalence relation is... |
| ertrd 8697 | A transitivity relation fo... |
| ertr2d 8698 | A transitivity relation fo... |
| ertr3d 8699 | A transitivity relation fo... |
| ertr4d 8700 | A transitivity relation fo... |
| erref 8701 | An equivalence relation is... |
| ercnv 8702 | The converse of an equival... |
| errn 8703 | The range and domain of an... |
| erssxp 8704 | An equivalence relation is... |
| erex 8705 | An equivalence relation is... |
| erexb 8706 | An equivalence relation is... |
| iserd 8707 | A reflexive, symmetric, tr... |
| iseri 8708 | A reflexive, symmetric, tr... |
| iseriALT 8709 | Alternate proof of ~ iseri... |
| brinxper 8710 | Conditions for a reflexive... |
| brdifun 8711 | Evaluate the incomparabili... |
| swoer 8712 | Incomparability under a st... |
| swoord1 8713 | The incomparability equiva... |
| swoord2 8714 | The incomparability equiva... |
| swoso 8715 | If the incomparability rel... |
| eqerlem 8716 | Lemma for ~ eqer . (Contr... |
| eqer 8717 | Equivalence relation invol... |
| ider 8718 | The identity relation is a... |
| 0er 8719 | The empty set is an equiva... |
| eceq1 8720 | Equality theorem for equiv... |
| eceq1d 8721 | Equality theorem for equiv... |
| eceq2 8722 | Equality theorem for equiv... |
| eceq2i 8723 | Equality theorem for the `... |
| eceq2d 8724 | Equality theorem for the `... |
| elecg 8725 | Membership in an equivalen... |
| ecref 8726 | All elements are in their ... |
| elec 8727 | Membership in an equivalen... |
| relelec 8728 | Membership in an equivalen... |
| elecres 8729 | Elementhood in the restric... |
| elecreseq 8730 | The restricted coset of ` ... |
| elecex 8731 | Condition for a coset to b... |
| ecss 8732 | An equivalence class is a ... |
| ecdmn0 8733 | A representative of a none... |
| ereldm 8734 | Equality of equivalence cl... |
| erth 8735 | Basic property of equivale... |
| erth2 8736 | Basic property of equivale... |
| erthi 8737 | Basic property of equivale... |
| erdisj 8738 | Equivalence classes do not... |
| ecidsn 8739 | An equivalence class modul... |
| qseq1 8740 | Equality theorem for quoti... |
| qseq2 8741 | Equality theorem for quoti... |
| qseq2i 8742 | Equality theorem for quoti... |
| qseq1d 8743 | Equality theorem for quoti... |
| qseq2d 8744 | Equality theorem for quoti... |
| qseq12 8745 | Equality theorem for quoti... |
| 0qs 8746 | Quotient set with the empt... |
| elqsg 8747 | Closed form of ~ elqs . (... |
| elqs 8748 | Membership in a quotient s... |
| elqsi 8749 | Membership in a quotient s... |
| elqsecl 8750 | Membership in a quotient s... |
| ecelqs 8751 | Membership of an equivalen... |
| ecelqsw 8752 | Membership of an equivalen... |
| ecelqsi 8753 | Membership of an equivalen... |
| ecopqsi 8754 | "Closure" law for equivale... |
| qsexg 8755 | A quotient set exists. (C... |
| qsex 8756 | A quotient set exists. (C... |
| uniqs 8757 | The union of a quotient se... |
| uniqsw 8758 | The union of a quotient se... |
| qsss 8759 | A quotient set is a set of... |
| uniqs2 8760 | The union of a quotient se... |
| snecg 8761 | The singleton of a coset i... |
| snec 8762 | The singleton of an equiva... |
| ecqs 8763 | Equivalence class in terms... |
| ecid 8764 | A set is equal to its cose... |
| qsid 8765 | A set is equal to its quot... |
| ectocld 8766 | Implicit substitution of c... |
| ectocl 8767 | Implicit substitution of c... |
| elqsn0 8768 | A quotient set does not co... |
| ecelqsdm 8769 | Membership of an equivalen... |
| ecelqsdmb 8770 | ` R ` -coset of ` B ` in a... |
| eceldmqs 8771 | ` R ` -coset in its domain... |
| xpider 8772 | A Cartesian square is an e... |
| iiner 8773 | The intersection of a none... |
| riiner 8774 | The relative intersection ... |
| erinxp 8775 | A restricted equivalence r... |
| ecinxp 8776 | Restrict the relation in a... |
| qsinxp 8777 | Restrict the equivalence r... |
| qsdisj 8778 | Members of a quotient set ... |
| qsdisj2 8779 | A quotient set is a disjoi... |
| qsel 8780 | If an element of a quotien... |
| uniinqs 8781 | Class union distributes ov... |
| qliftlem 8782 | Lemma for theorems about a... |
| qliftrel 8783 | ` F ` , a function lift, i... |
| qliftel 8784 | Elementhood in the relatio... |
| qliftel1 8785 | Elementhood in the relatio... |
| qliftfun 8786 | The function ` F ` is the ... |
| qliftfund 8787 | The function ` F ` is the ... |
| qliftfuns 8788 | The function ` F ` is the ... |
| qliftf 8789 | The domain and codomain of... |
| qliftval 8790 | The value of the function ... |
| ecoptocl 8791 | Implicit substitution of c... |
| 2ecoptocl 8792 | Implicit substitution of c... |
| 3ecoptocl 8793 | Implicit substitution of c... |
| brecop 8794 | Binary relation on a quoti... |
| brecop2 8795 | Binary relation on a quoti... |
| eroveu 8796 | Lemma for ~ erov and ~ ero... |
| erovlem 8797 | Lemma for ~ erov and ~ ero... |
| erov 8798 | The value of an operation ... |
| eroprf 8799 | Functionality of an operat... |
| erov2 8800 | The value of an operation ... |
| eroprf2 8801 | Functionality of an operat... |
| ecopoveq 8802 | This is the first of sever... |
| ecopovsym 8803 | Assuming the operation ` F... |
| ecopovtrn 8804 | Assuming that operation ` ... |
| ecopover 8805 | Assuming that operation ` ... |
| eceqoveq 8806 | Equality of equivalence re... |
| ecovcom 8807 | Lemma used to transfer a c... |
| ecovass 8808 | Lemma used to transfer an ... |
| ecovdi 8809 | Lemma used to transfer a d... |
| mapprc 8814 | When ` A ` is a proper cla... |
| pmex 8815 | The class of all partial f... |
| fnmap 8816 | Set exponentiation has a u... |
| fnpm 8817 | Partial function exponenti... |
| reldmmap 8818 | Set exponentiation is a we... |
| mapvalg 8819 | The value of set exponenti... |
| pmvalg 8820 | The value of the partial m... |
| mapval 8821 | The value of set exponenti... |
| elmapg 8822 | Membership relation for se... |
| elmapd 8823 | Deduction form of ~ elmapg... |
| elmapdd 8824 | Deduction associated with ... |
| mapdm0 8825 | The empty set is the only ... |
| elpmg 8826 | The predicate "is a partia... |
| elpm2g 8827 | The predicate "is a partia... |
| elpm2r 8828 | Sufficient condition for b... |
| elpmi 8829 | A partial function is a fu... |
| pmfun 8830 | A partial function is a fu... |
| elmapex 8831 | Eliminate antecedent for m... |
| elmapi 8832 | A mapping is a function, f... |
| mapfset 8833 | If ` B ` is a set, the val... |
| mapssfset 8834 | The value of the set expon... |
| mapfoss 8835 | The value of the set expon... |
| fsetsspwxp 8836 | The class of all functions... |
| fset0 8837 | The set of functions from ... |
| fsetdmprc0 8838 | The set of functions with ... |
| fsetex 8839 | The set of functions betwe... |
| f1setex 8840 | The set of injections betw... |
| fosetex 8841 | The set of surjections bet... |
| f1osetex 8842 | The set of bijections betw... |
| fsetfcdm 8843 | The class of functions wit... |
| fsetfocdm 8844 | The class of functions wit... |
| fsetprcnex 8845 | The class of all functions... |
| fsetcdmex 8846 | The class of all functions... |
| fsetexb 8847 | The class of all functions... |
| elmapfn 8848 | A mapping is a function wi... |
| elmapfun 8849 | A mapping is always a func... |
| elmapssres 8850 | A restricted mapping is a ... |
| elmapssresd 8851 | A restricted mapping is a ... |
| fpmg 8852 | A total function is a part... |
| pmss12g 8853 | Subset relation for the se... |
| pmresg 8854 | Elementhood of a restricte... |
| elmap 8855 | Membership relation for se... |
| mapval2 8856 | Alternate expression for t... |
| elpm 8857 | The predicate "is a partia... |
| elpm2 8858 | The predicate "is a partia... |
| fpm 8859 | A total function is a part... |
| mapsspm 8860 | Set exponentiation is a su... |
| pmsspw 8861 | Partial maps are a subset ... |
| mapsspw 8862 | Set exponentiation is a su... |
| mapfvd 8863 | The value of a function th... |
| elmapresaun 8864 | ~ fresaun transposed to ma... |
| fvmptmap 8865 | Special case of ~ fvmpt fo... |
| map0e 8866 | Set exponentiation with an... |
| map0b 8867 | Set exponentiation with an... |
| map0g 8868 | Set exponentiation is empt... |
| 0map0sn0 8869 | The set of mappings of the... |
| mapsnd 8870 | The value of set exponenti... |
| map0 8871 | Set exponentiation is empt... |
| mapsn 8872 | The value of set exponenti... |
| mapss 8873 | Subset inheritance for set... |
| fdiagfn 8874 | Functionality of the diago... |
| fvdiagfn 8875 | Functionality of the diago... |
| mapsnconst 8876 | Every singleton map is a c... |
| mapsncnv 8877 | Expression for the inverse... |
| mapsnf1o2 8878 | Explicit bijection between... |
| mapsnf1o3 8879 | Explicit bijection in the ... |
| ralxpmap 8880 | Quantification over functi... |
| dfixp 8883 | Eliminate the expression `... |
| ixpsnval 8884 | The value of an infinite C... |
| elixp2 8885 | Membership in an infinite ... |
| fvixp 8886 | Projection of a factor of ... |
| ixpfn 8887 | A nuple is a function. (C... |
| elixp 8888 | Membership in an infinite ... |
| elixpconst 8889 | Membership in an infinite ... |
| ixpconstg 8890 | Infinite Cartesian product... |
| ixpconst 8891 | Infinite Cartesian product... |
| ixpeq1 8892 | Equality theorem for infin... |
| ixpeq1d 8893 | Equality theorem for infin... |
| ss2ixp 8894 | Subclass theorem for infin... |
| ixpeq2 8895 | Equality theorem for infin... |
| ixpeq2dva 8896 | Equality theorem for infin... |
| ixpeq2dv 8897 | Equality theorem for infin... |
| cbvixp 8898 | Change bound variable in a... |
| cbvixpv 8899 | Change bound variable in a... |
| nfixpw 8900 | Bound-variable hypothesis ... |
| nfixp 8901 | Bound-variable hypothesis ... |
| nfixp1 8902 | The index variable in an i... |
| ixpprc 8903 | A cartesian product of pro... |
| ixpf 8904 | A member of an infinite Ca... |
| uniixp 8905 | The union of an infinite C... |
| ixpexg 8906 | The existence of an infini... |
| ixpin 8907 | The intersection of two in... |
| ixpiin 8908 | The indexed intersection o... |
| ixpint 8909 | The intersection of a coll... |
| ixp0x 8910 | An infinite Cartesian prod... |
| ixpssmap2g 8911 | An infinite Cartesian prod... |
| ixpssmapg 8912 | An infinite Cartesian prod... |
| 0elixp 8913 | Membership of the empty se... |
| ixpn0 8914 | The infinite Cartesian pro... |
| ixp0 8915 | The infinite Cartesian pro... |
| ixpssmap 8916 | An infinite Cartesian prod... |
| resixp 8917 | Restriction of an element ... |
| undifixp 8918 | Union of two projections o... |
| mptelixpg 8919 | Condition for an explicit ... |
| resixpfo 8920 | Restriction of elements of... |
| elixpsn 8921 | Membership in a class of s... |
| ixpsnf1o 8922 | A bijection between a clas... |
| mapsnf1o 8923 | A bijection between a set ... |
| boxriin 8924 | A rectangular subset of a ... |
| boxcutc 8925 | The relative complement of... |
| relen 8934 | Equinumerosity is a relati... |
| reldom 8935 | Dominance is a relation. ... |
| relsdom 8936 | Strict dominance is a rela... |
| encv 8937 | If two classes are equinum... |
| breng 8938 | Equinumerosity relation. ... |
| bren 8939 | Equinumerosity relation. ... |
| brdom2g 8940 | Dominance relation. This ... |
| brdomg 8941 | Dominance relation. (Cont... |
| brdomi 8942 | Dominance relation. (Cont... |
| brdom 8943 | Dominance relation. (Cont... |
| domen 8944 | Dominance in terms of equi... |
| domeng 8945 | Dominance in terms of equi... |
| ctex 8946 | A countable set is a set. ... |
| f1oen4g 8947 | The domain and range of a ... |
| f1dom4g 8948 | The domain of a one-to-one... |
| f1oen3g 8949 | The domain and range of a ... |
| f1dom3g 8950 | The domain of a one-to-one... |
| f1oen2g 8951 | The domain and range of a ... |
| f1dom2g 8952 | The domain of a one-to-one... |
| f1oeng 8953 | The domain and range of a ... |
| f1domg 8954 | The domain of a one-to-one... |
| f1oen 8955 | The domain and range of a ... |
| f1dom 8956 | The domain of a one-to-one... |
| brsdom 8957 | Strict dominance relation,... |
| isfi 8958 | Express " ` A ` is finite"... |
| enssdom 8959 | Equinumerosity implies dom... |
| enssdomOLD 8960 | Obsolete version of ~ enss... |
| dfdom2 8961 | Alternate definition of do... |
| endom 8962 | Equinumerosity implies dom... |
| sdomdom 8963 | Strict dominance implies d... |
| sdomnen 8964 | Strict dominance implies n... |
| brdom2 8965 | Dominance in terms of stri... |
| bren2 8966 | Equinumerosity expressed i... |
| enrefg 8967 | Equinumerosity is reflexiv... |
| enref 8968 | Equinumerosity is reflexiv... |
| eqeng 8969 | Equality implies equinumer... |
| domrefg 8970 | Dominance is reflexive. (... |
| en2d 8971 | Equinumerosity inference f... |
| en3d 8972 | Equinumerosity inference f... |
| en2i 8973 | Equinumerosity inference f... |
| en3i 8974 | Equinumerosity inference f... |
| dom2lem 8975 | A mapping (first hypothesi... |
| dom2d 8976 | A mapping (first hypothesi... |
| dom3d 8977 | A mapping (first hypothesi... |
| dom2 8978 | A mapping (first hypothesi... |
| dom3 8979 | A mapping (first hypothesi... |
| idssen 8980 | Equality implies equinumer... |
| domssl 8981 | If ` A ` is a subset of ` ... |
| domssr 8982 | If ` C ` is a superset of ... |
| ssdomg 8983 | A set dominates its subset... |
| ener 8984 | Equinumerosity is an equiv... |
| ensymb 8985 | Symmetry of equinumerosity... |
| ensym 8986 | Symmetry of equinumerosity... |
| ensymi 8987 | Symmetry of equinumerosity... |
| ensymd 8988 | Symmetry of equinumerosity... |
| entr 8989 | Transitivity of equinumero... |
| domtr 8990 | Transitivity of dominance ... |
| entri 8991 | A chained equinumerosity i... |
| entr2i 8992 | A chained equinumerosity i... |
| entr3i 8993 | A chained equinumerosity i... |
| entr4i 8994 | A chained equinumerosity i... |
| endomtr 8995 | Transitivity of equinumero... |
| domentr 8996 | Transitivity of dominance ... |
| f1imaeng 8997 | If a function is one-to-on... |
| f1imaen2g 8998 | If a function is one-to-on... |
| f1imaen3g 8999 | If a set function is one-t... |
| f1imaen 9000 | If a function is one-to-on... |
| en0 9001 | The empty set is equinumer... |
| en0ALT 9002 | Shorter proof of ~ en0 , d... |
| en0r 9003 | The empty set is equinumer... |
| ensn1 9004 | A singleton is equinumerou... |
| ensn1g 9005 | A singleton is equinumerou... |
| enpr1g 9006 | ` { A , A } ` has only one... |
| en1 9007 | A set is equinumerous to o... |
| en1b 9008 | A set is equinumerous to o... |
| reuen1 9009 | Two ways to express "exact... |
| euen1 9010 | Two ways to express "exact... |
| euen1b 9011 | Two ways to express " ` A ... |
| en1uniel 9012 | A singleton contains its s... |
| 2dom 9013 | A set that dominates ordin... |
| fundmen 9014 | A function is equinumerous... |
| fundmeng 9015 | A function is equinumerous... |
| cnven 9016 | A relational set is equinu... |
| cnvct 9017 | If a set is countable, so ... |
| fndmeng 9018 | A function is equinumerate... |
| mapsnend 9019 | Set exponentiation to a si... |
| mapsnen 9020 | Set exponentiation to a si... |
| snmapen 9021 | Set exponentiation: a sing... |
| snmapen1 9022 | Set exponentiation: a sing... |
| map1 9023 | Set exponentiation: ordina... |
| en2sn 9024 | Two singletons are equinum... |
| 0fi 9025 | The empty set is finite. ... |
| snfi 9026 | A singleton is finite. (C... |
| fiprc 9027 | The class of finite sets i... |
| unen 9028 | Equinumerosity of union of... |
| enrefnn 9029 | Equinumerosity is reflexiv... |
| en2prd 9030 | Two proper unordered pairs... |
| enpr2d 9031 | A pair with distinct eleme... |
| ssct 9032 | Any subset of a countable ... |
| difsnen 9033 | All decrements of a set ar... |
| domdifsn 9034 | Dominance over a set with ... |
| xpsnen 9035 | A set is equinumerous to i... |
| xpsneng 9036 | A set is equinumerous to i... |
| xp1en 9037 | One times a cardinal numbe... |
| endisj 9038 | Any two sets are equinumer... |
| undom 9039 | Dominance law for union. ... |
| xpcomf1o 9040 | The canonical bijection fr... |
| xpcomco 9041 | Composition with the bijec... |
| xpcomen 9042 | Commutative law for equinu... |
| xpcomeng 9043 | Commutative law for equinu... |
| xpsnen2g 9044 | A set is equinumerous to i... |
| xpassen 9045 | Associative law for equinu... |
| xpdom2 9046 | Dominance law for Cartesia... |
| xpdom2g 9047 | Dominance law for Cartesia... |
| xpdom1g 9048 | Dominance law for Cartesia... |
| xpdom3 9049 | A set is dominated by its ... |
| xpdom1 9050 | Dominance law for Cartesia... |
| domunsncan 9051 | A singleton cancellation l... |
| omxpenlem 9052 | Lemma for ~ omxpen . (Con... |
| omxpen 9053 | The cardinal and ordinal p... |
| omf1o 9054 | Construct an explicit bije... |
| pw2f1olem 9055 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9056 | The power set of a set is ... |
| pw2eng 9057 | The power set of a set is ... |
| pw2en 9058 | The power set of a set is ... |
| fopwdom 9059 | Covering implies injection... |
| enfixsn 9060 | Given two equipollent sets... |
| sbthlem1 9061 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9062 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9063 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9064 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9065 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9066 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9067 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9068 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9069 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9070 | Lemma for ~ sbth . (Contr... |
| sbth 9071 | Schroeder-Bernstein Theore... |
| sbthb 9072 | Schroeder-Bernstein Theore... |
| sbthcl 9073 | Schroeder-Bernstein Theore... |
| dfsdom2 9074 | Alternate definition of st... |
| brsdom2 9075 | Alternate definition of st... |
| sdomnsym 9076 | Strict dominance is asymme... |
| domnsym 9077 | Theorem 22(i) of [Suppes] ... |
| 0domg 9078 | Any set dominates the empt... |
| dom0 9079 | A set dominated by the emp... |
| 0sdomg 9080 | A set strictly dominates t... |
| 0dom 9081 | Any set dominates the empt... |
| 0sdom 9082 | A set strictly dominates t... |
| sdom0 9083 | The empty set does not str... |
| sdomdomtr 9084 | Transitivity of strict dom... |
| sdomentr 9085 | Transitivity of strict dom... |
| domsdomtr 9086 | Transitivity of dominance ... |
| ensdomtr 9087 | Transitivity of equinumero... |
| sdomirr 9088 | Strict dominance is irrefl... |
| sdomtr 9089 | Strict dominance is transi... |
| sdomn2lp 9090 | Strict dominance has no 2-... |
| enen1 9091 | Equality-like theorem for ... |
| enen2 9092 | Equality-like theorem for ... |
| domen1 9093 | Equality-like theorem for ... |
| domen2 9094 | Equality-like theorem for ... |
| sdomen1 9095 | Equality-like theorem for ... |
| sdomen2 9096 | Equality-like theorem for ... |
| domtriord 9097 | Dominance is trichotomous ... |
| sdomel 9098 | For ordinals, strict domin... |
| sdomdif 9099 | The difference of a set fr... |
| onsdominel 9100 | An ordinal with more eleme... |
| domunsn 9101 | Dominance over a set with ... |
| fodomr 9102 | There exists a mapping fro... |
| pwdom 9103 | Injection of sets implies ... |
| canth2 9104 | Cantor's Theorem. No set ... |
| canth2g 9105 | Cantor's theorem with the ... |
| 2pwuninel 9106 | The power set of the power... |
| 2pwne 9107 | No set equals the power se... |
| disjen 9108 | A stronger form of ~ pwuni... |
| disjenex 9109 | Existence version of ~ dis... |
| domss2 9110 | A corollary of ~ disjenex ... |
| domssex2 9111 | A corollary of ~ disjenex ... |
| domssex 9112 | Weakening of ~ domssex2 to... |
| xpf1o 9113 | Construct a bijection on a... |
| xpen 9114 | Equinumerosity law for Car... |
| mapen 9115 | Two set exponentiations ar... |
| mapdom1 9116 | Order-preserving property ... |
| mapxpen 9117 | Equinumerosity law for dou... |
| xpmapenlem 9118 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9119 | Equinumerosity law for set... |
| mapunen 9120 | Equinumerosity law for set... |
| map2xp 9121 | A cardinal power with expo... |
| mapdom2 9122 | Order-preserving property ... |
| mapdom3 9123 | Set exponentiation dominat... |
| pwen 9124 | If two sets are equinumero... |
| ssenen 9125 | Equinumerosity of equinume... |
| limenpsi 9126 | A limit ordinal is equinum... |
| limensuci 9127 | A limit ordinal is equinum... |
| limensuc 9128 | A limit ordinal is equinum... |
| infensuc 9129 | Any infinite ordinal is eq... |
| dif1enlem 9130 | Lemma for ~ rexdif1en and ... |
| rexdif1en 9131 | If a set is equinumerous t... |
| dif1en 9132 | If a set ` A ` is equinume... |
| dif1ennn 9133 | If a set ` A ` is equinume... |
| findcard 9134 | Schema for induction on th... |
| findcard2 9135 | Schema for induction on th... |
| findcard2s 9136 | Variation of ~ findcard2 r... |
| findcard2d 9137 | Deduction version of ~ fin... |
| nnfi 9138 | Natural numbers are finite... |
| pssnn 9139 | A proper subset of a natur... |
| ssnnfi 9140 | A subset of a natural numb... |
| unfi 9141 | The union of two finite se... |
| unfid 9142 | The union of two finite se... |
| ssfi 9143 | A subset of a finite set i... |
| ssfiALT 9144 | Shorter proof of ~ ssfi us... |
| diffi 9145 | If ` A ` is finite, ` ( A ... |
| cnvfi 9146 | If a set is finite, its co... |
| pwssfi 9147 | Every element of the power... |
| fnfi 9148 | A version of ~ fnex for fi... |
| f1oenfi 9149 | If the domain of a one-to-... |
| f1oenfirn 9150 | If the range of a one-to-o... |
| f1domfi 9151 | If the codomain of a one-t... |
| f1domfi2 9152 | If the domain of a one-to-... |
| enreffi 9153 | Equinumerosity is reflexiv... |
| ensymfib 9154 | Symmetry of equinumerosity... |
| entrfil 9155 | Transitivity of equinumero... |
| enfii 9156 | A set equinumerous to a fi... |
| enfi 9157 | Equinumerous sets have the... |
| enfiALT 9158 | Shorter proof of ~ enfi us... |
| domfi 9159 | A set dominated by a finit... |
| entrfi 9160 | Transitivity of equinumero... |
| entrfir 9161 | Transitivity of equinumero... |
| domtrfil 9162 | Transitivity of dominance ... |
| domtrfi 9163 | Transitivity of dominance ... |
| domtrfir 9164 | Transitivity of dominance ... |
| f1imaenfi 9165 | If a function is one-to-on... |
| ssdomfi 9166 | A finite set dominates its... |
| ssdomfi2 9167 | A set dominates its finite... |
| sbthfilem 9168 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9169 | Schroeder-Bernstein Theore... |
| domnsymfi 9170 | If a set dominates a finit... |
| sdomdomtrfi 9171 | Transitivity of strict dom... |
| domsdomtrfi 9172 | Transitivity of dominance ... |
| sucdom2 9173 | Strict dominance of a set ... |
| phplem1 9174 | Lemma for Pigeonhole Princ... |
| phplem2 9175 | Lemma for Pigeonhole Princ... |
| nneneq 9176 | Two equinumerous natural n... |
| php 9177 | Pigeonhole Principle. A n... |
| php2 9178 | Corollary of Pigeonhole Pr... |
| php3 9179 | Corollary of Pigeonhole Pr... |
| php4 9180 | Corollary of the Pigeonhol... |
| php5 9181 | Corollary of the Pigeonhol... |
| phpeqd 9182 | Corollary of the Pigeonhol... |
| nndomog 9183 | Cardinal ordering agrees w... |
| onomeneq 9184 | An ordinal number equinume... |
| onfin 9185 | An ordinal number is finit... |
| ordfin 9186 | A generalization of ~ onfi... |
| onfin2 9187 | A set is a natural number ... |
| nndomo 9188 | Cardinal ordering agrees w... |
| nnsdomo 9189 | Cardinal ordering agrees w... |
| sucdom 9190 | Strict dominance of a set ... |
| snnen2o 9191 | A singleton ` { A } ` is n... |
| 0sdom1dom 9192 | Strict dominance over 0 is... |
| 0sdom1domALT 9193 | Alternate proof of ~ 0sdom... |
| 1sdom2 9194 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9195 | Alternate proof of ~ 1sdom... |
| sdom1 9196 | A set has less than one me... |
| modom 9197 | Two ways to express "at mo... |
| modom2 9198 | Two ways to express "at mo... |
| rex2dom 9199 | A set that has at least 2 ... |
| 1sdom2dom 9200 | Strict dominance over 1 is... |
| 1sdom 9201 | A set that strictly domina... |
| unxpdomlem1 9202 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9203 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9204 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9205 | Cartesian product dominate... |
| unxpdom2 9206 | Corollary of ~ unxpdom . ... |
| sucxpdom 9207 | Cartesian product dominate... |
| pssinf 9208 | A set equinumerous to a pr... |
| fisseneq 9209 | A finite set is equal to i... |
| ominf 9210 | The set of natural numbers... |
| isinf 9211 | Any set that is not finite... |
| fineqvlem 9212 | Lemma for ~ fineqv . (Con... |
| fineqv 9213 | If the Axiom of Infinity i... |
| xpfir 9214 | The components of a nonemp... |
| ssfid 9215 | A subset of a finite set i... |
| infi 9216 | The intersection of two se... |
| rabfi 9217 | A restricted class built f... |
| finresfin 9218 | The restriction of a finit... |
| f1finf1o 9219 | Any injection from one fin... |
| nfielex 9220 | If a class is not finite, ... |
| en1eqsn 9221 | A set with one element is ... |
| en1eqsnbi 9222 | A set containing an elemen... |
| dif1ennnALT 9223 | Alternate proof of ~ dif1e... |
| enp1ilem 9224 | Lemma for uses of ~ enp1i ... |
| enp1i 9225 | Proof induction for ~ en2 ... |
| en2 9226 | A set equinumerous to ordi... |
| en3 9227 | A set equinumerous to ordi... |
| en4 9228 | A set equinumerous to ordi... |
| findcard3 9229 | Schema for strong inductio... |
| ac6sfi 9230 | A version of ~ ac6s for fi... |
| frfi 9231 | A partial order is well-fo... |
| fimax2g 9232 | A finite set has a maximum... |
| fimaxg 9233 | A finite set has a maximum... |
| fisupg 9234 | Lemma showing existence an... |
| wofi 9235 | A total order on a finite ... |
| ordunifi 9236 | The maximum of a finite co... |
| nnunifi 9237 | The union (supremum) of a ... |
| unblem1 9238 | Lemma for ~ unbnn . After... |
| unblem2 9239 | Lemma for ~ unbnn . The v... |
| unblem3 9240 | Lemma for ~ unbnn . The v... |
| unblem4 9241 | Lemma for ~ unbnn . The f... |
| unbnn 9242 | Any unbounded subset of na... |
| unbnn2 9243 | Version of ~ unbnn that do... |
| isfinite2 9244 | Any set strictly dominated... |
| nnsdomg 9245 | Omega strictly dominates a... |
| isfiniteg 9246 | A set is finite iff it is ... |
| infsdomnn 9247 | An infinite set strictly d... |
| infn0 9248 | An infinite set is not emp... |
| infn0ALT 9249 | Shorter proof of ~ infn0 u... |
| fin2inf 9250 | This (useless) theorem, wh... |
| unfilem1 9251 | Lemma for proving that the... |
| unfilem2 9252 | Lemma for proving that the... |
| unfilem3 9253 | Lemma for proving that the... |
| unfir 9254 | If a union is finite, the ... |
| unfib 9255 | A union is finite if and o... |
| unfi2 9256 | The union of two finite se... |
| difinf 9257 | An infinite set ` A ` minu... |
| fodomfi 9258 | An onto function implies d... |
| fofi 9259 | If an onto function has a ... |
| f1fi 9260 | If a 1-to-1 function has a... |
| imafi 9261 | Images of finite sets are ... |
| imafiOLD 9262 | Obsolete version of ~ imaf... |
| pwfir 9263 | If the power set of a set ... |
| pwfilem 9264 | Lemma for ~ pwfi . (Contr... |
| pwfi 9265 | The power set of a finite ... |
| xpfi 9266 | The Cartesian product of t... |
| 3xpfi 9267 | The Cartesian product of t... |
| domunfican 9268 | A finite set union cancell... |
| infcntss 9269 | Every infinite set has a d... |
| prfi 9270 | An unordered pair is finit... |
| prfiALT 9271 | Shorter proof of ~ prfi us... |
| tpfi 9272 | An unordered triple is fin... |
| fiint 9273 | Equivalent ways of stating... |
| fodomfir 9274 | There exists a mapping fro... |
| fodomfib 9275 | Equivalence of an onto map... |
| fodomfibOLD 9276 | Obsolete version of ~ fodo... |
| fofinf1o 9277 | Any surjection from one fi... |
| rneqdmfinf1o 9278 | Any function from a finite... |
| fidomdm 9279 | Any finite set dominates i... |
| dmfi 9280 | The domain of a finite set... |
| fundmfibi 9281 | A function is finite if an... |
| resfnfinfin 9282 | The restriction of a funct... |
| residfi 9283 | A restricted identity func... |
| cnvfiALT 9284 | Shorter proof of ~ cnvfi u... |
| rnfi 9285 | The range of a finite set ... |
| f1dmvrnfibi 9286 | A one-to-one function whos... |
| f1vrnfibi 9287 | A one-to-one function whic... |
| iunfi 9288 | The finite union of finite... |
| unifi 9289 | The finite union of finite... |
| unifi2 9290 | The finite union of finite... |
| infssuni 9291 | If an infinite set ` A ` i... |
| unirnffid 9292 | The union of the range of ... |
| mapfi 9293 | Set exponentiation of fini... |
| ixpfi 9294 | A Cartesian product of fin... |
| ixpfi2 9295 | A Cartesian product of fin... |
| mptfi 9296 | A finite mapping set is fi... |
| abrexfi 9297 | An image set from a finite... |
| cnvimamptfin 9298 | A preimage of a mapping wi... |
| elfpw 9299 | Membership in a class of f... |
| unifpw 9300 | A set is the union of its ... |
| f1opwfi 9301 | A one-to-one mapping induc... |
| fissuni 9302 | A finite subset of a union... |
| fipreima 9303 | Given a finite subset ` A ... |
| finsschain 9304 | A finite subset of the uni... |
| indexfi 9305 | If for every element of a ... |
| imafi2 9306 | The image by a finite set ... |
| unifi3 9307 | If a union is finite, then... |
| tfsnfin2 9308 | A transfinite sequence is ... |
| relfsupp 9311 | The property of a function... |
| relprcnfsupp 9312 | A proper class is never fi... |
| isfsupp 9313 | The property of a class to... |
| isfsuppd 9314 | Deduction form of ~ isfsup... |
| funisfsupp 9315 | The property of a function... |
| fsuppimp 9316 | Implications of a class be... |
| fsuppimpd 9317 | A finitely supported funct... |
| fsuppfund 9318 | A finitely supported funct... |
| fisuppfi 9319 | A function on a finite set... |
| fidmfisupp 9320 | A function with a finite d... |
| finnzfsuppd 9321 | If a function is zero outs... |
| fdmfisuppfi 9322 | The support of a function ... |
| fdmfifsupp 9323 | A function with a finite d... |
| fsuppmptdm 9324 | A mapping with a finite do... |
| fndmfisuppfi 9325 | The support of a function ... |
| fndmfifsupp 9326 | A function with a finite d... |
| suppeqfsuppbi 9327 | If two functions have the ... |
| suppssfifsupp 9328 | If the support of a functi... |
| fsuppsssupp 9329 | If the support of a functi... |
| fsuppsssuppgd 9330 | If the support of a functi... |
| fsuppss 9331 | A subset of a finitely sup... |
| fsuppssov1 9332 | Formula building theorem f... |
| fsuppxpfi 9333 | The cartesian product of t... |
| fczfsuppd 9334 | A constant function with v... |
| fsuppun 9335 | The union of two finitely ... |
| fsuppunfi 9336 | The union of the support o... |
| fsuppunbi 9337 | If the union of two classe... |
| 0fsupp 9338 | The empty set is a finitel... |
| snopfsupp 9339 | A singleton containing an ... |
| funsnfsupp 9340 | Finite support for a funct... |
| fsuppres 9341 | The restriction of a finit... |
| fmptssfisupp 9342 | The restriction of a mappi... |
| ressuppfi 9343 | If the support of the rest... |
| resfsupp 9344 | If the restriction of a fu... |
| resfifsupp 9345 | The restriction of a funct... |
| ffsuppbi 9346 | Two ways of saying that a ... |
| fsuppmptif 9347 | A function mapping an argu... |
| sniffsupp 9348 | A function mapping all but... |
| fsuppcolem 9349 | Lemma for ~ fsuppco . For... |
| fsuppco 9350 | The composition of a 1-1 f... |
| fsuppco2 9351 | The composition of a funct... |
| fsuppcor 9352 | The composition of a funct... |
| mapfienlem1 9353 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9354 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9355 | Lemma 3 for ~ mapfien . (... |
| mapfien 9356 | A bijection of the base se... |
| mapfien2 9357 | Equinumerousity relation f... |
| fival 9360 | The set of all the finite ... |
| elfi 9361 | Specific properties of an ... |
| elfi2 9362 | The empty intersection nee... |
| elfir 9363 | Sufficient condition for a... |
| intrnfi 9364 | Sufficient condition for t... |
| iinfi 9365 | An indexed intersection of... |
| inelfi 9366 | The intersection of two se... |
| ssfii 9367 | Any element of a set ` A `... |
| fi0 9368 | The set of finite intersec... |
| fieq0 9369 | A set is empty iff the cla... |
| fiin 9370 | The elements of ` ( fi `` ... |
| dffi2 9371 | The set of finite intersec... |
| fiss 9372 | Subset relationship for fu... |
| inficl 9373 | A set which is closed unde... |
| fipwuni 9374 | The set of finite intersec... |
| fisn 9375 | A singleton is closed unde... |
| fiuni 9376 | The union of the finite in... |
| fipwss 9377 | If a set is a family of su... |
| elfiun 9378 | A finite intersection of e... |
| dffi3 9379 | The set of finite intersec... |
| fifo 9380 | Describe a surjection from... |
| marypha1lem 9381 | Core induction for Philip ... |
| marypha1 9382 | (Philip) Hall's marriage t... |
| marypha2lem1 9383 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9384 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9385 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9386 | Lemma for ~ marypha2 . Pr... |
| marypha2 9387 | Version of ~ marypha1 usin... |
| dfsup2 9392 | Quantifier-free definition... |
| supeq1 9393 | Equality theorem for supre... |
| supeq1d 9394 | Equality deduction for sup... |
| supeq1i 9395 | Equality inference for sup... |
| supeq2 9396 | Equality theorem for supre... |
| supeq3 9397 | Equality theorem for supre... |
| supeq123d 9398 | Equality deduction for sup... |
| nfsup 9399 | Hypothesis builder for sup... |
| supmo 9400 | Any class ` B ` has at mos... |
| supexd 9401 | A supremum is a set. (Con... |
| supeu 9402 | A supremum is unique. Sim... |
| supval2 9403 | Alternate expression for t... |
| eqsup 9404 | Sufficient condition for a... |
| eqsupd 9405 | Sufficient condition for a... |
| supcl 9406 | A supremum belongs to its ... |
| supub 9407 | A supremum is an upper bou... |
| suplub 9408 | A supremum is the least up... |
| suplub2 9409 | Bidirectional form of ~ su... |
| supnub 9410 | An upper bound is not less... |
| supssd 9411 | Inequality deduction for s... |
| supex 9412 | A supremum is a set. (Con... |
| sup00 9413 | The supremum under an empt... |
| sup0riota 9414 | The supremum of an empty s... |
| sup0 9415 | The supremum of an empty s... |
| supmax 9416 | The greatest element of a ... |
| fisup2g 9417 | A finite set satisfies the... |
| fisupcl 9418 | A nonempty finite set cont... |
| supgtoreq 9419 | The supremum of a finite s... |
| suppr 9420 | The supremum of a pair. (... |
| supsn 9421 | The supremum of a singleto... |
| supisolem 9422 | Lemma for ~ supiso . (Con... |
| supisoex 9423 | Lemma for ~ supiso . (Con... |
| supiso 9424 | Image of a supremum under ... |
| infeq1 9425 | Equality theorem for infim... |
| infeq1d 9426 | Equality deduction for inf... |
| infeq1i 9427 | Equality inference for inf... |
| infeq2 9428 | Equality theorem for infim... |
| infeq3 9429 | Equality theorem for infim... |
| infeq123d 9430 | Equality deduction for inf... |
| nfinf 9431 | Hypothesis builder for inf... |
| infexd 9432 | An infimum is a set. (Con... |
| eqinf 9433 | Sufficient condition for a... |
| eqinfd 9434 | Sufficient condition for a... |
| infval 9435 | Alternate expression for t... |
| infcllem 9436 | Lemma for ~ infcl , ~ infl... |
| infcl 9437 | An infimum belongs to its ... |
| inflb 9438 | An infimum is a lower boun... |
| infglb 9439 | An infimum is the greatest... |
| infglbb 9440 | Bidirectional form of ~ in... |
| infnlb 9441 | A lower bound is not great... |
| infssd 9442 | Inequality deduction for i... |
| infex 9443 | An infimum is a set. (Con... |
| infmin 9444 | The smallest element of a ... |
| infmo 9445 | Any class ` B ` has at mos... |
| infeu 9446 | An infimum is unique. (Co... |
| fimin2g 9447 | A finite set has a minimum... |
| fiming 9448 | A finite set has a minimum... |
| fiinfg 9449 | Lemma showing existence an... |
| fiinf2g 9450 | A finite set satisfies the... |
| fiinfcl 9451 | A nonempty finite set cont... |
| infltoreq 9452 | The infimum of a finite se... |
| infpr 9453 | The infimum of a pair. (C... |
| infsupprpr 9454 | The infimum of a proper pa... |
| infsn 9455 | The infimum of a singleton... |
| inf00 9456 | The infimum regarding an e... |
| infempty 9457 | The infimum of an empty se... |
| infiso 9458 | Image of an infimum under ... |
| dfoi 9461 | Rewrite ~ df-oi with abbre... |
| oieq1 9462 | Equality theorem for ordin... |
| oieq2 9463 | Equality theorem for ordin... |
| nfoi 9464 | Hypothesis builder for ord... |
| ordiso2 9465 | Generalize ~ ordiso to pro... |
| ordiso 9466 | Order-isomorphic ordinal n... |
| ordtypecbv 9467 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9468 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9469 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9470 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9471 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9472 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9473 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9474 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9475 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9476 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9477 | Lemma for ~ ordtype . Usi... |
| oi0 9478 | Definition of the ordinal ... |
| oicl 9479 | The order type of the well... |
| oif 9480 | The order isomorphism of t... |
| oiiso2 9481 | The order isomorphism of t... |
| ordtype 9482 | For any set-like well-orde... |
| oiiniseg 9483 | ` ran F ` is an initial se... |
| ordtype2 9484 | For any set-like well-orde... |
| oiexg 9485 | The order isomorphism on a... |
| oion 9486 | The order type of the well... |
| oiiso 9487 | The order isomorphism of t... |
| oien 9488 | The order type of a well-o... |
| oieu 9489 | Uniqueness of the unique o... |
| oismo 9490 | When ` A ` is a subclass o... |
| oiid 9491 | The order type of an ordin... |
| hartogslem1 9492 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9493 | Lemma for ~ hartogs . (Co... |
| hartogs 9494 | The class of ordinals domi... |
| wofib 9495 | The only sets which are we... |
| wemaplem1 9496 | Value of the lexicographic... |
| wemaplem2 9497 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9498 | Lemma for ~ wemapso . Tra... |
| wemappo 9499 | Construct lexicographic or... |
| wemapsolem 9500 | Lemma for ~ wemapso . (Co... |
| wemapso 9501 | Construct lexicographic or... |
| wemapso2lem 9502 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9503 | An alternative to having a... |
| card2on 9504 | The alternate definition o... |
| card2inf 9505 | The alternate definition o... |
| harf 9508 | Functionality of the Harto... |
| harcl 9509 | Values of the Hartogs func... |
| harval 9510 | Function value of the Hart... |
| elharval 9511 | The Hartogs number of a se... |
| harndom 9512 | The Hartogs number of a se... |
| harword 9513 | Weak ordering property of ... |
| relwdom 9516 | Weak dominance is a relati... |
| brwdom 9517 | Property of weak dominance... |
| brwdomi 9518 | Property of weak dominance... |
| brwdomn0 9519 | Weak dominance over nonemp... |
| 0wdom 9520 | Any set weakly dominates t... |
| fowdom 9521 | An onto function implies w... |
| wdomref 9522 | Reflexivity of weak domina... |
| brwdom2 9523 | Alternate characterization... |
| domwdom 9524 | Weak dominance is implied ... |
| wdomtr 9525 | Transitivity of weak domin... |
| wdomen1 9526 | Equality-like theorem for ... |
| wdomen2 9527 | Equality-like theorem for ... |
| wdompwdom 9528 | Weak dominance strengthens... |
| canthwdom 9529 | Cantor's Theorem, stated u... |
| wdom2d 9530 | Deduce weak dominance from... |
| wdomd 9531 | Deduce weak dominance from... |
| brwdom3 9532 | Condition for weak dominan... |
| brwdom3i 9533 | Weak dominance implies exi... |
| unwdomg 9534 | Weak dominance of a (disjo... |
| xpwdomg 9535 | Weak dominance of a Cartes... |
| wdomima2g 9536 | A set is weakly dominant o... |
| wdomimag 9537 | A set is weakly dominant o... |
| unxpwdom2 9538 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9539 | If a Cartesian product is ... |
| ixpiunwdom 9540 | Describe an onto function ... |
| harwdom 9541 | The value of the Hartogs f... |
| axreg2 9543 | Axiom of Regularity expres... |
| zfregcl 9544 | The Axiom of Regularity wi... |
| zfregclOLD 9545 | Obsolete version of ~ zfre... |
| zfreg 9546 | The Axiom of Regularity us... |
| elirrv 9547 | The membership relation is... |
| elirrvOLD 9548 | Obsolete version of ~ elir... |
| elirrvOLDOLD 9549 | Obsolete version of ~ elir... |
| elirr 9550 | No class is a member of it... |
| elneq 9551 | A class is not equal to an... |
| nelaneq 9552 | A class is not an element ... |
| nelaneqOLD 9553 | Obsolete version of ~ nela... |
| nelaneqOLDOLD 9554 | Obsolete version of ~ nela... |
| epinid0 9555 | The membership relation an... |
| sucprcreg 9556 | A class is equal to its su... |
| sucprcregOLD 9557 | Obsolete version of ~ sucp... |
| ruv 9558 | The Russell class is equal... |
| ruALT 9559 | Alternate proof of ~ ru , ... |
| disjcsn 9560 | A class is disjoint from i... |
| zfregfr 9561 | The membership relation is... |
| elirrvALT 9562 | Alternate proof of ~ elirr... |
| en2lp 9563 | No class has 2-cycle membe... |
| elnanel 9564 | Two classes are not elemen... |
| cnvepnep 9565 | The membership (epsilon) r... |
| epnsym 9566 | The membership (epsilon) r... |
| elnotel 9567 | A class cannot be an eleme... |
| elnel 9568 | A class cannot be an eleme... |
| en3lplem1 9569 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9570 | Lemma for ~ en3lp . (Cont... |
| en3lp 9571 | No class has 3-cycle membe... |
| preleqg 9572 | Equality of two unordered ... |
| preleq 9573 | Equality of two unordered ... |
| preleqALT 9574 | Alternate proof of ~ prele... |
| opthreg 9575 | Theorem for alternate repr... |
| suc11reg 9576 | The successor operation be... |
| dford2 9577 | Assuming ~ ax-reg , an ord... |
| inf0 9578 | Existence of ` _om ` impli... |
| inf1 9579 | Variation of Axiom of Infi... |
| inf2 9580 | Variation of Axiom of Infi... |
| inf3lema 9581 | Lemma for our Axiom of Inf... |
| inf3lemb 9582 | Lemma for our Axiom of Inf... |
| inf3lemc 9583 | Lemma for our Axiom of Inf... |
| inf3lemd 9584 | Lemma for our Axiom of Inf... |
| inf3lem1 9585 | Lemma for our Axiom of Inf... |
| inf3lem2 9586 | Lemma for our Axiom of Inf... |
| inf3lem3 9587 | Lemma for our Axiom of Inf... |
| inf3lem4 9588 | Lemma for our Axiom of Inf... |
| inf3lem5 9589 | Lemma for our Axiom of Inf... |
| inf3lem6 9590 | Lemma for our Axiom of Inf... |
| inf3lem7 9591 | Lemma for our Axiom of Inf... |
| inf3 9592 | Our Axiom of Infinity ~ ax... |
| infeq5i 9593 | Half of ~ infeq5 . (Contr... |
| infeq5 9594 | The statement "there exist... |
| zfinf 9596 | Axiom of Infinity expresse... |
| axinf2 9597 | A standard version of Axio... |
| zfinf2 9599 | A standard version of the ... |
| omex 9600 | The existence of omega (th... |
| axinf 9601 | The first version of the A... |
| inf5 9602 | The statement "there exist... |
| omelon 9603 | Omega is an ordinal number... |
| dfom3 9604 | The class of natural numbe... |
| elom3 9605 | A simplification of ~ elom... |
| dfom4 9606 | A simplification of ~ df-o... |
| dfom5 9607 | ` _om ` is the smallest li... |
| oancom 9608 | Ordinal addition is not co... |
| isfinite 9609 | A set is finite iff it is ... |
| fict 9610 | A finite set is countable ... |
| nnsdom 9611 | A natural number is strict... |
| omenps 9612 | Omega is equinumerous to a... |
| omensuc 9613 | The set of natural numbers... |
| infdifsn 9614 | Removing a singleton from ... |
| infdiffi 9615 | Removing a finite set from... |
| unbnn3 9616 | Any unbounded subset of na... |
| noinfep 9617 | Using the Axiom of Regular... |
| cantnffval 9620 | The value of the Cantor no... |
| cantnfdm 9621 | The domain of the Cantor n... |
| cantnfvalf 9622 | Lemma for ~ cantnf . The ... |
| cantnfs 9623 | Elementhood in the set of ... |
| cantnfcl 9624 | Basic properties of the or... |
| cantnfval 9625 | The value of the Cantor no... |
| cantnfval2 9626 | Alternate expression for t... |
| cantnfsuc 9627 | The value of the recursive... |
| cantnfle 9628 | A lower bound on the ` CNF... |
| cantnflt 9629 | An upper bound on the part... |
| cantnflt2 9630 | An upper bound on the ` CN... |
| cantnff 9631 | The ` CNF ` function is a ... |
| cantnf0 9632 | The value of the zero func... |
| cantnfrescl 9633 | A function is finitely sup... |
| cantnfres 9634 | The ` CNF ` function respe... |
| cantnfp1lem1 9635 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9636 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9637 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9638 | If ` F ` is created by add... |
| oemapso 9639 | The relation ` T ` is a st... |
| oemapval 9640 | Value of the relation ` T ... |
| oemapvali 9641 | If ` F < G ` , then there ... |
| cantnflem1a 9642 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9643 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9644 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9645 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9646 | Lemma for ~ cantnf . This... |
| cantnflem2 9647 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9648 | Lemma for ~ cantnf . Here... |
| cantnflem4 9649 | Lemma for ~ cantnf . Comp... |
| cantnf 9650 | The Cantor Normal Form the... |
| oemapwe 9651 | The lexicographic order on... |
| cantnffval2 9652 | An alternate definition of... |
| cantnff1o 9653 | Simplify the isomorphism o... |
| wemapwe 9654 | Construct lexicographic or... |
| oef1o 9655 | A bijection of the base se... |
| cnfcomlem 9656 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9657 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9658 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9659 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9660 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9661 | Any infinite ordinal ` B `... |
| cnfcom3clem 9662 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9663 | Wrap the construction of ~... |
| ttrcleq 9666 | Equality theorem for trans... |
| nfttrcld 9667 | Bound variable hypothesis ... |
| nfttrcl 9668 | Bound variable hypothesis ... |
| relttrcl 9669 | The transitive closure of ... |
| brttrcl 9670 | Characterization of elemen... |
| brttrcl2 9671 | Characterization of elemen... |
| ssttrcl 9672 | If ` R ` is a relation, th... |
| ttrcltr 9673 | The transitive closure of ... |
| ttrclresv 9674 | The transitive closure of ... |
| ttrclco 9675 | Composition law for the tr... |
| cottrcl 9676 | Composition law for the tr... |
| ttrclss 9677 | If ` R ` is a subclass of ... |
| dmttrcl 9678 | The domain of a transitive... |
| rnttrcl 9679 | The range of a transitive ... |
| ttrclexg 9680 | If ` R ` is a set, then so... |
| dfttrcl2 9681 | When ` R ` is a set and a ... |
| ttrclselem1 9682 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9683 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9684 | If ` R ` is set-like over ... |
| trcl 9685 | For any set ` A ` , show t... |
| tz9.1 9686 | Every set has a transitive... |
| tz9.1c 9687 | Alternate expression for t... |
| epfrs 9688 | The strong form of the Axi... |
| zfregs 9689 | The strong form of the Axi... |
| zfregs2 9690 | Alternate strong form of t... |
| tcvalg 9693 | Value of the transitive cl... |
| tcid 9694 | Defining property of the t... |
| tctr 9695 | Defining property of the t... |
| tcmin 9696 | Defining property of the t... |
| tc2 9697 | A variant of the definitio... |
| tcsni 9698 | The transitive closure of ... |
| tcss 9699 | The transitive closure fun... |
| tcel 9700 | The transitive closure fun... |
| tcidm 9701 | The transitive closure fun... |
| tc0 9702 | The transitive closure of ... |
| tc00 9703 | The transitive closure is ... |
| setind 9704 | Set (epsilon) induction. ... |
| setind2 9705 | Set (epsilon) induction, s... |
| setinds 9706 | Principle of set induction... |
| setinds2f 9707 | ` _E ` induction schema, u... |
| setinds2 9708 | ` _E ` induction schema, u... |
| frmin 9709 | Every (possibly proper) su... |
| frind 9710 | A subclass of a well-found... |
| frinsg 9711 | Well-Founded Induction Sch... |
| frins 9712 | Well-Founded Induction Sch... |
| frins2f 9713 | Well-Founded Induction sch... |
| frins2 9714 | Well-Founded Induction sch... |
| frins3 9715 | Well-Founded Induction sch... |
| frr3g 9716 | Functions defined by well-... |
| frrlem15 9717 | Lemma for general well-fou... |
| frrlem16 9718 | Lemma for general well-fou... |
| frr1 9719 | Law of general well-founde... |
| frr2 9720 | Law of general well-founde... |
| frr3 9721 | Law of general well-founde... |
| r1funlim 9726 | The cumulative hierarchy o... |
| r1fnon 9727 | The cumulative hierarchy o... |
| r10 9728 | Value of the cumulative hi... |
| r1sucg 9729 | Value of the cumulative hi... |
| r1suc 9730 | Value of the cumulative hi... |
| r1limg 9731 | Value of the cumulative hi... |
| r1lim 9732 | Value of the cumulative hi... |
| r1fin 9733 | The first ` _om ` levels o... |
| r1sdom 9734 | Each stage in the cumulati... |
| r111 9735 | The cumulative hierarchy i... |
| r1tr 9736 | The cumulative hierarchy o... |
| r1tr2 9737 | The union of a cumulative ... |
| r1ordg 9738 | Ordering relation for the ... |
| r1ord3g 9739 | Ordering relation for the ... |
| r1ord 9740 | Ordering relation for the ... |
| r1ord2 9741 | Ordering relation for the ... |
| r1ord3 9742 | Ordering relation for the ... |
| r1sssuc 9743 | The value of the cumulativ... |
| r1pwss 9744 | Each set of the cumulative... |
| r1sscl 9745 | Each set of the cumulative... |
| r1val1 9746 | The value of the cumulativ... |
| tz9.12lem1 9747 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9748 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9749 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9750 | A set is well-founded if a... |
| tz9.13 9751 | Every set is well-founded,... |
| tz9.13g 9752 | Every set is well-founded,... |
| rankwflemb 9753 | Two ways of saying a set i... |
| rankf 9754 | The domain and codomain of... |
| rankon 9755 | The rank of a set is an or... |
| r1elwf 9756 | Any member of the cumulati... |
| rankvalb 9757 | Value of the rank function... |
| rankr1ai 9758 | One direction of ~ rankr1a... |
| rankvaln 9759 | Value of the rank function... |
| rankidb 9760 | Identity law for the rank ... |
| rankdmr1 9761 | A rank is a member of the ... |
| rankr1ag 9762 | A version of ~ rankr1a tha... |
| rankr1bg 9763 | A relationship between ran... |
| r1rankidb 9764 | Any set is a subset of the... |
| r1elssi 9765 | The range of the ` R1 ` fu... |
| r1elss 9766 | The range of the ` R1 ` fu... |
| pwwf 9767 | A power set is well-founde... |
| sswf 9768 | A subset of a well-founded... |
| snwf 9769 | A singleton is well-founde... |
| unwf 9770 | A binary union is well-fou... |
| prwf 9771 | An unordered pair is well-... |
| opwf 9772 | An ordered pair is well-fo... |
| unir1 9773 | The cumulative hierarchy o... |
| jech9.3 9774 | Every set belongs to some ... |
| rankwflem 9775 | Every set is well-founded,... |
| rankval 9776 | Value of the rank function... |
| rankvalg 9777 | Value of the rank function... |
| rankval2 9778 | Value of an alternate defi... |
| uniwf 9779 | A union is well-founded if... |
| rankr1clem 9780 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9781 | A relationship between the... |
| rankidn 9782 | A relationship between the... |
| rankpwi 9783 | The rank of a power set. ... |
| rankelb 9784 | The membership relation is... |
| wfelirr 9785 | A well-founded set is not ... |
| rankval3b 9786 | The value of the rank func... |
| ranksnb 9787 | The rank of a singleton. ... |
| rankonidlem 9788 | Lemma for ~ rankonid . (C... |
| rankonid 9789 | The rank of an ordinal num... |
| onwf 9790 | The ordinals are all well-... |
| onssr1 9791 | Initial segments of the or... |
| rankr1g 9792 | A relationship between the... |
| rankid 9793 | Identity law for the rank ... |
| rankr1 9794 | A relationship between the... |
| ssrankr1 9795 | A relationship between an ... |
| rankr1a 9796 | A relationship between ran... |
| r1val2 9797 | The value of the cumulativ... |
| r1val3 9798 | The value of the cumulativ... |
| rankel 9799 | The membership relation is... |
| rankval3 9800 | The value of the rank func... |
| bndrank 9801 | Any class whose elements h... |
| unbndrank 9802 | The elements of a proper c... |
| rankpw 9803 | The rank of a power set. ... |
| ranklim 9804 | The rank of a set belongs ... |
| r1pw 9805 | A stronger property of ` R... |
| r1pwALT 9806 | Alternate shorter proof of... |
| r1pwcl 9807 | The cumulative hierarchy o... |
| rankssb 9808 | The subset relation is inh... |
| rankss 9809 | The subset relation is inh... |
| rankunb 9810 | The rank of the union of t... |
| rankprb 9811 | The rank of an unordered p... |
| rankopb 9812 | The rank of an ordered pai... |
| rankuni2b 9813 | The value of the rank func... |
| ranksn 9814 | The rank of a singleton. ... |
| rankuni2 9815 | The rank of a union. Part... |
| rankun 9816 | The rank of the union of t... |
| rankpr 9817 | The rank of an unordered p... |
| rankop 9818 | The rank of an ordered pai... |
| r1rankid 9819 | Any set is a subset of the... |
| rankeq0b 9820 | A set is empty iff its ran... |
| rankeq0 9821 | A set is empty iff its ran... |
| rankr1id 9822 | The rank of the hierarchy ... |
| rankuni 9823 | The rank of a union. Part... |
| rankr1b 9824 | A relationship between ran... |
| ranksuc 9825 | The rank of a successor. ... |
| rankuniss 9826 | Upper bound of the rank of... |
| rankval4 9827 | The rank of a set is the s... |
| rankbnd 9828 | The rank of a set is bound... |
| rankbnd2 9829 | The rank of a set is bound... |
| rankc1 9830 | A relationship that can be... |
| rankc2 9831 | A relationship that can be... |
| rankelun 9832 | Rank membership is inherit... |
| rankelpr 9833 | Rank membership is inherit... |
| rankelop 9834 | Rank membership is inherit... |
| rankxpl 9835 | A lower bound on the rank ... |
| rankxpu 9836 | An upper bound on the rank... |
| rankfu 9837 | An upper bound on the rank... |
| rankmapu 9838 | An upper bound on the rank... |
| rankxplim 9839 | The rank of a Cartesian pr... |
| rankxplim2 9840 | If the rank of a Cartesian... |
| rankxplim3 9841 | The rank of a Cartesian pr... |
| rankxpsuc 9842 | The rank of a Cartesian pr... |
| tcwf 9843 | The transitive closure fun... |
| tcrank 9844 | This theorem expresses two... |
| scottex 9845 | Scott's trick collects all... |
| scott0 9846 | Scott's trick collects all... |
| scottexs 9847 | Theorem scheme version of ... |
| scott0s 9848 | Theorem scheme version of ... |
| cplem1 9849 | Lemma for the Collection P... |
| cplem2 9850 | Lemma for the Collection P... |
| cp 9851 | Collection Principle. Thi... |
| bnd 9852 | A very strong generalizati... |
| bnd2 9853 | A variant of the Boundedne... |
| kardex 9854 | The collection of all sets... |
| karden 9855 | If we allow the Axiom of R... |
| htalem 9856 | Lemma for defining an emul... |
| hta 9857 | A ZFC emulation of Hilbert... |
| djueq12 9864 | Equality theorem for disjo... |
| djueq1 9865 | Equality theorem for disjo... |
| djueq2 9866 | Equality theorem for disjo... |
| nfdju 9867 | Bound-variable hypothesis ... |
| djuex 9868 | The disjoint union of sets... |
| djuexb 9869 | The disjoint union of two ... |
| djulcl 9870 | Left closure of disjoint u... |
| djurcl 9871 | Right closure of disjoint ... |
| djulf1o 9872 | The left injection functio... |
| djurf1o 9873 | The right injection functi... |
| inlresf 9874 | The left injection restric... |
| inlresf1 9875 | The left injection restric... |
| inrresf 9876 | The right injection restri... |
| inrresf1 9877 | The right injection restri... |
| djuin 9878 | The images of any classes ... |
| djur 9879 | A member of a disjoint uni... |
| djuss 9880 | A disjoint union is a subc... |
| djuunxp 9881 | The union of a disjoint un... |
| djuexALT 9882 | Alternate proof of ~ djuex... |
| eldju1st 9883 | The first component of an ... |
| eldju2ndl 9884 | The second component of an... |
| eldju2ndr 9885 | The second component of an... |
| djuun 9886 | The disjoint union of two ... |
| 1stinl 9887 | The first component of the... |
| 2ndinl 9888 | The second component of th... |
| 1stinr 9889 | The first component of the... |
| 2ndinr 9890 | The second component of th... |
| updjudhf 9891 | The mapping of an element ... |
| updjudhcoinlf 9892 | The composition of the map... |
| updjudhcoinrg 9893 | The composition of the map... |
| updjud 9894 | Universal property of the ... |
| cardf2 9903 | The cardinality function i... |
| cardon 9904 | The cardinal number of a s... |
| isnum2 9905 | A way to express well-orde... |
| isnumi 9906 | A set equinumerous to an o... |
| ennum 9907 | Equinumerous sets are equi... |
| finnum 9908 | Every finite set is numera... |
| onenon 9909 | Every ordinal number is nu... |
| tskwe 9910 | A Tarski set is well-order... |
| xpnum 9911 | The cartesian product of n... |
| cardval3 9912 | An alternate definition of... |
| cardid2 9913 | Any numerable set is equin... |
| isnum3 9914 | A set is numerable iff it ... |
| oncardval 9915 | The value of the cardinal ... |
| oncardid 9916 | Any ordinal number is equi... |
| cardonle 9917 | The cardinal of an ordinal... |
| card0 9918 | The cardinality of the emp... |
| cardidm 9919 | The cardinality function i... |
| oncard 9920 | A set is a cardinal number... |
| ficardom 9921 | The cardinal number of a f... |
| ficardid 9922 | A finite set is equinumero... |
| cardnn 9923 | The cardinality of a natur... |
| cardnueq0 9924 | The empty set is the only ... |
| cardne 9925 | No member of a cardinal nu... |
| carden2a 9926 | If two sets have equal non... |
| carden2b 9927 | If two sets are equinumero... |
| card1 9928 | A set has cardinality one ... |
| cardsn 9929 | A singleton has cardinalit... |
| carddomi2 9930 | Two sets have the dominanc... |
| sdomsdomcardi 9931 | A set strictly dominates i... |
| cardlim 9932 | An infinite cardinal is a ... |
| cardsdomelir 9933 | A cardinal strictly domina... |
| cardsdomel 9934 | A cardinal strictly domina... |
| iscard 9935 | Two ways to express the pr... |
| iscard2 9936 | Two ways to express the pr... |
| carddom2 9937 | Two numerable sets have th... |
| harcard 9938 | The class of ordinal numbe... |
| cardprclem 9939 | Lemma for ~ cardprc . (Co... |
| cardprc 9940 | The class of all cardinal ... |
| carduni 9941 | The union of a set of card... |
| cardiun 9942 | The indexed union of a set... |
| cardennn 9943 | If ` A ` is equinumerous t... |
| cardsucinf 9944 | The cardinality of the suc... |
| cardsucnn 9945 | The cardinality of the suc... |
| cardom 9946 | The set of natural numbers... |
| carden2 9947 | Two numerable sets are equ... |
| cardsdom2 9948 | A numerable set is strictl... |
| domtri2 9949 | Trichotomy of dominance fo... |
| nnsdomel 9950 | Strict dominance and eleme... |
| cardval2 9951 | An alternate version of th... |
| isinffi 9952 | An infinite set contains s... |
| fidomtri 9953 | Trichotomy of dominance wi... |
| fidomtri2 9954 | Trichotomy of dominance wi... |
| harsdom 9955 | The Hartogs number of a we... |
| onsdom 9956 | Any well-orderable set is ... |
| harval2 9957 | An alternate expression fo... |
| harsucnn 9958 | The next cardinal after a ... |
| cardmin2 9959 | The smallest ordinal that ... |
| pm54.43lem 9960 | In Theorem *54.43 of [Whit... |
| pm54.43 9961 | Theorem *54.43 of [Whitehe... |
| enpr2 9962 | An unordered pair with dis... |
| pr2ne 9963 | If an unordered pair has t... |
| prdom2 9964 | An unordered pair has at m... |
| en2eqpr 9965 | Building a set with two el... |
| en2eleq 9966 | Express a set of pair card... |
| en2other2 9967 | Taking the other element t... |
| dif1card 9968 | The cardinality of a nonem... |
| leweon 9969 | Lexicographical order is a... |
| r0weon 9970 | A set-like well-ordering o... |
| infxpenlem 9971 | Lemma for ~ infxpen . (Co... |
| infxpen 9972 | Every infinite ordinal is ... |
| xpomen 9973 | The Cartesian product of o... |
| xpct 9974 | The cartesian product of t... |
| infxpidm2 9975 | Every infinite well-ordera... |
| infxpenc 9976 | A canonical version of ~ i... |
| infxpenc2lem1 9977 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 9978 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 9979 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 9980 | Existence form of ~ infxpe... |
| iunmapdisj 9981 | The union ` U_ n e. C ( A ... |
| fseqenlem1 9982 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 9983 | Lemma for ~ fseqen . (Con... |
| fseqdom 9984 | One half of ~ fseqen . (C... |
| fseqen 9985 | A set that is equinumerous... |
| infpwfidom 9986 | The collection of finite s... |
| dfac8alem 9987 | Lemma for ~ dfac8a . If t... |
| dfac8a 9988 | Numeration theorem: every ... |
| dfac8b 9989 | The well-ordering theorem:... |
| dfac8clem 9990 | Lemma for ~ dfac8c . (Con... |
| dfac8c 9991 | If the union of a set is w... |
| ac10ct 9992 | A proof of the well-orderi... |
| ween 9993 | A set is numerable iff it ... |
| ac5num 9994 | A version of ~ ac5b with t... |
| ondomen 9995 | If a set is dominated by a... |
| numdom 9996 | A set dominated by a numer... |
| ssnum 9997 | A subset of a numerable se... |
| onssnum 9998 | All subsets of the ordinal... |
| indcardi 9999 | Indirect strong induction ... |
| acnrcl 10000 | Reverse closure for the ch... |
| acneq 10001 | Equality theorem for the c... |
| isacn 10002 | The property of being a ch... |
| acni 10003 | The property of being a ch... |
| acni2 10004 | The property of being a ch... |
| acni3 10005 | The property of being a ch... |
| acnlem 10006 | Construct a mapping satisf... |
| numacn 10007 | A well-orderable set has c... |
| finacn 10008 | Every set has finite choic... |
| acndom 10009 | A set with long choice seq... |
| acnnum 10010 | A set ` X ` which has choi... |
| acnen 10011 | The class of choice sets o... |
| acndom2 10012 | A set smaller than one wit... |
| acnen2 10013 | The class of sets with cho... |
| fodomacn 10014 | A version of ~ fodom that ... |
| fodomnum 10015 | A version of ~ fodom that ... |
| fonum 10016 | A surjection maps numerabl... |
| numwdom 10017 | A surjection maps numerabl... |
| fodomfi2 10018 | Onto functions define domi... |
| wdomfil 10019 | Weak dominance agrees with... |
| infpwfien 10020 | Any infinite well-orderabl... |
| inffien 10021 | The set of finite intersec... |
| wdomnumr 10022 | Weak dominance agrees with... |
| alephfnon 10023 | The aleph function is a fu... |
| aleph0 10024 | The first infinite cardina... |
| alephlim 10025 | Value of the aleph functio... |
| alephsuc 10026 | Value of the aleph functio... |
| alephon 10027 | An aleph is an ordinal num... |
| alephcard 10028 | Every aleph is a cardinal ... |
| alephnbtwn 10029 | No cardinal can be sandwic... |
| alephnbtwn2 10030 | No set has equinumerosity ... |
| alephordilem1 10031 | Lemma for ~ alephordi . (... |
| alephordi 10032 | Strict ordering property o... |
| alephord 10033 | Ordering property of the a... |
| alephord2 10034 | Ordering property of the a... |
| alephord2i 10035 | Ordering property of the a... |
| alephord3 10036 | Ordering property of the a... |
| alephsucdom 10037 | A set dominated by an alep... |
| alephsuc2 10038 | An alternate representatio... |
| alephdom 10039 | Relationship between inclu... |
| alephgeom 10040 | Every aleph is greater tha... |
| alephislim 10041 | Every aleph is a limit ord... |
| aleph11 10042 | The aleph function is one-... |
| alephf1 10043 | The aleph function is a on... |
| alephsdom 10044 | If an ordinal is smaller t... |
| alephdom2 10045 | A dominated initial ordina... |
| alephle 10046 | The argument of the aleph ... |
| cardaleph 10047 | Given any transfinite card... |
| cardalephex 10048 | Every transfinite cardinal... |
| infenaleph 10049 | An infinite numerable set ... |
| isinfcard 10050 | Two ways to express the pr... |
| iscard3 10051 | Two ways to express the pr... |
| cardnum 10052 | Two ways to express the cl... |
| alephinit 10053 | An infinite initial ordina... |
| carduniima 10054 | The union of the image of ... |
| cardinfima 10055 | If a mapping to cardinals ... |
| alephiso 10056 | Aleph is an order isomorph... |
| alephprc 10057 | The class of all transfini... |
| alephsson 10058 | The class of transfinite c... |
| unialeph 10059 | The union of the class of ... |
| alephsmo 10060 | The aleph function is stri... |
| alephf1ALT 10061 | Alternate proof of ~ aleph... |
| alephfplem1 10062 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10063 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10064 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10065 | Lemma for ~ alephfp . (Co... |
| alephfp 10066 | The aleph function has a f... |
| alephfp2 10067 | The aleph function has at ... |
| alephval3 10068 | An alternate way to expres... |
| alephsucpw2 10069 | The power set of an aleph ... |
| mappwen 10070 | Power rule for cardinal ar... |
| finnisoeu 10071 | A finite totally ordered s... |
| iunfictbso 10072 | Countability of a countabl... |
| aceq1 10075 | Equivalence of two version... |
| aceq0 10076 | Equivalence of two version... |
| aceq2 10077 | Equivalence of two version... |
| aceq3lem 10078 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10079 | Equivalence of two version... |
| dfac4 10080 | Equivalence of two version... |
| dfac5lem1 10081 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10082 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10083 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10084 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10085 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4OLD 10086 | Obsolete version of ~ dfac... |
| dfac5 10087 | Equivalence of two version... |
| dfac2a 10088 | Our Axiom of Choice (in th... |
| dfac2b 10089 | Axiom of Choice (first for... |
| dfac2 10090 | Axiom of Choice (first for... |
| dfac7 10091 | Equivalence of the Axiom o... |
| dfac0 10092 | Equivalence of two version... |
| dfac1 10093 | Equivalence of two version... |
| dfac8 10094 | A proof of the equivalency... |
| dfac9 10095 | Equivalence of the axiom o... |
| dfac10 10096 | Axiom of Choice equivalent... |
| dfac10c 10097 | Axiom of Choice equivalent... |
| dfac10b 10098 | Axiom of Choice equivalent... |
| acacni 10099 | A choice equivalent: every... |
| dfacacn 10100 | A choice equivalent: every... |
| dfac13 10101 | The axiom of choice holds ... |
| dfac12lem1 10102 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10103 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10104 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10105 | The axiom of choice holds ... |
| dfac12k 10106 | Equivalence of ~ dfac12 an... |
| dfac12a 10107 | The axiom of choice holds ... |
| dfac12 10108 | The axiom of choice holds ... |
| kmlem1 10109 | Lemma for 5-quantifier AC ... |
| kmlem2 10110 | Lemma for 5-quantifier AC ... |
| kmlem3 10111 | Lemma for 5-quantifier AC ... |
| kmlem4 10112 | Lemma for 5-quantifier AC ... |
| kmlem5 10113 | Lemma for 5-quantifier AC ... |
| kmlem6 10114 | Lemma for 5-quantifier AC ... |
| kmlem7 10115 | Lemma for 5-quantifier AC ... |
| kmlem8 10116 | Lemma for 5-quantifier AC ... |
| kmlem9 10117 | Lemma for 5-quantifier AC ... |
| kmlem10 10118 | Lemma for 5-quantifier AC ... |
| kmlem11 10119 | Lemma for 5-quantifier AC ... |
| kmlem12 10120 | Lemma for 5-quantifier AC ... |
| kmlem13 10121 | Lemma for 5-quantifier AC ... |
| kmlem14 10122 | Lemma for 5-quantifier AC ... |
| kmlem15 10123 | Lemma for 5-quantifier AC ... |
| kmlem16 10124 | Lemma for 5-quantifier AC ... |
| dfackm 10125 | Equivalence of the Axiom o... |
| undjudom 10126 | Cardinal addition dominate... |
| endjudisj 10127 | Equinumerosity of a disjoi... |
| djuen 10128 | Disjoint unions of equinum... |
| djuenun 10129 | Disjoint union is equinume... |
| dju1en 10130 | Cardinal addition with car... |
| dju1dif 10131 | Adding and subtracting one... |
| dju1p1e2 10132 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10133 | Alternate proof of ~ dju1p... |
| dju0en 10134 | Cardinal addition with car... |
| xp2dju 10135 | Two times a cardinal numbe... |
| djucomen 10136 | Commutative law for cardin... |
| djuassen 10137 | Associative law for cardin... |
| xpdjuen 10138 | Cardinal multiplication di... |
| mapdjuen 10139 | Sum of exponents law for c... |
| pwdjuen 10140 | Sum of exponents law for c... |
| djudom1 10141 | Ordering law for cardinal ... |
| djudom2 10142 | Ordering law for cardinal ... |
| djudoml 10143 | A set is dominated by its ... |
| djuxpdom 10144 | Cartesian product dominate... |
| djufi 10145 | The disjoint union of two ... |
| cdainflem 10146 | Any partition of omega int... |
| djuinf 10147 | A set is infinite iff the ... |
| infdju1 10148 | An infinite set is equinum... |
| pwdju1 10149 | The sum of a powerset with... |
| pwdjuidm 10150 | If the natural numbers inj... |
| djulepw 10151 | If ` A ` is idempotent und... |
| onadju 10152 | The cardinal and ordinal s... |
| cardadju 10153 | The cardinal sum is equinu... |
| djunum 10154 | The disjoint union of two ... |
| unnum 10155 | The union of two numerable... |
| nnadju 10156 | The cardinal and ordinal s... |
| nnadjuALT 10157 | Shorter proof of ~ nnadju ... |
| ficardadju 10158 | The disjoint union of fini... |
| ficardun 10159 | The cardinality of the uni... |
| ficardun2 10160 | The cardinality of the uni... |
| pwsdompw 10161 | Lemma for ~ domtriom . Th... |
| unctb 10162 | The union of two countable... |
| infdjuabs 10163 | Absorption law for additio... |
| infunabs 10164 | An infinite set is equinum... |
| infdju 10165 | The sum of two cardinal nu... |
| infdif 10166 | The cardinality of an infi... |
| infdif2 10167 | Cardinality ordering for a... |
| infxpdom 10168 | Dominance law for multipli... |
| infxpabs 10169 | Absorption law for multipl... |
| infunsdom1 10170 | The union of two sets that... |
| infunsdom 10171 | The union of two sets that... |
| infxp 10172 | Absorption law for multipl... |
| pwdjudom 10173 | A property of dominance ov... |
| infpss 10174 | Every infinite set has an ... |
| infmap2 10175 | An exponentiation law for ... |
| ackbij2lem1 10176 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10177 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10178 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10179 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10180 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10181 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10182 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10183 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10184 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10185 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10186 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10187 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10188 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10189 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10190 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10191 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10192 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10193 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10194 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10195 | The Ackermann bijection, p... |
| ackbij1b 10196 | The Ackermann bijection, p... |
| ackbij2lem2 10197 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10198 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10199 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10200 | The Ackermann bijection, p... |
| r1om 10201 | The set of hereditarily fi... |
| fictb 10202 | A set is countable iff its... |
| cflem 10203 | A lemma used to simplify c... |
| cflemOLD 10204 | Obsolete version of ~ cfle... |
| cfval 10205 | Value of the cofinality fu... |
| cff 10206 | Cofinality is a function o... |
| cfub 10207 | An upper bound on cofinali... |
| cflm 10208 | Value of the cofinality fu... |
| cf0 10209 | Value of the cofinality fu... |
| cardcf 10210 | Cofinality is a cardinal n... |
| cflecard 10211 | Cofinality is bounded by t... |
| cfle 10212 | Cofinality is bounded by i... |
| cfon 10213 | The cofinality of any set ... |
| cfonOLD 10214 | Obsolete version of ~ cfon... |
| cfeq0 10215 | Only the ordinal zero has ... |
| cfsuc 10216 | Value of the cofinality fu... |
| cff1 10217 | There is always a map from... |
| cfflb 10218 | If there is a cofinal map ... |
| cfval2 10219 | Another expression for the... |
| coflim 10220 | A simpler expression for t... |
| cflim3 10221 | Another expression for the... |
| cflim2 10222 | The cofinality function is... |
| cfom 10223 | Value of the cofinality fu... |
| cfss 10224 | There is a cofinal subset ... |
| cfslb 10225 | Any cofinal subset of ` A ... |
| cfslbn 10226 | Any subset of ` A ` smalle... |
| cfslb2n 10227 | Any small collection of sm... |
| cofsmo 10228 | Any cofinal map implies th... |
| cfsmolem 10229 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10230 | The map in ~ cff1 can be a... |
| cfcoflem 10231 | Lemma for ~ cfcof , showin... |
| coftr 10232 | If there is a cofinal map ... |
| cfcof 10233 | If there is a cofinal map ... |
| cfidm 10234 | The cofinality function is... |
| alephsing 10235 | The cofinality of a limit ... |
| sornom 10236 | The range of a single-step... |
| isfin1a 10251 | Definition of a Ia-finite ... |
| fin1ai 10252 | Property of a Ia-finite se... |
| isfin2 10253 | Definition of a II-finite ... |
| fin2i 10254 | Property of a II-finite se... |
| isfin3 10255 | Definition of a III-finite... |
| isfin4 10256 | Definition of a IV-finite ... |
| fin4i 10257 | Infer that a set is IV-inf... |
| isfin5 10258 | Definition of a V-finite s... |
| isfin6 10259 | Definition of a VI-finite ... |
| isfin7 10260 | Definition of a VII-finite... |
| sdom2en01 10261 | A set with less than two e... |
| infpssrlem1 10262 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10263 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10264 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10265 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10266 | Lemma for ~ infpssr . (Co... |
| infpssr 10267 | Dedekind infinity implies ... |
| fin4en1 10268 | Dedekind finite is a cardi... |
| ssfin4 10269 | Dedekind finite sets have ... |
| domfin4 10270 | A set dominated by a Dedek... |
| ominf4 10271 | ` _om ` is Dedekind infini... |
| infpssALT 10272 | Alternate proof of ~ infps... |
| isfin4-2 10273 | Alternate definition of IV... |
| isfin4p1 10274 | Alternate definition of IV... |
| fin23lem7 10275 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10276 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10277 | A II-finite set contains m... |
| isfin2-2 10278 | ` Fin2 ` expressed in term... |
| ssfin2 10279 | A subset of a II-finite se... |
| enfin2i 10280 | II-finiteness is a cardina... |
| fin23lem24 10281 | Lemma for ~ fin23 . In a ... |
| fincssdom 10282 | In a chain of finite sets,... |
| fin23lem25 10283 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10284 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10285 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10286 | Lemma for ~ fin23 but coul... |
| fin23lem27 10287 | The mapping constructed in... |
| isfin3ds 10288 | Property of a III-finite s... |
| ssfin3ds 10289 | A subset of a III-finite s... |
| fin23lem12 10290 | The beginning of the proof... |
| fin23lem13 10291 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10292 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10293 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10294 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10295 | Lemma for ~ fin23 . The f... |
| fin23lem20 10296 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10297 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10298 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10299 | Lemma for ~ fin23 . The r... |
| fin23lem29 10300 | Lemma for ~ fin23 . The r... |
| fin23lem30 10301 | Lemma for ~ fin23 . The r... |
| fin23lem31 10302 | Lemma for ~ fin23 . The r... |
| fin23lem32 10303 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10304 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10305 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10306 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10307 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10308 | Lemma for ~ fin23 . The c... |
| fin23lem39 10309 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10310 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10311 | Lemma for ~ fin23 . A set... |
| isf32lem1 10312 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10313 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10314 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10315 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10316 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10317 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10318 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10319 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10320 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10321 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10322 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10323 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10324 | One half of ~ isfin3-2 . ... |
| isf33lem 10325 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10326 | Weakly Dedekind-infinite s... |
| isfin3-3 10327 | Weakly Dedekind-infinite s... |
| fin33i 10328 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10329 | Lemma for ~ compsscnv . (... |
| compsscnv 10330 | Complementation on a power... |
| isf34lem1 10331 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10332 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10333 | Complementation is an anti... |
| isf34lem3 10334 | Lemma for ~ isfin3-4 . (C... |
| compss 10335 | Express image under of the... |
| isf34lem4 10336 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10337 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10338 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10339 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10340 | Inference from ~ isfin3-4 ... |
| isfin3-4 10341 | Weakly Dedekind-infinite s... |
| fin11a 10342 | Every I-finite set is Ia-f... |
| enfin1ai 10343 | Ia-finiteness is a cardina... |
| isfin1-2 10344 | A set is finite in the usu... |
| isfin1-3 10345 | A set is I-finite iff ever... |
| isfin1-4 10346 | A set is I-finite iff ever... |
| dffin1-5 10347 | Compact quantifier-free ve... |
| fin23 10348 | Every II-finite set (every... |
| fin34 10349 | Every III-finite set is IV... |
| isfin5-2 10350 | Alternate definition of V-... |
| fin45 10351 | Every IV-finite set is V-f... |
| fin56 10352 | Every V-finite set is VI-f... |
| fin17 10353 | Every I-finite set is VII-... |
| fin67 10354 | Every VI-finite set is VII... |
| isfin7-2 10355 | A set is VII-finite iff it... |
| fin71num 10356 | A well-orderable set is VI... |
| dffin7-2 10357 | Class form of ~ isfin7-2 .... |
| dfacfin7 10358 | Axiom of Choice equivalent... |
| fin1a2lem1 10359 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10360 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10361 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10362 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10363 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10364 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10365 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10366 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10367 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10368 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10369 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10370 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10371 | Lemma for ~ fin1a2 . (Con... |
| fin12 10372 | Weak theorem which skips I... |
| fin1a2s 10373 | An II-infinite set can hav... |
| fin1a2 10374 | Every Ia-finite set is II-... |
| itunifval 10375 | Function value of iterated... |
| itunifn 10376 | Functionality of the itera... |
| ituni0 10377 | A zero-fold iterated union... |
| itunisuc 10378 | Successor iterated union. ... |
| itunitc1 10379 | Each union iterate is a me... |
| itunitc 10380 | The union of all union ite... |
| ituniiun 10381 | Unwrap an iterated union f... |
| hsmexlem7 10382 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10383 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10384 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10385 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10386 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10387 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10388 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10389 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10390 | Lemma for ~ hsmex . (Cont... |
| hsmex 10391 | The collection of heredita... |
| hsmex2 10392 | The set of hereditary size... |
| hsmex3 10393 | The set of hereditary size... |
| axcc2lem 10395 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10396 | A possibly more useful ver... |
| axcc3 10397 | A possibly more useful ver... |
| axcc4 10398 | A version of ~ axcc3 that ... |
| acncc 10399 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10400 | Relax the constraint on ~ ... |
| domtriomlem 10401 | Lemma for ~ domtriom . (C... |
| domtriom 10402 | Trichotomy of equinumerosi... |
| fin41 10403 | Under countable choice, th... |
| dominf 10404 | A nonempty set that is a s... |
| dcomex 10406 | The Axiom of Dependent Cho... |
| axdc2lem 10407 | Lemma for ~ axdc2 . We co... |
| axdc2 10408 | An apparent strengthening ... |
| axdc3lem 10409 | The class ` S ` of finite ... |
| axdc3lem2 10410 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10411 | Simple substitution lemma ... |
| axdc3lem4 10412 | Lemma for ~ axdc3 . We ha... |
| axdc3 10413 | Dependent Choice. Axiom D... |
| axdc4lem 10414 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10415 | A more general version of ... |
| axcclem 10416 | Lemma for ~ axcc . (Contr... |
| axcc 10417 | Although CC can be proven ... |
| zfac 10419 | Axiom of Choice expressed ... |
| ac2 10420 | Axiom of Choice equivalent... |
| ac3 10421 | Axiom of Choice using abbr... |
| axac3 10423 | This theorem asserts that ... |
| ackm 10424 | A remarkable equivalent to... |
| axac2 10425 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10426 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10427 | Apply a choice equivalent.... |
| cardeqv 10428 | All sets are well-orderabl... |
| numth3 10429 | All sets are well-orderabl... |
| numth2 10430 | Numeration theorem: any se... |
| numth 10431 | Numeration theorem: every ... |
| ac7 10432 | An Axiom of Choice equival... |
| ac7g 10433 | An Axiom of Choice equival... |
| ac4 10434 | Equivalent of Axiom of Cho... |
| ac4c 10435 | Equivalent of Axiom of Cho... |
| ac5 10436 | An Axiom of Choice equival... |
| ac5b 10437 | Equivalent of Axiom of Cho... |
| ac6num 10438 | A version of ~ ac6 which t... |
| ac6 10439 | Equivalent of Axiom of Cho... |
| ac6c4 10440 | Equivalent of Axiom of Cho... |
| ac6c5 10441 | Equivalent of Axiom of Cho... |
| ac9 10442 | An Axiom of Choice equival... |
| ac6s 10443 | Equivalent of Axiom of Cho... |
| ac6n 10444 | Equivalent of Axiom of Cho... |
| ac6s2 10445 | Generalization of the Axio... |
| ac6s3 10446 | Generalization of the Axio... |
| ac6sg 10447 | ~ ac6s with sethood as ant... |
| ac6sf 10448 | Version of ~ ac6 with boun... |
| ac6s4 10449 | Generalization of the Axio... |
| ac6s5 10450 | Generalization of the Axio... |
| ac8 10451 | An Axiom of Choice equival... |
| ac9s 10452 | An Axiom of Choice equival... |
| numthcor 10453 | Any set is strictly domina... |
| weth 10454 | Well-ordering theorem: any... |
| zorn2lem1 10455 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10456 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10457 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10458 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10459 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10460 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10461 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10462 | Zorn's Lemma of [Monk1] p.... |
| zorng 10463 | Zorn's Lemma. If the unio... |
| zornn0g 10464 | Variant of Zorn's lemma ~ ... |
| zorn2 10465 | Zorn's Lemma of [Monk1] p.... |
| zorn 10466 | Zorn's Lemma. If the unio... |
| zornn0 10467 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10468 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10469 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10470 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10471 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10472 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10473 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10474 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10475 | The Teichmüller-Tukey... |
| ttukeyg 10476 | The Teichmüller-Tukey... |
| ttukey 10477 | The Teichmüller-Tukey... |
| axdclem 10478 | Lemma for ~ axdc . (Contr... |
| axdclem2 10479 | Lemma for ~ axdc . Using ... |
| axdc 10480 | This theorem derives ~ ax-... |
| fodomg 10481 | An onto function implies d... |
| fodom 10482 | An onto function implies d... |
| dmct 10483 | The domain of a countable ... |
| rnct 10484 | The range of a countable s... |
| fodomb 10485 | Equivalence of an onto map... |
| wdomac 10486 | When assuming AC, weak and... |
| brdom3 10487 | Equivalence to a dominance... |
| brdom5 10488 | An equivalence to a domina... |
| brdom4 10489 | An equivalence to a domina... |
| brdom7disj 10490 | An equivalence to a domina... |
| brdom6disj 10491 | An equivalence to a domina... |
| fin71ac 10492 | Once we allow AC, the "str... |
| imadomg 10493 | An image of a function und... |
| fimact 10494 | The image by a function of... |
| fnrndomg 10495 | The range of a function is... |
| fnct 10496 | If the domain of a functio... |
| mptct 10497 | A countable mapping set is... |
| iunfo 10498 | Existence of an onto funct... |
| iundom2g 10499 | An upper bound for the car... |
| iundomg 10500 | An upper bound for the car... |
| iundom 10501 | An upper bound for the car... |
| unidom 10502 | An upper bound for the car... |
| uniimadom 10503 | An upper bound for the car... |
| uniimadomf 10504 | An upper bound for the car... |
| cardval 10505 | The value of the cardinal ... |
| cardid 10506 | Any set is equinumerous to... |
| cardidg 10507 | Any set is equinumerous to... |
| cardidd 10508 | Any set is equinumerous to... |
| cardf 10509 | The cardinality function i... |
| carden 10510 | Two sets are equinumerous ... |
| cardeq0 10511 | Only the empty set has car... |
| unsnen 10512 | Equinumerosity of a set wi... |
| carddom 10513 | Two sets have the dominanc... |
| cardsdom 10514 | Two sets have the strict d... |
| domtri 10515 | Trichotomy law for dominan... |
| entric 10516 | Trichotomy of equinumerosi... |
| entri2 10517 | Trichotomy of dominance an... |
| entri3 10518 | Trichotomy of dominance. ... |
| sdomsdomcard 10519 | A set strictly dominates i... |
| canth3 10520 | Cantor's theorem in terms ... |
| infxpidm 10521 | Every infinite class is eq... |
| ondomon 10522 | The class of ordinals domi... |
| cardmin 10523 | The smallest ordinal that ... |
| ficard 10524 | A set is finite iff its ca... |
| infinfg 10525 | Equivalence between two in... |
| infinf 10526 | Equivalence between two in... |
| unirnfdomd 10527 | The union of the range of ... |
| konigthlem 10528 | Lemma for ~ konigth . (Co... |
| konigth 10529 | Konig's Theorem. If ` m (... |
| alephsucpw 10530 | The power set of an aleph ... |
| aleph1 10531 | The set exponentiation of ... |
| alephval2 10532 | An alternate way to expres... |
| dominfac 10533 | A nonempty set that is a s... |
| iunctb 10534 | The countable union of cou... |
| unictb 10535 | The countable union of cou... |
| infmap 10536 | An exponentiation law for ... |
| alephadd 10537 | The sum of two alephs is t... |
| alephmul 10538 | The product of two alephs ... |
| alephexp1 10539 | An exponentiation law for ... |
| alephsuc3 10540 | An alternate representatio... |
| alephexp2 10541 | An expression equinumerous... |
| alephreg 10542 | A successor aleph is regul... |
| pwcfsdom 10543 | A corollary of Konig's The... |
| cfpwsdom 10544 | A corollary of Konig's The... |
| alephom 10545 | From ~ canth2 , we know th... |
| smobeth 10546 | The beth function is stric... |
| nd1 10547 | A lemma for proving condit... |
| nd2 10548 | A lemma for proving condit... |
| nd3 10549 | A lemma for proving condit... |
| nd4 10550 | A lemma for proving condit... |
| axextnd 10551 | A version of the Axiom of ... |
| axrepndlem1 10552 | Lemma for the Axiom of Rep... |
| axrepndlem2 10553 | Lemma for the Axiom of Rep... |
| axrepnd 10554 | A version of the Axiom of ... |
| axunndlem1 10555 | Lemma for the Axiom of Uni... |
| axunnd 10556 | A version of the Axiom of ... |
| axpowndlem1 10557 | Lemma for the Axiom of Pow... |
| axpowndlem2 10558 | Lemma for the Axiom of Pow... |
| axpowndlem3 10559 | Lemma for the Axiom of Pow... |
| axpowndlem4 10560 | Lemma for the Axiom of Pow... |
| axpownd 10561 | A version of the Axiom of ... |
| axregndlem1 10562 | Lemma for the Axiom of Reg... |
| axregndlem2 10563 | Lemma for the Axiom of Reg... |
| axregnd 10564 | A version of the Axiom of ... |
| axinfndlem1 10565 | Lemma for the Axiom of Inf... |
| axinfnd 10566 | A version of the Axiom of ... |
| axacndlem1 10567 | Lemma for the Axiom of Cho... |
| axacndlem2 10568 | Lemma for the Axiom of Cho... |
| axacndlem3 10569 | Lemma for the Axiom of Cho... |
| axacndlem4 10570 | Lemma for the Axiom of Cho... |
| axacndlem5 10571 | Lemma for the Axiom of Cho... |
| axacnd 10572 | A version of the Axiom of ... |
| zfcndext 10573 | Axiom of Extensionality ~ ... |
| zfcndrep 10574 | Axiom of Replacement ~ ax-... |
| zfcndun 10575 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10576 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10577 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10578 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10579 | Axiom of Choice ~ ax-ac , ... |
| elgch 10582 | Elementhood in the collect... |
| fingch 10583 | A finite set is a GCH-set.... |
| gchi 10584 | The only GCH-sets which ha... |
| gchen1 10585 | If ` A <_ B < ~P A ` , and... |
| gchen2 10586 | If ` A < B <_ ~P A ` , and... |
| gchor 10587 | If ` A <_ B <_ ~P A ` , an... |
| engch 10588 | The property of being a GC... |
| gchdomtri 10589 | Under certain conditions, ... |
| fpwwe2cbv 10590 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10591 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10592 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10593 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10594 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10595 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10596 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10597 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10598 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10599 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10600 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10601 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10602 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10603 | Given any function ` F ` f... |
| fpwwecbv 10604 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10605 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10606 | Given any function ` F ` f... |
| canth4 10607 | An "effective" form of Can... |
| canthnumlem 10608 | Lemma for ~ canthnum . (C... |
| canthnum 10609 | The set of well-orderable ... |
| canthwelem 10610 | Lemma for ~ canthwe . (Co... |
| canthwe 10611 | The set of well-orders of ... |
| canthp1lem1 10612 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10613 | Lemma for ~ canthp1 . (Co... |
| canthp1 10614 | A slightly stronger form o... |
| finngch 10615 | The exclusion of finite se... |
| gchdju1 10616 | An infinite GCH-set is ide... |
| gchinf 10617 | An infinite GCH-set is Ded... |
| pwfseqlem1 10618 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10619 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10620 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10621 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10622 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10623 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10624 | The powerset of a Dedekind... |
| pwxpndom2 10625 | The powerset of a Dedekind... |
| pwxpndom 10626 | The powerset of a Dedekind... |
| pwdjundom 10627 | The powerset of a Dedekind... |
| gchdjuidm 10628 | An infinite GCH-set is ide... |
| gchxpidm 10629 | An infinite GCH-set is ide... |
| gchpwdom 10630 | A relationship between dom... |
| gchaleph 10631 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10632 | If ` ( aleph `` A ) ` and ... |
| hargch 10633 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10634 | If ` ( aleph `` suc A ) ` ... |
| gch2 10635 | It is sufficient to requir... |
| gch3 10636 | An equivalent formulation ... |
| gch-kn 10637 | The equivalence of two ver... |
| gchaclem 10638 | Lemma for ~ gchac (obsolet... |
| gchhar 10639 | A "local" form of ~ gchac ... |
| gchacg 10640 | A "local" form of ~ gchac ... |
| gchac 10641 | The Generalized Continuum ... |
| elwina 10646 | Conditions of weak inacces... |
| elina 10647 | Conditions of strong inacc... |
| winaon 10648 | A weakly inaccessible card... |
| inawinalem 10649 | Lemma for ~ inawina . (Co... |
| inawina 10650 | Every strongly inaccessibl... |
| omina 10651 | ` _om ` is a strongly inac... |
| winacard 10652 | A weakly inaccessible card... |
| winainflem 10653 | A weakly inaccessible card... |
| winainf 10654 | A weakly inaccessible card... |
| winalim 10655 | A weakly inaccessible card... |
| winalim2 10656 | A nontrivial weakly inacce... |
| winafp 10657 | A nontrivial weakly inacce... |
| winafpi 10658 | This theorem, which states... |
| gchina 10659 | Assuming the GCH, weakly a... |
| iswun 10664 | Properties of a weak unive... |
| wuntr 10665 | A weak universe is transit... |
| wununi 10666 | A weak universe is closed ... |
| wunpw 10667 | A weak universe is closed ... |
| wunelss 10668 | The elements of a weak uni... |
| wunpr 10669 | A weak universe is closed ... |
| wunun 10670 | A weak universe is closed ... |
| wuntp 10671 | A weak universe is closed ... |
| wunss 10672 | A weak universe is closed ... |
| wunin 10673 | A weak universe is closed ... |
| wundif 10674 | A weak universe is closed ... |
| wunint 10675 | A weak universe is closed ... |
| wunsn 10676 | A weak universe is closed ... |
| wunsuc 10677 | A weak universe is closed ... |
| wun0 10678 | A weak universe contains t... |
| wunr1om 10679 | A weak universe is infinit... |
| wunom 10680 | A weak universe contains a... |
| wunfi 10681 | A weak universe contains a... |
| wunop 10682 | A weak universe is closed ... |
| wunot 10683 | A weak universe is closed ... |
| wunxp 10684 | A weak universe is closed ... |
| wunpm 10685 | A weak universe is closed ... |
| wunmap 10686 | A weak universe is closed ... |
| wunf 10687 | A weak universe is closed ... |
| wundm 10688 | A weak universe is closed ... |
| wunrn 10689 | A weak universe is closed ... |
| wuncnv 10690 | A weak universe is closed ... |
| wunres 10691 | A weak universe is closed ... |
| wunfv 10692 | A weak universe is closed ... |
| wunco 10693 | A weak universe is closed ... |
| wuntpos 10694 | A weak universe is closed ... |
| intwun 10695 | The intersection of a coll... |
| r1limwun 10696 | Each limit stage in the cu... |
| r1wunlim 10697 | The weak universes in the ... |
| wunex2 10698 | Construct a weak universe ... |
| wunex 10699 | Construct a weak universe ... |
| uniwun 10700 | Every set is contained in ... |
| wunex3 10701 | Construct a weak universe ... |
| wuncval 10702 | Value of the weak universe... |
| wuncid 10703 | The weak universe closure ... |
| wunccl 10704 | The weak universe closure ... |
| wuncss 10705 | The weak universe closure ... |
| wuncidm 10706 | The weak universe closure ... |
| wuncval2 10707 | Our earlier expression for... |
| eltskg 10710 | Properties of a Tarski cla... |
| eltsk2g 10711 | Properties of a Tarski cla... |
| tskpwss 10712 | First axiom of a Tarski cl... |
| tskpw 10713 | Second axiom of a Tarski c... |
| tsken 10714 | Third axiom of a Tarski cl... |
| 0tsk 10715 | The empty set is a (transi... |
| tsksdom 10716 | An element of a Tarski cla... |
| tskssel 10717 | A part of a Tarski class s... |
| tskss 10718 | The subsets of an element ... |
| tskin 10719 | The intersection of two el... |
| tsksn 10720 | A singleton of an element ... |
| tsktrss 10721 | A transitive element of a ... |
| tsksuc 10722 | If an element of a Tarski ... |
| tsk0 10723 | A nonempty Tarski class co... |
| tsk1 10724 | One is an element of a non... |
| tsk2 10725 | Two is an element of a non... |
| 2domtsk 10726 | If a Tarski class is not e... |
| tskr1om 10727 | A nonempty Tarski class is... |
| tskr1om2 10728 | A nonempty Tarski class co... |
| tskinf 10729 | A nonempty Tarski class is... |
| tskpr 10730 | If ` A ` and ` B ` are mem... |
| tskop 10731 | If ` A ` and ` B ` are mem... |
| tskxpss 10732 | A Cartesian product of two... |
| tskwe2 10733 | A Tarski class is well-ord... |
| inttsk 10734 | The intersection of a coll... |
| inar1 10735 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10736 | Alternate proof of ~ r1om ... |
| rankcf 10737 | Any set must be at least a... |
| inatsk 10738 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10739 | The set of hereditarily fi... |
| tskord 10740 | A Tarski class contains al... |
| tskcard 10741 | An even more direct relati... |
| r1tskina 10742 | There is a direct relation... |
| tskuni 10743 | The union of an element of... |
| tskwun 10744 | A nonempty transitive Tars... |
| tskint 10745 | The intersection of an ele... |
| tskun 10746 | The union of two elements ... |
| tskxp 10747 | The Cartesian product of t... |
| tskmap 10748 | Set exponentiation is an e... |
| tskurn 10749 | A transitive Tarski class ... |
| elgrug 10752 | Properties of a Grothendie... |
| grutr 10753 | A Grothendieck universe is... |
| gruelss 10754 | A Grothendieck universe is... |
| grupw 10755 | A Grothendieck universe co... |
| gruss 10756 | Any subset of an element o... |
| grupr 10757 | A Grothendieck universe co... |
| gruurn 10758 | A Grothendieck universe co... |
| gruiun 10759 | If ` B ( x ) ` is a family... |
| gruuni 10760 | A Grothendieck universe co... |
| grurn 10761 | A Grothendieck universe co... |
| gruima 10762 | A Grothendieck universe co... |
| gruel 10763 | Any element of an element ... |
| grusn 10764 | A Grothendieck universe co... |
| gruop 10765 | A Grothendieck universe co... |
| gruun 10766 | A Grothendieck universe co... |
| gruxp 10767 | A Grothendieck universe co... |
| grumap 10768 | A Grothendieck universe co... |
| gruixp 10769 | A Grothendieck universe co... |
| gruiin 10770 | A Grothendieck universe co... |
| gruf 10771 | A Grothendieck universe co... |
| gruen 10772 | A Grothendieck universe co... |
| gruwun 10773 | A nonempty Grothendieck un... |
| intgru 10774 | The intersection of a fami... |
| ingru 10775 | The intersection of a univ... |
| wfgru 10776 | The wellfounded part of a ... |
| grudomon 10777 | Each ordinal that is compa... |
| gruina 10778 | If a Grothendieck universe... |
| grur1a 10779 | A characterization of Grot... |
| grur1 10780 | A characterization of Grot... |
| grutsk1 10781 | Grothendieck universes are... |
| grutsk 10782 | Grothendieck universes are... |
| axgroth5 10784 | The Tarski-Grothendieck ax... |
| axgroth2 10785 | Alternate version of the T... |
| grothpw 10786 | Derive the Axiom of Power ... |
| grothpwex 10787 | Derive the Axiom of Power ... |
| axgroth6 10788 | The Tarski-Grothendieck ax... |
| grothomex 10789 | The Tarski-Grothendieck Ax... |
| grothac 10790 | The Tarski-Grothendieck Ax... |
| axgroth3 10791 | Alternate version of the T... |
| axgroth4 10792 | Alternate version of the T... |
| grothprimlem 10793 | Lemma for ~ grothprim . E... |
| grothprim 10794 | The Tarski-Grothendieck Ax... |
| grothtsk 10795 | The Tarski-Grothendieck Ax... |
| inaprc 10796 | An equivalent to the Tarsk... |
| tskmval 10799 | Value of our tarski map. ... |
| tskmid 10800 | The set ` A ` is an elemen... |
| tskmcl 10801 | A Tarski class that contai... |
| sstskm 10802 | Being a part of ` ( tarski... |
| eltskm 10803 | Belonging to ` ( tarskiMap... |
| elni 10836 | Membership in the class of... |
| elni2 10837 | Membership in the class of... |
| pinn 10838 | A positive integer is a na... |
| pion 10839 | A positive integer is an o... |
| piord 10840 | A positive integer is ordi... |
| niex 10841 | The class of positive inte... |
| 0npi 10842 | The empty set is not a pos... |
| 1pi 10843 | Ordinal 'one' is a positiv... |
| addpiord 10844 | Positive integer addition ... |
| mulpiord 10845 | Positive integer multiplic... |
| mulidpi 10846 | 1 is an identity element f... |
| ltpiord 10847 | Positive integer 'less tha... |
| ltsopi 10848 | Positive integer 'less tha... |
| ltrelpi 10849 | Positive integer 'less tha... |
| dmaddpi 10850 | Domain of addition on posi... |
| dmmulpi 10851 | Domain of multiplication o... |
| addclpi 10852 | Closure of addition of pos... |
| mulclpi 10853 | Closure of multiplication ... |
| addcompi 10854 | Addition of positive integ... |
| addasspi 10855 | Addition of positive integ... |
| mulcompi 10856 | Multiplication of positive... |
| mulasspi 10857 | Multiplication of positive... |
| distrpi 10858 | Multiplication of positive... |
| addcanpi 10859 | Addition cancellation law ... |
| mulcanpi 10860 | Multiplication cancellatio... |
| addnidpi 10861 | There is no identity eleme... |
| ltexpi 10862 | Ordering on positive integ... |
| ltapi 10863 | Ordering property of addit... |
| ltmpi 10864 | Ordering property of multi... |
| 1lt2pi 10865 | One is less than two (one ... |
| nlt1pi 10866 | No positive integer is les... |
| indpi 10867 | Principle of Finite Induct... |
| enqbreq 10879 | Equivalence relation for p... |
| enqbreq2 10880 | Equivalence relation for p... |
| enqer 10881 | The equivalence relation f... |
| enqex 10882 | The equivalence relation f... |
| nqex 10883 | The class of positive frac... |
| 0nnq 10884 | The empty set is not a pos... |
| elpqn 10885 | Each positive fraction is ... |
| ltrelnq 10886 | Positive fraction 'less th... |
| pinq 10887 | The representatives of pos... |
| 1nq 10888 | The positive fraction 'one... |
| nqereu 10889 | There is a unique element ... |
| nqerf 10890 | Corollary of ~ nqereu : th... |
| nqercl 10891 | Corollary of ~ nqereu : cl... |
| nqerrel 10892 | Any member of ` ( N. X. N.... |
| nqerid 10893 | Corollary of ~ nqereu : th... |
| enqeq 10894 | Corollary of ~ nqereu : if... |
| nqereq 10895 | The function ` /Q ` acts a... |
| addpipq2 10896 | Addition of positive fract... |
| addpipq 10897 | Addition of positive fract... |
| addpqnq 10898 | Addition of positive fract... |
| mulpipq2 10899 | Multiplication of positive... |
| mulpipq 10900 | Multiplication of positive... |
| mulpqnq 10901 | Multiplication of positive... |
| ordpipq 10902 | Ordering of positive fract... |
| ordpinq 10903 | Ordering of positive fract... |
| addpqf 10904 | Closure of addition on pos... |
| addclnq 10905 | Closure of addition on pos... |
| mulpqf 10906 | Closure of multiplication ... |
| mulclnq 10907 | Closure of multiplication ... |
| addnqf 10908 | Domain of addition on posi... |
| mulnqf 10909 | Domain of multiplication o... |
| addcompq 10910 | Addition of positive fract... |
| addcomnq 10911 | Addition of positive fract... |
| mulcompq 10912 | Multiplication of positive... |
| mulcomnq 10913 | Multiplication of positive... |
| adderpqlem 10914 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10915 | Lemma for ~ mulerpq . (Co... |
| adderpq 10916 | Addition is compatible wit... |
| mulerpq 10917 | Multiplication is compatib... |
| addassnq 10918 | Addition of positive fract... |
| mulassnq 10919 | Multiplication of positive... |
| mulcanenq 10920 | Lemma for distributive law... |
| distrnq 10921 | Multiplication of positive... |
| 1nqenq 10922 | The equivalence class of r... |
| mulidnq 10923 | Multiplication identity el... |
| recmulnq 10924 | Relationship between recip... |
| recidnq 10925 | A positive fraction times ... |
| recclnq 10926 | Closure law for positive f... |
| recrecnq 10927 | Reciprocal of reciprocal o... |
| dmrecnq 10928 | Domain of reciprocal on po... |
| ltsonq 10929 | 'Less than' is a strict or... |
| lterpq 10930 | Compatibility of ordering ... |
| ltanq 10931 | Ordering property of addit... |
| ltmnq 10932 | Ordering property of multi... |
| 1lt2nq 10933 | One is less than two (one ... |
| ltaddnq 10934 | The sum of two fractions i... |
| ltexnq 10935 | Ordering on positive fract... |
| halfnq 10936 | One-half of any positive f... |
| nsmallnq 10937 | The is no smallest positiv... |
| ltbtwnnq 10938 | There exists a number betw... |
| ltrnq 10939 | Ordering property of recip... |
| archnq 10940 | For any fraction, there is... |
| npex 10946 | The class of positive real... |
| elnp 10947 | Membership in positive rea... |
| elnpi 10948 | Membership in positive rea... |
| prn0 10949 | A positive real is not emp... |
| prpssnq 10950 | A positive real is a subse... |
| elprnq 10951 | A positive real is a set o... |
| 0npr 10952 | The empty set is not a pos... |
| prcdnq 10953 | A positive real is closed ... |
| prub 10954 | A positive fraction not in... |
| prnmax 10955 | A positive real has no lar... |
| npomex 10956 | A simplifying observation,... |
| prnmadd 10957 | A positive real has no lar... |
| ltrelpr 10958 | Positive real 'less than' ... |
| genpv 10959 | Value of general operation... |
| genpelv 10960 | Membership in value of gen... |
| genpprecl 10961 | Pre-closure law for genera... |
| genpdm 10962 | Domain of general operatio... |
| genpn0 10963 | The result of an operation... |
| genpss 10964 | The result of an operation... |
| genpnnp 10965 | The result of an operation... |
| genpcd 10966 | Downward closure of an ope... |
| genpnmax 10967 | An operation on positive r... |
| genpcl 10968 | Closure of an operation on... |
| genpass 10969 | Associativity of an operat... |
| plpv 10970 | Value of addition on posit... |
| mpv 10971 | Value of multiplication on... |
| dmplp 10972 | Domain of addition on posi... |
| dmmp 10973 | Domain of multiplication o... |
| nqpr 10974 | The canonical embedding of... |
| 1pr 10975 | The positive real number '... |
| addclprlem1 10976 | Lemma to prove downward cl... |
| addclprlem2 10977 | Lemma to prove downward cl... |
| addclpr 10978 | Closure of addition on pos... |
| mulclprlem 10979 | Lemma to prove downward cl... |
| mulclpr 10980 | Closure of multiplication ... |
| addcompr 10981 | Addition of positive reals... |
| addasspr 10982 | Addition of positive reals... |
| mulcompr 10983 | Multiplication of positive... |
| mulasspr 10984 | Multiplication of positive... |
| distrlem1pr 10985 | Lemma for distributive law... |
| distrlem4pr 10986 | Lemma for distributive law... |
| distrlem5pr 10987 | Lemma for distributive law... |
| distrpr 10988 | Multiplication of positive... |
| 1idpr 10989 | 1 is an identity element f... |
| ltprord 10990 | Positive real 'less than' ... |
| psslinpr 10991 | Proper subset is a linear ... |
| ltsopr 10992 | Positive real 'less than' ... |
| prlem934 10993 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 10994 | The sum of two positive re... |
| ltaddpr2 10995 | The sum of two positive re... |
| ltexprlem1 10996 | Lemma for Proposition 9-3.... |
| ltexprlem2 10997 | Lemma for Proposition 9-3.... |
| ltexprlem3 10998 | Lemma for Proposition 9-3.... |
| ltexprlem4 10999 | Lemma for Proposition 9-3.... |
| ltexprlem5 11000 | Lemma for Proposition 9-3.... |
| ltexprlem6 11001 | Lemma for Proposition 9-3.... |
| ltexprlem7 11002 | Lemma for Proposition 9-3.... |
| ltexpri 11003 | Proposition 9-3.5(iv) of [... |
| ltaprlem 11004 | Lemma for Proposition 9-3.... |
| ltapr 11005 | Ordering property of addit... |
| addcanpr 11006 | Addition cancellation law ... |
| prlem936 11007 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 11008 | Lemma for Proposition 9-3.... |
| reclem3pr 11009 | Lemma for Proposition 9-3.... |
| reclem4pr 11010 | Lemma for Proposition 9-3.... |
| recexpr 11011 | The reciprocal of a positi... |
| suplem1pr 11012 | The union of a nonempty, b... |
| suplem2pr 11013 | The union of a set of posi... |
| supexpr 11014 | The union of a nonempty, b... |
| enrer 11023 | The equivalence relation f... |
| nrex1 11024 | The class of signed reals ... |
| enrbreq 11025 | Equivalence relation for s... |
| enreceq 11026 | Equivalence class equality... |
| enrex 11027 | The equivalence relation f... |
| ltrelsr 11028 | Signed real 'less than' is... |
| addcmpblnr 11029 | Lemma showing compatibilit... |
| mulcmpblnrlem 11030 | Lemma used in lemma showin... |
| mulcmpblnr 11031 | Lemma showing compatibilit... |
| prsrlem1 11032 | Decomposing signed reals i... |
| addsrmo 11033 | There is at most one resul... |
| mulsrmo 11034 | There is at most one resul... |
| addsrpr 11035 | Addition of signed reals i... |
| mulsrpr 11036 | Multiplication of signed r... |
| ltsrpr 11037 | Ordering of signed reals i... |
| gt0srpr 11038 | Greater than zero in terms... |
| 0nsr 11039 | The empty set is not a sig... |
| 0r 11040 | The constant ` 0R ` is a s... |
| 1sr 11041 | The constant ` 1R ` is a s... |
| m1r 11042 | The constant ` -1R ` is a ... |
| addclsr 11043 | Closure of addition on sig... |
| mulclsr 11044 | Closure of multiplication ... |
| dmaddsr 11045 | Domain of addition on sign... |
| dmmulsr 11046 | Domain of multiplication o... |
| addcomsr 11047 | Addition of signed reals i... |
| addasssr 11048 | Addition of signed reals i... |
| mulcomsr 11049 | Multiplication of signed r... |
| mulasssr 11050 | Multiplication of signed r... |
| distrsr 11051 | Multiplication of signed r... |
| m1p1sr 11052 | Minus one plus one is zero... |
| m1m1sr 11053 | Minus one times minus one ... |
| ltsosr 11054 | Signed real 'less than' is... |
| 0lt1sr 11055 | 0 is less than 1 for signe... |
| 1ne0sr 11056 | 1 and 0 are distinct for s... |
| 0idsr 11057 | The signed real number 0 i... |
| 1idsr 11058 | 1 is an identity element f... |
| 00sr 11059 | A signed real times 0 is 0... |
| ltasr 11060 | Ordering property of addit... |
| pn0sr 11061 | A signed real plus its neg... |
| negexsr 11062 | Existence of negative sign... |
| recexsrlem 11063 | The reciprocal of a positi... |
| addgt0sr 11064 | The sum of two positive si... |
| mulgt0sr 11065 | The product of two positiv... |
| sqgt0sr 11066 | The square of a nonzero si... |
| recexsr 11067 | The reciprocal of a nonzer... |
| mappsrpr 11068 | Mapping from positive sign... |
| ltpsrpr 11069 | Mapping of order from posi... |
| map2psrpr 11070 | Equivalence for positive s... |
| supsrlem 11071 | Lemma for supremum theorem... |
| supsr 11072 | A nonempty, bounded set of... |
| opelcn 11089 | Ordered pair membership in... |
| opelreal 11090 | Ordered pair membership in... |
| elreal 11091 | Membership in class of rea... |
| elreal2 11092 | Ordered pair membership in... |
| 0ncn 11093 | The empty set is not a com... |
| ltrelre 11094 | 'Less than' is a relation ... |
| addcnsr 11095 | Addition of complex number... |
| mulcnsr 11096 | Multiplication of complex ... |
| eqresr 11097 | Equality of real numbers i... |
| addresr 11098 | Addition of real numbers i... |
| mulresr 11099 | Multiplication of real num... |
| ltresr 11100 | Ordering of real subset of... |
| ltresr2 11101 | Ordering of real subset of... |
| dfcnqs 11102 | Technical trick to permit ... |
| addcnsrec 11103 | Technical trick to permit ... |
| mulcnsrec 11104 | Technical trick to permit ... |
| axaddf 11105 | Addition is an operation o... |
| axmulf 11106 | Multiplication is an opera... |
| axcnex 11107 | The complex numbers form a... |
| axresscn 11108 | The real numbers are a sub... |
| ax1cn 11109 | 1 is a complex number. Ax... |
| axicn 11110 | ` _i ` is a complex number... |
| axaddcl 11111 | Closure law for addition o... |
| axaddrcl 11112 | Closure law for addition i... |
| axmulcl 11113 | Closure law for multiplica... |
| axmulrcl 11114 | Closure law for multiplica... |
| axmulcom 11115 | Multiplication of complex ... |
| axaddass 11116 | Addition of complex number... |
| axmulass 11117 | Multiplication of complex ... |
| axdistr 11118 | Distributive law for compl... |
| axi2m1 11119 | i-squared equals -1 (expre... |
| ax1ne0 11120 | 1 and 0 are distinct. Axi... |
| ax1rid 11121 | ` 1 ` is an identity eleme... |
| axrnegex 11122 | Existence of negative of r... |
| axrrecex 11123 | Existence of reciprocal of... |
| axcnre 11124 | A complex number can be ex... |
| axpre-lttri 11125 | Ordering on reals satisfie... |
| axpre-lttrn 11126 | Ordering on reals is trans... |
| axpre-ltadd 11127 | Ordering property of addit... |
| axpre-mulgt0 11128 | The product of two positiv... |
| axpre-sup 11129 | A nonempty, bounded-above ... |
| wuncn 11130 | A weak universe containing... |
| cnex 11156 | Alias for ~ ax-cnex . See... |
| addcl 11157 | Alias for ~ ax-addcl , for... |
| readdcl 11158 | Alias for ~ ax-addrcl , fo... |
| mulcl 11159 | Alias for ~ ax-mulcl , for... |
| remulcl 11160 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11161 | Alias for ~ ax-mulcom , fo... |
| addass 11162 | Alias for ~ ax-addass , fo... |
| mulass 11163 | Alias for ~ ax-mulass , fo... |
| adddi 11164 | Alias for ~ ax-distr , for... |
| recn 11165 | A real number is a complex... |
| reex 11166 | The real numbers form a se... |
| reelprrecn 11167 | Reals are a subset of the ... |
| cnelprrecn 11168 | Complex numbers are a subs... |
| mpoaddf 11169 | Addition is an operation o... |
| mpomulf 11170 | Multiplication is an opera... |
| elimne0 11171 | Hypothesis for weak deduct... |
| adddir 11172 | Distributive law for compl... |
| 0cn 11173 | Zero is a complex number. ... |
| 0cnd 11174 | Zero is a complex number, ... |
| c0ex 11175 | Zero is a set. (Contribut... |
| 0elpr01 11176 | 0 is an element of ` { 0 ,... |
| 1cnd 11177 | One is a complex number, d... |
| 1ex 11178 | One is a set. (Contribute... |
| 1elpr01 11179 | 1 is an element of ` { 0 ,... |
| cnre 11180 | Alias for ~ ax-cnre , for ... |
| mulrid 11181 | The number 1 is an identit... |
| mullid 11182 | Identity law for multiplic... |
| 1re 11183 | The number 1 is real. Thi... |
| 1red 11184 | The number 1 is real, dedu... |
| 0re 11185 | The number 0 is real. Rem... |
| 0red 11186 | The number 0 is real, dedu... |
| pr01ssre 11187 | The pair ` { 0 , 1 } ` is ... |
| mulridi 11188 | Identity law for multiplic... |
| mullidi 11189 | Identity law for multiplic... |
| addcli 11190 | Closure law for addition. ... |
| mulcli 11191 | Closure law for multiplica... |
| mulcomi 11192 | Commutative law for multip... |
| mulcomli 11193 | Commutative law for multip... |
| addassi 11194 | Associative law for additi... |
| mulassi 11195 | Associative law for multip... |
| adddii 11196 | Distributive law (left-dis... |
| adddiri 11197 | Distributive law (right-di... |
| recni 11198 | A real number is a complex... |
| readdcli 11199 | Closure law for addition o... |
| remulcli 11200 | Closure law for multiplica... |
| mulridd 11201 | Identity law for multiplic... |
| mullidd 11202 | Identity law for multiplic... |
| addcld 11203 | Closure law for addition. ... |
| mulcld 11204 | Closure law for multiplica... |
| mulcomd 11205 | Commutative law for multip... |
| addassd 11206 | Associative law for additi... |
| mulassd 11207 | Associative law for multip... |
| adddid 11208 | Distributive law (left-dis... |
| adddird 11209 | Distributive law (right-di... |
| adddirp1d 11210 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11211 | Join AB+CB into (A+C) on L... |
| recnd 11212 | Deduction from real number... |
| readdcld 11213 | Closure law for addition o... |
| remulcld 11214 | Closure law for multiplica... |
| pnfnre 11225 | Plus infinity is not a rea... |
| pnfnre2 11226 | Plus infinity is not a rea... |
| mnfnre 11227 | Minus infinity is not a re... |
| ressxr 11228 | The standard reals are a s... |
| rexpssxrxp 11229 | The Cartesian product of s... |
| rexr 11230 | A standard real is an exte... |
| 0xr 11231 | Zero is an extended real. ... |
| renepnf 11232 | No (finite) real equals pl... |
| renemnf 11233 | No real equals minus infin... |
| rexrd 11234 | A standard real is an exte... |
| renepnfd 11235 | No (finite) real equals pl... |
| renemnfd 11236 | No real equals minus infin... |
| pnfex 11237 | Plus infinity exists. (Co... |
| pnfxr 11238 | Plus infinity belongs to t... |
| pnfnemnf 11239 | Plus and minus infinity ar... |
| mnfnepnf 11240 | Minus and plus infinity ar... |
| mnfxr 11241 | Minus infinity belongs to ... |
| rexri 11242 | A standard real is an exte... |
| 1xr 11243 | ` 1 ` is an extended real ... |
| renfdisj 11244 | The reals and the infiniti... |
| ltrelxr 11245 | "Less than" is a relation ... |
| ltrel 11246 | "Less than" is a relation.... |
| lerelxr 11247 | "Less than or equal to" is... |
| lerel 11248 | "Less than or equal to" is... |
| xrlenlt 11249 | "Less than or equal to" ex... |
| xrlenltd 11250 | "Less than or equal to" ex... |
| xrltnle 11251 | "Less than" expressed in t... |
| xrltnled 11252 | 'Less than' in terms of 'l... |
| xrnltled 11253 | "Not less than" implies "l... |
| ssxr 11254 | The three (non-exclusive) ... |
| ltxrlt 11255 | The standard less-than ` <... |
| axlttri 11256 | Ordering on reals satisfie... |
| axlttrn 11257 | Ordering on reals is trans... |
| axltadd 11258 | Ordering property of addit... |
| axmulgt0 11259 | The product of two positiv... |
| axsup 11260 | A nonempty, bounded-above ... |
| lttr 11261 | Alias for ~ axlttrn , for ... |
| mulgt0 11262 | The product of two positiv... |
| lenlt 11263 | 'Less than or equal to' ex... |
| ltnle 11264 | 'Less than' expressed in t... |
| ltso 11265 | 'Less than' is a strict or... |
| gtso 11266 | 'Greater than' is a strict... |
| lttri2 11267 | Consequence of trichotomy.... |
| lttri3 11268 | Trichotomy law for 'less t... |
| lttri4 11269 | Trichotomy law for 'less t... |
| letri3 11270 | Trichotomy law. (Contribu... |
| leloe 11271 | 'Less than or equal to' ex... |
| eqlelt 11272 | Equality in terms of 'less... |
| ltle 11273 | 'Less than' implies 'less ... |
| leltne 11274 | 'Less than or equal to' im... |
| lelttr 11275 | Transitive law. (Contribu... |
| leltletr 11276 | Transitive law, weaker for... |
| ltletr 11277 | Transitive law. (Contribu... |
| ltleletr 11278 | Transitive law, weaker for... |
| letr 11279 | Transitive law. (Contribu... |
| ltnr 11280 | 'Less than' is irreflexive... |
| leid 11281 | 'Less than or equal to' is... |
| ltne 11282 | 'Less than' implies not eq... |
| ltnsym 11283 | 'Less than' is not symmetr... |
| ltnsym2 11284 | 'Less than' is antisymmetr... |
| letric 11285 | Trichotomy law. (Contribu... |
| ltlen 11286 | 'Less than' expressed in t... |
| eqle 11287 | Equality implies 'less tha... |
| eqled 11288 | Equality implies 'less tha... |
| ltadd2 11289 | Addition to both sides of ... |
| ne0gt0 11290 | A nonzero nonnegative numb... |
| lecasei 11291 | Ordering elimination by ca... |
| lelttric 11292 | Trichotomy law. (Contribu... |
| ltlecasei 11293 | Ordering elimination by ca... |
| ltnri 11294 | 'Less than' is irreflexive... |
| eqlei 11295 | Equality implies 'less tha... |
| eqlei2 11296 | Equality implies 'less tha... |
| gtneii 11297 | 'Less than' implies not eq... |
| ltneii 11298 | 'Greater than' implies not... |
| lttri2i 11299 | Consequence of trichotomy.... |
| lttri3i 11300 | Consequence of trichotomy.... |
| letri3i 11301 | Consequence of trichotomy.... |
| leloei 11302 | 'Less than or equal to' in... |
| ltleni 11303 | 'Less than' expressed in t... |
| ltnsymi 11304 | 'Less than' is not symmetr... |
| lenlti 11305 | 'Less than or equal to' in... |
| ltnlei 11306 | 'Less than' in terms of 'l... |
| ltlei 11307 | 'Less than' implies 'less ... |
| ltleii 11308 | 'Less than' implies 'less ... |
| ltnei 11309 | 'Less than' implies not eq... |
| letrii 11310 | Trichotomy law for 'less t... |
| lttri 11311 | 'Less than' is transitive.... |
| lelttri 11312 | 'Less than or equal to', '... |
| ltletri 11313 | 'Less than', 'less than or... |
| letri 11314 | 'Less than or equal to' is... |
| le2tri3i 11315 | Extended trichotomy law fo... |
| ltadd2i 11316 | Addition to both sides of ... |
| mulgt0i 11317 | The product of two positiv... |
| mulgt0ii 11318 | The product of two positiv... |
| ltnrd 11319 | 'Less than' is irreflexive... |
| gtned 11320 | 'Less than' implies not eq... |
| ltned 11321 | 'Greater than' implies not... |
| ne0gt0d 11322 | A nonzero nonnegative numb... |
| lttrid 11323 | Ordering on reals satisfie... |
| lttri2d 11324 | Consequence of trichotomy.... |
| lttri3d 11325 | Consequence of trichotomy.... |
| lttri4d 11326 | Trichotomy law for 'less t... |
| letri3d 11327 | Consequence of trichotomy.... |
| leloed 11328 | 'Less than or equal to' in... |
| eqleltd 11329 | Equality in terms of 'less... |
| ltlend 11330 | 'Less than' expressed in t... |
| lenltd 11331 | 'Less than or equal to' in... |
| ltnled 11332 | 'Less than' in terms of 'l... |
| ltled 11333 | 'Less than' implies 'less ... |
| ltnsymd 11334 | 'Less than' implies 'less ... |
| nltled 11335 | 'Not less than ' implies '... |
| lensymd 11336 | 'Less than or equal to' im... |
| letrid 11337 | Trichotomy law for 'less t... |
| leltned 11338 | 'Less than or equal to' im... |
| leneltd 11339 | 'Less than or equal to' an... |
| mulgt0d 11340 | The product of two positiv... |
| ltadd2d 11341 | Addition to both sides of ... |
| letrd 11342 | Transitive law deduction f... |
| lelttrd 11343 | Transitive law deduction f... |
| ltadd2dd 11344 | Addition to both sides of ... |
| ltletrd 11345 | Transitive law deduction f... |
| lttrd 11346 | Transitive law deduction f... |
| lelttrdi 11347 | If a number is less than a... |
| dedekind 11348 | The Dedekind cut theorem. ... |
| dedekindle 11349 | The Dedekind cut theorem, ... |
| mul12 11350 | Commutative/associative la... |
| mul32 11351 | Commutative/associative la... |
| mul31 11352 | Commutative/associative la... |
| mul4 11353 | Rearrangement of 4 factors... |
| mul4r 11354 | Rearrangement of 4 factors... |
| muladd11 11355 | A simple product of sums e... |
| 1p1times 11356 | Two times a number. (Cont... |
| peano2cn 11357 | A theorem for complex numb... |
| peano2re 11358 | A theorem for reals analog... |
| readdcan 11359 | Cancellation law for addit... |
| 00id 11360 | ` 0 ` is its own additive ... |
| mul02lem1 11361 | Lemma for ~ mul02 . If an... |
| mul02lem2 11362 | Lemma for ~ mul02 . Zero ... |
| mul02 11363 | Multiplication by ` 0 ` . ... |
| mul01 11364 | Multiplication by ` 0 ` . ... |
| addrid 11365 | ` 0 ` is an additive ident... |
| cnegex 11366 | Existence of the negative ... |
| cnegex2 11367 | Existence of a left invers... |
| addlid 11368 | ` 0 ` is a left identity f... |
| addcan 11369 | Cancellation law for addit... |
| addcan2 11370 | Cancellation law for addit... |
| addcom 11371 | Addition commutes. This u... |
| addridi 11372 | ` 0 ` is an additive ident... |
| addlidi 11373 | ` 0 ` is a left identity f... |
| mul02i 11374 | Multiplication by 0. Theo... |
| mul01i 11375 | Multiplication by ` 0 ` . ... |
| addcomi 11376 | Addition commutes. Based ... |
| addcomli 11377 | Addition commutes. (Contr... |
| addcani 11378 | Cancellation law for addit... |
| addcan2i 11379 | Cancellation law for addit... |
| mul12i 11380 | Commutative/associative la... |
| mul32i 11381 | Commutative/associative la... |
| mul4i 11382 | Rearrangement of 4 factors... |
| mul02d 11383 | Multiplication by 0. Theo... |
| mul01d 11384 | Multiplication by ` 0 ` . ... |
| addridd 11385 | ` 0 ` is an additive ident... |
| addlidd 11386 | ` 0 ` is a left identity f... |
| addcomd 11387 | Addition commutes. Based ... |
| addcand 11388 | Cancellation law for addit... |
| addcan2d 11389 | Cancellation law for addit... |
| addcanad 11390 | Cancelling a term on the l... |
| addcan2ad 11391 | Cancelling a term on the r... |
| addneintrd 11392 | Introducing a term on the ... |
| addneintr2d 11393 | Introducing a term on the ... |
| mul12d 11394 | Commutative/associative la... |
| mul32d 11395 | Commutative/associative la... |
| mul31d 11396 | Commutative/associative la... |
| mul4d 11397 | Rearrangement of 4 factors... |
| muladd11r 11398 | A simple product of sums e... |
| comraddd 11399 | Commute RHS addition, in d... |
| comraddi 11400 | Commute RHS addition. See... |
| ltaddneg 11401 | Adding a negative number t... |
| ltaddnegr 11402 | Adding a negative number t... |
| add12 11403 | Commutative/associative la... |
| add32 11404 | Commutative/associative la... |
| add32r 11405 | Commutative/associative la... |
| add4 11406 | Rearrangement of 4 terms i... |
| add42 11407 | Rearrangement of 4 terms i... |
| add12i 11408 | Commutative/associative la... |
| add32i 11409 | Commutative/associative la... |
| add4i 11410 | Rearrangement of 4 terms i... |
| add42i 11411 | Rearrangement of 4 terms i... |
| add12d 11412 | Commutative/associative la... |
| add32d 11413 | Commutative/associative la... |
| add4d 11414 | Rearrangement of 4 terms i... |
| add42d 11415 | Rearrangement of 4 terms i... |
| 0cnALT 11420 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11421 | Alternate proof of ~ 0cnAL... |
| negeu 11422 | Existential uniqueness of ... |
| subval 11423 | Value of subtraction, whic... |
| negeq 11424 | Equality theorem for negat... |
| negeqi 11425 | Equality inference for neg... |
| negeqd 11426 | Equality deduction for neg... |
| nfnegd 11427 | Deduction version of ~ nfn... |
| nfneg 11428 | Bound-variable hypothesis ... |
| csbnegg 11429 | Move class substitution in... |
| negex 11430 | A negative is a set. (Con... |
| subcl 11431 | Closure law for subtractio... |
| negcl 11432 | Closure law for negative. ... |
| negicn 11433 | ` -u _i ` is a complex num... |
| subf 11434 | Subtraction is an operatio... |
| subadd 11435 | Relationship between subtr... |
| subadd2 11436 | Relationship between subtr... |
| subsub23 11437 | Swap subtrahend and result... |
| pncan 11438 | Cancellation law for subtr... |
| pncan2 11439 | Cancellation law for subtr... |
| pncan3 11440 | Subtraction and addition o... |
| npcan 11441 | Cancellation law for subtr... |
| addsubass 11442 | Associative-type law for a... |
| addsub 11443 | Law for addition and subtr... |
| subadd23 11444 | Commutative/associative la... |
| addsub12 11445 | Commutative/associative la... |
| 2addsub 11446 | Law for subtraction and ad... |
| addsubeq4 11447 | Relation between sums and ... |
| pncan3oi 11448 | Subtraction and addition o... |
| mvrraddi 11449 | Move the right term in a s... |
| mvrladdi 11450 | Move the left term in a su... |
| mvlladdi 11451 | Move the left term in a su... |
| subid 11452 | Subtraction of a number fr... |
| subid1 11453 | Identity law for subtracti... |
| npncan 11454 | Cancellation law for subtr... |
| nppcan 11455 | Cancellation law for subtr... |
| nnpcan 11456 | Cancellation law for subtr... |
| nppcan3 11457 | Cancellation law for subtr... |
| subcan2 11458 | Cancellation law for subtr... |
| subeq0 11459 | If the difference between ... |
| npncan2 11460 | Cancellation law for subtr... |
| subsub2 11461 | Law for double subtraction... |
| nncan 11462 | Cancellation law for subtr... |
| subsub 11463 | Law for double subtraction... |
| nppcan2 11464 | Cancellation law for subtr... |
| subsub3 11465 | Law for double subtraction... |
| subsub4 11466 | Law for double subtraction... |
| sub32 11467 | Swap the second and third ... |
| nnncan 11468 | Cancellation law for subtr... |
| nnncan1 11469 | Cancellation law for subtr... |
| nnncan2 11470 | Cancellation law for subtr... |
| npncan3 11471 | Cancellation law for subtr... |
| pnpcan 11472 | Cancellation law for mixed... |
| pnpcan2 11473 | Cancellation law for mixed... |
| pnncan 11474 | Cancellation law for mixed... |
| ppncan 11475 | Cancellation law for mixed... |
| addsub4 11476 | Rearrangement of 4 terms i... |
| subadd4 11477 | Rearrangement of 4 terms i... |
| sub4 11478 | Rearrangement of 4 terms i... |
| neg0 11479 | Minus 0 equals 0. (Contri... |
| negid 11480 | Addition of a number and i... |
| negsub 11481 | Relationship between subtr... |
| subneg 11482 | Relationship between subtr... |
| negneg 11483 | A number is equal to the n... |
| neg11 11484 | Negative is one-to-one. (... |
| negcon1 11485 | Negative contraposition la... |
| negcon2 11486 | Negative contraposition la... |
| negeq0 11487 | A number is zero iff its n... |
| subcan 11488 | Cancellation law for subtr... |
| negsubdi 11489 | Distribution of negative o... |
| negdi 11490 | Distribution of negative o... |
| negdi2 11491 | Distribution of negative o... |
| negsubdi2 11492 | Distribution of negative o... |
| neg2sub 11493 | Relationship between subtr... |
| renegcli 11494 | Closure law for negative o... |
| resubcli 11495 | Closure law for subtractio... |
| renegcl 11496 | Closure law for negative o... |
| resubcl 11497 | Closure law for subtractio... |
| negreb 11498 | The negative of a real is ... |
| peano2cnm 11499 | "Reverse" second Peano pos... |
| peano2rem 11500 | "Reverse" second Peano pos... |
| negcli 11501 | Closure law for negative. ... |
| negidi 11502 | Addition of a number and i... |
| negnegi 11503 | A number is equal to the n... |
| subidi 11504 | Subtraction of a number fr... |
| subid1i 11505 | Identity law for subtracti... |
| negne0bi 11506 | A number is nonzero iff it... |
| negrebi 11507 | The negative of a real is ... |
| negne0i 11508 | The negative of a nonzero ... |
| subcli 11509 | Closure law for subtractio... |
| pncan3i 11510 | Subtraction and addition o... |
| negsubi 11511 | Relationship between subtr... |
| subnegi 11512 | Relationship between subtr... |
| subeq0i 11513 | If the difference between ... |
| neg11i 11514 | Negative is one-to-one. (... |
| negcon1i 11515 | Negative contraposition la... |
| negcon2i 11516 | Negative contraposition la... |
| negdii 11517 | Distribution of negative o... |
| negsubdii 11518 | Distribution of negative o... |
| negsubdi2i 11519 | Distribution of negative o... |
| subaddi 11520 | Relationship between subtr... |
| subadd2i 11521 | Relationship between subtr... |
| subaddrii 11522 | Relationship between subtr... |
| subsub23i 11523 | Swap subtrahend and result... |
| addsubassi 11524 | Associative-type law for s... |
| addsubi 11525 | Law for subtraction and ad... |
| subcani 11526 | Cancellation law for subtr... |
| subcan2i 11527 | Cancellation law for subtr... |
| pnncani 11528 | Cancellation law for mixed... |
| addsub4i 11529 | Rearrangement of 4 terms i... |
| 0reALT 11530 | Alternate proof of ~ 0re .... |
| negcld 11531 | Closure law for negative. ... |
| subidd 11532 | Subtraction of a number fr... |
| subid1d 11533 | Identity law for subtracti... |
| negidd 11534 | Addition of a number and i... |
| negnegd 11535 | A number is equal to the n... |
| negeq0d 11536 | A number is zero iff its n... |
| negne0bd 11537 | A number is nonzero iff it... |
| negcon1d 11538 | Contraposition law for una... |
| negcon1ad 11539 | Contraposition law for una... |
| neg11ad 11540 | The negatives of two compl... |
| negned 11541 | If two complex numbers are... |
| negne0d 11542 | The negative of a nonzero ... |
| negrebd 11543 | The negative of a real is ... |
| subcld 11544 | Closure law for subtractio... |
| pncand 11545 | Cancellation law for subtr... |
| pncan2d 11546 | Cancellation law for subtr... |
| pncan3d 11547 | Subtraction and addition o... |
| npcand 11548 | Cancellation law for subtr... |
| nncand 11549 | Cancellation law for subtr... |
| negsubd 11550 | Relationship between subtr... |
| subnegd 11551 | Relationship between subtr... |
| subeq0d 11552 | If the difference between ... |
| subne0d 11553 | Two unequal numbers have n... |
| subeq0ad 11554 | The difference of two comp... |
| subne0ad 11555 | If the difference of two c... |
| neg11d 11556 | If the difference between ... |
| negdid 11557 | Distribution of negative o... |
| negdi2d 11558 | Distribution of negative o... |
| negsubdid 11559 | Distribution of negative o... |
| negsubdi2d 11560 | Distribution of negative o... |
| neg2subd 11561 | Relationship between subtr... |
| subaddd 11562 | Relationship between subtr... |
| subadd2d 11563 | Relationship between subtr... |
| addsubassd 11564 | Associative-type law for s... |
| addsubd 11565 | Law for subtraction and ad... |
| subadd23d 11566 | Commutative/associative la... |
| addsub12d 11567 | Commutative/associative la... |
| npncand 11568 | Cancellation law for subtr... |
| nppcand 11569 | Cancellation law for subtr... |
| nppcan2d 11570 | Cancellation law for subtr... |
| nppcan3d 11571 | Cancellation law for subtr... |
| subsubd 11572 | Law for double subtraction... |
| subsub2d 11573 | Law for double subtraction... |
| subsub3d 11574 | Law for double subtraction... |
| subsub4d 11575 | Law for double subtraction... |
| sub32d 11576 | Swap the second and third ... |
| nnncand 11577 | Cancellation law for subtr... |
| nnncan1d 11578 | Cancellation law for subtr... |
| nnncan2d 11579 | Cancellation law for subtr... |
| npncan3d 11580 | Cancellation law for subtr... |
| pnpcand 11581 | Cancellation law for mixed... |
| pnpcan2d 11582 | Cancellation law for mixed... |
| pnncand 11583 | Cancellation law for mixed... |
| ppncand 11584 | Cancellation law for mixed... |
| subcand 11585 | Cancellation law for subtr... |
| subcan2d 11586 | Cancellation law for subtr... |
| subcanad 11587 | Cancellation law for subtr... |
| subneintrd 11588 | Introducing subtraction on... |
| subcan2ad 11589 | Cancellation law for subtr... |
| subneintr2d 11590 | Introducing subtraction on... |
| addsub4d 11591 | Rearrangement of 4 terms i... |
| subadd4d 11592 | Rearrangement of 4 terms i... |
| sub4d 11593 | Rearrangement of 4 terms i... |
| 2addsubd 11594 | Law for subtraction and ad... |
| addsubeq4d 11595 | Relation between sums and ... |
| subsubadd23 11596 | Swap the second and the th... |
| addsubsub23 11597 | Swap the second and the th... |
| subeqxfrd 11598 | Transfer two terms of a su... |
| mvlraddd 11599 | Move the right term in a s... |
| mvlladdd 11600 | Move the left term in a su... |
| mvrraddd 11601 | Move the right term in a s... |
| mvrladdd 11602 | Move the left term in a su... |
| assraddsubd 11603 | Associate RHS addition-sub... |
| subaddeqd 11604 | Transfer two terms of a su... |
| addlsub 11605 | Left-subtraction: Subtrac... |
| addrsub 11606 | Right-subtraction: Subtra... |
| subexsub 11607 | A subtraction law: Exchan... |
| addid0 11608 | If adding a number to a an... |
| addn0nid 11609 | Adding a nonzero number to... |
| pnpncand 11610 | Addition/subtraction cance... |
| subeqrev 11611 | Reverse the order of subtr... |
| addeq0 11612 | Two complex numbers add up... |
| pncan1 11613 | Cancellation law for addit... |
| npcan1 11614 | Cancellation law for subtr... |
| subeq0bd 11615 | If two complex numbers are... |
| renegcld 11616 | Closure law for negative o... |
| resubcld 11617 | Closure law for subtractio... |
| negn0 11618 | The image under negation o... |
| negf1o 11619 | Negation is an isomorphism... |
| kcnktkm1cn 11620 | k times k minus 1 is a com... |
| muladd 11621 | Product of two sums. (Con... |
| subdi 11622 | Distribution of multiplica... |
| subdir 11623 | Distribution of multiplica... |
| ine0 11624 | The imaginary unit ` _i ` ... |
| mulneg1 11625 | Product with negative is n... |
| mulneg2 11626 | The product with a negativ... |
| mulneg12 11627 | Swap the negative sign in ... |
| mul2neg 11628 | Product of two negatives. ... |
| submul2 11629 | Convert a subtraction to a... |
| mulm1 11630 | Product with minus one is ... |
| addneg1mul 11631 | Addition with product with... |
| mulsub 11632 | Product of two differences... |
| mulsub2 11633 | Swap the order of subtract... |
| mulm1i 11634 | Product with minus one is ... |
| mulneg1i 11635 | Product with negative is n... |
| mulneg2i 11636 | Product with negative is n... |
| mul2negi 11637 | Product of two negatives. ... |
| subdii 11638 | Distribution of multiplica... |
| subdiri 11639 | Distribution of multiplica... |
| muladdi 11640 | Product of two sums. (Con... |
| mulm1d 11641 | Product with minus one is ... |
| mulneg1d 11642 | Product with negative is n... |
| mulneg2d 11643 | Product with negative is n... |
| mul2negd 11644 | Product of two negatives. ... |
| subdid 11645 | Distribution of multiplica... |
| subdird 11646 | Distribution of multiplica... |
| muladdd 11647 | Product of two sums. (Con... |
| mulsubd 11648 | Product of two differences... |
| muls1d 11649 | Multiplication by one minu... |
| mulsubfacd 11650 | Multiplication followed by... |
| addmulsub 11651 | The product of a sum and a... |
| subaddmulsub 11652 | The difference with a prod... |
| mulsubaddmulsub 11653 | A special difference of a ... |
| gt0ne0 11654 | Positive implies nonzero. ... |
| lt0ne0 11655 | A number which is less tha... |
| ltadd1 11656 | Addition to both sides of ... |
| leadd1 11657 | Addition to both sides of ... |
| leadd2 11658 | Addition to both sides of ... |
| ltsubadd 11659 | 'Less than' relationship b... |
| ltsubadd2 11660 | 'Less than' relationship b... |
| lesubadd 11661 | 'Less than or equal to' re... |
| lesubadd2 11662 | 'Less than or equal to' re... |
| ltaddsub 11663 | 'Less than' relationship b... |
| ltaddsub2 11664 | 'Less than' relationship b... |
| leaddsub 11665 | 'Less than or equal to' re... |
| leaddsub2 11666 | 'Less than or equal to' re... |
| suble 11667 | Swap subtrahends in an ine... |
| lesub 11668 | Swap subtrahends in an ine... |
| ltsub23 11669 | 'Less than' relationship b... |
| ltsub13 11670 | 'Less than' relationship b... |
| le2add 11671 | Adding both sides of two '... |
| ltleadd 11672 | Adding both sides of two o... |
| leltadd 11673 | Adding both sides of two o... |
| lt2add 11674 | Adding both sides of two '... |
| addgt0 11675 | The sum of 2 positive numb... |
| addgegt0 11676 | The sum of nonnegative and... |
| addgtge0 11677 | The sum of nonnegative and... |
| addge0 11678 | The sum of 2 nonnegative n... |
| ltaddpos 11679 | Adding a positive number t... |
| ltaddpos2 11680 | Adding a positive number t... |
| ltsubpos 11681 | Subtracting a positive num... |
| posdif 11682 | Comparison of two numbers ... |
| lesub1 11683 | Subtraction from both side... |
| lesub2 11684 | Subtraction of both sides ... |
| ltsub1 11685 | Subtraction from both side... |
| ltsub2 11686 | Subtraction of both sides ... |
| lt2sub 11687 | Subtracting both sides of ... |
| le2sub 11688 | Subtracting both sides of ... |
| ltneg 11689 | Negative of both sides of ... |
| ltnegcon1 11690 | Contraposition of negative... |
| ltnegcon2 11691 | Contraposition of negative... |
| leneg 11692 | Negative of both sides of ... |
| lenegcon1 11693 | Contraposition of negative... |
| lenegcon2 11694 | Contraposition of negative... |
| lt0neg1 11695 | Comparison of a number and... |
| lt0neg2 11696 | Comparison of a number and... |
| le0neg1 11697 | Comparison of a number and... |
| le0neg2 11698 | Comparison of a number and... |
| addge01 11699 | A number is less than or e... |
| addge02 11700 | A number is less than or e... |
| add20 11701 | Two nonnegative numbers ar... |
| subge0 11702 | Nonnegative subtraction. ... |
| suble0 11703 | Nonpositive subtraction. ... |
| leaddle0 11704 | The sum of a real number a... |
| subge02 11705 | Nonnegative subtraction. ... |
| lesub0 11706 | Lemma to show a nonnegativ... |
| mulge0 11707 | The product of two nonnega... |
| mullt0 11708 | The product of two negativ... |
| msqgt0 11709 | A nonzero square is positi... |
| msqge0 11710 | A square is nonnegative. ... |
| 0lt1 11711 | 0 is less than 1. Theorem... |
| 0le1 11712 | 0 is less than or equal to... |
| relin01 11713 | An interval law for less t... |
| ltordlem 11714 | Lemma for ~ ltord1 . (Con... |
| ltord1 11715 | Infer an ordering relation... |
| leord1 11716 | Infer an ordering relation... |
| eqord1 11717 | A strictly increasing real... |
| ltord2 11718 | Infer an ordering relation... |
| leord2 11719 | Infer an ordering relation... |
| eqord2 11720 | A strictly decreasing real... |
| wloglei 11721 | Form of ~ wlogle where bot... |
| wlogle 11722 | If the predicate ` ch ( x ... |
| leidi 11723 | 'Less than or equal to' is... |
| gt0ne0i 11724 | Positive means nonzero (us... |
| gt0ne0ii 11725 | Positive implies nonzero. ... |
| msqgt0i 11726 | A nonzero square is positi... |
| msqge0i 11727 | A square is nonnegative. ... |
| addgt0i 11728 | Addition of 2 positive num... |
| addge0i 11729 | Addition of 2 nonnegative ... |
| addgegt0i 11730 | Addition of nonnegative an... |
| addgt0ii 11731 | Addition of 2 positive num... |
| add20i 11732 | Two nonnegative numbers ar... |
| ltnegi 11733 | Negative of both sides of ... |
| lenegi 11734 | Negative of both sides of ... |
| ltnegcon2i 11735 | Contraposition of negative... |
| mulge0i 11736 | The product of two nonnega... |
| lesub0i 11737 | Lemma to show a nonnegativ... |
| ltaddposi 11738 | Adding a positive number t... |
| posdifi 11739 | Comparison of two numbers ... |
| ltnegcon1i 11740 | Contraposition of negative... |
| lenegcon1i 11741 | Contraposition of negative... |
| subge0i 11742 | Nonnegative subtraction. ... |
| ltadd1i 11743 | Addition to both sides of ... |
| leadd1i 11744 | Addition to both sides of ... |
| leadd2i 11745 | Addition to both sides of ... |
| ltsubaddi 11746 | 'Less than' relationship b... |
| lesubaddi 11747 | 'Less than or equal to' re... |
| ltsubadd2i 11748 | 'Less than' relationship b... |
| lesubadd2i 11749 | 'Less than or equal to' re... |
| ltaddsubi 11750 | 'Less than' relationship b... |
| lt2addi 11751 | Adding both side of two in... |
| le2addi 11752 | Adding both side of two in... |
| gt0ne0d 11753 | Positive implies nonzero. ... |
| lt0ne0d 11754 | Something less than zero i... |
| leidd 11755 | 'Less than or equal to' is... |
| msqgt0d 11756 | A nonzero square is positi... |
| msqge0d 11757 | A square is nonnegative. ... |
| lt0neg1d 11758 | Comparison of a number and... |
| lt0neg2d 11759 | Comparison of a number and... |
| le0neg1d 11760 | Comparison of a number and... |
| le0neg2d 11761 | Comparison of a number and... |
| addgegt0d 11762 | Addition of nonnegative an... |
| addgtge0d 11763 | Addition of positive and n... |
| addgt0d 11764 | Addition of 2 positive num... |
| addge0d 11765 | Addition of 2 nonnegative ... |
| mulge0d 11766 | The product of two nonnega... |
| ltnegd 11767 | Negative of both sides of ... |
| lenegd 11768 | Negative of both sides of ... |
| ltnegcon1d 11769 | Contraposition of negative... |
| ltnegcon2d 11770 | Contraposition of negative... |
| lenegcon1d 11771 | Contraposition of negative... |
| lenegcon2d 11772 | Contraposition of negative... |
| ltaddposd 11773 | Adding a positive number t... |
| ltaddpos2d 11774 | Adding a positive number t... |
| ltsubposd 11775 | Subtracting a positive num... |
| posdifd 11776 | Comparison of two numbers ... |
| addge01d 11777 | A number is less than or e... |
| addge02d 11778 | A number is less than or e... |
| subge0d 11779 | Nonnegative subtraction. ... |
| suble0d 11780 | Nonpositive subtraction. ... |
| subge02d 11781 | Nonnegative subtraction. ... |
| ltadd1d 11782 | Addition to both sides of ... |
| leadd1d 11783 | Addition to both sides of ... |
| leadd2d 11784 | Addition to both sides of ... |
| ltsubaddd 11785 | 'Less than' relationship b... |
| lesubaddd 11786 | 'Less than or equal to' re... |
| ltsubadd2d 11787 | 'Less than' relationship b... |
| lesubadd2d 11788 | 'Less than or equal to' re... |
| ltaddsubd 11789 | 'Less than' relationship b... |
| ltaddsub2d 11790 | 'Less than' relationship b... |
| leaddsub2d 11791 | 'Less than or equal to' re... |
| subled 11792 | Swap subtrahends in an ine... |
| lesubd 11793 | Swap subtrahends in an ine... |
| ltsub23d 11794 | 'Less than' relationship b... |
| ltsub13d 11795 | 'Less than' relationship b... |
| lesub1d 11796 | Subtraction from both side... |
| lesub2d 11797 | Subtraction of both sides ... |
| ltsub1d 11798 | Subtraction from both side... |
| ltsub2d 11799 | Subtraction of both sides ... |
| ltadd1dd 11800 | Addition to both sides of ... |
| ltsub1dd 11801 | Subtraction from both side... |
| ltsub2dd 11802 | Subtraction of both sides ... |
| leadd1dd 11803 | Addition to both sides of ... |
| leadd2dd 11804 | Addition to both sides of ... |
| lesub1dd 11805 | Subtraction from both side... |
| lesub2dd 11806 | Subtraction of both sides ... |
| lesub3d 11807 | The result of subtracting ... |
| le2addd 11808 | Adding both side of two in... |
| le2subd 11809 | Subtracting both sides of ... |
| ltleaddd 11810 | Adding both sides of two o... |
| leltaddd 11811 | Adding both sides of two o... |
| lt2addd 11812 | Adding both side of two in... |
| lt2subd 11813 | Subtracting both sides of ... |
| possumd 11814 | Condition for a positive s... |
| sublt0d 11815 | When a subtraction gives a... |
| ltaddsublt 11816 | Addition and subtraction o... |
| 1le1 11817 | One is less than or equal ... |
| ixi 11818 | ` _i ` times itself is min... |
| recextlem1 11819 | Lemma for ~ recex . (Cont... |
| recextlem2 11820 | Lemma for ~ recex . (Cont... |
| recex 11821 | Existence of reciprocal of... |
| mulcand 11822 | Cancellation law for multi... |
| mulcan2d 11823 | Cancellation law for multi... |
| mulcanad 11824 | Cancellation of a nonzero ... |
| mulcan2ad 11825 | Cancellation of a nonzero ... |
| mulcan 11826 | Cancellation law for multi... |
| mulcan2 11827 | Cancellation law for multi... |
| mulcani 11828 | Cancellation law for multi... |
| mul0or 11829 | If a product is zero, one ... |
| mulne0b 11830 | The product of two nonzero... |
| mulne0 11831 | The product of two nonzero... |
| mulne0i 11832 | The product of two nonzero... |
| muleqadd 11833 | Property of numbers whose ... |
| receu 11834 | Existential uniqueness of ... |
| mulnzcnf 11835 | Multiplication maps nonzer... |
| mul0ori 11836 | If a product is zero, one ... |
| mul0ord 11837 | If a product is zero, one ... |
| msq0i 11838 | A number is zero iff its s... |
| msq0d 11839 | A number is zero iff its s... |
| mulne0bd 11840 | The product of two nonzero... |
| mulne0d 11841 | The product of two nonzero... |
| mulcan1g 11842 | A generalized form of the ... |
| mulcan2g 11843 | A generalized form of the ... |
| mulne0bad 11844 | A factor of a nonzero comp... |
| mulne0bbd 11845 | A factor of a nonzero comp... |
| 1div0 11848 | You can't divide by zero, ... |
| divval 11849 | Value of division: if ` A ... |
| divmul 11850 | Relationship between divis... |
| divmul2 11851 | Relationship between divis... |
| divmul3 11852 | Relationship between divis... |
| divcl 11853 | Closure law for division. ... |
| reccl 11854 | Closure law for reciprocal... |
| divcan2 11855 | A cancellation law for div... |
| divcan1 11856 | A cancellation law for div... |
| diveq0 11857 | A ratio is zero iff the nu... |
| divne0b 11858 | The ratio of nonzero numbe... |
| divne0 11859 | The ratio of nonzero numbe... |
| recne0 11860 | The reciprocal of a nonzer... |
| recid 11861 | Multiplication of a number... |
| recid2 11862 | Multiplication of a number... |
| divrec 11863 | Relationship between divis... |
| divrec2 11864 | Relationship between divis... |
| divass 11865 | An associative law for div... |
| div23 11866 | A commutative/associative ... |
| div32 11867 | A commutative/associative ... |
| div13 11868 | A commutative/associative ... |
| div12 11869 | A commutative/associative ... |
| divmulass 11870 | An associative law for div... |
| divmulasscom 11871 | An associative/commutative... |
| divdir 11872 | Distribution of division o... |
| divcan3 11873 | A cancellation law for div... |
| divcan4 11874 | A cancellation law for div... |
| div11 11875 | One-to-one relationship fo... |
| div11OLD 11876 | Obsolete version of ~ div1... |
| diveq1 11877 | Equality in terms of unit ... |
| divid 11878 | A number divided by itself... |
| dividOLD 11879 | Obsolete version of ~ divi... |
| div0 11880 | Division into zero is zero... |
| div0OLD 11881 | Obsolete version of ~ div0... |
| div1 11882 | A number divided by 1 is i... |
| 1div1e1 11883 | 1 divided by 1 is 1. (Con... |
| divneg 11884 | Move negative sign inside ... |
| muldivdir 11885 | Distribution of division o... |
| divsubdir 11886 | Distribution of division o... |
| muldivdid 11887 | Distribution of division o... |
| subdivcomb1 11888 | Bring a term in a subtract... |
| subdivcomb2 11889 | Bring a term in a subtract... |
| recrec 11890 | A number is equal to the r... |
| rec11 11891 | Reciprocal is one-to-one. ... |
| rec11r 11892 | Mutual reciprocals. (Cont... |
| divmuldiv 11893 | Multiplication of two rati... |
| divdivdiv 11894 | Division of two ratios. T... |
| divcan5 11895 | Cancellation of common fac... |
| divmul13 11896 | Swap the denominators in t... |
| divmul24 11897 | Swap the numerators in the... |
| divmuleq 11898 | Cross-multiply in an equal... |
| recdiv 11899 | The reciprocal of a ratio.... |
| divcan6 11900 | Cancellation of inverted f... |
| divdiv32 11901 | Swap denominators in a div... |
| divcan7 11902 | Cancel equal divisors in a... |
| dmdcan 11903 | Cancellation law for divis... |
| divdiv1 11904 | Division into a fraction. ... |
| divdiv2 11905 | Division by a fraction. (... |
| recdiv2 11906 | Division into a reciprocal... |
| ddcan 11907 | Cancellation in a double d... |
| divadddiv 11908 | Addition of two ratios. T... |
| divsubdiv 11909 | Subtraction of two ratios.... |
| conjmul 11910 | Two numbers whose reciproc... |
| rereccl 11911 | Closure law for reciprocal... |
| redivcl 11912 | Closure law for division o... |
| eqneg 11913 | A number equal to its nega... |
| eqnegd 11914 | A complex number equals it... |
| eqnegad 11915 | If a complex number equals... |
| div2neg 11916 | Quotient of two negatives.... |
| divneg2 11917 | Move negative sign inside ... |
| recclzi 11918 | Closure law for reciprocal... |
| recne0zi 11919 | The reciprocal of a nonzer... |
| recidzi 11920 | Multiplication of a number... |
| div1i 11921 | A number divided by 1 is i... |
| eqnegi 11922 | A number equal to its nega... |
| reccli 11923 | Closure law for reciprocal... |
| recidi 11924 | Multiplication of a number... |
| recreci 11925 | A number is equal to the r... |
| dividi 11926 | A number divided by itself... |
| div0i 11927 | Division into zero is zero... |
| divclzi 11928 | Closure law for division. ... |
| divcan1zi 11929 | A cancellation law for div... |
| divcan2zi 11930 | A cancellation law for div... |
| divreczi 11931 | Relationship between divis... |
| divcan3zi 11932 | A cancellation law for div... |
| divcan4zi 11933 | A cancellation law for div... |
| rec11i 11934 | Reciprocal is one-to-one. ... |
| divcli 11935 | Closure law for division. ... |
| divcan2i 11936 | A cancellation law for div... |
| divcan1i 11937 | A cancellation law for div... |
| divreci 11938 | Relationship between divis... |
| divcan3i 11939 | A cancellation law for div... |
| divcan4i 11940 | A cancellation law for div... |
| divne0i 11941 | The ratio of nonzero numbe... |
| rec11ii 11942 | Reciprocal is one-to-one. ... |
| divasszi 11943 | An associative law for div... |
| divmulzi 11944 | Relationship between divis... |
| divdirzi 11945 | Distribution of division o... |
| divdiv23zi 11946 | Swap denominators in a div... |
| divmuli 11947 | Relationship between divis... |
| divdiv32i 11948 | Swap denominators in a div... |
| divassi 11949 | An associative law for div... |
| divdiri 11950 | Distribution of division o... |
| div23i 11951 | A commutative/associative ... |
| div11i 11952 | One-to-one relationship fo... |
| divmuldivi 11953 | Multiplication of two rati... |
| divmul13i 11954 | Swap denominators of two r... |
| divadddivi 11955 | Addition of two ratios. T... |
| divdivdivi 11956 | Division of two ratios. T... |
| rerecclzi 11957 | Closure law for reciprocal... |
| rereccli 11958 | Closure law for reciprocal... |
| redivclzi 11959 | Closure law for division o... |
| redivcli 11960 | Closure law for division o... |
| div1d 11961 | A number divided by 1 is i... |
| reccld 11962 | Closure law for reciprocal... |
| recne0d 11963 | The reciprocal of a nonzer... |
| recidd 11964 | Multiplication of a number... |
| recid2d 11965 | Multiplication of a number... |
| recrecd 11966 | A number is equal to the r... |
| dividd 11967 | A number divided by itself... |
| div0d 11968 | Division into zero is zero... |
| divcld 11969 | Closure law for division. ... |
| divcan1d 11970 | A cancellation law for div... |
| divcan2d 11971 | A cancellation law for div... |
| divrecd 11972 | Relationship between divis... |
| divrec2d 11973 | Relationship between divis... |
| divcan3d 11974 | A cancellation law for div... |
| divcan4d 11975 | A cancellation law for div... |
| diveq0d 11976 | A ratio is zero iff the nu... |
| diveq1d 11977 | Equality in terms of unit ... |
| diveq1ad 11978 | The quotient of two comple... |
| diveq0ad 11979 | A fraction of complex numb... |
| divne1d 11980 | If two complex numbers are... |
| divne0bd 11981 | A ratio is zero iff the nu... |
| divnegd 11982 | Move negative sign inside ... |
| divneg2d 11983 | Move negative sign inside ... |
| div2negd 11984 | Quotient of two negatives.... |
| divne0d 11985 | The ratio of nonzero numbe... |
| recdivd 11986 | The reciprocal of a ratio.... |
| recdiv2d 11987 | Division into a reciprocal... |
| divcan6d 11988 | Cancellation of inverted f... |
| ddcand 11989 | Cancellation in a double d... |
| rec11d 11990 | Reciprocal is one-to-one. ... |
| divmuld 11991 | Relationship between divis... |
| div32d 11992 | A commutative/associative ... |
| div13d 11993 | A commutative/associative ... |
| divdiv32d 11994 | Swap denominators in a div... |
| divcan5d 11995 | Cancellation of common fac... |
| divcan5rd 11996 | Cancellation of common fac... |
| divcan7d 11997 | Cancel equal divisors in a... |
| dmdcand 11998 | Cancellation law for divis... |
| dmdcan2d 11999 | Cancellation law for divis... |
| divdiv1d 12000 | Division into a fraction. ... |
| divdiv2d 12001 | Division by a fraction. (... |
| divmul2d 12002 | Relationship between divis... |
| divmul3d 12003 | Relationship between divis... |
| divassd 12004 | An associative law for div... |
| div12d 12005 | A commutative/associative ... |
| div23d 12006 | A commutative/associative ... |
| divdird 12007 | Distribution of division o... |
| divsubdird 12008 | Distribution of division o... |
| div11d 12009 | One-to-one relationship fo... |
| divmuldivd 12010 | Multiplication of two rati... |
| divmul13d 12011 | Swap denominators of two r... |
| divmul24d 12012 | Swap the numerators in the... |
| divadddivd 12013 | Addition of two ratios. T... |
| divsubdivd 12014 | Subtraction of two ratios.... |
| divmuleqd 12015 | Cross-multiply in an equal... |
| divdivdivd 12016 | Division of two ratios. T... |
| diveq1bd 12017 | If two complex numbers are... |
| div2sub 12018 | Swap the order of subtract... |
| div2subd 12019 | Swap subtrahend and minuen... |
| rereccld 12020 | Closure law for reciprocal... |
| redivcld 12021 | Closure law for division o... |
| subrecd 12022 | Subtraction of reciprocals... |
| subrec 12023 | Subtraction of reciprocals... |
| subreci 12024 | Subtraction of reciprocals... |
| mvllmuld 12025 | Move the left term in a pr... |
| mvllmuli 12026 | Move the left term in a pr... |
| ldiv 12027 | Left-division. (Contribut... |
| rdiv 12028 | Right-division. (Contribu... |
| mdiv 12029 | A division law. (Contribu... |
| lineq 12030 | Solution of a (scalar) lin... |
| elimgt0 12031 | Hypothesis for weak deduct... |
| elimge0 12032 | Hypothesis for weak deduct... |
| ltp1 12033 | A number is less than itse... |
| lep1 12034 | A number is less than or e... |
| ltm1 12035 | A number minus 1 is less t... |
| lem1 12036 | A number minus 1 is less t... |
| letrp1 12037 | A transitive property of '... |
| p1le 12038 | A transitive property of p... |
| recgt0 12039 | The reciprocal of a positi... |
| prodgt0 12040 | Infer that a multiplicand ... |
| prodgt02 12041 | Infer that a multiplier is... |
| ltmul1a 12042 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12043 | Multiplication of both sid... |
| ltmul2 12044 | Multiplication of both sid... |
| lemul1 12045 | Multiplication of both sid... |
| lemul2 12046 | Multiplication of both sid... |
| lemul1a 12047 | Multiplication of both sid... |
| lemul2a 12048 | Multiplication of both sid... |
| ltmul12a 12049 | Comparison of product of t... |
| lemul12b 12050 | Comparison of product of t... |
| lemul12a 12051 | Comparison of product of t... |
| mulgt1OLD 12052 | Obsolete version of ~ mulg... |
| ltmulgt11 12053 | Multiplication by a number... |
| ltmulgt12 12054 | Multiplication by a number... |
| mulgt1 12055 | The product of two numbers... |
| lemulge11 12056 | Multiplication by a number... |
| lemulge12 12057 | Multiplication by a number... |
| ltdiv1 12058 | Division of both sides of ... |
| lediv1 12059 | Division of both sides of ... |
| gt0div 12060 | Division of a positive num... |
| ge0div 12061 | Division of a nonnegative ... |
| divgt0 12062 | The ratio of two positive ... |
| divge0 12063 | The ratio of nonnegative a... |
| mulge0b 12064 | A condition for multiplica... |
| mulle0b 12065 | A condition for multiplica... |
| mulsuble0b 12066 | A condition for multiplica... |
| ltmuldiv 12067 | 'Less than' relationship b... |
| ltmuldiv2 12068 | 'Less than' relationship b... |
| ltdivmul 12069 | 'Less than' relationship b... |
| ledivmul 12070 | 'Less than or equal to' re... |
| ltdivmul2 12071 | 'Less than' relationship b... |
| lt2mul2div 12072 | 'Less than' relationship b... |
| ledivmul2 12073 | 'Less than or equal to' re... |
| lemuldiv 12074 | 'Less than or equal' relat... |
| lemuldiv2 12075 | 'Less than or equal' relat... |
| ltrec 12076 | The reciprocal of both sid... |
| lerec 12077 | The reciprocal of both sid... |
| lt2msq1 12078 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12079 | Two nonnegative numbers co... |
| ltdiv2 12080 | Division of a positive num... |
| ltrec1 12081 | Reciprocal swap in a 'less... |
| lerec2 12082 | Reciprocal swap in a 'less... |
| ledivdiv 12083 | Invert ratios of positive ... |
| lediv2 12084 | Division of a positive num... |
| ltdiv23 12085 | Swap denominator with othe... |
| lediv23 12086 | Swap denominator with othe... |
| lediv12a 12087 | Comparison of ratio of two... |
| lediv2a 12088 | Division of both sides of ... |
| reclt1 12089 | The reciprocal of a positi... |
| recgt1 12090 | The reciprocal of a positi... |
| recgt1i 12091 | The reciprocal of a number... |
| recp1lt1 12092 | Construct a number less th... |
| recreclt 12093 | Given a positive number ` ... |
| le2msq 12094 | The square function on non... |
| msq11 12095 | The square of a nonnegativ... |
| ledivp1 12096 | "Less than or equal to" an... |
| squeeze0 12097 | If a nonnegative number is... |
| ltp1i 12098 | A number is less than itse... |
| recgt0i 12099 | The reciprocal of a positi... |
| recgt0ii 12100 | The reciprocal of a positi... |
| prodgt0i 12101 | Infer that a multiplicand ... |
| divgt0i 12102 | The ratio of two positive ... |
| divge0i 12103 | The ratio of nonnegative a... |
| ltreci 12104 | The reciprocal of both sid... |
| lereci 12105 | The reciprocal of both sid... |
| lt2msqi 12106 | The square function on non... |
| le2msqi 12107 | The square function on non... |
| msq11i 12108 | The square of a nonnegativ... |
| divgt0i2i 12109 | The ratio of two positive ... |
| ltrecii 12110 | The reciprocal of both sid... |
| divgt0ii 12111 | The ratio of two positive ... |
| ltmul1i 12112 | Multiplication of both sid... |
| ltdiv1i 12113 | Division of both sides of ... |
| ltmuldivi 12114 | 'Less than' relationship b... |
| ltmul2i 12115 | Multiplication of both sid... |
| lemul1i 12116 | Multiplication of both sid... |
| lemul2i 12117 | Multiplication of both sid... |
| ltdiv23i 12118 | Swap denominator with othe... |
| ledivp1i 12119 | "Less than or equal to" an... |
| ltdivp1i 12120 | Less-than and division rel... |
| ltdiv23ii 12121 | Swap denominator with othe... |
| ltmul1ii 12122 | Multiplication of both sid... |
| ltdiv1ii 12123 | Division of both sides of ... |
| ltp1d 12124 | A number is less than itse... |
| lep1d 12125 | A number is less than or e... |
| ltm1d 12126 | A number minus 1 is less t... |
| lem1d 12127 | A number minus 1 is less t... |
| recgt0d 12128 | The reciprocal of a positi... |
| divgt0d 12129 | The ratio of two positive ... |
| mulgt1d 12130 | The product of two numbers... |
| lemulge11d 12131 | Multiplication by a number... |
| lemulge12d 12132 | Multiplication by a number... |
| lemul1ad 12133 | Multiplication of both sid... |
| lemul2ad 12134 | Multiplication of both sid... |
| ltmul12ad 12135 | Comparison of product of t... |
| lemul12ad 12136 | Comparison of product of t... |
| lemul12bd 12137 | Comparison of product of t... |
| fimaxre 12138 | A finite set of real numbe... |
| fimaxre2 12139 | A nonempty finite set of r... |
| fimaxre3 12140 | A nonempty finite set of r... |
| fiminre 12141 | A nonempty finite set of r... |
| fiminre2 12142 | A nonempty finite set of r... |
| negfi 12143 | The negation of a finite s... |
| lbreu 12144 | If a set of reals contains... |
| lbcl 12145 | If a set of reals contains... |
| lble 12146 | If a set of reals contains... |
| lbinf 12147 | If a set of reals contains... |
| lbinfcl 12148 | If a set of reals contains... |
| lbinfle 12149 | If a set of reals contains... |
| sup2 12150 | A nonempty, bounded-above ... |
| sup3 12151 | A version of the completen... |
| infm3lem 12152 | Lemma for ~ infm3 . (Cont... |
| infm3 12153 | The completeness axiom for... |
| suprcl 12154 | Closure of supremum of a n... |
| suprub 12155 | A member of a nonempty bou... |
| suprubd 12156 | Natural deduction form of ... |
| suprcld 12157 | Natural deduction form of ... |
| suprlub 12158 | The supremum of a nonempty... |
| suprnub 12159 | An upper bound is not less... |
| suprleub 12160 | The supremum of a nonempty... |
| supaddc 12161 | The supremum function dist... |
| supadd 12162 | The supremum function dist... |
| supmul1 12163 | The supremum function dist... |
| supmullem1 12164 | Lemma for ~ supmul . (Con... |
| supmullem2 12165 | Lemma for ~ supmul . (Con... |
| supmul 12166 | The supremum function dist... |
| sup3ii 12167 | A version of the completen... |
| suprclii 12168 | Closure of supremum of a n... |
| suprubii 12169 | A member of a nonempty bou... |
| suprlubii 12170 | The supremum of a nonempty... |
| suprnubii 12171 | An upper bound is not less... |
| suprleubii 12172 | The supremum of a nonempty... |
| riotaneg 12173 | The negative of the unique... |
| negiso 12174 | Negation is an order anti-... |
| dfinfre 12175 | The infimum of a set of re... |
| infrecl 12176 | Closure of infimum of a no... |
| infrenegsup 12177 | The infimum of a set of re... |
| infregelb 12178 | Any lower bound of a nonem... |
| infrelb 12179 | If a nonempty set of real ... |
| infrefilb 12180 | The infimum of a finite se... |
| supfirege 12181 | The supremum of a finite s... |
| neg1cn 12182 | -1 is a complex number. (... |
| neg1rr 12183 | -1 is a real number. (Con... |
| neg1ne0 12184 | -1 is nonzero. (Contribut... |
| neg1lt0 12185 | -1 is less than 0. (Contr... |
| negneg1e1 12186 | ` -u -u 1 ` is 1. (Contri... |
| inelr 12187 | The imaginary unit ` _i ` ... |
| rimul 12188 | A real number times the im... |
| cru 12189 | The representation of comp... |
| crne0 12190 | The real representation of... |
| creur 12191 | The real part of a complex... |
| creui 12192 | The imaginary part of a co... |
| cju 12193 | The complex conjugate of a... |
| ofsubeq0 12194 | Function analogue of ~ sub... |
| ofnegsub 12195 | Function analogue of ~ neg... |
| ofsubge0 12196 | Function analogue of ~ sub... |
| indv 12199 | Value of the indicator fun... |
| indval 12200 | Value of the indicator fun... |
| indval0 12201 | The indicator function gen... |
| indval2 12202 | Alternate value of the ind... |
| indf 12203 | An indicator function as a... |
| indfval 12204 | Value of the indicator fun... |
| fvindre 12205 | The range of the indicator... |
| ind1 12206 | Value of the indicator fun... |
| ind0 12207 | Value of the indicator fun... |
| ind1a 12208 | Value of the indicator fun... |
| indconst0 12209 | Indicator of the empty set... |
| indconst1 12210 | Indicator of the whole set... |
| indpi1 12211 | Preimage of the singleton ... |
| nnexALT 12214 | Alternate proof of ~ nnex ... |
| peano5nni 12215 | Peano's inductive postulat... |
| nnssre 12216 | The positive integers are ... |
| nnsscn 12217 | The positive integers are ... |
| nnex 12218 | The set of positive intege... |
| nnre 12219 | A positive integer is a re... |
| nncn 12220 | A positive integer is a co... |
| nnrei 12221 | A positive integer is a re... |
| nncni 12222 | A positive integer is a co... |
| 1nn 12223 | Peano postulate: 1 is a po... |
| peano2nn 12224 | Peano postulate: a success... |
| dfnn2 12225 | Alternate definition of th... |
| dfnn3 12226 | Alternate definition of th... |
| nnred 12227 | A positive integer is a re... |
| nncnd 12228 | A positive integer is a co... |
| peano2nnd 12229 | Peano postulate: a success... |
| nnind 12230 | Principle of Mathematical ... |
| nnindALT 12231 | Principle of Mathematical ... |
| nnindd 12232 | Principle of Mathematical ... |
| nn1m1nn 12233 | Every positive integer is ... |
| nn1suc 12234 | If a statement holds for 1... |
| nnaddcl 12235 | Closure of addition of pos... |
| nnmulcl 12236 | Closure of multiplication ... |
| nnmulcli 12237 | Closure of multiplication ... |
| nnadd1com 12238 | Addition with 1 is commuta... |
| nnaddcom 12239 | Addition is commutative fo... |
| nnaddcomli 12240 | Version of ~ addcomli for ... |
| nnmtmip 12241 | "Minus times minus is plus... |
| nn2ge 12242 | There exists a positive in... |
| nnge1 12243 | A positive integer is one ... |
| nngt1ne1 12244 | A positive integer is grea... |
| nnle1eq1 12245 | A positive integer is less... |
| nngt0 12246 | A positive integer is posi... |
| nnnlt1 12247 | A positive integer is not ... |
| nnnle0 12248 | A positive integer is not ... |
| nnne0 12249 | A positive integer is nonz... |
| nnneneg 12250 | No positive integer is equ... |
| 0nnn 12251 | Zero is not a positive int... |
| 0nnnALT 12252 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12253 | Alternate version of ~ nnn... |
| nngt0i 12254 | A positive integer is posi... |
| nnne0i 12255 | A positive integer is nonz... |
| nndivre 12256 | The quotient of a real and... |
| nnrecre 12257 | The reciprocal of a positi... |
| nnrecgt0 12258 | The reciprocal of a positi... |
| nnsub 12259 | Subtraction of positive in... |
| nnsubi 12260 | Subtraction of positive in... |
| nndiv 12261 | Two ways to express " ` A ... |
| nndivtr 12262 | Transitive property of div... |
| nnge1d 12263 | A positive integer is one ... |
| nngt0d 12264 | A positive integer is posi... |
| nnne0d 12265 | A positive integer is nonz... |
| nnrecred 12266 | The reciprocal of a positi... |
| nnaddcld 12267 | Closure of addition of pos... |
| nnmulcld 12268 | Closure of multiplication ... |
| nndivred 12269 | A positive integer is one ... |
| 1t1e1ALT 12270 | Alternate proof of ~ 1t1e1... |
| nnadddir 12271 | Right-distributivity for n... |
| nnmul1com 12272 | Multiplication with 1 is c... |
| nnmulcom 12273 | Multiplication is commutat... |
| 1eltp012 12290 | 1 is an element of ` { 0 ,... |
| 0ne1 12291 | Zero is different from one... |
| 1m1e0 12292 | One minus one equals zero.... |
| 2nn 12293 | 2 is a positive integer. ... |
| 2re 12294 | The number 2 is real. (Co... |
| 2cn 12295 | The number 2 is a complex ... |
| 2cnALT 12296 | Alternate proof of ~ 2cn .... |
| 2ex 12297 | The number 2 is a set. (C... |
| 2cnd 12298 | The number 2 is a complex ... |
| 3nn 12299 | 3 is a positive integer. ... |
| 3re 12300 | The number 3 is real. (Co... |
| 3cn 12301 | The number 3 is a complex ... |
| 3ex 12302 | The number 3 is a set. (C... |
| 4nn 12303 | 4 is a positive integer. ... |
| 4re 12304 | The number 4 is real. (Co... |
| 4cn 12305 | The number 4 is a complex ... |
| 5nn 12306 | 5 is a positive integer. ... |
| 5re 12307 | The number 5 is real. (Co... |
| 5cn 12308 | The number 5 is a complex ... |
| 6nn 12309 | 6 is a positive integer. ... |
| 6re 12310 | The number 6 is real. (Co... |
| 6cn 12311 | The number 6 is a complex ... |
| 7nn 12312 | 7 is a positive integer. ... |
| 7re 12313 | The number 7 is real. (Co... |
| 7cn 12314 | The number 7 is a complex ... |
| 8nn 12315 | 8 is a positive integer. ... |
| 8re 12316 | The number 8 is real. (Co... |
| 8cn 12317 | The number 8 is a complex ... |
| 9nn 12318 | 9 is a positive integer. ... |
| 9re 12319 | The number 9 is real. (Co... |
| 9cn 12320 | The number 9 is a complex ... |
| 0le0 12321 | Zero is nonnegative. (Con... |
| 0le2 12322 | The number 0 is less than ... |
| 0le2OLD 12323 | Obsolete version of ~ 0le2... |
| 2pos 12324 | The number 2 is positive. ... |
| 2posOLD 12325 | Obsolete version of ~ 2pos... |
| 2ne0 12326 | The number 2 is nonzero. ... |
| 2thalfe1 12327 | 2 times one half equals 1.... |
| 3pos 12328 | The number 3 is positive. ... |
| 3ne0 12329 | The number 3 is nonzero. ... |
| 4pos 12330 | The number 4 is positive. ... |
| 4ne0 12331 | The number 4 is nonzero. ... |
| 5pos 12332 | The number 5 is positive. ... |
| 6pos 12333 | The number 6 is positive. ... |
| 7pos 12334 | The number 7 is positive. ... |
| 8pos 12335 | The number 8 is positive. ... |
| 9pos 12336 | The number 9 is positive. ... |
| 1pneg1e0 12337 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12338 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12339 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12340 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12341 | Function value at ` N + 1 ... |
| 1p0e1 12342 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12343 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12344 | 2 - 1 = 1. The result is ... |
| 2m1e1OLD 12345 | Obsolete version of ~ 2m1e... |
| 1e2m1 12346 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12347 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12348 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12349 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12350 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12351 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12352 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12353 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12354 | Two plus two equals four. ... |
| 2times 12355 | Two times a number. (Cont... |
| times2 12356 | A number times 2. (Contri... |
| 2timesi 12357 | Two times a number. (Cont... |
| times2i 12358 | A number times 2. (Contri... |
| 2txmxeqx 12359 | Two times a complex number... |
| 2div2e1 12360 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12361 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12362 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12363 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12364 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12365 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12366 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12367 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12368 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12369 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12370 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12371 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12372 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12373 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12374 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12375 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12376 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12377 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12378 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12379 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12380 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12381 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12382 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12383 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12384 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12385 | 3 times 2 equals 6. (Cont... |
| 2t3e6 12386 | 2 times 3 equals 6. (Cont... |
| 3t3e9 12387 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12388 | 4 times 2 equals 8. (Cont... |
| 2t4e8 12389 | 2 times 4 equals 8. (Cont... |
| 2t0e0 12390 | 2 times 0 equals 0. (Cont... |
| 4div2e2 12391 | One half of four is two. ... |
| 1lt2 12392 | 1 is less than 2. (Contri... |
| 2lt3 12393 | 2 is less than 3. (Contri... |
| 2le3 12394 | 2 is less than or equal to... |
| 1lt3 12395 | 1 is less than 3. (Contri... |
| 3lt4 12396 | 3 is less than 4. (Contri... |
| 2lt4 12397 | 2 is less than 4. (Contri... |
| 1lt4 12398 | 1 is less than 4. (Contri... |
| 4lt5 12399 | 4 is less than 5. (Contri... |
| 3lt5 12400 | 3 is less than 5. (Contri... |
| 2lt5 12401 | 2 is less than 5. (Contri... |
| 1lt5 12402 | 1 is less than 5. (Contri... |
| 5lt6 12403 | 5 is less than 6. (Contri... |
| 4lt6 12404 | 4 is less than 6. (Contri... |
| 3lt6 12405 | 3 is less than 6. (Contri... |
| 2lt6 12406 | 2 is less than 6. (Contri... |
| 1lt6 12407 | 1 is less than 6. (Contri... |
| 6lt7 12408 | 6 is less than 7. (Contri... |
| 5lt7 12409 | 5 is less than 7. (Contri... |
| 4lt7 12410 | 4 is less than 7. (Contri... |
| 3lt7 12411 | 3 is less than 7. (Contri... |
| 2lt7 12412 | 2 is less than 7. (Contri... |
| 1lt7 12413 | 1 is less than 7. (Contri... |
| 7lt8 12414 | 7 is less than 8. (Contri... |
| 6lt8 12415 | 6 is less than 8. (Contri... |
| 5lt8 12416 | 5 is less than 8. (Contri... |
| 4lt8 12417 | 4 is less than 8. (Contri... |
| 3lt8 12418 | 3 is less than 8. (Contri... |
| 2lt8 12419 | 2 is less than 8. (Contri... |
| 1lt8 12420 | 1 is less than 8. (Contri... |
| 8lt9 12421 | 8 is less than 9. (Contri... |
| 7lt9 12422 | 7 is less than 9. (Contri... |
| 6lt9 12423 | 6 is less than 9. (Contri... |
| 5lt9 12424 | 5 is less than 9. (Contri... |
| 4lt9 12425 | 4 is less than 9. (Contri... |
| 3lt9 12426 | 3 is less than 9. (Contri... |
| 2lt9 12427 | 2 is less than 9. (Contri... |
| 1lt9 12428 | 1 is less than 9. (Contri... |
| 0ne2 12429 | 0 is not equal to 2. (Con... |
| 1ne2 12430 | 1 is not equal to 2. (Con... |
| 1le2 12431 | 1 is less than or equal to... |
| 2cnne0 12432 | 2 is a nonzero complex num... |
| 2rene0 12433 | 2 is a nonzero real number... |
| 1le3 12434 | 1 is less than or equal to... |
| neg1mulneg1e1 12435 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12436 | One-half is real. (Contri... |
| halfcn 12437 | One-half is a complex numb... |
| halfgt0 12438 | One-half is greater than z... |
| halfge0 12439 | One-half is not negative. ... |
| halflt1 12440 | One-half is less than one.... |
| 2halves 12441 | Two halves make a whole. ... |
| 1mhlfehlf 12442 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12443 | An eighth of four thirds i... |
| halfthird 12444 | Half minus a third. (Cont... |
| halfpm6th 12445 | One half plus or minus one... |
| it0e0 12446 | i times 0 equals 0. (Cont... |
| 2mulicn 12447 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12448 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12449 | Closure of half of a numbe... |
| rehalfcl 12450 | Real closure of half. (Co... |
| half0 12451 | Half of a number is zero i... |
| halfpos2 12452 | A number is positive iff i... |
| halfpos 12453 | A positive number is great... |
| halfnneg2 12454 | A number is nonnegative if... |
| halfaddsubcl 12455 | Closure of half-sum and ha... |
| halfaddsub 12456 | Sum and difference of half... |
| subhalfhalf 12457 | Subtracting the half of a ... |
| lt2halves 12458 | A sum is less than the who... |
| addltmul 12459 | Sum is less than product f... |
| nominpos 12460 | There is no smallest posit... |
| avglt1 12461 | Ordering property for aver... |
| avglt2 12462 | Ordering property for aver... |
| avgle1 12463 | Ordering property for aver... |
| avgle2 12464 | Ordering property for aver... |
| avgle 12465 | The average of two numbers... |
| 2timesd 12466 | Two times a number. (Cont... |
| times2d 12467 | A number times 2. (Contri... |
| halfcld 12468 | Closure of half of a numbe... |
| 2halvesd 12469 | Two halves make a whole. ... |
| rehalfcld 12470 | Real closure of half. (Co... |
| lt2halvesd 12471 | A sum is less than the who... |
| rehalfcli 12472 | Half a real number is real... |
| lt2addmuld 12473 | If two real numbers are le... |
| add1p1 12474 | Adding two times 1 to a nu... |
| sub1m1 12475 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12476 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12477 | A complex number increased... |
| div4p1lem1div2 12478 | An integer greater than 5,... |
| nnunb 12479 | The set of positive intege... |
| arch 12480 | Archimedean property of re... |
| nnrecl 12481 | There exists a positive in... |
| bndndx 12482 | A bounded real sequence ` ... |
| elnn0 12485 | Nonnegative integers expre... |
| nnssnn0 12486 | Positive naturals are a su... |
| nn0ssre 12487 | Nonnegative integers are a... |
| nn0sscn 12488 | Nonnegative integers are a... |
| nn0ex 12489 | The set of nonnegative int... |
| nnnn0 12490 | A positive integer is a no... |
| nnnn0i 12491 | A positive integer is a no... |
| nn0re 12492 | A nonnegative integer is a... |
| nn0cn 12493 | A nonnegative integer is a... |
| nn0rei 12494 | A nonnegative integer is a... |
| nn0cni 12495 | A nonnegative integer is a... |
| dfn2 12496 | The set of positive intege... |
| elnnne0 12497 | The positive integer prope... |
| 0nn0 12498 | 0 is a nonnegative integer... |
| 1nn0 12499 | 1 is a nonnegative integer... |
| 2nn0 12500 | 2 is a nonnegative integer... |
| 3nn0 12501 | 3 is a nonnegative integer... |
| 4nn0 12502 | 4 is a nonnegative integer... |
| 5nn0 12503 | 5 is a nonnegative integer... |
| 6nn0 12504 | 6 is a nonnegative integer... |
| 7nn0 12505 | 7 is a nonnegative integer... |
| 8nn0 12506 | 8 is a nonnegative integer... |
| 9nn0 12507 | 9 is a nonnegative integer... |
| nn0ge0 12508 | A nonnegative integer is g... |
| nn0nlt0 12509 | A nonnegative integer is n... |
| nn0ge0i 12510 | Nonnegative integers are n... |
| nn0le0eq0 12511 | A nonnegative integer is l... |
| nn0p1gt0 12512 | A nonnegative integer incr... |
| nnnn0addcl 12513 | A positive integer plus a ... |
| nn0nnaddcl 12514 | A nonnegative integer plus... |
| 0mnnnnn0 12515 | The result of subtracting ... |
| un0addcl 12516 | If ` S ` is closed under a... |
| un0mulcl 12517 | If ` S ` is closed under m... |
| nn0addcl 12518 | Closure of addition of non... |
| nn0mulcl 12519 | Closure of multiplication ... |
| nn0addcli 12520 | Closure of addition of non... |
| nn0mulcli 12521 | Closure of multiplication ... |
| nn0p1nn 12522 | A nonnegative integer plus... |
| peano2nn0 12523 | Second Peano postulate for... |
| nnm1nn0 12524 | A positive integer minus 1... |
| elnn0nn 12525 | The nonnegative integer pr... |
| elnnnn0 12526 | The positive integer prope... |
| elnnnn0b 12527 | The positive integer prope... |
| elnnnn0c 12528 | The positive integer prope... |
| nn0addge1 12529 | A number is less than or e... |
| nn0addge2 12530 | A number is less than or e... |
| nn0addge1i 12531 | A number is less than or e... |
| nn0addge2i 12532 | A number is less than or e... |
| nn0sub 12533 | Subtraction of nonnegative... |
| ltsubnn0 12534 | Subtracting a nonnegative ... |
| nn0negleid 12535 | A nonnegative integer is g... |
| difgtsumgt 12536 | If the difference of a rea... |
| nn0le2x 12537 | A nonnegative integer is l... |
| nn0le2xi 12538 | A nonnegative integer is l... |
| nn0lele2xi 12539 | 'Less than or equal to' im... |
| fcdmnn0supp 12540 | Two ways to write the supp... |
| fcdmnn0fsupp 12541 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12542 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12543 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12544 | A positive integer is a no... |
| nn0red 12545 | A nonnegative integer is a... |
| nn0cnd 12546 | A nonnegative integer is a... |
| nn0ge0d 12547 | A nonnegative integer is g... |
| nn0addcld 12548 | Closure of addition of non... |
| nn0mulcld 12549 | Closure of multiplication ... |
| nn0readdcl 12550 | Closure law for addition o... |
| nn0n0n1ge2 12551 | A nonnegative integer whic... |
| nn0n0n1ge2b 12552 | A nonnegative integer is n... |
| nn0ge2m1nn 12553 | If a nonnegative integer i... |
| nn0ge2m1nn0 12554 | If a nonnegative integer i... |
| nn0nndivcl 12555 | Closure law for dividing o... |
| elxnn0 12558 | An extended nonnegative in... |
| nn0ssxnn0 12559 | The standard nonnegative i... |
| nn0xnn0 12560 | A standard nonnegative int... |
| xnn0xr 12561 | An extended nonnegative in... |
| 0xnn0 12562 | Zero is an extended nonneg... |
| pnf0xnn0 12563 | Positive infinity is an ex... |
| nn0nepnf 12564 | No standard nonnegative in... |
| nn0xnn0d 12565 | A standard nonnegative int... |
| nn0nepnfd 12566 | No standard nonnegative in... |
| xnn0nemnf 12567 | No extended nonnegative in... |
| xnn0xrnemnf 12568 | The extended nonnegative i... |
| xnn0nnn0pnf 12569 | An extended nonnegative in... |
| elz 12572 | Membership in the set of i... |
| nnnegz 12573 | The negative of a positive... |
| zre 12574 | An integer is a real. (Co... |
| zcn 12575 | An integer is a complex nu... |
| zrei 12576 | An integer is a real numbe... |
| zssre 12577 | The integers are a subset ... |
| zsscn 12578 | The integers are a subset ... |
| zex 12579 | The set of integers exists... |
| elnnz 12580 | Positive integer property ... |
| 0z 12581 | Zero is an integer. (Cont... |
| 0zd 12582 | Zero is an integer, deduct... |
| elnn0z 12583 | Nonnegative integer proper... |
| elznn0nn 12584 | Integer property expressed... |
| elznn0 12585 | Integer property expressed... |
| elznn 12586 | Integer property expressed... |
| zle0orge1 12587 | There is no integer in the... |
| elz2 12588 | Membership in the set of i... |
| dfz2 12589 | Alternative definition of ... |
| zexALT 12590 | Alternate proof of ~ zex .... |
| nnz 12591 | A positive integer is an i... |
| nnssz 12592 | Positive integers are a su... |
| nn0ssz 12593 | Nonnegative integers are a... |
| nn0z 12594 | A nonnegative integer is a... |
| nn0zd 12595 | A nonnegative integer is a... |
| nnzd 12596 | A positive integer is an i... |
| nnzi 12597 | A positive integer is an i... |
| nn0zi 12598 | A nonnegative integer is a... |
| elnnz1 12599 | Positive integer property ... |
| znnnlt1 12600 | An integer is not a positi... |
| nnzrab 12601 | Positive integers expresse... |
| nn0zrab 12602 | Nonnegative integers expre... |
| 1z 12603 | One is an integer. (Contr... |
| 1zzd 12604 | One is an integer, deducti... |
| 2z 12605 | 2 is an integer. (Contrib... |
| 3z 12606 | 3 is an integer. (Contrib... |
| 4z 12607 | 4 is an integer. (Contrib... |
| znegcl 12608 | Closure law for negative i... |
| neg1z 12609 | -1 is an integer. (Contri... |
| znegclb 12610 | A complex number is an int... |
| nn0negz 12611 | The negative of a nonnegat... |
| nn0negzi 12612 | The negative of a nonnegat... |
| zaddcl 12613 | Closure of addition of int... |
| peano2z 12614 | Second Peano postulate gen... |
| zsubcl 12615 | Closure of subtraction of ... |
| peano2zm 12616 | "Reverse" second Peano pos... |
| zletr 12617 | Transitive law of ordering... |
| zrevaddcl 12618 | Reverse closure law for ad... |
| znnsub 12619 | The positive difference of... |
| znn0sub 12620 | The nonnegative difference... |
| nzadd 12621 | The sum of a real number n... |
| zmulcl 12622 | Closure of multiplication ... |
| zltp1le 12623 | Integer ordering relation.... |
| zleltp1 12624 | Integer ordering relation.... |
| zlem1lt 12625 | Integer ordering relation.... |
| zltlem1 12626 | Integer ordering relation.... |
| zltlem1d 12627 | Integer ordering relation,... |
| zltp1led 12628 | Integer ordering relation,... |
| zgt0ge1 12629 | An integer greater than ` ... |
| nnleltp1 12630 | Positive integer ordering ... |
| nnltp1le 12631 | Positive integer ordering ... |
| nnaddm1cl 12632 | Closure of addition of pos... |
| nn0ltp1le 12633 | Nonnegative integer orderi... |
| nn0leltp1 12634 | Nonnegative integer orderi... |
| nn0ltlem1 12635 | Nonnegative integer orderi... |
| nn0sub2 12636 | Subtraction of nonnegative... |
| nn0lt10b 12637 | A nonnegative integer less... |
| nn0lt2 12638 | A nonnegative integer less... |
| nn0le2is012 12639 | A nonnegative integer whic... |
| nn0lem1lt 12640 | Nonnegative integer orderi... |
| nnlem1lt 12641 | Positive integer ordering ... |
| nnltlem1 12642 | Positive integer ordering ... |
| nnm1ge0 12643 | A positive integer decreas... |
| nn0ge0div 12644 | Division of a nonnegative ... |
| zdiv 12645 | Two ways to express " ` M ... |
| zdivadd 12646 | Property of divisibility: ... |
| zdivmul 12647 | Property of divisibility: ... |
| zextle 12648 | An extensionality-like pro... |
| zextlt 12649 | An extensionality-like pro... |
| recnz 12650 | The reciprocal of a number... |
| btwnnz 12651 | A number between an intege... |
| gtndiv 12652 | A larger number does not d... |
| halfnz 12653 | One-half is not an integer... |
| 3halfnz 12654 | Three halves is not an int... |
| suprzcl 12655 | The supremum of a bounded-... |
| prime 12656 | Two ways to express " ` A ... |
| msqznn 12657 | The square of a nonzero in... |
| zneo 12658 | No even integer equals an ... |
| nneo 12659 | A positive integer is even... |
| nneoi 12660 | A positive integer is even... |
| zeo 12661 | An integer is even or odd.... |
| zeo2 12662 | An integer is even or odd ... |
| peano2uz2 12663 | Second Peano postulate for... |
| peano5uzi 12664 | Peano's inductive postulat... |
| peano5uzti 12665 | Peano's inductive postulat... |
| dfuzi 12666 | An expression for the uppe... |
| uzind 12667 | Induction on the upper int... |
| uzind2 12668 | Induction on the upper int... |
| uzind3 12669 | Induction on the upper int... |
| nn0ind 12670 | Principle of Mathematical ... |
| nn0indALT 12671 | Principle of Mathematical ... |
| nn0indd 12672 | Principle of Mathematical ... |
| fzind 12673 | Induction on the integers ... |
| fnn0ind 12674 | Induction on the integers ... |
| nn0ind-raph 12675 | Principle of Mathematical ... |
| zindd 12676 | Principle of Mathematical ... |
| fzindd 12677 | Induction on the integers ... |
| btwnz 12678 | Any real number can be san... |
| zred 12679 | An integer is a real numbe... |
| zcnd 12680 | An integer is a complex nu... |
| znegcld 12681 | Closure law for negative i... |
| peano2zd 12682 | Deduction from second Pean... |
| zaddcld 12683 | Closure of addition of int... |
| zsubcld 12684 | Closure of subtraction of ... |
| zmulcld 12685 | Closure of multiplication ... |
| znnn0nn 12686 | The negative of a negative... |
| zadd2cl 12687 | Increasing an integer by 2... |
| zriotaneg 12688 | The negative of the unique... |
| suprfinzcl 12689 | The supremum of a nonempty... |
| 9p1e10 12692 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12693 | Version of the definition ... |
| decex 12694 | A decimal number is a set.... |
| deceq1 12695 | Equality theorem for the d... |
| deceq2 12696 | Equality theorem for the d... |
| deceq1i 12697 | Equality theorem for the d... |
| deceq2i 12698 | Equality theorem for the d... |
| deceq12i 12699 | Equality theorem for the d... |
| numnncl 12700 | Closure for a numeral (wit... |
| num0u 12701 | Add a zero in the units pl... |
| num0h 12702 | Add a zero in the higher p... |
| numcl 12703 | Closure for a decimal inte... |
| numsuc 12704 | The successor of a decimal... |
| deccl 12705 | Closure for a numeral. (C... |
| 11nn0 12706 | 11 is a nonnegative intege... |
| 12nn0 12707 | 12 is a nonnegative intege... |
| 16nn0 12708 | 16 is a nonnegative intege... |
| 25nn0 12709 | 25 is a nonnegative intege... |
| 10nn 12710 | 10 is a positive integer. ... |
| 10pos 12711 | The number 10 is positive.... |
| 10nn0 12712 | 10 is a nonnegative intege... |
| 10re 12713 | The number 10 is real. (C... |
| decnncl 12714 | Closure for a numeral. (C... |
| 11nn 12715 | 11 is a positive integer. ... |
| dec0u 12716 | Add a zero in the units pl... |
| dec0h 12717 | Add a zero in the higher p... |
| numnncl2 12718 | Closure for a decimal inte... |
| decnncl2 12719 | Closure for a decimal inte... |
| numlt 12720 | Comparing two decimal inte... |
| numltc 12721 | Comparing two decimal inte... |
| le9lt10 12722 | A "decimal digit" (i.e. a ... |
| declt 12723 | Comparing two decimal inte... |
| decltc 12724 | Comparing two decimal inte... |
| declth 12725 | Comparing two decimal inte... |
| decsuc 12726 | The successor of a decimal... |
| 3declth 12727 | Comparing two decimal inte... |
| 3decltc 12728 | Comparing two decimal inte... |
| decle 12729 | Comparing two decimal inte... |
| decleh 12730 | Comparing two decimal inte... |
| declei 12731 | Comparing a digit to a dec... |
| numlti 12732 | Comparing a digit to a dec... |
| declti 12733 | Comparing a digit to a dec... |
| decltdi 12734 | Comparing a digit to a dec... |
| numsucc 12735 | The successor of a decimal... |
| decsucc 12736 | The successor of a decimal... |
| 1e0p1 12737 | The successor of zero. (C... |
| dec10p 12738 | Ten plus an integer. (Con... |
| numma 12739 | Perform a multiply-add of ... |
| nummac 12740 | Perform a multiply-add of ... |
| numma2c 12741 | Perform a multiply-add of ... |
| numadd 12742 | Add two decimal integers `... |
| numaddc 12743 | Add two decimal integers `... |
| nummul1c 12744 | The product of a decimal i... |
| nummul2c 12745 | The product of a decimal i... |
| decma 12746 | Perform a multiply-add of ... |
| decmac 12747 | Perform a multiply-add of ... |
| decma2c 12748 | Perform a multiply-add of ... |
| decadd 12749 | Add two numerals ` M ` and... |
| decaddc 12750 | Add two numerals ` M ` and... |
| decaddc2 12751 | Add two numerals ` M ` and... |
| decrmanc 12752 | Perform a multiply-add of ... |
| decrmac 12753 | Perform a multiply-add of ... |
| decaddm10 12754 | The sum of two multiples o... |
| decaddi 12755 | Add two numerals ` M ` and... |
| decaddci 12756 | Add two numerals ` M ` and... |
| decaddci2 12757 | Add two numerals ` M ` and... |
| decsubi 12758 | Difference between a numer... |
| decmul1 12759 | The product of a numeral w... |
| decmul1c 12760 | The product of a numeral w... |
| decmul2c 12761 | The product of a numeral w... |
| decmulnc 12762 | The product of a numeral w... |
| 11multnc 12763 | The product of 11 (as nume... |
| decmul10add 12764 | A multiplication of a numb... |
| 6p5lem 12765 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12766 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12767 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12768 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12769 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12770 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12771 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12772 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12773 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12774 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12775 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12776 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12777 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12778 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12779 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12780 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12781 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12782 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12783 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12784 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12785 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12786 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12787 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12788 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12789 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12790 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12791 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12792 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12793 | 4 times 3 equals 12. (Con... |
| 4t4e16 12794 | 4 times 4 equals 16. (Con... |
| 5t2e10 12795 | 5 times 2 equals 10. (Con... |
| 5t3e15 12796 | 5 times 3 equals 15. (Con... |
| 5t4e20 12797 | 5 times 4 equals 20. (Con... |
| 5t5e25 12798 | 5 times 5 equals 25. (Con... |
| 6t2e12 12799 | 6 times 2 equals 12. (Con... |
| 6t3e18 12800 | 6 times 3 equals 18. (Con... |
| 6t4e24 12801 | 6 times 4 equals 24. (Con... |
| 6t5e30 12802 | 6 times 5 equals 30. (Con... |
| 6t6e36 12803 | 6 times 6 equals 36. (Con... |
| 7t2e14 12804 | 7 times 2 equals 14. (Con... |
| 7t3e21 12805 | 7 times 3 equals 21. (Con... |
| 7t4e28 12806 | 7 times 4 equals 28. (Con... |
| 7t5e35 12807 | 7 times 5 equals 35. (Con... |
| 7t6e42 12808 | 7 times 6 equals 42. (Con... |
| 7t7e49 12809 | 7 times 7 equals 49. (Con... |
| 8t2e16 12810 | 8 times 2 equals 16. (Con... |
| 8t3e24 12811 | 8 times 3 equals 24. (Con... |
| 8t4e32 12812 | 8 times 4 equals 32. (Con... |
| 8t5e40 12813 | 8 times 5 equals 40. (Con... |
| 8t6e48 12814 | 8 times 6 equals 48. (Con... |
| 8t7e56 12815 | 8 times 7 equals 56. (Con... |
| 8t8e64 12816 | 8 times 8 equals 64. (Con... |
| 9t2e18 12817 | 9 times 2 equals 18. (Con... |
| 9t3e27 12818 | 9 times 3 equals 27. (Con... |
| 9t4e36 12819 | 9 times 4 equals 36. (Con... |
| 9t5e45 12820 | 9 times 5 equals 45. (Con... |
| 9t6e54 12821 | 9 times 6 equals 54. (Con... |
| 9t7e63 12822 | 9 times 7 equals 63. (Con... |
| 9t8e72 12823 | 9 times 8 equals 72. (Con... |
| 9t9e81 12824 | 9 times 9 equals 81. (Con... |
| 9t11e99 12825 | 9 times 11 equals 99. (Co... |
| 9t11e99OLD 12826 | Obsolete version of ~ 9t11... |
| 9lt10 12827 | 9 is less than 10. (Contr... |
| 8lt10 12828 | 8 is less than 10. (Contr... |
| 7lt10 12829 | 7 is less than 10. (Contr... |
| 6lt10 12830 | 6 is less than 10. (Contr... |
| 5lt10 12831 | 5 is less than 10. (Contr... |
| 4lt10 12832 | 4 is less than 10. (Contr... |
| 3lt10 12833 | 3 is less than 10. (Contr... |
| 2lt10 12834 | 2 is less than 10. (Contr... |
| 1lt10 12835 | 1 is less than 10. (Contr... |
| 1lt10OLD 12836 | Obsolete version of ~ 1lt1... |
| decbin0 12837 | Decompose base 4 into base... |
| decbin2 12838 | Decompose base 4 into base... |
| decbin3 12839 | Decompose base 4 into base... |
| 5recm6rec 12840 | One fifth minus one sixth.... |
| uzval 12843 | The value of the upper int... |
| uzf 12844 | The domain and codomain of... |
| eluz1 12845 | Membership in the upper se... |
| eluzel2 12846 | Implication of membership ... |
| eluz2 12847 | Membership in an upper set... |
| eluzmn 12848 | Membership in an earlier u... |
| eluz1i 12849 | Membership in an upper set... |
| eluzuzle 12850 | An integer in an upper set... |
| eluzelz 12851 | A member of an upper set o... |
| eluzelre 12852 | A member of an upper set o... |
| eluzelcn 12853 | A member of an upper set o... |
| eluzle 12854 | Implication of membership ... |
| eluz 12855 | Membership in an upper set... |
| uzid 12856 | Membership of the least me... |
| uzidd 12857 | Membership of the least me... |
| uzn0 12858 | The upper integers are all... |
| uztrn 12859 | Transitive law for sets of... |
| uztrn2 12860 | Transitive law for sets of... |
| uzneg 12861 | Contraposition law for upp... |
| uzssz 12862 | An upper set of integers i... |
| uzssre 12863 | An upper set of integers i... |
| uzss 12864 | Subset relationship for tw... |
| uztric 12865 | Totality of the ordering r... |
| uz11 12866 | The upper integers functio... |
| eluzp1m1 12867 | Membership in the next upp... |
| eluzp1l 12868 | Strict ordering implied by... |
| eluzp1p1 12869 | Membership in the next upp... |
| eluzadd 12870 | Membership in a later uppe... |
| eluzsub 12871 | Membership in an earlier u... |
| eluzaddi 12872 | Membership in a later uppe... |
| eluzsubi 12873 | Membership in an earlier u... |
| subeluzsub 12874 | Membership of a difference... |
| uzm1 12875 | Choices for an element of ... |
| uznn0sub 12876 | The nonnegative difference... |
| uzin 12877 | Intersection of two upper ... |
| uzp1 12878 | Choices for an element of ... |
| nn0uz 12879 | Nonnegative integers expre... |
| nnuz 12880 | Positive integers expresse... |
| elnnuz 12881 | A positive integer express... |
| elnn0uz 12882 | A nonnegative integer expr... |
| 1eluzge0 12883 | 1 is an integer greater th... |
| 2eluzge0 12884 | 2 is an integer greater th... |
| 2eluzge1 12885 | 2 is an integer greater th... |
| 5eluz3 12886 | 5 is an integer greater th... |
| uzuzle23 12887 | An integer greater than or... |
| uzuzle24 12888 | An integer greater than or... |
| uzuzle34 12889 | An integer greater than or... |
| uzuzle35 12890 | An integer greater than or... |
| eluz2nn 12891 | An integer greater than or... |
| eluz3nn 12892 | An integer greater than or... |
| eluz4nn 12893 | An integer greater than or... |
| eluz5nn 12894 | An integer greater than or... |
| eluzge2nn0 12895 | If an integer is greater t... |
| eluz2n0 12896 | An integer greater than or... |
| uz3m2nn 12897 | An integer greater than or... |
| uznnssnn 12898 | The upper integers startin... |
| raluz 12899 | Restricted universal quant... |
| raluz2 12900 | Restricted universal quant... |
| rexuz 12901 | Restricted existential qua... |
| rexuz2 12902 | Restricted existential qua... |
| 2rexuz 12903 | Double existential quantif... |
| peano2uz 12904 | Second Peano postulate for... |
| peano2uzs 12905 | Second Peano postulate for... |
| peano2uzr 12906 | Reversed second Peano axio... |
| uzaddcl 12907 | Addition closure law for a... |
| nn0pzuz 12908 | The sum of a nonnegative i... |
| uzind4 12909 | Induction on the upper set... |
| uzind4ALT 12910 | Induction on the upper set... |
| uzind4s 12911 | Induction on the upper set... |
| uzind4s2 12912 | Induction on the upper set... |
| uzind4i 12913 | Induction on the upper int... |
| uzwo 12914 | Well-ordering principle: a... |
| uzwo2 12915 | Well-ordering principle: a... |
| nnwo 12916 | Well-ordering principle: a... |
| nnwof 12917 | Well-ordering principle: a... |
| nnwos 12918 | Well-ordering principle: a... |
| indstr 12919 | Strong Mathematical Induct... |
| eluznn0 12920 | Membership in a nonnegativ... |
| eluznn 12921 | Membership in a positive u... |
| eluz2b1 12922 | Two ways to say "an intege... |
| eluz2gt1 12923 | An integer greater than or... |
| eluz2b2 12924 | Two ways to say "an intege... |
| eluz2b3 12925 | Two ways to say "an intege... |
| uz2m1nn 12926 | One less than an integer g... |
| 1nuz2 12927 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12928 | A positive integer is eith... |
| uz2mulcl 12929 | Closure of multiplication ... |
| indstr2 12930 | Strong Mathematical Induct... |
| uzinfi 12931 | Extract the lower bound of... |
| nninf 12932 | The infimum of the set of ... |
| nn0inf 12933 | The infimum of the set of ... |
| infssuzle 12934 | The infimum of a subset of... |
| infssuzcl 12935 | The infimum of a subset of... |
| ublbneg 12936 | The image under negation o... |
| eqreznegel 12937 | Two ways to express the im... |
| supminf 12938 | The supremum of a bounded-... |
| lbzbi 12939 | If a set of reals is bound... |
| zsupss 12940 | Any nonempty bounded subse... |
| suprzcl2 12941 | The supremum of a bounded-... |
| suprzub 12942 | The supremum of a bounded-... |
| uzsupss 12943 | Any bounded subset of an u... |
| nn01to3 12944 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12945 | Alternate proof of ~ nn0ge... |
| uzwo3 12946 | Well-ordering principle: a... |
| zmin 12947 | There is a unique smallest... |
| zmax 12948 | There is a unique largest ... |
| zbtwnre 12949 | There is a unique integer ... |
| rebtwnz 12950 | There is a unique greatest... |
| elq 12953 | Membership in the set of r... |
| qmulz 12954 | If ` A ` is rational, then... |
| znq 12955 | The ratio of an integer an... |
| qre 12956 | A rational number is a rea... |
| zq 12957 | An integer is a rational n... |
| qred 12958 | A rational number is a rea... |
| zssq 12959 | The integers are a subset ... |
| nn0ssq 12960 | The nonnegative integers a... |
| nnssq 12961 | The positive integers are ... |
| qssre 12962 | The rationals are a subset... |
| qsscn 12963 | The rationals are a subset... |
| qex 12964 | The set of rational number... |
| nnq 12965 | A positive integer is rati... |
| qcn 12966 | A rational number is a com... |
| qexALT 12967 | Alternate proof of ~ qex .... |
| qaddcl 12968 | Closure of addition of rat... |
| qnegcl 12969 | Closure law for the negati... |
| qmulcl 12970 | Closure of multiplication ... |
| qsubcl 12971 | Closure of subtraction of ... |
| qreccl 12972 | Closure of reciprocal of r... |
| qdivcl 12973 | Closure of division of rat... |
| qrevaddcl 12974 | Reverse closure law for ad... |
| nnrecq 12975 | The reciprocal of a positi... |
| irradd 12976 | The sum of an irrational n... |
| irrmul 12977 | The product of an irration... |
| elpq 12978 | A positive rational is the... |
| elpqb 12979 | A class is a positive rati... |
| rpnnen1lem2 12980 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12981 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12982 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12983 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12984 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12985 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12986 | One half of ~ rpnnen , whe... |
| reexALT 12987 | Alternate proof of ~ reex ... |
| cnref1o 12988 | There is a natural one-to-... |
| cnexALT 12989 | The set of complex numbers... |
| xrex 12990 | The set of extended reals ... |
| mpoaddex 12991 | The addition operation is ... |
| addex 12992 | The addition operation is ... |
| mpomulex 12993 | The multiplication operati... |
| mulex 12994 | The multiplication operati... |
| elrp 12997 | Membership in the set of p... |
| elrpii 12998 | Membership in the set of p... |
| 1rp 12999 | 1 is a positive real. (Co... |
| 2rp 13000 | 2 is a positive real. (Co... |
| 3rp 13001 | 3 is a positive real. (Co... |
| 5rp 13002 | 5 is a positive real. (Co... |
| rpssre 13003 | The positive reals are a s... |
| rpre 13004 | A positive real is a real.... |
| rpxr 13005 | A positive real is an exte... |
| rpcn 13006 | A positive real is a compl... |
| nnrp 13007 | A positive integer is a po... |
| rpgt0 13008 | A positive real is greater... |
| rpge0 13009 | A positive real is greater... |
| rpregt0 13010 | A positive real is a posit... |
| rprege0 13011 | A positive real is a nonne... |
| rpne0 13012 | A positive real is nonzero... |
| rprene0 13013 | A positive real is a nonze... |
| rpcnne0 13014 | A positive real is a nonze... |
| neglt 13015 | The negative of a positive... |
| rpcndif0 13016 | A positive real number is ... |
| ralrp 13017 | Quantification over positi... |
| rexrp 13018 | Quantification over positi... |
| rpaddcl 13019 | Closure law for addition o... |
| rpmulcl 13020 | Closure law for multiplica... |
| rpmtmip 13021 | "Minus times minus is plus... |
| rpdivcl 13022 | Closure law for division o... |
| rpreccl 13023 | Closure law for reciprocat... |
| rphalfcl 13024 | Closure law for half of a ... |
| rpgecl 13025 | A number greater than or e... |
| rphalflt 13026 | Half of a positive real is... |
| rerpdivcl 13027 | Closure law for division o... |
| ge0p1rp 13028 | A nonnegative number plus ... |
| rpneg 13029 | Either a nonzero real or i... |
| negelrp 13030 | Elementhood of a negation ... |
| negelrpd 13031 | The negation of a negative... |
| 0nrp 13032 | Zero is not a positive rea... |
| ltsubrp 13033 | Subtracting a positive rea... |
| ltaddrp 13034 | Adding a positive number t... |
| difrp 13035 | Two ways to say one number... |
| elrpd 13036 | Membership in the set of p... |
| nnrpd 13037 | A positive integer is a po... |
| zgt1rpn0n1 13038 | An integer greater than 1 ... |
| rpred 13039 | A positive real is a real.... |
| rpxrd 13040 | A positive real is an exte... |
| rpcnd 13041 | A positive real is a compl... |
| rpgt0d 13042 | A positive real is greater... |
| rpge0d 13043 | A positive real is greater... |
| rpne0d 13044 | A positive real is nonzero... |
| rpregt0d 13045 | A positive real is real an... |
| rprege0d 13046 | A positive real is real an... |
| rprene0d 13047 | A positive real is a nonze... |
| rpcnne0d 13048 | A positive real is a nonze... |
| rpreccld 13049 | Closure law for reciprocat... |
| rprecred 13050 | Closure law for reciprocat... |
| rphalfcld 13051 | Closure law for half of a ... |
| reclt1d 13052 | The reciprocal of a positi... |
| recgt1d 13053 | The reciprocal of a positi... |
| rpaddcld 13054 | Closure law for addition o... |
| rpmulcld 13055 | Closure law for multiplica... |
| rpdivcld 13056 | Closure law for division o... |
| ltrecd 13057 | The reciprocal of both sid... |
| lerecd 13058 | The reciprocal of both sid... |
| ltrec1d 13059 | Reciprocal swap in a 'less... |
| lerec2d 13060 | Reciprocal swap in a 'less... |
| lediv2ad 13061 | Division of both sides of ... |
| ltdiv2d 13062 | Division of a positive num... |
| lediv2d 13063 | Division of a positive num... |
| ledivdivd 13064 | Invert ratios of positive ... |
| divge1 13065 | The ratio of a number over... |
| divlt1lt 13066 | A real number divided by a... |
| divle1le 13067 | A real number divided by a... |
| ledivge1le 13068 | If a number is less than o... |
| ge0p1rpd 13069 | A nonnegative number plus ... |
| rerpdivcld 13070 | Closure law for division o... |
| ltsubrpd 13071 | Subtracting a positive rea... |
| ltaddrpd 13072 | Adding a positive number t... |
| ltaddrp2d 13073 | Adding a positive number t... |
| ltmulgt11d 13074 | Multiplication by a number... |
| ltmulgt12d 13075 | Multiplication by a number... |
| gt0divd 13076 | Division of a positive num... |
| ge0divd 13077 | Division of a nonnegative ... |
| rpgecld 13078 | A number greater than or e... |
| divge0d 13079 | The ratio of nonnegative a... |
| ltmul1d 13080 | The ratio of nonnegative a... |
| ltmul2d 13081 | Multiplication of both sid... |
| lemul1d 13082 | Multiplication of both sid... |
| lemul2d 13083 | Multiplication of both sid... |
| ltdiv1d 13084 | Division of both sides of ... |
| lediv1d 13085 | Division of both sides of ... |
| ltmuldivd 13086 | 'Less than' relationship b... |
| ltmuldiv2d 13087 | 'Less than' relationship b... |
| lemuldivd 13088 | 'Less than or equal to' re... |
| lemuldiv2d 13089 | 'Less than or equal to' re... |
| ltdivmuld 13090 | 'Less than' relationship b... |
| ltdivmul2d 13091 | 'Less than' relationship b... |
| ledivmuld 13092 | 'Less than or equal to' re... |
| ledivmul2d 13093 | 'Less than or equal to' re... |
| ltmul1dd 13094 | The ratio of nonnegative a... |
| ltmul2dd 13095 | Multiplication of both sid... |
| ltdiv1dd 13096 | Division of both sides of ... |
| lediv1dd 13097 | Division of both sides of ... |
| lediv12ad 13098 | Comparison of ratio of two... |
| mul2lt0rlt0 13099 | If the result of a multipl... |
| mul2lt0rgt0 13100 | If the result of a multipl... |
| mul2lt0llt0 13101 | If the result of a multipl... |
| mul2lt0lgt0 13102 | If the result of a multipl... |
| mul2lt0bi 13103 | If the result of a multipl... |
| prodge0rd 13104 | Infer that a multiplicand ... |
| prodge0ld 13105 | Infer that a multiplier is... |
| ltdiv23d 13106 | Swap denominator with othe... |
| lediv23d 13107 | Swap denominator with othe... |
| lt2mul2divd 13108 | The ratio of nonnegative a... |
| nnledivrp 13109 | Division of a positive int... |
| nn0ledivnn 13110 | Division of a nonnegative ... |
| addlelt 13111 | If the sum of a real numbe... |
| ge2halflem1 13112 | Half of an integer greater... |
| ltxr 13119 | The 'less than' binary rel... |
| elxr 13120 | Membership in the set of e... |
| xrnemnf 13121 | An extended real other tha... |
| xrnepnf 13122 | An extended real other tha... |
| xrltnr 13123 | The extended real 'less th... |
| ltpnf 13124 | Any (finite) real is less ... |
| ltpnfd 13125 | Any (finite) real is less ... |
| 0ltpnf 13126 | Zero is less than plus inf... |
| mnflt 13127 | Minus infinity is less tha... |
| mnfltd 13128 | Minus infinity is less tha... |
| mnflt0 13129 | Minus infinity is less tha... |
| mnfltpnf 13130 | Minus infinity is less tha... |
| mnfltxr 13131 | Minus infinity is less tha... |
| pnfnlt 13132 | No extended real is greate... |
| nltmnf 13133 | No extended real is less t... |
| pnfge 13134 | Plus infinity is an upper ... |
| pnfged 13135 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13136 | An extended nonnegative in... |
| 0lepnf 13137 | 0 less than or equal to po... |
| xnn0ge0 13138 | An extended nonnegative in... |
| mnfle 13139 | Minus infinity is less tha... |
| mnfled 13140 | Minus infinity is less tha... |
| xrltnsym 13141 | Ordering on the extended r... |
| xrltnsym2 13142 | 'Less than' is antisymmetr... |
| xrlttri 13143 | Ordering on the extended r... |
| xrlttr 13144 | Ordering on the extended r... |
| xrltso 13145 | 'Less than' is a strict or... |
| xrlttri2 13146 | Trichotomy law for 'less t... |
| xrlttri3 13147 | Trichotomy law for 'less t... |
| xrleloe 13148 | 'Less than or equal' expre... |
| xrleltne 13149 | 'Less than or equal to' im... |
| xrltlen 13150 | 'Less than' expressed in t... |
| dfle2 13151 | Alternative definition of ... |
| dflt2 13152 | Alternative definition of ... |
| xrltle 13153 | 'Less than' implies 'less ... |
| xrltled 13154 | 'Less than' implies 'less ... |
| xrleid 13155 | 'Less than or equal to' is... |
| xrleidd 13156 | 'Less than or equal to' is... |
| xrletri 13157 | Trichotomy law for extende... |
| xrletri3 13158 | Trichotomy law for extende... |
| xrletrid 13159 | Trichotomy law for extende... |
| xrlelttr 13160 | Transitive law for orderin... |
| xrltletr 13161 | Transitive law for orderin... |
| xrletr 13162 | Transitive law for orderin... |
| xrlttrd 13163 | Transitive law for orderin... |
| xrlelttrd 13164 | Transitive law for orderin... |
| xrltletrd 13165 | Transitive law for orderin... |
| xrletrd 13166 | Transitive law for orderin... |
| xrltne 13167 | 'Less than' implies not eq... |
| xrgtned 13168 | 'Greater than' implies not... |
| nltpnft 13169 | An extended real is not le... |
| xgepnf 13170 | An extended real which is ... |
| ngtmnft 13171 | An extended real is not gr... |
| xlemnf 13172 | An extended real which is ... |
| xrrebnd 13173 | An extended real is real i... |
| xrre 13174 | A way of proving that an e... |
| xrre2 13175 | An extended real between t... |
| xrre3 13176 | A way of proving that an e... |
| ge0gtmnf 13177 | A nonnegative extended rea... |
| ge0nemnf 13178 | A nonnegative extended rea... |
| xrrege0 13179 | A nonnegative extended rea... |
| xrmax1 13180 | An extended real is less t... |
| xrmax2 13181 | An extended real is less t... |
| xrmin1 13182 | The minimum of two extende... |
| xrmin2 13183 | The minimum of two extende... |
| xrmaxeq 13184 | The maximum of two extende... |
| xrmineq 13185 | The minimum of two extende... |
| xrmaxlt 13186 | Two ways of saying the max... |
| xrltmin 13187 | Two ways of saying an exte... |
| xrmaxle 13188 | Two ways of saying the max... |
| xrlemin 13189 | Two ways of saying a numbe... |
| max1 13190 | A number is less than or e... |
| max1ALT 13191 | A number is less than or e... |
| max2 13192 | A number is less than or e... |
| 2resupmax 13193 | The supremum of two real n... |
| min1 13194 | The minimum of two numbers... |
| min2 13195 | The minimum of two numbers... |
| maxle 13196 | Two ways of saying the max... |
| lemin 13197 | Two ways of saying a numbe... |
| maxlt 13198 | Two ways of saying the max... |
| ltmin 13199 | Two ways of saying a numbe... |
| lemaxle 13200 | A real number which is les... |
| max0sub 13201 | Decompose a real number in... |
| ifle 13202 | An if statement transforms... |
| z2ge 13203 | There exists an integer gr... |
| qbtwnre 13204 | The rational numbers are d... |
| qbtwnxr 13205 | The rational numbers are d... |
| qsqueeze 13206 | If a nonnegative real is l... |
| qextltlem 13207 | Lemma for ~ qextlt and qex... |
| qextlt 13208 | An extensionality-like pro... |
| qextle 13209 | An extensionality-like pro... |
| xralrple 13210 | Show that ` A ` is less th... |
| alrple 13211 | Show that ` A ` is less th... |
| xnegeq 13212 | Equality of two extended n... |
| xnegex 13213 | A negative extended real e... |
| xnegpnf 13214 | Minus ` +oo ` . Remark of... |
| xnegmnf 13215 | Minus ` -oo ` . Remark of... |
| rexneg 13216 | Minus a real number. Rema... |
| xneg0 13217 | The negative of zero. (Co... |
| xnegcl 13218 | Closure of extended real n... |
| xnegneg 13219 | Extended real version of ~... |
| xneg11 13220 | Extended real version of ~... |
| xltnegi 13221 | Forward direction of ~ xlt... |
| xltneg 13222 | Extended real version of ~... |
| xleneg 13223 | Extended real version of ~... |
| xlt0neg1 13224 | Extended real version of ~... |
| xlt0neg2 13225 | Extended real version of ~... |
| xle0neg1 13226 | Extended real version of ~... |
| xle0neg2 13227 | Extended real version of ~... |
| xaddval 13228 | Value of the extended real... |
| xaddf 13229 | The extended real addition... |
| xmulval 13230 | Value of the extended real... |
| xaddpnf1 13231 | Addition of positive infin... |
| xaddpnf2 13232 | Addition of positive infin... |
| xaddmnf1 13233 | Addition of negative infin... |
| xaddmnf2 13234 | Addition of negative infin... |
| pnfaddmnf 13235 | Addition of positive and n... |
| mnfaddpnf 13236 | Addition of negative and p... |
| rexadd 13237 | The extended real addition... |
| rexsub 13238 | Extended real subtraction ... |
| rexaddd 13239 | The extended real addition... |
| xnn0xaddcl 13240 | The extended nonnegative i... |
| xaddnemnf 13241 | Closure of extended real a... |
| xaddnepnf 13242 | Closure of extended real a... |
| xnegid 13243 | Extended real version of ~... |
| xaddcl 13244 | The extended real addition... |
| xaddcom 13245 | The extended real addition... |
| xaddrid 13246 | Extended real version of ~... |
| xaddlid 13247 | Extended real version of ~... |
| xaddridd 13248 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13249 | Extended nonnegative integ... |
| xnn0lenn0nn0 13250 | An extended nonnegative in... |
| xnn0le2is012 13251 | An extended nonnegative in... |
| xnn0xadd0 13252 | The sum of two extended no... |
| xnegdi 13253 | Extended real version of ~... |
| xaddass 13254 | Associativity of extended ... |
| xaddass2 13255 | Associativity of extended ... |
| xpncan 13256 | Extended real version of ~... |
| xnpcan 13257 | Extended real version of ~... |
| xleadd1a 13258 | Extended real version of ~... |
| xleadd2a 13259 | Commuted form of ~ xleadd1... |
| xleadd1 13260 | Weakened version of ~ xlea... |
| xltadd1 13261 | Extended real version of ~... |
| xltadd2 13262 | Extended real version of ~... |
| xaddge0 13263 | The sum of nonnegative ext... |
| xle2add 13264 | Extended real version of ~... |
| xlt2add 13265 | Extended real version of ~... |
| xsubge0 13266 | Extended real version of ~... |
| xposdif 13267 | Extended real version of ~... |
| xlesubadd 13268 | Under certain conditions, ... |
| xmullem 13269 | Lemma for ~ rexmul . (Con... |
| xmullem2 13270 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13271 | Extended real multiplicati... |
| xmul01 13272 | Extended real version of ~... |
| xmul02 13273 | Extended real version of ~... |
| xmulneg1 13274 | Extended real version of ~... |
| xmulneg2 13275 | Extended real version of ~... |
| rexmul 13276 | The extended real multipli... |
| xmulf 13277 | The extended real multipli... |
| xmulcl 13278 | Closure of extended real m... |
| xmulpnf1 13279 | Multiplication by plus inf... |
| xmulpnf2 13280 | Multiplication by plus inf... |
| xmulmnf1 13281 | Multiplication by minus in... |
| xmulmnf2 13282 | Multiplication by minus in... |
| xmulpnf1n 13283 | Multiplication by plus inf... |
| xmulrid 13284 | Extended real version of ~... |
| xmullid 13285 | Extended real version of ~... |
| xmulm1 13286 | Extended real version of ~... |
| xmulasslem2 13287 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13288 | Extended real version of ~... |
| xmulge0 13289 | Extended real version of ~... |
| xmulasslem 13290 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13291 | Lemma for ~ xmulass . (Co... |
| xmulass 13292 | Associativity of the exten... |
| xlemul1a 13293 | Extended real version of ~... |
| xlemul2a 13294 | Extended real version of ~... |
| xlemul1 13295 | Extended real version of ~... |
| xlemul2 13296 | Extended real version of ~... |
| xltmul1 13297 | Extended real version of ~... |
| xltmul2 13298 | Extended real version of ~... |
| xadddilem 13299 | Lemma for ~ xadddi . (Con... |
| xadddi 13300 | Distributive property for ... |
| xadddir 13301 | Commuted version of ~ xadd... |
| xadddi2 13302 | The assumption that the mu... |
| xadddi2r 13303 | Commuted version of ~ xadd... |
| x2times 13304 | Extended real version of ~... |
| xnegcld 13305 | Closure of extended real n... |
| xaddcld 13306 | The extended real addition... |
| xmulcld 13307 | Closure of extended real m... |
| xadd4d 13308 | Rearrangement of 4 terms i... |
| xnn0add4d 13309 | Rearrangement of 4 terms i... |
| xrsupexmnf 13310 | Adding minus infinity to a... |
| xrinfmexpnf 13311 | Adding plus infinity to a ... |
| xrsupsslem 13312 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13313 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13314 | Any subset of extended rea... |
| xrinfmss 13315 | Any subset of extended rea... |
| xrinfmss2 13316 | Any subset of extended rea... |
| xrub 13317 | By quantifying only over r... |
| supxr 13318 | The supremum of a set of e... |
| supxr2 13319 | The supremum of a set of e... |
| supxrcl 13320 | The supremum of an arbitra... |
| supxrun 13321 | The supremum of the union ... |
| supxrmnf 13322 | Adding minus infinity to a... |
| supxrpnf 13323 | The supremum of a set of e... |
| supxrunb1 13324 | The supremum of an unbound... |
| supxrunb2 13325 | The supremum of an unbound... |
| supxrbnd1 13326 | The supremum of a bounded-... |
| supxrbnd2 13327 | The supremum of a bounded-... |
| xrsup0 13328 | The supremum of an empty s... |
| supxrub 13329 | A member of a set of exten... |
| supxrlub 13330 | The supremum of a set of e... |
| supxrleub 13331 | The supremum of a set of e... |
| supxrre 13332 | The real and extended real... |
| supxrbnd 13333 | The supremum of a bounded-... |
| supxrgtmnf 13334 | The supremum of a nonempty... |
| supxrre1 13335 | The supremum of a nonempty... |
| supxrre2 13336 | The supremum of a nonempty... |
| supxrss 13337 | Smaller sets of extended r... |
| xrsupssd 13338 | Inequality deduction for s... |
| infxrcl 13339 | The infimum of an arbitrar... |
| infxrlb 13340 | A member of a set of exten... |
| infxrgelb 13341 | The infimum of a set of ex... |
| infxrre 13342 | The real and extended real... |
| infxrmnf 13343 | The infinimum of a set of ... |
| xrinf0 13344 | The infimum of the empty s... |
| infxrss 13345 | Larger sets of extended re... |
| reltre 13346 | For all real numbers there... |
| rpltrp 13347 | For all positive real numb... |
| reltxrnmnf 13348 | For all extended real numb... |
| infmremnf 13349 | The infimum of the reals i... |
| infmrp1 13350 | The infimum of the positiv... |
| ixxval 13359 | Value of the interval func... |
| elixx1 13360 | Membership in an interval ... |
| ixxf 13361 | The set of intervals of ex... |
| ixxex 13362 | The set of intervals of ex... |
| ixxssxr 13363 | The set of intervals of ex... |
| elixx3g 13364 | Membership in a set of ope... |
| ixxssixx 13365 | An interval is a subset of... |
| ixxdisj 13366 | Split an interval into dis... |
| ixxun 13367 | Split an interval into two... |
| ixxin 13368 | Intersection of two interv... |
| ixxss1 13369 | Subset relationship for in... |
| ixxss2 13370 | Subset relationship for in... |
| ixxss12 13371 | Subset relationship for in... |
| ixxub 13372 | Extract the upper bound of... |
| ixxlb 13373 | Extract the lower bound of... |
| iooex 13374 | The set of open intervals ... |
| iooval 13375 | Value of the open interval... |
| ioo0 13376 | An empty open interval of ... |
| ioon0 13377 | An open interval of extend... |
| ndmioo 13378 | The open interval function... |
| iooid 13379 | An open interval with iden... |
| elioo3g 13380 | Membership in a set of ope... |
| elioore 13381 | A member of an open interv... |
| lbioo 13382 | An open interval does not ... |
| ubioo 13383 | An open interval does not ... |
| iooval2 13384 | Value of the open interval... |
| iooin 13385 | Intersection of two open i... |
| iooss1 13386 | Subset relationship for op... |
| iooss2 13387 | Subset relationship for op... |
| iocval 13388 | Value of the open-below, c... |
| icoval 13389 | Value of the closed-below,... |
| iccval 13390 | Value of the closed interv... |
| elioo1 13391 | Membership in an open inte... |
| elioo2 13392 | Membership in an open inte... |
| elioc1 13393 | Membership in an open-belo... |
| elico1 13394 | Membership in a closed-bel... |
| elicc1 13395 | Membership in a closed int... |
| iccid 13396 | A closed interval with ide... |
| ico0 13397 | An empty open interval of ... |
| ioc0 13398 | An empty open interval of ... |
| icc0 13399 | An empty closed interval o... |
| dfrp2 13400 | Alternate definition of th... |
| elicod 13401 | Membership in a left-close... |
| icogelb 13402 | An element of a left-close... |
| icogelbd 13403 | An element of a left-close... |
| elicore 13404 | A member of a left-closed ... |
| ubioc1 13405 | The upper bound belongs to... |
| lbico1 13406 | The lower bound belongs to... |
| iccleub 13407 | An element of a closed int... |
| iccgelb 13408 | An element of a closed int... |
| elioo5 13409 | Membership in an open inte... |
| eliooxr 13410 | A nonempty open interval s... |
| eliooord 13411 | Ordering implied by a memb... |
| elioo4g 13412 | Membership in an open inte... |
| ioossre 13413 | An open interval is a set ... |
| ioosscn 13414 | An open interval is a set ... |
| elioc2 13415 | Membership in an open-belo... |
| elico2 13416 | Membership in a closed-bel... |
| elicc2 13417 | Membership in a closed rea... |
| elicc2i 13418 | Inference for membership i... |
| elicc4 13419 | Membership in a closed rea... |
| iccss 13420 | Condition for a closed int... |
| iccssioo 13421 | Condition for a closed int... |
| icossico 13422 | Condition for a closed-bel... |
| iccss2 13423 | Condition for a closed int... |
| iccssico 13424 | Condition for a closed int... |
| iccssioo2 13425 | Condition for a closed int... |
| iccssico2 13426 | Condition for a closed int... |
| icossico2d 13427 | Condition for a closed-bel... |
| ioomax 13428 | The open interval from min... |
| iccmax 13429 | The closed interval from m... |
| ioopos 13430 | The set of positive reals ... |
| ioorp 13431 | The set of positive reals ... |
| iooshf 13432 | Shift the arguments of the... |
| iocssre 13433 | A closed-above interval wi... |
| icossre 13434 | A closed-below interval wi... |
| iccssre 13435 | A closed real interval is ... |
| iccssxr 13436 | A closed interval is a set... |
| iocssxr 13437 | An open-below, closed-abov... |
| icossxr 13438 | A closed-below, open-above... |
| ioossicc 13439 | An open interval is a subs... |
| iccssred 13440 | A closed real interval is ... |
| eliccxr 13441 | A member of a closed inter... |
| icossicc 13442 | A closed-below, open-above... |
| iocssicc 13443 | A closed-above, open-below... |
| ioossico 13444 | An open interval is a subs... |
| iocssioo 13445 | Condition for a closed int... |
| icossioo 13446 | Condition for a closed int... |
| ioossioo 13447 | Condition for an open inte... |
| iccsupr 13448 | A nonempty subset of a clo... |
| elioopnf 13449 | Membership in an unbounded... |
| elioomnf 13450 | Membership in an unbounded... |
| elicopnf 13451 | Membership in a closed unb... |
| repos 13452 | Two ways of saying that a ... |
| ioof 13453 | The set of open intervals ... |
| iccf 13454 | The set of closed interval... |
| unirnioo 13455 | The union of the range of ... |
| dfioo2 13456 | Alternate definition of th... |
| ioorebas 13457 | Open intervals are element... |
| xrge0neqmnf 13458 | A nonnegative extended rea... |
| xrge0nre 13459 | An extended real which is ... |
| elrege0 13460 | The predicate "is a nonneg... |
| nn0rp0 13461 | A nonnegative integer is a... |
| rge0ssre 13462 | Nonnegative real numbers a... |
| elxrge0 13463 | Elementhood in the set of ... |
| 0e0icopnf 13464 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13465 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13466 | The nonnegative reals are ... |
| ge0mulcl 13467 | The nonnegative reals are ... |
| ge0xaddcl 13468 | The nonnegative reals are ... |
| ge0xmulcl 13469 | The nonnegative extended r... |
| lbicc2 13470 | The lower bound of a close... |
| ubicc2 13471 | The upper bound of a close... |
| elicc01 13472 | Membership in the closed r... |
| elunitrn 13473 | The closed unit interval i... |
| elunitcn 13474 | The closed unit interval i... |
| 0elunit 13475 | Zero is an element of the ... |
| 1elunit 13476 | One is an element of the c... |
| iooneg 13477 | Membership in a negated op... |
| iccneg 13478 | Membership in a negated cl... |
| icoshft 13479 | A shifted real is a member... |
| icoshftf1o 13480 | Shifting a closed-below, o... |
| icoun 13481 | The union of two adjacent ... |
| icodisj 13482 | Adjacent left-closed right... |
| ioounsn 13483 | The union of an open inter... |
| snunioo 13484 | The closure of one end of ... |
| snunico 13485 | The closure of the open en... |
| snunioc 13486 | The closure of the open en... |
| prunioo 13487 | The closure of an open rea... |
| ioodisj 13488 | If the upper bound of one ... |
| ioojoin 13489 | Join two open intervals to... |
| difreicc 13490 | The class difference of ` ... |
| iccsplit 13491 | Split a closed interval in... |
| iccshftr 13492 | Membership in a shifted in... |
| iccshftri 13493 | Membership in a shifted in... |
| iccshftl 13494 | Membership in a shifted in... |
| iccshftli 13495 | Membership in a shifted in... |
| iccdil 13496 | Membership in a dilated in... |
| iccdili 13497 | Membership in a dilated in... |
| icccntr 13498 | Membership in a contracted... |
| icccntri 13499 | Membership in a contracted... |
| divelunit 13500 | A condition for a ratio to... |
| lincmb01cmp 13501 | A linear combination of tw... |
| iccf1o 13502 | Describe a bijection from ... |
| iccen 13503 | Any nontrivial closed inte... |
| xov1plusxeqvd 13504 | A complex number ` X ` is ... |
| unitssre 13505 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13506 | The closed unit interval i... |
| supicc 13507 | Supremum of a bounded set ... |
| supiccub 13508 | The supremum of a bounded ... |
| supicclub 13509 | The supremum of a bounded ... |
| supicclub2 13510 | The supremum of a bounded ... |
| zltaddlt1le 13511 | The sum of an integer and ... |
| xnn0xrge0 13512 | An extended nonnegative in... |
| nnge2recico01 13513 | The reciprocal of an integ... |
| fzval 13516 | The value of a finite set ... |
| fzval2 13517 | An alternative way of expr... |
| fzf 13518 | Establish the domain and c... |
| elfz1 13519 | Membership in a finite set... |
| elfz 13520 | Membership in a finite set... |
| elfz2 13521 | Membership in a finite set... |
| elfzd 13522 | Membership in a finite set... |
| elfz5 13523 | Membership in a finite set... |
| elfz4 13524 | Membership in a finite set... |
| elfzuzb 13525 | Membership in a finite set... |
| eluzfz 13526 | Membership in a finite set... |
| elfzuz 13527 | A member of a finite set o... |
| elfzuz3 13528 | Membership in a finite set... |
| elfzel2 13529 | Membership in a finite set... |
| elfzel1 13530 | Membership in a finite set... |
| elfzelz 13531 | A member of a finite set o... |
| elfzelzd 13532 | A member of a finite set o... |
| fzssz 13533 | A finite sequence of integ... |
| elfzle1 13534 | A member of a finite set o... |
| elfzle2 13535 | A member of a finite set o... |
| elfzuz2 13536 | Implication of membership ... |
| elfzle3 13537 | Membership in a finite set... |
| eluzfz1 13538 | Membership in a finite set... |
| eluzfz2 13539 | Membership in a finite set... |
| eluzfz2b 13540 | Membership in a finite set... |
| elfz3 13541 | Membership in a finite set... |
| elfz1eq 13542 | Membership in a finite set... |
| elfzubelfz 13543 | If there is a member in a ... |
| peano2fzr 13544 | A Peano-postulate-like the... |
| fzn0 13545 | Properties of a finite int... |
| fz0 13546 | A finite set of sequential... |
| fzn 13547 | A finite set of sequential... |
| fzen 13548 | A shifted finite set of se... |
| fz1n 13549 | A 1-based finite set of se... |
| 0nelfz1 13550 | 0 is not an element of a f... |
| 0fz1 13551 | Two ways to say a finite 1... |
| fz10 13552 | There are no integers betw... |
| uzsubsubfz 13553 | Membership of an integer g... |
| uzsubsubfz1 13554 | Membership of an integer g... |
| ige3m2fz 13555 | Membership of an integer g... |
| fzsplit2 13556 | Split a finite interval of... |
| fzsplit 13557 | Split a finite interval of... |
| fzdisj 13558 | Condition for two finite i... |
| fz01en 13559 | 0-based and 1-based finite... |
| elfznn 13560 | A member of a finite set o... |
| elfz1end 13561 | A nonempty finite range of... |
| fz1ssnn 13562 | A finite set of positive i... |
| fznn0sub 13563 | Subtraction closure for a ... |
| fzmmmeqm 13564 | Subtracting the difference... |
| fzaddel 13565 | Membership of a sum in a f... |
| fzadd2 13566 | Membership of a sum in a f... |
| fzsubel 13567 | Membership of a difference... |
| fzopth 13568 | A finite set of sequential... |
| fzass4 13569 | Two ways to express a nond... |
| fzss1 13570 | Subset relationship for fi... |
| fzss2 13571 | Subset relationship for fi... |
| fzssuz 13572 | A finite set of sequential... |
| fzsn 13573 | A finite interval of integ... |
| fzssp1 13574 | Subset relationship for fi... |
| fzssnn 13575 | Finite sets of sequential ... |
| ssfzunsnext 13576 | A subset of a finite seque... |
| ssfzunsn 13577 | A subset of a finite seque... |
| fzsuc 13578 | Join a successor to the en... |
| fzpred 13579 | Join a predecessor to the ... |
| fzpreddisj 13580 | A finite set of sequential... |
| elfzp1 13581 | Append an element to a fin... |
| fzp1ss 13582 | Subset relationship for fi... |
| fzelp1 13583 | Membership in a set of seq... |
| fzp1elp1 13584 | Add one to an element of a... |
| fznatpl1 13585 | Shift membership in a fini... |
| fzpr 13586 | A finite interval of integ... |
| fztp 13587 | A finite interval of integ... |
| fz12pr 13588 | An integer range between 1... |
| fzsuc2 13589 | Join a successor to the en... |
| fzp1disj 13590 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13591 | Remove a successor from th... |
| fzprval 13592 | Two ways of defining the f... |
| fztpval 13593 | Two ways of defining the f... |
| fzrev 13594 | Reversal of start and end ... |
| fzrev2 13595 | Reversal of start and end ... |
| fzrev2i 13596 | Reversal of start and end ... |
| fzrev3 13597 | The "complement" of a memb... |
| fzrev3i 13598 | The "complement" of a memb... |
| fznn 13599 | Finite set of sequential i... |
| elfz1b 13600 | Membership in a 1-based fi... |
| elfz1uz 13601 | Membership in a 1-based fi... |
| elfzm11 13602 | Membership in a finite set... |
| uzsplit 13603 | Express an upper integer s... |
| uzdisj 13604 | The first ` N ` elements o... |
| fseq1p1m1 13605 | Add/remove an item to/from... |
| fseq1m1p1 13606 | Add/remove an item to/from... |
| fz1sbc 13607 | Quantification over a one-... |
| elfzp1b 13608 | An integer is a member of ... |
| elfzm1b 13609 | An integer is a member of ... |
| elfzp12 13610 | Options for membership in ... |
| fzne1 13611 | Elementhood in a finite se... |
| fzdif1 13612 | Split the first element of... |
| fz0dif1 13613 | Split the first element of... |
| fzm1 13614 | Choices for an element of ... |
| fzneuz 13615 | No finite set of sequentia... |
| fznuz 13616 | Disjointness of the upper ... |
| uznfz 13617 | Disjointness of the upper ... |
| fzp1nel 13618 | One plus the upper bound o... |
| fzrevral 13619 | Reversal of scanning order... |
| fzrevral2 13620 | Reversal of scanning order... |
| fzrevral3 13621 | Reversal of scanning order... |
| fzshftral 13622 | Shift the scanning order i... |
| ige2m1fz1 13623 | Membership of an integer g... |
| ige2m1fz 13624 | Membership in a 0-based fi... |
| elfz2nn0 13625 | Membership in a finite set... |
| fznn0 13626 | Characterization of a fini... |
| elfznn0 13627 | A member of a finite set o... |
| elfz3nn0 13628 | The upper bound of a nonem... |
| fz0ssnn0 13629 | Finite sets of sequential ... |
| fz1ssfz0 13630 | Subset relationship for fi... |
| 0elfz 13631 | 0 is an element of a finit... |
| nn0fz0 13632 | A nonnegative integer is a... |
| elfz0add 13633 | An element of a finite set... |
| fz0sn 13634 | An integer range from 0 to... |
| fz0tp 13635 | An integer range from 0 to... |
| fz0to3un2pr 13636 | An integer range from 0 to... |
| fz0to4untppr 13637 | An integer range from 0 to... |
| fz0to5un2tp 13638 | An integer range from 0 to... |
| elfz0ubfz0 13639 | An element of a finite set... |
| elfz0fzfz0 13640 | A member of a finite set o... |
| fz0fzelfz0 13641 | If a member of a finite se... |
| fznn0sub2 13642 | Subtraction closure for a ... |
| uzsubfz0 13643 | Membership of an integer g... |
| fz0fzdiffz0 13644 | The difference of an integ... |
| elfzmlbm 13645 | Subtracting the lower boun... |
| elfzmlbp 13646 | Subtracting the lower boun... |
| fzctr 13647 | Lemma for theorems about t... |
| difelfzle 13648 | The difference of two inte... |
| difelfznle 13649 | The difference of two inte... |
| nn0split 13650 | Express the set of nonnega... |
| nn0disj 13651 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13652 | A finite set of sequential... |
| fvffz0 13653 | The function value of a fu... |
| 1fv 13654 | A function on a singleton.... |
| 4fvwrd4 13655 | The first four function va... |
| 2ffzeq 13656 | Two functions over 0-based... |
| preduz 13657 | The value of the predecess... |
| prednn 13658 | The value of the predecess... |
| prednn0 13659 | The value of the predecess... |
| predfz 13660 | Calculate the predecessor ... |
| fzof 13663 | Functionality of the half-... |
| elfzoel1 13664 | Reverse closure for half-o... |
| elfzoel2 13665 | Reverse closure for half-o... |
| elfzoelz 13666 | Reverse closure for half-o... |
| fzoval 13667 | Value of the half-open int... |
| elfzo 13668 | Membership in a half-open ... |
| elfzo2 13669 | Membership in a half-open ... |
| elfzod 13670 | Membership in a half-open ... |
| elfzouz 13671 | Membership in a half-open ... |
| nelfzo 13672 | An integer not being a mem... |
| fzolb 13673 | The left endpoint of a hal... |
| fzolb2 13674 | The left endpoint of a hal... |
| elfzole1 13675 | A member in a half-open in... |
| elfzolt2 13676 | A member in a half-open in... |
| elfzolt3 13677 | Membership in a half-open ... |
| elfzolt2b 13678 | A member in a half-open in... |
| elfzolt3b 13679 | Membership in a half-open ... |
| elfzop1le2 13680 | A member in a half-open in... |
| fzonel 13681 | A half-open range does not... |
| elfzouz2 13682 | The upper bound of a half-... |
| elfzofz 13683 | A half-open range is conta... |
| elfzo3 13684 | Express membership in a ha... |
| fzon0 13685 | A half-open integer interv... |
| fzossfz 13686 | A half-open range is conta... |
| fzossz 13687 | A half-open integer interv... |
| fzon 13688 | A half-open set of sequent... |
| fzo0n 13689 | A half-open range of nonne... |
| fzonlt0 13690 | A half-open integer range ... |
| fzo0 13691 | Half-open sets with equal ... |
| fzonnsub 13692 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13693 | If ` M < N ` then ` N - M ... |
| fzoss1 13694 | Subset relationship for ha... |
| fzoss2 13695 | Subset relationship for ha... |
| fzossrbm1 13696 | Subset of a half-open rang... |
| fzo0ss1 13697 | Subset relationship for ha... |
| fzossnn0 13698 | A half-open integer range ... |
| fzospliti 13699 | One direction of splitting... |
| fzosplit 13700 | Split a half-open integer ... |
| fzodisj 13701 | Abutting half-open integer... |
| fzouzsplit 13702 | Split an upper integer set... |
| fzouzdisj 13703 | A half-open integer range ... |
| fzoun 13704 | A half-open integer range ... |
| fzodisjsn 13705 | A half-open integer range ... |
| prinfzo0 13706 | The intersection of a half... |
| lbfzo0 13707 | An integer is strictly gre... |
| elfzo0 13708 | Membership in a half-open ... |
| elfzo0z 13709 | Membership in a half-open ... |
| nn0p1elfzo 13710 | A nonnegative integer incr... |
| elfzo0le 13711 | A member in a half-open ra... |
| elfzolem1 13712 | A member in a half-open in... |
| elfzo0subge1 13713 | The difference of the uppe... |
| elfzo0suble 13714 | The difference of the uppe... |
| elfzonn0 13715 | A member of a half-open ra... |
| fzonmapblen 13716 | The result of subtracting ... |
| fzofzim 13717 | If a nonnegative integer i... |
| fz1fzo0m1 13718 | Translation of one between... |
| fzossnn 13719 | Half-open integer ranges s... |
| elfzo1 13720 | Membership in a half-open ... |
| fzo1lb 13721 | 1 is the left endpoint of ... |
| 1elfzo1 13722 | 1 is in a half-open range ... |
| fzo1fzo0n0 13723 | An integer between 1 and a... |
| fzo0n0 13724 | A half-open integer range ... |
| fzoaddel 13725 | Translate membership in a ... |
| fzo0addel 13726 | Translate membership in a ... |
| fzo0addelr 13727 | Translate membership in a ... |
| fzoaddel2 13728 | Translate membership in a ... |
| elfzoextl 13729 | Membership of an integer i... |
| elfzoext 13730 | Membership of an integer i... |
| elincfzoext 13731 | Membership of an increased... |
| fzosubel 13732 | Translate membership in a ... |
| fzosubel2 13733 | Membership in a translated... |
| fzosubel3 13734 | Membership in a translated... |
| eluzgtdifelfzo 13735 | Membership of the differen... |
| ige2m2fzo 13736 | Membership of an integer g... |
| fzocatel 13737 | Translate membership in a ... |
| ubmelfzo 13738 | If an integer in a 1-based... |
| elfzodifsumelfzo 13739 | If an integer is in a half... |
| elfzom1elp1fzo 13740 | Membership of an integer i... |
| elfzom1elfzo 13741 | Membership in a half-open ... |
| fzval3 13742 | Expressing a closed intege... |
| fz0add1fz1 13743 | Translate membership in a ... |
| fzosn 13744 | Expressing a singleton as ... |
| elfzomin 13745 | Membership of an integer i... |
| zpnn0elfzo 13746 | Membership of an integer i... |
| zpnn0elfzo1 13747 | Membership of an integer i... |
| fzosplitsnm1 13748 | Removing a singleton from ... |
| elfzonlteqm1 13749 | If an element of a half-op... |
| fzonn0p1 13750 | A nonnegative integer is a... |
| fzossfzop1 13751 | A half-open range of nonne... |
| fzonn0p1p1 13752 | If a nonnegative integer i... |
| elfzom1p1elfzo 13753 | Increasing an element of a... |
| fzo0ssnn0 13754 | Half-open integer ranges s... |
| fzo01 13755 | Expressing the singleton o... |
| fzo12sn 13756 | A 1-based half-open intege... |
| fzo13pr 13757 | A 1-based half-open intege... |
| fzo0to2pr 13758 | A half-open integer range ... |
| fz01pr 13759 | An integer range between 0... |
| fzo0to3tp 13760 | A half-open integer range ... |
| fzo0to42pr 13761 | A half-open integer range ... |
| fzo1to4tp 13762 | A half-open integer range ... |
| fzo0sn0fzo1 13763 | A half-open range of nonne... |
| elfzo0l 13764 | A member of a half-open ra... |
| fzoend 13765 | The endpoint of a half-ope... |
| fzo0end 13766 | The endpoint of a zero-bas... |
| ssfzo12 13767 | Subset relationship for ha... |
| ssfzoulel 13768 | If a half-open integer ran... |
| ssfzo12bi 13769 | Subset relationship for ha... |
| fzoopth 13770 | A half-open integer range ... |
| ubmelm1fzo 13771 | The result of subtracting ... |
| fzofzp1 13772 | If a point is in a half-op... |
| fzofzp1b 13773 | If a point is in a half-op... |
| elfzom1b 13774 | An integer is a member of ... |
| elfzom1elp1fzo1 13775 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13776 | Membership of a positive i... |
| elfzonelfzo 13777 | If an element of a half-op... |
| elfzodif0 13778 | If an integer ` M ` is in ... |
| fzonfzoufzol 13779 | If an element of a half-op... |
| elfzomelpfzo 13780 | An integer increased by an... |
| elfznelfzo 13781 | A value in a finite set of... |
| elfznelfzob 13782 | A value in a finite set of... |
| peano2fzor 13783 | A Peano-postulate-like the... |
| fzosplitsn 13784 | Extending a half-open rang... |
| fzosplitpr 13785 | Extending a half-open inte... |
| fzosplitprm1 13786 | Extending a half-open inte... |
| fzosplitsni 13787 | Membership in a half-open ... |
| fzisfzounsn 13788 | A finite interval of integ... |
| elfzr 13789 | A member of a finite inter... |
| elfzlmr 13790 | A member of a finite inter... |
| elfz0lmr 13791 | A member of a finite inter... |
| fzone1 13792 | Elementhood in a half-open... |
| fzom1ne1 13793 | Elementhood in a half-open... |
| fzostep1 13794 | Two possibilities for a nu... |
| fzoshftral 13795 | Shift the scanning order i... |
| fzind2 13796 | Induction on the integers ... |
| fvinim0ffz 13797 | The function values for th... |
| injresinjlem 13798 | Lemma for ~ injresinj . (... |
| injresinj 13799 | A function whose restricti... |
| subfzo0 13800 | The difference between two... |
| fvf1tp 13801 | Values of a one-to-one fun... |
| flval 13806 | Value of the floor (greate... |
| flcl 13807 | The floor (greatest intege... |
| reflcl 13808 | The floor (greatest intege... |
| fllelt 13809 | A basic property of the fl... |
| flcld 13810 | The floor (greatest intege... |
| flle 13811 | A basic property of the fl... |
| flltp1 13812 | A basic property of the fl... |
| fllep1 13813 | A basic property of the fl... |
| fraclt1 13814 | The fractional part of a r... |
| fracle1 13815 | The fractional part of a r... |
| fracge0 13816 | The fractional part of a r... |
| flge 13817 | The floor function value i... |
| fllt 13818 | The floor function value i... |
| flflp1 13819 | Move floor function betwee... |
| flid 13820 | An integer is its own floo... |
| flidm 13821 | The floor function is idem... |
| flidz 13822 | A real number equals its f... |
| flltnz 13823 | The floor of a non-integer... |
| flwordi 13824 | Ordering relation for the ... |
| flword2 13825 | Ordering relation for the ... |
| flval2 13826 | An alternate way to define... |
| flval3 13827 | An alternate way to define... |
| flbi 13828 | A condition equivalent to ... |
| flbi2 13829 | A condition equivalent to ... |
| adddivflid 13830 | The floor of a sum of an i... |
| ico01fl0 13831 | The floor of a real number... |
| flge0nn0 13832 | The floor of a number grea... |
| flge1nn 13833 | The floor of a number grea... |
| fldivnn0 13834 | The floor function of a di... |
| refldivcl 13835 | The floor function of a di... |
| divfl0 13836 | The floor of a fraction is... |
| fladdz 13837 | An integer can be moved in... |
| flzadd 13838 | An integer can be moved in... |
| flmulnn0 13839 | Move a nonnegative integer... |
| btwnzge0 13840 | A real bounded between an ... |
| 2tnp1ge0ge0 13841 | Two times an integer plus ... |
| flhalf 13842 | Ordering relation for the ... |
| fldivle 13843 | The floor function of a di... |
| fldivnn0le 13844 | The floor function of a di... |
| flltdivnn0lt 13845 | The floor function of a di... |
| ltdifltdiv 13846 | If the dividend of a divis... |
| fldiv4p1lem1div2 13847 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13848 | The floor of an integer gr... |
| fldiv4lem1div2 13849 | The floor of a positive in... |
| ceilval 13850 | The value of the ceiling f... |
| dfceil2 13851 | Alternative definition of ... |
| ceilval2 13852 | The value of the ceiling f... |
| ceicl 13853 | The ceiling function retur... |
| ceilcl 13854 | Closure of the ceiling fun... |
| ceilcld 13855 | Closure of the ceiling fun... |
| ceige 13856 | The ceiling of a real numb... |
| ceilge 13857 | The ceiling of a real numb... |
| ceilged 13858 | The ceiling of a real numb... |
| ceim1l 13859 | One less than the ceiling ... |
| ceilm1lt 13860 | One less than the ceiling ... |
| ceile 13861 | The ceiling of a real numb... |
| ceille 13862 | The ceiling of a real numb... |
| ceilid 13863 | An integer is its own ceil... |
| ceilidz 13864 | A real number equals its c... |
| flleceil 13865 | The floor of a real number... |
| fleqceilz 13866 | A real number is an intege... |
| quoremz 13867 | Quotient and remainder of ... |
| quoremnn0 13868 | Quotient and remainder of ... |
| quoremnn0ALT 13869 | Alternate proof of ~ quore... |
| intfrac2 13870 | Decompose a real into inte... |
| intfracq 13871 | Decompose a rational numbe... |
| fldiv 13872 | Cancellation of the embedd... |
| fldiv2 13873 | Cancellation of an embedde... |
| fznnfl 13874 | Finite set of sequential i... |
| uzsup 13875 | An upper set of integers i... |
| ioopnfsup 13876 | An upper set of reals is u... |
| icopnfsup 13877 | An upper set of reals is u... |
| rpsup 13878 | The positive reals are unb... |
| resup 13879 | The real numbers are unbou... |
| xrsup 13880 | The extended real numbers ... |
| modval 13883 | The value of the modulo op... |
| modvalr 13884 | The value of the modulo op... |
| modcl 13885 | Closure law for the modulo... |
| flpmodeq 13886 | Partition of a division in... |
| modcld 13887 | Closure law for the modulo... |
| mod0 13888 | ` A mod B ` is zero iff ` ... |
| mulmod0 13889 | The product of an integer ... |
| negmod0 13890 | ` A ` is divisible by ` B ... |
| modge0 13891 | The modulo operation is no... |
| modlt 13892 | The modulo operation is le... |
| modelico 13893 | Modular reduction produces... |
| moddiffl 13894 | Value of the modulo operat... |
| moddifz 13895 | The modulo operation diffe... |
| modfrac 13896 | The fractional part of a n... |
| flmod 13897 | The floor function express... |
| intfrac 13898 | Break a number into its in... |
| zmod10 13899 | An integer modulo 1 is 0. ... |
| zmod1congr 13900 | Two arbitrary integers are... |
| modmulnn 13901 | Move a positive integer in... |
| modvalp1 13902 | The value of the modulo op... |
| zmodcl 13903 | Closure law for the modulo... |
| zmodcld 13904 | Closure law for the modulo... |
| zmodfz 13905 | An integer mod ` B ` lies ... |
| zmodfzo 13906 | An integer mod ` B ` lies ... |
| zmodfzp1 13907 | An integer mod ` B ` lies ... |
| modid 13908 | Identity law for modulo. ... |
| modid0 13909 | A positive real number mod... |
| modid2 13910 | Identity law for modulo. ... |
| zmodid2 13911 | Identity law for modulo re... |
| zmodidfzo 13912 | Identity law for modulo re... |
| zmodidfzoimp 13913 | Identity law for modulo re... |
| 0mod 13914 | Special case: 0 modulo a p... |
| 1mod 13915 | Special case: 1 modulo a r... |
| modabs 13916 | Absorption law for modulo.... |
| modabs2 13917 | Absorption law for modulo.... |
| modcyc 13918 | The modulo operation is pe... |
| modcyc2 13919 | The modulo operation is pe... |
| modadd1 13920 | Addition property of the m... |
| modaddb 13921 | Addition property of the m... |
| modaddid 13922 | The sums of two nonnegativ... |
| modaddabs 13923 | Absorption law for modulo.... |
| modaddmod 13924 | The sum of a real number m... |
| muladdmodid 13925 | The sum of a positive real... |
| mulp1mod1 13926 | The product of an integer ... |
| muladdmod 13927 | A real number is the sum o... |
| modmuladd 13928 | Decomposition of an intege... |
| modmuladdim 13929 | Implication of a decomposi... |
| modmuladdnn0 13930 | Implication of a decomposi... |
| negmod 13931 | The negation of a number m... |
| m1modnnsub1 13932 | Minus one modulo a positiv... |
| m1modge3gt1 13933 | Minus one modulo an intege... |
| addmodid 13934 | The sum of a positive inte... |
| addmodidr 13935 | The sum of a positive inte... |
| modadd2mod 13936 | The sum of a real number m... |
| modm1p1mod0 13937 | If a real number modulo a ... |
| modltm1p1mod 13938 | If a real number modulo a ... |
| modmul1 13939 | Multiplication property of... |
| modmul12d 13940 | Multiplication property of... |
| modnegd 13941 | Negation property of the m... |
| modadd12d 13942 | Additive property of the m... |
| modsub12d 13943 | Subtraction property of th... |
| modsubmod 13944 | The difference of a real n... |
| modsubmodmod 13945 | The difference of a real n... |
| 2txmodxeq0 13946 | Two times a positive real ... |
| 2submod 13947 | If a real number is betwee... |
| modifeq2int 13948 | If a nonnegative integer i... |
| modaddmodup 13949 | The sum of an integer modu... |
| modaddmodlo 13950 | The sum of an integer modu... |
| modmulmod 13951 | The product of a real numb... |
| modmulmodr 13952 | The product of an integer ... |
| modaddmulmod 13953 | The sum of a real number a... |
| moddi 13954 | Distribute multiplication ... |
| modsubdir 13955 | Distribute the modulo oper... |
| modeqmodmin 13956 | A real number equals the d... |
| modirr 13957 | A number modulo an irratio... |
| modfzo0difsn 13958 | For a number within a half... |
| modsumfzodifsn 13959 | The sum of a number within... |
| modlteq 13960 | Two nonnegative integers l... |
| addmodlteq 13961 | Two nonnegative integers l... |
| om2uz0i 13962 | The mapping ` G ` is a one... |
| om2uzsuci 13963 | The value of ` G ` (see ~ ... |
| om2uzuzi 13964 | The value ` G ` (see ~ om2... |
| om2uzlti 13965 | Less-than relation for ` G... |
| om2uzlt2i 13966 | The mapping ` G ` (see ~ o... |
| om2uzrani 13967 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13968 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13969 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13970 | An alternative definition ... |
| om2uzrdg 13971 | A helper lemma for the val... |
| uzrdglem 13972 | A helper lemma for the val... |
| uzrdgfni 13973 | The recursive definition g... |
| uzrdg0i 13974 | Initial value of a recursi... |
| uzrdgsuci 13975 | Successor value of a recur... |
| ltweuz 13976 | ` < ` is a well-founded re... |
| ltwenn 13977 | Less than well-orders the ... |
| ltwefz 13978 | Less than well-orders a se... |
| uzenom 13979 | An upper integer set is de... |
| uzinf 13980 | An upper integer set is in... |
| nnnfi 13981 | The set of positive intege... |
| uzrdgxfr 13982 | Transfer the value of the ... |
| fzennn 13983 | The cardinality of a finit... |
| fzen2 13984 | The cardinality of a finit... |
| cardfz 13985 | The cardinality of a finit... |
| hashgf1o 13986 | ` G ` maps ` _om ` one-to-... |
| fzfi 13987 | A finite interval of integ... |
| fzfid 13988 | Commonly used special case... |
| fzofi 13989 | Half-open integer sets are... |
| fsequb 13990 | The values of a finite rea... |
| fsequb2 13991 | The values of a finite rea... |
| fseqsupcl 13992 | The values of a finite rea... |
| fseqsupubi 13993 | The values of a finite rea... |
| nn0ennn 13994 | The nonnegative integers a... |
| nnenom 13995 | The set of positive intege... |
| nnct 13996 | ` NN ` is countable. (Con... |
| uzindi 13997 | Indirect strong induction ... |
| axdc4uzlem 13998 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13999 | A version of ~ axdc4 that ... |
| ssnn0fi 14000 | A subset of the nonnegativ... |
| rabssnn0fi 14001 | A subset of the nonnegativ... |
| uzsinds 14002 | Strong (or "total") induct... |
| nnsinds 14003 | Strong (or "total") induct... |
| nn0sinds 14004 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 14005 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 14006 | If all functions of a fini... |
| fsuppmapnn0fiubex 14007 | If all functions of a fini... |
| fsuppmapnn0fiub0 14008 | If all functions of a fini... |
| suppssfz 14009 | Condition for a function o... |
| fsuppmapnn0ub 14010 | If a function over the non... |
| fsuppmapnn0fz 14011 | If a function over the non... |
| mptnn0fsupp 14012 | A mapping from the nonnega... |
| mptnn0fsuppd 14013 | A mapping from the nonnega... |
| mptnn0fsuppr 14014 | A finitely supported mappi... |
| f13idfv 14015 | A one-to-one function with... |
| seqex 14018 | Existence of the sequence ... |
| seqeq1 14019 | Equality theorem for the s... |
| seqeq2 14020 | Equality theorem for the s... |
| seqeq3 14021 | Equality theorem for the s... |
| seqeq1d 14022 | Equality deduction for the... |
| seqeq2d 14023 | Equality deduction for the... |
| seqeq3d 14024 | Equality deduction for the... |
| seqeq123d 14025 | Equality deduction for the... |
| nfseq 14026 | Hypothesis builder for the... |
| seqval 14027 | Value of the sequence buil... |
| seqfn 14028 | The sequence builder funct... |
| seq1 14029 | Value of the sequence buil... |
| seq1i 14030 | Value of the sequence buil... |
| seqp1 14031 | Value of the sequence buil... |
| seqexw 14032 | Weak version of ~ seqex th... |
| seqp1d 14033 | Value of the sequence buil... |
| seqm1 14034 | Value of the sequence buil... |
| seqcl2 14035 | Closure properties of the ... |
| seqf2 14036 | Range of the recursive seq... |
| seqcl 14037 | Closure properties of the ... |
| seqf 14038 | Range of the recursive seq... |
| seqfveq2 14039 | Equality of sequences. (C... |
| seqfeq2 14040 | Equality of sequences. (C... |
| seqfveq 14041 | Equality of sequences. (C... |
| seqfeq 14042 | Equality of sequences. (C... |
| seqshft2 14043 | Shifting the index set of ... |
| seqres 14044 | Restricting its characteri... |
| serf 14045 | An infinite series of comp... |
| serfre 14046 | An infinite series of real... |
| monoord 14047 | Ordering relation for a mo... |
| monoord2 14048 | Ordering relation for a mo... |
| sermono 14049 | The partial sums in an inf... |
| seqsplit 14050 | Split a sequence into two ... |
| seq1p 14051 | Removing the first term fr... |
| seqcaopr3 14052 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 14053 | The sum of two infinite se... |
| seqcaopr 14054 | The sum of two infinite se... |
| seqf1olem2a 14055 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 14056 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 14057 | Lemma for ~ seqf1o . (Con... |
| seqf1o 14058 | Rearrange a sum via an arb... |
| seradd 14059 | The sum of two infinite se... |
| sersub 14060 | The difference of two infi... |
| seqid3 14061 | A sequence that consists e... |
| seqid 14062 | Discarding the first few t... |
| seqid2 14063 | The last few partial sums ... |
| seqhomo 14064 | Apply a homomorphism to a ... |
| seqz 14065 | If the operation ` .+ ` ha... |
| seqfeq4 14066 | Equality of series under d... |
| seqfeq3 14067 | Equality of series under d... |
| seqdistr 14068 | The distributive property ... |
| ser0 14069 | The value of the partial s... |
| ser0f 14070 | A zero-valued infinite ser... |
| serge0 14071 | A finite sum of nonnegativ... |
| serle 14072 | Comparison of partial sums... |
| ser1const 14073 | Value of the partial serie... |
| seqof 14074 | Distribute function operat... |
| seqof2 14075 | Distribute function operat... |
| expval 14078 | Value of exponentiation to... |
| expnnval 14079 | Value of exponentiation to... |
| exp0 14080 | Value of a complex number ... |
| 0exp0e1 14081 | The zeroth power of zero e... |
| exp1 14082 | Value of a complex number ... |
| expp1 14083 | Value of a complex number ... |
| expneg 14084 | Value of a complex number ... |
| expneg2 14085 | Value of a complex number ... |
| expn1 14086 | A complex number raised to... |
| expcllem 14087 | Lemma for proving nonnegat... |
| expcl2lem 14088 | Lemma for proving integer ... |
| nnexpcl 14089 | Closure of exponentiation ... |
| nn0expcl 14090 | Closure of exponentiation ... |
| zexpcl 14091 | Closure of exponentiation ... |
| qexpcl 14092 | Closure of exponentiation ... |
| reexpcl 14093 | Closure of exponentiation ... |
| expcl 14094 | Closure law for nonnegativ... |
| rpexpcl 14095 | Closure law for integer ex... |
| qexpclz 14096 | Closure of integer exponen... |
| reexpclz 14097 | Closure of integer exponen... |
| expclzlem 14098 | Lemma for ~ expclz . (Con... |
| expclz 14099 | Closure law for integer ex... |
| m1expcl2 14100 | Closure of integer exponen... |
| m1expcl 14101 | Closure of exponentiation ... |
| zexpcld 14102 | Closure of exponentiation ... |
| nn0expcli 14103 | Closure of exponentiation ... |
| nn0sqcl 14104 | The square of a nonnegativ... |
| expm1t 14105 | Exponentiation in terms of... |
| 1exp 14106 | Value of 1 raised to an in... |
| expeq0 14107 | A positive integer power i... |
| expne0 14108 | A positive integer power i... |
| expne0i 14109 | An integer power is nonzer... |
| expgt0 14110 | A positive real raised to ... |
| expnegz 14111 | Value of a nonzero complex... |
| 0exp 14112 | Value of zero raised to a ... |
| expge0 14113 | A nonnegative real raised ... |
| expge1 14114 | A real greater than or equ... |
| expgt1 14115 | A real greater than 1 rais... |
| mulexp 14116 | Nonnegative integer expone... |
| mulexpz 14117 | Integer exponentiation of ... |
| exprec 14118 | Integer exponentiation of ... |
| expadd 14119 | Sum of exponents law for n... |
| expaddzlem 14120 | Lemma for ~ expaddz . (Co... |
| expaddz 14121 | Sum of exponents law for i... |
| expmul 14122 | Product of exponents law f... |
| expmulz 14123 | Product of exponents law f... |
| m1expeven 14124 | Exponentiation of negative... |
| expsub 14125 | Exponent subtraction law f... |
| expp1z 14126 | Value of a nonzero complex... |
| expm1 14127 | Value of a nonzero complex... |
| expdiv 14128 | Nonnegative integer expone... |
| sqval 14129 | Value of the square of a c... |
| sqneg 14130 | The square of the negative... |
| sqnegd 14131 | The square of the negative... |
| sqsubswap 14132 | Swap the order of subtract... |
| sqcl 14133 | Closure of square. (Contr... |
| sqmul 14134 | Distribution of squaring o... |
| sqeq0 14135 | A complex number is zero i... |
| sqdiv 14136 | Distribution of squaring o... |
| sqdivid 14137 | The square of a nonzero co... |
| sqne0 14138 | A complex number is nonzer... |
| resqcl 14139 | Closure of squaring in rea... |
| resqcld 14140 | Closure of squaring in rea... |
| sqgt0 14141 | The square of a nonzero re... |
| sqn0rp 14142 | The square of a nonzero re... |
| nnsqcl 14143 | The positive naturals are ... |
| zsqcl 14144 | Integers are closed under ... |
| qsqcl 14145 | The square of a rational i... |
| sq11 14146 | The square function is one... |
| nn0sq11 14147 | The square function is one... |
| lt2sq 14148 | The square function is inc... |
| le2sq 14149 | The square function is non... |
| le2sq2 14150 | The square function is non... |
| sqge0 14151 | The square of a real is no... |
| sqge0d 14152 | The square of a real is no... |
| zsqcl2 14153 | The square of an integer i... |
| 0expd 14154 | Value of zero raised to a ... |
| exp0d 14155 | Value of a complex number ... |
| exp1d 14156 | Value of a complex number ... |
| expeq0d 14157 | If a positive integer powe... |
| sqvald 14158 | Value of square. Inferenc... |
| sqcld 14159 | Closure of square. (Contr... |
| sqeq0d 14160 | A number is zero iff its s... |
| expcld 14161 | Closure law for nonnegativ... |
| expp1d 14162 | Value of a complex number ... |
| expaddd 14163 | Sum of exponents law for n... |
| expmuld 14164 | Product of exponents law f... |
| sqrecd 14165 | Square of reciprocal is re... |
| expclzd 14166 | Closure law for integer ex... |
| expne0d 14167 | A nonnegative integer powe... |
| expnegd 14168 | Value of a nonzero complex... |
| exprecd 14169 | An integer power of a reci... |
| expp1zd 14170 | Value of a nonzero complex... |
| expm1d 14171 | Value of a nonzero complex... |
| expsubd 14172 | Exponent subtraction law f... |
| sqmuld 14173 | Distribution of squaring o... |
| sqdivd 14174 | Distribution of squaring o... |
| expdivd 14175 | Nonnegative integer expone... |
| mulexpd 14176 | Nonnegative integer expone... |
| znsqcld 14177 | The square of a nonzero in... |
| reexpcld 14178 | Closure of exponentiation ... |
| expge0d 14179 | A nonnegative real raised ... |
| expge1d 14180 | A real greater than or equ... |
| ltexp2a 14181 | Exponent ordering relation... |
| expmordi 14182 | Base ordering relationship... |
| rpexpmord 14183 | Base ordering relationship... |
| expcan 14184 | Cancellation law for integ... |
| ltexp2 14185 | Strict ordering law for ex... |
| leexp2 14186 | Ordering law for exponenti... |
| leexp2a 14187 | Weak ordering relationship... |
| ltexp2r 14188 | The integer powers of a fi... |
| leexp2r 14189 | Weak ordering relationship... |
| leexp1a 14190 | Weak base ordering relatio... |
| leexp1ad 14191 | Weak base ordering relatio... |
| exple1 14192 | A real between 0 and 1 inc... |
| expubnd 14193 | An upper bound on ` A ^ N ... |
| sumsqeq0 14194 | The sum of two squres of r... |
| sqvali 14195 | Value of square. Inferenc... |
| sqcli 14196 | Closure of square. (Contr... |
| sqeq0i 14197 | A complex number is zero i... |
| sqrecii 14198 | The square of a reciprocal... |
| sqmuli 14199 | Distribution of squaring o... |
| sqdivi 14200 | Distribution of squaring o... |
| resqcli 14201 | Closure of square in reals... |
| sqgt0i 14202 | The square of a nonzero re... |
| sqge0i 14203 | The square of a real is no... |
| lt2sqi 14204 | The square function on non... |
| le2sqi 14205 | The square function on non... |
| sq11i 14206 | The square function is one... |
| sq0 14207 | The square of 0 is 0. (Co... |
| sq0i 14208 | If a number is zero, then ... |
| sq0id 14209 | If a number is zero, then ... |
| sq1 14210 | The square of 1 is 1. (Co... |
| neg1sqe1 14211 | The square of ` -u 1 ` is ... |
| sq2 14212 | The square of 2 is 4. (Co... |
| sq3 14213 | The square of 3 is 9. (Co... |
| sq4e2t8 14214 | The square of 4 is 2 times... |
| cu2 14215 | The cube of 2 is 8. (Cont... |
| irec 14216 | The reciprocal of ` _i ` .... |
| i2 14217 | ` _i ` squared. (Contribu... |
| i3 14218 | ` _i ` cubed. (Contribute... |
| i4 14219 | ` _i ` to the fourth power... |
| nnlesq 14220 | A positive integer is less... |
| zzlesq 14221 | An integer is less than or... |
| iexpcyc 14222 | Taking ` _i ` to the ` K `... |
| expnass 14223 | A counterexample showing t... |
| sqlecan 14224 | Cancel one factor of a squ... |
| subsq 14225 | Factor the difference of t... |
| subsq2 14226 | Express the difference of ... |
| binom2i 14227 | The square of a binomial. ... |
| subsqi 14228 | Factor the difference of t... |
| sqeqori 14229 | The squares of two complex... |
| subsq0i 14230 | The two solutions to the d... |
| sqeqor 14231 | The squares of two complex... |
| binom2 14232 | The square of a binomial. ... |
| binom2d 14233 | Deduction form of ~ binom2... |
| binom21 14234 | Special case of ~ binom2 w... |
| binom2sub 14235 | Expand the square of a sub... |
| binom2sub1 14236 | Special case of ~ binom2su... |
| binom2subi 14237 | Expand the square of a sub... |
| mulbinom2 14238 | The square of a binomial w... |
| binom3 14239 | The cube of a binomial. (... |
| sq01 14240 | If a complex number equals... |
| zesq 14241 | An integer is even iff its... |
| nnesq 14242 | A positive integer is even... |
| crreczi 14243 | Reciprocal of a complex nu... |
| bernneq 14244 | Bernoulli's inequality, du... |
| bernneq2 14245 | Variation of Bernoulli's i... |
| bernneq3 14246 | A corollary of ~ bernneq .... |
| expnbnd 14247 | Exponentiation with a base... |
| expnlbnd 14248 | The reciprocal of exponent... |
| expnlbnd2 14249 | The reciprocal of exponent... |
| expmulnbnd 14250 | Exponentiation with a base... |
| digit2 14251 | Two ways to express the ` ... |
| digit1 14252 | Two ways to express the ` ... |
| modexp 14253 | Exponentiation property of... |
| discr1 14254 | A nonnegative quadratic fo... |
| discr 14255 | If a quadratic polynomial ... |
| expnngt1 14256 | If an integer power with a... |
| expnngt1b 14257 | An integer power with an i... |
| sqoddm1div8 14258 | A squared odd number minus... |
| nnsqcld 14259 | The naturals are closed un... |
| nnexpcld 14260 | Closure of exponentiation ... |
| nn0expcld 14261 | Closure of exponentiation ... |
| rpexpcld 14262 | Closure law for exponentia... |
| ltexp2rd 14263 | The power of a positive nu... |
| reexpclzd 14264 | Closure of exponentiation ... |
| sqgt0d 14265 | The square of a nonzero re... |
| ltexp2d 14266 | Ordering relationship for ... |
| leexp2d 14267 | Ordering law for exponenti... |
| expcand 14268 | Ordering relationship for ... |
| leexp2ad 14269 | Ordering relationship for ... |
| leexp2rd 14270 | Ordering relationship for ... |
| lt2sqd 14271 | The square function on non... |
| le2sqd 14272 | The square function on non... |
| sq11d 14273 | The square function is one... |
| ltexp1d 14274 | Elevating to a positive po... |
| ltexp1dd 14275 | Raising both sides of 'les... |
| exp11nnd 14276 | The function elevating non... |
| mulsubdivbinom2 14277 | The square of a binomial w... |
| muldivbinom2 14278 | The square of a binomial w... |
| sq10 14279 | The square of 10 is 100. ... |
| sq10e99m1 14280 | The square of 10 is 99 plu... |
| 3dec 14281 | A "decimal constructor" wh... |
| nn0le2msqi 14282 | The square function on non... |
| nn0opthlem1 14283 | A rather pretty lemma for ... |
| nn0opthlem2 14284 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14285 | An ordered pair theorem fo... |
| nn0opth2i 14286 | An ordered pair theorem fo... |
| nn0opth2 14287 | An ordered pair theorem fo... |
| facnn 14290 | Value of the factorial fun... |
| fac0 14291 | The factorial of 0. (Cont... |
| fac1 14292 | The factorial of 1. (Cont... |
| facp1 14293 | The factorial of a success... |
| fac2 14294 | The factorial of 2. (Cont... |
| fac3 14295 | The factorial of 3. (Cont... |
| fac4 14296 | The factorial of 4. (Cont... |
| facnn2 14297 | Value of the factorial fun... |
| faccl 14298 | Closure of the factorial f... |
| faccld 14299 | Closure of the factorial f... |
| facmapnn 14300 | The factorial function res... |
| facne0 14301 | The factorial function is ... |
| facdiv 14302 | A positive integer divides... |
| facndiv 14303 | No positive integer (great... |
| facwordi 14304 | Ordering property of facto... |
| faclbnd 14305 | A lower bound for the fact... |
| faclbnd2 14306 | A lower bound for the fact... |
| faclbnd3 14307 | A lower bound for the fact... |
| faclbnd4lem1 14308 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14309 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14310 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14311 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14312 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14313 | The factorial function gro... |
| faclbnd6 14314 | Geometric lower bound for ... |
| facubnd 14315 | An upper bound for the fac... |
| facavg 14316 | The product of two factori... |
| bcval 14319 | Value of the binomial coef... |
| bcval2 14320 | Value of the binomial coef... |
| bcval3 14321 | Value of the binomial coef... |
| bcval4 14322 | Value of the binomial coef... |
| bcrpcl 14323 | Closure of the binomial co... |
| bccmpl 14324 | "Complementing" its second... |
| bcn0 14325 | ` N ` choose 0 is 1. Rema... |
| bc0k 14326 | The binomial coefficient "... |
| bcnn 14327 | ` N ` choose ` N ` is 1. ... |
| bcn1 14328 | Binomial coefficient: ` N ... |
| bcnp1n 14329 | Binomial coefficient: ` N ... |
| bcm1k 14330 | The proportion of one bino... |
| bcp1n 14331 | The proportion of one bino... |
| bcp1nk 14332 | The proportion of one bino... |
| bcval5 14333 | Write out the top and bott... |
| bcn2 14334 | Binomial coefficient: ` N ... |
| bcp1m1 14335 | Compute the binomial coeff... |
| bcpasc 14336 | Pascal's rule for the bino... |
| bccl 14337 | A binomial coefficient, in... |
| bccl2 14338 | A binomial coefficient, in... |
| bcn2m1 14339 | Compute the binomial coeff... |
| bcn2p1 14340 | Compute the binomial coeff... |
| permnn 14341 | The number of permutations... |
| bcnm1 14342 | The binomial coefficient o... |
| 4bc3eq4 14343 | The value of four choose t... |
| 4bc2eq6 14344 | The value of four choose t... |
| hashkf 14347 | The finite part of the siz... |
| hashgval 14348 | The value of the ` # ` fun... |
| hashginv 14349 | The converse of ` G ` maps... |
| hashinf 14350 | The value of the ` # ` fun... |
| hashbnd 14351 | If ` A ` has size bounded ... |
| hashfxnn0 14352 | The size function is a fun... |
| hashf 14353 | The size function maps all... |
| hashxnn0 14354 | The value of the hash func... |
| hashresfn 14355 | Restriction of the domain ... |
| dmhashres 14356 | Restriction of the domain ... |
| hashnn0pnf 14357 | The value of the hash func... |
| hashnnn0genn0 14358 | If the size of a set is no... |
| hashnemnf 14359 | The size of a set is never... |
| hashv01gt1 14360 | The size of a set is eithe... |
| hashfz1 14361 | The set ` ( 1 ... N ) ` ha... |
| hashen 14362 | Two finite sets have the s... |
| hasheni 14363 | Equinumerous sets have the... |
| hasheqf1o 14364 | The size of two finite set... |
| fiinfnf1o 14365 | There is no bijection betw... |
| hasheqf1oi 14366 | The size of two sets is eq... |
| hashf1rn 14367 | The size of a finite set w... |
| hasheqf1od 14368 | The size of two sets is eq... |
| fz1eqb 14369 | Two possibly-empty 1-based... |
| hashcard 14370 | The size function of the c... |
| hashcl 14371 | Closure of the ` # ` funct... |
| hashxrcl 14372 | Extended real closure of t... |
| hashclb 14373 | Reverse closure of the ` #... |
| nfile 14374 | The size of any infinite s... |
| hashvnfin 14375 | A set of finite size is a ... |
| hashnfinnn0 14376 | The size of an infinite se... |
| isfinite4 14377 | A finite set is equinumero... |
| hasheq0 14378 | Two ways of saying a set i... |
| hashneq0 14379 | Two ways of saying a set i... |
| hashgt0n0 14380 | If the size of a set is gr... |
| hashnncl 14381 | Positive natural closure o... |
| hash0 14382 | The empty set has size zer... |
| hashelne0d 14383 | A set with an element has ... |
| hashsng 14384 | The size of a singleton. ... |
| hashen1 14385 | A set has size 1 if and on... |
| hash1elsn 14386 | A set of size 1 with a kno... |
| hashrabrsn 14387 | The size of a restricted c... |
| hashrabsn01 14388 | The size of a restricted c... |
| hashrabsn1 14389 | If the size of a restricte... |
| hashfn 14390 | A function is equinumerous... |
| fseq1hash 14391 | The value of the size func... |
| hashgadd 14392 | ` G ` maps ordinal additio... |
| hashgval2 14393 | A short expression for the... |
| hashdom 14394 | Dominance relation for the... |
| hashdomi 14395 | Non-strict order relation ... |
| hashsdom 14396 | Strict dominance relation ... |
| hashun 14397 | The size of the union of d... |
| hashun2 14398 | The size of the union of f... |
| hashun3 14399 | The size of the union of f... |
| hashinfxadd 14400 | The extended real addition... |
| hashunx 14401 | The size of the union of d... |
| hashge0 14402 | The cardinality of a set i... |
| hashgt0 14403 | The cardinality of a nonem... |
| hashge1 14404 | The cardinality of a nonem... |
| 1elfz0hash 14405 | 1 is an element of the fin... |
| hashnn0n0nn 14406 | If a nonnegative integer i... |
| hashunsng 14407 | The size of the union of a... |
| hashunsngx 14408 | The size of the union of a... |
| hashunsnggt 14409 | The size of a set is great... |
| hashprg 14410 | The size of an unordered p... |
| elprchashprn2 14411 | If one element of an unord... |
| hashprb 14412 | The size of an unordered p... |
| hashprdifel 14413 | The elements of an unorder... |
| prhash2ex 14414 | There is (at least) one se... |
| hashle00 14415 | If the size of a set is le... |
| hashgt0elex 14416 | If the size of a set is gr... |
| hashgt0elexb 14417 | The size of a set is great... |
| hashp1i 14418 | Size of a finite ordinal. ... |
| hash1 14419 | Size of a finite ordinal. ... |
| hash2 14420 | Size of a finite ordinal. ... |
| hash3 14421 | Size of a finite ordinal. ... |
| hash4 14422 | Size of a finite ordinal. ... |
| pr0hash2ex 14423 | There is (at least) one se... |
| hashss 14424 | The size of a subset is le... |
| prsshashgt1 14425 | The size of a superset of ... |
| hashin 14426 | The size of the intersecti... |
| hashssdif 14427 | The size of the difference... |
| hashdif 14428 | The size of the difference... |
| hashdifsn 14429 | The size of the difference... |
| hashdifpr 14430 | The size of the difference... |
| hashsn01 14431 | The size of a singleton is... |
| hashsnle1 14432 | The size of a singleton is... |
| hashsnlei 14433 | Get an upper bound on a co... |
| hash1snb 14434 | The size of a set is 1 if ... |
| euhash1 14435 | The size of a set is 1 in ... |
| hash1n0 14436 | If the size of a set is 1 ... |
| hashgt12el 14437 | In a set with more than on... |
| hashgt12el2 14438 | In a set with more than on... |
| hashgt23el 14439 | A set with more than two e... |
| hashunlei 14440 | Get an upper bound on a co... |
| hashsslei 14441 | Get an upper bound on a co... |
| hashfz 14442 | Value of the numeric cardi... |
| fzsdom2 14443 | Condition for finite range... |
| hashfzo 14444 | Cardinality of a half-open... |
| hashfzo0 14445 | Cardinality of a half-open... |
| hashfzp1 14446 | Value of the numeric cardi... |
| hashfz0 14447 | Value of the numeric cardi... |
| hashxplem 14448 | Lemma for ~ hashxp . (Con... |
| hashxp 14449 | The size of the Cartesian ... |
| hashmap 14450 | The size of the set expone... |
| hashpw 14451 | The size of the power set ... |
| hashfun 14452 | A finite set is a function... |
| hashres 14453 | The number of elements of ... |
| hashreshashfun 14454 | The number of elements of ... |
| hashimarn 14455 | The size of the image of a... |
| hashimarni 14456 | If the size of the image o... |
| hashfundm 14457 | The size of a set function... |
| hashf1dmrn 14458 | The size of the domain of ... |
| hashf1dmcdm 14459 | The size of the domain of ... |
| resunimafz0 14460 | TODO-AV: Revise using ` F... |
| fnfz0hash 14461 | The size of a function on ... |
| ffz0hash 14462 | The size of a function on ... |
| fnfz0hashnn0 14463 | The size of a function on ... |
| ffzo0hash 14464 | The size of a function on ... |
| fnfzo0hash 14465 | The size of a function on ... |
| fnfzo0hashnn0 14466 | The value of the size func... |
| hashbclem 14467 | Lemma for ~ hashbc : induc... |
| hashbc 14468 | The binomial coefficient c... |
| hashfacen 14469 | The number of bijections b... |
| hashf1lem1 14470 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 14471 | Lemma for ~ hashf1 . (Con... |
| hashf1 14472 | The permutation number ` |... |
| hashfac 14473 | A factorial counts the num... |
| leiso 14474 | Two ways to write a strict... |
| leisorel 14475 | Version of ~ isorel for st... |
| fz1isolem 14476 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14477 | Any finite ordered set has... |
| ishashinf 14478 | Any set that is not finite... |
| seqcoll 14479 | The function ` F ` contain... |
| seqcoll2 14480 | The function ` F ` contain... |
| phphashd 14481 | Corollary of the Pigeonhol... |
| phphashrd 14482 | Corollary of the Pigeonhol... |
| hashprlei 14483 | An unordered pair has at m... |
| hash2pr 14484 | A set of size two is an un... |
| hash2prde 14485 | A set of size two is an un... |
| hash2exprb 14486 | A set of size two is an un... |
| hash2prb 14487 | A set of size two is a pro... |
| prprrab 14488 | The set of proper pairs of... |
| nehash2 14489 | The cardinality of a set w... |
| hash2prd 14490 | A set of size two is an un... |
| hash2pwpr 14491 | If the size of a subset of... |
| hashle2pr 14492 | A nonempty set of size les... |
| hashle2prv 14493 | A nonempty subset of a pow... |
| pr2pwpr 14494 | The set of subsets of a pa... |
| hashge2el2dif 14495 | A set with size at least 2... |
| hashge2el2difr 14496 | A set with at least 2 diff... |
| hashge2el2difb 14497 | A set has size at least 2 ... |
| hashdmpropge2 14498 | The size of the domain of ... |
| hashtplei 14499 | An unordered triple has at... |
| hashtpg 14500 | The size of an unordered t... |
| hash7g 14501 | The size of an unordered s... |
| hashge3el3dif 14502 | A set with size at least 3... |
| elss2prb 14503 | An element of the set of s... |
| hash2sspr 14504 | A subset of size two is an... |
| exprelprel 14505 | If there is an element of ... |
| hash3tr 14506 | A set of size three is an ... |
| hash1to3 14507 | If the size of a set is be... |
| hash3tpde 14508 | A set of size three is an ... |
| hash3tpexb 14509 | A set of size three is an ... |
| hash3tpb 14510 | A set of size three is a p... |
| tpf1ofv0 14511 | The value of a one-to-one ... |
| tpf1ofv1 14512 | The value of a one-to-one ... |
| tpf1ofv2 14513 | The value of a one-to-one ... |
| tpf 14514 | A function into a (proper)... |
| tpfo 14515 | A function onto a (proper)... |
| tpf1o 14516 | A bijection onto a (proper... |
| fundmge2nop0 14517 | A function with a domain c... |
| fundmge2nop 14518 | A function with a domain c... |
| fun2dmnop0 14519 | A function with a domain c... |
| fun2dmnop 14520 | A function with a domain c... |
| hashdifsnp1 14521 | If the size of a set is a ... |
| fi1uzind 14522 | Properties of an ordered p... |
| brfi1uzind 14523 | Properties of a binary rel... |
| brfi1ind 14524 | Properties of a binary rel... |
| brfi1indALT 14525 | Alternate proof of ~ brfi1... |
| opfi1uzind 14526 | Properties of an ordered p... |
| opfi1ind 14527 | Properties of an ordered p... |
| iswrd 14530 | Property of being a word o... |
| wrdval 14531 | Value of the set of words ... |
| iswrdi 14532 | A zero-based sequence is a... |
| wrdf 14533 | A word is a zero-based seq... |
| wrdfd 14534 | A word is a zero-based seq... |
| iswrdb 14535 | A word over an alphabet is... |
| wrddm 14536 | The indices of a word (i.e... |
| sswrd 14537 | The set of words respects ... |
| snopiswrd 14538 | A singleton of an ordered ... |
| wrdexg 14539 | The set of words over a se... |
| wrdexb 14540 | The set of words over a se... |
| wrdexi 14541 | The set of words over a se... |
| wrdsymbcl 14542 | A symbol within a word ove... |
| wrdfn 14543 | A word is a function with ... |
| wrdv 14544 | A word over an alphabet is... |
| wrdlndm 14545 | The length of a word is no... |
| iswrdsymb 14546 | An arbitrary word is a wor... |
| wrdfin 14547 | A word is a finite set. (... |
| lencl 14548 | The length of a word is a ... |
| lennncl 14549 | The length of a nonempty w... |
| wrdffz 14550 | A word is a function from ... |
| wrdeq 14551 | Equality theorem for the s... |
| wrdeqi 14552 | Equality theorem for the s... |
| iswrddm0 14553 | A function with empty doma... |
| wrd0 14554 | The empty set is a word (t... |
| 0wrd0 14555 | The empty word is the only... |
| ffz0iswrd 14556 | A sequence with zero-based... |
| wrdsymb 14557 | A word is a word over the ... |
| nfwrd 14558 | Hypothesis builder for ` W... |
| csbwrdg 14559 | Class substitution for the... |
| wrdnval 14560 | Words of a fixed length ar... |
| wrdmap 14561 | Words as a mapping. (Cont... |
| hashwrdn 14562 | If there is only a finite ... |
| wrdnfi 14563 | If there is only a finite ... |
| wrdsymb0 14564 | A symbol at a position "ou... |
| wrdlenge1n0 14565 | A word with length at leas... |
| len0nnbi 14566 | The length of a word is a ... |
| wrdlenge2n0 14567 | A word with length at leas... |
| wrdsymb1 14568 | The first symbol of a none... |
| wrdlen1 14569 | A word of length 1 starts ... |
| fstwrdne 14570 | The first symbol of a none... |
| fstwrdne0 14571 | The first symbol of a none... |
| eqwrd 14572 | Two words are equal iff th... |
| elovmpowrd 14573 | Implications for the value... |
| elovmptnn0wrd 14574 | Implications for the value... |
| wrdred1 14575 | A word truncated by a symb... |
| wrdred1hash 14576 | The length of a word trunc... |
| lsw 14579 | Extract the last symbol of... |
| lsw0 14580 | The last symbol of an empt... |
| lsw0g 14581 | The last symbol of an empt... |
| lsw1 14582 | The last symbol of a word ... |
| lswcl 14583 | Closure of the last symbol... |
| lswlgt0cl 14584 | The last symbol of a nonem... |
| ccatfn 14587 | The concatenation operator... |
| ccatfval 14588 | Value of the concatenation... |
| ccatcl 14589 | The concatenation of two w... |
| ccatlen 14590 | The length of a concatenat... |
| ccat0 14591 | The concatenation of two w... |
| ccatval1 14592 | Value of a symbol in the l... |
| ccatval2 14593 | Value of a symbol in the r... |
| ccatval3 14594 | Value of a symbol in the r... |
| elfzelfzccat 14595 | An element of a finite set... |
| ccatvalfn 14596 | The concatenation of two w... |
| ccatdmss 14597 | The domain of a concatenat... |
| ccatsymb 14598 | The symbol at a given posi... |
| ccatfv0 14599 | The first symbol of a conc... |
| ccatval1lsw 14600 | The last symbol of the lef... |
| ccatval21sw 14601 | The first symbol of the ri... |
| ccatlid 14602 | Concatenation of a word by... |
| ccatrid 14603 | Concatenation of a word by... |
| ccatass 14604 | Associative law for concat... |
| ccatrn 14605 | The range of a concatenate... |
| ccatidid 14606 | Concatenation of the empty... |
| lswccatn0lsw 14607 | The last symbol of a word ... |
| lswccat0lsw 14608 | The last symbol of a word ... |
| ccatalpha 14609 | A concatenation of two arb... |
| ccatrcl1 14610 | Reverse closure of a conca... |
| ids1 14613 | Identity function protecti... |
| s1val 14614 | Value of a singleton word.... |
| s1rn 14615 | The range of a singleton w... |
| s1eq 14616 | Equality theorem for a sin... |
| s1eqd 14617 | Equality theorem for a sin... |
| s1cl 14618 | A singleton word is a word... |
| s1cld 14619 | A singleton word is a word... |
| s1prc 14620 | Value of a singleton word ... |
| s1cli 14621 | A singleton word is a word... |
| s1len 14622 | Length of a singleton word... |
| s1nz 14623 | A singleton word is not th... |
| s1dm 14624 | The domain of a singleton ... |
| s1dmALT 14625 | Alternate version of ~ s1d... |
| s1fv 14626 | Sole symbol of a singleton... |
| lsws1 14627 | The last symbol of a singl... |
| eqs1 14628 | A word of length 1 is a si... |
| wrdl1exs1 14629 | A word of length 1 is a si... |
| wrdl1s1 14630 | A word of length 1 is a si... |
| s111 14631 | The singleton word functio... |
| ccatws1cl 14632 | The concatenation of a wor... |
| ccatws1clv 14633 | The concatenation of a wor... |
| ccat2s1cl 14634 | The concatenation of two s... |
| ccats1alpha 14635 | A concatenation of a word ... |
| ccatws1len 14636 | The length of the concaten... |
| ccatws1lenp1b 14637 | The length of a word is ` ... |
| wrdlenccats1lenm1 14638 | The length of a word is th... |
| ccat2s1len 14639 | The length of the concaten... |
| ccatw2s1cl 14640 | The concatenation of a wor... |
| ccatw2s1len 14641 | The length of the concaten... |
| ccats1val1 14642 | Value of a symbol in the l... |
| ccats1val2 14643 | Value of the symbol concat... |
| ccat1st1st 14644 | The first symbol of a word... |
| ccat2s1p1 14645 | Extract the first of two c... |
| ccat2s1p2 14646 | Extract the second of two ... |
| ccatw2s1ass 14647 | Associative law for a conc... |
| ccatws1n0 14648 | The concatenation of a wor... |
| ccatws1ls 14649 | The last symbol of the con... |
| lswccats1 14650 | The last symbol of a word ... |
| lswccats1fst 14651 | The last symbol of a nonem... |
| ccatw2s1p1 14652 | Extract the symbol of the ... |
| ccatw2s1p2 14653 | Extract the second of two ... |
| ccat2s1fvw 14654 | Extract a symbol of a word... |
| ccat2s1fst 14655 | The first symbol of the co... |
| swrdnznd 14658 | The value of a subword ope... |
| swrdval 14659 | Value of a subword. (Cont... |
| swrd00 14660 | A zero length substring. ... |
| swrdcl 14661 | Closure of the subword ext... |
| swrdval2 14662 | Value of the subword extra... |
| swrdlen 14663 | Length of an extracted sub... |
| swrdfv 14664 | A symbol in an extracted s... |
| swrdfv0 14665 | The first symbol in an ext... |
| swrdf 14666 | A subword of a word is a f... |
| swrdvalfn 14667 | Value of the subword extra... |
| swrdrn 14668 | The range of a subword of ... |
| swrdlend 14669 | The value of the subword e... |
| swrdnd 14670 | The value of the subword e... |
| swrdnd2 14671 | Value of the subword extra... |
| swrdnnn0nd 14672 | The value of a subword ope... |
| swrdnd0 14673 | The value of a subword ope... |
| swrd0 14674 | A subword of an empty set ... |
| swrdrlen 14675 | Length of a right-anchored... |
| swrdlen2 14676 | Length of an extracted sub... |
| swrdfv2 14677 | A symbol in an extracted s... |
| swrdwrdsymb 14678 | A subword is a word over t... |
| swrdsb0eq 14679 | Two subwords with the same... |
| swrdsbslen 14680 | Two subwords with the same... |
| swrdspsleq 14681 | Two words have a common su... |
| swrds1 14682 | Extract a single symbol fr... |
| swrdlsw 14683 | Extract the last single sy... |
| ccatswrd 14684 | Joining two adjacent subwo... |
| swrdccat2 14685 | Recover the right half of ... |
| pfxnndmnd 14688 | The value of a prefix oper... |
| pfxval 14689 | Value of a prefix operatio... |
| pfx00 14690 | The zero length prefix is ... |
| pfx0 14691 | A prefix of an empty set i... |
| pfxval0 14692 | Value of a prefix operatio... |
| pfxcl 14693 | Closure of the prefix extr... |
| pfxmpt 14694 | Value of the prefix extrac... |
| pfxres 14695 | Value of the prefix extrac... |
| pfxf 14696 | A prefix of a word is a fu... |
| pfxfn 14697 | Value of the prefix extrac... |
| pfxfv 14698 | A symbol in a prefix of a ... |
| pfxlen 14699 | Length of a prefix. (Cont... |
| pfxid 14700 | A word is a prefix of itse... |
| pfxrn 14701 | The range of a prefix of a... |
| pfxn0 14702 | A prefix consisting of at ... |
| pfxnd 14703 | The value of a prefix oper... |
| pfxnd0 14704 | The value of a prefix oper... |
| pfxwrdsymb 14705 | A prefix of a word is a wo... |
| addlenpfx 14706 | The sum of the lengths of ... |
| pfxfv0 14707 | The first symbol of a pref... |
| pfxtrcfv 14708 | A symbol in a word truncat... |
| pfxtrcfv0 14709 | The first symbol in a word... |
| pfxfvlsw 14710 | The last symbol in a nonem... |
| pfxeq 14711 | The prefixes of two words ... |
| pfxtrcfvl 14712 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14713 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14714 | Two (nonempty) words are e... |
| disjwrdpfx 14715 | Sets of words are disjoint... |
| ccatpfx 14716 | Concatenating a prefix wit... |
| pfxccat1 14717 | Recover the left half of a... |
| pfx1 14718 | The prefix of length one o... |
| swrdswrdlem 14719 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14720 | A subword of a subword is ... |
| pfxswrd 14721 | A prefix of a subword is a... |
| swrdpfx 14722 | A subword of a prefix is a... |
| pfxpfx 14723 | A prefix of a prefix is a ... |
| pfxpfxid 14724 | A prefix of a prefix with ... |
| pfxcctswrd 14725 | The concatenation of the p... |
| lenpfxcctswrd 14726 | The length of the concaten... |
| lenrevpfxcctswrd 14727 | The length of the concaten... |
| pfxlswccat 14728 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14729 | The last symbol of a word ... |
| ccats1pfxeqrex 14730 | There exists a symbol such... |
| ccatopth 14731 | An ~ opth -like theorem fo... |
| ccatopth2 14732 | An ~ opth -like theorem fo... |
| ccatlcan 14733 | Concatenation of words is ... |
| ccatrcan 14734 | Concatenation of words is ... |
| wrdeqs1cat 14735 | Decompose a nonempty word ... |
| cats1un 14736 | Express a word with an ext... |
| wrdind 14737 | Perform induction over the... |
| wrd2ind 14738 | Perform induction over the... |
| swrdccatfn 14739 | The subword of a concatena... |
| swrdccatin1 14740 | The subword of a concatena... |
| pfxccatin12lem4 14741 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14742 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14743 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14744 | The subword of a concatena... |
| pfxccatin12lem2c 14745 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14746 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14747 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14748 | The subword of a concatena... |
| pfxccat3 14749 | The subword of a concatena... |
| swrdccat 14750 | The subword of a concatena... |
| pfxccatpfx1 14751 | A prefix of a concatenatio... |
| pfxccatpfx2 14752 | A prefix of a concatenatio... |
| pfxccat3a 14753 | A prefix of a concatenatio... |
| swrdccat3blem 14754 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14755 | A suffix of a concatenatio... |
| pfxccatid 14756 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14757 | A word is a prefix of a wo... |
| swrdccatin1d 14758 | The subword of a concatena... |
| swrdccatin2d 14759 | The subword of a concatena... |
| pfxccatin12d 14760 | The subword of a concatena... |
| reuccatpfxs1lem 14761 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14762 | There is a unique word hav... |
| reuccatpfxs1v 14763 | There is a unique word hav... |
| splval 14766 | Value of the substring rep... |
| splcl 14767 | Closure of the substring r... |
| splid 14768 | Splicing a subword for the... |
| spllen 14769 | The length of a splice. (... |
| splfv1 14770 | Symbols to the left of a s... |
| splfv2a 14771 | Symbols within the replace... |
| splval2 14772 | Value of a splice, assumin... |
| revval 14775 | Value of the word reversin... |
| revcl 14776 | The reverse of a word is a... |
| revlen 14777 | The reverse of a word has ... |
| revfv 14778 | Reverse of a word at a poi... |
| rev0 14779 | The empty word is its own ... |
| revs1 14780 | Singleton words are their ... |
| revccat 14781 | Antiautomorphic property o... |
| revrev 14782 | Reversal is an involution ... |
| reps 14785 | Construct a function mappi... |
| repsundef 14786 | A function mapping a half-... |
| repsconst 14787 | Construct a function mappi... |
| repsf 14788 | The constructed function m... |
| repswsymb 14789 | The symbols of a "repeated... |
| repsw 14790 | A function mapping a half-... |
| repswlen 14791 | The length of a "repeated ... |
| repsw0 14792 | The "repeated symbol word"... |
| repsdf2 14793 | Alternative definition of ... |
| repswsymball 14794 | All the symbols of a "repe... |
| repswsymballbi 14795 | A word is a "repeated symb... |
| repswfsts 14796 | The first symbol of a none... |
| repswlsw 14797 | The last symbol of a nonem... |
| repsw1 14798 | The "repeated symbol word"... |
| repswswrd 14799 | A subword of a "repeated s... |
| repswpfx 14800 | A prefix of a repeated sym... |
| repswccat 14801 | The concatenation of two "... |
| repswrevw 14802 | The reverse of a "repeated... |
| cshfn 14805 | Perform a cyclical shift f... |
| cshword 14806 | Perform a cyclical shift f... |
| cshnz 14807 | A cyclical shift is the em... |
| 0csh0 14808 | Cyclically shifting an emp... |
| cshw0 14809 | A word cyclically shifted ... |
| cshwmodn 14810 | Cyclically shifting a word... |
| cshwsublen 14811 | Cyclically shifting a word... |
| cshwn 14812 | A word cyclically shifted ... |
| cshwcl 14813 | A cyclically shifted word ... |
| cshwlen 14814 | The length of a cyclically... |
| cshwf 14815 | A cyclically shifted word ... |
| cshwfn 14816 | A cyclically shifted word ... |
| cshwrn 14817 | The range of a cyclically ... |
| cshwidxmod 14818 | The symbol at a given inde... |
| cshwidxmodr 14819 | The symbol at a given inde... |
| cshwidx0mod 14820 | The symbol at index 0 of a... |
| cshwidx0 14821 | The symbol at index 0 of a... |
| cshwidxm1 14822 | The symbol at index ((n-N)... |
| cshwidxm 14823 | The symbol at index (n-N) ... |
| cshwidxn 14824 | The symbol at index (n-1) ... |
| cshf1 14825 | Cyclically shifting a word... |
| cshinj 14826 | If a word is injectiv (reg... |
| repswcshw 14827 | A cyclically shifted "repe... |
| 2cshw 14828 | Cyclically shifting a word... |
| 2cshwid 14829 | Cyclically shifting a word... |
| lswcshw 14830 | The last symbol of a word ... |
| 2cshwcom 14831 | Cyclically shifting a word... |
| cshwleneq 14832 | If the results of cyclical... |
| 3cshw 14833 | Cyclically shifting a word... |
| cshweqdif2 14834 | If cyclically shifting two... |
| cshweqdifid 14835 | If cyclically shifting a w... |
| cshweqrep 14836 | If cyclically shifting a w... |
| cshw1 14837 | If cyclically shifting a w... |
| cshw1repsw 14838 | If cyclically shifting a w... |
| cshwsexa 14839 | The class of (different!) ... |
| 2cshwcshw 14840 | If a word is a cyclically ... |
| scshwfzeqfzo 14841 | For a nonempty word the se... |
| cshwcshid 14842 | A cyclically shifted word ... |
| cshwcsh2id 14843 | A cyclically shifted word ... |
| cshimadifsn 14844 | The image of a cyclically ... |
| cshimadifsn0 14845 | The image of a cyclically ... |
| wrdco 14846 | Mapping a word by a functi... |
| lenco 14847 | Length of a mapped word is... |
| s1co 14848 | Mapping of a singleton wor... |
| revco 14849 | Mapping of words (i.e., a ... |
| ccatco 14850 | Mapping of words commutes ... |
| cshco 14851 | Mapping of words commutes ... |
| swrdco 14852 | Mapping of words commutes ... |
| pfxco 14853 | Mapping of words commutes ... |
| lswco 14854 | Mapping of (nonempty) word... |
| repsco 14855 | Mapping of words commutes ... |
| cats1cld 14870 | Closure of concatenation w... |
| cats1co 14871 | Closure of concatenation w... |
| cats1cli 14872 | Closure of concatenation w... |
| cats1fvn 14873 | The last symbol of a conca... |
| cats1fv 14874 | A symbol other than the la... |
| cats1len 14875 | The length of concatenatio... |
| cats1cat 14876 | Closure of concatenation w... |
| cats2cat 14877 | Closure of concatenation o... |
| s2eqd 14878 | Equality theorem for a dou... |
| s3eqd 14879 | Equality theorem for a len... |
| s4eqd 14880 | Equality theorem for a len... |
| s5eqd 14881 | Equality theorem for a len... |
| s6eqd 14882 | Equality theorem for a len... |
| s7eqd 14883 | Equality theorem for a len... |
| s8eqd 14884 | Equality theorem for a len... |
| s3eq2 14885 | Equality theorem for a len... |
| s2cld 14886 | A doubleton word is a word... |
| s3cld 14887 | A length 3 string is a wor... |
| s4cld 14888 | A length 4 string is a wor... |
| s5cld 14889 | A length 5 string is a wor... |
| s6cld 14890 | A length 6 string is a wor... |
| s7cld 14891 | A length 7 string is a wor... |
| s8cld 14892 | A length 8 string is a wor... |
| s2cl 14893 | A doubleton word is a word... |
| s3cl 14894 | A length 3 string is a wor... |
| s2cli 14895 | A doubleton word is a word... |
| s3cli 14896 | A length 3 string is a wor... |
| s4cli 14897 | A length 4 string is a wor... |
| s5cli 14898 | A length 5 string is a wor... |
| s6cli 14899 | A length 6 string is a wor... |
| s7cli 14900 | A length 7 string is a wor... |
| s8cli 14901 | A length 8 string is a wor... |
| s2fv0 14902 | Extract the first symbol f... |
| s2fv1 14903 | Extract the second symbol ... |
| s2len 14904 | The length of a doubleton ... |
| s2dm 14905 | The domain of a doubleton ... |
| s3fv0 14906 | Extract the first symbol f... |
| s3fv1 14907 | Extract the second symbol ... |
| s3fv2 14908 | Extract the third symbol f... |
| s3len 14909 | The length of a length 3 s... |
| s4fv0 14910 | Extract the first symbol f... |
| s4fv1 14911 | Extract the second symbol ... |
| s4fv2 14912 | Extract the third symbol f... |
| s4fv3 14913 | Extract the fourth symbol ... |
| s4len 14914 | The length of a length 4 s... |
| s5len 14915 | The length of a length 5 s... |
| s6len 14916 | The length of a length 6 s... |
| s7len 14917 | The length of a length 7 s... |
| s8len 14918 | The length of a length 8 s... |
| lsws2 14919 | The last symbol of a doubl... |
| lsws3 14920 | The last symbol of a 3 let... |
| lsws4 14921 | The last symbol of a 4 let... |
| s2prop 14922 | A length 2 word is an unor... |
| s2dmALT 14923 | Alternate version of ~ s2d... |
| s3tpop 14924 | A length 3 word is an unor... |
| s4prop 14925 | A length 4 word is a union... |
| s3fn 14926 | A length 3 word is a funct... |
| funcnvs1 14927 | The converse of a singleto... |
| funcnvs2 14928 | The converse of a length 2... |
| funcnvs3 14929 | The converse of a length 3... |
| funcnvs4 14930 | The converse of a length 4... |
| s2f1o 14931 | A length 2 word with mutua... |
| f1oun2prg 14932 | A union of unordered pairs... |
| s4f1o 14933 | A length 4 word with mutua... |
| s4dom 14934 | The domain of a length 4 w... |
| s2co 14935 | Mapping a doubleton word b... |
| s3co 14936 | Mapping a length 3 string ... |
| s0s1 14937 | Concatenation of fixed len... |
| s1s2 14938 | Concatenation of fixed len... |
| s1s3 14939 | Concatenation of fixed len... |
| s1s4 14940 | Concatenation of fixed len... |
| s1s5 14941 | Concatenation of fixed len... |
| s1s6 14942 | Concatenation of fixed len... |
| s1s7 14943 | Concatenation of fixed len... |
| s2s2 14944 | Concatenation of fixed len... |
| s4s2 14945 | Concatenation of fixed len... |
| s4s3 14946 | Concatenation of fixed len... |
| s4s4 14947 | Concatenation of fixed len... |
| s3s4 14948 | Concatenation of fixed len... |
| s2s5 14949 | Concatenation of fixed len... |
| s5s2 14950 | Concatenation of fixed len... |
| s2eq2s1eq 14951 | Two length 2 words are equ... |
| s2eq2seq 14952 | Two length 2 words are equ... |
| s3eqs2s1eq 14953 | Two length 3 words are equ... |
| s3eq3seq 14954 | Two length 3 words are equ... |
| swrds2 14955 | Extract two adjacent symbo... |
| swrds2m 14956 | Extract two adjacent symbo... |
| wrdlen2i 14957 | Implications of a word of ... |
| wrd2pr2op 14958 | A word of length two repre... |
| wrdlen2 14959 | A word of length two. (Co... |
| wrdlen2s2 14960 | A word of length two as do... |
| wrdl2exs2 14961 | A word of length two is a ... |
| pfx2 14962 | A prefix of length two. (... |
| wrd3tpop 14963 | A word of length three rep... |
| wrdlen3s3 14964 | A word of length three as ... |
| repsw2 14965 | The "repeated symbol word"... |
| repsw3 14966 | The "repeated symbol word"... |
| swrd2lsw 14967 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14968 | Two words of length at lea... |
| ccatw2s1ccatws2 14969 | The concatenation of a wor... |
| ccat2s1fvwALT 14970 | Alternate proof of ~ ccat2... |
| wwlktovf 14971 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14972 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14973 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14974 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14975 | There is a bijection betwe... |
| eqwrds3 14976 | A word is equal with a len... |
| wrdl3s3 14977 | A word of length 3 is a le... |
| s2rn 14978 | Range of a length 2 string... |
| s3rn 14979 | Range of a length 3 string... |
| s7rn 14980 | Range of a length 7 string... |
| s7f1o 14981 | A length 7 word with mutua... |
| s3sndisj 14982 | The singletons consisting ... |
| s3iunsndisj 14983 | The union of singletons co... |
| ofccat 14984 | Letterwise operations on w... |
| ofs1 14985 | Letterwise operations on a... |
| ofs2 14986 | Letterwise operations on a... |
| coss12d 14987 | Subset deduction for compo... |
| trrelssd 14988 | The composition of subclas... |
| xpcogend 14989 | The most interesting case ... |
| xpcoidgend 14990 | If two classes are not dis... |
| cotr2g 14991 | Two ways of saying that th... |
| cotr2 14992 | Two ways of saying a relat... |
| cotr3 14993 | Two ways of saying a relat... |
| coemptyd 14994 | Deduction about compositio... |
| xptrrel 14995 | The cross product is alway... |
| 0trrel 14996 | The empty class is a trans... |
| cleq1lem 14997 | Equality implies bijection... |
| cleq1 14998 | Equality of relations impl... |
| clsslem 14999 | The closure of a subclass ... |
| trcleq1 15004 | Equality of relations impl... |
| trclsslem 15005 | The transitive closure (as... |
| trcleq2lem 15006 | Equality implies bijection... |
| cvbtrcl 15007 | Change of bound variable i... |
| trcleq12lem 15008 | Equality implies bijection... |
| trclexlem 15009 | Existence of relation impl... |
| trclublem 15010 | If a relation exists then ... |
| trclubi 15011 | The Cartesian product of t... |
| trclubgi 15012 | The union with the Cartesi... |
| trclub 15013 | The Cartesian product of t... |
| trclubg 15014 | The union with the Cartesi... |
| trclfv 15015 | The transitive closure of ... |
| brintclab 15016 | Two ways to express a bina... |
| brtrclfv 15017 | Two ways of expressing the... |
| brcnvtrclfv 15018 | Two ways of expressing the... |
| brtrclfvcnv 15019 | Two ways of expressing the... |
| brcnvtrclfvcnv 15020 | Two ways of expressing the... |
| trclfvss 15021 | The transitive closure (as... |
| trclfvub 15022 | The transitive closure of ... |
| trclfvlb 15023 | The transitive closure of ... |
| trclfvcotr 15024 | The transitive closure of ... |
| trclfvlb2 15025 | The transitive closure of ... |
| trclfvlb3 15026 | The transitive closure of ... |
| cotrtrclfv 15027 | The transitive closure of ... |
| trclidm 15028 | The transitive closure of ... |
| trclun 15029 | Transitive closure of a un... |
| trclfvg 15030 | The value of the transitiv... |
| trclfvcotrg 15031 | The value of the transitiv... |
| reltrclfv 15032 | The transitive closure of ... |
| dmtrclfv 15033 | The domain of the transiti... |
| reldmrelexp 15036 | The domain of the repeated... |
| relexp0g 15037 | A relation composed zero t... |
| relexp0 15038 | A relation composed zero t... |
| relexp0d 15039 | A relation composed zero t... |
| relexpsucnnr 15040 | A reduction for relation e... |
| relexp1g 15041 | A relation composed once i... |
| dfid5 15042 | Identity relation is equal... |
| dfid6 15043 | Identity relation expresse... |
| relexp1d 15044 | A relation composed once i... |
| relexpsucnnl 15045 | A reduction for relation e... |
| relexpsucl 15046 | A reduction for relation e... |
| relexpsucr 15047 | A reduction for relation e... |
| relexpsucrd 15048 | A reduction for relation e... |
| relexpsucld 15049 | A reduction for relation e... |
| relexpcnv 15050 | Commutation of converse an... |
| relexpcnvd 15051 | Commutation of converse an... |
| relexp0rel 15052 | The exponentiation of a cl... |
| relexprelg 15053 | The exponentiation of a cl... |
| relexprel 15054 | The exponentiation of a re... |
| relexpreld 15055 | The exponentiation of a re... |
| relexpnndm 15056 | The domain of an exponenti... |
| relexpdmg 15057 | The domain of an exponenti... |
| relexpdm 15058 | The domain of an exponenti... |
| relexpdmd 15059 | The domain of an exponenti... |
| relexpnnrn 15060 | The range of an exponentia... |
| relexprng 15061 | The range of an exponentia... |
| relexprn 15062 | The range of an exponentia... |
| relexprnd 15063 | The range of an exponentia... |
| relexpfld 15064 | The field of an exponentia... |
| relexpfldd 15065 | The field of an exponentia... |
| relexpaddnn 15066 | Relation composition becom... |
| relexpuzrel 15067 | The exponentiation of a cl... |
| relexpaddg 15068 | Relation composition becom... |
| relexpaddd 15069 | Relation composition becom... |
| rtrclreclem1 15072 | The reflexive, transitive ... |
| dfrtrclrec2 15073 | If two elements are connec... |
| rtrclreclem2 15074 | The reflexive, transitive ... |
| rtrclreclem3 15075 | The reflexive, transitive ... |
| rtrclreclem4 15076 | The reflexive, transitive ... |
| dfrtrcl2 15077 | The two definitions ` t* `... |
| relexpindlem 15078 | Principle of transitive in... |
| relexpind 15079 | Principle of transitive in... |
| rtrclind 15080 | Principle of transitive in... |
| shftlem 15083 | Two ways to write a shifte... |
| shftuz 15084 | A shift of the upper integ... |
| shftfval 15085 | The value of the sequence ... |
| shftdm 15086 | Domain of a relation shift... |
| shftfib 15087 | Value of a fiber of the re... |
| shftfn 15088 | Functionality and domain o... |
| shftval 15089 | Value of a sequence shifte... |
| shftval2 15090 | Value of a sequence shifte... |
| shftval3 15091 | Value of a sequence shifte... |
| shftval4 15092 | Value of a sequence shifte... |
| shftval5 15093 | Value of a shifted sequenc... |
| shftf 15094 | Functionality of a shifted... |
| 2shfti 15095 | Composite shift operations... |
| shftidt2 15096 | Identity law for the shift... |
| shftidt 15097 | Identity law for the shift... |
| shftcan1 15098 | Cancellation law for the s... |
| shftcan2 15099 | Cancellation law for the s... |
| seqshft 15100 | Shifting the index set of ... |
| sgnval 15103 | Value of the signum functi... |
| sgn0 15104 | The signum of 0 is 0. (Co... |
| sgnp 15105 | The signum of a positive e... |
| sgnrrp 15106 | The signum of a positive r... |
| sgn1 15107 | The signum of 1 is 1. (Co... |
| sgnpnf 15108 | The signum of ` +oo ` is 1... |
| sgnn 15109 | The signum of a negative e... |
| sgnmnf 15110 | The signum of ` -oo ` is -... |
| sgndm 15111 | The domain of the signum f... |
| sgncl 15112 | Closure of the signum: Th... |
| sgnrn 15113 | The range of the signum fu... |
| sgnfo 15114 | The signum function as ont... |
| sgnneg 15115 | Negation of the signum. (... |
| sgn3da 15116 | A conditional containing a... |
| sgnclre 15117 | Closure of the signum for ... |
| sgn0bi 15118 | Zero signum. (Contributed... |
| sgnnbi 15119 | Negative signum. (Contrib... |
| sgnpbi 15120 | Positive signum. (Contrib... |
| sgnsub 15121 | Signum of a difference wit... |
| sgnmul 15122 | Signum of a product. (Con... |
| sgnmulrp2 15123 | Multiplication by a positi... |
| sgnmulsgn 15124 | If two real numbers are of... |
| cjval 15131 | The value of the conjugate... |
| cjth 15132 | The defining property of t... |
| cjf 15133 | Domain and codomain of the... |
| cjcl 15134 | The conjugate of a complex... |
| reval 15135 | The value of the real part... |
| imval 15136 | The value of the imaginary... |
| imre 15137 | The imaginary part of a co... |
| reim 15138 | The real part of a complex... |
| recl 15139 | The real part of a complex... |
| imcl 15140 | The imaginary part of a co... |
| ref 15141 | Domain and codomain of the... |
| imf 15142 | Domain and codomain of the... |
| crre 15143 | The real part of a complex... |
| crim 15144 | The real part of a complex... |
| replim 15145 | Reconstruct a complex numb... |
| remim 15146 | Value of the conjugate of ... |
| reim0 15147 | The imaginary part of a re... |
| reim0b 15148 | A number is real iff its i... |
| rereb 15149 | A number is real iff it eq... |
| mulre 15150 | A product with a nonzero r... |
| rere 15151 | A real number equals its r... |
| cjreb 15152 | A number is real iff it eq... |
| recj 15153 | Real part of a complex con... |
| reneg 15154 | Real part of negative. (C... |
| readd 15155 | Real part distributes over... |
| resub 15156 | Real part distributes over... |
| remullem 15157 | Lemma for ~ remul , ~ immu... |
| remul 15158 | Real part of a product. (... |
| remul2 15159 | Real part of a product. (... |
| rediv 15160 | Real part of a division. ... |
| imcj 15161 | Imaginary part of a comple... |
| imneg 15162 | The imaginary part of a ne... |
| imadd 15163 | Imaginary part distributes... |
| imsub 15164 | Imaginary part distributes... |
| immul 15165 | Imaginary part of a produc... |
| immul2 15166 | Imaginary part of a produc... |
| imdiv 15167 | Imaginary part of a divisi... |
| cjre 15168 | A real number equals its c... |
| cjcj 15169 | The conjugate of the conju... |
| cjadd 15170 | Complex conjugate distribu... |
| cjmul 15171 | Complex conjugate distribu... |
| ipcnval 15172 | Standard inner product on ... |
| cjmulrcl 15173 | A complex number times its... |
| cjmulval 15174 | A complex number times its... |
| cjmulge0 15175 | A complex number times its... |
| cjneg 15176 | Complex conjugate of negat... |
| addcj 15177 | A number plus its conjugat... |
| cjsub 15178 | Complex conjugate distribu... |
| cjexp 15179 | Complex conjugate of posit... |
| imval2 15180 | The imaginary part of a nu... |
| re0 15181 | The real part of zero. (C... |
| im0 15182 | The imaginary part of zero... |
| re1 15183 | The real part of one. (Co... |
| im1 15184 | The imaginary part of one.... |
| rei 15185 | The real part of ` _i ` . ... |
| imi 15186 | The imaginary part of ` _i... |
| cj0 15187 | The conjugate of zero. (C... |
| cji 15188 | The complex conjugate of t... |
| cjreim 15189 | The conjugate of a represe... |
| cjreim2 15190 | The conjugate of the repre... |
| cj11 15191 | Complex conjugate is a one... |
| cjne0 15192 | A number is nonzero iff it... |
| cjdiv 15193 | Complex conjugate distribu... |
| cnrecnv 15194 | The inverse to the canonic... |
| sqeqd 15195 | A deduction for showing tw... |
| recli 15196 | The real part of a complex... |
| imcli 15197 | The imaginary part of a co... |
| cjcli 15198 | Closure law for complex co... |
| replimi 15199 | Construct a complex number... |
| cjcji 15200 | The conjugate of the conju... |
| reim0bi 15201 | A number is real iff its i... |
| rerebi 15202 | A real number equals its r... |
| cjrebi 15203 | A number is real iff it eq... |
| recji 15204 | Real part of a complex con... |
| imcji 15205 | Imaginary part of a comple... |
| cjmulrcli 15206 | A complex number times its... |
| cjmulvali 15207 | A complex number times its... |
| cjmulge0i 15208 | A complex number times its... |
| renegi 15209 | Real part of negative. (C... |
| imnegi 15210 | Imaginary part of negative... |
| cjnegi 15211 | Complex conjugate of negat... |
| addcji 15212 | A number plus its conjugat... |
| readdi 15213 | Real part distributes over... |
| imaddi 15214 | Imaginary part distributes... |
| remuli 15215 | Real part of a product. (... |
| immuli 15216 | Imaginary part of a produc... |
| cjaddi 15217 | Complex conjugate distribu... |
| cjmuli 15218 | Complex conjugate distribu... |
| ipcni 15219 | Standard inner product on ... |
| cjdivi 15220 | Complex conjugate distribu... |
| crrei 15221 | The real part of a complex... |
| crimi 15222 | The imaginary part of a co... |
| recld 15223 | The real part of a complex... |
| imcld 15224 | The imaginary part of a co... |
| cjcld 15225 | Closure law for complex co... |
| replimd 15226 | Construct a complex number... |
| remimd 15227 | Value of the conjugate of ... |
| cjcjd 15228 | The conjugate of the conju... |
| reim0bd 15229 | A number is real iff its i... |
| rerebd 15230 | A real number equals its r... |
| cjrebd 15231 | A number is real iff it eq... |
| cjne0d 15232 | A number is nonzero iff it... |
| recjd 15233 | Real part of a complex con... |
| imcjd 15234 | Imaginary part of a comple... |
| cjmulrcld 15235 | A complex number times its... |
| cjmulvald 15236 | A complex number times its... |
| cjmulge0d 15237 | A complex number times its... |
| renegd 15238 | Real part of negative. (C... |
| imnegd 15239 | Imaginary part of negative... |
| cjnegd 15240 | Complex conjugate of negat... |
| addcjd 15241 | A number plus its conjugat... |
| cjexpd 15242 | Complex conjugate of posit... |
| readdd 15243 | Real part distributes over... |
| imaddd 15244 | Imaginary part distributes... |
| resubd 15245 | Real part distributes over... |
| imsubd 15246 | Imaginary part distributes... |
| remuld 15247 | Real part of a product. (... |
| immuld 15248 | Imaginary part of a produc... |
| cjaddd 15249 | Complex conjugate distribu... |
| cjmuld 15250 | Complex conjugate distribu... |
| ipcnd 15251 | Standard inner product on ... |
| cjdivd 15252 | Complex conjugate distribu... |
| rered 15253 | A real number equals its r... |
| reim0d 15254 | The imaginary part of a re... |
| cjred 15255 | A real number equals its c... |
| remul2d 15256 | Real part of a product. (... |
| immul2d 15257 | Imaginary part of a produc... |
| redivd 15258 | Real part of a division. ... |
| imdivd 15259 | Imaginary part of a divisi... |
| crred 15260 | The real part of a complex... |
| crimd 15261 | The imaginary part of a co... |
| sqrtval 15266 | Value of square root funct... |
| absval 15267 | The absolute value (modulu... |
| rennim 15268 | A real number does not lie... |
| cnpart 15269 | The specification of restr... |
| sqrt0 15270 | The square root of zero is... |
| 01sqrexlem1 15271 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15272 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15273 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15274 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15275 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15276 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15277 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15278 | Existence of a square root... |
| resqrex 15279 | Existence of a square root... |
| sqrmo 15280 | Uniqueness for the square ... |
| resqreu 15281 | Existence and uniqueness f... |
| resqrtcl 15282 | Closure of the square root... |
| resqrtthlem 15283 | Lemma for ~ resqrtth . (C... |
| resqrtth 15284 | Square root theorem over t... |
| remsqsqrt 15285 | Square of square root. (C... |
| sqrtge0 15286 | The square root function i... |
| sqrtgt0 15287 | The square root function i... |
| sqrtmul 15288 | Square root distributes ov... |
| sqrtle 15289 | Square root is monotonic. ... |
| sqrtlt 15290 | Square root is strictly mo... |
| sqrt11 15291 | The square root function i... |
| sqrt00 15292 | A square root is zero iff ... |
| rpsqrtcl 15293 | The square root of a posit... |
| sqrtdiv 15294 | Square root distributes ov... |
| sqrtneglem 15295 | The square root of a negat... |
| sqrtneg 15296 | The square root of a negat... |
| sqrtsq2 15297 | Relationship between squar... |
| sqrtsq 15298 | Square root of square. (C... |
| sqrtmsq 15299 | Square root of square. (C... |
| sqrt1 15300 | The square root of 1 is 1.... |
| sqrt4 15301 | The square root of 4 is 2.... |
| sqrt9 15302 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15303 | The square root of 2 is bo... |
| sqrtm1 15304 | The imaginary unit is the ... |
| nn0sqeq1 15305 | A natural number with squa... |
| absneg 15306 | Absolute value of the nega... |
| abscl 15307 | Real closure of absolute v... |
| abscj 15308 | The absolute value of a nu... |
| absvalsq 15309 | Square of value of absolut... |
| absvalsq2 15310 | Square of value of absolut... |
| sqabsadd 15311 | Square of absolute value o... |
| sqabssub 15312 | Square of absolute value o... |
| absval2 15313 | Value of absolute value fu... |
| abs0 15314 | The absolute value of 0. ... |
| absi 15315 | The absolute value of the ... |
| absge0 15316 | Absolute value is nonnegat... |
| absrpcl 15317 | The absolute value of a no... |
| abs00 15318 | The absolute value of a nu... |
| abs00ad 15319 | A complex number is zero i... |
| abs00bd 15320 | If a complex number is zer... |
| absreimsq 15321 | Square of the absolute val... |
| absreim 15322 | Absolute value of a number... |
| absmul 15323 | Absolute value distributes... |
| absdiv 15324 | Absolute value distributes... |
| absid 15325 | A nonnegative number is it... |
| abs1 15326 | The absolute value of one ... |
| absnid 15327 | For a negative number, its... |
| leabs 15328 | A real number is less than... |
| absor 15329 | The absolute value of a re... |
| absre 15330 | Absolute value of a real n... |
| absresq 15331 | Square of the absolute val... |
| absmod0 15332 | ` A ` is divisible by ` B ... |
| absexp 15333 | Absolute value of positive... |
| absexpz 15334 | Absolute value of integer ... |
| abssq 15335 | Square can be moved in and... |
| sqabs 15336 | The squares of two reals a... |
| absrele 15337 | The absolute value of a co... |
| absimle 15338 | The absolute value of a co... |
| max0add 15339 | The sum of the positive an... |
| absz 15340 | A real number is an intege... |
| nn0abscl 15341 | The absolute value of an i... |
| zabscl 15342 | The absolute value of an i... |
| zabs0b 15343 | An integer has an absolute... |
| abslt 15344 | Absolute value and 'less t... |
| absle 15345 | Absolute value and 'less t... |
| abssubne0 15346 | If the absolute value of a... |
| absdiflt 15347 | The absolute value of a di... |
| absdifle 15348 | The absolute value of a di... |
| elicc4abs 15349 | Membership in a symmetric ... |
| lenegsq 15350 | Comparison to a nonnegativ... |
| releabs 15351 | The real part of a number ... |
| recval 15352 | Reciprocal expressed with ... |
| absidm 15353 | The absolute value functio... |
| absgt0 15354 | The absolute value of a no... |
| nnabscl 15355 | The absolute value of a no... |
| abssub 15356 | Swapping order of subtract... |
| abssubge0 15357 | Absolute value of a nonneg... |
| abssuble0 15358 | Absolute value of a nonpos... |
| absmax 15359 | The maximum of two numbers... |
| abstri 15360 | Triangle inequality for ab... |
| abs3dif 15361 | Absolute value of differen... |
| abs2dif 15362 | Difference of absolute val... |
| abs2dif2 15363 | Difference of absolute val... |
| abs2difabs 15364 | Absolute value of differen... |
| abs1m 15365 | For any complex number, th... |
| recan 15366 | Cancellation law involving... |
| absf 15367 | Mapping domain and codomai... |
| abs3lem 15368 | Lemma involving absolute v... |
| abslem2 15369 | Lemma involving absolute v... |
| rddif 15370 | The difference between a r... |
| absrdbnd 15371 | Bound on the absolute valu... |
| fzomaxdiflem 15372 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15373 | A bound on the separation ... |
| uzin2 15374 | The upper integers are clo... |
| rexanuz 15375 | Combine two different uppe... |
| rexanre 15376 | Combine two different uppe... |
| rexfiuz 15377 | Combine finitely many diff... |
| rexuz3 15378 | Restrict the base of the u... |
| rexanuz2 15379 | Combine two different uppe... |
| r19.29uz 15380 | A version of ~ 19.29 for u... |
| r19.2uz 15381 | A version of ~ r19.2z for ... |
| rexuzre 15382 | Convert an upper real quan... |
| rexico 15383 | Restrict the base of an up... |
| cau3lem 15384 | Lemma for ~ cau3 . (Contr... |
| cau3 15385 | Convert between three-quan... |
| cau4 15386 | Change the base of a Cauch... |
| caubnd2 15387 | A Cauchy sequence of compl... |
| caubnd 15388 | A Cauchy sequence of compl... |
| sqreulem 15389 | Lemma for ~ sqreu : write ... |
| sqreu 15390 | Existence and uniqueness f... |
| sqrtcl 15391 | Closure of the square root... |
| sqrtthlem 15392 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15393 | Mapping domain and codomai... |
| sqrtth 15394 | Square root theorem over t... |
| sqrtrege0 15395 | The square root function m... |
| eqsqrtor 15396 | Solve an equation containi... |
| eqsqrtd 15397 | A deduction for showing th... |
| eqsqrt2d 15398 | A deduction for showing th... |
| amgm2 15399 | Arithmetic-geometric mean ... |
| sqrtthi 15400 | Square root theorem. Theo... |
| sqrtcli 15401 | The square root of a nonne... |
| sqrtgt0i 15402 | The square root of a posit... |
| sqrtmsqi 15403 | Square root of square. (C... |
| sqrtsqi 15404 | Square root of square. (C... |
| sqsqrti 15405 | Square of square root. (C... |
| sqrtge0i 15406 | The square root of a nonne... |
| absidi 15407 | A nonnegative number is it... |
| absnidi 15408 | A negative number is the n... |
| leabsi 15409 | A real number is less than... |
| absori 15410 | The absolute value of a re... |
| absrei 15411 | Absolute value of a real n... |
| sqrtpclii 15412 | The square root of a posit... |
| sqrtgt0ii 15413 | The square root of a posit... |
| sqrt11i 15414 | The square root function i... |
| sqrtmuli 15415 | Square root distributes ov... |
| sqrtmulii 15416 | Square root distributes ov... |
| sqrtmsq2i 15417 | Relationship between squar... |
| sqrtlei 15418 | Square root is monotonic. ... |
| sqrtlti 15419 | Square root is strictly mo... |
| abslti 15420 | Absolute value and 'less t... |
| abslei 15421 | Absolute value and 'less t... |
| cnsqrt00 15422 | A square root of a complex... |
| absvalsqi 15423 | Square of value of absolut... |
| absvalsq2i 15424 | Square of value of absolut... |
| abscli 15425 | Real closure of absolute v... |
| absge0i 15426 | Absolute value is nonnegat... |
| absval2i 15427 | Value of absolute value fu... |
| abs00i 15428 | The absolute value of a nu... |
| absgt0i 15429 | The absolute value of a no... |
| absnegi 15430 | Absolute value of negative... |
| abscji 15431 | The absolute value of a nu... |
| releabsi 15432 | The real part of a number ... |
| abssubi 15433 | Swapping order of subtract... |
| absmuli 15434 | Absolute value distributes... |
| sqabsaddi 15435 | Square of absolute value o... |
| sqabssubi 15436 | Square of absolute value o... |
| absdivzi 15437 | Absolute value distributes... |
| abstrii 15438 | Triangle inequality for ab... |
| abs3difi 15439 | Absolute value of differen... |
| abs3lemi 15440 | Lemma involving absolute v... |
| rpsqrtcld 15441 | The square root of a posit... |
| sqrtgt0d 15442 | The square root of a posit... |
| absnidd 15443 | A negative number is the n... |
| leabsd 15444 | A real number is less than... |
| absord 15445 | The absolute value of a re... |
| absred 15446 | Absolute value of a real n... |
| resqrtcld 15447 | The square root of a nonne... |
| sqrtmsqd 15448 | Square root of square. (C... |
| sqrtsqd 15449 | Square root of square. (C... |
| sqrtge0d 15450 | The square root of a nonne... |
| sqrtnegd 15451 | The square root of a negat... |
| absidd 15452 | A nonnegative number is it... |
| sqrtdivd 15453 | Square root distributes ov... |
| sqrtmuld 15454 | Square root distributes ov... |
| sqrtsq2d 15455 | Relationship between squar... |
| sqrtled 15456 | Square root is monotonic. ... |
| sqrtltd 15457 | Square root is strictly mo... |
| sqr11d 15458 | The square root function i... |
| nn0absid 15459 | A nonnegative integer is i... |
| nn0absidi 15460 | A nonnegative integer is i... |
| absltd 15461 | Absolute value and 'less t... |
| absled 15462 | Absolute value and 'less t... |
| abssubge0d 15463 | Absolute value of a nonneg... |
| abssuble0d 15464 | Absolute value of a nonpos... |
| absdifltd 15465 | The absolute value of a di... |
| absdifled 15466 | The absolute value of a di... |
| icodiamlt 15467 | Two elements in a half-ope... |
| abscld 15468 | Real closure of absolute v... |
| sqrtcld 15469 | Closure of the square root... |
| sqrtrege0d 15470 | The real part of the squar... |
| sqsqrtd 15471 | Square root theorem. Theo... |
| msqsqrtd 15472 | Square root theorem. Theo... |
| sqr00d 15473 | A square root is zero iff ... |
| absvalsqd 15474 | Square of value of absolut... |
| absvalsq2d 15475 | Square of value of absolut... |
| absge0d 15476 | Absolute value is nonnegat... |
| absval2d 15477 | Value of absolute value fu... |
| abs00d 15478 | The absolute value of a nu... |
| absne0d 15479 | The absolute value of a nu... |
| absrpcld 15480 | The absolute value of a no... |
| absnegd 15481 | Absolute value of negative... |
| abscjd 15482 | The absolute value of a nu... |
| releabsd 15483 | The real part of a number ... |
| absexpd 15484 | Absolute value of positive... |
| abssubd 15485 | Swapping order of subtract... |
| absmuld 15486 | Absolute value distributes... |
| absdivd 15487 | Absolute value distributes... |
| abstrid 15488 | Triangle inequality for ab... |
| abs2difd 15489 | Difference of absolute val... |
| abs2dif2d 15490 | Difference of absolute val... |
| abs2difabsd 15491 | Absolute value of differen... |
| abs3difd 15492 | Absolute value of differen... |
| abs3lemd 15493 | Lemma involving absolute v... |
| reusq0 15494 | A complex number is the sq... |
| bhmafibid1cn 15495 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15496 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15497 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15498 | The Brahmagupta-Fibonacci ... |
| limsupgord 15501 | Ordering property of the s... |
| limsupcl 15502 | Closure of the superior li... |
| limsupval 15503 | The superior limit of an i... |
| limsupgf 15504 | Closure of the superior li... |
| limsupgval 15505 | Value of the superior limi... |
| limsupgle 15506 | The defining property of t... |
| limsuple 15507 | The defining property of t... |
| limsuplt 15508 | The defining property of t... |
| limsupval2 15509 | The superior limit, relati... |
| limsupgre 15510 | If a sequence of real numb... |
| limsupbnd1 15511 | If a sequence is eventuall... |
| limsupbnd2 15512 | If a sequence is eventuall... |
| climrel 15521 | The limit relation is a re... |
| rlimrel 15522 | The limit relation is a re... |
| clim 15523 | Express the predicate: Th... |
| rlim 15524 | Express the predicate: Th... |
| rlim2 15525 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15526 | Use strictly less-than in ... |
| rlim3 15527 | Restrict the range of the ... |
| climcl 15528 | Closure of the limit of a ... |
| rlimpm 15529 | Closure of a function with... |
| rlimf 15530 | Closure of a function with... |
| rlimss 15531 | Domain closure of a functi... |
| rlimcl 15532 | Closure of the limit of a ... |
| clim2 15533 | Express the predicate: Th... |
| clim2c 15534 | Express the predicate ` F ... |
| clim0 15535 | Express the predicate ` F ... |
| clim0c 15536 | Express the predicate ` F ... |
| rlim0 15537 | Express the predicate ` B ... |
| rlim0lt 15538 | Use strictly less-than in ... |
| climi 15539 | Convergence of a sequence ... |
| climi2 15540 | Convergence of a sequence ... |
| climi0 15541 | Convergence of a sequence ... |
| rlimi 15542 | Convergence at infinity of... |
| rlimi2 15543 | Convergence at infinity of... |
| ello1 15544 | Elementhood in the set of ... |
| ello12 15545 | Elementhood in the set of ... |
| ello12r 15546 | Sufficient condition for e... |
| lo1f 15547 | An eventually upper bounde... |
| lo1dm 15548 | An eventually upper bounde... |
| lo1bdd 15549 | The defining property of a... |
| ello1mpt 15550 | Elementhood in the set of ... |
| ello1mpt2 15551 | Elementhood in the set of ... |
| ello1d 15552 | Sufficient condition for e... |
| lo1bdd2 15553 | If an eventually bounded f... |
| lo1bddrp 15554 | Refine ~ o1bdd2 to give a ... |
| elo1 15555 | Elementhood in the set of ... |
| elo12 15556 | Elementhood in the set of ... |
| elo12r 15557 | Sufficient condition for e... |
| o1f 15558 | An eventually bounded func... |
| o1dm 15559 | An eventually bounded func... |
| o1bdd 15560 | The defining property of a... |
| lo1o1 15561 | A function is eventually b... |
| lo1o12 15562 | A function is eventually b... |
| elo1mpt 15563 | Elementhood in the set of ... |
| elo1mpt2 15564 | Elementhood in the set of ... |
| elo1d 15565 | Sufficient condition for e... |
| o1lo1 15566 | A real function is eventua... |
| o1lo12 15567 | A lower bounded real funct... |
| o1lo1d 15568 | A real eventually bounded ... |
| icco1 15569 | Derive eventual boundednes... |
| o1bdd2 15570 | If an eventually bounded f... |
| o1bddrp 15571 | Refine ~ o1bdd2 to give a ... |
| climconst 15572 | An (eventually) constant s... |
| rlimconst 15573 | A constant sequence conver... |
| rlimclim1 15574 | Forward direction of ~ rli... |
| rlimclim 15575 | A sequence on an upper int... |
| climrlim2 15576 | Produce a real limit from ... |
| climconst2 15577 | A constant sequence conver... |
| climz 15578 | The zero sequence converge... |
| rlimuni 15579 | A real function whose doma... |
| rlimdm 15580 | Two ways to express that a... |
| climuni 15581 | An infinite sequence of co... |
| fclim 15582 | The limit relation is func... |
| climdm 15583 | Two ways to express that a... |
| climeu 15584 | An infinite sequence of co... |
| climreu 15585 | An infinite sequence of co... |
| climmo 15586 | An infinite sequence of co... |
| rlimres 15587 | The restriction of a funct... |
| lo1res 15588 | The restriction of an even... |
| o1res 15589 | The restriction of an even... |
| rlimres2 15590 | The restriction of a funct... |
| lo1res2 15591 | The restriction of a funct... |
| o1res2 15592 | The restriction of a funct... |
| lo1resb 15593 | The restriction of a funct... |
| rlimresb 15594 | The restriction of a funct... |
| o1resb 15595 | The restriction of a funct... |
| climeq 15596 | Two functions that are eve... |
| lo1eq 15597 | Two functions that are eve... |
| rlimeq 15598 | Two functions that are eve... |
| o1eq 15599 | Two functions that are eve... |
| climmpt 15600 | Exhibit a function ` G ` w... |
| 2clim 15601 | If two sequences converge ... |
| climmpt2 15602 | Relate an integer limit on... |
| climshftlem 15603 | A shifted function converg... |
| climres 15604 | A function restricted to u... |
| climshft 15605 | A shifted function converg... |
| serclim0 15606 | The zero series converges ... |
| rlimcld2 15607 | If ` D ` is a closed set i... |
| rlimrege0 15608 | The limit of a sequence of... |
| rlimrecl 15609 | The limit of a real sequen... |
| rlimge0 15610 | The limit of a sequence of... |
| climshft2 15611 | A shifted function converg... |
| climrecl 15612 | The limit of a convergent ... |
| climge0 15613 | A nonnegative sequence con... |
| climabs0 15614 | Convergence to zero of the... |
| o1co 15615 | Sufficient condition for t... |
| o1compt 15616 | Sufficient condition for t... |
| rlimcn1 15617 | Image of a limit under a c... |
| rlimcn1b 15618 | Image of a limit under a c... |
| rlimcn3 15619 | Image of a limit under a c... |
| rlimcn2 15620 | Image of a limit under a c... |
| climcn1 15621 | Image of a limit under a c... |
| climcn2 15622 | Image of a limit under a c... |
| addcn2 15623 | Complex number addition is... |
| subcn2 15624 | Complex number subtraction... |
| mulcn2 15625 | Complex number multiplicat... |
| reccn2 15626 | The reciprocal function is... |
| cn1lem 15627 | A sufficient condition for... |
| abscn2 15628 | The absolute value functio... |
| cjcn2 15629 | The complex conjugate func... |
| recn2 15630 | The real part function is ... |
| imcn2 15631 | The imaginary part functio... |
| climcn1lem 15632 | The limit of a continuous ... |
| climabs 15633 | Limit of the absolute valu... |
| climcj 15634 | Limit of the complex conju... |
| climre 15635 | Limit of the real part of ... |
| climim 15636 | Limit of the imaginary par... |
| rlimmptrcl 15637 | Reverse closure for a real... |
| rlimabs 15638 | Limit of the absolute valu... |
| rlimcj 15639 | Limit of the complex conju... |
| rlimre 15640 | Limit of the real part of ... |
| rlimim 15641 | Limit of the imaginary par... |
| o1of2 15642 | Show that a binary operati... |
| o1add 15643 | The sum of two eventually ... |
| o1mul 15644 | The product of two eventua... |
| o1sub 15645 | The difference of two even... |
| rlimo1 15646 | Any function with a finite... |
| rlimdmo1 15647 | A convergent function is e... |
| o1rlimmul 15648 | The product of an eventual... |
| o1const 15649 | A constant function is eve... |
| lo1const 15650 | A constant function is eve... |
| lo1mptrcl 15651 | Reverse closure for an eve... |
| o1mptrcl 15652 | Reverse closure for an eve... |
| o1add2 15653 | The sum of two eventually ... |
| o1mul2 15654 | The product of two eventua... |
| o1sub2 15655 | The product of two eventua... |
| lo1add 15656 | The sum of two eventually ... |
| lo1mul 15657 | The product of an eventual... |
| lo1mul2 15658 | The product of an eventual... |
| o1dif 15659 | If the difference of two f... |
| lo1sub 15660 | The difference of an event... |
| climadd 15661 | Limit of the sum of two co... |
| climmul 15662 | Limit of the product of tw... |
| climsub 15663 | Limit of the difference of... |
| climaddc1 15664 | Limit of a constant ` C ` ... |
| climaddc2 15665 | Limit of a constant ` C ` ... |
| climmulc2 15666 | Limit of a sequence multip... |
| climsubc1 15667 | Limit of a constant ` C ` ... |
| climsubc2 15668 | Limit of a constant ` C ` ... |
| climle 15669 | Comparison of the limits o... |
| climsqz 15670 | Convergence of a sequence ... |
| climsqz2 15671 | Convergence of a sequence ... |
| rlimadd 15672 | Limit of the sum of two co... |
| rlimsub 15673 | Limit of the difference of... |
| rlimmul 15674 | Limit of the product of tw... |
| rlimdiv 15675 | Limit of the quotient of t... |
| rlimneg 15676 | Limit of the negative of a... |
| rlimle 15677 | Comparison of the limits o... |
| rlimsqzlem 15678 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15679 | Convergence of a sequence ... |
| rlimsqz2 15680 | Convergence of a sequence ... |
| lo1le 15681 | Transfer eventual upper bo... |
| o1le 15682 | Transfer eventual boundedn... |
| rlimno1 15683 | A function whose inverse c... |
| clim2ser 15684 | The limit of an infinite s... |
| clim2ser2 15685 | The limit of an infinite s... |
| iserex 15686 | An infinite series converg... |
| isermulc2 15687 | Multiplication of an infin... |
| climlec2 15688 | Comparison of a constant t... |
| iserle 15689 | Comparison of the limits o... |
| iserge0 15690 | The limit of an infinite s... |
| climub 15691 | The limit of a monotonic s... |
| climserle 15692 | The partial sums of a conv... |
| isershft 15693 | Index shift of the limit o... |
| isercolllem1 15694 | Lemma for ~ isercoll . (C... |
| isercolllem2 15695 | Lemma for ~ isercoll . (C... |
| isercolllem3 15696 | Lemma for ~ isercoll . (C... |
| isercoll 15697 | Rearrange an infinite seri... |
| isercoll2 15698 | Generalize ~ isercoll so t... |
| climsup 15699 | A bounded monotonic sequen... |
| climcau 15700 | A converging sequence of c... |
| climbdd 15701 | A converging sequence of c... |
| caucvgrlem 15702 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15703 | A Cauchy sequence of real ... |
| caucvgrlem2 15704 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15705 | A Cauchy sequence of compl... |
| caurcvg 15706 | A Cauchy sequence of real ... |
| caurcvg2 15707 | A Cauchy sequence of real ... |
| caucvg 15708 | A Cauchy sequence of compl... |
| caucvgb 15709 | A function is convergent i... |
| serf0 15710 | If an infinite series conv... |
| iseraltlem1 15711 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15712 | Lemma for ~ iseralt . The... |
| iseraltlem3 15713 | Lemma for ~ iseralt . Fro... |
| iseralt 15714 | The alternating series tes... |
| sumex 15717 | A sum is a set. (Contribu... |
| sumeq1 15718 | Equality theorem for a sum... |
| nfsum1 15719 | Bound-variable hypothesis ... |
| nfsum 15720 | Bound-variable hypothesis ... |
| sumeq2w 15721 | Equality theorem for sum, ... |
| sumeq2ii 15722 | Equality theorem for sum, ... |
| sumeq2 15723 | Equality theorem for sum. ... |
| cbvsum 15724 | Change bound variable in a... |
| cbvsumv 15725 | Change bound variable in a... |
| sumeq1i 15726 | Equality inference for sum... |
| sumeq2i 15727 | Equality inference for sum... |
| sumeq12i 15728 | Equality inference for sum... |
| sumeq1d 15729 | Equality deduction for sum... |
| sumeq2d 15730 | Equality deduction for sum... |
| sumeq2dv 15731 | Equality deduction for sum... |
| sumeq2sdv 15732 | Equality deduction for sum... |
| sumeq2sdvOLD 15733 | Obsolete version of ~ sume... |
| 2sumeq2dv 15734 | Equality deduction for dou... |
| sumeq12dv 15735 | Equality deduction for sum... |
| sumeq12rdv 15736 | Equality deduction for sum... |
| sum2id 15737 | The second class argument ... |
| sumfc 15738 | A lemma to facilitate conv... |
| fz1f1o 15739 | A lemma for working with f... |
| sumrblem 15740 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15741 | The sequence of partial su... |
| sumrb 15742 | Rebase the starting point ... |
| summolem3 15743 | Lemma for ~ summo . (Cont... |
| summolem2a 15744 | Lemma for ~ summo . (Cont... |
| summolem2 15745 | Lemma for ~ summo . (Cont... |
| summo 15746 | A sum has at most one limi... |
| zsum 15747 | Series sum with index set ... |
| isum 15748 | Series sum with an upper i... |
| fsum 15749 | The value of a sum over a ... |
| sum0 15750 | Any sum over the empty set... |
| sumz 15751 | Any sum of zero over a sum... |
| fsumf1o 15752 | Re-index a finite sum usin... |
| sumss 15753 | Change the index set to a ... |
| fsumss 15754 | Change the index set to a ... |
| sumss2 15755 | Change the index set of a ... |
| fsumcvg2 15756 | The sequence of partial su... |
| fsumsers 15757 | Special case of series sum... |
| fsumcvg3 15758 | A finite sum is convergent... |
| fsumser 15759 | A finite sum expressed in ... |
| fsumcl2lem 15760 | - Lemma for finite sum clo... |
| fsumcllem 15761 | - Lemma for finite sum clo... |
| fsumcl 15762 | Closure of a finite sum of... |
| fsumrecl 15763 | Closure of a finite sum of... |
| fsumzcl 15764 | Closure of a finite sum of... |
| fsumnn0cl 15765 | Closure of a finite sum of... |
| fsumrpcl 15766 | Closure of a finite sum of... |
| fsumclf 15767 | Closure of a finite sum of... |
| fsumzcl2 15768 | A finite sum with integer ... |
| fsumadd 15769 | The sum of two finite sums... |
| fsumsplit 15770 | Split a sum into two parts... |
| fsumsplitf 15771 | Split a sum into two parts... |
| sumsnf 15772 | A sum of a singleton is th... |
| fsumsplitsn 15773 | Separate out a term in a f... |
| fsumsplit1 15774 | Separate out a term in a f... |
| sumsn 15775 | A sum of a singleton is th... |
| fsum1 15776 | The finite sum of ` A ( k ... |
| sumpr 15777 | A sum over a pair is the s... |
| sumtp 15778 | A sum over a triple is the... |
| sumsns 15779 | A sum of a singleton is th... |
| fsumm1 15780 | Separate out the last term... |
| fzosump1 15781 | Separate out the last term... |
| fsum1p 15782 | Separate out the first ter... |
| fsummsnunz 15783 | A finite sum all of whose ... |
| fsumsplitsnun 15784 | Separate out a term in a f... |
| fsump1 15785 | The addition of the next t... |
| isumclim 15786 | An infinite sum equals the... |
| isumclim2 15787 | A converging series conver... |
| isumclim3 15788 | The sequence of partial fi... |
| sumnul 15789 | The sum of a non-convergen... |
| isumcl 15790 | The sum of a converging in... |
| isummulc2 15791 | An infinite sum multiplied... |
| isummulc1 15792 | An infinite sum multiplied... |
| isumdivc 15793 | An infinite sum divided by... |
| isumrecl 15794 | The sum of a converging in... |
| isumge0 15795 | An infinite sum of nonnega... |
| isumadd 15796 | Addition of infinite sums.... |
| sumsplit 15797 | Split a sum into two parts... |
| fsump1i 15798 | Optimized version of ~ fsu... |
| fsum2dlem 15799 | Lemma for ~ fsum2d - induc... |
| fsum2d 15800 | Write a double sum as a su... |
| fsumxp 15801 | Combine two sums into a si... |
| fsumcnv 15802 | Transform a region of summ... |
| fsumcom2 15803 | Interchange order of summa... |
| fsumcom 15804 | Interchange order of summa... |
| fsum0diaglem 15805 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15806 | Two ways to express "the s... |
| mptfzshft 15807 | 1-1 onto function in maps-... |
| fsumrev 15808 | Reversal of a finite sum. ... |
| fsumshft 15809 | Index shift of a finite su... |
| fsumshftm 15810 | Negative index shift of a ... |
| fsumrev2 15811 | Reversal of a finite sum. ... |
| fsum0diag2 15812 | Two ways to express "the s... |
| fsummulc2 15813 | A finite sum multiplied by... |
| fsummulc1 15814 | A finite sum multiplied by... |
| fsumdivc 15815 | A finite sum divided by a ... |
| fsumneg 15816 | Negation of a finite sum. ... |
| fsumsub 15817 | Split a finite sum over a ... |
| fsum2mul 15818 | Separate the nested sum of... |
| fsumconst 15819 | The sum of constant terms ... |
| fsumconst1 15820 | The sum of 1 over a finite... |
| fsumdifsnconst 15821 | The sum of constant terms ... |
| modfsummodslem1 15822 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15823 | Induction step for ~ modfs... |
| modfsummod 15824 | A finite sum modulo a posi... |
| fsumge0 15825 | If all of the terms of a f... |
| fsumless 15826 | A shorter sum of nonnegati... |
| fsumge1 15827 | A sum of nonnegative numbe... |
| fsum00 15828 | A sum of nonnegative numbe... |
| fsumle 15829 | If all of the terms of fin... |
| fsumlt 15830 | If every term in one finit... |
| fsumabs 15831 | Generalized triangle inequ... |
| telfsumo 15832 | Sum of a telescoping serie... |
| telfsumo2 15833 | Sum of a telescoping serie... |
| telfsum 15834 | Sum of a telescoping serie... |
| telfsum2 15835 | Sum of a telescoping serie... |
| fsumparts 15836 | Summation by parts. (Cont... |
| fsumrelem 15837 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15838 | The real part of a sum. (... |
| fsumim 15839 | The imaginary part of a su... |
| fsumcj 15840 | The complex conjugate of a... |
| fsumrlim 15841 | Limit of a finite sum of c... |
| fsumo1 15842 | The finite sum of eventual... |
| o1fsum 15843 | If ` A ( k ) ` is O(1), th... |
| seqabs 15844 | Generalized triangle inequ... |
| iserabs 15845 | Generalized triangle inequ... |
| cvgcmp 15846 | A comparison test for conv... |
| cvgcmpub 15847 | An upper bound for the lim... |
| cvgcmpce 15848 | A comparison test for conv... |
| abscvgcvg 15849 | An absolutely convergent s... |
| climfsum 15850 | Limit of a finite sum of c... |
| fsumiun 15851 | Sum over a disjoint indexe... |
| hashiun 15852 | The cardinality of a disjo... |
| hash2iun 15853 | The cardinality of a neste... |
| hash2iun1dif1 15854 | The cardinality of a neste... |
| hashrabrex 15855 | The number of elements in ... |
| hashuni 15856 | The cardinality of a disjo... |
| qshash 15857 | The cardinality of a set w... |
| indsum 15858 | Finite sum of a product wi... |
| indsumhash 15859 | The finite sum of the indi... |
| ackbijnn 15860 | Translate the Ackermann bi... |
| binomlem 15861 | Lemma for ~ binom (binomia... |
| binom 15862 | The binomial theorem: ` ( ... |
| binom1p 15863 | Special case of the binomi... |
| binom11 15864 | Special case of the binomi... |
| binom1dif 15865 | A summation for the differ... |
| bcxmaslem1 15866 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15867 | Parallel summation (Christ... |
| incexclem 15868 | Lemma for ~ incexc . (Con... |
| incexc 15869 | The inclusion/exclusion pr... |
| incexc2 15870 | The inclusion/exclusion pr... |
| isumshft 15871 | Index shift of an infinite... |
| isumsplit 15872 | Split off the first ` N ` ... |
| isum1p 15873 | The infinite sum of a conv... |
| isumnn0nn 15874 | Sum from 0 to infinity in ... |
| isumrpcl 15875 | The infinite sum of positi... |
| isumle 15876 | Comparison of two infinite... |
| isumless 15877 | A finite sum of nonnegativ... |
| isumsup2 15878 | An infinite sum of nonnega... |
| isumsup 15879 | An infinite sum of nonnega... |
| isumltss 15880 | A partial sum of a series ... |
| climcndslem1 15881 | Lemma for ~ climcnds : bou... |
| climcndslem2 15882 | Lemma for ~ climcnds : bou... |
| climcnds 15883 | The Cauchy condensation te... |
| divrcnv 15884 | The sequence of reciprocal... |
| divcnv 15885 | The sequence of reciprocal... |
| flo1 15886 | The floor function satisfi... |
| divcnvshft 15887 | Limit of a ratio function.... |
| supcvg 15888 | Extract a sequence ` f ` i... |
| infcvgaux1i 15889 | Auxiliary theorem for appl... |
| infcvgaux2i 15890 | Auxiliary theorem for appl... |
| harmonic 15891 | The harmonic series ` H ` ... |
| arisum 15892 | Arithmetic series sum of t... |
| arisum2 15893 | Arithmetic series sum of t... |
| trireciplem 15894 | Lemma for ~ trirecip . Sh... |
| trirecip 15895 | The sum of the reciprocals... |
| expcnv 15896 | A sequence of powers of a ... |
| explecnv 15897 | A sequence of terms conver... |
| geoserg 15898 | The value of the finite ge... |
| geoser 15899 | The value of the finite ge... |
| pwdif 15900 | The difference of two numb... |
| pwm1geoser 15901 | The n-th power of a number... |
| geolim 15902 | The partial sums in the in... |
| geolim2 15903 | The partial sums in the ge... |
| georeclim 15904 | The limit of a geometric s... |
| geo2sum 15905 | The value of the finite ge... |
| geo2sum2 15906 | The value of the finite ge... |
| geo2lim 15907 | The value of the infinite ... |
| geomulcvg 15908 | The geometric series conve... |
| geoisum 15909 | The infinite sum of ` 1 + ... |
| geoisumr 15910 | The infinite sum of recipr... |
| geoisum1 15911 | The infinite sum of ` A ^ ... |
| geoisum1c 15912 | The infinite sum of ` A x.... |
| 0.999... 15913 | The recurring decimal 0.99... |
| geoihalfsum 15914 | Prove that the infinite ge... |
| cvgrat 15915 | Ratio test for convergence... |
| mertenslem1 15916 | Lemma for ~ mertens . (Co... |
| mertenslem2 15917 | Lemma for ~ mertens . (Co... |
| mertens 15918 | Mertens' theorem. If ` A ... |
| prodf 15919 | An infinite product of com... |
| clim2prod 15920 | The limit of an infinite p... |
| clim2div 15921 | The limit of an infinite p... |
| prodfmul 15922 | The product of two infinit... |
| prodf1 15923 | The value of the partial p... |
| prodf1f 15924 | A one-valued infinite prod... |
| prodfclim1 15925 | The constant one product c... |
| prodfn0 15926 | No term of a nonzero infin... |
| prodfrec 15927 | The reciprocal of an infin... |
| prodfdiv 15928 | The quotient of two infini... |
| ntrivcvg 15929 | A non-trivially converging... |
| ntrivcvgn0 15930 | A product that converges t... |
| ntrivcvgfvn0 15931 | Any value of a product seq... |
| ntrivcvgtail 15932 | A tail of a non-trivially ... |
| ntrivcvgmullem 15933 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15934 | The product of two non-tri... |
| prodex 15937 | A product is a set. (Cont... |
| prodeq1f 15938 | Equality theorem for a pro... |
| prodeq1 15939 | Equality theorem for a pro... |
| nfcprod1 15940 | Bound-variable hypothesis ... |
| nfcprod 15941 | Bound-variable hypothesis ... |
| prodeq2w 15942 | Equality theorem for produ... |
| prodeq2ii 15943 | Equality theorem for produ... |
| prodeq2 15944 | Equality theorem for produ... |
| cbvprod 15945 | Change bound variable in a... |
| cbvprodv 15946 | Change bound variable in a... |
| cbvprodi 15947 | Change bound variable in a... |
| prodeq1i 15948 | Equality inference for pro... |
| prodeq1iOLD 15949 | Obsolete version of ~ prod... |
| prodeq2i 15950 | Equality inference for pro... |
| prodeq12i 15951 | Equality inference for pro... |
| prodeq1d 15952 | Equality deduction for pro... |
| prodeq2d 15953 | Equality deduction for pro... |
| prodeq2dv 15954 | Equality deduction for pro... |
| prodeq2sdv 15955 | Equality deduction for pro... |
| prodeq2sdvOLD 15956 | Obsolete version of ~ prod... |
| 2cprodeq2dv 15957 | Equality deduction for dou... |
| prodeq12dv 15958 | Equality deduction for pro... |
| prodeq12rdv 15959 | Equality deduction for pro... |
| prod2id 15960 | The second class argument ... |
| prodrblem 15961 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15962 | The sequence of partial pr... |
| prodrblem2 15963 | Lemma for ~ prodrb . (Con... |
| prodrb 15964 | Rebase the starting point ... |
| prodmolem3 15965 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15966 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15967 | Lemma for ~ prodmo . (Con... |
| prodmo 15968 | A product has at most one ... |
| zprod 15969 | Series product with index ... |
| iprod 15970 | Series product with an upp... |
| zprodn0 15971 | Nonzero series product wit... |
| iprodn0 15972 | Nonzero series product wit... |
| fprod 15973 | The value of a product ove... |
| fprodntriv 15974 | A non-triviality lemma for... |
| prod0 15975 | A product over the empty s... |
| prod1 15976 | Any product of one over a ... |
| prodfc 15977 | A lemma to facilitate conv... |
| fprodf1o 15978 | Re-index a finite product ... |
| prodss 15979 | Change the index set to a ... |
| fprodss 15980 | Change the index set to a ... |
| fprodser 15981 | A finite product expressed... |
| fprodcl2lem 15982 | Finite product closure lem... |
| fprodcllem 15983 | Finite product closure lem... |
| fprodcl 15984 | Closure of a finite produc... |
| fprodrecl 15985 | Closure of a finite produc... |
| fprodzcl 15986 | Closure of a finite produc... |
| fprodnncl 15987 | Closure of a finite produc... |
| fprodrpcl 15988 | Closure of a finite produc... |
| fprodnn0cl 15989 | Closure of a finite produc... |
| fprodcllemf 15990 | Finite product closure lem... |
| fprodreclf 15991 | Closure of a finite produc... |
| fprodmul 15992 | The product of two finite ... |
| fproddiv 15993 | The quotient of two finite... |
| prodsn 15994 | A product of a singleton i... |
| fprod1 15995 | A finite product of only o... |
| prodsnf 15996 | A product of a singleton i... |
| climprod1 15997 | The limit of a product ove... |
| fprodsplit 15998 | Split a finite product int... |
| fprodm1 15999 | Separate out the last term... |
| fprod1p 16000 | Separate out the first ter... |
| fprodp1 16001 | Multiply in the last term ... |
| fprodm1s 16002 | Separate out the last term... |
| fprodp1s 16003 | Multiply in the last term ... |
| prodsns 16004 | A product of the singleton... |
| fprodfac 16005 | Factorial using product no... |
| fprodabs 16006 | The absolute value of a fi... |
| fprodeq0 16007 | Any finite product contain... |
| fprodshft 16008 | Shift the index of a finit... |
| fprodrev 16009 | Reversal of a finite produ... |
| fprodconst 16010 | The product of constant te... |
| fprodn0 16011 | A finite product of nonzer... |
| fprod2dlem 16012 | Lemma for ~ fprod2d - indu... |
| fprod2d 16013 | Write a double product as ... |
| fprodxp 16014 | Combine two products into ... |
| fprodcnv 16015 | Transform a product region... |
| fprodcom2 16016 | Interchange order of multi... |
| fprodcom 16017 | Interchange product order.... |
| fprod0diag 16018 | Two ways to express "the p... |
| fproddivf 16019 | The quotient of two finite... |
| fprodsplitf 16020 | Split a finite product int... |
| fprodsplitsn 16021 | Separate out a term in a f... |
| fprodsplit1f 16022 | Separate out a term in a f... |
| fprodn0f 16023 | A finite product of nonzer... |
| fprodclf 16024 | Closure of a finite produc... |
| fprodge0 16025 | If all the terms of a fini... |
| fprodeq0g 16026 | Any finite product contain... |
| fprodge1 16027 | If all of the terms of a f... |
| fprodle 16028 | If all the terms of two fi... |
| fprodmodd 16029 | If all factors of two fini... |
| iprodclim 16030 | An infinite product equals... |
| iprodclim2 16031 | A converging product conve... |
| iprodclim3 16032 | The sequence of partial fi... |
| iprodcl 16033 | The product of a non-trivi... |
| iprodrecl 16034 | The product of a non-trivi... |
| iprodmul 16035 | Multiplication of infinite... |
| risefacval 16040 | The value of the rising fa... |
| fallfacval 16041 | The value of the falling f... |
| risefacval2 16042 | One-based value of rising ... |
| fallfacval2 16043 | One-based value of falling... |
| fallfacval3 16044 | A product representation o... |
| risefaccllem 16045 | Lemma for rising factorial... |
| fallfaccllem 16046 | Lemma for falling factoria... |
| risefaccl 16047 | Closure law for rising fac... |
| fallfaccl 16048 | Closure law for falling fa... |
| rerisefaccl 16049 | Closure law for rising fac... |
| refallfaccl 16050 | Closure law for falling fa... |
| nnrisefaccl 16051 | Closure law for rising fac... |
| zrisefaccl 16052 | Closure law for rising fac... |
| zfallfaccl 16053 | Closure law for falling fa... |
| nn0risefaccl 16054 | Closure law for rising fac... |
| rprisefaccl 16055 | Closure law for rising fac... |
| risefallfac 16056 | A relationship between ris... |
| fallrisefac 16057 | A relationship between fal... |
| risefall0lem 16058 | Lemma for ~ risefac0 and ~... |
| risefac0 16059 | The value of the rising fa... |
| fallfac0 16060 | The value of the falling f... |
| risefacp1 16061 | The value of the rising fa... |
| fallfacp1 16062 | The value of the falling f... |
| risefacp1d 16063 | The value of the rising fa... |
| fallfacp1d 16064 | The value of the falling f... |
| risefac1 16065 | The value of rising factor... |
| fallfac1 16066 | The value of falling facto... |
| risefacfac 16067 | Relate rising factorial to... |
| fallfacfwd 16068 | The forward difference of ... |
| 0fallfac 16069 | The value of the zero fall... |
| 0risefac 16070 | The value of the zero risi... |
| binomfallfaclem1 16071 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 16072 | Lemma for ~ binomfallfac .... |
| binomfallfac 16073 | A version of the binomial ... |
| binomrisefac 16074 | A version of the binomial ... |
| fallfacval4 16075 | Represent the falling fact... |
| bcfallfac 16076 | Binomial coefficient in te... |
| fallfacfac 16077 | Relate falling factorial t... |
| bpolylem 16080 | Lemma for ~ bpolyval . (C... |
| bpolyval 16081 | The value of the Bernoulli... |
| bpoly0 16082 | The value of the Bernoulli... |
| bpoly1 16083 | The value of the Bernoulli... |
| bpolycl 16084 | Closure law for Bernoulli ... |
| bpolysum 16085 | A sum for Bernoulli polyno... |
| bpolydiflem 16086 | Lemma for ~ bpolydif . (C... |
| bpolydif 16087 | Calculate the difference b... |
| fsumkthpow 16088 | A closed-form expression f... |
| bpoly2 16089 | The Bernoulli polynomials ... |
| bpoly3 16090 | The Bernoulli polynomials ... |
| bpoly4 16091 | The Bernoulli polynomials ... |
| fsumcube 16092 | Express the sum of cubes i... |
| eftcl 16105 | Closure of a term in the s... |
| reeftcl 16106 | The terms of the series ex... |
| eftabs 16107 | The absolute value of a te... |
| eftval 16108 | The value of a term in the... |
| efcllem 16109 | Lemma for ~ efcl . The se... |
| ef0lem 16110 | The series defining the ex... |
| efval 16111 | Value of the exponential f... |
| esum 16112 | Value of Euler's constant ... |
| eff 16113 | Domain and codomain of the... |
| efcl 16114 | Closure law for the expone... |
| efcld 16115 | Closure law for the expone... |
| efval2 16116 | Value of the exponential f... |
| efcvg 16117 | The series that defines th... |
| efcvgfsum 16118 | Exponential function conve... |
| reefcl 16119 | The exponential function i... |
| reefcld 16120 | The exponential function i... |
| ere 16121 | Euler's constant ` _e ` = ... |
| ege2le3 16122 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16123 | Value of the exponential f... |
| efcj 16124 | The exponential of a compl... |
| efaddlem 16125 | Lemma for ~ efadd (exponen... |
| efadd 16126 | Sum of exponents law for e... |
| fprodefsum 16127 | Move the exponential funct... |
| efcan 16128 | Cancellation law for expon... |
| efne0d 16129 | The exponential of a compl... |
| efne0 16130 | The exponential of a compl... |
| efne0OLD 16131 | Obsolete version of ~ efne... |
| efneg 16132 | The exponential of the opp... |
| eff2 16133 | The exponential function m... |
| efsub 16134 | Difference of exponents la... |
| efexp 16135 | The exponential of an inte... |
| efzval 16136 | Value of the exponential f... |
| efgt0 16137 | The exponential of a real ... |
| rpefcl 16138 | The exponential of a real ... |
| rpefcld 16139 | The exponential of a real ... |
| eftlcvg 16140 | The tail series of the exp... |
| eftlcl 16141 | Closure of the sum of an i... |
| reeftlcl 16142 | Closure of the sum of an i... |
| eftlub 16143 | An upper bound on the abso... |
| efsep 16144 | Separate out the next term... |
| effsumlt 16145 | The partial sums of the se... |
| eft0val 16146 | The value of the first ter... |
| ef4p 16147 | Separate out the first fou... |
| efgt1p2 16148 | The exponential of a posit... |
| efgt1p 16149 | The exponential of a posit... |
| efgt1 16150 | The exponential of a posit... |
| eflt 16151 | The exponential function o... |
| efle 16152 | The exponential function o... |
| reef11 16153 | The exponential function o... |
| reeff1 16154 | The exponential function m... |
| eflegeo 16155 | The exponential function o... |
| sinval 16156 | Value of the sine function... |
| cosval 16157 | Value of the cosine functi... |
| sinf 16158 | Domain and codomain of the... |
| cosf 16159 | Domain and codomain of the... |
| sincl 16160 | Closure of the sine functi... |
| coscl 16161 | Closure of the cosine func... |
| tanval 16162 | Value of the tangent funct... |
| tancl 16163 | The closure of the tangent... |
| sincld 16164 | Closure of the sine functi... |
| coscld 16165 | Closure of the cosine func... |
| tancld 16166 | Closure of the tangent fun... |
| tanval2 16167 | Express the tangent functi... |
| tanval3 16168 | Express the tangent functi... |
| resinval 16169 | The sine of a real number ... |
| recosval 16170 | The cosine of a real numbe... |
| efi4p 16171 | Separate out the first fou... |
| resin4p 16172 | Separate out the first fou... |
| recos4p 16173 | Separate out the first fou... |
| resincl 16174 | The sine of a real number ... |
| recoscl 16175 | The cosine of a real numbe... |
| retancl 16176 | The closure of the tangent... |
| resincld 16177 | Closure of the sine functi... |
| recoscld 16178 | Closure of the cosine func... |
| retancld 16179 | Closure of the tangent fun... |
| sinneg 16180 | The sine of a negative is ... |
| cosneg 16181 | The cosines of a number an... |
| tanneg 16182 | The tangent of a negative ... |
| sin0 16183 | Value of the sine function... |
| cos0 16184 | Value of the cosine functi... |
| tan0 16185 | The value of the tangent f... |
| efival 16186 | The exponential function i... |
| efmival 16187 | The exponential function i... |
| sinhval 16188 | Value of the hyperbolic si... |
| coshval 16189 | Value of the hyperbolic co... |
| resinhcl 16190 | The hyperbolic sine of a r... |
| rpcoshcl 16191 | The hyperbolic cosine of a... |
| recoshcl 16192 | The hyperbolic cosine of a... |
| retanhcl 16193 | The hyperbolic tangent of ... |
| tanhlt1 16194 | The hyperbolic tangent of ... |
| tanhbnd 16195 | The hyperbolic tangent of ... |
| efeul 16196 | Eulerian representation of... |
| efieq 16197 | The exponentials of two im... |
| sinadd 16198 | Addition formula for sine.... |
| cosadd 16199 | Addition formula for cosin... |
| tanaddlem 16200 | A useful intermediate step... |
| tanadd 16201 | Addition formula for tange... |
| sinsub 16202 | Sine of difference. (Cont... |
| cossub 16203 | Cosine of difference. (Co... |
| addsin 16204 | Sum of sines. (Contribute... |
| subsin 16205 | Difference of sines. (Con... |
| sinmul 16206 | Product of sines can be re... |
| cosmul 16207 | Product of cosines can be ... |
| addcos 16208 | Sum of cosines. (Contribu... |
| subcos 16209 | Difference of cosines. (C... |
| sincossq 16210 | Sine squared plus cosine s... |
| sin2t 16211 | Double-angle formula for s... |
| cos2t 16212 | Double-angle formula for c... |
| cos2tsin 16213 | Double-angle formula for c... |
| sinbnd 16214 | The sine of a real number ... |
| cosbnd 16215 | The cosine of a real numbe... |
| sinbnd2 16216 | The sine of a real number ... |
| cosbnd2 16217 | The cosine of a real numbe... |
| ef01bndlem 16218 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16219 | Bounds on the sine of a po... |
| cos01bnd 16220 | Bounds on the cosine of a ... |
| cos1bnd 16221 | Bounds on the cosine of 1.... |
| cos2bnd 16222 | Bounds on the cosine of 2.... |
| sinltx 16223 | The sine of a positive rea... |
| sin01gt0 16224 | The sine of a positive rea... |
| cos01gt0 16225 | The cosine of a positive r... |
| sin02gt0 16226 | The sine of a positive rea... |
| sincos1sgn 16227 | The signs of the sine and ... |
| sincos2sgn 16228 | The signs of the sine and ... |
| sin4lt0 16229 | The sine of 4 is negative.... |
| absefi 16230 | The absolute value of the ... |
| absef 16231 | The absolute value of the ... |
| absefib 16232 | A complex number is real i... |
| efieq1re 16233 | A number whose imaginary e... |
| demoivre 16234 | De Moivre's Formula. Proo... |
| demoivreALT 16235 | Alternate proof of ~ demoi... |
| eirrlem 16238 | Lemma for ~ eirr . (Contr... |
| eirr 16239 | ` _e ` is irrational. (Co... |
| egt2lt3 16240 | Euler's constant ` _e ` = ... |
| epos 16241 | Euler's constant ` _e ` is... |
| epr 16242 | Euler's constant ` _e ` is... |
| ene0 16243 | ` _e ` is not 0. (Contrib... |
| ene1 16244 | ` _e ` is not 1. (Contrib... |
| xpnnen 16245 | The Cartesian product of t... |
| znnen 16246 | The set of integers and th... |
| qnnen 16247 | The rational numbers are c... |
| rpnnen2lem1 16248 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16249 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16250 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16251 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16252 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16253 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16254 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16255 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16256 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16257 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16258 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16259 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16260 | The other half of ~ rpnnen... |
| rpnnen 16261 | The cardinality of the con... |
| rexpen 16262 | The real numbers are equin... |
| cpnnen 16263 | The complex numbers are eq... |
| rucALT 16264 | Alternate proof of ~ ruc .... |
| ruclem1 16265 | Lemma for ~ ruc (the reals... |
| ruclem2 16266 | Lemma for ~ ruc . Orderin... |
| ruclem3 16267 | Lemma for ~ ruc . The con... |
| ruclem4 16268 | Lemma for ~ ruc . Initial... |
| ruclem6 16269 | Lemma for ~ ruc . Domain ... |
| ruclem7 16270 | Lemma for ~ ruc . Success... |
| ruclem8 16271 | Lemma for ~ ruc . The int... |
| ruclem9 16272 | Lemma for ~ ruc . The fir... |
| ruclem10 16273 | Lemma for ~ ruc . Every f... |
| ruclem11 16274 | Lemma for ~ ruc . Closure... |
| ruclem12 16275 | Lemma for ~ ruc . The sup... |
| ruclem13 16276 | Lemma for ~ ruc . There i... |
| ruc 16277 | The set of positive intege... |
| resdomq 16278 | The set of rationals is st... |
| aleph1re 16279 | There are at least aleph-o... |
| aleph1irr 16280 | There are at least aleph-o... |
| cnso 16281 | The complex numbers can be... |
| sqrt2irrlem 16282 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16283 | The square root of 2 is ir... |
| sqrt2re 16284 | The square root of 2 exist... |
| sqrt2irr0 16285 | The square root of 2 is an... |
| nthruc 16286 | The sequence ` NN ` , ` ZZ... |
| nthruz 16287 | The sequence ` NN ` , ` NN... |
| divides 16290 | Define the divides relatio... |
| dvdsval2 16291 | One nonzero integer divide... |
| dvdsval3 16292 | One nonzero integer divide... |
| dvdszrcl 16293 | Reverse closure for the di... |
| dvdsmod0 16294 | If a positive integer divi... |
| p1modz1 16295 | If a number greater than 1... |
| dvdsmodexp 16296 | If a positive integer divi... |
| nndivdvds 16297 | Strong form of ~ dvdsval2 ... |
| nndivides 16298 | Definition of the divides ... |
| moddvds 16299 | Two ways to say ` A == B `... |
| modm1div 16300 | An integer greater than on... |
| addmulmodb 16301 | An integer plus a product ... |
| dvds0lem 16302 | A lemma to assist theorems... |
| dvds1lem 16303 | A lemma to assist theorems... |
| dvds2lem 16304 | A lemma to assist theorems... |
| iddvds 16305 | An integer divides itself.... |
| 1dvds 16306 | 1 divides any integer. Th... |
| dvds0 16307 | Any integer divides 0. Th... |
| negdvdsb 16308 | An integer divides another... |
| dvdsnegb 16309 | An integer divides another... |
| absdvdsb 16310 | An integer divides another... |
| dvdsabsb 16311 | An integer divides another... |
| 0dvds 16312 | Only 0 is divisible by 0. ... |
| dvdsmul1 16313 | An integer divides a multi... |
| dvdsmul2 16314 | An integer divides a multi... |
| iddvdsexp 16315 | An integer divides a posit... |
| muldvds1 16316 | If a product divides an in... |
| muldvds2 16317 | If a product divides an in... |
| dvdscmul 16318 | Multiplication by a consta... |
| dvdsmulc 16319 | Multiplication by a consta... |
| dvdscmulr 16320 | Cancellation law for the d... |
| dvdsmulcr 16321 | Cancellation law for the d... |
| summodnegmod 16322 | The sum of two integers mo... |
| difmod0 16323 | The difference of two inte... |
| modmulconst 16324 | Constant multiplication in... |
| dvds2ln 16325 | If an integer divides each... |
| dvds2add 16326 | If an integer divides each... |
| dvds2sub 16327 | If an integer divides each... |
| dvds2addd 16328 | Deduction form of ~ dvds2a... |
| dvds2subd 16329 | Deduction form of ~ dvds2s... |
| dvdstr 16330 | The divides relation is tr... |
| dvdstrd 16331 | The divides relation is tr... |
| dvdsmultr1 16332 | If an integer divides anot... |
| dvdsmultr1d 16333 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16334 | If an integer divides anot... |
| dvdsmultr2d 16335 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16336 | If an integer divides eith... |
| dvdssub2 16337 | If an integer divides a di... |
| dvdsadd 16338 | An integer divides another... |
| dvdsaddr 16339 | An integer divides another... |
| dvdssub 16340 | An integer divides another... |
| dvdssubr 16341 | An integer divides another... |
| dvdsadd2b 16342 | Adding a multiple of the b... |
| dvdsaddre2b 16343 | Adding a multiple of the b... |
| fsumdvds 16344 | If every term in a sum is ... |
| dvdslelem 16345 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16346 | The divisors of a positive... |
| dvdsleabs 16347 | The divisors of a nonzero ... |
| dvdsleabs2 16348 | Transfer divisibility to a... |
| dvdsabseq 16349 | If two integers divide eac... |
| dvdseq 16350 | If two nonnegative integer... |
| divconjdvds 16351 | If a nonzero integer ` M `... |
| dvdsdivcl 16352 | The complement of a diviso... |
| dvdsflip 16353 | An involution of the divis... |
| dvdsssfz1 16354 | The set of divisors of a n... |
| dvds1 16355 | The only nonnegative integ... |
| alzdvds 16356 | Only 0 is divisible by all... |
| dvdsext 16357 | Poset extensionality for d... |
| fzm1ndvds 16358 | No number between ` 1 ` an... |
| fzo0dvdseq 16359 | Zero is the only one of th... |
| fzocongeq 16360 | Two different elements of ... |
| addmodlteqALT 16361 | Two nonnegative integers l... |
| dvdsfac 16362 | A positive integer divides... |
| dvdsexp2im 16363 | If an integer divides anot... |
| dvdsexp 16364 | A power divides a power wi... |
| dvdsmod 16365 | Any number ` K ` whose mod... |
| mulmoddvds 16366 | If an integer is divisible... |
| 3dvds 16367 | A rule for divisibility by... |
| 3dvdsdec 16368 | A decimal number is divisi... |
| 3dvds2dec 16369 | A decimal number is divisi... |
| fprodfvdvdsd 16370 | A finite product of intege... |
| fproddvdsd 16371 | A finite product of intege... |
| evenelz 16372 | An even number is an integ... |
| zeo3 16373 | An integer is even or odd.... |
| zeo4 16374 | An integer is even or odd ... |
| zeneo 16375 | No even integer equals an ... |
| odd2np1lem 16376 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16377 | An integer is odd iff it i... |
| even2n 16378 | An integer is even iff it ... |
| oddm1even 16379 | An integer is odd iff its ... |
| oddp1even 16380 | An integer is odd iff its ... |
| oexpneg 16381 | The exponential of the neg... |
| mod2eq0even 16382 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16383 | An integer is 1 modulo 2 i... |
| oddnn02np1 16384 | A nonnegative integer is o... |
| oddge22np1 16385 | An integer greater than on... |
| evennn02n 16386 | A nonnegative integer is e... |
| evennn2n 16387 | A positive integer is even... |
| 2tp1odd 16388 | A number which is twice an... |
| mulsucdiv2z 16389 | An integer multiplied with... |
| sqoddm1div8z 16390 | A squared odd number minus... |
| 2teven 16391 | A number which is twice an... |
| zeo5 16392 | An integer is either even ... |
| evend2 16393 | An integer is even iff its... |
| oddp1d2 16394 | An integer is odd iff its ... |
| zob 16395 | Alternate characterization... |
| oddm1d2 16396 | An integer is odd iff its ... |
| ltoddhalfle 16397 | An integer is less than ha... |
| halfleoddlt 16398 | An integer is greater than... |
| opoe 16399 | The sum of two odds is eve... |
| omoe 16400 | The difference of two odds... |
| opeo 16401 | The sum of an odd and an e... |
| omeo 16402 | The difference of an odd a... |
| z0even 16403 | 2 divides 0. That means 0... |
| n2dvds1 16404 | 2 does not divide 1. That... |
| n2dvdsm1 16405 | 2 does not divide -1. Tha... |
| z2even 16406 | 2 divides 2. That means 2... |
| n2dvds3 16407 | 2 does not divide 3. That... |
| z4even 16408 | 2 divides 4. That means 4... |
| 4dvdseven 16409 | An integer which is divisi... |
| m1expe 16410 | Exponentiation of -1 by an... |
| m1expo 16411 | Exponentiation of -1 by an... |
| m1exp1 16412 | Exponentiation of negative... |
| nn0enne 16413 | A positive integer is an e... |
| nn0ehalf 16414 | The half of an even nonneg... |
| nnehalf 16415 | The half of an even positi... |
| nn0onn 16416 | An odd nonnegative integer... |
| nn0o1gt2 16417 | An odd nonnegative integer... |
| nno 16418 | An alternate characterizat... |
| nn0o 16419 | An alternate characterizat... |
| nn0ob 16420 | Alternate characterization... |
| nn0oddm1d2 16421 | A positive integer is odd ... |
| nnoddm1d2 16422 | A positive integer is odd ... |
| sumeven 16423 | If every term in a sum is ... |
| sumodd 16424 | If every term in a sum is ... |
| evensumodd 16425 | If every term in a sum wit... |
| oddsumodd 16426 | If every term in a sum wit... |
| pwp1fsum 16427 | The n-th power of a number... |
| oddpwp1fsum 16428 | An odd power of a number i... |
| divalglem0 16429 | Lemma for ~ divalg . (Con... |
| divalglem1 16430 | Lemma for ~ divalg . (Con... |
| divalglem2 16431 | Lemma for ~ divalg . (Con... |
| divalglem4 16432 | Lemma for ~ divalg . (Con... |
| divalglem5 16433 | Lemma for ~ divalg . (Con... |
| divalglem6 16434 | Lemma for ~ divalg . (Con... |
| divalglem7 16435 | Lemma for ~ divalg . (Con... |
| divalglem8 16436 | Lemma for ~ divalg . (Con... |
| divalglem9 16437 | Lemma for ~ divalg . (Con... |
| divalglem10 16438 | Lemma for ~ divalg . (Con... |
| divalg 16439 | The division algorithm (th... |
| divalgb 16440 | Express the division algor... |
| divalg2 16441 | The division algorithm (th... |
| divalgmod 16442 | The result of the ` mod ` ... |
| divalgmodcl 16443 | The result of the ` mod ` ... |
| modremain 16444 | The result of the modulo o... |
| ndvdssub 16445 | Corollary of the division ... |
| ndvdsadd 16446 | Corollary of the division ... |
| ndvdsp1 16447 | Special case of ~ ndvdsadd... |
| ndvdsi 16448 | A quick test for non-divis... |
| 5ndvds3 16449 | 5 does not divide 3. (Con... |
| 5ndvds6 16450 | 5 does not divide 6. (Con... |
| flodddiv4 16451 | The floor of an odd intege... |
| fldivndvdslt 16452 | The floor of an integer di... |
| flodddiv4lt 16453 | The floor of an odd number... |
| flodddiv4t2lthalf 16454 | The floor of an odd number... |
| bitsfval 16459 | Expand the definition of t... |
| bitsval 16460 | Expand the definition of t... |
| bitsval2 16461 | Expand the definition of t... |
| bitsss 16462 | The set of bits of an inte... |
| bitsf 16463 | The ` bits ` function is a... |
| bits0 16464 | Value of the zeroth bit. ... |
| bits0e 16465 | The zeroth bit of an even ... |
| bits0o 16466 | The zeroth bit of an odd n... |
| bitsp1 16467 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16468 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16469 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16470 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16471 | The bits of a number are a... |
| bitsmod 16472 | Truncating the bit sequenc... |
| bitsfi 16473 | Every number is associated... |
| bitscmp 16474 | The bit complement of ` N ... |
| 0bits 16475 | The bits of zero. (Contri... |
| m1bits 16476 | The bits of negative one. ... |
| bitsinv1lem 16477 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16478 | There is an explicit inver... |
| bitsinv2 16479 | There is an explicit inver... |
| bitsf1ocnv 16480 | The ` bits ` function rest... |
| bitsf1o 16481 | The ` bits ` function rest... |
| bitsf1 16482 | The ` bits ` function is a... |
| 2ebits 16483 | The bits of a power of two... |
| bitsinv 16484 | The inverse of the ` bits ... |
| bitsinvp1 16485 | Recursive definition of th... |
| sadadd2lem2 16486 | The core of the proof of ~... |
| sadfval 16488 | Define the addition of two... |
| sadcf 16489 | The carry sequence is a se... |
| sadc0 16490 | The initial element of the... |
| sadcp1 16491 | The carry sequence (which ... |
| sadval 16492 | The full adder sequence is... |
| sadcaddlem 16493 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16494 | Non-recursive definition o... |
| sadadd2lem 16495 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16496 | Sum of initial segments of... |
| sadadd3 16497 | Sum of initial segments of... |
| sadcl 16498 | The sum of two sequences i... |
| sadcom 16499 | The adder sequence functio... |
| saddisjlem 16500 | Lemma for ~ sadadd . (Con... |
| saddisj 16501 | The sum of disjoint sequen... |
| sadaddlem 16502 | Lemma for ~ sadadd . (Con... |
| sadadd 16503 | For sequences that corresp... |
| sadid1 16504 | The adder sequence functio... |
| sadid2 16505 | The adder sequence functio... |
| sadasslem 16506 | Lemma for ~ sadass . (Con... |
| sadass 16507 | Sequence addition is assoc... |
| sadeq 16508 | Any element of a sequence ... |
| bitsres 16509 | Restrict the bits of a num... |
| bitsuz 16510 | The bits of a number are a... |
| bitsshft 16511 | Shifting a bit sequence to... |
| smufval 16513 | The multiplication of two ... |
| smupf 16514 | The sequence of partial su... |
| smup0 16515 | The initial element of the... |
| smupp1 16516 | The initial element of the... |
| smuval 16517 | Define the addition of two... |
| smuval2 16518 | The partial sum sequence s... |
| smupvallem 16519 | If ` A ` only has elements... |
| smucl 16520 | The product of two sequenc... |
| smu01lem 16521 | Lemma for ~ smu01 and ~ sm... |
| smu01 16522 | Multiplication of a sequen... |
| smu02 16523 | Multiplication of a sequen... |
| smupval 16524 | Rewrite the elements of th... |
| smup1 16525 | Rewrite ~ smupp1 using onl... |
| smueqlem 16526 | Any element of a sequence ... |
| smueq 16527 | Any element of a sequence ... |
| smumullem 16528 | Lemma for ~ smumul . (Con... |
| smumul 16529 | For sequences that corresp... |
| gcdval 16532 | The value of the ` gcd ` o... |
| gcd0val 16533 | The value, by convention, ... |
| gcdn0val 16534 | The value of the ` gcd ` o... |
| gcdcllem1 16535 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16536 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16537 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16538 | Closure of the ` gcd ` ope... |
| gcddvds 16539 | The gcd of two integers di... |
| dvdslegcd 16540 | An integer which divides b... |
| nndvdslegcd 16541 | A positive integer which d... |
| gcdcl 16542 | Closure of the ` gcd ` ope... |
| gcdnncl 16543 | Closure of the ` gcd ` ope... |
| gcdcld 16544 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16545 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16546 | An integer is the product ... |
| divgcdz 16547 | An integer divided by the ... |
| gcdf 16548 | Domain and codomain of the... |
| gcdcom 16549 | The ` gcd ` operator is co... |
| gcdcomd 16550 | The ` gcd ` operator is co... |
| divgcdnn 16551 | A positive integer divided... |
| divgcdnnr 16552 | A positive integer divided... |
| gcdeq0 16553 | The gcd of two integers is... |
| gcdn0gt0 16554 | The gcd of two integers is... |
| gcd0id 16555 | The gcd of 0 and an intege... |
| gcdid0 16556 | The gcd of an integer and ... |
| nn0gcdid0 16557 | The gcd of a nonnegative i... |
| gcdneg 16558 | Negating one operand of th... |
| neggcd 16559 | Negating one operand of th... |
| gcdaddmlem 16560 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16561 | Adding a multiple of one o... |
| gcdadd 16562 | The GCD of two numbers is ... |
| gcdid 16563 | The gcd of a number and it... |
| gcd1 16564 | The gcd of a number with 1... |
| gcdabs1 16565 | ` gcd ` of the absolute va... |
| gcdabs2 16566 | ` gcd ` of the absolute va... |
| gcdabs 16567 | The gcd of two integers is... |
| modgcd 16568 | The gcd remains unchanged ... |
| 1gcd 16569 | The GCD of one and an inte... |
| gcdmultipled 16570 | The greatest common diviso... |
| gcdmultiplez 16571 | The GCD of a multiple of a... |
| gcdmultiple 16572 | The GCD of a multiple of a... |
| dvdsgcdidd 16573 | The greatest common diviso... |
| 6gcd4e2 16574 | The greatest common diviso... |
| bezoutlem1 16575 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16576 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16577 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16578 | Lemma for ~ bezout . (Con... |
| bezout 16579 | Bézout's identity: ... |
| dvdsgcd 16580 | An integer which divides e... |
| dvdsgcdb 16581 | Biconditional form of ~ dv... |
| dfgcd2 16582 | Alternate definition of th... |
| gcdass 16583 | Associative law for ` gcd ... |
| mulgcd 16584 | Distribute multiplication ... |
| absmulgcd 16585 | Distribute absolute value ... |
| mulgcdr 16586 | Reverse distribution law f... |
| gcddiv 16587 | Division law for GCD. (Con... |
| gcdzeq 16588 | A positive integer ` A ` i... |
| gcdeq 16589 | ` A ` is equal to its gcd ... |
| dvdssqim 16590 | Unidirectional form of ~ d... |
| dvdsexpim 16591 | If two numbers are divisib... |
| dvdsmulgcd 16592 | A divisibility equivalent ... |
| rpmulgcd 16593 | If ` K ` and ` M ` are rel... |
| rplpwr 16594 | If ` A ` and ` B ` are rel... |
| rprpwr 16595 | If ` A ` and ` B ` are rel... |
| rppwr 16596 | If ` A ` and ` B ` are rel... |
| nn0rppwr 16597 | If ` A ` and ` B ` are rel... |
| sqgcd 16598 | Square distributes over gc... |
| expgcd 16599 | Exponentiation distributes... |
| nn0expgcd 16600 | Exponentiation distributes... |
| zexpgcd 16601 | Exponentiation distributes... |
| dvdssqlem 16602 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16603 | Two numbers are divisible ... |
| bezoutr 16604 | Partial converse to ~ bezo... |
| bezoutr1 16605 | Converse of ~ bezout for w... |
| nn0seqcvgd 16606 | A strictly-decreasing nonn... |
| seq1st 16607 | A sequence whose iteration... |
| algr0 16608 | The value of the algorithm... |
| algrf 16609 | An algorithm is a step fun... |
| algrp1 16610 | The value of the algorithm... |
| alginv 16611 | If ` I ` is an invariant o... |
| algcvg 16612 | One way to prove that an a... |
| algcvgblem 16613 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16614 | Two ways of expressing tha... |
| algcvga 16615 | The countdown function ` C... |
| algfx 16616 | If ` F ` reaches a fixed p... |
| eucalgval2 16617 | The value of the step func... |
| eucalgval 16618 | Euclid's Algorithm ~ eucal... |
| eucalgf 16619 | Domain and codomain of the... |
| eucalginv 16620 | The invariant of the step ... |
| eucalglt 16621 | The second member of the s... |
| eucalgcvga 16622 | Once Euclid's Algorithm ha... |
| eucalg 16623 | Euclid's Algorithm compute... |
| lcmval 16628 | Value of the ` lcm ` opera... |
| lcmcom 16629 | The ` lcm ` operator is co... |
| lcm0val 16630 | The value, by convention, ... |
| lcmn0val 16631 | The value of the ` lcm ` o... |
| lcmcllem 16632 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16633 | Closure of the ` lcm ` ope... |
| dvdslcm 16634 | The lcm of two integers is... |
| lcmledvds 16635 | A positive integer which b... |
| lcmeq0 16636 | The lcm of two integers is... |
| lcmcl 16637 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16638 | The greatest common diviso... |
| lcmneg 16639 | Negating one operand of th... |
| neglcm 16640 | Negating one operand of th... |
| lcmabs 16641 | The lcm of two integers is... |
| lcmgcdlem 16642 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16643 | The product of two numbers... |
| lcmdvds 16644 | The lcm of two integers di... |
| lcmid 16645 | The lcm of an integer and ... |
| lcm1 16646 | The lcm of an integer and ... |
| lcmgcdnn 16647 | The product of two positiv... |
| lcmgcdeq 16648 | Two integers' absolute val... |
| lcmdvdsb 16649 | Biconditional form of ~ lc... |
| lcmass 16650 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16651 | The least common multiple ... |
| 6lcm4e12 16652 | The least common multiple ... |
| absproddvds 16653 | The absolute value of the ... |
| absprodnn 16654 | The absolute value of the ... |
| fissn0dvds 16655 | For each finite subset of ... |
| fissn0dvdsn0 16656 | For each finite subset of ... |
| lcmfval 16657 | Value of the ` _lcm ` func... |
| lcmf0val 16658 | The value, by convention, ... |
| lcmfn0val 16659 | The value of the ` _lcm ` ... |
| lcmfnnval 16660 | The value of the ` _lcm ` ... |
| lcmfcllem 16661 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16662 | Closure of the ` _lcm ` fu... |
| lcmfpr 16663 | The value of the ` _lcm ` ... |
| lcmfcl 16664 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16665 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16666 | The least common multiple ... |
| dvdslcmf 16667 | The least common multiple ... |
| lcmfledvds 16668 | A positive integer which i... |
| lcmf 16669 | Characterization of the le... |
| lcmf0 16670 | The least common multiple ... |
| lcmfsn 16671 | The least common multiple ... |
| lcmftp 16672 | The least common multiple ... |
| lcmfunsnlem1 16673 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16674 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16675 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16676 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16677 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16678 | The least common multiple ... |
| lcmfdvdsb 16679 | Biconditional form of ~ lc... |
| lcmfunsn 16680 | The ` _lcm ` function for ... |
| lcmfun 16681 | The ` _lcm ` function for ... |
| lcmfass 16682 | Associative law for the ` ... |
| lcmf2a3a4e12 16683 | The least common multiple ... |
| lcmflefac 16684 | The least common multiple ... |
| coprmgcdb 16685 | Two positive integers are ... |
| ncoprmgcdne1b 16686 | Two positive integers are ... |
| ncoprmgcdgt1b 16687 | Two positive integers are ... |
| coprmdvds1 16688 | If two positive integers a... |
| coprmdvds 16689 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16690 | If an integer is divisible... |
| mulgcddvds 16691 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16692 | If ` M ` is relatively pri... |
| qredeq 16693 | Two equal reduced fraction... |
| qredeu 16694 | Every rational number has ... |
| rpmul 16695 | If ` K ` is relatively pri... |
| rpdvds 16696 | If ` K ` is relatively pri... |
| coprmprod 16697 | The product of the element... |
| coprmproddvdslem 16698 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16699 | If a positive integer is d... |
| congr 16700 | Definition of congruence b... |
| divgcdcoprm0 16701 | Integers divided by gcd ar... |
| divgcdcoprmex 16702 | Integers divided by gcd ar... |
| cncongr1 16703 | One direction of the bicon... |
| cncongr2 16704 | The other direction of the... |
| cncongr 16705 | Cancellability of Congruen... |
| cncongrcoprm 16706 | Corollary 1 of Cancellabil... |
| isprm 16709 | The predicate "is a prime ... |
| prmnn 16710 | A prime number is a positi... |
| prmz 16711 | A prime number is an integ... |
| prmssnn 16712 | The prime numbers are a su... |
| prmex 16713 | The set of prime numbers e... |
| 0nprm 16714 | 0 is not a prime number. ... |
| 1nprm 16715 | 1 is not a prime number. ... |
| 1idssfct 16716 | The positive divisors of a... |
| isprm2lem 16717 | Lemma for ~ isprm2 . (Con... |
| isprm2 16718 | The predicate "is a prime ... |
| isprm3 16719 | The predicate "is a prime ... |
| isprm4 16720 | The predicate "is a prime ... |
| prmind2 16721 | A variation on ~ prmind as... |
| prmind 16722 | Perform induction over the... |
| dvdsprime 16723 | If ` M ` divides a prime, ... |
| nprm 16724 | A product of two integers ... |
| nprmi 16725 | An inference for composite... |
| dvdsnprmd 16726 | If a number is divisible b... |
| prm2orodd 16727 | A prime number is either 2... |
| 2prm 16728 | 2 is a prime number. (Con... |
| 2mulprm 16729 | A multiple of two is prime... |
| 3prm 16730 | 3 is a prime number. (Con... |
| 4nprm 16731 | 4 is not a prime number. ... |
| prmuz2 16732 | A prime number is an integ... |
| prmssuz2 16733 | The primes are integers gr... |
| prmgt1 16734 | A prime number is an integ... |
| prmm2nn0 16735 | Subtracting 2 from a prime... |
| oddprmgt2 16736 | An odd prime is greater th... |
| oddprmge3 16737 | An odd prime is greater th... |
| ge2nprmge4 16738 | A composite integer greate... |
| sqnprm 16739 | A square is never prime. ... |
| dvdsprm 16740 | An integer greater than or... |
| exprmfct 16741 | Every integer greater than... |
| prmdvdsfz 16742 | Each integer greater than ... |
| nprmdvds1 16743 | No prime number divides 1.... |
| isprm5 16744 | One need only check prime ... |
| isprm7 16745 | One need only check prime ... |
| maxprmfct 16746 | The set of prime factors o... |
| divgcdodd 16747 | Either ` A / ( A gcd B ) `... |
| coprm 16748 | A prime number either divi... |
| prmrp 16749 | Unequal prime numbers are ... |
| euclemma 16750 | Euclid's lemma. A prime n... |
| isprm6 16751 | A number is prime iff it s... |
| prmdvdsexp 16752 | A prime divides a positive... |
| prmdvdsexpb 16753 | A prime divides a positive... |
| prmdvdsexpr 16754 | If a prime divides a nonne... |
| prmdvdssq 16755 | Condition for a prime divi... |
| prmexpb 16756 | Two positive prime powers ... |
| prmfac1 16757 | The factorial of a number ... |
| dvdszzq 16758 | Divisibility for an intege... |
| rpexp 16759 | If two numbers ` A ` and `... |
| rpexp1i 16760 | Relative primality passes ... |
| rpexp12i 16761 | Relative primality passes ... |
| prmndvdsfaclt 16762 | A prime number does not di... |
| prmdvdsbc 16763 | Condition for a prime numb... |
| prmdvdsncoprmbd 16764 | Two positive integers are ... |
| ncoprmlnprm 16765 | If two positive integers a... |
| cncongrprm 16766 | Corollary 2 of Cancellabil... |
| isevengcd2 16767 | The predicate "is an even ... |
| isoddgcd1 16768 | The predicate "is an odd n... |
| 3lcm2e6 16769 | The least common multiple ... |
| qnumval 16774 | Value of the canonical num... |
| qdenval 16775 | Value of the canonical den... |
| qnumdencl 16776 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16777 | The canonical numerator of... |
| qdencl 16778 | The canonical denominator ... |
| fnum 16779 | Canonical numerator define... |
| fden 16780 | Canonical denominator defi... |
| qnumdenbi 16781 | Two numbers are the canoni... |
| qnumdencoprm 16782 | The canonical representati... |
| qeqnumdivden 16783 | Recover a rational number ... |
| qmuldeneqnum 16784 | Multiplying a rational by ... |
| divnumden 16785 | Calculate the reduced form... |
| divdenle 16786 | Reducing a quotient never ... |
| qnumgt0 16787 | A rational is positive iff... |
| qgt0numnn 16788 | A rational is positive iff... |
| nn0gcdsq 16789 | Squaring commutes with GCD... |
| zgcdsq 16790 | ~ nn0gcdsq extended to int... |
| numdensq 16791 | Squaring a rational square... |
| numsq 16792 | Square commutes with canon... |
| densq 16793 | Square commutes with canon... |
| qden1elz 16794 | A rational is an integer i... |
| zsqrtelqelz 16795 | If an integer has a ration... |
| nonsq 16796 | Any integer strictly betwe... |
| numdenexp 16797 | Elevating a rational numbe... |
| numexp 16798 | Elevating to a nonnegative... |
| denexp 16799 | Elevating to a nonnegative... |
| phival 16804 | Value of the Euler ` phi `... |
| phicl2 16805 | Bounds and closure for the... |
| phicl 16806 | Closure for the value of t... |
| phibndlem 16807 | Lemma for ~ phibnd . (Con... |
| phibnd 16808 | A slightly tighter bound o... |
| phicld 16809 | Closure for the value of t... |
| phi1 16810 | Value of the Euler ` phi `... |
| dfphi2 16811 | Alternate definition of th... |
| hashdvds 16812 | The number of numbers in a... |
| phiprmpw 16813 | Value of the Euler ` phi `... |
| phiprm 16814 | Value of the Euler ` phi `... |
| crth 16815 | The Chinese Remainder Theo... |
| phimullem 16816 | Lemma for ~ phimul . (Con... |
| phimul 16817 | The Euler ` phi ` function... |
| eulerthlem1 16818 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16819 | Lemma for ~ eulerth . (Co... |
| eulerth 16820 | Euler's theorem, a general... |
| fermltl 16821 | Fermat's little theorem. ... |
| prmdiv 16822 | Show an explicit expressio... |
| prmdiveq 16823 | The modular inverse of ` A... |
| prmdivdiv 16824 | The (modular) inverse of t... |
| hashgcdlem 16825 | A correspondence between e... |
| dvdsfi 16826 | A natural number has finit... |
| hashgcdeq 16827 | Number of initial positive... |
| phisum 16828 | The divisor sum identity o... |
| odzval 16829 | Value of the order functio... |
| odzcllem 16830 | - Lemma for ~ odzcl , show... |
| odzcl 16831 | The order of a group eleme... |
| odzid 16832 | Any element raised to the ... |
| odzdvds 16833 | The only powers of ` A ` t... |
| odzphi 16834 | The order of any group ele... |
| modprm1div 16835 | A prime number divides an ... |
| m1dvdsndvds 16836 | If an integer minus 1 is d... |
| modprminv 16837 | Show an explicit expressio... |
| modprminveq 16838 | The modular inverse of ` A... |
| vfermltl 16839 | Variant of Fermat's little... |
| vfermltlALT 16840 | Alternate proof of ~ vferm... |
| powm2modprm 16841 | If an integer minus 1 is d... |
| reumodprminv 16842 | For any prime number and f... |
| modprm0 16843 | For two positive integers ... |
| nnnn0modprm0 16844 | For a positive integer and... |
| modprmn0modprm0 16845 | For an integer not being 0... |
| coprimeprodsq 16846 | If three numbers are copri... |
| coprimeprodsq2 16847 | If three numbers are copri... |
| oddprm 16848 | A prime not equal to ` 2 `... |
| nnoddn2prm 16849 | A prime not equal to ` 2 `... |
| oddn2prm 16850 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16851 | A number is a prime number... |
| prm23lt5 16852 | A prime less than 5 is eit... |
| prm23ge5 16853 | A prime is either 2 or 3 o... |
| pythagtriplem1 16854 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16855 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16856 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16857 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16858 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16859 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16860 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16861 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16862 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16863 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16864 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16865 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16866 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16867 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16868 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16869 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16870 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16871 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16872 | Parameterize the Pythagore... |
| iserodd 16873 | Collect the odd terms in a... |
| pclem 16876 | - Lemma for the prime powe... |
| pcprecl 16877 | Closure of the prime power... |
| pcprendvds 16878 | Non-divisibility property ... |
| pcprendvds2 16879 | Non-divisibility property ... |
| pcpre1 16880 | Value of the prime power p... |
| pcpremul 16881 | Multiplicative property of... |
| pcval 16882 | The value of the prime pow... |
| pceulem 16883 | Lemma for ~ pceu . (Contr... |
| pceu 16884 | Uniqueness for the prime p... |
| pczpre 16885 | Connect the prime count pr... |
| pczcl 16886 | Closure of the prime power... |
| pccl 16887 | Closure of the prime power... |
| pccld 16888 | Closure of the prime power... |
| pcmul 16889 | Multiplication property of... |
| pcdiv 16890 | Division property of the p... |
| pcqmul 16891 | Multiplication property of... |
| pc0 16892 | The value of the prime pow... |
| pc1 16893 | Value of the prime count f... |
| pcqcl 16894 | Closure of the general pri... |
| pcqdiv 16895 | Division property of the p... |
| pcrec 16896 | Prime power of a reciproca... |
| pcexp 16897 | Prime power of an exponent... |
| pcxnn0cl 16898 | Extended nonnegative integ... |
| pcxcl 16899 | Extended real closure of t... |
| pcge0 16900 | The prime count of an inte... |
| pczdvds 16901 | Defining property of the p... |
| pcdvds 16902 | Defining property of the p... |
| pczndvds 16903 | Defining property of the p... |
| pcndvds 16904 | Defining property of the p... |
| pczndvds2 16905 | The remainder after dividi... |
| pcndvds2 16906 | The remainder after dividi... |
| pcdvdsb 16907 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16908 | There are a positive numbe... |
| pceq0 16909 | There are zero powers of a... |
| pcidlem 16910 | The prime count of a prime... |
| pcid 16911 | The prime count of a prime... |
| pcneg 16912 | The prime count of a negat... |
| pcabs 16913 | The prime count of an abso... |
| pcdvdstr 16914 | The prime count increases ... |
| pcgcd1 16915 | The prime count of a GCD i... |
| pcgcd 16916 | The prime count of a GCD i... |
| pc2dvds 16917 | A characterization of divi... |
| pc11 16918 | The prime count function, ... |
| pcz 16919 | The prime count function c... |
| pcprmpw2 16920 | Self-referential expressio... |
| pcprmpw 16921 | Self-referential expressio... |
| dvdsprmpweq 16922 | If a positive integer divi... |
| dvdsprmpweqnn 16923 | If an integer greater than... |
| dvdsprmpweqle 16924 | If a positive integer divi... |
| difsqpwdvds 16925 | If the difference of two s... |
| pcaddlem 16926 | Lemma for ~ pcadd . The o... |
| pcadd 16927 | An inequality for the prim... |
| pcadd2 16928 | The inequality of ~ pcadd ... |
| pcmptcl 16929 | Closure for the prime powe... |
| pcmpt 16930 | Construct a function with ... |
| pcmpt2 16931 | Dividing two prime count m... |
| pcmptdvds 16932 | The partial products of th... |
| pcprod 16933 | The product of the primes ... |
| sumhash 16934 | The sum of 1 over a set is... |
| fldivp1 16935 | The difference between the... |
| pcfaclem 16936 | Lemma for ~ pcfac . (Cont... |
| pcfac 16937 | Calculate the prime count ... |
| pcbc 16938 | Calculate the prime count ... |
| qexpz 16939 | If a power of a rational n... |
| expnprm 16940 | A second or higher power o... |
| oddprmdvds 16941 | Every positive integer whi... |
| prmpwdvds 16942 | A relation involving divis... |
| pockthlem 16943 | Lemma for ~ pockthg . (Co... |
| pockthg 16944 | The generalized Pocklingto... |
| pockthi 16945 | Pocklington's theorem, whi... |
| unbenlem 16946 | Lemma for ~ unben . (Cont... |
| unben 16947 | An unbounded set of positi... |
| infpnlem1 16948 | Lemma for ~ infpn . The s... |
| infpnlem2 16949 | Lemma for ~ infpn . For a... |
| infpn 16950 | There exist infinitely man... |
| infpn2 16951 | There exist infinitely man... |
| prmunb 16952 | The primes are unbounded. ... |
| prminf 16953 | There are an infinite numb... |
| prmreclem1 16954 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16955 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16956 | Lemma for ~ prmrec . The ... |
| prmreclem4 16957 | Lemma for ~ prmrec . Show... |
| prmreclem5 16958 | Lemma for ~ prmrec . Here... |
| prmreclem6 16959 | Lemma for ~ prmrec . If t... |
| prmrec 16960 | The sum of the reciprocals... |
| 1arithlem1 16961 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16962 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16963 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16964 | Lemma for ~ 1arith . (Con... |
| 1arith 16965 | Fundamental theorem of ari... |
| 1arith2 16966 | Fundamental theorem of ari... |
| elgz 16969 | Elementhood in the gaussia... |
| gzcn 16970 | A gaussian integer is a co... |
| zgz 16971 | An integer is a gaussian i... |
| igz 16972 | ` _i ` is a gaussian integ... |
| gznegcl 16973 | The gaussian integers are ... |
| gzcjcl 16974 | The gaussian integers are ... |
| gzaddcl 16975 | The gaussian integers are ... |
| gzmulcl 16976 | The gaussian integers are ... |
| gzreim 16977 | Construct a gaussian integ... |
| gzsubcl 16978 | The gaussian integers are ... |
| gzabssqcl 16979 | The squared norm of a gaus... |
| 4sqlem5 16980 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16981 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16982 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16983 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16984 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16985 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16986 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16987 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16988 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16989 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16990 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16991 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16992 | Euler's four-square identi... |
| 4sqlem11 16993 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16994 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16995 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16996 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16997 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16998 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16999 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 17000 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 17001 | Lemma for ~ 4sq . The pro... |
| 4sq 17002 | Lagrange's four-square the... |
| vdwapfval 17009 | Define the arithmetic prog... |
| vdwapf 17010 | The arithmetic progression... |
| vdwapval 17011 | Value of the arithmetic pr... |
| vdwapun 17012 | Remove the first element o... |
| vdwapid1 17013 | The first element of an ar... |
| vdwap0 17014 | Value of a length-1 arithm... |
| vdwap1 17015 | Value of a length-1 arithm... |
| vdwmc 17016 | The predicate " The ` <. R... |
| vdwmc2 17017 | Expand out the definition ... |
| vdwpc 17018 | The predicate " The colori... |
| vdwlem1 17019 | Lemma for ~ vdw . (Contri... |
| vdwlem2 17020 | Lemma for ~ vdw . (Contri... |
| vdwlem3 17021 | Lemma for ~ vdw . (Contri... |
| vdwlem4 17022 | Lemma for ~ vdw . (Contri... |
| vdwlem5 17023 | Lemma for ~ vdw . (Contri... |
| vdwlem6 17024 | Lemma for ~ vdw . (Contri... |
| vdwlem7 17025 | Lemma for ~ vdw . (Contri... |
| vdwlem8 17026 | Lemma for ~ vdw . (Contri... |
| vdwlem9 17027 | Lemma for ~ vdw . (Contri... |
| vdwlem10 17028 | Lemma for ~ vdw . Set up ... |
| vdwlem11 17029 | Lemma for ~ vdw . (Contri... |
| vdwlem12 17030 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 17031 | Lemma for ~ vdw . Main in... |
| vdw 17032 | Van der Waerden's theorem.... |
| vdwnnlem1 17033 | Corollary of ~ vdw , and l... |
| vdwnnlem2 17034 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 17035 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 17036 | Van der Waerden's theorem,... |
| ramtlecl 17038 | The set ` T ` of numbers w... |
| hashbcval 17040 | Value of the "binomial set... |
| hashbccl 17041 | The binomial set is a fini... |
| hashbcss 17042 | Subset relation for the bi... |
| hashbc0 17043 | The set of subsets of size... |
| hashbc2 17044 | The size of the binomial s... |
| 0hashbc 17045 | There are no subsets of th... |
| ramval 17046 | The value of the Ramsey nu... |
| ramcl2lem 17047 | Lemma for extended real cl... |
| ramtcl 17048 | The Ramsey number has the ... |
| ramtcl2 17049 | The Ramsey number is an in... |
| ramtub 17050 | The Ramsey number is a low... |
| ramub 17051 | The Ramsey number is a low... |
| ramub2 17052 | It is sufficient to check ... |
| rami 17053 | The defining property of a... |
| ramcl2 17054 | The Ramsey number is eithe... |
| ramxrcl 17055 | The Ramsey number is an ex... |
| ramubcl 17056 | If the Ramsey number is up... |
| ramlb 17057 | Establish a lower bound on... |
| 0ram 17058 | The Ramsey number when ` M... |
| 0ram2 17059 | The Ramsey number when ` M... |
| ram0 17060 | The Ramsey number when ` R... |
| 0ramcl 17061 | Lemma for ~ ramcl : Exist... |
| ramz2 17062 | The Ramsey number when ` F... |
| ramz 17063 | The Ramsey number when ` F... |
| ramub1lem1 17064 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 17065 | Lemma for ~ ramub1 . (Con... |
| ramub1 17066 | Inductive step for Ramsey'... |
| ramcl 17067 | Ramsey's theorem: the Rams... |
| ramsey 17068 | Ramsey's theorem with the ... |
| prmoval 17071 | Value of the primorial fun... |
| prmocl 17072 | Closure of the primorial f... |
| prmone0 17073 | The primorial function is ... |
| prmo0 17074 | The primorial of 0. (Cont... |
| prmo1 17075 | The primorial of 1. (Cont... |
| prmop1 17076 | The primorial of a success... |
| prmonn2 17077 | Value of the primorial fun... |
| prmo2 17078 | The primorial of 2. (Cont... |
| prmo3 17079 | The primorial of 3. (Cont... |
| prmdvdsprmo 17080 | The primorial of a number ... |
| prmdvdsprmop 17081 | The primorial of a number ... |
| fvprmselelfz 17082 | The value of the prime sel... |
| fvprmselgcd1 17083 | The greatest common diviso... |
| prmolefac 17084 | The primorial of a positiv... |
| prmodvdslcmf 17085 | The primorial of a nonnega... |
| prmolelcmf 17086 | The primorial of a positiv... |
| prmgaplem1 17087 | Lemma for ~ prmgap : The ... |
| prmgaplem2 17088 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 17089 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 17090 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 17091 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 17092 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 17093 | Lemma for ~ prmgap : for e... |
| prmgaplem6 17094 | Lemma for ~ prmgap : for e... |
| prmgaplem7 17095 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 17096 | Lemma for ~ prmgap . (Con... |
| prmgap 17097 | The prime gap theorem: for... |
| prmgaplcm 17098 | Alternate proof of ~ prmga... |
| prmgapprmolem 17099 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 17100 | Alternate proof of ~ prmga... |
| dec2dvds 17101 | Divisibility by two is obv... |
| dec5dvds 17102 | Divisibility by five is ob... |
| dec5dvds2 17103 | Divisibility by five is ob... |
| dec5nprm 17104 | A decimal number greater t... |
| dec2nprm 17105 | A decimal number greater t... |
| modxai 17106 | Add exponents in a power m... |
| mod2xi 17107 | Double exponents in a powe... |
| modxp1i 17108 | Add one to an exponent in ... |
| mod2xnegi 17109 | Version of ~ mod2xi with a... |
| modsubi 17110 | Subtract from within a mod... |
| gcdi 17111 | Calculate a GCD via Euclid... |
| gcdmodi 17112 | Calculate a GCD via Euclid... |
| numexp0 17113 | Calculate an integer power... |
| numexp1 17114 | Calculate an integer power... |
| numexpp1 17115 | Calculate an integer power... |
| numexp2x 17116 | Double an integer power. ... |
| decsplit0b 17117 | Split a decimal number int... |
| decsplit0 17118 | Split a decimal number int... |
| decsplit1 17119 | Split a decimal number int... |
| decsplit 17120 | Split a decimal number int... |
| karatsuba 17121 | The Karatsuba multiplicati... |
| 2exp4 17122 | Two to the fourth power is... |
| 2exp5 17123 | Two to the fifth power is ... |
| 2exp6 17124 | Two to the sixth power is ... |
| 2exp7 17125 | Two to the seventh power i... |
| 2exp8 17126 | Two to the eighth power is... |
| 2exp11 17127 | Two to the eleventh power ... |
| 2exp16 17128 | Two to the sixteenth power... |
| 3exp3 17129 | Three to the third power i... |
| 2expltfac 17130 | The factorial grows faster... |
| cshwsidrepsw 17131 | If cyclically shifting a w... |
| cshwsidrepswmod0 17132 | If cyclically shifting a w... |
| cshwshashlem1 17133 | If cyclically shifting a w... |
| cshwshashlem2 17134 | If cyclically shifting a w... |
| cshwshashlem3 17135 | If cyclically shifting a w... |
| cshwsdisj 17136 | The singletons resulting b... |
| cshwsiun 17137 | The set of (different!) wo... |
| cshwsex 17138 | The class of (different!) ... |
| cshws0 17139 | The size of the set of (di... |
| cshwrepswhash1 17140 | The size of the set of (di... |
| cshwshashnsame 17141 | If a word (not consisting ... |
| cshwshash 17142 | If a word has a length bei... |
| prmlem0 17143 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 17144 | A quick proof skeleton to ... |
| prmlem1 17145 | A quick proof skeleton to ... |
| 5prm 17146 | 5 is a prime number. (Con... |
| 6nprm 17147 | 6 is not a prime number. ... |
| 7prm 17148 | 7 is a prime number. (Con... |
| 8nprm 17149 | 8 is not a prime number. ... |
| 9nprm 17150 | 9 is not a prime number. ... |
| 10nprm 17151 | 10 is not a prime number. ... |
| 10nprmOLD 17152 | Obsolete version of ~ 10np... |
| 11prm 17153 | 11 is a prime number. (Co... |
| 13prm 17154 | 13 is a prime number. (Co... |
| 17prm 17155 | 17 is a prime number. (Co... |
| 19prm 17156 | 19 is a prime number. (Co... |
| 23prm 17157 | 23 is a prime number. (Co... |
| prmlem2 17158 | Our last proving session g... |
| 37prm 17159 | 37 is a prime number. (Co... |
| 43prm 17160 | 43 is a prime number. (Co... |
| 83prm 17161 | 83 is a prime number. (Co... |
| 139prm 17162 | 139 is a prime number. (C... |
| 163prm 17163 | 163 is a prime number. (C... |
| 317prm 17164 | 317 is a prime number. (C... |
| 631prm 17165 | 631 is a prime number. (C... |
| prmo4 17166 | The primorial of 4. (Cont... |
| prmo5 17167 | The primorial of 5. (Cont... |
| prmo6 17168 | The primorial of 6. (Cont... |
| 1259lem1 17169 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17170 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17171 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17172 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17173 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17174 | 1259 is a prime number. (... |
| 2503lem1 17175 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17176 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17177 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17178 | 2503 is a prime number. (... |
| 4001lem1 17179 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17180 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17181 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17182 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17183 | 4001 is a prime number. (... |
| brstruct 17186 | The structure relation is ... |
| isstruct2 17187 | The property of being a st... |
| structex 17188 | A structure is a set. (Co... |
| structn0fun 17189 | A structure without the em... |
| isstruct 17190 | The property of being a st... |
| structcnvcnv 17191 | Two ways to express the re... |
| structfung 17192 | The converse of the conver... |
| structfun 17193 | Convert between two kinds ... |
| structfn 17194 | Convert between two kinds ... |
| strleun 17195 | Combine two structures int... |
| strle1 17196 | Make a structure from a si... |
| strle2 17197 | Make a structure from a pa... |
| strle3 17198 | Make a structure from a tr... |
| sbcie2s 17199 | A special version of class... |
| sbcie3s 17200 | A special version of class... |
| reldmsets 17203 | The structure override ope... |
| setsvalg 17204 | Value of the structure rep... |
| setsval 17205 | Value of the structure rep... |
| fvsetsid 17206 | The value of the structure... |
| fsets 17207 | The structure replacement ... |
| setsdm 17208 | The domain of a structure ... |
| setsfun 17209 | A structure with replaceme... |
| setsfun0 17210 | A structure with replaceme... |
| setsn0fun 17211 | The value of the structure... |
| setsstruct2 17212 | An extensible structure wi... |
| setsexstruct2 17213 | An extensible structure wi... |
| setsstruct 17214 | An extensible structure wi... |
| wunsets 17215 | Closure of structure repla... |
| setsres 17216 | The structure replacement ... |
| setsabs 17217 | Replacing the same compone... |
| setscom 17218 | Different components can b... |
| sloteq 17221 | Equality theorem for the `... |
| slotfn 17222 | A slot is a function on se... |
| strfvnd 17223 | Deduction version of ~ str... |
| strfvn 17224 | Value of a structure compo... |
| strfvss 17225 | A structure component extr... |
| wunstr 17226 | Closure of a structure ind... |
| str0 17227 | All components of the empt... |
| strfvi 17228 | Structure slot extractors ... |
| fveqprc 17229 | Lemma for showing the equa... |
| oveqprc 17230 | Lemma for showing the equa... |
| wunndx 17233 | Closure of the index extra... |
| ndxarg 17234 | Get the numeric argument f... |
| ndxid 17235 | A structure component extr... |
| strndxid 17236 | The value of a structure c... |
| setsidvald 17237 | Value of the structure rep... |
| strfvd 17238 | Deduction version of ~ str... |
| strfv2d 17239 | Deduction version of ~ str... |
| strfv2 17240 | A variation on ~ strfv to ... |
| strfv 17241 | Extract a structure compon... |
| strfv3 17242 | Variant on ~ strfv for lar... |
| strssd 17243 | Deduction version of ~ str... |
| strss 17244 | Propagate component extrac... |
| setsid 17245 | Value of the structure rep... |
| setsnid 17246 | Value of the structure rep... |
| baseval 17249 | Value of the base set extr... |
| baseid 17250 | Utility theorem: index-ind... |
| basfn 17251 | The base set extractor is ... |
| base0 17252 | The base set of the empty ... |
| elbasfv 17253 | Utility theorem: reverse c... |
| elbasov 17254 | Utility theorem: reverse c... |
| strov2rcl 17255 | Partial reverse closure fo... |
| basendx 17256 | Index value of the base se... |
| basendxnn 17257 | The index value of the bas... |
| basndxelwund 17258 | The index of the base set ... |
| basprssdmsets 17259 | The pair of the base index... |
| opelstrbas 17260 | The base set of a structur... |
| 1strstr 17261 | A constructed one-slot str... |
| 1strbas 17262 | The base set of a construc... |
| 1strwunbndx 17263 | A constructed one-slot str... |
| 1strwun 17264 | A constructed one-slot str... |
| 2strstr 17265 | A constructed two-slot str... |
| 2strbas 17266 | The base set of a construc... |
| 2strop 17267 | The other slot of a constr... |
| reldmress 17270 | The structure restriction ... |
| ressval 17271 | Value of structure restric... |
| ressid2 17272 | General behavior of trivia... |
| ressval2 17273 | Value of nontrivial struct... |
| ressbas 17274 | Base set of a structure re... |
| ressbasssg 17275 | The base set of a restrict... |
| ressbas2 17276 | Base set of a structure re... |
| ressbasss 17277 | The base set of a restrict... |
| ressbasssOLD 17278 | Obsolete version of ~ ress... |
| ressbasss2 17279 | The base set of a restrict... |
| resseqnbas 17280 | The components of an exten... |
| ress0 17281 | All restrictions of the nu... |
| ressid 17282 | Behavior of trivial restri... |
| ressinbas 17283 | Restriction only cares abo... |
| ressval3d 17284 | Value of structure restric... |
| ressress 17285 | Restriction composition la... |
| ressabs 17286 | Restriction absorption law... |
| wunress 17287 | Closure of structure restr... |
| plusgndx 17314 | Index value of the ~ df-pl... |
| plusgid 17315 | Utility theorem: index-ind... |
| plusgndxnn 17316 | The index of the slot for ... |
| basendxltplusgndx 17317 | The index of the slot for ... |
| basendxnplusgndx 17318 | The slot for the base set ... |
| grpstr 17319 | A constructed group is a s... |
| grpbase 17320 | The base set of a construc... |
| grpplusg 17321 | The operation of a constru... |
| ressplusg 17322 | ` +g ` is unaffected by re... |
| grpbasex 17323 | The base of an explicitly ... |
| grpplusgx 17324 | The operation of an explic... |
| mulrndx 17325 | Index value of the ~ df-mu... |
| mulridx 17326 | Utility theorem: index-ind... |
| basendxnmulrndx 17327 | The slot for the base set ... |
| plusgndxnmulrndx 17328 | The slot for the group (ad... |
| rngstr 17329 | A constructed ring is a st... |
| rngbase 17330 | The base set of a construc... |
| rngplusg 17331 | The additive operation of ... |
| rngmulr 17332 | The multiplicative operati... |
| starvndx 17333 | Index value of the ~ df-st... |
| starvid 17334 | Utility theorem: index-ind... |
| starvndxnbasendx 17335 | The slot for the involutio... |
| starvndxnplusgndx 17336 | The slot for the involutio... |
| starvndxnmulrndx 17337 | The slot for the involutio... |
| ressmulr 17338 | ` .r ` is unaffected by re... |
| ressstarv 17339 | ` *r ` is unaffected by re... |
| srngstr 17340 | A constructed star ring is... |
| srngbase 17341 | The base set of a construc... |
| srngplusg 17342 | The addition operation of ... |
| srngmulr 17343 | The multiplication operati... |
| srnginvl 17344 | The involution function of... |
| scandx 17345 | Index value of the ~ df-sc... |
| scaid 17346 | Utility theorem: index-ind... |
| scandxnbasendx 17347 | The slot for the scalar is... |
| scandxnplusgndx 17348 | The slot for the scalar fi... |
| scandxnmulrndx 17349 | The slot for the scalar fi... |
| vscandx 17350 | Index value of the ~ df-vs... |
| vscaid 17351 | Utility theorem: index-ind... |
| vscandxnbasendx 17352 | The slot for the scalar pr... |
| vscandxnplusgndx 17353 | The slot for the scalar pr... |
| vscandxnmulrndx 17354 | The slot for the scalar pr... |
| vscandxnscandx 17355 | The slot for the scalar pr... |
| lmodstr 17356 | A constructed left module ... |
| lmodbase 17357 | The base set of a construc... |
| lmodplusg 17358 | The additive operation of ... |
| lmodsca 17359 | The set of scalars of a co... |
| lmodvsca 17360 | The scalar product operati... |
| ipndx 17361 | Index value of the ~ df-ip... |
| ipid 17362 | Utility theorem: index-ind... |
| ipndxnbasendx 17363 | The slot for the inner pro... |
| ipndxnplusgndx 17364 | The slot for the inner pro... |
| ipndxnmulrndx 17365 | The slot for the inner pro... |
| slotsdifipndx 17366 | The slot for the scalar is... |
| ipsstr 17367 | Lemma to shorten proofs of... |
| ipsbase 17368 | The base set of a construc... |
| ipsaddg 17369 | The additive operation of ... |
| ipsmulr 17370 | The multiplicative operati... |
| ipssca 17371 | The set of scalars of a co... |
| ipsvsca 17372 | The scalar product operati... |
| ipsip 17373 | The multiplicative operati... |
| resssca 17374 | ` Scalar ` is unaffected b... |
| ressvsca 17375 | ` .s ` is unaffected by re... |
| ressip 17376 | The inner product is unaff... |
| phlstr 17377 | A constructed pre-Hilbert ... |
| phlbase 17378 | The base set of a construc... |
| phlplusg 17379 | The additive operation of ... |
| phlsca 17380 | The ring of scalars of a c... |
| phlvsca 17381 | The scalar product operati... |
| phlip 17382 | The inner product (Hermiti... |
| tsetndx 17383 | Index value of the ~ df-ts... |
| tsetid 17384 | Utility theorem: index-ind... |
| tsetndxnn 17385 | The index of the slot for ... |
| basendxlttsetndx 17386 | The index of the slot for ... |
| tsetndxnbasendx 17387 | The slot for the topology ... |
| tsetndxnplusgndx 17388 | The slot for the topology ... |
| tsetndxnmulrndx 17389 | The slot for the topology ... |
| tsetndxnstarvndx 17390 | The slot for the topology ... |
| slotstnscsi 17391 | The slots ` Scalar ` , ` .... |
| topgrpstr 17392 | A constructed topological ... |
| topgrpbas 17393 | The base set of a construc... |
| topgrpplusg 17394 | The additive operation of ... |
| topgrptset 17395 | The topology of a construc... |
| resstset 17396 | ` TopSet ` is unaffected b... |
| plendx 17397 | Index value of the ~ df-pl... |
| pleid 17398 | Utility theorem: self-refe... |
| plendxnn 17399 | The index value of the ord... |
| basendxltplendx 17400 | The index value of the ` B... |
| plendxnbasendx 17401 | The slot for the order is ... |
| plendxnplusgndx 17402 | The slot for the "less tha... |
| plendxnmulrndx 17403 | The slot for the "less tha... |
| plendxnscandx 17404 | The slot for the "less tha... |
| plendxnvscandx 17405 | The slot for the "less tha... |
| slotsdifplendx 17406 | The index of the slot for ... |
| otpsstr 17407 | Functionality of a topolog... |
| otpsbas 17408 | The base set of a topologi... |
| otpstset 17409 | The open sets of a topolog... |
| otpsle 17410 | The order of a topological... |
| ressle 17411 | ` le ` is unaffected by re... |
| ocndx 17412 | Index value of the ~ df-oc... |
| ocid 17413 | Utility theorem: index-ind... |
| basendxnocndx 17414 | The slot for the orthocomp... |
| plendxnocndx 17415 | The slot for the orthocomp... |
| dsndx 17416 | Index value of the ~ df-ds... |
| dsid 17417 | Utility theorem: index-ind... |
| dsndxnn 17418 | The index of the slot for ... |
| basendxltdsndx 17419 | The index of the slot for ... |
| dsndxnbasendx 17420 | The slot for the distance ... |
| dsndxnplusgndx 17421 | The slot for the distance ... |
| dsndxnmulrndx 17422 | The slot for the distance ... |
| slotsdnscsi 17423 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17424 | The slot for the distance ... |
| slotsdifdsndx 17425 | The index of the slot for ... |
| unifndx 17426 | Index value of the ~ df-un... |
| unifid 17427 | Utility theorem: index-ind... |
| unifndxnn 17428 | The index of the slot for ... |
| basendxltunifndx 17429 | The index of the slot for ... |
| unifndxnbasendx 17430 | The slot for the uniform s... |
| unifndxntsetndx 17431 | The slot for the uniform s... |
| slotsdifunifndx 17432 | The index of the slot for ... |
| ressunif 17433 | ` UnifSet ` is unaffected ... |
| odrngstr 17434 | Functionality of an ordere... |
| odrngbas 17435 | The base set of an ordered... |
| odrngplusg 17436 | The addition operation of ... |
| odrngmulr 17437 | The multiplication operati... |
| odrngtset 17438 | The open sets of an ordere... |
| odrngle 17439 | The order of an ordered me... |
| odrngds 17440 | The metric of an ordered m... |
| ressds 17441 | ` dist ` is unaffected by ... |
| homndx 17442 | Index value of the ~ df-ho... |
| homid 17443 | Utility theorem: index-ind... |
| ccondx 17444 | Index value of the ~ df-cc... |
| ccoid 17445 | Utility theorem: index-ind... |
| slotsbhcdif 17446 | The slots ` Base ` , ` Hom... |
| slotsdifplendx2 17447 | The index of the slot for ... |
| slotsdifocndx 17448 | The index of the slot for ... |
| resshom 17449 | ` Hom ` is unaffected by r... |
| ressco 17450 | ` comp ` is unaffected by ... |
| restfn 17455 | The subspace topology oper... |
| topnfn 17456 | The topology extractor fun... |
| restval 17457 | The subspace topology indu... |
| elrest 17458 | The predicate "is an open ... |
| elrestr 17459 | Sufficient condition for b... |
| 0rest 17460 | Value of the structure res... |
| restid2 17461 | The subspace topology over... |
| restsspw 17462 | The subspace topology is a... |
| firest 17463 | The finite intersections o... |
| restid 17464 | The subspace topology of t... |
| topnval 17465 | Value of the topology extr... |
| topnid 17466 | Value of the topology extr... |
| topnpropd 17467 | The topology extractor fun... |
| reldmprds 17479 | The structure product is a... |
| prdsbasex 17481 | Lemma for structure produc... |
| imasvalstr 17482 | An image structure value i... |
| prdsvalstr 17483 | Structure product value is... |
| prdsbaslem 17484 | Lemma for ~ prdsbas and si... |
| prdsvallem 17485 | Lemma for ~ prdsval . (Co... |
| prdsval 17486 | Value of the structure pro... |
| prdssca 17487 | Scalar ring of a structure... |
| prdsbas 17488 | Base set of a structure pr... |
| prdsplusg 17489 | Addition in a structure pr... |
| prdsmulr 17490 | Multiplication in a struct... |
| prdsvsca 17491 | Scalar multiplication in a... |
| prdsip 17492 | Inner product in a structu... |
| prdsle 17493 | Structure product weak ord... |
| prdsless 17494 | Closure of the order relat... |
| prdsds 17495 | Structure product distance... |
| prdsdsfn 17496 | Structure product distance... |
| prdstset 17497 | Structure product topology... |
| prdshom 17498 | Structure product hom-sets... |
| prdsco 17499 | Structure product composit... |
| prdsbas2 17500 | The base set of a structur... |
| prdsbasmpt 17501 | A constructed tuple is a p... |
| prdsbasfn 17502 | Points in the structure pr... |
| prdsbasprj 17503 | Each point in a structure ... |
| prdsplusgval 17504 | Value of a componentwise s... |
| prdsplusgfval 17505 | Value of a structure produ... |
| prdsmulrval 17506 | Value of a componentwise r... |
| prdsmulrfval 17507 | Value of a structure produ... |
| prdsleval 17508 | Value of the product order... |
| prdsdsval 17509 | Value of the metric in a s... |
| prdsvscaval 17510 | Scalar multiplication in a... |
| prdsvscafval 17511 | Scalar multiplication of a... |
| prdsbas3 17512 | The base set of an indexed... |
| prdsbasmpt2 17513 | A constructed tuple is a p... |
| prdsbascl 17514 | An element of the base has... |
| prdsdsval2 17515 | Value of the metric in a s... |
| prdsdsval3 17516 | Value of the metric in a s... |
| pwsval 17517 | Value of a structure power... |
| pwsbas 17518 | Base set of a structure po... |
| pwselbasb 17519 | Membership in the base set... |
| pwselbas 17520 | An element of a structure ... |
| pwselbasr 17521 | The reverse direction of ~... |
| pwsplusgval 17522 | Value of addition in a str... |
| pwsmulrval 17523 | Value of multiplication in... |
| pwsle 17524 | Ordering in a structure po... |
| pwsleval 17525 | Ordering in a structure po... |
| pwsvscafval 17526 | Scalar multiplication in a... |
| pwsvscaval 17527 | Scalar multiplication of a... |
| pwssca 17528 | The ring of scalars of a s... |
| pwsdiagel 17529 | Membership of diagonal ele... |
| pwssnf1o 17530 | Triviality of singleton po... |
| imasval 17543 | Value of an image structur... |
| imasbas 17544 | The base set of an image s... |
| imasds 17545 | The distance function of a... |
| imasdsfn 17546 | The distance function is a... |
| imasdsval 17547 | The distance function of a... |
| imasdsval2 17548 | The distance function of a... |
| imasplusg 17549 | The group operation in an ... |
| imasmulr 17550 | The ring multiplication in... |
| imassca 17551 | The scalar field of an ima... |
| imasvsca 17552 | The scalar multiplication ... |
| imasip 17553 | The inner product of an im... |
| imastset 17554 | The topology of an image s... |
| imasle 17555 | The ordering of an image s... |
| f1ocpbllem 17556 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17557 | An injection is compatible... |
| f1ovscpbl 17558 | An injection is compatible... |
| f1olecpbl 17559 | An injection is compatible... |
| imasaddfnlem 17560 | The image structure operat... |
| imasaddvallem 17561 | The operation of an image ... |
| imasaddflem 17562 | The image set operations a... |
| imasaddfn 17563 | The image structure's grou... |
| imasaddval 17564 | The value of an image stru... |
| imasaddf 17565 | The image structure's grou... |
| imasmulfn 17566 | The image structure's ring... |
| imasmulval 17567 | The value of an image stru... |
| imasmulf 17568 | The image structure's ring... |
| imasvscafn 17569 | The image structure's scal... |
| imasvscaval 17570 | The value of an image stru... |
| imasvscaf 17571 | The image structure's scal... |
| imasless 17572 | The order relation defined... |
| imasleval 17573 | The value of the image str... |
| qusval 17574 | Value of a quotient struct... |
| quslem 17575 | The function in ~ qusval i... |
| qusin 17576 | Restrict the equivalence r... |
| qusbas 17577 | Base set of a quotient str... |
| quss 17578 | The scalar field of a quot... |
| divsfval 17579 | Value of the function in ~... |
| ercpbllem 17580 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17581 | Translate the function com... |
| erlecpbl 17582 | Translate the relation com... |
| qusaddvallem 17583 | Value of an operation defi... |
| qusaddflem 17584 | The operation of a quotien... |
| qusaddval 17585 | The addition in a quotient... |
| qusaddf 17586 | The addition in a quotient... |
| qusmulval 17587 | The multiplication in a qu... |
| qusmulf 17588 | The multiplication in a qu... |
| fnpr2o 17589 | Function with a domain of ... |
| fnpr2ob 17590 | Biconditional version of ~... |
| fvpr0o 17591 | The value of a function wi... |
| fvpr1o 17592 | The value of a function wi... |
| fvprif 17593 | The value of the pair func... |
| xpsfrnel 17594 | Elementhood in the target ... |
| xpsfeq 17595 | A function on ` 2o ` is de... |
| xpsfrnel2 17596 | Elementhood in the target ... |
| xpscf 17597 | Equivalent condition for t... |
| xpsfval 17598 | The value of the function ... |
| xpsff1o 17599 | The function appearing in ... |
| xpsfrn 17600 | A short expression for the... |
| xpsff1o2 17601 | The function appearing in ... |
| xpsval 17602 | Value of the binary struct... |
| xpsrnbas 17603 | The indexed structure prod... |
| xpsbas 17604 | The base set of the binary... |
| xpsaddlem 17605 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17606 | Value of the addition oper... |
| xpsmul 17607 | Value of the multiplicatio... |
| xpssca 17608 | Value of the scalar field ... |
| xpsvsca 17609 | Value of the scalar multip... |
| xpsless 17610 | Closure of the ordering in... |
| xpsle 17611 | Value of the ordering in a... |
| ismre 17620 | Property of being a Moore ... |
| fnmre 17621 | The Moore collection gener... |
| mresspw 17622 | A Moore collection is a su... |
| mress 17623 | A Moore-closed subset is a... |
| mre1cl 17624 | In any Moore collection th... |
| mreintcl 17625 | A nonempty collection of c... |
| mreiincl 17626 | A nonempty indexed interse... |
| mrerintcl 17627 | The relative intersection ... |
| mreriincl 17628 | The relative intersection ... |
| mreincl 17629 | Two closed sets have a clo... |
| mreuni 17630 | Since the entire base set ... |
| mreunirn 17631 | Two ways to express the no... |
| ismred 17632 | Properties that determine ... |
| ismred2 17633 | Properties that determine ... |
| mremre 17634 | The Moore collections of s... |
| submre 17635 | The subcollection of a clo... |
| xrsle 17636 | The ordering of the extend... |
| xrge0le 17637 | The "less than or equal to... |
| xrsbas 17638 | The base set of the extend... |
| xrge0base 17639 | The base of the extended n... |
| mrcflem 17640 | The domain and codomain of... |
| fnmrc 17641 | Moore-closure is a well-be... |
| mrcfval 17642 | Value of the function expr... |
| mrcf 17643 | The Moore closure is a fun... |
| mrcval 17644 | Evaluation of the Moore cl... |
| mrccl 17645 | The Moore closure of a set... |
| mrcsncl 17646 | The Moore closure of a sin... |
| mrcid 17647 | The closure of a closed se... |
| mrcssv 17648 | The closure of a set is a ... |
| mrcidb 17649 | A set is closed iff it is ... |
| mrcss 17650 | Closure preserves subset o... |
| mrcssid 17651 | The closure of a set is a ... |
| mrcidb2 17652 | A set is closed iff it con... |
| mrcidm 17653 | The closure operation is i... |
| mrcsscl 17654 | The closure is the minimal... |
| mrcuni 17655 | Idempotence of closure und... |
| mrcun 17656 | Idempotence of closure und... |
| mrcssvd 17657 | The Moore closure of a set... |
| mrcssd 17658 | Moore closure preserves su... |
| mrcssidd 17659 | A set is contained in its ... |
| mrcidmd 17660 | Moore closure is idempoten... |
| mressmrcd 17661 | In a Moore system, if a se... |
| submrc 17662 | In a closure system which ... |
| mrieqvlemd 17663 | In a Moore system, if ` Y ... |
| mrisval 17664 | Value of the set of indepe... |
| ismri 17665 | Criterion for a set to be ... |
| ismri2 17666 | Criterion for a subset of ... |
| ismri2d 17667 | Criterion for a subset of ... |
| ismri2dd 17668 | Definition of independence... |
| mriss 17669 | An independent set of a Mo... |
| mrissd 17670 | An independent set of a Mo... |
| ismri2dad 17671 | Consequence of a set in a ... |
| mrieqvd 17672 | In a Moore system, a set i... |
| mrieqv2d 17673 | In a Moore system, a set i... |
| mrissmrcd 17674 | In a Moore system, if an i... |
| mrissmrid 17675 | In a Moore system, subsets... |
| mreexd 17676 | In a Moore system, the clo... |
| mreexmrid 17677 | In a Moore system whose cl... |
| mreexexlemd 17678 | This lemma is used to gene... |
| mreexexlem2d 17679 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17680 | Base case of the induction... |
| mreexexlem4d 17681 | Induction step of the indu... |
| mreexexd 17682 | Exchange-type theorem. In... |
| mreexdomd 17683 | In a Moore system whose cl... |
| mreexfidimd 17684 | In a Moore system whose cl... |
| isacs 17685 | A set is an algebraic clos... |
| acsmre 17686 | Algebraic closure systems ... |
| isacs2 17687 | In the definition of an al... |
| acsfiel 17688 | A set is closed in an alge... |
| acsfiel2 17689 | A set is closed in an alge... |
| acsmred 17690 | An algebraic closure syste... |
| isacs1i 17691 | A closure system determine... |
| mreacs 17692 | Algebraicity is a composab... |
| acsfn 17693 | Algebraicity of a conditio... |
| acsfn0 17694 | Algebraicity of a point cl... |
| acsfn1 17695 | Algebraicity of a one-argu... |
| acsfn1c 17696 | Algebraicity of a one-argu... |
| acsfn2 17697 | Algebraicity of a two-argu... |
| iscat 17706 | The predicate "is a catego... |
| iscatd 17707 | Properties that determine ... |
| catidex 17708 | Each object in a category ... |
| catideu 17709 | Each object in a category ... |
| cidfval 17710 | Each object in a category ... |
| cidval 17711 | Each object in a category ... |
| cidffn 17712 | The identity arrow constru... |
| cidfn 17713 | The identity arrow operato... |
| catidd 17714 | Deduce the identity arrow ... |
| iscatd2 17715 | Version of ~ iscatd with a... |
| catidcl 17716 | Each object in a category ... |
| catlid 17717 | Left identity property of ... |
| catrid 17718 | Right identity property of... |
| catcocl 17719 | Closure of a composition a... |
| catass 17720 | Associativity of compositi... |
| catcone0 17721 | Composition of non-empty h... |
| 0catg 17722 | Any structure with an empt... |
| 0cat 17723 | The empty set is a categor... |
| homffval 17724 | Value of the functionalize... |
| fnhomeqhomf 17725 | If the Hom-set operation i... |
| homfval 17726 | Value of the functionalize... |
| homffn 17727 | The functionalized Hom-set... |
| homfeq 17728 | Condition for two categori... |
| homfeqd 17729 | If two structures have the... |
| homfeqbas 17730 | Deduce equality of base se... |
| homfeqval 17731 | Value of the functionalize... |
| comfffval 17732 | Value of the functionalize... |
| comffval 17733 | Value of the functionalize... |
| comfval 17734 | Value of the functionalize... |
| comfffval2 17735 | Value of the functionalize... |
| comffval2 17736 | Value of the functionalize... |
| comfval2 17737 | Value of the functionalize... |
| comfffn 17738 | The functionalized composi... |
| comffn 17739 | The functionalized composi... |
| comfeq 17740 | Condition for two categori... |
| comfeqd 17741 | Condition for two categori... |
| comfeqval 17742 | Equality of two compositio... |
| catpropd 17743 | Two structures with the sa... |
| cidpropd 17744 | Two structures with the sa... |
| oppcval 17747 | Value of the opposite cate... |
| oppchomfval 17748 | Hom-sets of the opposite c... |
| oppchom 17749 | Hom-sets of the opposite c... |
| oppccofval 17750 | Composition in the opposit... |
| oppcco 17751 | Composition in the opposit... |
| oppcbas 17752 | Base set of an opposite ca... |
| oppccatid 17753 | Lemma for ~ oppccat . (Co... |
| oppchomf 17754 | Hom-sets of the opposite c... |
| oppcid 17755 | Identity function of an op... |
| oppccat 17756 | An opposite category is a ... |
| 2oppcbas 17757 | The double opposite catego... |
| 2oppchomf 17758 | The double opposite catego... |
| 2oppccomf 17759 | The double opposite catego... |
| oppchomfpropd 17760 | If two categories have the... |
| oppccomfpropd 17761 | If two categories have the... |
| oppccatf 17762 | ` oppCat ` restricted to `... |
| monfval 17767 | Definition of a monomorphi... |
| ismon 17768 | Definition of a monomorphi... |
| ismon2 17769 | Write out the monomorphism... |
| monhom 17770 | A monomorphism is a morphi... |
| moni 17771 | Property of a monomorphism... |
| monpropd 17772 | If two categories have the... |
| oppcmon 17773 | A monomorphism in the oppo... |
| oppcepi 17774 | An epimorphism in the oppo... |
| isepi 17775 | Definition of an epimorphi... |
| isepi2 17776 | Write out the epimorphism ... |
| epihom 17777 | An epimorphism is a morphi... |
| epii 17778 | Property of an epimorphism... |
| sectffval 17785 | Value of the section opera... |
| sectfval 17786 | Value of the section relat... |
| sectss 17787 | The section relation is a ... |
| issect 17788 | The property " ` F ` is a ... |
| issect2 17789 | Property of being a sectio... |
| sectcan 17790 | If ` G ` is a section of `... |
| sectco 17791 | Composition of two section... |
| isofval 17792 | Function value of the func... |
| invffval 17793 | Value of the inverse relat... |
| invfval 17794 | Value of the inverse relat... |
| isinv 17795 | Value of the inverse relat... |
| invss 17796 | The inverse relation is a ... |
| invsym 17797 | The inverse relation is sy... |
| invsym2 17798 | The inverse relation is sy... |
| invfun 17799 | The inverse relation is a ... |
| isoval 17800 | The isomorphisms are the d... |
| inviso1 17801 | If ` G ` is an inverse to ... |
| inviso2 17802 | If ` G ` is an inverse to ... |
| invf 17803 | The inverse relation is a ... |
| invf1o 17804 | The inverse relation is a ... |
| invinv 17805 | The inverse of the inverse... |
| invco 17806 | The composition of two iso... |
| dfiso2 17807 | Alternate definition of an... |
| dfiso3 17808 | Alternate definition of an... |
| inveq 17809 | If there are two inverses ... |
| isofn 17810 | The function value of the ... |
| isohom 17811 | An isomorphism is a homomo... |
| isoco 17812 | The composition of two iso... |
| oppcsect 17813 | A section in the opposite ... |
| oppcsect2 17814 | A section in the opposite ... |
| oppcinv 17815 | An inverse in the opposite... |
| oppciso 17816 | An isomorphism in the oppo... |
| sectmon 17817 | If ` F ` is a section of `... |
| monsect 17818 | If ` F ` is a monomorphism... |
| sectepi 17819 | If ` F ` is a section of `... |
| episect 17820 | If ` F ` is an epimorphism... |
| sectid 17821 | The identity is a section ... |
| invid 17822 | The inverse of the identit... |
| idiso 17823 | The identity is an isomorp... |
| idinv 17824 | The inverse of the identit... |
| invisoinvl 17825 | The inverse of an isomorph... |
| invisoinvr 17826 | The inverse of an isomorph... |
| invcoisoid 17827 | The inverse of an isomorph... |
| isocoinvid 17828 | The inverse of an isomorph... |
| rcaninv 17829 | Right cancellation of an i... |
| cicfval 17832 | The set of isomorphic obje... |
| brcic 17833 | The relation "is isomorphi... |
| cic 17834 | Objects ` X ` and ` Y ` in... |
| brcici 17835 | Prove that two objects are... |
| cicref 17836 | Isomorphism is reflexive. ... |
| ciclcl 17837 | Isomorphism implies the le... |
| cicrcl 17838 | Isomorphism implies the ri... |
| cicsym 17839 | Isomorphism is symmetric. ... |
| cictr 17840 | Isomorphism is transitive.... |
| cicer 17841 | Isomorphism is an equivale... |
| sscrel 17848 | The subcategory subset rel... |
| brssc 17849 | The subcategory subset rel... |
| sscpwex 17850 | An analogue of ~ pwex for ... |
| subcrcl 17851 | Reverse closure for the su... |
| sscfn1 17852 | The subcategory subset rel... |
| sscfn2 17853 | The subcategory subset rel... |
| ssclem 17854 | Lemma for ~ ssc1 and simil... |
| isssc 17855 | Value of the subcategory s... |
| ssc1 17856 | Infer subset relation on o... |
| ssc2 17857 | Infer subset relation on m... |
| sscres 17858 | Any function restricted to... |
| sscid 17859 | The subcategory subset rel... |
| ssctr 17860 | The subcategory subset rel... |
| ssceq 17861 | The subcategory subset rel... |
| rescval 17862 | Value of the category rest... |
| rescval2 17863 | Value of the category rest... |
| rescbas 17864 | Base set of the category r... |
| reschom 17865 | Hom-sets of the category r... |
| reschomf 17866 | Hom-sets of the category r... |
| rescco 17867 | Composition in the categor... |
| rescabs 17868 | Restriction absorption law... |
| rescabs2 17869 | Restriction absorption law... |
| issubc 17870 | Elementhood in the set of ... |
| issubc2 17871 | Elementhood in the set of ... |
| 0ssc 17872 | For any category ` C ` , t... |
| 0subcat 17873 | For any category ` C ` , t... |
| catsubcat 17874 | For any category ` C ` , `... |
| subcssc 17875 | An element in the set of s... |
| subcfn 17876 | An element in the set of s... |
| subcss1 17877 | The objects of a subcatego... |
| subcss2 17878 | The morphisms of a subcate... |
| subcidcl 17879 | The identity of the origin... |
| subccocl 17880 | A subcategory is closed un... |
| subccatid 17881 | A subcategory is a categor... |
| subcid 17882 | The identity in a subcateg... |
| subccat 17883 | A subcategory is a categor... |
| issubc3 17884 | Alternate definition of a ... |
| fullsubc 17885 | The full subcategory gener... |
| fullresc 17886 | The category formed by str... |
| resscat 17887 | A category restricted to a... |
| subsubc 17888 | A subcategory of a subcate... |
| relfunc 17897 | The set of functors is a r... |
| funcrcl 17898 | Reverse closure for a func... |
| isfunc 17899 | Value of the set of functo... |
| isfuncd 17900 | Deduce that an operation i... |
| funcf1 17901 | The object part of a funct... |
| funcixp 17902 | The morphism part of a fun... |
| funcf2 17903 | The morphism part of a fun... |
| funcfn2 17904 | The morphism part of a fun... |
| funcid 17905 | A functor maps each identi... |
| funcco 17906 | A functor maps composition... |
| funcsect 17907 | The image of a section und... |
| funcinv 17908 | The image of an inverse un... |
| funciso 17909 | The image of an isomorphis... |
| funcoppc 17910 | A functor on categories yi... |
| idfuval 17911 | Value of the identity func... |
| idfu2nd 17912 | Value of the morphism part... |
| idfu2 17913 | Value of the morphism part... |
| idfu1st 17914 | Value of the object part o... |
| idfu1 17915 | Value of the object part o... |
| idfucl 17916 | The identity functor is a ... |
| cofuval 17917 | Value of the composition o... |
| cofu1st 17918 | Value of the object part o... |
| cofu1 17919 | Value of the object part o... |
| cofu2nd 17920 | Value of the morphism part... |
| cofu2 17921 | Value of the morphism part... |
| cofuval2 17922 | Value of the composition o... |
| cofucl 17923 | The composition of two fun... |
| cofuass 17924 | Functor composition is ass... |
| cofulid 17925 | The identity functor is a ... |
| cofurid 17926 | The identity functor is a ... |
| resfval 17927 | Value of the functor restr... |
| resfval2 17928 | Value of the functor restr... |
| resf1st 17929 | Value of the functor restr... |
| resf2nd 17930 | Value of the functor restr... |
| funcres 17931 | A functor restricted to a ... |
| funcres2b 17932 | Condition for a functor to... |
| funcres2 17933 | A functor into a restricte... |
| idfusubc0 17934 | The identity functor for a... |
| idfusubc 17935 | The identity functor for a... |
| wunfunc 17936 | A weak universe is closed ... |
| funcpropd 17937 | If two categories have the... |
| funcres2c 17938 | Condition for a functor to... |
| fullfunc 17943 | A full functor is a functo... |
| fthfunc 17944 | A faithful functor is a fu... |
| relfull 17945 | The set of full functors i... |
| relfth 17946 | The set of faithful functo... |
| isfull 17947 | Value of the set of full f... |
| isfull2 17948 | Equivalent condition for a... |
| fullfo 17949 | The morphism map of a full... |
| fulli 17950 | The morphism map of a full... |
| isfth 17951 | Value of the set of faithf... |
| isfth2 17952 | Equivalent condition for a... |
| isffth2 17953 | A fully faithful functor i... |
| fthf1 17954 | The morphism map of a fait... |
| fthi 17955 | The morphism map of a fait... |
| ffthf1o 17956 | The morphism map of a full... |
| fullpropd 17957 | If two categories have the... |
| fthpropd 17958 | If two categories have the... |
| fulloppc 17959 | The opposite functor of a ... |
| fthoppc 17960 | The opposite functor of a ... |
| ffthoppc 17961 | The opposite functor of a ... |
| fthsect 17962 | A faithful functor reflect... |
| fthinv 17963 | A faithful functor reflect... |
| fthmon 17964 | A faithful functor reflect... |
| fthepi 17965 | A faithful functor reflect... |
| ffthiso 17966 | A fully faithful functor r... |
| fthres2b 17967 | Condition for a faithful f... |
| fthres2c 17968 | Condition for a faithful f... |
| fthres2 17969 | A faithful functor into a ... |
| idffth 17970 | The identity functor is a ... |
| cofull 17971 | The composition of two ful... |
| cofth 17972 | The composition of two fai... |
| coffth 17973 | The composition of two ful... |
| rescfth 17974 | The inclusion functor from... |
| ressffth 17975 | The inclusion functor from... |
| fullres2c 17976 | Condition for a full funct... |
| ffthres2c 17977 | Condition for a fully fait... |
| inclfusubc 17978 | The "inclusion functor" fr... |
| fnfuc 17983 | The ` FuncCat ` operation ... |
| natfval 17984 | Value of the function givi... |
| isnat 17985 | Property of being a natura... |
| isnat2 17986 | Property of being a natura... |
| natffn 17987 | The natural transformation... |
| natrcl 17988 | Reverse closure for a natu... |
| nat1st2nd 17989 | Rewrite the natural transf... |
| natixp 17990 | A natural transformation i... |
| natcl 17991 | A component of a natural t... |
| natfn 17992 | A natural transformation i... |
| nati 17993 | Naturality property of a n... |
| wunnat 17994 | A weak universe is closed ... |
| catstr 17995 | A category structure is a ... |
| fucval 17996 | Value of the functor categ... |
| fuccofval 17997 | Value of the functor categ... |
| fucbas 17998 | The objects of the functor... |
| fuchom 17999 | The morphisms in the funct... |
| fucco 18000 | Value of the composition o... |
| fuccoval 18001 | Value of the functor categ... |
| fuccocl 18002 | The composition of two nat... |
| fucidcl 18003 | The identity natural trans... |
| fuclid 18004 | Left identity of natural t... |
| fucrid 18005 | Right identity of natural ... |
| fucass 18006 | Associativity of natural t... |
| fuccatid 18007 | The functor category is a ... |
| fuccat 18008 | The functor category is a ... |
| fucid 18009 | The identity morphism in t... |
| fucsect 18010 | Two natural transformation... |
| fucinv 18011 | Two natural transformation... |
| invfuc 18012 | If ` V ( x ) ` is an inver... |
| fuciso 18013 | A natural transformation i... |
| natpropd 18014 | If two categories have the... |
| fucpropd 18015 | If two categories have the... |
| initofn 18022 | ` InitO ` is a function on... |
| termofn 18023 | ` TermO ` is a function on... |
| zeroofn 18024 | ` ZeroO ` is a function on... |
| initorcl 18025 | Reverse closure for an ini... |
| termorcl 18026 | Reverse closure for a term... |
| zeroorcl 18027 | Reverse closure for a zero... |
| initoval 18028 | The value of the initial o... |
| termoval 18029 | The value of the terminal ... |
| zerooval 18030 | The value of the zero obje... |
| isinito 18031 | The predicate "is an initi... |
| istermo 18032 | The predicate "is a termin... |
| iszeroo 18033 | The predicate "is a zero o... |
| isinitoi 18034 | Implication of a class bei... |
| istermoi 18035 | Implication of a class bei... |
| initoid 18036 | For an initial object, the... |
| termoid 18037 | For a terminal object, the... |
| dfinito2 18038 | An initial object is a ter... |
| dftermo2 18039 | A terminal object is an in... |
| dfinito3 18040 | An alternate definition of... |
| dftermo3 18041 | An alternate definition of... |
| initoo 18042 | An initial object is an ob... |
| termoo 18043 | A terminal object is an ob... |
| iszeroi 18044 | Implication of a class bei... |
| 2initoinv 18045 | Morphisms between two init... |
| initoeu1 18046 | Initial objects are essent... |
| initoeu1w 18047 | Initial objects are essent... |
| initoeu2lem0 18048 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 18049 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 18050 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 18051 | Initial objects are essent... |
| 2termoinv 18052 | Morphisms between two term... |
| termoeu1 18053 | Terminal objects are essen... |
| termoeu1w 18054 | Terminal objects are essen... |
| homarcl 18063 | Reverse closure for an arr... |
| homafval 18064 | Value of the disjointified... |
| homaf 18065 | Functionality of the disjo... |
| homaval 18066 | Value of the disjointified... |
| elhoma 18067 | Value of the disjointified... |
| elhomai 18068 | Produce an arrow from a mo... |
| elhomai2 18069 | Produce an arrow from a mo... |
| homarcl2 18070 | Reverse closure for the do... |
| homarel 18071 | An arrow is an ordered pai... |
| homa1 18072 | The first component of an ... |
| homahom2 18073 | The second component of an... |
| homahom 18074 | The second component of an... |
| homadm 18075 | The domain of an arrow wit... |
| homacd 18076 | The codomain of an arrow w... |
| homadmcd 18077 | Decompose an arrow into do... |
| arwval 18078 | The set of arrows is the u... |
| arwrcl 18079 | The first component of an ... |
| arwhoma 18080 | An arrow is contained in t... |
| homarw 18081 | A hom-set is a subset of t... |
| arwdm 18082 | The domain of an arrow is ... |
| arwcd 18083 | The codomain of an arrow i... |
| dmaf 18084 | The domain function is a f... |
| cdaf 18085 | The codomain function is a... |
| arwhom 18086 | The second component of an... |
| arwdmcd 18087 | Decompose an arrow into do... |
| idafval 18092 | Value of the identity arro... |
| idaval 18093 | Value of the identity arro... |
| ida2 18094 | Morphism part of the ident... |
| idahom 18095 | Domain and codomain of the... |
| idadm 18096 | Domain of the identity arr... |
| idacd 18097 | Codomain of the identity a... |
| idaf 18098 | The identity arrow functio... |
| coafval 18099 | The value of the compositi... |
| eldmcoa 18100 | A pair ` <. G , F >. ` is ... |
| dmcoass 18101 | The domain of composition ... |
| homdmcoa 18102 | If ` F : X --> Y ` and ` G... |
| coaval 18103 | Value of composition for c... |
| coa2 18104 | The morphism part of arrow... |
| coahom 18105 | The composition of two com... |
| coapm 18106 | Composition of arrows is a... |
| arwlid 18107 | Left identity of a categor... |
| arwrid 18108 | Right identity of a catego... |
| arwass 18109 | Associativity of compositi... |
| setcval 18112 | Value of the category of s... |
| setcbas 18113 | Set of objects of the cate... |
| setchomfval 18114 | Set of arrows of the categ... |
| setchom 18115 | Set of arrows of the categ... |
| elsetchom 18116 | A morphism of sets is a fu... |
| setccofval 18117 | Composition in the categor... |
| setcco 18118 | Composition in the categor... |
| setccatid 18119 | Lemma for ~ setccat . (Co... |
| setccat 18120 | The category of sets is a ... |
| setcid 18121 | The identity arrow in the ... |
| setcmon 18122 | A monomorphism of sets is ... |
| setcepi 18123 | An epimorphism of sets is ... |
| setcsect 18124 | A section in the category ... |
| setcinv 18125 | An inverse in the category... |
| setciso 18126 | An isomorphism in the cate... |
| resssetc 18127 | The restriction of the cat... |
| funcsetcres2 18128 | A functor into a smaller c... |
| setc2obas 18129 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18130 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18131 | The category of sets in a ... |
| cat1 18132 | The definition of category... |
| catcval 18135 | Value of the category of c... |
| catcbas 18136 | Set of objects of the cate... |
| catchomfval 18137 | Set of arrows of the categ... |
| catchom 18138 | Set of arrows of the categ... |
| catccofval 18139 | Composition in the categor... |
| catcco 18140 | Composition in the categor... |
| catccatid 18141 | Lemma for ~ catccat . (Co... |
| catcid 18142 | The identity arrow in the ... |
| catccat 18143 | The category of categories... |
| resscatc 18144 | The restriction of the cat... |
| catcisolem 18145 | Lemma for ~ catciso . (Co... |
| catciso 18146 | A functor is an isomorphis... |
| catcbascl 18147 | An element of the base set... |
| catcslotelcl 18148 | A slot entry of an element... |
| catcbaselcl 18149 | The base set of an element... |
| catchomcl 18150 | The Hom-set of an element ... |
| catcccocl 18151 | The composition operation ... |
| catcoppccl 18152 | The category of categories... |
| catcfuccl 18153 | The category of categories... |
| fncnvimaeqv 18154 | The inverse images of the ... |
| bascnvimaeqv 18155 | The inverse image of the u... |
| estrcval 18158 | Value of the category of e... |
| estrcbas 18159 | Set of objects of the cate... |
| estrchomfval 18160 | Set of morphisms ("arrows"... |
| estrchom 18161 | The morphisms between exte... |
| elestrchom 18162 | A morphism between extensi... |
| estrccofval 18163 | Composition in the categor... |
| estrcco 18164 | Composition in the categor... |
| estrcbasbas 18165 | An element of the base set... |
| estrccatid 18166 | Lemma for ~ estrccat . (C... |
| estrccat 18167 | The category of extensible... |
| estrcid 18168 | The identity arrow in the ... |
| estrchomfn 18169 | The Hom-set operation in t... |
| estrchomfeqhom 18170 | The functionalized Hom-set... |
| estrreslem1 18171 | Lemma 1 for ~ estrres . (... |
| estrreslem2 18172 | Lemma 2 for ~ estrres . (... |
| estrres 18173 | Any restriction of a categ... |
| funcestrcsetclem1 18174 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18175 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18176 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18177 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18178 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18179 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18180 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18181 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18182 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18183 | The "natural forgetful fun... |
| fthestrcsetc 18184 | The "natural forgetful fun... |
| fullestrcsetc 18185 | The "natural forgetful fun... |
| equivestrcsetc 18186 | The "natural forgetful fun... |
| setc1strwun 18187 | A constructed one-slot str... |
| funcsetcestrclem1 18188 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18189 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18190 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18191 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18192 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18193 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18194 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18195 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18196 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18197 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18198 | The "embedding functor" fr... |
| fthsetcestrc 18199 | The "embedding functor" fr... |
| fullsetcestrc 18200 | The "embedding functor" fr... |
| embedsetcestrc 18201 | The "embedding functor" fr... |
| fnxpc 18210 | The binary product of cate... |
| xpcval 18211 | Value of the binary produc... |
| xpcbas 18212 | Set of objects of the bina... |
| xpchomfval 18213 | Set of morphisms of the bi... |
| xpchom 18214 | Set of morphisms of the bi... |
| relxpchom 18215 | A hom-set in the binary pr... |
| xpccofval 18216 | Value of composition in th... |
| xpcco 18217 | Value of composition in th... |
| xpcco1st 18218 | Value of composition in th... |
| xpcco2nd 18219 | Value of composition in th... |
| xpchom2 18220 | Value of the set of morphi... |
| xpcco2 18221 | Value of composition in th... |
| xpccatid 18222 | The product of two categor... |
| xpcid 18223 | The identity morphism in t... |
| xpccat 18224 | The product of two categor... |
| 1stfval 18225 | Value of the first project... |
| 1stf1 18226 | Value of the first project... |
| 1stf2 18227 | Value of the first project... |
| 2ndfval 18228 | Value of the first project... |
| 2ndf1 18229 | Value of the first project... |
| 2ndf2 18230 | Value of the first project... |
| 1stfcl 18231 | The first projection funct... |
| 2ndfcl 18232 | The second projection func... |
| prfval 18233 | Value of the pairing funct... |
| prf1 18234 | Value of the pairing funct... |
| prf2fval 18235 | Value of the pairing funct... |
| prf2 18236 | Value of the pairing funct... |
| prfcl 18237 | The pairing of functors ` ... |
| prf1st 18238 | Cancellation of pairing wi... |
| prf2nd 18239 | Cancellation of pairing wi... |
| 1st2ndprf 18240 | Break a functor into a pro... |
| catcxpccl 18241 | The category of categories... |
| xpcpropd 18242 | If two categories have the... |
| evlfval 18251 | Value of the evaluation fu... |
| evlf2 18252 | Value of the evaluation fu... |
| evlf2val 18253 | Value of the evaluation na... |
| evlf1 18254 | Value of the evaluation fu... |
| evlfcllem 18255 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18256 | The evaluation functor is ... |
| curfval 18257 | Value of the curry functor... |
| curf1fval 18258 | Value of the object part o... |
| curf1 18259 | Value of the object part o... |
| curf11 18260 | Value of the double evalua... |
| curf12 18261 | The partially evaluated cu... |
| curf1cl 18262 | The partially evaluated cu... |
| curf2 18263 | Value of the curry functor... |
| curf2val 18264 | Value of a component of th... |
| curf2cl 18265 | The curry functor at a mor... |
| curfcl 18266 | The curry functor of a fun... |
| curfpropd 18267 | If two categories have the... |
| uncfval 18268 | Value of the uncurry funct... |
| uncfcl 18269 | The uncurry operation take... |
| uncf1 18270 | Value of the uncurry funct... |
| uncf2 18271 | Value of the uncurry funct... |
| curfuncf 18272 | Cancellation of curry with... |
| uncfcurf 18273 | Cancellation of uncurry wi... |
| diagval 18274 | Define the diagonal functo... |
| diagcl 18275 | The diagonal functor is a ... |
| diag1cl 18276 | The constant functor of ` ... |
| diag11 18277 | Value of the constant func... |
| diag12 18278 | Value of the constant func... |
| diag2 18279 | Value of the diagonal func... |
| diag2cl 18280 | The diagonal functor at a ... |
| curf2ndf 18281 | As shown in ~ diagval , th... |
| hofval 18286 | Value of the Hom functor, ... |
| hof1fval 18287 | The object part of the Hom... |
| hof1 18288 | The object part of the Hom... |
| hof2fval 18289 | The morphism part of the H... |
| hof2val 18290 | The morphism part of the H... |
| hof2 18291 | The morphism part of the H... |
| hofcllem 18292 | Lemma for ~ hofcl . (Cont... |
| hofcl 18293 | Closure of the Hom functor... |
| oppchofcl 18294 | Closure of the opposite Ho... |
| yonval 18295 | Value of the Yoneda embedd... |
| yoncl 18296 | The Yoneda embedding is a ... |
| yon1cl 18297 | The Yoneda embedding at an... |
| yon11 18298 | Value of the Yoneda embedd... |
| yon12 18299 | Value of the Yoneda embedd... |
| yon2 18300 | Value of the Yoneda embedd... |
| hofpropd 18301 | If two categories have the... |
| yonpropd 18302 | If two categories have the... |
| oppcyon 18303 | Value of the opposite Yone... |
| oyoncl 18304 | The opposite Yoneda embedd... |
| oyon1cl 18305 | The opposite Yoneda embedd... |
| yonedalem1 18306 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18307 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18308 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18309 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18310 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18311 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18312 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18313 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18314 | Lemma for ~ yoneda . (Con... |
| yonedainv 18315 | The Yoneda Lemma with expl... |
| yonffthlem 18316 | Lemma for ~ yonffth . (Co... |
| yoneda 18317 | The Yoneda Lemma. There i... |
| yonffth 18318 | The Yoneda Lemma. The Yon... |
| yoniso 18319 | If the codomain is recover... |
| oduval 18322 | Value of an order dual str... |
| oduleval 18323 | Value of the less-equal re... |
| oduleg 18324 | Truth of the less-equal re... |
| odubas 18325 | Base set of an order dual ... |
| isprs 18330 | Property of being a preord... |
| prslem 18331 | Lemma for ~ prsref and ~ p... |
| prsref 18332 | "Less than or equal to" is... |
| prstr 18333 | "Less than or equal to" is... |
| oduprs 18334 | Being a proset is a self-d... |
| isdrs 18335 | Property of being a direct... |
| drsdir 18336 | Direction of a directed se... |
| drsprs 18337 | A directed set is a proset... |
| drsbn0 18338 | The base of a directed set... |
| drsdirfi 18339 | Any _finite_ number of ele... |
| isdrs2 18340 | Directed sets may be defin... |
| ispos 18348 | The predicate "is a poset"... |
| ispos2 18349 | A poset is an antisymmetri... |
| posprs 18350 | A poset is a proset. (Con... |
| posi 18351 | Lemma for poset properties... |
| posref 18352 | A poset ordering is reflex... |
| posasymb 18353 | A poset ordering is asymme... |
| postr 18354 | A poset ordering is transi... |
| 0pos 18355 | Technical lemma to simplif... |
| isposd 18356 | Properties that determine ... |
| isposi 18357 | Properties that determine ... |
| isposix 18358 | Properties that determine ... |
| pospropd 18359 | Posethood is determined on... |
| odupos 18360 | Being a poset is a self-du... |
| oduposb 18361 | Being a poset is a self-du... |
| pltfval 18363 | Value of the less-than rel... |
| pltval 18364 | Less-than relation. ( ~ d... |
| pltle 18365 | "Less than" implies "less ... |
| pltne 18366 | The "less than" relation i... |
| pltirr 18367 | The "less than" relation i... |
| pleval2i 18368 | One direction of ~ pleval2... |
| pleval2 18369 | "Less than or equal to" in... |
| pltnle 18370 | "Less than" implies not co... |
| pltval3 18371 | Alternate expression for t... |
| pltnlt 18372 | The less-than relation imp... |
| pltn2lp 18373 | The less-than relation has... |
| plttr 18374 | The less-than relation is ... |
| pltletr 18375 | Transitive law for chained... |
| plelttr 18376 | Transitive law for chained... |
| pospo 18377 | Write a poset structure in... |
| lubfval 18382 | Value of the least upper b... |
| lubdm 18383 | Domain of the least upper ... |
| lubfun 18384 | The LUB is a function. (C... |
| lubeldm 18385 | Member of the domain of th... |
| lubelss 18386 | A member of the domain of ... |
| lubeu 18387 | Unique existence proper of... |
| lubval 18388 | Value of the least upper b... |
| lubcl 18389 | The least upper bound func... |
| lubprop 18390 | Properties of greatest low... |
| luble 18391 | The greatest lower bound i... |
| lublecllem 18392 | Lemma for ~ lublecl and ~ ... |
| lublecl 18393 | The set of all elements le... |
| lubid 18394 | The LUB of elements less t... |
| glbfval 18395 | Value of the greatest lowe... |
| glbdm 18396 | Domain of the greatest low... |
| glbfun 18397 | The GLB is a function. (C... |
| glbeldm 18398 | Member of the domain of th... |
| glbelss 18399 | A member of the domain of ... |
| glbeu 18400 | Unique existence proper of... |
| glbval 18401 | Value of the greatest lowe... |
| glbcl 18402 | The least upper bound func... |
| glbprop 18403 | Properties of greatest low... |
| glble 18404 | The greatest lower bound i... |
| joinfval 18405 | Value of join function for... |
| joinfval2 18406 | Value of join function for... |
| joindm 18407 | Domain of join function fo... |
| joindef 18408 | Two ways to say that a joi... |
| joinval 18409 | Join value. Since both si... |
| joincl 18410 | Closure of join of element... |
| joindmss 18411 | Subset property of domain ... |
| joinval2lem 18412 | Lemma for ~ joinval2 and ~... |
| joinval2 18413 | Value of join for a poset ... |
| joineu 18414 | Uniqueness of join of elem... |
| joinlem 18415 | Lemma for join properties.... |
| lejoin1 18416 | A join's first argument is... |
| lejoin2 18417 | A join's second argument i... |
| joinle 18418 | A join is less than or equ... |
| meetfval 18419 | Value of meet function for... |
| meetfval2 18420 | Value of meet function for... |
| meetdm 18421 | Domain of meet function fo... |
| meetdef 18422 | Two ways to say that a mee... |
| meetval 18423 | Meet value. Since both si... |
| meetcl 18424 | Closure of meet of element... |
| meetdmss 18425 | Subset property of domain ... |
| meetval2lem 18426 | Lemma for ~ meetval2 and ~... |
| meetval2 18427 | Value of meet for a poset ... |
| meeteu 18428 | Uniqueness of meet of elem... |
| meetlem 18429 | Lemma for meet properties.... |
| lemeet1 18430 | A meet's first argument is... |
| lemeet2 18431 | A meet's second argument i... |
| meetle 18432 | A meet is less than or equ... |
| joincomALT 18433 | The join of a poset is com... |
| joincom 18434 | The join of a poset is com... |
| meetcomALT 18435 | The meet of a poset is com... |
| meetcom 18436 | The meet of a poset is com... |
| join0 18437 | Lemma for ~ odumeet . (Co... |
| meet0 18438 | Lemma for ~ odujoin . (Co... |
| odulub 18439 | Least upper bounds in a du... |
| odujoin 18440 | Joins in a dual order are ... |
| oduglb 18441 | Greatest lower bounds in a... |
| odumeet 18442 | Meets in a dual order are ... |
| poslubmo 18443 | Least upper bounds in a po... |
| posglbmo 18444 | Greatest lower bounds in a... |
| poslubd 18445 | Properties which determine... |
| poslubdg 18446 | Properties which determine... |
| posglbdg 18447 | Properties which determine... |
| istos 18450 | The predicate "is a toset"... |
| tosso 18451 | Write the totally ordered ... |
| tospos 18452 | A Toset is a Poset. (Cont... |
| tleile 18453 | In a Toset, any two elemen... |
| tltnle 18454 | In a Toset, "less than" is... |
| p0val 18459 | Value of poset zero. (Con... |
| p1val 18460 | Value of poset zero. (Con... |
| p0le 18461 | Any element is less than o... |
| ple1 18462 | Any element is less than o... |
| resspos 18463 | The restriction of a Poset... |
| resstos 18464 | The restriction of a Toset... |
| islat 18467 | The predicate "is a lattic... |
| odulatb 18468 | Being a lattice is self-du... |
| odulat 18469 | Being a lattice is self-du... |
| latcl2 18470 | The join and meet of any t... |
| latlem 18471 | Lemma for lattice properti... |
| latpos 18472 | A lattice is a poset. (Co... |
| latjcl 18473 | Closure of join operation ... |
| latmcl 18474 | Closure of meet operation ... |
| latref 18475 | A lattice ordering is refl... |
| latasymb 18476 | A lattice ordering is asym... |
| latasym 18477 | A lattice ordering is asym... |
| lattr 18478 | A lattice ordering is tran... |
| latasymd 18479 | Deduce equality from latti... |
| lattrd 18480 | A lattice ordering is tran... |
| latjcom 18481 | The join of a lattice comm... |
| latlej1 18482 | A join's first argument is... |
| latlej2 18483 | A join's second argument i... |
| latjle12 18484 | A join is less than or equ... |
| latleeqj1 18485 | "Less than or equal to" in... |
| latleeqj2 18486 | "Less than or equal to" in... |
| latjlej1 18487 | Add join to both sides of ... |
| latjlej2 18488 | Add join to both sides of ... |
| latjlej12 18489 | Add join to both sides of ... |
| latnlej 18490 | An idiom to express that a... |
| latnlej1l 18491 | An idiom to express that a... |
| latnlej1r 18492 | An idiom to express that a... |
| latnlej2 18493 | An idiom to express that a... |
| latnlej2l 18494 | An idiom to express that a... |
| latnlej2r 18495 | An idiom to express that a... |
| latjidm 18496 | Lattice join is idempotent... |
| latmcom 18497 | The join of a lattice comm... |
| latmle1 18498 | A meet is less than or equ... |
| latmle2 18499 | A meet is less than or equ... |
| latlem12 18500 | An element is less than or... |
| latleeqm1 18501 | "Less than or equal to" in... |
| latleeqm2 18502 | "Less than or equal to" in... |
| latmlem1 18503 | Add meet to both sides of ... |
| latmlem2 18504 | Add meet to both sides of ... |
| latmlem12 18505 | Add join to both sides of ... |
| latnlemlt 18506 | Negation of "less than or ... |
| latnle 18507 | Equivalent expressions for... |
| latmidm 18508 | Lattice meet is idempotent... |
| latabs1 18509 | Lattice absorption law. F... |
| latabs2 18510 | Lattice absorption law. F... |
| latledi 18511 | An ortholattice is distrib... |
| latmlej11 18512 | Ordering of a meet and joi... |
| latmlej12 18513 | Ordering of a meet and joi... |
| latmlej21 18514 | Ordering of a meet and joi... |
| latmlej22 18515 | Ordering of a meet and joi... |
| lubsn 18516 | The least upper bound of a... |
| latjass 18517 | Lattice join is associativ... |
| latj12 18518 | Swap 1st and 2nd members o... |
| latj32 18519 | Swap 2nd and 3rd members o... |
| latj13 18520 | Swap 1st and 3rd members o... |
| latj31 18521 | Swap 2nd and 3rd members o... |
| latjrot 18522 | Rotate lattice join of 3 c... |
| latj4 18523 | Rearrangement of lattice j... |
| latj4rot 18524 | Rotate lattice join of 4 c... |
| latjjdi 18525 | Lattice join distributes o... |
| latjjdir 18526 | Lattice join distributes o... |
| mod1ile 18527 | The weak direction of the ... |
| mod2ile 18528 | The weak direction of the ... |
| latmass 18529 | Lattice meet is associativ... |
| latdisdlem 18530 | Lemma for ~ latdisd . (Co... |
| latdisd 18531 | In a lattice, joins distri... |
| isclat 18534 | The predicate "is a comple... |
| clatpos 18535 | A complete lattice is a po... |
| clatlem 18536 | Lemma for properties of a ... |
| clatlubcl 18537 | Any subset of the base set... |
| clatlubcl2 18538 | Any subset of the base set... |
| clatglbcl 18539 | Any subset of the base set... |
| clatglbcl2 18540 | Any subset of the base set... |
| oduclatb 18541 | Being a complete lattice i... |
| clatl 18542 | A complete lattice is a la... |
| isglbd 18543 | Properties that determine ... |
| lublem 18544 | Lemma for the least upper ... |
| lubub 18545 | The LUB of a complete latt... |
| lubl 18546 | The LUB of a complete latt... |
| lubss 18547 | Subset law for least upper... |
| lubel 18548 | An element of a set is les... |
| lubun 18549 | The LUB of a union. (Cont... |
| clatglb 18550 | Properties of greatest low... |
| clatglble 18551 | The greatest lower bound i... |
| clatleglb 18552 | Two ways of expressing "le... |
| clatglbss 18553 | Subset law for greatest lo... |
| isdlat 18556 | Property of being a distri... |
| dlatmjdi 18557 | In a distributive lattice,... |
| dlatl 18558 | A distributive lattice is ... |
| odudlatb 18559 | The dual of a distributive... |
| dlatjmdi 18560 | In a distributive lattice,... |
| ipostr 18563 | The structure of ~ df-ipo ... |
| ipoval 18564 | Value of the inclusion pos... |
| ipobas 18565 | Base set of the inclusion ... |
| ipolerval 18566 | Relation of the inclusion ... |
| ipotset 18567 | Topology of the inclusion ... |
| ipole 18568 | Weak order condition of th... |
| ipolt 18569 | Strict order condition of ... |
| ipopos 18570 | The inclusion poset on a f... |
| isipodrs 18571 | Condition for a family of ... |
| ipodrscl 18572 | Direction by inclusion as ... |
| ipodrsfi 18573 | Finite upper bound propert... |
| fpwipodrs 18574 | The finite subsets of any ... |
| ipodrsima 18575 | The monotone image of a di... |
| isacs3lem 18576 | An algebraic closure syste... |
| acsdrsel 18577 | An algebraic closure syste... |
| isacs4lem 18578 | In a closure system in whi... |
| isacs5lem 18579 | If closure commutes with d... |
| acsdrscl 18580 | In an algebraic closure sy... |
| acsficl 18581 | A closure in an algebraic ... |
| isacs5 18582 | A closure system is algebr... |
| isacs4 18583 | A closure system is algebr... |
| isacs3 18584 | A closure system is algebr... |
| acsficld 18585 | In an algebraic closure sy... |
| acsficl2d 18586 | In an algebraic closure sy... |
| acsfiindd 18587 | In an algebraic closure sy... |
| acsmapd 18588 | In an algebraic closure sy... |
| acsmap2d 18589 | In an algebraic closure sy... |
| acsinfd 18590 | In an algebraic closure sy... |
| acsdomd 18591 | In an algebraic closure sy... |
| acsinfdimd 18592 | In an algebraic closure sy... |
| acsexdimd 18593 | In an algebraic closure sy... |
| mrelatglb 18594 | Greatest lower bounds in a... |
| mrelatglb0 18595 | The empty intersection in ... |
| mrelatlub 18596 | Least upper bounds in a Mo... |
| mreclatBAD 18597 | A Moore space is a complet... |
| isps 18602 | The predicate "is a poset"... |
| psrel 18603 | A poset is a relation. (C... |
| psref2 18604 | A poset is antisymmetric a... |
| pstr2 18605 | A poset is transitive. (C... |
| pslem 18606 | Lemma for ~ psref and othe... |
| psdmrn 18607 | The domain and range of a ... |
| psref 18608 | A poset is reflexive. (Co... |
| psrn 18609 | The range of a poset equal... |
| psasym 18610 | A poset is antisymmetric. ... |
| pstr 18611 | A poset is transitive. (C... |
| cnvps 18612 | The converse of a poset is... |
| cnvpsb 18613 | The converse of a poset is... |
| psss 18614 | Any subset of a partially ... |
| psssdm2 18615 | Field of a subposet. (Con... |
| psssdm 18616 | Field of a subposet. (Con... |
| istsr 18617 | The predicate is a toset. ... |
| istsr2 18618 | The predicate is a toset. ... |
| tsrlin 18619 | A toset is a linear order.... |
| tsrlemax 18620 | Two ways of saying a numbe... |
| tsrps 18621 | A toset is a poset. (Cont... |
| cnvtsr 18622 | The converse of a toset is... |
| tsrss 18623 | Any subset of a totally or... |
| ledm 18624 | The domain of ` <_ ` is ` ... |
| lern 18625 | The range of ` <_ ` is ` R... |
| lefld 18626 | The field of the 'less or ... |
| letsr 18627 | The "less than or equal to... |
| isdir 18632 | A condition for a relation... |
| reldir 18633 | A direction is a relation.... |
| dirdm 18634 | A direction's domain is eq... |
| dirref 18635 | A direction is reflexive. ... |
| dirtr 18636 | A direction is transitive.... |
| dirge 18637 | For any two elements of a ... |
| tsrdir 18638 | A totally ordered set is a... |
| ischn 18641 | Property of being a chain.... |
| chnwrd 18642 | A chain is an ordered sequ... |
| chnltm1 18643 | Basic property of a chain.... |
| pfxchn 18644 | A prefix of a chain is sti... |
| nfchnd 18645 | Bound-variable hypothesis ... |
| chneq1 18646 | Equality theorem for chain... |
| chneq2 18647 | Equality theorem for chain... |
| chneq12 18648 | Equality theorem for chain... |
| chnrss 18649 | Chains under a relation ar... |
| chndss 18650 | Chains with an alphabet ar... |
| chnrdss 18651 | Subset theorem for chains.... |
| chnexg 18652 | Chains with a set given fo... |
| nulchn 18653 | Empty set is an increasing... |
| s1chn 18654 | A singleton word is always... |
| chnind 18655 | Induction over a chain. S... |
| chnub 18656 | In a chain, the last eleme... |
| chnlt 18657 | Compare any two elements i... |
| chnso 18658 | A chain induces a total or... |
| chnccats1 18659 | Extend a chain with a sing... |
| chnccat 18660 | Concatenate two chains. (... |
| chnrev 18661 | Reverse of a chain is chai... |
| chnflenfi 18662 | There is a finite number o... |
| chnf 18663 | A chain is a zero-based fi... |
| chnpof1 18664 | A chain under relation whi... |
| chnpoadomd 18665 | A chain under relation whi... |
| chnpolleha 18666 | A chain under relation whi... |
| chnpolfz 18667 | Provided that chain's rela... |
| chnfi 18668 | There is a finite number o... |
| chninf 18669 | There is an infinite numbe... |
| chnfibg 18670 | Given a partial order, the... |
| ex-chn1 18671 | Example: a doubleton of tw... |
| ex-chn2 18672 | Example: sequence <" ZZ NN... |
| ismgm 18677 | The predicate "is a magma"... |
| ismgmn0 18678 | The predicate "is a magma"... |
| mgmcl 18679 | Closure of the operation o... |
| isnmgm 18680 | A condition for a structur... |
| mgmsscl 18681 | If the base set of a magma... |
| plusffval 18682 | The group addition operati... |
| plusfval 18683 | The group addition operati... |
| plusfeq 18684 | If the addition operation ... |
| plusffn 18685 | The group addition operati... |
| mgmplusf 18686 | The group addition functio... |
| mgmpropd 18687 | If two structures have the... |
| ismgmd 18688 | Deduce a magma from its pr... |
| issstrmgm 18689 | Characterize a substructur... |
| intopsn 18690 | The internal operation for... |
| mgmb1mgm1 18691 | The only magma with a base... |
| mgm0 18692 | Any set with an empty base... |
| mgm0b 18693 | The structure with an empt... |
| mgm1 18694 | The structure with one ele... |
| opifismgm 18695 | A structure with a group a... |
| mgmidmo 18696 | A two-sided identity eleme... |
| grpidval 18697 | The value of the identity ... |
| grpidpropd 18698 | If two structures have the... |
| fn0g 18699 | The group zero extractor i... |
| 0g0 18700 | The identity element funct... |
| ismgmid 18701 | The identity element of a ... |
| mgmidcl 18702 | The identity element of a ... |
| mgmlrid 18703 | The identity element of a ... |
| ismgmid2 18704 | Show that a given element ... |
| lidrideqd 18705 | If there is a left and rig... |
| lidrididd 18706 | If there is a left and rig... |
| grpidd 18707 | Deduce the identity elemen... |
| mgmidsssn0 18708 | Property of the set of ide... |
| grpinvalem 18709 | Lemma for ~ grpinva . (Co... |
| grpinva 18710 | Deduce right inverse from ... |
| grprida 18711 | Deduce right identity from... |
| gsumvalx 18712 | Expand out the substitutio... |
| gsumval 18713 | Expand out the substitutio... |
| gsumpropd 18714 | The group sum depends only... |
| gsumpropd2lem 18715 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18716 | A stronger version of ~ gs... |
| gsummgmpropd 18717 | A stronger version of ~ gs... |
| gsumress 18718 | The group sum in a substru... |
| gsumval1 18719 | Value of the group sum ope... |
| gsum0 18720 | Value of the empty group s... |
| gsumval2a 18721 | Value of the group sum ope... |
| gsumval2 18722 | Value of the group sum ope... |
| gsumsplit1r 18723 | Splitting off the rightmos... |
| gsumprval 18724 | Value of the group sum ope... |
| gsumpr12val 18725 | Value of the group sum ope... |
| mgmhmrcl 18730 | Reverse closure of a magma... |
| submgmrcl 18731 | Reverse closure for submag... |
| ismgmhm 18732 | Property of a magma homomo... |
| mgmhmf 18733 | A magma homomorphism is a ... |
| mgmhmpropd 18734 | Magma homomorphism depends... |
| mgmhmlin 18735 | A magma homomorphism prese... |
| mgmhmf1o 18736 | A magma homomorphism is bi... |
| idmgmhm 18737 | The identity homomorphism ... |
| issubmgm 18738 | Expand definition of a sub... |
| issubmgm2 18739 | Submagmas are subsets that... |
| rabsubmgmd 18740 | Deduction for proving that... |
| submgmss 18741 | Submagmas are subsets of t... |
| submgmid 18742 | Every magma is trivially a... |
| submgmcl 18743 | Submagmas are closed under... |
| submgmmgm 18744 | Submagmas are themselves m... |
| submgmbas 18745 | The base set of a submagma... |
| subsubmgm 18746 | A submagma of a submagma i... |
| resmgmhm 18747 | Restriction of a magma hom... |
| resmgmhm2 18748 | One direction of ~ resmgmh... |
| resmgmhm2b 18749 | Restriction of the codomai... |
| mgmhmco 18750 | The composition of magma h... |
| mgmhmima 18751 | The homomorphic image of a... |
| mgmhmeql 18752 | The equalizer of two magma... |
| submgmacs 18753 | Submagmas are an algebraic... |
| issgrp 18756 | The predicate "is a semigr... |
| issgrpv 18757 | The predicate "is a semigr... |
| issgrpn0 18758 | The predicate "is a semigr... |
| isnsgrp 18759 | A condition for a structur... |
| sgrpmgm 18760 | A semigroup is a magma. (... |
| sgrpass 18761 | A semigroup operation is a... |
| sgrpcl 18762 | Closure of the operation o... |
| sgrp0 18763 | Any set with an empty base... |
| sgrp0b 18764 | The structure with an empt... |
| sgrp1 18765 | The structure with one ele... |
| issgrpd 18766 | Deduce a semigroup from it... |
| sgrppropd 18767 | If two structures are sets... |
| prdsplusgsgrpcl 18768 | Structure product pointwis... |
| prdssgrpd 18769 | The product of a family of... |
| ismnddef 18772 | The predicate "is a monoid... |
| ismnd 18773 | The predicate "is a monoid... |
| isnmnd 18774 | A condition for a structur... |
| sgrpidmnd 18775 | A semigroup with an identi... |
| mndsgrp 18776 | A monoid is a semigroup. ... |
| mndmgm 18777 | A monoid is a magma. (Con... |
| mndcl 18778 | Closure of the operation o... |
| mndass 18779 | A monoid operation is asso... |
| mndid 18780 | A monoid has a two-sided i... |
| mndideu 18781 | The two-sided identity ele... |
| mnd32g 18782 | Commutative/associative la... |
| mnd12g 18783 | Commutative/associative la... |
| mnd4g 18784 | Commutative/associative la... |
| mndidcl 18785 | The identity element of a ... |
| mndbn0 18786 | The base set of a monoid i... |
| hashfinmndnn 18787 | A finite monoid has positi... |
| mndplusf 18788 | The group addition operati... |
| mndlrid 18789 | A monoid's identity elemen... |
| mndlid 18790 | The identity element of a ... |
| mndrid 18791 | The identity element of a ... |
| ismndd 18792 | Deduce a monoid from its p... |
| mndpfo 18793 | The addition operation of ... |
| mndfo 18794 | The addition operation of ... |
| mndpropd 18795 | If two structures have the... |
| mndprop 18796 | If two structures have the... |
| issubmnd 18797 | Characterize a submonoid b... |
| ress0g 18798 | ` 0g ` is unaffected by re... |
| submnd0 18799 | The zero of a submonoid is... |
| mndinvmod 18800 | Uniqueness of an inverse e... |
| mndpsuppss 18801 | The support of a mapping o... |
| mndpsuppfi 18802 | The support of a mapping o... |
| mndpfsupp 18803 | A mapping of a scalar mult... |
| prdsplusgcl 18804 | Structure product pointwis... |
| prdsidlem 18805 | Characterization of identi... |
| prdsmndd 18806 | The product of a family of... |
| prds0g 18807 | The identity in a product ... |
| pwsmnd 18808 | The structure power of a m... |
| pws0g 18809 | The identity in a structur... |
| imasmnd2 18810 | The image structure of a m... |
| imasmnd 18811 | The image structure of a m... |
| imasmndf1 18812 | The image of a monoid unde... |
| xpsmnd 18813 | The binary product of mono... |
| xpsmnd0 18814 | The identity element of a ... |
| mnd1 18815 | The (smallest) structure r... |
| mnd1id 18816 | The singleton element of a... |
| ismhm 18821 | Property of a monoid homom... |
| ismhmd 18822 | Deduction version of ~ ism... |
| mhmrcl1 18823 | Reverse closure of a monoi... |
| mhmrcl2 18824 | Reverse closure of a monoi... |
| mhmf 18825 | A monoid homomorphism is a... |
| ismhm0 18826 | Property of a monoid homom... |
| mhmismgmhm 18827 | Each monoid homomorphism i... |
| mhmpropd 18828 | Monoid homomorphism depend... |
| mhmlin 18829 | A monoid homomorphism comm... |
| mhm0 18830 | A monoid homomorphism pres... |
| idmhm 18831 | The identity homomorphism ... |
| mhmf1o 18832 | A monoid homomorphism is b... |
| mndvcl 18833 | Tuple-wise additive closur... |
| mndvass 18834 | Tuple-wise associativity i... |
| mndvlid 18835 | Tuple-wise left identity i... |
| mndvrid 18836 | Tuple-wise right identity ... |
| mhmvlin 18837 | Tuple extension of monoid ... |
| submrcl 18838 | Reverse closure for submon... |
| issubm 18839 | Expand definition of a sub... |
| issubm2 18840 | Submonoids are subsets tha... |
| issubmndb 18841 | The submonoid predicate. ... |
| issubmd 18842 | Deduction for proving a su... |
| mndissubm 18843 | If the base set of a monoi... |
| resmndismnd 18844 | If the base set of a monoi... |
| submss 18845 | Submonoids are subsets of ... |
| submid 18846 | Every monoid is trivially ... |
| subm0cl 18847 | Submonoids contain zero. ... |
| submcl 18848 | Submonoids are closed unde... |
| submmnd 18849 | Submonoids are themselves ... |
| submbas 18850 | The base set of a submonoi... |
| subm0 18851 | Submonoids have the same i... |
| subsubm 18852 | A submonoid of a submonoid... |
| 0subm 18853 | The zero submonoid of an a... |
| insubm 18854 | The intersection of two su... |
| 0mhm 18855 | The constant zero linear f... |
| resmhm 18856 | Restriction of a monoid ho... |
| resmhm2 18857 | One direction of ~ resmhm2... |
| resmhm2b 18858 | Restriction of the codomai... |
| mhmco 18859 | The composition of monoid ... |
| mhmimalem 18860 | Lemma for ~ mhmima and sim... |
| mhmima 18861 | The homomorphic image of a... |
| mhmeql 18862 | The equalizer of two monoi... |
| submacs 18863 | Submonoids are an algebrai... |
| mndind 18864 | Induction in a monoid. In... |
| prdspjmhm 18865 | A projection from a produc... |
| pwspjmhm 18866 | A projection from a struct... |
| pwsdiagmhm 18867 | Diagonal monoid homomorphi... |
| pwsco1mhm 18868 | Right composition with a f... |
| pwsco2mhm 18869 | Left composition with a mo... |
| gsumvallem2 18870 | Lemma for properties of th... |
| gsumsubm 18871 | Evaluate a group sum in a ... |
| gsumz 18872 | Value of a group sum over ... |
| gsumwsubmcl 18873 | Closure of the composite i... |
| gsumws1 18874 | A singleton composite reco... |
| gsumwcl 18875 | Closure of the composite o... |
| gsumsgrpccat 18876 | Homomorphic property of no... |
| gsumccat 18877 | Homomorphic property of co... |
| gsumws2 18878 | Valuation of a pair in a m... |
| gsumccatsn 18879 | Homomorphic property of co... |
| gsumspl 18880 | The primary purpose of the... |
| gsumwmhm 18881 | Behavior of homomorphisms ... |
| gsumwspan 18882 | The submonoid generated by... |
| frmdval 18887 | Value of the free monoid c... |
| frmdbas 18888 | The base set of a free mon... |
| frmdelbas 18889 | An element of the base set... |
| frmdplusg 18890 | The monoid operation of a ... |
| frmdadd 18891 | Value of the monoid operat... |
| vrmdfval 18892 | The canonical injection fr... |
| vrmdval 18893 | The value of the generatin... |
| vrmdf 18894 | The mapping from the index... |
| frmdmnd 18895 | A free monoid is a monoid.... |
| frmd0 18896 | The identity of the free m... |
| frmdsssubm 18897 | The set of words taking va... |
| frmdgsum 18898 | Any word in a free monoid ... |
| frmdss2 18899 | A subset of generators is ... |
| frmdup1 18900 | Any assignment of the gene... |
| frmdup2 18901 | The evaluation map has the... |
| frmdup3lem 18902 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18903 | Universal property of the ... |
| efmnd 18906 | The monoid of endofunction... |
| efmndbas 18907 | The base set of the monoid... |
| efmndbasabf 18908 | The base set of the monoid... |
| elefmndbas 18909 | Two ways of saying a funct... |
| elefmndbas2 18910 | Two ways of saying a funct... |
| efmndbasf 18911 | Elements in the monoid of ... |
| efmndhash 18912 | The monoid of endofunction... |
| efmndbasfi 18913 | The monoid of endofunction... |
| efmndfv 18914 | The function value of an e... |
| efmndtset 18915 | The topology of the monoid... |
| efmndplusg 18916 | The group operation of a m... |
| efmndov 18917 | The value of the group ope... |
| efmndcl 18918 | The group operation of the... |
| efmndtopn 18919 | The topology of the monoid... |
| symggrplem 18920 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18921 | The monoid of endofunction... |
| efmndsgrp 18922 | The monoid of endofunction... |
| ielefmnd 18923 | The identity function rest... |
| efmndid 18924 | The identity function rest... |
| efmndmnd 18925 | The monoid of endofunction... |
| efmnd0nmnd 18926 | Even the monoid of endofun... |
| efmndbas0 18927 | The base set of the monoid... |
| efmnd1hash 18928 | The monoid of endofunction... |
| efmnd1bas 18929 | The monoid of endofunction... |
| efmnd2hash 18930 | The monoid of endofunction... |
| submefmnd 18931 | If the base set of a monoi... |
| sursubmefmnd 18932 | The set of surjective endo... |
| injsubmefmnd 18933 | The set of injective endof... |
| idressubmefmnd 18934 | The singleton containing o... |
| idresefmnd 18935 | The structure with the sin... |
| smndex1ibas 18936 | The modulo function ` I ` ... |
| smndex1iidm 18937 | The modulo function ` I ` ... |
| smndex1gbas 18938 | The constant functions ` (... |
| smndex1gbasOLD 18939 | Obsolete version of ~ smnd... |
| smndex1gid 18940 | The composition of a const... |
| smndex1gidOLD 18941 | Obsolete version of ~ smnd... |
| smndex1igid 18942 | The composition of the mod... |
| smndex1igidOLD 18943 | Obsolete version of ~ smnd... |
| smndex1basss 18944 | The modulo function ` I ` ... |
| smndex1bas 18945 | The base set of the monoid... |
| smndex1mgm 18946 | The monoid of endofunction... |
| smndex1sgrp 18947 | The monoid of endofunction... |
| smndex1mndlem 18948 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18949 | The monoid of endofunction... |
| smndex1id 18950 | The modulo function ` I ` ... |
| smndex1n0mnd 18951 | The identity of the monoid... |
| nsmndex1 18952 | The base set ` B ` of the ... |
| smndex2dbas 18953 | The doubling function ` D ... |
| smndex2dnrinv 18954 | The doubling function ` D ... |
| smndex2hbas 18955 | The halving functions ` H ... |
| smndex2dlinvh 18956 | The halving functions ` H ... |
| mgm2nsgrplem1 18957 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18958 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18959 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18960 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18961 | A small magma (with two el... |
| sgrp2nmndlem1 18962 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18963 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18964 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18965 | A small semigroup (with tw... |
| sgrp2rid2ex 18966 | A small semigroup (with tw... |
| sgrp2nmndlem4 18967 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18968 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18969 | A small semigroup (with tw... |
| mgmnsgrpex 18970 | There is a magma which is ... |
| sgrpnmndex 18971 | There is a semigroup which... |
| sgrpssmgm 18972 | The class of all semigroup... |
| mndsssgrp 18973 | The class of all monoids i... |
| pwmndgplus 18974 | The operation of the monoi... |
| pwmndid 18975 | The identity of the monoid... |
| pwmnd 18976 | The power set of a class `... |
| isgrp 18983 | The predicate "is a group"... |
| grpmnd 18984 | A group is a monoid. (Con... |
| grpcl 18985 | Closure of the operation o... |
| grpass 18986 | A group operation is assoc... |
| grpinvex 18987 | Every member of a group ha... |
| grpideu 18988 | The two-sided identity ele... |
| grpassd 18989 | A group operation is assoc... |
| grpmndd 18990 | A group is a monoid. (Con... |
| grpcld 18991 | Closure of the operation o... |
| grpplusf 18992 | The group addition operati... |
| grpplusfo 18993 | The group addition operati... |
| resgrpplusfrn 18994 | The underlying set of a gr... |
| grppropd 18995 | If two structures have the... |
| grpprop 18996 | If two structures have the... |
| grppropstr 18997 | Generalize a specific 2-el... |
| grpss 18998 | Show that a structure exte... |
| isgrpd2e 18999 | Deduce a group from its pr... |
| isgrpd2 19000 | Deduce a group from its pr... |
| isgrpde 19001 | Deduce a group from its pr... |
| isgrpd 19002 | Deduce a group from its pr... |
| isgrpi 19003 | Properties that determine ... |
| grpsgrp 19004 | A group is a semigroup. (... |
| grpmgmd 19005 | A group is a magma, deduct... |
| dfgrp2 19006 | Alternate definition of a ... |
| dfgrp2e 19007 | Alternate definition of a ... |
| isgrpix 19008 | Properties that determine ... |
| grpidcl 19009 | The identity element of a ... |
| grpbn0 19010 | The base set of a group is... |
| grplid 19011 | The identity element of a ... |
| grprid 19012 | The identity element of a ... |
| grplidd 19013 | The identity element of a ... |
| grpridd 19014 | The identity element of a ... |
| grpn0 19015 | A group is not empty. (Co... |
| hashfingrpnn 19016 | A finite group has positiv... |
| grprcan 19017 | Right cancellation law for... |
| grpinveu 19018 | The left inverse element o... |
| grpid 19019 | Two ways of saying that an... |
| isgrpid2 19020 | Properties showing that an... |
| grpidd2 19021 | Deduce the identity elemen... |
| grpinvfval 19022 | The inverse function of a ... |
| grpinvfvalALT 19023 | Shorter proof of ~ grpinvf... |
| grpinvval 19024 | The inverse of a group ele... |
| grpinvfn 19025 | Functionality of the group... |
| grpinvfvi 19026 | The group inverse function... |
| grpsubfval 19027 | Group subtraction (divisio... |
| grpsubfvalALT 19028 | Shorter proof of ~ grpsubf... |
| grpsubval 19029 | Group subtraction (divisio... |
| grpinvf 19030 | The group inversion operat... |
| grpinvcl 19031 | A group element's inverse ... |
| grpinvcld 19032 | A group element's inverse ... |
| grplinv 19033 | The left inverse of a grou... |
| grprinv 19034 | The right inverse of a gro... |
| grpinvid1 19035 | The inverse of a group ele... |
| grpinvid2 19036 | The inverse of a group ele... |
| isgrpinv 19037 | Properties showing that a ... |
| grplinvd 19038 | The left inverse of a grou... |
| grprinvd 19039 | The right inverse of a gro... |
| grplrinv 19040 | In a group, every member h... |
| grpidinv2 19041 | A group's properties using... |
| grpidinv 19042 | A group has a left and rig... |
| grpinvid 19043 | The inverse of the identit... |
| grplcan 19044 | Left cancellation law for ... |
| grpasscan1 19045 | An associative cancellatio... |
| grpasscan2 19046 | An associative cancellatio... |
| grpidrcan 19047 | If right adding an element... |
| grpidlcan 19048 | If left adding an element ... |
| grpinvinv 19049 | Double inverse law for gro... |
| grpinvcnv 19050 | The group inverse is its o... |
| grpinv11 19051 | The group inverse is one-t... |
| grpinv11OLD 19052 | Obsolete version of ~ grpi... |
| grpinvf1o 19053 | The group inverse is a one... |
| grpinvnz 19054 | The inverse of a nonzero g... |
| grpinvnzcl 19055 | The inverse of a nonzero g... |
| grpsubinv 19056 | Subtraction of an inverse.... |
| grplmulf1o 19057 | Left multiplication by a g... |
| grpraddf1o 19058 | Right addition by a group ... |
| grpinvpropd 19059 | If two structures have the... |
| grpidssd 19060 | If the base set of a group... |
| grpinvssd 19061 | If the base set of a group... |
| grpinvadd 19062 | The inverse of the group o... |
| grpsubf 19063 | Functionality of group sub... |
| grpsubcl 19064 | Closure of group subtracti... |
| grpsubrcan 19065 | Right cancellation law for... |
| grpinvsub 19066 | Inverse of a group subtrac... |
| grpinvval2 19067 | A ~ df-neg -like equation ... |
| grpsubid 19068 | Subtraction of a group ele... |
| grpsubid1 19069 | Subtraction of the identit... |
| grpsubeq0 19070 | If the difference between ... |
| grpsubadd0sub 19071 | Subtraction expressed as a... |
| grpsubadd 19072 | Relationship between group... |
| grpsubsub 19073 | Double group subtraction. ... |
| grpaddsubass 19074 | Associative-type law for g... |
| grppncan 19075 | Cancellation law for subtr... |
| grpnpcan 19076 | Cancellation law for subtr... |
| grpsubsub4 19077 | Double group subtraction (... |
| grppnpcan2 19078 | Cancellation law for mixed... |
| grpnpncan 19079 | Cancellation law for group... |
| grpnpncan0 19080 | Cancellation law for group... |
| grpnnncan2 19081 | Cancellation law for group... |
| dfgrp3lem 19082 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 19083 | Alternate definition of a ... |
| dfgrp3e 19084 | Alternate definition of a ... |
| grplactfval 19085 | The left group action of e... |
| grplactval 19086 | The value of the left grou... |
| grplactcnv 19087 | The left group action of e... |
| grplactf1o 19088 | The left group action of e... |
| grpsubpropd 19089 | Weak property deduction fo... |
| grpsubpropd2 19090 | Strong property deduction ... |
| grp1 19091 | The (smallest) structure r... |
| grp1inv 19092 | The inverse function of th... |
| prdsinvlem 19093 | Characterization of invers... |
| prdsgrpd 19094 | The product of a family of... |
| prdsinvgd 19095 | Negation in a product of g... |
| pwsgrp 19096 | A structure power of a gro... |
| pwsinvg 19097 | Negation in a structure po... |
| pwssub 19098 | Subtraction in a structure... |
| imasgrp2 19099 | The image structure of a g... |
| imasgrp 19100 | The image structure of a g... |
| imasgrpf1 19101 | The image of a group under... |
| qusgrp2 19102 | Prove that a quotient stru... |
| xpsgrp 19103 | The binary product of grou... |
| xpsinv 19104 | Value of the negation oper... |
| xpsgrpsub 19105 | Value of the subtraction o... |
| mhmlem 19106 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 19107 | A surjective monoid morphi... |
| mhmmnd 19108 | The image of a monoid ` G ... |
| mhmfmhm 19109 | The function fulfilling th... |
| ghmgrp 19110 | The image of a group ` G `... |
| mulgfval 19113 | Group multiple (exponentia... |
| mulgfvalALT 19114 | Shorter proof of ~ mulgfva... |
| mulgval 19115 | Value of the group multipl... |
| mulgfn 19116 | Functionality of the group... |
| mulgfvi 19117 | The group multiple operati... |
| mulg0 19118 | Group multiple (exponentia... |
| mulgnn 19119 | Group multiple (exponentia... |
| ressmulgnn 19120 | Values for the group multi... |
| ressmulgnn0 19121 | Values for the group multi... |
| ressmulgnnd 19122 | Values for the group multi... |
| mulgnngsum 19123 | Group multiple (exponentia... |
| mulgnn0gsum 19124 | Group multiple (exponentia... |
| mulg1 19125 | Group multiple (exponentia... |
| mulgnnp1 19126 | Group multiple (exponentia... |
| mulg2 19127 | Group multiple (exponentia... |
| mulgnegnn 19128 | Group multiple (exponentia... |
| mulgnn0p1 19129 | Group multiple (exponentia... |
| mulgnnsubcl 19130 | Closure of the group multi... |
| mulgnn0subcl 19131 | Closure of the group multi... |
| mulgsubcl 19132 | Closure of the group multi... |
| mulgnncl 19133 | Closure of the group multi... |
| mulgnn0cl 19134 | Closure of the group multi... |
| mulgcl 19135 | Closure of the group multi... |
| mulgneg 19136 | Group multiple (exponentia... |
| mulgnegneg 19137 | The inverse of a negative ... |
| mulgm1 19138 | Group multiple (exponentia... |
| mulgnn0cld 19139 | Closure of the group multi... |
| mulgcld 19140 | Deduction associated with ... |
| mulgaddcomlem 19141 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19142 | The group multiple operato... |
| mulginvcom 19143 | The group multiple operato... |
| mulginvinv 19144 | The group multiple operato... |
| mulgnn0z 19145 | A group multiple of the id... |
| mulgz 19146 | A group multiple of the id... |
| mulgnndir 19147 | Sum of group multiples, fo... |
| mulgnn0dir 19148 | Sum of group multiples, ge... |
| mulgdirlem 19149 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19150 | Sum of group multiples, ge... |
| mulgp1 19151 | Group multiple (exponentia... |
| mulgneg2 19152 | Group multiple (exponentia... |
| mulgnnass 19153 | Product of group multiples... |
| mulgnn0ass 19154 | Product of group multiples... |
| mulgass 19155 | Product of group multiples... |
| mulgassr 19156 | Reversed product of group ... |
| mulgmodid 19157 | Casting out multiples of t... |
| mulgsubdir 19158 | Distribution of group mult... |
| mhmmulg 19159 | A homomorphism of monoids ... |
| mulgpropd 19160 | Two structures with the sa... |
| submmulgcl 19161 | Closure of the group multi... |
| submmulg 19162 | A group multiple is the sa... |
| pwsmulg 19163 | Value of a group multiple ... |
| issubg 19170 | The subgroup predicate. (... |
| subgss 19171 | A subgroup is a subset. (... |
| subgid 19172 | A group is a subgroup of i... |
| subggrp 19173 | A subgroup is a group. (C... |
| subgbas 19174 | The base of the restricted... |
| subgrcl 19175 | Reverse closure for the su... |
| subg0 19176 | A subgroup of a group must... |
| subginv 19177 | The inverse of an element ... |
| subg0cl 19178 | The group identity is an e... |
| subginvcl 19179 | The inverse of an element ... |
| subgcl 19180 | A subgroup is closed under... |
| subgsubcl 19181 | A subgroup is closed under... |
| subgsub 19182 | The subtraction of element... |
| subgmulgcl 19183 | Closure of the group multi... |
| subgmulg 19184 | A group multiple is the sa... |
| issubg2 19185 | Characterize the subgroups... |
| issubgrpd2 19186 | Prove a subgroup by closur... |
| issubgrpd 19187 | Prove a subgroup by closur... |
| issubg3 19188 | A subgroup is a symmetric ... |
| issubg4 19189 | A subgroup is a nonempty s... |
| grpissubg 19190 | If the base set of a group... |
| resgrpisgrp 19191 | If the base set of a group... |
| subgsubm 19192 | A subgroup is a submonoid.... |
| subsubg 19193 | A subgroup of a subgroup i... |
| subgint 19194 | The intersection of a none... |
| 0subg 19195 | The zero subgroup of an ar... |
| trivsubgd 19196 | The only subgroup of a tri... |
| trivsubgsnd 19197 | The only subgroup of a tri... |
| isnsg 19198 | Property of being a normal... |
| isnsg2 19199 | Weaken the condition of ~ ... |
| nsgbi 19200 | Defining property of a nor... |
| nsgsubg 19201 | A normal subgroup is a sub... |
| nsgconj 19202 | The conjugation of an elem... |
| isnsg3 19203 | A subgroup is normal iff t... |
| subgacs 19204 | Subgroups are an algebraic... |
| nsgacs 19205 | Normal subgroups form an a... |
| elnmz 19206 | Elementhood in the normali... |
| nmzbi 19207 | Defining property of the n... |
| nmzsubg 19208 | The normalizer N_G(S) of a... |
| ssnmz 19209 | A subgroup is a subset of ... |
| isnsg4 19210 | A subgroup is normal iff i... |
| nmznsg 19211 | Any subgroup is a normal s... |
| 0nsg 19212 | The zero subgroup is norma... |
| nsgid 19213 | The whole group is a norma... |
| 0idnsgd 19214 | The whole group and the ze... |
| trivnsgd 19215 | The only normal subgroup o... |
| triv1nsgd 19216 | A trivial group has exactl... |
| 1nsgtrivd 19217 | A group with exactly one n... |
| releqg 19218 | The left coset equivalence... |
| eqgfval 19219 | Value of the subgroup left... |
| eqgval 19220 | Value of the subgroup left... |
| eqger 19221 | The subgroup coset equival... |
| eqglact 19222 | A left coset can be expres... |
| eqgid 19223 | The left coset containing ... |
| eqgen 19224 | Each coset is equipotent t... |
| eqgcpbl 19225 | The subgroup coset equival... |
| eqg0el 19226 | Equivalence class of a quo... |
| quselbas 19227 | Membership in the base set... |
| quseccl0 19228 | Closure of the quotient ma... |
| qusgrp 19229 | If ` Y ` is a normal subgr... |
| quseccl 19230 | Closure of the quotient ma... |
| qusadd 19231 | Value of the group operati... |
| qus0 19232 | Value of the group identit... |
| qusinv 19233 | Value of the group inverse... |
| qussub 19234 | Value of the group subtrac... |
| ecqusaddd 19235 | Addition of equivalence cl... |
| ecqusaddcl 19236 | Closure of the addition in... |
| lagsubg2 19237 | Lagrange's theorem for fin... |
| lagsubg 19238 | Lagrange's theorem for Gro... |
| eqg0subg 19239 | The coset equivalence rela... |
| eqg0subgecsn 19240 | The equivalence classes mo... |
| qus0subgbas 19241 | The base set of a quotient... |
| qus0subgadd 19242 | The addition in a quotient... |
| cycsubmel 19243 | Characterization of an ele... |
| cycsubmcl 19244 | The set of nonnegative int... |
| cycsubm 19245 | The set of nonnegative int... |
| cyccom 19246 | Condition for an operation... |
| cycsubmcom 19247 | The operation of a monoid ... |
| cycsubggend 19248 | The cyclic subgroup genera... |
| cycsubgcl 19249 | The set of integer powers ... |
| cycsubgss 19250 | The cyclic subgroup genera... |
| cycsubg 19251 | The cyclic group generated... |
| cycsubgcld 19252 | The cyclic subgroup genera... |
| cycsubg2 19253 | The subgroup generated by ... |
| cycsubg2cl 19254 | Any multiple of an element... |
| reldmghm 19257 | Lemma for group homomorphi... |
| isghm 19258 | Property of being a homomo... |
| isghm3 19259 | Property of a group homomo... |
| ghmgrp1 19260 | A group homomorphism is on... |
| ghmgrp2 19261 | A group homomorphism is on... |
| ghmf 19262 | A group homomorphism is a ... |
| ghmlin 19263 | A homomorphism of groups i... |
| ghmid 19264 | A homomorphism of groups p... |
| ghminv 19265 | A homomorphism of groups p... |
| ghmsub 19266 | Linearity of subtraction t... |
| isghmd 19267 | Deduction for a group homo... |
| ghmmhm 19268 | A group homomorphism is a ... |
| ghmmhmb 19269 | Group homomorphisms and mo... |
| ghmmulg 19270 | A group homomorphism prese... |
| ghmrn 19271 | The range of a homomorphis... |
| 0ghm 19272 | The constant zero linear f... |
| idghm 19273 | The identity homomorphism ... |
| resghm 19274 | Restriction of a homomorph... |
| resghm2 19275 | One direction of ~ resghm2... |
| resghm2b 19276 | Restriction of the codomai... |
| ghmghmrn 19277 | A group homomorphism from ... |
| ghmco 19278 | The composition of group h... |
| ghmima 19279 | The image of a subgroup un... |
| ghmpreima 19280 | The inverse image of a sub... |
| ghmeql 19281 | The equalizer of two group... |
| ghmnsgima 19282 | The image of a normal subg... |
| ghmnsgpreima 19283 | The inverse image of a nor... |
| ghmker 19284 | The kernel of a homomorphi... |
| ghmeqker 19285 | Two source points map to t... |
| pwsdiagghm 19286 | Diagonal homomorphism into... |
| f1ghm0to0 19287 | If a group homomorphism ` ... |
| ghmf1 19288 | Two ways of saying a group... |
| kerf1ghm 19289 | A group homomorphism ` F `... |
| ghmf1o 19290 | A bijective group homomorp... |
| conjghm 19291 | Conjugation is an automorp... |
| conjsubg 19292 | A conjugated subgroup is a... |
| conjsubgen 19293 | A conjugated subgroup is e... |
| conjnmz 19294 | A subgroup is unchanged un... |
| conjnmzb 19295 | Alternative condition for ... |
| conjnsg 19296 | A normal subgroup is uncha... |
| qusghm 19297 | If ` Y ` is a normal subgr... |
| ghmpropd 19298 | Group homomorphism depends... |
| gimfn 19303 | The group isomorphism func... |
| isgim 19304 | An isomorphism of groups i... |
| gimf1o 19305 | An isomorphism of groups i... |
| gimghm 19306 | An isomorphism of groups i... |
| isgim2 19307 | A group isomorphism is a h... |
| subggim 19308 | Behavior of subgroups unde... |
| gimcnv 19309 | The converse of a group is... |
| gimco 19310 | The composition of group i... |
| gim0to0 19311 | A group isomorphism maps t... |
| brgic 19312 | The relation "is isomorphi... |
| brgici 19313 | Prove isomorphic by an exp... |
| gicref 19314 | Isomorphism is reflexive. ... |
| giclcl 19315 | Isomorphism implies the le... |
| gicrcl 19316 | Isomorphism implies the ri... |
| gicsym 19317 | Isomorphism is symmetric. ... |
| gictr 19318 | Isomorphism is transitive.... |
| gicer 19319 | Isomorphism is an equivale... |
| gicen 19320 | Isomorphic groups have equ... |
| gicsubgen 19321 | A less trivial example of ... |
| ghmqusnsglem1 19322 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsglem2 19323 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsg 19324 | The mapping ` H ` induced ... |
| ghmquskerlem1 19325 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19326 | In the case of theorem ~ g... |
| ghmquskerlem2 19327 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19328 | The mapping ` H ` induced ... |
| ghmqusker 19329 | A surjective group homomor... |
| gicqusker 19330 | The image ` H ` of a group... |
| isga 19333 | The predicate "is a (left)... |
| gagrp 19334 | The left argument of a gro... |
| gaset 19335 | The right argument of a gr... |
| gagrpid 19336 | The identity of the group ... |
| gaf 19337 | The mapping of the group a... |
| gafo 19338 | A group action is onto its... |
| gaass 19339 | An "associative" property ... |
| ga0 19340 | The action of a group on t... |
| gaid 19341 | The trivial action of a gr... |
| subgga 19342 | A subgroup acts on its par... |
| gass 19343 | A subset of a group action... |
| gasubg 19344 | The restriction of a group... |
| gaid2 19345 | A group operation is a lef... |
| galcan 19346 | The action of a particular... |
| gacan 19347 | Group inverses cancel in a... |
| gapm 19348 | The action of a particular... |
| gaorb 19349 | The orbit equivalence rela... |
| gaorber 19350 | The orbit equivalence rela... |
| gastacl 19351 | The stabilizer subgroup in... |
| gastacos 19352 | Write the coset relation f... |
| orbstafun 19353 | Existence and uniqueness f... |
| orbstaval 19354 | Value of the function at a... |
| orbsta 19355 | The Orbit-Stabilizer theor... |
| orbsta2 19356 | Relation between the size ... |
| cntrval 19361 | Substitute definition of t... |
| cntzfval 19362 | First level substitution f... |
| cntzval 19363 | Definition substitution fo... |
| elcntz 19364 | Elementhood in the central... |
| cntzel 19365 | Membership in a centralize... |
| cntzsnval 19366 | Special substitution for t... |
| elcntzsn 19367 | Value of the centralizer o... |
| sscntz 19368 | A centralizer expression f... |
| cntzrcl 19369 | Reverse closure for elemen... |
| cntzssv 19370 | The centralizer is uncondi... |
| cntzi 19371 | Membership in a centralize... |
| elcntr 19372 | Elementhood in the center ... |
| cntrss 19373 | The center is a subset of ... |
| cntri 19374 | Defining property of the c... |
| resscntz 19375 | Centralizer in a substruct... |
| cntzsgrpcl 19376 | Centralizers are closed un... |
| cntz2ss 19377 | Centralizers reverse the s... |
| cntzrec 19378 | Reciprocity relationship f... |
| cntziinsn 19379 | Express any centralizer as... |
| cntzsubm 19380 | Centralizers in a monoid a... |
| cntzsubg 19381 | Centralizers in a group ar... |
| cntzidss 19382 | If the elements of ` S ` c... |
| cntzmhm 19383 | Centralizers in a monoid a... |
| cntzmhm2 19384 | Centralizers in a monoid a... |
| cntrsubgnsg 19385 | A central subgroup is norm... |
| cntrnsg 19386 | The center of a group is a... |
| oppgval 19389 | Value of the opposite grou... |
| oppgplusfval 19390 | Value of the addition oper... |
| oppgplus 19391 | Value of the addition oper... |
| setsplusg 19392 | The other components of an... |
| oppgbas 19393 | Base set of an opposite gr... |
| oppgtset 19394 | Topology of an opposite gr... |
| oppgtopn 19395 | Topology of an opposite gr... |
| oppgmnd 19396 | The opposite of a monoid i... |
| oppgmndb 19397 | Bidirectional form of ~ op... |
| oppgid 19398 | Zero in a monoid is a symm... |
| oppggrp 19399 | The opposite of a group is... |
| oppggrpb 19400 | Bidirectional form of ~ op... |
| oppginv 19401 | Inverses in a group are a ... |
| invoppggim 19402 | The inverse is an antiauto... |
| oppggic 19403 | Every group is (naturally)... |
| oppgsubm 19404 | Being a submonoid is a sym... |
| oppgsubg 19405 | Being a subgroup is a symm... |
| oppgcntz 19406 | A centralizer in a group i... |
| oppgcntr 19407 | The center of a group is t... |
| gsumwrev 19408 | A sum in an opposite monoi... |
| oppgle 19409 | less-than relation of an o... |
| oppglt 19410 | less-than relation of an o... |
| symgval 19413 | The value of the symmetric... |
| symgbas 19414 | The base set of the symmet... |
| elsymgbas2 19415 | Two ways of saying a funct... |
| elsymgbas 19416 | Two ways of saying a funct... |
| symgbasf1o 19417 | Elements in the symmetric ... |
| symgbasf 19418 | A permutation (element of ... |
| symgbasmap 19419 | A permutation (element of ... |
| symghash 19420 | The symmetric group on ` n... |
| symgbasfi 19421 | The symmetric group on a f... |
| symgfv 19422 | The function value of a pe... |
| symgfvne 19423 | The function values of a p... |
| symgressbas 19424 | The symmetric group on ` A... |
| symgplusg 19425 | The group operation of a s... |
| symgov 19426 | The value of the group ope... |
| symgcl 19427 | The group operation of the... |
| idresperm 19428 | The identity function rest... |
| symgmov1 19429 | For a permutation of a set... |
| symgmov2 19430 | For a permutation of a set... |
| symgbas0 19431 | The base set of the symmet... |
| symg1hash 19432 | The symmetric group on a s... |
| symg1bas 19433 | The symmetric group on a s... |
| symg2hash 19434 | The symmetric group on a (... |
| symg2bas 19435 | The symmetric group on a p... |
| 0symgefmndeq 19436 | The symmetric group on the... |
| snsymgefmndeq 19437 | The symmetric group on a s... |
| symgpssefmnd 19438 | For a set ` A ` with more ... |
| symgvalstruct 19439 | The value of the symmetric... |
| symgsubmefmnd 19440 | The symmetric group on a s... |
| symgtset 19441 | The topology of the symmet... |
| symggrp 19442 | The symmetric group on a s... |
| symgid 19443 | The group identity element... |
| symginv 19444 | The group inverse in the s... |
| symgsubmefmndALT 19445 | The symmetric group on a s... |
| galactghm 19446 | The currying of a group ac... |
| lactghmga 19447 | The converse of ~ galactgh... |
| symgtopn 19448 | The topology of the symmet... |
| symgga 19449 | The symmetric group induce... |
| pgrpsubgsymgbi 19450 | Every permutation group is... |
| pgrpsubgsymg 19451 | Every permutation group is... |
| idressubgsymg 19452 | The singleton containing o... |
| idrespermg 19453 | The structure with the sin... |
| cayleylem1 19454 | Lemma for ~ cayley . (Con... |
| cayleylem2 19455 | Lemma for ~ cayley . (Con... |
| cayley 19456 | Cayley's Theorem (construc... |
| cayleyth 19457 | Cayley's Theorem (existenc... |
| symgfix2 19458 | If a permutation does not ... |
| symgextf 19459 | The extension of a permuta... |
| symgextfv 19460 | The function value of the ... |
| symgextfve 19461 | The function value of the ... |
| symgextf1lem 19462 | Lemma for ~ symgextf1 . (... |
| symgextf1 19463 | The extension of a permuta... |
| symgextfo 19464 | The extension of a permuta... |
| symgextf1o 19465 | The extension of a permuta... |
| symgextsymg 19466 | The extension of a permuta... |
| symgextres 19467 | The restriction of the ext... |
| gsumccatsymgsn 19468 | Homomorphic property of co... |
| gsmsymgrfixlem1 19469 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19470 | The composition of permuta... |
| fvcosymgeq 19471 | The values of two composit... |
| gsmsymgreqlem1 19472 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19473 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19474 | Two combination of permuta... |
| symgfixelq 19475 | A permutation of a set fix... |
| symgfixels 19476 | The restriction of a permu... |
| symgfixelsi 19477 | The restriction of a permu... |
| symgfixf 19478 | The mapping of a permutati... |
| symgfixf1 19479 | The mapping of a permutati... |
| symgfixfolem1 19480 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19481 | The mapping of a permutati... |
| symgfixf1o 19482 | The mapping of a permutati... |
| f1omvdmvd 19485 | A permutation of any class... |
| f1omvdcnv 19486 | A permutation and its inve... |
| mvdco 19487 | Composing two permutations... |
| f1omvdconj 19488 | Conjugation of a permutati... |
| f1otrspeq 19489 | A transposition is charact... |
| f1omvdco2 19490 | If exactly one of two perm... |
| f1omvdco3 19491 | If a point is moved by exa... |
| pmtrfval 19492 | The function generating tr... |
| pmtrval 19493 | A generated transposition,... |
| pmtrfv 19494 | General value of mapping a... |
| pmtrprfv 19495 | In a transposition of two ... |
| pmtrprfv3 19496 | In a transposition of two ... |
| pmtrf 19497 | Functionality of a transpo... |
| pmtrmvd 19498 | A transposition moves prec... |
| pmtrrn 19499 | Transposing two points giv... |
| pmtrfrn 19500 | A transposition (as a kind... |
| pmtrffv 19501 | Mapping of a point under a... |
| pmtrrn2 19502 | For any transposition ther... |
| pmtrfinv 19503 | A transposition function i... |
| pmtrfmvdn0 19504 | A transposition moves at l... |
| pmtrff1o 19505 | A transposition function i... |
| pmtrfcnv 19506 | A transposition function i... |
| pmtrfb 19507 | An intrinsic characterizat... |
| pmtrfconj 19508 | Any conjugate of a transpo... |
| symgsssg 19509 | The symmetric group has su... |
| symgfisg 19510 | The symmetric group has a ... |
| symgtrf 19511 | Transpositions are element... |
| symggen 19512 | The span of the transposit... |
| symggen2 19513 | A finite permutation group... |
| symgtrinv 19514 | To invert a permutation re... |
| pmtr3ncomlem1 19515 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19516 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19517 | Transpositions over sets w... |
| pmtrdifellem1 19518 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19519 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19520 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19521 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19522 | A transposition of element... |
| pmtrdifwrdellem1 19523 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19524 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19525 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19526 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19527 | A sequence of transpositio... |
| pmtrdifwrdel2 19528 | A sequence of transpositio... |
| pmtrprfval 19529 | The transpositions on a pa... |
| pmtrprfvalrn 19530 | The range of the transposi... |
| psgnunilem1 19535 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19536 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19537 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19538 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19539 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19540 | Addition and subtraction o... |
| psgnuni 19541 | If the same permutation ca... |
| psgnfval 19542 | Function definition of the... |
| psgnfn 19543 | Functionality and domain o... |
| psgndmsubg 19544 | The finitary permutations ... |
| psgneldm 19545 | Property of being a finita... |
| psgneldm2 19546 | The finitary permutations ... |
| psgneldm2i 19547 | A sequence of transpositio... |
| psgneu 19548 | A finitary permutation has... |
| psgnval 19549 | Value of the permutation s... |
| psgnvali 19550 | A finitary permutation has... |
| psgnvalii 19551 | Any representation of a pe... |
| psgnpmtr 19552 | All transpositions are odd... |
| psgn0fv0 19553 | The permutation sign funct... |
| sygbasnfpfi 19554 | The class of non-fixed poi... |
| psgnfvalfi 19555 | Function definition of the... |
| psgnvalfi 19556 | Value of the permutation s... |
| psgnran 19557 | The range of the permutati... |
| gsmtrcl 19558 | The group sum of transposi... |
| psgnfitr 19559 | A permutation of a finite ... |
| psgnfieu 19560 | A permutation of a finite ... |
| pmtrsn 19561 | The value of the transposi... |
| psgnsn 19562 | The permutation sign funct... |
| psgnprfval 19563 | The permutation sign funct... |
| psgnprfval1 19564 | The permutation sign of th... |
| psgnprfval2 19565 | The permutation sign of th... |
| odfval 19574 | Value of the order functio... |
| odfvalALT 19575 | Shorter proof of ~ odfval ... |
| odval 19576 | Second substitution for th... |
| odlem1 19577 | The group element order is... |
| odcl 19578 | The order of a group eleme... |
| odf 19579 | Functionality of the group... |
| odid 19580 | Any element to the power o... |
| odlem2 19581 | Any positive annihilator o... |
| odmodnn0 19582 | Reduce the argument of a g... |
| mndodconglem 19583 | Lemma for ~ mndodcong . (... |
| mndodcong 19584 | If two multipliers are con... |
| mndodcongi 19585 | If two multipliers are con... |
| oddvdsnn0 19586 | The only multiples of ` A ... |
| odnncl 19587 | If a nonzero multiple of a... |
| odmod 19588 | Reduce the argument of a g... |
| oddvds 19589 | The only multiples of ` A ... |
| oddvdsi 19590 | Any group element is annih... |
| odcong 19591 | If two multipliers are con... |
| odeq 19592 | The ~ oddvds property uniq... |
| odval2 19593 | A non-conditional definiti... |
| odcld 19594 | The order of a group eleme... |
| odm1inv 19595 | The (order-1)th multiple o... |
| odmulgid 19596 | A relationship between the... |
| odmulg2 19597 | The order of a multiple di... |
| odmulg 19598 | Relationship between the o... |
| odmulgeq 19599 | A multiple of a point of f... |
| odbezout 19600 | If ` N ` is coprime to the... |
| od1 19601 | The order of the group ide... |
| odeq1 19602 | The group identity is the ... |
| odinv 19603 | The order of the inverse o... |
| odf1 19604 | The multiples of an elemen... |
| odinf 19605 | The multiples of an elemen... |
| dfod2 19606 | An alternative definition ... |
| odcl2 19607 | The order of an element of... |
| oddvds2 19608 | The order of an element of... |
| finodsubmsubg 19609 | A submonoid whose elements... |
| 0subgALT 19610 | A shorter proof of ~ 0subg... |
| submod 19611 | The order of an element is... |
| subgod 19612 | The order of an element is... |
| odsubdvds 19613 | The order of an element of... |
| odf1o1 19614 | An element with zero order... |
| odf1o2 19615 | An element with nonzero or... |
| odhash 19616 | An element of zero order g... |
| odhash2 19617 | If an element has nonzero ... |
| odhash3 19618 | An element which generates... |
| odngen 19619 | A cyclic subgroup of size ... |
| gexval 19620 | Value of the exponent of a... |
| gexlem1 19621 | The group element order is... |
| gexcl 19622 | The exponent of a group is... |
| gexid 19623 | Any element to the power o... |
| gexlem2 19624 | Any positive annihilator o... |
| gexdvdsi 19625 | Any group element is annih... |
| gexdvds 19626 | The only ` N ` that annihi... |
| gexdvds2 19627 | An integer divides the gro... |
| gexod 19628 | Any group element is annih... |
| gexcl3 19629 | If the order of every grou... |
| gexnnod 19630 | Every group element has fi... |
| gexcl2 19631 | The exponent of a finite g... |
| gexdvds3 19632 | The exponent of a finite g... |
| gex1 19633 | A group or monoid has expo... |
| ispgp 19634 | A group is a ` P ` -group ... |
| pgpprm 19635 | Reverse closure for the fi... |
| pgpgrp 19636 | Reverse closure for the se... |
| pgpfi1 19637 | A finite group with order ... |
| pgp0 19638 | The identity subgroup is a... |
| subgpgp 19639 | A subgroup of a p-group is... |
| sylow1lem1 19640 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19641 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19642 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19643 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19644 | Lemma for ~ sylow1 . Usin... |
| sylow1 19645 | Sylow's first theorem. If... |
| odcau 19646 | Cauchy's theorem for the o... |
| pgpfi 19647 | The converse to ~ pgpfi1 .... |
| pgpfi2 19648 | Alternate version of ~ pgp... |
| pgphash 19649 | The order of a p-group. (... |
| isslw 19650 | The property of being a Sy... |
| slwprm 19651 | Reverse closure for the fi... |
| slwsubg 19652 | A Sylow ` P ` -subgroup is... |
| slwispgp 19653 | Defining property of a Syl... |
| slwpss 19654 | A proper superset of a Syl... |
| slwpgp 19655 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19656 | Every ` P ` -subgroup is c... |
| slwn0 19657 | Every finite group contain... |
| subgslw 19658 | A Sylow subgroup that is c... |
| sylow2alem1 19659 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19660 | Lemma for ~ sylow2a . All... |
| sylow2a 19661 | A named lemma of Sylow's s... |
| sylow2blem1 19662 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19663 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19664 | Sylow's second theorem. P... |
| sylow2b 19665 | Sylow's second theorem. A... |
| slwhash 19666 | A sylow subgroup has cardi... |
| fislw 19667 | The sylow subgroups of a f... |
| sylow2 19668 | Sylow's second theorem. S... |
| sylow3lem1 19669 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19670 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19671 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19672 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19673 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19674 | Lemma for ~ sylow3 , secon... |
| sylow3 19675 | Sylow's third theorem. Th... |
| lsmfval 19680 | The subgroup sum function ... |
| lsmvalx 19681 | Subspace sum value (for a ... |
| lsmelvalx 19682 | Subspace sum membership (f... |
| lsmelvalix 19683 | Subspace sum membership (f... |
| oppglsm 19684 | The subspace sum operation... |
| lsmssv 19685 | Subgroup sum is a subset o... |
| lsmless1x 19686 | Subset implies subgroup su... |
| lsmless2x 19687 | Subset implies subgroup su... |
| lsmub1x 19688 | Subgroup sum is an upper b... |
| lsmub2x 19689 | Subgroup sum is an upper b... |
| lsmval 19690 | Subgroup sum value (for a ... |
| lsmelval 19691 | Subgroup sum membership (f... |
| lsmelvali 19692 | Subgroup sum membership (f... |
| lsmelvalm 19693 | Subgroup sum membership an... |
| lsmelvalmi 19694 | Membership of vector subtr... |
| lsmsubm 19695 | The sum of two commuting s... |
| lsmsubg 19696 | The sum of two commuting s... |
| lsmcom2 19697 | Subgroup sum commutes. (C... |
| smndlsmidm 19698 | The direct product is idem... |
| lsmub1 19699 | Subgroup sum is an upper b... |
| lsmub2 19700 | Subgroup sum is an upper b... |
| lsmunss 19701 | Union of subgroups is a su... |
| lsmless1 19702 | Subset implies subgroup su... |
| lsmless2 19703 | Subset implies subgroup su... |
| lsmless12 19704 | Subset implies subgroup su... |
| lsmidm 19705 | Subgroup sum is idempotent... |
| lsmlub 19706 | The least upper bound prop... |
| lsmss1 19707 | Subgroup sum with a subset... |
| lsmss1b 19708 | Subgroup sum with a subset... |
| lsmss2 19709 | Subgroup sum with a subset... |
| lsmss2b 19710 | Subgroup sum with a subset... |
| lsmass 19711 | Subgroup sum is associativ... |
| mndlsmidm 19712 | Subgroup sum is idempotent... |
| lsm01 19713 | Subgroup sum with the zero... |
| lsm02 19714 | Subgroup sum with the zero... |
| subglsm 19715 | The subgroup sum evaluated... |
| lssnle 19716 | Equivalent expressions for... |
| lsmmod 19717 | The modular law holds for ... |
| lsmmod2 19718 | Modular law dual for subgr... |
| lsmpropd 19719 | If two structures have the... |
| cntzrecd 19720 | Commute the "subgroups com... |
| lsmcntz 19721 | The "subgroups commute" pr... |
| lsmcntzr 19722 | The "subgroups commute" pr... |
| lsmdisj 19723 | Disjointness from a subgro... |
| lsmdisj2 19724 | Association of the disjoin... |
| lsmdisj3 19725 | Association of the disjoin... |
| lsmdisjr 19726 | Disjointness from a subgro... |
| lsmdisj2r 19727 | Association of the disjoin... |
| lsmdisj3r 19728 | Association of the disjoin... |
| lsmdisj2a 19729 | Association of the disjoin... |
| lsmdisj2b 19730 | Association of the disjoin... |
| lsmdisj3a 19731 | Association of the disjoin... |
| lsmdisj3b 19732 | Association of the disjoin... |
| subgdisj1 19733 | Vectors belonging to disjo... |
| subgdisj2 19734 | Vectors belonging to disjo... |
| subgdisjb 19735 | Vectors belonging to disjo... |
| pj1fval 19736 | The left projection functi... |
| pj1val 19737 | The left projection functi... |
| pj1eu 19738 | Uniqueness of a left proje... |
| pj1f 19739 | The left projection functi... |
| pj2f 19740 | The right projection funct... |
| pj1id 19741 | Any element of a direct su... |
| pj1eq 19742 | Any element of a direct su... |
| pj1lid 19743 | The left projection functi... |
| pj1rid 19744 | The left projection functi... |
| pj1ghm 19745 | The left projection functi... |
| pj1ghm2 19746 | The left projection functi... |
| lsmhash 19747 | The order of the direct pr... |
| efgmval 19754 | Value of the formal invers... |
| efgmf 19755 | The formal inverse operati... |
| efgmnvl 19756 | The inversion function on ... |
| efgrcl 19757 | Lemma for ~ efgval . (Con... |
| efglem 19758 | Lemma for ~ efgval . (Con... |
| efgval 19759 | Value of the free group co... |
| efger 19760 | Value of the free group co... |
| efgi 19761 | Value of the free group co... |
| efgi0 19762 | Value of the free group co... |
| efgi1 19763 | Value of the free group co... |
| efgtf 19764 | Value of the free group co... |
| efgtval 19765 | Value of the extension fun... |
| efgval2 19766 | Value of the free group co... |
| efgi2 19767 | Value of the free group co... |
| efgtlen 19768 | Value of the free group co... |
| efginvrel2 19769 | The inverse of the reverse... |
| efginvrel1 19770 | The inverse of the reverse... |
| efgsf 19771 | Value of the auxiliary fun... |
| efgsdm 19772 | Elementhood in the domain ... |
| efgsval 19773 | Value of the auxiliary fun... |
| efgsdmi 19774 | Property of the last link ... |
| efgsval2 19775 | Value of the auxiliary fun... |
| efgsrel 19776 | The start and end of any e... |
| efgs1 19777 | A singleton of an irreduci... |
| efgs1b 19778 | Every extension sequence e... |
| efgsp1 19779 | If ` F ` is an extension s... |
| efgsres 19780 | An initial segment of an e... |
| efgsfo 19781 | For any word, there is a s... |
| efgredlema 19782 | The reduced word that form... |
| efgredlemf 19783 | Lemma for ~ efgredleme . ... |
| efgredlemg 19784 | Lemma for ~ efgred . (Con... |
| efgredleme 19785 | Lemma for ~ efgred . (Con... |
| efgredlemd 19786 | The reduced word that form... |
| efgredlemc 19787 | The reduced word that form... |
| efgredlemb 19788 | The reduced word that form... |
| efgredlem 19789 | The reduced word that form... |
| efgred 19790 | The reduced word that form... |
| efgrelexlema 19791 | If two words ` A , B ` are... |
| efgrelexlemb 19792 | If two words ` A , B ` are... |
| efgrelex 19793 | If two words ` A , B ` are... |
| efgredeu 19794 | There is a unique reduced ... |
| efgred2 19795 | Two extension sequences ha... |
| efgcpbllema 19796 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19797 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19798 | Two extension sequences ha... |
| efgcpbl2 19799 | Two extension sequences ha... |
| frgpval 19800 | Value of the free group co... |
| frgpcpbl 19801 | Compatibility of the group... |
| frgp0 19802 | The free group is a group.... |
| frgpeccl 19803 | Closure of the quotient ma... |
| frgpgrp 19804 | The free group is a group.... |
| frgpadd 19805 | Addition in the free group... |
| frgpinv 19806 | The inverse of an element ... |
| frgpmhm 19807 | The "natural map" from wor... |
| vrgpfval 19808 | The canonical injection fr... |
| vrgpval 19809 | The value of the generatin... |
| vrgpf 19810 | The mapping from the index... |
| vrgpinv 19811 | The inverse of a generatin... |
| frgpuptf 19812 | Any assignment of the gene... |
| frgpuptinv 19813 | Any assignment of the gene... |
| frgpuplem 19814 | Any assignment of the gene... |
| frgpupf 19815 | Any assignment of the gene... |
| frgpupval 19816 | Any assignment of the gene... |
| frgpup1 19817 | Any assignment of the gene... |
| frgpup2 19818 | The evaluation map has the... |
| frgpup3lem 19819 | The evaluation map has the... |
| frgpup3 19820 | Universal property of the ... |
| 0frgp 19821 | The free group on zero gen... |
| isabl 19826 | The predicate "is an Abeli... |
| ablgrp 19827 | An Abelian group is a grou... |
| ablgrpd 19828 | An Abelian group is a grou... |
| ablcmn 19829 | An Abelian group is a comm... |
| ablcmnd 19830 | An Abelian group is a comm... |
| iscmn 19831 | The predicate "is a commut... |
| isabl2 19832 | The predicate "is an Abeli... |
| cmnpropd 19833 | If two structures have the... |
| ablpropd 19834 | If two structures have the... |
| ablprop 19835 | If two structures have the... |
| iscmnd 19836 | Properties that determine ... |
| isabld 19837 | Properties that determine ... |
| isabli 19838 | Properties that determine ... |
| cmnmnd 19839 | A commutative monoid is a ... |
| cmncom 19840 | A commutative monoid is co... |
| ablcom 19841 | An Abelian group operation... |
| cmn32 19842 | Commutative/associative la... |
| cmn4 19843 | Commutative/associative la... |
| cmn12 19844 | Commutative/associative la... |
| abl32 19845 | Commutative/associative la... |
| cmnmndd 19846 | A commutative monoid is a ... |
| cmnbascntr 19847 | The base set of a commutat... |
| rinvmod 19848 | Uniqueness of a right inve... |
| ablinvadd 19849 | The inverse of an Abelian ... |
| ablsub2inv 19850 | Abelian group subtraction ... |
| ablsubadd 19851 | Relationship between Abeli... |
| ablsub4 19852 | Commutative/associative su... |
| abladdsub4 19853 | Abelian group addition/sub... |
| abladdsub 19854 | Associative-type law for g... |
| ablsubadd23 19855 | Commutative/associative la... |
| ablsubaddsub 19856 | Double subtraction and add... |
| ablpncan2 19857 | Cancellation law for subtr... |
| ablpncan3 19858 | A cancellation law for Abe... |
| ablsubsub 19859 | Law for double subtraction... |
| ablsubsub4 19860 | Law for double subtraction... |
| ablpnpcan 19861 | Cancellation law for mixed... |
| ablnncan 19862 | Cancellation law for group... |
| ablsub32 19863 | Swap the second and third ... |
| ablnnncan 19864 | Cancellation law for group... |
| ablnnncan1 19865 | Cancellation law for group... |
| ablsubsub23 19866 | Swap subtrahend and result... |
| mulgnn0di 19867 | Group multiple of a sum, f... |
| mulgdi 19868 | Group multiple of a sum. ... |
| mulgmhm 19869 | The map from ` x ` to ` n ... |
| mulgghm 19870 | The map from ` x ` to ` n ... |
| mulgsubdi 19871 | Group multiple of a differ... |
| ghmfghm 19872 | The function fulfilling th... |
| ghmcmn 19873 | The image of a commutative... |
| ghmabl 19874 | The image of an abelian gr... |
| invghm 19875 | The inversion map is a gro... |
| eqgabl 19876 | Value of the subgroup cose... |
| qusecsub 19877 | Two subgroup cosets are eq... |
| subgabl 19878 | A subgroup of an abelian g... |
| subcmn 19879 | A submonoid of a commutati... |
| submcmn 19880 | A submonoid of a commutati... |
| submcmn2 19881 | A submonoid is commutative... |
| cntzcmn 19882 | The centralizer of any sub... |
| cntzcmnss 19883 | Any subset in a commutativ... |
| cntrcmnd 19884 | The center of a monoid is ... |
| cntrabl 19885 | The center of a group is a... |
| cntzspan 19886 | If the generators commute,... |
| cntzcmnf 19887 | Discharge the centralizer ... |
| ghmplusg 19888 | The pointwise sum of two l... |
| ablnsg 19889 | Every subgroup of an abeli... |
| odadd1 19890 | The order of a product in ... |
| odadd2 19891 | The order of a product in ... |
| odadd 19892 | The order of a product is ... |
| gex2abl 19893 | A group with exponent 2 (o... |
| gexexlem 19894 | Lemma for ~ gexex . (Cont... |
| gexex 19895 | In an abelian group with f... |
| torsubg 19896 | The set of all elements of... |
| oddvdssubg 19897 | The set of all elements wh... |
| lsmcomx 19898 | Subgroup sum commutes (ext... |
| ablcntzd 19899 | All subgroups in an abelia... |
| lsmcom 19900 | Subgroup sum commutes. (C... |
| lsmsubg2 19901 | The sum of two subgroups i... |
| lsm4 19902 | Commutative/associative la... |
| prdscmnd 19903 | The product of a family of... |
| prdsabld 19904 | The product of a family of... |
| pwscmn 19905 | The structure power on a c... |
| pwsabl 19906 | The structure power on an ... |
| qusabl 19907 | If ` Y ` is a subgroup of ... |
| abl1 19908 | The (smallest) structure r... |
| abln0 19909 | Abelian groups (and theref... |
| cnaddablx 19910 | The complex numbers are an... |
| cnaddabl 19911 | The complex numbers are an... |
| cnaddid 19912 | The group identity element... |
| cnaddinv 19913 | Value of the group inverse... |
| zaddablx 19914 | The integers are an Abelia... |
| frgpnabllem1 19915 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19916 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19917 | The free group on two or m... |
| imasabl 19918 | The image structure of an ... |
| iscyg 19921 | Definition of a cyclic gro... |
| iscyggen 19922 | The property of being a cy... |
| iscyggen2 19923 | The property of being a cy... |
| iscyg2 19924 | A cyclic group is a group ... |
| cyggeninv 19925 | The inverse of a cyclic ge... |
| cyggenod 19926 | An element is the generato... |
| cyggenod2 19927 | In an infinite cyclic grou... |
| iscyg3 19928 | Definition of a cyclic gro... |
| iscygd 19929 | Definition of a cyclic gro... |
| iscygodd 19930 | Show that a group with an ... |
| cycsubmcmn 19931 | The set of nonnegative int... |
| cyggrp 19932 | A cyclic group is a group.... |
| cygabl 19933 | A cyclic group is abelian.... |
| cygctb 19934 | A cyclic group is countabl... |
| 0cyg 19935 | The trivial group is cycli... |
| prmcyg 19936 | A group with prime order i... |
| lt6abl 19937 | A group with fewer than ` ... |
| ghmcyg 19938 | The image of a cyclic grou... |
| cyggex2 19939 | The exponent of a cyclic g... |
| cyggex 19940 | The exponent of a finite c... |
| cyggexb 19941 | A finite abelian group is ... |
| giccyg 19942 | Cyclicity is a group prope... |
| cycsubgcyg 19943 | The cyclic subgroup genera... |
| cycsubgcyg2 19944 | The cyclic subgroup genera... |
| gsumval3a 19945 | Value of the group sum ope... |
| gsumval3eu 19946 | The group sum as defined i... |
| gsumval3lem1 19947 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19948 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19949 | Value of the group sum ope... |
| gsumcllem 19950 | Lemma for ~ gsumcl and rel... |
| gsumzres 19951 | Extend a finite group sum ... |
| gsumzcl2 19952 | Closure of a finite group ... |
| gsumzcl 19953 | Closure of a finite group ... |
| gsumzf1o 19954 | Re-index a finite group su... |
| gsumres 19955 | Extend a finite group sum ... |
| gsumcl2 19956 | Closure of a finite group ... |
| gsumcl 19957 | Closure of a finite group ... |
| gsumf1o 19958 | Re-index a finite group su... |
| gsumreidx 19959 | Re-index a finite group su... |
| gsumzsubmcl 19960 | Closure of a group sum in ... |
| gsumsubmcl 19961 | Closure of a group sum in ... |
| gsumsubgcl 19962 | Closure of a group sum in ... |
| gsumzaddlem 19963 | The sum of two group sums.... |
| gsumzadd 19964 | The sum of two group sums.... |
| gsumadd 19965 | The sum of two group sums.... |
| gsummptfsadd 19966 | The sum of two group sums ... |
| gsummptfidmadd 19967 | The sum of two group sums ... |
| gsummptfidmadd2 19968 | The sum of two group sums ... |
| gsumzsplit 19969 | Split a group sum into two... |
| gsumsplit 19970 | Split a group sum into two... |
| gsumsplit2 19971 | Split a group sum into two... |
| gsummptfidmsplit 19972 | Split a group sum expresse... |
| gsummptfidmsplitres 19973 | Split a group sum expresse... |
| gsummptfzsplit 19974 | Split a group sum expresse... |
| gsummptfzsplitl 19975 | Split a group sum expresse... |
| gsumconst 19976 | Sum of a constant series. ... |
| gsumconstf 19977 | Sum of a constant series. ... |
| gsummptshft 19978 | Index shift of a finite gr... |
| gsumzmhm 19979 | Apply a group homomorphism... |
| gsummhm 19980 | Apply a group homomorphism... |
| gsummhm2 19981 | Apply a group homomorphism... |
| gsummptmhm 19982 | Apply a group homomorphism... |
| gsummulglem 19983 | Lemma for ~ gsummulg and ~... |
| gsummulg 19984 | Nonnegative multiple of a ... |
| gsummulgz 19985 | Integer multiple of a grou... |
| gsumzoppg 19986 | The opposite of a group su... |
| gsumzinv 19987 | Inverse of a group sum. (... |
| gsuminv 19988 | Inverse of a group sum. (... |
| gsummptfidminv 19989 | Inverse of a group sum exp... |
| gsumsub 19990 | The difference of two grou... |
| gsummptfssub 19991 | The difference of two grou... |
| gsummptfidmsub 19992 | The difference of two grou... |
| gsumsnfd 19993 | Group sum of a singleton, ... |
| gsumsnd 19994 | Group sum of a singleton, ... |
| gsumsnf 19995 | Group sum of a singleton, ... |
| gsumsn 19996 | Group sum of a singleton. ... |
| gsumpr 19997 | Group sum of a pair. (Con... |
| gsumzunsnd 19998 | Append an element to a fin... |
| gsumunsnfd 19999 | Append an element to a fin... |
| gsumunsnd 20000 | Append an element to a fin... |
| gsumunsnf 20001 | Append an element to a fin... |
| gsumunsn 20002 | Append an element to a fin... |
| gsumdifsnd 20003 | Extract a summand from a f... |
| gsumpt 20004 | Sum of a family that is no... |
| gsummptf1o 20005 | Re-index a finite group su... |
| gsummptun 20006 | Group sum of a disjoint un... |
| gsummpt1n0 20007 | If only one summand in a f... |
| gsummptif1n0 20008 | If only one summand in a f... |
| gsummptcl 20009 | Closure of a finite group ... |
| gsummptfif1o 20010 | Re-index a finite group su... |
| gsummptfzcl 20011 | Closure of a finite group ... |
| gsum2dlem1 20012 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 20013 | Lemma for ~ gsum2d . (Con... |
| gsum2d 20014 | Write a sum over a two-dim... |
| gsum2d2lem 20015 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 20016 | Write a group sum over a t... |
| gsumcom2 20017 | Two-dimensional commutatio... |
| gsumxp 20018 | Write a group sum over a c... |
| gsumcom 20019 | Commute the arguments of a... |
| gsumcom3 20020 | A commutative law for fini... |
| gsumcom3fi 20021 | A commutative law for fini... |
| gsumxp2 20022 | Write a group sum over a c... |
| prdsgsum 20023 | Finite commutative sums in... |
| pwsgsum 20024 | Finite commutative sums in... |
| fsfnn0gsumfsffz 20025 | Replacing a finitely suppo... |
| nn0gsumfz 20026 | Replacing a finitely suppo... |
| nn0gsumfz0 20027 | Replacing a finitely suppo... |
| gsummptnn0fz 20028 | A final group sum over a f... |
| gsummptnn0fzfv 20029 | A final group sum over a f... |
| telgsumfzslem 20030 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 20031 | Telescoping group sum rang... |
| telgsumfz 20032 | Telescoping group sum rang... |
| telgsumfz0s 20033 | Telescoping finite group s... |
| telgsumfz0 20034 | Telescoping finite group s... |
| telgsums 20035 | Telescoping finitely suppo... |
| telgsum 20036 | Telescoping finitely suppo... |
| reldmdprd 20041 | The domain of the internal... |
| dmdprd 20042 | The domain of definition o... |
| dmdprdd 20043 | Show that a given family i... |
| dprddomprc 20044 | A family of subgroups inde... |
| dprddomcld 20045 | If a family of subgroups i... |
| dprdval0prc 20046 | The internal direct produc... |
| dprdval 20047 | The value of the internal ... |
| eldprd 20048 | A class ` A ` is an intern... |
| dprdgrp 20049 | Reverse closure for the in... |
| dprdf 20050 | The function ` S ` is a fa... |
| dprdf2 20051 | The function ` S ` is a fa... |
| dprdcntz 20052 | The function ` S ` is a fa... |
| dprddisj 20053 | The function ` S ` is a fa... |
| dprdw 20054 | The property of being a fi... |
| dprdwd 20055 | A mapping being a finitely... |
| dprdff 20056 | A finitely supported funct... |
| dprdfcl 20057 | A finitely supported funct... |
| dprdffsupp 20058 | A finitely supported funct... |
| dprdfcntz 20059 | A function on the elements... |
| dprdssv 20060 | The internal direct produc... |
| dprdfid 20061 | A function mapping all but... |
| eldprdi 20062 | The domain of definition o... |
| dprdfinv 20063 | Take the inverse of a grou... |
| dprdfadd 20064 | Take the sum of group sums... |
| dprdfsub 20065 | Take the difference of gro... |
| dprdfeq0 20066 | The zero function is the o... |
| dprdf11 20067 | Two group sums over a dire... |
| dprdsubg 20068 | The internal direct produc... |
| dprdub 20069 | Each factor is a subset of... |
| dprdlub 20070 | The direct product is smal... |
| dprdspan 20071 | The direct product is the ... |
| dprdres 20072 | Restriction of a direct pr... |
| dprdss 20073 | Create a direct product by... |
| dprdz 20074 | A family consisting entire... |
| dprd0 20075 | The empty family is an int... |
| dprdf1o 20076 | Rearrange the index set of... |
| dprdf1 20077 | Rearrange the index set of... |
| subgdmdprd 20078 | A direct product in a subg... |
| subgdprd 20079 | A direct product in a subg... |
| dprdsn 20080 | A singleton family is an i... |
| dmdprdsplitlem 20081 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 20082 | The function ` S ` is a fa... |
| dprddisj2 20083 | The function ` S ` is a fa... |
| dprd2dlem2 20084 | The direct product of a co... |
| dprd2dlem1 20085 | The direct product of a co... |
| dprd2da 20086 | The direct product of a co... |
| dprd2db 20087 | The direct product of a co... |
| dprd2d2 20088 | The direct product of a co... |
| dmdprdsplit2lem 20089 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 20090 | The direct product splits ... |
| dmdprdsplit 20091 | The direct product splits ... |
| dprdsplit 20092 | The direct product is the ... |
| dmdprdpr 20093 | A singleton family is an i... |
| dprdpr 20094 | A singleton family is an i... |
| dpjlem 20095 | Lemma for theorems about d... |
| dpjcntz 20096 | The two subgroups that app... |
| dpjdisj 20097 | The two subgroups that app... |
| dpjlsm 20098 | The two subgroups that app... |
| dpjfval 20099 | Value of the direct produc... |
| dpjval 20100 | Value of the direct produc... |
| dpjf 20101 | The ` X ` -th index projec... |
| dpjidcl 20102 | The key property of projec... |
| dpjeq 20103 | Decompose a group sum into... |
| dpjid 20104 | The key property of projec... |
| dpjlid 20105 | The ` X ` -th index projec... |
| dpjrid 20106 | The ` Y ` -th index projec... |
| dpjghm 20107 | The direct product is the ... |
| dpjghm2 20108 | The direct product is the ... |
| ablfacrplem 20109 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 20110 | A finite abelian group who... |
| ablfacrp2 20111 | The factors ` K , L ` of ~... |
| ablfac1lem 20112 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 20113 | The factors of ~ ablfac1b ... |
| ablfac1b 20114 | Any abelian group is the d... |
| ablfac1c 20115 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20116 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20117 | The factorization of ~ abl... |
| pgpfac1lem1 20118 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20119 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20120 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20121 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20122 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20123 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20124 | Factorization of a finite ... |
| pgpfaclem1 20125 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20126 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20127 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20128 | Full factorization of a fi... |
| ablfaclem1 20129 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20130 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20131 | Lemma for ~ ablfac . (Con... |
| ablfac 20132 | The Fundamental Theorem of... |
| ablfac2 20133 | Choose generators for each... |
| issimpg 20136 | The predicate "is a simple... |
| issimpgd 20137 | Deduce a simple group from... |
| simpggrp 20138 | A simple group is a group.... |
| simpggrpd 20139 | A simple group is a group.... |
| simpg2nsg 20140 | A simple group has two nor... |
| trivnsimpgd 20141 | Trivial groups are not sim... |
| simpgntrivd 20142 | Simple groups are nontrivi... |
| simpgnideld 20143 | A simple group contains a ... |
| simpgnsgd 20144 | The only normal subgroups ... |
| simpgnsgeqd 20145 | A normal subgroup of a sim... |
| 2nsgsimpgd 20146 | If any normal subgroup of ... |
| simpgnsgbid 20147 | A nontrivial group is simp... |
| ablsimpnosubgd 20148 | A subgroup of an abelian s... |
| ablsimpg1gend 20149 | An abelian simple group is... |
| ablsimpgcygd 20150 | An abelian simple group is... |
| ablsimpgfindlem1 20151 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20152 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20153 | Relationship between the o... |
| ablsimpgfind 20154 | An abelian simple group is... |
| fincygsubgd 20155 | The subgroup referenced in... |
| fincygsubgodd 20156 | Calculate the order of a s... |
| fincygsubgodexd 20157 | A finite cyclic group has ... |
| prmgrpsimpgd 20158 | A group of prime order is ... |
| ablsimpgprmd 20159 | An abelian simple group ha... |
| ablsimpgd 20160 | An abelian group is simple... |
| isomnd 20165 | A (left) ordered monoid is... |
| isogrp 20166 | A (left-)ordered group is ... |
| ogrpgrp 20167 | A left-ordered group is a ... |
| omndmnd 20168 | A left-ordered monoid is a... |
| omndtos 20169 | A left-ordered monoid is a... |
| omndadd 20170 | In an ordered monoid, the ... |
| omndaddr 20171 | In a right ordered monoid,... |
| omndadd2d 20172 | In a commutative left orde... |
| omndadd2rd 20173 | In a left- and right- orde... |
| submomnd 20174 | A submonoid of an ordered ... |
| omndmul2 20175 | In an ordered monoid, the ... |
| omndmul3 20176 | In an ordered monoid, the ... |
| omndmul 20177 | In a commutative ordered m... |
| ogrpinv0le 20178 | In an ordered group, the o... |
| ogrpsub 20179 | In an ordered group, the o... |
| ogrpaddlt 20180 | In an ordered group, stric... |
| ogrpaddltbi 20181 | In a right ordered group, ... |
| ogrpaddltrd 20182 | In a right ordered group, ... |
| ogrpaddltrbid 20183 | In a right ordered group, ... |
| ogrpsublt 20184 | In an ordered group, stric... |
| ogrpinv0lt 20185 | In an ordered group, the o... |
| ogrpinvlt 20186 | In an ordered group, the o... |
| gsumle 20187 | A finite sum in an ordered... |
| fnmgp 20190 | The multiplicative group o... |
| mgpval 20191 | Value of the multiplicatio... |
| mgpplusg 20192 | Value of the group operati... |
| mgpbas 20193 | Base set of the multiplica... |
| mgpsca 20194 | The multiplication monoid ... |
| mgptset 20195 | Topology component of the ... |
| mgptopn 20196 | Topology of the multiplica... |
| mgpds 20197 | Distance function of the m... |
| mgpress 20198 | Subgroup commutes with the... |
| prdsmgp 20199 | The multiplicative monoid ... |
| isrng 20202 | The predicate "is a non-un... |
| rngabl 20203 | A non-unital ring is an (a... |
| rngmgp 20204 | A non-unital ring is a sem... |
| rngmgpf 20205 | Restricted functionality o... |
| rnggrp 20206 | A non-unital ring is a (ad... |
| rngass 20207 | Associative law for the mu... |
| rngdi 20208 | Distributive law for the m... |
| rngdir 20209 | Distributive law for the m... |
| rngacl 20210 | Closure of the addition op... |
| rng0cl 20211 | The zero element of a non-... |
| rngcl 20212 | Closure of the multiplicat... |
| rnglz 20213 | The zero of a non-unital r... |
| rngrz 20214 | The zero of a non-unital r... |
| rngmneg1 20215 | Negation of a product in a... |
| rngmneg2 20216 | Negation of a product in a... |
| rngm2neg 20217 | Double negation of a produ... |
| rngansg 20218 | Every additive subgroup of... |
| rngsubdi 20219 | Ring multiplication distri... |
| rngsubdir 20220 | Ring multiplication distri... |
| isrngd 20221 | Properties that determine ... |
| rngpropd 20222 | If two structures have the... |
| prdsmulrngcl 20223 | Closure of the multiplicat... |
| prdsrngd 20224 | A product of non-unital ri... |
| imasrng 20225 | The image structure of a n... |
| imasrngf1 20226 | The image of a non-unital ... |
| xpsrngd 20227 | A product of two non-unita... |
| qusrng 20228 | The quotient structure of ... |
| rng1zrlem 20229 | Lemma for ~ rng1zr and ~ s... |
| rng1zr 20230 | The only ring with a base ... |
| rngen1zr 20231 | The only ring with one ele... |
| rngen1zr0 20232 | The only ring with one ele... |
| ringidval 20235 | The value of the unity ele... |
| dfur2 20236 | The multiplicative identit... |
| ringurd 20237 | Deduce the unity element o... |
| issrg 20240 | The predicate "is a semiri... |
| srgcmn 20241 | A semiring is a commutativ... |
| srgmnd 20242 | A semiring is a monoid. (... |
| srgmgp 20243 | A semiring is a monoid und... |
| srgdilem 20244 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20245 | Closure of the multiplicat... |
| srgass 20246 | Associative law for the mu... |
| srgideu 20247 | The unity element of a sem... |
| srgfcl 20248 | Functionality of the multi... |
| srgdi 20249 | Distributive law for the m... |
| srgdir 20250 | Distributive law for the m... |
| srgidcl 20251 | The unity element of a sem... |
| srg0cl 20252 | The zero element of a semi... |
| srgidmlem 20253 | Lemma for ~ srglidm and ~ ... |
| srglidm 20254 | The unity element of a sem... |
| srgridm 20255 | The unity element of a sem... |
| issrgid 20256 | Properties showing that an... |
| srgacl 20257 | Closure of the addition op... |
| srgcom 20258 | Commutativity of the addit... |
| srgrz 20259 | The zero of a semiring is ... |
| srglz 20260 | The zero of a semiring is ... |
| srgisid 20261 | In a semiring, the only le... |
| o2timesd 20262 | An element of a ring-like ... |
| rglcom4d 20263 | Restricted commutativity o... |
| srgo2times 20264 | A semiring element plus it... |
| srgcom4lem 20265 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20266 | Restricted commutativity o... |
| srg1zr 20267 | The only semiring with a b... |
| srgen1zr0 20268 | The only semiring with one... |
| srgmulgass 20269 | An associative property be... |
| srgpcomp 20270 | If two elements of a semir... |
| srgpcompp 20271 | If two elements of a semir... |
| srgpcomppsc 20272 | If two elements of a semir... |
| srglmhm 20273 | Left-multiplication in a s... |
| srgrmhm 20274 | Right-multiplication in a ... |
| srgsummulcr 20275 | A finite semiring sum mult... |
| sgsummulcl 20276 | A finite semiring sum mult... |
| srg1expzeq1 20277 | The exponentiation (by a n... |
| srgbinomlem1 20278 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20279 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20280 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20281 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20282 | Lemma for ~ srgbinom . In... |
| srgbinom 20283 | The binomial theorem for c... |
| csrgbinom 20284 | The binomial theorem for c... |
| isring 20289 | The predicate "is a (unita... |
| ringgrp 20290 | A ring is a group. (Contr... |
| ringmgp 20291 | A ring is a monoid under m... |
| iscrng 20292 | A commutative ring is a ri... |
| crngmgp 20293 | A commutative ring's multi... |
| ringgrpd 20294 | A ring is a group. (Contr... |
| ringmnd 20295 | A ring is a monoid under a... |
| ringmgm 20296 | A ring is a magma. (Contr... |
| crngring 20297 | A commutative ring is a ri... |
| crngringd 20298 | A commutative ring is a ri... |
| crnggrpd 20299 | A commutative ring is a gr... |
| mgpf 20300 | Restricted functionality o... |
| ringdilem 20301 | Properties of a unital rin... |
| ringcl 20302 | Closure of the multiplicat... |
| crngcom 20303 | A commutative ring's multi... |
| iscrng2 20304 | A commutative ring is a ri... |
| ringass 20305 | Associative law for multip... |
| ringideu 20306 | The unity element of a rin... |
| crngcomd 20307 | Multiplication is commutat... |
| crngbascntr 20308 | The base set of a commutat... |
| ringassd 20309 | Associative law for multip... |
| crng12d 20310 | Commutative/associative la... |
| crng32d 20311 | Commutative/associative la... |
| ringcld 20312 | Closure of the multiplicat... |
| ringdi 20313 | Distributive law for the m... |
| ringdir 20314 | Distributive law for the m... |
| ringdid 20315 | Distributive law for the m... |
| ringdird 20316 | Distributive law for the m... |
| ringidcl 20317 | The unity element of a rin... |
| ringidcld 20318 | The unity element of a rin... |
| ring0cl 20319 | The zero element of a ring... |
| ringidmlem 20320 | Lemma for ~ ringlidm and ~... |
| ringlidm 20321 | The unity element of a rin... |
| ringridm 20322 | The unity element of a rin... |
| isringid 20323 | Properties showing that an... |
| ringlidmd 20324 | The unity element of a rin... |
| ringridmd 20325 | The unity element of a rin... |
| ringid 20326 | The multiplication operati... |
| ringo2times 20327 | A ring element plus itself... |
| ringadd2 20328 | A ring element plus itself... |
| ringidss 20329 | A subset of the multiplica... |
| ringacl 20330 | Closure of the addition op... |
| ringcomlem 20331 | Lemma for ~ ringcom . Thi... |
| ringcom 20332 | Commutativity of the addit... |
| ringabl 20333 | A ring is an Abelian group... |
| ringcmn 20334 | A ring is a commutative mo... |
| ringabld 20335 | A ring is an Abelian group... |
| ringcmnd 20336 | A ring is a commutative mo... |
| ringrng 20337 | A unital ring is a non-uni... |
| ringssrng 20338 | The unital rings are non-u... |
| isringrng 20339 | The predicate "is a unital... |
| ringpropd 20340 | If two structures have the... |
| crngpropd 20341 | If two structures have the... |
| ringprop 20342 | If two structures have the... |
| isringd 20343 | Properties that determine ... |
| iscrngd 20344 | Properties that determine ... |
| ringlz 20345 | The zero of a unital ring ... |
| ringrz 20346 | The zero of a unital ring ... |
| ringlzd 20347 | The zero of a unital ring ... |
| ringrzd 20348 | The zero of a unital ring ... |
| ringsrg 20349 | Any ring is also a semirin... |
| ring1eq0 20350 | If one and zero are equal,... |
| ring1ne0 20351 | If a ring has at least two... |
| ringinvnz1ne0 20352 | In a unital ring, a left i... |
| ringinvnzdiv 20353 | In a unital ring, a left i... |
| ringnegl 20354 | Negation in a ring is the ... |
| ringnegr 20355 | Negation in a ring is the ... |
| ringmneg1 20356 | Negation of a product in a... |
| ringmneg2 20357 | Negation of a product in a... |
| ringm2neg 20358 | Double negation of a produ... |
| ringsubdi 20359 | Ring multiplication distri... |
| ringsubdir 20360 | Ring multiplication distri... |
| mulgass2 20361 | An associative property be... |
| ring1 20362 | The (smallest) structure r... |
| ringn0 20363 | Rings exist. (Contributed... |
| ringlghm 20364 | Left-multiplication in a r... |
| ringrghm 20365 | Right-multiplication in a ... |
| gsummulc1 20366 | A finite ring sum multipli... |
| gsummulc2 20367 | A finite ring sum multipli... |
| gsummgp0 20368 | If one factor in a finite ... |
| gsumdixp 20369 | Distribute a binary produc... |
| prdsmulrcl 20370 | A structure product of rin... |
| prdsringd 20371 | A product of rings is a ri... |
| prdscrngd 20372 | A product of commutative r... |
| prds1 20373 | Value of the ring unity in... |
| pwsring 20374 | A structure power of a rin... |
| pws1 20375 | Value of the ring unity in... |
| pwscrng 20376 | A structure power of a com... |
| pwsmgp 20377 | The multiplicative group o... |
| pwspjmhmmgpd 20378 | The projection given by ~ ... |
| pwsexpg 20379 | Value of a group exponenti... |
| pwsgprod 20380 | Finite products in a power... |
| imasring 20381 | The image structure of a r... |
| imasringf1 20382 | The image of a ring under ... |
| xpsringd 20383 | A product of two rings is ... |
| xpsring1d 20384 | The multiplicative identit... |
| qusring2 20385 | The quotient structure of ... |
| crngbinom 20386 | The binomial theorem for c... |
| opprval 20389 | Value of the opposite ring... |
| opprmulfval 20390 | Value of the multiplicatio... |
| opprmul 20391 | Value of the multiplicatio... |
| crngoppr 20392 | In a commutative ring, the... |
| opprlem 20393 | Lemma for ~ opprbas and ~ ... |
| opprbas 20394 | Base set of an opposite ri... |
| oppradd 20395 | Addition operation of an o... |
| opprrng 20396 | An opposite non-unital rin... |
| opprrngb 20397 | A class is a non-unital ri... |
| opprring 20398 | An opposite ring is a ring... |
| opprringb 20399 | Bidirectional form of ~ op... |
| oppr0 20400 | Additive identity of an op... |
| oppr1 20401 | Multiplicative identity of... |
| opprneg 20402 | The negative function in a... |
| opprsubg 20403 | Being a subgroup is a symm... |
| mulgass3 20404 | An associative property be... |
| reldvdsr 20411 | The divides relation is a ... |
| dvdsrval 20412 | Value of the divides relat... |
| dvdsr 20413 | Value of the divides relat... |
| dvdsr2 20414 | Value of the divides relat... |
| dvdsrmul 20415 | A left-multiple of ` X ` i... |
| dvdsrcl 20416 | Closure of a dividing elem... |
| dvdsrcl2 20417 | Closure of a dividing elem... |
| dvdsrid 20418 | An element in a (unital) r... |
| dvdsrtr 20419 | Divisibility is transitive... |
| dvdsrmul1 20420 | The divisibility relation ... |
| dvdsrneg 20421 | An element divides its neg... |
| dvdsr01 20422 | In a ring, zero is divisib... |
| dvdsr02 20423 | Only zero is divisible by ... |
| isunit 20424 | Property of being a unit o... |
| 1unit 20425 | The multiplicative identit... |
| unitcl 20426 | A unit is an element of th... |
| unitss 20427 | The set of units is contai... |
| opprunit 20428 | Being a unit is a symmetri... |
| crngunit 20429 | Property of being a unit i... |
| dvdsunit 20430 | A divisor of a unit is a u... |
| unitmulcl 20431 | The product of units is a ... |
| unitmulclb 20432 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20433 | The base set of the group ... |
| unitgrp 20434 | The group of units is a gr... |
| unitabl 20435 | The group of units of a co... |
| unitgrpid 20436 | The identity of the group ... |
| unitsubm 20437 | The group of units is a su... |
| invrfval 20440 | Multiplicative inverse fun... |
| unitinvcl 20441 | The inverse of a unit exis... |
| unitinvinv 20442 | The inverse of the inverse... |
| ringinvcl 20443 | The inverse of a unit is a... |
| unitlinv 20444 | A unit times its inverse i... |
| unitrinv 20445 | A unit times its inverse i... |
| 1rinv 20446 | The inverse of the ring un... |
| 0unit 20447 | The additive identity is a... |
| unitnegcl 20448 | The negative of a unit is ... |
| ringunitnzdiv 20449 | In a unitary ring, a unit ... |
| ring1nzdiv 20450 | In a unitary ring, the rin... |
| dvrfval 20453 | Division operation in a ri... |
| dvrval 20454 | Division operation in a ri... |
| dvrcl 20455 | Closure of division operat... |
| unitdvcl 20456 | The units are closed under... |
| dvrid 20457 | A ring element divided by ... |
| dvr1 20458 | A ring element divided by ... |
| dvrass 20459 | An associative law for div... |
| dvrcan1 20460 | A cancellation law for div... |
| dvrcan3 20461 | A cancellation law for div... |
| dvreq1 20462 | Equality in terms of ratio... |
| dvrdir 20463 | Distributive law for the d... |
| rdivmuldivd 20464 | Multiplication of two rati... |
| ringinvdv 20465 | Write the inverse function... |
| rngidpropd 20466 | The ring unity depends onl... |
| dvdsrpropd 20467 | The divisibility relation ... |
| unitpropd 20468 | The set of units depends o... |
| invrpropd 20469 | The ring inverse function ... |
| isirred 20470 | An irreducible element of ... |
| isnirred 20471 | The property of being a no... |
| isirred2 20472 | Expand out the class diffe... |
| opprirred 20473 | Irreducibility is symmetri... |
| irredn0 20474 | The additive identity is n... |
| irredcl 20475 | An irreducible element is ... |
| irrednu 20476 | An irreducible element is ... |
| irredn1 20477 | The multiplicative identit... |
| irredrmul 20478 | The product of an irreduci... |
| irredlmul 20479 | The product of a unit and ... |
| irredmul 20480 | If product of two elements... |
| irredneg 20481 | The negative of an irreduc... |
| irrednegb 20482 | An element is irreducible ... |
| rnghmrcl 20489 | Reverse closure of a non-u... |
| rnghmfn 20490 | The mapping of two non-uni... |
| rnghmval 20491 | The set of the non-unital ... |
| isrnghm 20492 | A function is a non-unital... |
| isrnghmmul 20493 | A function is a non-unital... |
| rnghmmgmhm 20494 | A non-unital ring homomorp... |
| rnghmval2 20495 | The non-unital ring homomo... |
| isrngim 20496 | An isomorphism of non-unit... |
| rngimrcl 20497 | Reverse closure for an iso... |
| rnghmghm 20498 | A non-unital ring homomorp... |
| rnghmf 20499 | A ring homomorphism is a f... |
| rnghmmul 20500 | A homomorphism of non-unit... |
| isrnghm2d 20501 | Demonstration of non-unita... |
| isrnghmd 20502 | Demonstration of non-unita... |
| rnghmf1o 20503 | A non-unital ring homomorp... |
| isrngim2 20504 | An isomorphism of non-unit... |
| rngimf1o 20505 | An isomorphism of non-unit... |
| rngimrnghm 20506 | An isomorphism of non-unit... |
| rngimcnv 20507 | The converse of an isomorp... |
| rnghmco 20508 | The composition of non-uni... |
| idrnghm 20509 | The identity homomorphism ... |
| c0mgm 20510 | The constant mapping to ze... |
| c0mhm 20511 | The constant mapping to ze... |
| c0ghm 20512 | The constant mapping to ze... |
| c0snmgmhm 20513 | The constant mapping to ze... |
| c0snmhm 20514 | The constant mapping to ze... |
| c0snghm 20515 | The constant mapping to ze... |
| rngisomfv1 20516 | If there is a non-unital r... |
| rngisom1 20517 | If there is a non-unital r... |
| rngisomring 20518 | If there is a non-unital r... |
| rngisomring1 20519 | If there is a non-unital r... |
| dfrhm2 20525 | The property of a ring hom... |
| rhmrcl1 20527 | Reverse closure of a ring ... |
| rhmrcl2 20528 | Reverse closure of a ring ... |
| isrhm 20529 | A function is a ring homom... |
| rhmmhm 20530 | A ring homomorphism is a h... |
| rhmisrnghm 20531 | Each unital ring homomorph... |
| rimrcl 20532 | Reverse closure for an iso... |
| isrim0 20533 | A ring isomorphism is a ho... |
| rhmghm 20534 | A ring homomorphism is an ... |
| rhmf 20535 | A ring homomorphism is a f... |
| rimcnv 20536 | The converse of a ring iso... |
| rhmmul 20537 | A homomorphism of rings pr... |
| isrhm2d 20538 | Demonstration of ring homo... |
| isrhmd 20539 | Demonstration of ring homo... |
| rhm1 20540 | Ring homomorphisms are req... |
| idrhm 20541 | The identity homomorphism ... |
| rhmf1o 20542 | A ring homomorphism is bij... |
| isrim 20543 | An isomorphism of rings is... |
| rimf1o 20544 | An isomorphism of rings is... |
| rimrhm 20545 | A ring isomorphism is a ho... |
| rimrcl1 20546 | Reverse closure of a ring ... |
| rimrcl2 20547 | Reverse closure of a ring ... |
| rimgim 20548 | An isomorphism of rings is... |
| rimisrngim 20549 | Each unital ring isomorphi... |
| rhmfn 20550 | The mapping of two rings t... |
| rhmval 20551 | The ring homomorphisms bet... |
| rhmco 20552 | The composition of ring ho... |
| pwsco1rhm 20553 | Right composition with a f... |
| pwsco2rhm 20554 | Left composition with a ri... |
| brric 20555 | The relation "is isomorphi... |
| brrici 20556 | Prove isomorphic by an exp... |
| ricsym 20557 | Ring isomorphism is symmet... |
| brric2 20558 | The relation "is isomorphi... |
| ricgic 20559 | If two rings are (ring) is... |
| rhmdvdsr 20560 | A ring homomorphism preser... |
| rhmopp 20561 | A ring homomorphism is als... |
| elrhmunit 20562 | Ring homomorphisms preserv... |
| rhmunitinv 20563 | Ring homomorphisms preserv... |
| isnzr 20566 | Property of a nonzero ring... |
| nzrnz 20567 | One and zero are different... |
| nzrring 20568 | A nonzero ring is a ring. ... |
| nzrringOLD 20569 | Obsolete version of ~ nzrr... |
| isnzr2 20570 | Equivalent characterizatio... |
| isnzr2hash 20571 | Equivalent characterizatio... |
| nzrpropd 20572 | If two structures have the... |
| opprnzrb 20573 | The opposite of a nonzero ... |
| opprnzr 20574 | The opposite of a nonzero ... |
| ringelnzr 20575 | A ring is nonzero if it ha... |
| nzrunit 20576 | A unit is nonzero in any n... |
| 0ringnnzr 20577 | A ring is a zero ring iff ... |
| 0ring 20578 | If a ring has only one ele... |
| 0ringdif 20579 | A zero ring is a ring whic... |
| 0ringbas 20580 | The base set of a zero rin... |
| 0ring01eq 20581 | In a ring with only one el... |
| 01eq0ring 20582 | If the zero and the identi... |
| 01eq0ringOLD 20583 | Obsolete version of ~ 01eq... |
| 0ring01eqbi 20584 | In a unital ring the zero ... |
| 0ring1eq0 20585 | In a zero ring, a ring whi... |
| c0rhm 20586 | The constant mapping to ze... |
| c0rnghm 20587 | The constant mapping to ze... |
| zrrnghm 20588 | The constant mapping to ze... |
| nrhmzr 20589 | There is no ring homomorph... |
| islring 20592 | The predicate "is a local ... |
| lringnzr 20593 | A local ring is a nonzero ... |
| lringring 20594 | A local ring is a ring. (... |
| lringnz 20595 | A local ring is a nonzero ... |
| lringuplu 20596 | If the sum of two elements... |
| issubrng 20599 | The subring of non-unital ... |
| subrngss 20600 | A subring is a subset. (C... |
| subrngid 20601 | Every non-unital ring is a... |
| subrngrng 20602 | A subring is a non-unital ... |
| subrngrcl 20603 | Reverse closure for a subr... |
| subrngsubg 20604 | A subring is a subgroup. ... |
| subrngringnsg 20605 | A subring is a normal subg... |
| subrngbas 20606 | Base set of a subring stru... |
| subrng0 20607 | A subring always has the s... |
| subrngacl 20608 | A subring is closed under ... |
| subrngmcl 20609 | A subring is closed under ... |
| issubrng2 20610 | Characterize the subrings ... |
| opprsubrng 20611 | Being a subring is a symme... |
| subrngint 20612 | The intersection of a none... |
| subrngin 20613 | The intersection of two su... |
| subrngmre 20614 | The subrings of a non-unit... |
| subsubrng 20615 | A subring of a subring is ... |
| subsubrng2 20616 | The set of subrings of a s... |
| rhmimasubrnglem 20617 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20618 | The homomorphic image of a... |
| cntzsubrng 20619 | Centralizers in a non-unit... |
| subrngpropd 20620 | If two structures have the... |
| issubrg 20623 | The subring predicate. (C... |
| subrgss 20624 | A subring is a subset. (C... |
| subrgid 20625 | Every ring is a subring of... |
| subrgring 20626 | A subring is a ring. (Con... |
| subrgcrng 20627 | A subring of a commutative... |
| subrgrcl 20628 | Reverse closure for a subr... |
| subrgsubg 20629 | A subring is a subgroup. ... |
| subrgsubrng 20630 | A subring of a unital ring... |
| subrg0 20631 | A subring always has the s... |
| subrg1cl 20632 | A subring contains the mul... |
| subrgbas 20633 | Base set of a subring stru... |
| subrg1 20634 | A subring always has the s... |
| subrgacl 20635 | A subring is closed under ... |
| subrgmcl 20636 | A subring is closed under ... |
| subrgsubm 20637 | A subring is a submonoid o... |
| subrgdvds 20638 | If an element divides anot... |
| subrguss 20639 | A unit of a subring is a u... |
| subrginv 20640 | A subring always has the s... |
| subrgdv 20641 | A subring always has the s... |
| subrgunit 20642 | An element of a ring is a ... |
| subrgugrp 20643 | The units of a subring for... |
| issubrg2 20644 | Characterize the subrings ... |
| opprsubrg 20645 | Being a subring is a symme... |
| subrgnzr 20646 | A subring of a nonzero rin... |
| subrgint 20647 | The intersection of a none... |
| subrgin 20648 | The intersection of two su... |
| subrgmre 20649 | The subrings of a ring are... |
| subsubrg 20650 | A subring of a subring is ... |
| subsubrg2 20651 | The set of subrings of a s... |
| issubrg3 20652 | A subring is an additive s... |
| resrhm 20653 | Restriction of a ring homo... |
| resrhm2b 20654 | Restriction of the codomai... |
| rhmeql 20655 | The equalizer of two ring ... |
| rhmima 20656 | The homomorphic image of a... |
| rnrhmsubrg 20657 | The range of a ring homomo... |
| cntzsubr 20658 | Centralizers in a ring are... |
| pwsdiagrhm 20659 | Diagonal homomorphism into... |
| subrgpropd 20660 | If two structures have the... |
| rhmpropd 20661 | Ring homomorphism depends ... |
| rgspnval 20664 | Value of the ring-span of ... |
| rgspncl 20665 | The ring-span of a set is ... |
| rgspnssid 20666 | The ring-span of a set con... |
| rgspnmin 20667 | The ring-span is contained... |
| rngcval 20670 | Value of the category of n... |
| rnghmresfn 20671 | The class of non-unital ri... |
| rnghmresel 20672 | An element of the non-unit... |
| rngcbas 20673 | Set of objects of the cate... |
| rngchomfval 20674 | Set of arrows of the categ... |
| rngchom 20675 | Set of arrows of the categ... |
| elrngchom 20676 | A morphism of non-unital r... |
| rngchomfeqhom 20677 | The functionalized Hom-set... |
| rngccofval 20678 | Composition in the categor... |
| rngcco 20679 | Composition in the categor... |
| dfrngc2 20680 | Alternate definition of th... |
| rnghmsscmap2 20681 | The non-unital ring homomo... |
| rnghmsscmap 20682 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20683 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20684 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20685 | The non-unital ring homomo... |
| rngccat 20686 | The category of non-unital... |
| rngcid 20687 | The identity arrow in the ... |
| rngcsect 20688 | A section in the category ... |
| rngcinv 20689 | An inverse in the category... |
| rngciso 20690 | An isomorphism in the cate... |
| rngcifuestrc 20691 | The "inclusion functor" fr... |
| funcrngcsetc 20692 | The "natural forgetful fun... |
| funcrngcsetcALT 20693 | Alternate proof of ~ funcr... |
| zrinitorngc 20694 | The zero ring is an initia... |
| zrtermorngc 20695 | The zero ring is a termina... |
| zrzeroorngc 20696 | The zero ring is a zero ob... |
| ringcval 20699 | Value of the category of u... |
| rhmresfn 20700 | The class of unital ring h... |
| rhmresel 20701 | An element of the unital r... |
| ringcbas 20702 | Set of objects of the cate... |
| ringchomfval 20703 | Set of arrows of the categ... |
| ringchom 20704 | Set of arrows of the categ... |
| elringchom 20705 | A morphism of unital rings... |
| ringchomfeqhom 20706 | The functionalized Hom-set... |
| ringccofval 20707 | Composition in the categor... |
| ringcco 20708 | Composition in the categor... |
| dfringc2 20709 | Alternate definition of th... |
| rhmsscmap2 20710 | The unital ring homomorphi... |
| rhmsscmap 20711 | The unital ring homomorphi... |
| rhmsubcsetclem1 20712 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20713 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20714 | The unital ring homomorphi... |
| ringccat 20715 | The category of unital rin... |
| ringcid 20716 | The identity arrow in the ... |
| rhmsscrnghm 20717 | The unital ring homomorphi... |
| rhmsubcrngclem1 20718 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20719 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20720 | The unital ring homomorphi... |
| rngcresringcat 20721 | The restriction of the cat... |
| ringcsect 20722 | A section in the category ... |
| ringcinv 20723 | An inverse in the category... |
| ringciso 20724 | An isomorphism in the cate... |
| ringcbasbas 20725 | An element of the base set... |
| funcringcsetc 20726 | The "natural forgetful fun... |
| zrtermoringc 20727 | The zero ring is a termina... |
| zrninitoringc 20728 | The zero ring is not an in... |
| srhmsubclem1 20729 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20730 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20731 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20732 | According to ~ df-subc , t... |
| sringcat 20733 | The restriction of the cat... |
| crhmsubc 20734 | According to ~ df-subc , t... |
| cringcat 20735 | The restriction of the cat... |
| rngcrescrhm 20736 | The category of non-unital... |
| rhmsubclem1 20737 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20738 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20739 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20740 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20741 | According to ~ df-subc , t... |
| rhmsubccat 20742 | The restriction of the cat... |
| rrgval 20749 | Value of the set or left-r... |
| isrrg 20750 | Membership in the set of l... |
| rrgeq0i 20751 | Property of a left-regular... |
| rrgeq0 20752 | Left-multiplication by a l... |
| rrgsupp 20753 | Left multiplication by a l... |
| rrgss 20754 | Left-regular elements are ... |
| unitrrg 20755 | Units are regular elements... |
| rrgnz 20756 | In a nonzero ring, the zer... |
| isdomn 20757 | Expand definition of a dom... |
| domnnzr 20758 | A domain is a nonzero ring... |
| domnring 20759 | A domain is a ring. (Cont... |
| domneq0 20760 | In a domain, a product is ... |
| domnmuln0 20761 | In a domain, a product of ... |
| isdomn5 20762 | The equivalence between th... |
| isdomn2 20763 | A ring is a domain iff all... |
| isdomn2OLD 20764 | Obsolete version of ~ isdo... |
| domnrrg 20765 | In a domain, a nonzero ele... |
| isdomn6 20766 | A ring is a domain iff the... |
| isdomn3 20767 | Nonzero elements form a mu... |
| isdomn4 20768 | A ring is a domain iff it ... |
| opprdomnb 20769 | A class is a domain if and... |
| opprdomn 20770 | The opposite of a domain i... |
| isdomn4r 20771 | A ring is a domain iff it ... |
| domnlcanb 20772 | Left-cancellation law for ... |
| domnlcan 20773 | Left-cancellation law for ... |
| domnrcanb 20774 | Right-cancellation law for... |
| domnrcan 20775 | Right-cancellation law for... |
| domneq0r 20776 | Right multiplication by a ... |
| isidom 20777 | An integral domain is a co... |
| idomdomd 20778 | An integral domain is a do... |
| idomcringd 20779 | An integral domain is a co... |
| idomringd 20780 | An integral domain is a ri... |
| isdrng 20785 | The predicate "is a divisi... |
| drngunit 20786 | Elementhood in the set of ... |
| drngui 20787 | The set of units of a divi... |
| drngring 20788 | A division ring is a ring.... |
| drngringd 20789 | A division ring is a ring.... |
| drnggrpd 20790 | A division ring is a group... |
| drnggrp 20791 | A division ring is a group... |
| isfld 20792 | A field is a commutative d... |
| flddrngd 20793 | A field is a division ring... |
| fldcrngd 20794 | A field is a commutative r... |
| isdrng2 20795 | A division ring can equiva... |
| drngprop 20796 | If two structures have the... |
| drngmgp 20797 | A division ring contains a... |
| drngid 20798 | A division ring's unity is... |
| drngunz 20799 | A division ring's unity is... |
| drngnzr 20800 | A division ring is a nonze... |
| drngdomn 20801 | A division ring is a domai... |
| drngmcl 20802 | The product of two nonzero... |
| drngmclOLD 20803 | Obsolete version of ~ drng... |
| drngid2 20804 | Properties showing that an... |
| drnginvrcl 20805 | Closure of the multiplicat... |
| drnginvrn0 20806 | The multiplicative inverse... |
| drnginvrcld 20807 | Closure of the multiplicat... |
| drnginvrl 20808 | Property of the multiplica... |
| drnginvrr 20809 | Property of the multiplica... |
| drnginvrld 20810 | Property of the multiplica... |
| drnginvrrd 20811 | Property of the multiplica... |
| drngmul0or 20812 | A product is zero iff one ... |
| drngmul0orOLD 20813 | Obsolete version of ~ drng... |
| drngmulne0 20814 | A product is nonzero iff b... |
| drngmuleq0 20815 | An element is zero iff its... |
| opprdrng 20816 | The opposite of a division... |
| isdrngd 20817 | Properties that characteri... |
| isdrngrd 20818 | Properties that characteri... |
| isdrngdOLD 20819 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20820 | Obsolete version of ~ isdr... |
| drngpropd 20821 | If two structures have the... |
| fldpropd 20822 | If two structures have the... |
| fldidom 20823 | A field is an integral dom... |
| fidomndrnglem 20824 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20825 | A finite domain is a divis... |
| fiidomfld 20826 | A finite integral domain i... |
| rng1nnzr 20827 | The (smallest) structure r... |
| ring1zr 20828 | The only unital ring with ... |
| ringen1zr0 20829 | The only unital ring with ... |
| rng1nfld 20830 | The zero ring is not a fie... |
| issubdrg 20831 | Characterize the subfields... |
| drhmsubc 20832 | According to ~ df-subc , t... |
| drngcat 20833 | The restriction of the cat... |
| fldcat 20834 | The restriction of the cat... |
| fldc 20835 | The restriction of the cat... |
| fldhmsubc 20836 | According to ~ df-subc , t... |
| issdrg 20839 | Property of a division sub... |
| sdrgrcl 20840 | Reverse closure for a sub-... |
| sdrgdrng 20841 | A sub-division-ring is a d... |
| sdrgsubrg 20842 | A sub-division-ring is a s... |
| sdrgid 20843 | Every division ring is a d... |
| sdrgss 20844 | A division subring is a su... |
| sdrgbas 20845 | Base set of a sub-division... |
| issdrg2 20846 | Property of a division sub... |
| sdrgunit 20847 | A unit of a sub-division-r... |
| imadrhmcl 20848 | The image of a (nontrivial... |
| fldsdrgfld 20849 | A sub-division-ring of a f... |
| acsfn1p 20850 | Construction of a closure ... |
| subrgacs 20851 | Closure property of subrin... |
| sdrgacs 20852 | Closure property of divisi... |
| cntzsdrg 20853 | Centralizers in division r... |
| subdrgint 20854 | The intersection of a none... |
| sdrgint 20855 | The intersection of a none... |
| primefld 20856 | The smallest sub division ... |
| primefld0cl 20857 | The prime field contains t... |
| primefld1cl 20858 | The prime field contains t... |
| abvfval 20861 | Value of the set of absolu... |
| isabv 20862 | Elementhood in the set of ... |
| isabvd 20863 | Properties that determine ... |
| abvrcl 20864 | Reverse closure for the ab... |
| abvfge0 20865 | An absolute value is a fun... |
| abvf 20866 | An absolute value is a fun... |
| abvcl 20867 | An absolute value is a fun... |
| abvge0 20868 | The absolute value of a nu... |
| abveq0 20869 | The value of an absolute v... |
| abvne0 20870 | The absolute value of a no... |
| abvgt0 20871 | The absolute value of a no... |
| abvmul 20872 | An absolute value distribu... |
| abvtri 20873 | An absolute value satisfie... |
| abv0 20874 | The absolute value of zero... |
| abv1z 20875 | The absolute value of one ... |
| abv1 20876 | The absolute value of one ... |
| abvneg 20877 | The absolute value of a ne... |
| abvsubtri 20878 | An absolute value satisfie... |
| abvrec 20879 | The absolute value distrib... |
| abvdiv 20880 | The absolute value distrib... |
| abvdom 20881 | Any ring with an absolute ... |
| abvres 20882 | The restriction of an abso... |
| abvtrivd 20883 | The trivial absolute value... |
| abvtrivg 20884 | The trivial absolute value... |
| abvtriv 20885 | The trivial absolute value... |
| abvpropd 20886 | If two structures have the... |
| abvn0b 20887 | Another characterization o... |
| staffval 20892 | The functionalization of t... |
| stafval 20893 | The functionalization of t... |
| staffn 20894 | The functionalization is e... |
| issrng 20895 | The predicate "is a star r... |
| srngrhm 20896 | The involution function in... |
| srngring 20897 | A star ring is a ring. (C... |
| srngcnv 20898 | The involution function in... |
| srngf1o 20899 | The involution function in... |
| srngcl 20900 | The involution function in... |
| srngnvl 20901 | The involution function in... |
| srngadd 20902 | The involution function in... |
| srngmul 20903 | The involution function in... |
| srng1 20904 | The conjugate of the ring ... |
| srng0 20905 | The conjugate of the ring ... |
| issrngd 20906 | Properties that determine ... |
| idsrngd 20907 | A commutative ring is a st... |
| isorng 20912 | An ordered ring is a ring ... |
| orngring 20913 | An ordered ring is a ring.... |
| orngogrp 20914 | An ordered ring is an orde... |
| isofld 20915 | An ordered field is a fiel... |
| orngmul 20916 | In an ordered ring, the or... |
| orngsqr 20917 | In an ordered ring, all sq... |
| ornglmulle 20918 | In an ordered ring, multip... |
| orngrmulle 20919 | In an ordered ring, multip... |
| ornglmullt 20920 | In an ordered ring, multip... |
| orngrmullt 20921 | In an ordered ring, multip... |
| orngmullt 20922 | In an ordered ring, the st... |
| ofldfld 20923 | An ordered field is a fiel... |
| ofldtos 20924 | An ordered field is a tota... |
| orng0le1 20925 | In an ordered ring, the ri... |
| ofldlt1 20926 | In an ordered field, the r... |
| suborng 20927 | Every subring of an ordere... |
| subofld 20928 | Every subfield of an order... |
| islmod 20933 | The predicate "is a left m... |
| lmodlema 20934 | Lemma for properties of a ... |
| islmodd 20935 | Properties that determine ... |
| lmodgrp 20936 | A left module is a group. ... |
| lmodring 20937 | The scalar component of a ... |
| lmodfgrp 20938 | The scalar component of a ... |
| lmodgrpd 20939 | A left module is a group. ... |
| lmodbn0 20940 | The base set of a left mod... |
| lmodacl 20941 | Closure of ring addition f... |
| lmodmcl 20942 | Closure of ring multiplica... |
| lmodsn0 20943 | The set of scalars in a le... |
| lmodvacl 20944 | Closure of vector addition... |
| lmodass 20945 | Left module vector sum is ... |
| lmodlcan 20946 | Left cancellation law for ... |
| lmodvscl 20947 | Closure of scalar product ... |
| lmodvscld 20948 | Closure of scalar product ... |
| scaffval 20949 | The scalar multiplication ... |
| scafval 20950 | The scalar multiplication ... |
| scafeq 20951 | If the scalar multiplicati... |
| scaffn 20952 | The scalar multiplication ... |
| lmodscaf 20953 | The scalar multiplication ... |
| lmodvsdi 20954 | Distributive law for scala... |
| lmodvsdir 20955 | Distributive law for scala... |
| lmodvsass 20956 | Associative law for scalar... |
| lmod0cl 20957 | The ring zero in a left mo... |
| lmod1cl 20958 | The ring unity in a left m... |
| lmodvs1 20959 | Scalar product with the ri... |
| lmod0vcl 20960 | The zero vector is a vecto... |
| lmod0vlid 20961 | Left identity law for the ... |
| lmod0vrid 20962 | Right identity law for the... |
| lmod0vid 20963 | Identity equivalent to the... |
| lmod0vs 20964 | Zero times a vector is the... |
| lmodvs0 20965 | Anything times the zero ve... |
| lmodvsmmulgdi 20966 | Distributive law for a gro... |
| lmodfopnelem1 20967 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20968 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20969 | The (functionalized) opera... |
| lcomf 20970 | A linear-combination sum i... |
| lcomfsupp 20971 | A linear-combination sum i... |
| lmodvnegcl 20972 | Closure of vector negative... |
| lmodvnegid 20973 | Addition of a vector with ... |
| lmodvneg1 20974 | Minus 1 times a vector is ... |
| lmodvsneg 20975 | Multiplication of a vector... |
| lmodvsubcl 20976 | Closure of vector subtract... |
| lmodcom 20977 | Left module vector sum is ... |
| lmodabl 20978 | A left module is an abelia... |
| lmodcmn 20979 | A left module is a commuta... |
| lmodnegadd 20980 | Distribute negation throug... |
| lmod4 20981 | Commutative/associative la... |
| lmodvsubadd 20982 | Relationship between vecto... |
| lmodvaddsub4 20983 | Vector addition/subtractio... |
| lmodvpncan 20984 | Addition/subtraction cance... |
| lmodvnpcan 20985 | Cancellation law for vecto... |
| lmodvsubval2 20986 | Value of vector subtractio... |
| lmodsubvs 20987 | Subtraction of a scalar pr... |
| lmodsubdi 20988 | Scalar multiplication dist... |
| lmodsubdir 20989 | Scalar multiplication dist... |
| lmodsubeq0 20990 | If the difference between ... |
| lmodsubid 20991 | Subtraction of a vector fr... |
| lmodvsghm 20992 | Scalar multiplication of t... |
| lmodprop2d 20993 | If two structures have the... |
| lmodpropd 20994 | If two structures have the... |
| gsumvsmul 20995 | Pull a scalar multiplicati... |
| mptscmfsupp0 20996 | A mapping to a scalar prod... |
| mptscmfsuppd 20997 | A function mapping to a sc... |
| rmodislmodlem 20998 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20999 | The right module ` R ` ind... |
| lssset 21002 | The set of all (not necess... |
| islss 21003 | The predicate "is a subspa... |
| islssd 21004 | Properties that determine ... |
| lssss 21005 | A subspace is a set of vec... |
| lssel 21006 | A subspace member is a vec... |
| lss1 21007 | The set of vectors in a le... |
| lssuni 21008 | The union of all subspaces... |
| lssn0 21009 | A subspace is not empty. ... |
| 00lss 21010 | The empty structure has no... |
| lsscl 21011 | Closure property of a subs... |
| lssvacl 21012 | Closure of vector addition... |
| lssvsubcl 21013 | Closure of vector subtract... |
| lssvancl1 21014 | Non-closure: if one vector... |
| lssvancl2 21015 | Non-closure: if one vector... |
| lss0cl 21016 | The zero vector belongs to... |
| lsssn0 21017 | The singleton of the zero ... |
| lss0ss 21018 | The zero subspace is inclu... |
| lssle0 21019 | No subspace is smaller tha... |
| lssne0 21020 | A nonzero subspace has a n... |
| lssvneln0 21021 | A vector ` X ` which doesn... |
| lssneln0 21022 | A vector ` X ` which doesn... |
| lssssr 21023 | Conclude subspace ordering... |
| lssvscl 21024 | Closure of scalar product ... |
| lssvnegcl 21025 | Closure of negative vector... |
| lsssubg 21026 | All subspaces are subgroup... |
| lsssssubg 21027 | All subspaces are subgroup... |
| islss3 21028 | A linear subspace of a mod... |
| lsslmod 21029 | A submodule is a module. ... |
| lsslss 21030 | The subspaces of a subspac... |
| islss4 21031 | A linear subspace is a sub... |
| lss1d 21032 | One-dimensional subspace (... |
| lssintcl 21033 | The intersection of a none... |
| lssincl 21034 | The intersection of two su... |
| lssmre 21035 | The subspaces of a module ... |
| lssacs 21036 | Submodules are an algebrai... |
| prdsvscacl 21037 | Pointwise scalar multiplic... |
| prdslmodd 21038 | The product of a family of... |
| pwslmod 21039 | A structure power of a lef... |
| lspfval 21042 | The span function for a le... |
| lspf 21043 | The span function on a lef... |
| lspval 21044 | The span of a set of vecto... |
| lspcl 21045 | The span of a set of vecto... |
| lspsncl 21046 | The span of a singleton is... |
| lspprcl 21047 | The span of a pair is a su... |
| lsptpcl 21048 | The span of an unordered t... |
| lspsnsubg 21049 | The span of a singleton is... |
| 00lsp 21050 | ~ fvco4i lemma for linear ... |
| lspid 21051 | The span of a subspace is ... |
| lspssv 21052 | A span is a set of vectors... |
| lspss 21053 | Span preserves subset orde... |
| lspssid 21054 | A set of vectors is a subs... |
| lspidm 21055 | The span of a set of vecto... |
| lspun 21056 | The span of union is the s... |
| lspssp 21057 | If a set of vectors is a s... |
| mrclsp 21058 | Moore closure generalizes ... |
| lspsnss 21059 | The span of the singleton ... |
| ellspsn3 21060 | A member of the span of th... |
| lspprss 21061 | The span of a pair of vect... |
| lspsnid 21062 | A vector belongs to the sp... |
| ellspsn6 21063 | Relationship between a vec... |
| ellspsn5b 21064 | Relationship between a vec... |
| ellspsn5 21065 | Relationship between a vec... |
| lspprid1 21066 | A member of a pair of vect... |
| lspprid2 21067 | A member of a pair of vect... |
| lspprvacl 21068 | The sum of two vectors bel... |
| lssats2 21069 | A way to express atomistic... |
| ellspsni 21070 | A scalar product with a ve... |
| lspsn 21071 | Span of the singleton of a... |
| ellspsn 21072 | Member of span of the sing... |
| lspsnvsi 21073 | Span of a scalar product o... |
| lspsnss2 21074 | Comparable spans of single... |
| lspsnneg 21075 | Negation does not change t... |
| lspsnsub 21076 | Swapping subtraction order... |
| lspsn0 21077 | Span of the singleton of t... |
| lsp0 21078 | Span of the empty set. (C... |
| lspuni0 21079 | Union of the span of the e... |
| lspun0 21080 | The span of a union with t... |
| lspsneq0 21081 | Span of the singleton is t... |
| lspsneq0b 21082 | Equal singleton spans impl... |
| lmodindp1 21083 | Two independent (non-colin... |
| lsslsp 21084 | Spans in submodules corres... |
| lss0v 21085 | The zero vector in a submo... |
| lsspropd 21086 | If two structures have the... |
| lsppropd 21087 | If two structures have the... |
| reldmlmhm 21094 | Lemma for module homomorph... |
| lmimfn 21095 | Lemma for module isomorphi... |
| islmhm 21096 | Property of being a homomo... |
| islmhm3 21097 | Property of a module homom... |
| lmhmlem 21098 | Non-quantified consequence... |
| lmhmsca 21099 | A homomorphism of left mod... |
| lmghm 21100 | A homomorphism of left mod... |
| lmhmlmod2 21101 | A homomorphism of left mod... |
| lmhmlmod1 21102 | A homomorphism of left mod... |
| lmhmf 21103 | A homomorphism of left mod... |
| lmhmlin 21104 | A homomorphism of left mod... |
| lmodvsinv 21105 | Multiplication of a vector... |
| lmodvsinv2 21106 | Multiplying a negated vect... |
| islmhm2 21107 | A one-equation proof of li... |
| islmhmd 21108 | Deduction for a module hom... |
| 0lmhm 21109 | The constant zero linear f... |
| idlmhm 21110 | The identity function on a... |
| invlmhm 21111 | The negative function on a... |
| lmhmco 21112 | The composition of two mod... |
| lmhmplusg 21113 | The pointwise sum of two l... |
| lmhmvsca 21114 | The pointwise scalar produ... |
| lmhmf1o 21115 | A bijective module homomor... |
| lmhmima 21116 | The image of a subspace un... |
| lmhmpreima 21117 | The inverse image of a sub... |
| lmhmlsp 21118 | Homomorphisms preserve spa... |
| lmhmrnlss 21119 | The range of a homomorphis... |
| lmhmkerlss 21120 | The kernel of a homomorphi... |
| reslmhm 21121 | Restriction of a homomorph... |
| reslmhm2 21122 | Expansion of the codomain ... |
| reslmhm2b 21123 | Expansion of the codomain ... |
| lmhmeql 21124 | The equalizer of two modul... |
| lspextmo 21125 | A linear function is compl... |
| pwsdiaglmhm 21126 | Diagonal homomorphism into... |
| pwssplit0 21127 | Splitting for structure po... |
| pwssplit1 21128 | Splitting for structure po... |
| pwssplit2 21129 | Splitting for structure po... |
| pwssplit3 21130 | Splitting for structure po... |
| islmim 21131 | An isomorphism of left mod... |
| lmimf1o 21132 | An isomorphism of left mod... |
| lmimlmhm 21133 | An isomorphism of modules ... |
| lmimgim 21134 | An isomorphism of modules ... |
| islmim2 21135 | An isomorphism of left mod... |
| lmimcnv 21136 | The converse of a bijectiv... |
| brlmic 21137 | The relation "is isomorphi... |
| brlmici 21138 | Prove isomorphic by an exp... |
| lmiclcl 21139 | Isomorphism implies the le... |
| lmicrcl 21140 | Isomorphism implies the ri... |
| lmicsym 21141 | Module isomorphism is symm... |
| lmhmpropd 21142 | Module homomorphism depend... |
| islbs 21145 | The predicate " ` B ` is a... |
| lbsss 21146 | A basis is a set of vector... |
| lbsel 21147 | An element of a basis is a... |
| lbssp 21148 | The span of a basis is the... |
| lbsind 21149 | A basis is linearly indepe... |
| lbsind2 21150 | A basis is linearly indepe... |
| lbspss 21151 | No proper subset of a basi... |
| lsmcl 21152 | The sum of two subspaces i... |
| lsmspsn 21153 | Member of subspace sum of ... |
| lsmelval2 21154 | Subspace sum membership in... |
| lsmsp 21155 | Subspace sum in terms of s... |
| lsmsp2 21156 | Subspace sum of spans of s... |
| lsmssspx 21157 | Subspace sum (in its exten... |
| lsmpr 21158 | The span of a pair of vect... |
| lsppreli 21159 | A vector expressed as a su... |
| lsmelpr 21160 | Two ways to say that a vec... |
| lsppr0 21161 | The span of a vector paire... |
| lsppr 21162 | Span of a pair of vectors.... |
| lspprel 21163 | Member of the span of a pa... |
| lspprabs 21164 | Absorption of vector sum i... |
| lspvadd 21165 | The span of a vector sum i... |
| lspsntri 21166 | Triangle-type inequality f... |
| lspsntrim 21167 | Triangle-type inequality f... |
| lbspropd 21168 | If two structures have the... |
| pj1lmhm 21169 | The left projection functi... |
| pj1lmhm2 21170 | The left projection functi... |
| islvec 21173 | The predicate "is a left v... |
| lvecdrng 21174 | The set of scalars of a le... |
| lveclmod 21175 | A left vector space is a l... |
| lveclmodd 21176 | A vector space is a left m... |
| lvecgrpd 21177 | A vector space is a group.... |
| lsslvec 21178 | A vector subspace is a vec... |
| lmhmlvec 21179 | The property for modules t... |
| lvecvs0or 21180 | If a scalar product is zer... |
| lvecvsn0 21181 | A scalar product is nonzer... |
| lssvs0or 21182 | If a scalar product belong... |
| lvecvscan 21183 | Cancellation law for scala... |
| lvecvscan2 21184 | Cancellation law for scala... |
| lvecinv 21185 | Invert coefficient of scal... |
| lspsnvs 21186 | A nonzero scalar product d... |
| lspsneleq 21187 | Membership relation that i... |
| lspsncmp 21188 | Comparable spans of nonzer... |
| lspsnne1 21189 | Two ways to express that v... |
| lspsnne2 21190 | Two ways to express that v... |
| lspsnnecom 21191 | Swap two vectors with diff... |
| lspabs2 21192 | Absorption law for span of... |
| lspabs3 21193 | Absorption law for span of... |
| lspsneq 21194 | Equal spans of singletons ... |
| lspsneu 21195 | Nonzero vectors with equal... |
| ellspsn4 21196 | A member of the span of th... |
| lspdisj 21197 | The span of a vector not i... |
| lspdisjb 21198 | A nonzero vector is not in... |
| lspdisj2 21199 | Unequal spans are disjoint... |
| lspfixed 21200 | Show membership in the spa... |
| lspexch 21201 | Exchange property for span... |
| lspexchn1 21202 | Exchange property for span... |
| lspexchn2 21203 | Exchange property for span... |
| lspindpi 21204 | Partial independence prope... |
| lspindp1 21205 | Alternate way to say 3 vec... |
| lspindp2l 21206 | Alternate way to say 3 vec... |
| lspindp2 21207 | Alternate way to say 3 vec... |
| lspindp3 21208 | Independence of 2 vectors ... |
| lspindp4 21209 | (Partial) independence of ... |
| lvecindp 21210 | Compute the ` X ` coeffici... |
| lvecindp2 21211 | Sums of independent vector... |
| lspsnsubn0 21212 | Unequal singleton spans im... |
| lsmcv 21213 | Subspace sum has the cover... |
| lspsolvlem 21214 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21215 | If ` X ` is in the span of... |
| lssacsex 21216 | In a vector space, subspac... |
| lspsnat 21217 | There is no subspace stric... |
| lspsncv0 21218 | The span of a singleton co... |
| lsppratlem1 21219 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21220 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21221 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21222 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21223 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21224 | Lemma for ~ lspprat . Neg... |
| lspprat 21225 | A proper subspace of the s... |
| islbs2 21226 | An equivalent formulation ... |
| islbs3 21227 | An equivalent formulation ... |
| lbsacsbs 21228 | Being a basis in a vector ... |
| lvecdim 21229 | The dimension theorem for ... |
| lbsextlem1 21230 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21231 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21232 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21233 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21234 | For any linearly independe... |
| lbsext 21235 | For any linearly independe... |
| lbsexg 21236 | Every vector space has a b... |
| lbsex 21237 | Every vector space has a b... |
| lvecprop2d 21238 | If two structures have the... |
| lvecpropd 21239 | If two structures have the... |
| sraval 21244 | Lemma for ~ srabase throug... |
| sralem 21245 | Lemma for ~ srabase and si... |
| srabase 21246 | Base set of a subring alge... |
| sraaddg 21247 | Additive operation of a su... |
| sramulr 21248 | Multiplicative operation o... |
| srasca 21249 | The set of scalars of a su... |
| sravsca 21250 | The scalar product operati... |
| sraip 21251 | The inner product operatio... |
| sratset 21252 | Topology component of a su... |
| sratopn 21253 | Topology component of a su... |
| srads 21254 | Distance function of a sub... |
| sraring 21255 | Condition for a subring al... |
| sralmod 21256 | The subring algebra is a l... |
| sralmod0 21257 | The subring module inherit... |
| issubrgd 21258 | Prove a subring by closure... |
| rlmfn 21259 | ` ringLMod ` is a function... |
| rlmval 21260 | Value of the ring module. ... |
| rlmval2 21261 | Value of the ring module e... |
| rlmbas 21262 | Base set of the ring modul... |
| rlmplusg 21263 | Vector addition in the rin... |
| rlm0 21264 | Zero vector in the ring mo... |
| rlmsub 21265 | Subtraction in the ring mo... |
| rlmmulr 21266 | Ring multiplication in the... |
| rlmsca 21267 | Scalars in the ring module... |
| rlmsca2 21268 | Scalars in the ring module... |
| rlmvsca 21269 | Scalar multiplication in t... |
| rlmtopn 21270 | Topology component of the ... |
| rlmds 21271 | Metric component of the ri... |
| rlmlmod 21272 | The ring module is a modul... |
| rlmlvec 21273 | The ring module over a div... |
| rlmlsm 21274 | Subgroup sum of the ring m... |
| rlmvneg 21275 | Vector negation in the rin... |
| rlmscaf 21276 | Functionalized scalar mult... |
| ixpsnbasval 21277 | The value of an infinite C... |
| lidlval 21282 | Value of the set of ring i... |
| rspval 21283 | Value of the ring span fun... |
| lidlss 21284 | An ideal is a subset of th... |
| lidlssbas 21285 | The base set of the restri... |
| lidlbas 21286 | A (left) ideal of a ring i... |
| islidl 21287 | Predicate of being a (left... |
| rnglidlmcl 21288 | A (left) ideal containing ... |
| rngridlmcl 21289 | A right ideal (which is a ... |
| dflidl2rng 21290 | Alternate (the usual textb... |
| isridlrng 21291 | A right ideal is a left id... |
| lidl0cl 21292 | An ideal contains 0. (Con... |
| lidlacl 21293 | An ideal is closed under a... |
| lidlnegcl 21294 | An ideal contains negative... |
| lidlsubg 21295 | An ideal is a subgroup of ... |
| lidlsubcl 21296 | An ideal is closed under s... |
| lidlmcl 21297 | An ideal is closed under l... |
| lidl1el 21298 | An ideal contains 1 iff it... |
| dflidl2 21299 | Alternate (the usual textb... |
| lidl0ALT 21300 | Alternate proof for ~ lidl... |
| rnglidl0 21301 | Every non-unital ring cont... |
| lidl0 21302 | Every ring contains a zero... |
| lidl1ALT 21303 | Alternate proof for ~ lidl... |
| rnglidl1 21304 | The base set of every non-... |
| lidl1 21305 | Every ring contains a unit... |
| lidlacs 21306 | The ideal system is an alg... |
| rspcl 21307 | The span of a set of ring ... |
| rspssid 21308 | The span of a set of ring ... |
| rsp1 21309 | The span of the identity e... |
| rsp0 21310 | The span of the zero eleme... |
| rspssp 21311 | The ideal span of a set of... |
| elrspsn 21312 | Membership in a principal ... |
| mrcrsp 21313 | Moore closure generalizes ... |
| lidlnz 21314 | A nonzero ideal contains a... |
| drngnidl 21315 | A division ring has only t... |
| lidlrsppropd 21316 | The left ideals and ring s... |
| rnglidlmmgm 21317 | The multiplicative group o... |
| rnglidlmsgrp 21318 | The multiplicative group o... |
| rnglidlrng 21319 | A (left) ideal of a non-un... |
| lidlnsg 21320 | An ideal is a normal subgr... |
| 2idlval 21323 | Definition of a two-sided ... |
| isridl 21324 | A right ideal is a left id... |
| 2idlelb 21325 | Membership in a two-sided ... |
| 2idllidld 21326 | A two-sided ideal is a lef... |
| 2idlridld 21327 | A two-sided ideal is a rig... |
| df2idl2rng 21328 | Alternate (the usual textb... |
| df2idl2 21329 | Alternate (the usual textb... |
| ridl0 21330 | Every ring contains a zero... |
| ridl1 21331 | Every ring contains a unit... |
| 2idl0 21332 | Every ring contains a zero... |
| 2idl1 21333 | Every ring contains a unit... |
| 2idlss 21334 | A two-sided ideal is a sub... |
| 2idlbas 21335 | The base set of a two-side... |
| 2idlelbas 21336 | The base set of a two-side... |
| rng2idlsubrng 21337 | A two-sided ideal of a non... |
| rng2idlnsg 21338 | A two-sided ideal of a non... |
| rng2idl0 21339 | The zero (additive identit... |
| rng2idlsubgsubrng 21340 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21341 | A two-sided ideal of a non... |
| rng2idlsubg0 21342 | The zero (additive identit... |
| 2idlcpblrng 21343 | The coset equivalence rela... |
| 2idlcpbl 21344 | The coset equivalence rela... |
| qus2idrng 21345 | The quotient of a non-unit... |
| qus1 21346 | The multiplicative identit... |
| qusring 21347 | If ` S ` is a two-sided id... |
| qusrhm 21348 | If ` S ` is a two-sided id... |
| rhmpreimaidl 21349 | The preimage of an ideal b... |
| kerlidl 21350 | The kernel of a ring homom... |
| qusmul2idl 21351 | Value of the ring operatio... |
| crngridl 21352 | In a commutative ring, the... |
| crng2idl 21353 | In a commutative ring, a t... |
| qusmulrng 21354 | Value of the multiplicatio... |
| quscrng 21355 | The quotient of a commutat... |
| qusmulcrng 21356 | Value of the ring operatio... |
| rhmqusnsg 21357 | The mapping ` J ` induced ... |
| rngqiprng1elbas 21358 | The ring unity of a two-si... |
| rngqiprngghmlem1 21359 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21360 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21361 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21362 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21363 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21364 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21365 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21366 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21367 | The base set of the produc... |
| rngqiprng 21368 | The product of the quotien... |
| rngqiprngimf 21369 | ` F ` is a function from (... |
| rngqiprngimfv 21370 | The value of the function ... |
| rngqiprngghm 21371 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21372 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21373 | ` F ` is a function from (... |
| rngqiprnglin 21374 | ` F ` is linear with respe... |
| rngqiprngho 21375 | ` F ` is a homomorphism of... |
| rngqiprngim 21376 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21377 | The unity of a two-sided i... |
| rngringbdlem1 21378 | In a unital ring, the quot... |
| rngringbdlem2 21379 | A non-unital ring is unita... |
| rngringbd 21380 | A non-unital ring is unita... |
| ring2idlqus 21381 | For every unital ring ther... |
| ring2idlqusb 21382 | A non-unital ring is unita... |
| rngqiprngfulem1 21383 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21384 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21385 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21386 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21387 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21388 | The ring unity of the prod... |
| rngqiprngfu 21389 | The function value of ` F ... |
| rngqiprngu 21390 | If a non-unital ring has a... |
| ring2idlqus1 21391 | If a non-unital ring has a... |
| lpival 21396 | Value of the set of princi... |
| islpidl 21397 | Property of being a princi... |
| lpi0 21398 | The zero ideal is always p... |
| lpi1 21399 | The unit ideal is always p... |
| islpir 21400 | Principal ideal rings are ... |
| lpiss 21401 | Principal ideals are a sub... |
| islpir2 21402 | Principal ideal rings are ... |
| lpirring 21403 | Principal ideal rings are ... |
| drnglpir 21404 | Division rings are princip... |
| rspsn 21405 | Membership in principal id... |
| lidldvgen 21406 | An element generates an id... |
| lpigen 21407 | An ideal is principal iff ... |
| cnfldstr 21428 | The field of complex numbe... |
| cnfldex 21429 | The field of complex numbe... |
| cnfldbas 21430 | The base set of the field ... |
| mpocnfldadd 21431 | The addition operation of ... |
| cnfldadd 21432 | The addition operation of ... |
| mpocnfldmul 21433 | The multiplication operati... |
| cnfldmul 21434 | The multiplication operati... |
| cnfldcj 21435 | The conjugation operation ... |
| cnfldtset 21436 | The topology component of ... |
| cnfldle 21437 | The ordering of the field ... |
| cnfldds 21438 | The metric of the field of... |
| cnfldunif 21439 | The uniform structure comp... |
| cnfldfun 21440 | The field of complex numbe... |
| cnfldfunALT 21441 | The field of complex numbe... |
| xrsstr 21442 | The extended real structur... |
| xrsex 21443 | The extended real structur... |
| xrsadd 21444 | The addition operation of ... |
| xrsmul 21445 | The multiplication operati... |
| xrstset 21446 | The topology component of ... |
| cncrng 21447 | The complex numbers form a... |
| cnring 21448 | The complex numbers form a... |
| xrsmcmn 21449 | The "multiplicative group"... |
| cnfld0 21450 | Zero is the zero element o... |
| cnfld1 21451 | One is the unity element o... |
| cnfldneg 21452 | The additive inverse in th... |
| cnfldplusf 21453 | The functionalized additio... |
| cnfldsub 21454 | The subtraction operator i... |
| cndrng 21455 | The complex numbers form a... |
| cnflddiv 21456 | The division operation in ... |
| cnfldinv 21457 | The multiplicative inverse... |
| cnfldmulg 21458 | The group multiple functio... |
| cnfldexp 21459 | The exponentiation operato... |
| cnsrng 21460 | The complex numbers form a... |
| xrsmgm 21461 | The "additive group" of th... |
| xrsnsgrp 21462 | The "additive group" of th... |
| xrsmgmdifsgrp 21463 | The "additive group" of th... |
| xrsds 21464 | The metric of the extended... |
| xrsdsval 21465 | The metric of the extended... |
| xrsdsreval 21466 | The metric of the extended... |
| xrsdsreclblem 21467 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21468 | The metric of the extended... |
| cnsubmlem 21469 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21470 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21471 | Lemma for ~ resubdrg and f... |
| cnsubdrglem 21472 | Lemma for ~ resubdrg and f... |
| qsubdrg 21473 | The rational numbers form ... |
| zsubrg 21474 | The integers form a subrin... |
| gzsubrg 21475 | The gaussian integers form... |
| nn0subm 21476 | The nonnegative integers f... |
| rege0subm 21477 | The nonnegative reals form... |
| absabv 21478 | The regular absolute value... |
| zsssubrg 21479 | The integers are a subset ... |
| qsssubdrg 21480 | The rational numbers are a... |
| cnsubrg 21481 | There are no subrings of t... |
| cnmgpabl 21482 | The unit group of the comp... |
| cnmgpid 21483 | The group identity element... |
| cnmsubglem 21484 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21485 | The positive reals form a ... |
| gzrngunitlem 21486 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21487 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21488 | Relate a group sum on ` CC... |
| regsumfsum 21489 | Relate a group sum on ` ( ... |
| expmhm 21490 | Exponentiation is a monoid... |
| nn0srg 21491 | The nonnegative integers f... |
| rge0srg 21492 | The nonnegative real numbe... |
| xrge0plusg 21493 | The additive law of the ex... |
| xrs1mnd 21494 | The extended real numbers,... |
| xrs10 21495 | The zero of the extended r... |
| xrs1cmn 21496 | The extended real numbers ... |
| xrge0subm 21497 | The nonnegative extended r... |
| xrge0cmn 21498 | The nonnegative extended r... |
| xrge0omnd 21499 | The nonnegative extended r... |
| zringcrng 21502 | The ring of integers is a ... |
| zringring 21503 | The ring of integers is a ... |
| zringrng 21504 | The ring of integers is a ... |
| zringabl 21505 | The ring of integers is an... |
| zringgrp 21506 | The ring of integers is an... |
| zringbas 21507 | The integers are the base ... |
| zringplusg 21508 | The addition operation of ... |
| zringsub 21509 | The subtraction of element... |
| zringmulg 21510 | The multiplication (group ... |
| zringmulr 21511 | The multiplication operati... |
| zring0 21512 | The zero element of the ri... |
| zring1 21513 | The unity element of the r... |
| zringnzr 21514 | The ring of integers is a ... |
| dvdsrzring 21515 | Ring divisibility in the r... |
| zringlpirlem1 21516 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21517 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21518 | Lemma for ~ zringlpir . A... |
| zringinvg 21519 | The additive inverse of an... |
| zringunit 21520 | The units of ` ZZ ` are th... |
| zringlpir 21521 | The integers are a princip... |
| zringndrg 21522 | The integers are not a div... |
| zringcyg 21523 | The integers are a cyclic ... |
| zringsubgval 21524 | Subtraction in the ring of... |
| zringmpg 21525 | The multiplicative group o... |
| prmirredlem 21526 | A positive integer is irre... |
| dfprm2 21527 | The positive irreducible e... |
| prmirred 21528 | The irreducible elements o... |
| expghm 21529 | Exponentiation is a group ... |
| mulgghm2 21530 | The powers of a group elem... |
| mulgrhm 21531 | The powers of the element ... |
| mulgrhm2 21532 | The powers of the element ... |
| irinitoringc 21533 | The ring of integers is an... |
| nzerooringczr 21534 | There is no zero object in... |
| pzriprnglem1 21535 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21536 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21537 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21538 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21539 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21540 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21541 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21542 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21543 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21544 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21545 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21546 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21547 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21548 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21549 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21550 | The ring unity of the ring... |
| pzriprng 21551 | The non-unital ring ` ( ZZ... |
| pzriprng1 21552 | The ring unity of the ring... |
| zrhval 21561 | Define the unique homomorp... |
| zrhval2 21562 | Alternate value of the ` Z... |
| zrhmulg 21563 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21564 | The ` ZRHom ` homomorphism... |
| zrhrhm 21565 | The ` ZRHom ` homomorphism... |
| zrh1 21566 | Interpretation of 1 in a r... |
| zrh0 21567 | Interpretation of 0 in a r... |
| zrhpropd 21568 | The ` ZZ ` ring homomorphi... |
| zlmval 21569 | Augment an abelian group w... |
| zlmlem 21570 | Lemma for ~ zlmbas and ~ z... |
| zlmbas 21571 | Base set of a ` ZZ ` -modu... |
| zlmplusg 21572 | Group operation of a ` ZZ ... |
| zlmmulr 21573 | Ring operation of a ` ZZ `... |
| zlmsca 21574 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21575 | Scalar multiplication oper... |
| zlmlmod 21576 | The ` ZZ ` -module operati... |
| chrval 21577 | Definition substitution of... |
| chrcl 21578 | Closure of the characteris... |
| chrid 21579 | The canonical ` ZZ ` ring ... |
| chrdvds 21580 | The ` ZZ ` ring homomorphi... |
| chrcong 21581 | If two integers are congru... |
| dvdschrmulg 21582 | In a ring, any multiple of... |
| fermltlchr 21583 | A generalization of Fermat... |
| chrnzr 21584 | Nonzero rings are precisel... |
| chrrhm 21585 | The characteristic restric... |
| domnchr 21586 | The characteristic of a do... |
| znlidl 21587 | The set ` n ZZ ` is an ide... |
| zncrng2 21588 | Making a commutative ring ... |
| znval 21589 | The value of the ` Z/nZ ` ... |
| znle 21590 | The value of the ` Z/nZ ` ... |
| znval2 21591 | Self-referential expressio... |
| znbaslem 21592 | Lemma for ~ znbas . (Cont... |
| znbas2 21593 | The base set of ` Z/nZ ` i... |
| znadd 21594 | The additive structure of ... |
| znmul 21595 | The multiplicative structu... |
| znzrh 21596 | The ` ZZ ` ring homomorphi... |
| znbas 21597 | The base set of ` Z/nZ ` s... |
| zncrng 21598 | ` Z/nZ ` is a commutative ... |
| znzrh2 21599 | The ` ZZ ` ring homomorphi... |
| znzrhval 21600 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21601 | The ` ZZ ` ring homomorphi... |
| zncyg 21602 | The group ` ZZ / n ZZ ` is... |
| zndvds 21603 | Express equality of equiva... |
| zndvds0 21604 | Special case of ~ zndvds w... |
| znf1o 21605 | The function ` F ` enumera... |
| zzngim 21606 | The ` ZZ ` ring homomorphi... |
| znle2 21607 | The ordering of the ` Z/nZ... |
| znleval 21608 | The ordering of the ` Z/nZ... |
| znleval2 21609 | The ordering of the ` Z/nZ... |
| zntoslem 21610 | Lemma for ~ zntos . (Cont... |
| zntos 21611 | The ` Z/nZ ` structure is ... |
| znhash 21612 | The ` Z/nZ ` structure has... |
| znfi 21613 | The ` Z/nZ ` structure is ... |
| znfld 21614 | The ` Z/nZ ` structure is ... |
| znidomb 21615 | The ` Z/nZ ` structure is ... |
| znchr 21616 | Cyclic rings are defined b... |
| znunit 21617 | The units of ` Z/nZ ` are ... |
| znunithash 21618 | The size of the unit group... |
| znrrg 21619 | The regular elements of ` ... |
| cygznlem1 21620 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21621 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21622 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21623 | A cyclic group with ` n ` ... |
| cygzn 21624 | A cyclic group with ` n ` ... |
| cygth 21625 | The "fundamental theorem o... |
| cyggic 21626 | Cyclic groups are isomorph... |
| frgpcyg 21627 | A free group is cyclic iff... |
| freshmansdream 21628 | For a prime number ` P ` ,... |
| frobrhm 21629 | In a commutative ring with... |
| ofldchr 21630 | The characteristic of an o... |
| cnmsgnsubg 21631 | The signs form a multiplic... |
| cnmsgnbas 21632 | The base set of the sign s... |
| cnmsgngrp 21633 | The group of signs under m... |
| psgnghm 21634 | The sign is a homomorphism... |
| psgnghm2 21635 | The sign is a homomorphism... |
| psgninv 21636 | The sign of a permutation ... |
| psgnco 21637 | Multiplicativity of the pe... |
| zrhpsgnmhm 21638 | Embedding of permutation s... |
| zrhpsgninv 21639 | The embedded sign of a per... |
| evpmss 21640 | Even permutations are perm... |
| psgnevpmb 21641 | A class is an even permuta... |
| psgnodpm 21642 | A permutation which is odd... |
| psgnevpm 21643 | A permutation which is eve... |
| psgnodpmr 21644 | If a permutation has sign ... |
| zrhpsgnevpm 21645 | The sign of an even permut... |
| zrhpsgnodpm 21646 | The sign of an odd permuta... |
| cofipsgn 21647 | Composition of any class `... |
| zrhpsgnelbas 21648 | Embedding of permutation s... |
| zrhcopsgnelbas 21649 | Embedding of permutation s... |
| evpmodpmf1o 21650 | The function for performin... |
| pmtrodpm 21651 | A transposition is an odd ... |
| psgnfix1 21652 | A permutation of a finite ... |
| psgnfix2 21653 | A permutation of a finite ... |
| psgndiflemB 21654 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21655 | Lemma 2 for ~ psgndif . (... |
| psgndif 21656 | Embedding of permutation s... |
| copsgndif 21657 | Embedding of permutation s... |
| rebase 21660 | The base of the field of r... |
| remulg 21661 | The multiplication (group ... |
| resubdrg 21662 | The real numbers form a di... |
| resubgval 21663 | Subtraction in the field o... |
| replusg 21664 | The addition operation of ... |
| remulr 21665 | The multiplication operati... |
| re0g 21666 | The zero element of the fi... |
| re1r 21667 | The unity element of the f... |
| rele2 21668 | The ordering relation of t... |
| relt 21669 | The ordering relation of t... |
| reds 21670 | The distance of the field ... |
| redvr 21671 | The division operation of ... |
| retos 21672 | The real numbers are a tot... |
| refld 21673 | The real numbers form a fi... |
| refldcj 21674 | The conjugation operation ... |
| resrng 21675 | The real numbers form a st... |
| regsumsupp 21676 | The group sum over the rea... |
| rzgrp 21677 | The quotient group ` RR / ... |
| isphl 21682 | The predicate "is a genera... |
| phllvec 21683 | A pre-Hilbert space is a l... |
| phllmod 21684 | A pre-Hilbert space is a l... |
| phlsrng 21685 | The scalar ring of a pre-H... |
| phllmhm 21686 | The inner product of a pre... |
| ipcl 21687 | Closure of the inner produ... |
| ipcj 21688 | Conjugate of an inner prod... |
| iporthcom 21689 | Orthogonality (meaning inn... |
| ip0l 21690 | Inner product with a zero ... |
| ip0r 21691 | Inner product with a zero ... |
| ipeq0 21692 | The inner product of a vec... |
| ipdir 21693 | Distributive law for inner... |
| ipdi 21694 | Distributive law for inner... |
| ip2di 21695 | Distributive law for inner... |
| ipsubdir 21696 | Distributive law for inner... |
| ipsubdi 21697 | Distributive law for inner... |
| ip2subdi 21698 | Distributive law for inner... |
| ipass 21699 | Associative law for inner ... |
| ipassr 21700 | "Associative" law for seco... |
| ipassr2 21701 | "Associative" law for inne... |
| ipffval 21702 | The inner product operatio... |
| ipfval 21703 | The inner product operatio... |
| ipfeq 21704 | If the inner product opera... |
| ipffn 21705 | The inner product operatio... |
| phlipf 21706 | The inner product operatio... |
| ip2eq 21707 | Two vectors are equal iff ... |
| isphld 21708 | Properties that determine ... |
| phlpropd 21709 | If two structures have the... |
| ssipeq 21710 | The inner product on a sub... |
| phssipval 21711 | The inner product on a sub... |
| phssip 21712 | The inner product (as a fu... |
| phlssphl 21713 | A subspace of an inner pro... |
| ocvfval 21720 | The orthocomplement operat... |
| ocvval 21721 | Value of the orthocompleme... |
| elocv 21722 | Elementhood in the orthoco... |
| ocvi 21723 | Property of a member of th... |
| ocvss 21724 | The orthocomplement of a s... |
| ocvocv 21725 | A set is contained in its ... |
| ocvlss 21726 | The orthocomplement of a s... |
| ocv2ss 21727 | Orthocomplements reverse s... |
| ocvin 21728 | An orthocomplement has tri... |
| ocvsscon 21729 | Two ways to say that ` S `... |
| ocvlsp 21730 | The orthocomplement of a l... |
| ocv0 21731 | The orthocomplement of the... |
| ocvz 21732 | The orthocomplement of the... |
| ocv1 21733 | The orthocomplement of the... |
| unocv 21734 | The orthocomplement of a u... |
| iunocv 21735 | The orthocomplement of an ... |
| cssval 21736 | The set of closed subspace... |
| iscss 21737 | The predicate "is a closed... |
| cssi 21738 | Property of a closed subsp... |
| cssss 21739 | A closed subspace is a sub... |
| iscss2 21740 | It is sufficient to prove ... |
| ocvcss 21741 | The orthocomplement of any... |
| cssincl 21742 | The zero subspace is a clo... |
| css0 21743 | The zero subspace is a clo... |
| css1 21744 | The whole space is a close... |
| csslss 21745 | A closed subspace of a pre... |
| lsmcss 21746 | A subset of a pre-Hilbert ... |
| cssmre 21747 | The closed subspaces of a ... |
| mrccss 21748 | The Moore closure correspo... |
| thlval 21749 | Value of the Hilbert latti... |
| thlbas 21750 | Base set of the Hilbert la... |
| thlle 21751 | Ordering on the Hilbert la... |
| thlleval 21752 | Ordering on the Hilbert la... |
| thloc 21753 | Orthocomplement on the Hil... |
| pjfval 21760 | The value of the projectio... |
| pjdm 21761 | A subspace is in the domai... |
| pjpm 21762 | The projection map is a pa... |
| pjfval2 21763 | Value of the projection ma... |
| pjval 21764 | Value of the projection ma... |
| pjdm2 21765 | A subspace is in the domai... |
| pjff 21766 | A projection is a linear o... |
| pjf 21767 | A projection is a function... |
| pjf2 21768 | A projection is a function... |
| pjfo 21769 | A projection is a surjecti... |
| pjcss 21770 | A projection subspace is a... |
| ocvpj 21771 | The orthocomplement of a p... |
| ishil 21772 | The predicate "is a Hilber... |
| ishil2 21773 | The predicate "is a Hilber... |
| isobs 21774 | The predicate "is an ortho... |
| obsip 21775 | The inner product of two e... |
| obsipid 21776 | A basis element has length... |
| obsrcl 21777 | Reverse closure for an ort... |
| obsss 21778 | An orthonormal basis is a ... |
| obsne0 21779 | A basis element is nonzero... |
| obsocv 21780 | An orthonormal basis has t... |
| obs2ocv 21781 | The double orthocomplement... |
| obselocv 21782 | A basis element is in the ... |
| obs2ss 21783 | A basis has no proper subs... |
| obslbs 21784 | An orthogonal basis is a l... |
| reldmdsmm 21787 | The direct sum is a well-b... |
| dsmmval 21788 | Value of the module direct... |
| dsmmbase 21789 | Base set of the module dir... |
| dsmmval2 21790 | Self-referential definitio... |
| dsmmbas2 21791 | Base set of the direct sum... |
| dsmmfi 21792 | For finite products, the d... |
| dsmmelbas 21793 | Membership in the finitely... |
| dsmm0cl 21794 | The all-zero vector is con... |
| dsmmacl 21795 | The finite hull is closed ... |
| prdsinvgd2 21796 | Negation of a single coord... |
| dsmmsubg 21797 | The finite hull of a produ... |
| dsmmlss 21798 | The finite hull of a produ... |
| dsmmlmod 21799 | The direct sum of a family... |
| frlmval 21802 | Value of the "free module"... |
| frlmlmod 21803 | The free module is a modul... |
| frlmpws 21804 | The free module as a restr... |
| frlmlss 21805 | The base set of the free m... |
| frlmpwsfi 21806 | The finite free module is ... |
| frlmsca 21807 | The ring of scalars of a f... |
| frlm0 21808 | Zero in a free module (rin... |
| frlmbas 21809 | Base set of the free modul... |
| frlmelbas 21810 | Membership in the base set... |
| frlmrcl 21811 | If a free module is inhabi... |
| frlmbasfsupp 21812 | Elements of the free modul... |
| frlmbasmap 21813 | Elements of the free modul... |
| frlmbasf 21814 | Elements of the free modul... |
| frlmlvec 21815 | The free module over a div... |
| frlmfibas 21816 | The base set of the finite... |
| elfrlmbasn0 21817 | If the dimension of a free... |
| frlmplusgval 21818 | Addition in a free module.... |
| frlmsubgval 21819 | Subtraction in a free modu... |
| frlmvscafval 21820 | Scalar multiplication in a... |
| frlmvplusgvalc 21821 | Coordinates of a sum with ... |
| frlmvscaval 21822 | Coordinates of a scalar mu... |
| frlmplusgvalb 21823 | Addition in a free module ... |
| frlmvscavalb 21824 | Scalar multiplication in a... |
| frlmvplusgscavalb 21825 | Addition combined with sca... |
| frlmgsum 21826 | Finite commutative sums in... |
| frlmsplit2 21827 | Restriction is homomorphic... |
| frlmsslss 21828 | A subset of a free module ... |
| frlmsslss2 21829 | A subset of a free module ... |
| frlmbas3 21830 | An element of the base set... |
| mpofrlmd 21831 | Elements of the free modul... |
| frlmip 21832 | The inner product of a fre... |
| frlmipval 21833 | The inner product of a fre... |
| frlmphllem 21834 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21835 | Conditions for a free modu... |
| uvcfval 21838 | Value of the unit-vector g... |
| uvcval 21839 | Value of a single unit vec... |
| uvcvval 21840 | Value of a unit vector coo... |
| uvcvvcl 21841 | A coordinate of a unit vec... |
| uvcvvcl2 21842 | A unit vector coordinate i... |
| uvcvv1 21843 | The unit vector is one at ... |
| uvcvv0 21844 | The unit vector is zero at... |
| uvcff 21845 | Domain and codomain of the... |
| uvcf1 21846 | In a nonzero ring, each un... |
| uvcresum 21847 | Any element of a free modu... |
| frlmssuvc1 21848 | A scalar multiple of a uni... |
| frlmssuvc2 21849 | A nonzero scalar multiple ... |
| frlmsslsp 21850 | A subset of a free module ... |
| frlmlbs 21851 | The unit vectors comprise ... |
| frlmup1 21852 | Any assignment of unit vec... |
| frlmup2 21853 | The evaluation map has the... |
| frlmup3 21854 | The range of such an evalu... |
| frlmup4 21855 | Universal property of the ... |
| ellspd 21856 | The elements of the span o... |
| elfilspd 21857 | Simplified version of ~ el... |
| rellindf 21862 | The independent-family pre... |
| islinds 21863 | Property of an independent... |
| linds1 21864 | An independent set of vect... |
| linds2 21865 | An independent set of vect... |
| islindf 21866 | Property of an independent... |
| islinds2 21867 | Expanded property of an in... |
| islindf2 21868 | Property of an independent... |
| lindff 21869 | Functional property of a l... |
| lindfind 21870 | A linearly independent fam... |
| lindsind 21871 | A linearly independent set... |
| lindfind2 21872 | In a linearly independent ... |
| lindsind2 21873 | In a linearly independent ... |
| lindff1 21874 | A linearly independent fam... |
| lindfrn 21875 | The range of an independen... |
| f1lindf 21876 | Rearranging and deleting e... |
| lindfres 21877 | Any restriction of an inde... |
| lindsss 21878 | Any subset of an independe... |
| f1linds 21879 | A family constructed from ... |
| islindf3 21880 | In a nonzero ring, indepen... |
| lindfmm 21881 | Linear independence of a f... |
| lindsmm 21882 | Linear independence of a s... |
| lindsmm2 21883 | The monomorphic image of a... |
| lsslindf 21884 | Linear independence is unc... |
| lsslinds 21885 | Linear independence is unc... |
| islbs4 21886 | A basis is an independent ... |
| lbslinds 21887 | A basis is independent. (... |
| islinds3 21888 | A subset is linearly indep... |
| islinds4 21889 | A set is independent in a ... |
| lmimlbs 21890 | The isomorphic image of a ... |
| lmiclbs 21891 | Having a basis is an isomo... |
| islindf4 21892 | A family is independent if... |
| islindf5 21893 | A family is independent if... |
| indlcim 21894 | An independent, spanning f... |
| lbslcic 21895 | A module with a basis is i... |
| lmisfree 21896 | A module has a basis iff i... |
| lvecisfrlm 21897 | Every vector space is isom... |
| lmimco 21898 | The composition of two iso... |
| lmictra 21899 | Module isomorphism is tran... |
| uvcf1o 21900 | In a nonzero ring, the map... |
| uvcendim 21901 | In a nonzero ring, the num... |
| frlmisfrlm 21902 | A free module is isomorphi... |
| frlmiscvec 21903 | Every free module is isomo... |
| isassa 21910 | The properties of an assoc... |
| assalem 21911 | The properties of an assoc... |
| assaass 21912 | Left-associative property ... |
| assaassr 21913 | Right-associative property... |
| assalmod 21914 | An associative algebra is ... |
| assaring 21915 | An associative algebra is ... |
| assasca 21916 | The scalars of an associat... |
| assa2ass 21917 | Left- and right-associativ... |
| assa2ass2 21918 | Left- and right-associativ... |
| isassad 21919 | Sufficient condition for b... |
| issubassa3 21920 | A subring that is also a s... |
| issubassa 21921 | The subalgebras of an asso... |
| sraassab 21922 | A subring algebra is an as... |
| sraassa 21923 | The subring algebra over a... |
| rlmassa 21924 | The ring module over a com... |
| assapropd 21925 | If two structures have the... |
| aspval 21926 | Value of the algebraic clo... |
| asplss 21927 | The algebraic span of a se... |
| aspid 21928 | The algebraic span of a su... |
| aspsubrg 21929 | The algebraic span of a se... |
| aspss 21930 | Span preserves subset orde... |
| aspssid 21931 | A set of vectors is a subs... |
| asclfval 21932 | Function value of the alge... |
| asclval 21933 | Value of a mapped algebra ... |
| asclfn 21934 | Unconditional functionalit... |
| asclf 21935 | The algebra scalar lifting... |
| asclghm 21936 | The algebra scalar lifting... |
| asclelbas 21937 | Lifted scalars are in the ... |
| ascl0 21938 | The scalar 0 embedded into... |
| ascl1 21939 | The scalar 1 embedded into... |
| asclmul1 21940 | Left multiplication by a l... |
| asclmul2 21941 | Right multiplication by a ... |
| ascldimul 21942 | The algebra scalar lifting... |
| asclinvg 21943 | The group inverse (negatio... |
| asclrhm 21944 | The algebra scalar lifting... |
| rnascl 21945 | The set of lifted scalars ... |
| issubassa2 21946 | A subring of a unital alge... |
| rnasclsubrg 21947 | The scalar multiples of th... |
| rnasclmulcl 21948 | (Vector) multiplication is... |
| rnasclassa 21949 | The scalar multiples of th... |
| ressascl 21950 | The lifting of scalars is ... |
| asclpropd 21951 | If two structures have the... |
| aspval2 21952 | The algebraic closure is t... |
| assamulgscmlem1 21953 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21954 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21955 | Exponentiation of a scalar... |
| asclmulg 21956 | Apply group multiplication... |
| zlmassa 21957 | The ` ZZ ` -module operati... |
| reldmpsr 21968 | The multivariate power ser... |
| psrval 21969 | Value of the multivariate ... |
| psrvalstr 21970 | The multivariate power ser... |
| psrbag 21971 | Elementhood in the set of ... |
| psrbagf 21972 | A finite bag is a function... |
| psrbagfsupp 21973 | Finite bags have finite su... |
| snifpsrbag 21974 | A bag containing one eleme... |
| fczpsrbag 21975 | The constant function equa... |
| psrbaglesupp 21976 | The support of a dominated... |
| psrbaglecl 21977 | The set of finite bags is ... |
| psrbagaddcl 21978 | The sum of two finite bags... |
| psrbagcon 21979 | The analogue of the statem... |
| psrbaglefi 21980 | There are finitely many ba... |
| psrbagconcl 21981 | The complement of a bag is... |
| psrbagleadd1 21982 | The analogue of " ` X <_ F... |
| psrbagconf1o 21983 | Bag complementation is a b... |
| psrbagres 21984 | Restrict a bag of variable... |
| gsumbagdiaglem 21985 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21986 | Two-dimensional commutatio... |
| psrass1lem 21987 | A group sum commutation us... |
| psrbas 21988 | The base set of the multiv... |
| psrelbas 21989 | An element of the set of p... |
| psrelbasfun 21990 | An element of the set of p... |
| psrplusg 21991 | The addition operation of ... |
| psradd 21992 | The addition operation of ... |
| psraddcl 21993 | Closure of the power serie... |
| rhmpsrlem1 21994 | Lemma for ~ rhmpsr et al. ... |
| rhmpsrlem2 21995 | Lemma for ~ rhmpsr et al. ... |
| psrmulr 21996 | The multiplication operati... |
| psrmulfval 21997 | The multiplication operati... |
| psrmulval 21998 | The multiplication operati... |
| psrmulcllem 21999 | Closure of the power serie... |
| psrmulcl 22000 | Closure of the power serie... |
| psrsca 22001 | The scalar field of the mu... |
| psrvscafval 22002 | The scalar multiplication ... |
| psrvsca 22003 | The scalar multiplication ... |
| psrvscaval 22004 | The scalar multiplication ... |
| psrvscacl 22005 | Closure of the power serie... |
| psr0cl 22006 | The zero element of the ri... |
| psr0lid 22007 | The zero element of the ri... |
| psrnegcl 22008 | The negative function in t... |
| psrlinv 22009 | The negative function in t... |
| psrgrp 22010 | The ring of power series i... |
| psr0 22011 | The zero element of the ri... |
| psrneg 22012 | The negative function of t... |
| psrlmod 22013 | The ring of power series i... |
| psr1cl 22014 | The identity element of th... |
| psrlidm 22015 | The identity element of th... |
| psrridm 22016 | The identity element of th... |
| psrass1 22017 | Associative identity for t... |
| psrdi 22018 | Distributive law for the r... |
| psrdir 22019 | Distributive law for the r... |
| psrass23l 22020 | Associative identity for t... |
| psrcom 22021 | Commutative law for the ri... |
| psrass23 22022 | Associative identities for... |
| psrring 22023 | The ring of power series i... |
| psr1 22024 | The identity element of th... |
| psrcrng 22025 | The ring of power series i... |
| psrassa 22026 | The ring of power series i... |
| resspsrbas 22027 | A restricted power series ... |
| resspsradd 22028 | A restricted power series ... |
| resspsrmul 22029 | A restricted power series ... |
| resspsrvsca 22030 | A restricted power series ... |
| subrgpsr 22031 | A subring of the base ring... |
| psrascl 22032 | Value of the scalar inject... |
| psrasclcl 22033 | A scalar is lifted into a ... |
| mvrfval 22034 | Value of the generating el... |
| mvrval 22035 | Value of the generating el... |
| mvrval2 22036 | Value of the generating el... |
| mvrid 22037 | The ` X i ` -th coefficien... |
| mvrf 22038 | The power series variable ... |
| mvrf1 22039 | The power series variable ... |
| mvrcl2 22040 | A power series variable is... |
| reldmmpl 22041 | The multivariate polynomia... |
| mplval 22042 | Value of the set of multiv... |
| mplbas 22043 | Base set of the set of mul... |
| mplelbas 22044 | Property of being a polyno... |
| mvrcl 22045 | A power series variable is... |
| mvrf2 22046 | The power series/polynomia... |
| mplrcl 22047 | Reverse closure for the po... |
| mplelsfi 22048 | A polynomial treated as a ... |
| mplval2 22049 | Self-referential expressio... |
| mplbasss 22050 | The set of polynomials is ... |
| mplelf 22051 | A polynomial is defined as... |
| mplsubglem 22052 | If ` A ` is an ideal of se... |
| mpllsslem 22053 | If ` A ` is an ideal of su... |
| mplsubglem2 22054 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 22055 | The set of polynomials is ... |
| mpllss 22056 | The set of polynomials is ... |
| mplsubrglem 22057 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 22058 | The set of polynomials is ... |
| mpl0 22059 | The zero polynomial. (Con... |
| mplplusg 22060 | Value of addition in a pol... |
| mplmulr 22061 | Value of multiplication in... |
| mpladd 22062 | The addition operation on ... |
| mplneg 22063 | The negative function on m... |
| mplmul 22064 | The multiplication operati... |
| mpl1 22065 | The identity element of th... |
| mplsca 22066 | The scalar field of a mult... |
| mplvsca2 22067 | The scalar multiplication ... |
| mplvsca 22068 | The scalar multiplication ... |
| mplvscaval 22069 | The scalar multiplication ... |
| mplgrp 22070 | The polynomial ring is a g... |
| mpllmod 22071 | The polynomial ring is a l... |
| mplring 22072 | The polynomial ring is a r... |
| mpllvec 22073 | The polynomial ring is a v... |
| mplcrng 22074 | The polynomial ring is a c... |
| mplassa 22075 | The polynomial ring is an ... |
| mplringd 22076 | The polynomial ring is a r... |
| mplcrngd 22077 | The polynomial ring is a c... |
| mpllmodd 22078 | The polynomial ring is a l... |
| mplascl0 22079 | The zero scalar as a polyn... |
| mplascl1 22080 | The one scalar as a polyno... |
| ressmplbas2 22081 | The base set of a restrict... |
| ressmplbas 22082 | A restricted polynomial al... |
| ressmpladd 22083 | A restricted polynomial al... |
| ressmplmul 22084 | A restricted polynomial al... |
| ressmplvsca 22085 | A restricted power series ... |
| subrgmpl 22086 | A subring of the base ring... |
| mplsubrgcl 22087 | An element of a polynomial... |
| subrgmvr 22088 | The variables in a subring... |
| subrgmvrf 22089 | The variables in a polynom... |
| mplmon 22090 | A monomial is a polynomial... |
| mplmonmul 22091 | The product of two monomia... |
| mplcoe1 22092 | Decompose a polynomial int... |
| mplcoe3 22093 | Decompose a monomial in on... |
| mplcoe5lem 22094 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 22095 | Decompose a monomial into ... |
| mplcoe2 22096 | Decompose a monomial into ... |
| mplbas2 22097 | An alternative expression ... |
| ltbval 22098 | Value of the well-order on... |
| ltbwe 22099 | The finite bag order is a ... |
| reldmopsr 22100 | Lemma for ordered power se... |
| opsrval 22101 | The value of the "ordered ... |
| opsrle 22102 | An alternative expression ... |
| opsrval2 22103 | Self-referential expressio... |
| opsrbaslem 22104 | Get a component of the ord... |
| opsrbas 22105 | The base set of the ordere... |
| opsrplusg 22106 | The addition operation of ... |
| opsrmulr 22107 | The multiplication operati... |
| opsrvsca 22108 | The scalar product operati... |
| opsrsca 22109 | The scalar ring of the ord... |
| opsrtoslem1 22110 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 22111 | Lemma for ~ opsrtos . (Co... |
| opsrtos 22112 | The ordered power series s... |
| opsrso 22113 | The ordered power series s... |
| opsrcrng 22114 | The ring of ordered power ... |
| opsrassa 22115 | The ring of ordered power ... |
| mplmon2 22116 | Express a scaled monomial.... |
| psrbag0 22117 | The empty bag is a bag. (... |
| psrbagsn 22118 | A singleton bag is a bag. ... |
| mplascl 22119 | Value of the scalar inject... |
| mplasclf 22120 | The scalar injection is a ... |
| subrgascl 22121 | The scalar injection funct... |
| subrgasclcl 22122 | The scalars in a polynomia... |
| mplmon2cl 22123 | A scaled monomial is a pol... |
| mplmon2mul 22124 | Product of scaled monomial... |
| mplind 22125 | Prove a property of polyno... |
| mplcoe4 22126 | Decompose a polynomial int... |
| evlslem4 22131 | The support of a tensor pr... |
| psrbagev1 22132 | A bag of multipliers provi... |
| psrbagev2 22133 | Closure of a sum using a b... |
| evlslem2 22134 | A linear function on the p... |
| evlslem3 22135 | Lemma for ~ evlseu . Poly... |
| evlslem6 22136 | Lemma for ~ evlseu . Fini... |
| evlslem1 22137 | Lemma for ~ evlseu , give ... |
| evlseu 22138 | For a given interpretation... |
| reldmevls 22139 | Well-behaved binary operat... |
| mpfrcl 22140 | Reverse closure for the se... |
| evlsval 22141 | Value of the polynomial ev... |
| evlsval2 22142 | Characterizing properties ... |
| evlsrhm 22143 | Polynomial evaluation is a... |
| evlsval3 22144 | Give a formula for the pol... |
| evlsvval 22145 | Give a formula for the eva... |
| evlsvvvallem 22146 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 22147 | Lemma for theorems using ~... |
| evlsvvval 22148 | Give a formula for the eva... |
| evlssca 22149 | Polynomial evaluation maps... |
| evlsvar 22150 | Polynomial evaluation maps... |
| evlsgsumadd 22151 | Polynomial evaluation maps... |
| evlsgsummul 22152 | Polynomial evaluation maps... |
| evlspw 22153 | Polynomial evaluation for ... |
| evlsvarpw 22154 | Polynomial evaluation for ... |
| evlval 22155 | Value of the simple/same r... |
| evlrhm 22156 | The simple evaluation map ... |
| evlcl 22157 | A polynomial over the ring... |
| evladdval 22158 | Polynomial evaluation buil... |
| evlmulval 22159 | Polynomial evaluation buil... |
| evlsscasrng 22160 | The evaluation of a scalar... |
| evlsca 22161 | Simple polynomial evaluati... |
| evlsvarsrng 22162 | The evaluation of the vari... |
| evlvar 22163 | Simple polynomial evaluati... |
| mpfconst 22164 | Constants are multivariate... |
| mpfproj 22165 | Projections are multivaria... |
| mpfsubrg 22166 | Polynomial functions are a... |
| mpff 22167 | Polynomial functions are f... |
| mpfaddcl 22168 | The sum of multivariate po... |
| mpfmulcl 22169 | The product of multivariat... |
| mpfind 22170 | Prove a property of polyno... |
| selvffval 22173 | Value of the "variable sel... |
| selvfval 22174 | Value of the "variable sel... |
| selvval 22175 | Value of the "variable sel... |
| mhmcompl 22176 | The composition of a monoi... |
| mplmapghm 22177 | The function ` H ` mapping... |
| mhmcoaddmpl 22178 | Show that the ring homomor... |
| rhmcomulmpl 22179 | Show that the ring homomor... |
| evlscl 22180 | A polynomial over the ring... |
| evlsscaval 22181 | Polynomial evaluation buil... |
| evlsvarval 22182 | Polynomial evaluation buil... |
| evlsexpval 22183 | Polynomial evaluation buil... |
| evlsaddval 22184 | Polynomial evaluation buil... |
| evlsmulval 22185 | Polynomial evaluation buil... |
| evlsmaprhm 22186 | The function ` F ` mapping... |
| evlsevl 22187 | Evaluation in a subring is... |
| evlvvval 22188 | Give a formula for the eva... |
| selvcllem1 22189 | ` T ` is an associative al... |
| selvcllem2 22190 | ` D ` is a ring homomorphi... |
| selvcllem3 22191 | The third argument passed ... |
| selvcllemh 22192 | Apply the third argument (... |
| selvcllem4 22193 | The fourth argument passed... |
| selvcllem5 22194 | The fifth argument passed ... |
| selvcl 22195 | Closure of the "variable s... |
| selvval2 22196 | Value of the "variable sel... |
| selvvvval 22197 | Recover the original polyn... |
| selvadd 22198 | The "variable selection" f... |
| selvmul 22199 | The "variable selection" f... |
| reldmmhp 22204 | The domain of the homogene... |
| mhpfval 22205 | Value of the "homogeneous ... |
| mhpval 22206 | Value of the "homogeneous ... |
| ismhp 22207 | Property of being a homoge... |
| ismhp2 22208 | Deduce a homogeneous polyn... |
| ismhp3 22209 | A polynomial is homogeneou... |
| mhprcl 22210 | Reverse closure for homoge... |
| mhpmpl 22211 | A homogeneous polynomial i... |
| mhpdeg 22212 | All nonzero terms of a hom... |
| mhp0cl 22213 | The zero polynomial is hom... |
| mhpsclcl 22214 | A scalar (or constant) pol... |
| mhpvarcl 22215 | A power series variable is... |
| mhpmulcl 22216 | A product of homogeneous p... |
| mhppwdeg 22217 | Degree of a homogeneous po... |
| mhpaddcl 22218 | Homogeneous polynomials ar... |
| mhpinvcl 22219 | Homogeneous polynomials ar... |
| mhpsubg 22220 | Homogeneous polynomials fo... |
| mhpvscacl 22221 | Homogeneous polynomials ar... |
| mhplss 22222 | Homogeneous polynomials fo... |
| psdffval 22224 | Value of the power series ... |
| psdfval 22225 | Give a map between power s... |
| psdval 22226 | Evaluate the partial deriv... |
| psdcoef 22227 | Coefficient of a term of t... |
| psdcl 22228 | The derivative of a power ... |
| psdmplcl 22229 | The derivative of a polyno... |
| psdadd 22230 | The derivative of a sum is... |
| psdvsca 22231 | The derivative of a scaled... |
| psdmullem 22232 | Lemma for ~ psdmul . Tran... |
| psdmul 22233 | Product rule for power ser... |
| psd1 22234 | The derivative of one is z... |
| psdascl 22235 | The derivative of a consta... |
| psdmvr 22236 | The partial derivative of ... |
| psdpw 22237 | Power rule for partial der... |
| psr1baslem 22249 | The set of finite bags on ... |
| psr1val 22250 | Value of the ring of univa... |
| psr1crng 22251 | The ring of univariate pow... |
| psr1assa 22252 | The ring of univariate pow... |
| psr1tos 22253 | The ordered power series s... |
| psr1bas2 22254 | The base set of the ring o... |
| psr1bas 22255 | The base set of the ring o... |
| vr1val 22256 | The value of the generator... |
| vr1cl2 22257 | The variable ` X ` is a me... |
| ply1val 22258 | The value of the set of un... |
| ply1bas 22259 | The value of the base set ... |
| ply1lss 22260 | Univariate polynomials for... |
| ply1subrg 22261 | Univariate polynomials for... |
| ply1crng 22262 | The ring of univariate pol... |
| ply1assa 22263 | The ring of univariate pol... |
| psr1bascl 22264 | A univariate power series ... |
| psr1basf 22265 | Univariate power series ba... |
| ply1basf 22266 | Univariate polynomial base... |
| ply1bascl 22267 | A univariate polynomial is... |
| ply1bascl2 22268 | A univariate polynomial is... |
| coe1fval 22269 | Value of the univariate po... |
| coe1fv 22270 | Value of an evaluated coef... |
| fvcoe1 22271 | Value of a multivariate co... |
| coe1fval3 22272 | Univariate power series co... |
| coe1f2 22273 | Functionality of univariat... |
| coe1fval2 22274 | Univariate polynomial coef... |
| coe1f 22275 | Functionality of univariat... |
| coe1fvalcl 22276 | A coefficient of a univari... |
| coe1sfi 22277 | Finite support of univaria... |
| coe1fsupp 22278 | The coefficient vector of ... |
| mptcoe1fsupp 22279 | A mapping involving coeffi... |
| coe1ae0 22280 | The coefficient vector of ... |
| vr1cl 22281 | The generator of a univari... |
| opsr0 22282 | Zero in the ordered power ... |
| opsr1 22283 | One in the ordered power s... |
| psr1plusg 22284 | Value of addition in a uni... |
| psr1vsca 22285 | Value of scalar multiplica... |
| psr1mulr 22286 | Value of multiplication in... |
| ply1plusg 22287 | Value of addition in a uni... |
| ply1vsca 22288 | Value of scalar multiplica... |
| ply1mulr 22289 | Value of multiplication in... |
| ply1ass23l 22290 | Associative identity with ... |
| ressply1bas2 22291 | The base set of a restrict... |
| ressply1bas 22292 | A restricted polynomial al... |
| ressply1add 22293 | A restricted polynomial al... |
| ressply1mul 22294 | A restricted polynomial al... |
| ressply1vsca 22295 | A restricted power series ... |
| subrgply1 22296 | A subring of the base ring... |
| gsumply1subr 22297 | Evaluate a group sum in a ... |
| psrbaspropd 22298 | Property deduction for pow... |
| psrplusgpropd 22299 | Property deduction for pow... |
| mplbaspropd 22300 | Property deduction for pol... |
| psropprmul 22301 | Reversing multiplication i... |
| ply1opprmul 22302 | Reversing multiplication i... |
| 00ply1bas 22303 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22304 | Protection compatibility o... |
| ply1plusgfvi 22305 | Protection compatibility o... |
| ply1baspropd 22306 | Property deduction for uni... |
| ply1plusgpropd 22307 | Property deduction for uni... |
| opsrring 22308 | Ordered power series form ... |
| opsrlmod 22309 | Ordered power series form ... |
| psr1ring 22310 | Univariate power series fo... |
| ply1ring 22311 | Univariate polynomials for... |
| psr1lmod 22312 | Univariate power series fo... |
| psr1sca 22313 | Scalars of a univariate po... |
| psr1sca2 22314 | Scalars of a univariate po... |
| ply1lmod 22315 | Univariate polynomials for... |
| ply1sca 22316 | Scalars of a univariate po... |
| ply1sca2 22317 | Scalars of a univariate po... |
| ply1ascl0 22318 | The zero scalar as a polyn... |
| ply1ascl1 22319 | The multiplicative identit... |
| ply1mpl0 22320 | The univariate polynomial ... |
| ply10s0 22321 | Zero times a univariate po... |
| ply1mpl1 22322 | The univariate polynomial ... |
| ply1ascl 22323 | The univariate polynomial ... |
| subrg1ascl 22324 | The scalar injection funct... |
| subrg1asclcl 22325 | The scalars in a polynomia... |
| subrgvr1 22326 | The variables in a subring... |
| subrgvr1cl 22327 | The variables in a polynom... |
| coe1z 22328 | The coefficient vector of ... |
| coe1add 22329 | The coefficient vector of ... |
| coe1addfv 22330 | A particular coefficient o... |
| coe1subfv 22331 | A particular coefficient o... |
| coe1mul2lem1 22332 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22333 | An equivalence for ~ coe1m... |
| coe1mul2 22334 | The coefficient vector of ... |
| coe1mul 22335 | The coefficient vector of ... |
| ply1moncl 22336 | Closure of the expression ... |
| ply1tmcl 22337 | Closure of the expression ... |
| coe1tm 22338 | Coefficient vector of a po... |
| coe1tmfv1 22339 | Nonzero coefficient of a p... |
| coe1tmfv2 22340 | Zero coefficient of a poly... |
| coe1tmmul2 22341 | Coefficient vector of a po... |
| coe1tmmul 22342 | Coefficient vector of a po... |
| coe1tmmul2fv 22343 | Function value of a right-... |
| coe1pwmul 22344 | Coefficient vector of a po... |
| coe1pwmulfv 22345 | Function value of a right-... |
| ply1scltm 22346 | A scalar is a term with ze... |
| coe1sclmul 22347 | Coefficient vector of a po... |
| coe1sclmulfv 22348 | A single coefficient of a ... |
| coe1sclmul2 22349 | Coefficient vector of a po... |
| ply1sclf 22350 | A scalar polynomial is a p... |
| ply1sclcl 22351 | The value of the algebra s... |
| coe1scl 22352 | Coefficient vector of a sc... |
| ply1sclid 22353 | Recover the base scalar fr... |
| ply1sclf1 22354 | The polynomial scalar func... |
| ply1scl0 22355 | The zero scalar is zero. ... |
| ply1scln0 22356 | Nonzero scalars create non... |
| ply1scl1 22357 | The one scalar is the unit... |
| coe1id 22358 | Coefficient vector of the ... |
| ply1idvr1 22359 | The identity of a polynomi... |
| ply1idvr1OLD 22360 | Obsolete version of ~ ply1... |
| cply1mul 22361 | The product of two constan... |
| ply1coefsupp 22362 | The decomposition of a uni... |
| ply1coe 22363 | Decompose a univariate pol... |
| eqcoe1ply1eq 22364 | Two polynomials over the s... |
| ply1coe1eq 22365 | Two polynomials over the s... |
| cply1coe0 22366 | All but the first coeffici... |
| cply1coe0bi 22367 | A polynomial is constant (... |
| coe1fzgsumdlem 22368 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22369 | Value of an evaluated coef... |
| ply1scleq 22370 | Equality of a constant pol... |
| ply1chr 22371 | The characteristic of a po... |
| gsumsmonply1 22372 | A finite group sum of scal... |
| gsummoncoe1 22373 | A coefficient of the polyn... |
| gsumply1eq 22374 | Two univariate polynomials... |
| lply1binom 22375 | The binomial theorem for l... |
| lply1binomsc 22376 | The binomial theorem for l... |
| ply1fermltlchr 22377 | Fermat's little theorem fo... |
| reldmevls1 22382 | Well-behaved binary operat... |
| ply1frcl 22383 | Reverse closure for the se... |
| evls1fval 22384 | Value of the univariate po... |
| evls1val 22385 | Value of the univariate po... |
| evls1rhmlem 22386 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22387 | Polynomial evaluation is a... |
| evls1sca 22388 | Univariate polynomial eval... |
| evls1gsumadd 22389 | Univariate polynomial eval... |
| evls1gsummul 22390 | Univariate polynomial eval... |
| evls1pw 22391 | Univariate polynomial eval... |
| evls1varpw 22392 | Univariate polynomial eval... |
| evl1fval 22393 | Value of the simple/same r... |
| evl1val 22394 | Value of the simple/same r... |
| evl1fval1lem 22395 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22396 | Value of the simple/same r... |
| evl1rhm 22397 | Polynomial evaluation is a... |
| fveval1fvcl 22398 | The function value of the ... |
| evl1sca 22399 | Polynomial evaluation maps... |
| evl1scad 22400 | Polynomial evaluation buil... |
| evl1var 22401 | Polynomial evaluation maps... |
| evl1vard 22402 | Polynomial evaluation buil... |
| evls1var 22403 | Univariate polynomial eval... |
| evls1scasrng 22404 | The evaluation of a scalar... |
| evls1varsrng 22405 | The evaluation of the vari... |
| evl1addd 22406 | Polynomial evaluation buil... |
| evl1subd 22407 | Polynomial evaluation buil... |
| evl1muld 22408 | Polynomial evaluation buil... |
| evl1vsd 22409 | Polynomial evaluation buil... |
| evl1expd 22410 | Polynomial evaluation buil... |
| pf1const 22411 | Constants are polynomial f... |
| pf1id 22412 | The identity is a polynomi... |
| pf1subrg 22413 | Polynomial functions are a... |
| pf1rcl 22414 | Reverse closure for the se... |
| pf1f 22415 | Polynomial functions are f... |
| mpfpf1 22416 | Convert a multivariate pol... |
| pf1mpf 22417 | Convert a univariate polyn... |
| pf1addcl 22418 | The sum of multivariate po... |
| pf1mulcl 22419 | The product of multivariat... |
| pf1ind 22420 | Prove a property of polyno... |
| evl1gsumdlem 22421 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22422 | Polynomial evaluation buil... |
| evl1gsumadd 22423 | Univariate polynomial eval... |
| evl1gsumaddval 22424 | Value of a univariate poly... |
| evl1gsummul 22425 | Univariate polynomial eval... |
| evl1varpw 22426 | Univariate polynomial eval... |
| evl1varpwval 22427 | Value of a univariate poly... |
| evl1scvarpw 22428 | Univariate polynomial eval... |
| evl1scvarpwval 22429 | Value of a univariate poly... |
| evl1gsummon 22430 | Value of a univariate poly... |
| evls1scafv 22431 | Value of the univariate po... |
| evls1expd 22432 | Univariate polynomial eval... |
| evls1varpwval 22433 | Univariate polynomial eval... |
| evls1fpws 22434 | Evaluation of a univariate... |
| ressply1evl 22435 | Evaluation of a univariate... |
| evls1addd 22436 | Univariate polynomial eval... |
| evls1muld 22437 | Univariate polynomial eval... |
| evls1vsca 22438 | Univariate polynomial eval... |
| asclply1subcl 22439 | Closure of the algebra sca... |
| evls1fvcl 22440 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 22441 | The function ` F ` mapping... |
| evls1maplmhm 22442 | The function ` F ` mapping... |
| evls1maprnss 22443 | The function ` F ` mapping... |
| evl1maprhm 22444 | The function ` F ` mapping... |
| rhmmpl 22445 | Provide a ring homomorphis... |
| ply1vscl 22446 | Closure of scalar multipli... |
| mhmcoply1 22447 | The composition of a monoi... |
| rhmply1 22448 | Provide a ring homomorphis... |
| rhmply1vr1 22449 | A ring homomorphism betwee... |
| rhmply1vsca 22450 | Apply a ring homomorphism ... |
| rhmply1mon 22451 | Apply a ring homomorphism ... |
| mamufval 22454 | Functional value of the ma... |
| mamuval 22455 | Multiplication of two matr... |
| mamufv 22456 | A cell in the multiplicati... |
| mamudm 22457 | The domain of the matrix m... |
| mamufacex 22458 | Every solution of the equa... |
| mamures 22459 | Rows in a matrix product a... |
| grpvlinv 22460 | Tuple-wise left inverse in... |
| grpvrinv 22461 | Tuple-wise right inverse i... |
| ringvcl 22462 | Tuple-wise multiplication ... |
| mamucl 22463 | Operation closure of matri... |
| mamuass 22464 | Matrix multiplication is a... |
| mamudi 22465 | Matrix multiplication dist... |
| mamudir 22466 | Matrix multiplication dist... |
| mamuvs1 22467 | Matrix multiplication dist... |
| mamuvs2 22468 | Matrix multiplication dist... |
| matbas0pc 22471 | There is no matrix with a ... |
| matbas0 22472 | There is no matrix for a n... |
| matval 22473 | Value of the matrix algebr... |
| matrcl 22474 | Reverse closure for the ma... |
| matbas 22475 | The matrix ring has the sa... |
| matplusg 22476 | The matrix ring has the sa... |
| matsca 22477 | The matrix ring has the sa... |
| matvsca 22478 | The matrix ring has the sa... |
| mat0 22479 | The matrix ring has the sa... |
| matinvg 22480 | The matrix ring has the sa... |
| mat0op 22481 | Value of a zero matrix as ... |
| matsca2 22482 | The scalars of the matrix ... |
| matbas2 22483 | The base set of the matrix... |
| matbas2i 22484 | A matrix is a function. (... |
| matbas2d 22485 | The base set of the matrix... |
| eqmat 22486 | Two square matrices of the... |
| matecl 22487 | Each entry (according to W... |
| matecld 22488 | Each entry (according to W... |
| matplusg2 22489 | Addition in the matrix rin... |
| matvsca2 22490 | Scalar multiplication in t... |
| matlmod 22491 | The matrix ring is a linea... |
| matgrp 22492 | The matrix ring is a group... |
| matvscl 22493 | Closure of the scalar mult... |
| matsubg 22494 | The matrix ring has the sa... |
| matplusgcell 22495 | Addition in the matrix rin... |
| matsubgcell 22496 | Subtraction in the matrix ... |
| matinvgcell 22497 | Additive inversion in the ... |
| matvscacell 22498 | Scalar multiplication in t... |
| matgsum 22499 | Finite commutative sums in... |
| matmulr 22500 | Multiplication in the matr... |
| mamumat1cl 22501 | The identity matrix (as op... |
| mat1comp 22502 | The components of the iden... |
| mamulid 22503 | The identity matrix (as op... |
| mamurid 22504 | The identity matrix (as op... |
| matring 22505 | Existence of the matrix ri... |
| matassa 22506 | Existence of the matrix al... |
| matmulcell 22507 | Multiplication in the matr... |
| mpomatmul 22508 | Multiplication of two N x ... |
| mat1 22509 | Value of an identity matri... |
| mat1ov 22510 | Entries of an identity mat... |
| mat1bas 22511 | The identity matrix is a m... |
| matsc 22512 | The identity matrix multip... |
| ofco2 22513 | Distribution law for the f... |
| oftpos 22514 | The transposition of the v... |
| mattposcl 22515 | The transpose of a square ... |
| mattpostpos 22516 | The transpose of the trans... |
| mattposvs 22517 | The transposition of a mat... |
| mattpos1 22518 | The transposition of the i... |
| tposmap 22519 | The transposition of an I ... |
| mamutpos 22520 | Behavior of transposes in ... |
| mattposm 22521 | Multiplying two transposed... |
| matgsumcl 22522 | Closure of a group sum ove... |
| madetsumid 22523 | The identity summand in th... |
| matepmcl 22524 | Each entry of a matrix wit... |
| matepm2cl 22525 | Each entry of a matrix wit... |
| madetsmelbas 22526 | A summand of the determina... |
| madetsmelbas2 22527 | A summand of the determina... |
| mat0dimbas0 22528 | The empty set is the one a... |
| mat0dim0 22529 | The zero of the algebra of... |
| mat0dimid 22530 | The identity of the algebr... |
| mat0dimscm 22531 | The scalar multiplication ... |
| mat0dimcrng 22532 | The algebra of matrices wi... |
| mat1dimelbas 22533 | A matrix with dimension 1 ... |
| mat1dimbas 22534 | A matrix with dimension 1 ... |
| mat1dim0 22535 | The zero of the algebra of... |
| mat1dimid 22536 | The identity of the algebr... |
| mat1dimscm 22537 | The scalar multiplication ... |
| mat1dimmul 22538 | The ring multiplication in... |
| mat1dimcrng 22539 | The algebra of matrices wi... |
| mat1f1o 22540 | There is a 1-1 function fr... |
| mat1rhmval 22541 | The value of the ring homo... |
| mat1rhmelval 22542 | The value of the ring homo... |
| mat1rhmcl 22543 | The value of the ring homo... |
| mat1f 22544 | There is a function from a... |
| mat1ghm 22545 | There is a group homomorph... |
| mat1mhm 22546 | There is a monoid homomorp... |
| mat1rhm 22547 | There is a ring homomorphi... |
| mat1rngiso 22548 | There is a ring isomorphis... |
| mat1ric 22549 | A ring is isomorphic to th... |
| dmatval 22554 | The set of ` N ` x ` N ` d... |
| dmatel 22555 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22556 | An ` N ` x ` N ` diagonal ... |
| dmatid 22557 | The identity matrix is a d... |
| dmatelnd 22558 | An extradiagonal entry of ... |
| dmatmul 22559 | The product of two diagona... |
| dmatsubcl 22560 | The difference of two diag... |
| dmatsgrp 22561 | The set of diagonal matric... |
| dmatmulcl 22562 | The product of two diagona... |
| dmatsrng 22563 | The set of diagonal matric... |
| dmatcrng 22564 | The subring of diagonal ma... |
| dmatscmcl 22565 | The multiplication of a di... |
| scmatval 22566 | The set of ` N ` x ` N ` s... |
| scmatel 22567 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22568 | A scalar matrix can be exp... |
| scmatscmide 22569 | An entry of a scalar matri... |
| scmatscmiddistr 22570 | Distributive law for scala... |
| scmatmat 22571 | An ` N ` x ` N ` scalar ma... |
| scmate 22572 | An entry of an ` N ` x ` N... |
| scmatmats 22573 | The set of an ` N ` x ` N ... |
| scmateALT 22574 | Alternate proof of ~ scmat... |
| scmatscm 22575 | The multiplication of a ma... |
| scmatid 22576 | The identity matrix is a s... |
| scmatdmat 22577 | A scalar matrix is a diago... |
| scmataddcl 22578 | The sum of two scalar matr... |
| scmatsubcl 22579 | The difference of two scal... |
| scmatmulcl 22580 | The product of two scalar ... |
| scmatsgrp 22581 | The set of scalar matrices... |
| scmatsrng 22582 | The set of scalar matrices... |
| scmatcrng 22583 | The subring of scalar matr... |
| scmatsgrp1 22584 | The set of scalar matrices... |
| scmatsrng1 22585 | The set of scalar matrices... |
| smatvscl 22586 | Closure of the scalar mult... |
| scmatlss 22587 | The set of scalar matrices... |
| scmatstrbas 22588 | The set of scalar matrices... |
| scmatrhmval 22589 | The value of the ring homo... |
| scmatrhmcl 22590 | The value of the ring homo... |
| scmatf 22591 | There is a function from a... |
| scmatfo 22592 | There is a function from a... |
| scmatf1 22593 | There is a 1-1 function fr... |
| scmatf1o 22594 | There is a bijection betwe... |
| scmatghm 22595 | There is a group homomorph... |
| scmatmhm 22596 | There is a monoid homomorp... |
| scmatrhm 22597 | There is a ring homomorphi... |
| scmatrngiso 22598 | There is a ring isomorphis... |
| scmatric 22599 | A ring is isomorphic to ev... |
| mat0scmat 22600 | The empty matrix over a ri... |
| mat1scmat 22601 | A 1-dimensional matrix ove... |
| mvmulfval 22604 | Functional value of the ma... |
| mvmulval 22605 | Multiplication of a vector... |
| mvmulfv 22606 | A cell/element in the vect... |
| mavmulval 22607 | Multiplication of a vector... |
| mavmulfv 22608 | A cell/element in the vect... |
| mavmulcl 22609 | Multiplication of an NxN m... |
| 1mavmul 22610 | Multiplication of the iden... |
| mavmulass 22611 | Associativity of the multi... |
| mavmuldm 22612 | The domain of the matrix v... |
| mavmulsolcl 22613 | Every solution of the equa... |
| mavmul0 22614 | Multiplication of a 0-dime... |
| mavmul0g 22615 | The result of the 0-dimens... |
| mvmumamul1 22616 | The multiplication of an M... |
| mavmumamul1 22617 | The multiplication of an N... |
| marrepfval 22622 | First substitution for the... |
| marrepval0 22623 | Second substitution for th... |
| marrepval 22624 | Third substitution for the... |
| marrepeval 22625 | An entry of a matrix with ... |
| marrepcl 22626 | Closure of the row replace... |
| marepvfval 22627 | First substitution for the... |
| marepvval0 22628 | Second substitution for th... |
| marepvval 22629 | Third substitution for the... |
| marepveval 22630 | An entry of a matrix with ... |
| marepvcl 22631 | Closure of the column repl... |
| ma1repvcl 22632 | Closure of the column repl... |
| ma1repveval 22633 | An entry of an identity ma... |
| mulmarep1el 22634 | Element by element multipl... |
| mulmarep1gsum1 22635 | The sum of element by elem... |
| mulmarep1gsum2 22636 | The sum of element by elem... |
| 1marepvmarrepid 22637 | Replacing the ith row by 0... |
| submabas 22640 | Any subset of the index se... |
| submafval 22641 | First substitution for a s... |
| submaval0 22642 | Second substitution for a ... |
| submaval 22643 | Third substitution for a s... |
| submaeval 22644 | An entry of a submatrix of... |
| 1marepvsma1 22645 | The submatrix of the ident... |
| mdetfval 22648 | First substitution for the... |
| mdetleib 22649 | Full substitution of our d... |
| mdetleib2 22650 | Leibniz' formula can also ... |
| nfimdetndef 22651 | The determinant is not def... |
| mdetfval1 22652 | First substitution of an a... |
| mdetleib1 22653 | Full substitution of an al... |
| mdet0pr 22654 | The determinant function f... |
| mdet0f1o 22655 | The determinant function f... |
| mdet0fv0 22656 | The determinant of the emp... |
| mdetf 22657 | Functionality of the deter... |
| mdetcl 22658 | The determinant evaluates ... |
| m1detdiag 22659 | The determinant of a 1-dim... |
| mdetdiaglem 22660 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22661 | The determinant of a diago... |
| mdetdiagid 22662 | The determinant of a diago... |
| mdet1 22663 | The determinant of the ide... |
| mdetrlin 22664 | The determinant function i... |
| mdetrsca 22665 | The determinant function i... |
| mdetrsca2 22666 | The determinant function i... |
| mdetr0 22667 | The determinant of a matri... |
| mdet0 22668 | The determinant of the zer... |
| mdetrlin2 22669 | The determinant function i... |
| mdetralt 22670 | The determinant function i... |
| mdetralt2 22671 | The determinant function i... |
| mdetero 22672 | The determinant function i... |
| mdettpos 22673 | Determinant is invariant u... |
| mdetunilem1 22674 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22675 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22676 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22677 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22678 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22679 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22680 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22681 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22682 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22683 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22684 | According to the definitio... |
| mdetmul 22685 | Multiplicativity of the de... |
| m2detleiblem1 22686 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22687 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22688 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22689 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22690 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22691 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22692 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22693 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22698 | Lemma for ~ maducoeval2 . ... |
| madufval 22699 | First substitution for the... |
| maduval 22700 | Second substitution for th... |
| maducoeval 22701 | An entry of the adjunct (c... |
| maducoeval2 22702 | An entry of the adjunct (c... |
| maduf 22703 | Creating the adjunct of ma... |
| madutpos 22704 | The adjuct of a transposed... |
| madugsum 22705 | The determinant of a matri... |
| madurid 22706 | Multiplying a matrix with ... |
| madulid 22707 | Multiplying the adjunct of... |
| minmar1fval 22708 | First substitution for the... |
| minmar1val0 22709 | Second substitution for th... |
| minmar1val 22710 | Third substitution for the... |
| minmar1eval 22711 | An entry of a matrix for a... |
| minmar1marrep 22712 | The minor matrix is a spec... |
| minmar1cl 22713 | Closure of the row replace... |
| maducoevalmin1 22714 | The coefficients of an adj... |
| symgmatr01lem 22715 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22716 | Applying a permutation tha... |
| gsummatr01lem1 22717 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22718 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22719 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22720 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22721 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22722 | Replacing a row of a squar... |
| smadiadetlem0 22723 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22724 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22725 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22726 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22727 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22728 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22729 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22730 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22731 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22732 | The determinant of a subma... |
| smadiadetglem1 22733 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22734 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22735 | The determinant of a squar... |
| smadiadetg0 22736 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22737 | The determinant of a squar... |
| invrvald 22738 | If a matrix multiplied wit... |
| matinv 22739 | The inverse of a matrix is... |
| matunit 22740 | A matrix is a unit in the ... |
| slesolvec 22741 | Every solution of a system... |
| slesolinv 22742 | The solution of a system o... |
| slesolinvbi 22743 | The solution of a system o... |
| slesolex 22744 | Every system of linear equ... |
| cramerimplem1 22745 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22746 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22747 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22748 | One direction of Cramer's ... |
| cramerlem1 22749 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22750 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22751 | Lemma 3 for ~ cramer . (C... |
| cramer0 22752 | Special case of Cramer's r... |
| cramer 22753 | Cramer's rule. According ... |
| pmatring 22754 | The set of polynomial matr... |
| pmatlmod 22755 | The set of polynomial matr... |
| pmatassa 22756 | The set of polynomial matr... |
| pmat0op 22757 | The zero polynomial matrix... |
| pmat1op 22758 | The identity polynomial ma... |
| pmat1ovd 22759 | Entries of the identity po... |
| pmat0opsc 22760 | The zero polynomial matrix... |
| pmat1opsc 22761 | The identity polynomial ma... |
| pmat1ovscd 22762 | Entries of the identity po... |
| pmatcoe1fsupp 22763 | For a polynomial matrix th... |
| 1pmatscmul 22764 | The scalar product of the ... |
| cpmat 22771 | Value of the constructor o... |
| cpmatpmat 22772 | A constant polynomial matr... |
| cpmatel 22773 | Property of a constant pol... |
| cpmatelimp 22774 | Implication of a set being... |
| cpmatel2 22775 | Another property of a cons... |
| cpmatelimp2 22776 | Another implication of a s... |
| 1elcpmat 22777 | The identity of the ring o... |
| cpmatacl 22778 | The set of all constant po... |
| cpmatinvcl 22779 | The set of all constant po... |
| cpmatmcllem 22780 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22781 | The set of all constant po... |
| cpmatsubgpmat 22782 | The set of all constant po... |
| cpmatsrgpmat 22783 | The set of all constant po... |
| 0elcpmat 22784 | The zero of the ring of al... |
| mat2pmatfval 22785 | Value of the matrix transf... |
| mat2pmatval 22786 | The result of a matrix tra... |
| mat2pmatvalel 22787 | A (matrix) element of the ... |
| mat2pmatbas 22788 | The result of a matrix tra... |
| mat2pmatbas0 22789 | The result of a matrix tra... |
| mat2pmatf 22790 | The matrix transformation ... |
| mat2pmatf1 22791 | The matrix transformation ... |
| mat2pmatghm 22792 | The transformation of matr... |
| mat2pmatmul 22793 | The transformation of matr... |
| mat2pmat1 22794 | The transformation of the ... |
| mat2pmatmhm 22795 | The transformation of matr... |
| mat2pmatrhm 22796 | The transformation of matr... |
| mat2pmatlin 22797 | The transformation of matr... |
| 0mat2pmat 22798 | The transformed zero matri... |
| idmatidpmat 22799 | The transformed identity m... |
| d0mat2pmat 22800 | The transformed empty set ... |
| d1mat2pmat 22801 | The transformation of a ma... |
| mat2pmatscmxcl 22802 | A transformed matrix multi... |
| m2cpm 22803 | The result of a matrix tra... |
| m2cpmf 22804 | The matrix transformation ... |
| m2cpmf1 22805 | The matrix transformation ... |
| m2cpmghm 22806 | The transformation of matr... |
| m2cpmmhm 22807 | The transformation of matr... |
| m2cpmrhm 22808 | The transformation of matr... |
| m2pmfzmap 22809 | The transformed values of ... |
| m2pmfzgsumcl 22810 | Closure of the sum of scal... |
| cpm2mfval 22811 | Value of the inverse matri... |
| cpm2mval 22812 | The result of an inverse m... |
| cpm2mvalel 22813 | A (matrix) element of the ... |
| cpm2mf 22814 | The inverse matrix transfo... |
| m2cpminvid 22815 | The inverse transformation... |
| m2cpminvid2lem 22816 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22817 | The transformation applied... |
| m2cpmfo 22818 | The matrix transformation ... |
| m2cpmf1o 22819 | The matrix transformation ... |
| m2cpmrngiso 22820 | The transformation of matr... |
| matcpmric 22821 | The ring of matrices over ... |
| m2cpminv 22822 | The inverse matrix transfo... |
| m2cpminv0 22823 | The inverse matrix transfo... |
| decpmatval0 22826 | The matrix consisting of t... |
| decpmatval 22827 | The matrix consisting of t... |
| decpmate 22828 | An entry of the matrix con... |
| decpmatcl 22829 | Closure of the decompositi... |
| decpmataa0 22830 | The matrix consisting of t... |
| decpmatfsupp 22831 | The mapping to the matrice... |
| decpmatid 22832 | The matrix consisting of t... |
| decpmatmullem 22833 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22834 | The matrix consisting of t... |
| decpmatmulsumfsupp 22835 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22836 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22837 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22838 | Write a polynomial matrix ... |
| pmatcollpw2lem 22839 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22840 | Write a polynomial matrix ... |
| monmatcollpw 22841 | The matrix consisting of t... |
| pmatcollpwlem 22842 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22843 | Write a polynomial matrix ... |
| pmatcollpwfi 22844 | Write a polynomial matrix ... |
| pmatcollpw3lem 22845 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22846 | Write a polynomial matrix ... |
| pmatcollpw3fi 22847 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22848 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22849 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22850 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22851 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22852 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22853 | Write a scalar matrix over... |
| pm2mpf1lem 22856 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22857 | Value of the transformatio... |
| pm2mpfval 22858 | A polynomial matrix transf... |
| pm2mpcl 22859 | The transformation of poly... |
| pm2mpf 22860 | The transformation of poly... |
| pm2mpf1 22861 | The transformation of poly... |
| pm2mpcoe1 22862 | A coefficient of the polyn... |
| idpm2idmp 22863 | The transformation of the ... |
| mptcoe1matfsupp 22864 | The mapping extracting the... |
| mply1topmatcllem 22865 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22866 | A polynomial over matrices... |
| mply1topmatcl 22867 | A polynomial over matrices... |
| mp2pm2mplem1 22868 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22869 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22870 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22871 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22872 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22873 | A polynomial over matrices... |
| pm2mpghmlem2 22874 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22875 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22876 | The transformation of poly... |
| pm2mpf1o 22877 | The transformation of poly... |
| pm2mpghm 22878 | The transformation of poly... |
| pm2mpgrpiso 22879 | The transformation of poly... |
| pm2mpmhmlem1 22880 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22881 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22882 | The transformation of poly... |
| pm2mprhm 22883 | The transformation of poly... |
| pm2mprngiso 22884 | The transformation of poly... |
| pmmpric 22885 | The ring of polynomial mat... |
| monmat2matmon 22886 | The transformation of a po... |
| pm2mp 22887 | The transformation of a su... |
| chmatcl 22890 | Closure of the characteris... |
| chmatval 22891 | The entries of the charact... |
| chpmatfval 22892 | Value of the characteristi... |
| chpmatval 22893 | The characteristic polynom... |
| chpmatply1 22894 | The characteristic polynom... |
| chpmatval2 22895 | The characteristic polynom... |
| chpmat0d 22896 | The characteristic polynom... |
| chpmat1dlem 22897 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22898 | The characteristic polynom... |
| chpdmatlem0 22899 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22900 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22901 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22902 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22903 | The characteristic polynom... |
| chpscmat 22904 | The characteristic polynom... |
| chpscmat0 22905 | The characteristic polynom... |
| chpscmatgsumbin 22906 | The characteristic polynom... |
| chpscmatgsummon 22907 | The characteristic polynom... |
| chp0mat 22908 | The characteristic polynom... |
| chpidmat 22909 | The characteristic polynom... |
| chmaidscmat 22910 | The characteristic polynom... |
| fvmptnn04if 22911 | The function values of a m... |
| fvmptnn04ifa 22912 | The function value of a ma... |
| fvmptnn04ifb 22913 | The function value of a ma... |
| fvmptnn04ifc 22914 | The function value of a ma... |
| fvmptnn04ifd 22915 | The function value of a ma... |
| chfacfisf 22916 | The "characteristic factor... |
| chfacfisfcpmat 22917 | The "characteristic factor... |
| chfacffsupp 22918 | The "characteristic factor... |
| chfacfscmulcl 22919 | Closure of a scaled value ... |
| chfacfscmul0 22920 | A scaled value of the "cha... |
| chfacfscmulfsupp 22921 | A mapping of scaled values... |
| chfacfscmulgsum 22922 | Breaking up a sum of value... |
| chfacfpmmulcl 22923 | Closure of the value of th... |
| chfacfpmmul0 22924 | The value of the "characte... |
| chfacfpmmulfsupp 22925 | A mapping of values of the... |
| chfacfpmmulgsum 22926 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22927 | Breaking up a sum of value... |
| cayhamlem1 22928 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22929 | The right-hand fundamental... |
| cpmidgsum 22930 | Representation of the iden... |
| cpmidgsumm2pm 22931 | Representation of the iden... |
| cpmidpmatlem1 22932 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22933 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22934 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22935 | Representation of the iden... |
| cpmadugsumlemB 22936 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22937 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22938 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22939 | The product of the charact... |
| cpmadugsum 22940 | The product of the charact... |
| cpmidgsum2 22941 | Representation of the iden... |
| cpmidg2sum 22942 | Equality of two sums repre... |
| cpmadumatpolylem1 22943 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22944 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22945 | The product of the charact... |
| cayhamlem2 22946 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22947 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22948 | The coefficients of the ch... |
| cayhamlem3 22949 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22950 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22951 | The Cayley-Hamilton theore... |
| cayleyhamilton 22952 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22953 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22954 | The Cayley-Hamilton theore... |
| istopg 22957 | Express the predicate " ` ... |
| istop2g 22958 | Express the predicate " ` ... |
| uniopn 22959 | The union of a subset of a... |
| iunopn 22960 | The indexed union of a sub... |
| inopn 22961 | The intersection of two op... |
| fitop 22962 | A topology is closed under... |
| fiinopn 22963 | The intersection of a none... |
| iinopn 22964 | The intersection of a none... |
| unopn 22965 | The union of two open sets... |
| 0opn 22966 | The empty set is an open s... |
| 0ntop 22967 | The empty set is not a top... |
| topopn 22968 | The underlying set of a to... |
| eltopss 22969 | A member of a topology is ... |
| riinopn 22970 | A finite indexed relative ... |
| rintopn 22971 | A finite relative intersec... |
| istopon 22974 | Property of being a topolo... |
| topontop 22975 | A topology on a given base... |
| toponuni 22976 | The base set of a topology... |
| topontopi 22977 | A topology on a given base... |
| toponunii 22978 | The base set of a topology... |
| toptopon 22979 | Alternative definition of ... |
| toptopon2 22980 | A topology is the same thi... |
| topontopon 22981 | A topology on a set is a t... |
| funtopon 22982 | The class ` TopOn ` is a f... |
| toponrestid 22983 | Given a topology on a set,... |
| toponsspwpw 22984 | The set of topologies on a... |
| dmtopon 22985 | The domain of ` TopOn ` is... |
| fntopon 22986 | The class ` TopOn ` is a f... |
| toprntopon 22987 | A topology is the same thi... |
| toponmax 22988 | The base set of a topology... |
| toponss 22989 | A member of a topology is ... |
| toponcom 22990 | If ` K ` is a topology on ... |
| toponcomb 22991 | Biconditional form of ~ to... |
| topgele 22992 | The topologies over the sa... |
| topsn 22993 | The only topology on a sin... |
| istps 22996 | Express the predicate "is ... |
| istps2 22997 | Express the predicate "is ... |
| tpsuni 22998 | The base set of a topologi... |
| tpstop 22999 | The topology extractor on ... |
| tpspropd 23000 | A topological space depend... |
| tpsprop2d 23001 | A topological space depend... |
| topontopn 23002 | Express the predicate "is ... |
| tsettps 23003 | If the topology component ... |
| istpsi 23004 | Properties that determine ... |
| eltpsg 23005 | Properties that determine ... |
| eltpsi 23006 | Properties that determine ... |
| isbasisg 23009 | Express the predicate "the... |
| isbasis2g 23010 | Express the predicate "the... |
| isbasis3g 23011 | Express the predicate "the... |
| basis1 23012 | Property of a basis. (Con... |
| basis2 23013 | Property of a basis. (Con... |
| fiinbas 23014 | If a set is closed under f... |
| basdif0 23015 | A basis is not affected by... |
| baspartn 23016 | A disjoint system of sets ... |
| tgval 23017 | The topology generated by ... |
| tgval2 23018 | Definition of a topology g... |
| eltg 23019 | Membership in a topology g... |
| eltg2 23020 | Membership in a topology g... |
| eltg2b 23021 | Membership in a topology g... |
| eltg4i 23022 | An open set in a topology ... |
| eltg3i 23023 | The union of a set of basi... |
| eltg3 23024 | Membership in a topology g... |
| tgval3 23025 | Alternate expression for t... |
| tg1 23026 | Property of a member of a ... |
| tg2 23027 | Property of a member of a ... |
| bastg 23028 | A member of a basis is a s... |
| unitg 23029 | The topology generated by ... |
| tgss 23030 | Subset relation for genera... |
| tgcl 23031 | Show that a basis generate... |
| tgclb 23032 | The property ~ tgcl can be... |
| tgtopon 23033 | A basis generates a topolo... |
| topbas 23034 | A topology is its own basi... |
| tgtop 23035 | A topology is its own basi... |
| eltop 23036 | Membership in a topology, ... |
| eltop2 23037 | Membership in a topology. ... |
| eltop3 23038 | Membership in a topology. ... |
| fibas 23039 | A collection of finite int... |
| tgdom 23040 | A space has no more open s... |
| tgiun 23041 | The indexed union of a set... |
| tgidm 23042 | The topology generator fun... |
| bastop 23043 | Two ways to express that a... |
| tgtop11 23044 | The topology generation fu... |
| 0top 23045 | The singleton of the empty... |
| en1top 23046 | ` { (/) } ` is the only to... |
| en2top 23047 | If a topology has two elem... |
| tgss3 23048 | A criterion for determinin... |
| tgss2 23049 | A criterion for determinin... |
| basgen 23050 | Given a topology ` J ` , s... |
| basgen2 23051 | Given a topology ` J ` , s... |
| 2basgen 23052 | Conditions that determine ... |
| tgfiss 23053 | If a subbase is included i... |
| tgdif0 23054 | A generated topology is no... |
| bastop1 23055 | A subset of a topology is ... |
| bastop2 23056 | A version of ~ bastop1 tha... |
| distop 23057 | The discrete topology on a... |
| topnex 23058 | The class of all topologie... |
| distopon 23059 | The discrete topology on a... |
| sn0topon 23060 | The singleton of the empty... |
| sn0top 23061 | The singleton of the empty... |
| indislem 23062 | A lemma to eliminate some ... |
| indistopon 23063 | The indiscrete topology on... |
| indistop 23064 | The indiscrete topology on... |
| indisuni 23065 | The base set of the indisc... |
| fctop 23066 | The finite complement topo... |
| fctop2 23067 | The finite complement topo... |
| cctop 23068 | The countable complement t... |
| ppttop 23069 | The particular point topol... |
| pptbas 23070 | The particular point topol... |
| epttop 23071 | The excluded point topolog... |
| indistpsx 23072 | The indiscrete topology on... |
| indistps 23073 | The indiscrete topology on... |
| indistps2 23074 | The indiscrete topology on... |
| indistpsALT 23075 | The indiscrete topology on... |
| indistps2ALT 23076 | The indiscrete topology on... |
| distps 23077 | The discrete topology on a... |
| fncld 23084 | The closed-set generator i... |
| cldval 23085 | The set of closed sets of ... |
| ntrfval 23086 | The interior function on t... |
| clsfval 23087 | The closure function on th... |
| cldrcl 23088 | Reverse closure of the clo... |
| iscld 23089 | The predicate "the class `... |
| iscld2 23090 | A subset of the underlying... |
| cldss 23091 | A closed set is a subset o... |
| cldss2 23092 | The set of closed sets is ... |
| cldopn 23093 | The complement of a closed... |
| isopn2 23094 | A subset of the underlying... |
| opncld 23095 | The complement of an open ... |
| difopn 23096 | The difference of a closed... |
| topcld 23097 | The underlying set of a to... |
| ntrval 23098 | The interior of a subset o... |
| clsval 23099 | The closure of a subset of... |
| 0cld 23100 | The empty set is closed. ... |
| iincld 23101 | The indexed intersection o... |
| intcld 23102 | The intersection of a set ... |
| uncld 23103 | The union of two closed se... |
| cldcls 23104 | A closed subset equals its... |
| incld 23105 | The intersection of two cl... |
| riincld 23106 | An indexed relative inters... |
| iuncld 23107 | A finite indexed union of ... |
| unicld 23108 | A finite union of closed s... |
| clscld 23109 | The closure of a subset of... |
| clsf 23110 | The closure function is a ... |
| ntropn 23111 | The interior of a subset o... |
| clsval2 23112 | Express closure in terms o... |
| ntrval2 23113 | Interior expressed in term... |
| ntrdif 23114 | An interior of a complemen... |
| clsdif 23115 | A closure of a complement ... |
| clsss 23116 | Subset relationship for cl... |
| ntrss 23117 | Subset relationship for in... |
| sscls 23118 | A subset of a topology's u... |
| ntrss2 23119 | A subset includes its inte... |
| ssntr 23120 | An open subset of a set is... |
| clsss3 23121 | The closure of a subset of... |
| ntrss3 23122 | The interior of a subset o... |
| ntrin 23123 | A pairwise intersection of... |
| cmclsopn 23124 | The complement of a closur... |
| cmntrcld 23125 | The complement of an inter... |
| iscld3 23126 | A subset is closed iff it ... |
| iscld4 23127 | A subset is closed iff it ... |
| isopn3 23128 | A subset is open iff it eq... |
| clsidm 23129 | The closure operation is i... |
| ntridm 23130 | The interior operation is ... |
| clstop 23131 | The closure of a topology'... |
| ntrtop 23132 | The interior of a topology... |
| 0ntr 23133 | A subset with an empty int... |
| clsss2 23134 | If a subset is included in... |
| elcls 23135 | Membership in a closure. ... |
| elcls2 23136 | Membership in a closure. ... |
| clsndisj 23137 | Any open set containing a ... |
| ntrcls0 23138 | A subset whose closure has... |
| ntreq0 23139 | Two ways to say that a sub... |
| cldmre 23140 | The closed sets of a topol... |
| mrccls 23141 | Moore closure generalizes ... |
| cls0 23142 | The closure of the empty s... |
| ntr0 23143 | The interior of the empty ... |
| isopn3i 23144 | An open subset equals its ... |
| elcls3 23145 | Membership in a closure in... |
| opncldf1 23146 | A bijection useful for con... |
| opncldf2 23147 | The values of the open-clo... |
| opncldf3 23148 | The values of the converse... |
| isclo 23149 | A set ` A ` is clopen iff ... |
| isclo2 23150 | A set ` A ` is clopen iff ... |
| discld 23151 | The open sets of a discret... |
| sn0cld 23152 | The closed sets of the top... |
| indiscld 23153 | The closed sets of an indi... |
| mretopd 23154 | A Moore collection which i... |
| toponmre 23155 | The topologies over a give... |
| cldmreon 23156 | The closed sets of a topol... |
| iscldtop 23157 | A family is the closed set... |
| mreclatdemoBAD 23158 | The closed subspaces of a ... |
| neifval 23161 | Value of the neighborhood ... |
| neif 23162 | The neighborhood function ... |
| neiss2 23163 | A set with a neighborhood ... |
| neival 23164 | Value of the set of neighb... |
| isnei 23165 | The predicate "the class `... |
| neiint 23166 | An intuitive definition of... |
| isneip 23167 | The predicate "the class `... |
| neii1 23168 | A neighborhood is included... |
| neisspw 23169 | The neighborhoods of any s... |
| neii2 23170 | Property of a neighborhood... |
| neiss 23171 | Any neighborhood of a set ... |
| ssnei 23172 | A set is included in any o... |
| elnei 23173 | A point belongs to any of ... |
| 0nnei 23174 | The empty set is not a nei... |
| neips 23175 | A neighborhood of a set is... |
| opnneissb 23176 | An open set is a neighborh... |
| opnssneib 23177 | Any superset of an open se... |
| ssnei2 23178 | Any subset ` M ` of ` X ` ... |
| neindisj 23179 | Any neighborhood of an ele... |
| opnneiss 23180 | An open set is a neighborh... |
| opnneip 23181 | An open set is a neighborh... |
| opnnei 23182 | A set is open iff it is a ... |
| tpnei 23183 | The underlying set of a to... |
| neiuni 23184 | The union of the neighborh... |
| neindisj2 23185 | A point ` P ` belongs to t... |
| topssnei 23186 | A finer topology has more ... |
| innei 23187 | The intersection of two ne... |
| opnneiid 23188 | Only an open set is a neig... |
| neissex 23189 | For any neighborhood ` N `... |
| 0nei 23190 | The empty set is a neighbo... |
| neipeltop 23191 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23192 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23193 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23194 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23195 | If, to each element ` P ` ... |
| lpfval 23200 | The limit point function o... |
| lpval 23201 | The set of limit points of... |
| islp 23202 | The predicate "the class `... |
| lpsscls 23203 | The limit points of a subs... |
| lpss 23204 | The limit points of a subs... |
| lpdifsn 23205 | ` P ` is a limit point of ... |
| lpss3 23206 | Subset relationship for li... |
| islp2 23207 | The predicate " ` P ` is a... |
| islp3 23208 | The predicate " ` P ` is a... |
| maxlp 23209 | A point is a limit point o... |
| clslp 23210 | The closure of a subset of... |
| islpi 23211 | A point belonging to a set... |
| cldlp 23212 | A subset of a topological ... |
| isperf 23213 | Definition of a perfect sp... |
| isperf2 23214 | Definition of a perfect sp... |
| isperf3 23215 | A perfect space is a topol... |
| perflp 23216 | The limit points of a perf... |
| perfi 23217 | Property of a perfect spac... |
| perftop 23218 | A perfect space is a topol... |
| restrcl 23219 | Reverse closure for the su... |
| restbas 23220 | A subspace topology basis ... |
| tgrest 23221 | A subspace can be generate... |
| resttop 23222 | A subspace topology is a t... |
| resttopon 23223 | A subspace topology is a t... |
| restuni 23224 | The underlying set of a su... |
| stoig 23225 | The topological space buil... |
| restco 23226 | Composition of subspaces. ... |
| restabs 23227 | Equivalence of being a sub... |
| restin 23228 | When the subspace region i... |
| restuni2 23229 | The underlying set of a su... |
| resttopon2 23230 | The underlying set of a su... |
| rest0 23231 | The subspace topology indu... |
| restsn 23232 | The only subspace topology... |
| restsn2 23233 | The subspace topology indu... |
| restcld 23234 | A closed set of a subspace... |
| restcldi 23235 | A closed set is closed in ... |
| restcldr 23236 | A set which is closed in t... |
| restopnb 23237 | If ` B ` is an open subset... |
| ssrest 23238 | If ` K ` is a finer topolo... |
| restopn2 23239 | If ` A ` is open, then ` B... |
| restdis 23240 | A subspace of a discrete t... |
| restfpw 23241 | The restriction of the set... |
| neitr 23242 | The neighborhood of a trac... |
| restcls 23243 | A closure in a subspace to... |
| restntr 23244 | An interior in a subspace ... |
| restlp 23245 | The limit points of a subs... |
| restperf 23246 | Perfection of a subspace. ... |
| perfopn 23247 | An open subset of a perfec... |
| resstopn 23248 | The topology of a restrict... |
| resstps 23249 | A restricted topological s... |
| ordtbaslem 23250 | Lemma for ~ ordtbas . In ... |
| ordtval 23251 | Value of the order topolog... |
| ordtuni 23252 | Value of the order topolog... |
| ordtbas2 23253 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23254 | In a total order, the fini... |
| ordttopon 23255 | Value of the order topolog... |
| ordtopn1 23256 | An upward ray ` ( P , +oo ... |
| ordtopn2 23257 | A downward ray ` ( -oo , P... |
| ordtopn3 23258 | An open interval ` ( A , B... |
| ordtcld1 23259 | A downward ray ` ( -oo , P... |
| ordtcld2 23260 | An upward ray ` [ P , +oo ... |
| ordtcld3 23261 | A closed interval ` [ A , ... |
| ordttop 23262 | The order topology is a to... |
| ordtcnv 23263 | The order dual generates t... |
| ordtrest 23264 | The subspace topology of a... |
| ordtrest2lem 23265 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23266 | An interval-closed set ` A... |
| letopon 23267 | The topology of the extend... |
| letop 23268 | The topology of the extend... |
| letopuni 23269 | The topology of the extend... |
| xrstopn 23270 | The topology component of ... |
| xrstps 23271 | The extended real number s... |
| leordtvallem1 23272 | Lemma for ~ leordtval . (... |
| leordtvallem2 23273 | Lemma for ~ leordtval . (... |
| leordtval2 23274 | The topology of the extend... |
| leordtval 23275 | The topology of the extend... |
| iccordt 23276 | A closed interval is close... |
| iocpnfordt 23277 | An unbounded above open in... |
| icomnfordt 23278 | An unbounded above open in... |
| iooordt 23279 | An open interval is open i... |
| reordt 23280 | The real numbers are an op... |
| lecldbas 23281 | The set of closed interval... |
| pnfnei 23282 | A neighborhood of ` +oo ` ... |
| mnfnei 23283 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23284 | The restriction of the les... |
| ordtresticc 23285 | The restriction of the les... |
| lmrel 23292 | The topological space conv... |
| lmrcl 23293 | Reverse closure for the co... |
| lmfval 23294 | The relation "sequence ` f... |
| cnfval 23295 | The set of all continuous ... |
| cnpfval 23296 | The function mapping the p... |
| iscn 23297 | The predicate "the class `... |
| cnpval 23298 | The set of all functions f... |
| iscnp 23299 | The predicate "the class `... |
| iscn2 23300 | The predicate "the class `... |
| iscnp2 23301 | The predicate "the class `... |
| cntop1 23302 | Reverse closure for a cont... |
| cntop2 23303 | Reverse closure for a cont... |
| cnptop1 23304 | Reverse closure for a func... |
| cnptop2 23305 | Reverse closure for a func... |
| iscnp3 23306 | The predicate "the class `... |
| cnprcl 23307 | Reverse closure for a func... |
| cnf 23308 | A continuous function is a... |
| cnpf 23309 | A continuous function at p... |
| cnpcl 23310 | The value of a continuous ... |
| cnf2 23311 | A continuous function is a... |
| cnpf2 23312 | A continuous function at p... |
| cnprcl2 23313 | Reverse closure for a func... |
| tgcn 23314 | The continuity predicate w... |
| tgcnp 23315 | The "continuous at a point... |
| subbascn 23316 | The continuity predicate w... |
| ssidcn 23317 | The identity function is a... |
| cnpimaex 23318 | Property of a function con... |
| idcn 23319 | A restricted identity func... |
| lmbr 23320 | Express the binary relatio... |
| lmbr2 23321 | Express the binary relatio... |
| lmbrf 23322 | Express the binary relatio... |
| lmconst 23323 | A constant sequence conver... |
| lmcvg 23324 | Convergence property of a ... |
| iscnp4 23325 | The predicate "the class `... |
| cnpnei 23326 | A condition for continuity... |
| cnima 23327 | An open subset of the codo... |
| cnco 23328 | The composition of two con... |
| cnpco 23329 | The composition of a funct... |
| cnclima 23330 | A closed subset of the cod... |
| iscncl 23331 | A characterization of a co... |
| cncls2i 23332 | Property of the preimage o... |
| cnntri 23333 | Property of the preimage o... |
| cnclsi 23334 | Property of the image of a... |
| cncls2 23335 | Continuity in terms of clo... |
| cncls 23336 | Continuity in terms of clo... |
| cnntr 23337 | Continuity in terms of int... |
| cnss1 23338 | If the topology ` K ` is f... |
| cnss2 23339 | If the topology ` K ` is f... |
| cncnpi 23340 | A continuous function is c... |
| cnsscnp 23341 | The set of continuous func... |
| cncnp 23342 | A continuous function is c... |
| cncnp2 23343 | A continuous function is c... |
| cnnei 23344 | Continuity in terms of nei... |
| cnconst2 23345 | A constant function is con... |
| cnconst 23346 | A constant function is con... |
| cnrest 23347 | Continuity of a restrictio... |
| cnrest2 23348 | Equivalence of continuity ... |
| cnrest2r 23349 | Equivalence of continuity ... |
| cnpresti 23350 | One direction of ~ cnprest... |
| cnprest 23351 | Equivalence of continuity ... |
| cnprest2 23352 | Equivalence of point-conti... |
| cndis 23353 | Every function is continuo... |
| cnindis 23354 | Every function is continuo... |
| cnpdis 23355 | If ` A ` is an isolated po... |
| paste 23356 | Pasting lemma. If ` A ` a... |
| lmfpm 23357 | If ` F ` converges, then `... |
| lmfss 23358 | Inclusion of a function ha... |
| lmcl 23359 | Closure of a limit. (Cont... |
| lmss 23360 | Limit on a subspace. (Con... |
| sslm 23361 | A finer topology has fewer... |
| lmres 23362 | A function converges iff i... |
| lmff 23363 | If ` F ` converges, there ... |
| lmcls 23364 | Any convergent sequence of... |
| lmcld 23365 | Any convergent sequence of... |
| lmcnp 23366 | The image of a convergent ... |
| lmcn 23367 | The image of a convergent ... |
| ist0 23382 | The predicate "is a T_0 sp... |
| ist1 23383 | The predicate "is a T_1 sp... |
| ishaus 23384 | The predicate "is a Hausdo... |
| iscnrm 23385 | The property of being comp... |
| t0sep 23386 | Any two topologically indi... |
| t0dist 23387 | Any two distinct points in... |
| t1sncld 23388 | In a T_1 space, singletons... |
| t1ficld 23389 | In a T_1 space, finite set... |
| hausnei 23390 | Neighborhood property of a... |
| t0top 23391 | A T_0 space is a topologic... |
| t1top 23392 | A T_1 space is a topologic... |
| haustop 23393 | A Hausdorff space is a top... |
| isreg 23394 | The predicate "is a regula... |
| regtop 23395 | A regular space is a topol... |
| regsep 23396 | In a regular space, every ... |
| isnrm 23397 | The predicate "is a normal... |
| nrmtop 23398 | A normal space is a topolo... |
| cnrmtop 23399 | A completely normal space ... |
| iscnrm2 23400 | The property of being comp... |
| ispnrm 23401 | The property of being perf... |
| pnrmnrm 23402 | A perfectly normal space i... |
| pnrmtop 23403 | A perfectly normal space i... |
| pnrmcld 23404 | A closed set in a perfectl... |
| pnrmopn 23405 | An open set in a perfectly... |
| ist0-2 23406 | The predicate "is a T_0 sp... |
| ist0-3 23407 | The predicate "is a T_0 sp... |
| cnt0 23408 | The preimage of a T_0 topo... |
| ist1-2 23409 | An alternate characterizat... |
| t1t0 23410 | A T_1 space is a T_0 space... |
| ist1-3 23411 | A space is T_1 iff every p... |
| cnt1 23412 | The preimage of a T_1 topo... |
| ishaus2 23413 | Express the predicate " ` ... |
| haust1 23414 | A Hausdorff space is a T_1... |
| hausnei2 23415 | The Hausdorff condition st... |
| cnhaus 23416 | The preimage of a Hausdorf... |
| nrmsep3 23417 | In a normal space, given a... |
| nrmsep2 23418 | In a normal space, any two... |
| nrmsep 23419 | In a normal space, disjoin... |
| isnrm2 23420 | An alternate characterizat... |
| isnrm3 23421 | A topological space is nor... |
| cnrmi 23422 | A subspace of a completely... |
| cnrmnrm 23423 | A completely normal space ... |
| restcnrm 23424 | A subspace of a completely... |
| resthauslem 23425 | Lemma for ~ resthaus and s... |
| lpcls 23426 | The limit points of the cl... |
| perfcls 23427 | A subset of a perfect spac... |
| restt0 23428 | A subspace of a T_0 topolo... |
| restt1 23429 | A subspace of a T_1 topolo... |
| resthaus 23430 | A subspace of a Hausdorff ... |
| t1sep2 23431 | Any two points in a T_1 sp... |
| t1sep 23432 | Any two distinct points in... |
| sncld 23433 | A singleton is closed in a... |
| sshauslem 23434 | Lemma for ~ sshaus and sim... |
| sst0 23435 | A topology finer than a T_... |
| sst1 23436 | A topology finer than a T_... |
| sshaus 23437 | A topology finer than a Ha... |
| regsep2 23438 | In a regular space, a clos... |
| isreg2 23439 | A topological space is reg... |
| dnsconst 23440 | If a continuous mapping to... |
| ordtt1 23441 | The order topology is T_1 ... |
| lmmo 23442 | A sequence in a Hausdorff ... |
| lmfun 23443 | The convergence relation i... |
| dishaus 23444 | A discrete topology is Hau... |
| ordthauslem 23445 | Lemma for ~ ordthaus . (C... |
| ordthaus 23446 | The order topology of a to... |
| xrhaus 23447 | The topology of the extend... |
| iscmp 23450 | The predicate "is a compac... |
| cmpcov 23451 | An open cover of a compact... |
| cmpcov2 23452 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23453 | Combine ~ cmpcov with ~ ac... |
| cncmp 23454 | Compactness is respected b... |
| fincmp 23455 | A finite topology is compa... |
| 0cmp 23456 | The singleton of the empty... |
| cmptop 23457 | A compact topology is a to... |
| rncmp 23458 | The image of a compact set... |
| imacmp 23459 | The image of a compact set... |
| discmp 23460 | A discrete topology is com... |
| cmpsublem 23461 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23462 | Two equivalent ways of des... |
| tgcmp 23463 | A topology generated by a ... |
| cmpcld 23464 | A closed subset of a compa... |
| uncmp 23465 | The union of two compact s... |
| fiuncmp 23466 | A finite union of compact ... |
| sscmp 23467 | A subset of a compact topo... |
| hauscmplem 23468 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23469 | A compact subspace of a T2... |
| cmpfi 23470 | If a topology is compact a... |
| cmpfii 23471 | In a compact topology, a s... |
| bwth 23472 | The glorious Bolzano-Weier... |
| isconn 23475 | The predicate ` J ` is a c... |
| isconn2 23476 | The predicate ` J ` is a c... |
| connclo 23477 | The only nonempty clopen s... |
| conndisj 23478 | If a topology is connected... |
| conntop 23479 | A connected topology is a ... |
| indisconn 23480 | The indiscrete topology (o... |
| dfconn2 23481 | An alternate definition of... |
| connsuba 23482 | Connectedness for a subspa... |
| connsub 23483 | Two equivalent ways of say... |
| cnconn 23484 | Connectedness is respected... |
| nconnsubb 23485 | Disconnectedness for a sub... |
| connsubclo 23486 | If a clopen set meets a co... |
| connima 23487 | The image of a connected s... |
| conncn 23488 | A continuous function from... |
| iunconnlem 23489 | Lemma for ~ iunconn . (Co... |
| iunconn 23490 | The indexed union of conne... |
| unconn 23491 | The union of two connected... |
| clsconn 23492 | The closure of a connected... |
| conncompid 23493 | The connected component co... |
| conncompconn 23494 | The connected component co... |
| conncompss 23495 | The connected component co... |
| conncompcld 23496 | The connected component co... |
| conncompclo 23497 | The connected component co... |
| t1connperf 23498 | A connected T_1 space is p... |
| is1stc 23503 | The predicate "is a first-... |
| is1stc2 23504 | An equivalent way of sayin... |
| 1stctop 23505 | A first-countable topology... |
| 1stcclb 23506 | A property of points in a ... |
| 1stcfb 23507 | For any point ` A ` in a f... |
| is2ndc 23508 | The property of being seco... |
| 2ndctop 23509 | A second-countable topolog... |
| 2ndci 23510 | A countable basis generate... |
| 2ndcsb 23511 | Having a countable subbase... |
| 2ndcredom 23512 | A second-countable space h... |
| 2ndc1stc 23513 | A second-countable space i... |
| 1stcrestlem 23514 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23515 | A subspace of a first-coun... |
| 2ndcrest 23516 | A subspace of a second-cou... |
| 2ndcctbss 23517 | If a topology is second-co... |
| 2ndcdisj 23518 | Any disjoint family of ope... |
| 2ndcdisj2 23519 | Any disjoint collection of... |
| 2ndcomap 23520 | A surjective continuous op... |
| 2ndcsep 23521 | A second-countable topolog... |
| dis2ndc 23522 | A discrete space is second... |
| 1stcelcls 23523 | A point belongs to the clo... |
| 1stccnp 23524 | A mapping is continuous at... |
| 1stccn 23525 | A mapping ` X --> Y ` , wh... |
| islly 23530 | The property of being a lo... |
| isnlly 23531 | The property of being an n... |
| llyeq 23532 | Equality theorem for the `... |
| nllyeq 23533 | Equality theorem for the `... |
| llytop 23534 | A locally ` A ` space is a... |
| nllytop 23535 | A locally ` A ` space is a... |
| llyi 23536 | The property of a locally ... |
| nllyi 23537 | The property of an n-local... |
| nlly2i 23538 | Eliminate the neighborhood... |
| llynlly 23539 | A locally ` A ` space is n... |
| llyssnlly 23540 | A locally ` A ` space is n... |
| llyss 23541 | The "locally" predicate re... |
| nllyss 23542 | The "n-locally" predicate ... |
| subislly 23543 | The property of a subspace... |
| restnlly 23544 | If the property ` A ` pass... |
| restlly 23545 | If the property ` A ` pass... |
| islly2 23546 | An alternative expression ... |
| llyrest 23547 | An open subspace of a loca... |
| nllyrest 23548 | An open subspace of an n-l... |
| loclly 23549 | If ` A ` is a local proper... |
| llyidm 23550 | Idempotence of the "locall... |
| nllyidm 23551 | Idempotence of the "n-loca... |
| toplly 23552 | A topology is locally a to... |
| topnlly 23553 | A topology is n-locally a ... |
| hauslly 23554 | A Hausdorff space is local... |
| hausnlly 23555 | A Hausdorff space is n-loc... |
| hausllycmp 23556 | A compact Hausdorff space ... |
| cldllycmp 23557 | A closed subspace of a loc... |
| lly1stc 23558 | First-countability is a lo... |
| dislly 23559 | The discrete space ` ~P X ... |
| disllycmp 23560 | A discrete space is locall... |
| dis1stc 23561 | A discrete space is first-... |
| hausmapdom 23562 | If ` X ` is a first-counta... |
| hauspwdom 23563 | Simplify the cardinal ` A ... |
| refrel 23570 | Refinement is a relation. ... |
| isref 23571 | The property of being a re... |
| refbas 23572 | A refinement covers the sa... |
| refssex 23573 | Every set in a refinement ... |
| ssref 23574 | A subcover is a refinement... |
| refref 23575 | Reflexivity of refinement.... |
| reftr 23576 | Refinement is transitive. ... |
| refun0 23577 | Adding the empty set prese... |
| isptfin 23578 | The statement "is a point-... |
| islocfin 23579 | The statement "is a locall... |
| finptfin 23580 | A finite cover is a point-... |
| ptfinfin 23581 | A point covered by a point... |
| finlocfin 23582 | A finite cover of a topolo... |
| locfintop 23583 | A locally finite cover cov... |
| locfinbas 23584 | A locally finite cover mus... |
| locfinnei 23585 | A point covered by a local... |
| lfinpfin 23586 | A locally finite cover is ... |
| lfinun 23587 | Adding a finite set preser... |
| locfincmp 23588 | For a compact space, the l... |
| unisngl 23589 | Taking the union of the se... |
| dissnref 23590 | The set of singletons is a... |
| dissnlocfin 23591 | The set of singletons is l... |
| locfindis 23592 | The locally finite covers ... |
| locfincf 23593 | A locally finite cover in ... |
| comppfsc 23594 | A space where every open c... |
| kgenval 23597 | Value of the compact gener... |
| elkgen 23598 | Value of the compact gener... |
| kgeni 23599 | Property of the open sets ... |
| kgentopon 23600 | The compact generator gene... |
| kgenuni 23601 | The base set of the compac... |
| kgenftop 23602 | The compact generator gene... |
| kgenf 23603 | The compact generator is a... |
| kgentop 23604 | A compactly generated spac... |
| kgenss 23605 | The compact generator gene... |
| kgenhaus 23606 | The compact generator gene... |
| kgencmp 23607 | The compact generator topo... |
| kgencmp2 23608 | The compact generator topo... |
| kgenidm 23609 | The compact generator is i... |
| iskgen2 23610 | A space is compactly gener... |
| iskgen3 23611 | Derive the usual definitio... |
| llycmpkgen2 23612 | A locally compact space is... |
| cmpkgen 23613 | A compact space is compact... |
| llycmpkgen 23614 | A locally compact space is... |
| 1stckgenlem 23615 | The one-point compactifica... |
| 1stckgen 23616 | A first-countable space is... |
| kgen2ss 23617 | The compact generator pres... |
| kgencn 23618 | A function from a compactl... |
| kgencn2 23619 | A function ` F : J --> K `... |
| kgencn3 23620 | The set of continuous func... |
| kgen2cn 23621 | A continuous function is a... |
| txval 23626 | Value of the binary topolo... |
| txuni2 23627 | The underlying set of the ... |
| txbasex 23628 | The basis for the product ... |
| txbas 23629 | The set of Cartesian produ... |
| eltx 23630 | A set in a product is open... |
| txtop 23631 | The product of two topolog... |
| ptval 23632 | The value of the product t... |
| ptpjpre1 23633 | The preimage of a projecti... |
| elpt 23634 | Elementhood in the bases o... |
| elptr 23635 | A basic open set in the pr... |
| elptr2 23636 | A basic open set in the pr... |
| ptbasid 23637 | The base set of the produc... |
| ptuni2 23638 | The base set for the produ... |
| ptbasin 23639 | The basis for a product to... |
| ptbasin2 23640 | The basis for a product to... |
| ptbas 23641 | The basis for a product to... |
| ptpjpre2 23642 | The basis for a product to... |
| ptbasfi 23643 | The basis for the product ... |
| pttop 23644 | The product topology is a ... |
| ptopn 23645 | A basic open set in the pr... |
| ptopn2 23646 | A sub-basic open set in th... |
| xkotf 23647 | Functionality of function ... |
| xkobval 23648 | Alternative expression for... |
| xkoval 23649 | Value of the compact-open ... |
| xkotop 23650 | The compact-open topology ... |
| xkoopn 23651 | A basic open set of the co... |
| txtopi 23652 | The product of two topolog... |
| txtopon 23653 | The underlying set of the ... |
| txuni 23654 | The underlying set of the ... |
| txunii 23655 | The underlying set of the ... |
| ptuni 23656 | The base set for the produ... |
| ptunimpt 23657 | Base set of a product topo... |
| pttopon 23658 | The base set for the produ... |
| pttoponconst 23659 | The base set for a product... |
| ptuniconst 23660 | The base set for a product... |
| xkouni 23661 | The base set of the compac... |
| xkotopon 23662 | The base set of the compac... |
| ptval2 23663 | The value of the product t... |
| txopn 23664 | The product of two open se... |
| txcld 23665 | The product of two closed ... |
| txcls 23666 | Closure of a rectangle in ... |
| txss12 23667 | Subset property of the top... |
| txbasval 23668 | It is sufficient to consid... |
| neitx 23669 | The Cartesian product of t... |
| txcnpi 23670 | Continuity of a two-argume... |
| tx1cn 23671 | Continuity of the first pr... |
| tx2cn 23672 | Continuity of the second p... |
| ptpjcn 23673 | Continuity of a projection... |
| ptpjopn 23674 | The projection map is an o... |
| ptcld 23675 | A closed box in the produc... |
| ptcldmpt 23676 | A closed box in the produc... |
| ptclsg 23677 | The closure of a box in th... |
| ptcls 23678 | The closure of a box in th... |
| dfac14lem 23679 | Lemma for ~ dfac14 . By e... |
| dfac14 23680 | Theorem ~ ptcls is an equi... |
| xkoccn 23681 | The "constant function" fu... |
| txcnp 23682 | If two functions are conti... |
| ptcnplem 23683 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23684 | If every projection of a f... |
| upxp 23685 | Universal property of the ... |
| txcnmpt 23686 | A map into the product of ... |
| uptx 23687 | Universal property of the ... |
| txcn 23688 | A map into the product of ... |
| ptcn 23689 | If every projection of a f... |
| prdstopn 23690 | Topology of a structure pr... |
| prdstps 23691 | A structure product of top... |
| pwstps 23692 | A structure power of a top... |
| txrest 23693 | The subspace of a topologi... |
| txdis 23694 | The topological product of... |
| txindislem 23695 | Lemma for ~ txindis . (Co... |
| txindis 23696 | The topological product of... |
| txdis1cn 23697 | A function is jointly cont... |
| txlly 23698 | If the property ` A ` is p... |
| txnlly 23699 | If the property ` A ` is p... |
| pthaus 23700 | The product of a collectio... |
| ptrescn 23701 | Restriction is a continuou... |
| txtube 23702 | The "tube lemma". If ` X ... |
| txcmplem1 23703 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23704 | Lemma for ~ txcmp . (Cont... |
| txcmp 23705 | The topological product of... |
| txcmpb 23706 | The topological product of... |
| hausdiag 23707 | A topology is Hausdorff if... |
| hauseqlcld 23708 | In a Hausdorff topology, t... |
| txhaus 23709 | The topological product of... |
| txlm 23710 | Two sequences converge iff... |
| lmcn2 23711 | The image of a convergent ... |
| tx1stc 23712 | The topological product of... |
| tx2ndc 23713 | The topological product of... |
| txkgen 23714 | The topological product of... |
| xkohaus 23715 | If the codomain space is H... |
| xkoptsub 23716 | The compact-open topology ... |
| xkopt 23717 | The compact-open topology ... |
| xkopjcn 23718 | Continuity of a projection... |
| xkoco1cn 23719 | If ` F ` is a continuous f... |
| xkoco2cn 23720 | If ` F ` is a continuous f... |
| xkococnlem 23721 | Continuity of the composit... |
| xkococn 23722 | Continuity of the composit... |
| cnmptid 23723 | The identity function is c... |
| cnmptc 23724 | A constant function is con... |
| cnmpt11 23725 | The composition of continu... |
| cnmpt11f 23726 | The composition of continu... |
| cnmpt1t 23727 | The composition of continu... |
| cnmpt12f 23728 | The composition of continu... |
| cnmpt12 23729 | The composition of continu... |
| cnmpt1st 23730 | The projection onto the fi... |
| cnmpt2nd 23731 | The projection onto the se... |
| cnmpt2c 23732 | A constant function is con... |
| cnmpt21 23733 | The composition of continu... |
| cnmpt21f 23734 | The composition of continu... |
| cnmpt2t 23735 | The composition of continu... |
| cnmpt22 23736 | The composition of continu... |
| cnmpt22f 23737 | The composition of continu... |
| cnmpt1res 23738 | The restriction of a conti... |
| cnmpt2res 23739 | The restriction of a conti... |
| cnmptcom 23740 | The argument converse of a... |
| cnmptkc 23741 | The curried first projecti... |
| cnmptkp 23742 | The evaluation of the inne... |
| cnmptk1 23743 | The composition of a curri... |
| cnmpt1k 23744 | The composition of a one-a... |
| cnmptkk 23745 | The composition of two cur... |
| xkofvcn 23746 | Joint continuity of the fu... |
| cnmptk1p 23747 | The evaluation of a currie... |
| cnmptk2 23748 | The uncurrying of a currie... |
| xkoinjcn 23749 | Continuity of "injection",... |
| cnmpt2k 23750 | The currying of a two-argu... |
| txconn 23751 | The topological product of... |
| imasnopn 23752 | If a relation graph is ope... |
| imasncld 23753 | If a relation graph is clo... |
| imasncls 23754 | If a relation graph is clo... |
| qtopval 23757 | Value of the quotient topo... |
| qtopval2 23758 | Value of the quotient topo... |
| elqtop 23759 | Value of the quotient topo... |
| qtopres 23760 | The quotient topology is u... |
| qtoptop2 23761 | The quotient topology is a... |
| qtoptop 23762 | The quotient topology is a... |
| elqtop2 23763 | Value of the quotient topo... |
| qtopuni 23764 | The base set of the quotie... |
| elqtop3 23765 | Value of the quotient topo... |
| qtoptopon 23766 | The base set of the quotie... |
| qtopid 23767 | A quotient map is a contin... |
| idqtop 23768 | The quotient topology indu... |
| qtopcmplem 23769 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23770 | A quotient of a compact sp... |
| qtopconn 23771 | A quotient of a connected ... |
| qtopkgen 23772 | A quotient of a compactly ... |
| basqtop 23773 | An injection maps bases to... |
| tgqtop 23774 | An injection maps generate... |
| qtopcld 23775 | The property of being a cl... |
| qtopcn 23776 | Universal property of a qu... |
| qtopss 23777 | A surjective continuous fu... |
| qtopeu 23778 | Universal property of the ... |
| qtoprest 23779 | If ` A ` is a saturated op... |
| qtopomap 23780 | If ` F ` is a surjective c... |
| qtopcmap 23781 | If ` F ` is a surjective c... |
| imastopn 23782 | The topology of an image s... |
| imastps 23783 | The image of a topological... |
| qustps 23784 | A quotient structure is a ... |
| kqfval 23785 | Value of the function appe... |
| kqfeq 23786 | Two points in the Kolmogor... |
| kqffn 23787 | The topological indistingu... |
| kqval 23788 | Value of the quotient topo... |
| kqtopon 23789 | The Kolmogorov quotient is... |
| kqid 23790 | The topological indistingu... |
| ist0-4 23791 | The topological indistingu... |
| kqfvima 23792 | When the image set is open... |
| kqsat 23793 | Any open set is saturated ... |
| kqdisj 23794 | A version of ~ imain for t... |
| kqcldsat 23795 | Any closed set is saturate... |
| kqopn 23796 | The topological indistingu... |
| kqcld 23797 | The topological indistingu... |
| kqt0lem 23798 | Lemma for ~ kqt0 . (Contr... |
| isr0 23799 | The property " ` J ` is an... |
| r0cld 23800 | The analogue of the T_1 ax... |
| regr1lem 23801 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23802 | A Kolmogorov quotient of a... |
| kqreglem1 23803 | A Kolmogorov quotient of a... |
| kqreglem2 23804 | If the Kolmogorov quotient... |
| kqnrmlem1 23805 | A Kolmogorov quotient of a... |
| kqnrmlem2 23806 | If the Kolmogorov quotient... |
| kqtop 23807 | The Kolmogorov quotient is... |
| kqt0 23808 | The Kolmogorov quotient is... |
| kqf 23809 | The Kolmogorov quotient is... |
| r0sep 23810 | The separation property of... |
| nrmr0reg 23811 | A normal R_0 space is also... |
| regr1 23812 | A regular space is R_1, wh... |
| kqreg 23813 | The Kolmogorov quotient of... |
| kqnrm 23814 | The Kolmogorov quotient of... |
| hmeofn 23819 | The set of homeomorphisms ... |
| hmeofval 23820 | The set of all the homeomo... |
| ishmeo 23821 | The predicate F is a homeo... |
| hmeocn 23822 | A homeomorphism is continu... |
| hmeocnvcn 23823 | The converse of a homeomor... |
| hmeocnv 23824 | The converse of a homeomor... |
| hmeof1o2 23825 | A homeomorphism is a 1-1-o... |
| hmeof1o 23826 | A homeomorphism is a 1-1-o... |
| hmeoima 23827 | The image of an open set b... |
| hmeoopn 23828 | Homeomorphisms preserve op... |
| hmeocld 23829 | Homeomorphisms preserve cl... |
| hmeocls 23830 | Homeomorphisms preserve cl... |
| hmeontr 23831 | Homeomorphisms preserve in... |
| hmeoimaf1o 23832 | The function mapping open ... |
| hmeores 23833 | The restriction of a homeo... |
| hmeoco 23834 | The composite of two homeo... |
| idhmeo 23835 | The identity function is a... |
| hmeocnvb 23836 | The converse of a homeomor... |
| hmeoqtop 23837 | A homeomorphism is a quoti... |
| hmph 23838 | Express the predicate ` J ... |
| hmphi 23839 | If there is a homeomorphis... |
| hmphtop 23840 | Reverse closure for the ho... |
| hmphtop1 23841 | The relation "being homeom... |
| hmphtop2 23842 | The relation "being homeom... |
| hmphref 23843 | "Is homeomorphic to" is re... |
| hmphsym 23844 | "Is homeomorphic to" is sy... |
| hmphtr 23845 | "Is homeomorphic to" is tr... |
| hmpher 23846 | "Is homeomorphic to" is an... |
| hmphen 23847 | Homeomorphisms preserve th... |
| hmphsymb 23848 | "Is homeomorphic to" is sy... |
| haushmphlem 23849 | Lemma for ~ haushmph and s... |
| cmphmph 23850 | Compactness is a topologic... |
| connhmph 23851 | Connectedness is a topolog... |
| t0hmph 23852 | T_0 is a topological prope... |
| t1hmph 23853 | T_1 is a topological prope... |
| haushmph 23854 | Hausdorff-ness is a topolo... |
| reghmph 23855 | Regularity is a topologica... |
| nrmhmph 23856 | Normality is a topological... |
| hmph0 23857 | A topology homeomorphic to... |
| hmphdis 23858 | Homeomorphisms preserve to... |
| hmphindis 23859 | Homeomorphisms preserve to... |
| indishmph 23860 | Equinumerous sets equipped... |
| hmphen2 23861 | Homeomorphisms preserve th... |
| cmphaushmeo 23862 | A continuous bijection fro... |
| ordthmeolem 23863 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23864 | An order isomorphism is a ... |
| txhmeo 23865 | Lift a pair of homeomorphi... |
| txswaphmeolem 23866 | Show inverse for the "swap... |
| txswaphmeo 23867 | There is a homeomorphism f... |
| pt1hmeo 23868 | The canonical homeomorphis... |
| ptuncnv 23869 | Exhibit the converse funct... |
| ptunhmeo 23870 | Define a homeomorphism fro... |
| xpstopnlem1 23871 | The function ` F ` used in... |
| xpstps 23872 | A binary product of topolo... |
| xpstopnlem2 23873 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23874 | The topology on a binary p... |
| ptcmpfi 23875 | A topological product of f... |
| xkocnv 23876 | The inverse of the "curryi... |
| xkohmeo 23877 | The Exponential Law for to... |
| qtopf1 23878 | If a quotient map is injec... |
| qtophmeo 23879 | If two functions on a base... |
| t0kq 23880 | A topological space is T_0... |
| kqhmph 23881 | A topological space is T_0... |
| ist1-5lem 23882 | Lemma for ~ ist1-5 and sim... |
| t1r0 23883 | A T_1 space is R_0. That ... |
| ist1-5 23884 | A topological space is T_1... |
| ishaus3 23885 | A topological space is Hau... |
| nrmreg 23886 | A normal T_1 space is regu... |
| reghaus 23887 | A regular T_0 space is Hau... |
| nrmhaus 23888 | A T_1 normal space is Haus... |
| elmptrab 23889 | Membership in a one-parame... |
| elmptrab2 23890 | Membership in a one-parame... |
| isfbas 23891 | The predicate " ` F ` is a... |
| fbasne0 23892 | There are no empty filter ... |
| 0nelfb 23893 | No filter base contains th... |
| fbsspw 23894 | A filter base on a set is ... |
| fbelss 23895 | An element of the filter b... |
| fbdmn0 23896 | The domain of a filter bas... |
| isfbas2 23897 | The predicate " ` F ` is a... |
| fbasssin 23898 | A filter base contains sub... |
| fbssfi 23899 | A filter base contains sub... |
| fbssint 23900 | A filter base contains sub... |
| fbncp 23901 | A filter base does not con... |
| fbun 23902 | A necessary and sufficient... |
| fbfinnfr 23903 | No filter base containing ... |
| opnfbas 23904 | The collection of open sup... |
| trfbas2 23905 | Conditions for the trace o... |
| trfbas 23906 | Conditions for the trace o... |
| isfil 23909 | The predicate "is a filter... |
| filfbas 23910 | A filter is a filter base.... |
| 0nelfil 23911 | The empty set doesn't belo... |
| fileln0 23912 | An element of a filter is ... |
| filsspw 23913 | A filter is a subset of th... |
| filelss 23914 | An element of a filter is ... |
| filss 23915 | A filter is closed under t... |
| filin 23916 | A filter is closed under t... |
| filtop 23917 | The underlying set belongs... |
| isfil2 23918 | Derive the standard axioms... |
| isfildlem 23919 | Lemma for ~ isfild . (Con... |
| isfild 23920 | Sufficient condition for a... |
| filfi 23921 | A filter is closed under t... |
| filinn0 23922 | The intersection of two el... |
| filintn0 23923 | A filter has the finite in... |
| filn0 23924 | The empty set is not a fil... |
| infil 23925 | The intersection of two fi... |
| snfil 23926 | A singleton is a filter. ... |
| fbasweak 23927 | A filter base on any set i... |
| snfbas 23928 | Condition for a singleton ... |
| fsubbas 23929 | A condition for a set to g... |
| fbasfip 23930 | A filter base has the fini... |
| fbunfip 23931 | A helpful lemma for showin... |
| fgval 23932 | The filter generating clas... |
| elfg 23933 | A condition for elements o... |
| ssfg 23934 | A filter base is a subset ... |
| fgss 23935 | A bigger base generates a ... |
| fgss2 23936 | A condition for a filter t... |
| fgfil 23937 | A filter generates itself.... |
| elfilss 23938 | An element belongs to a fi... |
| filfinnfr 23939 | No filter containing a fin... |
| fgcl 23940 | A generated filter is a fi... |
| fgabs 23941 | Absorption law for filter ... |
| neifil 23942 | The neighborhoods of a non... |
| filunibas 23943 | Recover the base set from ... |
| filunirn 23944 | Two ways to express a filt... |
| filconn 23945 | A filter gives rise to a c... |
| fbasrn 23946 | Given a filter on a domain... |
| filuni 23947 | The union of a nonempty se... |
| trfil1 23948 | Conditions for the trace o... |
| trfil2 23949 | Conditions for the trace o... |
| trfil3 23950 | Conditions for the trace o... |
| trfilss 23951 | If ` A ` is a member of th... |
| fgtr 23952 | If ` A ` is a member of th... |
| trfg 23953 | The trace operation and th... |
| trnei 23954 | The trace, over a set ` A ... |
| cfinfil 23955 | Relative complements of th... |
| csdfil 23956 | The set of all elements wh... |
| supfil 23957 | The supersets of a nonempt... |
| zfbas 23958 | The set of upper sets of i... |
| uzrest 23959 | The restriction of the set... |
| uzfbas 23960 | The set of upper sets of i... |
| isufil 23965 | The property of being an u... |
| ufilfil 23966 | An ultrafilter is a filter... |
| ufilss 23967 | For any subset of the base... |
| ufilb 23968 | The complement is in an ul... |
| ufilmax 23969 | Any filter finer than an u... |
| isufil2 23970 | The maximal property of an... |
| ufprim 23971 | An ultrafilter is a prime ... |
| trufil 23972 | Conditions for the trace o... |
| filssufilg 23973 | A filter is contained in s... |
| filssufil 23974 | A filter is contained in s... |
| isufl 23975 | Define the (strong) ultraf... |
| ufli 23976 | Property of a set that sat... |
| numufl 23977 | Consequence of ~ filssufil... |
| fiufl 23978 | A finite set satisfies the... |
| acufl 23979 | The axiom of choice implie... |
| ssufl 23980 | If ` Y ` is a subset of ` ... |
| ufileu 23981 | If the ultrafilter contain... |
| filufint 23982 | A filter is equal to the i... |
| uffix 23983 | Lemma for ~ fixufil and ~ ... |
| fixufil 23984 | The condition describing a... |
| uffixfr 23985 | An ultrafilter is either f... |
| uffix2 23986 | A classification of fixed ... |
| uffixsn 23987 | The singleton of the gener... |
| ufildom1 23988 | An ultrafilter is generate... |
| uffinfix 23989 | An ultrafilter containing ... |
| cfinufil 23990 | An ultrafilter is free iff... |
| ufinffr 23991 | An infinite subset is cont... |
| ufilen 23992 | Any infinite set has an ul... |
| ufildr 23993 | An ultrafilter gives rise ... |
| fin1aufil 23994 | There are no definable fre... |
| fmval 24005 | Introduce a function that ... |
| fmfil 24006 | A mapping filter is a filt... |
| fmf 24007 | Pushing-forward via a func... |
| fmss 24008 | A finer filter produces a ... |
| elfm 24009 | An element of a mapping fi... |
| elfm2 24010 | An element of a mapping fi... |
| fmfg 24011 | The image filter of a filt... |
| elfm3 24012 | An alternate formulation o... |
| imaelfm 24013 | An image of a filter eleme... |
| rnelfmlem 24014 | Lemma for ~ rnelfm . (Con... |
| rnelfm 24015 | A condition for a filter t... |
| fmfnfmlem1 24016 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 24017 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 24018 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 24019 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 24020 | A filter finer than an ima... |
| fmufil 24021 | An image filter of an ultr... |
| fmid 24022 | The filter map applied to ... |
| fmco 24023 | Composition of image filte... |
| ufldom 24024 | The ultrafilter lemma prop... |
| flimval 24025 | The set of limit points of... |
| elflim2 24026 | The predicate "is a limit ... |
| flimtop 24027 | Reverse closure for the li... |
| flimneiss 24028 | A filter contains the neig... |
| flimnei 24029 | A filter contains all of t... |
| flimelbas 24030 | A limit point of a filter ... |
| flimfil 24031 | Reverse closure for the li... |
| flimtopon 24032 | Reverse closure for the li... |
| elflim 24033 | The predicate "is a limit ... |
| flimss2 24034 | A limit point of a filter ... |
| flimss1 24035 | A limit point of a filter ... |
| neiflim 24036 | A point is a limit point o... |
| flimopn 24037 | The condition for being a ... |
| fbflim 24038 | A condition for a filter t... |
| fbflim2 24039 | A condition for a filter b... |
| flimclsi 24040 | The convergent points of a... |
| hausflimlem 24041 | If ` A ` and ` B ` are bot... |
| hausflimi 24042 | One direction of ~ hausfli... |
| hausflim 24043 | A condition for a topology... |
| flimcf 24044 | Fineness is properly chara... |
| flimrest 24045 | The set of limit points in... |
| flimclslem 24046 | Lemma for ~ flimcls . (Co... |
| flimcls 24047 | Closure in terms of filter... |
| flimsncls 24048 | If ` A ` is a limit point ... |
| hauspwpwf1 24049 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 24050 | If ` X ` is a Hausdorff sp... |
| flffval 24051 | Given a topology and a fil... |
| flfval 24052 | Given a function from a fi... |
| flfnei 24053 | The property of being a li... |
| flfneii 24054 | A neighborhood of a limit ... |
| isflf 24055 | The property of being a li... |
| flfelbas 24056 | A limit point of a functio... |
| flffbas 24057 | Limit points of a function... |
| flftg 24058 | Limit points of a function... |
| hausflf 24059 | If a function has its valu... |
| hausflf2 24060 | If a convergent function h... |
| cnpflfi 24061 | Forward direction of ~ cnp... |
| cnpflf2 24062 | ` F ` is continuous at poi... |
| cnpflf 24063 | Continuity of a function a... |
| cnflf 24064 | A function is continuous i... |
| cnflf2 24065 | A function is continuous i... |
| flfcnp 24066 | A continuous function pres... |
| lmflf 24067 | The topological limit rela... |
| txflf 24068 | Two sequences converge in ... |
| flfcnp2 24069 | The image of a convergent ... |
| fclsval 24070 | The set of all cluster poi... |
| isfcls 24071 | A cluster point of a filte... |
| fclsfil 24072 | Reverse closure for the cl... |
| fclstop 24073 | Reverse closure for the cl... |
| fclstopon 24074 | Reverse closure for the cl... |
| isfcls2 24075 | A cluster point of a filte... |
| fclsopn 24076 | Write the cluster point co... |
| fclsopni 24077 | An open neighborhood of a ... |
| fclselbas 24078 | A cluster point is in the ... |
| fclsneii 24079 | A neighborhood of a cluste... |
| fclssscls 24080 | The set of cluster points ... |
| fclsnei 24081 | Cluster points in terms of... |
| supnfcls 24082 | The filter of supersets of... |
| fclsbas 24083 | Cluster points in terms of... |
| fclsss1 24084 | A finer topology has fewer... |
| fclsss2 24085 | A finer filter has fewer c... |
| fclsrest 24086 | The set of cluster points ... |
| fclscf 24087 | Characterization of finene... |
| flimfcls 24088 | A limit point is a cluster... |
| fclsfnflim 24089 | A filter clusters at a poi... |
| flimfnfcls 24090 | A filter converges to a po... |
| fclscmpi 24091 | Forward direction of ~ fcl... |
| fclscmp 24092 | A space is compact iff eve... |
| uffclsflim 24093 | The cluster points of an u... |
| ufilcmp 24094 | A space is compact iff eve... |
| fcfval 24095 | The set of cluster points ... |
| isfcf 24096 | The property of being a cl... |
| fcfnei 24097 | The property of being a cl... |
| fcfelbas 24098 | A cluster point of a funct... |
| fcfneii 24099 | A neighborhood of a cluste... |
| flfssfcf 24100 | A limit point of a functio... |
| uffcfflf 24101 | If the domain filter is an... |
| cnpfcfi 24102 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 24103 | A function ` F ` is contin... |
| cnfcf 24104 | Continuity of a function i... |
| flfcntr 24105 | A continuous function's va... |
| alexsublem 24106 | Lemma for ~ alexsub . (Co... |
| alexsub 24107 | The Alexander Subbase Theo... |
| alexsubb 24108 | Biconditional form of the ... |
| alexsubALTlem1 24109 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 24110 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 24111 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 24112 | Lemma for ~ alexsubALT . ... |
| alexsubALT 24113 | The Alexander Subbase Theo... |
| ptcmplem1 24114 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 24115 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 24116 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 24117 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 24118 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 24119 | Tychonoff's theorem: The ... |
| ptcmp 24120 | Tychonoff's theorem: The ... |
| cnextval 24123 | The function applying cont... |
| cnextfval 24124 | The continuous extension o... |
| cnextrel 24125 | In the general case, a con... |
| cnextfun 24126 | If the target space is Hau... |
| cnextfvval 24127 | The value of the continuou... |
| cnextf 24128 | Extension by continuity. ... |
| cnextcn 24129 | Extension by continuity. ... |
| cnextfres1 24130 | ` F ` and its extension by... |
| cnextfres 24131 | ` F ` and its extension by... |
| istmd 24136 | The predicate "is a topolo... |
| tmdmnd 24137 | A topological monoid is a ... |
| tmdtps 24138 | A topological monoid is a ... |
| istgp 24139 | The predicate "is a topolo... |
| tgpgrp 24140 | A topological group is a g... |
| tgptmd 24141 | A topological group is a t... |
| tgptps 24142 | A topological group is a t... |
| tmdtopon 24143 | The topology of a topologi... |
| tgptopon 24144 | The topology of a topologi... |
| tmdcn 24145 | In a topological monoid, t... |
| tgpcn 24146 | In a topological group, th... |
| tgpinv 24147 | In a topological group, th... |
| grpinvhmeo 24148 | The inverse function in a ... |
| cnmpt1plusg 24149 | Continuity of the group su... |
| cnmpt2plusg 24150 | Continuity of the group su... |
| tmdcn2 24151 | Write out the definition o... |
| tgpsubcn 24152 | In a topological group, th... |
| istgp2 24153 | A group with a topology is... |
| tmdmulg 24154 | In a topological monoid, t... |
| tgpmulg 24155 | In a topological group, th... |
| tgpmulg2 24156 | In a topological monoid, t... |
| tmdgsum 24157 | In a topological monoid, t... |
| tmdgsum2 24158 | For any neighborhood ` U `... |
| oppgtmd 24159 | The opposite of a topologi... |
| oppgtgp 24160 | The opposite of a topologi... |
| distgp 24161 | Any group equipped with th... |
| indistgp 24162 | Any group equipped with th... |
| efmndtmd 24163 | The monoid of endofunction... |
| tmdlactcn 24164 | The left group action of e... |
| tgplacthmeo 24165 | The left group action of e... |
| submtmd 24166 | A submonoid of a topologic... |
| subgtgp 24167 | A subgroup of a topologica... |
| symgtgp 24168 | The symmetric group is a t... |
| subgntr 24169 | A subgroup of a topologica... |
| opnsubg 24170 | An open subgroup of a topo... |
| clssubg 24171 | The closure of a subgroup ... |
| clsnsg 24172 | The closure of a normal su... |
| cldsubg 24173 | A subgroup of finite index... |
| tgpconncompeqg 24174 | The connected component co... |
| tgpconncomp 24175 | The identity component, th... |
| tgpconncompss 24176 | The identity component is ... |
| ghmcnp 24177 | A group homomorphism on to... |
| snclseqg 24178 | The coset of the closure o... |
| tgphaus 24179 | A topological group is Hau... |
| tgpt1 24180 | Hausdorff and T1 are equiv... |
| tgpt0 24181 | Hausdorff and T0 are equiv... |
| qustgpopn 24182 | A quotient map in a topolo... |
| qustgplem 24183 | Lemma for ~ qustgp . (Con... |
| qustgp 24184 | The quotient of a topologi... |
| qustgphaus 24185 | The quotient of a topologi... |
| prdstmdd 24186 | The product of a family of... |
| prdstgpd 24187 | The product of a family of... |
| tsmsfbas 24190 | The collection of all sets... |
| tsmslem1 24191 | The finite partial sums of... |
| tsmsval2 24192 | Definition of the topologi... |
| tsmsval 24193 | Definition of the topologi... |
| tsmspropd 24194 | The group sum depends only... |
| eltsms 24195 | The property of being a su... |
| tsmsi 24196 | The property of being a su... |
| tsmscl 24197 | A sum in a topological gro... |
| haustsms 24198 | In a Hausdorff topological... |
| haustsms2 24199 | In a Hausdorff topological... |
| tsmscls 24200 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24201 | The convergent points of a... |
| tsmsid 24202 | If a sum is finite, the us... |
| haustsmsid 24203 | In a Hausdorff topological... |
| tsms0 24204 | The sum of zero is zero. ... |
| tsmssubm 24205 | Evaluate an infinite group... |
| tsmsres 24206 | Extend an infinite group s... |
| tsmsf1o 24207 | Re-index an infinite group... |
| tsmsmhm 24208 | Apply a continuous group h... |
| tsmsadd 24209 | The sum of two infinite gr... |
| tsmsinv 24210 | Inverse of an infinite gro... |
| tsmssub 24211 | The difference of two infi... |
| tgptsmscls 24212 | A sum in a topological gro... |
| tgptsmscld 24213 | The set of limit points to... |
| tsmssplit 24214 | Split a topological group ... |
| tsmsxplem1 24215 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24216 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24217 | Write a sum over a two-dim... |
| istrg 24226 | Express the predicate " ` ... |
| trgtmd 24227 | The multiplicative monoid ... |
| istdrg 24228 | Express the predicate " ` ... |
| tdrgunit 24229 | The unit group of a topolo... |
| trgtgp 24230 | A topological ring is a to... |
| trgtmd2 24231 | A topological ring is a to... |
| trgtps 24232 | A topological ring is a to... |
| trgring 24233 | A topological ring is a ri... |
| trggrp 24234 | A topological ring is a gr... |
| tdrgtrg 24235 | A topological division rin... |
| tdrgdrng 24236 | A topological division rin... |
| tdrgring 24237 | A topological division rin... |
| tdrgtmd 24238 | A topological division rin... |
| tdrgtps 24239 | A topological division rin... |
| istdrg2 24240 | A topological-ring divisio... |
| mulrcn 24241 | The functionalization of t... |
| invrcn2 24242 | The multiplicative inverse... |
| invrcn 24243 | The multiplicative inverse... |
| cnmpt1mulr 24244 | Continuity of ring multipl... |
| cnmpt2mulr 24245 | Continuity of ring multipl... |
| dvrcn 24246 | The division function is c... |
| istlm 24247 | The predicate " ` W ` is a... |
| vscacn 24248 | The scalar multiplication ... |
| tlmtmd 24249 | A topological module is a ... |
| tlmtps 24250 | A topological module is a ... |
| tlmlmod 24251 | A topological module is a ... |
| tlmtrg 24252 | The scalar ring of a topol... |
| tlmscatps 24253 | The scalar ring of a topol... |
| istvc 24254 | A topological vector space... |
| tvctdrg 24255 | The scalar field of a topo... |
| cnmpt1vsca 24256 | Continuity of scalar multi... |
| cnmpt2vsca 24257 | Continuity of scalar multi... |
| tlmtgp 24258 | A topological vector space... |
| tvctlm 24259 | A topological vector space... |
| tvclmod 24260 | A topological vector space... |
| tvclvec 24261 | A topological vector space... |
| ustfn 24264 | The defined uniform struct... |
| ustval 24265 | The class of all uniform s... |
| isust 24266 | The predicate " ` U ` is a... |
| ustssxp 24267 | Entourages are subsets of ... |
| ustssel 24268 | A uniform structure is upw... |
| ustbasel 24269 | The full set is always an ... |
| ustincl 24270 | A uniform structure is clo... |
| ustdiag 24271 | The diagonal set is includ... |
| ustinvel 24272 | If ` V ` is an entourage, ... |
| ustexhalf 24273 | For each entourage ` V ` t... |
| ustrel 24274 | The elements of uniform st... |
| ustfilxp 24275 | A uniform structure on a n... |
| ustne0 24276 | A uniform structure cannot... |
| ustssco 24277 | In an uniform structure, a... |
| ustexsym 24278 | In an uniform structure, f... |
| ustex2sym 24279 | In an uniform structure, f... |
| ustex3sym 24280 | In an uniform structure, f... |
| ustref 24281 | Any element of the base se... |
| ust0 24282 | The unique uniform structu... |
| ustn0 24283 | The empty set is not an un... |
| ustund 24284 | If two intersecting sets `... |
| ustelimasn 24285 | Any point ` A ` is near en... |
| ustneism 24286 | For a point ` A ` in ` X `... |
| ustbas2 24287 | Second direction for ~ ust... |
| ustuni 24288 | The set union of a uniform... |
| ustbas 24289 | Recover the base of an uni... |
| ustimasn 24290 | Lemma for ~ ustuqtop . (C... |
| trust 24291 | The trace of a uniform str... |
| utopval 24294 | The topology induced by a ... |
| elutop 24295 | Open sets in the topology ... |
| utoptop 24296 | The topology induced by a ... |
| utopbas 24297 | The base of the topology i... |
| utoptopon 24298 | Topology induced by a unif... |
| restutop 24299 | Restriction of a topology ... |
| restutopopn 24300 | The restriction of the top... |
| ustuqtoplem 24301 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24302 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24303 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24304 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24305 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24306 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24307 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24308 | For a given uniform struct... |
| utopsnneiplem 24309 | The neighborhoods of a poi... |
| utopsnneip 24310 | The neighborhoods of a poi... |
| utopsnnei 24311 | Images of singletons by en... |
| utop2nei 24312 | For any symmetrical entour... |
| utop3cls 24313 | Relation between a topolog... |
| utopreg 24314 | All Hausdorff uniform spac... |
| ussval 24321 | The uniform structure on u... |
| ussid 24322 | In case the base of the ` ... |
| isusp 24323 | The predicate ` W ` is a u... |
| ressuss 24324 | Value of the uniform struc... |
| ressust 24325 | The uniform structure of a... |
| ressusp 24326 | The restriction of a unifo... |
| tusval 24327 | The value of the uniform s... |
| tuslem 24328 | Lemma for ~ tusbas , ~ tus... |
| tusbas 24329 | The base set of a construc... |
| tusunif 24330 | The uniform structure of a... |
| tususs 24331 | The uniform structure of a... |
| tustopn 24332 | The topology induced by a ... |
| tususp 24333 | A constructed uniform spac... |
| tustps 24334 | A constructed uniform spac... |
| uspreg 24335 | If a uniform space is Haus... |
| ucnval 24338 | The set of all uniformly c... |
| isucn 24339 | The predicate " ` F ` is a... |
| isucn2 24340 | The predicate " ` F ` is a... |
| ucnimalem 24341 | Reformulate the ` G ` func... |
| ucnima 24342 | An equivalent statement of... |
| ucnprima 24343 | The preimage by a uniforml... |
| iducn 24344 | The identity is uniformly ... |
| cstucnd 24345 | A constant function is uni... |
| ucncn 24346 | Uniform continuity implies... |
| iscfilu 24349 | The predicate " ` F ` is a... |
| cfilufbas 24350 | A Cauchy filter base is a ... |
| cfiluexsm 24351 | For a Cauchy filter base a... |
| fmucndlem 24352 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24353 | The image of a Cauchy filt... |
| cfilufg 24354 | The filter generated by a ... |
| trcfilu 24355 | Condition for the trace of... |
| cfiluweak 24356 | A Cauchy filter base is al... |
| neipcfilu 24357 | In an uniform space, a nei... |
| iscusp 24360 | The predicate " ` W ` is a... |
| cuspusp 24361 | A complete uniform space i... |
| cuspcvg 24362 | In a complete uniform spac... |
| iscusp2 24363 | The predicate " ` W ` is a... |
| cnextucn 24364 | Extension by continuity. ... |
| ucnextcn 24365 | Extension by continuity. ... |
| ispsmet 24366 | Express the predicate " ` ... |
| psmetdmdm 24367 | Recover the base set from ... |
| psmetf 24368 | The distance function of a... |
| psmetcl 24369 | Closure of the distance fu... |
| psmet0 24370 | The distance function of a... |
| psmettri2 24371 | Triangle inequality for th... |
| psmetsym 24372 | The distance function of a... |
| psmettri 24373 | Triangle inequality for th... |
| psmetge0 24374 | The distance function of a... |
| psmetxrge0 24375 | The distance function of a... |
| psmetres2 24376 | Restriction of a pseudomet... |
| psmetlecl 24377 | Real closure of an extende... |
| distspace 24378 | A set ` X ` together with ... |
| ismet 24385 | Express the predicate " ` ... |
| isxmet 24386 | Express the predicate " ` ... |
| ismeti 24387 | Properties that determine ... |
| isxmetd 24388 | Properties that determine ... |
| isxmet2d 24389 | It is safe to only require... |
| metflem 24390 | Lemma for ~ metf and other... |
| xmetf 24391 | Mapping of the distance fu... |
| metf 24392 | Mapping of the distance fu... |
| xmetcl 24393 | Closure of the distance fu... |
| metcl 24394 | Closure of the distance fu... |
| ismet2 24395 | An extended metric is a me... |
| metxmet 24396 | A metric is an extended me... |
| xmetdmdm 24397 | Recover the base set from ... |
| metdmdm 24398 | Recover the base set from ... |
| xmetunirn 24399 | Two ways to express an ext... |
| xmeteq0 24400 | The value of an extended m... |
| meteq0 24401 | The value of a metric is z... |
| xmettri2 24402 | Triangle inequality for th... |
| mettri2 24403 | Triangle inequality for th... |
| xmet0 24404 | The distance function of a... |
| met0 24405 | The distance function of a... |
| xmetge0 24406 | The distance function of a... |
| metge0 24407 | The distance function of a... |
| xmetlecl 24408 | Real closure of an extende... |
| xmetsym 24409 | The distance function of a... |
| xmetpsmet 24410 | An extended metric is a ps... |
| xmettpos 24411 | The distance function of a... |
| metsym 24412 | The distance function of a... |
| xmettri 24413 | Triangle inequality for th... |
| mettri 24414 | Triangle inequality for th... |
| xmettri3 24415 | Triangle inequality for th... |
| mettri3 24416 | Triangle inequality for th... |
| xmetrtri 24417 | One half of the reverse tr... |
| xmetrtri2 24418 | The reverse triangle inequ... |
| metrtri 24419 | Reverse triangle inequalit... |
| xmetgt0 24420 | The distance function of a... |
| metgt0 24421 | The distance function of a... |
| metn0 24422 | A metric space is nonempty... |
| xmetres2 24423 | Restriction of an extended... |
| metreslem 24424 | Lemma for ~ metres . (Con... |
| metres2 24425 | Lemma for ~ metres . (Con... |
| xmetres 24426 | A restriction of an extend... |
| metres 24427 | A restriction of a metric ... |
| 0met 24428 | The empty metric. (Contri... |
| prdsdsf 24429 | The product metric is a fu... |
| prdsxmetlem 24430 | The product metric is an e... |
| prdsxmet 24431 | The product metric is an e... |
| prdsmet 24432 | The product metric is a me... |
| ressprdsds 24433 | Restriction of a product m... |
| resspwsds 24434 | Restriction of a power met... |
| imasdsf1olem 24435 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24436 | The distance function is t... |
| imasf1oxmet 24437 | The image of an extended m... |
| imasf1omet 24438 | The image of a metric is a... |
| xpsdsfn 24439 | Closure of the metric in a... |
| xpsdsfn2 24440 | Closure of the metric in a... |
| xpsxmetlem 24441 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24442 | A product metric of extend... |
| xpsdsval 24443 | Value of the metric in a b... |
| xpsmet 24444 | The direct product of two ... |
| blfvalps 24445 | The value of the ball func... |
| blfval 24446 | The value of the ball func... |
| blvalps 24447 | The ball around a point ` ... |
| blval 24448 | The ball around a point ` ... |
| elblps 24449 | Membership in a ball. (Co... |
| elbl 24450 | Membership in a ball. (Co... |
| elbl2ps 24451 | Membership in a ball. (Co... |
| elbl2 24452 | Membership in a ball. (Co... |
| elbl3ps 24453 | Membership in a ball, with... |
| elbl3 24454 | Membership in a ball, with... |
| blcomps 24455 | Commute the arguments to t... |
| blcom 24456 | Commute the arguments to t... |
| xblpnfps 24457 | The infinity ball in an ex... |
| xblpnf 24458 | The infinity ball in an ex... |
| blpnf 24459 | The infinity ball in a sta... |
| bldisj 24460 | Two balls are disjoint if ... |
| blgt0 24461 | A nonempty ball implies th... |
| bl2in 24462 | Two balls are disjoint if ... |
| xblss2ps 24463 | One ball is contained in a... |
| xblss2 24464 | One ball is contained in a... |
| blss2ps 24465 | One ball is contained in a... |
| blss2 24466 | One ball is contained in a... |
| blhalf 24467 | A ball of radius ` R / 2 `... |
| blfps 24468 | Mapping of a ball. (Contr... |
| blf 24469 | Mapping of a ball. (Contr... |
| blrnps 24470 | Membership in the range of... |
| blrn 24471 | Membership in the range of... |
| xblcntrps 24472 | A ball contains its center... |
| xblcntr 24473 | A ball contains its center... |
| blcntrps 24474 | A ball contains its center... |
| blcntr 24475 | A ball contains its center... |
| xbln0 24476 | A ball is nonempty iff the... |
| bln0 24477 | A ball is not empty. (Con... |
| blelrnps 24478 | A ball belongs to the set ... |
| blelrn 24479 | A ball belongs to the set ... |
| blssm 24480 | A ball is a subset of the ... |
| unirnblps 24481 | The union of the set of ba... |
| unirnbl 24482 | The union of the set of ba... |
| blin 24483 | The intersection of two ba... |
| ssblps 24484 | The size of a ball increas... |
| ssbl 24485 | The size of a ball increas... |
| blssps 24486 | Any point ` P ` in a ball ... |
| blss 24487 | Any point ` P ` in a ball ... |
| blssexps 24488 | Two ways to express the ex... |
| blssex 24489 | Two ways to express the ex... |
| ssblex 24490 | A nested ball exists whose... |
| blin2 24491 | Given any two balls and a ... |
| blbas 24492 | The balls of a metric spac... |
| blres 24493 | A ball in a restricted met... |
| xmeterval 24494 | Value of the "finitely sep... |
| xmeter 24495 | The "finitely separated" r... |
| xmetec 24496 | The equivalence classes un... |
| blssec 24497 | A ball centered at ` P ` i... |
| blpnfctr 24498 | The infinity ball in an ex... |
| xmetresbl 24499 | An extended metric restric... |
| mopnval 24500 | An open set is a subset of... |
| mopntopon 24501 | The set of open sets of a ... |
| mopntop 24502 | The set of open sets of a ... |
| mopnuni 24503 | The union of all open sets... |
| elmopn 24504 | The defining property of a... |
| mopnfss 24505 | The family of open sets of... |
| mopnm 24506 | The base set of a metric s... |
| elmopn2 24507 | A defining property of an ... |
| mopnss 24508 | An open set of a metric sp... |
| isxms 24509 | Express the predicate " ` ... |
| isxms2 24510 | Express the predicate " ` ... |
| isms 24511 | Express the predicate " ` ... |
| isms2 24512 | Express the predicate " ` ... |
| xmstopn 24513 | The topology component of ... |
| mstopn 24514 | The topology component of ... |
| xmstps 24515 | An extended metric space i... |
| msxms 24516 | A metric space is an exten... |
| mstps 24517 | A metric space is a topolo... |
| xmsxmet 24518 | The distance function, sui... |
| msmet 24519 | The distance function, sui... |
| msf 24520 | The distance function of a... |
| xmsxmet2 24521 | The distance function, sui... |
| msmet2 24522 | The distance function, sui... |
| mscl 24523 | Closure of the distance fu... |
| xmscl 24524 | Closure of the distance fu... |
| xmsge0 24525 | The distance function in a... |
| xmseq0 24526 | The distance between two p... |
| xmssym 24527 | The distance function in a... |
| xmstri2 24528 | Triangle inequality for th... |
| mstri2 24529 | Triangle inequality for th... |
| xmstri 24530 | Triangle inequality for th... |
| mstri 24531 | Triangle inequality for th... |
| xmstri3 24532 | Triangle inequality for th... |
| mstri3 24533 | Triangle inequality for th... |
| msrtri 24534 | Reverse triangle inequalit... |
| xmspropd 24535 | Property deduction for an ... |
| mspropd 24536 | Property deduction for a m... |
| setsmsbas 24537 | The base set of a construc... |
| setsmsds 24538 | The distance function of a... |
| setsmstset 24539 | The topology of a construc... |
| setsmstopn 24540 | The topology of a construc... |
| setsxms 24541 | The constructed metric spa... |
| setsms 24542 | The constructed metric spa... |
| tmsval 24543 | For any metric there is an... |
| tmslem 24544 | Lemma for ~ tmsbas , ~ tms... |
| tmsbas 24545 | The base set of a construc... |
| tmsds 24546 | The metric of a constructe... |
| tmstopn 24547 | The topology of a construc... |
| tmsxms 24548 | The constructed metric spa... |
| tmsms 24549 | The constructed metric spa... |
| imasf1obl 24550 | The image of a metric spac... |
| imasf1oxms 24551 | The image of a metric spac... |
| imasf1oms 24552 | The image of a metric spac... |
| prdsbl 24553 | A ball in the product metr... |
| mopni 24554 | An open set of a metric sp... |
| mopni2 24555 | An open set of a metric sp... |
| mopni3 24556 | An open set of a metric sp... |
| blssopn 24557 | The balls of a metric spac... |
| unimopn 24558 | The union of a collection ... |
| mopnin 24559 | The intersection of two op... |
| mopn0 24560 | The empty set is an open s... |
| rnblopn 24561 | A ball of a metric space i... |
| blopn 24562 | A ball of a metric space i... |
| neibl 24563 | The neighborhoods around a... |
| blnei 24564 | A ball around a point is a... |
| lpbl 24565 | Every ball around a limit ... |
| blsscls2 24566 | A smaller closed ball is c... |
| blcld 24567 | A "closed ball" in a metri... |
| blcls 24568 | The closure of an open bal... |
| blsscls 24569 | If two concentric balls ha... |
| metss 24570 | Two ways of saying that me... |
| metequiv 24571 | Two ways of saying that tw... |
| metequiv2 24572 | If there is a sequence of ... |
| metss2lem 24573 | Lemma for ~ metss2 . (Con... |
| metss2 24574 | If the metric ` D ` is "st... |
| comet 24575 | The composition of an exte... |
| stdbdmetval 24576 | Value of the standard boun... |
| stdbdxmet 24577 | The standard bounded metri... |
| stdbdmet 24578 | The standard bounded metri... |
| stdbdbl 24579 | The standard bounded metri... |
| stdbdmopn 24580 | The standard bounded metri... |
| mopnex 24581 | The topology generated by ... |
| methaus 24582 | The topology generated by ... |
| met1stc 24583 | The topology generated by ... |
| met2ndci 24584 | A separable metric space (... |
| met2ndc 24585 | A metric space is second-c... |
| metrest 24586 | Two alternate formulations... |
| ressxms 24587 | The restriction of a metri... |
| ressms 24588 | The restriction of a metri... |
| prdsmslem1 24589 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24590 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24591 | Lemma for ~ prdsxms . The... |
| prdsxms 24592 | The indexed product struct... |
| prdsms 24593 | The indexed product struct... |
| pwsxms 24594 | A power of an extended met... |
| pwsms 24595 | A power of a metric space ... |
| xpsxms 24596 | A binary product of metric... |
| xpsms 24597 | A binary product of metric... |
| tmsxps 24598 | Express the product of two... |
| tmsxpsmopn 24599 | Express the product of two... |
| tmsxpsval 24600 | Value of the product of tw... |
| tmsxpsval2 24601 | Value of the product of tw... |
| metcnp3 24602 | Two ways to express that `... |
| metcnp 24603 | Two ways to say a mapping ... |
| metcnp2 24604 | Two ways to say a mapping ... |
| metcn 24605 | Two ways to say a mapping ... |
| metcnpi 24606 | Epsilon-delta property of ... |
| metcnpi2 24607 | Epsilon-delta property of ... |
| metcnpi3 24608 | Epsilon-delta property of ... |
| txmetcnp 24609 | Continuity of a binary ope... |
| txmetcn 24610 | Continuity of a binary ope... |
| metuval 24611 | Value of the uniform struc... |
| metustel 24612 | Define a filter base ` F `... |
| metustss 24613 | Range of the elements of t... |
| metustrel 24614 | Elements of the filter bas... |
| metustto 24615 | Any two elements of the fi... |
| metustid 24616 | The identity diagonal is i... |
| metustsym 24617 | Elements of the filter bas... |
| metustexhalf 24618 | For any element ` A ` of t... |
| metustfbas 24619 | The filter base generated ... |
| metust 24620 | The uniform structure gene... |
| cfilucfil 24621 | Given a metric ` D ` and a... |
| metuust 24622 | The uniform structure gene... |
| cfilucfil2 24623 | Given a metric ` D ` and a... |
| blval2 24624 | The ball around a point ` ... |
| elbl4 24625 | Membership in a ball, alte... |
| metuel 24626 | Elementhood in the uniform... |
| metuel2 24627 | Elementhood in the uniform... |
| metustbl 24628 | The "section" image of an ... |
| psmetutop 24629 | The topology induced by a ... |
| xmetutop 24630 | The topology induced by a ... |
| xmsusp 24631 | If the uniform set of a me... |
| restmetu 24632 | The uniform structure gene... |
| metucn 24633 | Uniform continuity in metr... |
| dscmet 24634 | The discrete metric on any... |
| dscopn 24635 | The discrete metric genera... |
| nrmmetd 24636 | Show that a group norm gen... |
| abvmet 24637 | An absolute value ` F ` ge... |
| nmfval 24650 | The value of the norm func... |
| nmval 24651 | The value of the norm as t... |
| nmfval0 24652 | The value of the norm func... |
| nmfval2 24653 | The value of the norm func... |
| nmval2 24654 | The value of the norm on a... |
| nmf2 24655 | The norm on a metric group... |
| nmpropd 24656 | Weak property deduction fo... |
| nmpropd2 24657 | Strong property deduction ... |
| isngp 24658 | The property of being a no... |
| isngp2 24659 | The property of being a no... |
| isngp3 24660 | The property of being a no... |
| ngpgrp 24661 | A normed group is a group.... |
| ngpms 24662 | A normed group is a metric... |
| ngpxms 24663 | A normed group is an exten... |
| ngptps 24664 | A normed group is a topolo... |
| ngpmet 24665 | The (induced) metric of a ... |
| ngpds 24666 | Value of the distance func... |
| ngpdsr 24667 | Value of the distance func... |
| ngpds2 24668 | Write the distance between... |
| ngpds2r 24669 | Write the distance between... |
| ngpds3 24670 | Write the distance between... |
| ngpds3r 24671 | Write the distance between... |
| ngprcan 24672 | Cancel right addition insi... |
| ngplcan 24673 | Cancel left addition insid... |
| isngp4 24674 | Express the property of be... |
| ngpinvds 24675 | Two elements are the same ... |
| ngpsubcan 24676 | Cancel right subtraction i... |
| nmf 24677 | The norm on a normed group... |
| nmcl 24678 | The norm of a normed group... |
| nmge0 24679 | The norm of a normed group... |
| nmeq0 24680 | The identity is the only e... |
| nmne0 24681 | The norm of a nonzero elem... |
| nmrpcl 24682 | The norm of a nonzero elem... |
| nminv 24683 | The norm of a negated elem... |
| nmmtri 24684 | The triangle inequality fo... |
| nmsub 24685 | The norm of the difference... |
| nmrtri 24686 | Reverse triangle inequalit... |
| nm2dif 24687 | Inequality for the differe... |
| nmtri 24688 | The triangle inequality fo... |
| nmtri2 24689 | Triangle inequality for th... |
| ngpi 24690 | The properties of a normed... |
| nm0 24691 | Norm of the identity eleme... |
| nmgt0 24692 | The norm of a nonzero elem... |
| sgrim 24693 | The induced metric on a su... |
| sgrimval 24694 | The induced metric on a su... |
| subgnm 24695 | The norm in a subgroup. (... |
| subgnm2 24696 | A substructure assigns the... |
| subgngp 24697 | A normed group restricted ... |
| ngptgp 24698 | A normed abelian group is ... |
| ngppropd 24699 | Property deduction for a n... |
| reldmtng 24700 | The function ` toNrmGrp ` ... |
| tngval 24701 | Value of the function whic... |
| tnglem 24702 | Lemma for ~ tngbas and sim... |
| tngbas 24703 | The base set of a structur... |
| tngplusg 24704 | The group addition of a st... |
| tng0 24705 | The group identity of a st... |
| tngmulr 24706 | The ring multiplication of... |
| tngsca 24707 | The scalar ring of a struc... |
| tngvsca 24708 | The scalar multiplication ... |
| tngip 24709 | The inner product operatio... |
| tngds 24710 | The metric function of a s... |
| tngtset 24711 | The topology generated by ... |
| tngtopn 24712 | The topology generated by ... |
| tngnm 24713 | The topology generated by ... |
| tngngp2 24714 | A norm turns a group into ... |
| tngngpd 24715 | Derive the axioms for a no... |
| tngngp 24716 | Derive the axioms for a no... |
| tnggrpr 24717 | If a structure equipped wi... |
| tngngp3 24718 | Alternate definition of a ... |
| nrmtngdist 24719 | The augmentation of a norm... |
| nrmtngnrm 24720 | The augmentation of a norm... |
| tngngpim 24721 | The induced metric of a no... |
| isnrg 24722 | A normed ring is a ring wi... |
| nrgabv 24723 | The norm of a normed ring ... |
| nrgngp 24724 | A normed ring is a normed ... |
| nrgring 24725 | A normed ring is a ring. ... |
| nmmul 24726 | The norm of a product in a... |
| nrgdsdi 24727 | Distribute a distance calc... |
| nrgdsdir 24728 | Distribute a distance calc... |
| nm1 24729 | The norm of one in a nonze... |
| unitnmn0 24730 | The norm of a unit is nonz... |
| nminvr 24731 | The norm of an inverse in ... |
| nmdvr 24732 | The norm of a division in ... |
| nrgdomn 24733 | A nonzero normed ring is a... |
| nrgtgp 24734 | A normed ring is a topolog... |
| subrgnrg 24735 | A normed ring restricted t... |
| tngnrg 24736 | Given any absolute value o... |
| isnlm 24737 | A normed (left) module is ... |
| nmvs 24738 | Defining property of a nor... |
| nlmngp 24739 | A normed module is a norme... |
| nlmlmod 24740 | A normed module is a left ... |
| nlmnrg 24741 | The scalar component of a ... |
| nlmngp2 24742 | The scalar component of a ... |
| nlmdsdi 24743 | Distribute a distance calc... |
| nlmdsdir 24744 | Distribute a distance calc... |
| nlmmul0or 24745 | If a scalar product is zer... |
| sranlm 24746 | The subring algebra over a... |
| nlmvscnlem2 24747 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24748 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24749 | The scalar multiplication ... |
| rlmnlm 24750 | The ring module over a nor... |
| rlmnm 24751 | The norm function in the r... |
| nrgtrg 24752 | A normed ring is a topolog... |
| nrginvrcnlem 24753 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24754 | The ring inverse function ... |
| nrgtdrg 24755 | A normed division ring is ... |
| nlmtlm 24756 | A normed module is a topol... |
| isnvc 24757 | A normed vector space is j... |
| nvcnlm 24758 | A normed vector space is a... |
| nvclvec 24759 | A normed vector space is a... |
| nvclmod 24760 | A normed vector space is a... |
| isnvc2 24761 | A normed vector space is j... |
| nvctvc 24762 | A normed vector space is a... |
| lssnlm 24763 | A subspace of a normed mod... |
| lssnvc 24764 | A subspace of a normed vec... |
| rlmnvc 24765 | The ring module over a nor... |
| ngpocelbl 24766 | Membership of an off-cente... |
| nmoffn 24773 | The function producing ope... |
| reldmnghm 24774 | Lemma for normed group hom... |
| reldmnmhm 24775 | Lemma for module homomorph... |
| nmofval 24776 | Value of the operator norm... |
| nmoval 24777 | Value of the operator norm... |
| nmogelb 24778 | Property of the operator n... |
| nmolb 24779 | Any upper bound on the val... |
| nmolb2d 24780 | Any upper bound on the val... |
| nmof 24781 | The operator norm is a fun... |
| nmocl 24782 | The operator norm of an op... |
| nmoge0 24783 | The operator norm of an op... |
| nghmfval 24784 | A normed group homomorphis... |
| isnghm 24785 | A normed group homomorphis... |
| isnghm2 24786 | A normed group homomorphis... |
| isnghm3 24787 | A normed group homomorphis... |
| bddnghm 24788 | A bounded group homomorphi... |
| nghmcl 24789 | A normed group homomorphis... |
| nmoi 24790 | The operator norm achieves... |
| nmoix 24791 | The operator norm is a bou... |
| nmoi2 24792 | The operator norm is a bou... |
| nmoleub 24793 | The operator norm, defined... |
| nghmrcl1 24794 | Reverse closure for a norm... |
| nghmrcl2 24795 | Reverse closure for a norm... |
| nghmghm 24796 | A normed group homomorphis... |
| nmo0 24797 | The operator norm of the z... |
| nmoeq0 24798 | The operator norm is zero ... |
| nmoco 24799 | An upper bound on the oper... |
| nghmco 24800 | The composition of normed ... |
| nmotri 24801 | Triangle inequality for th... |
| nghmplusg 24802 | The sum of two bounded lin... |
| 0nghm 24803 | The zero operator is a nor... |
| nmoid 24804 | The operator norm of the i... |
| idnghm 24805 | The identity operator is a... |
| nmods 24806 | Upper bound for the distan... |
| nghmcn 24807 | A normed group homomorphis... |
| isnmhm 24808 | A normed module homomorphi... |
| nmhmrcl1 24809 | Reverse closure for a norm... |
| nmhmrcl2 24810 | Reverse closure for a norm... |
| nmhmlmhm 24811 | A normed module homomorphi... |
| nmhmnghm 24812 | A normed module homomorphi... |
| nmhmghm 24813 | A normed module homomorphi... |
| isnmhm2 24814 | A normed module homomorphi... |
| nmhmcl 24815 | A normed module homomorphi... |
| idnmhm 24816 | The identity operator is a... |
| 0nmhm 24817 | The zero operator is a bou... |
| nmhmco 24818 | The composition of bounded... |
| nmhmplusg 24819 | The sum of two bounded lin... |
| qtopbaslem 24820 | The set of open intervals ... |
| qtopbas 24821 | The set of open intervals ... |
| retopbas 24822 | A basis for the standard t... |
| retop 24823 | The standard topology on t... |
| uniretop 24824 | The underlying set of the ... |
| retopon 24825 | The standard topology on t... |
| retps 24826 | The standard topological s... |
| iooretop 24827 | Open intervals are open se... |
| icccld 24828 | Closed intervals are close... |
| icopnfcld 24829 | Right-unbounded closed int... |
| iocmnfcld 24830 | Left-unbounded closed inte... |
| qdensere 24831 | ` QQ ` is dense in the sta... |
| cnmetdval 24832 | Value of the distance func... |
| cnmet 24833 | The absolute value metric ... |
| cnxmet 24834 | The absolute value metric ... |
| cnbl0 24835 | Two ways to write the open... |
| cnblcld 24836 | Two ways to write the clos... |
| cnfldms 24837 | The complex number field i... |
| cnfldxms 24838 | The complex number field i... |
| cnfldtps 24839 | The complex number field i... |
| cnfldnm 24840 | The norm of the field of c... |
| cnngp 24841 | The complex numbers form a... |
| cnnrg 24842 | The complex numbers form a... |
| cnfldtopn 24843 | The topology of the comple... |
| cnfldtopon 24844 | The topology of the comple... |
| cnfldtop 24845 | The topology of the comple... |
| cnfldhaus 24846 | The topology of the comple... |
| unicntop 24847 | The underlying set of the ... |
| cnopn 24848 | The set of complex numbers... |
| cnn0opn 24849 | The set of nonzero complex... |
| zringnrg 24850 | The ring of integers is a ... |
| remetdval 24851 | Value of the distance func... |
| remet 24852 | The absolute value metric ... |
| rexmet 24853 | The absolute value metric ... |
| bl2ioo 24854 | A ball in terms of an open... |
| ioo2bl 24855 | An open interval of reals ... |
| ioo2blex 24856 | An open interval of reals ... |
| blssioo 24857 | The balls of the standard ... |
| tgioo 24858 | The topology generated by ... |
| qdensere2 24859 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24860 | An open ball in the comple... |
| rehaus 24861 | The standard topology on t... |
| tgqioo 24862 | The topology generated by ... |
| re2ndc 24863 | The standard topology on t... |
| resubmet 24864 | The subspace topology indu... |
| tgioo2 24865 | The standard topology on t... |
| rerest 24866 | The subspace topology indu... |
| tgioo4 24867 | The standard topology on t... |
| tgioo3 24868 | The standard topology on t... |
| xrtgioo 24869 | The topology on the extend... |
| xrrest 24870 | The subspace topology indu... |
| xrrest2 24871 | The subspace topology indu... |
| xrsxmet 24872 | The metric on the extended... |
| xrsdsre 24873 | The metric on the extended... |
| xrsblre 24874 | Any ball of the metric of ... |
| xrsmopn 24875 | The metric on the extended... |
| zcld 24876 | The integers are a closed ... |
| recld2 24877 | The real numbers are a clo... |
| zcld2 24878 | The integers are a closed ... |
| zdis 24879 | The integers are a discret... |
| sszcld 24880 | Every subset of the intege... |
| reperflem 24881 | A subset of the real numbe... |
| reperf 24882 | The real numbers are a per... |
| cnperf 24883 | The complex numbers are a ... |
| iccntr 24884 | The interior of a closed i... |
| icccmplem1 24885 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24886 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24887 | Lemma for ~ icccmp . (Con... |
| icccmp 24888 | A closed interval in ` RR ... |
| reconnlem1 24889 | Lemma for ~ reconn . Conn... |
| reconnlem2 24890 | Lemma for ~ reconn . (Con... |
| reconn 24891 | A subset of the reals is c... |
| retopconn 24892 | Corollary of ~ reconn . T... |
| iccconn 24893 | A closed interval is conne... |
| opnreen 24894 | Every nonempty open set is... |
| rectbntr0 24895 | A countable subset of the ... |
| xrge0gsumle 24896 | A finite sum in the nonneg... |
| xrge0tsms 24897 | Any finite or infinite sum... |
| xrge0tsms2 24898 | Any finite or infinite sum... |
| metdcnlem 24899 | The metric function of a m... |
| xmetdcn2 24900 | The metric function of an ... |
| xmetdcn 24901 | The metric function of an ... |
| metdcn2 24902 | The metric function of a m... |
| metdcn 24903 | The metric function of a m... |
| msdcn 24904 | The metric function of a m... |
| cnmpt1ds 24905 | Continuity of the metric f... |
| cnmpt2ds 24906 | Continuity of the metric f... |
| nmcn 24907 | The norm of a normed group... |
| ngnmcncn 24908 | The norm of a normed group... |
| abscn 24909 | The absolute value functio... |
| metdsval 24910 | Value of the "distance to ... |
| metdsf 24911 | The distance from a point ... |
| metdsge 24912 | The distance from the poin... |
| metds0 24913 | If a point is in a set, it... |
| metdstri 24914 | A generalization of the tr... |
| metdsle 24915 | The distance from a point ... |
| metdsre 24916 | The distance from a point ... |
| metdseq0 24917 | The distance from a point ... |
| metdscnlem 24918 | Lemma for ~ metdscn . (Co... |
| metdscn 24919 | The function ` F ` which g... |
| metdscn2 24920 | The function ` F ` which g... |
| metnrmlem1a 24921 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24922 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24923 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24924 | Lemma for ~ metnrm . (Con... |
| metnrm 24925 | A metric space is normal. ... |
| metreg 24926 | A metric space is regular.... |
| addcnlem 24927 | Lemma for ~ addcn , ~ subc... |
| addcn 24928 | Complex number addition is... |
| subcn 24929 | Complex number subtraction... |
| mulcn 24930 | Complex number multiplicat... |
| mpomulcn 24931 | Complex number multiplicat... |
| divcn 24932 | Complex number division is... |
| cnfldtgp 24933 | The complex numbers form a... |
| fsumcn 24934 | A finite sum of functions ... |
| fsum2cn 24935 | Version of ~ fsumcn for tw... |
| expcn 24936 | The power function on comp... |
| divccn 24937 | Division by a nonzero cons... |
| sqcn 24938 | The square function on com... |
| iitopon 24943 | The unit interval is a top... |
| iitop 24944 | The unit interval is a top... |
| iiuni 24945 | The base set of the unit i... |
| dfii2 24946 | Alternate definition of th... |
| dfii3 24947 | Alternate definition of th... |
| dfii4 24948 | Alternate definition of th... |
| dfii5 24949 | The unit interval expresse... |
| iicmp 24950 | The unit interval is compa... |
| iiconn 24951 | The unit interval is conne... |
| cncfval 24952 | The value of the continuou... |
| elcncf 24953 | Membership in the set of c... |
| elcncf2 24954 | Version of ~ elcncf with a... |
| cncfrss 24955 | Reverse closure of the con... |
| cncfrss2 24956 | Reverse closure of the con... |
| cncff 24957 | A continuous complex funct... |
| cncfi 24958 | Defining property of a con... |
| elcncf1di 24959 | Membership in the set of c... |
| elcncf1ii 24960 | Membership in the set of c... |
| rescncf 24961 | A continuous complex funct... |
| cncfcdm 24962 | Change the codomain of a c... |
| cncfss 24963 | The set of continuous func... |
| climcncf 24964 | Image of a limit under a c... |
| abscncf 24965 | Absolute value is continuo... |
| recncf 24966 | Real part is continuous. ... |
| imcncf 24967 | Imaginary part is continuo... |
| cjcncf 24968 | Complex conjugate is conti... |
| mulc1cncf 24969 | Multiplication by a consta... |
| divccncf 24970 | Division by a constant is ... |
| cncfco 24971 | The composition of two con... |
| cncfcompt2 24972 | Composition of continuous ... |
| cncfmet 24973 | Relate complex function co... |
| cncfcn 24974 | Relate complex function co... |
| cncfcn1 24975 | Relate complex function co... |
| cncfmptc 24976 | A constant function is a c... |
| cncfmptid 24977 | The identity function is a... |
| cncfmpt1f 24978 | Composition of continuous ... |
| cncfmpt2f 24979 | Composition of continuous ... |
| cncfmpt2ss 24980 | Composition of continuous ... |
| addccncf 24981 | Adding a constant is a con... |
| idcncf 24982 | The identity function is a... |
| sub1cncf 24983 | Subtracting a constant is ... |
| sub2cncf 24984 | Subtraction from a constan... |
| cdivcncf 24985 | Division with a constant n... |
| negcncf 24986 | The negative function is c... |
| negfcncf 24987 | The negative of a continuo... |
| abscncfALT 24988 | Absolute value is continuo... |
| cncfcnvcn 24989 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24990 | The power function on comp... |
| cnmptre 24991 | Lemma for ~ iirevcn and re... |
| cnmpopc 24992 | Piecewise definition of a ... |
| iirev 24993 | Reverse the unit interval.... |
| iirevcn 24994 | The reversion function is ... |
| iihalf1 24995 | Map the first half of ` II... |
| iihalf1cn 24996 | The first half function is... |
| iihalf2 24997 | Map the second half of ` I... |
| iihalf2cn 24998 | The second half function i... |
| elii1 24999 | Divide the unit interval i... |
| elii2 25000 | Divide the unit interval i... |
| iimulcl 25001 | The unit interval is close... |
| iimulcn 25002 | Multiplication is a contin... |
| icoopnst 25003 | A half-open interval start... |
| iocopnst 25004 | A half-open interval endin... |
| icchmeo 25005 | The natural bijection from... |
| icopnfcnv 25006 | Define a bijection from ` ... |
| icopnfhmeo 25007 | The defined bijection from... |
| iccpnfcnv 25008 | Define a bijection from ` ... |
| iccpnfhmeo 25009 | The defined bijection from... |
| xrhmeo 25010 | The bijection from ` [ -u ... |
| xrhmph 25011 | The extended reals are hom... |
| xrcmp 25012 | The topology of the extend... |
| xrconn 25013 | The topology of the extend... |
| icccvx 25014 | A linear combination of tw... |
| oprpiece1res1 25015 | Restriction to the first p... |
| oprpiece1res2 25016 | Restriction to the second ... |
| cnrehmeo 25017 | The canonical bijection fr... |
| cnheiborlem 25018 | Lemma for ~ cnheibor . (C... |
| cnheibor 25019 | Heine-Borel theorem for co... |
| cnllycmp 25020 | The topology on the comple... |
| rellycmp 25021 | The topology on the reals ... |
| bndth 25022 | The Boundedness Theorem. ... |
| evth 25023 | The Extreme Value Theorem.... |
| evth2 25024 | The Extreme Value Theorem,... |
| lebnumlem1 25025 | Lemma for ~ lebnum . The ... |
| lebnumlem2 25026 | Lemma for ~ lebnum . As a... |
| lebnumlem3 25027 | Lemma for ~ lebnum . By t... |
| lebnum 25028 | The Lebesgue number lemma,... |
| xlebnum 25029 | Generalize ~ lebnum to ext... |
| lebnumii 25030 | Specialize the Lebesgue nu... |
| ishtpy 25036 | Membership in the class of... |
| htpycn 25037 | A homotopy is a continuous... |
| htpyi 25038 | A homotopy evaluated at it... |
| ishtpyd 25039 | Deduction for membership i... |
| htpycom 25040 | Given a homotopy from ` F ... |
| htpyid 25041 | A homotopy from a function... |
| htpyco1 25042 | Compose a homotopy with a ... |
| htpyco2 25043 | Compose a homotopy with a ... |
| htpycc 25044 | Concatenate two homotopies... |
| isphtpy 25045 | Membership in the class of... |
| phtpyhtpy 25046 | A path homotopy is a homot... |
| phtpycn 25047 | A path homotopy is a conti... |
| phtpyi 25048 | Membership in the class of... |
| phtpy01 25049 | Two path-homotopic paths h... |
| isphtpyd 25050 | Deduction for membership i... |
| isphtpy2d 25051 | Deduction for membership i... |
| phtpycom 25052 | Given a homotopy from ` F ... |
| phtpyid 25053 | A homotopy from a path to ... |
| phtpyco2 25054 | Compose a path homotopy wi... |
| phtpycc 25055 | Concatenate two path homot... |
| phtpcrel 25057 | The path homotopy relation... |
| isphtpc 25058 | The relation "is path homo... |
| phtpcer 25059 | Path homotopy is an equiva... |
| phtpc01 25060 | Path homotopic paths have ... |
| reparphti 25061 | Lemma for ~ reparpht . (C... |
| reparpht 25062 | Reparametrization lemma. ... |
| phtpcco2 25063 | Compose a path homotopy wi... |
| pcofval 25074 | The value of the path conc... |
| pcoval 25075 | The concatenation of two p... |
| pcovalg 25076 | Evaluate the concatenation... |
| pcoval1 25077 | Evaluate the concatenation... |
| pco0 25078 | The starting point of a pa... |
| pco1 25079 | The ending point of a path... |
| pcoval2 25080 | Evaluate the concatenation... |
| pcocn 25081 | The concatenation of two p... |
| copco 25082 | The composition of a conca... |
| pcohtpylem 25083 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 25084 | Homotopy invariance of pat... |
| pcoptcl 25085 | A constant function is a p... |
| pcopt 25086 | Concatenation with a point... |
| pcopt2 25087 | Concatenation with a point... |
| pcoass 25088 | Order of concatenation doe... |
| pcorevcl 25089 | Closure for a reversed pat... |
| pcorevlem 25090 | Lemma for ~ pcorev . Prov... |
| pcorev 25091 | Concatenation with the rev... |
| pcorev2 25092 | Concatenation with the rev... |
| pcophtb 25093 | The path homotopy equivale... |
| om1val 25094 | The definition of the loop... |
| om1bas 25095 | The base set of the loop s... |
| om1elbas 25096 | Elementhood in the base se... |
| om1addcl 25097 | Closure of the group opera... |
| om1plusg 25098 | The group operation (which... |
| om1tset 25099 | The topology of the loop s... |
| om1opn 25100 | The topology of the loop s... |
| pi1val 25101 | The definition of the fund... |
| pi1bas 25102 | The base set of the fundam... |
| pi1blem 25103 | Lemma for ~ pi1buni . (Co... |
| pi1buni 25104 | Another way to write the l... |
| pi1bas2 25105 | The base set of the fundam... |
| pi1eluni 25106 | Elementhood in the base se... |
| pi1bas3 25107 | The base set of the fundam... |
| pi1cpbl 25108 | The group operation, loop ... |
| elpi1 25109 | The elements of the fundam... |
| elpi1i 25110 | The elements of the fundam... |
| pi1addf 25111 | The group operation of ` p... |
| pi1addval 25112 | The concatenation of two p... |
| pi1grplem 25113 | Lemma for ~ pi1grp . (Con... |
| pi1grp 25114 | The fundamental group is a... |
| pi1id 25115 | The identity element of th... |
| pi1inv 25116 | An inverse in the fundamen... |
| pi1xfrf 25117 | Functionality of the loop ... |
| pi1xfrval 25118 | The value of the loop tran... |
| pi1xfr 25119 | Given a path ` F ` and its... |
| pi1xfrcnvlem 25120 | Given a path ` F ` between... |
| pi1xfrcnv 25121 | Given a path ` F ` between... |
| pi1xfrgim 25122 | The mapping ` G ` between ... |
| pi1cof 25123 | Functionality of the loop ... |
| pi1coval 25124 | The value of the loop tran... |
| pi1coghm 25125 | The mapping ` G ` between ... |
| isclm 25128 | A subcomplex module is a l... |
| clmsca 25129 | The ring of scalars ` F ` ... |
| clmsubrg 25130 | The base set of the ring o... |
| clmlmod 25131 | A subcomplex module is a l... |
| clmgrp 25132 | A subcomplex module is an ... |
| clmabl 25133 | A subcomplex module is an ... |
| clmring 25134 | The scalar ring of a subco... |
| clmfgrp 25135 | The scalar ring of a subco... |
| clm0 25136 | The zero of the scalar rin... |
| clm1 25137 | The identity of the scalar... |
| clmadd 25138 | The addition of the scalar... |
| clmmul 25139 | The multiplication of the ... |
| clmcj 25140 | The conjugation of the sca... |
| isclmi 25141 | Reverse direction of ~ isc... |
| clmzss 25142 | The scalar ring of a subco... |
| clmsscn 25143 | The scalar ring of a subco... |
| clmsub 25144 | Subtraction in the scalar ... |
| clmneg 25145 | Negation in the scalar rin... |
| clmneg1 25146 | Minus one is in the scalar... |
| clmabs 25147 | Norm in the scalar ring of... |
| clmacl 25148 | Closure of ring addition f... |
| clmmcl 25149 | Closure of ring multiplica... |
| clmsubcl 25150 | Closure of ring subtractio... |
| lmhmclm 25151 | The domain of a linear ope... |
| clmvscl 25152 | Closure of scalar product ... |
| clmvsass 25153 | Associative law for scalar... |
| clmvscom 25154 | Commutative law for the sc... |
| clmvsdir 25155 | Distributive law for scala... |
| clmvsdi 25156 | Distributive law for scala... |
| clmvs1 25157 | Scalar product with ring u... |
| clmvs2 25158 | A vector plus itself is tw... |
| clm0vs 25159 | Zero times a vector is the... |
| clmopfne 25160 | The (functionalized) opera... |
| isclmp 25161 | The predicate "is a subcom... |
| isclmi0 25162 | Properties that determine ... |
| clmvneg1 25163 | Minus 1 times a vector is ... |
| clmvsneg 25164 | Multiplication of a vector... |
| clmmulg 25165 | The group multiple functio... |
| clmsubdir 25166 | Scalar multiplication dist... |
| clmpm1dir 25167 | Subtractive distributive l... |
| clmnegneg 25168 | Double negative of a vecto... |
| clmnegsubdi2 25169 | Distribution of negative o... |
| clmsub4 25170 | Rearrangement of 4 terms i... |
| clmvsrinv 25171 | A vector minus itself. (C... |
| clmvslinv 25172 | Minus a vector plus itself... |
| clmvsubval 25173 | Value of vector subtractio... |
| clmvsubval2 25174 | Value of vector subtractio... |
| clmvz 25175 | Two ways to express the ne... |
| zlmclm 25176 | The ` ZZ ` -module operati... |
| clmzlmvsca 25177 | The scalar product of a su... |
| nmoleub2lem 25178 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25179 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25180 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25181 | The operator norm is the s... |
| nmoleub2b 25182 | The operator norm is the s... |
| nmoleub3 25183 | The operator norm is the s... |
| nmhmcn 25184 | A linear operator over a n... |
| cmodscexp 25185 | The powers of ` _i ` belon... |
| cmodscmulexp 25186 | The scalar product of a ve... |
| cvslvec 25189 | A subcomplex vector space ... |
| cvsclm 25190 | A subcomplex vector space ... |
| iscvs 25191 | A subcomplex vector space ... |
| iscvsp 25192 | The predicate "is a subcom... |
| iscvsi 25193 | Properties that determine ... |
| cvsi 25194 | The properties of a subcom... |
| cvsunit 25195 | Unit group of the scalar r... |
| cvsdiv 25196 | Division of the scalar rin... |
| cvsdivcl 25197 | The scalar field of a subc... |
| cvsmuleqdivd 25198 | An equality involving rati... |
| cvsdiveqd 25199 | An equality involving rati... |
| cnlmodlem1 25200 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25201 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25202 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25203 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25204 | The set of complex numbers... |
| cnstrcvs 25205 | The set of complex numbers... |
| cnrbas 25206 | The set of complex numbers... |
| cnrlmod 25207 | The complex left module of... |
| cnrlvec 25208 | The complex left module of... |
| cncvs 25209 | The complex left module of... |
| recvs 25210 | The field of the real numb... |
| qcvs 25211 | The field of rational numb... |
| zclmncvs 25212 | The ring of integers as le... |
| isncvsngp 25213 | A normed subcomplex vector... |
| isncvsngpd 25214 | Properties that determine ... |
| ncvsi 25215 | The properties of a normed... |
| ncvsprp 25216 | Proportionality property o... |
| ncvsge0 25217 | The norm of a scalar produ... |
| ncvsm1 25218 | The norm of the opposite o... |
| ncvsdif 25219 | The norm of the difference... |
| ncvspi 25220 | The norm of a vector plus ... |
| ncvs1 25221 | From any nonzero vector of... |
| cnrnvc 25222 | The module of complex numb... |
| cnncvs 25223 | The module of complex numb... |
| cnnm 25224 | The norm of the normed sub... |
| ncvspds 25225 | Value of the distance func... |
| cnindmet 25226 | The metric induced on the ... |
| cnncvsaddassdemo 25227 | Derive the associative law... |
| cnncvsmulassdemo 25228 | Derive the associative law... |
| cnncvsabsnegdemo 25229 | Derive the absolute value ... |
| iscph 25234 | A subcomplex pre-Hilbert s... |
| cphphl 25235 | A subcomplex pre-Hilbert s... |
| cphnlm 25236 | A subcomplex pre-Hilbert s... |
| cphngp 25237 | A subcomplex pre-Hilbert s... |
| cphlmod 25238 | A subcomplex pre-Hilbert s... |
| cphlvec 25239 | A subcomplex pre-Hilbert s... |
| cphnvc 25240 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25241 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25242 | Lemma for ~ cphreccl . (C... |
| cphsca 25243 | A subcomplex pre-Hilbert s... |
| cphsubrg 25244 | The scalar field of a subc... |
| cphreccl 25245 | The scalar field of a subc... |
| cphdivcl 25246 | The scalar field of a subc... |
| cphcjcl 25247 | The scalar field of a subc... |
| cphsqrtcl 25248 | The scalar field of a subc... |
| cphabscl 25249 | The scalar field of a subc... |
| cphsqrtcl2 25250 | The scalar field of a subc... |
| cphsqrtcl3 25251 | If the scalar field of a s... |
| cphqss 25252 | The scalar field of a subc... |
| cphclm 25253 | A subcomplex pre-Hilbert s... |
| cphnmvs 25254 | Norm of a scalar product. ... |
| cphipcl 25255 | An inner product is a memb... |
| cphnmfval 25256 | The value of the norm in a... |
| cphnm 25257 | The square of the norm is ... |
| nmsq 25258 | The square of the norm is ... |
| cphnmf 25259 | The norm of a vector is a ... |
| cphnmcl 25260 | The norm of a vector is a ... |
| reipcl 25261 | An inner product of an ele... |
| ipge0 25262 | The inner product in a sub... |
| cphipcj 25263 | Conjugate of an inner prod... |
| cphipipcj 25264 | An inner product times its... |
| cphorthcom 25265 | Orthogonality (meaning inn... |
| cphip0l 25266 | Inner product with a zero ... |
| cphip0r 25267 | Inner product with a zero ... |
| cphipeq0 25268 | The inner product of a vec... |
| cphdir 25269 | Distributive law for inner... |
| cphdi 25270 | Distributive law for inner... |
| cph2di 25271 | Distributive law for inner... |
| cphsubdir 25272 | Distributive law for inner... |
| cphsubdi 25273 | Distributive law for inner... |
| cph2subdi 25274 | Distributive law for inner... |
| cphass 25275 | Associative law for inner ... |
| cphassr 25276 | "Associative" law for seco... |
| cph2ass 25277 | Move scalar multiplication... |
| cphassi 25278 | Associative law for the fi... |
| cphassir 25279 | "Associative" law for the ... |
| cphpyth 25280 | The pythagorean theorem fo... |
| tcphex 25281 | Lemma for ~ tcphbas and si... |
| tcphval 25282 | Define a function to augme... |
| tcphbas 25283 | The base set of a subcompl... |
| tchplusg 25284 | The addition operation of ... |
| tcphsub 25285 | The subtraction operation ... |
| tcphmulr 25286 | The ring operation of a su... |
| tcphsca 25287 | The scalar field of a subc... |
| tcphvsca 25288 | The scalar multiplication ... |
| tcphip 25289 | The inner product of a sub... |
| tcphtopn 25290 | The topology of a subcompl... |
| tcphphl 25291 | Augmentation of a subcompl... |
| tchnmfval 25292 | The norm of a subcomplex p... |
| tcphnmval 25293 | The norm of a subcomplex p... |
| cphtcphnm 25294 | The norm of a norm-augment... |
| tcphds 25295 | The distance of a pre-Hilb... |
| phclm 25296 | A pre-Hilbert space whose ... |
| tcphcphlem3 25297 | Lemma for ~ tcphcph : real... |
| ipcau2 25298 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25299 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25300 | Lemma for ~ tcphcph : homo... |
| tcphcph 25301 | The standard definition of... |
| ipcau 25302 | The Cauchy-Schwarz inequal... |
| nmparlem 25303 | Lemma for ~ nmpar . (Cont... |
| nmpar 25304 | A subcomplex pre-Hilbert s... |
| cphipval2 25305 | Value of the inner product... |
| 4cphipval2 25306 | Four times the inner produ... |
| cphipval 25307 | Value of the inner product... |
| ipcnlem2 25308 | The inner product operatio... |
| ipcnlem1 25309 | The inner product operatio... |
| ipcn 25310 | The inner product operatio... |
| cnmpt1ip 25311 | Continuity of inner produc... |
| cnmpt2ip 25312 | Continuity of inner produc... |
| csscld 25313 | A "closed subspace" in a s... |
| clsocv 25314 | The orthogonal complement ... |
| cphsscph 25315 | A subspace of a subcomplex... |
| lmmbr 25322 | Express the binary relatio... |
| lmmbr2 25323 | Express the binary relatio... |
| lmmbr3 25324 | Express the binary relatio... |
| lmmcvg 25325 | Convergence property of a ... |
| lmmbrf 25326 | Express the binary relatio... |
| lmnn 25327 | A condition that implies c... |
| cfilfval 25328 | The set of Cauchy filters ... |
| iscfil 25329 | The property of being a Ca... |
| iscfil2 25330 | The property of being a Ca... |
| cfilfil 25331 | A Cauchy filter is a filte... |
| cfili 25332 | Property of a Cauchy filte... |
| cfil3i 25333 | A Cauchy filter contains b... |
| cfilss 25334 | A filter finer than a Cauc... |
| fgcfil 25335 | The Cauchy filter conditio... |
| fmcfil 25336 | The Cauchy filter conditio... |
| iscfil3 25337 | A filter is Cauchy iff it ... |
| cfilfcls 25338 | Similar to ultrafilters ( ... |
| caufval 25339 | The set of Cauchy sequence... |
| iscau 25340 | Express the property " ` F... |
| iscau2 25341 | Express the property " ` F... |
| iscau3 25342 | Express the Cauchy sequenc... |
| iscau4 25343 | Express the property " ` F... |
| iscauf 25344 | Express the property " ` F... |
| caun0 25345 | A metric with a Cauchy seq... |
| caufpm 25346 | Inclusion of a Cauchy sequ... |
| caucfil 25347 | A Cauchy sequence predicat... |
| iscmet 25348 | The property " ` D ` is a ... |
| cmetcvg 25349 | The convergence of a Cauch... |
| cmetmet 25350 | A complete metric space is... |
| cmetmeti 25351 | A complete metric space is... |
| cmetcaulem 25352 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25353 | The convergence of a Cauch... |
| iscmet3lem3 25354 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25355 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25356 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25357 | The property " ` D ` is a ... |
| iscmet2 25358 | A metric ` D ` is complete... |
| cfilresi 25359 | A Cauchy filter on a metri... |
| cfilres 25360 | Cauchy filter on a metric ... |
| caussi 25361 | Cauchy sequence on a metri... |
| causs 25362 | Cauchy sequence on a metri... |
| equivcfil 25363 | If the metric ` D ` is "st... |
| equivcau 25364 | If the metric ` D ` is "st... |
| lmle 25365 | If the distance from each ... |
| nglmle 25366 | If the norm of each member... |
| lmclim 25367 | Relate a limit on the metr... |
| lmclimf 25368 | Relate a limit on the metr... |
| metelcls 25369 | A point belongs to the clo... |
| metcld 25370 | A subset of a metric space... |
| metcld2 25371 | A subset of a metric space... |
| caubl 25372 | Sufficient condition to en... |
| caublcls 25373 | The convergent point of a ... |
| metcnp4 25374 | Two ways to say a mapping ... |
| metcn4 25375 | Two ways to say a mapping ... |
| iscmet3i 25376 | Properties that determine ... |
| lmcau 25377 | Every convergent sequence ... |
| flimcfil 25378 | Every convergent filter in... |
| metsscmetcld 25379 | A complete subspace of a m... |
| cmetss 25380 | A subspace of a complete m... |
| equivcmet 25381 | If two metrics are strongl... |
| relcmpcmet 25382 | If ` D ` is a metric space... |
| cmpcmet 25383 | A compact metric space is ... |
| cfilucfil3 25384 | Given a metric ` D ` and a... |
| cfilucfil4 25385 | Given a metric ` D ` and a... |
| cncmet 25386 | The set of complex numbers... |
| recmet 25387 | The real numbers are a com... |
| bcthlem1 25388 | Lemma for ~ bcth . Substi... |
| bcthlem2 25389 | Lemma for ~ bcth . The ba... |
| bcthlem3 25390 | Lemma for ~ bcth . The li... |
| bcthlem4 25391 | Lemma for ~ bcth . Given ... |
| bcthlem5 25392 | Lemma for ~ bcth . The pr... |
| bcth 25393 | Baire's Category Theorem. ... |
| bcth2 25394 | Baire's Category Theorem, ... |
| bcth3 25395 | Baire's Category Theorem, ... |
| isbn 25402 | A Banach space is a normed... |
| bnsca 25403 | The scalar field of a Bana... |
| bnnvc 25404 | A Banach space is a normed... |
| bnnlm 25405 | A Banach space is a normed... |
| bnngp 25406 | A Banach space is a normed... |
| bnlmod 25407 | A Banach space is a left m... |
| bncms 25408 | A Banach space is a comple... |
| iscms 25409 | A complete metric space is... |
| cmscmet 25410 | The induced metric on a co... |
| bncmet 25411 | The induced metric on Bana... |
| cmsms 25412 | A complete metric space is... |
| cmspropd 25413 | Property deduction for a c... |
| cmssmscld 25414 | The restriction of a metri... |
| cmsss 25415 | The restriction of a compl... |
| lssbn 25416 | A subspace of a Banach spa... |
| cmetcusp1 25417 | If the uniform set of a co... |
| cmetcusp 25418 | The uniform space generate... |
| cncms 25419 | The field of complex numbe... |
| cnflduss 25420 | The uniform structure of t... |
| cnfldcusp 25421 | The field of complex numbe... |
| resscdrg 25422 | The real numbers are a sub... |
| cncdrg 25423 | The only complete subfield... |
| srabn 25424 | The subring algebra over a... |
| rlmbn 25425 | The ring module over a com... |
| ishl 25426 | The predicate "is a subcom... |
| hlbn 25427 | Every subcomplex Hilbert s... |
| hlcph 25428 | Every subcomplex Hilbert s... |
| hlphl 25429 | Every subcomplex Hilbert s... |
| hlcms 25430 | Every subcomplex Hilbert s... |
| hlprlem 25431 | Lemma for ~ hlpr . (Contr... |
| hlress 25432 | The scalar field of a subc... |
| hlpr 25433 | The scalar field of a subc... |
| ishl2 25434 | A Hilbert space is a compl... |
| cphssphl 25435 | A Banach subspace of a sub... |
| cmslssbn 25436 | A complete linear subspace... |
| cmscsscms 25437 | A closed subspace of a com... |
| bncssbn 25438 | A closed subspace of a Ban... |
| cssbn 25439 | A complete subspace of a n... |
| csschl 25440 | A complete subspace of a c... |
| cmslsschl 25441 | A complete linear subspace... |
| chlcsschl 25442 | A closed subspace of a sub... |
| retopn 25443 | The topology of the real n... |
| recms 25444 | The real numbers form a co... |
| reust 25445 | The Uniform structure of t... |
| recusp 25446 | The real numbers form a co... |
| rrxval 25451 | Value of the generalized E... |
| rrxbase 25452 | The base of the generalize... |
| rrxprds 25453 | Expand the definition of t... |
| rrxip 25454 | The inner product of the g... |
| rrxnm 25455 | The norm of the generalize... |
| rrxcph 25456 | Generalized Euclidean real... |
| rrxds 25457 | The distance over generali... |
| rrxvsca 25458 | The scalar product over ge... |
| rrxplusgvscavalb 25459 | The result of the addition... |
| rrxsca 25460 | The field of real numbers ... |
| rrx0 25461 | The zero ("origin") in a g... |
| rrx0el 25462 | The zero ("origin") in a g... |
| csbren 25463 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25464 | Triangle inequality in R^n... |
| rrxf 25465 | Euclidean vectors as funct... |
| rrxfsupp 25466 | Euclidean vectors are of f... |
| rrxsuppss 25467 | Support of Euclidean vecto... |
| rrxmvallem 25468 | Support of the function us... |
| rrxmval 25469 | The value of the Euclidean... |
| rrxmfval 25470 | The value of the Euclidean... |
| rrxmetlem 25471 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25472 | Euclidean space is a metri... |
| rrxdstprj1 25473 | The distance between two p... |
| rrxbasefi 25474 | The base of the generalize... |
| rrxdsfi 25475 | The distance over generali... |
| rrxmetfi 25476 | Euclidean space is a metri... |
| rrxdsfival 25477 | The value of the Euclidean... |
| ehlval 25478 | Value of the Euclidean spa... |
| ehlbase 25479 | The base of the Euclidean ... |
| ehl0base 25480 | The base of the Euclidean ... |
| ehl0 25481 | The Euclidean space of dim... |
| ehleudis 25482 | The Euclidean distance fun... |
| ehleudisval 25483 | The value of the Euclidean... |
| ehl1eudis 25484 | The Euclidean distance fun... |
| ehl1eudisval 25485 | The value of the Euclidean... |
| ehl2eudis 25486 | The Euclidean distance fun... |
| ehl2eudisval 25487 | The value of the Euclidean... |
| minveclem1 25488 | Lemma for ~ minvec . The ... |
| minveclem4c 25489 | Lemma for ~ minvec . The ... |
| minveclem2 25490 | Lemma for ~ minvec . Any ... |
| minveclem3a 25491 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25492 | Lemma for ~ minvec . The ... |
| minveclem3 25493 | Lemma for ~ minvec . The ... |
| minveclem4a 25494 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25495 | Lemma for ~ minvec . The ... |
| minveclem4 25496 | Lemma for ~ minvec . The ... |
| minveclem5 25497 | Lemma for ~ minvec . Disc... |
| minveclem6 25498 | Lemma for ~ minvec . Any ... |
| minveclem7 25499 | Lemma for ~ minvec . Sinc... |
| minvec 25500 | Minimizing vector theorem,... |
| pjthlem1 25501 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25502 | Lemma for ~ pjth . (Contr... |
| pjth 25503 | Projection Theorem: Any H... |
| pjth2 25504 | Projection Theorem with ab... |
| cldcss 25505 | Corollary of the Projectio... |
| cldcss2 25506 | Corollary of the Projectio... |
| hlhil 25507 | Corollary of the Projectio... |
| addcncf 25508 | The addition of two contin... |
| subcncf 25509 | The subtraction of two con... |
| mulcncf 25510 | The multiplication of two ... |
| divcncf 25511 | The quotient of two contin... |
| pmltpclem1 25512 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25513 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25514 | Any function on the reals ... |
| ivthlem1 25515 | Lemma for ~ ivth . The se... |
| ivthlem2 25516 | Lemma for ~ ivth . Show t... |
| ivthlem3 25517 | Lemma for ~ ivth , the int... |
| ivth 25518 | The intermediate value the... |
| ivth2 25519 | The intermediate value the... |
| ivthle 25520 | The intermediate value the... |
| ivthle2 25521 | The intermediate value the... |
| ivthicc 25522 | The interval between any t... |
| evthicc 25523 | Specialization of the Extr... |
| evthicc2 25524 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25525 | A continuous function on a... |
| ovolfcl 25530 | Closure for the interval e... |
| ovolfioo 25531 | Unpack the interval coveri... |
| ovolficc 25532 | Unpack the interval coveri... |
| ovolficcss 25533 | Any (closed) interval cove... |
| ovolfsval 25534 | The value of the interval ... |
| ovolfsf 25535 | Closure for the interval l... |
| ovolsf 25536 | Closure for the partial su... |
| ovolval 25537 | The value of the outer mea... |
| elovolmlem 25538 | Lemma for ~ elovolm and re... |
| elovolm 25539 | Elementhood in the set ` M... |
| elovolmr 25540 | Sufficient condition for e... |
| ovolmge0 25541 | The set ` M ` is composed ... |
| ovolcl 25542 | The volume of a set is an ... |
| ovollb 25543 | The outer volume is a lowe... |
| ovolgelb 25544 | The outer volume is the gr... |
| ovolge0 25545 | The volume of a set is alw... |
| ovolf 25546 | The domain and codomain of... |
| ovollecl 25547 | If an outer volume is boun... |
| ovolsslem 25548 | Lemma for ~ ovolss . (Con... |
| ovolss 25549 | The volume of a set is mon... |
| ovolsscl 25550 | If a set is contained in a... |
| ovolssnul 25551 | A subset of a nullset is n... |
| ovollb2lem 25552 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25553 | It is often more convenien... |
| ovolctb 25554 | The volume of a denumerabl... |
| ovolq 25555 | The rational numbers have ... |
| ovolctb2 25556 | The volume of a countable ... |
| ovol0 25557 | The empty set has 0 outer ... |
| ovolfi 25558 | A finite set has 0 outer L... |
| ovolsn 25559 | A singleton has 0 outer Le... |
| ovolunlem1a 25560 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25561 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25562 | Lemma for ~ ovolun . (Con... |
| ovolun 25563 | The Lebesgue outer measure... |
| ovolunnul 25564 | Adding a nullset does not ... |
| ovolfiniun 25565 | The Lebesgue outer measure... |
| ovoliunlem1 25566 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25567 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25568 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25569 | The Lebesgue outer measure... |
| ovoliun2 25570 | The Lebesgue outer measure... |
| ovoliunnul 25571 | A countable union of nulls... |
| shft2rab 25572 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25573 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25574 | Lemma for ~ ovolshft . (C... |
| ovolshft 25575 | The Lebesgue outer measure... |
| sca2rab 25576 | If ` B ` is a scale of ` A... |
| ovolscalem1 25577 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25578 | Lemma for ~ ovolshft . (C... |
| ovolsca 25579 | The Lebesgue outer measure... |
| ovolicc1 25580 | The measure of a closed in... |
| ovolicc2lem1 25581 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25582 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25583 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25584 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25585 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25586 | The measure of a closed in... |
| ovolicc 25587 | The measure of a closed in... |
| ovolicopnf 25588 | The measure of a right-unb... |
| ovolre 25589 | The measure of the real nu... |
| ismbl 25590 | The predicate " ` A ` is L... |
| ismbl2 25591 | From ~ ovolun , it suffice... |
| volres 25592 | A self-referencing abbrevi... |
| volf 25593 | The domain and codomain of... |
| mblvol 25594 | The volume of a measurable... |
| mblss 25595 | A measurable set is a subs... |
| mblsplit 25596 | The defining property of m... |
| volss 25597 | The Lebesgue measure is mo... |
| cmmbl 25598 | The complement of a measur... |
| nulmbl 25599 | A nullset is measurable. ... |
| nulmbl2 25600 | A set of outer measure zer... |
| unmbl 25601 | A union of measurable sets... |
| shftmbl 25602 | A shift of a measurable se... |
| 0mbl 25603 | The empty set is measurabl... |
| rembl 25604 | The set of all real number... |
| unidmvol 25605 | The union of the Lebesgue ... |
| inmbl 25606 | An intersection of measura... |
| difmbl 25607 | A difference of measurable... |
| finiunmbl 25608 | A finite union of measurab... |
| volun 25609 | The Lebesgue measure funct... |
| volinun 25610 | Addition of non-disjoint s... |
| volfiniun 25611 | The volume of a disjoint f... |
| iundisj 25612 | Rewrite a countable union ... |
| iundisj2 25613 | A disjoint union is disjoi... |
| voliunlem1 25614 | Lemma for ~ voliun . (Con... |
| voliunlem2 25615 | Lemma for ~ voliun . (Con... |
| voliunlem3 25616 | Lemma for ~ voliun . (Con... |
| iunmbl 25617 | The measurable sets are cl... |
| voliun 25618 | The Lebesgue measure funct... |
| volsuplem 25619 | Lemma for ~ volsup . (Con... |
| volsup 25620 | The volume of the limit of... |
| iunmbl2 25621 | The measurable sets are cl... |
| ioombl1lem1 25622 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25623 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25624 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25625 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25626 | An open right-unbounded in... |
| icombl1 25627 | A closed unbounded-above i... |
| icombl 25628 | A closed-below, open-above... |
| ioombl 25629 | An open real interval is m... |
| iccmbl 25630 | A closed real interval is ... |
| iccvolcl 25631 | A closed real interval has... |
| ovolioo 25632 | The measure of an open int... |
| volioo 25633 | The measure of an open int... |
| ioovolcl 25634 | An open real interval has ... |
| ovolfs2 25635 | Alternative expression for... |
| ioorcl2 25636 | An open interval with fini... |
| ioorf 25637 | Define a function from ope... |
| ioorval 25638 | Define a function from ope... |
| ioorinv2 25639 | The function ` F ` is an "... |
| ioorinv 25640 | The function ` F ` is an "... |
| ioorcl 25641 | The function ` F ` does no... |
| uniiccdif 25642 | A union of closed interval... |
| uniioovol 25643 | A disjoint union of open i... |
| uniiccvol 25644 | An almost-disjoint union o... |
| uniioombllem1 25645 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25646 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25647 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25648 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25649 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25650 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25651 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25652 | Lemma for ~ uniioombl . (... |
| uniioombl 25653 | A disjoint union of open i... |
| uniiccmbl 25654 | An almost-disjoint union o... |
| dyadf 25655 | The function ` F ` returns... |
| dyadval 25656 | Value of the dyadic ration... |
| dyadovol 25657 | Volume of a dyadic rationa... |
| dyadss 25658 | Two closed dyadic rational... |
| dyaddisjlem 25659 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25660 | Two closed dyadic rational... |
| dyadmaxlem 25661 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25662 | Any nonempty set of dyadic... |
| dyadmbllem 25663 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25664 | Any union of dyadic ration... |
| opnmbllem 25665 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25666 | All open sets are measurab... |
| opnmblALT 25667 | All open sets are measurab... |
| subopnmbl 25668 | Sets which are open in a m... |
| volsup2 25669 | The volume of ` A ` is the... |
| volcn 25670 | The function formed by res... |
| volivth 25671 | The Intermediate Value The... |
| vitalilem1 25672 | Lemma for ~ vitali . (Con... |
| vitalilem2 25673 | Lemma for ~ vitali . (Con... |
| vitalilem3 25674 | Lemma for ~ vitali . (Con... |
| vitalilem4 25675 | Lemma for ~ vitali . (Con... |
| vitalilem5 25676 | Lemma for ~ vitali . (Con... |
| vitali 25677 | If the reals can be well-o... |
| ismbf1 25688 | The predicate " ` F ` is a... |
| mbff 25689 | A measurable function is a... |
| mbfdm 25690 | The domain of a measurable... |
| mbfconstlem 25691 | Lemma for ~ mbfconst and r... |
| ismbf 25692 | The predicate " ` F ` is a... |
| ismbfcn 25693 | A complex function is meas... |
| mbfima 25694 | Definitional property of a... |
| mbfimaicc 25695 | The preimage of any closed... |
| mbfimasn 25696 | The preimage of a point un... |
| mbfconst 25697 | A constant function is mea... |
| mbf0 25698 | The empty function is meas... |
| mbfid 25699 | The identity function is m... |
| mbfmptcl 25700 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25701 | The domain of a measurable... |
| ismbfcn2 25702 | A complex function is meas... |
| ismbfd 25703 | Deduction to prove measura... |
| ismbf2d 25704 | Deduction to prove measura... |
| mbfeqalem1 25705 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25706 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25707 | If two functions are equal... |
| mbfres 25708 | The restriction of a measu... |
| mbfres2 25709 | Measurability of a piecewi... |
| mbfss 25710 | Change the domain of a mea... |
| mbfmulc2lem 25711 | Multiplication by a consta... |
| mbfmulc2re 25712 | Multiplication by a consta... |
| mbfmax 25713 | The maximum of two functio... |
| mbfneg 25714 | The negative of a measurab... |
| mbfpos 25715 | The positive part of a mea... |
| mbfposr 25716 | Converse to ~ mbfpos . (C... |
| mbfposb 25717 | A function is measurable i... |
| ismbf3d 25718 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25719 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25720 | The preimage of any open s... |
| mbfimaopn2 25721 | The preimage of any set op... |
| cncombf 25722 | The composition of a conti... |
| cnmbf 25723 | A continuous function is m... |
| mbfaddlem 25724 | The sum of two measurable ... |
| mbfadd 25725 | The sum of two measurable ... |
| mbfsub 25726 | The difference of two meas... |
| mbfmulc2 25727 | A complex constant times a... |
| mbfsup 25728 | The supremum of a sequence... |
| mbfinf 25729 | The infimum of a sequence ... |
| mbflimsup 25730 | The limit supremum of a se... |
| mbflimlem 25731 | The pointwise limit of a s... |
| mbflim 25732 | The pointwise limit of a s... |
| 0pval 25735 | The zero function evaluate... |
| 0plef 25736 | Two ways to say that the f... |
| 0pledm 25737 | Adjust the domain of the l... |
| isi1f 25738 | The predicate " ` F ` is a... |
| i1fmbf 25739 | Simple functions are measu... |
| i1ff 25740 | A simple function is a fun... |
| i1frn 25741 | A simple function has fini... |
| i1fima 25742 | Any preimage of a simple f... |
| i1fima2 25743 | Any preimage of a simple f... |
| i1fima2sn 25744 | Preimage of a singleton. ... |
| i1fd 25745 | A simplified set of assump... |
| i1f0rn 25746 | Any simple function takes ... |
| itg1val 25747 | The value of the integral ... |
| itg1val2 25748 | The value of the integral ... |
| itg1cl 25749 | Closure of the integral on... |
| itg1ge0 25750 | Closure of the integral on... |
| i1f0 25751 | The zero function is simpl... |
| itg10 25752 | The zero function has zero... |
| i1f1lem 25753 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25754 | Base case simple functions... |
| itg11 25755 | The integral of an indicat... |
| itg1addlem1 25756 | Decompose a preimage, whic... |
| i1faddlem 25757 | Decompose the preimage of ... |
| i1fmullem 25758 | Decompose the preimage of ... |
| i1fadd 25759 | The sum of two simple func... |
| i1fmul 25760 | The pointwise product of t... |
| itg1addlem2 25761 | Lemma for ~ itg1add . The... |
| itg1addlem3 25762 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25763 | Lemma for ~ itg1add . (Co... |
| itg1addlem5 25764 | Lemma for ~ itg1add . (Co... |
| itg1add 25765 | The integral of a sum of s... |
| i1fmulclem 25766 | Decompose the preimage of ... |
| i1fmulc 25767 | A nonnegative constant tim... |
| itg1mulc 25768 | The integral of a constant... |
| i1fres 25769 | The "restriction" of a sim... |
| i1fpos 25770 | The positive part of a sim... |
| i1fposd 25771 | Deduction form of ~ i1fpos... |
| i1fsub 25772 | The difference of two simp... |
| itg1sub 25773 | The integral of a differen... |
| itg10a 25774 | The integral of a simple f... |
| itg1ge0a 25775 | The integral of an almost ... |
| itg1lea 25776 | Approximate version of ~ i... |
| itg1le 25777 | If one simple function dom... |
| itg1climres 25778 | Restricting the simple fun... |
| mbfi1fseqlem1 25779 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25780 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25781 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25782 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25783 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25784 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25785 | A characterization of meas... |
| mbfi1flimlem 25786 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25787 | Any real measurable functi... |
| mbfmullem2 25788 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25789 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25790 | The product of two measura... |
| itg2lcl 25791 | The set of lower sums is a... |
| itg2val 25792 | Value of the integral on n... |
| itg2l 25793 | Elementhood in the set ` L... |
| itg2lr 25794 | Sufficient condition for e... |
| xrge0f 25795 | A real function is a nonne... |
| itg2cl 25796 | The integral of a nonnegat... |
| itg2ub 25797 | The integral of a nonnegat... |
| itg2leub 25798 | Any upper bound on the int... |
| itg2ge0 25799 | The integral of a nonnegat... |
| itg2itg1 25800 | The integral of a nonnegat... |
| itg20 25801 | The integral of the zero f... |
| itg2lecl 25802 | If an ` S.2 ` integral is ... |
| itg2le 25803 | If one function dominates ... |
| itg2const 25804 | Integral of a constant fun... |
| itg2const2 25805 | When the base set of a con... |
| itg2seq 25806 | Definitional property of t... |
| itg2uba 25807 | Approximate version of ~ i... |
| itg2lea 25808 | Approximate version of ~ i... |
| itg2eqa 25809 | Approximate equality of in... |
| itg2mulclem 25810 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25811 | The integral of a nonnegat... |
| itg2splitlem 25812 | Lemma for ~ itg2split . (... |
| itg2split 25813 | The ` S.2 ` integral split... |
| itg2monolem1 25814 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25815 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25816 | Lemma for ~ itg2mono . (C... |
| itg2mono 25817 | The Monotone Convergence T... |
| itg2i1fseqle 25818 | Subject to the conditions ... |
| itg2i1fseq 25819 | Subject to the conditions ... |
| itg2i1fseq2 25820 | In an extension to the res... |
| itg2i1fseq3 25821 | Special case of ~ itg2i1fs... |
| itg2addlem 25822 | Lemma for ~ itg2add . (Co... |
| itg2add 25823 | The ` S.2 ` integral is li... |
| itg2gt0 25824 | If the function ` F ` is s... |
| itg2cnlem1 25825 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25826 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25827 | A sort of absolute continu... |
| ibllem 25828 | Conditioned equality theor... |
| isibl 25829 | The predicate " ` F ` is i... |
| isibl2 25830 | The predicate " ` F ` is i... |
| iblmbf 25831 | An integrable function is ... |
| iblitg 25832 | If a function is integrabl... |
| dfitg 25833 | Evaluate the class substit... |
| itgex 25834 | An integral is a set. (Co... |
| itgeq1f 25835 | Equality theorem for an in... |
| itgeq1fOLD 25836 | Obsolete version of ~ itge... |
| itgeq1 25837 | Equality theorem for an in... |
| nfitg1 25838 | Bound-variable hypothesis ... |
| nfitg 25839 | Bound-variable hypothesis ... |
| cbvitg 25840 | Change bound variable in a... |
| cbvitgv 25841 | Change bound variable in a... |
| itgeq2 25842 | Equality theorem for an in... |
| itgresr 25843 | The domain of an integral ... |
| itg0 25844 | The integral of anything o... |
| itgz 25845 | The integral of zero on an... |
| itgeq2dv 25846 | Equality theorem for an in... |
| itgmpt 25847 | Change bound variable in a... |
| itgcl 25848 | The integral of an integra... |
| itgvallem 25849 | Substitution lemma. (Cont... |
| itgvallem3 25850 | Lemma for ~ itgposval and ... |
| ibl0 25851 | The zero function is integ... |
| iblcnlem1 25852 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25853 | Expand out the universal q... |
| itgcnlem 25854 | Expand out the sum in ~ df... |
| iblrelem 25855 | Integrability of a real fu... |
| iblposlem 25856 | Lemma for ~ iblpos . (Con... |
| iblpos 25857 | Integrability of a nonnega... |
| iblre 25858 | Integrability of a real fu... |
| itgrevallem1 25859 | Lemma for ~ itgposval and ... |
| itgposval 25860 | The integral of a nonnegat... |
| itgreval 25861 | Decompose the integral of ... |
| itgrecl 25862 | Real closure of an integra... |
| iblcn 25863 | Integrability of a complex... |
| itgcnval 25864 | Decompose the integral of ... |
| itgre 25865 | Real part of an integral. ... |
| itgim 25866 | Imaginary part of an integ... |
| iblneg 25867 | The negative of an integra... |
| itgneg 25868 | Negation of an integral. ... |
| iblss 25869 | A subset of an integrable ... |
| iblss2 25870 | Change the domain of an in... |
| itgitg2 25871 | Transfer an integral using... |
| i1fibl 25872 | A simple function is integ... |
| itgitg1 25873 | Transfer an integral using... |
| itgle 25874 | Monotonicity of an integra... |
| itgge0 25875 | The integral of a positive... |
| itgss 25876 | Expand the set of an integ... |
| itgss2 25877 | Expand the set of an integ... |
| itgeqa 25878 | Approximate equality of in... |
| itgss3 25879 | Expand the set of an integ... |
| itgioo 25880 | Equality of integrals on o... |
| itgless 25881 | Expand the integral of a n... |
| iblconst 25882 | A constant function is int... |
| itgconst 25883 | Integral of a constant fun... |
| ibladdlem 25884 | Lemma for ~ ibladd . (Con... |
| ibladd 25885 | Add two integrals over the... |
| iblsub 25886 | Subtract two integrals ove... |
| itgaddlem1 25887 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25888 | Lemma for ~ itgadd . (Con... |
| itgadd 25889 | Add two integrals over the... |
| itgsub 25890 | Subtract two integrals ove... |
| itgfsum 25891 | Take a finite sum of integ... |
| iblabslem 25892 | Lemma for ~ iblabs . (Con... |
| iblabs 25893 | The absolute value of an i... |
| iblabsr 25894 | A measurable function is i... |
| iblmulc2 25895 | Multiply an integral by a ... |
| itgmulc2lem1 25896 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25897 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25898 | Multiply an integral by a ... |
| itgabs 25899 | The triangle inequality fo... |
| itgsplit 25900 | The ` S. ` integral splits... |
| itgspliticc 25901 | The ` S. ` integral splits... |
| itgsplitioo 25902 | The ` S. ` integral splits... |
| bddmulibl 25903 | A bounded function times a... |
| bddibl 25904 | A bounded function is inte... |
| cniccibl 25905 | A continuous function on a... |
| bddiblnc 25906 | Choice-free proof of ~ bdd... |
| cnicciblnc 25907 | Choice-free proof of ~ cni... |
| itggt0 25908 | The integral of a strictly... |
| itgcn 25909 | Transfer ~ itg2cn to the f... |
| ditgeq1 25912 | Equality theorem for the d... |
| ditgeq2 25913 | Equality theorem for the d... |
| ditgeq3 25914 | Equality theorem for the d... |
| ditgeq3dv 25915 | Equality theorem for the d... |
| ditgex 25916 | A directed integral is a s... |
| ditg0 25917 | Value of the directed inte... |
| cbvditg 25918 | Change bound variable in a... |
| cbvditgv 25919 | Change bound variable in a... |
| ditgpos 25920 | Value of the directed inte... |
| ditgneg 25921 | Value of the directed inte... |
| ditgcl 25922 | Closure of a directed inte... |
| ditgswap 25923 | Reverse a directed integra... |
| ditgsplitlem 25924 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25925 | This theorem is the raison... |
| reldv 25934 | The derivative function is... |
| limcvallem 25935 | Lemma for ~ ellimc . (Con... |
| limcfval 25936 | Value and set bounds on th... |
| ellimc 25937 | Value of the limit predica... |
| limcrcl 25938 | Reverse closure for the li... |
| limccl 25939 | Closure of the limit opera... |
| limcdif 25940 | It suffices to consider fu... |
| ellimc2 25941 | Write the definition of a ... |
| limcnlp 25942 | If ` B ` is not a limit po... |
| ellimc3 25943 | Write the epsilon-delta de... |
| limcflflem 25944 | Lemma for ~ limcflf . (Co... |
| limcflf 25945 | The limit operator can be ... |
| limcmo 25946 | If ` B ` is a limit point ... |
| limcmpt 25947 | Express the limit operator... |
| limcmpt2 25948 | Express the limit operator... |
| limcresi 25949 | Any limit of ` F ` is also... |
| limcres 25950 | If ` B ` is an interior po... |
| cnplimc 25951 | A function is continuous a... |
| cnlimc 25952 | ` F ` is a continuous func... |
| cnlimci 25953 | If ` F ` is a continuous f... |
| cnmptlimc 25954 | If ` F ` is a continuous f... |
| limccnp 25955 | If the limit of ` F ` at `... |
| limccnp2 25956 | The image of a convergent ... |
| limcco 25957 | Composition of two limits.... |
| limciun 25958 | A point is a limit of ` F ... |
| limcun 25959 | A point is a limit of ` F ... |
| dvlem 25960 | Closure for a difference q... |
| dvfval 25961 | Value and set bounds on th... |
| eldv 25962 | The differentiable predica... |
| dvcl 25963 | The derivative function ta... |
| dvbssntr 25964 | The set of differentiable ... |
| dvbss 25965 | The set of differentiable ... |
| dvbsss 25966 | The set of differentiable ... |
| perfdvf 25967 | The derivative is a functi... |
| recnprss 25968 | Both ` RR ` and ` CC ` are... |
| recnperf 25969 | Both ` RR ` and ` CC ` are... |
| dvfg 25970 | Explicitly write out the f... |
| dvf 25971 | The derivative is a functi... |
| dvfcn 25972 | The derivative is a functi... |
| dvreslem 25973 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25974 | Lemma for ~ dvres2 . (Con... |
| dvres 25975 | Restriction of a derivativ... |
| dvres2 25976 | Restriction of the base se... |
| dvres3 25977 | Restriction of a complex d... |
| dvres3a 25978 | Restriction of a complex d... |
| dvidlem 25979 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25980 | Derivative of a function r... |
| dvconst 25981 | Derivative of a constant f... |
| dvid 25982 | Derivative of the identity... |
| dvcnp 25983 | The difference quotient is... |
| dvcnp2 25984 | A function is continuous a... |
| dvcn 25985 | A differentiable function ... |
| dvnfval 25986 | Value of the iterated deri... |
| dvnff 25987 | The iterated derivative is... |
| dvn0 25988 | Zero times iterated deriva... |
| dvnp1 25989 | Successor iterated derivat... |
| dvn1 25990 | One times iterated derivat... |
| dvnf 25991 | The N-times derivative is ... |
| dvnbss 25992 | The set of N-times differe... |
| dvnadd 25993 | The ` N ` -th derivative o... |
| dvn2bss 25994 | An N-times differentiable ... |
| dvnres 25995 | Multiple derivative versio... |
| cpnfval 25996 | Condition for n-times cont... |
| fncpn 25997 | The ` C^n ` object is a fu... |
| elcpn 25998 | Condition for n-times cont... |
| cpnord 25999 | ` C^n ` conditions are ord... |
| cpncn 26000 | A ` C^n ` function is cont... |
| cpnres 26001 | The restriction of a ` C^n... |
| dvaddbr 26002 | The sum rule for derivativ... |
| dvmulbr 26003 | The product rule for deriv... |
| dvadd 26004 | The sum rule for derivativ... |
| dvmul 26005 | The product rule for deriv... |
| dvaddf 26006 | The sum rule for everywher... |
| dvmulf 26007 | The product rule for every... |
| dvcmul 26008 | The product rule when one ... |
| dvcmulf 26009 | The product rule when one ... |
| dvcobr 26010 | The chain rule for derivat... |
| dvco 26011 | The chain rule for derivat... |
| dvcof 26012 | The chain rule for everywh... |
| dvcjbr 26013 | The derivative of the conj... |
| dvcj 26014 | The derivative of the conj... |
| dvfre 26015 | The derivative of a real f... |
| dvnfre 26016 | The ` N ` -th derivative o... |
| dvexp 26017 | Derivative of a power func... |
| dvexp2 26018 | Derivative of an exponenti... |
| dvrec 26019 | Derivative of the reciproc... |
| dvmptres3 26020 | Function-builder for deriv... |
| dvmptid 26021 | Function-builder for deriv... |
| dvmptc 26022 | Function-builder for deriv... |
| dvmptcl 26023 | Closure lemma for ~ dvmptc... |
| dvmptadd 26024 | Function-builder for deriv... |
| dvmptmul 26025 | Function-builder for deriv... |
| dvmptres2 26026 | Function-builder for deriv... |
| dvmptres 26027 | Function-builder for deriv... |
| dvmptcmul 26028 | Function-builder for deriv... |
| dvmptdivc 26029 | Function-builder for deriv... |
| dvmptneg 26030 | Function-builder for deriv... |
| dvmptsub 26031 | Function-builder for deriv... |
| dvmptcj 26032 | Function-builder for deriv... |
| dvmptre 26033 | Function-builder for deriv... |
| dvmptim 26034 | Function-builder for deriv... |
| dvmptntr 26035 | Function-builder for deriv... |
| dvmptco 26036 | Function-builder for deriv... |
| dvrecg 26037 | Derivative of the reciproc... |
| dvmptdiv 26038 | Function-builder for deriv... |
| dvmptfsum 26039 | Function-builder for deriv... |
| dvcnvlem 26040 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 26041 | A weak version of ~ dvcnvr... |
| dvexp3 26042 | Derivative of an exponenti... |
| dveflem 26043 | Derivative of the exponent... |
| dvef 26044 | Derivative of the exponent... |
| dvsincos 26045 | Derivative of the sine and... |
| dvsin 26046 | Derivative of the sine fun... |
| dvcos 26047 | Derivative of the cosine f... |
| dvferm1lem 26048 | Lemma for ~ dvferm . (Con... |
| dvferm1 26049 | One-sided version of ~ dvf... |
| dvferm2lem 26050 | Lemma for ~ dvferm . (Con... |
| dvferm2 26051 | One-sided version of ~ dvf... |
| dvferm 26052 | Fermat's theorem on statio... |
| rollelem 26053 | Lemma for ~ rolle . (Cont... |
| rolle 26054 | Rolle's theorem. If ` F `... |
| cmvth 26055 | Cauchy's Mean Value Theore... |
| mvth 26056 | The Mean Value Theorem. I... |
| dvlip 26057 | A function with derivative... |
| dvlipcn 26058 | A complex function with de... |
| dvlip2 26059 | Combine the results of ~ d... |
| c1liplem1 26060 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 26061 | C^1 functions are Lipschit... |
| c1lip2 26062 | C^1 functions are Lipschit... |
| c1lip3 26063 | C^1 functions are Lipschit... |
| dveq0 26064 | If a continuous function h... |
| dv11cn 26065 | Two functions defined on a... |
| dvgt0lem1 26066 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 26067 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 26068 | A function on a closed int... |
| dvlt0 26069 | A function on a closed int... |
| dvge0 26070 | A function on a closed int... |
| dvle 26071 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 26072 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 26073 | Lemma for ~ dvivth . (Con... |
| dvivth 26074 | Darboux' theorem, or the i... |
| dvne0 26075 | A function on a closed int... |
| dvne0f1 26076 | A function on a closed int... |
| lhop1lem 26077 | Lemma for ~ lhop1 . (Cont... |
| lhop1 26078 | L'Hôpital's Rule for... |
| lhop2 26079 | L'Hôpital's Rule for... |
| lhop 26080 | L'Hôpital's Rule. I... |
| dvcnvrelem1 26081 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 26082 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 26083 | The derivative rule for in... |
| dvcvx 26084 | A real function with stric... |
| dvfsumle 26085 | Compare a finite sum to an... |
| dvfsumge 26086 | Compare a finite sum to an... |
| dvfsumabs 26087 | Compare a finite sum to an... |
| dvmptrecl 26088 | Real closure of a derivati... |
| dvfsumrlimf 26089 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 26090 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 26091 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem3 26092 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 26093 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 26094 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 26095 | Compare a finite sum to an... |
| dvfsumrlim2 26096 | Compare a finite sum to an... |
| dvfsumrlim3 26097 | Conjoin the statements of ... |
| dvfsum2 26098 | The reverse of ~ dvfsumrli... |
| ftc1lem1 26099 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 26100 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 26101 | The Fundamental Theorem of... |
| ftc1lem3 26102 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 26103 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 26104 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 26105 | Lemma for ~ ftc1 . (Contr... |
| ftc1 26106 | The Fundamental Theorem of... |
| ftc1cn 26107 | Strengthen the assumptions... |
| ftc2 26108 | The Fundamental Theorem of... |
| ftc2ditglem 26109 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 26110 | Directed integral analogue... |
| itgparts 26111 | Integration by parts. If ... |
| itgsubstlem 26112 | Lemma for ~ itgsubst . (C... |
| itgsubst 26113 | Integration by ` u ` -subs... |
| itgpowd 26114 | The integral of a monomial... |
| reldmmdeg 26119 | Multivariate degree is a b... |
| tdeglem1 26120 | Functionality of the total... |
| tdeglem3 26121 | Additivity of the total de... |
| tdeglem4 26122 | There is only one multi-in... |
| tdeglem2 26123 | Simplification of total de... |
| mdegfval 26124 | Value of the multivariate ... |
| mdegval 26125 | Value of the multivariate ... |
| mdegleb 26126 | Property of being of limit... |
| mdeglt 26127 | If there is an upper limit... |
| mdegldg 26128 | A nonzero polynomial has s... |
| mdegxrcl 26129 | Closure of polynomial degr... |
| mdegxrf 26130 | Functionality of polynomia... |
| mdegcl 26131 | Sharp closure for multivar... |
| mdeg0 26132 | Degree of the zero polynom... |
| mdegnn0cl 26133 | Degree of a nonzero polyno... |
| degltlem1 26134 | Theorem on arithmetic of e... |
| degltp1le 26135 | Theorem on arithmetic of e... |
| mdegaddle 26136 | The degree of a sum is at ... |
| mdegvscale 26137 | The degree of a scalar mul... |
| mdegvsca 26138 | The degree of a scalar mul... |
| mdegle0 26139 | A polynomial has nonpositi... |
| mdegmullem 26140 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 26141 | The multivariate degree of... |
| deg1fval 26142 | Relate univariate polynomi... |
| deg1xrf 26143 | Functionality of univariat... |
| deg1xrcl 26144 | Closure of univariate poly... |
| deg1cl 26145 | Sharp closure of univariat... |
| mdegpropd 26146 | Property deduction for pol... |
| deg1fvi 26147 | Univariate polynomial degr... |
| deg1propd 26148 | Property deduction for pol... |
| deg1z 26149 | Degree of the zero univari... |
| deg1nn0cl 26150 | Degree of a nonzero univar... |
| deg1n0ima 26151 | Degree image of a set of p... |
| deg1nn0clb 26152 | A polynomial is nonzero if... |
| deg1lt0 26153 | A polynomial is zero iff i... |
| deg1ldg 26154 | A nonzero univariate polyn... |
| deg1ldgn 26155 | An index at which a polyno... |
| deg1ldgdomn 26156 | A nonzero univariate polyn... |
| deg1leb 26157 | Property of being of limit... |
| deg1val 26158 | Value of the univariate de... |
| deg1lt 26159 | If the degree of a univari... |
| deg1ge 26160 | Conversely, a nonzero coef... |
| coe1mul3 26161 | The coefficient vector of ... |
| coe1mul4 26162 | Value of the "leading" coe... |
| deg1addle 26163 | The degree of a sum is at ... |
| deg1addle2 26164 | If both factors have degre... |
| deg1add 26165 | Exact degree of a sum of t... |
| deg1vscale 26166 | The degree of a scalar tim... |
| deg1vsca 26167 | The degree of a scalar tim... |
| deg1invg 26168 | The degree of the negated ... |
| deg1suble 26169 | The degree of a difference... |
| deg1sub 26170 | Exact degree of a differen... |
| deg1mulle2 26171 | Produce a bound on the pro... |
| deg1sublt 26172 | Subtraction of two polynom... |
| deg1le0 26173 | A polynomial has nonpositi... |
| deg1sclle 26174 | A scalar polynomial has no... |
| deg1scl 26175 | A nonzero scalar polynomia... |
| deg1mul2 26176 | Degree of multiplication o... |
| deg1mul 26177 | Degree of multiplication o... |
| deg1mul3 26178 | Degree of multiplication o... |
| deg1mul3le 26179 | Degree of multiplication o... |
| deg1tmle 26180 | Limiting degree of a polyn... |
| deg1tm 26181 | Exact degree of a polynomi... |
| deg1pwle 26182 | Limiting degree of a varia... |
| deg1pw 26183 | Exact degree of a variable... |
| ply1nz 26184 | Univariate polynomials ove... |
| ply1nzb 26185 | Univariate polynomials are... |
| ply1domn 26186 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26187 | The ring of univariate pol... |
| ply1divmo 26198 | Uniqueness of a quotient i... |
| ply1divex 26199 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26200 | The division algorithm for... |
| ply1divalg2 26201 | Reverse the order of multi... |
| uc1pval 26202 | Value of the set of unitic... |
| isuc1p 26203 | Being a unitic polynomial.... |
| mon1pval 26204 | Value of the set of monic ... |
| ismon1p 26205 | Being a monic polynomial. ... |
| uc1pcl 26206 | Unitic polynomials are pol... |
| mon1pcl 26207 | Monic polynomials are poly... |
| uc1pn0 26208 | Unitic polynomials are not... |
| mon1pn0 26209 | Monic polynomials are not ... |
| uc1pdeg 26210 | Unitic polynomials have no... |
| uc1pldg 26211 | Unitic polynomials have un... |
| mon1pldg 26212 | Unitic polynomials have on... |
| mon1puc1p 26213 | Monic polynomials are unit... |
| uc1pmon1p 26214 | Make a unitic polynomial m... |
| deg1submon1p 26215 | The difference of two moni... |
| mon1pid 26216 | Monicity and degree of the... |
| q1pval 26217 | Value of the univariate po... |
| q1peqb 26218 | Characterizing property of... |
| q1pcl 26219 | Closure of the quotient by... |
| r1pval 26220 | Value of the polynomial re... |
| r1pcl 26221 | Closure of remainder follo... |
| r1pdeglt 26222 | The remainder has a degree... |
| r1pid 26223 | Express the original polyn... |
| r1pid2 26224 | Identity law for polynomia... |
| dvdsq1p 26225 | Divisibility in a polynomi... |
| dvdsr1p 26226 | Divisibility in a polynomi... |
| ply1remlem 26227 | A term of the form ` x - N... |
| ply1rem 26228 | The polynomial remainder t... |
| facth1 26229 | The factor theorem and its... |
| fta1glem1 26230 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26231 | Lemma for ~ fta1g . (Cont... |
| fta1g 26232 | The one-sided fundamental ... |
| fta1blem 26233 | Lemma for ~ fta1b . (Cont... |
| fta1b 26234 | The assumption that ` R ` ... |
| idomrootle 26235 | No element of an integral ... |
| drnguc1p 26236 | Over a division ring, all ... |
| ig1peu 26237 | There is a unique monic po... |
| ig1pval 26238 | Substitutions for the poly... |
| ig1pval2 26239 | Generator of the zero idea... |
| ig1pval3 26240 | Characterizing properties ... |
| ig1pcl 26241 | The monic generator of an ... |
| ig1pdvds 26242 | The monic generator of an ... |
| ig1prsp 26243 | Any ideal of polynomials o... |
| ply1lpir 26244 | The ring of polynomials ov... |
| ply1pid 26245 | The polynomials over a fie... |
| plyco0 26254 | Two ways to say that a fun... |
| plyval 26255 | Value of the polynomial se... |
| plybss 26256 | Reverse closure of the par... |
| elply 26257 | Definition of a polynomial... |
| elply2 26258 | The coefficient function c... |
| plyun0 26259 | The set of polynomials is ... |
| plyf 26260 | A polynomial is a function... |
| plyss 26261 | The polynomial set functio... |
| plyssc 26262 | Every polynomial ring is c... |
| elplyr 26263 | Sufficient condition for e... |
| elplyd 26264 | Sufficient condition for e... |
| ply1termlem 26265 | Lemma for ~ ply1term . (C... |
| ply1term 26266 | A one-term polynomial. (C... |
| plypow 26267 | A power is a polynomial. ... |
| plyconst 26268 | A constant function is a p... |
| ne0p 26269 | A test to show that a poly... |
| ply0 26270 | The zero function is a pol... |
| plyid 26271 | The identity function is a... |
| plyeq0lem 26272 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26273 | If a polynomial is zero at... |
| plypf1 26274 | Write the set of complex p... |
| plyaddlem1 26275 | Derive the coefficient fun... |
| plymullem1 26276 | Derive the coefficient fun... |
| plyaddlem 26277 | Lemma for ~ plyadd . (Con... |
| plymullem 26278 | Lemma for ~ plymul . (Con... |
| plyadd 26279 | The sum of two polynomials... |
| plymul 26280 | The product of two polynom... |
| plysub 26281 | The difference of two poly... |
| plyaddcl 26282 | The sum of two polynomials... |
| plymulcl 26283 | The product of two polynom... |
| plysubcl 26284 | The difference of two poly... |
| coeval 26285 | Value of the coefficient f... |
| coeeulem 26286 | Lemma for ~ coeeu . (Cont... |
| coeeu 26287 | Uniqueness of the coeffici... |
| coelem 26288 | Lemma for properties of th... |
| coeeq 26289 | If ` A ` satisfies the pro... |
| dgrval 26290 | Value of the degree functi... |
| dgrlem 26291 | Lemma for ~ dgrcl and simi... |
| coef 26292 | The domain and codomain of... |
| coef2 26293 | The domain and codomain of... |
| coef3 26294 | The domain and codomain of... |
| dgrcl 26295 | The degree of any polynomi... |
| dgrub 26296 | If the ` M ` -th coefficie... |
| dgrub2 26297 | All the coefficients above... |
| dgrlb 26298 | If all the coefficients ab... |
| coeidlem 26299 | Lemma for ~ coeid . (Cont... |
| coeid 26300 | Reconstruct a polynomial a... |
| coeid2 26301 | Reconstruct a polynomial a... |
| coeid3 26302 | Reconstruct a polynomial a... |
| plyco 26303 | The composition of two pol... |
| coeeq2 26304 | Compute the coefficient fu... |
| dgrle 26305 | Given an explicit expressi... |
| dgreq 26306 | If the highest term in a p... |
| 0dgr 26307 | A constant function has de... |
| 0dgrb 26308 | A function has degree zero... |
| dgrnznn 26309 | A nonzero polynomial with ... |
| coefv0 26310 | The result of evaluating a... |
| coeaddlem 26311 | Lemma for ~ coeadd and ~ d... |
| coemullem 26312 | Lemma for ~ coemul and ~ d... |
| coeadd 26313 | The coefficient function o... |
| coemul 26314 | A coefficient of a product... |
| coe11 26315 | The coefficient function i... |
| coemulhi 26316 | The leading coefficient of... |
| coemulc 26317 | The coefficient function i... |
| coe0 26318 | The coefficients of the ze... |
| coesub 26319 | The coefficient function o... |
| coe1termlem 26320 | The coefficient function o... |
| coe1term 26321 | The coefficient function o... |
| dgr1term 26322 | The degree of a monomial. ... |
| plycn 26323 | A polynomial is a continuo... |
| dgr0 26324 | The degree of the zero pol... |
| coeidp 26325 | The coefficients of the id... |
| dgrid 26326 | The degree of the identity... |
| dgreq0 26327 | The leading coefficient of... |
| dgrlt 26328 | Two ways to say that the d... |
| dgradd 26329 | The degree of a sum of pol... |
| dgradd2 26330 | The degree of a sum of pol... |
| dgrmul2 26331 | The degree of a product of... |
| dgrmul 26332 | The degree of a product of... |
| dgrmulc 26333 | Scalar multiplication by a... |
| dgrsub 26334 | The degree of a difference... |
| dgrcolem1 26335 | The degree of a compositio... |
| dgrcolem2 26336 | Lemma for ~ dgrco . (Cont... |
| dgrco 26337 | The degree of a compositio... |
| plycjlem 26338 | Lemma for ~ plycj and ~ co... |
| plycj 26339 | The double conjugation of ... |
| coecj 26340 | Double conjugation of a po... |
| plycjOLD 26341 | Obsolete version of ~ plyc... |
| coecjOLD 26342 | Obsolete version of ~ coec... |
| plyrecj 26343 | A polynomial with real coe... |
| plymul0or 26344 | Polynomial multiplication ... |
| ofmulrt 26345 | The set of roots of a prod... |
| plymul02 26346 | Product of a polynomial wi... |
| plyn0mulidp 26347 | Coefficients of a non-zero... |
| plymulidp 26348 | Coefficients of a polynomi... |
| plyreres 26349 | Real-coefficient polynomia... |
| dvply1 26350 | Derivative of a polynomial... |
| dvply2g 26351 | The derivative of a polyno... |
| dvply2 26352 | The derivative of a polyno... |
| dvnply2 26353 | Polynomials have polynomia... |
| dvnply 26354 | Polynomials have polynomia... |
| plycpn 26355 | Polynomials are smooth. (... |
| quotval 26358 | Value of the quotient func... |
| plydivlem1 26359 | Lemma for ~ plydivalg . (... |
| plydivlem2 26360 | Lemma for ~ plydivalg . (... |
| plydivlem3 26361 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26362 | Lemma for ~ plydivex . In... |
| plydivex 26363 | Lemma for ~ plydivalg . (... |
| plydiveu 26364 | Lemma for ~ plydivalg . (... |
| plydivalg 26365 | The division algorithm on ... |
| quotlem 26366 | Lemma for properties of th... |
| quotcl 26367 | The quotient of two polyno... |
| quotcl2 26368 | Closure of the quotient fu... |
| quotdgr 26369 | Remainder property of the ... |
| plyremlem 26370 | Closure of a linear factor... |
| plyrem 26371 | The polynomial remainder t... |
| facth 26372 | The factor theorem. If a ... |
| fta1lem 26373 | Lemma for ~ fta1 . (Contr... |
| fta1 26374 | The easy direction of the ... |
| quotcan 26375 | Exact division with a mult... |
| vieta1lem1 26376 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26377 | Lemma for ~ vieta1 : induc... |
| vieta1 26378 | The first-order Vieta's fo... |
| plyexmo 26379 | An infinite set of values ... |
| elaa 26382 | Elementhood in the set of ... |
| aacn 26383 | An algebraic number is a c... |
| aasscn 26384 | The algebraic numbers are ... |
| elqaalem1 26385 | Lemma for ~ elqaa . The f... |
| elqaalem2 26386 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26387 | Lemma for ~ elqaa . (Cont... |
| elqaa 26388 | The set of numbers generat... |
| qaa 26389 | Every rational number is a... |
| qssaa 26390 | The rational numbers are c... |
| iaa 26391 | The imaginary unit is alge... |
| aareccl 26392 | The reciprocal of an algeb... |
| aacjcl 26393 | The conjugate of an algebr... |
| aannenlem1 26394 | Lemma for ~ aannen . (Con... |
| aannenlem2 26395 | Lemma for ~ aannen . (Con... |
| aannenlem3 26396 | The algebraic numbers are ... |
| aannen 26397 | The algebraic numbers are ... |
| aalioulem1 26398 | Lemma for ~ aaliou . An i... |
| aalioulem2 26399 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26400 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26401 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26402 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26403 | Lemma for ~ aaliou . (Con... |
| aaliou 26404 | Liouville's theorem on dio... |
| geolim3 26405 | Geometric series convergen... |
| aaliou2 26406 | Liouville's approximation ... |
| aaliou2b 26407 | Liouville's approximation ... |
| aaliou3lem1 26408 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26409 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26410 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26411 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26412 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26413 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26414 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26415 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26416 | Example of a "Liouville nu... |
| aaliou3 26417 | Example of a "Liouville nu... |
| taylfvallem1 26422 | Lemma for ~ taylfval . (C... |
| taylfvallem 26423 | Lemma for ~ taylfval . (C... |
| taylfval 26424 | Define the Taylor polynomi... |
| eltayl 26425 | Value of the Taylor series... |
| taylf 26426 | The Taylor series defines ... |
| tayl0 26427 | The Taylor series is alway... |
| taylplem1 26428 | Lemma for ~ taylpfval and ... |
| taylplem2 26429 | Lemma for ~ taylpfval and ... |
| taylpfval 26430 | Define the Taylor polynomi... |
| taylpf 26431 | The Taylor polynomial is a... |
| taylpval 26432 | Value of the Taylor polyno... |
| taylply2 26433 | The Taylor polynomial is a... |
| taylply 26434 | The Taylor polynomial is a... |
| dvtaylp 26435 | The derivative of the Tayl... |
| dvntaylp 26436 | The ` M ` -th derivative o... |
| dvntaylp0 26437 | The first ` N ` derivative... |
| taylthlem1 26438 | Lemma for ~ taylth . This... |
| taylthlem2 26439 | Lemma for ~ taylth . (Con... |
| taylth 26440 | Taylor's theorem. The Tay... |
| ulmrel 26443 | The uniform limit relation... |
| ulmscl 26444 | Closure of the base set in... |
| ulmval 26445 | Express the predicate: Th... |
| ulmcl 26446 | Closure of a uniform limit... |
| ulmf 26447 | Closure of a uniform limit... |
| ulmpm 26448 | Closure of a uniform limit... |
| ulmf2 26449 | Closure of a uniform limit... |
| ulm2 26450 | Simplify ~ ulmval when ` F... |
| ulmi 26451 | The uniform limit property... |
| ulmclm 26452 | A uniform limit of functio... |
| ulmres 26453 | A sequence of functions co... |
| ulmshftlem 26454 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26455 | A sequence of functions co... |
| ulm0 26456 | Every function converges u... |
| ulmuni 26457 | A sequence of functions un... |
| ulmdm 26458 | Two ways to express that a... |
| ulmcaulem 26459 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26460 | A sequence of functions co... |
| ulmcau2 26461 | A sequence of functions co... |
| ulmss 26462 | A uniform limit of functio... |
| ulmbdd 26463 | A uniform limit of bounded... |
| ulmcn 26464 | A uniform limit of continu... |
| ulmdvlem1 26465 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26466 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26467 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26468 | If ` F ` is a sequence of ... |
| mtest 26469 | The Weierstrass M-test. I... |
| mtestbdd 26470 | Given the hypotheses of th... |
| mbfulm 26471 | A uniform limit of measura... |
| iblulm 26472 | A uniform limit of integra... |
| itgulm 26473 | A uniform limit of integra... |
| itgulm2 26474 | A uniform limit of integra... |
| pserval 26475 | Value of the function ` G ... |
| pserval2 26476 | Value of the function ` G ... |
| psergf 26477 | The sequence of terms in t... |
| radcnvlem1 26478 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26479 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26480 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26481 | Zero is always a convergen... |
| radcnvcl 26482 | The radius of convergence ... |
| radcnvlt1 26483 | If ` X ` is within the ope... |
| radcnvlt2 26484 | If ` X ` is within the ope... |
| radcnvle 26485 | If ` X ` is a convergent p... |
| dvradcnv 26486 | The radius of convergence ... |
| pserulm 26487 | If ` S ` is a region conta... |
| psercn2 26488 | Since by ~ pserulm the ser... |
| psercnlem2 26489 | Lemma for ~ psercn . (Con... |
| psercnlem1 26490 | Lemma for ~ psercn . (Con... |
| psercn 26491 | An infinite series converg... |
| pserdvlem1 26492 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26493 | Lemma for ~ pserdv . (Con... |
| pserdv 26494 | The derivative of a power ... |
| pserdv2 26495 | The derivative of a power ... |
| abelthlem1 26496 | Lemma for ~ abelth . (Con... |
| abelthlem2 26497 | Lemma for ~ abelth . The ... |
| abelthlem3 26498 | Lemma for ~ abelth . (Con... |
| abelthlem4 26499 | Lemma for ~ abelth . (Con... |
| abelthlem5 26500 | Lemma for ~ abelth . (Con... |
| abelthlem6 26501 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26502 | Lemma for ~ abelth . (Con... |
| abelthlem7 26503 | Lemma for ~ abelth . (Con... |
| abelthlem8 26504 | Lemma for ~ abelth . (Con... |
| abelthlem9 26505 | Lemma for ~ abelth . By a... |
| abelth 26506 | Abel's theorem. If the po... |
| abelth2 26507 | Abel's theorem, restricted... |
| efcn 26508 | The exponential function i... |
| sincn 26509 | Sine is continuous. (Cont... |
| coscn 26510 | Cosine is continuous. (Co... |
| reeff1olem 26511 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26512 | The real exponential funct... |
| reefiso 26513 | The exponential function o... |
| efcvx 26514 | The exponential function o... |
| reefgim 26515 | The exponential function i... |
| pilem1 26516 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26517 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26518 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26519 | ` _pi ` is between 2 and 4... |
| sinpi 26520 | The sine of ` _pi ` is 0. ... |
| pire 26521 | ` _pi ` is a real number. ... |
| 2pire 26522 | ` ( 2 x. _pi ) ` is a real... |
| picn 26523 | ` _pi ` is a complex numbe... |
| 2picn 26524 | ` ( 2 x. _pi ) ` is a comp... |
| pipos 26525 | ` _pi ` is positive. (Con... |
| pige0 26526 | ` _pi ` is nonnegative. (... |
| pine0 26527 | ` _pi ` is nonzero. (Cont... |
| pirp 26528 | ` _pi ` is a positive real... |
| negpicn 26529 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 26530 | Lemma for ~ sinhalfpi and ... |
| halfpire 26531 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26532 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26533 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26534 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26535 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26536 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26537 | The cosine of ` -u _pi / 2... |
| efhalfpi 26538 | The exponential of ` _i _p... |
| cospi 26539 | The cosine of ` _pi ` is `... |
| efipi 26540 | The exponential of ` _i x.... |
| eulerid 26541 | Euler's identity. (Contri... |
| sin2pi 26542 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26543 | The cosine of ` 2 _pi ` is... |
| ef2pi 26544 | The exponential of ` 2 _pi... |
| ef2kpi 26545 | If ` K ` is an integer, th... |
| efper 26546 | The exponential function i... |
| sinperlem 26547 | Lemma for ~ sinper and ~ c... |
| sinper 26548 | The sine function is perio... |
| cosper 26549 | The cosine function is per... |
| sin2kpi 26550 | If ` K ` is an integer, th... |
| cos2kpi 26551 | If ` K ` is an integer, th... |
| sin2pim 26552 | Sine of a number subtracte... |
| cos2pim 26553 | Cosine of a number subtrac... |
| sinmpi 26554 | Sine of a number less ` _p... |
| cosmpi 26555 | Cosine of a number less ` ... |
| sinppi 26556 | Sine of a number plus ` _p... |
| cosppi 26557 | Cosine of a number plus ` ... |
| efimpi 26558 | The exponential function a... |
| sinhalfpip 26559 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26560 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26561 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26562 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26563 | Ptolemy's Theorem. This t... |
| sincosq1lem 26564 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26565 | The signs of the sine and ... |
| sincosq2sgn 26566 | The signs of the sine and ... |
| sincosq3sgn 26567 | The signs of the sine and ... |
| sincosq4sgn 26568 | The signs of the sine and ... |
| coseq00topi 26569 | Location of the zeroes of ... |
| coseq0negpitopi 26570 | Location of the zeroes of ... |
| tanrpcl 26571 | Positive real closure of t... |
| tangtx 26572 | The tangent function is gr... |
| tanabsge 26573 | The tangent function is gr... |
| sinq12gt0 26574 | The sine of a number stric... |
| sinq12ge0 26575 | The sine of a number betwe... |
| sinq34lt0t 26576 | The sine of a number stric... |
| cosq14gt0 26577 | The cosine of a number str... |
| cosq14ge0 26578 | The cosine of a number bet... |
| sincosq1eq 26579 | Complementarity of the sin... |
| sincos4thpi 26580 | The sine and cosine of ` _... |
| tan4thpi 26581 | The tangent of ` _pi / 4 `... |
| tan4thpiOLD 26582 | Obsolete version of ~ tan4... |
| sincos6thpi 26583 | The sine and cosine of ` _... |
| sincos3rdpi 26584 | The sine and cosine of ` _... |
| pigt3 26585 | ` _pi ` is greater than 3.... |
| pige3 26586 | ` _pi ` is greater than or... |
| pige3ALT 26587 | Alternate proof of ~ pige3... |
| abssinper 26588 | The absolute value of sine... |
| sinkpi 26589 | The sine of an integer mul... |
| coskpi 26590 | The absolute value of the ... |
| sineq0 26591 | A complex number whose sin... |
| coseq1 26592 | A complex number whose cos... |
| cos02pilt1 26593 | Cosine is less than one be... |
| cosq34lt1 26594 | Cosine is less than one in... |
| efeq1 26595 | A complex number whose exp... |
| cosne0 26596 | The cosine function has no... |
| cosordlem 26597 | Lemma for ~ cosord . (Con... |
| cosord 26598 | Cosine is decreasing over ... |
| cos0pilt1 26599 | Cosine is between minus on... |
| cos11 26600 | Cosine is one-to-one over ... |
| sinord 26601 | Sine is increasing over th... |
| recosf1o 26602 | The cosine function is a b... |
| resinf1o 26603 | The sine function is a bij... |
| tanord1 26604 | The tangent function is st... |
| tanord 26605 | The tangent function is st... |
| tanregt0 26606 | The real part of the tange... |
| negpitopissre 26607 | The interval ` ( -u _pi (,... |
| efgh 26608 | The exponential function o... |
| efif1olem1 26609 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26610 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26611 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26612 | The exponential function o... |
| efif1o 26613 | The exponential function o... |
| efifo 26614 | The exponential function o... |
| eff1olem 26615 | The exponential function m... |
| eff1o 26616 | The exponential function m... |
| efabl 26617 | The image of a subgroup of... |
| efsubm 26618 | The image of a subgroup of... |
| circgrp 26619 | The circle group ` T ` is ... |
| circsubm 26620 | The circle group ` T ` is ... |
| logrn 26625 | The range of the natural l... |
| ellogrn 26626 | Write out the property ` A... |
| dflog2 26627 | The natural logarithm func... |
| relogrn 26628 | The range of the natural l... |
| logrncn 26629 | The range of the natural l... |
| eff1o2 26630 | The exponential function r... |
| logf1o 26631 | The natural logarithm func... |
| dfrelog 26632 | The natural logarithm func... |
| relogf1o 26633 | The natural logarithm func... |
| logrncl 26634 | Closure of the natural log... |
| logcl 26635 | Closure of the natural log... |
| logimcl 26636 | Closure of the imaginary p... |
| logcld 26637 | The logarithm of a nonzero... |
| logimcld 26638 | The imaginary part of the ... |
| logimclad 26639 | The imaginary part of the ... |
| abslogimle 26640 | The imaginary part of the ... |
| logrnaddcl 26641 | The range of the natural l... |
| relogcl 26642 | Closure of the natural log... |
| eflog 26643 | Relationship between the n... |
| logeq0im1 26644 | If the logarithm of a numb... |
| logccne0 26645 | The logarithm isn't 0 if i... |
| logne0 26646 | Logarithm of a non-1 posit... |
| reeflog 26647 | Relationship between the n... |
| logef 26648 | Relationship between the n... |
| relogef 26649 | Relationship between the n... |
| logeftb 26650 | Relationship between the n... |
| relogeftb 26651 | Relationship between the n... |
| log1 26652 | The natural logarithm of `... |
| loge 26653 | The natural logarithm of `... |
| logi 26654 | The natural logarithm of `... |
| logneg 26655 | The natural logarithm of a... |
| logm1 26656 | The natural logarithm of n... |
| lognegb 26657 | If a number has imaginary ... |
| relogoprlem 26658 | Lemma for ~ relogmul and ~... |
| relogmul 26659 | The natural logarithm of t... |
| relogdiv 26660 | The natural logarithm of t... |
| explog 26661 | Exponentiation of a nonzer... |
| reexplog 26662 | Exponentiation of a positi... |
| relogexp 26663 | The natural logarithm of p... |
| relog 26664 | Real part of a logarithm. ... |
| relogiso 26665 | The natural logarithm func... |
| reloggim 26666 | The natural logarithm is a... |
| logltb 26667 | The natural logarithm func... |
| logfac 26668 | The logarithm of a factori... |
| eflogeq 26669 | Solve an equation involvin... |
| logleb 26670 | Natural logarithm preserve... |
| rplogcl 26671 | Closure of the logarithm f... |
| logge0 26672 | The logarithm of a number ... |
| logcj 26673 | The natural logarithm dist... |
| efiarg 26674 | The exponential of the "ar... |
| cosargd 26675 | The cosine of the argument... |
| cosarg0d 26676 | The cosine of the argument... |
| argregt0 26677 | Closure of the argument of... |
| argrege0 26678 | Closure of the argument of... |
| argimgt0 26679 | Closure of the argument of... |
| argimlt0 26680 | Closure of the argument of... |
| logimul 26681 | Multiplying a number by ` ... |
| logneg2 26682 | The logarithm of the negat... |
| logmul2 26683 | Generalization of ~ relogm... |
| logdiv2 26684 | Generalization of ~ relogd... |
| abslogle 26685 | Bound on the magnitude of ... |
| tanarg 26686 | The basic relation between... |
| logdivlti 26687 | The ` log x / x ` function... |
| logdivlt 26688 | The ` log x / x ` function... |
| logdivle 26689 | The ` log x / x ` function... |
| relogcld 26690 | Closure of the natural log... |
| reeflogd 26691 | Relationship between the n... |
| relogmuld 26692 | The natural logarithm of t... |
| relogdivd 26693 | The natural logarithm of t... |
| logled 26694 | Natural logarithm preserve... |
| relogefd 26695 | Relationship between the n... |
| rplogcld 26696 | Closure of the logarithm f... |
| logge0d 26697 | The logarithm of a number ... |
| logge0b 26698 | The logarithm of a number ... |
| loggt0b 26699 | The logarithm of a number ... |
| logle1b 26700 | The logarithm of a number ... |
| loglt1b 26701 | The logarithm of a number ... |
| divlogrlim 26702 | The inverse logarithm func... |
| logno1 26703 | The logarithm function is ... |
| dvrelog 26704 | The derivative of the real... |
| relogcn 26705 | The real logarithm functio... |
| ellogdm 26706 | Elementhood in the "contin... |
| logdmn0 26707 | A number in the continuous... |
| logdmnrp 26708 | A number in the continuous... |
| logdmss 26709 | The continuity domain of `... |
| logcnlem2 26710 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26711 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26712 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26713 | Lemma for ~ logcn . (Cont... |
| logcn 26714 | The logarithm function is ... |
| dvloglem 26715 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26716 | The "continuous domain" of... |
| logf1o2 26717 | The logarithm maps its con... |
| dvlog 26718 | The derivative of the comp... |
| dvlog2lem 26719 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26720 | The derivative of the comp... |
| advlog 26721 | The antiderivative of the ... |
| advlogexp 26722 | The antiderivative of a po... |
| efopnlem1 26723 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26724 | Lemma for ~ efopn . (Cont... |
| efopn 26725 | The exponential map is an ... |
| logtayllem 26726 | Lemma for ~ logtayl . (Co... |
| logtayl 26727 | The Taylor series for ` -u... |
| logtaylsum 26728 | The Taylor series for ` -u... |
| logtayl2 26729 | Power series expression fo... |
| logccv 26730 | The natural logarithm func... |
| cxpval 26731 | Value of the complex power... |
| cxpef 26732 | Value of the complex power... |
| 0cxp 26733 | Value of the complex power... |
| cxpexpz 26734 | Relate the complex power f... |
| cxpexp 26735 | Relate the complex power f... |
| logcxp 26736 | Logarithm of a complex pow... |
| cxp0 26737 | Value of the complex power... |
| cxp1 26738 | Value of the complex power... |
| 1cxp 26739 | Value of the complex power... |
| ecxp 26740 | Write the exponential func... |
| cxpcl 26741 | Closure of the complex pow... |
| recxpcl 26742 | Real closure of the comple... |
| rpcxpcl 26743 | Positive real closure of t... |
| cxpne0 26744 | Complex exponentiation is ... |
| cxpeq0 26745 | Complex exponentiation is ... |
| cxpadd 26746 | Sum of exponents law for c... |
| cxpp1 26747 | Value of a nonzero complex... |
| cxpneg 26748 | Value of a complex number ... |
| cxpsub 26749 | Exponent subtraction law f... |
| cxpge0 26750 | Nonnegative exponentiation... |
| mulcxplem 26751 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26752 | Complex exponentiation of ... |
| cxprec 26753 | Complex exponentiation of ... |
| divcxp 26754 | Complex exponentiation of ... |
| cxpmul 26755 | Product of exponents law f... |
| cxpmul2 26756 | Product of exponents law f... |
| cxproot 26757 | The complex power function... |
| cxpmul2z 26758 | Generalize ~ cxpmul2 to ne... |
| abscxp 26759 | Absolute value of a power,... |
| abscxp2 26760 | Absolute value of a power,... |
| cxplt 26761 | Ordering property for comp... |
| cxple 26762 | Ordering property for comp... |
| cxplea 26763 | Ordering property for comp... |
| cxple2 26764 | Ordering property for comp... |
| cxplt2 26765 | Ordering property for comp... |
| cxple2a 26766 | Ordering property for comp... |
| cxplt3 26767 | Ordering property for comp... |
| cxple3 26768 | Ordering property for comp... |
| cxpsqrtlem 26769 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26770 | The complex exponential fu... |
| logsqrt 26771 | Logarithm of a square root... |
| cxp0d 26772 | Value of the complex power... |
| cxp1d 26773 | Value of the complex power... |
| 1cxpd 26774 | Value of the complex power... |
| cxpcld 26775 | Closure of the complex pow... |
| cxpmul2d 26776 | Product of exponents law f... |
| 0cxpd 26777 | Value of the complex power... |
| cxpexpzd 26778 | Relate the complex power f... |
| cxpefd 26779 | Value of the complex power... |
| cxpne0d 26780 | Complex exponentiation is ... |
| cxpp1d 26781 | Value of a nonzero complex... |
| cxpnegd 26782 | Value of a complex number ... |
| cxpmul2zd 26783 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26784 | Sum of exponents law for c... |
| cxpsubd 26785 | Exponent subtraction law f... |
| cxpltd 26786 | Ordering property for comp... |
| cxpled 26787 | Ordering property for comp... |
| cxplead 26788 | Ordering property for comp... |
| divcxpd 26789 | Complex exponentiation of ... |
| recxpcld 26790 | Positive real closure of t... |
| cxpge0d 26791 | Nonnegative exponentiation... |
| cxple2ad 26792 | Ordering property for comp... |
| cxplt2d 26793 | Ordering property for comp... |
| cxple2d 26794 | Ordering property for comp... |
| mulcxpd 26795 | Complex exponentiation of ... |
| recxpf1lem 26796 | Complex exponentiation on ... |
| cxpsqrtth 26797 | Square root theorem over t... |
| 2irrexpq 26798 | There exist irrational num... |
| cxprecd 26799 | Complex exponentiation of ... |
| rpcxpcld 26800 | Positive real closure of t... |
| logcxpd 26801 | Logarithm of a complex pow... |
| cxplt3d 26802 | Ordering property for comp... |
| cxple3d 26803 | Ordering property for comp... |
| cxpmuld 26804 | Product of exponents law f... |
| cxpgt0d 26805 | A positive real raised to ... |
| cxpcom 26806 | Commutative law for real e... |
| dvcxp1 26807 | The derivative of a comple... |
| dvcxp2 26808 | The derivative of a comple... |
| dvsqrt 26809 | The derivative of the real... |
| dvcncxp1 26810 | Derivative of complex powe... |
| dvcnsqrt 26811 | Derivative of square root ... |
| cxpcn 26812 | Domain of continuity of th... |
| cxpcn2 26813 | Continuity of the complex ... |
| cxpcn3lem 26814 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26815 | Extend continuity of the c... |
| resqrtcn 26816 | Continuity of the real squ... |
| sqrtcn 26817 | Continuity of the square r... |
| cxpaddlelem 26818 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26819 | Ordering property for comp... |
| abscxpbnd 26820 | Bound on the absolute valu... |
| root1id 26821 | Property of an ` N ` -th r... |
| root1eq1 26822 | The only powers of an ` N ... |
| root1cj 26823 | Within the ` N ` -th roots... |
| cxpeq 26824 | Solve an equation involvin... |
| zrtelqelz 26825 | If the ` N ` -th root of a... |
| zrtdvds 26826 | A positive integer root di... |
| rtprmirr 26827 | The root of a prime number... |
| loglesqrt 26828 | An upper bound on the loga... |
| logreclem 26829 | Symmetry of the natural lo... |
| logrec 26830 | Logarithm of a reciprocal ... |
| logbval 26833 | Define the value of the ` ... |
| logbcl 26834 | General logarithm closure.... |
| logbid1 26835 | General logarithm is 1 whe... |
| logb1 26836 | The logarithm of ` 1 ` to ... |
| elogb 26837 | The general logarithm of a... |
| logbchbase 26838 | Change of base for logarit... |
| relogbval 26839 | Value of the general logar... |
| relogbcl 26840 | Closure of the general log... |
| relogbzcl 26841 | Closure of the general log... |
| relogbreexp 26842 | Power law for the general ... |
| relogbzexp 26843 | Power law for the general ... |
| relogbmul 26844 | The logarithm of the produ... |
| relogbmulexp 26845 | The logarithm of the produ... |
| relogbdiv 26846 | The logarithm of the quoti... |
| relogbexp 26847 | Identity law for general l... |
| nnlogbexp 26848 | Identity law for general l... |
| logbrec 26849 | Logarithm of a reciprocal ... |
| logbleb 26850 | The general logarithm func... |
| logblt 26851 | The general logarithm func... |
| relogbcxp 26852 | Identity law for the gener... |
| cxplogb 26853 | Identity law for the gener... |
| relogbcxpb 26854 | The logarithm is the inver... |
| logbmpt 26855 | The general logarithm to a... |
| logbf 26856 | The general logarithm to a... |
| logbfval 26857 | The general logarithm of a... |
| relogbf 26858 | The general logarithm to a... |
| logblog 26859 | The general logarithm to t... |
| logbgt0b 26860 | The logarithm of a positiv... |
| logbgcd1irr 26861 | The logarithm of an intege... |
| 2logb9irr 26862 | Example for ~ logbgcd1irr ... |
| logbprmirr 26863 | The logarithm of a prime t... |
| 2logb3irr 26864 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26865 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26866 | The square root of two to ... |
| 2irrexpqALT 26867 | Alternate proof of ~ 2irre... |
| angval 26868 | Define the angle function,... |
| angcan 26869 | Cancel a constant multipli... |
| angneg 26870 | Cancel a negative sign in ... |
| angvald 26871 | The (signed) angle between... |
| angcld 26872 | The (signed) angle between... |
| angrteqvd 26873 | Two vectors are at a right... |
| cosangneg2d 26874 | The cosine of the angle be... |
| angrtmuld 26875 | Perpendicularity of two ve... |
| ang180lem1 26876 | Lemma for ~ ang180 . Show... |
| ang180lem2 26877 | Lemma for ~ ang180 . Show... |
| ang180lem3 26878 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26879 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26880 | Lemma for ~ ang180 : Redu... |
| ang180 26881 | The sum of angles ` m A B ... |
| lawcoslem1 26882 | Lemma for ~ lawcos . Here... |
| lawcos 26883 | Law of cosines (also known... |
| pythag 26884 | Pythagorean theorem. Give... |
| isosctrlem1 26885 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26886 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26887 | Lemma for ~ isosctr . Cor... |
| isosctr 26888 | Isosceles triangle theorem... |
| ssscongptld 26889 | If two triangles have equa... |
| affineequiv 26890 | Equivalence between two wa... |
| affineequiv2 26891 | Equivalence between two wa... |
| affineequiv3 26892 | Equivalence between two wa... |
| affineequiv4 26893 | Equivalence between two wa... |
| affineequivne 26894 | Equivalence between two wa... |
| angpieqvdlem 26895 | Equivalence used in the pr... |
| angpieqvdlem2 26896 | Equivalence used in ~ angp... |
| angpined 26897 | If the angle at ABC is ` _... |
| angpieqvd 26898 | The angle ABC is ` _pi ` i... |
| chordthmlem 26899 | If ` M ` is the midpoint o... |
| chordthmlem2 26900 | If M is the midpoint of AB... |
| chordthmlem3 26901 | If M is the midpoint of AB... |
| chordthmlem4 26902 | If P is on the segment AB ... |
| chordthmlem5 26903 | If P is on the segment AB ... |
| chordthm 26904 | The intersecting chords th... |
| heron 26905 | Heron's formula gives the ... |
| quad2 26906 | The quadratic equation, wi... |
| quad 26907 | The quadratic equation. (... |
| 1cubrlem 26908 | The cube roots of unity. ... |
| 1cubr 26909 | The cube roots of unity. ... |
| dcubic1lem 26910 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26911 | Reverse direction of ~ dcu... |
| dcubic1 26912 | Forward direction of ~ dcu... |
| dcubic 26913 | Solutions to the depressed... |
| mcubic 26914 | Solutions to a monic cubic... |
| cubic2 26915 | The solution to the genera... |
| cubic 26916 | The cubic equation, which ... |
| binom4 26917 | Work out a quartic binomia... |
| dquartlem1 26918 | Lemma for ~ dquart . (Con... |
| dquartlem2 26919 | Lemma for ~ dquart . (Con... |
| dquart 26920 | Solve a depressed quartic ... |
| quart1cl 26921 | Closure lemmas for ~ quart... |
| quart1lem 26922 | Lemma for ~ quart1 . (Con... |
| quart1 26923 | Depress a quartic equation... |
| quartlem1 26924 | Lemma for ~ quart . (Cont... |
| quartlem2 26925 | Closure lemmas for ~ quart... |
| quartlem3 26926 | Closure lemmas for ~ quart... |
| quartlem4 26927 | Closure lemmas for ~ quart... |
| quart 26928 | The quartic equation, writ... |
| asinlem 26935 | The argument to the logari... |
| asinlem2 26936 | The argument to the logari... |
| asinlem3a 26937 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26938 | The argument to the logari... |
| asinf 26939 | Domain and codomain of the... |
| asincl 26940 | Closure for the arcsin fun... |
| acosf 26941 | Domain and codoamin of the... |
| acoscl 26942 | Closure for the arccos fun... |
| atandm 26943 | Since the property is a li... |
| atandm2 26944 | This form of ~ atandm is a... |
| atandm3 26945 | A compact form of ~ atandm... |
| atandm4 26946 | A compact form of ~ atandm... |
| atanf 26947 | Domain and codoamin of the... |
| atancl 26948 | Closure for the arctan fun... |
| asinval 26949 | Value of the arcsin functi... |
| acosval 26950 | Value of the arccos functi... |
| atanval 26951 | Value of the arctan functi... |
| atanre 26952 | A real number is in the do... |
| asinneg 26953 | The arcsine function is od... |
| acosneg 26954 | The negative symmetry rela... |
| efiasin 26955 | The exponential of the arc... |
| sinasin 26956 | The arcsine function is an... |
| cosacos 26957 | The arccosine function is ... |
| asinsinlem 26958 | Lemma for ~ asinsin . (Co... |
| asinsin 26959 | The arcsine function compo... |
| acoscos 26960 | The arccosine function is ... |
| asin1 26961 | The arcsine of ` 1 ` is ` ... |
| acos1 26962 | The arccosine of ` 1 ` is ... |
| reasinsin 26963 | The arcsine function compo... |
| asinsinb 26964 | Relationship between sine ... |
| acoscosb 26965 | Relationship between cosin... |
| asinbnd 26966 | The arcsine function has r... |
| acosbnd 26967 | The arccosine function has... |
| asinrebnd 26968 | Bounds on the arcsine func... |
| asinrecl 26969 | The arcsine function is re... |
| acosrecl 26970 | The arccosine function is ... |
| cosasin 26971 | The cosine of the arcsine ... |
| sinacos 26972 | The sine of the arccosine ... |
| atandmneg 26973 | The domain of the arctange... |
| atanneg 26974 | The arctangent function is... |
| atan0 26975 | The arctangent of zero is ... |
| atandmcj 26976 | The arctangent function di... |
| atancj 26977 | The arctangent function di... |
| atanrecl 26978 | The arctangent function is... |
| efiatan 26979 | Value of the exponential o... |
| atanlogaddlem 26980 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26981 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26982 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26983 | A variation on ~ atanlogad... |
| efiatan2 26984 | Value of the exponential o... |
| 2efiatan 26985 | Value of the exponential o... |
| tanatan 26986 | The arctangent function is... |
| atandmtan 26987 | The tangent function has r... |
| cosatan 26988 | The cosine of an arctangen... |
| cosatanne0 26989 | The arctangent function ha... |
| atantan 26990 | The arctangent function is... |
| atantanb 26991 | Relationship between tange... |
| atanbndlem 26992 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26993 | The arctangent function is... |
| atanord 26994 | The arctangent function is... |
| atan1 26995 | The arctangent of ` 1 ` is... |
| bndatandm 26996 | A point in the open unit d... |
| atans 26997 | The "domain of continuity"... |
| atans2 26998 | It suffices to show that `... |
| atansopn 26999 | The domain of continuity o... |
| atansssdm 27000 | The domain of continuity o... |
| ressatans 27001 | The real number line is a ... |
| dvatan 27002 | The derivative of the arct... |
| atancn 27003 | The arctangent is a contin... |
| atantayl 27004 | The Taylor series for ` ar... |
| atantayl2 27005 | The Taylor series for ` ar... |
| atantayl3 27006 | The Taylor series for ` ar... |
| leibpilem1 27007 | Lemma for ~ leibpi . (Con... |
| leibpilem2 27008 | The Leibniz formula for ` ... |
| leibpi 27009 | The Leibniz formula for ` ... |
| leibpisum 27010 | The Leibniz formula for ` ... |
| log2cnv 27011 | Using the Taylor series fo... |
| log2tlbnd 27012 | Bound the error term in th... |
| log2ublem1 27013 | Lemma for ~ log2ub . The ... |
| log2ublem2 27014 | Lemma for ~ log2ub . (Con... |
| log2ublem3 27015 | Lemma for ~ log2ub . In d... |
| log2ub 27016 | ` log 2 ` is less than ` 2... |
| log2le1 27017 | ` log 2 ` is less than ` 1... |
| birthdaylem1 27018 | Lemma for ~ birthday . (C... |
| birthdaylem2 27019 | For general ` N ` and ` K ... |
| birthdaylem3 27020 | For general ` N ` and ` K ... |
| birthday 27021 | The Birthday Problem. The... |
| dmarea 27024 | The domain of the area fun... |
| areambl 27025 | The fibers of a measurable... |
| areass 27026 | A measurable region is a s... |
| dfarea 27027 | Rewrite ~ df-area self-ref... |
| areaf 27028 | Area measurement is a func... |
| areacl 27029 | The area of a measurable r... |
| areage0 27030 | The area of a measurable r... |
| areaval 27031 | The area of a measurable r... |
| rlimcnp 27032 | Relate a limit of a real-v... |
| rlimcnp2 27033 | Relate a limit of a real-v... |
| rlimcnp3 27034 | Relate a limit of a real-v... |
| xrlimcnp 27035 | Relate a limit of a real-v... |
| efrlim 27036 | The limit of the sequence ... |
| dfef2 27037 | The limit of the sequence ... |
| cxplim 27038 | A power to a negative expo... |
| sqrtlim 27039 | The inverse square root fu... |
| rlimcxp 27040 | Any power to a positive ex... |
| o1cxp 27041 | An eventually bounded func... |
| cxp2limlem 27042 | A linear factor grows slow... |
| cxp2lim 27043 | Any power grows slower tha... |
| cxploglim 27044 | The logarithm grows slower... |
| cxploglim2 27045 | Every power of the logarit... |
| divsqrtsumlem 27046 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 27047 | The function ` F ` used in... |
| divsqrsum 27048 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 27049 | A bound on the distance of... |
| divsqrtsumo1 27050 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 27051 | Closure of a 0-1 linear co... |
| scvxcvx 27052 | A strictly convex function... |
| jensenlem1 27053 | Lemma for ~ jensen . (Con... |
| jensenlem2 27054 | Lemma for ~ jensen . (Con... |
| jensen 27055 | Jensen's inequality, a fin... |
| amgmlem 27056 | Lemma for ~ amgm . (Contr... |
| amgm 27057 | Inequality of arithmetic a... |
| logdifbnd 27060 | Bound on the difference of... |
| logdiflbnd 27061 | Lower bound on the differe... |
| emcllem1 27062 | Lemma for ~ emcl . The se... |
| emcllem2 27063 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 27064 | Lemma for ~ emcl . The fu... |
| emcllem4 27065 | Lemma for ~ emcl . The di... |
| emcllem5 27066 | Lemma for ~ emcl . The pa... |
| emcllem6 27067 | Lemma for ~ emcl . By the... |
| emcllem7 27068 | Lemma for ~ emcl and ~ har... |
| emcl 27069 | Closure and bounds for the... |
| harmonicbnd 27070 | A bound on the harmonic se... |
| harmonicbnd2 27071 | A bound on the harmonic se... |
| emre 27072 | The Euler-Mascheroni const... |
| emgt0 27073 | The Euler-Mascheroni const... |
| harmonicbnd3 27074 | A bound on the harmonic se... |
| harmoniclbnd 27075 | A bound on the harmonic se... |
| harmonicubnd 27076 | A bound on the harmonic se... |
| harmonicbnd4 27077 | The asymptotic behavior of... |
| fsumharmonic 27078 | Bound a finite sum based o... |
| zetacvg 27081 | The zeta series is converg... |
| eldmgm 27088 | Elementhood in the set of ... |
| dmgmaddn0 27089 | If ` A ` is not a nonposit... |
| dmlogdmgm 27090 | If ` A ` is in the continu... |
| rpdmgm 27091 | A positive real number is ... |
| dmgmn0 27092 | If ` A ` is not a nonposit... |
| dmgmaddnn0 27093 | If ` A ` is not a nonposit... |
| dmgmdivn0 27094 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 27095 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 27096 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 27097 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 27098 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 27099 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 27100 | The series ` G ` is unifor... |
| lgamgulm 27101 | The series ` G ` is unifor... |
| lgamgulm2 27102 | Rewrite the limit of the s... |
| lgambdd 27103 | The log-Gamma function is ... |
| lgamucov 27104 | The ` U ` regions used in ... |
| lgamucov2 27105 | The ` U ` regions used in ... |
| lgamcvglem 27106 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 27107 | The log-Gamma function is ... |
| lgamf 27108 | The log-Gamma function is ... |
| gamf 27109 | The Gamma function is a co... |
| gamcl 27110 | The exponential of the log... |
| eflgam 27111 | The exponential of the log... |
| gamne0 27112 | The Gamma function is neve... |
| igamval 27113 | Value of the inverse Gamma... |
| igamz 27114 | Value of the inverse Gamma... |
| igamgam 27115 | Value of the inverse Gamma... |
| igamlgam 27116 | Value of the inverse Gamma... |
| igamf 27117 | Closure of the inverse Gam... |
| igamcl 27118 | Closure of the inverse Gam... |
| gamigam 27119 | The Gamma function is the ... |
| lgamcvg 27120 | The series ` G ` converges... |
| lgamcvg2 27121 | The series ` G ` converges... |
| gamcvg 27122 | The pointwise exponential ... |
| lgamp1 27123 | The functional equation of... |
| gamp1 27124 | The functional equation of... |
| gamcvg2lem 27125 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 27126 | An infinite product expres... |
| regamcl 27127 | The Gamma function is real... |
| relgamcl 27128 | The log-Gamma function is ... |
| rpgamcl 27129 | The log-Gamma function is ... |
| lgam1 27130 | The log-Gamma function at ... |
| gam1 27131 | The log-Gamma function at ... |
| facgam 27132 | The Gamma function general... |
| gamfac 27133 | The Gamma function general... |
| wilthlem1 27134 | The only elements that are... |
| wilthlem2 27135 | Lemma for ~ wilth : induct... |
| wilthlem3 27136 | Lemma for ~ wilth . Here ... |
| wilth 27137 | Wilson's theorem. A numbe... |
| wilthimp 27138 | The forward implication of... |
| ftalem1 27139 | Lemma for ~ fta : "growth... |
| ftalem2 27140 | Lemma for ~ fta . There e... |
| ftalem3 27141 | Lemma for ~ fta . There e... |
| ftalem4 27142 | Lemma for ~ fta : Closure... |
| ftalem5 27143 | Lemma for ~ fta : Main pr... |
| ftalem6 27144 | Lemma for ~ fta : Dischar... |
| ftalem7 27145 | Lemma for ~ fta . Shift t... |
| fta 27146 | The Fundamental Theorem of... |
| basellem1 27147 | Lemma for ~ basel . Closu... |
| basellem2 27148 | Lemma for ~ basel . Show ... |
| basellem3 27149 | Lemma for ~ basel . Using... |
| basellem4 27150 | Lemma for ~ basel . By ~ ... |
| basellem5 27151 | Lemma for ~ basel . Using... |
| basellem6 27152 | Lemma for ~ basel . The f... |
| basellem7 27153 | Lemma for ~ basel . The f... |
| basellem8 27154 | Lemma for ~ basel . The f... |
| basellem9 27155 | Lemma for ~ basel . Since... |
| basel 27156 | The sum of the inverse squ... |
| efnnfsumcl 27169 | Finite sum closure in the ... |
| ppisval 27170 | The set of primes less tha... |
| ppisval2 27171 | The set of primes less tha... |
| ppifi 27172 | The set of primes less tha... |
| prmdvdsfi 27173 | The set of prime divisors ... |
| chtf 27174 | Domain and codoamin of the... |
| chtcl 27175 | Real closure of the Chebys... |
| chtval 27176 | Value of the Chebyshev fun... |
| efchtcl 27177 | The Chebyshev function is ... |
| chtge0 27178 | The Chebyshev function is ... |
| vmaval 27179 | Value of the von Mangoldt ... |
| isppw 27180 | Two ways to say that ` A `... |
| isppw2 27181 | Two ways to say that ` A `... |
| vmappw 27182 | Value of the von Mangoldt ... |
| vmaprm 27183 | Value of the von Mangoldt ... |
| vmacl 27184 | Closure for the von Mangol... |
| vmaf 27185 | Functionality of the von M... |
| efvmacl 27186 | The von Mangoldt is closed... |
| vmage0 27187 | The von Mangoldt function ... |
| chpval 27188 | Value of the second Chebys... |
| chpf 27189 | Functionality of the secon... |
| chpcl 27190 | Closure for the second Che... |
| efchpcl 27191 | The second Chebyshev funct... |
| chpge0 27192 | The second Chebyshev funct... |
| ppival 27193 | Value of the prime-countin... |
| ppival2 27194 | Value of the prime-countin... |
| ppival2g 27195 | Value of the prime-countin... |
| ppif 27196 | Domain and codomain of the... |
| ppicl 27197 | Real closure of the prime-... |
| muval 27198 | The value of the Möbi... |
| muval1 27199 | The value of the Möbi... |
| muval2 27200 | The value of the Möbi... |
| isnsqf 27201 | Two ways to say that a num... |
| issqf 27202 | Two ways to say that a num... |
| sqfpc 27203 | The prime count of a squar... |
| dvdssqf 27204 | A divisor of a squarefree ... |
| sqf11 27205 | A squarefree number is com... |
| muf 27206 | The Möbius function i... |
| mucl 27207 | Closure of the Möbius... |
| sgmval 27208 | The value of the divisor f... |
| sgmval2 27209 | The value of the divisor f... |
| 0sgm 27210 | The value of the sum-of-di... |
| sgmf 27211 | The divisor function is a ... |
| sgmcl 27212 | Closure of the divisor fun... |
| sgmnncl 27213 | Closure of the divisor fun... |
| mule1 27214 | The Möbius function t... |
| chtfl 27215 | The Chebyshev function doe... |
| chpfl 27216 | The second Chebyshev funct... |
| ppiprm 27217 | The prime-counting functio... |
| ppinprm 27218 | The prime-counting functio... |
| chtprm 27219 | The Chebyshev function at ... |
| chtnprm 27220 | The Chebyshev function at ... |
| chpp1 27221 | The second Chebyshev funct... |
| chtwordi 27222 | The Chebyshev function is ... |
| chpwordi 27223 | The second Chebyshev funct... |
| chtdif 27224 | The difference of the Cheb... |
| efchtdvds 27225 | The exponentiated Chebyshe... |
| ppifl 27226 | The prime-counting functio... |
| ppip1le 27227 | The prime-counting functio... |
| ppiwordi 27228 | The prime-counting functio... |
| ppidif 27229 | The difference of the prim... |
| ppi1 27230 | The prime-counting functio... |
| cht1 27231 | The Chebyshev function at ... |
| vma1 27232 | The von Mangoldt function ... |
| chp1 27233 | The second Chebyshev funct... |
| ppi1i 27234 | Inference form of ~ ppiprm... |
| ppi2i 27235 | Inference form of ~ ppinpr... |
| ppi2 27236 | The prime-counting functio... |
| ppi3 27237 | The prime-counting functio... |
| cht2 27238 | The Chebyshev function at ... |
| cht3 27239 | The Chebyshev function at ... |
| ppinncl 27240 | Closure of the prime-count... |
| chtrpcl 27241 | Closure of the Chebyshev f... |
| ppieq0 27242 | The prime-counting functio... |
| ppiltx 27243 | The prime-counting functio... |
| prmorcht 27244 | Relate the primorial (prod... |
| mumullem1 27245 | Lemma for ~ mumul . A mul... |
| mumullem2 27246 | Lemma for ~ mumul . The p... |
| mumul 27247 | The Möbius function i... |
| sqff1o 27248 | There is a bijection from ... |
| fsumdvdsdiaglem 27249 | A "diagonal commutation" o... |
| fsumdvdsdiag 27250 | A "diagonal commutation" o... |
| fsumdvdscom 27251 | A double commutation of di... |
| dvdsppwf1o 27252 | A bijection between the di... |
| dvdsflf1o 27253 | A bijection from the numbe... |
| dvdsflsumcom 27254 | A sum commutation from ` s... |
| fsumfldivdiaglem 27255 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27256 | The right-hand side of ~ d... |
| musum 27257 | The sum of the Möbius... |
| musumsum 27258 | Evaluate a collapsing sum ... |
| muinv 27259 | The Möbius inversion ... |
| mpodvdsmulf1o 27260 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27261 | Product of two divisor sum... |
| dvdsmulf1o 27262 | If ` M ` and ` N ` are two... |
| sgmppw 27263 | The value of the divisor f... |
| 0sgmppw 27264 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27265 | The sum of divisors for a ... |
| 1sgm2ppw 27266 | The sum of the divisors of... |
| sgmmul 27267 | The divisor function for f... |
| ppiublem1 27268 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27269 | A prime greater than ` 3 `... |
| ppiub 27270 | An upper bound on the prim... |
| vmalelog 27271 | The von Mangoldt function ... |
| chtlepsi 27272 | The first Chebyshev functi... |
| chprpcl 27273 | Closure of the second Cheb... |
| chpeq0 27274 | The second Chebyshev funct... |
| chteq0 27275 | The first Chebyshev functi... |
| chtleppi 27276 | Upper bound on the ` theta... |
| chtublem 27277 | Lemma for ~ chtub . (Cont... |
| chtub 27278 | An upper bound on the Cheb... |
| fsumvma 27279 | Rewrite a sum over the von... |
| fsumvma2 27280 | Apply ~ fsumvma for the co... |
| pclogsum 27281 | The logarithmic analogue o... |
| vmasum 27282 | The sum of the von Mangold... |
| logfac2 27283 | Another expression for the... |
| chpval2 27284 | Express the second Chebysh... |
| chpchtsum 27285 | The second Chebyshev funct... |
| chpub 27286 | An upper bound on the seco... |
| logfacubnd 27287 | A simple upper bound on th... |
| logfaclbnd 27288 | A lower bound on the logar... |
| logfacbnd3 27289 | Show the stronger statemen... |
| logfacrlim 27290 | Combine the estimates ~ lo... |
| logexprlim 27291 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27292 | Write out ~ logfacrlim as ... |
| mersenne 27293 | A Mersenne prime is a prim... |
| perfect1 27294 | Euclid's contribution to t... |
| perfectlem1 27295 | Lemma for ~ perfect . (Co... |
| perfectlem2 27296 | Lemma for ~ perfect . (Co... |
| perfect 27297 | The Euclid-Euler theorem, ... |
| dchrval 27300 | Value of the group of Diri... |
| dchrbas 27301 | Base set of the group of D... |
| dchrelbas 27302 | A Dirichlet character is a... |
| dchrelbas2 27303 | A Dirichlet character is a... |
| dchrelbas3 27304 | A Dirichlet character is a... |
| dchrelbasd 27305 | A Dirichlet character is a... |
| dchrrcl 27306 | Reverse closure for a Diri... |
| dchrmhm 27307 | A Dirichlet character is a... |
| dchrf 27308 | A Dirichlet character is a... |
| dchrelbas4 27309 | A Dirichlet character is a... |
| dchrzrh1 27310 | Value of a Dirichlet chara... |
| dchrzrhcl 27311 | A Dirichlet character take... |
| dchrzrhmul 27312 | A Dirichlet character is c... |
| dchrplusg 27313 | Group operation on the gro... |
| dchrmul 27314 | Group operation on the gro... |
| dchrmulcl 27315 | Closure of the group opera... |
| dchrn0 27316 | A Dirichlet character is n... |
| dchr1cl 27317 | Closure of the principal D... |
| dchrmullid 27318 | Left identity for the prin... |
| dchrinvcl 27319 | Closure of the group inver... |
| dchrabl 27320 | The set of Dirichlet chara... |
| dchrfi 27321 | The group of Dirichlet cha... |
| dchrghm 27322 | A Dirichlet character rest... |
| dchr1 27323 | Value of the principal Dir... |
| dchreq 27324 | A Dirichlet character is d... |
| dchrresb 27325 | A Dirichlet character is d... |
| dchrabs 27326 | A Dirichlet character take... |
| dchrinv 27327 | The inverse of a Dirichlet... |
| dchrabs2 27328 | A Dirichlet character take... |
| dchr1re 27329 | The principal Dirichlet ch... |
| dchrptlem1 27330 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27331 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27332 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27333 | For any element other than... |
| dchrsum2 27334 | An orthogonality relation ... |
| dchrsum 27335 | An orthogonality relation ... |
| sumdchr2 27336 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27337 | There are exactly ` phi ( ... |
| sumdchr 27338 | An orthogonality relation ... |
| dchr2sum 27339 | An orthogonality relation ... |
| sum2dchr 27340 | An orthogonality relation ... |
| bcctr 27341 | Value of the central binom... |
| pcbcctr 27342 | Prime count of a central b... |
| bcmono 27343 | The binomial coefficient i... |
| bcmax 27344 | The binomial coefficient t... |
| bcp1ctr 27345 | Ratio of two central binom... |
| bclbnd 27346 | A bound on the binomial co... |
| efexple 27347 | Convert a bound on a power... |
| bpos1lem 27348 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27349 | Bertrand's postulate, chec... |
| bposlem1 27350 | An upper bound on the prim... |
| bposlem2 27351 | There are no odd primes in... |
| bposlem3 27352 | Lemma for ~ bpos . Since ... |
| bposlem4 27353 | Lemma for ~ bpos . (Contr... |
| bposlem5 27354 | Lemma for ~ bpos . Bound ... |
| bposlem6 27355 | Lemma for ~ bpos . By usi... |
| bposlem7 27356 | Lemma for ~ bpos . The fu... |
| bposlem8 27357 | Lemma for ~ bpos . Evalua... |
| bposlem9 27358 | Lemma for ~ bpos . Derive... |
| bpos 27359 | Bertrand's postulate: ther... |
| zabsle1 27362 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27363 | When ` a ` is coprime to t... |
| lgslem2 27364 | The set ` Z ` of all integ... |
| lgslem3 27365 | The set ` Z ` of all integ... |
| lgslem4 27366 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27367 | Value of the Legendre symb... |
| lgsfval 27368 | Value of the function ` F ... |
| lgsfcl2 27369 | The function ` F ` is clos... |
| lgscllem 27370 | The Legendre symbol is an ... |
| lgsfcl 27371 | Closure of the function ` ... |
| lgsfle1 27372 | The function ` F ` has mag... |
| lgsval2lem 27373 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27374 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27375 | The Legendre symbol is an ... |
| lgs0 27376 | The Legendre symbol when t... |
| lgscl 27377 | The Legendre symbol is an ... |
| lgsle1 27378 | The Legendre symbol has ab... |
| lgsval2 27379 | The Legendre symbol at a p... |
| lgs2 27380 | The Legendre symbol at ` 2... |
| lgsval3 27381 | The Legendre symbol at an ... |
| lgsvalmod 27382 | The Legendre symbol is equ... |
| lgsval4 27383 | Restate ~ lgsval for nonze... |
| lgsfcl3 27384 | Closure of the function ` ... |
| lgsval4a 27385 | Same as ~ lgsval4 for posi... |
| lgscl1 27386 | The value of the Legendre ... |
| lgsneg 27387 | The Legendre symbol is eit... |
| lgsneg1 27388 | The Legendre symbol for no... |
| lgsmod 27389 | The Legendre (Jacobi) symb... |
| lgsdilem 27390 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27391 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27392 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27393 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27394 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27395 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27396 | The Legendre symbol is com... |
| lgsdirprm 27397 | The Legendre symbol is com... |
| lgsdir 27398 | The Legendre symbol is com... |
| lgsdilem2 27399 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27400 | The Legendre symbol is com... |
| lgsne0 27401 | The Legendre symbol is non... |
| lgsabs1 27402 | The Legendre symbol is non... |
| lgssq 27403 | The Legendre symbol at a s... |
| lgssq2 27404 | The Legendre symbol at a s... |
| lgsprme0 27405 | The Legendre symbol at any... |
| 1lgs 27406 | The Legendre symbol at ` 1... |
| lgs1 27407 | The Legendre symbol at ` 1... |
| lgsmodeq 27408 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27409 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27410 | Variation on ~ lgsdir vali... |
| lgsdinn0 27411 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27412 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27413 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27414 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27415 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27416 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27417 | The Legendre symbol for od... |
| lgsqrmod 27418 | If the Legendre symbol of ... |
| lgsqrmodndvds 27419 | If the Legendre symbol of ... |
| lgsdchrval 27420 | The Legendre symbol functi... |
| lgsdchr 27421 | The Legendre symbol functi... |
| gausslemma2dlem0a 27422 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27423 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27424 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27425 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27426 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27427 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27428 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27429 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27430 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27431 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27432 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27433 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27434 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27435 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27436 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27437 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27438 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27439 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27440 | Gauss' Lemma (see also the... |
| lgseisenlem1 27441 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27442 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27443 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27444 | Lemma for ~ lgseisen . (C... |
| lgseisen 27445 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27446 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27447 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27448 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27449 | The Law of Quadratic Recip... |
| lgsquad2lem1 27450 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27451 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27452 | Extend ~ lgsquad to coprim... |
| lgsquad3 27453 | Extend ~ lgsquad2 to integ... |
| m1lgs 27454 | The first supplement to th... |
| 2lgslem1a1 27455 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27456 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27457 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27458 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27459 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27460 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27461 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27462 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27463 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27464 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27465 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27466 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27467 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27468 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27469 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27470 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27471 | The Legendre symbol for ` ... |
| 2lgslem4 27472 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27473 | The second supplement to t... |
| 2lgsoddprmlem1 27474 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27475 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27476 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27477 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27478 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27479 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27480 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27481 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27482 | The second supplement to t... |
| 2sqlem1 27483 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27484 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27485 | Fibonacci's identity (actu... |
| 2sqlem3 27486 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27487 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27488 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27489 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27490 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27491 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27492 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27493 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27494 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27495 | Lemma for ~ 2sq . (Contri... |
| 2sq 27496 | All primes of the form ` 4... |
| 2sqblem 27497 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27498 | The converse to ~ 2sq . (... |
| 2sq2 27499 | ` 2 ` is the sum of square... |
| 2sqn0 27500 | If the sum of two squares ... |
| 2sqcoprm 27501 | If the sum of two squares ... |
| 2sqmod 27502 | Given two decompositions o... |
| 2sqmo 27503 | There exists at most one d... |
| 2sqnn0 27504 | All primes of the form ` 4... |
| 2sqnn 27505 | All primes of the form ` 4... |
| addsq2reu 27506 | For each complex number ` ... |
| addsqn2reu 27507 | For each complex number ` ... |
| addsqrexnreu 27508 | For each complex number, t... |
| addsqnreup 27509 | There is no unique decompo... |
| addsq2nreurex 27510 | For each complex number ` ... |
| addsqn2reurex2 27511 | For each complex number ` ... |
| 2sqreulem1 27512 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27513 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27514 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27515 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27516 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27517 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27518 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27519 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27520 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27521 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27522 | There exists a unique deco... |
| 2sqreunn 27523 | There exists a unique deco... |
| 2sqreult 27524 | There exists a unique deco... |
| 2sqreultb 27525 | There exists a unique deco... |
| 2sqreunnlt 27526 | There exists a unique deco... |
| 2sqreunnltb 27527 | There exists a unique deco... |
| 2sqreuop 27528 | There exists a unique deco... |
| 2sqreuopnn 27529 | There exists a unique deco... |
| 2sqreuoplt 27530 | There exists a unique deco... |
| 2sqreuopltb 27531 | There exists a unique deco... |
| 2sqreuopnnlt 27532 | There exists a unique deco... |
| 2sqreuopnnltb 27533 | There exists a unique deco... |
| 2sqreuopb 27534 | There exists a unique deco... |
| chebbnd1lem1 27535 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27536 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27537 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27538 | The Chebyshev bound: The ... |
| chtppilimlem1 27539 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27540 | Lemma for ~ chtppilim . (... |
| chtppilim 27541 | The ` theta ` function is ... |
| chto1ub 27542 | The ` theta ` function is ... |
| chebbnd2 27543 | The Chebyshev bound, part ... |
| chto1lb 27544 | The ` theta ` function is ... |
| chpchtlim 27545 | The ` psi ` and ` theta ` ... |
| chpo1ub 27546 | The ` psi ` function is up... |
| chpo1ubb 27547 | The ` psi ` function is up... |
| vmadivsum 27548 | The sum of the von Mangold... |
| vmadivsumb 27549 | Give a total bound on the ... |
| rplogsumlem1 27550 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27551 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27552 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27553 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27554 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27555 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27556 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27557 | Lemma for ~ dchrisum . Le... |
| dchrisum 27558 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27559 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27560 | The sum of the Möbius... |
| dchrvmasumlem1 27561 | An alternative expression ... |
| dchrvmasum2lem 27562 | Give an expression for ` l... |
| dchrvmasum2if 27563 | Combine the results of ~ d... |
| dchrvmasumlem2 27564 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27565 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27566 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27567 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27568 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27569 | An asymptotic approximatio... |
| dchrvmaeq0 27570 | The set ` W ` is the colle... |
| dchrisum0fval 27571 | Value of the function ` F ... |
| dchrisum0fmul 27572 | The function ` F ` , the d... |
| dchrisum0ff 27573 | The function ` F ` is a re... |
| dchrisum0flblem1 27574 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27575 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27576 | The divisor sum of a real ... |
| dchrisum0fno1 27577 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27578 | A partial result along the... |
| dchrisum0re 27579 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27580 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27581 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27582 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27583 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27584 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27585 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27586 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27587 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27588 | The sum of the Möbius... |
| dchrvmasumlem 27589 | The sum of the Möbius... |
| dchrmusum 27590 | The sum of the Möbius... |
| dchrvmasum 27591 | The sum of the von Mangold... |
| rpvmasum 27592 | The sum of the von Mangold... |
| rplogsum 27593 | The sum of ` log p / p ` o... |
| dirith2 27594 | Dirichlet's theorem: there... |
| dirith 27595 | Dirichlet's theorem: there... |
| mudivsum 27596 | Asymptotic formula for ` s... |
| mulogsumlem 27597 | Lemma for ~ mulogsum . (C... |
| mulogsum 27598 | Asymptotic formula for ... |
| logdivsum 27599 | Asymptotic analysis of ... |
| mulog2sumlem1 27600 | Asymptotic formula for ... |
| mulog2sumlem2 27601 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27602 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27603 | Asymptotic formula for ... |
| vmalogdivsum2 27604 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27605 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27606 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27607 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27608 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27609 | The Möbius inverse of... |
| log2sumbnd 27610 | Bound on the difference be... |
| selberglem1 27611 | Lemma for ~ selberg . Est... |
| selberglem2 27612 | Lemma for ~ selberg . (Co... |
| selberglem3 27613 | Lemma for ~ selberg . Est... |
| selberg 27614 | Selberg's symmetry formula... |
| selbergb 27615 | Convert eventual boundedne... |
| selberg2lem 27616 | Lemma for ~ selberg2 . Eq... |
| selberg2 27617 | Selberg's symmetry formula... |
| selberg2b 27618 | Convert eventual boundedne... |
| chpdifbndlem1 27619 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27620 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27621 | A bound on the difference ... |
| logdivbnd 27622 | A bound on a sum of logs, ... |
| selberg3lem1 27623 | Introduce a log weighting ... |
| selberg3lem2 27624 | Lemma for ~ selberg3 . Eq... |
| selberg3 27625 | Introduce a log weighting ... |
| selberg4lem1 27626 | Lemma for ~ selberg4 . Eq... |
| selberg4 27627 | The Selberg symmetry formu... |
| pntrval 27628 | Define the residual of the... |
| pntrf 27629 | Functionality of the resid... |
| pntrmax 27630 | There is a bound on the re... |
| pntrsumo1 27631 | A bound on a sum over ` R ... |
| pntrsumbnd 27632 | A bound on a sum over ` R ... |
| pntrsumbnd2 27633 | A bound on a sum over ` R ... |
| selbergr 27634 | Selberg's symmetry formula... |
| selberg3r 27635 | Selberg's symmetry formula... |
| selberg4r 27636 | Selberg's symmetry formula... |
| selberg34r 27637 | The sum of ~ selberg3r and... |
| pntsval 27638 | Define the "Selberg functi... |
| pntsf 27639 | Functionality of the Selbe... |
| selbergs 27640 | Selberg's symmetry formula... |
| selbergsb 27641 | Selberg's symmetry formula... |
| pntsval2 27642 | The Selberg function can b... |
| pntrlog2bndlem1 27643 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27644 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27645 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27646 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27647 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27648 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27649 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27650 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27651 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27652 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27653 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27654 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27655 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27656 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27657 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27658 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27659 | Lemma for ~ pnt . Establi... |
| pntlemd 27660 | Lemma for ~ pnt . Closure... |
| pntlemc 27661 | Lemma for ~ pnt . Closure... |
| pntlema 27662 | Lemma for ~ pnt . Closure... |
| pntlemb 27663 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27664 | Lemma for ~ pnt . Closure... |
| pntlemh 27665 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27666 | Lemma for ~ pnt . The "na... |
| pntlemq 27667 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27668 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27669 | Lemma for ~ pnt . The ind... |
| pntlemi 27670 | Lemma for ~ pnt . Elimina... |
| pntlemf 27671 | Lemma for ~ pnt . Add up ... |
| pntlemk 27672 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27673 | Lemma for ~ pnt . Combine... |
| pntleme 27674 | Lemma for ~ pnt . Package... |
| pntlem3 27675 | Lemma for ~ pnt . Equatio... |
| pntlemp 27676 | Lemma for ~ pnt . Wrappin... |
| pntleml 27677 | Lemma for ~ pnt . Equatio... |
| pnt3 27678 | The Prime Number Theorem, ... |
| pnt2 27679 | The Prime Number Theorem, ... |
| pnt 27680 | The Prime Number Theorem: ... |
| abvcxp 27681 | Raising an absolute value ... |
| padicfval 27682 | Value of the p-adic absolu... |
| padicval 27683 | Value of the p-adic absolu... |
| ostth2lem1 27684 | Lemma for ~ ostth2 , altho... |
| qrngbas 27685 | The base set of the field ... |
| qdrng 27686 | The rationals form a divis... |
| qrng0 27687 | The zero element of the fi... |
| qrng1 27688 | The unity element of the f... |
| qrngneg 27689 | The additive inverse in th... |
| qrngdiv 27690 | The division operation in ... |
| qabvle 27691 | By using induction on ` N ... |
| qabvexp 27692 | Induct the product rule ~ ... |
| ostthlem1 27693 | Lemma for ~ ostth . If tw... |
| ostthlem2 27694 | Lemma for ~ ostth . Refin... |
| qabsabv 27695 | The regular absolute value... |
| padicabv 27696 | The p-adic absolute value ... |
| padicabvf 27697 | The p-adic absolute value ... |
| padicabvcxp 27698 | All positive powers of the... |
| ostth1 27699 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27700 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27701 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27702 | Lemma for ~ ostth2 . (Con... |
| ostth2 27703 | - Lemma for ~ ostth : regu... |
| ostth3 27704 | - Lemma for ~ ostth : p-ad... |
| ostth 27705 | Ostrowski's theorem, which... |
| elno 27712 | Membership in the surreals... |
| ltsval 27713 | The value of the surreal l... |
| bdayval 27714 | The value of the birthday ... |
| nofun 27715 | A surreal is a function. ... |
| nodmon 27716 | The domain of a surreal is... |
| norn 27717 | The range of a surreal is ... |
| nofnbday 27718 | A surreal is a function ov... |
| nodmord 27719 | The domain of a surreal ha... |
| elno2 27720 | An alternative condition f... |
| elno3 27721 | Another condition for memb... |
| ltsval2 27722 | Alternate expression for s... |
| nofv 27723 | The function value of a su... |
| nosgnn0 27724 | ` (/) ` is not a surreal s... |
| nosgnn0i 27725 | If ` X ` is a surreal sign... |
| noreson 27726 | The restriction of a surre... |
| ltsintdifex 27727 |
If ` A |
| ltsres 27728 | If the restrictions of two... |
| noxp1o 27729 | The Cartesian product of a... |
| noseponlem 27730 | Lemma for ~ nosepon . Con... |
| nosepon 27731 | Given two unequal surreals... |
| noextend 27732 | Extending a surreal by one... |
| noextendseq 27733 | Extend a surreal by a sequ... |
| noextenddif 27734 | Calculate the place where ... |
| noextendlt 27735 | Extending a surreal with a... |
| noextendgt 27736 | Extending a surreal with a... |
| nolesgn2o 27737 | Given ` A ` less-than or e... |
| nolesgn2ores 27738 | Given ` A ` less-than or e... |
| nogesgn1o 27739 | Given ` A ` greater than o... |
| nogesgn1ores 27740 | Given ` A ` greater than o... |
| ltssolem1 27741 | Lemma for ~ ltsso . The "... |
| ltsso 27742 | Less-than totally orders t... |
| bdayfo 27743 | The birthday function maps... |
| fvnobday 27744 | The value of a surreal at ... |
| nosepnelem 27745 | Lemma for ~ nosepne . (Co... |
| nosepne 27746 | The value of two non-equal... |
| nosep1o 27747 | If the value of a surreal ... |
| nosep2o 27748 | If the value of a surreal ... |
| nosepdmlem 27749 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27750 | The first place two surrea... |
| nosepeq 27751 | The values of two surreals... |
| nosepssdm 27752 | Given two non-equal surrea... |
| nodenselem4 27753 | Lemma for ~ nodense . Sho... |
| nodenselem5 27754 | Lemma for ~ nodense . If ... |
| nodenselem6 27755 | The restriction of a surre... |
| nodenselem7 27756 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27757 | Lemma for ~ nodense . Giv... |
| nodense 27758 | Given two distinct surreal... |
| bdayimaon 27759 | Lemma for full-eta propert... |
| nolt02olem 27760 | Lemma for ~ nolt02o . If ... |
| nolt02o 27761 | Given ` A ` less-than ` B ... |
| nogt01o 27762 | Given ` A ` greater than `... |
| noresle 27763 | Restriction law for surrea... |
| nomaxmo 27764 | A class of surreals has at... |
| nominmo 27765 | A class of surreals has at... |
| nosupprefixmo 27766 | In any class of surreals, ... |
| noinfprefixmo 27767 | In any class of surreals, ... |
| nosupcbv 27768 | Lemma to change bound vari... |
| nosupno 27769 | The next several theorems ... |
| nosupdm 27770 | The domain of the surreal ... |
| nosupbday 27771 | Birthday bounding law for ... |
| nosupfv 27772 | The value of surreal supre... |
| nosupres 27773 | A restriction law for surr... |
| nosupbnd1lem1 27774 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27775 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27776 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27777 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27778 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27779 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27780 | Bounding law from below fo... |
| nosupbnd2lem1 27781 | Bounding law from above wh... |
| nosupbnd2 27782 | Bounding law from above fo... |
| noinfcbv 27783 | Change bound variables for... |
| noinfno 27784 | The next several theorems ... |
| noinfdm 27785 | Next, we calculate the dom... |
| noinfbday 27786 | Birthday bounding law for ... |
| noinffv 27787 | The value of surreal infim... |
| noinfres 27788 | The restriction of surreal... |
| noinfbnd1lem1 27789 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27790 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27791 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27792 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27793 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27794 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27795 | Bounding law from above fo... |
| noinfbnd2lem1 27796 | Bounding law from below wh... |
| noinfbnd2 27797 | Bounding law from below fo... |
| nosupinfsep 27798 | Given two sets of surreals... |
| noetasuplem1 27799 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27800 | Lemma for ~ noeta . The r... |
| noetasuplem3 27801 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27802 | Lemma for ~ noeta . When ... |
| noetainflem1 27803 | Lemma for ~ noeta . Estab... |
| noetainflem2 27804 | Lemma for ~ noeta . The r... |
| noetainflem3 27805 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27806 | Lemma for ~ noeta . If ` ... |
| noetalem1 27807 | Lemma for ~ noeta . Eithe... |
| noetalem2 27808 | Lemma for ~ noeta . The f... |
| noeta 27809 | The full-eta axiom for the... |
| ltsirr 27812 | Surreal less-than is irref... |
| ltstr 27813 | Surreal less-than is trans... |
| ltsasym 27814 | Surreal less-than is asymm... |
| ltslin 27815 | Surreal less-than obeys tr... |
| ltstrieq2 27816 | Trichotomy law for surreal... |
| ltstrine 27817 | Trichotomy law for surreal... |
| lenlts 27818 | Surreal less-than or equal... |
| ltnles 27819 | Surreal less-than in terms... |
| lesloe 27820 | Surreal less-than or equal... |
| lestri3 27821 | Trichotomy law for surreal... |
| lesnltd 27822 | Surreal less-than or equal... |
| ltsnled 27823 | Surreal less-than in terms... |
| lesloed 27824 | Surreal less-than or equal... |
| lestri3d 27825 | Trichotomy law for surreal... |
| ltlestr 27826 | Surreal transitive law. (... |
| leltstr 27827 | Surreal transitive law. (... |
| lestr 27828 | Surreal transitive law. (... |
| ltstrd 27829 | Surreal less-than is trans... |
| ltlestrd 27830 | Surreal less-than is trans... |
| leltstrd 27831 | Surreal less-than is trans... |
| lestrd 27832 | Surreal less-than or equal... |
| lesid 27833 | Surreal less-than or equal... |
| lestric 27834 | Surreal trichotomy law. (... |
| maxs1 27835 | A surreal is less than or ... |
| maxs2 27836 | A surreal is less than or ... |
| mins1 27837 | The minimum of two surreal... |
| mins2 27838 | The minimum of two surreal... |
| ltlesd 27839 | Surreal less-than implies ... |
| ltsne 27840 | Surreal less-than implies ... |
| ltlesnd 27841 | Surreal less-than in terms... |
| bdayfun 27842 | The birthday function is a... |
| bdayfn 27843 | The birthday function is a... |
| bdaydm 27844 | The birthday function's do... |
| bdaydmOLD 27845 | Obsolete version of ~ bday... |
| bdayrn 27846 | The birthday function's ra... |
| bdayon 27847 | The value of the birthday ... |
| nobdaymin 27848 | Any non-empty class of sur... |
| nocvxminlem 27849 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27850 | Given a nonempty convex cl... |
| noprc 27851 | The surreal numbers are a ... |
| noeta2 27856 | A version of ~ noeta with ... |
| brslts 27857 | Binary relation form of th... |
| sltsex1 27858 | The first argument of surr... |
| sltsex2 27859 | The second argument of sur... |
| sltsss1 27860 | The first argument of surr... |
| sltsss2 27861 | The second argument of sur... |
| sltssep 27862 | The separation property of... |
| sltsd 27863 | Deduce surreal set less-th... |
| sltssnb 27864 | Surreal set less-than of t... |
| sltssn 27865 | Surreal set less-than of t... |
| sltssepc 27866 | Two elements of separated ... |
| sltssepcd 27867 | Two elements of separated ... |
| ssslts1 27868 | Relation between surreal s... |
| ssslts2 27869 | Relation between surreal s... |
| nulslts 27870 | The empty set is less-than... |
| nulsgts 27871 | The empty set is greater t... |
| nulsltsd 27872 | The empty set is less-than... |
| nulsgtsd 27873 | The empty set is greater t... |
| conway 27874 | Conway's Simplicity Theore... |
| cutsval 27875 | The value of the surreal c... |
| cutcuts 27876 | Cut properties of the surr... |
| cutscl 27877 | Closure law for surreal cu... |
| cutscld 27878 | Closure law for surreal cu... |
| cutbday 27879 | The birthday of the surrea... |
| eqcuts 27880 | Condition for equality to ... |
| eqcuts2 27881 | Condition for equality to ... |
| sltstr 27882 | Transitive law for surreal... |
| sltsun1 27883 | Union law for surreal set ... |
| sltsun2 27884 | Union law for surreal set ... |
| cutsun12 27885 | Union law for surreal cuts... |
| dmcuts 27886 | The domain of the surreal ... |
| cutsf 27887 | Functionality statement fo... |
| etaslts 27888 | A restatement of ~ noeta u... |
| etaslts2 27889 | A version of ~ etaslts wit... |
| cutbdaybnd 27890 | An upper bound on the birt... |
| cutbdaybnd2 27891 | An upper bound on the birt... |
| cutbdaybnd2lim 27892 | An upper bound on the birt... |
| cutbdaylt 27893 | If a surreal lies in a gap... |
| lesrec 27894 | A comparison law for surre... |
| lesrecd 27895 | A comparison law for surre... |
| ltsrec 27896 | A comparison law for surre... |
| ltsrecd 27897 | A comparison law for surre... |
| sltsdisj 27898 | If ` A ` preceeds ` B ` , ... |
| eqcuts3 27899 | A variant of the simplicit... |
| 0no 27904 | Surreal zero is a surreal.... |
| 1no 27905 | Surreal one is a surreal. ... |
| bday0 27906 | Calculate the birthday of ... |
| 0lt1s 27907 | Surreal zero is less than ... |
| bday0b 27908 | The only surreal with birt... |
| bday1 27909 | The birthday of surreal on... |
| cuteq0 27910 | Condition for a surreal cu... |
| cutneg 27911 | The simplest number greate... |
| cuteq1 27912 | Condition for a surreal cu... |
| gt0ne0s 27913 | A positive surreal is not ... |
| gt0ne0sd 27914 | A positive surreal is not ... |
| 1ne0s 27915 | Surreal zero does not equa... |
| rightge0 27916 | A surreal is non-negative ... |
| madeval 27927 | The value of the made by f... |
| madeval2 27928 | Alternative characterizati... |
| oldval 27929 | The value of the old optio... |
| newval 27930 | The value of the new optio... |
| madef 27931 | The made function is a fun... |
| oldf 27932 | The older function is a fu... |
| newf 27933 | The new function is a func... |
| old0 27934 | No surreal is older than `... |
| madessno 27935 | Made sets are surreals. (... |
| oldssno 27936 | Old sets are surreals. (C... |
| newssno 27937 | New sets are surreals. (C... |
| madeno 27938 | An element of a made set i... |
| oldno 27939 | An element of an old set i... |
| newno 27940 | An element of a new set is... |
| madenod 27941 | An element of a made set i... |
| oldnod 27942 | An element of an old set i... |
| newnod 27943 | An element of a new set is... |
| leftval 27944 | The value of the left opti... |
| rightval 27945 | The value of the right opt... |
| elleft 27946 | Membership in the left set... |
| elright 27947 | Membership in the right se... |
| leftlt 27948 | A member of a surreal's le... |
| rightgt 27949 | A member of a surreal's ri... |
| leftf 27950 | The functionality of the l... |
| rightf 27951 | The functionality of the r... |
| elmade 27952 | Membership in the made fun... |
| elmade2 27953 | Membership in the made fun... |
| elold 27954 | Membership in an old set. ... |
| sltsleft 27955 | A surreal is greater than ... |
| sltsright 27956 | A surreal is less than its... |
| lltr 27957 | The left options of a surr... |
| made0 27958 | The only surreal made on d... |
| new0 27959 | The only surreal new on da... |
| old1 27960 | The only surreal older tha... |
| madess 27961 | If ` A ` is less than or e... |
| oldssmade 27962 | The older-than set is a su... |
| oldmade 27963 | An element of an old set i... |
| oldmaded 27964 | An element of an old set i... |
| oldss 27965 | If ` A ` is less than or e... |
| leftssold 27966 | The left options are a sub... |
| rightssold 27967 | The right options are a su... |
| leftssno 27968 | The left set of a surreal ... |
| rightssno 27969 | The right set of a surreal... |
| leftold 27970 | An element of a left set i... |
| rightold 27971 | An element of a right set ... |
| leftno 27972 | An element of a left set i... |
| rightno 27973 | An element of a right set ... |
| leftoldd 27974 | An element of a left set i... |
| leftnod 27975 | An element of a left set i... |
| rightoldd 27976 | An element of a right set ... |
| rightnod 27977 | An element of a right set ... |
| madecut 27978 | Given a section that is a ... |
| madeun 27979 | The made set is the union ... |
| madeoldsuc 27980 | The made set is the old se... |
| oldsuc 27981 | The value of the old set a... |
| oldlim 27982 | The value of the old set a... |
| madebdayim 27983 | If a surreal is a member o... |
| oldbdayim 27984 | If ` X ` is in the old set... |
| oldirr 27985 | No surreal is a member of ... |
| leftirr 27986 | No surreal is a member of ... |
| rightirr 27987 | No surreal is a member of ... |
| left0s 27988 | The left set of ` 0s ` is ... |
| right0s 27989 | The right set of ` 0s ` is... |
| left1s 27990 | The left set of ` 1s ` is ... |
| right1s 27991 | The right set of ` 1s ` is... |
| lrold 27992 | The union of the left and ... |
| madebdaylemold 27993 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27994 | Lemma for ~ madebday . If... |
| madebday 27995 | A surreal is part of the s... |
| oldbday 27996 | A surreal is part of the s... |
| newbday 27997 | A surreal is an element of... |
| newbdayim 27998 | One direction of the bicon... |
| lrcut 27999 | A surreal is equal to the ... |
| cutsfo 28000 | The surreal cut function i... |
| ltsn0 28001 | If ` X ` is less than ` Y ... |
| lruneq 28002 | If two surreals share a bi... |
| ltslpss 28003 | If two surreals share a bi... |
| leslss 28004 | If two surreals ` A ` and ... |
| 0elold 28005 | Zero is in the old set of ... |
| 0elleft 28006 | Zero is in the left set of... |
| 0elright 28007 | Zero is in the right set o... |
| madefi 28008 | The made set of an ordinal... |
| oldfi 28009 | The old set of an ordinal ... |
| bdayiun 28010 | The birthday of a surreal ... |
| bdayle 28011 | A condition for bounding a... |
| sltsbday 28012 | Birthday comparison rule f... |
| cofslts 28013 | If every element of ` A ` ... |
| coinitslts 28014 | If ` B ` is coinitial with... |
| cofcut1 28015 | If ` C ` is cofinal with `... |
| cofcut1d 28016 | If ` C ` is cofinal with `... |
| cofcut2 28017 | If ` A ` and ` C ` are mut... |
| cofcut2d 28018 | If ` A ` and ` C ` are mut... |
| cofcutr 28019 | If ` X ` is the cut of ` A... |
| cofcutr1d 28020 | If ` X ` is the cut of ` A... |
| cofcutr2d 28021 | If ` X ` is the cut of ` A... |
| cofcutrtime 28022 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 28023 | If ` X ` is a timely cut o... |
| cofcutrtime2d 28024 | If ` X ` is a timely cut o... |
| cofss 28025 | Cofinality for a subset. ... |
| coiniss 28026 | Coinitiality for a subset.... |
| cutlt 28027 | Eliminating all elements b... |
| cutpos 28028 | Reduce the elements of a c... |
| cutmax 28029 | If ` A ` has a maximum, th... |
| cutmin 28030 | If ` B ` has a minimum, th... |
| cutminmax 28031 | If the left set of ` X ` h... |
| lrrecval 28034 | The next step in the devel... |
| lrrecval2 28035 | Next, we establish an alte... |
| lrrecpo 28036 | Now, we establish that ` R... |
| lrrecse 28037 | Next, we show that ` R ` i... |
| lrrecfr 28038 | Now we show that ` R ` is ... |
| lrrecpred 28039 | Finally, we calculate the ... |
| noinds 28040 | Induction principle for a ... |
| norecfn 28041 | Surreal recursion over one... |
| norecov 28042 | Calculate the value of the... |
| noxpordpo 28045 | To get through most of the... |
| noxpordfr 28046 | Next we establish the foun... |
| noxpordse 28047 | Next we establish the set-... |
| noxpordpred 28048 | Next we calculate the pred... |
| no2indlesm 28049 | Double induction on surrea... |
| no2inds 28050 | Double induction on surrea... |
| norec2fn 28051 | The double-recursion opera... |
| norec2ov 28052 | The value of the double-re... |
| no3inds 28053 | Triple induction over surr... |
| addsfn 28056 | Surreal addition is a func... |
| addsval 28057 | The value of surreal addit... |
| addsval2 28058 | The value of surreal addit... |
| addsrid 28059 | Surreal addition to zero i... |
| addsridd 28060 | Surreal addition to zero i... |
| addscom 28061 | Surreal addition commutes.... |
| addscomd 28062 | Surreal addition commutes.... |
| addslid 28063 | Surreal addition to zero i... |
| addsproplem1 28064 | Lemma for surreal addition... |
| addsproplem2 28065 | Lemma for surreal addition... |
| addsproplem3 28066 | Lemma for surreal addition... |
| addsproplem4 28067 | Lemma for surreal addition... |
| addsproplem5 28068 | Lemma for surreal addition... |
| addsproplem6 28069 | Lemma for surreal addition... |
| addsproplem7 28070 | Lemma for surreal addition... |
| addsprop 28071 | Inductively show that surr... |
| addcutslem 28072 | Lemma for ~ addcuts . Sho... |
| addcuts 28073 | Demonstrate the cut proper... |
| addcuts2 28074 | Show that the cut involved... |
| addscld 28075 | Surreal numbers are closed... |
| addscl 28076 | Surreal numbers are closed... |
| addsf 28077 | Function statement for sur... |
| addsfo 28078 | Surreal addition is onto. ... |
| peano2no 28079 | A theorem for surreals tha... |
| ltadds1im 28080 | Surreal less-than is prese... |
| ltadds2im 28081 | Surreal less-than is prese... |
| leadds1im 28082 | Surreal less-than or equal... |
| leadds2im 28083 | Surreal less-than or equal... |
| leadds1 28084 | Addition to both sides of ... |
| leadds2 28085 | Addition to both sides of ... |
| ltadds2 28086 | Addition to both sides of ... |
| ltadds1 28087 | Addition to both sides of ... |
| addscan2 28088 | Cancellation law for surre... |
| addscan1 28089 | Cancellation law for surre... |
| leadds1d 28090 | Addition to both sides of ... |
| leadds2d 28091 | Addition to both sides of ... |
| ltadds2d 28092 | Addition to both sides of ... |
| ltadds1d 28093 | Addition to both sides of ... |
| addscan2d 28094 | Cancellation law for surre... |
| addscan1d 28095 | Cancellation law for surre... |
| addsuniflem 28096 | Lemma for ~ addsunif . St... |
| addsunif 28097 | Uniformity theorem for sur... |
| addsasslem1 28098 | Lemma for addition associa... |
| addsasslem2 28099 | Lemma for addition associa... |
| addsass 28100 | Surreal addition is associ... |
| addsassd 28101 | Surreal addition is associ... |
| adds32d 28102 | Commutative/associative la... |
| adds12d 28103 | Commutative/associative la... |
| adds4d 28104 | Rearrangement of four term... |
| adds42d 28105 | Rearrangement of four term... |
| ltaddspos1d 28106 | Addition of a positive num... |
| ltaddspos2d 28107 | Addition of a positive num... |
| lt2addsd 28108 | Adding both sides of two s... |
| addsgt0d 28109 | The sum of two positive su... |
| ltsp1d 28110 | A surreal is less than its... |
| addsge01d 28111 | A surreal is less-than or ... |
| addbdaylem 28112 | Lemma for ~ addbday . (Co... |
| addbday 28113 | The birthday of the sum of... |
| negsfn 28118 | Surreal negation is a func... |
| subsfn 28119 | Surreal subtraction is a f... |
| negsval 28120 | The value of the surreal n... |
| neg0s 28121 | Negative surreal zero is s... |
| neg1s 28122 | An expression for negative... |
| negsproplem1 28123 | Lemma for surreal negation... |
| negsproplem2 28124 | Lemma for surreal negation... |
| negsproplem3 28125 | Lemma for surreal negation... |
| negsproplem4 28126 | Lemma for surreal negation... |
| negsproplem5 28127 | Lemma for surreal negation... |
| negsproplem6 28128 | Lemma for surreal negation... |
| negsproplem7 28129 | Lemma for surreal negation... |
| negsprop 28130 | Show closure and ordering ... |
| negscl 28131 | The surreals are closed un... |
| negscld 28132 | The surreals are closed un... |
| ltnegsim 28133 | The forward direction of t... |
| negcut 28134 | The cut properties of surr... |
| negcut2 28135 | The cut that defines surre... |
| negsid 28136 | Surreal addition of a numb... |
| negsidd 28137 | Surreal addition of a numb... |
| negsex 28138 | Every surreal has a negati... |
| negnegs 28139 | A surreal is equal to the ... |
| ltnegs 28140 | Negative of both sides of ... |
| lenegs 28141 | Negative of both sides of ... |
| ltnegsd 28142 | Negative of both sides of ... |
| lenegsd 28143 | Negative of both sides of ... |
| negs11 28144 | Surreal negation is one-to... |
| negsdi 28145 | Distribution of surreal ne... |
| lt0negs2d 28146 | Comparison of a surreal an... |
| negsf 28147 | Function statement for sur... |
| negsfo 28148 | Function statement for sur... |
| negsf1o 28149 | Surreal negation is a bije... |
| negsunif 28150 | Uniformity property for su... |
| negbdaylem 28151 | Lemma for ~ negbday . Bou... |
| negbday 28152 | Negation of a surreal numb... |
| negleft 28153 | The left set of the negati... |
| negright 28154 | The right set of the negat... |
| subsval 28155 | The value of surreal subtr... |
| subsvald 28156 | The value of surreal subtr... |
| subscl 28157 | Closure law for surreal su... |
| subscld 28158 | Closure law for surreal su... |
| subsf 28159 | Function statement for sur... |
| subsfo 28160 | Surreal subtraction is an ... |
| negsval2 28161 | Surreal negation in terms ... |
| negsval2d 28162 | Surreal negation in terms ... |
| subsid1 28163 | Identity law for subtracti... |
| subsid 28164 | Subtraction of a surreal f... |
| subadds 28165 | Relationship between addit... |
| subaddsd 28166 | Relationship between addit... |
| pncans 28167 | Cancellation law for surre... |
| pncan3s 28168 | Subtraction and addition o... |
| pncan2s 28169 | Cancellation law for surre... |
| npcans 28170 | Cancellation law for surre... |
| ltsubs1 28171 | Subtraction from both side... |
| ltsubs2 28172 | Subtraction from both side... |
| ltsubs1d 28173 | Subtraction from both side... |
| ltsubs2d 28174 | Subtraction from both side... |
| negsubsdi2d 28175 | Distribution of negative o... |
| addsubsassd 28176 | Associative-type law for s... |
| addsubsd 28177 | Law for surreal addition a... |
| ltsubsubsbd 28178 | Equivalence for the surrea... |
| ltsubsubs2bd 28179 | Equivalence for the surrea... |
| ltsubsubs3bd 28180 | Equivalence for the surrea... |
| lesubsubsbd 28181 | Equivalence for the surrea... |
| lesubsubs2bd 28182 | Equivalence for the surrea... |
| lesubsubs3bd 28183 | Equivalence for the surrea... |
| ltsubaddsd 28184 | Surreal less-than relation... |
| ltsubadds2d 28185 | Surreal less-than relation... |
| ltaddsubsd 28186 | Surreal less-than relation... |
| ltaddsubs2d 28187 | Surreal less-than relation... |
| lesubaddsd 28188 | Surreal less-than or equal... |
| subsubs4d 28189 | Law for double surreal sub... |
| subsubs2d 28190 | Law for double surreal sub... |
| lesubsd 28191 | Swap subtrahends in a surr... |
| nncansd 28192 | Cancellation law for surre... |
| posdifsd 28193 | Comparison of two surreals... |
| ltsubsposd 28194 | Subtraction of a positive ... |
| subsge0d 28195 | Non-negative subtraction. ... |
| addsubs4d 28196 | Rearrangement of four term... |
| ltsm1d 28197 | A surreal is greater than ... |
| subscan1d 28198 | Cancellation law for surre... |
| subscan2d 28199 | Cancellation law for surre... |
| subseq0d 28200 | The difference between two... |
| mulsfn 28203 | Surreal multiplication is ... |
| mulsval 28204 | The value of surreal multi... |
| mulsval2lem 28205 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 28206 | The value of surreal multi... |
| muls01 28207 | Surreal multiplication by ... |
| mulsrid 28208 | Surreal one is a right ide... |
| mulsridd 28209 | Surreal one is a right ide... |
| mulsproplemcbv 28210 | Lemma for surreal multipli... |
| mulsproplem1 28211 | Lemma for surreal multipli... |
| mulsproplem2 28212 | Lemma for surreal multipli... |
| mulsproplem3 28213 | Lemma for surreal multipli... |
| mulsproplem4 28214 | Lemma for surreal multipli... |
| mulsproplem5 28215 | Lemma for surreal multipli... |
| mulsproplem6 28216 | Lemma for surreal multipli... |
| mulsproplem7 28217 | Lemma for surreal multipli... |
| mulsproplem8 28218 | Lemma for surreal multipli... |
| mulsproplem9 28219 | Lemma for surreal multipli... |
| mulsproplem10 28220 | Lemma for surreal multipli... |
| mulsproplem11 28221 | Lemma for surreal multipli... |
| mulsproplem12 28222 | Lemma for surreal multipli... |
| mulsproplem13 28223 | Lemma for surreal multipli... |
| mulsproplem14 28224 | Lemma for surreal multipli... |
| mulsprop 28225 | Surreals are closed under ... |
| mulcutlem 28226 | Lemma for ~ mulcut . Stat... |
| mulcut 28227 | Show the cut properties of... |
| mulcut2 28228 | Show that the cut involved... |
| mulscl 28229 | The surreals are closed un... |
| mulscld 28230 | The surreals are closed un... |
| ltmuls 28231 | An ordering relationship f... |
| ltmulsd 28232 | An ordering relationship f... |
| lemulsd 28233 | An ordering relationship f... |
| mulscom 28234 | Surreal multiplication com... |
| mulscomd 28235 | Surreal multiplication com... |
| muls02 28236 | Surreal multiplication by ... |
| mulslid 28237 | Surreal one is a left iden... |
| mulslidd 28238 | Surreal one is a left iden... |
| mulsgt0 28239 | The product of two positiv... |
| mulsgt0d 28240 | The product of two positiv... |
| mulsge0d 28241 | The product of two non-neg... |
| sltmuls1 28242 | One surreal set less-than ... |
| sltmuls2 28243 | One surreal set less-than ... |
| mulsuniflem 28244 | Lemma for ~ mulsunif . St... |
| mulsunif 28245 | Surreal multiplication has... |
| addsdilem1 28246 | Lemma for surreal distribu... |
| addsdilem2 28247 | Lemma for surreal distribu... |
| addsdilem3 28248 | Lemma for ~ addsdi . Show... |
| addsdilem4 28249 | Lemma for ~ addsdi . Show... |
| addsdi 28250 | Distributive law for surre... |
| addsdid 28251 | Distributive law for surre... |
| addsdird 28252 | Distributive law for surre... |
| subsdid 28253 | Distribution of surreal mu... |
| subsdird 28254 | Distribution of surreal mu... |
| mulnegs1d 28255 | Product with negative is n... |
| mulnegs2d 28256 | Product with negative is n... |
| mul2negsd 28257 | Surreal product of two neg... |
| mulsasslem1 28258 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28259 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28260 | Lemma for ~ mulsass . Dem... |
| mulsass 28261 | Associative law for surrea... |
| mulsassd 28262 | Associative law for surrea... |
| muls4d 28263 | Rearrangement of four surr... |
| mulsunif2lem 28264 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28265 | Alternate expression for s... |
| ltmuls2 28266 | Multiplication of both sid... |
| ltmuls2d 28267 | Multiplication of both sid... |
| ltmuls1d 28268 | Multiplication of both sid... |
| lemuls2d 28269 | Multiplication of both sid... |
| lemuls1d 28270 | Multiplication of both sid... |
| ltmulnegs1d 28271 | Multiplication of both sid... |
| ltmulnegs2d 28272 | Multiplication of both sid... |
| mulscan2dlem 28273 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28274 | Cancellation of surreal mu... |
| mulscan1d 28275 | Cancellation of surreal mu... |
| muls12d 28276 | Commutative/associative la... |
| lemuls1ad 28277 | Multiplication of both sid... |
| ltmuls12ad 28278 | Comparison of the product ... |
| divsmo 28279 | Uniqueness of surreal inve... |
| muls0ord 28280 | If a surreal product is ze... |
| mulsne0bd 28281 | The product of two nonzero... |
| divsval 28284 | The value of surreal divis... |
| norecdiv 28285 | If a surreal has a recipro... |
| noreceuw 28286 | If a surreal has a recipro... |
| recsne0 28287 | If a surreal has a recipro... |
| divmulsw 28288 | Relationship between surre... |
| divmulswd 28289 | Relationship between surre... |
| divsclw 28290 | Weak division closure law.... |
| divsclwd 28291 | Weak division closure law.... |
| divscan2wd 28292 | A weak cancellation law fo... |
| divscan1wd 28293 | A weak cancellation law fo... |
| ltdivmulswd 28294 | Surreal less-than relation... |
| ltdivmuls2wd 28295 | Surreal less-than relation... |
| ltmuldivswd 28296 | Surreal less-than relation... |
| ltmuldivs2wd 28297 | Surreal less-than relation... |
| divsasswd 28298 | An associative law for sur... |
| divs1 28299 | A surreal divided by one i... |
| divs1d 28300 | A surreal divided by one i... |
| precsexlemcbv 28301 | Lemma for surreal reciproc... |
| precsexlem1 28302 | Lemma for surreal reciproc... |
| precsexlem2 28303 | Lemma for surreal reciproc... |
| precsexlem3 28304 | Lemma for surreal reciproc... |
| precsexlem4 28305 | Lemma for surreal reciproc... |
| precsexlem5 28306 | Lemma for surreal reciproc... |
| precsexlem6 28307 | Lemma for surreal reciproc... |
| precsexlem7 28308 | Lemma for surreal reciproc... |
| precsexlem8 28309 | Lemma for surreal reciproc... |
| precsexlem9 28310 | Lemma for surreal reciproc... |
| precsexlem10 28311 | Lemma for surreal reciproc... |
| precsexlem11 28312 | Lemma for surreal reciproc... |
| precsex 28313 | Every positive surreal has... |
| recsex 28314 | A nonzero surreal has a re... |
| recsexd 28315 | A nonzero surreal has a re... |
| divmuls 28316 | Relationship between surre... |
| divmulsd 28317 | Relationship between surre... |
| divscl 28318 | Surreal division closure l... |
| divscld 28319 | Surreal division closure l... |
| divscan2d 28320 | A cancellation law for sur... |
| divscan1d 28321 | A cancellation law for sur... |
| ltdivmulsd 28322 | Surreal less-than relation... |
| ltdivmuls2d 28323 | Surreal less-than relation... |
| ltmuldivsd 28324 | Surreal less-than relation... |
| ltmuldivs2d 28325 | Surreal less-than relation... |
| divsassd 28326 | An associative law for sur... |
| divmuldivsd 28327 | Multiplication of two surr... |
| divdivs1d 28328 | Surreal division into a fr... |
| divsrecd 28329 | Relationship between surre... |
| divsdird 28330 | Distribution of surreal di... |
| divscan3d 28331 | A cancellation law for sur... |
| abssval 28334 | The value of surreal absol... |
| absscl 28335 | Closure law for surreal ab... |
| abssid 28336 | The absolute value of a no... |
| abs0s 28337 | The absolute value of surr... |
| abssnid 28338 | For a negative surreal, it... |
| absmuls 28339 | Surreal absolute value dis... |
| abssge0 28340 | The absolute value of a su... |
| abssor 28341 | The absolute value of a su... |
| absnegs 28342 | Surreal absolute value of ... |
| leabss 28343 | A surreal is less than or ... |
| abslts 28344 | Surreal absolute value and... |
| abssubs 28345 | Swapping order of surreal ... |
| elons 28348 | Membership in the class of... |
| onssno 28349 | The surreal ordinals are a... |
| onno 28350 | A surreal ordinal is a sur... |
| 0ons 28351 | Surreal zero is a surreal ... |
| 1ons 28352 | Surreal one is a surreal o... |
| elons2 28353 | A surreal is ordinal iff i... |
| elons2d 28354 | The cut of any set of surr... |
| onleft 28355 | The left set of a surreal ... |
| ltonold 28356 | The class of ordinals less... |
| ltonsex 28357 | The class of ordinals less... |
| oncutleft 28358 | A surreal ordinal is equal... |
| oncutlt 28359 | A surreal ordinal is the s... |
| bday11on 28360 | The birthday function is o... |
| onnolt 28361 | If a surreal ordinal is le... |
| onlts 28362 | Less-than is the same as b... |
| onles 28363 | Less-than or equal is the ... |
| onltsd 28364 | Less-than is the same as b... |
| onlesd 28365 | Less-than or equal is the ... |
| oniso 28366 | The birthday function rest... |
| onswe 28367 | Surreal less-than well-ord... |
| onsse 28368 | Surreal less-than is set-l... |
| onsis 28369 | Transfinite induction sche... |
| ons2ind 28370 | Double induction schema fo... |
| bdayons 28371 | The birthday of a surreal ... |
| onaddscl 28372 | The surreal ordinals are c... |
| onmulscl 28373 | The surreal ordinals are c... |
| addonbday 28374 | The birthday of the sum of... |
| peano2ons 28375 | The successor of a surreal... |
| onsbnd 28376 | The surreals of a given bi... |
| onsbnd2 28377 | The surreals of a given bi... |
| seqsex 28380 | Existence of the surreal s... |
| seqseq123d 28381 | Equality deduction for the... |
| nfseqs 28382 | Hypothesis builder for the... |
| seqsval 28383 | The value of the surreal s... |
| noseqex 28384 | The next several theorems ... |
| noseq0 28385 | The surreal ` A ` is a mem... |
| noseqp1 28386 | One plus an element of ` Z... |
| noseqind 28387 | Peano's inductive postulat... |
| noseqinds 28388 | Induction schema for surre... |
| noseqssno 28389 | A surreal sequence is a su... |
| noseqno 28390 | An element of a surreal se... |
| om2noseq0 28391 | The mapping ` G ` is a one... |
| om2noseqsuc 28392 | The value of ` G ` at a su... |
| om2noseqfo 28393 | Function statement for ` G... |
| om2noseqlt 28394 | Surreal less-than relation... |
| om2noseqlt2 28395 | The mapping ` G ` preserve... |
| om2noseqf1o 28396 | ` G ` is a bijection. (Co... |
| om2noseqiso 28397 | ` G ` is an isomorphism fr... |
| om2noseqoi 28398 | An alternative definition ... |
| om2noseqrdg 28399 | A helper lemma for the val... |
| noseqrdglem 28400 | A helper lemma for the val... |
| noseqrdgfn 28401 | The recursive definition g... |
| noseqrdg0 28402 | Initial value of a recursi... |
| noseqrdgsuc 28403 | Successor value of a recur... |
| seqsfn 28404 | The surreal sequence build... |
| seqs1 28405 | The value of the surreal s... |
| seqsp1 28406 | The value of the surreal s... |
| n0sexg 28411 | The set of all non-negativ... |
| n0sex 28412 | The set of all non-negativ... |
| nnsex 28413 | The set of all positive su... |
| peano5n0s 28414 | Peano's inductive postulat... |
| n0ssno 28415 | The non-negative surreal i... |
| nnssn0s 28416 | The positive surreal integ... |
| nnssno 28417 | The positive surreal integ... |
| n0no 28418 | A non-negative surreal int... |
| nnno 28419 | A positive surreal integer... |
| n0nod 28420 | A non-negative surreal int... |
| nnnod 28421 | A positive surreal integer... |
| nnn0s 28422 | A positive surreal integer... |
| nnn0sd 28423 | A positive surreal integer... |
| 0n0s 28424 | Peano postulate: ` 0s ` is... |
| peano2n0s 28425 | Peano postulate: the succe... |
| peano2n0sd 28426 | Peano postulate: the succe... |
| dfn0s2 28427 | Alternate definition of th... |
| n0sind 28428 | Principle of Mathematical ... |
| n0cut 28429 | A cut form for non-negativ... |
| n0cut2 28430 | A cut form for the success... |
| n0on 28431 | A surreal natural is a sur... |
| nnne0s 28432 | A surreal positive integer... |
| n0sge0 28433 | A non-negative integer is ... |
| nnsgt0 28434 | A positive integer is grea... |
| elnns 28435 | Membership in the positive... |
| elnns2 28436 | A positive surreal integer... |
| n0s0suc 28437 | A non-negative surreal int... |
| nnsge1 28438 | A positive surreal integer... |
| n0addscl 28439 | The non-negative surreal i... |
| n0mulscl 28440 | The non-negative surreal i... |
| nnaddscl 28441 | The positive surreal integ... |
| nnmulscl 28442 | The positive surreal integ... |
| 1n0s 28443 | Surreal one is a non-negat... |
| 1nns 28444 | Surreal one is a positive ... |
| peano2nns 28445 | Peano postulate for positi... |
| nnsrecgt0d 28446 | The reciprocal of a positi... |
| n0bday 28447 | A non-negative surreal int... |
| n0ssoldg 28448 | The non-negative surreal i... |
| n0ssold 28449 | The non-negative surreal i... |
| n0fincut 28450 | The simplest number greate... |
| onsfi 28451 | A surreal ordinal with a f... |
| eln0s2 28452 | A non-negative surreal int... |
| onltn0s 28453 | A surreal ordinal that is ... |
| n0cutlt 28454 | A non-negative surreal int... |
| seqn0sfn 28455 | The surreal sequence build... |
| eln0s 28456 | A non-negative surreal int... |
| n0s0m1 28457 | Every non-negative surreal... |
| n0subs 28458 | Subtraction of non-negativ... |
| n0subs2 28459 | Subtraction of non-negativ... |
| n0ltsp1le 28460 | Non-negative surreal order... |
| n0lesltp1 28461 | Non-negative surreal order... |
| n0lesm1lt 28462 | Non-negative surreal order... |
| n0lts1e0 28463 | A non-negative surreal int... |
| bdayn0p1 28464 | The birthday of ` A +s 1s ... |
| bdayn0sf1o 28465 | The birthday function rest... |
| n0p1nns 28466 | One plus a non-negative su... |
| dfnns2 28467 | Alternate definition of th... |
| nnsind 28468 | Principle of Mathematical ... |
| nn1m1nns 28469 | Every positive surreal int... |
| nnm1n0s 28470 | A positive surreal integer... |
| eucliddivs 28471 | Euclid's division lemma fo... |
| oldfib 28472 | The old set of an ordinal ... |
| zsex 28475 | The surreal integers form ... |
| zssno 28476 | The surreal integers are a... |
| zno 28477 | A surreal integer is a sur... |
| znod 28478 | A surreal integer is a sur... |
| elzs 28479 | Membership in the set of s... |
| nnzsubs 28480 | The difference of two surr... |
| nnzs 28481 | A positive surreal integer... |
| nnzsd 28482 | A positive surreal integer... |
| 0zs 28483 | Zero is a surreal integer.... |
| n0zs 28484 | A non-negative surreal int... |
| n0zsd 28485 | A non-negative surreal int... |
| 1zs 28486 | One is a surreal integer. ... |
| znegscl 28487 | The surreal integers are c... |
| znegscld 28488 | The surreal integers are c... |
| zaddscl 28489 | The surreal integers are c... |
| zaddscld 28490 | The surreal integers are c... |
| zsubscld 28491 | The surreal integers are c... |
| zmulscld 28492 | The surreal integers are c... |
| elzn0s 28493 | A surreal integer is a sur... |
| elzs2 28494 | A surreal integer is eithe... |
| eln0zs 28495 | Non-negative surreal integ... |
| elnnzs 28496 | Positive surreal integer p... |
| elznns 28497 | Surreal integer property e... |
| zn0subs 28498 | The non-negative differenc... |
| peano5uzs 28499 | Peano's inductive postulat... |
| uzsind 28500 | Induction on the upper sur... |
| zsbday 28501 | A surreal integer has a fi... |
| zcuts 28502 | A cut expression for surre... |
| zcuts0 28503 | Either the left or right s... |
| zsoring 28504 | The surreal integers form ... |
| 1p1e2s 28511 | One plus one is two. Surr... |
| no2times 28512 | Version of ~ 2times for su... |
| 2nns 28513 | Surreal two is a surreal n... |
| 2no 28514 | Surreal two is a surreal n... |
| 2ne0s 28515 | Surreal two is nonzero. (... |
| n0seo 28516 | A non-negative surreal int... |
| zseo 28517 | A surreal integer is eithe... |
| twocut 28518 | Two times the cut of zero ... |
| nohalf 28519 | An explicit expression for... |
| expsval 28520 | The value of surreal expon... |
| expnnsval 28521 | Value of surreal exponenti... |
| exps0 28522 | Surreal exponentiation to ... |
| exps1 28523 | Surreal exponentiation to ... |
| expsp1 28524 | Value of a surreal number ... |
| expscllem 28525 | Lemma for proving non-nega... |
| expscl 28526 | Closure law for surreal ex... |
| n0expscl 28527 | Closure law for non-negati... |
| nnexpscl 28528 | Closure law for positive s... |
| zexpscl 28529 | Closure law for surreal in... |
| expadds 28530 | Sum of exponents law for s... |
| expsne0 28531 | A non-negative surreal int... |
| expsgt0 28532 | A non-negative surreal int... |
| pw2recs 28533 | Any power of two has a mul... |
| pw2divscld 28534 | Division closure for power... |
| pw2divmulsd 28535 | Relationship between surre... |
| pw2divscan3d 28536 | Cancellation law for surre... |
| pw2divscan2d 28537 | A cancellation law for sur... |
| pw2divsassd 28538 | An associative law for div... |
| pw2divscan4d 28539 | Cancellation law for divis... |
| pw2gt0divsd 28540 | Division of a positive sur... |
| pw2ge0divsd 28541 | Divison of a non-negative ... |
| pw2divsrecd 28542 | Relationship between surre... |
| pw2divsdird 28543 | Distribution of surreal di... |
| pw2divsnegd 28544 | Move negative sign inside ... |
| pw2ltdivmulsd 28545 | Surreal less-than relation... |
| pw2ltmuldivs2d 28546 | Surreal less-than relation... |
| pw2ltsdiv1d 28547 | Surreal less-than relation... |
| avglts1d 28548 | Ordering property for aver... |
| avglts2d 28549 | Ordering property for aver... |
| pw2divs0d 28550 | Division into zero is zero... |
| pw2divsidd 28551 | Identity law for division ... |
| pw2ltdivmuls2d 28552 | Surreal less-than relation... |
| halfcut 28553 | Relate the cut of twice of... |
| addhalfcut 28554 | The cut of a surreal non-n... |
| pw2cut 28555 | Extend ~ halfcut to arbitr... |
| pw2cutp1 28556 | Simplify ~ pw2cut in the c... |
| pw2cut2 28557 | Cut expression for powers ... |
| bdaypw2n0bndlem 28558 | Lemma for ~ bdaypw2n0bnd .... |
| bdaypw2n0bnd 28559 | Upper bound for the birthd... |
| bdaypw2bnd 28560 | Birthday bounding rule for... |
| bdayfinbndcbv 28561 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem1 28562 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem2 28563 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbnd 28564 | Given a non-negative integ... |
| z12bdaylem1 28565 | Lemma for ~ z12bday . Pro... |
| z12bdaylem2 28566 | Lemma for ~ z12bday . Sho... |
| elz12s 28567 | Membership in the dyadic f... |
| elz12si 28568 | Inference form of membersh... |
| z12sex 28569 | The class of dyadic fracti... |
| zz12s 28570 | A surreal integer is a dya... |
| z12no 28571 | A dyadic is a surreal. (C... |
| z12addscl 28572 | The dyadics are closed und... |
| z12negscl 28573 | The dyadics are closed und... |
| z12subscl 28574 | The dyadics are closed und... |
| z12shalf 28575 | Half of a dyadic is a dyad... |
| z12negsclb 28576 | A surreal is a dyadic frac... |
| z12zsodd 28577 | A dyadic fraction is eithe... |
| z12sge0 28578 | An expression for non-nega... |
| z12bdaylem 28579 | Lemma for ~ z12bday . Han... |
| z12bday 28580 | A dyadic fraction has a fi... |
| bdayfinlem 28581 | Lemma for ~ bdayfin . Han... |
| bdayfin 28582 | A surreal has a finite bir... |
| dfz12s2 28583 | The set of dyadic fraction... |
| elreno 28586 | Membership in the set of s... |
| reno 28587 | A surreal real is a surrea... |
| renod 28588 | A surreal real is a surrea... |
| recut 28589 | The cut involved in defini... |
| elreno2 28590 | Alternate characterization... |
| 0reno 28591 | Surreal zero is a surreal ... |
| 1reno 28592 | Surreal one is a surreal r... |
| renegscl 28593 | The surreal reals are clos... |
| readdscl 28594 | The surreal reals are clos... |
| remulscllem1 28595 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28596 | Lemma for ~ remulscl . Bo... |
| remulscl 28597 | The surreal reals are clos... |
| itvndx 28608 | Index value of the Interva... |
| lngndx 28609 | Index value of the "line" ... |
| itvid 28610 | Utility theorem: index-ind... |
| lngid 28611 | Utility theorem: index-ind... |
| slotsinbpsd 28612 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28613 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28614 | The slot for the line is n... |
| trkgstr 28615 | Functionality of a Tarski ... |
| trkgbas 28616 | The base set of a Tarski g... |
| trkgdist 28617 | The measure of a distance ... |
| trkgitv 28618 | The congruence relation in... |
| istrkgc 28625 | Property of being a Tarski... |
| istrkgb 28626 | Property of being a Tarski... |
| istrkgcb 28627 | Property of being a Tarski... |
| istrkge 28628 | Property of fulfilling Euc... |
| istrkgl 28629 | Building lines from the se... |
| istrkgld 28630 | Property of fulfilling the... |
| istrkg2ld 28631 | Property of fulfilling the... |
| istrkg3ld 28632 | Property of fulfilling the... |
| axtgcgrrflx 28633 | Axiom of reflexivity of co... |
| axtgcgrid 28634 | Axiom of identity of congr... |
| axtgsegcon 28635 | Axiom of segment construct... |
| axtg5seg 28636 | Five segments axiom, Axiom... |
| axtgbtwnid 28637 | Identity of Betweenness. ... |
| axtgpasch 28638 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28639 | Axiom of Continuity. Axio... |
| axtgcont 28640 | Axiom of Continuity. Axio... |
| axtglowdim2 28641 | Lower dimension axiom for ... |
| axtgupdim2 28642 | Upper dimension axiom for ... |
| axtgeucl 28643 | Euclid's Axiom. Axiom A10... |
| tgjustf 28644 | Given any function ` F ` ,... |
| tgjustr 28645 | Given any equivalence rela... |
| tgjustc1 28646 | A justification for using ... |
| tgjustc2 28647 | A justification for using ... |
| tgcgrcomimp 28648 | Congruence commutes on the... |
| tgcgrcomr 28649 | Congruence commutes on the... |
| tgcgrcoml 28650 | Congruence commutes on the... |
| tgcgrcomlr 28651 | Congruence commutes on bot... |
| tgcgreqb 28652 | Congruence and equality. ... |
| tgcgreq 28653 | Congruence and equality. ... |
| tgcgrneq 28654 | Congruence and equality. ... |
| tgcgrtriv 28655 | Degenerate segments are co... |
| tgcgrextend 28656 | Link congruence over a pai... |
| tgsegconeq 28657 | Two points that satisfy th... |
| tgbtwntriv2 28658 | Betweenness always holds f... |
| tgbtwncom 28659 | Betweenness commutes. The... |
| tgbtwncomb 28660 | Betweenness commutes, bico... |
| tgbtwnne 28661 | Betweenness and inequality... |
| tgbtwntriv1 28662 | Betweenness always holds f... |
| tgbtwnswapid 28663 | If you can swap the first ... |
| tgbtwnintr 28664 | Inner transitivity law for... |
| tgbtwnexch3 28665 | Exchange the first endpoin... |
| tgbtwnouttr2 28666 | Outer transitivity law for... |
| tgbtwnexch2 28667 | Exchange the outer point o... |
| tgbtwnouttr 28668 | Outer transitivity law for... |
| tgbtwnexch 28669 | Outer transitivity law for... |
| tgtrisegint 28670 | A line segment between two... |
| tglowdim1 28671 | Lower dimension axiom for ... |
| tglowdim1i 28672 | Lower dimension axiom for ... |
| tgldimor 28673 | Excluded-middle like state... |
| tgldim0eq 28674 | In dimension zero, any two... |
| tgldim0itv 28675 | In dimension zero, any two... |
| tgldim0cgr 28676 | In dimension zero, any two... |
| tgbtwndiff 28677 | There is always a ` c ` di... |
| tgdim01 28678 | In geometries of dimension... |
| tgifscgr 28679 | Inner five segment congrue... |
| tgcgrsub 28680 | Removing identical parts f... |
| iscgrg 28683 | The congruence property fo... |
| iscgrgd 28684 | The property for two seque... |
| iscgrglt 28685 | The property for two seque... |
| trgcgrg 28686 | The property for two trian... |
| trgcgr 28687 | Triangle congruence. (Con... |
| ercgrg 28688 | The shape congruence relat... |
| tgcgrxfr 28689 | A line segment can be divi... |
| cgr3id 28690 | Reflexivity law for three-... |
| cgr3simp1 28691 | Deduce segment congruence ... |
| cgr3simp2 28692 | Deduce segment congruence ... |
| cgr3simp3 28693 | Deduce segment congruence ... |
| cgr3swap12 28694 | Permutation law for three-... |
| cgr3swap23 28695 | Permutation law for three-... |
| cgr3swap13 28696 | Permutation law for three-... |
| cgr3rotr 28697 | Permutation law for three-... |
| cgr3rotl 28698 | Permutation law for three-... |
| trgcgrcom 28699 | Commutative law for three-... |
| cgr3tr 28700 | Transitivity law for three... |
| tgbtwnxfr 28701 | A condition for extending ... |
| tgcgr4 28702 | Two quadrilaterals to be c... |
| isismt 28705 | Property of being an isome... |
| ismot 28706 | Property of being an isome... |
| motcgr 28707 | Property of a motion: dist... |
| idmot 28708 | The identity is a motion. ... |
| motf1o 28709 | Motions are bijections. (... |
| motcl 28710 | Closure of motions. (Cont... |
| motco 28711 | The composition of two mot... |
| cnvmot 28712 | The converse of a motion i... |
| motplusg 28713 | The operation for motions ... |
| motgrp 28714 | The motions of a geometry ... |
| motcgrg 28715 | Property of a motion: dist... |
| motcgr3 28716 | Property of a motion: dist... |
| tglng 28717 | Lines of a Tarski Geometry... |
| tglnfn 28718 | Lines as functions. (Cont... |
| tglnunirn 28719 | Lines are sets of points. ... |
| tglnpt 28720 | Lines are sets of points. ... |
| tglngne 28721 | It takes two different poi... |
| tglngval 28722 | The line going through poi... |
| tglnssp 28723 | Lines are subset of the ge... |
| tgellng 28724 | Property of lying on the l... |
| tgcolg 28725 | We choose the notation ` (... |
| btwncolg1 28726 | Betweenness implies coline... |
| btwncolg2 28727 | Betweenness implies coline... |
| btwncolg3 28728 | Betweenness implies coline... |
| colcom 28729 | Swapping the points defini... |
| colrot1 28730 | Rotating the points defini... |
| colrot2 28731 | Rotating the points defini... |
| ncolcom 28732 | Swapping non-colinear poin... |
| ncolrot1 28733 | Rotating non-colinear poin... |
| ncolrot2 28734 | Rotating non-colinear poin... |
| tgdim01ln 28735 | In geometries of dimension... |
| ncoltgdim2 28736 | If there are three non-col... |
| lnxfr 28737 | Transfer law for colineari... |
| lnext 28738 | Extend a line with a missi... |
| tgfscgr 28739 | Congruence law for the gen... |
| lncgr 28740 | Congruence rule for lines.... |
| lnid 28741 | Identity law for points on... |
| tgidinside 28742 | Law for finding a point in... |
| tgbtwnconn1lem1 28743 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 28744 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 28745 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 28746 | Connectivity law for betwe... |
| tgbtwnconn2 28747 | Another connectivity law f... |
| tgbtwnconn3 28748 | Inner connectivity law for... |
| tgbtwnconnln3 28749 | Derive colinearity from be... |
| tgbtwnconn22 28750 | Double connectivity law fo... |
| tgbtwnconnln1 28751 | Derive colinearity from be... |
| tgbtwnconnln2 28752 | Derive colinearity from be... |
| legval 28755 | Value of the less-than rel... |
| legov 28756 | Value of the less-than rel... |
| legov2 28757 | An equivalent definition o... |
| legid 28758 | Reflexivity of the less-th... |
| btwnleg 28759 | Betweenness implies less-t... |
| legtrd 28760 | Transitivity of the less-t... |
| legtri3 28761 | Equality from the less-tha... |
| legtrid 28762 | Trichotomy law for the les... |
| leg0 28763 | Degenerated (zero-length) ... |
| legeq 28764 | Deduce equality from "less... |
| legbtwn 28765 | Deduce betweenness from "l... |
| tgcgrsub2 28766 | Removing identical parts f... |
| ltgseg 28767 | The set ` E ` denotes the ... |
| ltgov 28768 | Strict "shorter than" geom... |
| legov3 28769 | An equivalent definition o... |
| legso 28770 | The "shorter than" relatio... |
| ishlg 28773 | Rays : Definition 6.1 of ... |
| hlcomb 28774 | The half-line relation com... |
| hlcomd 28775 | The half-line relation com... |
| hlne1 28776 | The half-line relation imp... |
| hlne2 28777 | The half-line relation imp... |
| hlln 28778 | The half-line relation imp... |
| hleqnid 28779 | The endpoint does not belo... |
| hlid 28780 | The half-line relation is ... |
| hltr 28781 | The half-line relation is ... |
| hlbtwn 28782 | Betweenness is a sufficien... |
| btwnhl1 28783 | Deduce half-line from betw... |
| btwnhl2 28784 | Deduce half-line from betw... |
| btwnhl 28785 | Swap betweenness for a hal... |
| lnhl 28786 | Either a point ` C ` on th... |
| hlcgrex 28787 | Construct a point on a hal... |
| hlcgreulem 28788 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 28789 | The point constructed in ~... |
| btwnlng1 28790 | Betweenness implies coline... |
| btwnlng2 28791 | Betweenness implies coline... |
| btwnlng3 28792 | Betweenness implies coline... |
| lncom 28793 | Swapping the points defini... |
| lnrot1 28794 | Rotating the points defini... |
| lnrot2 28795 | Rotating the points defini... |
| ncolne1 28796 | Non-colinear points are di... |
| ncolne2 28797 | Non-colinear points are di... |
| tgisline 28798 | The property of being a pr... |
| tglnne 28799 | It takes two different poi... |
| tglndim0 28800 | There are no lines in dime... |
| tgelrnln 28801 | The property of being a pr... |
| tglineeltr 28802 | Transitivity law for lines... |
| tglineelsb2 28803 | If ` S ` lies on PQ , then... |
| tglinerflx1 28804 | Reflexivity law for line m... |
| tglinerflx2 28805 | Reflexivity law for line m... |
| tglinecom 28806 | Commutativity law for line... |
| tglinethru 28807 | If ` A ` is a line contain... |
| tghilberti1 28808 | There is a line through an... |
| tghilberti2 28809 | There is at most one line ... |
| tglinethrueu 28810 | There is a unique line goi... |
| tglinesseq 28811 | If a line is a subset of a... |
| tglnne0 28812 | A line ` A ` has at least ... |
| tglineintmo 28813 | Two distinct lines interse... |
| tglineineq 28814 | Two distinct lines interse... |
| tglineinsn 28815 | If two distinct lines inte... |
| tglineneq 28816 | Given three non-colinear p... |
| tglineinteq 28817 | Two distinct lines interse... |
| ncolncol 28818 | Deduce non-colinearity fro... |
| coltr 28819 | A transitivity law for col... |
| coltr3 28820 | A transitivity law for col... |
| colline 28821 | Three points are colinear ... |
| tglowdim2l 28822 | Reformulation of the lower... |
| tglowdim2ln 28823 | There is always one point ... |
| tglnpt2 28824 | Find a second point on a l... |
| tglnpt3 28825 | Find a third point on a li... |
| tglnpt4 28826 | Find a second point on a l... |
| mirreu3 28829 | Existential uniqueness of ... |
| mirval 28830 | Value of the point inversi... |
| mirfv 28831 | Value of the point inversi... |
| mircgr 28832 | Property of the image by t... |
| mirbtwn 28833 | Property of the image by t... |
| ismir 28834 | Property of the image by t... |
| mirf 28835 | Point inversion as functio... |
| mircl 28836 | Closure of the point inver... |
| mirmir 28837 | The point inversion functi... |
| mircom 28838 | Variation on ~ mirmir . (... |
| mirreu 28839 | Any point has a unique ant... |
| mireq 28840 | Equality deduction for poi... |
| mirinv 28841 | The only invariant point o... |
| mirne 28842 | Mirror of non-center point... |
| mircinv 28843 | The center point is invari... |
| mirf1o 28844 | The point inversion functi... |
| miriso 28845 | The point inversion functi... |
| mirbtwni 28846 | Point inversion preserves ... |
| mirbtwnb 28847 | Point inversion preserves ... |
| mircgrs 28848 | Point inversion preserves ... |
| mirmir2 28849 | Point inversion of a point... |
| mirmot 28850 | Point investion is a motio... |
| mirln 28851 | If two points are on the s... |
| mirln2 28852 | If a point and its mirror ... |
| mirconn 28853 | Point inversion of connect... |
| mirhl 28854 | If two points ` X ` and ` ... |
| mirbtwnhl 28855 | If the center of the point... |
| mirhl2 28856 | Deduce half-line relation ... |
| mircgrextend 28857 | Link congruence over a pai... |
| mirtrcgr 28858 | Point inversion of one poi... |
| mirauto 28859 | Point inversion preserves ... |
| miduniq 28860 | Uniqueness of the middle p... |
| miduniq1 28861 | Uniqueness of the middle p... |
| miduniq2 28862 | If two point inversions co... |
| colmid 28863 | Colinearity and equidistan... |
| symquadlem 28864 | Lemma of the symmetrical q... |
| krippenlem 28865 | Lemma for ~ krippen . We ... |
| krippen 28866 | Krippenlemma (German for c... |
| midexlem 28867 | Lemma for the existence of... |
| israg 28872 | Property for 3 points A, B... |
| ragcom 28873 | Commutative rule for right... |
| ragcol 28874 | The right angle property i... |
| ragmir 28875 | Right angle property is pr... |
| mirrag 28876 | Right angle is conserved b... |
| ragtrivb 28877 | Trivial right angle. Theo... |
| ragflat2 28878 | Deduce equality from two r... |
| ragflat 28879 | Deduce equality from two r... |
| ragtriva 28880 | Trivial right angle. Theo... |
| ragflat3 28881 | Right angle and colinearit... |
| ragcgr 28882 | Right angle and colinearit... |
| motrag 28883 | Right angles are preserved... |
| ragncol 28884 | Right angle implies non-co... |
| perpln1 28885 | Derive a line from perpend... |
| perpln2 28886 | Derive a line from perpend... |
| isperp 28887 | Property for 2 lines A, B ... |
| perpcom 28888 | The "perpendicular" relati... |
| perpneq 28889 | Two perpendicular lines ar... |
| isperp2 28890 | Property for 2 lines A, B,... |
| isperp2d 28891 | One direction of ~ isperp2... |
| ragperp 28892 | Deduce that two lines are ... |
| footexALT 28893 | Alternative version of ~ f... |
| footexlem1 28894 | Lemma for ~ footex . (Con... |
| footexlem2 28895 | Lemma for ~ footex . (Con... |
| footex 28896 | From a point ` C ` outside... |
| foot 28897 | From a point ` C ` outside... |
| footne 28898 | Uniqueness of the foot poi... |
| footeq 28899 | Uniqueness of the foot poi... |
| hlperpnel 28900 | A point on a half-line whi... |
| perprag 28901 | Deduce a right angle from ... |
| perpdragALT 28902 | Deduce a right angle from ... |
| perpdrag 28903 | Deduce a right angle from ... |
| colperp 28904 | Deduce a perpendicularity ... |
| colperpexlem1 28905 | Lemma for ~ colperp . Fir... |
| colperpexlem2 28906 | Lemma for ~ colperpex . S... |
| colperpexlem3 28907 | Lemma for ~ colperpex . C... |
| colperpex 28908 | In dimension 2 and above, ... |
| mideulem2 28909 | Lemma for ~ opphllem , whi... |
| opphllem 28910 | Lemma 8.24 of [Schwabhause... |
| mideulem 28911 | Lemma for ~ mideu . We ca... |
| midex 28912 | Existence of the midpoint,... |
| mideu 28913 | Existence and uniqueness o... |
| islnopp 28914 | The property for two point... |
| islnoppd 28915 | Deduce that ` A ` and ` B ... |
| oppne1 28916 | Points lying on opposite s... |
| oppne2 28917 | Points lying on opposite s... |
| oppne3 28918 | Points lying on opposite s... |
| oppcom 28919 | Commutativity rule for "op... |
| opptgdim2 28920 | If two points opposite to ... |
| oppnid 28921 | The "opposite to a line" r... |
| opphllem1 28922 | Lemma for ~ opphl . (Cont... |
| opphllem2 28923 | Lemma for ~ opphl . Lemma... |
| opphllem3 28924 | Lemma for ~ opphl : We as... |
| opphllem4 28925 | Lemma for ~ opphl . (Cont... |
| opphllem5 28926 | Second part of Lemma 9.4 o... |
| opphllem6 28927 | First part of Lemma 9.4 of... |
| oppperpex 28928 | Restating ~ colperpex usin... |
| opphl 28929 | If two points ` A ` and ` ... |
| outpasch 28930 | Axiom of Pasch, outer form... |
| hlpasch 28931 | An application of the axio... |
| ishpg 28934 | Value of the half-plane re... |
| hpgbr 28935 | Half-planes : property for... |
| hpgne1 28936 | Points on the open half pl... |
| hpgne2 28937 | Points on the open half pl... |
| lnopp2hpgb 28938 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 28939 | If two points lie on the o... |
| hpgerlem 28940 | Lemma for the proof that t... |
| hpgid 28941 | The half-plane relation is... |
| hpgcom 28942 | The half-plane relation co... |
| hpgtr 28943 | The half-plane relation is... |
| colopp 28944 | Opposite sides of a line f... |
| colhp 28945 | Half-plane relation for co... |
| hphl 28946 | If two points are on the s... |
| midf 28951 | Midpoint as a function. (... |
| midcl 28952 | Closure of the midpoint. ... |
| ismidb 28953 | Property of the midpoint. ... |
| midbtwn 28954 | Betweenness of midpoint. ... |
| midcgr 28955 | Congruence of midpoint. (... |
| midid 28956 | Midpoint of a null segment... |
| midcom 28957 | Commutativity rule for the... |
| mirmid 28958 | Point inversion preserves ... |
| lmieu 28959 | Uniqueness of the line mir... |
| lmif 28960 | Line mirror as a function.... |
| lmicl 28961 | Closure of the line mirror... |
| islmib 28962 | Property of the line mirro... |
| lmicom 28963 | The line mirroring functio... |
| lmilmi 28964 | Line mirroring is an invol... |
| lmireu 28965 | Any point has a unique ant... |
| lmieq 28966 | Equality deduction for lin... |
| lmiinv 28967 | The invariants of the line... |
| lmicinv 28968 | The mirroring line is an i... |
| lmimid 28969 | If we have a right angle, ... |
| lmif1o 28970 | The line mirroring functio... |
| lmiisolem 28971 | Lemma for ~ lmiiso . (Con... |
| lmiiso 28972 | The line mirroring functio... |
| lmimot 28973 | Line mirroring is a motion... |
| hypcgrlem1 28974 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 28975 | Lemma for ~ hypcgr , case ... |
| hypcgr 28976 | If the catheti of two righ... |
| lmiopp 28977 | Line mirroring produces po... |
| lnperpex 28978 | Existence of a perpendicul... |
| trgcopy 28979 | Triangle construction: a c... |
| trgcopyeulem 28980 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 28981 | Triangle construction: a c... |
| tgplnfn 28984 | The plane generating funct... |
| tgelrnpln 28985 | The property of being a pl... |
| plngval 28986 | The plane defined by a lin... |
| isplng 28987 | The property of being a pl... |
| plngrnssp 28988 | Planes are sets of points.... |
| elplng 28989 | Elementhood in the plane d... |
| plngssp 28990 | Planes are sets of points.... |
| elplngid 28991 | The point ` R ` is itself ... |
| elplnglnid 28992 | The line ` A ` itself is a... |
| lnincplng 28993 | If two lines ` A ` and ` B... |
| plngcplem 28994 | Lemma for ~ plngcp . (Con... |
| plngcp 28995 | The plane defined by a lin... |
| plngrotlem1 28996 | Lemma for ~ plngrot . (Co... |
| plngrotlem2 28997 | Lemma for ~ plngrot . (Co... |
| plngrotlem3 28998 | Lemma for ~ plngrot . (Co... |
| plngrot 28999 | The plane defined by a lin... |
| lnssplnglem 29000 | Lemma for ~ lnssplng . (C... |
| lnssplng 29001 | A line defined by two poin... |
| plng3p 29002 | If ` H ` is a plane contai... |
| iscgra 29005 | Property for two angles AB... |
| iscgra1 29006 | A special version of ~ isc... |
| iscgrad 29007 | Sufficient conditions for ... |
| cgrane1 29008 | Angles imply inequality. ... |
| cgrane2 29009 | Angles imply inequality. ... |
| cgrane3 29010 | Angles imply inequality. ... |
| cgrane4 29011 | Angles imply inequality. ... |
| cgrahl1 29012 | Angle congruence is indepe... |
| cgrahl2 29013 | Angle congruence is indepe... |
| cgracgr 29014 | First direction of proposi... |
| cgraid 29015 | Angle congruence is reflex... |
| cgraswap 29016 | Swap rays in a congruence ... |
| cgrcgra 29017 | Triangle congruence implie... |
| cgracom 29018 | Angle congruence commutes.... |
| cgratr 29019 | Angle congruence is transi... |
| flatcgra 29020 | Flat angles are congruent.... |
| cgraswaplr 29021 | Swap both side of angle co... |
| cgrabtwn 29022 | Angle congruence preserves... |
| cgrahl 29023 | Angle congruence preserves... |
| cgracol 29024 | Angle congruence preserves... |
| cgrancol 29025 | Angle congruence preserves... |
| dfcgra2 29026 | This is the full statement... |
| sacgr 29027 | Supplementary angles of co... |
| oacgr 29028 | Vertical angle theorem. V... |
| acopy 29029 | Angle construction. Theor... |
| acopyeu 29030 | Angle construction. Theor... |
| isinag 29034 | Property for point ` X ` t... |
| isinagd 29035 | Sufficient conditions for ... |
| inagflat 29036 | Any point lies in a flat a... |
| inagswap 29037 | Swap the order of the half... |
| inagne1 29038 | Deduce inequality from the... |
| inagne2 29039 | Deduce inequality from the... |
| inagne3 29040 | Deduce inequality from the... |
| inaghl 29041 | The "point lie in angle" r... |
| isleag 29043 | Geometrical "less than" pr... |
| isleagd 29044 | Sufficient condition for "... |
| leagne1 29045 | Deduce inequality from the... |
| leagne2 29046 | Deduce inequality from the... |
| leagne3 29047 | Deduce inequality from the... |
| leagne4 29048 | Deduce inequality from the... |
| cgrg3col4 29049 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 29050 | First congruence theorem: ... |
| tgsas 29051 | First congruence theorem: ... |
| tgsas2 29052 | First congruence theorem: ... |
| tgsas3 29053 | First congruence theorem: ... |
| tgasa1 29054 | Second congruence theorem:... |
| tgasa 29055 | Second congruence theorem:... |
| tgsss1 29056 | Third congruence theorem: ... |
| tgsss2 29057 | Third congruence theorem: ... |
| tgsss3 29058 | Third congruence theorem: ... |
| dfcgrg2 29059 | Congruence for two triangl... |
| isoas 29060 | Congruence theorem for iso... |
| iseqlg 29063 | Property of a triangle bei... |
| iseqlgd 29064 | Condition for a triangle t... |
| brprlng 29067 | Property of two lines ` A ... |
| prlngd 29068 | Deduce parallelism between... |
| prlngref 29069 | Parallelism is reflexive. ... |
| prlngsym 29070 | Parallelism is symmetric. ... |
| f1otrgds 29071 | Convenient lemma for ~ f1o... |
| f1otrgitv 29072 | Convenient lemma for ~ f1o... |
| f1otrg 29073 | A bijection between bases ... |
| f1otrge 29074 | A bijection between bases ... |
| ttgval 29077 | Define a function to augme... |
| ttglem 29078 | Lemma for ~ ttgbas , ~ ttg... |
| ttgbas 29079 | The base set of a subcompl... |
| ttgplusg 29080 | The addition operation of ... |
| ttgsub 29081 | The subtraction operation ... |
| ttgvsca 29082 | The scalar product of a su... |
| ttgds 29083 | The metric of a subcomplex... |
| ttgitvval 29084 | Betweenness for a subcompl... |
| ttgelitv 29085 | Betweenness for a subcompl... |
| ttgbtwnid 29086 | Any subcomplex module equi... |
| ttgcontlem1 29087 | Lemma for % ttgcont . (Co... |
| xmstrkgc 29088 | Any metric space fulfills ... |
| cchhllem 29089 | Lemma for chlbas and chlvs... |
| elee 29096 | Membership in a Euclidean ... |
| mptelee 29097 | A condition for a mapping ... |
| mpteleeOLD 29098 | Obsolete version of ~ mpte... |
| eleenn 29099 | If ` A ` is in ` ( EE `` N... |
| eleei 29100 | The forward direction of ~... |
| eedimeq 29101 | A point belongs to at most... |
| brbtwn 29102 | The binary relation form o... |
| brcgr 29103 | The binary relation form o... |
| fveere 29104 | The function value of a po... |
| fveecn 29105 | The function value of a po... |
| eqeefv 29106 | Two points are equal iff t... |
| eqeelen 29107 | Two points are equal iff t... |
| brbtwn2 29108 | Alternate characterization... |
| colinearalglem1 29109 | Lemma for ~ colinearalg . ... |
| colinearalglem2 29110 | Lemma for ~ colinearalg . ... |
| colinearalglem3 29111 | Lemma for ~ colinearalg . ... |
| colinearalglem4 29112 | Lemma for ~ colinearalg . ... |
| colinearalg 29113 | An algebraic characterizat... |
| eleesub 29114 | Membership of a subtractio... |
| eleesubd 29115 | Membership of a subtractio... |
| axdimuniq 29116 | The unique dimension axiom... |
| axcgrrflx 29117 | ` A ` is as far from ` B `... |
| axcgrtr 29118 | Congruence is transitive. ... |
| axcgrid 29119 | If there is no distance be... |
| axsegconlem1 29120 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 29121 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 29122 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 29123 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 29124 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 29125 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 29126 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 29127 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 29128 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 29129 | Lemma for ~ axsegcon . Sh... |
| axsegcon 29130 | Any segment ` A B ` can be... |
| ax5seglem1 29131 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 29132 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 29133 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 29134 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 29135 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 29136 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 29137 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 29138 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 29139 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 29140 | Lemma for ~ ax5seg . Take... |
| ax5seg 29141 | The five segment axiom. T... |
| axbtwnid 29142 | Points are indivisible. T... |
| axpaschlem 29143 | Lemma for ~ axpasch . Set... |
| axpasch 29144 | The inner Pasch axiom. Ta... |
| axlowdimlem1 29145 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 29146 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 29147 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 29148 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 29149 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 29150 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 29151 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 29152 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 29153 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 29154 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 29155 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 29156 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 29157 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 29158 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 29159 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 29160 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 29161 | Lemma for ~ axlowdim . Es... |
| axlowdim1 29162 | The lower dimension axiom ... |
| axlowdim2 29163 | The lower two-dimensional ... |
| axlowdim 29164 | The general lower dimensio... |
| axeuclidlem 29165 | Lemma for ~ axeuclid . Ha... |
| axeuclid 29166 | Euclid's axiom. Take an a... |
| axcontlem1 29167 | Lemma for ~ axcont . Chan... |
| axcontlem2 29168 | Lemma for ~ axcont . The ... |
| axcontlem3 29169 | Lemma for ~ axcont . Give... |
| axcontlem4 29170 | Lemma for ~ axcont . Give... |
| axcontlem5 29171 | Lemma for ~ axcont . Comp... |
| axcontlem6 29172 | Lemma for ~ axcont . Stat... |
| axcontlem7 29173 | Lemma for ~ axcont . Give... |
| axcontlem8 29174 | Lemma for ~ axcont . A po... |
| axcontlem9 29175 | Lemma for ~ axcont . Give... |
| axcontlem10 29176 | Lemma for ~ axcont . Give... |
| axcontlem11 29177 | Lemma for ~ axcont . Elim... |
| axcontlem12 29178 | Lemma for ~ axcont . Elim... |
| axcont 29179 | The axiom of continuity. ... |
| eengv 29182 | The value of the Euclidean... |
| eengstr 29183 | The Euclidean geometry as ... |
| eengbas 29184 | The Base of the Euclidean ... |
| ebtwntg 29185 | The betweenness relation u... |
| ecgrtg 29186 | The congruence relation us... |
| elntg 29187 | The line definition in the... |
| elntg2 29188 | The line definition in the... |
| eengtrkg 29189 | The geometry structure for... |
| eengtrkge 29190 | The geometry structure for... |
| edgfid 29193 | Utility theorem: index-ind... |
| edgfndx 29194 | Index value of the ~ df-ed... |
| edgfndxnn 29195 | The index value of the edg... |
| edgfndxid 29196 | The value of the edge func... |
| basendxltedgfndx 29197 | The index value of the ` B... |
| basendxnedgfndx 29198 | The slots ` Base ` and ` .... |
| vtxval 29203 | The set of vertices of a g... |
| iedgval 29204 | The set of indexed edges o... |
| 1vgrex 29205 | A graph with at least one ... |
| opvtxval 29206 | The set of vertices of a g... |
| opvtxfv 29207 | The set of vertices of a g... |
| opvtxov 29208 | The set of vertices of a g... |
| opiedgval 29209 | The set of indexed edges o... |
| opiedgfv 29210 | The set of indexed edges o... |
| opiedgov 29211 | The set of indexed edges o... |
| opvtxfvi 29212 | The set of vertices of a g... |
| opiedgfvi 29213 | The set of indexed edges o... |
| funvtxdmge2val 29214 | The set of vertices of an ... |
| funiedgdmge2val 29215 | The set of indexed edges o... |
| funvtxdm2val 29216 | The set of vertices of an ... |
| funiedgdm2val 29217 | The set of indexed edges o... |
| funvtxval0 29218 | The set of vertices of an ... |
| basvtxval 29219 | The set of vertices of a g... |
| edgfiedgval 29220 | The set of indexed edges o... |
| funvtxval 29221 | The set of vertices of a g... |
| funiedgval 29222 | The set of indexed edges o... |
| structvtxvallem 29223 | Lemma for ~ structvtxval a... |
| structvtxval 29224 | The set of vertices of an ... |
| structiedg0val 29225 | The set of indexed edges o... |
| structgrssvtxlem 29226 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 29227 | The set of vertices of a g... |
| structgrssiedg 29228 | The set of indexed edges o... |
| struct2grstr 29229 | A graph represented as an ... |
| struct2grvtx 29230 | The set of vertices of a g... |
| struct2griedg 29231 | The set of indexed edges o... |
| graop 29232 | Any representation of a gr... |
| grastruct 29233 | Any representation of a gr... |
| gropd 29234 | If any representation of a... |
| grstructd 29235 | If any representation of a... |
| gropeld 29236 | If any representation of a... |
| grstructeld 29237 | If any representation of a... |
| setsvtx 29238 | The vertices of a structur... |
| setsiedg 29239 | The (indexed) edges of a s... |
| snstrvtxval 29240 | The set of vertices of a g... |
| snstriedgval 29241 | The set of indexed edges o... |
| vtxval0 29242 | Degenerated case 1 for ver... |
| iedgval0 29243 | Degenerated case 1 for edg... |
| vtxvalsnop 29244 | Degenerated case 2 for ver... |
| iedgvalsnop 29245 | Degenerated case 2 for edg... |
| vtxval3sn 29246 | Degenerated case 3 for ver... |
| iedgval3sn 29247 | Degenerated case 3 for edg... |
| vtxvalprc 29248 | Degenerated case 4 for ver... |
| iedgvalprc 29249 | Degenerated case 4 for edg... |
| edgval 29252 | The edges of a graph. (Co... |
| iedgedg 29253 | An indexed edge is an edge... |
| edgopval 29254 | The edges of a graph repre... |
| edgov 29255 | The edges of a graph repre... |
| edgstruct 29256 | The edges of a graph repre... |
| edgiedgb 29257 | A set is an edge iff it is... |
| edg0iedg0 29258 | There is no edge in a grap... |
| isuhgr 29263 | The predicate "is an undir... |
| isushgr 29264 | The predicate "is an undir... |
| uhgrf 29265 | The edge function of an un... |
| ushgrf 29266 | The edge function of an un... |
| uhgrss 29267 | An edge is a subset of ver... |
| uhgreq12g 29268 | If two sets have the same ... |
| uhgrfun 29269 | The edge function of an un... |
| uhgrn0 29270 | An edge is a nonempty subs... |
| lpvtx 29271 | The endpoints of a loop (w... |
| ushgruhgr 29272 | An undirected simple hyper... |
| isuhgrop 29273 | The property of being an u... |
| uhgr0e 29274 | The empty graph, with vert... |
| uhgr0vb 29275 | The null graph, with no ve... |
| uhgr0 29276 | The null graph represented... |
| uhgrun 29277 | The union ` U ` of two (un... |
| uhgrunop 29278 | The union of two (undirect... |
| ushgrun 29279 | The union ` U ` of two (un... |
| ushgrunop 29280 | The union of two (undirect... |
| uhgrstrrepe 29281 | Replacing (or adding) the ... |
| incistruhgr 29282 | An _incidence structure_ `... |
| isupgr 29287 | The property of being an u... |
| wrdupgr 29288 | The property of being an u... |
| upgrf 29289 | The edge function of an un... |
| upgrfn 29290 | The edge function of an un... |
| upgrss 29291 | An edge is a subset of ver... |
| upgrn0 29292 | An edge is a nonempty subs... |
| upgrle 29293 | An edge of an undirected p... |
| upgrfi 29294 | An edge is a finite subset... |
| upgrex 29295 | An edge is an unordered pa... |
| upgrbi 29296 | Show that an unordered pai... |
| upgrop 29297 | A pseudograph represented ... |
| isumgr 29298 | The property of being an u... |
| isumgrs 29299 | The simplified property of... |
| wrdumgr 29300 | The property of being an u... |
| umgrf 29301 | The edge function of an un... |
| umgrfn 29302 | The edge function of an un... |
| umgredg2 29303 | An edge of a multigraph ha... |
| umgrbi 29304 | Show that an unordered pai... |
| upgruhgr 29305 | An undirected pseudograph ... |
| umgrupgr 29306 | An undirected multigraph i... |
| umgruhgr 29307 | An undirected multigraph i... |
| upgrle2 29308 | An edge of an undirected p... |
| umgrnloopv 29309 | In a multigraph, there is ... |
| umgredgprv 29310 | In a multigraph, an edge i... |
| umgrnloop 29311 | In a multigraph, there is ... |
| umgrnloop0 29312 | A multigraph has no loops.... |
| umgr0e 29313 | The empty graph, with vert... |
| upgr0e 29314 | The empty graph, with vert... |
| upgr1elem 29315 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 29316 | A pseudograph with one edg... |
| upgr0eop 29317 | The empty graph, with vert... |
| upgr1eop 29318 | A pseudograph with one edg... |
| upgr0eopALT 29319 | Alternate proof of ~ upgr0... |
| upgr1eopALT 29320 | Alternate proof of ~ upgr1... |
| upgrun 29321 | The union ` U ` of two pse... |
| upgrunop 29322 | The union of two pseudogra... |
| umgrun 29323 | The union ` U ` of two mul... |
| umgrunop 29324 | The union of two multigrap... |
| umgrislfupgrlem 29325 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 29326 | A multigraph is a loop-fre... |
| lfgredgge2 29327 | An edge of a loop-free gra... |
| lfgrnloop 29328 | A loop-free graph has no l... |
| uhgredgiedgb 29329 | In a hypergraph, a set is ... |
| uhgriedg0edg0 29330 | A hypergraph has no edges ... |
| uhgredgn0 29331 | An edge of a hypergraph is... |
| edguhgr 29332 | An edge of a hypergraph is... |
| uhgredgrnv 29333 | An edge of a hypergraph co... |
| uhgredgss 29334 | The set of edges of a hype... |
| upgredgss 29335 | The set of edges of a pseu... |
| umgredgss 29336 | The set of edges of a mult... |
| edgupgr 29337 | Properties of an edge of a... |
| edgumgr 29338 | Properties of an edge of a... |
| uhgrvtxedgiedgb 29339 | In a hypergraph, a vertex ... |
| upgredg 29340 | For each edge in a pseudog... |
| umgredg 29341 | For each edge in a multigr... |
| upgrpredgv 29342 | An edge of a pseudograph a... |
| umgrpredgv 29343 | An edge of a multigraph al... |
| upgredg2vtx 29344 | For a vertex incident to a... |
| upgredgpr 29345 | If a proper pair (of verti... |
| edglnl 29346 | The edges incident with a ... |
| numedglnl 29347 | The number of edges incide... |
| umgredgne 29348 | An edge of a multigraph al... |
| umgrnloop2 29349 | A multigraph has no loops.... |
| umgredgnlp 29350 | An edge of a multigraph is... |
| isuspgr 29355 | The property of being a si... |
| isusgr 29356 | The property of being a si... |
| uspgrf 29357 | The edge function of a sim... |
| usgrf 29358 | The edge function of a sim... |
| isusgrs 29359 | The property of being a si... |
| usgrfs 29360 | The edge function of a sim... |
| usgrfun 29361 | The edge function of a sim... |
| usgredgss 29362 | The set of edges of a simp... |
| edgusgr 29363 | An edge of a simple graph ... |
| isuspgrop 29364 | The property of being an u... |
| isusgrop 29365 | The property of being an u... |
| usgrop 29366 | A simple graph represented... |
| isausgr 29367 | The property of an ordered... |
| ausgrusgrb 29368 | The equivalence of the def... |
| usgrausgri 29369 | A simple graph represented... |
| ausgrumgri 29370 | If an alternatively define... |
| ausgrusgri 29371 | The equivalence of the def... |
| usgrausgrb 29372 | The equivalence of the def... |
| usgredgop 29373 | An edge of a simple graph ... |
| usgrf1o 29374 | The edge function of a sim... |
| usgrf1 29375 | The edge function of a sim... |
| uspgrf1oedg 29376 | The edge function of a sim... |
| usgrss 29377 | An edge is a subset of ver... |
| uspgredgiedg 29378 | In a simple pseudograph, f... |
| uspgriedgedg 29379 | In a simple pseudograph, f... |
| uspgrushgr 29380 | A simple pseudograph is an... |
| uspgrupgr 29381 | A simple pseudograph is an... |
| uspgrupgrushgr 29382 | A graph is a simple pseudo... |
| usgruspgr 29383 | A simple graph is a simple... |
| usgrumgr 29384 | A simple graph is an undir... |
| usgrumgruspgr 29385 | A graph is a simple graph ... |
| usgruspgrb 29386 | A class is a simple graph ... |
| uspgruhgr 29387 | An undirected simple pseud... |
| usgrupgr 29388 | A simple graph is an undir... |
| usgruhgr 29389 | A simple graph is an undir... |
| usgrislfuspgr 29390 | A simple graph is a loop-f... |
| uspgrun 29391 | The union ` U ` of two sim... |
| uspgrunop 29392 | The union of two simple ps... |
| usgrun 29393 | The union ` U ` of two sim... |
| usgrunop 29394 | The union of two simple gr... |
| usgredg2 29395 | The value of the "edge fun... |
| usgredg2ALT 29396 | Alternate proof of ~ usgre... |
| usgredgprv 29397 | In a simple graph, an edge... |
| usgredgprvALT 29398 | Alternate proof of ~ usgre... |
| usgredgppr 29399 | An edge of a simple graph ... |
| usgrpredgv 29400 | An edge of a simple graph ... |
| edgssv2 29401 | An edge of a simple graph ... |
| usgredg 29402 | For each edge in a simple ... |
| usgrnloopv 29403 | In a simple graph, there i... |
| usgrnloopvALT 29404 | Alternate proof of ~ usgrn... |
| usgrnloop 29405 | In a simple graph, there i... |
| usgrnloopALT 29406 | Alternate proof of ~ usgrn... |
| usgrnloop0 29407 | A simple graph has no loop... |
| usgrnloop0ALT 29408 | Alternate proof of ~ usgrn... |
| usgredgne 29409 | An edge of a simple graph ... |
| usgrf1oedg 29410 | The edge function of a sim... |
| uhgr2edg 29411 | If a vertex is adjacent to... |
| umgr2edg 29412 | If a vertex is adjacent to... |
| usgr2edg 29413 | If a vertex is adjacent to... |
| umgr2edg1 29414 | If a vertex is adjacent to... |
| usgr2edg1 29415 | If a vertex is adjacent to... |
| umgrvad2edg 29416 | If a vertex is adjacent to... |
| umgr2edgneu 29417 | If a vertex is adjacent to... |
| usgrsizedg 29418 | In a simple graph, the siz... |
| usgredg3 29419 | The value of the "edge fun... |
| usgredg4 29420 | For a vertex incident to a... |
| usgredgreu 29421 | For a vertex incident to a... |
| usgredg2vtx 29422 | For a vertex incident to a... |
| uspgredg2vtxeu 29423 | For a vertex incident to a... |
| usgredg2vtxeu 29424 | For a vertex incident to a... |
| usgredg2vtxeuALT 29425 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29426 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29427 | In a simple pseudograph, t... |
| usgredg2vlem1 29428 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29429 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29430 | In a simple graph, the map... |
| usgriedgleord 29431 | Alternate version of ~ usg... |
| ushgredgedg 29432 | In a simple hypergraph the... |
| usgredgedg 29433 | In a simple graph there is... |
| ushgredgedgloop 29434 | In a simple hypergraph the... |
| uspgredgleord 29435 | In a simple pseudograph th... |
| usgredgleord 29436 | In a simple graph the numb... |
| usgredgleordALT 29437 | Alternate proof for ~ usgr... |
| usgrstrrepe 29438 | Replacing (or adding) the ... |
| usgr0e 29439 | The empty graph, with vert... |
| usgr0vb 29440 | The null graph, with no ve... |
| uhgr0v0e 29441 | The null graph, with no ve... |
| uhgr0vsize0 29442 | The size of a hypergraph w... |
| uhgr0edgfi 29443 | A graph of order 0 (i.e. w... |
| usgr0v 29444 | The null graph, with no ve... |
| uhgr0vusgr 29445 | The null graph, with no ve... |
| usgr0 29446 | The null graph represented... |
| uspgr1e 29447 | A simple pseudograph with ... |
| usgr1e 29448 | A simple graph with one ed... |
| usgr0eop 29449 | The empty graph, with vert... |
| uspgr1eop 29450 | A simple pseudograph with ... |
| uspgr1ewop 29451 | A simple pseudograph with ... |
| uspgr1v1eop 29452 | A simple pseudograph with ... |
| usgr1eop 29453 | A simple graph with (at le... |
| uspgr2v1e2w 29454 | A simple pseudograph with ... |
| usgr2v1e2w 29455 | A simple graph with two ve... |
| edg0usgr 29456 | A class without edges is a... |
| lfuhgr1v0e 29457 | A loop-free hypergraph wit... |
| usgr1vr 29458 | A simple graph with one ve... |
| usgr1v 29459 | A class with one (or no) v... |
| usgr1v0edg 29460 | A class with one (or no) v... |
| usgrexmpldifpr 29461 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29462 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29463 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29464 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29465 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29466 | ` G ` is a simple graph of... |
| griedg0prc 29467 | The class of empty graphs ... |
| griedg0ssusgr 29468 | The class of all simple gr... |
| usgrprc 29469 | The class of simple graphs... |
| relsubgr 29472 | The class of the subgraph ... |
| subgrv 29473 | If a class is a subgraph o... |
| issubgr 29474 | The property of a set to b... |
| issubgr2 29475 | The property of a set to b... |
| subgrprop 29476 | The properties of a subgra... |
| subgrprop2 29477 | The properties of a subgra... |
| uhgrissubgr 29478 | The property of a hypergra... |
| subgrprop3 29479 | The properties of a subgra... |
| egrsubgr 29480 | An empty graph consisting ... |
| 0grsubgr 29481 | The null graph (represente... |
| 0uhgrsubgr 29482 | The null graph (as hypergr... |
| uhgrsubgrself 29483 | A hypergraph is a subgraph... |
| subgrfun 29484 | The edge function of a sub... |
| subgruhgrfun 29485 | The edge function of a sub... |
| subgreldmiedg 29486 | An element of the domain o... |
| subgruhgredgd 29487 | An edge of a subgraph of a... |
| subumgredg2 29488 | An edge of a subgraph of a... |
| subuhgr 29489 | A subgraph of a hypergraph... |
| subupgr 29490 | A subgraph of a pseudograp... |
| subumgr 29491 | A subgraph of a multigraph... |
| subusgr 29492 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29493 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29494 | A spanning subgraph ` S ` ... |
| uhgrspan 29495 | A spanning subgraph ` S ` ... |
| upgrspan 29496 | A spanning subgraph ` S ` ... |
| umgrspan 29497 | A spanning subgraph ` S ` ... |
| usgrspan 29498 | A spanning subgraph ` S ` ... |
| uhgrspanop 29499 | A spanning subgraph of a h... |
| upgrspanop 29500 | A spanning subgraph of a p... |
| umgrspanop 29501 | A spanning subgraph of a m... |
| usgrspanop 29502 | A spanning subgraph of a s... |
| uhgrspan1lem1 29503 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29504 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29505 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29506 | The induced subgraph ` S `... |
| upgrreslem 29507 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29508 | Lemma for ~ umgrres and ~ ... |
| upgrres 29509 | A subgraph obtained by rem... |
| umgrres 29510 | A subgraph obtained by rem... |
| usgrres 29511 | A subgraph obtained by rem... |
| upgrres1lem1 29512 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29513 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29514 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29515 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29516 | A pseudograph obtained by ... |
| umgrres1 29517 | A multigraph obtained by r... |
| usgrres1 29518 | Restricting a simple graph... |
| isfusgr 29521 | The property of being a fi... |
| fusgrvtxfi 29522 | A finite simple graph has ... |
| isfusgrf1 29523 | The property of being a fi... |
| isfusgrcl 29524 | The property of being a fi... |
| fusgrusgr 29525 | A finite simple graph is a... |
| opfusgr 29526 | A finite simple graph repr... |
| usgredgffibi 29527 | The number of edges in a s... |
| fusgredgfi 29528 | In a finite simple graph t... |
| usgr1v0e 29529 | The size of a (finite) sim... |
| usgrfilem 29530 | In a finite simple graph, ... |
| fusgrfisbase 29531 | Induction base for ~ fusgr... |
| fusgrfisstep 29532 | Induction step in ~ fusgrf... |
| fusgrfis 29533 | A finite simple graph is o... |
| fusgrfupgrfs 29534 | A finite simple graph is a... |
| nbgrprc0 29537 | The set of neighbors is em... |
| nbgrcl 29538 | If a class ` X ` has at le... |
| nbgrval 29539 | The set of neighbors of a ... |
| dfnbgr2 29540 | Alternate definition of th... |
| dfnbgr3 29541 | Alternate definition of th... |
| nbgrnvtx0 29542 | If a class ` X ` is not a ... |
| nbgrel 29543 | Characterization of a neig... |
| nbgrisvtx 29544 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29545 | The neighbors of a vertex ... |
| nbuhgr 29546 | The set of neighbors of a ... |
| nbupgr 29547 | The set of neighbors of a ... |
| nbupgrel 29548 | A neighbor of a vertex in ... |
| nbumgrvtx 29549 | The set of neighbors of a ... |
| nbumgr 29550 | The set of neighbors of an... |
| nbusgrvtx 29551 | The set of neighbors of a ... |
| nbusgr 29552 | The set of neighbors of an... |
| nbgr2vtx1edg 29553 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29554 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29555 | If a hypergraph has two ve... |
| nbusgreledg 29556 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29557 | A vertex which is not endp... |
| nbgr0vtx 29558 | In a null graph (with no v... |
| nbgr0edglem 29559 | Lemma for ~ nbgr0edg and ~... |
| nbgr0edg 29560 | In an empty graph (with no... |
| nbgr1vtx 29561 | In a graph with one vertex... |
| nbgrnself 29562 | A vertex in a graph is not... |
| nbgrnself2 29563 | A class ` X ` is not a nei... |
| nbgrssovtx 29564 | The neighbors of a vertex ... |
| nbgrssvwo2 29565 | The neighbors of a vertex ... |
| nbgrsym 29566 | In a graph, the neighborho... |
| nbupgrres 29567 | The neighborhood of a vert... |
| usgrnbcnvfv 29568 | Applying the edge function... |
| nbusgredgeu 29569 | For each neighbor of a ver... |
| edgnbusgreu 29570 | For each edge incident to ... |
| nbusgredgeu0 29571 | For each neighbor of a ver... |
| nbusgrf1o0 29572 | The mapping of neighbors o... |
| nbusgrf1o1 29573 | The set of neighbors of a ... |
| nbusgrf1o 29574 | The set of neighbors of a ... |
| nbedgusgr 29575 | The number of neighbors of... |
| edgusgrnbfin 29576 | The number of neighbors of... |
| nbusgrfi 29577 | The class of neighbors of ... |
| nbfiusgrfi 29578 | The class of neighbors of ... |
| hashnbusgrnn0 29579 | The number of neighbors of... |
| nbfusgrlevtxm1 29580 | The number of neighbors of... |
| nbfusgrlevtxm2 29581 | If there is a vertex which... |
| nbusgrvtxm1 29582 | If the number of neighbors... |
| nb3grprlem1 29583 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29584 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29585 | The neighbors of a vertex ... |
| nb3grpr2 29586 | The neighbors of a vertex ... |
| nb3gr2nb 29587 | If the neighbors of two ve... |
| uvtxval 29590 | The set of all universal v... |
| uvtxel 29591 | A universal vertex, i.e. a... |
| uvtxisvtx 29592 | A universal vertex is a ve... |
| uvtxssvtx 29593 | The set of the universal v... |
| vtxnbuvtx 29594 | A universal vertex has all... |
| uvtxnbgrss 29595 | A universal vertex has all... |
| uvtxnbgrvtx 29596 | A universal vertex is neig... |
| uvtx0 29597 | There is no universal vert... |
| isuvtx 29598 | The set of all universal v... |
| uvtxel1 29599 | Characterization of a univ... |
| uvtx01vtx 29600 | If a graph/class has no ed... |
| uvtx2vtx1edg 29601 | If a graph has two vertice... |
| uvtx2vtx1edgb 29602 | If a hypergraph has two ve... |
| uvtxnbgr 29603 | A universal vertex has all... |
| uvtxnbgrb 29604 | A vertex is universal iff ... |
| uvtxusgr 29605 | The set of all universal v... |
| uvtxusgrel 29606 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 29607 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 29608 | If the number of neighbors... |
| uvtxnbvtxm1 29609 | A universal vertex has ` n... |
| nbupgruvtxres 29610 | The neighborhood of a univ... |
| uvtxupgrres 29611 | A universal vertex is univ... |
| cplgruvtxb 29616 | A graph ` G ` is complete ... |
| prcliscplgr 29617 | A proper class (representi... |
| iscplgr 29618 | The property of being a co... |
| iscplgrnb 29619 | A graph is complete iff al... |
| iscplgredg 29620 | A graph ` G ` is complete ... |
| iscusgr 29621 | The property of being a co... |
| cusgrusgr 29622 | A complete simple graph is... |
| cusgrcplgr 29623 | A complete simple graph is... |
| iscusgrvtx 29624 | A simple graph is complete... |
| cusgruvtxb 29625 | A simple graph is complete... |
| iscusgredg 29626 | A simple graph is complete... |
| cusgredg 29627 | In a complete simple graph... |
| cplgr0 29628 | The null graph (with no ve... |
| cusgr0 29629 | The null graph (with no ve... |
| cplgr0v 29630 | A null graph (with no vert... |
| cusgr0v 29631 | A graph with no vertices a... |
| cplgr1vlem 29632 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 29633 | A graph with one vertex is... |
| cusgr1v 29634 | A graph with one vertex an... |
| cplgr2v 29635 | An undirected hypergraph w... |
| cplgr2vpr 29636 | An undirected hypergraph w... |
| nbcplgr 29637 | In a complete graph, each ... |
| cplgr3v 29638 | A pseudograph with three (... |
| cusgr3vnbpr 29639 | The neighbors of a vertex ... |
| cplgrop 29640 | A complete graph represent... |
| cusgrop 29641 | A complete simple graph re... |
| cusgrexilem1 29642 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 29643 | Lemma for ~ usgrexi . (Co... |
| usgrexi 29644 | An arbitrary set regarded ... |
| cusgrexilem2 29645 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 29646 | An arbitrary set ` V ` reg... |
| cusgrexg 29647 | For each set there is a se... |
| structtousgr 29648 | Any (extensible) structure... |
| structtocusgr 29649 | Any (extensible) structure... |
| cffldtocusgr 29650 | The field of complex numbe... |
| cusgrres 29651 | Restricting a complete sim... |
| cusgrsizeindb0 29652 | Base case of the induction... |
| cusgrsizeindb1 29653 | Base case of the induction... |
| cusgrsizeindslem 29654 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 29655 | Part 1 of induction step i... |
| cusgrsize2inds 29656 | Induction step in ~ cusgrs... |
| cusgrsize 29657 | The size of a finite compl... |
| cusgrfilem1 29658 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 29659 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 29660 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 29661 | If the size of a complete ... |
| usgredgsscusgredg 29662 | A simple graph is a subgra... |
| usgrsscusgr 29663 | A simple graph is a subgra... |
| sizusglecusglem1 29664 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 29665 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 29666 | The size of a simple graph... |
| fusgrmaxsize 29667 | The maximum size of a fini... |
| vtxdgfval 29670 | The value of the vertex de... |
| vtxdgval 29671 | The degree of a vertex. (... |
| vtxdgfival 29672 | The degree of a vertex for... |
| vtxdgop 29673 | The vertex degree expresse... |
| vtxdgf 29674 | The vertex degree function... |
| vtxdgelxnn0 29675 | The degree of a vertex is ... |
| vtxdg0v 29676 | The degree of a vertex in ... |
| vtxdg0e 29677 | The degree of a vertex in ... |
| vtxdgfisnn0 29678 | The degree of a vertex in ... |
| vtxdgfisf 29679 | The vertex degree function... |
| vtxdeqd 29680 | Equality theorem for the v... |
| vtxduhgr0e 29681 | The degree of a vertex in ... |
| vtxdlfuhgr1v 29682 | The degree of the vertex i... |
| vdumgr0 29683 | A vertex in a multigraph h... |
| vtxdun 29684 | The degree of a vertex in ... |
| vtxdfiun 29685 | The degree of a vertex in ... |
| vtxduhgrun 29686 | The degree of a vertex in ... |
| vtxduhgrfiun 29687 | The degree of a vertex in ... |
| vtxdlfgrval 29688 | The value of the vertex de... |
| vtxdumgrval 29689 | The value of the vertex de... |
| vtxdusgrval 29690 | The value of the vertex de... |
| vtxd0nedgb 29691 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 29692 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 29693 | The value of the vertex de... |
| vtxdusgrfvedg 29694 | The value of the vertex de... |
| vtxduhgr0nedg 29695 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 29696 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 29697 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 29698 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 29699 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 29700 | The vertex degree function... |
| vtxdgfusgr 29701 | In a finite simple graph, ... |
| fusgrn0degnn0 29702 | In a nonempty, finite grap... |
| 1loopgruspgr 29703 | A graph with one edge whic... |
| 1loopgredg 29704 | The set of edges in a grap... |
| 1loopgrnb0 29705 | In a graph (simple pseudog... |
| 1loopgrvd2 29706 | The vertex degree of a one... |
| 1loopgrvd0 29707 | The vertex degree of a one... |
| 1hevtxdg0 29708 | The vertex degree of verte... |
| 1hevtxdg1 29709 | The vertex degree of verte... |
| 1hegrvtxdg1 29710 | The vertex degree of a gra... |
| 1hegrvtxdg1r 29711 | The vertex degree of a gra... |
| 1egrvtxdg1 29712 | The vertex degree of a one... |
| 1egrvtxdg1r 29713 | The vertex degree of a one... |
| 1egrvtxdg0 29714 | The vertex degree of a one... |
| p1evtxdeqlem 29715 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 29716 | If an edge ` E ` which doe... |
| p1evtxdp1 29717 | If an edge ` E ` (not bein... |
| uspgrloopvtx 29718 | The set of vertices in a g... |
| uspgrloopvtxel 29719 | A vertex in a graph (simpl... |
| uspgrloopiedg 29720 | The set of edges in a grap... |
| uspgrloopedg 29721 | The set of edges in a grap... |
| uspgrloopnb0 29722 | In a graph (simple pseudog... |
| uspgrloopvd2 29723 | The vertex degree of a one... |
| umgr2v2evtx 29724 | The set of vertices in a m... |
| umgr2v2evtxel 29725 | A vertex in a multigraph w... |
| umgr2v2eiedg 29726 | The edge function in a mul... |
| umgr2v2eedg 29727 | The set of edges in a mult... |
| umgr2v2e 29728 | A multigraph with two edge... |
| umgr2v2enb1 29729 | In a multigraph with two e... |
| umgr2v2evd2 29730 | In a multigraph with two e... |
| hashnbusgrvd 29731 | In a simple graph, the num... |
| usgruvtxvdb 29732 | In a finite simple graph w... |
| vdiscusgrb 29733 | A finite simple graph with... |
| vdiscusgr 29734 | In a finite complete simpl... |
| vtxdusgradjvtx 29735 | The degree of a vertex in ... |
| usgrvd0nedg 29736 | If a vertex in a simple gr... |
| uhgrvd00 29737 | If every vertex in a hyper... |
| usgrvd00 29738 | If every vertex in a simpl... |
| vdegp1ai 29739 | The induction step for a v... |
| vdegp1bi 29740 | The induction step for a v... |
| vdegp1ci 29741 | The induction step for a v... |
| vtxdginducedm1lem1 29742 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 29743 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 29744 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 29745 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 29746 | The degree of a vertex ` v... |
| vtxdginducedm1fi 29747 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 29748 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 29749 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 29750 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 29751 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 29752 | Induction step of ~ finsum... |
| finsumvtxdg2size 29753 | The sum of the degrees of ... |
| fusgr1th 29754 | The sum of the degrees of ... |
| finsumvtxdgeven 29755 | The sum of the degrees of ... |
| vtxdgoddnumeven 29756 | The number of vertices of ... |
| fusgrvtxdgonume 29757 | The number of vertices of ... |
| isrgr 29762 | The property of a class be... |
| rgrprop 29763 | The properties of a k-regu... |
| isrusgr 29764 | The property of being a k-... |
| rusgrprop 29765 | The properties of a k-regu... |
| rusgrrgr 29766 | A k-regular simple graph i... |
| rusgrusgr 29767 | A k-regular simple graph i... |
| finrusgrfusgr 29768 | A finite regular simple gr... |
| isrusgr0 29769 | The property of being a k-... |
| rusgrprop0 29770 | The properties of a k-regu... |
| usgreqdrusgr 29771 | If all vertices in a simpl... |
| fusgrregdegfi 29772 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 29773 | If all vertices in a nonem... |
| frusgrnn0 29774 | In a nonempty finite k-reg... |
| 0edg0rgr 29775 | A graph is 0-regular if it... |
| uhgr0edg0rgr 29776 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 29777 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 29778 | A simple graph is 0-regula... |
| 0vtxrgr 29779 | A null graph (with no vert... |
| 0vtxrusgr 29780 | A graph with no vertices a... |
| 0uhgrrusgr 29781 | The null graph as hypergra... |
| 0grrusgr 29782 | The null graph represented... |
| 0grrgr 29783 | The null graph represented... |
| cusgrrusgr 29784 | A complete simple graph wi... |
| cusgrm1rusgr 29785 | A finite simple graph with... |
| rusgrpropnb 29786 | The properties of a k-regu... |
| rusgrpropedg 29787 | The properties of a k-regu... |
| rusgrpropadjvtx 29788 | The properties of a k-regu... |
| rusgrnumwrdl2 29789 | In a k-regular simple grap... |
| rusgr1vtxlem 29790 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 29791 | If a k-regular simple grap... |
| rgrusgrprc 29792 | The class of 0-regular sim... |
| rusgrprc 29793 | The class of 0-regular sim... |
| rgrprc 29794 | The class of 0-regular gra... |
| rgrprcx 29795 | The class of 0-regular gra... |
| rgrx0ndm 29796 | 0 is not in the domain of ... |
| rgrx0nd 29797 | The potentially alternativ... |
| ewlksfval 29804 | The set of s-walks of edge... |
| isewlk 29805 | Conditions for a function ... |
| ewlkprop 29806 | Properties of an s-walk of... |
| ewlkinedg 29807 | The intersection (common v... |
| ewlkle 29808 | An s-walk of edges is also... |
| upgrewlkle2 29809 | In a pseudograph, there is... |
| wkslem1 29810 | Lemma 1 for walks to subst... |
| wkslem2 29811 | Lemma 2 for walks to subst... |
| wksfval 29812 | The set of walks (in an un... |
| iswlk 29813 | Properties of a pair of fu... |
| wlkprop 29814 | Properties of a walk. (Co... |
| wlkv 29815 | The classes involved in a ... |
| iswlkg 29816 | Generalization of ~ iswlk ... |
| wlkf 29817 | The mapping enumerating th... |
| wlkcl 29818 | A walk has length ` # ( F ... |
| wlkp 29819 | The mapping enumerating th... |
| wlkpwrd 29820 | The sequence of vertices o... |
| wlklenvp1 29821 | The number of vertices of ... |
| wksv 29822 | The class of walks is a se... |
| wlkn0 29823 | The sequence of vertices o... |
| wlklenvm1 29824 | The number of edges of a w... |
| ifpsnprss 29825 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 29826 | Each pair of adjacent vert... |
| wlkvtxiedg 29827 | The vertices of a walk are... |
| relwlk 29828 | The set ` ( Walks `` G ) `... |
| wlkvv 29829 | If there is at least one w... |
| wlkop 29830 | A walk is an ordered pair.... |
| wlkcpr 29831 | A walk as class with two c... |
| wlk2f 29832 | If there is a walk ` W ` t... |
| wlkcomp 29833 | A walk expressed by proper... |
| wlkcompim 29834 | Implications for the prope... |
| wlkelwrd 29835 | The components of a walk a... |
| wlkeq 29836 | Conditions for two walks (... |
| edginwlk 29837 | The value of the edge func... |
| upgredginwlk 29838 | The value of the edge func... |
| iedginwlk 29839 | The value of the edge func... |
| wlkl1loop 29840 | A walk of length 1 from a ... |
| wlk1walk 29841 | A walk is a 1-walk "on the... |
| wlk1ewlk 29842 | A walk is an s-walk "on th... |
| upgriswlk 29843 | Properties of a pair of fu... |
| upgrwlkedg 29844 | The edges of a walk in a p... |
| upgrwlkcompim 29845 | Implications for the prope... |
| wlkvtxedg 29846 | The vertices of a walk are... |
| upgrwlkvtxedg 29847 | The pairs of connected ver... |
| uspgr2wlkeq 29848 | Conditions for two walks w... |
| uspgr2wlkeq2 29849 | Conditions for two walks w... |
| uspgr2wlkeqi 29850 | Conditions for two walks w... |
| umgrwlknloop 29851 | In a multigraph, each walk... |
| wlkv0 29852 | If there is a walk in the ... |
| g0wlk0 29853 | There is no walk in a null... |
| 0wlk0 29854 | There is no walk for the e... |
| wlk0prc 29855 | There is no walk in a null... |
| wlklenvclwlk 29856 | The number of vertices in ... |
| wlkson 29857 | The set of walks between t... |
| iswlkon 29858 | Properties of a pair of fu... |
| wlkonprop 29859 | Properties of a walk betwe... |
| wlkpvtx 29860 | A walk connects vertices. ... |
| wlkepvtx 29861 | The endpoints of a walk ar... |
| wlkoniswlk 29862 | A walk between two vertice... |
| wlkonwlk 29863 | A walk is a walk between i... |
| wlkonwlk1l 29864 | A walk is a walk from its ... |
| wlksoneq1eq2 29865 | Two walks with identical s... |
| wlkonl1iedg 29866 | If there is a walk between... |
| wlkon2n0 29867 | The length of a walk betwe... |
| 2wlklem 29868 | Lemma for theorems for wal... |
| upgr2wlk 29869 | Properties of a pair of fu... |
| wlkreslem 29870 | Lemma for ~ wlkres . (Con... |
| wlkres 29871 | The restriction ` <. H , Q... |
| redwlklem 29872 | Lemma for ~ redwlk . (Con... |
| redwlk 29873 | A walk ending at the last ... |
| wlkp1lem1 29874 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 29875 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 29876 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 29877 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 29878 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 29879 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 29880 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 29881 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 29882 | Append one path segment (e... |
| wlkdlem1 29883 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 29884 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 29885 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 29886 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 29887 | Two words representing a w... |
| lfgrwlkprop 29888 | Two adjacent vertices in a... |
| lfgriswlk 29889 | Conditions for a pair of f... |
| lfgrwlknloop 29890 | In a loop-free graph, each... |
| reltrls 29895 | The set ` ( Trails `` G ) ... |
| trlsfval 29896 | The set of trails (in an u... |
| istrl 29897 | Conditions for a pair of c... |
| trliswlk 29898 | A trail is a walk. (Contr... |
| trlf1 29899 | The enumeration ` F ` of a... |
| trlreslem 29900 | Lemma for ~ trlres . Form... |
| trlres 29901 | The restriction ` <. H , Q... |
| upgrtrls 29902 | The set of trails in a pse... |
| upgristrl 29903 | Properties of a pair of fu... |
| upgrf1istrl 29904 | Properties of a pair of a ... |
| wksonproplem 29905 | Lemma for theorems for pro... |
| trlsonfval 29906 | The set of trails between ... |
| istrlson 29907 | Properties of a pair of fu... |
| trlsonprop 29908 | Properties of a trail betw... |
| trlsonistrl 29909 | A trail between two vertic... |
| trlsonwlkon 29910 | A trail between two vertic... |
| trlontrl 29911 | A trail is a trail between... |
| relpths 29920 | The set ` ( Paths `` G ) `... |
| pthsfval 29921 | The set of paths (in an un... |
| spthsfval 29922 | The set of simple paths (i... |
| ispth 29923 | Conditions for a pair of c... |
| isspth 29924 | Conditions for a pair of c... |
| pthistrl 29925 | A path is a trail (in an u... |
| spthispth 29926 | A simple path is a path (i... |
| pthiswlk 29927 | A path is a walk (in an un... |
| spthiswlk 29928 | A simple path is a walk (i... |
| pthdivtx 29929 | The inner vertices of a pa... |
| pthdadjvtx 29930 | The adjacent vertices of a... |
| dfpth2 29931 | Alternate definition for a... |
| pthdifv 29932 | The vertices of a path are... |
| 2pthnloop 29933 | A path of length at least ... |
| upgr2pthnlp 29934 | A path of length at least ... |
| spthdifv 29935 | The vertices of a simple p... |
| spthdep 29936 | A simple path (at least of... |
| pthdepisspth 29937 | A path with different star... |
| upgrwlkdvdelem 29938 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 29939 | In a pseudograph, all edge... |
| upgrspthswlk 29940 | The set of simple paths in... |
| upgrwlkdvspth 29941 | A walk consisting of diffe... |
| pthsonfval 29942 | The set of paths between t... |
| spthson 29943 | The set of simple paths be... |
| ispthson 29944 | Properties of a pair of fu... |
| isspthson 29945 | Properties of a pair of fu... |
| pthsonprop 29946 | Properties of a path betwe... |
| spthonprop 29947 | Properties of a simple pat... |
| pthonispth 29948 | A path between two vertice... |
| pthontrlon 29949 | A path between two vertice... |
| pthonpth 29950 | A path is a path between i... |
| isspthonpth 29951 | A pair of functions is a s... |
| spthonisspth 29952 | A simple path between to v... |
| spthonpthon 29953 | A simple path between two ... |
| spthonepeq 29954 | The endpoints of a simple ... |
| uhgrwkspthlem1 29955 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 29956 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 29957 | Any walk of length 1 betwe... |
| usgr2wlkneq 29958 | The vertices and edges are... |
| usgr2wlkspthlem1 29959 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 29960 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 29961 | In a simple graph, any wal... |
| usgr2trlncl 29962 | In a simple graph, any tra... |
| usgr2trlspth 29963 | In a simple graph, any tra... |
| usgr2pthspth 29964 | In a simple graph, any pat... |
| usgr2pthlem 29965 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 29966 | In a simple graph, there i... |
| usgr2pth0 29967 | In a simply graph, there i... |
| pthdlem1 29968 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 29969 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 29970 | Lemma 2 for ~ pthd . (Con... |
| pthd 29971 | Two words representing a t... |
| clwlks 29974 | The set of closed walks (i... |
| isclwlk 29975 | A pair of functions repres... |
| clwlkiswlk 29976 | A closed walk is a walk (i... |
| clwlkwlk 29977 | Closed walks are walks (in... |
| clwlkswks 29978 | Closed walks are walks (in... |
| isclwlke 29979 | Properties of a pair of fu... |
| isclwlkupgr 29980 | Properties of a pair of fu... |
| clwlkcomp 29981 | A closed walk expressed by... |
| clwlkcompim 29982 | Implications for the prope... |
| upgrclwlkcompim 29983 | Implications for the prope... |
| clwlkcompbp 29984 | Basic properties of the co... |
| clwlkl1loop 29985 | A closed walk of length 1 ... |
| crcts 29990 | The set of circuits (in an... |
| cycls 29991 | The set of cycles (in an u... |
| iscrct 29992 | Sufficient and necessary c... |
| iscycl 29993 | Sufficient and necessary c... |
| crctprop 29994 | The properties of a circui... |
| cyclprop 29995 | The properties of a cycle:... |
| crctisclwlk 29996 | A circuit is a closed walk... |
| crctistrl 29997 | A circuit is a trail. (Co... |
| crctiswlk 29998 | A circuit is a walk. (Con... |
| cyclispth 29999 | A cycle is a path. (Contr... |
| cycliswlk 30000 | A cycle is a walk. (Contr... |
| cycliscrct 30001 | A cycle is a circuit. (Co... |
| cyclnumvtx 30002 | The number of vertices of ... |
| cyclnspth 30003 | A (non-trivial) cycle is n... |
| pthisspthorcycl 30004 | A path is either a simple ... |
| pthspthcyc 30005 | A pair ` <. F , P >. ` rep... |
| cyclispthon 30006 | A cycle is a path starting... |
| lfgrn1cycl 30007 | In a loop-free graph there... |
| usgr2trlncrct 30008 | In a simple graph, any tra... |
| umgrn1cycl 30009 | In a multigraph graph (wit... |
| uspgrn2crct 30010 | In a simple pseudograph th... |
| usgrn2cycl 30011 | In a simple graph there ar... |
| crctcshwlkn0lem1 30012 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 30013 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 30014 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 30015 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 30016 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 30017 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 30018 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 30019 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 30020 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 30021 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 30022 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 30023 | Cyclically shifting the in... |
| crctcshwlk 30024 | Cyclically shifting the in... |
| crctcshtrl 30025 | Cyclically shifting the in... |
| crctcsh 30026 | Cyclically shifting the in... |
| wwlks 30037 | The set of walks (in an un... |
| iswwlks 30038 | A word over the set of ver... |
| wwlksn 30039 | The set of walks (in an un... |
| iswwlksn 30040 | A word over the set of ver... |
| wwlksnprcl 30041 | Derivation of the length o... |
| iswwlksnx 30042 | Properties of a word to re... |
| wwlkbp 30043 | Basic properties of a walk... |
| wwlknbp 30044 | Basic properties of a walk... |
| wwlknp 30045 | Properties of a set being ... |
| wwlknbp1 30046 | Other basic properties of ... |
| wwlknvtx 30047 | The symbols of a word ` W ... |
| wwlknllvtx 30048 | If a word ` W ` represents... |
| wwlknlsw 30049 | If a word represents a wal... |
| wspthsn 30050 | The set of simple paths of... |
| iswspthn 30051 | An element of the set of s... |
| wspthnp 30052 | Properties of a set being ... |
| wwlksnon 30053 | The set of walks of a fixe... |
| wspthsnon 30054 | The set of simple paths of... |
| iswwlksnon 30055 | The set of walks of a fixe... |
| wwlksnon0 30056 | Sufficient conditions for ... |
| wwlksonvtx 30057 | If a word ` W ` represents... |
| iswspthsnon 30058 | The set of simple paths of... |
| wwlknon 30059 | An element of the set of w... |
| wspthnon 30060 | An element of the set of s... |
| wspthnonp 30061 | Properties of a set being ... |
| wspthneq1eq2 30062 | Two simple paths with iden... |
| wwlksn0s 30063 | The set of all walks as wo... |
| wwlkssswrd 30064 | Walks (represented by word... |
| wwlksn0 30065 | A walk of length 0 is repr... |
| 0enwwlksnge1 30066 | In graphs without edges, t... |
| wwlkswwlksn 30067 | A walk of a fixed length a... |
| wwlkssswwlksn 30068 | The walks of a fixed lengt... |
| wlkiswwlks1 30069 | The sequence of vertices i... |
| wlklnwwlkln1 30070 | The sequence of vertices i... |
| wlkiswwlks2lem1 30071 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 30072 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 30073 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 30074 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 30075 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 30076 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 30077 | A walk as word corresponds... |
| wlkiswwlks 30078 | A walk as word corresponds... |
| wlkiswwlksupgr2 30079 | A walk as word corresponds... |
| wlkiswwlkupgr 30080 | A walk as word corresponds... |
| wlkswwlksf1o 30081 | The mapping of (ordinary) ... |
| wlkswwlksen 30082 | The set of walks as words ... |
| wwlksm1edg 30083 | Removing the trailing edge... |
| wlklnwwlkln2lem 30084 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 30085 | A walk of length ` N ` as ... |
| wlklnwwlkn 30086 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 30087 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 30088 | A walk of length ` N ` as ... |
| wlknewwlksn 30089 | If a walk in a pseudograph... |
| wlknwwlksnbij 30090 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 30091 | In a simple pseudograph, t... |
| wlknwwlksneqs 30092 | The set of walks of a fixe... |
| wwlkseq 30093 | Equality of two walks (as ... |
| wwlksnred 30094 | Reduction of a walk (as wo... |
| wwlksnext 30095 | Extension of a walk (as wo... |
| wwlksnextbi 30096 | Extension of a walk (as wo... |
| wwlksnredwwlkn 30097 | For each walk (as word) of... |
| wwlksnredwwlkn0 30098 | For each walk (as word) of... |
| wwlksnextwrd 30099 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 30100 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 30101 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 30102 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 30103 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 30104 | There is a bijection betwe... |
| wwlksnexthasheq 30105 | The number of the extensio... |
| disjxwwlksn 30106 | Sets of walks (as words) e... |
| wwlksnndef 30107 | Conditions for ` WWalksN `... |
| wwlksnfi 30108 | The number of walks repres... |
| wlksnfi 30109 | The number of walks of fix... |
| wlksnwwlknvbij 30110 | There is a bijection betwe... |
| wwlksnextproplem1 30111 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 30112 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 30113 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 30114 | Adding additional properti... |
| disjxwwlkn 30115 | Sets of walks (as words) e... |
| hashwwlksnext 30116 | Number of walks (as words)... |
| wwlksnwwlksnon 30117 | A walk of fixed length is ... |
| wspthsnwspthsnon 30118 | A simple path of fixed len... |
| wspthsnonn0vne 30119 | If the set of simple paths... |
| wspthsswwlkn 30120 | The set of simple paths of... |
| wspthnfi 30121 | In a finite graph, the set... |
| wwlksnonfi 30122 | In a finite graph, the set... |
| wspthsswwlknon 30123 | The set of simple paths of... |
| wspthnonfi 30124 | In a finite graph, the set... |
| wspniunwspnon 30125 | The set of nonempty simple... |
| wspn0 30126 | If there are no vertices, ... |
| 2wlkdlem1 30127 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 30128 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 30129 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 30130 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 30131 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 30132 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 30133 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 30134 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 30135 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 30136 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 30137 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 30138 | Construction of a walk fro... |
| 2wlkond 30139 | A walk of length 2 from on... |
| 2trld 30140 | Construction of a trail fr... |
| 2trlond 30141 | A trail of length 2 from o... |
| 2pthd 30142 | A path of length 2 from on... |
| 2spthd 30143 | A simple path of length 2 ... |
| 2pthond 30144 | A simple path of length 2 ... |
| 2pthon3v 30145 | For a vertex adjacent to t... |
| umgr2adedgwlklem 30146 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 30147 | In a multigraph, two adjac... |
| umgr2adedgwlkon 30148 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 30149 | Alternate proof for ~ umgr... |
| umgr2adedgspth 30150 | In a multigraph, two adjac... |
| umgr2wlk 30151 | In a multigraph, there is ... |
| umgr2wlkon 30152 | For each pair of adjacent ... |
| elwwlks2s3 30153 | A walk of length 2 as word... |
| midwwlks2s3 30154 | There is a vertex between ... |
| wwlks2onv 30155 | If a length 3 string repre... |
| elwwlks2ons3im 30156 | A walk as word of length 2... |
| elwwlks2ons3 30157 | For each walk of length 2 ... |
| s3wwlks2on 30158 | A length 3 string which re... |
| sps3wwlks2on 30159 | A length 3 string which re... |
| usgrwwlks2on 30160 | A walk of length 2 between... |
| umgrwwlks2on 30161 | A walk of length 2 between... |
| wwlks2onsym 30162 | There is a walk of length ... |
| elwwlks2on 30163 | A walk of length 2 between... |
| elwspths2on 30164 | A simple path of length 2 ... |
| elwspths2onw 30165 | A simple path of length 2 ... |
| wpthswwlks2on 30166 | For two different vertices... |
| 2wspdisj 30167 | All simple paths of length... |
| 2wspiundisj 30168 | All simple paths of length... |
| usgr2wspthons3 30169 | A simple path of length 2 ... |
| usgr2wspthon 30170 | A simple path of length 2 ... |
| elwwlks2 30171 | A walk of length 2 between... |
| elwspths2spth 30172 | A simple path of length 2 ... |
| rusgrnumwwlkl1 30173 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 30174 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 30175 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 30176 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 30177 | Induction base 1 for ~ rus... |
| rusgr0edg 30178 | Special case for graphs wi... |
| rusgrnumwwlks 30179 | Induction step for ~ rusgr... |
| rusgrnumwwlk 30180 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 30181 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 30182 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 30183 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 30184 | In a ` K `-regular graph, ... |
| clwwlk 30187 | The set of closed walks (i... |
| isclwwlk 30188 | Properties of a word to re... |
| clwwlkbp 30189 | Basic properties of a clos... |
| clwwlkgt0 30190 | There is no empty closed w... |
| clwwlksswrd 30191 | Closed walks (represented ... |
| clwwlk1loop 30192 | A closed walk of length 1 ... |
| clwwlkccatlem 30193 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 30194 | The concatenation of two w... |
| umgrclwwlkge2 30195 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 30196 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 30197 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 30198 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 30199 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 30200 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 30201 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 30202 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 30203 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 30204 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 30205 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 30206 | A closed walk as word of l... |
| clwlkclwwlk2 30207 | A closed walk corresponds ... |
| clwlkclwwlkflem 30208 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 30209 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 30210 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 30211 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 30212 | ` F ` is a function from t... |
| clwlkclwwlkfo 30213 | ` F ` is a function from t... |
| clwlkclwwlkf1 30214 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 30215 | ` F ` is a bijection betwe... |
| clwlkclwwlken 30216 | The set of the nonempty cl... |
| clwwisshclwwslemlem 30217 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 30218 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 30219 | Cyclically shifting a clos... |
| clwwisshclwwsn 30220 | Cyclically shifting a clos... |
| erclwwlkrel 30221 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 30222 | Two classes are equivalent... |
| erclwwlkeqlen 30223 | If two classes are equival... |
| erclwwlkref 30224 | ` .~ ` is a reflexive rela... |
| erclwwlksym 30225 | ` .~ ` is a symmetric rela... |
| erclwwlktr 30226 | ` .~ ` is a transitive rel... |
| erclwwlk 30227 | ` .~ ` is an equivalence r... |
| clwwlkn 30230 | The set of closed walks of... |
| isclwwlkn 30231 | A word over the set of ver... |
| clwwlkn0 30232 | There is no closed walk of... |
| clwwlkneq0 30233 | Sufficient conditions for ... |
| clwwlkclwwlkn 30234 | A closed walk of a fixed l... |
| clwwlksclwwlkn 30235 | The closed walks of a fixe... |
| clwwlknlen 30236 | The length of a word repre... |
| clwwlknnn 30237 | The length of a closed wal... |
| clwwlknwrd 30238 | A closed walk of a fixed l... |
| clwwlknbp 30239 | Basic properties of a clos... |
| isclwwlknx 30240 | Characterization of a word... |
| clwwlknp 30241 | Properties of a set being ... |
| clwwlknwwlksn 30242 | A word representing a clos... |
| clwwlknlbonbgr1 30243 | The last but one vertex in... |
| clwwlkinwwlk 30244 | If the initial vertex of a... |
| clwwlkn1 30245 | A closed walk of length 1 ... |
| loopclwwlkn1b 30246 | The singleton word consist... |
| clwwlkn1loopb 30247 | A word represents a closed... |
| clwwlkn2 30248 | A closed walk of length 2 ... |
| clwwlknfi 30249 | If there is only a finite ... |
| clwwlkel 30250 | Obtaining a closed walk (a... |
| clwwlkf 30251 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 30252 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 30253 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 30254 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 30255 | F is a 1-1 onto function, ... |
| clwwlken 30256 | The set of closed walks of... |
| clwwlknwwlkncl 30257 | Obtaining a closed walk (a... |
| clwwlkwwlksb 30258 | A nonempty word over verti... |
| clwwlknwwlksnb 30259 | A word over vertices repre... |
| clwwlkext2edg 30260 | If a word concatenated wit... |
| wwlksext2clwwlk 30261 | If a word represents a wal... |
| wwlksubclwwlk 30262 | Any prefix of a word repre... |
| clwwnisshclwwsn 30263 | Cyclically shifting a clos... |
| eleclclwwlknlem1 30264 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 30265 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 30266 | The set of cyclical shifts... |
| clwwlknccat 30267 | The concatenation of two w... |
| umgr2cwwk2dif 30268 | If a word represents a clo... |
| umgr2cwwkdifex 30269 | If a word represents a clo... |
| erclwwlknrel 30270 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 30271 | Two classes are equivalent... |
| erclwwlkneqlen 30272 | If two classes are equival... |
| erclwwlknref 30273 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 30274 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 30275 | ` .~ ` is a transitive rel... |
| erclwwlkn 30276 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 30277 | The quotient set of the se... |
| hashclwwlkn0 30278 | The number of closed walks... |
| eclclwwlkn1 30279 | An equivalence class accor... |
| eleclclwwlkn 30280 | A member of an equivalence... |
| hashecclwwlkn1 30281 | The size of every equivale... |
| umgrhashecclwwlk 30282 | The size of every equivale... |
| fusgrhashclwwlkn 30283 | The size of the set of clo... |
| clwwlkndivn 30284 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 30285 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 30286 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 30287 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 30288 | There is a one-to-one onto... |
| clwlkssizeeq 30289 | The size of the set of clo... |
| clwlksndivn 30290 | The size of the set of clo... |
| clwwlknonmpo 30293 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 30294 | The set of closed walks on... |
| isclwwlknon 30295 | A word over the set of ver... |
| clwwlk0on0 30296 | There is no word over the ... |
| clwwlknon0 30297 | Sufficient conditions for ... |
| clwwlknonfin 30298 | In a finite graph ` G ` , ... |
| clwwlknonel 30299 | Characterization of a word... |
| clwwlknonccat 30300 | The concatenation of two w... |
| clwwlknon1 30301 | The set of closed walks on... |
| clwwlknon1loop 30302 | If there is a loop at vert... |
| clwwlknon1nloop 30303 | If there is no loop at ver... |
| clwwlknon1sn 30304 | The set of (closed) walks ... |
| clwwlknon1le1 30305 | There is at most one (clos... |
| clwwlknon2 30306 | The set of closed walks on... |
| clwwlknon2x 30307 | The set of closed walks on... |
| s2elclwwlknon2 30308 | Sufficient conditions of a... |
| clwwlknon2num 30309 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 30310 | A word over vertices repre... |
| clwwlknonex2lem1 30311 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 30312 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 30313 | Extending a closed walk ` ... |
| clwwlknonex2e 30314 | Extending a closed walk ` ... |
| clwwlknondisj 30315 | The sets of closed walks o... |
| clwwlknun 30316 | The set of closed walks of... |
| clwwlkvbij 30317 | There is a bijection betwe... |
| 0ewlk 30318 | The empty set (empty seque... |
| 1ewlk 30319 | A sequence of 1 edge is an... |
| 0wlk 30320 | A pair of an empty set (of... |
| is0wlk 30321 | A pair of an empty set (of... |
| 0wlkonlem1 30322 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 30323 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 30324 | A walk of length 0 from a ... |
| 0wlkons1 30325 | A walk of length 0 from a ... |
| 0trl 30326 | A pair of an empty set (of... |
| is0trl 30327 | A pair of an empty set (of... |
| 0trlon 30328 | A trail of length 0 from a... |
| 0pth 30329 | A pair of an empty set (of... |
| 0spth 30330 | A pair of an empty set (of... |
| 0pthon 30331 | A path of length 0 from a ... |
| 0pthon1 30332 | A path of length 0 from a ... |
| 0pthonv 30333 | For each vertex there is a... |
| 0clwlk 30334 | A pair of an empty set (of... |
| 0clwlkv 30335 | Any vertex (more precisely... |
| 0clwlk0 30336 | There is no closed walk in... |
| 0crct 30337 | A pair of an empty set (of... |
| 0cycl 30338 | A pair of an empty set (of... |
| 1pthdlem1 30339 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 30340 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 30341 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 30342 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 30343 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 30344 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 30345 | In a graph with two vertic... |
| 1trld 30346 | In a graph with two vertic... |
| 1pthd 30347 | In a graph with two vertic... |
| 1pthond 30348 | In a graph with two vertic... |
| upgr1wlkdlem1 30349 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 30350 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 30351 | In a pseudograph with two ... |
| upgr1trld 30352 | In a pseudograph with two ... |
| upgr1pthd 30353 | In a pseudograph with two ... |
| upgr1pthond 30354 | In a pseudograph with two ... |
| lppthon 30355 | A loop (which is an edge a... |
| lp1cycl 30356 | A loop (which is an edge a... |
| 1pthon2v 30357 | For each pair of adjacent ... |
| 1pthon2ve 30358 | For each pair of adjacent ... |
| wlk2v2elem1 30359 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 30360 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 30361 | In a graph with two vertic... |
| ntrl2v2e 30362 | A walk which is not a trai... |
| 3wlkdlem1 30363 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 30364 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 30365 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 30366 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 30367 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 30368 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 30369 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 30370 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 30371 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 30372 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 30373 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 30374 | Construction of a walk fro... |
| 3wlkond 30375 | A walk of length 3 from on... |
| 3trld 30376 | Construction of a trail fr... |
| 3trlond 30377 | A trail of length 3 from o... |
| 3pthd 30378 | A path of length 3 from on... |
| 3pthond 30379 | A path of length 3 from on... |
| 3spthd 30380 | A simple path of length 3 ... |
| 3spthond 30381 | A simple path of length 3 ... |
| 3cycld 30382 | Construction of a 3-cycle ... |
| 3cyclpd 30383 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 30384 | If there is a cycle of len... |
| uhgr3cyclexlem 30385 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 30386 | If there are three differe... |
| umgr3cyclex 30387 | If there are three (differ... |
| umgr3v3e3cycl 30388 | If and only if there is a ... |
| upgr4cycl4dv4e 30389 | If there is a cycle of len... |
| dfconngr1 30392 | Alternative definition of ... |
| isconngr 30393 | The property of being a co... |
| isconngr1 30394 | The property of being a co... |
| cusconngr 30395 | A complete hypergraph is c... |
| 0conngr 30396 | A graph without vertices i... |
| 0vconngr 30397 | A graph without vertices i... |
| 1conngr 30398 | A graph with (at most) one... |
| conngrv2edg 30399 | A vertex in a connected gr... |
| vdn0conngrumgrv2 30400 | A vertex in a connected mu... |
| releupth 30403 | The set ` ( EulerPaths `` ... |
| eupths 30404 | The Eulerian paths on the ... |
| iseupth 30405 | The property " ` <. F , P ... |
| iseupthf1o 30406 | The property " ` <. F , P ... |
| eupthi 30407 | Properties of an Eulerian ... |
| eupthf1o 30408 | The ` F ` function in an E... |
| eupthfi 30409 | Any graph with an Eulerian... |
| eupthseg 30410 | The ` N ` -th edge in an e... |
| upgriseupth 30411 | The property " ` <. F , P ... |
| upgreupthi 30412 | Properties of an Eulerian ... |
| upgreupthseg 30413 | The ` N ` -th edge in an e... |
| eupthcl 30414 | An Eulerian path has lengt... |
| eupthistrl 30415 | An Eulerian path is a trai... |
| eupthiswlk 30416 | An Eulerian path is a walk... |
| eupthpf 30417 | The ` P ` function in an E... |
| eupth0 30418 | There is an Eulerian path ... |
| eupthres 30419 | The restriction ` <. H , Q... |
| eupthp1 30420 | Append one path segment to... |
| eupth2eucrct 30421 | Append one path segment to... |
| eupth2lem1 30422 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30423 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30424 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30425 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30426 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30427 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30428 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30429 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30430 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30431 | The effect on vertex degre... |
| eupth2lem3lem1 30432 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30433 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30434 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30435 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30436 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30437 | Formerly part of proof of ... |
| eupth2lem3lem7 30438 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30439 | Formerly part of proof of ... |
| eupth2lem3 30440 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30441 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30442 | Lemma for ~ eupth2 (induct... |
| eupth2 30443 | The only vertices of odd d... |
| eulerpathpr 30444 | A graph with an Eulerian p... |
| eulerpath 30445 | A pseudograph with an Eule... |
| eulercrct 30446 | A pseudograph with an Eule... |
| eucrctshift 30447 | Cyclically shifting the in... |
| eucrct2eupth1 30448 | Removing one edge ` ( I ``... |
| eucrct2eupth 30449 | Removing one edge ` ( I ``... |
| konigsbergvtx 30450 | The set of vertices of the... |
| konigsbergiedg 30451 | The indexed edges of the K... |
| konigsbergiedgw 30452 | The indexed edges of the K... |
| konigsbergssiedgwpr 30453 | Each subset of the indexed... |
| konigsbergssiedgw 30454 | Each subset of the indexed... |
| konigsbergumgr 30455 | The Königsberg graph ... |
| konigsberglem1 30456 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30457 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30458 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30459 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30460 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30461 | The Königsberg Bridge... |
| isfrgr 30464 | The property of being a fr... |
| frgrusgr 30465 | A friendship graph is a si... |
| frgr0v 30466 | Any null graph (set with n... |
| frgr0vb 30467 | Any null graph (without ve... |
| frgruhgr0v 30468 | Any null graph (without ve... |
| frgr0 30469 | The null graph (graph with... |
| frcond1 30470 | The friendship condition: ... |
| frcond2 30471 | The friendship condition: ... |
| frgreu 30472 | Variant of ~ frcond2 : An... |
| frcond3 30473 | The friendship condition, ... |
| frcond4 30474 | The friendship condition, ... |
| frgr1v 30475 | Any graph with (at most) o... |
| nfrgr2v 30476 | Any graph with two (differ... |
| frgr3vlem1 30477 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30478 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30479 | Any graph with three verti... |
| 1vwmgr 30480 | Every graph with one verte... |
| 3vfriswmgrlem 30481 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30482 | Every friendship graph wit... |
| 1to2vfriswmgr 30483 | Every friendship graph wit... |
| 1to3vfriswmgr 30484 | Every friendship graph wit... |
| 1to3vfriendship 30485 | The friendship theorem for... |
| 2pthfrgrrn 30486 | Between any two (different... |
| 2pthfrgrrn2 30487 | Between any two (different... |
| 2pthfrgr 30488 | Between any two (different... |
| 3cyclfrgrrn1 30489 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30490 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30491 | Every vertex in a friendsh... |
| 3cyclfrgr 30492 | Every vertex in a friendsh... |
| 4cycl2v2nb 30493 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30494 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30495 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30496 | A graph with a 4-cycle is ... |
| frgrnbnb 30497 | If two neighbors ` U ` and... |
| frgrconngr 30498 | A friendship graph is conn... |
| vdgn0frgrv2 30499 | A vertex in a friendship g... |
| vdgn1frgrv2 30500 | Any vertex in a friendship... |
| vdgn1frgrv3 30501 | Any vertex in a friendship... |
| vdgfrgrgt2 30502 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30503 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30504 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30505 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30506 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30507 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30508 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30509 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30510 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30511 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30512 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30513 | In a friendship graph, two... |
| frgrwopreglem4a 30514 | In a friendship graph any ... |
| frgrwopreglem5a 30515 | If a friendship graph has ... |
| frgrwopreglem1 30516 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30517 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30518 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30519 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30520 | According to statement 5 i... |
| frgrwopregbsn 30521 | According to statement 5 i... |
| frgrwopreg1 30522 | According to statement 5 i... |
| frgrwopreg2 30523 | According to statement 5 i... |
| frgrwopreglem5lem 30524 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30525 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30526 | Alternate direct proof of ... |
| frgrwopreg 30527 | In a friendship graph ther... |
| frgrregorufr0 30528 | In a friendship graph ther... |
| frgrregorufr 30529 | If there is a vertex havin... |
| frgrregorufrg 30530 | If there is a vertex havin... |
| frgr2wwlkeu 30531 | For two different vertices... |
| frgr2wwlkn0 30532 | In a friendship graph, the... |
| frgr2wwlk1 30533 | In a friendship graph, the... |
| frgr2wsp1 30534 | In a friendship graph, the... |
| frgr2wwlkeqm 30535 | If there is a (simple) pat... |
| frgrhash2wsp 30536 | The number of simple paths... |
| fusgreg2wsplem 30537 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30538 | The set of paths of length... |
| fusgreghash2wspv 30539 | According to statement 7 i... |
| fusgreg2wsp 30540 | In a finite simple graph, ... |
| 2wspmdisj 30541 | The sets of paths of lengt... |
| fusgreghash2wsp 30542 | In a finite k-regular grap... |
| frrusgrord0lem 30543 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30544 | If a nonempty finite frien... |
| frrusgrord 30545 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30546 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30547 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30548 | If the initial vertex of a... |
| clwwnonrepclwwnon 30549 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30550 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30551 | Value of operation ` C ` ,... |
| 2clwwlk2 30552 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30553 | Characterization of an ele... |
| 2clwwlk2clwwlk 30554 | An element of the value of... |
| numclwwlk1lem2foalem 30555 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30556 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30557 | Characterization of an ele... |
| numclwwlk1lem2foa 30558 | Going forth and back from ... |
| numclwwlk1lem2f 30559 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30560 | Value of the function ` T ... |
| numclwwlk1lem2f1 30561 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30562 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30563 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30564 | The set of double loops of... |
| numclwwlk1 30565 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30566 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30567 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30568 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30569 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30570 | The sets of the two repres... |
| wlkl0 30571 | There is exactly one walk ... |
| clwlknon2num 30572 | There are k walks of lengt... |
| numclwlk1lem1 30573 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 30574 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 30575 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 30576 | Value of operation ` H ` ,... |
| numclwwlkovh 30577 | Value of operation ` H ` ,... |
| numclwwlkovq 30578 | Value of operation ` Q ` ,... |
| numclwwlkqhash 30579 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 30580 | In a friendship graph, for... |
| numclwlk2lem2f 30581 | ` R ` is a function mappin... |
| numclwlk2lem2fv 30582 | Value of the function ` R ... |
| numclwlk2lem2f1o 30583 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 30584 | In a friendship graph, the... |
| numclwwlk2 30585 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 30586 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 30587 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 30588 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 30589 | Statement 12 in [Huneke] p... |
| numclwwlk4 30590 | The total number of closed... |
| numclwwlk5lem 30591 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 30592 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 30593 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 30594 | For a prime divisor ` P ` ... |
| numclwwlk7 30595 | Statement 14 in [Huneke] p... |
| numclwwlk8 30596 | The size of the set of clo... |
| frgrreggt1 30597 | If a finite nonempty frien... |
| frgrreg 30598 | If a finite nonempty frien... |
| frgrregord013 30599 | If a finite friendship gra... |
| frgrregord13 30600 | If a nonempty finite frien... |
| frgrogt3nreg 30601 | If a finite friendship gra... |
| friendshipgt3 30602 | The friendship theorem for... |
| friendship 30603 | The friendship theorem: I... |
| conventions 30604 |
H... |
| conventions-labels 30605 |
... |
| conventions-comments 30606 |
... |
| natded 30607 | Here are typical n... |
| ex-natded5.2 30608 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 30609 | A more efficient proof of ... |
| ex-natded5.2i 30610 | The same as ~ ex-natded5.2... |
| ex-natded5.3 30611 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 30612 | A more efficient proof of ... |
| ex-natded5.3i 30613 | The same as ~ ex-natded5.3... |
| ex-natded5.5 30614 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 30615 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 30616 | A more efficient proof of ... |
| ex-natded5.8 30617 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 30618 | A more efficient proof of ... |
| ex-natded5.13 30619 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 30620 | A more efficient proof of ... |
| ex-natded9.20 30621 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 30622 | A more efficient proof of ... |
| ex-natded9.26 30623 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 30624 | A more efficient proof of ... |
| ex-or 30625 | Example for ~ df-or . Exa... |
| ex-an 30626 | Example for ~ df-an . Exa... |
| ex-dif 30627 | Example for ~ df-dif . Ex... |
| ex-un 30628 | Example for ~ df-un . Exa... |
| ex-in 30629 | Example for ~ df-in . Exa... |
| ex-uni 30630 | Example for ~ df-uni . Ex... |
| ex-ss 30631 | Example for ~ df-ss . Exa... |
| ex-pss 30632 | Example for ~ df-pss . Ex... |
| ex-pw 30633 | Example for ~ df-pw . Exa... |
| ex-pr 30634 | Example for ~ df-pr . (Co... |
| ex-br 30635 | Example for ~ df-br . Exa... |
| ex-opab 30636 | Example for ~ df-opab . E... |
| ex-eprel 30637 | Example for ~ df-eprel . ... |
| ex-id 30638 | Example for ~ df-id . Exa... |
| ex-po 30639 | Example for ~ df-po . Exa... |
| ex-xp 30640 | Example for ~ df-xp . Exa... |
| ex-cnv 30641 | Example for ~ df-cnv . Ex... |
| ex-co 30642 | Example for ~ df-co . Exa... |
| ex-dm 30643 | Example for ~ df-dm . Exa... |
| ex-rn 30644 | Example for ~ df-rn . Exa... |
| ex-res 30645 | Example for ~ df-res . Ex... |
| ex-ima 30646 | Example for ~ df-ima . Ex... |
| ex-fv 30647 | Example for ~ df-fv . Exa... |
| ex-1st 30648 | Example for ~ df-1st . Ex... |
| ex-2nd 30649 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 30650 | Example for ~ df-dec , 100... |
| ex-fl 30651 | Example for ~ df-fl . Exa... |
| ex-ceil 30652 | Example for ~ df-ceil . (... |
| ex-mod 30653 | Example for ~ df-mod . (C... |
| ex-exp 30654 | Example for ~ df-exp . (C... |
| ex-fac 30655 | Example for ~ df-fac . (C... |
| ex-bc 30656 | Example for ~ df-bc . (Co... |
| ex-hash 30657 | Example for ~ df-hash . (... |
| ex-sqrt 30658 | Example for ~ df-sqrt . (... |
| ex-abs 30659 | Example for ~ df-abs . (C... |
| ex-dvds 30660 | Example for ~ df-dvds : 3 ... |
| ex-gcd 30661 | Example for ~ df-gcd . (C... |
| ex-lcm 30662 | Example for ~ df-lcm . (C... |
| ex-prmo 30663 | Example for ~ df-prmo : ` ... |
| aevdemo 30664 | Proof illustrating the com... |
| ex-ind-dvds 30665 | Example of a proof by indu... |
| ex-fpar 30666 | Formalized example provide... |
| avril1 30667 | Poisson d'Avril's Theorem.... |
| 2bornot2b 30668 | The law of excluded middle... |
| helloworld 30669 | The classic "Hello world" ... |
| 1p1e2apr1 30670 | One plus one equals two. ... |
| eqid1 30671 | Law of identity (reflexivi... |
| 1div0apr 30672 | Division by zero is forbid... |
| topnfbey 30673 | Nothing seems to be imposs... |
| 9p10ne21 30674 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 30675 | 9 + 10 equals 21. This as... |
| nrt2irr 30677 | The ` N ` -th root of 2 is... |
| nowisdomv 30678 | One's wisdom on matters of... |
| isplig 30681 | The predicate "is a planar... |
| ispligb 30682 | The predicate "is a planar... |
| tncp 30683 | In any planar incidence ge... |
| l2p 30684 | For any line in a planar i... |
| lpni 30685 | For any line in a planar i... |
| nsnlplig 30686 | There is no "one-point lin... |
| nsnlpligALT 30687 | Alternate version of ~ nsn... |
| n0lplig 30688 | There is no "empty line" i... |
| n0lpligALT 30689 | Alternate version of ~ n0l... |
| eulplig 30690 | Through two distinct point... |
| pliguhgr 30691 | Any planar incidence geome... |
| dummylink 30692 | Alias for ~ a1ii that may ... |
| id1 30693 | Alias for ~ idALT that may... |
| isgrpo 30702 | The predicate "is a group ... |
| isgrpoi 30703 | Properties that determine ... |
| grpofo 30704 | A group operation maps ont... |
| grpocl 30705 | Closure law for a group op... |
| grpolidinv 30706 | A group has a left identit... |
| grpon0 30707 | The base set of a group is... |
| grpoass 30708 | A group operation is assoc... |
| grpoidinvlem1 30709 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 30710 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 30711 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 30712 | Lemma for ~ grpoidinv . (... |
| grpoidinv 30713 | A group has a left and rig... |
| grpoideu 30714 | The left identity element ... |
| grporndm 30715 | A group's range in terms o... |
| 0ngrp 30716 | The empty set is not a gro... |
| gidval 30717 | The value of the identity ... |
| grpoidval 30718 | Lemma for ~ grpoidcl and o... |
| grpoidcl 30719 | The identity element of a ... |
| grpoidinv2 30720 | A group's properties using... |
| grpolid 30721 | The identity element of a ... |
| grporid 30722 | The identity element of a ... |
| grporcan 30723 | Right cancellation law for... |
| grpoinveu 30724 | The left inverse element o... |
| grpoid 30725 | Two ways of saying that an... |
| grporn 30726 | The range of a group opera... |
| grpoinvfval 30727 | The inverse function of a ... |
| grpoinvval 30728 | The inverse of a group ele... |
| grpoinvcl 30729 | A group element's inverse ... |
| grpoinv 30730 | The properties of a group ... |
| grpolinv 30731 | The left inverse of a grou... |
| grporinv 30732 | The right inverse of a gro... |
| grpoinvid1 30733 | The inverse of a group ele... |
| grpoinvid2 30734 | The inverse of a group ele... |
| grpolcan 30735 | Left cancellation law for ... |
| grpo2inv 30736 | Double inverse law for gro... |
| grpoinvf 30737 | Mapping of the inverse fun... |
| grpoinvop 30738 | The inverse of the group o... |
| grpodivfval 30739 | Group division (or subtrac... |
| grpodivval 30740 | Group division (or subtrac... |
| grpodivinv 30741 | Group division by an inver... |
| grpoinvdiv 30742 | Inverse of a group divisio... |
| grpodivf 30743 | Mapping for group division... |
| grpodivcl 30744 | Closure of group division ... |
| grpodivdiv 30745 | Double group division. (C... |
| grpomuldivass 30746 | Associative-type law for m... |
| grpodivid 30747 | Division of a group member... |
| grponpcan 30748 | Cancellation law for group... |
| isablo 30751 | The predicate "is an Abeli... |
| ablogrpo 30752 | An Abelian group operation... |
| ablocom 30753 | An Abelian group operation... |
| ablo32 30754 | Commutative/associative la... |
| ablo4 30755 | Commutative/associative la... |
| isabloi 30756 | Properties that determine ... |
| ablomuldiv 30757 | Law for group multiplicati... |
| ablodivdiv 30758 | Law for double group divis... |
| ablodivdiv4 30759 | Law for double group divis... |
| ablodiv32 30760 | Swap the second and third ... |
| ablonncan 30761 | Cancellation law for group... |
| ablonnncan1 30762 | Cancellation law for group... |
| vcrel 30765 | The class of all complex v... |
| vciOLD 30766 | Obsolete version of ~ cvsi... |
| vcsm 30767 | Functionality of th scalar... |
| vccl 30768 | Closure of the scalar prod... |
| vcidOLD 30769 | Identity element for the s... |
| vcdi 30770 | Distributive law for the s... |
| vcdir 30771 | Distributive law for the s... |
| vcass 30772 | Associative law for the sc... |
| vc2OLD 30773 | A vector plus itself is tw... |
| vcablo 30774 | Vector addition is an Abel... |
| vcgrp 30775 | Vector addition is a group... |
| vclcan 30776 | Left cancellation law for ... |
| vczcl 30777 | The zero vector is a vecto... |
| vc0rid 30778 | The zero vector is a right... |
| vc0 30779 | Zero times a vector is the... |
| vcz 30780 | Anything times the zero ve... |
| vcm 30781 | Minus 1 times a vector is ... |
| isvclem 30782 | Lemma for ~ isvcOLD . (Co... |
| vcex 30783 | The components of a comple... |
| isvcOLD 30784 | The predicate "is a comple... |
| isvciOLD 30785 | Properties that determine ... |
| cnaddabloOLD 30786 | Obsolete version of ~ cnad... |
| cnidOLD 30787 | Obsolete version of ~ cnad... |
| cncvcOLD 30788 | Obsolete version of ~ cncv... |
| nvss 30798 | Structure of the class of ... |
| nvvcop 30799 | A normed complex vector sp... |
| nvrel 30807 | The class of all normed co... |
| vafval 30808 | Value of the function for ... |
| bafval 30809 | Value of the function for ... |
| smfval 30810 | Value of the function for ... |
| 0vfval 30811 | Value of the function for ... |
| nmcvfval 30812 | Value of the norm function... |
| nvop2 30813 | A normed complex vector sp... |
| nvvop 30814 | The vector space component... |
| isnvlem 30815 | Lemma for ~ isnv . (Contr... |
| nvex 30816 | The components of a normed... |
| isnv 30817 | The predicate "is a normed... |
| isnvi 30818 | Properties that determine ... |
| nvi 30819 | The properties of a normed... |
| nvvc 30820 | The vector space component... |
| nvablo 30821 | The vector addition operat... |
| nvgrp 30822 | The vector addition operat... |
| nvgf 30823 | Mapping for the vector add... |
| nvsf 30824 | Mapping for the scalar mul... |
| nvgcl 30825 | Closure law for the vector... |
| nvcom 30826 | The vector addition (group... |
| nvass 30827 | The vector addition (group... |
| nvadd32 30828 | Commutative/associative la... |
| nvrcan 30829 | Right cancellation law for... |
| nvadd4 30830 | Rearrangement of 4 terms i... |
| nvscl 30831 | Closure law for the scalar... |
| nvsid 30832 | Identity element for the s... |
| nvsass 30833 | Associative law for the sc... |
| nvscom 30834 | Commutative law for the sc... |
| nvdi 30835 | Distributive law for the s... |
| nvdir 30836 | Distributive law for the s... |
| nv2 30837 | A vector plus itself is tw... |
| vsfval 30838 | Value of the function for ... |
| nvzcl 30839 | Closure law for the zero v... |
| nv0rid 30840 | The zero vector is a right... |
| nv0lid 30841 | The zero vector is a left ... |
| nv0 30842 | Zero times a vector is the... |
| nvsz 30843 | Anything times the zero ve... |
| nvinv 30844 | Minus 1 times a vector is ... |
| nvinvfval 30845 | Function for the negative ... |
| nvm 30846 | Vector subtraction in term... |
| nvmval 30847 | Value of vector subtractio... |
| nvmval2 30848 | Value of vector subtractio... |
| nvmfval 30849 | Value of the function for ... |
| nvmf 30850 | Mapping for the vector sub... |
| nvmcl 30851 | Closure law for the vector... |
| nvnnncan1 30852 | Cancellation law for vecto... |
| nvmdi 30853 | Distributive law for scala... |
| nvnegneg 30854 | Double negative of a vecto... |
| nvmul0or 30855 | If a scalar product is zer... |
| nvrinv 30856 | A vector minus itself. (C... |
| nvlinv 30857 | Minus a vector plus itself... |
| nvpncan2 30858 | Cancellation law for vecto... |
| nvpncan 30859 | Cancellation law for vecto... |
| nvaddsub 30860 | Commutative/associative la... |
| nvnpcan 30861 | Cancellation law for a nor... |
| nvaddsub4 30862 | Rearrangement of 4 terms i... |
| nvmeq0 30863 | The difference between two... |
| nvmid 30864 | A vector minus itself is t... |
| nvf 30865 | Mapping for the norm funct... |
| nvcl 30866 | The norm of a normed compl... |
| nvcli 30867 | The norm of a normed compl... |
| nvs 30868 | Proportionality property o... |
| nvsge0 30869 | The norm of a scalar produ... |
| nvm1 30870 | The norm of the negative o... |
| nvdif 30871 | The norm of the difference... |
| nvpi 30872 | The norm of a vector plus ... |
| nvz0 30873 | The norm of a zero vector ... |
| nvz 30874 | The norm of a vector is ze... |
| nvtri 30875 | Triangle inequality for th... |
| nvmtri 30876 | Triangle inequality for th... |
| nvabs 30877 | Norm difference property o... |
| nvge0 30878 | The norm of a normed compl... |
| nvgt0 30879 | A nonzero norm is positive... |
| nv1 30880 | From any nonzero vector, c... |
| nvop 30881 | A complex inner product sp... |
| cnnv 30882 | The set of complex numbers... |
| cnnvg 30883 | The vector addition (group... |
| cnnvba 30884 | The base set of the normed... |
| cnnvs 30885 | The scalar product operati... |
| cnnvnm 30886 | The norm operation of the ... |
| cnnvm 30887 | The vector subtraction ope... |
| elimnv 30888 | Hypothesis elimination lem... |
| elimnvu 30889 | Hypothesis elimination lem... |
| imsval 30890 | Value of the induced metri... |
| imsdval 30891 | Value of the induced metri... |
| imsdval2 30892 | Value of the distance func... |
| nvnd 30893 | The norm of a normed compl... |
| imsdf 30894 | Mapping for the induced me... |
| imsmetlem 30895 | Lemma for ~ imsmet . (Con... |
| imsmet 30896 | The induced metric of a no... |
| imsxmet 30897 | The induced metric of a no... |
| cnims 30898 | The metric induced on the ... |
| vacn 30899 | Vector addition is jointly... |
| nmcvcn 30900 | The norm of a normed compl... |
| nmcnc 30901 | The norm of a normed compl... |
| smcnlem 30902 | Lemma for ~ smcn . (Contr... |
| smcn 30903 | Scalar multiplication is j... |
| vmcn 30904 | Vector subtraction is join... |
| dipfval 30907 | The inner product function... |
| ipval 30908 | Value of the inner product... |
| ipval2lem2 30909 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 30910 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 30911 | Lemma for ~ ipval3 . (Con... |
| ipval2 30912 | Expansion of the inner pro... |
| 4ipval2 30913 | Four times the inner produ... |
| ipval3 30914 | Expansion of the inner pro... |
| ipidsq 30915 | The inner product of a vec... |
| ipnm 30916 | Norm expressed in terms of... |
| dipcl 30917 | An inner product is a comp... |
| ipf 30918 | Mapping for the inner prod... |
| dipcj 30919 | The complex conjugate of a... |
| ipipcj 30920 | An inner product times its... |
| diporthcom 30921 | Orthogonality (meaning inn... |
| dip0r 30922 | Inner product with a zero ... |
| dip0l 30923 | Inner product with a zero ... |
| ipz 30924 | The inner product of a vec... |
| dipcn 30925 | Inner product is jointly c... |
| sspval 30928 | The set of all subspaces o... |
| isssp 30929 | The predicate "is a subspa... |
| sspid 30930 | A normed complex vector sp... |
| sspnv 30931 | A subspace is a normed com... |
| sspba 30932 | The base set of a subspace... |
| sspg 30933 | Vector addition on a subsp... |
| sspgval 30934 | Vector addition on a subsp... |
| ssps 30935 | Scalar multiplication on a... |
| sspsval 30936 | Scalar multiplication on a... |
| sspmlem 30937 | Lemma for ~ sspm and other... |
| sspmval 30938 | Vector addition on a subsp... |
| sspm 30939 | Vector subtraction on a su... |
| sspz 30940 | The zero vector of a subsp... |
| sspn 30941 | The norm on a subspace is ... |
| sspnval 30942 | The norm on a subspace in ... |
| sspimsval 30943 | The induced metric on a su... |
| sspims 30944 | The induced metric on a su... |
| lnoval 30957 | The set of linear operator... |
| islno 30958 | The predicate "is a linear... |
| lnolin 30959 | Basic linearity property o... |
| lnof 30960 | A linear operator is a map... |
| lno0 30961 | The value of a linear oper... |
| lnocoi 30962 | The composition of two lin... |
| lnoadd 30963 | Addition property of a lin... |
| lnosub 30964 | Subtraction property of a ... |
| lnomul 30965 | Scalar multiplication prop... |
| nvo00 30966 | Two ways to express a zero... |
| nmoofval 30967 | The operator norm function... |
| nmooval 30968 | The operator norm function... |
| nmosetre 30969 | The set in the supremum of... |
| nmosetn0 30970 | The set in the supremum of... |
| nmoxr 30971 | The norm of an operator is... |
| nmooge0 30972 | The norm of an operator is... |
| nmorepnf 30973 | The norm of an operator is... |
| nmoreltpnf 30974 | The norm of any operator i... |
| nmogtmnf 30975 | The norm of an operator is... |
| nmoolb 30976 | A lower bound for an opera... |
| nmoubi 30977 | An upper bound for an oper... |
| nmoub3i 30978 | An upper bound for an oper... |
| nmoub2i 30979 | An upper bound for an oper... |
| nmobndi 30980 | Two ways to express that a... |
| nmounbi 30981 | Two ways two express that ... |
| nmounbseqi 30982 | An unbounded operator dete... |
| nmounbseqiALT 30983 | Alternate shorter proof of... |
| nmobndseqi 30984 | A bounded sequence determi... |
| nmobndseqiALT 30985 | Alternate shorter proof of... |
| bloval 30986 | The class of bounded linea... |
| isblo 30987 | The predicate "is a bounde... |
| isblo2 30988 | The predicate "is a bounde... |
| bloln 30989 | A bounded operator is a li... |
| blof 30990 | A bounded operator is an o... |
| nmblore 30991 | The norm of a bounded oper... |
| 0ofval 30992 | The zero operator between ... |
| 0oval 30993 | Value of the zero operator... |
| 0oo 30994 | The zero operator is an op... |
| 0lno 30995 | The zero operator is linea... |
| nmoo0 30996 | The operator norm of the z... |
| 0blo 30997 | The zero operator is a bou... |
| nmlno0lem 30998 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 30999 | The norm of a linear opera... |
| nmlno0 31000 | The norm of a linear opera... |
| nmlnoubi 31001 | An upper bound for the ope... |
| nmlnogt0 31002 | The norm of a nonzero line... |
| lnon0 31003 | The domain of a nonzero li... |
| nmblolbii 31004 | A lower bound for the norm... |
| nmblolbi 31005 | A lower bound for the norm... |
| isblo3i 31006 | The predicate "is a bounde... |
| blo3i 31007 | Properties that determine ... |
| blometi 31008 | Upper bound for the distan... |
| blocnilem 31009 | Lemma for ~ blocni and ~ l... |
| blocni 31010 | A linear operator is conti... |
| lnocni 31011 | If a linear operator is co... |
| blocn 31012 | A linear operator is conti... |
| blocn2 31013 | A bounded linear operator ... |
| ajfval 31014 | The adjoint function. (Co... |
| hmoval 31015 | The set of Hermitian (self... |
| ishmo 31016 | The predicate "is a hermit... |
| phnv 31019 | Every complex inner produc... |
| phrel 31020 | The class of all complex i... |
| phnvi 31021 | Every complex inner produc... |
| isphg 31022 | The predicate "is a comple... |
| phop 31023 | A complex inner product sp... |
| cncph 31024 | The set of complex numbers... |
| elimph 31025 | Hypothesis elimination lem... |
| elimphu 31026 | Hypothesis elimination lem... |
| isph 31027 | The predicate "is an inner... |
| phpar2 31028 | The parallelogram law for ... |
| phpar 31029 | The parallelogram law for ... |
| ip0i 31030 | A slight variant of Equati... |
| ip1ilem 31031 | Lemma for ~ ip1i . (Contr... |
| ip1i 31032 | Equation 6.47 of [Ponnusam... |
| ip2i 31033 | Equation 6.48 of [Ponnusam... |
| ipdirilem 31034 | Lemma for ~ ipdiri . (Con... |
| ipdiri 31035 | Distributive law for inner... |
| ipasslem1 31036 | Lemma for ~ ipassi . Show... |
| ipasslem2 31037 | Lemma for ~ ipassi . Show... |
| ipasslem3 31038 | Lemma for ~ ipassi . Show... |
| ipasslem4 31039 | Lemma for ~ ipassi . Show... |
| ipasslem5 31040 | Lemma for ~ ipassi . Show... |
| ipasslem7 31041 | Lemma for ~ ipassi . Show... |
| ipasslem8 31042 | Lemma for ~ ipassi . By ~... |
| ipasslem9 31043 | Lemma for ~ ipassi . Conc... |
| ipasslem10 31044 | Lemma for ~ ipassi . Show... |
| ipasslem11 31045 | Lemma for ~ ipassi . Show... |
| ipassi 31046 | Associative law for inner ... |
| dipdir 31047 | Distributive law for inner... |
| dipdi 31048 | Distributive law for inner... |
| ip2dii 31049 | Inner product of two sums.... |
| dipass 31050 | Associative law for inner ... |
| dipassr 31051 | "Associative" law for seco... |
| dipassr2 31052 | "Associative" law for inne... |
| dipsubdir 31053 | Distributive law for inner... |
| dipsubdi 31054 | Distributive law for inner... |
| pythi 31055 | The Pythagorean theorem fo... |
| siilem1 31056 | Lemma for ~ sii . (Contri... |
| siilem2 31057 | Lemma for ~ sii . (Contri... |
| siii 31058 | Inference from ~ sii . (C... |
| sii 31059 | Obsolete version of ~ ipca... |
| ipblnfi 31060 | A function ` F ` generated... |
| ip2eqi 31061 | Two vectors are equal iff ... |
| phoeqi 31062 | A condition implying that ... |
| ajmoi 31063 | Every operator has at most... |
| ajfuni 31064 | The adjoint function is a ... |
| ajfun 31065 | The adjoint function is a ... |
| ajval 31066 | Value of the adjoint funct... |
| iscbn 31069 | A complex Banach space is ... |
| cbncms 31070 | The induced metric on comp... |
| bnnv 31071 | Every complex Banach space... |
| bnrel 31072 | The class of all complex B... |
| bnsscmcl 31073 | A subspace of a Banach spa... |
| cnbn 31074 | The set of complex numbers... |
| ubthlem1 31075 | Lemma for ~ ubth . The fu... |
| ubthlem2 31076 | Lemma for ~ ubth . Given ... |
| ubthlem3 31077 | Lemma for ~ ubth . Prove ... |
| ubth 31078 | Uniform Boundedness Theore... |
| minvecolem1 31079 | Lemma for ~ minveco . The... |
| minvecolem2 31080 | Lemma for ~ minveco . Any... |
| minvecolem3 31081 | Lemma for ~ minveco . The... |
| minvecolem4a 31082 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 31083 | Lemma for ~ minveco . The... |
| minvecolem4c 31084 | Lemma for ~ minveco . The... |
| minvecolem4 31085 | Lemma for ~ minveco . The... |
| minvecolem5 31086 | Lemma for ~ minveco . Dis... |
| minvecolem6 31087 | Lemma for ~ minveco . Any... |
| minvecolem7 31088 | Lemma for ~ minveco . Sin... |
| minveco 31089 | Minimizing vector theorem,... |
| ishlo 31092 | The predicate "is a comple... |
| hlobn 31093 | Every complex Hilbert spac... |
| hlph 31094 | Every complex Hilbert spac... |
| hlrel 31095 | The class of all complex H... |
| hlnv 31096 | Every complex Hilbert spac... |
| hlnvi 31097 | Every complex Hilbert spac... |
| hlvc 31098 | Every complex Hilbert spac... |
| hlcmet 31099 | The induced metric on a co... |
| hlmet 31100 | The induced metric on a co... |
| hlpar2 31101 | The parallelogram law sati... |
| hlpar 31102 | The parallelogram law sati... |
| hlex 31103 | The base set of a Hilbert ... |
| hladdf 31104 | Mapping for Hilbert space ... |
| hlcom 31105 | Hilbert space vector addit... |
| hlass 31106 | Hilbert space vector addit... |
| hl0cl 31107 | The Hilbert space zero vec... |
| hladdid 31108 | Hilbert space addition wit... |
| hlmulf 31109 | Mapping for Hilbert space ... |
| hlmulid 31110 | Hilbert space scalar multi... |
| hlmulass 31111 | Hilbert space scalar multi... |
| hldi 31112 | Hilbert space scalar multi... |
| hldir 31113 | Hilbert space scalar multi... |
| hlmul0 31114 | Hilbert space scalar multi... |
| hlipf 31115 | Mapping for Hilbert space ... |
| hlipcj 31116 | Conjugate law for Hilbert ... |
| hlipdir 31117 | Distributive law for Hilbe... |
| hlipass 31118 | Associative law for Hilber... |
| hlipgt0 31119 | The inner product of a Hil... |
| hlcompl 31120 | Completeness of a Hilbert ... |
| cnchl 31121 | The set of complex numbers... |
| htthlem 31122 | Lemma for ~ htth . The co... |
| htth 31123 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 31179 | The group (addition) opera... |
| h2hsm 31180 | The scalar product operati... |
| h2hnm 31181 | The norm function of Hilbe... |
| h2hvs 31182 | The vector subtraction ope... |
| h2hmetdval 31183 | Value of the distance func... |
| h2hcau 31184 | The Cauchy sequences of Hi... |
| h2hlm 31185 | The limit sequences of Hil... |
| axhilex-zf 31186 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 31187 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 31188 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 31189 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 31190 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 31191 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 31192 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 31193 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 31194 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 31195 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 31196 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 31197 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 31198 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 31199 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 31200 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 31201 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 31202 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 31203 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 31216 | The Hilbert space scalar p... |
| hvaddcl 31217 | Closure of vector addition... |
| hvmulcl 31218 | Closure of scalar multipli... |
| hvmulcli 31219 | Closure inference for scal... |
| hvsubf 31220 | Mapping domain and codomai... |
| hvsubval 31221 | Value of vector subtractio... |
| hvsubcl 31222 | Closure of vector subtract... |
| hvaddcli 31223 | Closure of vector addition... |
| hvcomi 31224 | Commutation of vector addi... |
| hvsubvali 31225 | Value of vector subtractio... |
| hvsubcli 31226 | Closure of vector subtract... |
| ifhvhv0 31227 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 31228 | Addition with the zero vec... |
| hvmul0 31229 | Scalar multiplication with... |
| hvmul0or 31230 | If a scalar product is zer... |
| hvsubid 31231 | Subtraction of a vector fr... |
| hvnegid 31232 | Addition of negative of a ... |
| hv2neg 31233 | Two ways to express the ne... |
| hvaddlidi 31234 | Addition with the zero vec... |
| hvnegidi 31235 | Addition of negative of a ... |
| hv2negi 31236 | Two ways to express the ne... |
| hvm1neg 31237 | Convert minus one times a ... |
| hvaddsubval 31238 | Value of vector addition i... |
| hvadd32 31239 | Commutative/associative la... |
| hvadd12 31240 | Commutative/associative la... |
| hvadd4 31241 | Hilbert vector space addit... |
| hvsub4 31242 | Hilbert vector space addit... |
| hvaddsub12 31243 | Commutative/associative la... |
| hvpncan 31244 | Addition/subtraction cance... |
| hvpncan2 31245 | Addition/subtraction cance... |
| hvaddsubass 31246 | Associativity of sum and d... |
| hvpncan3 31247 | Subtraction and addition o... |
| hvmulcom 31248 | Scalar multiplication comm... |
| hvsubass 31249 | Hilbert vector space assoc... |
| hvsub32 31250 | Hilbert vector space commu... |
| hvmulassi 31251 | Scalar multiplication asso... |
| hvmulcomi 31252 | Scalar multiplication comm... |
| hvmul2negi 31253 | Double negative in scalar ... |
| hvsubdistr1 31254 | Scalar multiplication dist... |
| hvsubdistr2 31255 | Scalar multiplication dist... |
| hvdistr1i 31256 | Scalar multiplication dist... |
| hvsubdistr1i 31257 | Scalar multiplication dist... |
| hvassi 31258 | Hilbert vector space assoc... |
| hvadd32i 31259 | Hilbert vector space commu... |
| hvsubassi 31260 | Hilbert vector space assoc... |
| hvsub32i 31261 | Hilbert vector space commu... |
| hvadd12i 31262 | Hilbert vector space commu... |
| hvadd4i 31263 | Hilbert vector space addit... |
| hvsubsub4i 31264 | Hilbert vector space addit... |
| hvsubsub4 31265 | Hilbert vector space addit... |
| hv2times 31266 | Two times a vector. (Cont... |
| hvnegdii 31267 | Distribution of negative o... |
| hvsubeq0i 31268 | If the difference between ... |
| hvsubcan2i 31269 | Vector cancellation law. ... |
| hvaddcani 31270 | Cancellation law for vecto... |
| hvsubaddi 31271 | Relationship between vecto... |
| hvnegdi 31272 | Distribution of negative o... |
| hvsubeq0 31273 | If the difference between ... |
| hvaddeq0 31274 | If the sum of two vectors ... |
| hvaddcan 31275 | Cancellation law for vecto... |
| hvaddcan2 31276 | Cancellation law for vecto... |
| hvmulcan 31277 | Cancellation law for scala... |
| hvmulcan2 31278 | Cancellation law for scala... |
| hvsubcan 31279 | Cancellation law for vecto... |
| hvsubcan2 31280 | Cancellation law for vecto... |
| hvsub0 31281 | Subtraction of a zero vect... |
| hvsubadd 31282 | Relationship between vecto... |
| hvaddsub4 31283 | Hilbert vector space addit... |
| hicl 31285 | Closure of inner product. ... |
| hicli 31286 | Closure inference for inne... |
| his5 31291 | Associative law for inner ... |
| his52 31292 | Associative law for inner ... |
| his35 31293 | Move scalar multiplication... |
| his35i 31294 | Move scalar multiplication... |
| his7 31295 | Distributive law for inner... |
| hiassdi 31296 | Distributive/associative l... |
| his2sub 31297 | Distributive law for inner... |
| his2sub2 31298 | Distributive law for inner... |
| hire 31299 | A necessary and sufficient... |
| hiidrcl 31300 | Real closure of inner prod... |
| hi01 31301 | Inner product with the 0 v... |
| hi02 31302 | Inner product with the 0 v... |
| hiidge0 31303 | Inner product with self is... |
| his6 31304 | Zero inner product with se... |
| his1i 31305 | Conjugate law for inner pr... |
| abshicom 31306 | Commuted inner products ha... |
| hial0 31307 | A vector whose inner produ... |
| hial02 31308 | A vector whose inner produ... |
| hisubcomi 31309 | Two vector subtractions si... |
| hi2eq 31310 | Lemma used to prove equali... |
| hial2eq 31311 | Two vectors whose inner pr... |
| hial2eq2 31312 | Two vectors whose inner pr... |
| orthcom 31313 | Orthogonality commutes. (... |
| normlem0 31314 | Lemma used to derive prope... |
| normlem1 31315 | Lemma used to derive prope... |
| normlem2 31316 | Lemma used to derive prope... |
| normlem3 31317 | Lemma used to derive prope... |
| normlem4 31318 | Lemma used to derive prope... |
| normlem5 31319 | Lemma used to derive prope... |
| normlem6 31320 | Lemma used to derive prope... |
| normlem7 31321 | Lemma used to derive prope... |
| normlem8 31322 | Lemma used to derive prope... |
| normlem9 31323 | Lemma used to derive prope... |
| normlem7tALT 31324 | Lemma used to derive prope... |
| bcseqi 31325 | Equality case of Bunjakova... |
| normlem9at 31326 | Lemma used to derive prope... |
| dfhnorm2 31327 | Alternate definition of th... |
| normf 31328 | The norm function maps fro... |
| normval 31329 | The value of the norm of a... |
| normcl 31330 | Real closure of the norm o... |
| normge0 31331 | The norm of a vector is no... |
| normgt0 31332 | The norm of nonzero vector... |
| norm0 31333 | The norm of a zero vector.... |
| norm-i 31334 | Theorem 3.3(i) of [Beran] ... |
| normne0 31335 | A norm is nonzero iff its ... |
| normcli 31336 | Real closure of the norm o... |
| normsqi 31337 | The square of a norm. (Co... |
| norm-i-i 31338 | Theorem 3.3(i) of [Beran] ... |
| normsq 31339 | The square of a norm. (Co... |
| normsub0i 31340 | Two vectors are equal iff ... |
| normsub0 31341 | Two vectors are equal iff ... |
| norm-ii-i 31342 | Triangle inequality for no... |
| norm-ii 31343 | Triangle inequality for no... |
| norm-iii-i 31344 | Theorem 3.3(iii) of [Beran... |
| norm-iii 31345 | Theorem 3.3(iii) of [Beran... |
| normsubi 31346 | Negative doesn't change th... |
| normpythi 31347 | Analogy to Pythagorean the... |
| normsub 31348 | Swapping order of subtract... |
| normneg 31349 | The norm of a vector equal... |
| normpyth 31350 | Analogy to Pythagorean the... |
| normpyc 31351 | Corollary to Pythagorean t... |
| norm3difi 31352 | Norm of differences around... |
| norm3adifii 31353 | Norm of differences around... |
| norm3lem 31354 | Lemma involving norm of di... |
| norm3dif 31355 | Norm of differences around... |
| norm3dif2 31356 | Norm of differences around... |
| norm3lemt 31357 | Lemma involving norm of di... |
| norm3adifi 31358 | Norm of differences around... |
| normpari 31359 | Parallelogram law for norm... |
| normpar 31360 | Parallelogram law for norm... |
| normpar2i 31361 | Corollary of parallelogram... |
| polid2i 31362 | Generalized polarization i... |
| polidi 31363 | Polarization identity. Re... |
| polid 31364 | Polarization identity. Re... |
| hilablo 31365 | Hilbert space vector addit... |
| hilid 31366 | The group identity element... |
| hilvc 31367 | Hilbert space is a complex... |
| hilnormi 31368 | Hilbert space norm in term... |
| hilhhi 31369 | Deduce the structure of Hi... |
| hhnv 31370 | Hilbert space is a normed ... |
| hhva 31371 | The group (addition) opera... |
| hhba 31372 | The base set of Hilbert sp... |
| hh0v 31373 | The zero vector of Hilbert... |
| hhsm 31374 | The scalar product operati... |
| hhvs 31375 | The vector subtraction ope... |
| hhnm 31376 | The norm function of Hilbe... |
| hhims 31377 | The induced metric of Hilb... |
| hhims2 31378 | Hilbert space distance met... |
| hhmet 31379 | The induced metric of Hilb... |
| hhxmet 31380 | The induced metric of Hilb... |
| hhmetdval 31381 | Value of the distance func... |
| hhip 31382 | The inner product operatio... |
| hhph 31383 | The Hilbert space of the H... |
| bcsiALT 31384 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 31385 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 31386 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 31387 | Corollary of the Bunjakova... |
| bcs3 31388 | Corollary of the Bunjakova... |
| hcau 31389 | Member of the set of Cauch... |
| hcauseq 31390 | A Cauchy sequences on a Hi... |
| hcaucvg 31391 | A Cauchy sequence on a Hil... |
| seq1hcau 31392 | A sequence on a Hilbert sp... |
| hlimi 31393 | Express the predicate: Th... |
| hlimseqi 31394 | A sequence with a limit on... |
| hlimveci 31395 | Closure of the limit of a ... |
| hlimconvi 31396 | Convergence of a sequence ... |
| hlim2 31397 | The limit of a sequence on... |
| hlimadd 31398 | Limit of the sum of two se... |
| hilmet 31399 | The Hilbert space norm det... |
| hilxmet 31400 | The Hilbert space norm det... |
| hilmetdval 31401 | Value of the distance func... |
| hilims 31402 | Hilbert space distance met... |
| hhcau 31403 | The Cauchy sequences of Hi... |
| hhlm 31404 | The limit sequences of Hil... |
| hhcmpl 31405 | Lemma used for derivation ... |
| hilcompl 31406 | Lemma used for derivation ... |
| hhcms 31408 | The Hilbert space induced ... |
| hhhl 31409 | The Hilbert space structur... |
| hilcms 31410 | The Hilbert space norm det... |
| hilhl 31411 | The Hilbert space of the H... |
| issh 31413 | Subspace ` H ` of a Hilber... |
| issh2 31414 | Subspace ` H ` of a Hilber... |
| shss 31415 | A subspace is a subset of ... |
| shel 31416 | A member of a subspace of ... |
| shex 31417 | The set of subspaces of a ... |
| shssii 31418 | A closed subspace of a Hil... |
| sheli 31419 | A member of a subspace of ... |
| shelii 31420 | A member of a subspace of ... |
| sh0 31421 | The zero vector belongs to... |
| shaddcl 31422 | Closure of vector addition... |
| shmulcl 31423 | Closure of vector scalar m... |
| issh3 31424 | Subspace ` H ` of a Hilber... |
| shsubcl 31425 | Closure of vector subtract... |
| isch 31427 | Closed subspace ` H ` of a... |
| isch2 31428 | Closed subspace ` H ` of a... |
| chsh 31429 | A closed subspace is a sub... |
| chsssh 31430 | Closed subspaces are subsp... |
| chex 31431 | The set of closed subspace... |
| chshii 31432 | A closed subspace is a sub... |
| ch0 31433 | The zero vector belongs to... |
| chss 31434 | A closed subspace of a Hil... |
| chel 31435 | A member of a closed subsp... |
| chssii 31436 | A closed subspace of a Hil... |
| cheli 31437 | A member of a closed subsp... |
| chelii 31438 | A member of a closed subsp... |
| chlimi 31439 | The limit property of a cl... |
| hlim0 31440 | The zero sequence in Hilbe... |
| hlimcaui 31441 | If a sequence in Hilbert s... |
| hlimf 31442 | Function-like behavior of ... |
| hlimuni 31443 | A Hilbert space sequence c... |
| hlimreui 31444 | The limit of a Hilbert spa... |
| hlimeui 31445 | The limit of a Hilbert spa... |
| isch3 31446 | A Hilbert subspace is clos... |
| chcompl 31447 | Completeness of a closed s... |
| helch 31448 | The Hilbert lattice one (w... |
| ifchhv 31449 | Prove ` if ( A e. CH , A ,... |
| helsh 31450 | Hilbert space is a subspac... |
| shsspwh 31451 | Subspaces are subsets of H... |
| chsspwh 31452 | Closed subspaces are subse... |
| hsn0elch 31453 | The zero subspace belongs ... |
| norm1 31454 | From any nonzero Hilbert s... |
| norm1exi 31455 | A normalized vector exists... |
| norm1hex 31456 | A normalized vector can ex... |
| elch0 31459 | Membership in zero for clo... |
| h0elch 31460 | The zero subspace is a clo... |
| h0elsh 31461 | The zero subspace is a sub... |
| hhssva 31462 | The vector addition operat... |
| hhsssm 31463 | The scalar multiplication ... |
| hhssnm 31464 | The norm operation on a su... |
| issubgoilem 31465 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31466 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31467 | Abelian group property of ... |
| hhssablo 31468 | Abelian group property of ... |
| hhssnv 31469 | Normed complex vector spac... |
| hhssnvt 31470 | Normed complex vector spac... |
| hhsst 31471 | A member of ` SH ` is a su... |
| hhshsslem1 31472 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31473 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31474 | The predicate " ` H ` is a... |
| hhsssh2 31475 | The predicate " ` H ` is a... |
| hhssba 31476 | The base set of a subspace... |
| hhssvs 31477 | The vector subtraction ope... |
| hhssvsf 31478 | Mapping of the vector subt... |
| hhssims 31479 | Induced metric of a subspa... |
| hhssims2 31480 | Induced metric of a subspa... |
| hhssmet 31481 | Induced metric of a subspa... |
| hhssmetdval 31482 | Value of the distance func... |
| hhsscms 31483 | The induced metric of a cl... |
| hhssbnOLD 31484 | Obsolete version of ~ cssb... |
| ocval 31485 | Value of orthogonal comple... |
| ocel 31486 | Membership in orthogonal c... |
| shocel 31487 | Membership in orthogonal c... |
| ocsh 31488 | The orthogonal complement ... |
| shocsh 31489 | The orthogonal complement ... |
| ocss 31490 | An orthogonal complement i... |
| shocss 31491 | An orthogonal complement i... |
| occon 31492 | Contraposition law for ort... |
| occon2 31493 | Double contraposition for ... |
| occon2i 31494 | Double contraposition for ... |
| oc0 31495 | The zero vector belongs to... |
| ocorth 31496 | Members of a subset and it... |
| shocorth 31497 | Members of a subspace and ... |
| ococss 31498 | Inclusion in complement of... |
| shococss 31499 | Inclusion in complement of... |
| shorth 31500 | Members of orthogonal subs... |
| ocin 31501 | Intersection of a Hilbert ... |
| occon3 31502 | Hilbert lattice contraposi... |
| ocnel 31503 | A nonzero vector in the co... |
| chocvali 31504 | Value of the orthogonal co... |
| shuni 31505 | Two subspaces with trivial... |
| chocunii 31506 | Lemma for uniqueness part ... |
| pjhthmo 31507 | Projection Theorem, unique... |
| occllem 31508 | Lemma for ~ occl . (Contr... |
| occl 31509 | Closure of complement of H... |
| shoccl 31510 | Closure of complement of H... |
| choccl 31511 | Closure of complement of H... |
| choccli 31512 | Closure of ` CH ` orthocom... |
| shsval 31517 | Value of subspace sum of t... |
| shsss 31518 | The subspace sum is a subs... |
| shsel 31519 | Membership in the subspace... |
| shsel3 31520 | Membership in the subspace... |
| shseli 31521 | Membership in subspace sum... |
| shscli 31522 | Closure of subspace sum. ... |
| shscl 31523 | Closure of subspace sum. ... |
| shscom 31524 | Commutative law for subspa... |
| shsva 31525 | Vector sum belongs to subs... |
| shsel1 31526 | A subspace sum contains a ... |
| shsel2 31527 | A subspace sum contains a ... |
| shsvs 31528 | Vector subtraction belongs... |
| shsub1 31529 | Subspace sum is an upper b... |
| shsub2 31530 | Subspace sum is an upper b... |
| choc0 31531 | The orthocomplement of the... |
| choc1 31532 | The orthocomplement of the... |
| chocnul 31533 | Orthogonal complement of t... |
| shintcli 31534 | Closure of intersection of... |
| shintcl 31535 | The intersection of a none... |
| chintcli 31536 | The intersection of a none... |
| chintcl 31537 | The intersection (infimum)... |
| spanval 31538 | Value of the linear span o... |
| hsupval 31539 | Value of supremum of set o... |
| chsupval 31540 | The value of the supremum ... |
| spancl 31541 | The span of a subset of Hi... |
| elspancl 31542 | A member of a span is a ve... |
| shsupcl 31543 | Closure of the subspace su... |
| hsupcl 31544 | Closure of supremum of set... |
| chsupcl 31545 | Closure of supremum of sub... |
| hsupss 31546 | Subset relation for suprem... |
| chsupss 31547 | Subset relation for suprem... |
| hsupunss 31548 | The union of a set of Hilb... |
| chsupunss 31549 | The union of a set of clos... |
| spanss2 31550 | A subset of Hilbert space ... |
| shsupunss 31551 | The union of a set of subs... |
| spanid 31552 | A subspace of Hilbert spac... |
| spanss 31553 | Ordering relationship for ... |
| spanssoc 31554 | The span of a subset of Hi... |
| sshjval 31555 | Value of join for subsets ... |
| shjval 31556 | Value of join in ` SH ` . ... |
| chjval 31557 | Value of join in ` CH ` . ... |
| chjvali 31558 | Value of join in ` CH ` . ... |
| sshjval3 31559 | Value of join for subsets ... |
| sshjcl 31560 | Closure of join for subset... |
| shjcl 31561 | Closure of join in ` SH ` ... |
| chjcl 31562 | Closure of join in ` CH ` ... |
| shjcom 31563 | Commutative law for Hilber... |
| shless 31564 | Subset implies subset of s... |
| shlej1 31565 | Add disjunct to both sides... |
| shlej2 31566 | Add disjunct to both sides... |
| shincli 31567 | Closure of intersection of... |
| shscomi 31568 | Commutative law for subspa... |
| shsvai 31569 | Vector sum belongs to subs... |
| shsel1i 31570 | A subspace sum contains a ... |
| shsel2i 31571 | A subspace sum contains a ... |
| shsvsi 31572 | Vector subtraction belongs... |
| shunssi 31573 | Union is smaller than subs... |
| shunssji 31574 | Union is smaller than Hilb... |
| shsleji 31575 | Subspace sum is smaller th... |
| shjcomi 31576 | Commutative law for join i... |
| shsub1i 31577 | Subspace sum is an upper b... |
| shsub2i 31578 | Subspace sum is an upper b... |
| shub1i 31579 | Hilbert lattice join is an... |
| shjcli 31580 | Closure of ` CH ` join. (... |
| shjshcli 31581 | ` SH ` closure of join. (... |
| shlessi 31582 | Subset implies subset of s... |
| shlej1i 31583 | Add disjunct to both sides... |
| shlej2i 31584 | Add disjunct to both sides... |
| shslej 31585 | Subspace sum is smaller th... |
| shincl 31586 | Closure of intersection of... |
| shub1 31587 | Hilbert lattice join is an... |
| shub2 31588 | A subspace is a subset of ... |
| shsidmi 31589 | Idempotent law for Hilbert... |
| shslubi 31590 | The least upper bound law ... |
| shlesb1i 31591 | Hilbert lattice ordering i... |
| shsval2i 31592 | An alternate way to expres... |
| shsval3i 31593 | An alternate way to expres... |
| shmodsi 31594 | The modular law holds for ... |
| shmodi 31595 | The modular law is implied... |
| pjhthlem1 31596 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 31597 | Lemma for ~ pjhth . (Cont... |
| pjhth 31598 | Projection Theorem: Any H... |
| pjhtheu 31599 | Projection Theorem: Any H... |
| pjhfval 31601 | The value of the projectio... |
| pjhval 31602 | Value of a projection. (C... |
| pjpreeq 31603 | Equality with a projection... |
| pjeq 31604 | Equality with a projection... |
| axpjcl 31605 | Closure of a projection in... |
| pjhcl 31606 | Closure of a projection in... |
| omlsilem 31607 | Lemma for orthomodular law... |
| omlsii 31608 | Subspace inference form of... |
| omlsi 31609 | Subspace form of orthomodu... |
| ococi 31610 | Complement of complement o... |
| ococ 31611 | Complement of complement o... |
| dfch2 31612 | Alternate definition of th... |
| ococin 31613 | The double complement is t... |
| hsupval2 31614 | Alternate definition of su... |
| chsupval2 31615 | The value of the supremum ... |
| sshjval2 31616 | Value of join in the set o... |
| chsupid 31617 | A subspace is the supremum... |
| chsupsn 31618 | Value of supremum of subse... |
| shlub 31619 | Hilbert lattice join is th... |
| shlubi 31620 | Hilbert lattice join is th... |
| pjhtheu2 31621 | Uniqueness of ` y ` for th... |
| pjcli 31622 | Closure of a projection in... |
| pjhcli 31623 | Closure of a projection in... |
| pjpjpre 31624 | Decomposition of a vector ... |
| axpjpj 31625 | Decomposition of a vector ... |
| pjclii 31626 | Closure of a projection in... |
| pjhclii 31627 | Closure of a projection in... |
| pjpj0i 31628 | Decomposition of a vector ... |
| pjpji 31629 | Decomposition of a vector ... |
| pjpjhth 31630 | Projection Theorem: Any H... |
| pjpjhthi 31631 | Projection Theorem: Any H... |
| pjop 31632 | Orthocomplement projection... |
| pjpo 31633 | Projection in terms of ort... |
| pjopi 31634 | Orthocomplement projection... |
| pjpoi 31635 | Projection in terms of ort... |
| pjoc1i 31636 | Projection of a vector in ... |
| pjchi 31637 | Projection of a vector in ... |
| pjoccl 31638 | The part of a vector that ... |
| pjoc1 31639 | Projection of a vector in ... |
| pjomli 31640 | Subspace form of orthomodu... |
| pjoml 31641 | Subspace form of orthomodu... |
| pjococi 31642 | Proof of orthocomplement t... |
| pjoc2i 31643 | Projection of a vector in ... |
| pjoc2 31644 | Projection of a vector in ... |
| sh0le 31645 | The zero subspace is the s... |
| ch0le 31646 | The zero subspace is the s... |
| shle0 31647 | No subspace is smaller tha... |
| chle0 31648 | No Hilbert lattice element... |
| chnlen0 31649 | A Hilbert lattice element ... |
| ch0pss 31650 | The zero subspace is a pro... |
| orthin 31651 | The intersection of orthog... |
| ssjo 31652 | The lattice join of a subs... |
| shne0i 31653 | A nonzero subspace has a n... |
| shs0i 31654 | Hilbert subspace sum with ... |
| shs00i 31655 | Two subspaces are zero iff... |
| ch0lei 31656 | The closed subspace zero i... |
| chle0i 31657 | No Hilbert closed subspace... |
| chne0i 31658 | A nonzero closed subspace ... |
| chocini 31659 | Intersection of a closed s... |
| chj0i 31660 | Join with lattice zero in ... |
| chm1i 31661 | Meet with lattice one in `... |
| chjcli 31662 | Closure of ` CH ` join. (... |
| chsleji 31663 | Subspace sum is smaller th... |
| chseli 31664 | Membership in subspace sum... |
| chincli 31665 | Closure of Hilbert lattice... |
| chsscon3i 31666 | Hilbert lattice contraposi... |
| chsscon1i 31667 | Hilbert lattice contraposi... |
| chsscon2i 31668 | Hilbert lattice contraposi... |
| chcon2i 31669 | Hilbert lattice contraposi... |
| chcon1i 31670 | Hilbert lattice contraposi... |
| chcon3i 31671 | Hilbert lattice contraposi... |
| chunssji 31672 | Union is smaller than ` CH... |
| chjcomi 31673 | Commutative law for join i... |
| chub1i 31674 | ` CH ` join is an upper bo... |
| chub2i 31675 | ` CH ` join is an upper bo... |
| chlubi 31676 | Hilbert lattice join is th... |
| chlubii 31677 | Hilbert lattice join is th... |
| chlej1i 31678 | Add join to both sides of ... |
| chlej2i 31679 | Add join to both sides of ... |
| chlej12i 31680 | Add join to both sides of ... |
| chlejb1i 31681 | Hilbert lattice ordering i... |
| chdmm1i 31682 | De Morgan's law for meet i... |
| chdmm2i 31683 | De Morgan's law for meet i... |
| chdmm3i 31684 | De Morgan's law for meet i... |
| chdmm4i 31685 | De Morgan's law for meet i... |
| chdmj1i 31686 | De Morgan's law for join i... |
| chdmj2i 31687 | De Morgan's law for join i... |
| chdmj3i 31688 | De Morgan's law for join i... |
| chdmj4i 31689 | De Morgan's law for join i... |
| chnlei 31690 | Equivalent expressions for... |
| chjassi 31691 | Associative law for Hilber... |
| chj00i 31692 | Two Hilbert lattice elemen... |
| chjoi 31693 | The join of a closed subsp... |
| chj1i 31694 | Join with Hilbert lattice ... |
| chm0i 31695 | Meet with Hilbert lattice ... |
| chm0 31696 | Meet with Hilbert lattice ... |
| shjshsi 31697 | Hilbert lattice join equal... |
| shjshseli 31698 | A closed subspace sum equa... |
| chne0 31699 | A nonzero closed subspace ... |
| chocin 31700 | Intersection of a closed s... |
| chssoc 31701 | A closed subspace less tha... |
| chj0 31702 | Join with Hilbert lattice ... |
| chslej 31703 | Subspace sum is smaller th... |
| chincl 31704 | Closure of Hilbert lattice... |
| chsscon3 31705 | Hilbert lattice contraposi... |
| chsscon1 31706 | Hilbert lattice contraposi... |
| chsscon2 31707 | Hilbert lattice contraposi... |
| chpsscon3 31708 | Hilbert lattice contraposi... |
| chpsscon1 31709 | Hilbert lattice contraposi... |
| chpsscon2 31710 | Hilbert lattice contraposi... |
| chjcom 31711 | Commutative law for Hilber... |
| chub1 31712 | Hilbert lattice join is gr... |
| chub2 31713 | Hilbert lattice join is gr... |
| chlub 31714 | Hilbert lattice join is th... |
| chlej1 31715 | Add join to both sides of ... |
| chlej2 31716 | Add join to both sides of ... |
| chlejb1 31717 | Hilbert lattice ordering i... |
| chlejb2 31718 | Hilbert lattice ordering i... |
| chnle 31719 | Equivalent expressions for... |
| chjo 31720 | The join of a closed subsp... |
| chabs1 31721 | Hilbert lattice absorption... |
| chabs2 31722 | Hilbert lattice absorption... |
| chabs1i 31723 | Hilbert lattice absorption... |
| chabs2i 31724 | Hilbert lattice absorption... |
| chjidm 31725 | Idempotent law for Hilbert... |
| chjidmi 31726 | Idempotent law for Hilbert... |
| chj12i 31727 | A rearrangement of Hilbert... |
| chj4i 31728 | Rearrangement of the join ... |
| chjjdiri 31729 | Hilbert lattice join distr... |
| chdmm1 31730 | De Morgan's law for meet i... |
| chdmm2 31731 | De Morgan's law for meet i... |
| chdmm3 31732 | De Morgan's law for meet i... |
| chdmm4 31733 | De Morgan's law for meet i... |
| chdmj1 31734 | De Morgan's law for join i... |
| chdmj2 31735 | De Morgan's law for join i... |
| chdmj3 31736 | De Morgan's law for join i... |
| chdmj4 31737 | De Morgan's law for join i... |
| chjass 31738 | Associative law for Hilber... |
| chj12 31739 | A rearrangement of Hilbert... |
| chj4 31740 | Rearrangement of the join ... |
| ledii 31741 | An ortholattice is distrib... |
| lediri 31742 | An ortholattice is distrib... |
| lejdii 31743 | An ortholattice is distrib... |
| lejdiri 31744 | An ortholattice is distrib... |
| ledi 31745 | An ortholattice is distrib... |
| spansn0 31746 | The span of the singleton ... |
| span0 31747 | The span of the empty set ... |
| elspani 31748 | Membership in the span of ... |
| spanuni 31749 | The span of a union is the... |
| spanun 31750 | The span of a union is the... |
| sshhococi 31751 | The join of two Hilbert sp... |
| hne0 31752 | Hilbert space has a nonzer... |
| chsup0 31753 | The supremum of the empty ... |
| h1deoi 31754 | Membership in orthocomplem... |
| h1dei 31755 | Membership in 1-dimensiona... |
| h1did 31756 | A generating vector belong... |
| h1dn0 31757 | A nonzero vector generates... |
| h1de2i 31758 | Membership in 1-dimensiona... |
| h1de2bi 31759 | Membership in 1-dimensiona... |
| h1de2ctlem 31760 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 31761 | Membership in 1-dimensiona... |
| spansni 31762 | The span of a singleton in... |
| elspansni 31763 | Membership in the span of ... |
| spansn 31764 | The span of a singleton in... |
| spansnch 31765 | The span of a Hilbert spac... |
| spansnsh 31766 | The span of a Hilbert spac... |
| spansnchi 31767 | The span of a singleton in... |
| spansnid 31768 | A vector belongs to the sp... |
| spansnmul 31769 | A scalar product with a ve... |
| elspansncl 31770 | A member of a span of a si... |
| elspansn 31771 | Membership in the span of ... |
| elspansn2 31772 | Membership in the span of ... |
| spansncol 31773 | The singletons of collinea... |
| spansneleqi 31774 | Membership relation implie... |
| spansneleq 31775 | Membership relation that i... |
| spansnss 31776 | The span of the singleton ... |
| elspansn3 31777 | A member of the span of th... |
| elspansn4 31778 | A span membership conditio... |
| elspansn5 31779 | A vector belonging to both... |
| spansnss2 31780 | The span of the singleton ... |
| normcan 31781 | Cancellation-type law that... |
| pjspansn 31782 | A projection on the span o... |
| spansnpji 31783 | A subset of Hilbert space ... |
| spanunsni 31784 | The span of the union of a... |
| spanpr 31785 | The span of a pair of vect... |
| h1datomi 31786 | A 1-dimensional subspace i... |
| h1datom 31787 | A 1-dimensional subspace i... |
| cmbr 31789 | Binary relation expressing... |
| pjoml2i 31790 | Variation of orthomodular ... |
| pjoml3i 31791 | Variation of orthomodular ... |
| pjoml4i 31792 | Variation of orthomodular ... |
| pjoml5i 31793 | The orthomodular law. Rem... |
| pjoml6i 31794 | An equivalent of the ortho... |
| cmbri 31795 | Binary relation expressing... |
| cmcmlem 31796 | Commutation is symmetric. ... |
| cmcmi 31797 | Commutation is symmetric. ... |
| cmcm2i 31798 | Commutation with orthocomp... |
| cmcm3i 31799 | Commutation with orthocomp... |
| cmcm4i 31800 | Commutation with orthocomp... |
| cmbr2i 31801 | Alternate definition of th... |
| cmcmii 31802 | Commutation is symmetric. ... |
| cmcm2ii 31803 | Commutation with orthocomp... |
| cmcm3ii 31804 | Commutation with orthocomp... |
| cmbr3i 31805 | Alternate definition for t... |
| cmbr4i 31806 | Alternate definition for t... |
| lecmi 31807 | Comparable Hilbert lattice... |
| lecmii 31808 | Comparable Hilbert lattice... |
| cmj1i 31809 | A Hilbert lattice element ... |
| cmj2i 31810 | A Hilbert lattice element ... |
| cmm1i 31811 | A Hilbert lattice element ... |
| cmm2i 31812 | A Hilbert lattice element ... |
| cmbr3 31813 | Alternate definition for t... |
| cm0 31814 | The zero Hilbert lattice e... |
| cmidi 31815 | The commutes relation is r... |
| pjoml2 31816 | Variation of orthomodular ... |
| pjoml3 31817 | Variation of orthomodular ... |
| pjoml5 31818 | The orthomodular law. Rem... |
| cmcm 31819 | Commutation is symmetric. ... |
| cmcm3 31820 | Commutation with orthocomp... |
| cmcm2 31821 | Commutation with orthocomp... |
| lecm 31822 | Comparable Hilbert lattice... |
| fh1 31823 | Foulis-Holland Theorem. I... |
| fh2 31824 | Foulis-Holland Theorem. I... |
| cm2j 31825 | A lattice element that com... |
| fh1i 31826 | Foulis-Holland Theorem. I... |
| fh2i 31827 | Foulis-Holland Theorem. I... |
| fh3i 31828 | Variation of the Foulis-Ho... |
| fh4i 31829 | Variation of the Foulis-Ho... |
| cm2ji 31830 | A lattice element that com... |
| cm2mi 31831 | A lattice element that com... |
| qlax1i 31832 | One of the equations showi... |
| qlax2i 31833 | One of the equations showi... |
| qlax3i 31834 | One of the equations showi... |
| qlax4i 31835 | One of the equations showi... |
| qlax5i 31836 | One of the equations showi... |
| qlaxr1i 31837 | One of the conditions show... |
| qlaxr2i 31838 | One of the conditions show... |
| qlaxr4i 31839 | One of the conditions show... |
| qlaxr5i 31840 | One of the conditions show... |
| qlaxr3i 31841 | A variation of the orthomo... |
| chscllem1 31842 | Lemma for ~ chscl . (Cont... |
| chscllem2 31843 | Lemma for ~ chscl . (Cont... |
| chscllem3 31844 | Lemma for ~ chscl . (Cont... |
| chscllem4 31845 | Lemma for ~ chscl . (Cont... |
| chscl 31846 | The subspace sum of two cl... |
| osumi 31847 | If two closed subspaces of... |
| osumcori 31848 | Corollary of ~ osumi . (C... |
| osumcor2i 31849 | Corollary of ~ osumi , sho... |
| osum 31850 | If two closed subspaces of... |
| spansnji 31851 | The subspace sum of a clos... |
| spansnj 31852 | The subspace sum of a clos... |
| spansnscl 31853 | The subspace sum of a clos... |
| sumspansn 31854 | The sum of two vectors bel... |
| spansnm0i 31855 | The meet of different one-... |
| nonbooli 31856 | A Hilbert lattice with two... |
| spansncvi 31857 | Hilbert space has the cove... |
| spansncv 31858 | Hilbert space has the cove... |
| 5oalem1 31859 | Lemma for orthoarguesian l... |
| 5oalem2 31860 | Lemma for orthoarguesian l... |
| 5oalem3 31861 | Lemma for orthoarguesian l... |
| 5oalem4 31862 | Lemma for orthoarguesian l... |
| 5oalem5 31863 | Lemma for orthoarguesian l... |
| 5oalem6 31864 | Lemma for orthoarguesian l... |
| 5oalem7 31865 | Lemma for orthoarguesian l... |
| 5oai 31866 | Orthoarguesian law 5OA. Th... |
| 3oalem1 31867 | Lemma for 3OA (weak) ortho... |
| 3oalem2 31868 | Lemma for 3OA (weak) ortho... |
| 3oalem3 31869 | Lemma for 3OA (weak) ortho... |
| 3oalem4 31870 | Lemma for 3OA (weak) ortho... |
| 3oalem5 31871 | Lemma for 3OA (weak) ortho... |
| 3oalem6 31872 | Lemma for 3OA (weak) ortho... |
| 3oai 31873 | 3OA (weak) orthoarguesian ... |
| pjorthi 31874 | Projection components on o... |
| pjch1 31875 | Property of identity proje... |
| pjo 31876 | The orthogonal projection.... |
| pjcompi 31877 | Component of a projection.... |
| pjidmi 31878 | A projection is idempotent... |
| pjadjii 31879 | A projection is self-adjoi... |
| pjaddii 31880 | Projection of vector sum i... |
| pjinormii 31881 | The inner product of a pro... |
| pjmulii 31882 | Projection of (scalar) pro... |
| pjsubii 31883 | Projection of vector diffe... |
| pjsslem 31884 | Lemma for subset relations... |
| pjss2i 31885 | Subset relationship for pr... |
| pjssmii 31886 | Projection meet property. ... |
| pjssge0ii 31887 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 31888 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 31889 | The projection on a subspa... |
| pjadji 31890 | A projection is self-adjoi... |
| pjaddi 31891 | Projection of vector sum i... |
| pjinormi 31892 | The inner product of a pro... |
| pjsubi 31893 | Projection of vector diffe... |
| pjmuli 31894 | Projection of scalar produ... |
| pjige0i 31895 | The inner product of a pro... |
| pjige0 31896 | The inner product of a pro... |
| pjcjt2 31897 | The projection on a subspa... |
| pj0i 31898 | The projection of the zero... |
| pjch 31899 | Projection of a vector in ... |
| pjid 31900 | The projection of a vector... |
| pjvec 31901 | The set of vectors belongi... |
| pjocvec 31902 | The set of vectors belongi... |
| pjocini 31903 | Membership of projection i... |
| pjini 31904 | Membership of projection i... |
| pjjsi 31905 | A sufficient condition for... |
| pjfni 31906 | Functionality of a project... |
| pjrni 31907 | The range of a projection.... |
| pjfoi 31908 | A projection maps onto its... |
| pjfi 31909 | The mapping of a projectio... |
| pjvi 31910 | The value of a projection ... |
| pjhfo 31911 | A projection maps onto its... |
| pjrn 31912 | The range of a projection.... |
| pjhf 31913 | The mapping of a projectio... |
| pjfn 31914 | Functionality of a project... |
| pjsumi 31915 | The projection on a subspa... |
| pj11i 31916 | One-to-one correspondence ... |
| pjdsi 31917 | Vector decomposition into ... |
| pjds3i 31918 | Vector decomposition into ... |
| pj11 31919 | One-to-one correspondence ... |
| pjmfn 31920 | Functionality of the proje... |
| pjmf1 31921 | The projector function map... |
| pjoi0 31922 | The inner product of proje... |
| pjoi0i 31923 | The inner product of proje... |
| pjopythi 31924 | Pythagorean theorem for pr... |
| pjopyth 31925 | Pythagorean theorem for pr... |
| pjnormi 31926 | The norm of the projection... |
| pjpythi 31927 | Pythagorean theorem for pr... |
| pjneli 31928 | If a vector does not belon... |
| pjnorm 31929 | The norm of the projection... |
| pjpyth 31930 | Pythagorean theorem for pr... |
| pjnel 31931 | If a vector does not belon... |
| pjnorm2 31932 | A vector belongs to the su... |
| mayete3i 31933 | Mayet's equation E_3. Par... |
| mayetes3i 31934 | Mayet's equation E^*_3, de... |
| hosmval 31940 | Value of the sum of two Hi... |
| hommval 31941 | Value of the scalar produc... |
| hodmval 31942 | Value of the difference of... |
| hfsmval 31943 | Value of the sum of two Hi... |
| hfmmval 31944 | Value of the scalar produc... |
| hosval 31945 | Value of the sum of two Hi... |
| homval 31946 | Value of the scalar produc... |
| hodval 31947 | Value of the difference of... |
| hfsval 31948 | Value of the sum of two Hi... |
| hfmval 31949 | Value of the scalar produc... |
| hoscl 31950 | Closure of the sum of two ... |
| homcl 31951 | Closure of the scalar prod... |
| hodcl 31952 | Closure of the difference ... |
| ho0val 31955 | Value of the zero Hilbert ... |
| ho0f 31956 | Functionality of the zero ... |
| df0op2 31957 | Alternate definition of Hi... |
| dfiop2 31958 | Alternate definition of Hi... |
| hoif 31959 | Functionality of the Hilbe... |
| hoival 31960 | The value of the Hilbert s... |
| hoico1 31961 | Composition with the Hilbe... |
| hoico2 31962 | Composition with the Hilbe... |
| hoaddcl 31963 | The sum of Hilbert space o... |
| homulcl 31964 | The scalar product of a Hi... |
| hoeq 31965 | Equality of Hilbert space ... |
| hoeqi 31966 | Equality of Hilbert space ... |
| hoscli 31967 | Closure of Hilbert space o... |
| hodcli 31968 | Closure of Hilbert space o... |
| hocoi 31969 | Composition of Hilbert spa... |
| hococli 31970 | Closure of composition of ... |
| hocofi 31971 | Mapping of composition of ... |
| hocofni 31972 | Functionality of compositi... |
| hoaddcli 31973 | Mapping of sum of Hilbert ... |
| hosubcli 31974 | Mapping of difference of H... |
| hoaddfni 31975 | Functionality of sum of Hi... |
| hosubfni 31976 | Functionality of differenc... |
| hoaddcomi 31977 | Commutativity of sum of Hi... |
| hosubcl 31978 | Mapping of difference of H... |
| hoaddcom 31979 | Commutativity of sum of Hi... |
| hodsi 31980 | Relationship between Hilbe... |
| hoaddassi 31981 | Associativity of sum of Hi... |
| hoadd12i 31982 | Commutative/associative la... |
| hoadd32i 31983 | Commutative/associative la... |
| hocadddiri 31984 | Distributive law for Hilbe... |
| hocsubdiri 31985 | Distributive law for Hilbe... |
| ho2coi 31986 | Double composition of Hilb... |
| hoaddass 31987 | Associativity of sum of Hi... |
| hoadd32 31988 | Commutative/associative la... |
| hoadd4 31989 | Rearrangement of 4 terms i... |
| hocsubdir 31990 | Distributive law for Hilbe... |
| hoaddridi 31991 | Sum of a Hilbert space ope... |
| hodidi 31992 | Difference of a Hilbert sp... |
| ho0coi 31993 | Composition of the zero op... |
| hoid1i 31994 | Composition of Hilbert spa... |
| hoid1ri 31995 | Composition of Hilbert spa... |
| hoaddrid 31996 | Sum of a Hilbert space ope... |
| hodid 31997 | Difference of a Hilbert sp... |
| hon0 31998 | A Hilbert space operator i... |
| hodseqi 31999 | Subtraction and addition o... |
| ho0subi 32000 | Subtraction of Hilbert spa... |
| honegsubi 32001 | Relationship between Hilbe... |
| ho0sub 32002 | Subtraction of Hilbert spa... |
| hosubid1 32003 | The zero operator subtract... |
| honegsub 32004 | Relationship between Hilbe... |
| homullid 32005 | An operator equals its sca... |
| homco1 32006 | Associative law for scalar... |
| homulass 32007 | Scalar product associative... |
| hoadddi 32008 | Scalar product distributiv... |
| hoadddir 32009 | Scalar product reverse dis... |
| homul12 32010 | Swap first and second fact... |
| honegneg 32011 | Double negative of a Hilbe... |
| hosubneg 32012 | Relationship between opera... |
| hosubdi 32013 | Scalar product distributiv... |
| honegdi 32014 | Distribution of negative o... |
| honegsubdi 32015 | Distribution of negative o... |
| honegsubdi2 32016 | Distribution of negative o... |
| hosubsub2 32017 | Law for double subtraction... |
| hosub4 32018 | Rearrangement of 4 terms i... |
| hosubadd4 32019 | Rearrangement of 4 terms i... |
| hoaddsubass 32020 | Associative-type law for a... |
| hoaddsub 32021 | Law for operator addition ... |
| hosubsub 32022 | Law for double subtraction... |
| hosubsub4 32023 | Law for double subtraction... |
| ho2times 32024 | Two times a Hilbert space ... |
| hoaddsubassi 32025 | Associativity of sum and d... |
| hoaddsubi 32026 | Law for sum and difference... |
| hosd1i 32027 | Hilbert space operator sum... |
| hosd2i 32028 | Hilbert space operator sum... |
| hopncani 32029 | Hilbert space operator can... |
| honpcani 32030 | Hilbert space operator can... |
| hosubeq0i 32031 | If the difference between ... |
| honpncani 32032 | Hilbert space operator can... |
| ho01i 32033 | A condition implying that ... |
| ho02i 32034 | A condition implying that ... |
| hoeq1 32035 | A condition implying that ... |
| hoeq2 32036 | A condition implying that ... |
| adjmo 32037 | Every Hilbert space operat... |
| adjsym 32038 | Symmetry property of an ad... |
| eigrei 32039 | A necessary and sufficient... |
| eigre 32040 | A necessary and sufficient... |
| eigposi 32041 | A sufficient condition (fi... |
| eigorthi 32042 | A necessary and sufficient... |
| eigorth 32043 | A necessary and sufficient... |
| nmopval 32061 | Value of the norm of a Hil... |
| elcnop 32062 | Property defining a contin... |
| ellnop 32063 | Property defining a linear... |
| lnopf 32064 | A linear Hilbert space ope... |
| elbdop 32065 | Property defining a bounde... |
| bdopln 32066 | A bounded linear Hilbert s... |
| bdopf 32067 | A bounded linear Hilbert s... |
| nmopsetretALT 32068 | The set in the supremum of... |
| nmopsetretHIL 32069 | The set in the supremum of... |
| nmopsetn0 32070 | The set in the supremum of... |
| nmopxr 32071 | The norm of a Hilbert spac... |
| nmoprepnf 32072 | The norm of a Hilbert spac... |
| nmopgtmnf 32073 | The norm of a Hilbert spac... |
| nmopreltpnf 32074 | The norm of a Hilbert spac... |
| nmopre 32075 | The norm of a bounded oper... |
| elbdop2 32076 | Property defining a bounde... |
| elunop 32077 | Property defining a unitar... |
| elhmop 32078 | Property defining a Hermit... |
| hmopf 32079 | A Hermitian operator is a ... |
| hmopex 32080 | The class of Hermitian ope... |
| nmfnval 32081 | Value of the norm of a Hil... |
| nmfnsetre 32082 | The set in the supremum of... |
| nmfnsetn0 32083 | The set in the supremum of... |
| nmfnxr 32084 | The norm of any Hilbert sp... |
| nmfnrepnf 32085 | The norm of a Hilbert spac... |
| nlfnval 32086 | Value of the null space of... |
| elcnfn 32087 | Property defining a contin... |
| ellnfn 32088 | Property defining a linear... |
| lnfnf 32089 | A linear Hilbert space fun... |
| dfadj2 32090 | Alternate definition of th... |
| funadj 32091 | Functionality of the adjoi... |
| dmadjss 32092 | The domain of the adjoint ... |
| dmadjop 32093 | A member of the domain of ... |
| adjeu 32094 | Elementhood in the domain ... |
| adjval 32095 | Value of the adjoint funct... |
| adjval2 32096 | Value of the adjoint funct... |
| cnvadj 32097 | The adjoint function equal... |
| funcnvadj 32098 | The converse of the adjoin... |
| adj1o 32099 | The adjoint function maps ... |
| dmadjrn 32100 | The adjoint of an operator... |
| eigvecval 32101 | The set of eigenvectors of... |
| eigvalfval 32102 | The eigenvalues of eigenve... |
| specval 32103 | The value of the spectrum ... |
| speccl 32104 | The spectrum of an operato... |
| hhlnoi 32105 | The linear operators of Hi... |
| hhnmoi 32106 | The norm of an operator in... |
| hhbloi 32107 | A bounded linear operator ... |
| hh0oi 32108 | The zero operator in Hilbe... |
| hhcno 32109 | The continuous operators o... |
| hhcnf 32110 | The continuous functionals... |
| dmadjrnb 32111 | The adjoint of an operator... |
| nmoplb 32112 | A lower bound for an opera... |
| nmopub 32113 | An upper bound for an oper... |
| nmopub2tALT 32114 | An upper bound for an oper... |
| nmopub2tHIL 32115 | An upper bound for an oper... |
| nmopge0 32116 | The norm of any Hilbert sp... |
| nmopgt0 32117 | A linear Hilbert space ope... |
| cnopc 32118 | Basic continuity property ... |
| lnopl 32119 | Basic linearity property o... |
| unop 32120 | Basic inner product proper... |
| unopf1o 32121 | A unitary operator in Hilb... |
| unopnorm 32122 | A unitary operator is idem... |
| cnvunop 32123 | The inverse (converse) of ... |
| unopadj 32124 | The inverse (converse) of ... |
| unoplin 32125 | A unitary operator is line... |
| counop 32126 | The composition of two uni... |
| hmop 32127 | Basic inner product proper... |
| hmopre 32128 | The inner product of the v... |
| nmfnlb 32129 | A lower bound for a functi... |
| nmfnleub 32130 | An upper bound for the nor... |
| nmfnleub2 32131 | An upper bound for the nor... |
| nmfnge0 32132 | The norm of any Hilbert sp... |
| elnlfn 32133 | Membership in the null spa... |
| elnlfn2 32134 | Membership in the null spa... |
| cnfnc 32135 | Basic continuity property ... |
| lnfnl 32136 | Basic linearity property o... |
| adjcl 32137 | Closure of the adjoint of ... |
| adj1 32138 | Property of an adjoint Hil... |
| adj2 32139 | Property of an adjoint Hil... |
| adjeq 32140 | A property that determines... |
| adjadj 32141 | Double adjoint. Theorem 3... |
| adjvalval 32142 | Value of the value of the ... |
| unopadj2 32143 | The adjoint of a unitary o... |
| hmopadj 32144 | A Hermitian operator is se... |
| hmdmadj 32145 | Every Hermitian operator h... |
| hmopadj2 32146 | An operator is Hermitian i... |
| hmoplin 32147 | A Hermitian operator is li... |
| brafval 32148 | The bra of a vector, expre... |
| braval 32149 | A bra-ket juxtaposition, e... |
| braadd 32150 | Linearity property of bra ... |
| bramul 32151 | Linearity property of bra ... |
| brafn 32152 | The bra function is a func... |
| bralnfn 32153 | The Dirac bra function is ... |
| bracl 32154 | Closure of the bra functio... |
| bra0 32155 | The Dirac bra of the zero ... |
| brafnmul 32156 | Anti-linearity property of... |
| kbfval 32157 | The outer product of two v... |
| kbop 32158 | The outer product of two v... |
| kbval 32159 | The value of the operator ... |
| kbmul 32160 | Multiplication property of... |
| kbpj 32161 | If a vector ` A ` has norm... |
| eleigvec 32162 | Membership in the set of e... |
| eleigvec2 32163 | Membership in the set of e... |
| eleigveccl 32164 | Closure of an eigenvector ... |
| eigvalval 32165 | The eigenvalue of an eigen... |
| eigvalcl 32166 | An eigenvalue is a complex... |
| eigvec1 32167 | Property of an eigenvector... |
| eighmre 32168 | The eigenvalues of a Hermi... |
| eighmorth 32169 | Eigenvectors of a Hermitia... |
| nmopnegi 32170 | Value of the norm of the n... |
| lnop0 32171 | The value of a linear Hilb... |
| lnopmul 32172 | Multiplicative property of... |
| lnopli 32173 | Basic scalar product prope... |
| lnopfi 32174 | A linear Hilbert space ope... |
| lnop0i 32175 | The value of a linear Hilb... |
| lnopaddi 32176 | Additive property of a lin... |
| lnopmuli 32177 | Multiplicative property of... |
| lnopaddmuli 32178 | Sum/product property of a ... |
| lnopsubi 32179 | Subtraction property for a... |
| lnopsubmuli 32180 | Subtraction/product proper... |
| lnopmulsubi 32181 | Product/subtraction proper... |
| homco2 32182 | Move a scalar product out ... |
| idunop 32183 | The identity function (res... |
| 0cnop 32184 | The identically zero funct... |
| 0cnfn 32185 | The identically zero funct... |
| idcnop 32186 | The identity function (res... |
| idhmop 32187 | The Hilbert space identity... |
| 0hmop 32188 | The identically zero funct... |
| 0lnop 32189 | The identically zero funct... |
| 0lnfn 32190 | The identically zero funct... |
| nmop0 32191 | The norm of the zero opera... |
| nmfn0 32192 | The norm of the identicall... |
| hmopbdoptHIL 32193 | A Hermitian operator is a ... |
| hoddii 32194 | Distributive law for Hilbe... |
| hoddi 32195 | Distributive law for Hilbe... |
| nmop0h 32196 | The norm of any operator o... |
| idlnop 32197 | The identity function (res... |
| 0bdop 32198 | The identically zero opera... |
| adj0 32199 | Adjoint of the zero operat... |
| nmlnop0iALT 32200 | A linear operator with a z... |
| nmlnop0iHIL 32201 | A linear operator with a z... |
| nmlnopgt0i 32202 | A linear Hilbert space ope... |
| nmlnop0 32203 | A linear operator with a z... |
| nmlnopne0 32204 | A linear operator with a n... |
| lnopmi 32205 | The scalar product of a li... |
| lnophsi 32206 | The sum of two linear oper... |
| lnophdi 32207 | The difference of two line... |
| lnopcoi 32208 | The composition of two lin... |
| lnopco0i 32209 | The composition of a linea... |
| lnopeq0lem1 32210 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 32211 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 32212 | A condition implying that ... |
| lnopeqi 32213 | Two linear Hilbert space o... |
| lnopeq 32214 | Two linear Hilbert space o... |
| lnopunilem1 32215 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 32216 | Lemma for ~ lnopunii . (C... |
| lnopunii 32217 | If a linear operator (whos... |
| elunop2 32218 | An operator is unitary iff... |
| nmopun 32219 | Norm of a unitary Hilbert ... |
| unopbd 32220 | A unitary operator is a bo... |
| lnophmlem1 32221 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 32222 | Lemma for ~ lnophmi . (Co... |
| lnophmi 32223 | A linear operator is Hermi... |
| lnophm 32224 | A linear operator is Hermi... |
| hmops 32225 | The sum of two Hermitian o... |
| hmopm 32226 | The scalar product of a He... |
| hmopd 32227 | The difference of two Herm... |
| hmopco 32228 | The composition of two com... |
| nmbdoplbi 32229 | A lower bound for the norm... |
| nmbdoplb 32230 | A lower bound for the norm... |
| nmcexi 32231 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 32232 | The norm of a continuous l... |
| nmcoplbi 32233 | A lower bound for the norm... |
| nmcopex 32234 | The norm of a continuous l... |
| nmcoplb 32235 | A lower bound for the norm... |
| nmophmi 32236 | The norm of the scalar pro... |
| bdophmi 32237 | The scalar product of a bo... |
| lnconi 32238 | Lemma for ~ lnopconi and ~... |
| lnopconi 32239 | A condition equivalent to ... |
| lnopcon 32240 | A condition equivalent to ... |
| lnopcnbd 32241 | A linear operator is conti... |
| lncnopbd 32242 | A continuous linear operat... |
| lncnbd 32243 | A continuous linear operat... |
| lnopcnre 32244 | A linear operator is conti... |
| lnfnli 32245 | Basic property of a linear... |
| lnfnfi 32246 | A linear Hilbert space fun... |
| lnfn0i 32247 | The value of a linear Hilb... |
| lnfnaddi 32248 | Additive property of a lin... |
| lnfnmuli 32249 | Multiplicative property of... |
| lnfnaddmuli 32250 | Sum/product property of a ... |
| lnfnsubi 32251 | Subtraction property for a... |
| lnfn0 32252 | The value of a linear Hilb... |
| lnfnmul 32253 | Multiplicative property of... |
| nmbdfnlbi 32254 | A lower bound for the norm... |
| nmbdfnlb 32255 | A lower bound for the norm... |
| nmcfnexi 32256 | The norm of a continuous l... |
| nmcfnlbi 32257 | A lower bound for the norm... |
| nmcfnex 32258 | The norm of a continuous l... |
| nmcfnlb 32259 | A lower bound of the norm ... |
| lnfnconi 32260 | A condition equivalent to ... |
| lnfncon 32261 | A condition equivalent to ... |
| lnfncnbd 32262 | A linear functional is con... |
| imaelshi 32263 | The image of a subspace un... |
| rnelshi 32264 | The range of a linear oper... |
| nlelshi 32265 | The null space of a linear... |
| nlelchi 32266 | The null space of a contin... |
| riesz3i 32267 | A continuous linear functi... |
| riesz4i 32268 | A continuous linear functi... |
| riesz4 32269 | A continuous linear functi... |
| riesz1 32270 | Part 1 of the Riesz repres... |
| riesz2 32271 | Part 2 of the Riesz repres... |
| cnlnadjlem1 32272 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 32273 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 32274 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 32275 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 32276 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 32277 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 32278 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 32279 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 32280 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 32281 | Every continuous linear op... |
| cnlnadjeui 32282 | Every continuous linear op... |
| cnlnadjeu 32283 | Every continuous linear op... |
| cnlnadj 32284 | Every continuous linear op... |
| cnlnssadj 32285 | Every continuous linear Hi... |
| bdopssadj 32286 | Every bounded linear Hilbe... |
| bdopadj 32287 | Every bounded linear Hilbe... |
| adjbdln 32288 | The adjoint of a bounded l... |
| adjbdlnb 32289 | An operator is bounded and... |
| adjbd1o 32290 | The mapping of adjoints of... |
| adjlnop 32291 | The adjoint of an operator... |
| adjsslnop 32292 | Every operator with an adj... |
| nmopadjlei 32293 | Property of the norm of an... |
| nmopadjlem 32294 | Lemma for ~ nmopadji . (C... |
| nmopadji 32295 | Property of the norm of an... |
| adjeq0 32296 | An operator is zero iff it... |
| adjmul 32297 | The adjoint of the scalar ... |
| adjadd 32298 | The adjoint of the sum of ... |
| nmoptrii 32299 | Triangle inequality for th... |
| nmopcoi 32300 | Upper bound for the norm o... |
| bdophsi 32301 | The sum of two bounded lin... |
| bdophdi 32302 | The difference between two... |
| bdopcoi 32303 | The composition of two bou... |
| nmoptri2i 32304 | Triangle-type inequality f... |
| adjcoi 32305 | The adjoint of a compositi... |
| nmopcoadji 32306 | The norm of an operator co... |
| nmopcoadj2i 32307 | The norm of an operator co... |
| nmopcoadj0i 32308 | An operator composed with ... |
| unierri 32309 | If we approximate a chain ... |
| branmfn 32310 | The norm of the bra functi... |
| brabn 32311 | The bra of a vector is a b... |
| rnbra 32312 | The set of bras equals the... |
| bra11 32313 | The bra function maps vect... |
| bracnln 32314 | A bra is a continuous line... |
| cnvbraval 32315 | Value of the converse of t... |
| cnvbracl 32316 | Closure of the converse of... |
| cnvbrabra 32317 | The converse bra of the br... |
| bracnvbra 32318 | The bra of the converse br... |
| bracnlnval 32319 | The vector that a continuo... |
| cnvbramul 32320 | Multiplication property of... |
| kbass1 32321 | Dirac bra-ket associative ... |
| kbass2 32322 | Dirac bra-ket associative ... |
| kbass3 32323 | Dirac bra-ket associative ... |
| kbass4 32324 | Dirac bra-ket associative ... |
| kbass5 32325 | Dirac bra-ket associative ... |
| kbass6 32326 | Dirac bra-ket associative ... |
| leopg 32327 | Ordering relation for posi... |
| leop 32328 | Ordering relation for oper... |
| leop2 32329 | Ordering relation for oper... |
| leop3 32330 | Operator ordering in terms... |
| leoppos 32331 | Binary relation defining a... |
| leoprf2 32332 | The ordering relation for ... |
| leoprf 32333 | The ordering relation for ... |
| leopsq 32334 | The square of a Hermitian ... |
| 0leop 32335 | The zero operator is a pos... |
| idleop 32336 | The identity operator is a... |
| leopadd 32337 | The sum of two positive op... |
| leopmuli 32338 | The scalar product of a no... |
| leopmul 32339 | The scalar product of a po... |
| leopmul2i 32340 | Scalar product applied to ... |
| leoptri 32341 | The positive operator orde... |
| leoptr 32342 | The positive operator orde... |
| leopnmid 32343 | A bounded Hermitian operat... |
| nmopleid 32344 | A nonzero, bounded Hermiti... |
| opsqrlem1 32345 | Lemma for opsqri . (Contr... |
| opsqrlem2 32346 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 32347 | Lemma for opsqri . (Contr... |
| opsqrlem4 32348 | Lemma for opsqri . (Contr... |
| opsqrlem5 32349 | Lemma for opsqri . (Contr... |
| opsqrlem6 32350 | Lemma for opsqri . (Contr... |
| pjhmopi 32351 | A projector is a Hermitian... |
| pjlnopi 32352 | A projector is a linear op... |
| pjnmopi 32353 | The operator norm of a pro... |
| pjbdlni 32354 | A projector is a bounded l... |
| pjhmop 32355 | A projection is a Hermitia... |
| hmopidmchi 32356 | An idempotent Hermitian op... |
| hmopidmpji 32357 | An idempotent Hermitian op... |
| hmopidmch 32358 | An idempotent Hermitian op... |
| hmopidmpj 32359 | An idempotent Hermitian op... |
| pjsdii 32360 | Distributive law for Hilbe... |
| pjddii 32361 | Distributive law for Hilbe... |
| pjsdi2i 32362 | Chained distributive law f... |
| pjcoi 32363 | Composition of projections... |
| pjcocli 32364 | Closure of composition of ... |
| pjcohcli 32365 | Closure of composition of ... |
| pjadjcoi 32366 | Adjoint of composition of ... |
| pjcofni 32367 | Functionality of compositi... |
| pjss1coi 32368 | Subset relationship for pr... |
| pjss2coi 32369 | Subset relationship for pr... |
| pjssmi 32370 | Projection meet property. ... |
| pjssge0i 32371 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 32372 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 32373 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 32374 | Composition of projections... |
| pjscji 32375 | The projection of orthogon... |
| pjssumi 32376 | The projection on a subspa... |
| pjssposi 32377 | Projector ordering can be ... |
| pjordi 32378 | The definition of projecto... |
| pjssdif2i 32379 | The projection subspace of... |
| pjssdif1i 32380 | A necessary and sufficient... |
| pjimai 32381 | The image of a projection.... |
| pjidmcoi 32382 | A projection is idempotent... |
| pjoccoi 32383 | Composition of projections... |
| pjtoi 32384 | Subspace sum of projection... |
| pjoci 32385 | Projection of orthocomplem... |
| pjidmco 32386 | A projection operator is i... |
| dfpjop 32387 | Definition of projection o... |
| pjhmopidm 32388 | Two ways to express the se... |
| elpjidm 32389 | A projection operator is i... |
| elpjhmop 32390 | A projection operator is H... |
| 0leopj 32391 | A projector is a positive ... |
| pjadj2 32392 | A projector is self-adjoin... |
| pjadj3 32393 | A projector is self-adjoin... |
| elpjch 32394 | Reconstruction of the subs... |
| elpjrn 32395 | Reconstruction of the subs... |
| pjinvari 32396 | A closed subspace ` H ` wi... |
| pjin1i 32397 | Lemma for Theorem 1.22 of ... |
| pjin2i 32398 | Lemma for Theorem 1.22 of ... |
| pjin3i 32399 | Lemma for Theorem 1.22 of ... |
| pjclem1 32400 | Lemma for projection commu... |
| pjclem2 32401 | Lemma for projection commu... |
| pjclem3 32402 | Lemma for projection commu... |
| pjclem4a 32403 | Lemma for projection commu... |
| pjclem4 32404 | Lemma for projection commu... |
| pjci 32405 | Two subspaces commute iff ... |
| pjcmul1i 32406 | A necessary and sufficient... |
| pjcmul2i 32407 | The projection subspace of... |
| pjcohocli 32408 | Closure of composition of ... |
| pjadj2coi 32409 | Adjoint of double composit... |
| pj2cocli 32410 | Closure of double composit... |
| pj3lem1 32411 | Lemma for projection tripl... |
| pj3si 32412 | Stronger projection triple... |
| pj3i 32413 | Projection triplet theorem... |
| pj3cor1i 32414 | Projection triplet corolla... |
| pjs14i 32415 | Theorem S-14 of Watanabe, ... |
| isst 32418 | Property of a state. (Con... |
| ishst 32419 | Property of a complex Hilb... |
| sticl 32420 | ` [ 0 , 1 ] ` closure of t... |
| stcl 32421 | Real closure of the value ... |
| hstcl 32422 | Closure of the value of a ... |
| hst1a 32423 | Unit value of a Hilbert-sp... |
| hstel2 32424 | Properties of a Hilbert-sp... |
| hstorth 32425 | Orthogonality property of ... |
| hstosum 32426 | Orthogonal sum property of... |
| hstoc 32427 | Sum of a Hilbert-space-val... |
| hstnmoc 32428 | Sum of norms of a Hilbert-... |
| stge0 32429 | The value of a state is no... |
| stle1 32430 | The value of a state is le... |
| hstle1 32431 | The norm of the value of a... |
| hst1h 32432 | The norm of a Hilbert-spac... |
| hst0h 32433 | The norm of a Hilbert-spac... |
| hstpyth 32434 | Pythagorean property of a ... |
| hstle 32435 | Ordering property of a Hil... |
| hstles 32436 | Ordering property of a Hil... |
| hstoh 32437 | A Hilbert-space-valued sta... |
| hst0 32438 | A Hilbert-space-valued sta... |
| sthil 32439 | The value of a state at th... |
| stj 32440 | The value of a state on a ... |
| sto1i 32441 | The state of a subspace pl... |
| sto2i 32442 | The state of the orthocomp... |
| stge1i 32443 | If a state is greater than... |
| stle0i 32444 | If a state is less than or... |
| stlei 32445 | Ordering law for states. ... |
| stlesi 32446 | Ordering law for states. ... |
| stji1i 32447 | Join of components of Sasa... |
| stm1i 32448 | State of component of unit... |
| stm1ri 32449 | State of component of unit... |
| stm1addi 32450 | Sum of states whose meet i... |
| staddi 32451 | If the sum of 2 states is ... |
| stm1add3i 32452 | Sum of states whose meet i... |
| stadd3i 32453 | If the sum of 3 states is ... |
| st0 32454 | The state of the zero subs... |
| strlem1 32455 | Lemma for strong state the... |
| strlem2 32456 | Lemma for strong state the... |
| strlem3a 32457 | Lemma for strong state the... |
| strlem3 32458 | Lemma for strong state the... |
| strlem4 32459 | Lemma for strong state the... |
| strlem5 32460 | Lemma for strong state the... |
| strlem6 32461 | Lemma for strong state the... |
| stri 32462 | Strong state theorem. The... |
| strb 32463 | Strong state theorem (bidi... |
| hstrlem2 32464 | Lemma for strong set of CH... |
| hstrlem3a 32465 | Lemma for strong set of CH... |
| hstrlem3 32466 | Lemma for strong set of CH... |
| hstrlem4 32467 | Lemma for strong set of CH... |
| hstrlem5 32468 | Lemma for strong set of CH... |
| hstrlem6 32469 | Lemma for strong set of CH... |
| hstri 32470 | Hilbert space admits a str... |
| hstrbi 32471 | Strong CH-state theorem (b... |
| largei 32472 | A Hilbert lattice admits a... |
| jplem1 32473 | Lemma for Jauch-Piron theo... |
| jplem2 32474 | Lemma for Jauch-Piron theo... |
| jpi 32475 | The function ` S ` , that ... |
| golem1 32476 | Lemma for Godowski's equat... |
| golem2 32477 | Lemma for Godowski's equat... |
| goeqi 32478 | Godowski's equation, shown... |
| stcltr1i 32479 | Property of a strong class... |
| stcltr2i 32480 | Property of a strong class... |
| stcltrlem1 32481 | Lemma for strong classical... |
| stcltrlem2 32482 | Lemma for strong classical... |
| stcltrthi 32483 | Theorem for classically st... |
| cvbr 32487 | Binary relation expressing... |
| cvbr2 32488 | Binary relation expressing... |
| cvcon3 32489 | Contraposition law for the... |
| cvpss 32490 | The covers relation implie... |
| cvnbtwn 32491 | The covers relation implie... |
| cvnbtwn2 32492 | The covers relation implie... |
| cvnbtwn3 32493 | The covers relation implie... |
| cvnbtwn4 32494 | The covers relation implie... |
| cvnsym 32495 | The covers relation is not... |
| cvnref 32496 | The covers relation is not... |
| cvntr 32497 | The covers relation is not... |
| spansncv2 32498 | Hilbert space has the cove... |
| mdbr 32499 | Binary relation expressing... |
| mdi 32500 | Consequence of the modular... |
| mdbr2 32501 | Binary relation expressing... |
| mdbr3 32502 | Binary relation expressing... |
| mdbr4 32503 | Binary relation expressing... |
| dmdbr 32504 | Binary relation expressing... |
| dmdmd 32505 | The dual modular pair prop... |
| mddmd 32506 | The modular pair property ... |
| dmdi 32507 | Consequence of the dual mo... |
| dmdbr2 32508 | Binary relation expressing... |
| dmdi2 32509 | Consequence of the dual mo... |
| dmdbr3 32510 | Binary relation expressing... |
| dmdbr4 32511 | Binary relation expressing... |
| dmdi4 32512 | Consequence of the dual mo... |
| dmdbr5 32513 | Binary relation expressing... |
| mddmd2 32514 | Relationship between modul... |
| mdsl0 32515 | A sublattice condition tha... |
| ssmd1 32516 | Ordering implies the modul... |
| ssmd2 32517 | Ordering implies the modul... |
| ssdmd1 32518 | Ordering implies the dual ... |
| ssdmd2 32519 | Ordering implies the dual ... |
| dmdsl3 32520 | Sublattice mapping for a d... |
| mdsl3 32521 | Sublattice mapping for a m... |
| mdslle1i 32522 | Order preservation of the ... |
| mdslle2i 32523 | Order preservation of the ... |
| mdslj1i 32524 | Join preservation of the o... |
| mdslj2i 32525 | Meet preservation of the r... |
| mdsl1i 32526 | If the modular pair proper... |
| mdsl2i 32527 | If the modular pair proper... |
| mdsl2bi 32528 | If the modular pair proper... |
| cvmdi 32529 | The covering property impl... |
| mdslmd1lem1 32530 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 32531 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 32532 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 32533 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 32534 | Preservation of the modula... |
| mdslmd2i 32535 | Preservation of the modula... |
| mdsldmd1i 32536 | Preservation of the dual m... |
| mdslmd3i 32537 | Modular pair conditions th... |
| mdslmd4i 32538 | Modular pair condition tha... |
| csmdsymi 32539 | Cross-symmetry implies M-s... |
| mdexchi 32540 | An exchange lemma for modu... |
| cvmd 32541 | The covering property impl... |
| cvdmd 32542 | The covering property impl... |
| ela 32544 | Atoms in a Hilbert lattice... |
| elat2 32545 | Expanded membership relati... |
| elatcv0 32546 | A Hilbert lattice element ... |
| atcv0 32547 | An atom covers the zero su... |
| atssch 32548 | Atoms are a subset of the ... |
| atelch 32549 | An atom is a Hilbert latti... |
| atne0 32550 | An atom is not the Hilbert... |
| atss 32551 | A lattice element smaller ... |
| atsseq 32552 | Two atoms in a subset rela... |
| atcveq0 32553 | A Hilbert lattice element ... |
| h1da 32554 | A 1-dimensional subspace i... |
| spansna 32555 | The span of the singleton ... |
| sh1dle 32556 | A 1-dimensional subspace i... |
| ch1dle 32557 | A 1-dimensional subspace i... |
| atom1d 32558 | The 1-dimensional subspace... |
| superpos 32559 | Superposition Principle. ... |
| chcv1 32560 | The Hilbert lattice has th... |
| chcv2 32561 | The Hilbert lattice has th... |
| chjatom 32562 | The join of a closed subsp... |
| shatomici 32563 | The lattice of Hilbert sub... |
| hatomici 32564 | The Hilbert lattice is ato... |
| hatomic 32565 | A Hilbert lattice is atomi... |
| shatomistici 32566 | The lattice of Hilbert sub... |
| hatomistici 32567 | ` CH ` is atomistic, i.e. ... |
| chpssati 32568 | Two Hilbert lattice elemen... |
| chrelati 32569 | The Hilbert lattice is rel... |
| chrelat2i 32570 | A consequence of relative ... |
| cvati 32571 | If a Hilbert lattice eleme... |
| cvbr4i 32572 | An alternate way to expres... |
| cvexchlem 32573 | Lemma for ~ cvexchi . (Co... |
| cvexchi 32574 | The Hilbert lattice satisf... |
| chrelat2 32575 | A consequence of relative ... |
| chrelat3 32576 | A consequence of relative ... |
| chrelat3i 32577 | A consequence of the relat... |
| chrelat4i 32578 | A consequence of relative ... |
| cvexch 32579 | The Hilbert lattice satisf... |
| cvp 32580 | The Hilbert lattice satisf... |
| atnssm0 32581 | The meet of a Hilbert latt... |
| atnemeq0 32582 | The meet of distinct atoms... |
| atssma 32583 | The meet with an atom's su... |
| atcv0eq 32584 | Two atoms covering the zer... |
| atcv1 32585 | Two atoms covering the zer... |
| atexch 32586 | The Hilbert lattice satisf... |
| atomli 32587 | An assertion holding in at... |
| atoml2i 32588 | An assertion holding in at... |
| atordi 32589 | An ordering law for a Hilb... |
| atcvatlem 32590 | Lemma for ~ atcvati . (Co... |
| atcvati 32591 | A nonzero Hilbert lattice ... |
| atcvat2i 32592 | A Hilbert lattice element ... |
| atord 32593 | An ordering law for a Hilb... |
| atcvat2 32594 | A Hilbert lattice element ... |
| chirredlem1 32595 | Lemma for ~ chirredi . (C... |
| chirredlem2 32596 | Lemma for ~ chirredi . (C... |
| chirredlem3 32597 | Lemma for ~ chirredi . (C... |
| chirredlem4 32598 | Lemma for ~ chirredi . (C... |
| chirredi 32599 | The Hilbert lattice is irr... |
| chirred 32600 | The Hilbert lattice is irr... |
| atcvat3i 32601 | A condition implying that ... |
| atcvat4i 32602 | A condition implying exist... |
| atdmd 32603 | Two Hilbert lattice elemen... |
| atmd 32604 | Two Hilbert lattice elemen... |
| atmd2 32605 | Two Hilbert lattice elemen... |
| atabsi 32606 | Absorption of an incompara... |
| atabs2i 32607 | Absorption of an incompara... |
| mdsymlem1 32608 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 32609 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 32610 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 32611 | Lemma for ~ mdsymi . This... |
| mdsymlem5 32612 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 32613 | Lemma for ~ mdsymi . This... |
| mdsymlem7 32614 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 32615 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 32616 | M-symmetry of the Hilbert ... |
| mdsym 32617 | M-symmetry of the Hilbert ... |
| dmdsym 32618 | Dual M-symmetry of the Hil... |
| atdmd2 32619 | Two Hilbert lattice elemen... |
| sumdmdii 32620 | If the subspace sum of two... |
| cmmdi 32621 | Commuting subspaces form a... |
| cmdmdi 32622 | Commuting subspaces form a... |
| sumdmdlem 32623 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 32624 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 32625 | The subspace sum of two Hi... |
| dmdbr4ati 32626 | Dual modular pair property... |
| dmdbr5ati 32627 | Dual modular pair property... |
| dmdbr6ati 32628 | Dual modular pair property... |
| dmdbr7ati 32629 | Dual modular pair property... |
| mdoc1i 32630 | Orthocomplements form a mo... |
| mdoc2i 32631 | Orthocomplements form a mo... |
| dmdoc1i 32632 | Orthocomplements form a du... |
| dmdoc2i 32633 | Orthocomplements form a du... |
| mdcompli 32634 | A condition equivalent to ... |
| dmdcompli 32635 | A condition equivalent to ... |
| mddmdin0i 32636 | If dual modular implies mo... |
| cdjreui 32637 | A member of the sum of dis... |
| cdj1i 32638 | Two ways to express " ` A ... |
| cdj3lem1 32639 | A property of " ` A ` and ... |
| cdj3lem2 32640 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 32641 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 32642 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 32643 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 32644 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 32645 | Lemma for ~ cdj3i . The s... |
| cdj3i 32646 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 32647 | (_This theorem is a dummy ... |
| sa-abvi 32648 | A theorem about the univer... |
| xfree 32649 | A partial converse to ~ 19... |
| xfree2 32650 | A partial converse to ~ 19... |
| addltmulALT 32651 | A proof readability experi... |
| ad11antr 32652 | Deduction adding 11 conjun... |
| simp-12l 32653 | Simplification of a conjun... |
| simp-12r 32654 | Simplification of a conjun... |
| an52ds 32655 | Inference exchanging the l... |
| an62ds 32656 | Inference exchanging the l... |
| an72ds 32657 | Inference exchanging the l... |
| an82ds 32658 | Inference exchanging the l... |
| syl22anbrc 32659 | Syllogism inference. (Con... |
| bian1dOLD 32660 | Obsolete version of ~ bian... |
| or3di 32661 | Distributive law for disju... |
| or3dir 32662 | Distributive law for disju... |
| 3o1cs 32663 | Deduction eliminating disj... |
| 3o2cs 32664 | Deduction eliminating disj... |
| 3o3cs 32665 | Deduction eliminating disj... |
| 13an22anass 32666 | Associative law for four c... |
| sbc2iedf 32667 | Conversion of implicit sub... |
| rspc2daf 32668 | Double restricted speciali... |
| ralcom4f 32669 | Commutation of restricted ... |
| rexcom4f 32670 | Commutation of restricted ... |
| 19.9d2rf 32671 | A deduction version of one... |
| 19.9d2r 32672 | A deduction version of one... |
| r19.29ffa 32673 | A commonly used pattern ba... |
| reu6dv 32674 | A condition which implies ... |
| eqtrb 32675 | A transposition of equalit... |
| eqelbid 32676 | A variable elimination law... |
| opsbc2ie 32677 | Conversion of implicit sub... |
| opreu2reuALT 32678 | Correspondence between uni... |
| 2reucom 32681 | Double restricted existent... |
| 2reu2rex1 32682 | Double restricted existent... |
| 2reureurex 32683 | Double restricted existent... |
| 2reu2reu2 32684 | Double restricted existent... |
| opreu2reu1 32685 | Equivalent definition of t... |
| sq2reunnltb 32686 | There exists a unique deco... |
| addsqnot2reu 32687 | For each complex number ` ... |
| sbceqbidf 32688 | Equality theorem for class... |
| sbcies 32689 | A special version of class... |
| mo5f 32690 | Alternate definition of "a... |
| nmo 32691 | Negation of "at most one".... |
| reuxfrdf 32692 | Transfer existential uniqu... |
| rexunirn 32693 | Restricted existential qua... |
| rmoxfrd 32694 | Transfer "at most one" res... |
| rmoun 32695 | "At most one" restricted e... |
| rmounid 32696 | A case where an "at most o... |
| riotaeqbidva 32697 | Equivalent wff's yield equ... |
| dmrab 32698 | Domain of a restricted cla... |
| difrab2 32699 | Difference of two restrict... |
| rabexgfGS 32700 | Separation Scheme in terms... |
| rabsnel 32701 | Truth implied by equality ... |
| rabsspr 32702 | Conditions for a restricte... |
| rabsstp 32703 | Conditions for a restricte... |
| 3unrab 32704 | Union of three restricted ... |
| foresf1o 32705 | From a surjective function... |
| rabfodom 32706 | Domination relation for re... |
| rabrexfi 32707 | Conditions for a class abs... |
| abrexdomjm 32708 | An indexed set is dominate... |
| abrexdom2jm 32709 | An indexed set is dominate... |
| abrexexd 32710 | Existence of a class abstr... |
| elabreximd 32711 | Class substitution in an i... |
| elabreximdv 32712 | Class substitution in an i... |
| abrexss 32713 | A necessary condition for ... |
| nelun 32714 | Negated membership for a u... |
| snsssng 32715 | If a singleton is a subset... |
| n0nsnel 32716 | If a class with one elemen... |
| inin 32717 | Intersection with an inter... |
| difininv 32718 | Condition for the intersec... |
| difeq 32719 | Rewriting an equation with... |
| eqdif 32720 | If both set differences of... |
| indifbi 32721 | Two ways to express equali... |
| diffib 32722 | Case where ~ diffi is a bi... |
| difxp1ss 32723 | Difference law for Cartesi... |
| difxp2ss 32724 | Difference law for Cartesi... |
| indifundif 32725 | A remarkable equation with... |
| elpwincl1 32726 | Closure of intersection wi... |
| elpwdifcl 32727 | Closure of class differenc... |
| elpwiuncl 32728 | Closure of indexed union w... |
| elpreq 32729 | Equality wihin a pair. (C... |
| prssad 32730 | If a pair is a subset of a... |
| prssbd 32731 | If a pair is a subset of a... |
| nelpr 32732 | A set ` A ` not in a pair ... |
| inpr0 32733 | Rewrite an empty intersect... |
| neldifpr1 32734 | The first element of a pai... |
| neldifpr2 32735 | The second element of a pa... |
| unidifsnel 32736 | The other element of a pai... |
| unidifsnne 32737 | The other element of a pai... |
| tpssg 32738 | An unordered triple of ele... |
| tpssd 32739 | Deduction version of tpssi... |
| tpssad 32740 | If an ordered triple is a ... |
| tpssbd 32741 | If an ordered triple is a ... |
| tpsscd 32742 | If an ordered triple is a ... |
| ifeqeqx 32743 | An equality theorem tailor... |
| elimifd 32744 | Elimination of a condition... |
| elim2if 32745 | Elimination of two conditi... |
| elim2ifim 32746 | Elimination of two conditi... |
| ifeq3da 32747 | Given an expression ` C ` ... |
| ifnetrue 32748 | Deduce truth from a condit... |
| ifnefals 32749 | Deduce falsehood from a co... |
| ifnebib 32750 | The converse of ~ ifbi hol... |
| ififcom 32751 | Commute two nested conditi... |
| uniinn0 32752 | Sufficient and necessary c... |
| uniin1 32753 | Union of intersection. Ge... |
| uniin2 32754 | Union of intersection. Ge... |
| difuncomp 32755 | Express a class difference... |
| elpwunicl 32756 | Closure of a set union wit... |
| cbviunf 32757 | Rule used to change the bo... |
| iuneq12daf 32758 | Equality deduction for ind... |
| iunin1f 32759 | Indexed union of intersect... |
| ssiun3 32760 | Subset equivalence for an ... |
| ssiun2sf 32761 | Subset relationship for an... |
| iuninc 32762 | The union of an increasing... |
| iundifdifd 32763 | The intersection of a set ... |
| iundifdif 32764 | The intersection of a set ... |
| iunrdx 32765 | Re-index an indexed union.... |
| iunpreima 32766 | Preimage of an indexed uni... |
| iunrnmptss 32767 | A subset relation for an i... |
| iunxunsn 32768 | Appending a set to an inde... |
| iunxunpr 32769 | Appending two sets to an i... |
| iunxpssiun1 32770 | Provide an upper bound for... |
| iinabrex 32771 | Rewriting an indexed inter... |
| disjnf 32772 | In case ` x ` is not free ... |
| cbvdisjf 32773 | Change bound variables in ... |
| disjss1f 32774 | A subset of a disjoint col... |
| disjeq1f 32775 | Equality theorem for disjo... |
| disjxun0 32776 | Simplify a disjoint union.... |
| disjdifprg 32777 | A trivial partition into a... |
| disjdifprg2 32778 | A trivial partition of a s... |
| disji2f 32779 | Property of a disjoint col... |
| disjif 32780 | Property of a disjoint col... |
| disjorf 32781 | Two ways to say that a col... |
| disjorsf 32782 | Two ways to say that a col... |
| disjif2 32783 | Property of a disjoint col... |
| disjabrex 32784 | Rewriting a disjoint colle... |
| disjabrexf 32785 | Rewriting a disjoint colle... |
| disjpreima 32786 | A preimage of a disjoint s... |
| disjrnmpt 32787 | Rewriting a disjoint colle... |
| disjin 32788 | If a collection is disjoin... |
| disjin2 32789 | If a collection is disjoin... |
| disjxpin 32790 | Derive a disjunction over ... |
| iundisjf 32791 | Rewrite a countable union ... |
| iundisj2f 32792 | A disjoint union is disjoi... |
| disjrdx 32793 | Re-index a disjunct collec... |
| disjex 32794 | Two ways to say that two c... |
| disjexc 32795 | A variant of ~ disjex , ap... |
| disjunsn 32796 | Append an element to a dis... |
| disjun0 32797 | Adding the empty element p... |
| disjiunel 32798 | A set of elements B of a d... |
| disjuniel 32799 | A set of elements B of a d... |
| xpdisjres 32800 | Restriction of a constant ... |
| opeldifid 32801 | Ordered pair elementhood o... |
| difres 32802 | Case when class difference... |
| imadifxp 32803 | Image of the difference wi... |
| relfi 32804 | A relation (set) is finite... |
| 0res 32805 | Restriction of the empty f... |
| fcoinver 32806 | Build an equivalence relat... |
| fcoinvbr 32807 | Binary relation for the eq... |
| breq1dd 32808 | Equality deduction for a b... |
| breq2dd 32809 | Equality deduction for a b... |
| brabgaf 32810 | The law of concretion for ... |
| brelg 32811 | Two things in a binary rel... |
| br8d 32812 | Substitution for an eight-... |
| fnfvor 32813 | Relation between two funct... |
| ofrco 32814 | Function relation between ... |
| opabdm 32815 | Domain of an ordered-pair ... |
| opabrn 32816 | Range of an ordered-pair c... |
| opabssi 32817 | Sufficient condition for a... |
| opabid2ss 32818 | One direction of ~ opabid2... |
| ssrelf 32819 | A subclass relationship de... |
| eqrelrd2 32820 | A version of ~ eqrelrdv2 w... |
| erbr3b 32821 | Biconditional for equivale... |
| iunsnima 32822 | Image of a singleton by an... |
| iunsnima2 32823 | Version of ~ iunsnima with... |
| fconst7v 32824 | An alternative way to expr... |
| constcof 32825 | Composition with a constan... |
| ac6sf2 32826 | Alternate version of ~ ac6... |
| ac6mapd 32827 | Axiom of choice equivalent... |
| fnresin 32828 | Restriction of a function ... |
| fresunsn 32829 | Recover the original funct... |
| f1o3d 32830 | Describe an implicit one-t... |
| eldmne0 32831 | A function of nonempty dom... |
| f1rnen 32832 | Equinumerosity of the rang... |
| f1oeq3dd 32833 | Equality deduction for one... |
| rinvf1o 32834 | Sufficient conditions for ... |
| fresf1o 32835 | Conditions for a restricti... |
| nfpconfp 32836 | The set of fixed points of... |
| fmptco1f1o 32837 | The action of composing (t... |
| cofmpt2 32838 | Express composition of a m... |
| f1mptrn 32839 | Express injection for a ma... |
| dfimafnf 32840 | Alternate definition of th... |
| funimass4f 32841 | Membership relation for th... |
| suppss2f 32842 | Show that the support of a... |
| ofrn 32843 | The range of the function ... |
| ofrn2 32844 | The range of the function ... |
| off2 32845 | The function operation pro... |
| ofresid 32846 | Applying an operation rest... |
| unipreima 32847 | Preimage of a class union.... |
| opfv 32848 | Value of a function produc... |
| xppreima 32849 | The preimage of a Cartesia... |
| 2ndimaxp 32850 | Image of a cartesian produ... |
| dmdju 32851 | Domain of a disjoint union... |
| djussxp2 32852 | Stronger version of ~ djus... |
| 2ndresdju 32853 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 32854 | The ` 2nd ` function restr... |
| xppreima2 32855 | The preimage of a Cartesia... |
| abfmpunirn 32856 | Membership in a union of a... |
| rabfmpunirn 32857 | Membership in a union of a... |
| abfmpeld 32858 | Membership in an element o... |
| abfmpel 32859 | Membership in an element o... |
| fmptdF 32860 | Domain and codomain of the... |
| fmptcof2 32861 | Composition of two functio... |
| fcomptf 32862 | Express composition of two... |
| acunirnmpt 32863 | Axiom of choice for the un... |
| acunirnmpt2 32864 | Axiom of choice for the un... |
| acunirnmpt2f 32865 | Axiom of choice for the un... |
| aciunf1lem 32866 | Choice in an index union. ... |
| aciunf1 32867 | Choice in an index union. ... |
| ofoprabco 32868 | Function operation as a co... |
| ofpreima 32869 | Express the preimage of a ... |
| ofpreima2 32870 | Express the preimage of a ... |
| funcnv5mpt 32871 | Two ways to say that a fun... |
| funcnv4mpt 32872 | Two ways to say that a fun... |
| preimane 32873 | Different elements have di... |
| fnpreimac 32874 | Choose a set ` x ` contain... |
| fgreu 32875 | Exactly one point of a fun... |
| fcnvgreu 32876 | If the converse of a relat... |
| rnmposs 32877 | The range of an operation ... |
| mptssALT 32878 | Deduce subset relation of ... |
| dfcnv2 32879 | Alternative definition of ... |
| partfun2 32880 | Rewrite a function defined... |
| rnressnsn 32881 | The range of a restriction... |
| mpomptxf 32882 | Express a two-argument fun... |
| of0r 32883 | Function operation with th... |
| elmaprd 32884 | Deduction associated with ... |
| suppovss 32885 | A bound for the support of... |
| elsuppfnd 32886 | Deduce membership in the s... |
| fisuppov1 32887 | Formula building theorem f... |
| suppun2 32888 | The support of a union is ... |
| fdifsupp 32889 | Express the support of a f... |
| suppiniseg 32890 | Relation between the suppo... |
| fsuppinisegfi 32891 | The initial segment ` ( ``... |
| fressupp 32892 | The restriction of a funct... |
| fdifsuppconst 32893 | A function is a zero const... |
| ressupprn 32894 | The range of a function re... |
| supppreima 32895 | Express the support of a f... |
| fsupprnfi 32896 | Finite support implies fin... |
| mptiffisupp 32897 | Conditions for a mapping f... |
| cosnopne 32898 | Composition of two ordered... |
| cosnop 32899 | Composition of two ordered... |
| cnvprop 32900 | Converse of a pair of orde... |
| brprop 32901 | Binary relation for a pair... |
| mptprop 32902 | Rewrite pairs of ordered p... |
| coprprop 32903 | Composition of two pairs o... |
| fmptunsnop 32904 | Two ways to express a func... |
| gtiso 32905 | Two ways to write a strict... |
| isoun 32906 | Infer an isomorphism from ... |
| disjdsct 32907 | A disjoint collection is d... |
| df1stres 32908 | Definition for a restricti... |
| df2ndres 32909 | Definition for a restricti... |
| 1stpreimas 32910 | The preimage of a singleto... |
| 1stpreima 32911 | The preimage by ` 1st ` is... |
| 2ndpreima 32912 | The preimage by ` 2nd ` is... |
| curry2ima 32913 | The image of a curried fun... |
| preiman0 32914 | The preimage of a nonempty... |
| intimafv 32915 | The intersection of an ima... |
| snct 32916 | A singleton is countable. ... |
| prct 32917 | An unordered pair is count... |
| mpocti 32918 | An operation is countable ... |
| abrexct 32919 | An image set of a countabl... |
| mptctf 32920 | A countable mapping set is... |
| abrexctf 32921 | An image set of a countabl... |
| padct 32922 | Index a countable set with... |
| f1od2 32923 | Sufficient condition for a... |
| fcobij 32924 | Composing functions with a... |
| fcobijfs 32925 | Composing finitely support... |
| fcobijfs2 32926 | Composing finitely support... |
| suppss3 32927 | Deduce a function's suppor... |
| fsuppcurry1 32928 | Finite support of a currie... |
| fsuppcurry2 32929 | Finite support of a currie... |
| offinsupp1 32930 | Finite support for a funct... |
| ffs2 32931 | Rewrite a function's suppo... |
| ffsrn 32932 | The range of a finitely su... |
| cocnvf1o 32933 | Composing with the inverse... |
| resf1o 32934 | Restriction of functions t... |
| maprnin 32935 | Restricting the range of t... |
| fpwrelmapffslem 32936 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 32937 | Define a canonical mapping... |
| fpwrelmapffs 32938 | Define a canonical mapping... |
| sgnval2 32939 | Value of the signum of a r... |
| creq0 32940 | The real representation of... |
| 1nei 32941 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 32942 | An integer unit times itse... |
| nnmulge 32943 | Multiplying by a positive ... |
| submuladdd 32944 | The product of a differenc... |
| binom2subadd 32945 | The difference of the squa... |
| cjsubd 32946 | Complex conjugate distribu... |
| re0cj 32947 | The conjugate of a pure im... |
| receqid 32948 | Real numbers equal to thei... |
| pythagreim 32949 | A simplified version of th... |
| efiargd 32950 | The exponential of the "ar... |
| arginv 32951 | The argument of the invers... |
| argcj 32952 | The argument of the conjug... |
| quad3d 32953 | Variant of quadratic equat... |
| lt2addrd 32954 | If the right-hand side of ... |
| nn0mnfxrd 32955 | Nonnegative integers or mi... |
| xrlelttric 32956 | Trichotomy law for extende... |
| xaddeq0 32957 | Two extended reals which a... |
| rexmul2 32958 | If the result ` A ` of an ... |
| xrinfm 32959 | The extended real numbers ... |
| le2halvesd 32960 | A sum is less than the who... |
| xraddge02 32961 | A number is less than or e... |
| xrge0addge 32962 | A number is less than or e... |
| xlt2addrd 32963 | If the right-hand side of ... |
| xrge0infss 32964 | Any subset of nonnegative ... |
| xrge0infssd 32965 | Inequality deduction for i... |
| xrge0addcld 32966 | Nonnegative extended reals... |
| xrge0subcld 32967 | Condition for closure of n... |
| infxrge0lb 32968 | A member of a set of nonne... |
| infxrge0glb 32969 | The infimum of a set of no... |
| infxrge0gelb 32970 | The infimum of a set of no... |
| xrofsup 32971 | The supremum is preserved ... |
| supxrnemnf 32972 | The supremum of a nonempty... |
| xnn0gt0 32973 | Nonzero extended nonnegati... |
| xnn01gt 32974 | An extended nonnegative in... |
| nn0xmulclb 32975 | Finite multiplication in t... |
| xnn0nn0d 32976 | Conditions for an extended... |
| xnn0nnd 32977 | Conditions for an extended... |
| joiniooico 32978 | Disjoint joining an open i... |
| ubico 32979 | A right-open interval does... |
| xeqlelt 32980 | Equality in terms of 'less... |
| eliccelico 32981 | Relate elementhood to a cl... |
| elicoelioo 32982 | Relate elementhood to a cl... |
| iocinioc2 32983 | Intersection between two o... |
| xrdifh 32984 | Class difference of a half... |
| iocinif 32985 | Relate intersection of two... |
| difioo 32986 | The difference between two... |
| difico 32987 | The difference between two... |
| uzssico 32988 | Upper integer sets are a s... |
| fz2ssnn0 32989 | A finite set of sequential... |
| nndiffz1 32990 | Upper set of the positive ... |
| ssnnssfz 32991 | For any finite subset of `... |
| fzm1ne1 32992 | Elementhood of an integer ... |
| fzspl 32993 | Split the last element of ... |
| fzdif2 32994 | Split the last element of ... |
| fzodif2 32995 | Split the last element of ... |
| fzodif1 32996 | Set difference of two half... |
| fzsplit3 32997 | Split a finite interval of... |
| nn0diffz0 32998 | Upper set of the nonnegati... |
| bcm1n 32999 | The proportion of one bino... |
| iundisjfi 33000 | Rewrite a countable union ... |
| iundisj2fi 33001 | A disjoint union is disjoi... |
| iundisjcnt 33002 | Rewrite a countable union ... |
| iundisj2cnt 33003 | A countable disjoint union... |
| f1ocnt 33004 | Given a countable set ` A ... |
| fz1nnct 33005 | NN and integer ranges star... |
| fz1nntr 33006 | NN and integer ranges star... |
| fzo0opth 33007 | Equality for a half open i... |
| nn0difffzod 33008 | A nonnegative integer that... |
| suppssnn0 33009 | Show that the support of a... |
| hashunif 33010 | The cardinality of a disjo... |
| hashxpe 33011 | The size of the Cartesian ... |
| hashgt1 33012 | Restate "set contains at l... |
| hashpss 33013 | The size of a proper subse... |
| hashne0 33014 | Deduce that the size of a ... |
| hashimaf1 33015 | Taking the image of a set ... |
| elq2 33016 | Elementhood in the rationa... |
| znumd 33017 | Numerator of an integer. ... |
| zdend 33018 | Denominator of an integer.... |
| numdenneg 33019 | Numerator and denominator ... |
| divnumden2 33020 | Calculate the reduced form... |
| expgt0b 33021 | A real number ` A ` raised... |
| nn0split01 33022 | Split 0 and 1 from the non... |
| nn0disj01 33023 | The pair ` { 0 , 1 } ` doe... |
| nnindf 33024 | Principle of Mathematical ... |
| nn0min 33025 | Extracting the minimum pos... |
| subne0nn 33026 | A nonnegative difference i... |
| ltesubnnd 33027 | Subtracting an integer num... |
| fprodeq02 33028 | If one of the factors is z... |
| fprodex01 33029 | A product of factors equal... |
| prodpr 33030 | A product over a pair is t... |
| prodtp 33031 | A product over a triple is... |
| fsumub 33032 | An upper bound for a term ... |
| fsumiunle 33033 | Upper bound for a sum of n... |
| dfdec100 33034 | Split the hundreds from a ... |
| sgnsgn 33035 | Signum is idempotent. (Co... |
| sgnmulsgp 33036 | If two real numbers are of... |
| nexple 33037 | A lower bound for an expon... |
| 2exple2exp 33038 | If a nonnegative integer `... |
| expevenpos 33039 | Even powers are positive. ... |
| oexpled 33040 | Odd power monomials are mo... |
| indsumin 33041 | Finite sum of a product wi... |
| prodindf 33042 | The product of indicators ... |
| indsn 33043 | The indicator function of ... |
| indf1o 33044 | The bijection between a po... |
| indpreima 33045 | A function with range ` { ... |
| indf1ofs 33046 | The bijection between fini... |
| indsupp 33047 | The support of the indicat... |
| indfsd 33048 | The indicator function of ... |
| indfsid 33049 | Conditions for a function ... |
| dp2eq1 33052 | Equality theorem for the d... |
| dp2eq2 33053 | Equality theorem for the d... |
| dp2eq1i 33054 | Equality theorem for the d... |
| dp2eq2i 33055 | Equality theorem for the d... |
| dp2eq12i 33056 | Equality theorem for the d... |
| dp20u 33057 | Add a zero in the tenths (... |
| dp20h 33058 | Add a zero in the unit pla... |
| dp2cl 33059 | Closure for the decimal fr... |
| dp2clq 33060 | Closure for a decimal frac... |
| rpdp2cl 33061 | Closure for a decimal frac... |
| rpdp2cl2 33062 | Closure for a decimal frac... |
| dp2lt10 33063 | Decimal fraction builds re... |
| dp2lt 33064 | Comparing two decimal frac... |
| dp2ltsuc 33065 | Comparing a decimal fracti... |
| dp2ltc 33066 | Comparing two decimal expa... |
| dpval 33069 | Define the value of the de... |
| dpcl 33070 | Prove that the closure of ... |
| dpfrac1 33071 | Prove a simple equivalence... |
| dpval2 33072 | Value of the decimal point... |
| dpval3 33073 | Value of the decimal point... |
| dpmul10 33074 | Multiply by 10 a decimal e... |
| decdiv10 33075 | Divide a decimal number by... |
| dpmul100 33076 | Multiply by 100 a decimal ... |
| dp3mul10 33077 | Multiply by 10 a decimal e... |
| dpmul1000 33078 | Multiply by 1000 a decimal... |
| dpval3rp 33079 | Value of the decimal point... |
| dp0u 33080 | Add a zero in the tenths p... |
| dp0h 33081 | Remove a zero in the units... |
| rpdpcl 33082 | Closure of the decimal poi... |
| dplt 33083 | Comparing two decimal expa... |
| dplti 33084 | Comparing a decimal expans... |
| dpgti 33085 | Comparing a decimal expans... |
| dpltc 33086 | Comparing two decimal inte... |
| dpexpp1 33087 | Add one zero to the mantis... |
| 0dp2dp 33088 | Multiply by 10 a decimal e... |
| dpadd2 33089 | Addition with one decimal,... |
| dpadd 33090 | Addition with one decimal.... |
| dpadd3 33091 | Addition with two decimals... |
| dpmul 33092 | Multiplication with one de... |
| dpmul4 33093 | An upper bound to multipli... |
| threehalves 33094 | Example theorem demonstrat... |
| 1mhdrd 33095 | Example theorem demonstrat... |
| xdivval 33098 | Value of division: the (un... |
| xrecex 33099 | Existence of reciprocal of... |
| xmulcand 33100 | Cancellation law for exten... |
| xreceu 33101 | Existential uniqueness of ... |
| xdivcld 33102 | Closure law for the extend... |
| xdivcl 33103 | Closure law for the extend... |
| xdivmul 33104 | Relationship between divis... |
| rexdiv 33105 | The extended real division... |
| xdivrec 33106 | Relationship between divis... |
| xdivid 33107 | A number divided by itself... |
| xdiv0 33108 | Division into zero is zero... |
| xdiv0rp 33109 | Division into zero is zero... |
| eliccioo 33110 | Membership in a closed int... |
| elxrge02 33111 | Elementhood in the set of ... |
| xdivpnfrp 33112 | Plus infinity divided by a... |
| rpxdivcld 33113 | Closure law for extended d... |
| xrpxdivcld 33114 | Closure law for extended d... |
| wrdres 33115 | Condition for the restrict... |
| wrdsplex 33116 | Existence of a split of a ... |
| wrdfsupp 33117 | A word has finite support.... |
| wrdpmcl 33118 | Closure of a word with per... |
| pfx1s2 33119 | The prefix of length 1 of ... |
| pfxrn2 33120 | The range of a prefix of a... |
| pfxrn3 33121 | Express the range of a pre... |
| pfxf1 33122 | Condition for a prefix to ... |
| s1f1 33123 | Conditions for a length 1 ... |
| s2rnOLD 33124 | Obsolete version of ~ s2rn... |
| s2f1 33125 | Conditions for a length 2 ... |
| s3rnOLD 33126 | Obsolete version of ~ s2rn... |
| s3f1 33127 | Conditions for a length 3 ... |
| s3clhash 33128 | Closure of the words of le... |
| ccatf1 33129 | Conditions for a concatena... |
| pfxlsw2ccat 33130 | Reconstruct a word from it... |
| ccatws1f1o 33131 | Conditions for the concate... |
| ccatws1f1olast 33132 | Two ways to reorder symbol... |
| wrdt2ind 33133 | Perform an induction over ... |
| swrdrn2 33134 | The range of a subword is ... |
| swrdrn3 33135 | Express the range of a sub... |
| swrdf1 33136 | Condition for a subword to... |
| swrdrndisj 33137 | Condition for the range of... |
| splfv3 33138 | Symbols to the right of a ... |
| 1cshid 33139 | Cyclically shifting a sing... |
| cshw1s2 33140 | Cyclically shifting a leng... |
| cshwrnid 33141 | Cyclically shifting a word... |
| cshf1o 33142 | Condition for the cyclic s... |
| ressplusf 33143 | The group operation functi... |
| ressnm 33144 | The norm in a restricted s... |
| abvpropd2 33145 | Weaker version of ~ abvpro... |
| ressprs 33146 | The restriction of a prose... |
| posrasymb 33147 | A poset ordering is asymme... |
| odutos 33148 | Being a toset is a self-du... |
| tlt2 33149 | In a Toset, two elements m... |
| tlt3 33150 | In a Toset, two elements m... |
| trleile 33151 | In a Toset, two elements m... |
| toslublem 33152 | Lemma for ~ toslub and ~ x... |
| toslub 33153 | In a toset, the lowest upp... |
| tosglblem 33154 | Lemma for ~ tosglb and ~ x... |
| tosglb 33155 | Same theorem as ~ toslub ,... |
| clatp0cl 33156 | The poset zero of a comple... |
| clatp1cl 33157 | The poset one of a complet... |
| mntoval 33162 | Operation value of the mon... |
| ismnt 33163 | Express the statement " ` ... |
| ismntd 33164 | Property of being a monoto... |
| mntf 33165 | A monotone function is a f... |
| mgcoval 33166 | Operation value of the mon... |
| mgcval 33167 | Monotone Galois connection... |
| mgcf1 33168 | The lower adjoint ` F ` of... |
| mgcf2 33169 | The upper adjoint ` G ` of... |
| mgccole1 33170 | An inequality for the kern... |
| mgccole2 33171 | Inequality for the closure... |
| mgcmnt1 33172 | The lower adjoint ` F ` of... |
| mgcmnt2 33173 | The upper adjoint ` G ` of... |
| mgcmntco 33174 | A Galois connection like s... |
| dfmgc2lem 33175 | Lemma for dfmgc2, backward... |
| dfmgc2 33176 | Alternate definition of th... |
| mgcmnt1d 33177 | Galois connection implies ... |
| mgcmnt2d 33178 | Galois connection implies ... |
| mgccnv 33179 | The inverse Galois connect... |
| pwrssmgc 33180 | Given a function ` F ` , e... |
| mgcf1olem1 33181 | Property of a Galois conne... |
| mgcf1olem2 33182 | Property of a Galois conne... |
| mgcf1o 33183 | Given a Galois connection,... |
| xrs0 33186 | The zero of the extended r... |
| xrslt 33187 | The "strictly less than" r... |
| xrsinvgval 33188 | The inversion operation in... |
| xrsmulgzz 33189 | The "multiple" function in... |
| xrstos 33190 | The extended real numbers ... |
| xrsclat 33191 | The extended real numbers ... |
| xrsp0 33192 | The poset 0 of the extende... |
| xrsp1 33193 | The poset 1 of the extende... |
| xrge00 33194 | The zero of the extended n... |
| xrge0mulgnn0 33195 | The group multiple functio... |
| xrge0addass 33196 | Associativity of extended ... |
| xrge0addgt0 33197 | The sum of nonnegative and... |
| xrge0adddir 33198 | Right-distributivity of ex... |
| xrge0adddi 33199 | Left-distributivity of ext... |
| xrge0npcan 33200 | Extended nonnegative real ... |
| fsumrp0cl 33201 | Closure of a finite sum of... |
| mndcld 33202 | Closure of the operation o... |
| mndassd 33203 | A monoid operation is asso... |
| mndlrinv 33204 | In a monoid, if an element... |
| mndlrinvb 33205 | In a monoid, if an element... |
| mndlactf1 33206 | If an element ` X ` of a m... |
| mndlactfo 33207 | An element ` X ` of a mono... |
| mndractf1 33208 | If an element ` X ` of a m... |
| mndractfo 33209 | An element ` X ` of a mono... |
| mndlactf1o 33210 | An element ` X ` of a mono... |
| mndractf1o 33211 | An element ` X ` of a mono... |
| cmn4d 33212 | Commutative/associative la... |
| cmn246135 33213 | Rearrange terms in a commu... |
| cmn145236 33214 | Rearrange terms in a commu... |
| submcld 33215 | Submonoids are closed unde... |
| abliso 33216 | The image of an Abelian gr... |
| lmhmghmd 33217 | A module homomorphism is a... |
| mhmimasplusg 33218 | Value of the operation of ... |
| lmhmimasvsca 33219 | Value of the scalar produc... |
| grpidcld 33220 | The identity element of a ... |
| grpinvinvd 33221 | Double inverse law for gro... |
| grpsubcld 33222 | Closure of group subtracti... |
| subgcld 33223 | A subgroup is closed under... |
| subgsubcld 33224 | A subgroup is closed under... |
| subgmulgcld 33225 | Closure of the group multi... |
| ressmulgnn0d 33226 | Values for the group multi... |
| ablcomd 33227 | An abelian group operation... |
| gsumsubg 33228 | The group sum in a subgrou... |
| gsumsra 33229 | The group sum in a subring... |
| gsummpt2co 33230 | Split a finite sum into a ... |
| gsummpt2d 33231 | Express a finite sum over ... |
| lmodvslmhm 33232 | Scalar multiplication in a... |
| gsumvsmul1 33233 | Pull a scalar multiplicati... |
| gsummptres 33234 | Extend a finite group sum ... |
| gsummptres2 33235 | Extend a finite group sum ... |
| gsummptfsres 33236 | Extend a finitely supporte... |
| gsummptf1od 33237 | Re-index a finite group su... |
| gsummptrev 33238 | Revert ordering in a group... |
| gsummptp1 33239 | Reindex a zero-based sum a... |
| gsummptfzsplitra 33240 | Split a group sum expresse... |
| gsummptfzsplitla 33241 | Split a group sum expresse... |
| gsummptfsf1o 33242 | Re-index a finite group su... |
| gsumfs2d 33243 | Express a finite sum over ... |
| gsumzresunsn 33244 | Append an element to a fin... |
| gsumpart 33245 | Express a group sum as a d... |
| gsumtp 33246 | Group sum of an unordered ... |
| gsumzrsum 33247 | Relate a group sum on ` ZZ... |
| gsummulgc2 33248 | A finite group sum multipl... |
| gsumhashmul 33249 | Express a group sum by gro... |
| gsummulsubdishift1 33250 | Distribute a subtraction o... |
| gsummulsubdishift2 33251 | Distribute a subtraction o... |
| gsummulsubdishift1s 33252 | Distribute a subtraction o... |
| gsummulsubdishift2s 33253 | Distribute a subtraction o... |
| suppgsumssiun 33254 | The support of a function ... |
| xrge0tsmsd 33255 | Any finite or infinite sum... |
| xrge0tsmsbi 33256 | Any limit of a finite or i... |
| xrge0tsmseq 33257 | Any limit of a finite or i... |
| gsumwun 33258 | In a commutative ring, a g... |
| gsumwrd2dccatlem 33259 | Lemma for ~ gsumwrd2dccat ... |
| gsumwrd2dccat 33260 | Rewrite a sum ranging over... |
| cntzun 33261 | The centralizer of a union... |
| cntzsnid 33262 | The centralizer of the ide... |
| cntrcrng 33263 | The center of a ring is a ... |
| symgfcoeu 33264 | Uniqueness property of per... |
| symgcom 33265 | Two permutations ` X ` and... |
| symgcom2 33266 | Two permutations ` X ` and... |
| symgcntz 33267 | All elements of a (finite)... |
| odpmco 33268 | The composition of two odd... |
| symgsubg 33269 | The value of the group sub... |
| pmtrprfv2 33270 | In a transposition of two ... |
| pmtrcnel 33271 | Composing a permutation ` ... |
| pmtrcnel2 33272 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 33273 | Composing a permutation ` ... |
| fzo0pmtrlast 33274 | Reorder a half-open intege... |
| wrdpmtrlast 33275 | Reorder a word, so that th... |
| pmtridf1o 33276 | Transpositions of ` X ` an... |
| pmtridfv1 33277 | Value at X of the transpos... |
| pmtridfv2 33278 | Value at Y of the transpos... |
| psgnid 33279 | Permutation sign of the id... |
| psgndmfi 33280 | For a finite base set, the... |
| pmtrto1cl 33281 | Useful lemma for the follo... |
| psgnfzto1stlem 33282 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 33283 | Value of our permutation `... |
| fzto1st1 33284 | Special case where the per... |
| fzto1st 33285 | The function moving one el... |
| fzto1stinvn 33286 | Value of the inverse of ou... |
| psgnfzto1st 33287 | The permutation sign for m... |
| tocycval 33290 | Value of the cycle builder... |
| tocycfv 33291 | Function value of a permut... |
| tocycfvres1 33292 | A cyclic permutation is a ... |
| tocycfvres2 33293 | A cyclic permutation is th... |
| cycpmfvlem 33294 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 33295 | Value of a cycle function ... |
| cycpmfv2 33296 | Value of a cycle function ... |
| cycpmfv3 33297 | Values outside of the orbi... |
| cycpmcl 33298 | Cyclic permutations are pe... |
| tocycf 33299 | The permutation cycle buil... |
| tocyc01 33300 | Permutation cycles built f... |
| cycpm2tr 33301 | A cyclic permutation of 2 ... |
| cycpm2cl 33302 | Closure for the 2-cycles. ... |
| cyc2fv1 33303 | Function value of a 2-cycl... |
| cyc2fv2 33304 | Function value of a 2-cycl... |
| trsp2cyc 33305 | Exhibit the word a transpo... |
| cycpmco2f1 33306 | The word U used in ~ cycpm... |
| cycpmco2rn 33307 | The orbit of the compositi... |
| cycpmco2lem1 33308 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 33309 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 33310 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 33311 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 33312 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 33313 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 33314 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 33315 | The composition of a cycli... |
| cyc2fvx 33316 | Function value of a 2-cycl... |
| cycpm3cl 33317 | Closure of the 3-cycles in... |
| cycpm3cl2 33318 | Closure of the 3-cycles in... |
| cyc3fv1 33319 | Function value of a 3-cycl... |
| cyc3fv2 33320 | Function value of a 3-cycl... |
| cyc3fv3 33321 | Function value of a 3-cycl... |
| cyc3co2 33322 | Represent a 3-cycle as a c... |
| cycpmconjvlem 33323 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 33324 | A formula for computing co... |
| cycpmrn 33325 | The range of the word used... |
| tocyccntz 33326 | All elements of a (finite)... |
| evpmval 33327 | Value of the set of even p... |
| cnmsgn0g 33328 | The neutral element of the... |
| evpmsubg 33329 | The alternating group is a... |
| evpmid 33330 | The identity is an even pe... |
| altgnsg 33331 | The alternating group ` ( ... |
| cyc3evpm 33332 | 3-Cycles are even permutat... |
| cyc3genpmlem 33333 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 33334 | The alternating group ` A ... |
| cycpmgcl 33335 | Cyclic permutations are pe... |
| cycpmconjslem1 33336 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 33337 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 33338 | All cycles of the same len... |
| cyc3conja 33339 | All 3-cycles are conjugate... |
| sgnsv 33342 | The sign mapping. (Contri... |
| sgnsval 33343 | The sign value. (Contribu... |
| sgnsf 33344 | The sign function. (Contr... |
| fxpval 33347 | Value of the set of fixed ... |
| fxpss 33348 | The set of fixed points is... |
| fxpgaval 33349 | Value of the set of fixed ... |
| isfxp 33350 | Property of being a fixed ... |
| fxpgaeq 33351 | A fixed point ` X ` is inv... |
| conjga 33352 | Group conjugation induces ... |
| cntrval2 33353 | Express the center ` Z ` o... |
| fxpsubm 33354 | Provided the group action ... |
| fxpsubg 33355 | The fixed points of a grou... |
| fxpsubrg 33356 | The fixed points of a grou... |
| fxpsdrg 33357 | The fixed points of a grou... |
| inftmrel 33362 | The infinitesimal relation... |
| isinftm 33363 | Express ` x ` is infinites... |
| isarchi 33364 | Express the predicate " ` ... |
| pnfinf 33365 | Plus infinity is an infini... |
| xrnarchi 33366 | The completed real line is... |
| isarchi2 33367 | Alternative way to express... |
| submarchi 33368 | A submonoid is archimedean... |
| isarchi3 33369 | This is the usual definiti... |
| archirng 33370 | Property of Archimedean or... |
| archirngz 33371 | Property of Archimedean le... |
| archiexdiv 33372 | In an Archimedean group, g... |
| archiabllem1a 33373 | Lemma for ~ archiabl : In... |
| archiabllem1b 33374 | Lemma for ~ archiabl . (C... |
| archiabllem1 33375 | Archimedean ordered groups... |
| archiabllem2a 33376 | Lemma for ~ archiabl , whi... |
| archiabllem2c 33377 | Lemma for ~ archiabl . (C... |
| archiabllem2b 33378 | Lemma for ~ archiabl . (C... |
| archiabllem2 33379 | Archimedean ordered groups... |
| archiabl 33380 | Archimedean left- and righ... |
| isarchiofld 33381 | Axiom of Archimedes : a ch... |
| isslmd 33384 | The predicate "is a semimo... |
| slmdlema 33385 | Lemma for properties of a ... |
| lmodslmd 33386 | Left semimodules generaliz... |
| slmdcmn 33387 | A semimodule is a commutat... |
| slmdmnd 33388 | A semimodule is a monoid. ... |
| slmdsrg 33389 | The scalar component of a ... |
| slmdbn0 33390 | The base set of a semimodu... |
| slmdacl 33391 | Closure of ring addition f... |
| slmdmcl 33392 | Closure of ring multiplica... |
| slmdsn0 33393 | The set of scalars in a se... |
| slmdvacl 33394 | Closure of vector addition... |
| slmdass 33395 | Semiring left module vecto... |
| slmdvscl 33396 | Closure of scalar product ... |
| slmdvsdi 33397 | Distributive law for scala... |
| slmdvsdir 33398 | Distributive law for scala... |
| slmdvsass 33399 | Associative law for scalar... |
| slmd0cl 33400 | The ring zero in a semimod... |
| slmd1cl 33401 | The ring unity in a semiri... |
| slmdvs1 33402 | Scalar product with ring u... |
| slmd0vcl 33403 | The zero vector is a vecto... |
| slmd0vlid 33404 | Left identity law for the ... |
| slmd0vrid 33405 | Right identity law for the... |
| slmd0vs 33406 | Zero times a vector is the... |
| slmdvs0 33407 | Anything times the zero ve... |
| gsumvsca1 33408 | Scalar product of a finite... |
| gsumvsca2 33409 | Scalar product of a finite... |
| prmsimpcyc 33410 | A group of prime order is ... |
| ringrngd 33411 | A unital ring is a non-uni... |
| ringdi22 33412 | Expand the product of two ... |
| urpropd 33413 | Sufficient condition for r... |
| subrgmcld 33414 | A subring is closed under ... |
| ress1r 33415 | ` 1r ` is unaffected by re... |
| ringm1expp1 33416 | Ring exponentiation of min... |
| ringinvval 33417 | The ring inverse expressed... |
| dvrcan5 33418 | Cancellation law for commo... |
| subrgchr 33419 | If ` A ` is a subring of `... |
| rmfsupp2 33420 | A mapping of a multiplicat... |
| unitnz 33421 | In a nonzero ring, a unit ... |
| isunit2 33422 | Alternate definition of be... |
| isunit3 33423 | Alternate definition of be... |
| isunitc 33424 | Characterize units in a co... |
| elrgspnlem1 33425 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem2 33426 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem3 33427 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem4 33428 | Lemma for ~ elrgspn . (Co... |
| elrgspn 33429 | Membership in the subring ... |
| elrgspnsubrunlem1 33430 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrunlem2 33431 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrun 33432 | Membership in the ring spa... |
| irrednzr 33433 | A ring with an irreducible... |
| 0ringsubrg 33434 | A subring of a zero ring i... |
| 0ringcring 33435 | The zero ring is commutati... |
| reldmrloc 33440 | Ring localization is a pro... |
| erlval 33441 | Value of the ring localiza... |
| rlocval 33442 | Expand the value of the ri... |
| erlcl1 33443 | Closure for the ring local... |
| erlcl2 33444 | Closure for the ring local... |
| erldi 33445 | Main property of the ring ... |
| erlbrd 33446 | Deduce the ring localizati... |
| erlbr2d 33447 | Deduce the ring localizati... |
| erler 33448 | The relation used to build... |
| erld2 33449 | Main property of the ring ... |
| elrlocbasi 33450 | Membership in the basis of... |
| rlocbas 33451 | The base set of a ring loc... |
| rlocaddval 33452 | Value of the addition in t... |
| rlocmulval 33453 | Value of the addition in t... |
| rloccring 33454 | The ring localization ` L ... |
| rloc0g 33455 | The zero of a ring localiz... |
| rloc1r 33456 | The multiplicative identit... |
| rlocf1 33457 | The embedding ` F ` of a r... |
| rlocinvunit 33458 | In the localization of a r... |
| rlocisunit 33459 | Characterize the units of ... |
| domnmuln0rd 33460 | In a domain, factors of a ... |
| domnprodn0 33461 | In a domain, a finite prod... |
| domnprodeq0 33462 | A product over a domain is... |
| domnpropd 33463 | If two structures have the... |
| idompropd 33464 | If two structures have the... |
| idomrcan 33465 | Right-cancellation law for... |
| domnlcanOLD 33466 | Obsolete version of ~ domn... |
| domnlcanbOLD 33467 | Obsolete version of ~ domn... |
| idomrcanOLD 33468 | Obsolete version of ~ idom... |
| 1rrg 33469 | The multiplicative identit... |
| rrgsubm 33470 | The left regular elements ... |
| subrdom 33471 | A subring of a domain is a... |
| subridom 33472 | A subring of an integral d... |
| subrfld 33473 | A subring of a field is an... |
| ricnzr1 33474 | A ring isomorphism maps a ... |
| ricdomn1 33475 | A ring isomorphism maps a ... |
| ricdomn 33476 | A ring is a domain if and ... |
| eufndx 33479 | Index value of the Euclide... |
| eufid 33480 | Utility theorem: index-ind... |
| ringinveu 33483 | If a ring unit element ` X... |
| isdrng4 33484 | A division ring is a ring ... |
| rndrhmcl 33485 | The image of a division ri... |
| qfld 33486 | The field of rational numb... |
| subsdrg 33487 | A subring of a sub-divisio... |
| sdrgdvcl 33488 | A sub-division-ring is clo... |
| sdrginvcl 33489 | A sub-division-ring is clo... |
| primefldchr 33490 | The characteristic of a pr... |
| fracval 33493 | Value of the field of frac... |
| fracbas 33494 | The base of the field of f... |
| fracerl 33495 | Rewrite the ring localizat... |
| fracf1 33496 | The embedding of a commuta... |
| fracfld 33497 | The field of fractions of ... |
| idomsubr 33498 | Every integral domain is i... |
| fldgenval 33501 | Value of the field generat... |
| fldgenssid 33502 | The field generated by a s... |
| fldgensdrg 33503 | A generated subfield is a ... |
| fldgenssv 33504 | A generated subfield is a ... |
| fldgenss 33505 | Generated subfields preser... |
| fldgenidfld 33506 | The subfield generated by ... |
| fldgenssp 33507 | The field generated by a s... |
| fldgenid 33508 | The subfield of a field ` ... |
| fldgenfld 33509 | A generated subfield is a ... |
| primefldgen1 33510 | The prime field of a divis... |
| 1fldgenq 33511 | The field of rational numb... |
| rhmdvd 33512 | A ring homomorphism preser... |
| kerunit 33513 | If a unit element lies in ... |
| reldmresv 33516 | The scalar restriction is ... |
| resvval 33517 | Value of structure restric... |
| resvid2 33518 | General behavior of trivia... |
| resvval2 33519 | Value of nontrivial struct... |
| resvsca 33520 | Base set of a structure re... |
| resvlem 33521 | Other elements of a scalar... |
| resvbas 33522 | ` Base ` is unaffected by ... |
| resvplusg 33523 | ` +g ` is unaffected by sc... |
| resvvsca 33524 | ` .s ` is unaffected by sc... |
| resvmulr 33525 | ` .r ` is unaffected by sc... |
| resv0g 33526 | ` 0g ` is unaffected by sc... |
| resv1r 33527 | ` 1r ` is unaffected by sc... |
| resvcmn 33528 | Scalar restriction preserv... |
| gzcrng 33529 | The gaussian integers form... |
| cnfldfld 33530 | The complex numbers form a... |
| reofld 33531 | The real numbers form an o... |
| nn0omnd 33532 | The nonnegative integers f... |
| gsumind 33533 | The group sum of an indica... |
| rearchi 33534 | The field of the real numb... |
| nn0archi 33535 | The monoid of the nonnegat... |
| xrge0slmod 33536 | The extended nonnegative r... |
| qusker 33537 | The kernel of a quotient m... |
| eqgvscpbl 33538 | The left coset equivalence... |
| qusvscpbl 33539 | The quotient map distribut... |
| qusvsval 33540 | Value of the scalar multip... |
| imaslmod 33541 | The image structure of a l... |
| imasmhm 33542 | Given a function ` F ` wit... |
| imasghm 33543 | Given a function ` F ` wit... |
| imasrhm 33544 | Given a function ` F ` wit... |
| imaslmhm 33545 | Given a function ` F ` wit... |
| quslmod 33546 | If ` G ` is a submodule in... |
| quslmhm 33547 | If ` G ` is a submodule of... |
| quslvec 33548 | If ` S ` is a vector subsp... |
| ecxpid 33549 | The equivalence class of a... |
| qsxpid 33550 | The quotient set of a cart... |
| qusxpid 33551 | The Group quotient equival... |
| qustriv 33552 | The quotient of a group ` ... |
| qustrivr 33553 | Converse of ~ qustriv . (... |
| znfermltl 33554 | Fermat's little theorem in... |
| islinds5 33555 | A set is linearly independ... |
| ellspds 33556 | Variation on ~ ellspd . (... |
| 0ellsp 33557 | Zero is in all spans. (Co... |
| 0nellinds 33558 | The group identity cannot ... |
| rspsnid 33559 | A principal ideal contains... |
| elrsp 33560 | Write the elements of a ri... |
| ellpi 33561 | Elementhood in a left prin... |
| lpirlidllpi 33562 | In a principal ideal ring,... |
| rspidlid 33563 | The ideal span of an ideal... |
| pidlnz 33564 | A principal ideal generate... |
| lbslsp 33565 | Any element of a left modu... |
| lindssn 33566 | Any singleton of a nonzero... |
| lindflbs 33567 | Conditions for an independ... |
| islbs5 33568 | An equivalent formulation ... |
| linds2eq 33569 | Deduce equality of element... |
| lindfpropd 33570 | Property deduction for lin... |
| lindspropd 33571 | Property deduction for lin... |
| dvdsruassoi 33572 | If two elements ` X ` and ... |
| dvdsruasso 33573 | Two elements ` X ` and ` Y... |
| dvdsruasso2 33574 | A reformulation of ~ dvdsr... |
| dvdsrspss 33575 | In a ring, an element ` X ... |
| rspsnasso 33576 | Two elements ` X ` and ` Y... |
| unitprodclb 33577 | A finite product is a unit... |
| elgrplsmsn 33578 | Membership in a sumset wit... |
| lsmsnorb 33579 | The sumset of a group with... |
| lsmsnorb2 33580 | The sumset of a single ele... |
| elringlsm 33581 | Membership in a product of... |
| elringlsmd 33582 | Membership in a product of... |
| ringlsmss 33583 | Closure of the product of ... |
| ringlsmss1 33584 | The product of an ideal ` ... |
| ringlsmss2 33585 | The product with an ideal ... |
| lsmsnpridl 33586 | The product of the ring wi... |
| lsmsnidl 33587 | The product of the ring wi... |
| lsmidllsp 33588 | The sum of two ideals is t... |
| lsmidl 33589 | The sum of two ideals is a... |
| lsmssass 33590 | Group sum is associative, ... |
| grplsm0l 33591 | Sumset with the identity s... |
| grplsmid 33592 | The direct sum of an eleme... |
| quslsm 33593 | Express the image by the q... |
| qusbas2 33594 | Alternate definition of th... |
| qus0g 33595 | The identity element of a ... |
| qusima 33596 | The image of a subgroup by... |
| qusrn 33597 | The natural map from eleme... |
| nsgqus0 33598 | A normal subgroup ` N ` is... |
| nsgmgclem 33599 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 33600 | There is a monotone Galois... |
| nsgqusf1olem1 33601 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 33602 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 33603 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 33604 | The canonical projection h... |
| lmhmqusker 33605 | A surjective module homomo... |
| lmicqusker 33606 | The image ` H ` of a modul... |
| lidlmcld 33607 | An ideal is closed under l... |
| intlidl 33608 | The intersection of a none... |
| 0ringidl 33609 | The zero ideal is the only... |
| pidlnzb 33610 | A principal ideal is nonze... |
| lidlunitel 33611 | If an ideal ` I ` contains... |
| unitpidl1 33612 | The ideal ` I ` generated ... |
| rhmquskerlem 33613 | The mapping ` J ` induced ... |
| rhmqusker 33614 | A surjective ring homomorp... |
| ricqusker 33615 | The image ` H ` of a ring ... |
| elrspunidl 33616 | Elementhood in the span of... |
| elrspunsn 33617 | Membership to the span of ... |
| lidlincl 33618 | Ideals are closed under in... |
| idlinsubrg 33619 | The intersection between a... |
| rhmimaidl 33620 | The image of an ideal ` I ... |
| drngidl 33621 | A nonzero ring is a divisi... |
| drngidlhash 33622 | A ring is a division ring ... |
| prmidlval 33625 | The class of prime ideals ... |
| isprmidl 33626 | The predicate "is a prime ... |
| prmidlnr 33627 | A prime ideal is a proper ... |
| prmidl 33628 | The main property of a pri... |
| prmidl2 33629 | A condition that shows an ... |
| idlmulssprm 33630 | Let ` P ` be a prime ideal... |
| pridln1 33631 | A proper ideal cannot cont... |
| prmidlidl 33632 | A prime ideal is an ideal.... |
| prmidlssidl 33633 | Prime ideals as a subset o... |
| cringm4 33634 | Commutative/associative la... |
| isprmidlc 33635 | The predicate "is prime id... |
| prmidlc 33636 | Property of a prime ideal ... |
| prmidlprop 33637 | Property of prime ideals. ... |
| 0ringprmidl 33638 | The trivial ring does not ... |
| prmidl0 33639 | The zero ideal of a commut... |
| rhmpreimaprmidl 33640 | The preimage of a prime id... |
| qsidomlem1 33641 | If the quotient ring of a ... |
| qsidomlem2 33642 | A quotient by a prime idea... |
| qsidom 33643 | An ideal ` I ` in the comm... |
| qsnzr 33644 | A quotient of a nonzero ri... |
| ssdifidllem 33645 | Lemma for ~ ssdifidl : Th... |
| ssdifidl 33646 | Let ` R ` be a ring, and l... |
| ssdifidlprm 33647 | If the set ` S ` of ~ ssdi... |
| prmidlsubm 33648 | The complement of a prime ... |
| mxidlval 33651 | The set of maximal ideals ... |
| ismxidl 33652 | The predicate "is a maxima... |
| mxidlidl 33653 | A maximal ideal is an idea... |
| mxidlnr 33654 | A maximal ideal is proper.... |
| mxidlmax 33655 | A maximal ideal is a maxim... |
| mxidln1 33656 | One is not contained in an... |
| mxidlnzr 33657 | A ring with a maximal idea... |
| mxidlmaxv 33658 | An ideal ` I ` strictly co... |
| crngmxidl 33659 | In a commutative ring, max... |
| mxidlprm 33660 | Every maximal ideal is pri... |
| mxidlirredi 33661 | In an integral domain, the... |
| mxidlirred 33662 | In a principal ideal domai... |
| ssmxidllem 33663 | The set ` P ` used in the ... |
| ssmxidl 33664 | Let ` R ` be a ring, and l... |
| drnglidl1ne0 33665 | In a nonzero ring, the zer... |
| drng0mxidl 33666 | In a division ring, the ze... |
| drngmxidl 33667 | The zero ideal is the only... |
| drngmxidlr 33668 | If a ring's only maximal i... |
| krull 33669 | Krull's theorem: Any nonz... |
| mxidlnzrb 33670 | A ring is nonzero if and o... |
| krullndrng 33671 | Krull's theorem for non-di... |
| opprabs 33672 | The opposite ring of the o... |
| oppreqg 33673 | Group coset equivalence re... |
| opprnsg 33674 | Normal subgroups of the op... |
| opprlidlabs 33675 | The ideals of the opposite... |
| oppr2idl 33676 | Two sided ideal of the opp... |
| opprmxidlabs 33677 | The maximal ideal of the o... |
| opprqusbas 33678 | The base of the quotient o... |
| opprqusplusg 33679 | The group operation of the... |
| opprqus0g 33680 | The group identity element... |
| opprqusmulr 33681 | The multiplication operati... |
| opprqus1r 33682 | The ring unity of the quot... |
| opprqusdrng 33683 | The quotient of the opposi... |
| qsdrngilem 33684 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 33685 | A quotient by a maximal le... |
| qsdrnglem2 33686 | Lemma for ~ qsdrng . (Con... |
| qsdrng 33687 | An ideal ` M ` is both lef... |
| qsfld 33688 | An ideal ` M ` in the comm... |
| mxidlprmALT 33689 | Every maximal ideal is pri... |
| drnglring 33690 | A division ring is a local... |
| dflring2 33691 | Alternate definition of a ... |
| dflringlem 33692 | Lemma for ~ dflring3 . If... |
| dflringlem2 33693 | Lemma for ~ dflring3 . In... |
| dflringlem3 33694 | Lemma for ~ dflring3 . In... |
| dflring3 33695 | Alternate definition of a ... |
| dflring4 33696 | Alternate definition of a ... |
| fldlring 33697 | A field is a local ring. ... |
| idlsrgstr 33700 | A constructed semiring of ... |
| idlsrgval 33701 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 33702 | Base of the ideals of a ri... |
| idlsrgplusg 33703 | Additive operation of the ... |
| idlsrg0g 33704 | The zero ideal is the addi... |
| idlsrgmulr 33705 | Multiplicative operation o... |
| idlsrgtset 33706 | Topology component of the ... |
| idlsrgmulrval 33707 | Value of the ring multipli... |
| idlsrgmulrcl 33708 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 33709 | In a commutative ring, the... |
| idlsrgmulrss2 33710 | The product of two ideals ... |
| idlsrgmulrssin 33711 | In a commutative ring, the... |
| idlsrgmnd 33712 | The ideals of a ring form ... |
| idlsrgcmnd 33713 | The ideals of a ring form ... |
| rprmval 33714 | The prime elements of a ri... |
| isrprm 33715 | Property for ` P ` to be a... |
| rprmcl 33716 | A ring prime is an element... |
| rprmdvds 33717 | If a ring prime ` Q ` divi... |
| rprmnz 33718 | A ring prime is nonzero. ... |
| rprmnunit 33719 | A ring prime is not a unit... |
| rsprprmprmidl 33720 | In a commutative ring, ide... |
| rsprprmprmidlb 33721 | An ideal generated by a si... |
| rprmndvdsr1 33722 | A ring prime element does ... |
| rprmasso 33723 | In an integral domain, the... |
| rprmasso2 33724 | In an integral domain, if ... |
| rprmasso3 33725 | In an integral domain, if ... |
| unitmulrprm 33726 | A ring unit multiplied by ... |
| rprmndvdsru 33727 | A ring prime element does ... |
| rprmirredlem 33728 | Lemma for ~ rprmirred . (... |
| rprmirred 33729 | In an integral domain, rin... |
| rprmirredb 33730 | In a principal ideal domai... |
| rprmdvdspow 33731 | If a prime element divides... |
| rprmdvdsprod 33732 | If a prime element ` Q ` d... |
| 1arithidomlem1 33733 | Lemma for ~ 1arithidom . ... |
| 1arithidomlem2 33734 | Lemma for ~ 1arithidom : i... |
| 1arithidom 33735 | Uniqueness of prime factor... |
| isufd 33738 | The property of being a Un... |
| ufdprmidl 33739 | In a unique factorization ... |
| ufdidom 33740 | A nonzero unique factoriza... |
| pidufd 33741 | Every principal ideal doma... |
| 1arithufdlem1 33742 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem2 33743 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem3 33744 | Lemma for ~ 1arithufd . I... |
| 1arithufdlem4 33745 | Lemma for ~ 1arithufd . N... |
| 1arithufd 33746 | Existence of a factorizati... |
| dfufd2lem 33747 | Lemma for ~ dfufd2 . (Con... |
| dfufd2 33748 | Alternative definition of ... |
| zringidom 33749 | The ring of integers is an... |
| zringpid 33750 | The ring of integers is a ... |
| dfprm3 33751 | The (positive) prime eleme... |
| zringfrac 33752 | The field of fractions of ... |
| assaassd 33753 | Left-associative property ... |
| assaassrd 33754 | Right-associative property... |
| 0ringmon1p 33755 | There are no monic polynom... |
| fply1 33756 | Conditions for a function ... |
| ply1lvec 33757 | In a division ring, the un... |
| evls1fn 33758 | Functionality of the subri... |
| evls1dm 33759 | The domain of the subring ... |
| evls1fvf 33760 | The subring evaluation fun... |
| evl1fvf 33761 | The univariate polynomial ... |
| evl1fpws 33762 | Evaluation of a univariate... |
| ressply1evls1 33763 | Subring evaluation of a un... |
| ressdeg1 33764 | The degree of a univariate... |
| ressply10g 33765 | A restricted polynomial al... |
| ressply1mon1p 33766 | The monic polynomials of a... |
| ressply1invg 33767 | An element of a restricted... |
| ressply1sub 33768 | A restricted polynomial al... |
| ressasclcl 33769 | Closure of the univariate ... |
| evls1subd 33770 | Univariate polynomial eval... |
| deg1le0eq0 33771 | A polynomial with nonposit... |
| ply1asclunit 33772 | A nonzero scalar polynomia... |
| ply1unit 33773 | In a field ` F ` , a polyn... |
| evl1deg1 33774 | Evaluation of a univariate... |
| evl1deg2 33775 | Evaluation of a univariate... |
| evl1deg3 33776 | Evaluation of a univariate... |
| evls1monply1 33777 | Subring evaluation of a sc... |
| ply1dg1rt 33778 | Express the root ` - B / A... |
| ply1dg1rtn0 33779 | Polynomials of degree 1 ov... |
| ply1mulrtss 33780 | The roots of a factor ` F ... |
| deg1prod 33781 | Degree of a product of pol... |
| ply1dg3rt0irred 33782 | If a cubic polynomial over... |
| m1pmeq 33783 | If two monic polynomials `... |
| ply1fermltl 33784 | Fermat's little theorem fo... |
| coe1mon 33785 | Coefficient vector of a mo... |
| ply1moneq 33786 | Two monomials are equal if... |
| ply1coedeg 33787 | Decompose a univariate pol... |
| coe1zfv 33788 | The coefficients of the ze... |
| coe1vr1 33789 | Polynomial coefficient of ... |
| deg1vr 33790 | The degree of the variable... |
| vr1nz 33791 | A univariate polynomial va... |
| ply1degltel 33792 | Characterize elementhood i... |
| ply1degleel 33793 | Characterize elementhood i... |
| ply1degltlss 33794 | The space ` S ` of the uni... |
| gsummoncoe1fzo 33795 | A coefficient of the polyn... |
| gsummoncoe1fz 33796 | A coefficient of the polyn... |
| ply1gsumz 33797 | If a polynomial given as a... |
| deg1addlt 33798 | If both factors have degre... |
| ig1pnunit 33799 | The polynomial ideal gener... |
| ig1pmindeg 33800 | The polynomial ideal gener... |
| q1pdir 33801 | Distribution of univariate... |
| q1pvsca 33802 | Scalar multiplication prop... |
| r1pvsca 33803 | Scalar multiplication prop... |
| r1p0 33804 | Polynomial remainder opera... |
| r1pcyc 33805 | The polynomial remainder o... |
| r1padd1 33806 | Addition property of the p... |
| r1pid2OLD 33807 | Obsolete version of ~ r1pi... |
| r1plmhm 33808 | The univariate polynomial ... |
| r1pquslmic 33809 | The univariate polynomial ... |
| psrbasfsupp 33810 | Rewrite a finite support f... |
| psrnzr 33811 | The ring of power series o... |
| mplnzr 33812 | The multivariate polynomia... |
| 0mplrim 33813 | Build a ring isomorphism b... |
| 0mplric 33814 | Multivariate polynomials w... |
| mplasclco 33815 | Case where composing an al... |
| selvascl 33816 | The "variable selection" f... |
| selvply1rhmlema 33817 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlemb 33818 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem1 33819 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem2 33820 | Lemma for ~ selvply1rhm : ... |
| selvply1rhmlem3 33821 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem4 33822 | Lemma for ~ selvply1rhm : ... |
| selvply1rhmlem5 33823 | Lemma for ~ selvply1rhm . ... |
| selvply1rhm 33824 | Build a ring homomorphism ... |
| selvply1rhm0 33825 | The ring homomorphism ` H ... |
| mplidomlem 33826 | Lemma for ~ mplidom . (Co... |
| mplidom 33827 | The multivariate polynomia... |
| extvval 33830 | Value of the "variable ext... |
| extvfval 33831 | The "variable extension" f... |
| extvfv 33832 | The "variable extension" f... |
| extvfvv 33833 | The "variable extension" f... |
| extvfvvcl 33834 | Closure for the "variable ... |
| extvfvcl 33835 | Closure for the "variable ... |
| extvfvalf 33836 | The "variable extension" f... |
| mvrvalind 33837 | Value of the generating el... |
| mplmulmvr 33838 | Multiply a polynomial ` F ... |
| evlscaval 33839 | Polynomial evaluation for ... |
| evlvarval 33840 | Polynomial evaluation buil... |
| evlextv 33841 | Evaluating a variable-exte... |
| mplvrpmlem 33842 | Lemma for ~ mplvrpmga and ... |
| mplvrpmfgalem 33843 | Permuting variables in a m... |
| mplvrpmga 33844 | The action of permuting va... |
| mplvrpmmhm 33845 | The action of permuting va... |
| mplvrpmrhm 33846 | The action of permuting va... |
| psrgsum 33847 | Finite commutative sums of... |
| psrmon 33848 | A monomial is a power seri... |
| psrmonmul 33849 | The product of two power s... |
| psrmonmul2 33850 | The product of two power s... |
| psrmonprod 33851 | Finite product of bags of ... |
| mplgsum 33852 | Finite commutative sums of... |
| mplmonprod 33853 | Finite product of monomial... |
| splyval 33858 | The symmetric polynomials ... |
| splysubrg 33859 | The symmetric polynomials ... |
| issply 33860 | Conditions for being a sym... |
| esplyval 33861 | The elementary polynomials... |
| esplyfval 33862 | The ` K ` -th elementary p... |
| esplyfval0 33863 | The ` 0 ` -th elementary s... |
| esplyfval2 33864 | When ` K ` is out-of-bound... |
| esplylem 33865 | Lemma for ~ esplyfv and ot... |
| esplympl 33866 | Elementary symmetric polyn... |
| esplymhp 33867 | The ` K ` -th elementary s... |
| esplyfv1 33868 | Coefficient for the ` K ` ... |
| esplyfv 33869 | Coefficient for the ` K ` ... |
| esplysply 33870 | The ` K ` -th elementary s... |
| esplyfval3 33871 | Alternate expression for t... |
| esplyfval1 33872 | The first elementary symme... |
| esplyfvaln 33873 | The last elementary symmet... |
| esplyind 33874 | A recursive formula for th... |
| esplyindfv 33875 | A recursive formula for th... |
| esplyfvn 33876 | Express the last elementar... |
| vietadeg1 33877 | The degree of a product of... |
| vietalem 33878 | Lemma for ~ vieta : induct... |
| vieta 33879 | Vieta's Formulas: Coeffic... |
| sra1r 33880 | The unity element of a sub... |
| sradrng 33881 | Condition for a subring al... |
| sraidom 33882 | Condition for a subring al... |
| srasubrg 33883 | A subring of the original ... |
| sralvec 33884 | Given a sub division ring ... |
| srafldlvec 33885 | Given a subfield ` F ` of ... |
| resssra 33886 | The subring algebra of a r... |
| lsssra 33887 | A subring is a subspace of... |
| srapwov 33888 | The "power" operation on a... |
| drgext0g 33889 | The additive neutral eleme... |
| drgextvsca 33890 | The scalar multiplication ... |
| drgext0gsca 33891 | The additive neutral eleme... |
| drgextsubrg 33892 | The scalar field is a subr... |
| drgextlsp 33893 | The scalar field is a subs... |
| drgextgsum 33894 | Group sum in a division ri... |
| lvecdimfi 33895 | Finite version of ~ lvecdi... |
| exsslsb 33896 | Any finite generating set ... |
| lbslelsp 33897 | The size of a basis ` X ` ... |
| dimval 33900 | The dimension of a vector ... |
| dimvalfi 33901 | The dimension of a vector ... |
| dimcl 33902 | Closure of the vector spac... |
| lmimdim 33903 | Module isomorphisms preser... |
| lmicdim 33904 | Module isomorphisms preser... |
| lvecdim0i 33905 | A vector space of dimensio... |
| lvecdim0 33906 | A vector space of dimensio... |
| lssdimle 33907 | The dimension of a linear ... |
| dimpropd 33908 | If two structures have the... |
| rlmdim 33909 | The left vector space indu... |
| frlmdim 33910 | Dimension of a free left m... |
| tnglvec 33911 | Augmenting a structure wit... |
| tngdim 33912 | Dimension of a left vector... |
| rrxdim 33913 | Dimension of the generaliz... |
| matdim 33914 | Dimension of the space of ... |
| lbslsat 33915 | A nonzero vector ` X ` is ... |
| lsatdim 33916 | A line, spanned by a nonze... |
| drngdimgt0 33917 | The dimension of a vector ... |
| lmhmlvec2 33918 | A homomorphism of left vec... |
| kerlmhm 33919 | The kernel of a vector spa... |
| imlmhm 33920 | The image of a vector spac... |
| ply1degltdimlem 33921 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 33922 | The space ` S ` of the uni... |
| lindsunlem 33923 | Lemma for ~ lindsun . (Co... |
| lindsun 33924 | Condition for the union of... |
| lbsdiflsp0 33925 | The linear spans of two di... |
| dimkerim 33926 | Given a linear map ` F ` b... |
| qusdimsum 33927 | Let ` W ` be a vector spac... |
| fedgmullem1 33928 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 33929 | Lemma for ~ fedgmul . (Co... |
| fedgmul 33930 | The multiplicativity formu... |
| dimlssid 33931 | If the dimension of a line... |
| lvecendof1f1o 33932 | If an endomorphism ` U ` o... |
| lactlmhm 33933 | In an associative algebra ... |
| assalactf1o 33934 | In an associative algebra ... |
| assarrginv 33935 | If an element ` X ` of an ... |
| assafld 33936 | If an algebra ` A ` of fin... |
| relfldext 33943 | The field extension is a r... |
| brfldext 33944 | The field extension relati... |
| ccfldextrr 33945 | The field of the complex n... |
| fldextfld1 33946 | A field extension is only ... |
| fldextfld2 33947 | A field extension is only ... |
| fldextsubrg 33948 | Field extension implies a ... |
| sdrgfldext 33949 | A field ` E ` and any sub-... |
| fldextress 33950 | Field extension implies a ... |
| brfinext 33951 | The finite field extension... |
| extdgval 33952 | Value of the field extensi... |
| fldextsdrg 33953 | Deduce sub-division-ring f... |
| fldextsralvec 33954 | The subring algebra associ... |
| extdgcl 33955 | Closure of the field exten... |
| extdggt0 33956 | Degrees of field extension... |
| fldexttr 33957 | Field extension is a trans... |
| fldextid 33958 | The field extension relati... |
| extdgid 33959 | A trivial field extension ... |
| fldsdrgfldext 33960 | A sub-division-ring of a f... |
| fldsdrgfldext2 33961 | A sub-sub-division-ring of... |
| extdgmul 33962 | The multiplicativity formu... |
| finextfldext 33963 | A finite field extension i... |
| finexttrb 33964 | The extension ` E ` of ` K... |
| extdg1id 33965 | If the degree of the exten... |
| extdg1b 33966 | The degree of the extensio... |
| fldgenfldext 33967 | A subfield ` F ` extended ... |
| fldextchr 33968 | The characteristic of a su... |
| evls1fldgencl 33969 | Closure of the subring pol... |
| ccfldsrarelvec 33970 | The subring algebra of the... |
| ccfldextdgrr 33971 | The degree of the field ex... |
| fldextrspunlsplem 33972 | Lemma for ~ fldextrspunlsp... |
| fldextrspunlsp 33973 | Lemma for ~ fldextrspunfld... |
| fldextrspunlem1 33974 | Lemma for ~ fldextrspunfld... |
| fldextrspunfld 33975 | The ring generated by the ... |
| fldextrspunlem2 33976 | Part of the proof of Propo... |
| fldextrspundgle 33977 | Inequality involving the d... |
| fldextrspundglemul 33978 | Given two field extensions... |
| fldextrspundgdvdslem 33979 | Lemma for ~ fldextrspundgd... |
| fldextrspundgdvds 33980 | Given two finite extension... |
| fldext2rspun 33981 | Given two field extensions... |
| irngval 33984 | The elements of a field ` ... |
| elirng 33985 | Property for an element ` ... |
| irngss 33986 | All elements of a subring ... |
| irngssv 33987 | An integral element is an ... |
| 0ringirng 33988 | A zero ring ` R ` has no i... |
| irngnzply1lem 33989 | In the case of a field ` E... |
| irngnzply1 33990 | In the case of a field ` E... |
| extdgfialglem1 33991 | Lemma for ~ extdgfialg . ... |
| extdgfialglem2 33992 | Lemma for ~ extdgfialg . ... |
| extdgfialg 33993 | A finite field extension `... |
| bralgext 33996 | Express the fact that a fi... |
| finextalg 33997 | A finite field extension i... |
| ply1annidllem 34000 | Write the set ` Q ` of pol... |
| ply1annidl 34001 | The set ` Q ` of polynomia... |
| ply1annnr 34002 | The set ` Q ` of polynomia... |
| ply1annig1p 34003 | The ideal ` Q ` of polynom... |
| minplyval 34004 | Expand the value of the mi... |
| minplycl 34005 | The minimal polynomial is ... |
| ply1annprmidl 34006 | The set ` Q ` of polynomia... |
| minplymindeg 34007 | The minimal polynomial of ... |
| minplyann 34008 | The minimal polynomial for... |
| minplyirredlem 34009 | Lemma for ~ minplyirred . ... |
| minplyirred 34010 | A nonzero minimal polynomi... |
| irngnminplynz 34011 | Integral elements have non... |
| minplym1p 34012 | A minimal polynomial is mo... |
| minplynzm1p 34013 | If a minimal polynomial is... |
| minplyelirng 34014 | If the minimal polynomial ... |
| irredminply 34015 | An irreducible, monic, ann... |
| algextdeglem1 34016 | Lemma for ~ algextdeg . (... |
| algextdeglem2 34017 | Lemma for ~ algextdeg . B... |
| algextdeglem3 34018 | Lemma for ~ algextdeg . T... |
| algextdeglem4 34019 | Lemma for ~ algextdeg . B... |
| algextdeglem5 34020 | Lemma for ~ algextdeg . T... |
| algextdeglem6 34021 | Lemma for ~ algextdeg . B... |
| algextdeglem7 34022 | Lemma for ~ algextdeg . T... |
| algextdeglem8 34023 | Lemma for ~ algextdeg . T... |
| algextdeg 34024 | The degree of an algebraic... |
| rtelextdg2lem 34025 | Lemma for ~ rtelextdg2 : ... |
| rtelextdg2 34026 | If an element ` X ` is a s... |
| fldext2chn 34027 | In a non-empty chain ` T `... |
| constrrtll 34030 | In the construction of con... |
| constrrtlc1 34031 | In the construction of con... |
| constrrtlc2 34032 | In the construction of con... |
| constrrtcclem 34033 | In the construction of con... |
| constrrtcc 34034 | In the construction of con... |
| isconstr 34035 | Property of being a constr... |
| constr0 34036 | The first step of the cons... |
| constrsuc 34037 | Membership in the successo... |
| constrlim 34038 | Limit step of the construc... |
| constrsscn 34039 | Closure of the constructib... |
| constrsslem 34040 | Lemma for ~ constrss . Th... |
| constr01 34041 | ` 0 ` and ` 1 ` are in all... |
| constrss 34042 | Constructed points are in ... |
| constrmon 34043 | The construction of constr... |
| constrconj 34044 | If a point ` X ` of the co... |
| constrfin 34045 | Each step of the construct... |
| constrelextdg2 34046 | If the ` N ` -th step ` ( ... |
| constrextdg2lem 34047 | Lemma for ~ constrextdg2 .... |
| constrextdg2 34048 | Any step ` ( C `` N ) ` of... |
| constrext2chnlem 34049 | Lemma for ~ constrext2chn ... |
| constrfiss 34050 | For any finite set ` A ` o... |
| constrllcllem 34051 | Constructible numbers are ... |
| constrlccllem 34052 | Constructible numbers are ... |
| constrcccllem 34053 | Constructible numbers are ... |
| constrcbvlem 34054 | Technical lemma for elimin... |
| constrllcl 34055 | Constructible numbers are ... |
| constrlccl 34056 | Constructible numbers are ... |
| constrcccl 34057 | Constructible numbers are ... |
| constrext2chn 34058 | If a constructible number ... |
| constrcn 34059 | Constructible numbers are ... |
| nn0constr 34060 | Nonnegative integers are c... |
| constraddcl 34061 | Constructive numbers are c... |
| constrnegcl 34062 | Constructible numbers are ... |
| zconstr 34063 | Integers are constructible... |
| constrdircl 34064 | Constructible numbers are ... |
| iconstr 34065 | The imaginary unit ` _i ` ... |
| constrremulcl 34066 | If two real numbers ` X ` ... |
| constrcjcl 34067 | Constructible numbers are ... |
| constrrecl 34068 | Constructible numbers are ... |
| constrimcl 34069 | Constructible numbers are ... |
| constrmulcl 34070 | Constructible numbers are ... |
| constrreinvcl 34071 | If a real number ` X ` is ... |
| constrinvcl 34072 | Constructible numbers are ... |
| constrcon 34073 | Contradiction of construct... |
| constrsdrg 34074 | Constructible numbers form... |
| constrfld 34075 | The constructible numbers ... |
| constrresqrtcl 34076 | If a positive real number ... |
| constrabscl 34077 | Constructible numbers are ... |
| constrsqrtcl 34078 | Constructible numbers are ... |
| 2sqr3minply 34079 | The polynomial ` ( ( X ^ 3... |
| 2sqr3nconstr 34080 | Doubling the cube is an im... |
| cos9thpiminplylem1 34081 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem2 34082 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem3 34083 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem4 34084 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem5 34085 | The constructed complex nu... |
| cos9thpiminplylem6 34086 | Evaluation of the polynomi... |
| cos9thpiminply 34087 | The polynomial ` ( ( X ^ 3... |
| cos9thpinconstrlem1 34088 | The complex number ` O ` ,... |
| cos9thpinconstrlem2 34089 | The complex number ` A ` i... |
| cos9thpinconstr 34090 | Trisecting an angle is an ... |
| trisecnconstr 34091 | Not all angles can be tris... |
| smatfval 34094 | Value of the submatrix. (... |
| smatrcl 34095 | Closure of the rectangular... |
| smatlem 34096 | Lemma for the next theorem... |
| smattl 34097 | Entries of a submatrix, to... |
| smattr 34098 | Entries of a submatrix, to... |
| smatbl 34099 | Entries of a submatrix, bo... |
| smatbr 34100 | Entries of a submatrix, bo... |
| smatcl 34101 | Closure of the square subm... |
| matmpo 34102 | Write a square matrix as a... |
| 1smat1 34103 | The submatrix of the ident... |
| submat1n 34104 | One case where the submatr... |
| submatres 34105 | Special case where the sub... |
| submateqlem1 34106 | Lemma for ~ submateq . (C... |
| submateqlem2 34107 | Lemma for ~ submateq . (C... |
| submateq 34108 | Sufficient condition for t... |
| submatminr1 34109 | If we take a submatrix by ... |
| lmatval 34112 | Value of the literal matri... |
| lmatfval 34113 | Entries of a literal matri... |
| lmatfvlem 34114 | Useful lemma to extract li... |
| lmatcl 34115 | Closure of the literal mat... |
| lmat22lem 34116 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 34117 | Entry of a 2x2 literal mat... |
| lmat22e12 34118 | Entry of a 2x2 literal mat... |
| lmat22e21 34119 | Entry of a 2x2 literal mat... |
| lmat22e22 34120 | Entry of a 2x2 literal mat... |
| lmat22det 34121 | The determinant of a liter... |
| mdetpmtr1 34122 | The determinant of a matri... |
| mdetpmtr2 34123 | The determinant of a matri... |
| mdetpmtr12 34124 | The determinant of a matri... |
| mdetlap1 34125 | A Laplace expansion of the... |
| madjusmdetlem1 34126 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 34127 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 34128 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 34129 | Lemma for ~ madjusmdet . ... |
| madjusmdet 34130 | Express the cofactor of th... |
| mdetlap 34131 | Laplace expansion of the d... |
| ist0cld 34132 | The predicate "is a T_0 sp... |
| txomap 34133 | Given two open maps ` F ` ... |
| qtopt1 34134 | If every equivalence class... |
| qtophaus 34135 | If an open map's graph in ... |
| circtopn 34136 | The topology of the unit c... |
| circcn 34137 | The function gluing the re... |
| reff 34138 | For any cover refinement, ... |
| locfinreflem 34139 | A locally finite refinemen... |
| locfinref 34140 | A locally finite refinemen... |
| iscref 34143 | The property that every op... |
| crefeq 34144 | Equality theorem for the "... |
| creftop 34145 | A space where every open c... |
| crefi 34146 | The property that every op... |
| crefdf 34147 | A formulation of ~ crefi e... |
| crefss 34148 | The "every open cover has ... |
| cmpcref 34149 | Equivalent definition of c... |
| cmpfiref 34150 | Every open cover of a Comp... |
| ldlfcntref 34153 | Every open cover of a Lind... |
| ispcmp 34156 | The predicate "is a paraco... |
| cmppcmp 34157 | Every compact space is par... |
| dispcmp 34158 | Every discrete space is pa... |
| pcmplfin 34159 | Given a paracompact topolo... |
| pcmplfinf 34160 | Given a paracompact topolo... |
| rspecval 34163 | Value of the spectrum of t... |
| rspecbas 34164 | The prime ideals form the ... |
| rspectset 34165 | Topology component of the ... |
| rspectopn 34166 | The topology component of ... |
| zarcls0 34167 | The closure of the identit... |
| zarcls1 34168 | The unit ideal ` B ` is th... |
| zarclsun 34169 | The union of two closed se... |
| zarclsiin 34170 | In a Zariski topology, the... |
| zarclsint 34171 | The intersection of a fami... |
| zarclssn 34172 | The closed points of Zaris... |
| zarcls 34173 | The open sets of the Zaris... |
| zartopn 34174 | The Zariski topology is a ... |
| zartop 34175 | The Zariski topology is a ... |
| zartopon 34176 | The points of the Zariski ... |
| zar0ring 34177 | The Zariski Topology of th... |
| zart0 34178 | The Zariski topology is T_... |
| zarmxt1 34179 | The Zariski topology restr... |
| zarcmplem 34180 | Lemma for ~ zarcmp . (Con... |
| zarcmp 34181 | The Zariski topology is co... |
| rspectps 34182 | The spectrum of a ring ` R... |
| rhmpreimacnlem 34183 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 34184 | The function mapping a pri... |
| metidval 34189 | Value of the metric identi... |
| metidss 34190 | As a relation, the metric ... |
| metidv 34191 | ` A ` and ` B ` identify b... |
| metideq 34192 | Basic property of the metr... |
| metider 34193 | The metric identification ... |
| pstmval 34194 | Value of the metric induce... |
| pstmfval 34195 | Function value of the metr... |
| pstmxmet 34196 | The metric induced by a ps... |
| hauseqcn 34197 | In a Hausdorff topology, t... |
| elunitge0 34198 | An element of the closed u... |
| unitssxrge0 34199 | The closed unit interval i... |
| unitdivcld 34200 | Necessary conditions for a... |
| iistmd 34201 | The closed unit interval f... |
| unicls 34202 | The union of the closed se... |
| tpr2tp 34203 | The usual topology on ` ( ... |
| tpr2uni 34204 | The usual topology on ` ( ... |
| xpinpreima 34205 | Rewrite the cartesian prod... |
| xpinpreima2 34206 | Rewrite the cartesian prod... |
| sqsscirc1 34207 | The complex square of side... |
| sqsscirc2 34208 | The complex square of side... |
| cnre2csqlem 34209 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 34210 | Image of a centered square... |
| tpr2rico 34211 | For any point of an open s... |
| cnvordtrestixx 34212 | The restriction of the 'gr... |
| prsdm 34213 | Domain of the relation of ... |
| prsrn 34214 | Range of the relation of a... |
| prsss 34215 | Relation of a subproset. ... |
| prsssdm 34216 | Domain of a subproset rela... |
| ordtprsval 34217 | Value of the order topolog... |
| ordtprsuni 34218 | Value of the order topolog... |
| ordtcnvNEW 34219 | The order dual generates t... |
| ordtrestNEW 34220 | The subspace topology of a... |
| ordtrest2NEWlem 34221 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 34222 | An interval-closed set ` A... |
| ordtconnlem1 34223 | Connectedness in the order... |
| ordtconn 34224 | Connectedness in the order... |
| mndpluscn 34225 | A mapping that is both a h... |
| mhmhmeotmd 34226 | Deduce a Topological Monoi... |
| rmulccn 34227 | Multiplication by a real c... |
| raddcn 34228 | Addition in the real numbe... |
| xrmulc1cn 34229 | The operation multiplying ... |
| fmcncfil 34230 | The image of a Cauchy filt... |
| xrge0hmph 34231 | The extended nonnegative r... |
| xrge0iifcnv 34232 | Define a bijection from ` ... |
| xrge0iifcv 34233 | The defined function's val... |
| xrge0iifiso 34234 | The defined bijection from... |
| xrge0iifhmeo 34235 | Expose a homeomorphism fro... |
| xrge0iifhom 34236 | The defined function from ... |
| xrge0iif1 34237 | Condition for the defined ... |
| xrge0iifmhm 34238 | The defined function from ... |
| xrge0pluscn 34239 | The addition operation of ... |
| xrge0mulc1cn 34240 | The operation multiplying ... |
| xrge0tps 34241 | The extended nonnegative r... |
| xrge0topn 34242 | The topology of the extend... |
| xrge0haus 34243 | The topology of the extend... |
| xrge0tmd 34244 | The extended nonnegative r... |
| xrge0tmdALT 34245 | Alternate proof of ~ xrge0... |
| lmlim 34246 | Relate a limit in a given ... |
| lmlimxrge0 34247 | Relate a limit in the nonn... |
| rge0scvg 34248 | Implication of convergence... |
| fsumcvg4 34249 | A serie with finite suppor... |
| pnfneige0 34250 | A neighborhood of ` +oo ` ... |
| lmxrge0 34251 | Express "sequence ` F ` co... |
| lmdvg 34252 | If a monotonic sequence of... |
| lmdvglim 34253 | If a monotonic real number... |
| pl1cn 34254 | A univariate polynomial is... |
| zringnm 34257 | The norm (function) for a ... |
| zzsnm 34258 | The norm of the ring of th... |
| zlm0 34259 | Zero of a ` ZZ ` -module. ... |
| zlm1 34260 | Unity element of a ` ZZ ` ... |
| zlmds 34261 | Distance in a ` ZZ ` -modu... |
| zlmtset 34262 | Topology in a ` ZZ ` -modu... |
| zlmnm 34263 | Norm of a ` ZZ ` -module (... |
| zhmnrg 34264 | The ` ZZ ` -module built f... |
| nmmulg 34265 | The norm of a group produc... |
| zrhnm 34266 | The norm of the image by `... |
| cnzh 34267 | The ` ZZ ` -module of ` CC... |
| rezh 34268 | The ` ZZ ` -module of ` RR... |
| qqhval 34271 | Value of the canonical hom... |
| zrhf1ker 34272 | The kernel of the homomorp... |
| zrhchr 34273 | The kernel of the homomorp... |
| zrhker 34274 | The kernel of the homomorp... |
| zrhunitpreima 34275 | The preimage by ` ZRHom ` ... |
| elzrhunit 34276 | Condition for the image by... |
| zrhneg 34277 | The canonical homomorphism... |
| zrhcntr 34278 | The canonical representati... |
| elzdif0 34279 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 34280 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 34281 | Value of the canonical hom... |
| qqhvval 34282 | Value of the canonical hom... |
| qqh0 34283 | The image of ` 0 ` by the ... |
| qqh1 34284 | The image of ` 1 ` by the ... |
| qqhf 34285 | ` QQHom ` as a function. ... |
| qqhvq 34286 | The image of a quotient by... |
| qqhghm 34287 | The ` QQHom ` homomorphism... |
| qqhrhm 34288 | The ` QQHom ` homomorphism... |
| qqhnm 34289 | The norm of the image by `... |
| qqhcn 34290 | The ` QQHom ` homomorphism... |
| qqhucn 34291 | The ` QQHom ` homomorphism... |
| rrhval 34295 | Value of the canonical hom... |
| rrhcn 34296 | If the topology of ` R ` i... |
| rrhf 34297 | If the topology of ` R ` i... |
| isrrext 34299 | Express the property " ` R... |
| rrextnrg 34300 | An extension of ` RR ` is ... |
| rrextdrg 34301 | An extension of ` RR ` is ... |
| rrextnlm 34302 | The norm of an extension o... |
| rrextchr 34303 | The ring characteristic of... |
| rrextcusp 34304 | An extension of ` RR ` is ... |
| rrexttps 34305 | An extension of ` RR ` is ... |
| rrexthaus 34306 | The topology of an extensi... |
| rrextust 34307 | The uniformity of an exten... |
| rerrext 34308 | The field of the real numb... |
| cnrrext 34309 | The field of the complex n... |
| qqtopn 34310 | The topology of the field ... |
| rrhfe 34311 | If ` R ` is an extension o... |
| rrhcne 34312 | If ` R ` is an extension o... |
| rrhqima 34313 | The ` RRHom ` homomorphism... |
| rrh0 34314 | The image of ` 0 ` by the ... |
| xrhval 34317 | The value of the embedding... |
| zrhre 34318 | The ` ZRHom ` homomorphism... |
| qqhre 34319 | The ` QQHom ` homomorphism... |
| rrhre 34320 | The ` RRHom ` homomorphism... |
| relmntop 34323 | Manifold is a relation. (... |
| ismntoplly 34324 | Property of being a manifo... |
| ismntop 34325 | Property of being a manifo... |
| esumex 34328 | An extended sum is a set b... |
| esumcl 34329 | Closure for extended sum i... |
| esumeq12dvaf 34330 | Equality deduction for ext... |
| esumeq12dva 34331 | Equality deduction for ext... |
| esumeq12d 34332 | Equality deduction for ext... |
| esumeq1 34333 | Equality theorem for an ex... |
| esumeq1d 34334 | Equality theorem for an ex... |
| esumeq2 34335 | Equality theorem for exten... |
| esumeq2d 34336 | Equality deduction for ext... |
| esumeq2dv 34337 | Equality deduction for ext... |
| esumeq2sdv 34338 | Equality deduction for ext... |
| nfesum1 34339 | Bound-variable hypothesis ... |
| nfesum2 34340 | Bound-variable hypothesis ... |
| cbvesum 34341 | Change bound variable in a... |
| cbvesumv 34342 | Change bound variable in a... |
| esumid 34343 | Identify the extended sum ... |
| esumgsum 34344 | A finite extended sum is t... |
| esumval 34345 | Develop the value of the e... |
| esumel 34346 | The extended sum is a limi... |
| esumnul 34347 | Extended sum over the empt... |
| esum0 34348 | Extended sum of zero. (Co... |
| esumf1o 34349 | Re-index an extended sum u... |
| esumc 34350 | Convert from the collectio... |
| esumrnmpt 34351 | Rewrite an extended sum in... |
| esumsplit 34352 | Split an extended sum into... |
| esummono 34353 | Extended sum is monotonic.... |
| esumpad 34354 | Extend an extended sum by ... |
| esumpad2 34355 | Remove zeroes from an exte... |
| esumadd 34356 | Addition of infinite sums.... |
| esumle 34357 | If all of the terms of an ... |
| gsumesum 34358 | Relate a group sum on ` ( ... |
| esumlub 34359 | The extended sum is the lo... |
| esumaddf 34360 | Addition of infinite sums.... |
| esumlef 34361 | If all of the terms of an ... |
| esumcst 34362 | The extended sum of a cons... |
| esumsnf 34363 | The extended sum of a sing... |
| esumsn 34364 | The extended sum of a sing... |
| esumpr 34365 | Extended sum over a pair. ... |
| esumpr2 34366 | Extended sum over a pair, ... |
| esumrnmpt2 34367 | Rewrite an extended sum in... |
| esumfzf 34368 | Formulating a partial exte... |
| esumfsup 34369 | Formulating an extended su... |
| esumfsupre 34370 | Formulating an extended su... |
| esumss 34371 | Change the index set to a ... |
| esumpinfval 34372 | The value of the extended ... |
| esumpfinvallem 34373 | Lemma for ~ esumpfinval . ... |
| esumpfinval 34374 | The value of the extended ... |
| esumpfinvalf 34375 | Same as ~ esumpfinval , mi... |
| esumpinfsum 34376 | The value of the extended ... |
| esumpcvgval 34377 | The value of the extended ... |
| esumpmono 34378 | The partial sums in an ext... |
| esumcocn 34379 | Lemma for ~ esummulc2 and ... |
| esummulc1 34380 | An extended sum multiplied... |
| esummulc2 34381 | An extended sum multiplied... |
| esumdivc 34382 | An extended sum divided by... |
| hashf2 34383 | Lemma for ~ hasheuni . (C... |
| hasheuni 34384 | The cardinality of a disjo... |
| esumcvg 34385 | The sequence of partial su... |
| esumcvg2 34386 | Simpler version of ~ esumc... |
| esumcvgsum 34387 | The value of the extended ... |
| esumsup 34388 | Express an extended sum as... |
| esumgect 34389 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 34390 | All terms of a converging ... |
| esum2dlem 34391 | Lemma for ~ esum2d (finite... |
| esum2d 34392 | Write a double extended su... |
| esumiun 34393 | Sum over a nonnecessarily ... |
| ofceq 34396 | Equality theorem for funct... |
| ofcfval 34397 | Value of an operation appl... |
| ofcval 34398 | Evaluate a function/consta... |
| ofcfn 34399 | The function operation pro... |
| ofcfeqd2 34400 | Equality theorem for funct... |
| ofcfval3 34401 | General value of ` ( F oFC... |
| ofcf 34402 | The function/constant oper... |
| ofcfval2 34403 | The function operation exp... |
| ofcfval4 34404 | The function/constant oper... |
| ofcc 34405 | Left operation by a consta... |
| ofcof 34406 | Relate function operation ... |
| sigaex 34409 | Lemma for ~ issiga and ~ i... |
| sigaval 34410 | The set of sigma-algebra w... |
| issiga 34411 | An alternative definition ... |
| isrnsiga 34412 | The property of being a si... |
| 0elsiga 34413 | A sigma-algebra contains t... |
| baselsiga 34414 | A sigma-algebra contains i... |
| sigasspw 34415 | A sigma-algebra is a set o... |
| sigaclcu 34416 | A sigma-algebra is closed ... |
| sigaclcuni 34417 | A sigma-algebra is closed ... |
| sigaclfu 34418 | A sigma-algebra is closed ... |
| sigaclcu2 34419 | A sigma-algebra is closed ... |
| sigaclfu2 34420 | A sigma-algebra is closed ... |
| sigaclcu3 34421 | A sigma-algebra is closed ... |
| issgon 34422 | Property of being a sigma-... |
| sgon 34423 | A sigma-algebra is a sigma... |
| elsigass 34424 | An element of a sigma-alge... |
| elrnsiga 34425 | Dropping the base informat... |
| isrnsigau 34426 | The property of being a si... |
| unielsiga 34427 | A sigma-algebra contains i... |
| dmvlsiga 34428 | Lebesgue-measurable subset... |
| pwsiga 34429 | Any power set forms a sigm... |
| prsiga 34430 | The smallest possible sigm... |
| sigaclci 34431 | A sigma-algebra is closed ... |
| difelsiga 34432 | A sigma-algebra is closed ... |
| unelsiga 34433 | A sigma-algebra is closed ... |
| inelsiga 34434 | A sigma-algebra is closed ... |
| sigainb 34435 | Building a sigma-algebra f... |
| insiga 34436 | The intersection of a coll... |
| sigagenval 34439 | Value of the generated sig... |
| sigagensiga 34440 | A generated sigma-algebra ... |
| sgsiga 34441 | A generated sigma-algebra ... |
| unisg 34442 | The sigma-algebra generate... |
| dmsigagen 34443 | A sigma-algebra can be gen... |
| sssigagen 34444 | A set is a subset of the s... |
| sssigagen2 34445 | A subset of the generating... |
| elsigagen 34446 | Any element of a set is al... |
| elsigagen2 34447 | Any countable union of ele... |
| sigagenss 34448 | The generated sigma-algebr... |
| sigagenss2 34449 | Sufficient condition for i... |
| sigagenid 34450 | The sigma-algebra generate... |
| ispisys 34451 | The property of being a pi... |
| ispisys2 34452 | The property of being a pi... |
| inelpisys 34453 | Pi-systems are closed unde... |
| sigapisys 34454 | All sigma-algebras are pi-... |
| isldsys 34455 | The property of being a la... |
| pwldsys 34456 | The power set of the unive... |
| unelldsys 34457 | Lambda-systems are closed ... |
| sigaldsys 34458 | All sigma-algebras are lam... |
| ldsysgenld 34459 | The intersection of all la... |
| sigapildsyslem 34460 | Lemma for ~ sigapildsys . ... |
| sigapildsys 34461 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 34462 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 34463 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 34464 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 34465 | The lambda system ` E ` ge... |
| dynkin 34466 | Dynkin's lambda-pi theorem... |
| isros 34467 | The property of being a ri... |
| rossspw 34468 | A ring of sets is a collec... |
| 0elros 34469 | A ring of sets contains th... |
| unelros 34470 | A ring of sets is closed u... |
| difelros 34471 | A ring of sets is closed u... |
| inelros 34472 | A ring of sets is closed u... |
| fiunelros 34473 | A ring of sets is closed u... |
| issros 34474 | The property of being a se... |
| srossspw 34475 | A semiring of sets is a co... |
| 0elsros 34476 | A semiring of sets contain... |
| inelsros 34477 | A semiring of sets is clos... |
| diffiunisros 34478 | In semiring of sets, compl... |
| rossros 34479 | Rings of sets are semiring... |
| brsiga 34482 | The Borel Algebra on real ... |
| brsigarn 34483 | The Borel Algebra is a sig... |
| brsigasspwrn 34484 | The Borel Algebra is a set... |
| unibrsiga 34485 | The union of the Borel Alg... |
| cldssbrsiga 34486 | A Borel Algebra contains a... |
| sxval 34489 | Value of the product sigma... |
| sxsiga 34490 | A product sigma-algebra is... |
| sxsigon 34491 | A product sigma-algebra is... |
| sxuni 34492 | The base set of a product ... |
| elsx 34493 | The cartesian product of t... |
| measbase 34496 | The base set of a measure ... |
| measval 34497 | The value of the ` measure... |
| ismeas 34498 | The property of being a me... |
| isrnmeas 34499 | The property of being a me... |
| dmmeas 34500 | The domain of a measure is... |
| measbasedom 34501 | The base set of a measure ... |
| measfrge0 34502 | A measure is a function ov... |
| measfn 34503 | A measure is a function on... |
| measvxrge0 34504 | The values of a measure ar... |
| measvnul 34505 | The measure of the empty s... |
| measge0 34506 | A measure is nonnegative. ... |
| measle0 34507 | If the measure of a given ... |
| measvun 34508 | The measure of a countable... |
| measxun2 34509 | The measure the union of t... |
| measun 34510 | The measure the union of t... |
| measvunilem 34511 | Lemma for ~ measvuni . (C... |
| measvunilem0 34512 | Lemma for ~ measvuni . (C... |
| measvuni 34513 | The measure of a countable... |
| measssd 34514 | A measure is monotone with... |
| measunl 34515 | A measure is sub-additive ... |
| measiuns 34516 | The measure of the union o... |
| measiun 34517 | A measure is sub-additive.... |
| meascnbl 34518 | A measure is continuous fr... |
| measinblem 34519 | Lemma for ~ measinb . (Co... |
| measinb 34520 | Building a measure restric... |
| measres 34521 | Building a measure restric... |
| measinb2 34522 | Building a measure restric... |
| measdivcst 34523 | Division of a measure by a... |
| measdivcstALTV 34524 | Alternate version of ~ mea... |
| cntmeas 34525 | The Counting measure is a ... |
| pwcntmeas 34526 | The counting measure is a ... |
| cntnevol 34527 | Counting and Lebesgue meas... |
| voliune 34528 | The Lebesgue measure funct... |
| volfiniune 34529 | The Lebesgue measure funct... |
| volmeas 34530 | The Lebesgue measure is a ... |
| ddeval1 34533 | Value of the delta measure... |
| ddeval0 34534 | Value of the delta measure... |
| ddemeas 34535 | The Dirac delta measure is... |
| relae 34539 | 'almost everywhere' is a r... |
| brae 34540 | 'almost everywhere' relati... |
| braew 34541 | 'almost everywhere' relati... |
| truae 34542 | A truth holds almost every... |
| aean 34543 | A conjunction holds almost... |
| faeval 34545 | Value of the 'almost every... |
| relfae 34546 | The 'almost everywhere' bu... |
| brfae 34547 | 'almost everywhere' relati... |
| ismbfm 34550 | The predicate " ` F ` is a... |
| elunirnmbfm 34551 | The property of being a me... |
| mbfmfun 34552 | A measurable function is a... |
| mbfmf 34553 | A measurable function as a... |
| mbfmcnvima 34554 | The preimage by a measurab... |
| isanmbfm 34555 | The predicate to be a meas... |
| mbfmbfmOLD 34556 | A measurable function to a... |
| mbfmbfm 34557 | A measurable function to a... |
| mbfmcst 34558 | A constant function is mea... |
| 1stmbfm 34559 | The first projection map i... |
| 2ndmbfm 34560 | The second projection map ... |
| imambfm 34561 | If the sigma-algebra in th... |
| cnmbfm 34562 | A continuous function is m... |
| mbfmco 34563 | The composition of two mea... |
| mbfmco2 34564 | The pair building of two m... |
| mbfmvolf 34565 | Measurable functions with ... |
| elmbfmvol2 34566 | Measurable functions with ... |
| mbfmcnt 34567 | All functions are measurab... |
| br2base 34568 | The base set for the gener... |
| dya2ub 34569 | An upper bound for a dyadi... |
| sxbrsigalem0 34570 | The closed half-spaces of ... |
| sxbrsigalem3 34571 | The sigma-algebra generate... |
| dya2iocival 34572 | The function ` I ` returns... |
| dya2iocress 34573 | Dyadic intervals are subse... |
| dya2iocbrsiga 34574 | Dyadic intervals are Borel... |
| dya2icobrsiga 34575 | Dyadic intervals are Borel... |
| dya2icoseg 34576 | For any point and any clos... |
| dya2icoseg2 34577 | For any point and any open... |
| dya2iocrfn 34578 | The function returning dya... |
| dya2iocct 34579 | The dyadic rectangle set i... |
| dya2iocnrect 34580 | For any point of an open r... |
| dya2iocnei 34581 | For any point of an open s... |
| dya2iocuni 34582 | Every open set of ` ( RR X... |
| dya2iocucvr 34583 | The dyadic rectangular set... |
| sxbrsigalem1 34584 | The Borel algebra on ` ( R... |
| sxbrsigalem2 34585 | The sigma-algebra generate... |
| sxbrsigalem4 34586 | The Borel algebra on ` ( R... |
| sxbrsigalem5 34587 | First direction for ~ sxbr... |
| sxbrsigalem6 34588 | First direction for ~ sxbr... |
| sxbrsiga 34589 | The product sigma-algebra ... |
| omsval 34592 | Value of the function mapp... |
| omsfval 34593 | Value of the outer measure... |
| omscl 34594 | A closure lemma for the co... |
| omsf 34595 | A constructed outer measur... |
| oms0 34596 | A constructed outer measur... |
| omsmon 34597 | A constructed outer measur... |
| omssubaddlem 34598 | For any small margin ` E `... |
| omssubadd 34599 | A constructed outer measur... |
| carsgval 34602 | Value of the Caratheodory ... |
| carsgcl 34603 | Closure of the Caratheodor... |
| elcarsg 34604 | Property of being a Carath... |
| baselcarsg 34605 | The universe set, ` O ` , ... |
| 0elcarsg 34606 | The empty set is Caratheod... |
| carsguni 34607 | The union of all Caratheod... |
| elcarsgss 34608 | Caratheodory measurable se... |
| difelcarsg 34609 | The Caratheodory measurabl... |
| inelcarsg 34610 | The Caratheodory measurabl... |
| unelcarsg 34611 | The Caratheodory-measurabl... |
| difelcarsg2 34612 | The Caratheodory-measurabl... |
| carsgmon 34613 | Utility lemma: Apply mono... |
| carsgsigalem 34614 | Lemma for the following th... |
| fiunelcarsg 34615 | The Caratheodory measurabl... |
| carsgclctunlem1 34616 | Lemma for ~ carsgclctun . ... |
| carsggect 34617 | The outer measure is count... |
| carsgclctunlem2 34618 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 34619 | Lemma for ~ carsgclctun . ... |
| carsgclctun 34620 | The Caratheodory measurabl... |
| carsgsiga 34621 | The Caratheodory measurabl... |
| omsmeas 34622 | The restriction of a const... |
| pmeasmono 34623 | This theorem's hypotheses ... |
| pmeasadd 34624 | A premeasure on a ring of ... |
| itgeq12dv 34625 | Equality theorem for an in... |
| sitgval 34631 | Value of the simple functi... |
| issibf 34632 | The predicate " ` F ` is a... |
| sibf0 34633 | The constant zero function... |
| sibfmbl 34634 | A simple function is measu... |
| sibff 34635 | A simple function is a fun... |
| sibfrn 34636 | A simple function has fini... |
| sibfima 34637 | Any preimage of a singleto... |
| sibfinima 34638 | The measure of the interse... |
| sibfof 34639 | Applying function operatio... |
| sitgfval 34640 | Value of the Bochner integ... |
| sitgclg 34641 | Closure of the Bochner int... |
| sitgclbn 34642 | Closure of the Bochner int... |
| sitgclcn 34643 | Closure of the Bochner int... |
| sitgclre 34644 | Closure of the Bochner int... |
| sitg0 34645 | The integral of the consta... |
| sitgf 34646 | The integral for simple fu... |
| sitgaddlemb 34647 | Lemma for * sitgadd . (Co... |
| sitmval 34648 | Value of the simple functi... |
| sitmfval 34649 | Value of the integral dist... |
| sitmcl 34650 | Closure of the integral di... |
| sitmf 34651 | The integral metric as a f... |
| oddpwdc 34653 | Lemma for ~ eulerpart . T... |
| oddpwdcv 34654 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 34655 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 34656 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 34657 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 34658 | Lemma for ~ eulerpart . (... |
| eulerpartlems 34659 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 34660 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 34661 | Lemma for ~ eulerpart . (... |
| eulerpartleme 34662 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 34663 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 34664 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 34665 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 34666 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 34667 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 34668 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 34669 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 34670 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 34671 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 34672 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 34673 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 34674 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 34675 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 34676 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 34677 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 34678 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 34679 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 34680 | Lemma for ~ eulerpart . (... |
| eulerpart 34681 | Euler's theorem on partiti... |
| subiwrd 34684 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 34685 | Length of a subword of an ... |
| iwrdsplit 34686 | Lemma for ~ sseqp1 . (Con... |
| sseqval 34687 | Value of the strong sequen... |
| sseqfv1 34688 | Value of the strong sequen... |
| sseqfn 34689 | A strong recursive sequenc... |
| sseqmw 34690 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 34691 | A strong recursive sequenc... |
| sseqfres 34692 | The first elements in the ... |
| sseqfv2 34693 | Value of the strong sequen... |
| sseqp1 34694 | Value of the strong sequen... |
| fiblem 34697 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 34698 | Value of the Fibonacci seq... |
| fib1 34699 | Value of the Fibonacci seq... |
| fibp1 34700 | Value of the Fibonacci seq... |
| fib2 34701 | Value of the Fibonacci seq... |
| fib3 34702 | Value of the Fibonacci seq... |
| fib4 34703 | Value of the Fibonacci seq... |
| fib5 34704 | Value of the Fibonacci seq... |
| fib6 34705 | Value of the Fibonacci seq... |
| elprob 34708 | The property of being a pr... |
| domprobmeas 34709 | A probability measure is a... |
| domprobsiga 34710 | The domain of a probabilit... |
| probtot 34711 | The probability of the uni... |
| prob01 34712 | A probability is an elemen... |
| probnul 34713 | The probability of the emp... |
| unveldomd 34714 | The universe is an element... |
| unveldom 34715 | The universe is an element... |
| nuleldmp 34716 | The empty set is an elemen... |
| probcun 34717 | The probability of the uni... |
| probun 34718 | The probability of the uni... |
| probdif 34719 | The probability of the dif... |
| probinc 34720 | A probability law is incre... |
| probdsb 34721 | The probability of the com... |
| probmeasd 34722 | A probability measure is a... |
| probvalrnd 34723 | The value of a probability... |
| probtotrnd 34724 | The probability of the uni... |
| totprobd 34725 | Law of total probability, ... |
| totprob 34726 | Law of total probability. ... |
| probfinmeasb 34727 | Build a probability measur... |
| probfinmeasbALTV 34728 | Alternate version of ~ pro... |
| probmeasb 34729 | Build a probability from a... |
| cndprobval 34732 | The value of the condition... |
| cndprobin 34733 | An identity linking condit... |
| cndprob01 34734 | The conditional probabilit... |
| cndprobtot 34735 | The conditional probabilit... |
| cndprobnul 34736 | The conditional probabilit... |
| cndprobprob 34737 | The conditional probabilit... |
| bayesth 34738 | Bayes Theorem. (Contribut... |
| rrvmbfm 34741 | A real-valued random varia... |
| isrrvv 34742 | Elementhood to the set of ... |
| rrvvf 34743 | A real-valued random varia... |
| rrvfn 34744 | A real-valued random varia... |
| rrvdm 34745 | The domain of a random var... |
| rrvrnss 34746 | The range of a random vari... |
| rrvf2 34747 | A real-valued random varia... |
| rrvdmss 34748 | The domain of a random var... |
| rrvfinvima 34749 | For a real-value random va... |
| 0rrv 34750 | The constant function equa... |
| rrvadd 34751 | The sum of two random vari... |
| rrvmulc 34752 | A random variable multipli... |
| rrvsum 34753 | An indexed sum of random v... |
| boolesineq 34754 | Boole's inequality (union ... |
| orvcval 34757 | Value of the preimage mapp... |
| orvcval2 34758 | Another way to express the... |
| elorvc 34759 | Elementhood of a preimage.... |
| orvcval4 34760 | The value of the preimage ... |
| orvcoel 34761 | If the relation produces o... |
| orvccel 34762 | If the relation produces c... |
| elorrvc 34763 | Elementhood of a preimage ... |
| orrvcval4 34764 | The value of the preimage ... |
| orrvcoel 34765 | If the relation produces o... |
| orrvccel 34766 | If the relation produces c... |
| orvcgteel 34767 | Preimage maps produced by ... |
| orvcelval 34768 | Preimage maps produced by ... |
| orvcelel 34769 | Preimage maps produced by ... |
| dstrvval 34770 | The value of the distribut... |
| dstrvprob 34771 | The distribution of a rand... |
| orvclteel 34772 | Preimage maps produced by ... |
| dstfrvel 34773 | Elementhood of preimage ma... |
| dstfrvunirn 34774 | The limit of all preimage ... |
| orvclteinc 34775 | Preimage maps produced by ... |
| dstfrvinc 34776 | A cumulative distribution ... |
| dstfrvclim1 34777 | The limit of the cumulativ... |
| coinfliplem 34778 | Division in the extended r... |
| coinflipprob 34779 | The ` P ` we defined for c... |
| coinflipspace 34780 | The space of our coin-flip... |
| coinflipuniv 34781 | The universe of our coin-f... |
| coinfliprv 34782 | The ` X ` we defined for c... |
| coinflippv 34783 | The probability of heads i... |
| coinflippvt 34784 | The probability of tails i... |
| ballotlemoex 34785 | ` O ` is a set. (Contribu... |
| ballotlem1 34786 | The size of the universe i... |
| ballotlemelo 34787 | Elementhood in ` O ` . (C... |
| ballotlem2 34788 | The probability that the f... |
| ballotlemfval 34789 | The value of ` F ` . (Con... |
| ballotlemfelz 34790 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 34791 | If the ` J ` th ballot is ... |
| ballotlemfc0 34792 | ` F ` takes value 0 betwee... |
| ballotlemfcc 34793 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 34794 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 34795 | ` ( F `` C ) ` always star... |
| ballotleme 34796 | Elements of ` E ` . (Cont... |
| ballotlemodife 34797 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 34798 | If the first pick is a vot... |
| ballotlem5 34799 | If A is not ahead througho... |
| ballotlemi 34800 | Value of ` I ` for a given... |
| ballotlemiex 34801 | Properties of ` ( I `` C )... |
| ballotlemi1 34802 | The first tie cannot be re... |
| ballotlemii 34803 | The first tie cannot be re... |
| ballotlemsup 34804 | The set of zeroes of ` F `... |
| ballotlemimin 34805 | ` ( I `` C ) ` is the firs... |
| ballotlemic 34806 | If the first vote is for B... |
| ballotlem1c 34807 | If the first vote is for A... |
| ballotlemsval 34808 | Value of ` S ` . (Contrib... |
| ballotlemsv 34809 | Value of ` S ` evaluated a... |
| ballotlemsgt1 34810 | ` S ` maps values less tha... |
| ballotlemsdom 34811 | Domain of ` S ` for a give... |
| ballotlemsel1i 34812 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 34813 | The defined ` S ` is a bij... |
| ballotlemsi 34814 | The image by ` S ` of the ... |
| ballotlemsima 34815 | The image by ` S ` of an i... |
| ballotlemieq 34816 | If two countings share the... |
| ballotlemrval 34817 | Value of ` R ` . (Contrib... |
| ballotlemscr 34818 | The image of ` ( R `` C ) ... |
| ballotlemrv 34819 | Value of ` R ` evaluated a... |
| ballotlemrv1 34820 | Value of ` R ` before the ... |
| ballotlemrv2 34821 | Value of ` R ` after the t... |
| ballotlemro 34822 | Range of ` R ` is included... |
| ballotlemgval 34823 | Expand the value of ` .^ `... |
| ballotlemgun 34824 | A property of the defined ... |
| ballotlemfg 34825 | Express the value of ` ( F... |
| ballotlemfrc 34826 | Express the value of ` ( F... |
| ballotlemfrci 34827 | Reverse counting preserves... |
| ballotlemfrceq 34828 | Value of ` F ` for a rever... |
| ballotlemfrcn0 34829 | Value of ` F ` for a rever... |
| ballotlemrc 34830 | Range of ` R ` . (Contrib... |
| ballotlemirc 34831 | Applying ` R ` does not ch... |
| ballotlemrinv0 34832 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 34833 | ` R ` is its own inverse :... |
| ballotlem1ri 34834 | When the vote on the first... |
| ballotlem7 34835 | ` R ` is a bijection betwe... |
| ballotlem8 34836 | There are as many counting... |
| ballotth 34837 | Bertrand's ballot problem ... |
| fzssfzo 34838 | Condition for an integer i... |
| gsumncl 34839 | Closure of a group sum in ... |
| gsumnunsn 34840 | Closure of a group sum in ... |
| ccatmulgnn0dir 34841 | Concatenation of words fol... |
| ofcccat 34842 | Letterwise operations on w... |
| ofcs1 34843 | Letterwise operations on a... |
| ofcs2 34844 | Letterwise operations on a... |
| plyrecld 34845 | Closure of a polynomial wi... |
| signsplypnf 34846 | The quotient of a polynomi... |
| signsply0 34847 | Lemma for the rule of sign... |
| signspval 34848 | The value of the skipping ... |
| signsw0glem 34849 | Neutral element property o... |
| signswbase 34850 | The base of ` W ` is the u... |
| signswplusg 34851 | The operation of ` W ` . ... |
| signsw0g 34852 | The neutral element of ` W... |
| signswmnd 34853 | ` W ` is a monoid structur... |
| signswrid 34854 | The zero-skipping operatio... |
| signswlid 34855 | The zero-skipping operatio... |
| signswn0 34856 | The zero-skipping operatio... |
| signswch 34857 | The zero-skipping operatio... |
| signslema 34858 | Computational part of ~~? ... |
| signstfv 34859 | Value of the zero-skipping... |
| signstfval 34860 | Value of the zero-skipping... |
| signstcl 34861 | Closure of the zero skippi... |
| signstf 34862 | The zero skipping sign wor... |
| signstlen 34863 | Length of the zero skippin... |
| signstf0 34864 | Sign of a single letter wo... |
| signstfvn 34865 | Zero-skipping sign in a wo... |
| signsvtn0 34866 | If the last letter is nonz... |
| signstfvp 34867 | Zero-skipping sign in a wo... |
| signstfvneq0 34868 | In case the first letter i... |
| signstfvcl 34869 | Closure of the zero skippi... |
| signstfvc 34870 | Zero-skipping sign in a wo... |
| signstres 34871 | Restriction of a zero skip... |
| signstfveq0a 34872 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 34873 | In case the last letter is... |
| signsvvfval 34874 | The value of ` V ` , which... |
| signsvvf 34875 | ` V ` is a function. (Con... |
| signsvf0 34876 | There is no change of sign... |
| signsvf1 34877 | In a single-letter word, w... |
| signsvfn 34878 | Number of changes in a wor... |
| signsvtp 34879 | Adding a letter of the sam... |
| signsvtn 34880 | Adding a letter of a diffe... |
| signsvfpn 34881 | Adding a letter of the sam... |
| signsvfnn 34882 | Adding a letter of a diffe... |
| signlem0 34883 | Adding a zero as the highe... |
| signshf 34884 | ` H ` , corresponding to t... |
| signshwrd 34885 | ` H ` , corresponding to t... |
| signshlen 34886 | Length of ` H ` , correspo... |
| signshnz 34887 | ` H ` is not the empty wor... |
| iblidicc 34888 | The identity function is i... |
| rpsqrtcn 34889 | Continuity of the real pos... |
| divsqrtid 34890 | A real number divided by i... |
| cxpcncf1 34891 | The power function on comp... |
| efmul2picn 34892 | Multiplying by ` ( _i x. (... |
| fct2relem 34893 | Lemma for ~ ftc2re . (Con... |
| ftc2re 34894 | The Fundamental Theorem of... |
| fdvposlt 34895 | Functions with a positive ... |
| fdvneggt 34896 | Functions with a negative ... |
| fdvposle 34897 | Functions with a nonnegati... |
| fdvnegge 34898 | Functions with a nonpositi... |
| prodfzo03 34899 | A product of three factors... |
| actfunsnf1o 34900 | The action ` F ` of extend... |
| actfunsnrndisj 34901 | The action ` F ` of extend... |
| itgexpif 34902 | The basis for the circle m... |
| fsum2dsub 34903 | Lemma for ~ breprexp - Re-... |
| reprval 34906 | Value of the representatio... |
| repr0 34907 | There is exactly one repre... |
| reprf 34908 | Members of the representat... |
| reprsum 34909 | Sums of values of the memb... |
| reprle 34910 | Upper bound to the terms i... |
| reprsuc 34911 | Express the representation... |
| reprfi 34912 | Bounded representations ar... |
| reprss 34913 | Representations with terms... |
| reprinrn 34914 | Representations with term ... |
| reprlt 34915 | There are no representatio... |
| hashreprin 34916 | Express a sum of represent... |
| reprgt 34917 | There are no representatio... |
| reprinfz1 34918 | For the representation of ... |
| reprfi2 34919 | Corollary of ~ reprinfz1 .... |
| reprfz1 34920 | Corollary of ~ reprinfz1 .... |
| hashrepr 34921 | Develop the number of repr... |
| reprpmtf1o 34922 | Transposing ` 0 ` and ` X ... |
| reprdifc 34923 | Express the representation... |
| chpvalz 34924 | Value of the second Chebys... |
| chtvalz 34925 | Value of the Chebyshev fun... |
| breprexplema 34926 | Lemma for ~ breprexp (indu... |
| breprexplemb 34927 | Lemma for ~ breprexp (clos... |
| breprexplemc 34928 | Lemma for ~ breprexp (indu... |
| breprexp 34929 | Express the ` S ` th power... |
| breprexpnat 34930 | Express the ` S ` th power... |
| vtsval 34933 | Value of the Vinogradov tr... |
| vtscl 34934 | Closure of the Vinogradov ... |
| vtsprod 34935 | Express the Vinogradov tri... |
| circlemeth 34936 | The Hardy, Littlewood and ... |
| circlemethnat 34937 | The Hardy, Littlewood and ... |
| circlevma 34938 | The Circle Method, where t... |
| circlemethhgt 34939 | The circle method, where t... |
| hgt750lemc 34943 | An upper bound to the summ... |
| hgt750lemd 34944 | An upper bound to the summ... |
| hgt749d 34945 | A deduction version of ~ a... |
| logdivsqrle 34946 | Conditions for ` ( ( log `... |
| hgt750lem 34947 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 34948 | Decimal multiplication gal... |
| hgt750lemf 34949 | Lemma for the statement 7.... |
| hgt750lemg 34950 | Lemma for the statement 7.... |
| oddprm2 34951 | Two ways to write the set ... |
| hgt750lemb 34952 | An upper bound on the cont... |
| hgt750lema 34953 | An upper bound on the cont... |
| hgt750leme 34954 | An upper bound on the cont... |
| tgoldbachgnn 34955 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 34956 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 34957 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 34958 | Odd integers greater than ... |
| tgoldbachgt 34959 | Odd integers greater than ... |
| istrkg2d 34962 | Property of fulfilling dim... |
| axtglowdim2ALTV 34963 | Alternate version of ~ axt... |
| axtgupdim2ALTV 34964 | Alternate version of ~ axt... |
| cgranbtwn 34965 | Null angle implies between... |
| btwnlng13 34966 | If ` Z ` is between ` X ` ... |
| morleylemrneab 34967 | Lemma for morley . (Contr... |
| afsval 34970 | Value of the AFS relation ... |
| brafs 34971 | Binary relation form of th... |
| tg5segofs 34972 | Rephrase ~ axtg5seg using ... |
| lpadval 34975 | Value of the ` leftpad ` f... |
| lpadlem1 34976 | Lemma for the ` leftpad ` ... |
| lpadlem3 34977 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 34978 | Length of a left-padded wo... |
| lpadlem2 34979 | Lemma for the ` leftpad ` ... |
| lpadlen2 34980 | Length of a left-padded wo... |
| lpadmax 34981 | Length of a left-padded wo... |
| lpadleft 34982 | The contents of prefix of ... |
| lpadright 34983 | The suffix of a left-padde... |
| bnj170 34996 | ` /\ ` -manipulation. (Co... |
| bnj240 34997 | ` /\ ` -manipulation. (Co... |
| bnj248 34998 | ` /\ ` -manipulation. (Co... |
| bnj250 34999 | ` /\ ` -manipulation. (Co... |
| bnj251 35000 | ` /\ ` -manipulation. (Co... |
| bnj252 35001 | ` /\ ` -manipulation. (Co... |
| bnj253 35002 | ` /\ ` -manipulation. (Co... |
| bnj255 35003 | ` /\ ` -manipulation. (Co... |
| bnj256 35004 | ` /\ ` -manipulation. (Co... |
| bnj257 35005 | ` /\ ` -manipulation. (Co... |
| bnj258 35006 | ` /\ ` -manipulation. (Co... |
| bnj268 35007 | ` /\ ` -manipulation. (Co... |
| bnj290 35008 | ` /\ ` -manipulation. (Co... |
| bnj291 35009 | ` /\ ` -manipulation. (Co... |
| bnj312 35010 | ` /\ ` -manipulation. (Co... |
| bnj334 35011 | ` /\ ` -manipulation. (Co... |
| bnj345 35012 | ` /\ ` -manipulation. (Co... |
| bnj422 35013 | ` /\ ` -manipulation. (Co... |
| bnj432 35014 | ` /\ ` -manipulation. (Co... |
| bnj446 35015 | ` /\ ` -manipulation. (Co... |
| bnj23 35016 | First-order logic and set ... |
| bnj31 35017 | First-order logic and set ... |
| bnj62 35018 | First-order logic and set ... |
| bnj89 35019 | First-order logic and set ... |
| bnj90 35020 | First-order logic and set ... |
| bnj101 35021 | First-order logic and set ... |
| bnj105 35022 | First-order logic and set ... |
| bnj115 35023 | First-order logic and set ... |
| bnj132 35024 | First-order logic and set ... |
| bnj133 35025 | First-order logic and set ... |
| bnj156 35026 | First-order logic and set ... |
| bnj158 35027 | First-order logic and set ... |
| bnj168 35028 | First-order logic and set ... |
| bnj206 35029 | First-order logic and set ... |
| bnj216 35030 | First-order logic and set ... |
| bnj219 35031 | First-order logic and set ... |
| bnj226 35032 | First-order logic and set ... |
| bnj228 35033 | First-order logic and set ... |
| bnj519 35034 | First-order logic and set ... |
| bnj524 35035 | First-order logic and set ... |
| bnj525 35036 | First-order logic and set ... |
| bnj534 35037 | First-order logic and set ... |
| bnj538 35038 | First-order logic and set ... |
| bnj529 35039 | First-order logic and set ... |
| bnj551 35040 | First-order logic and set ... |
| bnj563 35041 | First-order logic and set ... |
| bnj564 35042 | First-order logic and set ... |
| bnj593 35043 | First-order logic and set ... |
| bnj596 35044 | First-order logic and set ... |
| bnj610 35045 | Pass from equality ( ` x =... |
| bnj642 35046 | ` /\ ` -manipulation. (Co... |
| bnj643 35047 | ` /\ ` -manipulation. (Co... |
| bnj645 35048 | ` /\ ` -manipulation. (Co... |
| bnj658 35049 | ` /\ ` -manipulation. (Co... |
| bnj667 35050 | ` /\ ` -manipulation. (Co... |
| bnj705 35051 | ` /\ ` -manipulation. (Co... |
| bnj706 35052 | ` /\ ` -manipulation. (Co... |
| bnj707 35053 | ` /\ ` -manipulation. (Co... |
| bnj708 35054 | ` /\ ` -manipulation. (Co... |
| bnj721 35055 | ` /\ ` -manipulation. (Co... |
| bnj832 35056 | ` /\ ` -manipulation. (Co... |
| bnj835 35057 | ` /\ ` -manipulation. (Co... |
| bnj836 35058 | ` /\ ` -manipulation. (Co... |
| bnj837 35059 | ` /\ ` -manipulation. (Co... |
| bnj769 35060 | ` /\ ` -manipulation. (Co... |
| bnj770 35061 | ` /\ ` -manipulation. (Co... |
| bnj771 35062 | ` /\ ` -manipulation. (Co... |
| bnj887 35063 | ` /\ ` -manipulation. (Co... |
| bnj918 35064 | First-order logic and set ... |
| bnj919 35065 | First-order logic and set ... |
| bnj923 35066 | First-order logic and set ... |
| bnj927 35067 | First-order logic and set ... |
| bnj931 35068 | First-order logic and set ... |
| bnj937 35069 | First-order logic and set ... |
| bnj941 35070 | First-order logic and set ... |
| bnj945 35071 | Technical lemma for ~ bnj6... |
| bnj946 35072 | First-order logic and set ... |
| bnj951 35073 | ` /\ ` -manipulation. (Co... |
| bnj956 35074 | First-order logic and set ... |
| bnj976 35075 | First-order logic and set ... |
| bnj982 35076 | First-order logic and set ... |
| bnj1019 35077 | First-order logic and set ... |
| bnj1023 35078 | First-order logic and set ... |
| bnj1095 35079 | First-order logic and set ... |
| bnj1096 35080 | First-order logic and set ... |
| bnj1098 35081 | First-order logic and set ... |
| bnj1101 35082 | First-order logic and set ... |
| bnj1113 35083 | First-order logic and set ... |
| bnj1109 35084 | First-order logic and set ... |
| bnj1131 35085 | First-order logic and set ... |
| bnj1138 35086 | First-order logic and set ... |
| bnj1143 35087 | First-order logic and set ... |
| bnj1146 35088 | First-order logic and set ... |
| bnj1149 35089 | First-order logic and set ... |
| bnj1185 35090 | First-order logic and set ... |
| bnj1196 35091 | First-order logic and set ... |
| bnj1198 35092 | First-order logic and set ... |
| bnj1209 35093 | First-order logic and set ... |
| bnj1211 35094 | First-order logic and set ... |
| bnj1213 35095 | First-order logic and set ... |
| bnj1212 35096 | First-order logic and set ... |
| bnj1219 35097 | First-order logic and set ... |
| bnj1224 35098 | First-order logic and set ... |
| bnj1230 35099 | First-order logic and set ... |
| bnj1232 35100 | First-order logic and set ... |
| bnj1235 35101 | First-order logic and set ... |
| bnj1239 35102 | First-order logic and set ... |
| bnj1238 35103 | First-order logic and set ... |
| bnj1241 35104 | First-order logic and set ... |
| bnj1247 35105 | First-order logic and set ... |
| bnj1254 35106 | First-order logic and set ... |
| bnj1262 35107 | First-order logic and set ... |
| bnj1266 35108 | First-order logic and set ... |
| bnj1265 35109 | First-order logic and set ... |
| bnj1275 35110 | First-order logic and set ... |
| bnj1276 35111 | First-order logic and set ... |
| bnj1292 35112 | First-order logic and set ... |
| bnj1293 35113 | First-order logic and set ... |
| bnj1294 35114 | First-order logic and set ... |
| bnj1299 35115 | First-order logic and set ... |
| bnj1304 35116 | First-order logic and set ... |
| bnj1316 35117 | First-order logic and set ... |
| bnj1317 35118 | First-order logic and set ... |
| bnj1322 35119 | First-order logic and set ... |
| bnj1340 35120 | First-order logic and set ... |
| bnj1345 35121 | First-order logic and set ... |
| bnj1350 35122 | First-order logic and set ... |
| bnj1351 35123 | First-order logic and set ... |
| bnj1352 35124 | First-order logic and set ... |
| bnj1361 35125 | First-order logic and set ... |
| bnj1366 35126 | First-order logic and set ... |
| bnj1379 35127 | First-order logic and set ... |
| bnj1383 35128 | First-order logic and set ... |
| bnj1385 35129 | First-order logic and set ... |
| bnj1386 35130 | First-order logic and set ... |
| bnj1397 35131 | First-order logic and set ... |
| bnj1400 35132 | First-order logic and set ... |
| bnj1405 35133 | First-order logic and set ... |
| bnj1422 35134 | First-order logic and set ... |
| bnj1424 35135 | First-order logic and set ... |
| bnj1436 35136 | First-order logic and set ... |
| bnj1441 35137 | First-order logic and set ... |
| bnj1441g 35138 | First-order logic and set ... |
| bnj1454 35139 | First-order logic and set ... |
| bnj1459 35140 | First-order logic and set ... |
| bnj1464 35141 | Conversion of implicit sub... |
| bnj1465 35142 | First-order logic and set ... |
| bnj1468 35143 | Conversion of implicit sub... |
| bnj1476 35144 | First-order logic and set ... |
| bnj1502 35145 | First-order logic and set ... |
| bnj1503 35146 | First-order logic and set ... |
| bnj1517 35147 | First-order logic and set ... |
| bnj1521 35148 | First-order logic and set ... |
| bnj1533 35149 | First-order logic and set ... |
| bnj1534 35150 | First-order logic and set ... |
| bnj1536 35151 | First-order logic and set ... |
| bnj1538 35152 | First-order logic and set ... |
| bnj1541 35153 | First-order logic and set ... |
| bnj1542 35154 | First-order logic and set ... |
| bnj110 35155 | Well-founded induction res... |
| bnj157 35156 | Well-founded induction res... |
| bnj66 35157 | Technical lemma for ~ bnj6... |
| bnj91 35158 | First-order logic and set ... |
| bnj92 35159 | First-order logic and set ... |
| bnj93 35160 | Technical lemma for ~ bnj9... |
| bnj95 35161 | Technical lemma for ~ bnj1... |
| bnj96 35162 | Technical lemma for ~ bnj1... |
| bnj97 35163 | Technical lemma for ~ bnj1... |
| bnj98 35164 | Technical lemma for ~ bnj1... |
| bnj106 35165 | First-order logic and set ... |
| bnj118 35166 | First-order logic and set ... |
| bnj121 35167 | First-order logic and set ... |
| bnj124 35168 | Technical lemma for ~ bnj1... |
| bnj125 35169 | Technical lemma for ~ bnj1... |
| bnj126 35170 | Technical lemma for ~ bnj1... |
| bnj130 35171 | Technical lemma for ~ bnj1... |
| bnj149 35172 | Technical lemma for ~ bnj1... |
| bnj150 35173 | Technical lemma for ~ bnj1... |
| bnj151 35174 | Technical lemma for ~ bnj1... |
| bnj154 35175 | Technical lemma for ~ bnj1... |
| bnj155 35176 | Technical lemma for ~ bnj1... |
| bnj153 35177 | Technical lemma for ~ bnj8... |
| bnj207 35178 | Technical lemma for ~ bnj8... |
| bnj213 35179 | First-order logic and set ... |
| bnj222 35180 | Technical lemma for ~ bnj2... |
| bnj229 35181 | Technical lemma for ~ bnj5... |
| bnj517 35182 | Technical lemma for ~ bnj5... |
| bnj518 35183 | Technical lemma for ~ bnj8... |
| bnj523 35184 | Technical lemma for ~ bnj8... |
| bnj526 35185 | Technical lemma for ~ bnj8... |
| bnj528 35186 | Technical lemma for ~ bnj8... |
| bnj535 35187 | Technical lemma for ~ bnj8... |
| bnj539 35188 | Technical lemma for ~ bnj8... |
| bnj540 35189 | Technical lemma for ~ bnj8... |
| bnj543 35190 | Technical lemma for ~ bnj8... |
| bnj544 35191 | Technical lemma for ~ bnj8... |
| bnj545 35192 | Technical lemma for ~ bnj8... |
| bnj546 35193 | Technical lemma for ~ bnj8... |
| bnj548 35194 | Technical lemma for ~ bnj8... |
| bnj553 35195 | Technical lemma for ~ bnj8... |
| bnj554 35196 | Technical lemma for ~ bnj8... |
| bnj556 35197 | Technical lemma for ~ bnj8... |
| bnj557 35198 | Technical lemma for ~ bnj8... |
| bnj558 35199 | Technical lemma for ~ bnj8... |
| bnj561 35200 | Technical lemma for ~ bnj8... |
| bnj562 35201 | Technical lemma for ~ bnj8... |
| bnj570 35202 | Technical lemma for ~ bnj8... |
| bnj571 35203 | Technical lemma for ~ bnj8... |
| bnj605 35204 | Technical lemma. This lem... |
| bnj581 35205 | Technical lemma for ~ bnj5... |
| bnj589 35206 | Technical lemma for ~ bnj8... |
| bnj590 35207 | Technical lemma for ~ bnj8... |
| bnj591 35208 | Technical lemma for ~ bnj8... |
| bnj594 35209 | Technical lemma for ~ bnj8... |
| bnj580 35210 | Technical lemma for ~ bnj5... |
| bnj579 35211 | Technical lemma for ~ bnj8... |
| bnj602 35212 | Equality theorem for the `... |
| bnj607 35213 | Technical lemma for ~ bnj8... |
| bnj609 35214 | Technical lemma for ~ bnj8... |
| bnj611 35215 | Technical lemma for ~ bnj8... |
| bnj600 35216 | Technical lemma for ~ bnj8... |
| bnj601 35217 | Technical lemma for ~ bnj8... |
| bnj852 35218 | Technical lemma for ~ bnj6... |
| bnj864 35219 | Technical lemma for ~ bnj6... |
| bnj865 35220 | Technical lemma for ~ bnj6... |
| bnj873 35221 | Technical lemma for ~ bnj6... |
| bnj849 35222 | Technical lemma for ~ bnj6... |
| bnj882 35223 | Definition (using hypothes... |
| bnj18eq1 35224 | Equality theorem for trans... |
| bnj893 35225 | Property of ` _trCl ` . U... |
| bnj900 35226 | Technical lemma for ~ bnj6... |
| bnj906 35227 | Property of ` _trCl ` . (... |
| bnj908 35228 | Technical lemma for ~ bnj6... |
| bnj911 35229 | Technical lemma for ~ bnj6... |
| bnj916 35230 | Technical lemma for ~ bnj6... |
| bnj917 35231 | Technical lemma for ~ bnj6... |
| bnj934 35232 | Technical lemma for ~ bnj6... |
| bnj929 35233 | Technical lemma for ~ bnj6... |
| bnj938 35234 | Technical lemma for ~ bnj6... |
| bnj944 35235 | Technical lemma for ~ bnj6... |
| bnj953 35236 | Technical lemma for ~ bnj6... |
| bnj958 35237 | Technical lemma for ~ bnj6... |
| bnj1000 35238 | Technical lemma for ~ bnj8... |
| bnj965 35239 | Technical lemma for ~ bnj8... |
| bnj964 35240 | Technical lemma for ~ bnj6... |
| bnj966 35241 | Technical lemma for ~ bnj6... |
| bnj967 35242 | Technical lemma for ~ bnj6... |
| bnj969 35243 | Technical lemma for ~ bnj6... |
| bnj970 35244 | Technical lemma for ~ bnj6... |
| bnj910 35245 | Technical lemma for ~ bnj6... |
| bnj978 35246 | Technical lemma for ~ bnj6... |
| bnj981 35247 | Technical lemma for ~ bnj6... |
| bnj983 35248 | Technical lemma for ~ bnj6... |
| bnj984 35249 | Technical lemma for ~ bnj6... |
| bnj985v 35250 | Version of ~ bnj985 with a... |
| bnj985 35251 | Technical lemma for ~ bnj6... |
| bnj986 35252 | Technical lemma for ~ bnj6... |
| bnj996 35253 | Technical lemma for ~ bnj6... |
| bnj998 35254 | Technical lemma for ~ bnj6... |
| bnj999 35255 | Technical lemma for ~ bnj6... |
| bnj1001 35256 | Technical lemma for ~ bnj6... |
| bnj1006 35257 | Technical lemma for ~ bnj6... |
| bnj1014 35258 | Technical lemma for ~ bnj6... |
| bnj1015 35259 | Technical lemma for ~ bnj6... |
| bnj1018g 35260 | Version of ~ bnj1018 with ... |
| bnj1018 35261 | Technical lemma for ~ bnj6... |
| bnj1020 35262 | Technical lemma for ~ bnj6... |
| bnj1021 35263 | Technical lemma for ~ bnj6... |
| bnj907 35264 | Technical lemma for ~ bnj6... |
| bnj1029 35265 | Property of ` _trCl ` . (... |
| bnj1033 35266 | Technical lemma for ~ bnj6... |
| bnj1034 35267 | Technical lemma for ~ bnj6... |
| bnj1039 35268 | Technical lemma for ~ bnj6... |
| bnj1040 35269 | Technical lemma for ~ bnj6... |
| bnj1047 35270 | Technical lemma for ~ bnj6... |
| bnj1049 35271 | Technical lemma for ~ bnj6... |
| bnj1052 35272 | Technical lemma for ~ bnj6... |
| bnj1053 35273 | Technical lemma for ~ bnj6... |
| bnj1071 35274 | Technical lemma for ~ bnj6... |
| bnj1083 35275 | Technical lemma for ~ bnj6... |
| bnj1090 35276 | Technical lemma for ~ bnj6... |
| bnj1093 35277 | Technical lemma for ~ bnj6... |
| bnj1097 35278 | Technical lemma for ~ bnj6... |
| bnj1110 35279 | Technical lemma for ~ bnj6... |
| bnj1112 35280 | Technical lemma for ~ bnj6... |
| bnj1118 35281 | Technical lemma for ~ bnj6... |
| bnj1121 35282 | Technical lemma for ~ bnj6... |
| bnj1123 35283 | Technical lemma for ~ bnj6... |
| bnj1030 35284 | Technical lemma for ~ bnj6... |
| bnj1124 35285 | Property of ` _trCl ` . (... |
| bnj1133 35286 | Technical lemma for ~ bnj6... |
| bnj1128 35287 | Technical lemma for ~ bnj6... |
| bnj1127 35288 | Property of ` _trCl ` . (... |
| bnj1125 35289 | Property of ` _trCl ` . (... |
| bnj1145 35290 | Technical lemma for ~ bnj6... |
| bnj1147 35291 | Property of ` _trCl ` . (... |
| bnj1137 35292 | Property of ` _trCl ` . (... |
| bnj1148 35293 | Property of ` _pred ` . (... |
| bnj1136 35294 | Technical lemma for ~ bnj6... |
| bnj1152 35295 | Technical lemma for ~ bnj6... |
| bnj1154 35296 | Property of ` Fr ` . (Con... |
| bnj1171 35297 | Technical lemma for ~ bnj6... |
| bnj1172 35298 | Technical lemma for ~ bnj6... |
| bnj1173 35299 | Technical lemma for ~ bnj6... |
| bnj1174 35300 | Technical lemma for ~ bnj6... |
| bnj1175 35301 | Technical lemma for ~ bnj6... |
| bnj1176 35302 | Technical lemma for ~ bnj6... |
| bnj1177 35303 | Technical lemma for ~ bnj6... |
| bnj1186 35304 | Technical lemma for ~ bnj6... |
| bnj1190 35305 | Technical lemma for ~ bnj6... |
| bnj1189 35306 | Technical lemma for ~ bnj6... |
| bnj69 35307 | Existence of a minimal ele... |
| bnj1228 35308 | Existence of a minimal ele... |
| bnj1204 35309 | Well-founded induction. T... |
| bnj1234 35310 | Technical lemma for ~ bnj6... |
| bnj1245 35311 | Technical lemma for ~ bnj6... |
| bnj1256 35312 | Technical lemma for ~ bnj6... |
| bnj1259 35313 | Technical lemma for ~ bnj6... |
| bnj1253 35314 | Technical lemma for ~ bnj6... |
| bnj1279 35315 | Technical lemma for ~ bnj6... |
| bnj1286 35316 | Technical lemma for ~ bnj6... |
| bnj1280 35317 | Technical lemma for ~ bnj6... |
| bnj1296 35318 | Technical lemma for ~ bnj6... |
| bnj1309 35319 | Technical lemma for ~ bnj6... |
| bnj1307 35320 | Technical lemma for ~ bnj6... |
| bnj1311 35321 | Technical lemma for ~ bnj6... |
| bnj1318 35322 | Technical lemma for ~ bnj6... |
| bnj1326 35323 | Technical lemma for ~ bnj6... |
| bnj1321 35324 | Technical lemma for ~ bnj6... |
| bnj1364 35325 | Property of ` _FrSe ` . (... |
| bnj1371 35326 | Technical lemma for ~ bnj6... |
| bnj1373 35327 | Technical lemma for ~ bnj6... |
| bnj1374 35328 | Technical lemma for ~ bnj6... |
| bnj1384 35329 | Technical lemma for ~ bnj6... |
| bnj1388 35330 | Technical lemma for ~ bnj6... |
| bnj1398 35331 | Technical lemma for ~ bnj6... |
| bnj1413 35332 | Property of ` _trCl ` . (... |
| bnj1408 35333 | Technical lemma for ~ bnj1... |
| bnj1414 35334 | Property of ` _trCl ` . (... |
| bnj1415 35335 | Technical lemma for ~ bnj6... |
| bnj1416 35336 | Technical lemma for ~ bnj6... |
| bnj1418 35337 | Property of ` _pred ` . (... |
| bnj1417 35338 | Technical lemma for ~ bnj6... |
| bnj1421 35339 | Technical lemma for ~ bnj6... |
| bnj1444 35340 | Technical lemma for ~ bnj6... |
| bnj1445 35341 | Technical lemma for ~ bnj6... |
| bnj1446 35342 | Technical lemma for ~ bnj6... |
| bnj1447 35343 | Technical lemma for ~ bnj6... |
| bnj1448 35344 | Technical lemma for ~ bnj6... |
| bnj1449 35345 | Technical lemma for ~ bnj6... |
| bnj1442 35346 | Technical lemma for ~ bnj6... |
| bnj1450 35347 | Technical lemma for ~ bnj6... |
| bnj1423 35348 | Technical lemma for ~ bnj6... |
| bnj1452 35349 | Technical lemma for ~ bnj6... |
| bnj1466 35350 | Technical lemma for ~ bnj6... |
| bnj1467 35351 | Technical lemma for ~ bnj6... |
| bnj1463 35352 | Technical lemma for ~ bnj6... |
| bnj1489 35353 | Technical lemma for ~ bnj6... |
| bnj1491 35354 | Technical lemma for ~ bnj6... |
| bnj1312 35355 | Technical lemma for ~ bnj6... |
| bnj1493 35356 | Technical lemma for ~ bnj6... |
| bnj1497 35357 | Technical lemma for ~ bnj6... |
| bnj1498 35358 | Technical lemma for ~ bnj6... |
| bnj60 35359 | Well-founded recursion, pa... |
| bnj1514 35360 | Technical lemma for ~ bnj1... |
| bnj1518 35361 | Technical lemma for ~ bnj1... |
| bnj1519 35362 | Technical lemma for ~ bnj1... |
| bnj1520 35363 | Technical lemma for ~ bnj1... |
| bnj1501 35364 | Technical lemma for ~ bnj1... |
| bnj1500 35365 | Well-founded recursion, pa... |
| bnj1525 35366 | Technical lemma for ~ bnj1... |
| bnj1529 35367 | Technical lemma for ~ bnj1... |
| bnj1523 35368 | Technical lemma for ~ bnj1... |
| bnj1522 35369 | Well-founded recursion, pa... |
| nfan1c 35370 | Variant of ~ nfan and comm... |
| cbvex1v 35371 | Rule used to change bound ... |
| dvelimalcased 35372 | Eliminate a disjoint varia... |
| dvelimalcasei 35373 | Eliminate a disjoint varia... |
| dvelimexcased 35374 | Eliminate a disjoint varia... |
| dvelimexcasei 35375 | Eliminate a disjoint varia... |
| exdifsn 35376 | There exists an element in... |
| srcmpltd 35377 | If a statement is true for... |
| prsrcmpltd 35378 | If a statement is true for... |
| axnulALT2 35379 | Alternate proof of ~ axnul... |
| dff15 35380 | A one-to-one function in t... |
| f1resveqaeq 35381 | If a function restricted t... |
| f1resrcmplf1dlem 35382 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 35383 | If a function's restrictio... |
| funen1cnv 35384 | If a function is equinumer... |
| ordprcon 35385 | If an ordinal class is not... |
| xoromon 35386 | ` _om ` is either an ordin... |
| fissorduni 35387 | The union (supremum) of a ... |
| ordtypeon 35388 | A proper class with a set-... |
| fnrelpredd 35389 | A function that preserves ... |
| cardpred 35390 | The cardinality function p... |
| nummin 35391 | Every nonempty class of nu... |
| 1enumen 35392 | The Fundamental Theorem of... |
| 1enumcard 35393 | The Fundamental Theorem of... |
| r11 35394 | Value of the cumulative hi... |
| r12 35395 | Value of the cumulative hi... |
| r1wf 35396 | Each stage in the cumulati... |
| elwf 35397 | An element of a well-found... |
| r1elcl 35398 | Each set of the cumulative... |
| rankval2b 35399 | Value of an alternate defi... |
| rankval4b 35400 | The rank of a set is the s... |
| rankfilimbi 35401 | If all elements in a finit... |
| rankfilimb 35402 | The rank of a finite well-... |
| r1filimi 35403 | If all elements in a finit... |
| r1filim 35404 | A finite set appears in th... |
| r1omfi 35405 | Hereditarily finite sets a... |
| r1omhf 35406 | A set is hereditarily fini... |
| r1ssel 35407 | A set is a subset of the v... |
| axnulALT3 35408 | Alternate proof of ~ axnul... |
| axprALT2 35409 | Alternate proof of ~ axpr ... |
| r1omfv 35410 | Value of the cumulative hi... |
| trssfir1om 35411 | If every element in a tran... |
| r1omhfb 35412 | The class of all hereditar... |
| prcinf 35413 | Any proper class is litera... |
| fineqvrep 35414 | If all sets are finite, th... |
| fineqvpow 35415 | If all sets are finite, th... |
| fineqvac 35416 | If all sets are finite, th... |
| fineqvacALT 35417 | Shorter proof of ~ fineqva... |
| fineqvomon 35418 | If all sets are finite, th... |
| fineqvomonb 35419 | All sets are finite iff al... |
| omprcomonb 35420 | The class of all finite or... |
| fineqvnttrclselem1 35421 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem2 35422 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem3 35423 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclse 35424 | A counterexample demonstra... |
| fineqvinfep 35425 | A counterexample demonstra... |
| axreg 35427 | Derivation of ~ ax-reg fro... |
| axregscl 35428 | A version of ~ ax-regs wit... |
| axregszf 35429 | Derivation of ~ zfregs usi... |
| setindregs 35430 | Set (epsilon) induction. ... |
| setinds2regs 35431 | Principle of set induction... |
| noinfepfnregs 35432 | There are no infinite desc... |
| noinfepregs 35433 | There are no infinite desc... |
| tz9.1regs 35434 | Every set has a transitive... |
| unir1regs 35435 | The cumulative hierarchy o... |
| trssfir1omregs 35436 | If every element in a tran... |
| r1omhfbregs 35437 | The class of all hereditar... |
| fineqvr1ombregs 35438 | All sets are finite iff al... |
| axregs 35439 | Derivation of ~ ax-regs fr... |
| axsepg2 35440 | A generalization of ~ ax-s... |
| axsepg3 35441 | A generalization of ~ ax-s... |
| axsepg3ALT 35442 | Alternate proof of ~ axsep... |
| axsepg4 35443 | A generalization of ~ ax-s... |
| axsepg5 35444 | A generalization of ~ ax-s... |
| axnulg 35445 | A generalization of ~ ax-n... |
| axpowg 35446 | A generalization of ~ ax-p... |
| axpowg2 35447 | A generalization of ~ ax-p... |
| axpowg3 35448 | A generalization of ~ ax-p... |
| gblacfnacd 35449 | If ` G ` is a global choic... |
| onvf1odlem1 35450 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem2 35451 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem3 35452 | Lemma for ~ onvf1od . The... |
| onvf1odlem4 35453 | Lemma for ~ onvf1od . If ... |
| onvf1od 35454 | If ` G ` is a global choic... |
| vonf1wev 35455 | If ` F ` maps the universe... |
| vonf1owev 35456 | If ` F ` is a bijection fr... |
| vonf1owevOLD 35457 | Obsolete version of ~ vonf... |
| wevgblacfn 35458 | If ` R ` is a well-orderin... |
| vonf1osev 35459 | If ` F ` is a bijection fr... |
| wevonprcf1o 35460 | If ` R ` is a set-like wel... |
| vonf1oonf1 35461 | If ` F ` is a bijection fr... |
| vonf1oonfo 35462 | If ` F ` is a bijection fr... |
| onvfowev 35463 | If ` F ` maps the ordinals... |
| zltp1ne 35464 | Integer ordering relation.... |
| nnltp1ne 35465 | Positive integer ordering ... |
| nn0ltp1ne 35466 | Nonnegative integer orderi... |
| 0nn0m1nnn0 35467 | A number is zero if and on... |
| f1resfz0f1d 35468 | If a function with a seque... |
| fisshasheq 35469 | A finite set is equal to i... |
| revpfxsfxrev 35470 | The reverse of a prefix of... |
| swrdrevpfx 35471 | A subword expressed in ter... |
| 1enum 35472 | The Fundamental Theorem of... |
| lfuhgr 35473 | A hypergraph is loop-free ... |
| lfuhgr2 35474 | A hypergraph is loop-free ... |
| lfuhgr3 35475 | A hypergraph is loop-free ... |
| cplgredgex 35476 | Any two (distinct) vertice... |
| cusgredgex 35477 | Any two (distinct) vertice... |
| cusgredgex2 35478 | Any two distinct vertices ... |
| pfxwlk 35479 | A prefix of a walk is a wa... |
| revwlk 35480 | The reverse of a walk is a... |
| revwlkb 35481 | Two words represent a walk... |
| swrdwlk 35482 | Two matching subwords of a... |
| pthhashvtx 35483 | A graph containing a path ... |
| spthcycl 35484 | A walk is a trivial path i... |
| usgrgt2cycl 35485 | A non-trivial cycle in a s... |
| usgrcyclgt2v 35486 | A simple graph with a non-... |
| subgrwlk 35487 | If a walk exists in a subg... |
| subgrtrl 35488 | If a trail exists in a sub... |
| subgrpth 35489 | If a path exists in a subg... |
| subgrcycl 35490 | If a cycle exists in a sub... |
| cusgr3cyclex 35491 | Every complete simple grap... |
| loop1cycl 35492 | A hypergraph has a cycle o... |
| 2cycld 35493 | Construction of a 2-cycle ... |
| 2cycl2d 35494 | Construction of a 2-cycle ... |
| umgr2cycllem 35495 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 35496 | A multigraph with two dist... |
| dfacycgr1 35499 | An alternate definition of... |
| isacycgr 35500 | The property of being an a... |
| isacycgr1 35501 | The property of being an a... |
| acycgrcycl 35502 | Any cycle in an acyclic gr... |
| acycgr0v 35503 | A null graph (with no vert... |
| acycgr1v 35504 | A multigraph with one vert... |
| acycgr2v 35505 | A simple graph with two ve... |
| prclisacycgr 35506 | A proper class (representi... |
| acycgrislfgr 35507 | An acyclic hypergraph is a... |
| upgracycumgr 35508 | An acyclic pseudograph is ... |
| umgracycusgr 35509 | An acyclic multigraph is a... |
| upgracycusgr 35510 | An acyclic pseudograph is ... |
| cusgracyclt3v 35511 | A complete simple graph is... |
| pthacycspth 35512 | A path in an acyclic graph... |
| acycgrsubgr 35513 | The subgraph of an acyclic... |
| quartfull 35520 | The quartic equation, writ... |
| deranglem 35521 | Lemma for derangements. (... |
| derangval 35522 | Define the derangement fun... |
| derangf 35523 | The derangement number is ... |
| derang0 35524 | The derangement number of ... |
| derangsn 35525 | The derangement number of ... |
| derangenlem 35526 | One half of ~ derangen . ... |
| derangen 35527 | The derangement number is ... |
| subfacval 35528 | The subfactorial is define... |
| derangen2 35529 | Write the derangement numb... |
| subfacf 35530 | The subfactorial is a func... |
| subfaclefac 35531 | The subfactorial is less t... |
| subfac0 35532 | The subfactorial at zero. ... |
| subfac1 35533 | The subfactorial at one. ... |
| subfacp1lem1 35534 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 35535 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 35536 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 35537 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 35538 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 35539 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 35540 | Lemma for ~ subfacp1 . By... |
| subfacp1 35541 | A two-term recurrence for ... |
| subfacval2 35542 | A closed-form expression f... |
| subfaclim 35543 | The subfactorial converges... |
| subfacval3 35544 | Another closed form expres... |
| derangfmla 35545 | The derangements formula, ... |
| erdszelem1 35546 | Lemma for ~ erdsze . (Con... |
| erdszelem2 35547 | Lemma for ~ erdsze . (Con... |
| erdszelem3 35548 | Lemma for ~ erdsze . (Con... |
| erdszelem4 35549 | Lemma for ~ erdsze . (Con... |
| erdszelem5 35550 | Lemma for ~ erdsze . (Con... |
| erdszelem6 35551 | Lemma for ~ erdsze . (Con... |
| erdszelem7 35552 | Lemma for ~ erdsze . (Con... |
| erdszelem8 35553 | Lemma for ~ erdsze . (Con... |
| erdszelem9 35554 | Lemma for ~ erdsze . (Con... |
| erdszelem10 35555 | Lemma for ~ erdsze . (Con... |
| erdszelem11 35556 | Lemma for ~ erdsze . (Con... |
| erdsze 35557 | The Erdős-Szekeres th... |
| erdsze2lem1 35558 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 35559 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 35560 | Generalize the statement o... |
| kur14lem1 35561 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 35562 | Lemma for ~ kur14 . Write... |
| kur14lem3 35563 | Lemma for ~ kur14 . A clo... |
| kur14lem4 35564 | Lemma for ~ kur14 . Compl... |
| kur14lem5 35565 | Lemma for ~ kur14 . Closu... |
| kur14lem6 35566 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 35567 | Lemma for ~ kur14 : main p... |
| kur14lem8 35568 | Lemma for ~ kur14 . Show ... |
| kur14lem9 35569 | Lemma for ~ kur14 . Since... |
| kur14lem10 35570 | Lemma for ~ kur14 . Disch... |
| kur14 35571 | Kuratowski's closure-compl... |
| ispconn 35578 | The property of being a pa... |
| pconncn 35579 | The property of being a pa... |
| pconntop 35580 | A simply connected space i... |
| issconn 35581 | The property of being a si... |
| sconnpconn 35582 | A simply connected space i... |
| sconntop 35583 | A simply connected space i... |
| sconnpht 35584 | A closed path in a simply ... |
| cnpconn 35585 | An image of a path-connect... |
| pconnconn 35586 | A path-connected space is ... |
| txpconn 35587 | The topological product of... |
| ptpconn 35588 | The topological product of... |
| indispconn 35589 | The indiscrete topology (o... |
| connpconn 35590 | A connected and locally pa... |
| qtoppconn 35591 | A quotient of a path-conne... |
| pconnpi1 35592 | All fundamental groups in ... |
| sconnpht2 35593 | Any two paths in a simply ... |
| sconnpi1 35594 | A path-connected topologic... |
| txsconnlem 35595 | Lemma for ~ txsconn . (Co... |
| txsconn 35596 | The topological product of... |
| cvxpconn 35597 | A convex subset of the com... |
| cvxsconn 35598 | A convex subset of the com... |
| blsconn 35599 | An open ball in the comple... |
| cnllysconn 35600 | The topology of the comple... |
| resconn 35601 | A subset of ` RR ` is simp... |
| ioosconn 35602 | An open interval is simply... |
| iccsconn 35603 | A closed interval is simpl... |
| retopsconn 35604 | The real numbers are simpl... |
| iccllysconn 35605 | A closed interval is local... |
| rellysconn 35606 | The real numbers are local... |
| iisconn 35607 | The unit interval is simpl... |
| iillysconn 35608 | The unit interval is local... |
| iinllyconn 35609 | The unit interval is local... |
| fncvm 35612 | Lemma for covering maps. ... |
| cvmscbv 35613 | Change bound variables in ... |
| iscvm 35614 | The property of being a co... |
| cvmtop1 35615 | Reverse closure for a cove... |
| cvmtop2 35616 | Reverse closure for a cove... |
| cvmcn 35617 | A covering map is a contin... |
| cvmcov 35618 | Property of a covering map... |
| cvmsrcl 35619 | Reverse closure for an eve... |
| cvmsi 35620 | One direction of ~ cvmsval... |
| cvmsval 35621 | Elementhood in the set ` S... |
| cvmsss 35622 | An even covering is a subs... |
| cvmsn0 35623 | An even covering is nonemp... |
| cvmsuni 35624 | An even covering of ` U ` ... |
| cvmsdisj 35625 | An even covering of ` U ` ... |
| cvmshmeo 35626 | Every element of an even c... |
| cvmsf1o 35627 | ` F ` , localized to an el... |
| cvmscld 35628 | The sets of an even coveri... |
| cvmsss2 35629 | An open subset of an evenl... |
| cvmcov2 35630 | The covering map property ... |
| cvmseu 35631 | Every element in ` U. T ` ... |
| cvmsiota 35632 | Identify the unique elemen... |
| cvmopnlem 35633 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 35634 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 35635 | A covering map is an open ... |
| cvmliftmolem1 35636 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 35637 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 35638 | A lift of a continuous fun... |
| cvmliftmo 35639 | A lift of a continuous fun... |
| cvmliftlem1 35640 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 35641 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 35642 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 35643 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 35644 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 35645 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 35646 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 35647 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 35648 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 35649 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 35650 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 35651 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 35652 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 35653 | Lemma for ~ cvmlift . Dis... |
| cvmlift 35654 | One of the important prope... |
| cvmfo 35655 | A covering map is an onto ... |
| cvmliftiota 35656 | Write out a function ` H `... |
| cvmlift2lem1 35657 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 35658 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 35659 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 35660 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 35661 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 35662 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 35663 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 35664 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 35665 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 35666 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 35667 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 35668 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 35669 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 35670 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 35671 | A two-dimensional version ... |
| cvmliftphtlem 35672 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 35673 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 35674 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 35675 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 35676 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 35677 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 35678 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 35679 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 35680 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 35681 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 35682 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 35683 | A general version of ~ cvm... |
| snmlff 35684 | The function ` F ` from ~ ... |
| snmlfval 35685 | The function ` F ` from ~ ... |
| snmlval 35686 | The property " ` A ` is si... |
| snmlflim 35687 | If ` A ` is simply normal,... |
| goel 35702 | A "Godel-set of membership... |
| goelel3xp 35703 | A "Godel-set of membership... |
| goeleq12bg 35704 | Two "Godel-set of membersh... |
| gonafv 35705 | The "Godel-set for the She... |
| goaleq12d 35706 | Equality of the "Godel-set... |
| gonanegoal 35707 | The Godel-set for the Shef... |
| satf 35708 | The satisfaction predicate... |
| satfsucom 35709 | The satisfaction predicate... |
| satfn 35710 | The satisfaction predicate... |
| satom 35711 | The satisfaction predicate... |
| satfvsucom 35712 | The satisfaction predicate... |
| satfv0 35713 | The value of the satisfact... |
| satfvsuclem1 35714 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 35715 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 35716 | The value of the satisfact... |
| satfv1lem 35717 | Lemma for ~ satfv1 . (Con... |
| satfv1 35718 | The value of the satisfact... |
| satfsschain 35719 | The binary relation of a s... |
| satfvsucsuc 35720 | The satisfaction predicate... |
| satfbrsuc 35721 | The binary relation of a s... |
| satfrel 35722 | The value of the satisfact... |
| satfdmlem 35723 | Lemma for ~ satfdm . (Con... |
| satfdm 35724 | The domain of the satisfac... |
| satfrnmapom 35725 | The range of the satisfact... |
| satfv0fun 35726 | The value of the satisfact... |
| satf0 35727 | The satisfaction predicate... |
| satf0sucom 35728 | The satisfaction predicate... |
| satf00 35729 | The value of the satisfact... |
| satf0suclem 35730 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 35731 | The value of the satisfact... |
| satf0op 35732 | An element of a value of t... |
| satf0n0 35733 | The value of the satisfact... |
| sat1el2xp 35734 | The first component of an ... |
| fmlafv 35735 | The valid Godel formulas o... |
| fmla 35736 | The set of all valid Godel... |
| fmla0 35737 | The valid Godel formulas o... |
| fmla0xp 35738 | The valid Godel formulas o... |
| fmlasuc0 35739 | The valid Godel formulas o... |
| fmlafvel 35740 | A class is a valid Godel f... |
| fmlasuc 35741 | The valid Godel formulas o... |
| fmla1 35742 | The valid Godel formulas o... |
| isfmlasuc 35743 | The characterization of a ... |
| fmlasssuc 35744 | The Godel formulas of heig... |
| fmlaomn0 35745 | The empty set is not a God... |
| fmlan0 35746 | The empty set is not a God... |
| gonan0 35747 | The "Godel-set of NAND" is... |
| goaln0 35748 | The "Godel-set of universa... |
| gonarlem 35749 | Lemma for ~ gonar (inducti... |
| gonar 35750 | If the "Godel-set of NAND"... |
| goalrlem 35751 | Lemma for ~ goalr (inducti... |
| goalr 35752 | If the "Godel-set of unive... |
| fmla0disjsuc 35753 | The set of valid Godel for... |
| fmlasucdisj 35754 | The valid Godel formulas o... |
| satfdmfmla 35755 | The domain of the satisfac... |
| satffunlem 35756 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 35757 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 35758 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 35759 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 35760 | The domain of an ordered p... |
| satffunlem2lem2 35761 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 35762 | Lemma 1 for ~ satffun : in... |
| satffunlem2 35763 | Lemma 2 for ~ satffun : in... |
| satffun 35764 | The value of the satisfact... |
| satff 35765 | The satisfaction predicate... |
| satfun 35766 | The satisfaction predicate... |
| satfvel 35767 | An element of the value of... |
| satfv0fvfmla0 35768 | The value of the satisfact... |
| satefv 35769 | The simplified satisfactio... |
| sate0 35770 | The simplified satisfactio... |
| satef 35771 | The simplified satisfactio... |
| sate0fv0 35772 | A simplified satisfaction ... |
| satefvfmla0 35773 | The simplified satisfactio... |
| sategoelfvb 35774 | Characterization of a valu... |
| sategoelfv 35775 | Condition of a valuation `... |
| ex-sategoelel 35776 | Example of a valuation of ... |
| ex-sategoel 35777 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 35778 | The value of the satisfact... |
| 2goelgoanfmla1 35779 | Two Godel-sets of membersh... |
| satefvfmla1 35780 | The simplified satisfactio... |
| ex-sategoelelomsuc 35781 | Example of a valuation of ... |
| ex-sategoelel12 35782 | Example of a valuation of ... |
| prv 35783 | The "proves" relation on a... |
| elnanelprv 35784 | The wff ` ( A e. B -/\ B e... |
| prv0 35785 | Every wff encoded as ` U `... |
| prv1n 35786 | No wff encoded as a Godel-... |
| mvtval 35855 | The set of variable typeco... |
| mrexval 35856 | The set of "raw expression... |
| mexval 35857 | The set of expressions, wh... |
| mexval2 35858 | The set of expressions, wh... |
| mdvval 35859 | The set of disjoint variab... |
| mvrsval 35860 | The set of variables in an... |
| mvrsfpw 35861 | The set of variables in an... |
| mrsubffval 35862 | The substitution of some v... |
| mrsubfval 35863 | The substitution of some v... |
| mrsubval 35864 | The substitution of some v... |
| mrsubcv 35865 | The value of a substituted... |
| mrsubvr 35866 | The value of a substituted... |
| mrsubff 35867 | A substitution is a functi... |
| mrsubrn 35868 | Although it is defined for... |
| mrsubff1 35869 | When restricted to complet... |
| mrsubff1o 35870 | When restricted to complet... |
| mrsub0 35871 | The value of the substitut... |
| mrsubf 35872 | A substitution is a functi... |
| mrsubccat 35873 | Substitution distributes o... |
| mrsubcn 35874 | A substitution does not ch... |
| elmrsubrn 35875 | Characterization of the su... |
| mrsubco 35876 | The composition of two sub... |
| mrsubvrs 35877 | The set of variables in a ... |
| msubffval 35878 | A substitution applied to ... |
| msubfval 35879 | A substitution applied to ... |
| msubval 35880 | A substitution applied to ... |
| msubrsub 35881 | A substitution applied to ... |
| msubty 35882 | The type of a substituted ... |
| elmsubrn 35883 | Characterization of substi... |
| msubrn 35884 | Although it is defined for... |
| msubff 35885 | A substitution is a functi... |
| msubco 35886 | The composition of two sub... |
| msubf 35887 | A substitution is a functi... |
| mvhfval 35888 | Value of the function mapp... |
| mvhval 35889 | Value of the function mapp... |
| mpstval 35890 | A pre-statement is an orde... |
| elmpst 35891 | Property of being a pre-st... |
| msrfval 35892 | Value of the reduct of a p... |
| msrval 35893 | Value of the reduct of a p... |
| mpstssv 35894 | A pre-statement is an orde... |
| mpst123 35895 | Decompose a pre-statement ... |
| mpstrcl 35896 | The elements of a pre-stat... |
| msrf 35897 | The reduct of a pre-statem... |
| msrrcl 35898 | If ` X ` and ` Y ` have th... |
| mstaval 35899 | Value of the set of statem... |
| msrid 35900 | The reduct of a statement ... |
| msrfo 35901 | The reduct of a pre-statem... |
| mstapst 35902 | A statement is a pre-state... |
| elmsta 35903 | Property of being a statem... |
| ismfs 35904 | A formal system is a tuple... |
| mfsdisj 35905 | The constants and variable... |
| mtyf2 35906 | The type function maps var... |
| mtyf 35907 | The type function maps var... |
| mvtss 35908 | The set of variable typeco... |
| maxsta 35909 | An axiom is a statement. ... |
| mvtinf 35910 | Each variable typecode has... |
| msubff1 35911 | When restricted to complet... |
| msubff1o 35912 | When restricted to complet... |
| mvhf 35913 | The function mapping varia... |
| mvhf1 35914 | The function mapping varia... |
| msubvrs 35915 | The set of variables in a ... |
| mclsrcl 35916 | Reverse closure for the cl... |
| mclsssvlem 35917 | Lemma for ~ mclsssv . (Co... |
| mclsval 35918 | The function mapping varia... |
| mclsssv 35919 | The closure of a set of ex... |
| ssmclslem 35920 | Lemma for ~ ssmcls . (Con... |
| vhmcls 35921 | All variable hypotheses ar... |
| ssmcls 35922 | The original expressions a... |
| ss2mcls 35923 | The closure is monotonic u... |
| mclsax 35924 | The closure is closed unde... |
| mclsind 35925 | Induction theorem for clos... |
| mppspstlem 35926 | Lemma for ~ mppspst . (Co... |
| mppsval 35927 | Definition of a provable p... |
| elmpps 35928 | Definition of a provable p... |
| mppspst 35929 | A provable pre-statement i... |
| mthmval 35930 | A theorem is a pre-stateme... |
| elmthm 35931 | A theorem is a pre-stateme... |
| mthmi 35932 | A statement whose reduct i... |
| mthmsta 35933 | A theorem is a pre-stateme... |
| mppsthm 35934 | A provable pre-statement i... |
| mthmblem 35935 | Lemma for ~ mthmb . (Cont... |
| mthmb 35936 | If two statements have the... |
| mthmpps 35937 | Given a theorem, there is ... |
| mclsppslem 35938 | The closure is closed unde... |
| mclspps 35939 | The closure is closed unde... |
| rexxfr3d 35993 | Transfer existential quant... |
| rexxfr3dALT 35994 | Longer proof of ~ rexxfr3d... |
| rspssbasd 35995 | The span of a set of ring ... |
| ellcsrspsn 35996 | Membership in a left coset... |
| ply1divalg3 35997 | Uniqueness of polynomial r... |
| r1peuqusdeg1 35998 | Uniqueness of polynomial r... |
| problem1 36020 | Practice problem 1. Clues... |
| problem2 36021 | Practice problem 2. Clues... |
| problem3 36022 | Practice problem 3. Clues... |
| problem4 36023 | Practice problem 4. Clues... |
| problem5 36024 | Practice problem 5. Clues... |
| quad3 36025 | Variant of quadratic equat... |
| climuzcnv 36026 | Utility lemma to convert b... |
| sinccvglem 36027 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 36028 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 36029 | The circumference of a cir... |
| elfzm12 36030 | Membership in a curtailed ... |
| nn0seqcvg 36031 | A strictly-decreasing nonn... |
| lediv2aALT 36032 | Division of both sides of ... |
| abs2sqlei 36033 | The absolute values of two... |
| abs2sqlti 36034 | The absolute values of two... |
| abs2sqle 36035 | The absolute values of two... |
| abs2sqlt 36036 | The absolute values of two... |
| abs2difi 36037 | Difference of absolute val... |
| abs2difabsi 36038 | Absolute value of differen... |
| 2thALT 36039 | Alternate proof of ~ 2th .... |
| orbi2iALT 36040 | Alternate proof of ~ orbi2... |
| pm3.48ALT 36041 | Alternate proof of ~ pm3.4... |
| 3jcadALT 36042 | Alternate proof of ~ 3jcad... |
| currybi 36043 | Biconditional version of C... |
| antnest 36044 | Suppose ` ph ` , ` ps ` ar... |
| antnestlaw3lem 36045 | Lemma for ~ antnestlaw3 . ... |
| antnestlaw1 36046 | A law of nested antecedent... |
| antnestlaw2 36047 | A law of nested antecedent... |
| antnestlaw3 36048 | A law of nested antecedent... |
| antnestALT 36049 | Alternative proof of ~ ant... |
| axextprim 36056 | ~ ax-ext without distinct ... |
| axrepprim 36057 | ~ ax-rep without distinct ... |
| axunprim 36058 | ~ ax-un without distinct v... |
| axpowprim 36059 | ~ ax-pow without distinct ... |
| axregprim 36060 | ~ ax-reg without distinct ... |
| axinfprim 36061 | ~ ax-inf without distinct ... |
| axacprim 36062 | ~ ax-ac without distinct v... |
| untelirr 36063 | We call a class "untanged"... |
| untuni 36064 | The union of a class is un... |
| untsucf 36065 | If a class is untangled, t... |
| unt0 36066 | The null set is untangled.... |
| untint 36067 | If there is an untangled e... |
| efrunt 36068 | If ` A ` is well-founded b... |
| untangtr 36069 | A transitive class is unta... |
| 3jaodd 36070 | Double deduction form of ~... |
| 3orit 36071 | Closed form of ~ 3ori . (... |
| biimpexp 36072 | A biconditional in the ant... |
| nepss 36073 | Two classes are unequal if... |
| 3ccased 36074 | Triple disjunction form of... |
| dfso3 36075 | Expansion of the definitio... |
| brtpid1 36076 | A binary relation involvin... |
| brtpid2 36077 | A binary relation involvin... |
| brtpid3 36078 | A binary relation involvin... |
| iota5f 36079 | A method for computing iot... |
| jath 36080 | Closed form of ~ ja . Pro... |
| xpab 36081 | Cartesian product of two c... |
| nnuni 36082 | The union of a finite ordi... |
| sqdivzi 36083 | Distribution of square ove... |
| supfz 36084 | The supremum of a finite s... |
| inffz 36085 | The infimum of a finite se... |
| fz0n 36086 | The sequence ` ( 0 ... ( N... |
| shftvalg 36087 | Value of a sequence shifte... |
| divcnvlin 36088 | Limit of the ratio of two ... |
| climlec3 36089 | Comparison of a constant t... |
| iexpire 36090 | ` _i ` raised to itself is... |
| bcneg1 36091 | The binomial coefficient o... |
| bcm1nt 36092 | The proportion of one bino... |
| bcprod 36093 | A product identity for bin... |
| bccolsum 36094 | A column-sum rule for bino... |
| iprodefisumlem 36095 | Lemma for ~ iprodefisum . ... |
| iprodefisum 36096 | Applying the exponential f... |
| iprodgam 36097 | An infinite product versio... |
| faclimlem1 36098 | Lemma for ~ faclim . Clos... |
| faclimlem2 36099 | Lemma for ~ faclim . Show... |
| faclimlem3 36100 | Lemma for ~ faclim . Alge... |
| faclim 36101 | An infinite product expres... |
| iprodfac 36102 | An infinite product expres... |
| faclim2 36103 | Another factorial limit du... |
| gcd32 36104 | Swap the second and third ... |
| gcdabsorb 36105 | Absorption law for gcd. (... |
| dftr6 36106 | A potential definition of ... |
| coep 36107 | Composition with the membe... |
| coepr 36108 | Composition with the conve... |
| dffr5 36109 | A quantifier-free definiti... |
| dfso2 36110 | Quantifier-free definition... |
| br8 36111 | Substitution for an eight-... |
| br6 36112 | Substitution for a six-pla... |
| br4 36113 | Substitution for a four-pl... |
| cnvco1 36114 | Another distributive law o... |
| cnvco2 36115 | Another distributive law o... |
| eldm3 36116 | Quantifier-free definition... |
| elrn3 36117 | Quantifier-free definition... |
| pocnv 36118 | The converse of a partial ... |
| socnv 36119 | The converse of a strict o... |
| elintfv 36120 | Membership in an intersect... |
| funpsstri 36121 | A condition for subset tri... |
| fundmpss 36122 | If a class ` F ` is a prop... |
| funsseq 36123 | Given two functions with e... |
| fununiq 36124 | The uniqueness condition o... |
| funbreq 36125 | An equality condition for ... |
| br1steq 36126 | Uniqueness condition for t... |
| br2ndeq 36127 | Uniqueness condition for t... |
| dfdm5 36128 | Definition of domain in te... |
| dfrn5 36129 | Definition of range in ter... |
| opelco3 36130 | Alternate way of saying th... |
| elima4 36131 | Quantifier-free expression... |
| fv1stcnv 36132 | The value of the converse ... |
| fv2ndcnv 36133 | The value of the converse ... |
| elpotr 36134 | A class of transitive sets... |
| dford5reg 36135 | Given ~ ax-reg , an ordina... |
| dfon2lem1 36136 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 36137 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 36138 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 36139 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 36140 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 36141 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 36142 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 36143 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 36144 | Lemma for ~ dfon2 . A cla... |
| dfon2 36145 | ` On ` consists of all set... |
| rdgprc0 36146 | The value of the recursive... |
| rdgprc 36147 | The value of the recursive... |
| dfrdg2 36148 | Alternate definition of th... |
| dfrdg3 36149 | Generalization of ~ dfrdg2... |
| axextdfeq 36150 | A version of ~ ax-ext for ... |
| ax8dfeq 36151 | A version of ~ ax-8 for us... |
| axextdist 36152 | ~ ax-ext with distinctors ... |
| axextbdist 36153 | ~ axextb with distinctors ... |
| 19.12b 36154 | Version of ~ 19.12vv with ... |
| exnel 36155 | There is always a set not ... |
| distel 36156 | Distinctors in terms of me... |
| axextndbi 36157 | ~ axextnd as a bicondition... |
| hbntg 36158 | A more general form of ~ h... |
| hbimtg 36159 | A more general and closed ... |
| hbaltg 36160 | A more general and closed ... |
| hbng 36161 | A more general form of ~ h... |
| hbimg 36162 | A more general form of ~ h... |
| wsuceq123 36167 | Equality theorem for well-... |
| wsuceq1 36168 | Equality theorem for well-... |
| wsuceq2 36169 | Equality theorem for well-... |
| wsuceq3 36170 | Equality theorem for well-... |
| nfwsuc 36171 | Bound-variable hypothesis ... |
| wlimeq12 36172 | Equality theorem for the l... |
| wlimeq1 36173 | Equality theorem for the l... |
| wlimeq2 36174 | Equality theorem for the l... |
| nfwlim 36175 | Bound-variable hypothesis ... |
| elwlim 36176 | Membership in the limit cl... |
| wzel 36177 | The zero of a well-founded... |
| wsuclem 36178 | Lemma for the supremum pro... |
| wsucex 36179 | Existence theorem for well... |
| wsuccl 36180 | If ` X ` is a set with an ... |
| wsuclb 36181 | A well-founded successor i... |
| wlimss 36182 | The class of limit points ... |
| txpss3v 36231 | A tail Cartesian product i... |
| txprel 36232 | A tail Cartesian product i... |
| brtxp 36233 | Characterize a ternary rel... |
| brtxp2 36234 | The binary relation over a... |
| dfpprod2 36235 | Expanded definition of par... |
| pprodcnveq 36236 | A converse law for paralle... |
| pprodss4v 36237 | The parallel product is a ... |
| brpprod 36238 | Characterize a quaternary ... |
| brpprod3a 36239 | Condition for parallel pro... |
| brpprod3b 36240 | Condition for parallel pro... |
| relsset 36241 | The subset class is a bina... |
| brsset 36242 | For sets, the ` SSet ` bin... |
| idsset 36243 | ` _I ` is equal to the int... |
| eltrans 36244 | Membership in the class of... |
| dfon3 36245 | A quantifier-free definiti... |
| dfon4 36246 | Another quantifier-free de... |
| brtxpsd 36247 | Expansion of a common form... |
| brtxpsd2 36248 | Another common abbreviatio... |
| brtxpsd3 36249 | A third common abbreviatio... |
| relbigcup 36250 | The ` Bigcup ` relationshi... |
| brbigcup 36251 | Binary relation over ` Big... |
| dfbigcup2 36252 | ` Bigcup ` using maps-to n... |
| fobigcup 36253 | ` Bigcup ` maps the univer... |
| fnbigcup 36254 | ` Bigcup ` is a function o... |
| fvbigcup 36255 | For sets, ` Bigcup ` yield... |
| elfix 36256 | Membership in the fixpoint... |
| elfix2 36257 | Alternative membership in ... |
| dffix2 36258 | The fixpoints of a class i... |
| fixssdm 36259 | The fixpoints of a class a... |
| fixssrn 36260 | The fixpoints of a class a... |
| fixcnv 36261 | The fixpoints of a class a... |
| fixun 36262 | The fixpoint operator dist... |
| ellimits 36263 | Membership in the class of... |
| limitssson 36264 | The class of all limit ord... |
| dfom5b 36265 | A quantifier-free definiti... |
| sscoid 36266 | A condition for subset and... |
| dffun10 36267 | Another potential definiti... |
| elfuns 36268 | Membership in the class of... |
| elfunsg 36269 | Closed form of ~ elfuns . ... |
| brsingle 36270 | The binary relation form o... |
| elsingles 36271 | Membership in the class of... |
| fnsingle 36272 | The singleton relationship... |
| fvsingle 36273 | The value of the singleton... |
| dfsingles2 36274 | Alternate definition of th... |
| snelsingles 36275 | A singleton is a member of... |
| dfiota3 36276 | A definition of iota using... |
| dffv5 36277 | Another quantifier-free de... |
| unisnif 36278 | Express union of singleton... |
| brimage 36279 | Binary relation form of th... |
| brimageg 36280 | Closed form of ~ brimage .... |
| funimage 36281 | ` Image A ` is a function.... |
| fnimage 36282 | ` Image R ` is a function ... |
| imageval 36283 | The image functor in maps-... |
| fvimage 36284 | Value of the image functor... |
| brcart 36285 | Binary relation form of th... |
| brdomain 36286 | Binary relation form of th... |
| brrange 36287 | Binary relation form of th... |
| brdomaing 36288 | Closed form of ~ brdomain ... |
| brrangeg 36289 | Closed form of ~ brrange .... |
| brimg 36290 | Binary relation form of th... |
| brapply 36291 | Binary relation form of th... |
| brcup 36292 | Binary relation form of th... |
| brcap 36293 | Binary relation form of th... |
| lemsuccf 36294 | Lemma for unfolding differ... |
| brsuccf 36295 | Binary relation form of th... |
| dfsuccf2 36296 | Alternate definition of Sc... |
| funpartlem 36297 | Lemma for ~ funpartfun . ... |
| funpartfun 36298 | The functional part of ` F... |
| funpartss 36299 | The functional part of ` F... |
| funpartfv 36300 | The function value of the ... |
| fullfunfnv 36301 | The full functional part o... |
| fullfunfv 36302 | The function value of the ... |
| brfullfun 36303 | A binary relation form con... |
| brrestrict 36304 | Binary relation form of th... |
| dfrecs2 36305 | A quantifier-free definiti... |
| dfrdg4 36306 | A quantifier-free definiti... |
| dfint3 36307 | Quantifier-free definition... |
| imagesset 36308 | The Image functor applied ... |
| brub 36309 | Binary relation form of th... |
| brlb 36310 | Binary relation form of th... |
| altopex 36315 | Alternative ordered pairs ... |
| altopthsn 36316 | Two alternate ordered pair... |
| altopeq12 36317 | Equality for alternate ord... |
| altopeq1 36318 | Equality for alternate ord... |
| altopeq2 36319 | Equality for alternate ord... |
| altopth1 36320 | Equality of the first memb... |
| altopth2 36321 | Equality of the second mem... |
| altopthg 36322 | Alternate ordered pair the... |
| altopthbg 36323 | Alternate ordered pair the... |
| altopth 36324 | The alternate ordered pair... |
| altopthb 36325 | Alternate ordered pair the... |
| altopthc 36326 | Alternate ordered pair the... |
| altopthd 36327 | Alternate ordered pair the... |
| altxpeq1 36328 | Equality for alternate Car... |
| altxpeq2 36329 | Equality for alternate Car... |
| elaltxp 36330 | Membership in alternate Ca... |
| altopelaltxp 36331 | Alternate ordered pair mem... |
| altxpsspw 36332 | An inclusion rule for alte... |
| altxpexg 36333 | The alternate Cartesian pr... |
| rankaltopb 36334 | Compute the rank of an alt... |
| nfaltop 36335 | Bound-variable hypothesis ... |
| sbcaltop 36336 | Distribution of class subs... |
| cgrrflx2d 36339 | Deduction form of ~ axcgrr... |
| cgrtr4d 36340 | Deduction form of ~ axcgrt... |
| cgrtr4and 36341 | Deduction form of ~ axcgrt... |
| cgrrflx 36342 | Reflexivity law for congru... |
| cgrrflxd 36343 | Deduction form of ~ cgrrfl... |
| cgrcomim 36344 | Congruence commutes on the... |
| cgrcom 36345 | Congruence commutes betwee... |
| cgrcomand 36346 | Deduction form of ~ cgrcom... |
| cgrtr 36347 | Transitivity law for congr... |
| cgrtrand 36348 | Deduction form of ~ cgrtr ... |
| cgrtr3 36349 | Transitivity law for congr... |
| cgrtr3and 36350 | Deduction form of ~ cgrtr3... |
| cgrcoml 36351 | Congruence commutes on the... |
| cgrcomr 36352 | Congruence commutes on the... |
| cgrcomlr 36353 | Congruence commutes on bot... |
| cgrcomland 36354 | Deduction form of ~ cgrcom... |
| cgrcomrand 36355 | Deduction form of ~ cgrcom... |
| cgrcomlrand 36356 | Deduction form of ~ cgrcom... |
| cgrtriv 36357 | Degenerate segments are co... |
| cgrid2 36358 | Identity law for congruenc... |
| cgrdegen 36359 | Two congruent segments are... |
| brofs 36360 | Binary relation form of th... |
| 5segofs 36361 | Rephrase ~ ax5seg using th... |
| ofscom 36362 | The outer five segment pre... |
| cgrextend 36363 | Link congruence over a pai... |
| cgrextendand 36364 | Deduction form of ~ cgrext... |
| segconeq 36365 | Two points that satisfy th... |
| segconeu 36366 | Existential uniqueness ver... |
| btwntriv2 36367 | Betweenness always holds f... |
| btwncomim 36368 | Betweenness commutes. Imp... |
| btwncom 36369 | Betweenness commutes. (Co... |
| btwncomand 36370 | Deduction form of ~ btwnco... |
| btwntriv1 36371 | Betweenness always holds f... |
| btwnswapid 36372 | If you can swap the first ... |
| btwnswapid2 36373 | If you can swap arguments ... |
| btwnintr 36374 | Inner transitivity law for... |
| btwnexch3 36375 | Exchange the first endpoin... |
| btwnexch3and 36376 | Deduction form of ~ btwnex... |
| btwnouttr2 36377 | Outer transitivity law for... |
| btwnexch2 36378 | Exchange the outer point o... |
| btwnouttr 36379 | Outer transitivity law for... |
| btwnexch 36380 | Outer transitivity law for... |
| btwnexchand 36381 | Deduction form of ~ btwnex... |
| btwndiff 36382 | There is always a ` c ` di... |
| trisegint 36383 | A line segment between two... |
| funtransport 36386 | The ` TransportTo ` relati... |
| fvtransport 36387 | Calculate the value of the... |
| transportcl 36388 | Closure law for segment tr... |
| transportprops 36389 | Calculate the defining pro... |
| brifs 36398 | Binary relation form of th... |
| ifscgr 36399 | Inner five segment congrue... |
| cgrsub 36400 | Removing identical parts f... |
| brcgr3 36401 | Binary relation form of th... |
| cgr3permute3 36402 | Permutation law for three-... |
| cgr3permute1 36403 | Permutation law for three-... |
| cgr3permute2 36404 | Permutation law for three-... |
| cgr3permute4 36405 | Permutation law for three-... |
| cgr3permute5 36406 | Permutation law for three-... |
| cgr3tr4 36407 | Transitivity law for three... |
| cgr3com 36408 | Commutativity law for thre... |
| cgr3rflx 36409 | Identity law for three-pla... |
| cgrxfr 36410 | A line segment can be divi... |
| btwnxfr 36411 | A condition for extending ... |
| colinrel 36412 | Colinearity is a relations... |
| brcolinear2 36413 | Alternate colinearity bina... |
| brcolinear 36414 | The binary relation form o... |
| colinearex 36415 | The colinear predicate exi... |
| colineardim1 36416 | If ` A ` is colinear with ... |
| colinearperm1 36417 | Permutation law for coline... |
| colinearperm3 36418 | Permutation law for coline... |
| colinearperm2 36419 | Permutation law for coline... |
| colinearperm4 36420 | Permutation law for coline... |
| colinearperm5 36421 | Permutation law for coline... |
| colineartriv1 36422 | Trivial case of colinearit... |
| colineartriv2 36423 | Trivial case of colinearit... |
| btwncolinear1 36424 | Betweenness implies coline... |
| btwncolinear2 36425 | Betweenness implies coline... |
| btwncolinear3 36426 | Betweenness implies coline... |
| btwncolinear4 36427 | Betweenness implies coline... |
| btwncolinear5 36428 | Betweenness implies coline... |
| btwncolinear6 36429 | Betweenness implies coline... |
| colinearxfr 36430 | Transfer law for colineari... |
| lineext 36431 | Extend a line with a missi... |
| brofs2 36432 | Change some conditions for... |
| brifs2 36433 | Change some conditions for... |
| brfs 36434 | Binary relation form of th... |
| fscgr 36435 | Congruence law for the gen... |
| linecgr 36436 | Congruence rule for lines.... |
| linecgrand 36437 | Deduction form of ~ linecg... |
| lineid 36438 | Identity law for points on... |
| idinside 36439 | Law for finding a point in... |
| endofsegid 36440 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 36441 | Deduction form of ~ endofs... |
| btwnconn1lem1 36442 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 36443 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 36444 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 36445 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 36446 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 36447 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 36448 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 36449 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 36450 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 36451 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 36452 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 36453 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 36454 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 36455 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 36456 | Connectitivy law for betwe... |
| btwnconn2 36457 | Another connectivity law f... |
| btwnconn3 36458 | Inner connectivity law for... |
| midofsegid 36459 | If two points fall in the ... |
| segcon2 36460 | Generalization of ~ axsegc... |
| brsegle 36463 | Binary relation form of th... |
| brsegle2 36464 | Alternate characterization... |
| seglecgr12im 36465 | Substitution law for segme... |
| seglecgr12 36466 | Substitution law for segme... |
| seglerflx 36467 | Segment comparison is refl... |
| seglemin 36468 | Any segment is at least as... |
| segletr 36469 | Segment less than is trans... |
| segleantisym 36470 | Antisymmetry law for segme... |
| seglelin 36471 | Linearity law for segment ... |
| btwnsegle 36472 | If ` B ` falls between ` A... |
| colinbtwnle 36473 | Given three colinear point... |
| broutsideof 36476 | Binary relation form of ` ... |
| broutsideof2 36477 | Alternate form of ` Outsid... |
| outsidene1 36478 | Outsideness implies inequa... |
| outsidene2 36479 | Outsideness implies inequa... |
| btwnoutside 36480 | A principle linking outsid... |
| broutsideof3 36481 | Characterization of outsid... |
| outsideofrflx 36482 | Reflexivity of outsideness... |
| outsideofcom 36483 | Commutativity law for outs... |
| outsideoftr 36484 | Transitivity law for outsi... |
| outsideofeq 36485 | Uniqueness law for ` Outsi... |
| outsideofeu 36486 | Given a nondegenerate ray,... |
| outsidele 36487 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 36488 | Outside of implies colinea... |
| funray 36495 | Show that the ` Ray ` rela... |
| fvray 36496 | Calculate the value of the... |
| funline 36497 | Show that the ` Line ` rel... |
| linedegen 36498 | When ` Line ` is applied w... |
| fvline 36499 | Calculate the value of the... |
| liness 36500 | A line is a subset of the ... |
| fvline2 36501 | Alternate definition of a ... |
| lineunray 36502 | A line is composed of a po... |
| lineelsb2 36503 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 36504 | Reflexivity law for line m... |
| linecom 36505 | Commutativity law for line... |
| linerflx2 36506 | Reflexivity law for line m... |
| ellines 36507 | Membership in the set of a... |
| linethru 36508 | If ` A ` is a line contain... |
| hilbert1.1 36509 | There is a line through an... |
| hilbert1.2 36510 | There is at most one line ... |
| linethrueu 36511 | There is a unique line goi... |
| lineintmo 36512 | Two distinct lines interse... |
| fwddifval 36517 | Calculate the value of the... |
| fwddifnval 36518 | The value of the forward d... |
| fwddifn0 36519 | The value of the n-iterate... |
| fwddifnp1 36520 | The value of the n-iterate... |
| rankung 36521 | The rank of the union of t... |
| ranksng 36522 | The rank of a singleton. ... |
| rankelg 36523 | The membership relation is... |
| rankpwg 36524 | The rank of a power set. ... |
| rank0 36525 | The rank of the empty set ... |
| rankeq1o 36526 | The only set with rank ` 1... |
| elhf 36529 | Membership in the heredita... |
| elhf2 36530 | Alternate form of membersh... |
| elhf2g 36531 | Hereditarily finiteness vi... |
| 0hf 36532 | The empty set is a heredit... |
| hfun 36533 | The union of two HF sets i... |
| hfsn 36534 | The singleton of an HF set... |
| hfadj 36535 | Adjoining one HF element t... |
| hfelhf 36536 | Any member of an HF set is... |
| hftr 36537 | The class of all hereditar... |
| hfext 36538 | Extensionality for HF sets... |
| hfuni 36539 | The union of an HF set is ... |
| hfpw 36540 | The power class of an HF s... |
| hfninf 36541 | ` _om ` is not hereditaril... |
| nmulfn 36544 | Natural multiplication is ... |
| nmulprop 36545 | Show closure and value of ... |
| nmulcl 36546 | Closure law for natural mu... |
| nmulval 36547 | Show the value of natural ... |
| nmulcld 36548 | Closure law for natrual mu... |
| nmulcom 36549 | Natural multiplication com... |
| nmulr0 36550 | Natural multiplication by ... |
| nmull0 36551 | Natural multiplication by ... |
| rmoeqi 36552 | Equality inference for res... |
| rmoeqbii 36553 | Equality inference for res... |
| reueqi 36554 | Equality inference for res... |
| reueqbii 36555 | Equality inference for res... |
| sbceqbii 36556 | Formula-building inference... |
| disjeq1i 36557 | Equality theorem for disjo... |
| disjeq12i 36558 | Equality theorem for disjo... |
| rabeqbii 36559 | Equality theorem for restr... |
| iuneq12i 36560 | Equality theorem for index... |
| iineq1i 36561 | Equality theorem for index... |
| iineq12i 36562 | Equality theorem for index... |
| riotaeqbii 36563 | Equivalent wff's and equal... |
| riotaeqi 36564 | Equal domains yield equal ... |
| ixpeq1i 36565 | Equality inference for inf... |
| ixpeq12i 36566 | Equality inference for inf... |
| sumeq2si 36567 | Equality inference for sum... |
| sumeq12si 36568 | Equality inference for sum... |
| prodeq2si 36569 | Equality inference for pro... |
| prodeq12si 36570 | Equality inference for pro... |
| itgeq12i 36571 | Equality inference for an ... |
| itgeq1i 36572 | Equality inference for an ... |
| itgeq2i 36573 | Equality inference for an ... |
| ditgeq123i 36574 | Equality inference for the... |
| ditgeq12i 36575 | Equality inference for the... |
| ditgeq3i 36576 | Equality inference for the... |
| rmoeqdv 36577 | Formula-building rule for ... |
| rmoeqbidv 36578 | Formula-building rule for ... |
| sbequbidv 36579 | Deduction substituting bot... |
| disjeq12dv 36580 | Equality theorem for disjo... |
| ixpeq12dv 36581 | Equality theorem for infin... |
| sumeq12sdv 36582 | Equality deduction for sum... |
| prodeq12sdv 36583 | Equality deduction for pro... |
| itgeq12sdv 36584 | Equality theorem for an in... |
| itgeq2sdv 36585 | Equality theorem for an in... |
| ditgeq123dv 36586 | Equality theorem for the d... |
| ditgeq12d 36587 | Equality theorem for the d... |
| ditgeq3sdv 36588 | Equality theorem for the d... |
| in-ax8 36589 | A proof of ~ ax-8 that doe... |
| ss-ax8 36590 | A proof of ~ ax-8 that doe... |
| cbvralvw2 36591 | Change bound variable and ... |
| cbvrexvw2 36592 | Change bound variable and ... |
| cbvrmovw2 36593 | Change bound variable and ... |
| cbvreuvw2 36594 | Change bound variable and ... |
| cbvsbcvw2 36595 | Change bound variable of a... |
| cbvcsbvw2 36596 | Change bound variable of a... |
| cbviunvw2 36597 | Change bound variable and ... |
| cbviinvw2 36598 | Change bound variable and ... |
| cbvmptvw2 36599 | Change bound variable and ... |
| cbvdisjvw2 36600 | Change bound variable and ... |
| cbvriotavw2 36601 | Change bound variable and ... |
| cbvoprab1vw 36602 | Change the first bound var... |
| cbvoprab2vw 36603 | Change the second bound va... |
| cbvoprab123vw 36604 | Change all bound variables... |
| cbvoprab23vw 36605 | Change the second and thir... |
| cbvoprab13vw 36606 | Change the first and third... |
| cbvmpovw2 36607 | Change bound variables and... |
| cbvmpo1vw2 36608 | Change domains and the fir... |
| cbvmpo2vw2 36609 | Change domains and the sec... |
| cbvixpvw2 36610 | Change bound variable and ... |
| cbvsumvw2 36611 | Change bound variable and ... |
| cbvprodvw2 36612 | Change bound variable and ... |
| cbvitgvw2 36613 | Change bound variable and ... |
| cbvditgvw2 36614 | Change bound variable and ... |
| cbvmodavw 36615 | Change bound variable in t... |
| cbveudavw 36616 | Change bound variable in t... |
| cbvrmodavw 36617 | Change bound variable in t... |
| cbvreudavw 36618 | Change bound variable in t... |
| cbvsbdavw 36619 | Change bound variable in p... |
| cbvsbdavw2 36620 | Change bound variable in p... |
| cbvabdavw 36621 | Change bound variable in c... |
| cbvsbcdavw 36622 | Change bound variable of a... |
| cbvsbcdavw2 36623 | Change bound variable of a... |
| cbvcsbdavw 36624 | Change bound variable of a... |
| cbvcsbdavw2 36625 | Change bound variable of a... |
| cbvrabdavw 36626 | Change bound variable in r... |
| cbviundavw 36627 | Change bound variable in i... |
| cbviindavw 36628 | Change bound variable in i... |
| cbvopab1davw 36629 | Change the first bound var... |
| cbvopab2davw 36630 | Change the second bound va... |
| cbvopabdavw 36631 | Change bound variables in ... |
| cbvmptdavw 36632 | Change bound variable in a... |
| cbvdisjdavw 36633 | Change bound variable in a... |
| cbviotadavw 36634 | Change bound variable in a... |
| cbvriotadavw 36635 | Change bound variable in a... |
| cbvoprab1davw 36636 | Change the first bound var... |
| cbvoprab2davw 36637 | Change the second bound va... |
| cbvoprab3davw 36638 | Change the third bound var... |
| cbvoprab123davw 36639 | Change all bound variables... |
| cbvoprab12davw 36640 | Change the first and secon... |
| cbvoprab23davw 36641 | Change the second and thir... |
| cbvoprab13davw 36642 | Change the first and third... |
| cbvixpdavw 36643 | Change bound variable in a... |
| cbvsumdavw 36644 | Change bound variable in a... |
| cbvproddavw 36645 | Change bound variable in a... |
| cbvitgdavw 36646 | Change bound variable in a... |
| cbvditgdavw 36647 | Change bound variable in a... |
| cbvrmodavw2 36648 | Change bound variable and ... |
| cbvreudavw2 36649 | Change bound variable and ... |
| cbvrabdavw2 36650 | Change bound variable and ... |
| cbviundavw2 36651 | Change bound variable and ... |
| cbviindavw2 36652 | Change bound variable and ... |
| cbvmptdavw2 36653 | Change bound variable and ... |
| cbvdisjdavw2 36654 | Change bound variable and ... |
| cbvriotadavw2 36655 | Change bound variable and ... |
| cbvmpodavw2 36656 | Change bound variable and ... |
| cbvmpo1davw2 36657 | Change first bound variabl... |
| cbvmpo2davw2 36658 | Change second bound variab... |
| cbvixpdavw2 36659 | Change bound variable and ... |
| cbvsumdavw2 36660 | Change bound variable and ... |
| cbvproddavw2 36661 | Change bound variable and ... |
| cbvitgdavw2 36662 | Change bound variable and ... |
| cbvditgdavw2 36663 | Change bound variable and ... |
| mpomulnzcnf 36664 | Multiplication maps nonzer... |
| a1i14 36665 | Add two antecedents to a w... |
| a1i24 36666 | Add two antecedents to a w... |
| exp5d 36667 | An exportation inference. ... |
| exp5g 36668 | An exportation inference. ... |
| exp5k 36669 | An exportation inference. ... |
| exp56 36670 | An exportation inference. ... |
| exp58 36671 | An exportation inference. ... |
| exp510 36672 | An exportation inference. ... |
| exp511 36673 | An exportation inference. ... |
| exp512 36674 | An exportation inference. ... |
| 3com12d 36675 | Commutation in consequent.... |
| imp5p 36676 | A triple importation infer... |
| imp5q 36677 | A triple importation infer... |
| ecase13d 36678 | Deduction for elimination ... |
| subtr 36679 | Transitivity of implicit s... |
| subtr2 36680 | Transitivity of implicit s... |
| trer 36681 | A relation intersected wit... |
| elicc3 36682 | An equivalent membership c... |
| finminlem 36683 | A useful lemma about finit... |
| gtinf 36684 | Any number greater than an... |
| opnrebl 36685 | A set is open in the stand... |
| opnrebl2 36686 | A set is open in the stand... |
| nn0prpwlem 36687 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 36688 | Two nonnegative integers a... |
| topbnd 36689 | Two equivalent expressions... |
| opnbnd 36690 | A set is open iff it is di... |
| cldbnd 36691 | A set is closed iff it con... |
| ntruni 36692 | A union of interiors is a ... |
| clsun 36693 | A pairwise union of closur... |
| clsint2 36694 | The closure of an intersec... |
| opnregcld 36695 | A set is regularly closed ... |
| cldregopn 36696 | A set if regularly open if... |
| neiin 36697 | Two neighborhoods intersec... |
| hmeoclda 36698 | Homeomorphisms preserve cl... |
| hmeocldb 36699 | Homeomorphisms preserve cl... |
| ivthALT 36700 | An alternate proof of the ... |
| fnerel 36703 | Fineness is a relation. (... |
| isfne 36704 | The predicate " ` B ` is f... |
| isfne4 36705 | The predicate " ` B ` is f... |
| isfne4b 36706 | A condition for a topology... |
| isfne2 36707 | The predicate " ` B ` is f... |
| isfne3 36708 | The predicate " ` B ` is f... |
| fnebas 36709 | A finer cover covers the s... |
| fnetg 36710 | A finer cover generates a ... |
| fnessex 36711 | If ` B ` is finer than ` A... |
| fneuni 36712 | If ` B ` is finer than ` A... |
| fneint 36713 | If a cover is finer than a... |
| fness 36714 | A cover is finer than its ... |
| fneref 36715 | Reflexivity of the finenes... |
| fnetr 36716 | Transitivity of the finene... |
| fneval 36717 | Two covers are finer than ... |
| fneer 36718 | Fineness intersected with ... |
| topfne 36719 | Fineness for covers corres... |
| topfneec 36720 | A cover is equivalent to a... |
| topfneec2 36721 | A topology is precisely id... |
| fnessref 36722 | A cover is finer iff it ha... |
| refssfne 36723 | A cover is a refinement if... |
| neibastop1 36724 | A collection of neighborho... |
| neibastop2lem 36725 | Lemma for ~ neibastop2 . ... |
| neibastop2 36726 | In the topology generated ... |
| neibastop3 36727 | The topology generated by ... |
| topmtcl 36728 | The meet of a collection o... |
| topmeet 36729 | Two equivalent formulation... |
| topjoin 36730 | Two equivalent formulation... |
| fnemeet1 36731 | The meet of a collection o... |
| fnemeet2 36732 | The meet of equivalence cl... |
| fnejoin1 36733 | Join of equivalence classe... |
| fnejoin2 36734 | Join of equivalence classe... |
| fgmin 36735 | Minimality property of a g... |
| neifg 36736 | The neighborhood filter of... |
| tailfval 36737 | The tail function for a di... |
| tailval 36738 | The tail of an element in ... |
| eltail 36739 | An element of a tail. (Co... |
| tailf 36740 | The tail function of a dir... |
| tailini 36741 | A tail contains its initia... |
| tailfb 36742 | The collection of tails of... |
| filnetlem1 36743 | Lemma for ~ filnet . Chan... |
| filnetlem2 36744 | Lemma for ~ filnet . The ... |
| filnetlem3 36745 | Lemma for ~ filnet . (Con... |
| filnetlem4 36746 | Lemma for ~ filnet . (Con... |
| filnet 36747 | A filter has the same conv... |
| tb-ax1 36748 | The first of three axioms ... |
| tb-ax2 36749 | The second of three axioms... |
| tb-ax3 36750 | The third of three axioms ... |
| tbsyl 36751 | The weak syllogism from Ta... |
| re1ax2lem 36752 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 36753 | ~ ax-2 rederived from the ... |
| naim1 36754 | Constructor theorem for ` ... |
| naim2 36755 | Constructor theorem for ` ... |
| naim1i 36756 | Constructor rule for ` -/\... |
| naim2i 36757 | Constructor rule for ` -/\... |
| naim12i 36758 | Constructor rule for ` -/\... |
| nabi1i 36759 | Constructor rule for ` -/\... |
| nabi2i 36760 | Constructor rule for ` -/\... |
| nabi12i 36761 | Constructor rule for ` -/\... |
| df3nandALT1 36764 | The double nand expressed ... |
| df3nandALT2 36765 | The double nand expressed ... |
| andnand1 36766 | Double and in terms of dou... |
| imnand2 36767 | An ` -> ` nand relation. ... |
| nalfal 36768 | Not all sets hold ` F. ` a... |
| nexntru 36769 | There does not exist a set... |
| nexfal 36770 | There does not exist a set... |
| neufal 36771 | There does not exist exact... |
| neutru 36772 | There does not exist exact... |
| nmotru 36773 | There does not exist at mo... |
| mofal 36774 | There exist at most one se... |
| nrmo 36775 | "At most one" restricted e... |
| meran1 36776 | A single axiom for proposi... |
| meran2 36777 | A single axiom for proposi... |
| meran3 36778 | A single axiom for proposi... |
| waj-ax 36779 | A single axiom for proposi... |
| lukshef-ax2 36780 | A single axiom for proposi... |
| arg-ax 36781 | A single axiom for proposi... |
| negsym1 36782 | In the paper "On Variable ... |
| imsym1 36783 | A symmetry with ` -> ` . ... |
| bisym1 36784 | A symmetry with ` <-> ` . ... |
| consym1 36785 | A symmetry with ` /\ ` . ... |
| dissym1 36786 | A symmetry with ` \/ ` . ... |
| nandsym1 36787 | A symmetry with ` -/\ ` . ... |
| unisym1 36788 | A symmetry with ` A. ` . ... |
| exisym1 36789 | A symmetry with ` E. ` . ... |
| unqsym1 36790 | A symmetry with ` E! ` . ... |
| amosym1 36791 | A symmetry with ` E* ` . ... |
| subsym1 36792 | A symmetry with ` [ x / y ... |
| ontopbas 36793 | An ordinal number is a top... |
| onsstopbas 36794 | The class of ordinal numbe... |
| onpsstopbas 36795 | The class of ordinal numbe... |
| ontgval 36796 | The topology generated fro... |
| ontgsucval 36797 | The topology generated fro... |
| onsuctop 36798 | A successor ordinal number... |
| onsuctopon 36799 | One of the topologies on a... |
| ordtoplem 36800 | Membership of the class of... |
| ordtop 36801 | An ordinal is a topology i... |
| onsucconni 36802 | A successor ordinal number... |
| onsucconn 36803 | A successor ordinal number... |
| ordtopconn 36804 | An ordinal topology is con... |
| onintopssconn 36805 | An ordinal topology is con... |
| onsuct0 36806 | A successor ordinal number... |
| ordtopt0 36807 | An ordinal topology is T_0... |
| onsucsuccmpi 36808 | The successor of a success... |
| onsucsuccmp 36809 | The successor of a success... |
| limsucncmpi 36810 | The successor of a limit o... |
| limsucncmp 36811 | The successor of a limit o... |
| ordcmp 36812 | An ordinal topology is com... |
| ssoninhaus 36813 | The ordinal topologies ` 1... |
| onint1 36814 | The ordinal T_1 spaces are... |
| oninhaus 36815 | The ordinal Hausdorff spac... |
| fveleq 36816 | Please add description her... |
| findfvcl 36817 | Please add description her... |
| findreccl 36818 | Please add description her... |
| findabrcl 36819 | Please add description her... |
| nnssi2 36820 | Convert a theorem for real... |
| nnssi3 36821 | Convert a theorem for real... |
| nndivsub 36822 | Please add description her... |
| nndivlub 36823 | A factor of a positive int... |
| ee7.2aOLD 36826 | Lemma for Euclid's Element... |
| weiunval 36827 | Value of the relation cons... |
| weiunlem 36828 | Lemma for ~ weiunpo , ~ we... |
| weiunfrlem 36829 | Lemma for ~ weiunfr . (Co... |
| weiunpo 36830 | A partial ordering on an i... |
| weiunso 36831 | A strict ordering on an in... |
| weiunfr 36832 | A well-founded relation on... |
| weiunse 36833 | The relation constructed i... |
| weiunwe 36834 | A well-ordering on an inde... |
| numiunnum 36835 | An indexed union of sets i... |
| axtco 36836 | Axiom of Transitive Contai... |
| axtco1 36838 | Strong form of the Axiom o... |
| axtco2 36839 | Weak form of the Axiom of ... |
| axtco1from2 36840 | Strong form ~ axtco1 of th... |
| axtco1g 36841 | Strong form of the Axiom o... |
| axtco2g 36842 | Weak form of the Axiom of ... |
| axtcond 36843 | A version of the Axiom of ... |
| axuntco 36844 | Derivation of ~ ax-un from... |
| axnulregtco 36845 | Derivation of ~ ax-nul fro... |
| elALTtco 36846 | Derivation of ~ el from ~ ... |
| tz9.1ctco 36847 | Version of ~ tz9.1c derive... |
| tz9.1tco 36848 | Version of ~ tz9.1 derived... |
| tr0elw 36849 | Every nonempty transitive ... |
| tr0el 36850 | Every nonempty transitive ... |
| ttceq 36853 | Equality theorem for trans... |
| ttceqi 36854 | Equality inference for tra... |
| ttceqd 36855 | Equality deduction for tra... |
| nfttc 36856 | Bound-variable hypothesis ... |
| ttcid 36857 | The transitive closure con... |
| ttctr 36858 | The transitive closure of ... |
| ttctr2 36859 | The transitive closure of ... |
| ttctr3 36860 | The transitive closure of ... |
| ttcmin 36861 | The transitive closure of ... |
| ttcexrg 36862 | If the transitive closure ... |
| ttcss 36863 | A transitive closure conta... |
| ttcss2 36864 | The subclass relationship ... |
| ttcel 36865 | A transitive closure conta... |
| ttcel2 36866 | Elements turn into subclas... |
| ttctrid 36867 | The transitive closure of ... |
| ttcidm 36868 | The transitive closure ope... |
| ssttctr 36869 | Transitivity of ` A C_ TC+... |
| elttctr 36870 | Transitivity of ` A e. TC+... |
| dfttc2g 36871 | A shorter expression for t... |
| ttc0 36872 | The transitive closure of ... |
| ttc00 36873 | A class has an empty trans... |
| csbttc 36874 | Distribute proper substitu... |
| ttcuniun 36875 | Relationship between ` TC+... |
| ttciunun 36876 | Relationship between ` TC+... |
| ttcun 36877 | Distribute union of two cl... |
| ttcuni 36878 | Distribute union of a clas... |
| ttciun 36879 | Distribute indexed union t... |
| ttcpwss 36880 | The transitive closure of ... |
| ttcsnssg 36881 | The transitive closure is ... |
| ttcsnidg 36882 | The singleton transitive c... |
| ttcsnmin 36883 | The singleton transitive c... |
| ttcsng 36884 | Relationship between ` TC+... |
| ttcsnexg 36885 | If the transitive closure ... |
| ttcsnexbig 36886 | The transitive closure of ... |
| ttcsntrsucg 36887 | The singleton transitive c... |
| dfttc3gw 36888 | If the transitive closure ... |
| ttcwf 36889 | A set is well-founded iff ... |
| ttcwf2 36890 | If a transitive closure cl... |
| ttcwf3 36891 | The sets whose transitive ... |
| ttc0elw 36892 | If a transitive closure is... |
| dfttc4lem1 36893 | Lemma for ~ dfttc4 . (Con... |
| dfttc4lem2 36894 | Lemma for ~ dfttc4 . (Con... |
| dfttc4 36895 | An alternative expression ... |
| elttcirr 36896 | Irreflexivity of ` A e. TC... |
| ttcexg 36897 | The transitive closure of ... |
| ttcexbi 36898 | A class is a set iff its t... |
| dfttc3g 36899 | The transitive closure of ... |
| ttc0el 36900 | A transitive closure conta... |
| mh-setind 36901 | Principle of set induction... |
| mh-setindnd 36902 | A version of ~ mh-setind w... |
| regsfromregtco 36903 | Derivation of ~ ax-regs fr... |
| regsfromsetind 36904 | Derivation of ~ ax-regs fr... |
| regsfromunir1 36905 | Derivation of ~ ax-regs fr... |
| mh-inf3f1 36906 | A variant of ~ inf3 . If ... |
| mh-inf3sn 36907 | Version of ~ inf3 for the ... |
| mh-prprimbi 36908 | Shortest possible version ... |
| mh-unprimbi 36909 | Shortest possible version ... |
| mh-regprimbi 36910 | Shortest possible version ... |
| mh-infprim1bi 36911 | Shortest possible axiom of... |
| mh-infprim2bi 36912 | Shortest possible axiom of... |
| mh-infprim3bi 36913 | An axiom of infinity in pr... |
| dnival 36914 | Value of the "distance to ... |
| dnicld1 36915 | Closure theorem for the "d... |
| dnicld2 36916 | Closure theorem for the "d... |
| dnif 36917 | The "distance to nearest i... |
| dnizeq0 36918 | The distance to nearest in... |
| dnizphlfeqhlf 36919 | The distance to nearest in... |
| rddif2 36920 | Variant of ~ rddif . (Con... |
| dnibndlem1 36921 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 36922 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 36923 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 36924 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 36925 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 36926 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 36927 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 36928 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 36929 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 36930 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 36931 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 36932 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 36933 | Lemma for ~ dnibnd . (Con... |
| dnibnd 36934 | The "distance to nearest i... |
| dnicn 36935 | The "distance to nearest i... |
| knoppcnlem1 36936 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 36937 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 36938 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 36939 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 36940 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 36941 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 36942 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 36943 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 36944 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 36945 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 36946 | Lemma for ~ knoppcn . (Co... |
| knoppcn 36947 | The continuous nowhere dif... |
| knoppcld 36948 | Closure theorem for Knopp'... |
| unblimceq0lem 36949 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 36950 | If ` F ` is unbounded near... |
| unbdqndv1 36951 | If the difference quotient... |
| unbdqndv2lem1 36952 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 36953 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 36954 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 36955 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 36956 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 36957 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 36958 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 36959 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 36960 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 36961 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 36962 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 36963 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 36964 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 36965 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 36966 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 36967 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 36968 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 36969 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 36970 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 36971 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 36972 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 36973 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 36974 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 36975 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 36976 | Lemma for ~ knoppndv . (C... |
| knoppndv 36977 | The continuous nowhere dif... |
| knoppf 36978 | Knopp's function is a func... |
| knoppcn2 36979 | Variant of ~ knoppcn with ... |
| cnndvlem1 36980 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 36981 | Lemma for ~ cnndv . (Cont... |
| cnndv 36982 | There exists a continuous ... |
| bj-mp2c 36983 | A double _modus ponens_ in... |
| bj-mp2d 36984 | A double _modus ponens_ in... |
| bj-0 36985 | A syntactic theorem. See ... |
| bj-1 36986 | In this proof, the use of ... |
| bj-a1k 36987 | Weakening of ~ ax-1 . As ... |
| bj-poni 36988 | Inference associated with ... |
| bj-nnclav 36989 | When ` F. ` is substituted... |
| bj-nnclavi 36990 | Inference associated with ... |
| bj-nnclavc 36991 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 36992 | Inference associated with ... |
| bj-jarrii 36993 | Inference associated with ... |
| bj-imim21 36994 | The propositional function... |
| bj-imim21i 36995 | The propositional function... |
| bj-imim11 36996 | The propositional function... |
| bj-imim11i 36997 | The propositional function... |
| bj-peircestab 36998 | Over minimal implicational... |
| bj-stabpeirce 36999 | This minimal implicational... |
| bj-bisimpl 37000 | Implication from equivalen... |
| bj-bisimpr 37001 | Implication from equivalen... |
| bj-syl66ib 37002 | A mixed syllogism inferenc... |
| bj-orim2 37003 | Proof of ~ orim2 from the ... |
| bj-currypeirce 37004 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 37005 | Peirce's axiom ~ peirce im... |
| bj-animbi 37006 | Conjunction in terms of im... |
| bj-currypara 37007 | Curry's paradox. Note tha... |
| bj-con2com 37008 | A commuted form of the con... |
| bj-con2comi 37009 | Inference associated with ... |
| bj-nimn 37010 | If a formula is true, then... |
| bj-nimni 37011 | Inference associated with ... |
| bj-peircei 37012 | Inference associated with ... |
| bj-looinvi 37013 | Inference associated with ... |
| bj-looinvii 37014 | Inference associated with ... |
| bj-mt2bi 37015 | Version of ~ mt2 where the... |
| bj-fal 37016 | Shortening of ~ fal using ... |
| bj-ntrufal 37017 | The negation of a theorem ... |
| bj-dfnul2 37018 | Alternate definition of th... |
| bj-jaoi1 37019 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 37020 | Shortens ~ consensus (110>... |
| bj-dfbi4 37021 | Alternate definition of th... |
| bj-dfbi5 37022 | Alternate definition of th... |
| bj-dfbi6 37023 | Alternate definition of th... |
| bj-bijust0ALT 37024 | Alternate proof of ~ bijus... |
| bj-bijust00 37025 | A self-implication does no... |
| bj-consensus 37026 | Version of ~ consensus exp... |
| bj-consensusALT 37027 | Alternate proof of ~ bj-co... |
| bj-df-ifc 37028 | Candidate definition for t... |
| bj-dfif 37029 | Alternate definition of th... |
| bj-ififc 37030 | A biconditional connecting... |
| bj-imbi12 37031 | Uncurried (imported) form ... |
| bj-falor 37032 | Dual of ~ truan (which has... |
| bj-falor2 37033 | Dual of ~ truan . (Contri... |
| bj-bibibi 37034 | A property of the bicondit... |
| bj-imn3ani 37035 | Duplication of ~ bnj1224 .... |
| bj-andnotim 37036 | Two ways of expressing a c... |
| bj-bi3ant 37037 | This used to be in the mai... |
| bj-bisym 37038 | This used to be in the mai... |
| bj-bixor 37039 | Equivalence of two ternary... |
| bj-axdd2 37040 | This implication, proved u... |
| bj-axd2d 37041 | This implication, proved u... |
| bj-axtd 37042 | This implication, proved f... |
| bj-gl4 37043 | In a normal modal logic, t... |
| bj-axc4 37044 | Over minimal calculus, the... |
| prvlem1 37049 | An elementary property of ... |
| prvlem2 37050 | An elementary property of ... |
| bj-babygodel 37051 | See the section header com... |
| bj-babylob 37052 | See the section header com... |
| bj-godellob 37053 | Proof of Gödel's theo... |
| bj-exexalal 37054 | A lemma for changing bound... |
| bj-genr 37055 | Generalization rule on the... |
| bj-genl 37056 | Generalization rule on the... |
| bj-genan 37057 | Generalization rule on a c... |
| bj-mpgs 37058 | From a closed form theorem... |
| bj-almp 37059 | A quantified form of ~ ax-... |
| bj-sylggt 37060 | Stronger form of ~ sylgt ,... |
| bj-alrimg 37061 | The general form of the *a... |
| bj-sylgt2 37062 | Uncurried (imported) form ... |
| bj-nexdh 37063 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 37064 | Uncurried (imported) form ... |
| bj-alimii 37065 | Inference associated with ... |
| bj-ala1i 37066 | Add an antecedent in a uni... |
| bj-almpi 37067 | A quantified form of ~ mpi... |
| bj-almpig 37068 | A partially quantified for... |
| bj-alsyl 37069 | Syllogism under the univer... |
| bj-2alim 37070 | Closed form of ~ 2alimi . ... |
| bj-alimdh 37071 | General instance of ~ alim... |
| bj-alrimdh 37072 | Deduction form of Theorem ... |
| bj-alrimd 37073 | A slightly more general ~ ... |
| bj-exa1i 37074 | Add an antecedent in an ex... |
| bj-alanim 37075 | Closed form of ~ alanimi .... |
| bj-2albi 37076 | Closed form of ~ 2albii . ... |
| bj-notalbii 37077 | Equivalence of universal q... |
| bj-2exim 37078 | Closed form of ~ 2eximi . ... |
| bj-2exbi 37079 | Closed form of ~ 2exbii . ... |
| bj-3exbi 37080 | Closed form of ~ 3exbii . ... |
| bj-sylget 37081 | Dual statement of ~ sylgt ... |
| bj-sylget2 37082 | Uncurried (imported) form ... |
| bj-exlimg 37083 | The general form of the *e... |
| bj-sylge 37084 | Dual statement of ~ sylg (... |
| bj-exlimd 37085 | A slightly more general ~ ... |
| bj-nfimexal 37086 | A weak from of nonfreeness... |
| bj-exim 37087 | Theorem 19.22 of [Margaris... |
| bj-alexim 37088 | Closed form of ~ aleximi .... |
| bj-aleximiALT 37089 | Alternate proof of ~ alexi... |
| bj-hbxfrbi 37090 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 37091 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 37092 | Distribute quantifiers ove... |
| bj-exalimi 37093 | An inference for distribut... |
| bj-eximcom 37094 | A commuted form of ~ exim ... |
| bj-exalims 37095 | Distributing quantifiers o... |
| bj-exalimsi 37096 | An inference for distribut... |
| bj-axdd2ALT 37097 | Alternate proof of ~ bj-ax... |
| bj-ax12ig 37098 | A lemma used to prove a we... |
| bj-ax12i 37099 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 37100 | Closed form of ~ nfim and ... |
| bj-spimnfe 37101 | A universal specification ... |
| bj-spimenfa 37102 | An existential generalizat... |
| bj-spim 37103 | A lemma for universal spec... |
| bj-spime 37104 | A lemma for existential ge... |
| bj-cbvalimd0 37105 | A lemma for alpha-renaming... |
| bj-cbvalimdlem 37106 | A lemma for alpha-renaming... |
| bj-cbveximdlem 37107 | A lemma for alpha-renaming... |
| bj-cbvalimd 37108 | A lemma for alpha-renaming... |
| bj-cbveximd 37109 | A lemma for alpha-renaming... |
| bj-cbvalimdv 37110 | A lemma for alpha-renaming... |
| bj-cbveximdv 37111 | A lemma for alpha-renaming... |
| bj-spvw 37112 | Version of ~ spvw and ~ 19... |
| bj-spvew 37113 | Version of ~ 19.8v and ~ 1... |
| bj-alextruim 37114 | An equivalent expression f... |
| bj-exextruan 37115 | An equivalent expression f... |
| bj-cbvalvv 37116 | Universally quantifying ov... |
| bj-cbvexvv 37117 | Existentially quantifying ... |
| bj-cbvaw 37118 | Universally quantifying ov... |
| bj-cbvew 37119 | Existentially quantifying ... |
| bj-cbveaw 37120 | Universally quantifying ov... |
| bj-cbvaew 37121 | Exixtentially quantifying ... |
| bj-ax12wlem 37122 | A lemma used to prove a we... |
| bj-cbval 37123 | Changing a bound variable ... |
| bj-cbvex 37124 | Changing a bound variable ... |
| bj-df-sb 37127 | Proposed definition to rep... |
| bj-sbcex 37128 | Proof of ~ sbcex when taki... |
| bj-dfsbc 37129 | Proof of ~ df-sbc when tak... |
| bj-ssbeq 37130 | Substitution in an equalit... |
| bj-ssblem1 37131 | A lemma for the definiens ... |
| bj-ssblem2 37132 | An instance of ~ ax-11 pro... |
| bj-ax12v 37133 | A weaker form of ~ ax-12 a... |
| bj-ax12 37134 | Remove a DV condition from... |
| bj-ax12ssb 37135 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 37136 | Special case of ~ 19.41 pr... |
| bj-equsexval 37137 | Special case of ~ equsexv ... |
| bj-subst 37138 | Proof of ~ sbalex from cor... |
| bj-ssbid2 37139 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 37140 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 37141 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 37142 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 37143 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 37144 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 37145 | Proof of ~ ax6e (hence ~ a... |
| bj-spim0 37146 | A universal specialization... |
| bj-spimvwt 37147 | Closed form of ~ spimvw . ... |
| bj-spnfw 37148 | Theorem close to a closed ... |
| bj-cbvexiw 37149 | Change bound variable. Th... |
| bj-cbvexivw 37150 | Change bound variable. Th... |
| bj-modald 37151 | A short form of the axiom ... |
| bj-denot 37152 | A weakening of ~ ax-6 and ... |
| bj-eqs 37153 | A lemma for substitutions,... |
| bj-cbvexw 37154 | Change bound variable. Th... |
| bj-ax12w 37155 | The general statement that... |
| bj-ax89 37156 | A theorem which could be u... |
| bj-cleljusti 37157 | One direction of ~ cleljus... |
| bj-alcomexcom 37158 | Commutation of two existen... |
| bj-hbald 37159 | General statement that ~ h... |
| bj-hbalt 37160 | Closed form of (general in... |
| bj-hbal 37161 | More general instance of ~... |
| axc11n11 37162 | Proof of ~ axc11n from { ~... |
| axc11n11r 37163 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 37164 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 37165 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 37166 | Alternate proof of ~ bj-ax... |
| bj-sb 37167 | A weak variant of ~ sbid2 ... |
| bj-modalbe 37168 | The predicate-calculus ver... |
| bj-spst 37169 | Closed form of ~ sps . On... |
| bj-19.21bit 37170 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 37171 | Closed form of ~ 19.23bi .... |
| bj-nexrt 37172 | Closed form of ~ nexr . C... |
| bj-alrim 37173 | Closed form of ~ alrimi . ... |
| bj-alrim2 37174 | Uncurried (imported) form ... |
| bj-nfdt0 37175 | A theorem close to a close... |
| bj-nfdt 37176 | Closed form of ~ nf5d and ... |
| bj-nexdt 37177 | Closed form of ~ nexd . (... |
| bj-nexdvt 37178 | Closed form of ~ nexdv . ... |
| bj-alexbiex 37179 | Adding a second quantifier... |
| bj-exexbiex 37180 | Adding a second quantifier... |
| bj-alalbial 37181 | Adding a second quantifier... |
| bj-exalbial 37182 | Adding a second quantifier... |
| bj-19.9htbi 37183 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 37184 | Strengthening ~ hbnt by re... |
| bj-biexal1 37185 | A general FOL biconditiona... |
| bj-biexal2 37186 | When ` ph ` is substituted... |
| bj-biexal3 37187 | When ` ph ` is substituted... |
| bj-bialal 37188 | When ` ph ` is substituted... |
| bj-biexex 37189 | When ` ph ` is substituted... |
| bj-hbexd 37190 | A more general instance of... |
| bj-hbext 37191 | Closed form of ~ bj-hbex a... |
| bj-hbex 37192 | A more general instance of... |
| bj-nfalt 37193 | Closed form of ~ nfal . (... |
| bj-nfext 37194 | Closed form of ~ nfex . (... |
| bj-eeanvw 37195 | Version of ~ exdistrv with... |
| bj-modal4 37196 | First-order logic form of ... |
| bj-modal4e 37197 | First-order logic form of ... |
| bj-modalb 37198 | A short form of the axiom ... |
| bj-wnf1 37199 | When ` ph ` is substituted... |
| bj-wnf2 37200 | When ` ph ` is substituted... |
| bj-wnfanf 37201 | When ` ph ` is substituted... |
| bj-wnfenf 37202 | When ` ph ` is substituted... |
| bj-19.12 37203 | See ~ 19.12 . Could be la... |
| bj-substax12 37204 | Equivalent form of the axi... |
| bj-substw 37205 | Weak form of the LHS of ~ ... |
| bj-nnfa 37208 | Nonfreeness implies the eq... |
| bj-nnfad 37209 | Nonfreeness implies the eq... |
| bj-nnfai 37210 | Nonfreeness implies the eq... |
| bj-nnfe 37211 | Nonfreeness implies the eq... |
| bj-nnfed 37212 | Nonfreeness implies the eq... |
| bj-nnfei 37213 | Nonfreeness implies the eq... |
| bj-nnfea 37214 | Nonfreeness implies the eq... |
| bj-nnfead 37215 | Nonfreeness implies the eq... |
| bj-nnfeai 37216 | Nonfreeness implies the eq... |
| bj-alnnf 37217 | In deduction-style proofs,... |
| bj-alnnf2 37218 | If a proposition holds, th... |
| bj-dfnnf2 37219 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 37220 | New nonfreeness implies ol... |
| bj-nnfim1 37221 | A consequence of nonfreene... |
| bj-nnfim2 37222 | A consequence of nonfreene... |
| bj-nnftht 37223 | A variable is nonfree in a... |
| bj-nnfth 37224 | A variable is nonfree in a... |
| bj-nnf-alrim 37225 | Proof of the closed form o... |
| bj-stdpc5t 37226 | Alias of ~ bj-nnf-alrim fo... |
| bj-nnfbi 37227 | If two formulas are equiva... |
| bj-nnfbd0 37228 | If two formulas are equiva... |
| bj-nnfbii 37229 | If two formulas are equiva... |
| bj-nnfnt 37230 | A variable is nonfree in a... |
| bj-nnfnth 37231 | A variable is nonfree in t... |
| bj-nnfim 37232 | Nonfreeness in the anteced... |
| bj-nnfimd 37233 | Nonfreeness in the anteced... |
| bj-nnfan 37234 | Nonfreeness in both conjun... |
| bj-nnfand 37235 | Nonfreeness in both conjun... |
| bj-nnfor 37236 | Nonfreeness in both disjun... |
| bj-nnford 37237 | Nonfreeness in both disjun... |
| bj-nnfbit 37238 | Nonfreeness in both sides ... |
| bj-nnfbid 37239 | Nonfreeness in both sides ... |
| bj-nnf-exlim 37240 | Proof of the closed form o... |
| bj-19.21t 37241 | Statement ~ 19.21t proved ... |
| bj-19.23t 37242 | Statement ~ 19.23t proved ... |
| bj-19.36im 37243 | One direction of ~ 19.36 f... |
| bj-19.37im 37244 | One direction of ~ 19.37 f... |
| bj-19.42t 37245 | Closed form of ~ 19.42 fro... |
| bj-19.41t 37246 | Closed form of ~ 19.41 fro... |
| bj-pm11.53vw 37247 | Version of ~ pm11.53v with... |
| bj-nnfv 37248 | A non-occurring variable i... |
| bj-nnfbd 37249 | If two formulas are equiva... |
| bj-pm11.53a 37250 | A variant of ~ pm11.53v . ... |
| bj-equsvt 37251 | A variant of ~ equsv . (C... |
| bj-equsalvwd 37252 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 37253 | Variant of ~ equsexvw . (... |
| bj-nnf-spim 37254 | A universal specialization... |
| bj-nnf-spime 37255 | An existential generalizat... |
| bj-nnf-cbvaliv 37256 | The only DV conditions are... |
| bj-sbievwd 37257 | Variant of ~ sbievw . (Co... |
| bj-sbft 37258 | Version of ~ sbft using ` ... |
| bj-nnf-cbvali 37259 | Compared with ~ bj-nnf-cbv... |
| bj-nnf-cbval 37260 | Compared with ~ cbvalv1 , ... |
| bj-dfnnf3 37261 | Alternate definition of no... |
| bj-nfnnfTEMP 37262 | New nonfreeness is equival... |
| bj-wnfnf 37263 | When ` ph ` is substituted... |
| bj-nnfa1 37264 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 37265 | See ~ nfe1 . (Contributed... |
| bj-nnflemaa 37266 | One of four lemmas for non... |
| bj-nnflemee 37267 | One of four lemmas for non... |
| bj-nnflemae 37268 | One of four lemmas for non... |
| bj-nnflemea 37269 | One of four lemmas for non... |
| bj-nnfalt 37270 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 37271 | See ~ nfex and ~ bj-nfext ... |
| bj-pm11.53v 37272 | Version of ~ pm11.53v with... |
| bj-axc10 37273 | Alternate proof of ~ axc10... |
| bj-alequex 37274 | A fol lemma. See ~ aleque... |
| bj-spimt2 37275 | A step in the proof of ~ s... |
| bj-cbv3ta 37276 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 37277 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 37278 | A theorem close to a close... |
| bj-hbsb3 37279 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 37280 | A theorem close to a close... |
| bj-nfs1t2 37281 | A theorem close to a close... |
| bj-nfs1 37282 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 37283 | Version of ~ axc10 with a ... |
| bj-spimtv 37284 | Version of ~ spimt with a ... |
| bj-cbv3hv2 37285 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 37286 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 37287 | Version of ~ cbv2h with a ... |
| bj-cbv2v 37288 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 37289 | Version of ~ cbvald with a... |
| bj-cbvexdv 37290 | Version of ~ cbvexd with a... |
| bj-cbval2vv 37291 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 37292 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 37293 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 37294 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 37295 | Version of ~ cbvex4v with ... |
| bj-equsalhv 37296 | Version of ~ equsalh with ... |
| bj-axc11nv 37297 | Version of ~ axc11n with a... |
| bj-aecomsv 37298 | Version of ~ aecoms with a... |
| bj-axc11v 37299 | Version of ~ axc11 with a ... |
| bj-drnf2v 37300 | Version of ~ drnf2 with a ... |
| bj-equs45fv 37301 | Version of ~ equs45f with ... |
| bj-hbs1 37302 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 37303 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 37304 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 37305 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 37306 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 37307 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 37308 | Biconditional version of a... |
| bj-hbaeb 37309 | Biconditional version of ~... |
| bj-hbnaeb 37310 | Biconditional version of ~... |
| bj-dvv 37311 | A special instance of ~ bj... |
| bj-equsal1t 37312 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 37313 | Inference associated with ... |
| bj-equsal1 37314 | One direction of ~ equsal ... |
| bj-equsal2 37315 | One direction of ~ equsal ... |
| bj-equsal 37316 | Shorter proof of ~ equsal ... |
| stdpc5t 37317 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 37318 | More direct proof of ~ std... |
| 2stdpc5 37319 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 37320 | Proof of ~ 19.21t from ~ s... |
| exlimii 37321 | Inference associated with ... |
| ax11-pm 37322 | Proof of ~ ax-11 similar t... |
| ax6er 37323 | Commuted form of ~ ax6e . ... |
| exlimiieq1 37324 | Inferring a theorem when i... |
| exlimiieq2 37325 | Inferring a theorem when i... |
| ax11-pm2 37326 | Proof of ~ ax-11 from the ... |
| bj-sbsb 37327 | Biconditional showing two ... |
| bj-dfsb2 37328 | Alternate (dual) definitio... |
| bj-sbf3 37329 | Substitution has no effect... |
| bj-sbf4 37330 | Substitution has no effect... |
| bj-eu3f 37331 | Version of ~ eu3v where th... |
| bj-sblem1 37332 | Lemma for substitution. (... |
| bj-sblem2 37333 | Lemma for substitution. (... |
| bj-sblem 37334 | Lemma for substitution. (... |
| bj-sbievw1 37335 | Lemma for substitution. (... |
| bj-sbievw2 37336 | Lemma for substitution. (... |
| bj-sbievw 37337 | Lemma for substitution. C... |
| bj-sbievv 37338 | Version of ~ sbie with a s... |
| bj-moeub 37339 | Uniqueness is equivalent t... |
| bj-sbidmOLD 37340 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 37341 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 37342 | Curried (exported) form of... |
| bj-dvelimv 37343 | A version of ~ dvelim usin... |
| bj-nfeel2 37344 | Nonfreeness in a membershi... |
| bj-axc14nf 37345 | Proof of a version of ~ ax... |
| bj-axc14 37346 | Alternate proof of ~ axc14... |
| mobidvALT 37347 | Alternate proof of ~ mobid... |
| sbn1ALT 37348 | Alternate proof of ~ sbn1 ... |
| eliminable1 37349 | A theorem used to prove th... |
| eliminable2a 37350 | A theorem used to prove th... |
| eliminable2b 37351 | A theorem used to prove th... |
| eliminable2c 37352 | A theorem used to prove th... |
| eliminable3a 37353 | A theorem used to prove th... |
| eliminable3b 37354 | A theorem used to prove th... |
| eliminable-velab 37355 | A theorem used to prove th... |
| eliminable-veqab 37356 | A theorem used to prove th... |
| eliminable-abeqv 37357 | A theorem used to prove th... |
| eliminable-abeqab 37358 | A theorem used to prove th... |
| eliminable-abelv 37359 | A theorem used to prove th... |
| eliminable-abelab 37360 | A theorem used to prove th... |
| bj-denoteslem 37361 | Duplicate of ~ issettru an... |
| bj-denotesALTV 37362 | Moved to main as ~ iseqset... |
| bj-issettruALTV 37363 | Moved to main as ~ issettr... |
| bj-elabtru 37364 | This is as close as we can... |
| bj-issetwt 37365 | Closed form of ~ bj-issetw... |
| bj-issetw 37366 | The closest one can get to... |
| bj-issetiv 37367 | Version of ~ bj-isseti wit... |
| bj-isseti 37368 | Version of ~ isseti with a... |
| bj-ralvw 37369 | A weak version of ~ ralv n... |
| bj-rexvw 37370 | A weak version of ~ rexv n... |
| bj-rababw 37371 | A weak version of ~ rabab ... |
| bj-rexcom4bv 37372 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 37373 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 37374 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 37375 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 37376 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 37377 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 37378 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 37379 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 37380 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 37381 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 37382 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 37383 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 37384 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 37385 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 37386 | Remove from ~ spcimdv depe... |
| elelb 37387 | Equivalence between two co... |
| bj-pwvrelb 37388 | Characterization of the el... |
| bj-nfcsym 37389 | The nonfreeness quantifier... |
| bj-sbeqALT 37390 | Substitution in an equalit... |
| bj-sbeq 37391 | Distribute proper substitu... |
| bj-sbceqgALT 37392 | Distribute proper substitu... |
| bj-csbsnlem 37393 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 37394 | Substitution in a singleto... |
| bj-sbel1 37395 | Version of ~ sbcel1g when ... |
| bj-abv 37396 | The class of sets verifyin... |
| bj-abvALT 37397 | Alternate version of ~ bj-... |
| bj-ab0 37398 | The class of sets verifyin... |
| bj-abf 37399 | Shorter proof of ~ abf (wh... |
| bj-csbprc 37400 | More direct proof of ~ csb... |
| bj-exlimvmpi 37401 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 37402 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 37403 | Lemma for theorems of the ... |
| bj-exlimmpbir 37404 | Lemma for theorems of the ... |
| bj-vtoclf 37405 | Remove dependency on ~ ax-... |
| bj-vtocl 37406 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 37407 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 37408 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 37409 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 37410 | A version of ~ vtoclg with... |
| bj-rabeqbid 37411 | Version of ~ rabeqbidv wit... |
| bj-seex 37412 | Version of ~ seex with a d... |
| bj-nfcf 37413 | Version of ~ df-nfc with a... |
| bj-zfauscl 37414 | General version of ~ zfaus... |
| bj-elabd2ALT 37415 | Alternate proof of ~ elabd... |
| bj-unrab 37416 | Generalization of ~ unrab ... |
| bj-inrab 37417 | Generalization of ~ inrab ... |
| bj-inrab2 37418 | Shorter proof of ~ inrab .... |
| bj-inrab3 37419 | Generalization of ~ dfrab3... |
| bj-rabtr 37420 | Restricted class abstracti... |
| bj-rabtrALT 37421 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 37422 | Proof of ~ bj-rabtr found ... |
| bj-gabss 37425 | Inclusion of generalized c... |
| bj-gabssd 37426 | Inclusion of generalized c... |
| bj-gabeqd 37427 | Equality of generalized cl... |
| bj-gabeqis 37428 | Equality of generalized cl... |
| bj-elgab 37429 | Elements of a generalized ... |
| bj-gabima 37430 | Generalized class abstract... |
| bj-ru1 37433 | A version of Russell's par... |
| bj-ru 37434 | Remove dependency on ~ ax-... |
| currysetlem 37435 | Lemma for ~ currysetlem , ... |
| curryset 37436 | Curry's paradox in set the... |
| currysetlem1 37437 | Lemma for ~ currysetALT . ... |
| currysetlem2 37438 | Lemma for ~ currysetALT . ... |
| currysetlem3 37439 | Lemma for ~ currysetALT . ... |
| currysetALT 37440 | Alternate proof of ~ curry... |
| bj-n0i 37441 | Inference associated with ... |
| bj-disjsn01 37442 | Disjointness of the single... |
| bj-0nel1 37443 | The empty set does not bel... |
| bj-1nel0 37444 | ` 1o ` does not belong to ... |
| bj-xpimasn 37445 | The image of a singleton, ... |
| bj-xpima1sn 37446 | The image of a singleton b... |
| bj-xpima1snALT 37447 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 37448 | The image of a singleton b... |
| bj-xpnzex 37449 | If the first factor of a p... |
| bj-xpexg2 37450 | Curried (exported) form of... |
| bj-xpnzexb 37451 | If the first factor of a p... |
| bj-cleq 37452 | Substitution property for ... |
| bj-snsetex 37453 | The class of sets "whose s... |
| bj-clexab 37454 | Sethood of certain classes... |
| bj-sngleq 37457 | Substitution property for ... |
| bj-elsngl 37458 | Characterization of the el... |
| bj-snglc 37459 | Characterization of the el... |
| bj-snglss 37460 | The singletonization of a ... |
| bj-0nelsngl 37461 | The empty set is not a mem... |
| bj-snglinv 37462 | Inverse of singletonizatio... |
| bj-snglex 37463 | A class is a set if and on... |
| bj-tageq 37466 | Substitution property for ... |
| bj-eltag 37467 | Characterization of the el... |
| bj-0eltag 37468 | The empty set belongs to t... |
| bj-tagn0 37469 | The tagging of a class is ... |
| bj-tagss 37470 | The tagging of a class is ... |
| bj-snglsstag 37471 | The singletonization is in... |
| bj-sngltagi 37472 | The singletonization is in... |
| bj-sngltag 37473 | The singletonization and t... |
| bj-tagci 37474 | Characterization of the el... |
| bj-tagcg 37475 | Characterization of the el... |
| bj-taginv 37476 | Inverse of tagging. (Cont... |
| bj-tagex 37477 | A class is a set if and on... |
| bj-xtageq 37478 | The products of a given cl... |
| bj-xtagex 37479 | The product of a set and t... |
| bj-projeq 37482 | Substitution property for ... |
| bj-projeq2 37483 | Substitution property for ... |
| bj-projun 37484 | The class projection on a ... |
| bj-projex 37485 | Sethood of the class proje... |
| bj-projval 37486 | Value of the class project... |
| bj-1upleq 37489 | Substitution property for ... |
| bj-pr1eq 37492 | Substitution property for ... |
| bj-pr1un 37493 | The first projection prese... |
| bj-pr1val 37494 | Value of the first project... |
| bj-pr11val 37495 | Value of the first project... |
| bj-pr1ex 37496 | Sethood of the first proje... |
| bj-1uplth 37497 | The characteristic propert... |
| bj-1uplex 37498 | A monuple is a set if and ... |
| bj-1upln0 37499 | A monuple is nonempty. (C... |
| bj-2upleq 37502 | Substitution property for ... |
| bj-pr21val 37503 | Value of the first project... |
| bj-pr2eq 37506 | Substitution property for ... |
| bj-pr2un 37507 | The second projection pres... |
| bj-pr2val 37508 | Value of the second projec... |
| bj-pr22val 37509 | Value of the second projec... |
| bj-pr2ex 37510 | Sethood of the second proj... |
| bj-2uplth 37511 | The characteristic propert... |
| bj-2uplex 37512 | A couple is a set if and o... |
| bj-2upln0 37513 | A couple is nonempty. (Co... |
| bj-2upln1upl 37514 | A couple is never equal to... |
| bj-rcleqf 37515 | Relative version of ~ cleq... |
| bj-rcleq 37516 | Relative version of ~ dfcl... |
| bj-reabeq 37517 | Relative form of ~ eqabb .... |
| bj-disj2r 37518 | Relative version of ~ ssdi... |
| bj-sscon 37519 | Contraposition law for rel... |
| bj-abex 37520 | Two ways of stating that t... |
| bj-clex 37521 | Two ways of stating that a... |
| bj-axsn 37522 | Two ways of stating the ax... |
| bj-snexg 37524 | A singleton built on a set... |
| bj-snex 37525 | A singleton is a set. See... |
| bj-axbun 37526 | Two ways of stating the ax... |
| bj-unexg 37528 | Existence of binary unions... |
| bj-prexg 37529 | Existence of unordered pai... |
| bj-prex 37530 | Existence of unordered pai... |
| bj-axadj 37531 | Two ways of stating the ax... |
| bj-adjg1 37533 | Existence of the result of... |
| bj-snfromadj 37534 | Singleton from adjunction ... |
| bj-prfromadj 37535 | Unordered pair from adjunc... |
| bj-adjfrombun 37536 | Adjunction from singleton ... |
| eleq2w2ALT 37537 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 37538 | Alternate proof of ~ clel3... |
| bj-pw0ALT 37539 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 37540 | Quantitative version of ~ ... |
| bj-unirel 37541 | Quantitative version of ~ ... |
| bj-elpwg 37542 | If the intersection of two... |
| bj-velpwALT 37543 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 37544 | Alternate proof of ~ elpwg... |
| bj-vjust 37545 | Justification theorem for ... |
| bj-nul 37546 | Two formulations of the ax... |
| bj-nuliota 37547 | Definition of the empty se... |
| bj-nuliotaALT 37548 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 37549 | Alternate proof of ~ vtocl... |
| bj-elsn12g 37550 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 37551 | Biconditional version of ~... |
| bj-pwcfsdom 37552 | Remove hypothesis from ~ p... |
| bj-grur1 37553 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 37554 | The extension of a predica... |
| bj-dfid2ALT 37555 | Alternate version of ~ dfi... |
| bj-0nelopab 37556 | The empty set is never an ... |
| bj-brrelex12ALT 37557 | Two classes related by a b... |
| bj-epelg 37558 | The membership relation an... |
| bj-epelb 37559 | Two classes are related by... |
| bj-nsnid 37560 | A set does not contain the... |
| bj-rdg0gALT 37561 | Alternate proof of ~ rdg0g... |
| bj-axnul 37562 | Over the base theory ~ ax-... |
| bj-rep 37563 | Version of the axiom of re... |
| bj-axseprep 37564 | Axiom of separation (unive... |
| bj-axreprepsep 37565 | Strong axiom of replacemen... |
| bj-evaleq 37566 | Equality theorem for the `... |
| bj-evalfun 37567 | The evaluation at a class ... |
| bj-evalfn 37568 | The evaluation at a class ... |
| bj-evalf 37569 | The evaluation at a class ... |
| bj-evalval 37570 | Value of the evaluation at... |
| bj-evalid 37571 | The evaluation at a set of... |
| bj-ndxarg 37572 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 37573 | Closed general form of ~ s... |
| bj-rest00 37576 | An elementwise intersectio... |
| bj-restsn 37577 | An elementwise intersectio... |
| bj-restsnss 37578 | Special case of ~ bj-rests... |
| bj-restsnss2 37579 | Special case of ~ bj-rests... |
| bj-restsn0 37580 | An elementwise intersectio... |
| bj-restsn10 37581 | Special case of ~ bj-rests... |
| bj-restsnid 37582 | The elementwise intersecti... |
| bj-rest10 37583 | An elementwise intersectio... |
| bj-rest10b 37584 | Alternate version of ~ bj-... |
| bj-restn0 37585 | An elementwise intersectio... |
| bj-restn0b 37586 | Alternate version of ~ bj-... |
| bj-restpw 37587 | The elementwise intersecti... |
| bj-rest0 37588 | An elementwise intersectio... |
| bj-restb 37589 | An elementwise intersectio... |
| bj-restv 37590 | An elementwise intersectio... |
| bj-resta 37591 | An elementwise intersectio... |
| bj-restuni 37592 | The union of an elementwis... |
| bj-restuni2 37593 | The union of an elementwis... |
| bj-restreg 37594 | A reformulation of the axi... |
| bj-raldifsn 37595 | All elements in a set sati... |
| bj-0int 37596 | If ` A ` is a collection o... |
| bj-mooreset 37597 | A Moore collection is a se... |
| bj-ismoore 37600 | Characterization of Moore ... |
| bj-ismoored0 37601 | Necessary condition to be ... |
| bj-ismoored 37602 | Necessary condition to be ... |
| bj-ismoored2 37603 | Necessary condition to be ... |
| bj-ismooredr 37604 | Sufficient condition to be... |
| bj-ismooredr2 37605 | Sufficient condition to be... |
| bj-discrmoore 37606 | The powerclass ` ~P A ` is... |
| bj-0nmoore 37607 | The empty set is not a Moo... |
| bj-snmoore 37608 | A singleton is a Moore col... |
| bj-snmooreb 37609 | A singleton is a Moore col... |
| bj-prmoore 37610 | A pair formed of two neste... |
| bj-0nelmpt 37611 | The empty set is not an el... |
| bj-mptval 37612 | Value of a function given ... |
| bj-dfmpoa 37613 | An equivalent definition o... |
| bj-mpomptALT 37614 | Alternate proof of ~ mpomp... |
| setsstrset 37631 | Relation between ~ df-sets... |
| bj-nfald 37632 | Variant of ~ nfald . (Con... |
| bj-nfexd 37633 | Variant of ~ nfexd . (Con... |
| cgsex2gd 37634 | Implicit substitution infe... |
| copsex2gd 37635 | Implicit substitution infe... |
| copsex2d 37636 | Implicit substitution dedu... |
| copsex2b 37637 | Biconditional form of ~ co... |
| opelopabd 37638 | Membership of an ordered p... |
| opelopabb 37639 | Membership of an ordered p... |
| opelopabbv 37640 | Membership of an ordered p... |
| bj-opelrelex 37641 | The coordinates of an orde... |
| bj-opelresdm 37642 | If an ordered pair is in a... |
| bj-brresdm 37643 | If two classes are related... |
| brabd0 37644 | Expressing that two sets a... |
| brabd 37645 | Expressing that two sets a... |
| bj-brab2a1 37646 | "Unbounded" version of ~ b... |
| bj-opabssvv 37647 | A variant of ~ relopabiv (... |
| bj-funidres 37648 | The restricted identity re... |
| bj-opelidb 37649 | Characterization of the or... |
| bj-opelidb1 37650 | Characterization of the or... |
| bj-inexeqex 37651 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 37652 | If the intersection of two... |
| bj-opelid 37653 | Characterization of the or... |
| bj-ideqg 37654 | Characterization of the cl... |
| bj-ideqgALT 37655 | Alternate proof of ~ bj-id... |
| bj-ideqb 37656 | Characterization of classe... |
| bj-idres 37657 | Alternate expression for t... |
| bj-opelidres 37658 | Characterization of the or... |
| bj-idreseq 37659 | Sufficient condition for t... |
| bj-idreseqb 37660 | Characterization for two c... |
| bj-ideqg1 37661 | For sets, the identity rel... |
| bj-ideqg1ALT 37662 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 37663 | Characterization of the co... |
| bj-elid3 37664 | Characterization of the co... |
| bj-elid4 37665 | Characterization of the el... |
| bj-elid5 37666 | Characterization of the el... |
| bj-elid6 37667 | Characterization of the el... |
| bj-elid7 37668 | Characterization of the el... |
| bj-diagval 37671 | Value of the functionalize... |
| bj-diagval2 37672 | Value of the functionalize... |
| bj-eldiag 37673 | Characterization of the el... |
| bj-eldiag2 37674 | Characterization of the el... |
| bj-imdirvallem 37677 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 37678 | Value of the functionalize... |
| bj-imdirval2lem 37679 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 37680 | Value of the functionalize... |
| bj-imdirval3 37681 | Value of the functionalize... |
| bj-imdiridlem 37682 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 37683 | Functorial property of the... |
| bj-opelopabid 37684 | Membership in an ordered-p... |
| bj-opabco 37685 | Composition of ordered-pai... |
| bj-xpcossxp 37686 | The composition of two Car... |
| bj-imdirco 37687 | Functorial property of the... |
| bj-iminvval 37690 | Value of the functionalize... |
| bj-iminvval2 37691 | Value of the functionalize... |
| bj-iminvid 37692 | Functorial property of the... |
| bj-inftyexpitaufo 37699 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 37702 | An element of the circle a... |
| bj-inftyexpiinv 37705 | Utility theorem for the in... |
| bj-inftyexpiinj 37706 | Injectivity of the paramet... |
| bj-inftyexpidisj 37707 | An element of the circle a... |
| bj-ccinftydisj 37710 | The circle at infinity is ... |
| bj-elccinfty 37711 | A lemma for infinite exten... |
| bj-ccssccbar 37714 | Complex numbers are extend... |
| bj-ccinftyssccbar 37715 | Infinite extended complex ... |
| bj-pinftyccb 37718 | The class ` pinfty ` is an... |
| bj-pinftynrr 37719 | The extended complex numbe... |
| bj-minftyccb 37722 | The class ` minfty ` is an... |
| bj-minftynrr 37723 | The extended complex numbe... |
| bj-pinftynminfty 37724 | The extended complex numbe... |
| bj-rrhatsscchat 37733 | The real projective line i... |
| bj-imafv 37748 | If the direct image of a s... |
| bj-funun 37749 | Value of a function expres... |
| bj-fununsn1 37750 | Value of a function expres... |
| bj-fununsn2 37751 | Value of a function expres... |
| bj-fvsnun1 37752 | The value of a function wi... |
| bj-fvsnun2 37753 | The value of a function wi... |
| bj-fvmptunsn1 37754 | Value of a function expres... |
| bj-fvmptunsn2 37755 | Value of a function expres... |
| bj-iomnnom 37756 | The canonical bijection fr... |
| bj-smgrpssmgm 37765 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 37766 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 37767 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 37768 | Monoids are semigroups (el... |
| bj-cmnssmnd 37769 | Commutative monoids are mo... |
| bj-cmnssmndel 37770 | Commutative monoids are mo... |
| bj-grpssmnd 37771 | Groups are monoids. (Cont... |
| bj-grpssmndel 37772 | Groups are monoids (elemen... |
| bj-ablssgrp 37773 | Abelian groups are groups.... |
| bj-ablssgrpel 37774 | Abelian groups are groups ... |
| bj-ablsscmn 37775 | Abelian groups are commuta... |
| bj-ablsscmnel 37776 | Abelian groups are commuta... |
| bj-modssabl 37777 | (The additive groups of) m... |
| bj-vecssmod 37778 | Vector spaces are modules.... |
| bj-vecssmodel 37779 | Vector spaces are modules ... |
| bj-finsumval0 37782 | Value of a finite sum. (C... |
| bj-fvimacnv0 37783 | Variant of ~ fvimacnv wher... |
| bj-isvec 37784 | The predicate "is a vector... |
| bj-fldssdrng 37785 | Fields are division rings.... |
| bj-flddrng 37786 | Fields are division rings ... |
| bj-rrdrg 37787 | The field of real numbers ... |
| bj-isclm 37788 | The predicate "is a subcom... |
| bj-isrvec 37791 | The predicate "is a real v... |
| bj-rvecmod 37792 | Real vector spaces are mod... |
| bj-rvecssmod 37793 | Real vector spaces are mod... |
| bj-rvecrr 37794 | The field of scalars of a ... |
| bj-isrvecd 37795 | The predicate "is a real v... |
| bj-rvecvec 37796 | Real vector spaces are vec... |
| bj-isrvec2 37797 | The predicate "is a real v... |
| bj-rvecssvec 37798 | Real vector spaces are vec... |
| bj-rveccmod 37799 | Real vector spaces are sub... |
| bj-rvecsscmod 37800 | Real vector spaces are sub... |
| bj-rvecsscvec 37801 | Real vector spaces are sub... |
| bj-rveccvec 37802 | Real vector spaces are sub... |
| bj-rvecssabl 37803 | (The additive groups of) r... |
| bj-rvecabl 37804 | (The additive groups of) r... |
| bj-subcom 37805 | A consequence of commutati... |
| bj-lineqi 37806 | Solution of a (scalar) lin... |
| bj-bary1lem 37807 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 37808 | Lemma for ~ bj-bary1 : com... |
| bj-bary1 37809 | Barycentric coordinates in... |
| bj-endval 37812 | Value of the monoid of end... |
| bj-endbase 37813 | Base set of the monoid of ... |
| bj-endcomp 37814 | Composition law of the mon... |
| bj-endmnd 37815 | The monoid of endomorphism... |
| taupilem3 37816 | Lemma for tau-related theo... |
| taupilemrplb 37817 | A set of positive reals ha... |
| taupilem1 37818 | Lemma for ~ taupi . A pos... |
| taupilem2 37819 | Lemma for ~ taupi . The s... |
| taupi 37820 | Relationship between ` _ta... |
| dfgcd3 37821 | Alternate definition of th... |
| irrdifflemf 37822 | Lemma for ~ irrdiff . The... |
| irrdiff 37823 | The irrationals are exactl... |
| qdiff 37824 | The rationals are exactly ... |
| qdiffALT 37825 | Alternate proof of ~ qdiff... |
| iccioo01 37826 | The closed unit interval i... |
| csbrecsg 37827 | Move class substitution in... |
| csbrdgg 37828 | Move class substitution in... |
| csboprabg 37829 | Move class substitution in... |
| csbmpo123 37830 | Move class substitution in... |
| con1bii2 37831 | A contraposition inference... |
| con2bii2 37832 | A contraposition inference... |
| vtoclefex 37833 | Implicit substitution of a... |
| rnmptsn 37834 | The range of a function ma... |
| f1omptsnlem 37835 | This is the core of the pr... |
| f1omptsn 37836 | A function mapping to sing... |
| mptsnunlem 37837 | This is the core of the pr... |
| mptsnun 37838 | A class ` B ` is equal to ... |
| dissneqlem 37839 | This is the core of the pr... |
| dissneq 37840 | Any topology that contains... |
| exlimim 37841 | Closed form of ~ exlimimd ... |
| exlimimd 37842 | Existential elimination ru... |
| exellim 37843 | Closed form of ~ exellimdd... |
| exellimddv 37844 | Eliminate an antecedent wh... |
| topdifinfindis 37845 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 37846 | This is the core of the pr... |
| topdifinffin 37847 | Part of Exercise 3 of [Mun... |
| topdifinf 37848 | Part of Exercise 3 of [Mun... |
| topdifinfeq 37849 | Two different ways of defi... |
| icorempo 37850 | Closed-below, open-above i... |
| icoreresf 37851 | Closed-below, open-above i... |
| icoreval 37852 | Value of the closed-below,... |
| icoreelrnab 37853 | Elementhood in the set of ... |
| isbasisrelowllem1 37854 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 37855 | Lemma for ~ isbasisrelowl ... |
| icoreclin 37856 | The set of closed-below, o... |
| isbasisrelowl 37857 | The set of all closed-belo... |
| icoreunrn 37858 | The union of all closed-be... |
| istoprelowl 37859 | The set of all closed-belo... |
| icoreelrn 37860 | A class abstraction which ... |
| iooelexlt 37861 | An element of an open inte... |
| relowlssretop 37862 | The lower limit topology o... |
| relowlpssretop 37863 | The lower limit topology o... |
| sucneqond 37864 | Inequality of an ordinal s... |
| sucneqoni 37865 | Inequality of an ordinal s... |
| onsucuni3 37866 | If an ordinal number has a... |
| 1oequni2o 37867 | The ordinal number ` 1o ` ... |
| rdgsucuni 37868 | If an ordinal number has a... |
| rdgeqoa 37869 | If a recursive function wi... |
| elxp8 37870 | Membership in a Cartesian ... |
| cbveud 37871 | Deduction used to change b... |
| cbvreud 37872 | Deduction used to change b... |
| difunieq 37873 | The difference of unions i... |
| inunissunidif 37874 | Theorem about subsets of t... |
| rdgellim 37875 | Elementhood in a recursive... |
| rdglimss 37876 | A recursive definition at ... |
| rdgssun 37877 | In a recursive definition ... |
| exrecfnlem 37878 | Lemma for ~ exrecfn . (Co... |
| exrecfn 37879 | Theorem about the existenc... |
| exrecfnpw 37880 | For any base set, a set wh... |
| finorwe 37881 | If the Axiom of Infinity i... |
| dffinxpf 37884 | This theorem is the same a... |
| finxpeq1 37885 | Equality theorem for Carte... |
| finxpeq2 37886 | Equality theorem for Carte... |
| csbfinxpg 37887 | Distribute proper substitu... |
| finxpreclem1 37888 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 37889 | Lemma for ` ^^ ` recursion... |
| finxp0 37890 | The value of Cartesian exp... |
| finxp1o 37891 | The value of Cartesian exp... |
| finxpreclem3 37892 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 37893 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 37894 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 37895 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 37896 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 37897 | The value of Cartesian exp... |
| finxp2o 37898 | The value of Cartesian exp... |
| finxp3o 37899 | The value of Cartesian exp... |
| finxpnom 37900 | Cartesian exponentiation w... |
| finxp00 37901 | Cartesian exponentiation o... |
| iunctb2 37902 | Using the axiom of countab... |
| domalom 37903 | A class which dominates ev... |
| isinf2 37904 | The converse of ~ isinf . ... |
| ctbssinf 37905 | Using the axiom of choice,... |
| ralssiun 37906 | The index set of an indexe... |
| nlpineqsn 37907 | For every point ` p ` of a... |
| nlpfvineqsn 37908 | Given a subset ` A ` of ` ... |
| fvineqsnf1 37909 | A theorem about functions ... |
| fvineqsneu 37910 | A theorem about functions ... |
| fvineqsneq 37911 | A theorem about functions ... |
| pibp16 37912 | Property P000016 of pi-bas... |
| pibp19 37913 | Property P000019 of pi-bas... |
| pibp21 37914 | Property P000021 of pi-bas... |
| pibt1 37915 | Theorem T000001 of pi-base... |
| pibt2 37916 | Theorem T000002 of pi-base... |
| wl-section-prop 37917 | Intuitionistic logic is no... |
| wl-section-boot 37921 | In this section, I provide... |
| wl-luk-imim1i 37922 | Inference adding common co... |
| wl-luk-syl 37923 | An inference version of th... |
| wl-luk-imtrid 37924 | A syllogism rule of infere... |
| wl-luk-pm2.18d 37925 | Deduction based on reducti... |
| wl-luk-con4i 37926 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 37927 | Inference rule. Copy of ~... |
| wl-luk-a1i 37928 | Inference rule. Copy of ~... |
| wl-luk-mpi 37929 | A nested _modus ponens_ in... |
| wl-luk-imim2i 37930 | Inference adding common an... |
| wl-luk-imtrdi 37931 | A syllogism rule of infere... |
| wl-luk-ax3 37932 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 37933 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 37934 | This theorem, called "Asse... |
| wl-luk-com12 37935 | Inference that swaps (comm... |
| wl-luk-pm2.21 37936 | From a wff and its negatio... |
| wl-luk-con1i 37937 | A contraposition inference... |
| wl-luk-ja 37938 | Inference joining the ante... |
| wl-luk-imim2 37939 | A closed form of syllogism... |
| wl-luk-a1d 37940 | Deduction introducing an e... |
| wl-luk-ax2 37941 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 37942 | Principle of identity. Th... |
| wl-luk-notnotr 37943 | Converse of double negatio... |
| wl-luk-pm2.04 37944 | Swap antecedents. Theorem... |
| wl-section-impchain 37945 | An implication like ` ( ps... |
| wl-impchain-mp-x 37946 | This series of theorems pr... |
| wl-impchain-mp-0 37947 | This theorem is the start ... |
| wl-impchain-mp-1 37948 | This theorem is in fact a ... |
| wl-impchain-mp-2 37949 | This theorem is in fact a ... |
| wl-impchain-com-1.x 37950 | It is often convenient to ... |
| wl-impchain-com-1.1 37951 | A degenerate form of antec... |
| wl-impchain-com-1.2 37952 | This theorem is in fact a ... |
| wl-impchain-com-1.3 37953 | This theorem is in fact a ... |
| wl-impchain-com-1.4 37954 | This theorem is in fact a ... |
| wl-impchain-com-n.m 37955 | This series of theorems al... |
| wl-impchain-com-2.3 37956 | This theorem is in fact a ... |
| wl-impchain-com-2.4 37957 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 37958 | This theorem is in fact a ... |
| wl-impchain-a1-x 37959 | If an implication chain is... |
| wl-impchain-a1-1 37960 | Inference rule, a copy of ... |
| wl-impchain-a1-2 37961 | Inference rule, a copy of ... |
| wl-impchain-a1-3 37962 | Inference rule, a copy of ... |
| wl-ifp-ncond1 37963 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 37964 | If one case of an ` if- ` ... |
| wl-ifpimpr 37965 | If one case of an ` if- ` ... |
| wl-ifp4impr 37966 | If one case of an ` if- ` ... |
| wl-df-3xor 37967 | Alternative definition of ... |
| wl-df3xor2 37968 | Alternative definition of ... |
| wl-df3xor3 37969 | Alternative form of ~ wl-d... |
| wl-3xortru 37970 | If the first input is true... |
| wl-3xorfal 37971 | If the first input is fals... |
| wl-3xorbi 37972 | Triple xor can be replaced... |
| wl-3xorbi2 37973 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 37974 | Equivalence theorem for tr... |
| wl-3xorbi123i 37975 | Equivalence theorem for tr... |
| wl-3xorrot 37976 | Rotation law for triple xo... |
| wl-3xorcoma 37977 | Commutative law for triple... |
| wl-3xorcomb 37978 | Commutative law for triple... |
| wl-3xornot1 37979 | Flipping the first input f... |
| wl-3xornot 37980 | Triple xor distributes ove... |
| wl-1xor 37981 | In the recursive scheme ... |
| wl-2xor 37982 | In the recursive scheme ... |
| wl-df-3mintru2 37983 | Alternative definition of ... |
| wl-df2-3mintru2 37984 | The adder carry in disjunc... |
| wl-df3-3mintru2 37985 | The adder carry in conjunc... |
| wl-df4-3mintru2 37986 | An alternative definition ... |
| wl-1mintru1 37987 | Using the recursion formul... |
| wl-1mintru2 37988 | Using the recursion formul... |
| wl-2mintru1 37989 | Using the recursion formul... |
| wl-2mintru2 37990 | Using the recursion formul... |
| wl-df3maxtru1 37991 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 37993 | A version of ~ ax-wl-13v w... |
| wl-cleq-0 37994 |
Disclaimer: |
| wl-cleq-1 37995 |
Disclaimer: |
| wl-cleq-2 37996 |
Disclaimer: |
| wl-cleq-3 37997 |
Disclaimer: |
| wl-cleq-4 37998 |
Disclaimer: |
| wl-cleq-5 37999 |
Disclaimer: |
| wl-cleq-6 38000 |
Disclaimer: |
| wl-df-clab 38003 | Disclaimer: The material ... |
| wl-isseteq 38004 | A class equal to a set var... |
| wl-ax12v2cl 38005 | The class version of ~ ax1... |
| wl-df.clab 38006 | Define class abstractions,... |
| wl-df.cleq 38007 | Define the equality connec... |
| wl-dfcleq.basic 38008 | This theorem is a conserva... |
| wl-dfcleq.just 38009 | The hypotheses added to th... |
| wl-df.clel 38010 | Define the membership conn... |
| wl-dfclel.basic 38011 | This theorem gives a conse... |
| wl-dfclel.just 38012 | Add a hypothesis to ~ wl-d... |
| wl-dfcleq 38013 | The defining characterizat... |
| wl-dfclel 38014 | The defining characterizat... |
| wl-mps 38015 | Replacing a nested consequ... |
| wl-syls1 38016 | Replacing a nested consequ... |
| wl-syls2 38017 | Replacing a nested anteced... |
| wl-embant 38018 | A true wff can always be a... |
| wl-orel12 38019 | In a conjunctive normal fo... |
| wl-cases2-dnf 38020 | A particular instance of ~... |
| wl-cbvmotv 38021 | Change bound variable. Us... |
| wl-moteq 38022 | Change bound variable. Us... |
| wl-motae 38023 | Change bound variable. Us... |
| wl-moae 38024 | Two ways to express "at mo... |
| wl-euae 38025 | Two ways to express "exact... |
| wl-nax6im 38026 | The following series of th... |
| wl-hbae1 38027 | This specialization of ~ h... |
| wl-naevhba1v 38028 | An instance of ~ hbn1w app... |
| wl-spae 38029 | Prove an instance of ~ sp ... |
| wl-speqv 38030 | Under the assumption ` -. ... |
| wl-19.8eqv 38031 | Under the assumption ` -. ... |
| wl-19.2reqv 38032 | Under the assumption ` -. ... |
| wl-nfalv 38033 | If ` x ` is not present in... |
| wl-nfimf1 38034 | An antecedent is irrelevan... |
| wl-nfae1 38035 | Unlike ~ nfae , this speci... |
| wl-nfnae1 38036 | Unlike ~ nfnae , this spec... |
| wl-aetr 38037 | A transitive law for varia... |
| wl-axc11r 38038 | Same as ~ axc11r , but usi... |
| wl-dral1d 38039 | A version of ~ dral1 with ... |
| wl-cbvalnaed 38040 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 38041 | A more general version of ... |
| wl-exeq 38042 | The semantics of ` E. x y ... |
| wl-aleq 38043 | The semantics of ` A. x y ... |
| wl-nfeqfb 38044 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 38045 | If ` y ` is not free in ` ... |
| wl-equsalvw 38046 | Version of ~ equsalv with ... |
| wl-equsald 38047 | Deduction version of ~ equ... |
| wl-equsaldv 38048 | Deduction version of ~ equ... |
| wl-equsal 38049 | A useful equivalence relat... |
| wl-equsal1t 38050 | The expression ` x = y ` i... |
| wl-equsalcom 38051 | This simple equivalence ea... |
| wl-equsal1i 38052 | The antecedent ` x = y ` i... |
| wl-sbid2ft 38053 | A more general version of ... |
| wl-cbvalsbi 38054 | Change bounded variables i... |
| wl-sbrimt 38055 | Substitution with a variab... |
| wl-sblimt 38056 | Substitution with a variab... |
| wl-sb9v 38057 | Commutation of quantificat... |
| wl-sb8ft 38058 | Substitution of variable i... |
| wl-sb8eft 38059 | Substitution of variable i... |
| wl-sb8t 38060 | Substitution of variable i... |
| wl-sb8et 38061 | Substitution of variable i... |
| wl-sbhbt 38062 | Closed form of ~ sbhb . C... |
| wl-sbnf1 38063 | Two ways expressing that `... |
| wl-equsb3 38064 | ~ equsb3 with a distinctor... |
| wl-equsb4 38065 | Substitution applied to an... |
| wl-2sb6d 38066 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 38067 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 38068 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 38069 | Version of ~ sbcom2 with a... |
| wl-sbalnae 38070 | A theorem used in eliminat... |
| wl-sbal1 38071 | A theorem used in eliminat... |
| wl-sbal2 38072 | Move quantifier in and out... |
| wl-2spsbbi 38073 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 38074 | This theorem provides a ba... |
| wl-lem-nexmo 38075 | This theorem provides a ba... |
| wl-lem-moexsb 38076 | The antecedent ` A. x ( ph... |
| wl-alanbii 38077 | This theorem extends ~ ala... |
| wl-mo2df 38078 | Version of ~ mof with a co... |
| wl-mo2tf 38079 | Closed form of ~ mof with ... |
| wl-eudf 38080 | Version of ~ eu6 with a co... |
| wl-eutf 38081 | Closed form of ~ eu6 with ... |
| wl-euequf 38082 | ~ euequ proved with a dist... |
| wl-mo2t 38083 | Closed form of ~ mof . (C... |
| wl-mo3t 38084 | Closed form of ~ mo3 . (C... |
| wl-nfsbtv 38085 | Closed form of ~ nfsbv . ... |
| wl-sb8eut 38086 | Substitution of variable i... |
| wl-sb8eutv 38087 | Substitution of variable i... |
| wl-sb8mot 38088 | Substitution of variable i... |
| wl-sb8motv 38089 | Substitution of variable i... |
| wl-issetft 38090 | A closed form of ~ issetf ... |
| wl-axc11rc11 38091 | Proving ~ axc11r from ~ ax... |
| wl-clabv 38092 | Variant of ~ df-clab , whe... |
| wl-dfclab 38093 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 38094 | Using class abstraction in... |
| wl-clabt 38095 | Using class abstraction in... |
| wl-eujustlem1 38096 | Version of ~ cbvexvw with ... |
| rabiun 38097 | Abstraction restricted to ... |
| iundif1 38098 | Indexed union of class dif... |
| imadifss 38099 | The difference of images i... |
| cureq 38100 | Equality theorem for curry... |
| unceq 38101 | Equality theorem for uncur... |
| curf 38102 | Functional property of cur... |
| uncf 38103 | Functional property of unc... |
| curfv 38104 | Value of currying. (Contr... |
| uncov 38105 | Value of uncurrying. (Con... |
| curunc 38106 | Currying of uncurrying. (... |
| unccur 38107 | Uncurrying of currying. (... |
| phpreu 38108 | Theorem related to pigeonh... |
| finixpnum 38109 | A finite Cartesian product... |
| fin2solem 38110 | Lemma for ~ fin2so . (Con... |
| fin2so 38111 | Any totally ordered Tarski... |
| ltflcei 38112 | Theorem to move the floor ... |
| leceifl 38113 | Theorem to move the floor ... |
| sin2h 38114 | Half-angle rule for sine. ... |
| cos2h 38115 | Half-angle rule for cosine... |
| tan2h 38116 | Half-angle rule for tangen... |
| lindsadd 38117 | In a vector space, the uni... |
| lindsdom 38118 | A linearly independent set... |
| lindsenlbs 38119 | A maximal linearly indepen... |
| matunitlindflem1 38120 | One direction of ~ matunit... |
| matunitlindflem2 38121 | One direction of ~ matunit... |
| matunitlindf 38122 | A matrix over a field is i... |
| ptrest 38123 | Expressing a restriction o... |
| ptrecube 38124 | Any point in an open set o... |
| poimirlem1 38125 | Lemma for ~ poimir - the v... |
| poimirlem2 38126 | Lemma for ~ poimir - conse... |
| poimirlem3 38127 | Lemma for ~ poimir to add ... |
| poimirlem4 38128 | Lemma for ~ poimir connect... |
| poimirlem5 38129 | Lemma for ~ poimir to esta... |
| poimirlem6 38130 | Lemma for ~ poimir establi... |
| poimirlem7 38131 | Lemma for ~ poimir , simil... |
| poimirlem8 38132 | Lemma for ~ poimir , estab... |
| poimirlem9 38133 | Lemma for ~ poimir , estab... |
| poimirlem10 38134 | Lemma for ~ poimir establi... |
| poimirlem11 38135 | Lemma for ~ poimir connect... |
| poimirlem12 38136 | Lemma for ~ poimir connect... |
| poimirlem13 38137 | Lemma for ~ poimir - for a... |
| poimirlem14 38138 | Lemma for ~ poimir - for a... |
| poimirlem15 38139 | Lemma for ~ poimir , that ... |
| poimirlem16 38140 | Lemma for ~ poimir establi... |
| poimirlem17 38141 | Lemma for ~ poimir establi... |
| poimirlem18 38142 | Lemma for ~ poimir stating... |
| poimirlem19 38143 | Lemma for ~ poimir establi... |
| poimirlem20 38144 | Lemma for ~ poimir establi... |
| poimirlem21 38145 | Lemma for ~ poimir stating... |
| poimirlem22 38146 | Lemma for ~ poimir , that ... |
| poimirlem23 38147 | Lemma for ~ poimir , two w... |
| poimirlem24 38148 | Lemma for ~ poimir , two w... |
| poimirlem25 38149 | Lemma for ~ poimir stating... |
| poimirlem26 38150 | Lemma for ~ poimir showing... |
| poimirlem27 38151 | Lemma for ~ poimir showing... |
| poimirlem28 38152 | Lemma for ~ poimir , a var... |
| poimirlem29 38153 | Lemma for ~ poimir connect... |
| poimirlem30 38154 | Lemma for ~ poimir combini... |
| poimirlem31 38155 | Lemma for ~ poimir , assig... |
| poimirlem32 38156 | Lemma for ~ poimir , combi... |
| poimir 38157 | Poincare-Miranda theorem. ... |
| broucube 38158 | Brouwer - or as Kulpa call... |
| heicant 38159 | Heine-Cantor theorem: a co... |
| opnmbllem0 38160 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 38161 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 38162 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 38163 | The difference between two... |
| mblfinlem4 38164 | Backward direction of ~ is... |
| ismblfin 38165 | Measurability in terms of ... |
| ovoliunnfl 38166 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 38167 | Demonstration of ~ ovoliun... |
| voliunnfl 38168 | ~ voliun is incompatible w... |
| volsupnfl 38169 | ~ volsup is incompatible w... |
| mbfresfi 38170 | Measurability of a piecewi... |
| mbfposadd 38171 | If the sum of two measurab... |
| cnambfre 38172 | A real-valued, a.e. contin... |
| dvtanlem 38173 | Lemma for ~ dvtan - the do... |
| dvtan 38174 | Derivative of tangent. (C... |
| itg2addnclem 38175 | An alternate expression fo... |
| itg2addnclem2 38176 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 38177 | Lemma incomprehensible in ... |
| itg2addnc 38178 | Alternate proof of ~ itg2a... |
| itg2gt0cn 38179 | ~ itg2gt0 holds on functio... |
| ibladdnclem 38180 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 38181 | Choice-free analogue of ~ ... |
| itgaddnclem1 38182 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 38183 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 38184 | Choice-free analogue of ~ ... |
| iblsubnc 38185 | Choice-free analogue of ~ ... |
| itgsubnc 38186 | Choice-free analogue of ~ ... |
| iblabsnclem 38187 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 38188 | Choice-free analogue of ~ ... |
| iblmulc2nc 38189 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 38190 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 38191 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 38192 | Choice-free analogue of ~ ... |
| itgabsnc 38193 | Choice-free analogue of ~ ... |
| itggt0cn 38194 | ~ itggt0 holds for continu... |
| ftc1cnnclem 38195 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 38196 | Choice-free proof of ~ ftc... |
| ftc1anclem1 38197 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 38198 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 38199 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 38200 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 38201 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 38202 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 38203 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 38204 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 38205 | ~ ftc1a holds for function... |
| ftc2nc 38206 | Choice-free proof of ~ ftc... |
| asindmre 38207 | Real part of domain of dif... |
| dvasin 38208 | Derivative of arcsine. (C... |
| dvacos 38209 | Derivative of arccosine. ... |
| dvreasin 38210 | Real derivative of arcsine... |
| dvreacos 38211 | Real derivative of arccosi... |
| areacirclem1 38212 | Antiderivative of cross-se... |
| areacirclem2 38213 | Endpoint-inclusive continu... |
| areacirclem3 38214 | Integrability of cross-sec... |
| areacirclem4 38215 | Endpoint-inclusive continu... |
| areacirclem5 38216 | Finding the cross-section ... |
| areacirc 38217 | The area of a circle of ra... |
| unirep 38218 | Define a quantity whose de... |
| cover2 38219 | Two ways of expressing the... |
| cover2g 38220 | Two ways of expressing the... |
| brabg2 38221 | Relation by a binary relat... |
| opelopab3 38222 | Ordered pair membership in... |
| cocanfo 38223 | Cancellation of a surjecti... |
| brresi2 38224 | Restriction of a binary re... |
| fnopabeqd 38225 | Equality deduction for fun... |
| fvopabf4g 38226 | Function value of an opera... |
| fnopabco 38227 | Composition of a function ... |
| opropabco 38228 | Composition of an operator... |
| cocnv 38229 | Composition with a functio... |
| f1ocan1fv 38230 | Cancel a composition by a ... |
| f1ocan2fv 38231 | Cancel a composition by th... |
| inixp 38232 | Intersection of Cartesian ... |
| upixp 38233 | Universal property of the ... |
| abrexdom 38234 | An indexed set is dominate... |
| abrexdom2 38235 | An indexed set is dominate... |
| ac6gf 38236 | Axiom of Choice. (Contrib... |
| indexa 38237 | If for every element of an... |
| indexdom 38238 | If for every element of an... |
| frinfm 38239 | A subset of a well-founded... |
| welb 38240 | A nonempty subset of a wel... |
| supex2g 38241 | Existence of supremum. (C... |
| supclt 38242 | Closure of supremum. (Con... |
| supubt 38243 | Upper bound property of su... |
| filbcmb 38244 | Combine a finite set of lo... |
| fzmul 38245 | Membership of a product in... |
| sdclem2 38246 | Lemma for ~ sdc . (Contri... |
| sdclem1 38247 | Lemma for ~ sdc . (Contri... |
| sdc 38248 | Strong dependent choice. ... |
| fdc 38249 | Finite version of dependen... |
| fdc1 38250 | Variant of ~ fdc with no s... |
| seqpo 38251 | Two ways to say that a seq... |
| incsequz 38252 | An increasing sequence of ... |
| incsequz2 38253 | An increasing sequence of ... |
| nnubfi 38254 | A bounded above set of pos... |
| nninfnub 38255 | An infinite set of positiv... |
| subspopn 38256 | An open set is open in the... |
| neificl 38257 | Neighborhoods are closed u... |
| lpss2 38258 | Limit points of a subset a... |
| metf1o 38259 | Use a bijection with a met... |
| blssp 38260 | A ball in the subspace met... |
| mettrifi 38261 | Generalized triangle inequ... |
| lmclim2 38262 | A sequence in a metric spa... |
| geomcau 38263 | If the distance between co... |
| caures 38264 | The restriction of a Cauch... |
| caushft 38265 | A shifted Cauchy sequence ... |
| constcncf 38266 | A constant function is a c... |
| cnres2 38267 | The restriction of a conti... |
| cnresima 38268 | A continuous function is c... |
| cncfres 38269 | A continuous function on c... |
| istotbnd 38273 | The predicate "is a totall... |
| istotbnd2 38274 | The predicate "is a totall... |
| istotbnd3 38275 | A metric space is totally ... |
| totbndmet 38276 | The predicate "totally bou... |
| 0totbnd 38277 | The metric (there is only ... |
| sstotbnd2 38278 | Condition for a subset of ... |
| sstotbnd 38279 | Condition for a subset of ... |
| sstotbnd3 38280 | Use a net that is not nece... |
| totbndss 38281 | A subset of a totally boun... |
| equivtotbnd 38282 | If the metric ` M ` is "st... |
| isbnd 38284 | The predicate "is a bounde... |
| bndmet 38285 | A bounded metric space is ... |
| isbndx 38286 | A "bounded extended metric... |
| isbnd2 38287 | The predicate "is a bounde... |
| isbnd3 38288 | A metric space is bounded ... |
| isbnd3b 38289 | A metric space is bounded ... |
| bndss 38290 | A subset of a bounded metr... |
| blbnd 38291 | A ball is bounded. (Contr... |
| ssbnd 38292 | A subset of a metric space... |
| totbndbnd 38293 | A totally bounded metric s... |
| equivbnd 38294 | If the metric ` M ` is "st... |
| bnd2lem 38295 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 38296 | If balls are totally bound... |
| prdsbnd 38297 | The product metric over fi... |
| prdstotbnd 38298 | The product metric over fi... |
| prdsbnd2 38299 | If balls are totally bound... |
| cntotbnd 38300 | A subset of the complex nu... |
| cnpwstotbnd 38301 | A subset of ` A ^ I ` , wh... |
| ismtyval 38304 | The set of isometries betw... |
| isismty 38305 | The condition "is an isome... |
| ismtycnv 38306 | The inverse of an isometry... |
| ismtyima 38307 | The image of a ball under ... |
| ismtyhmeolem 38308 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 38309 | An isometry is a homeomorp... |
| ismtybndlem 38310 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 38311 | Isometries preserve bounde... |
| ismtyres 38312 | A restriction of an isomet... |
| heibor1lem 38313 | Lemma for ~ heibor1 . A c... |
| heibor1 38314 | One half of ~ heibor , tha... |
| heiborlem1 38315 | Lemma for ~ heibor . We w... |
| heiborlem2 38316 | Lemma for ~ heibor . Subs... |
| heiborlem3 38317 | Lemma for ~ heibor . Usin... |
| heiborlem4 38318 | Lemma for ~ heibor . Usin... |
| heiborlem5 38319 | Lemma for ~ heibor . The ... |
| heiborlem6 38320 | Lemma for ~ heibor . Sinc... |
| heiborlem7 38321 | Lemma for ~ heibor . Sinc... |
| heiborlem8 38322 | Lemma for ~ heibor . The ... |
| heiborlem9 38323 | Lemma for ~ heibor . Disc... |
| heiborlem10 38324 | Lemma for ~ heibor . The ... |
| heibor 38325 | Generalized Heine-Borel Th... |
| bfplem1 38326 | Lemma for ~ bfp . The seq... |
| bfplem2 38327 | Lemma for ~ bfp . Using t... |
| bfp 38328 | Banach fixed point theorem... |
| rrnval 38331 | The n-dimensional Euclidea... |
| rrnmval 38332 | The value of the Euclidean... |
| rrnmet 38333 | Euclidean space is a metri... |
| rrndstprj1 38334 | The distance between two p... |
| rrndstprj2 38335 | Bound on the distance betw... |
| rrncmslem 38336 | Lemma for ~ rrncms . (Con... |
| rrncms 38337 | Euclidean space is complet... |
| repwsmet 38338 | The supremum metric on ` R... |
| rrnequiv 38339 | The supremum metric on ` R... |
| rrntotbnd 38340 | A set in Euclidean space i... |
| rrnheibor 38341 | Heine-Borel theorem for Eu... |
| ismrer1 38342 | An isometry between ` RR `... |
| reheibor 38343 | Heine-Borel theorem for re... |
| iccbnd 38344 | A closed interval in ` RR ... |
| icccmpALT 38345 | A closed interval in ` RR ... |
| isass 38350 | The predicate "is an assoc... |
| isexid 38351 | The predicate ` G ` has a ... |
| ismgmOLD 38354 | Obsolete version of ~ ismg... |
| clmgmOLD 38355 | Obsolete version of ~ mgmc... |
| opidonOLD 38356 | Obsolete version of ~ mndp... |
| rngopidOLD 38357 | Obsolete version of ~ mndp... |
| opidon2OLD 38358 | Obsolete version of ~ mndp... |
| isexid2 38359 | If ` G e. ( Magma i^i ExId... |
| exidu1 38360 | Uniqueness of the left and... |
| idrval 38361 | The value of the identity ... |
| iorlid 38362 | A magma right and left ide... |
| cmpidelt 38363 | A magma right and left ide... |
| smgrpismgmOLD 38366 | Obsolete version of ~ sgrp... |
| issmgrpOLD 38367 | Obsolete version of ~ issg... |
| smgrpmgm 38368 | A semigroup is a magma. (... |
| smgrpassOLD 38369 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 38372 | Obsolete version of ~ mnds... |
| mndoisexid 38373 | A monoid has an identity e... |
| mndoismgmOLD 38374 | Obsolete version of ~ mndm... |
| mndomgmid 38375 | A monoid is a magma with a... |
| ismndo 38376 | The predicate "is a monoid... |
| ismndo1 38377 | The predicate "is a monoid... |
| ismndo2 38378 | The predicate "is a monoid... |
| grpomndo 38379 | A group is a monoid. (Con... |
| exidcl 38380 | Closure of the binary oper... |
| exidreslem 38381 | Lemma for ~ exidres and ~ ... |
| exidres 38382 | The restriction of a binar... |
| exidresid 38383 | The restriction of a binar... |
| ablo4pnp 38384 | A commutative/associative ... |
| grpoeqdivid 38385 | Two group elements are equ... |
| grposnOLD 38386 | The group operation for th... |
| elghomlem1OLD 38389 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 38390 | Obsolete as of 15-Mar-2020... |
| elghomOLD 38391 | Obsolete version of ~ isgh... |
| ghomlinOLD 38392 | Obsolete version of ~ ghml... |
| ghomidOLD 38393 | Obsolete version of ~ ghmi... |
| ghomf 38394 | Mapping property of a grou... |
| ghomco 38395 | The composition of two gro... |
| ghomdiv 38396 | Group homomorphisms preser... |
| grpokerinj 38397 | A group homomorphism is in... |
| relrngo 38400 | The class of all unital ri... |
| isrngo 38401 | The predicate "is a (unita... |
| isrngod 38402 | Conditions that determine ... |
| rngoi 38403 | The properties of a unital... |
| rngosm 38404 | Functionality of the multi... |
| rngocl 38405 | Closure of the multiplicat... |
| rngoid 38406 | The multiplication operati... |
| rngoideu 38407 | The unity element of a rin... |
| rngodi 38408 | Distributive law for the m... |
| rngodir 38409 | Distributive law for the m... |
| rngoass 38410 | Associative law for the mu... |
| rngo2 38411 | A ring element plus itself... |
| rngoablo 38412 | A ring's addition operatio... |
| rngoablo2 38413 | In a unital ring the addit... |
| rngogrpo 38414 | A ring's addition operatio... |
| rngone0 38415 | The base set of a ring is ... |
| rngogcl 38416 | Closure law for the additi... |
| rngocom 38417 | The addition operation of ... |
| rngoaass 38418 | The addition operation of ... |
| rngoa32 38419 | The addition operation of ... |
| rngoa4 38420 | Rearrangement of 4 terms i... |
| rngorcan 38421 | Right cancellation law for... |
| rngolcan 38422 | Left cancellation law for ... |
| rngo0cl 38423 | A ring has an additive ide... |
| rngo0rid 38424 | The additive identity of a... |
| rngo0lid 38425 | The additive identity of a... |
| rngolz 38426 | The zero of a unital ring ... |
| rngorz 38427 | The zero of a unital ring ... |
| rngosn3 38428 | Obsolete as of 25-Jan-2020... |
| rngosn4 38429 | Obsolete as of 25-Jan-2020... |
| rngosn6 38430 | Obsolete as of 25-Jan-2020... |
| rngonegcl 38431 | A ring is closed under neg... |
| rngoaddneg1 38432 | Adding the negative in a r... |
| rngoaddneg2 38433 | Adding the negative in a r... |
| rngosub 38434 | Subtraction in a ring, in ... |
| rngmgmbs4 38435 | The range of an internal o... |
| rngodm1dm2 38436 | In a unital ring the domai... |
| rngorn1 38437 | In a unital ring the range... |
| rngorn1eq 38438 | In a unital ring the range... |
| rngomndo 38439 | In a unital ring the multi... |
| rngoidmlem 38440 | The unity element of a rin... |
| rngolidm 38441 | The unity element of a rin... |
| rngoridm 38442 | The unity element of a rin... |
| rngo1cl 38443 | The unity element of a rin... |
| rngoueqz 38444 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 38445 | Negation in a ring is the ... |
| rngonegmn1r 38446 | Negation in a ring is the ... |
| rngoneglmul 38447 | Negation of a product in a... |
| rngonegrmul 38448 | Negation of a product in a... |
| rngosubdi 38449 | Ring multiplication distri... |
| rngosubdir 38450 | Ring multiplication distri... |
| zerdivemp1x 38451 | In a unital ring a left in... |
| isdivrngo 38454 | The predicate "is a divisi... |
| drngoi 38455 | The properties of a divisi... |
| gidsn 38456 | Obsolete as of 23-Jan-2020... |
| zrdivrng 38457 | The zero ring is not a div... |
| dvrunz 38458 | In a division ring the rin... |
| isgrpda 38459 | Properties that determine ... |
| isdrngo1 38460 | The predicate "is a divisi... |
| divrngcl 38461 | The product of two nonzero... |
| isdrngo2 38462 | A division ring is a ring ... |
| isdrngo3 38463 | A division ring is a ring ... |
| rngohomval 38468 | The set of ring homomorphi... |
| isrngohom 38469 | The predicate "is a ring h... |
| rngohomf 38470 | A ring homomorphism is a f... |
| rngohomcl 38471 | Closure law for a ring hom... |
| rngohom1 38472 | A ring homomorphism preser... |
| rngohomadd 38473 | Ring homomorphisms preserv... |
| rngohommul 38474 | Ring homomorphisms preserv... |
| rngogrphom 38475 | A ring homomorphism is a g... |
| rngohom0 38476 | A ring homomorphism preser... |
| rngohomsub 38477 | Ring homomorphisms preserv... |
| rngohomco 38478 | The composition of two rin... |
| rngokerinj 38479 | A ring homomorphism is inj... |
| rngoisoval 38481 | The set of ring isomorphis... |
| isrngoiso 38482 | The predicate "is a ring i... |
| rngoiso1o 38483 | A ring isomorphism is a bi... |
| rngoisohom 38484 | A ring isomorphism is a ri... |
| rngoisocnv 38485 | The inverse of a ring isom... |
| rngoisoco 38486 | The composition of two rin... |
| isriscg 38488 | The ring isomorphism relat... |
| isrisc 38489 | The ring isomorphism relat... |
| risc 38490 | The ring isomorphism relat... |
| risci 38491 | Determine that two rings a... |
| riscer 38492 | Ring isomorphism is an equ... |
| iscom2 38499 | A device to add commutativ... |
| iscrngo 38500 | The predicate "is a commut... |
| iscrngo2 38501 | The predicate "is a commut... |
| iscringd 38502 | Conditions that determine ... |
| flddivrng 38503 | A field is a division ring... |
| crngorngo 38504 | A commutative ring is a ri... |
| crngocom 38505 | The multiplication operati... |
| crngm23 38506 | Commutative/associative la... |
| crngm4 38507 | Commutative/associative la... |
| fldcrngo 38508 | A field is a commutative r... |
| isfld2 38509 | The predicate "is a field"... |
| crngohomfo 38510 | The image of a homomorphis... |
| idlval 38517 | The class of ideals of a r... |
| isidl 38518 | The predicate "is an ideal... |
| isidlc 38519 | The predicate "is an ideal... |
| idlss 38520 | An ideal of ` R ` is a sub... |
| idlcl 38521 | An element of an ideal is ... |
| idl0cl 38522 | An ideal contains ` 0 ` . ... |
| idladdcl 38523 | An ideal is closed under a... |
| idllmulcl 38524 | An ideal is closed under m... |
| idlrmulcl 38525 | An ideal is closed under m... |
| idlnegcl 38526 | An ideal is closed under n... |
| idlsubcl 38527 | An ideal is closed under s... |
| rngoidl 38528 | A ring ` R ` is an ` R ` i... |
| 0idl 38529 | The set containing only ` ... |
| 1idl 38530 | Two ways of expressing the... |
| 0rngo 38531 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 38532 | The only ideals in a divis... |
| intidl 38533 | The intersection of a none... |
| inidl 38534 | The intersection of two id... |
| unichnidl 38535 | The union of a nonempty ch... |
| keridl 38536 | The kernel of a ring homom... |
| pridlval 38537 | The class of prime ideals ... |
| ispridl 38538 | The predicate "is a prime ... |
| pridlidl 38539 | A prime ideal is an ideal.... |
| pridlnr 38540 | A prime ideal is a proper ... |
| pridl 38541 | The main property of a pri... |
| ispridl2 38542 | A condition that shows an ... |
| maxidlval 38543 | The set of maximal ideals ... |
| ismaxidl 38544 | The predicate "is a maxima... |
| maxidlidl 38545 | A maximal ideal is an idea... |
| maxidlnr 38546 | A maximal ideal is proper.... |
| maxidlmax 38547 | A maximal ideal is a maxim... |
| maxidln1 38548 | One is not contained in an... |
| maxidln0 38549 | A ring with a maximal idea... |
| isprrngo 38554 | The predicate "is a prime ... |
| prrngorngo 38555 | A prime ring is a ring. (... |
| smprngopr 38556 | A simple ring (one whose o... |
| divrngpr 38557 | A division ring is a prime... |
| isdmn 38558 | The predicate "is a domain... |
| isdmn2 38559 | The predicate "is a domain... |
| dmncrng 38560 | A domain is a commutative ... |
| dmnrngo 38561 | A domain is a ring. (Cont... |
| flddmn 38562 | A field is a domain. (Con... |
| igenval 38565 | The ideal generated by a s... |
| igenss 38566 | A set is a subset of the i... |
| igenidl 38567 | The ideal generated by a s... |
| igenmin 38568 | The ideal generated by a s... |
| igenidl2 38569 | The ideal generated by an ... |
| igenval2 38570 | The ideal generated by a s... |
| prnc 38571 | A principal ideal (an idea... |
| isfldidl 38572 | Determine if a ring is a f... |
| isfldidl2 38573 | Determine if a ring is a f... |
| ispridlc 38574 | The predicate "is a prime ... |
| pridlc 38575 | Property of a prime ideal ... |
| pridlc2 38576 | Property of a prime ideal ... |
| pridlc3 38577 | Property of a prime ideal ... |
| isdmn3 38578 | The predicate "is a domain... |
| dmnnzd 38579 | A domain has no zero-divis... |
| dmncan1 38580 | Cancellation law for domai... |
| dmncan2 38581 | Cancellation law for domai... |
| efald2 38582 | A proof by contradiction. ... |
| notbinot1 38583 | Simplification rule of neg... |
| bicontr 38584 | Biconditional of its own n... |
| impor 38585 | An equivalent formula for ... |
| orfa 38586 | The falsum ` F. ` can be r... |
| notbinot2 38587 | Commutation rule between n... |
| biimpor 38588 | A rewriting rule for bicon... |
| orfa1 38589 | Add a contradicting disjun... |
| orfa2 38590 | Remove a contradicting dis... |
| bifald 38591 | Infer the equivalence to a... |
| orsild 38592 | A lemma for not-or-not eli... |
| orsird 38593 | A lemma for not-or-not eli... |
| cnf1dd 38594 | A lemma for Conjunctive No... |
| cnf2dd 38595 | A lemma for Conjunctive No... |
| cnfn1dd 38596 | A lemma for Conjunctive No... |
| cnfn2dd 38597 | A lemma for Conjunctive No... |
| or32dd 38598 | A rearrangement of disjunc... |
| notornotel1 38599 | A lemma for not-or-not eli... |
| notornotel2 38600 | A lemma for not-or-not eli... |
| contrd 38601 | A proof by contradiction, ... |
| an12i 38602 | An inference from commutin... |
| exmid2 38603 | An excluded middle law. (... |
| selconj 38604 | An inference for selecting... |
| truconj 38605 | Add true as a conjunct. (... |
| orel 38606 | An inference for disjuncti... |
| negel 38607 | An inference for negation ... |
| botel 38608 | An inference for bottom el... |
| tradd 38609 | Add top ad a conjunct. (C... |
| gm-sbtru 38610 | Substitution does not chan... |
| sbfal 38611 | Substitution does not chan... |
| sbcani 38612 | Distribution of class subs... |
| sbcori 38613 | Distribution of class subs... |
| sbcimi 38614 | Distribution of class subs... |
| sbcni 38615 | Move class substitution in... |
| sbali 38616 | Discard class substitution... |
| sbexi 38617 | Discard class substitution... |
| sbcalf 38618 | Move universal quantifier ... |
| sbcexf 38619 | Move existential quantifie... |
| sbcalfi 38620 | Move universal quantifier ... |
| sbcexfi 38621 | Move existential quantifie... |
| spsbcdi 38622 | A lemma for eliminating a ... |
| alrimii 38623 | A lemma for introducing a ... |
| spesbcdi 38624 | A lemma for introducing an... |
| exlimddvf 38625 | A lemma for eliminating an... |
| exlimddvfi 38626 | A lemma for eliminating an... |
| sbceq1ddi 38627 | A lemma for eliminating in... |
| sbccom2lem 38628 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 38629 | Commutative law for double... |
| sbccom2f 38630 | Commutative law for double... |
| sbccom2fi 38631 | Commutative law for double... |
| csbcom2fi 38632 | Commutative law for double... |
| fald 38633 | Refutation of falsity, in ... |
| tsim1 38634 | A Tseitin axiom for logica... |
| tsim2 38635 | A Tseitin axiom for logica... |
| tsim3 38636 | A Tseitin axiom for logica... |
| tsbi1 38637 | A Tseitin axiom for logica... |
| tsbi2 38638 | A Tseitin axiom for logica... |
| tsbi3 38639 | A Tseitin axiom for logica... |
| tsbi4 38640 | A Tseitin axiom for logica... |
| tsxo1 38641 | A Tseitin axiom for logica... |
| tsxo2 38642 | A Tseitin axiom for logica... |
| tsxo3 38643 | A Tseitin axiom for logica... |
| tsxo4 38644 | A Tseitin axiom for logica... |
| tsan1 38645 | A Tseitin axiom for logica... |
| tsan2 38646 | A Tseitin axiom for logica... |
| tsan3 38647 | A Tseitin axiom for logica... |
| tsna1 38648 | A Tseitin axiom for logica... |
| tsna2 38649 | A Tseitin axiom for logica... |
| tsna3 38650 | A Tseitin axiom for logica... |
| tsor1 38651 | A Tseitin axiom for logica... |
| tsor2 38652 | A Tseitin axiom for logica... |
| tsor3 38653 | A Tseitin axiom for logica... |
| ts3an1 38654 | A Tseitin axiom for triple... |
| ts3an2 38655 | A Tseitin axiom for triple... |
| ts3an3 38656 | A Tseitin axiom for triple... |
| ts3or1 38657 | A Tseitin axiom for triple... |
| ts3or2 38658 | A Tseitin axiom for triple... |
| ts3or3 38659 | A Tseitin axiom for triple... |
| iuneq2f 38660 | Equality deduction for ind... |
| rabeq12f 38661 | Equality deduction for res... |
| csbeq12 38662 | Equality deduction for sub... |
| sbeqi 38663 | Equality deduction for sub... |
| ralbi12f 38664 | Equality deduction for res... |
| oprabbi 38665 | Equality deduction for cla... |
| mpobi123f 38666 | Equality deduction for map... |
| iuneq12f 38667 | Equality deduction for ind... |
| iineq12f 38668 | Equality deduction for ind... |
| opabbi 38669 | Equality deduction for cla... |
| mptbi12f 38670 | Equality deduction for map... |
| orcomdd 38671 | Commutativity of logic dis... |
| scottexf 38672 | A version of ~ scottex wit... |
| scott0f 38673 | A version of ~ scott0 with... |
| scottn0f 38674 | A version of ~ scott0f wit... |
| ac6s3f 38675 | Generalization of the Axio... |
| ac6s6 38676 | Generalization of the Axio... |
| ac6s6f 38677 | Generalization of the Axio... |
| el2v1 38733 | New way ( ~ elv , and the ... |
| el3v1 38734 | New way ( ~ elv , and the ... |
| el3v2 38735 | New way ( ~ elv , and the ... |
| el3v12 38736 | New way ( ~ elv , and the ... |
| el3v13 38737 | New way ( ~ elv , and the ... |
| el3v23 38738 | New way ( ~ elv , and the ... |
| anan 38739 | Multiple commutations in c... |
| triantru3 38740 | A wff is equivalent to its... |
| biorfd 38741 | A wff is equivalent to its... |
| eqbrtr 38742 | Substitution of equal clas... |
| eqbrb 38743 | Substitution of equal clas... |
| eqeltr 38744 | Substitution of equal clas... |
| eqelb 38745 | Substitution of equal clas... |
| eqeqan2d 38746 | Implication of introducing... |
| disjresin 38747 | The restriction to a disjo... |
| disjresdisj 38748 | The intersection of restri... |
| disjresdif 38749 | The difference between res... |
| disjresundif 38750 | Lemma for ~ ressucdifsn2 .... |
| inres2 38751 | Two ways of expressing the... |
| coideq 38752 | Equality theorem for compo... |
| nexmo1 38753 | If there is no case where ... |
| eqab2 38754 | Implication of a class abs... |
| r2alan 38755 | Double restricted universa... |
| ssrabi 38756 | Inference of restricted ab... |
| rabimbieq 38757 | Restricted equivalent wff'... |
| abeqin 38758 | Intersection with class ab... |
| abeqinbi 38759 | Intersection with class ab... |
| eqrabi 38760 | Class element of a restric... |
| rabeqel 38761 | Class element of a restric... |
| eqrelf 38762 | The equality connective be... |
| br1cnvinxp 38763 | Binary relation on the con... |
| releleccnv 38764 | Elementhood in a converse ... |
| releccnveq 38765 | Equality of converse ` R `... |
| xpv 38766 | Cartesian product of a cla... |
| vxp 38767 | Cartesian product of the u... |
| opelvvdif 38768 | Negated elementhood of ord... |
| vvdifopab 38769 | Ordered-pair class abstrac... |
| brvdif 38770 | Binary relation with unive... |
| brvdif2 38771 | Binary relation with unive... |
| brvvdif 38772 | Binary relation with the c... |
| brvbrvvdif 38773 | Binary relation with the c... |
| brcnvep 38774 | The converse of the binary... |
| elecALTV 38775 | Elementhood in the ` R ` -... |
| brcnvepres 38776 | Restricted converse epsilo... |
| brres2 38777 | Binary relation on a restr... |
| br1cnvres 38778 | Binary relation on the con... |
| elec1cnvres 38779 | Elementhood in the convers... |
| ec1cnvres 38780 | Converse restricted coset ... |
| eldmres 38781 | Elementhood in the domain ... |
| elrnres 38782 | Element of the range of a ... |
| eldmressnALTV 38783 | Element of the domain of a... |
| elrnressn 38784 | Element of the range of a ... |
| eldm4 38785 | Elementhood in a domain. ... |
| eldmres2 38786 | Elementhood in the domain ... |
| eldmres3 38787 | Elementhood in the domain ... |
| eceq1i 38788 | Equality theorem for ` C `... |
| ecres 38789 | Restricted coset of ` B ` ... |
| eccnvepres 38790 | Restricted converse epsilo... |
| eleccnvep 38791 | Elementhood in the convers... |
| eccnvep 38792 | The converse epsilon coset... |
| extep 38793 | Property of epsilon relati... |
| disjeccnvep 38794 | Property of the epsilon re... |
| eccnvepres2 38795 | The restricted converse ep... |
| eccnvepres3 38796 | Condition for a restricted... |
| eldmqsres 38797 | Elementhood in a restricte... |
| eldmqsres2 38798 | Elementhood in a restricte... |
| qsss1 38799 | Subclass theorem for quoti... |
| qseq1i 38800 | Equality theorem for quoti... |
| brinxprnres 38801 | Binary relation on a restr... |
| inxprnres 38802 | Restriction of a class as ... |
| dfres4 38803 | Alternate definition of th... |
| exan3 38804 | Equivalent expressions wit... |
| exanres 38805 | Equivalent expressions wit... |
| exanres3 38806 | Equivalent expressions wit... |
| exanres2 38807 | Equivalent expressions wit... |
| cnvepres 38808 | Restricted converse epsilo... |
| eqrel2 38809 | Equality of relations. (C... |
| rncnv 38810 | Range of converse is the d... |
| dfdm6 38811 | Alternate definition of do... |
| dfrn6 38812 | Alternate definition of ra... |
| rncnvepres 38813 | The range of the restricte... |
| dmecd 38814 | Equality of the coset of `... |
| dmec2d 38815 | Equality of the coset of `... |
| brid 38816 | Property of the identity b... |
| ideq2 38817 | For sets, the identity bin... |
| idresssidinxp 38818 | Condition for the identity... |
| idreseqidinxp 38819 | Condition for the identity... |
| extid 38820 | Property of identity relat... |
| inxpss 38821 | Two ways to say that an in... |
| idinxpss 38822 | Two ways to say that an in... |
| ref5 38823 | Two ways to say that an in... |
| inxpss3 38824 | Two ways to say that an in... |
| inxpss2 38825 | Two ways to say that inter... |
| inxpssidinxp 38826 | Two ways to say that inter... |
| idinxpssinxp 38827 | Two ways to say that inter... |
| idinxpssinxp2 38828 | Identity intersection with... |
| idinxpssinxp3 38829 | Identity intersection with... |
| idinxpssinxp4 38830 | Identity intersection with... |
| relcnveq3 38831 | Two ways of saying a relat... |
| relcnveq 38832 | Two ways of saying a relat... |
| relcnveq2 38833 | Two ways of saying a relat... |
| relcnveq4 38834 | Two ways of saying a relat... |
| qsresid 38835 | Simplification of a specia... |
| n0elqs 38836 | Two ways of expressing tha... |
| n0elqs2 38837 | Two ways of expressing tha... |
| rnresequniqs 38838 | The range of a restriction... |
| n0el2 38839 | Two ways of expressing tha... |
| cnvepresex 38840 | Sethood condition for the ... |
| cnvepima 38841 | The image of converse epsi... |
| inex3 38842 | Sufficient condition for t... |
| inxpex 38843 | Sufficient condition for a... |
| eqres 38844 | Converting a class constan... |
| brrabga 38845 | The law of concretion for ... |
| brcnvrabga 38846 | The law of concretion for ... |
| opideq 38847 | Equality conditions for or... |
| iss2 38848 | A subclass of the identity... |
| eldmcnv 38849 | Elementhood in a domain of... |
| dfrel5 38850 | Alternate definition of th... |
| dfrel6 38851 | Alternate definition of th... |
| cnvresrn 38852 | Converse restricted to ran... |
| relssinxpdmrn 38853 | Subset of restriction, spe... |
| cnvref4 38854 | Two ways to say that a rel... |
| cnvref5 38855 | Two ways to say that a rel... |
| ecin0 38856 | Two ways of saying that th... |
| ecinn0 38857 | Two ways of saying that th... |
| ineleq 38858 | Equivalence of restricted ... |
| inecmo 38859 | Equivalence of a double re... |
| inecmo2 38860 | Equivalence of a double re... |
| ineccnvmo 38861 | Equivalence of a double re... |
| alrmomorn 38862 | Equivalence of an "at most... |
| alrmomodm 38863 | Equivalence of an "at most... |
| ralmo 38864 | "At most one" can be restr... |
| ralrnmo 38865 | On the range, "at most one... |
| dmqsex 38866 | Sethood of the domain quot... |
| raldmqsmo 38867 | On the quotient carrier, "... |
| ralrmo3 38868 | Pull a restricted universa... |
| raldmqseu 38869 | Equivalence between "exact... |
| rsp3 38870 | From a restricted universa... |
| rsp3eq 38871 | From a restricted universa... |
| ineccnvmo2 38872 | Equivalence of a double un... |
| inecmo3 38873 | Equivalence of a double un... |
| moeu2 38874 | Uniqueness is equivalent t... |
| mopickr 38875 | "At most one" picks a vari... |
| moantr 38876 | Sufficient condition for t... |
| brabidgaw 38877 | The law of concretion for ... |
| brabidga 38878 | The law of concretion for ... |
| inxp2 38879 | Intersection with a Cartes... |
| opabf 38880 | A class abstraction of a c... |
| ec0 38881 | The empty-coset of a class... |
| brcnvin 38882 | Intersection with a conver... |
| ssdmral 38883 | Subclass of a domain. (Co... |
| xrnss3v 38885 | A range Cartesian product ... |
| xrnrel 38886 | A range Cartesian product ... |
| brxrn 38887 | Characterize a ternary rel... |
| brxrn2 38888 | A characterization of the ... |
| dfxrn2 38889 | Alternate definition of th... |
| brxrncnvep 38890 | The range product with con... |
| dmxrn 38891 | Domain of the range produc... |
| dmcnvep 38892 | Domain of converse epsilon... |
| dmxrncnvep 38893 | Domain of the range produc... |
| dmcnvepres 38894 | Domain of the restricted c... |
| dmuncnvepres 38895 | Domain of the union with t... |
| dmxrnuncnvepres 38896 | Domain of the combined rel... |
| ecun 38897 | The union coset of ` A ` .... |
| ecunres 38898 | The restricted union coset... |
| ecuncnvepres 38899 | The restricted union with ... |
| xrneq1 38900 | Equality theorem for the r... |
| xrneq1i 38901 | Equality theorem for the r... |
| xrneq1d 38902 | Equality theorem for the r... |
| xrneq2 38903 | Equality theorem for the r... |
| xrneq2i 38904 | Equality theorem for the r... |
| xrneq2d 38905 | Equality theorem for the r... |
| xrneq12 38906 | Equality theorem for the r... |
| xrneq12i 38907 | Equality theorem for the r... |
| xrneq12d 38908 | Equality theorem for the r... |
| elecxrn 38909 | Elementhood in the ` ( R |... |
| ecxrn 38910 | The ` ( R |X. S ) ` -coset... |
| relecxrn 38911 | The ` ( R |X. S ) ` -coset... |
| ecxrn2 38912 | The ` ( R |X. S ) ` -coset... |
| ecxrncnvep 38913 | The ` ( R |X. ``' _E ) ` -... |
| ecxrncnvep2 38914 | The ` ( R |X. ``' _E ) ` -... |
| disjressuc2 38915 | Double restricted quantifi... |
| disjecxrn 38916 | Two ways of saying that ` ... |
| disjecxrncnvep 38917 | Two ways of saying that co... |
| disjsuc2 38918 | Double restricted quantifi... |
| xrninxp 38919 | Intersection of a range Ca... |
| xrninxp2 38920 | Intersection of a range Ca... |
| xrninxpex 38921 | Sufficient condition for t... |
| inxpxrn 38922 | Two ways to express the in... |
| br1cnvxrn2 38923 | The converse of a binary r... |
| elec1cnvxrn2 38924 | Elementhood in the convers... |
| rnxrn 38925 | Range of the range Cartesi... |
| rnxrnres 38926 | Range of a range Cartesian... |
| rnxrncnvepres 38927 | Range of a range Cartesian... |
| rnxrnidres 38928 | Range of a range Cartesian... |
| xrnres 38929 | Two ways to express restri... |
| xrnres2 38930 | Two ways to express restri... |
| xrnres3 38931 | Two ways to express restri... |
| xrnres4 38932 | Two ways to express restri... |
| xrnresex 38933 | Sufficient condition for a... |
| xrnidresex 38934 | Sufficient condition for a... |
| xrncnvepresex 38935 | Sufficient condition for a... |
| dmxrncnvepres 38936 | Domain of the range produc... |
| dmxrncnvepres2 38937 | Domain of the range produc... |
| eldmxrncnvepres 38938 | Element of the domain of t... |
| eldmxrncnvepres2 38939 | Element of the domain of t... |
| eceldmqsxrncnvepres 38940 | An ` ( R |X. ( ``' _E |`` ... |
| eceldmqsxrncnvepres2 38941 | An ` ( R |X. ( ``' _E |`` ... |
| brin2 38942 | Binary relation on an inte... |
| brin3 38943 | Binary relation on an inte... |
| elrels2 38945 | The element of the relatio... |
| elrelsrel 38946 | The element of the relatio... |
| elrelsrelim 38947 | The element of the relatio... |
| elrels5 38948 | Equivalent expressions for... |
| elrels6 38949 | Equivalent expressions for... |
| dfqmap2 38951 | Alternate definition of th... |
| dfqmap3 38952 | Alternate definition of th... |
| ecqmap 38953 | ` QMap ` fibers are single... |
| ecqmap2 38954 | Fiber of ` QMap ` equals s... |
| qmapex 38955 | Quotient map exists if ` R... |
| relqmap 38956 | Quotient map is a relation... |
| dmqmap 38957 | ` QMap ` preserves the dom... |
| rnqmap 38958 | The range of the quotient ... |
| dfadjliftmap 38960 | Alternate (expanded) defin... |
| dfadjliftmap2 38961 | Alternate definition of th... |
| blockadjliftmap 38962 | A "two-stage" construction... |
| dfblockliftmap 38964 | Alternate definition of th... |
| dfblockliftmap2 38965 | Alternate definition of th... |
| dfsucmap3 38967 | Alternate definition of th... |
| dfsucmap2 38968 | Alternate definition of th... |
| dfsucmap4 38969 | Alternate definition of th... |
| brsucmap 38970 | Binary relation form of th... |
| relsucmap 38971 | The successor map is a rel... |
| dmsucmap 38972 | The domain of the successo... |
| dfsuccl2 38974 | Alternate definition of th... |
| mopre 38975 | There is at most one prede... |
| exeupre2 38976 | Whenever a predecessor exi... |
| dfsuccl3 38977 | Alternate definition of th... |
| dfsuccl4 38978 | Alternate definition that ... |
| dfpre 38980 | Alternate definition of th... |
| dfpre2 38981 | Alternate definition of th... |
| dfpre3 38982 | Alternate definition of th... |
| dfpred4 38983 | Alternate definition of th... |
| dfpre4 38984 | Alternate definition of th... |
| shiftstableeq2 38987 | Equality theorem for shift... |
| suceqsneq 38988 | One-to-one relationship be... |
| sucdifsn2 38989 | Absorption of union with a... |
| sucdifsn 38990 | The difference between the... |
| ressucdifsn2 38991 | The difference between res... |
| ressucdifsn 38992 | The difference between res... |
| sucmapsuc 38993 | A set is succeeded by its ... |
| sucmapleftuniq 38994 | Left uniqueness of the suc... |
| exeupre 38995 | Whenever a predecessor exi... |
| preex 38996 | The successor-predecessor ... |
| eupre2 38997 | Unique predecessor exists ... |
| eupre 38998 | Unique predecessor exists ... |
| presucmap 38999 | ` pre ` is really a predec... |
| preuniqval 39000 | Uniqueness/canonicity of `... |
| sucpre 39001 | ` suc ` is a right-inverse... |
| presuc 39002 | ` pre ` is a left-inverse ... |
| press 39003 | Predecessor is a subset of... |
| preel 39004 | Predecessor is a subset of... |
| dfcoss2 39007 | Alternate definition of th... |
| dfcoss3 39008 | Alternate definition of th... |
| dfcoss4 39009 | Alternate definition of th... |
| cosscnv 39010 | Class of cosets by the con... |
| coss1cnvres 39011 | Class of cosets by the con... |
| coss2cnvepres 39012 | Special case of ~ coss1cnv... |
| cossex 39013 | If ` A ` is a set then the... |
| cosscnvex 39014 | If ` A ` is a set then the... |
| 1cosscnvepresex 39015 | Sufficient condition for a... |
| 1cossxrncnvepresex 39016 | Sufficient condition for a... |
| relcoss 39017 | Cosets by ` R ` is a relat... |
| relcoels 39018 | Coelements on ` A ` is a r... |
| cossss 39019 | Subclass theorem for the c... |
| cosseq 39020 | Equality theorem for the c... |
| cosseqi 39021 | Equality theorem for the c... |
| cosseqd 39022 | Equality theorem for the c... |
| 1cossres 39023 | The class of cosets by a r... |
| dfcoels 39024 | Alternate definition of th... |
| brcoss 39025 | ` A ` and ` B ` are cosets... |
| brcoss2 39026 | Alternate form of the ` A ... |
| brcoss3 39027 | Alternate form of the ` A ... |
| brcosscnvcoss 39028 | For sets, the ` A ` and ` ... |
| brcoels 39029 | ` B ` and ` C ` are coelem... |
| cocossss 39030 | Two ways of saying that co... |
| cnvcosseq 39031 | The converse of cosets by ... |
| br2coss 39032 | Cosets by ` ,~ R ` binary ... |
| br1cossres 39033 | ` B ` and ` C ` are cosets... |
| br1cossres2 39034 | ` B ` and ` C ` are cosets... |
| brressn 39035 | Binary relation on a restr... |
| ressn2 39036 | A class ' R ' restricted t... |
| refressn 39037 | Any class ' R ' restricted... |
| antisymressn 39038 | Every class ' R ' restrict... |
| trressn 39039 | Any class ' R ' restricted... |
| relbrcoss 39040 | ` A ` and ` B ` are cosets... |
| br1cossinres 39041 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 39042 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 39043 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 39044 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 39045 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 39046 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 39047 | The domain of cosets is th... |
| dmcoss2 39048 | The domain of cosets is th... |
| rncossdmcoss 39049 | The range of cosets is the... |
| dm1cosscnvepres 39050 | The domain of cosets of th... |
| dmcoels 39051 | The domain of coelements i... |
| eldmcoss 39052 | Elementhood in the domain ... |
| eldmcoss2 39053 | Elementhood in the domain ... |
| eldm1cossres 39054 | Elementhood in the domain ... |
| eldm1cossres2 39055 | Elementhood in the domain ... |
| refrelcosslem 39056 | Lemma for the left side of... |
| refrelcoss3 39057 | The class of cosets by ` R... |
| refrelcoss2 39058 | The class of cosets by ` R... |
| symrelcoss3 39059 | The class of cosets by ` R... |
| symrelcoss2 39060 | The class of cosets by ` R... |
| cossssid 39061 | Equivalent expressions for... |
| cossssid2 39062 | Equivalent expressions for... |
| cossssid3 39063 | Equivalent expressions for... |
| cossssid4 39064 | Equivalent expressions for... |
| cossssid5 39065 | Equivalent expressions for... |
| brcosscnv 39066 | ` A ` and ` B ` are cosets... |
| brcosscnv2 39067 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 39068 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 39069 | Cosets by the converse ran... |
| cosscnvssid3 39070 | Equivalent expressions for... |
| cosscnvssid4 39071 | Equivalent expressions for... |
| cosscnvssid5 39072 | Equivalent expressions for... |
| coss0 39073 | Cosets by the empty set ar... |
| cossid 39074 | Cosets by the identity rel... |
| cosscnvid 39075 | Cosets by the converse ide... |
| trcoss 39076 | Sufficient condition for t... |
| eleccossin 39077 | Two ways of saying that th... |
| trcoss2 39078 | Equivalent expressions for... |
| cosselrels 39079 | Cosets of sets are element... |
| cnvelrels 39080 | The converse of a set is a... |
| cosscnvelrels 39081 | Cosets of converse sets ar... |
| dfssr2 39083 | Alternate definition of th... |
| relssr 39084 | The subset relation is a r... |
| brssr 39085 | The subset relation and su... |
| brssrid 39086 | Any set is a subset of its... |
| issetssr 39087 | Two ways of expressing set... |
| brssrres 39088 | Restricted subset binary r... |
| br1cnvssrres 39089 | Restricted converse subset... |
| brcnvssr 39090 | The converse of a subset r... |
| brcnvssrid 39091 | Any set is a converse subs... |
| br1cossxrncnvssrres 39092 | ` <. B , C >. ` and ` <. D... |
| extssr 39093 | Property of subset relatio... |
| dfrefrels2 39097 | Alternate definition of th... |
| dfrefrels3 39098 | Alternate definition of th... |
| dfrefrel2 39099 | Alternate definition of th... |
| dfrefrel3 39100 | Alternate definition of th... |
| dfrefrel5 39101 | Alternate definition of th... |
| elrefrels2 39102 | Element of the class of re... |
| elrefrels3 39103 | Element of the class of re... |
| elrefrelsrel 39104 | For sets, being an element... |
| refreleq 39105 | Equality theorem for refle... |
| refrelid 39106 | Identity relation is refle... |
| refrelcoss 39107 | The class of cosets by ` R... |
| refrelressn 39108 | Any class ' R ' restricted... |
| dfcnvrefrels2 39112 | Alternate definition of th... |
| dfcnvrefrels3 39113 | Alternate definition of th... |
| dfcnvrefrel2 39114 | Alternate definition of th... |
| dfcnvrefrel3 39115 | Alternate definition of th... |
| dfcnvrefrel4 39116 | Alternate definition of th... |
| dfcnvrefrel5 39117 | Alternate definition of th... |
| elcnvrefrels2 39118 | Element of the class of co... |
| elcnvrefrels3 39119 | Element of the class of co... |
| elcnvrefrelsrel 39120 | For sets, being an element... |
| cnvrefrelcoss2 39121 | Necessary and sufficient c... |
| cosselcnvrefrels2 39122 | Necessary and sufficient c... |
| cosselcnvrefrels3 39123 | Necessary and sufficient c... |
| cosselcnvrefrels4 39124 | Necessary and sufficient c... |
| cosselcnvrefrels5 39125 | Necessary and sufficient c... |
| dfsymrels2 39129 | Alternate definition of th... |
| dfsymrels3 39130 | Alternate definition of th... |
| elrelscnveq3 39131 | Two ways of saying a relat... |
| elrelscnveq 39132 | Two ways of saying a relat... |
| elrelscnveq2 39133 | Two ways of saying a relat... |
| elrelscnveq4 39134 | Two ways of saying a relat... |
| dfsymrels4 39135 | Alternate definition of th... |
| dfsymrels5 39136 | Alternate definition of th... |
| dfsymrel2 39137 | Alternate definition of th... |
| dfsymrel3 39138 | Alternate definition of th... |
| dfsymrel4 39139 | Alternate definition of th... |
| dfsymrel5 39140 | Alternate definition of th... |
| elsymrels2 39141 | Element of the class of sy... |
| elsymrels3 39142 | Element of the class of sy... |
| elsymrels4 39143 | Element of the class of sy... |
| elsymrels5 39144 | Element of the class of sy... |
| elsymrelsrel 39145 | For sets, being an element... |
| symreleq 39146 | Equality theorem for symme... |
| symrelim 39147 | Symmetric relation implies... |
| symrelcoss 39148 | The class of cosets by ` R... |
| idsymrel 39149 | The identity relation is s... |
| epnsymrel 39150 | The membership (epsilon) r... |
| symrefref2 39151 | Symmetry is a sufficient c... |
| symrefref3 39152 | Symmetry is a sufficient c... |
| refsymrels2 39153 | Elements of the class of r... |
| refsymrels3 39154 | Elements of the class of r... |
| refsymrel2 39155 | A relation which is reflex... |
| refsymrel3 39156 | A relation which is reflex... |
| elrefsymrels2 39157 | Elements of the class of r... |
| elrefsymrels3 39158 | Elements of the class of r... |
| elrefsymrelsrel 39159 | For sets, being an element... |
| dftrrels2 39163 | Alternate definition of th... |
| dftrrels3 39164 | Alternate definition of th... |
| dftrrel2 39165 | Alternate definition of th... |
| dftrrel3 39166 | Alternate definition of th... |
| eltrrels2 39167 | Element of the class of tr... |
| eltrrels3 39168 | Element of the class of tr... |
| eltrrelsrel 39169 | For sets, being an element... |
| trreleq 39170 | Equality theorem for the t... |
| trrelressn 39171 | Any class ' R ' restricted... |
| dfeqvrels2 39176 | Alternate definition of th... |
| dfeqvrels3 39177 | Alternate definition of th... |
| dfeqvrel2 39178 | Alternate definition of th... |
| dfeqvrel3 39179 | Alternate definition of th... |
| eleqvrels2 39180 | Element of the class of eq... |
| eleqvrels3 39181 | Element of the class of eq... |
| eleqvrelsrel 39182 | For sets, being an element... |
| elcoeleqvrels 39183 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 39184 | For sets, being an element... |
| eqvrelrel 39185 | An equivalence relation is... |
| eqvrelrefrel 39186 | An equivalence relation is... |
| eqvrelsymrel 39187 | An equivalence relation is... |
| eqvreltrrel 39188 | An equivalence relation is... |
| eqvrelim 39189 | Equivalence relation impli... |
| eqvreleq 39190 | Equality theorem for equiv... |
| eqvreleqi 39191 | Equality theorem for equiv... |
| eqvreleqd 39192 | Equality theorem for equiv... |
| eqvrelsym 39193 | An equivalence relation is... |
| eqvrelsymb 39194 | An equivalence relation is... |
| eqvreltr 39195 | An equivalence relation is... |
| eqvreltrd 39196 | A transitivity relation fo... |
| eqvreltr4d 39197 | A transitivity relation fo... |
| eqvrelref 39198 | An equivalence relation is... |
| eqvrelth 39199 | Basic property of equivale... |
| eqvrelcl 39200 | Elementhood in the field o... |
| eqvrelthi 39201 | Basic property of equivale... |
| eqvreldisj 39202 | Equivalence classes do not... |
| qsdisjALTV 39203 | Elements of a quotient set... |
| eqvrelqsel 39204 | If an element of a quotien... |
| eqvrelcoss 39205 | Two ways to express equiva... |
| eqvrelcoss3 39206 | Two ways to express equiva... |
| eqvrelcoss2 39207 | Two ways to express equiva... |
| eqvrelcoss4 39208 | Two ways to express equiva... |
| dfcoeleqvrels 39209 | Alternate definition of th... |
| dfcoeleqvrel 39210 | Alternate definition of th... |
| brredunds 39214 | Binary relation on the cla... |
| brredundsredund 39215 | For sets, binary relation ... |
| redundss3 39216 | Implication of redundancy ... |
| redundeq1 39217 | Equivalence of redundancy ... |
| redundpim3 39218 | Implication of redundancy ... |
| redundpbi1 39219 | Equivalence of redundancy ... |
| refrelsredund4 39220 | The naive version of the c... |
| refrelsredund2 39221 | The naive version of the c... |
| refrelsredund3 39222 | The naive version of the c... |
| refrelredund4 39223 | The naive version of the d... |
| refrelredund2 39224 | The naive version of the d... |
| refrelredund3 39225 | The naive version of the d... |
| dmqseq 39228 | Equality theorem for domai... |
| dmqseqi 39229 | Equality theorem for domai... |
| dmqseqd 39230 | Equality theorem for domai... |
| dmqseqeq1 39231 | Equality theorem for domai... |
| dmqseqeq1i 39232 | Equality theorem for domai... |
| dmqseqeq1d 39233 | Equality theorem for domai... |
| brdmqss 39234 | The domain quotient binary... |
| brdmqssqs 39235 | If ` A ` and ` R ` are set... |
| n0eldmqs 39236 | The empty set is not an el... |
| qseq 39237 | The quotient set equal to ... |
| n0eldmqseq 39238 | The empty set is not an el... |
| n0elim 39239 | Implication of that the em... |
| n0el3 39240 | Two ways of expressing tha... |
| cnvepresdmqss 39241 | The domain quotient binary... |
| cnvepresdmqs 39242 | The domain quotient predic... |
| unidmqs 39243 | The range of a relation is... |
| unidmqseq 39244 | The union of the domain qu... |
| dmqseqim 39245 | If the domain quotient of ... |
| dmqseqim2 39246 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 39247 | Elementhood in the domain ... |
| eldmqs1cossres 39248 | Elementhood in the domain ... |
| releldmqscoss 39249 | Elementhood in the domain ... |
| dmqscoelseq 39250 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 39251 | Two ways to express the eq... |
| brers 39256 | Binary equivalence relatio... |
| dferALTV2 39257 | Equivalence relation with ... |
| erALTVeq1 39258 | Equality theorem for equiv... |
| erALTVeq1i 39259 | Equality theorem for equiv... |
| erALTVeq1d 39260 | Equality theorem for equiv... |
| dfcomember 39261 | Alternate definition of th... |
| dfcomember2 39262 | Alternate definition of th... |
| dfcomember3 39263 | Alternate definition of th... |
| eqvreldmqs 39264 | Two ways to express comemb... |
| eqvreldmqs2 39265 | Two ways to express comemb... |
| brerser 39266 | Binary equivalence relatio... |
| erimeq2 39267 | Equivalence relation on it... |
| erimeq 39268 | Equivalence relation on it... |
| dffunsALTV 39272 | Alternate definition of th... |
| dffunsALTV2 39273 | Alternate definition of th... |
| dffunsALTV3 39274 | Alternate definition of th... |
| dffunsALTV4 39275 | Alternate definition of th... |
| dffunsALTV5 39276 | Alternate definition of th... |
| dffunALTV2 39277 | Alternate definition of th... |
| dffunALTV3 39278 | Alternate definition of th... |
| dffunALTV4 39279 | Alternate definition of th... |
| dffunALTV5 39280 | Alternate definition of th... |
| elfunsALTV 39281 | Elementhood in the class o... |
| elfunsALTV2 39282 | Elementhood in the class o... |
| elfunsALTV3 39283 | Elementhood in the class o... |
| elfunsALTV4 39284 | Elementhood in the class o... |
| elfunsALTV5 39285 | Elementhood in the class o... |
| elfunsALTVfunALTV 39286 | The element of the class o... |
| funALTVfun 39287 | Our definition of the func... |
| funALTVss 39288 | Subclass theorem for funct... |
| funALTVeq 39289 | Equality theorem for funct... |
| funALTVeqi 39290 | Equality inference for the... |
| funALTVeqd 39291 | Equality deduction for the... |
| dfdisjs 39297 | Alternate definition of th... |
| dfdisjs2 39298 | Alternate definition of th... |
| dfdisjs3 39299 | Alternate definition of th... |
| dfdisjs4 39300 | Alternate definition of th... |
| dfdisjs5 39301 | Alternate definition of th... |
| dfdisjALTV 39302 | Alternate definition of th... |
| dfdisjALTV2 39303 | Alternate definition of th... |
| dfdisjALTV3 39304 | Alternate definition of th... |
| dfdisjALTV4 39305 | Alternate definition of th... |
| dfdisjALTV5 39306 | Alternate definition of th... |
| dfdisjALTV5a 39307 | Alternate definition of th... |
| disjimeceqim 39308 | ` Disj ` implies coset-equ... |
| disjimeceqim2 39309 | ` Disj ` implies injectivi... |
| disjimeceqbi 39310 | ` Disj ` gives bicondition... |
| disjimeceqbi2 39311 | Injectivity of the block c... |
| disjimrmoeqec 39312 | Under ` Disj ` , every blo... |
| disjimdmqseq 39313 | Disjointness implies uniqu... |
| dfeldisj2 39314 | Alternate definition of th... |
| dfeldisj3 39315 | Alternate definition of th... |
| dfeldisj4 39316 | Alternate definition of th... |
| dfeldisj5 39317 | Alternate definition of th... |
| dfeldisj5a 39318 | Alternate definition of th... |
| eldisjim3 39319 | ` ElDisj ` elimination (tw... |
| eldisjdmqsim2 39320 | ElDisj of quotient implies... |
| eldisjdmqsim 39321 | Shared output implies equa... |
| suceldisj 39322 | Disjointness of successor ... |
| eldisjs 39323 | Elementhood in the class o... |
| eldisjs2 39324 | Elementhood in the class o... |
| eldisjs3 39325 | Elementhood in the class o... |
| eldisjs4 39326 | Elementhood in the class o... |
| eldisjs5 39327 | Elementhood in the class o... |
| eldisjsdisj 39328 | The element of the class o... |
| qmapeldisjs 39329 | When ` R ` is a set (e.g.,... |
| disjqmap2 39330 | Disjointness of ` QMap ` e... |
| disjqmap 39331 | Disjointness of ` QMap ` e... |
| eleldisjs 39332 | Elementhood in the disjoin... |
| eleldisjseldisj 39333 | The element of the disjoin... |
| disjrel 39334 | Disjoint relation is a rel... |
| disjss 39335 | Subclass theorem for disjo... |
| disjssi 39336 | Subclass theorem for disjo... |
| disjssd 39337 | Subclass theorem for disjo... |
| disjeq 39338 | Equality theorem for disjo... |
| disjeqi 39339 | Equality theorem for disjo... |
| disjeqd 39340 | Equality theorem for disjo... |
| disjdmqseqeq1 39341 | Lemma for the equality the... |
| eldisjss 39342 | Subclass theorem for disjo... |
| eldisjssi 39343 | Subclass theorem for disjo... |
| eldisjssd 39344 | Subclass theorem for disjo... |
| eldisjeq 39345 | Equality theorem for disjo... |
| eldisjeqi 39346 | Equality theorem for disjo... |
| eldisjeqd 39347 | Equality theorem for disjo... |
| disjres 39348 | Disjoint restriction. (Co... |
| eldisjn0elb 39349 | Two forms of disjoint elem... |
| disjxrn 39350 | Two ways of saying that a ... |
| disjxrnres5 39351 | Disjoint range Cartesian p... |
| disjorimxrn 39352 | Disjointness condition for... |
| disjimxrn 39353 | Disjointness condition for... |
| disjimres 39354 | Disjointness condition for... |
| disjimin 39355 | Disjointness condition for... |
| disjiminres 39356 | Disjointness condition for... |
| disjimxrnres 39357 | Disjointness condition for... |
| disjALTV0 39358 | The null class is disjoint... |
| disjALTVid 39359 | The class of identity rela... |
| disjALTVidres 39360 | The class of identity rela... |
| disjALTVinidres 39361 | The intersection with rest... |
| disjALTVxrnidres 39362 | The class of range Cartesi... |
| disjsuc 39363 | Disjoint range Cartesian p... |
| qmapeldisjsim 39364 | Injectivity of coset map f... |
| qmapeldisjsbi 39365 | Injectivity of coset map f... |
| rnqmapeleldisjsim 39366 | Element-disjointness of th... |
| dfantisymrel4 39368 | Alternate definition of th... |
| dfantisymrel5 39369 | Alternate definition of th... |
| antisymrelres 39370 | (Contributed by Peter Mazs... |
| antisymrelressn 39371 | (Contributed by Peter Mazs... |
| dfpart2 39376 | Alternate definition of th... |
| dfmembpart2 39377 | Alternate definition of th... |
| brparts 39378 | Binary partitions relation... |
| brparts2 39379 | Binary partitions relation... |
| brpartspart 39380 | Binary partition and the p... |
| parteq1 39381 | Equality theorem for parti... |
| parteq2 39382 | Equality theorem for parti... |
| parteq12 39383 | Equality theorem for parti... |
| parteq1i 39384 | Equality theorem for parti... |
| parteq1d 39385 | Equality theorem for parti... |
| partsuc2 39386 | Property of the partition.... |
| partsuc 39387 | Property of the partition.... |
| disjim 39388 | The "Divide et Aequivalere... |
| disjimi 39389 | Every disjoint relation ge... |
| detlem 39390 | If a relation is disjoint,... |
| eldisjim 39391 | If the elements of ` A ` a... |
| eldisjim2 39392 | Alternate form of ~ eldisj... |
| eqvrel0 39393 | The null class is an equiv... |
| det0 39394 | The cosets by the null cla... |
| eqvrelcoss0 39395 | The cosets by the null cla... |
| eqvrelid 39396 | The identity relation is a... |
| eqvrel1cossidres 39397 | The cosets by a restricted... |
| eqvrel1cossinidres 39398 | The cosets by an intersect... |
| eqvrel1cossxrnidres 39399 | The cosets by a range Cart... |
| detid 39400 | The cosets by the identity... |
| eqvrelcossid 39401 | The cosets by the identity... |
| detidres 39402 | The cosets by the restrict... |
| detinidres 39403 | The cosets by the intersec... |
| detxrnidres 39404 | The cosets by the range Ca... |
| disjlem14 39405 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 39406 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 39407 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 39408 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 39409 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 39410 | Lemma for ~ disjdmqseq via... |
| disjdmqs 39411 | If a relation is disjoint,... |
| disjdmqseq 39412 | If a relation is disjoint,... |
| eldisjn0el 39413 | Special case of ~ disjdmqs... |
| partim2 39414 | Disjoint relation on its n... |
| partim 39415 | Partition implies equivale... |
| partimeq 39416 | Partition implies that the... |
| eldisjlem19 39417 | Special case of ~ disjlem1... |
| membpartlem19 39418 | Together with ~ disjlem19 ... |
| petlem 39419 | If you can prove that the ... |
| petlemi 39420 | If you can prove disjointn... |
| pet02 39421 | Class ` A ` is a partition... |
| pet0 39422 | Class ` A ` is a partition... |
| petid2 39423 | Class ` A ` is a partition... |
| petid 39424 | A class is a partition by ... |
| petidres2 39425 | Class ` A ` is a partition... |
| petidres 39426 | A class is a partition by ... |
| petinidres2 39427 | Class ` A ` is a partition... |
| petinidres 39428 | A class is a partition by ... |
| petxrnidres2 39429 | Class ` A ` is a partition... |
| petxrnidres 39430 | A class is a partition by ... |
| eqvreldisj1 39431 | The elements of the quotie... |
| eqvreldisj2 39432 | The elements of the quotie... |
| eqvreldisj3 39433 | The elements of the quotie... |
| eqvreldisj4 39434 | Intersection with the conv... |
| eqvreldisj5 39435 | Range Cartesian product wi... |
| eqvrelqseqdisj2 39436 | Implication of ~ eqvreldis... |
| disjimeldisjdmqs 39437 | ` Disj ` implies element-d... |
| eldisjsim1 39438 | An element of the class of... |
| eldisjsim2 39439 | An element of the class of... |
| disjsssrels 39440 | The class of disjoint rela... |
| eldisjsim3 39441 | ` Disjs ` implies element-... |
| eldisjsim4 39442 | ` Disjs ` implies element-... |
| eldisjsim5 39443 | ` Disjs ` is closed under ... |
| eldisjs6 39444 | Elementhood in the class o... |
| eldisjs7 39445 | Elementhood in the class o... |
| dfdisjs6 39446 | Alternate definition of th... |
| dfdisjs7 39447 | Alternate definition of th... |
| fences3 39448 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 39449 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 39450 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 39451 | Lemma for the Partition-Eq... |
| mainer 39452 | The Main Theorem of Equiva... |
| partimcomember 39453 | Partition with general ` R... |
| mpet3 39454 | Member Partition-Equivalen... |
| cpet2 39455 | The conventional form of t... |
| cpet 39456 | The conventional form of M... |
| mpet 39457 | Member Partition-Equivalen... |
| mpet2 39458 | Member Partition-Equivalen... |
| mpets2 39459 | Member Partition-Equivalen... |
| mpets 39460 | Member Partition-Equivalen... |
| mainpart 39461 | Partition with general ` R... |
| fences 39462 | The Theorem of Fences by E... |
| fences2 39463 | The Theorem of Fences by E... |
| mainer2 39464 | The Main Theorem of Equiva... |
| mainerim 39465 | Every equivalence relation... |
| petincnvepres2 39466 | A partition-equivalence th... |
| petincnvepres 39467 | The shortest form of a par... |
| pet2 39468 | Partition-Equivalence Theo... |
| pet 39469 | Partition-Equivalence Theo... |
| pets 39470 | Partition-Equivalence Theo... |
| dmqsblocks 39471 | If the ~ pet span ` ( R |X... |
| dfpetparts2 39476 | Alternate definition of ` ... |
| dfpet2parts2 39477 | Grade stability applied to... |
| dfpeters2 39478 | Alternate definition of ` ... |
| typesafepets 39479 | Type-safe ~ pets scheme. ... |
| petseq 39480 | Generalized partition-equi... |
| pets2eq 39481 | Grade-stable generalized p... |
| prtlem60 39482 | Lemma for ~ prter3 . (Con... |
| bicomdd 39483 | Commute two sides of a bic... |
| jca2r 39484 | Inference conjoining the c... |
| jca3 39485 | Inference conjoining the c... |
| prtlem70 39486 | Lemma for ~ prter3 : a rea... |
| ibdr 39487 | Reverse of ~ ibd . (Contr... |
| prtlem100 39488 | Lemma for ~ prter3 . (Con... |
| prtlem5 39489 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 39490 | Lemma for ~ prter2 . (Con... |
| brabsb2 39491 | A closed form of ~ brabsb ... |
| eqbrrdv2 39492 | Other version of ~ eqbrrdi... |
| prtlem9 39493 | Lemma for ~ prter3 . (Con... |
| prtlem10 39494 | Lemma for ~ prter3 . (Con... |
| prtlem11 39495 | Lemma for ~ prter2 . (Con... |
| prtlem12 39496 | Lemma for ~ prtex and ~ pr... |
| prtlem13 39497 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 39498 | Lemma for ~ prtex , ~ prte... |
| prtlem400 39499 | Lemma for ~ prter2 and als... |
| erprt 39502 | The quotient set of an equ... |
| prtlem14 39503 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 39504 | Lemma for ~ prter1 and ~ p... |
| prtlem17 39505 | Lemma for ~ prter2 . (Con... |
| prtlem18 39506 | Lemma for ~ prter2 . (Con... |
| prtlem19 39507 | Lemma for ~ prter2 . (Con... |
| prter1 39508 | Every partition generates ... |
| prtex 39509 | The equivalence relation g... |
| prter2 39510 | The quotient set of the eq... |
| prter3 39511 | For every partition there ... |
| axc5 39522 | This theorem repeats ~ sp ... |
| ax4fromc4 39523 | Rederivation of Axiom ~ ax... |
| ax10fromc7 39524 | Rederivation of Axiom ~ ax... |
| ax6fromc10 39525 | Rederivation of Axiom ~ ax... |
| hba1-o 39526 | The setvar ` x ` is not fr... |
| axc4i-o 39527 | Inference version of ~ ax-... |
| equid1 39528 | Proof of ~ equid from our ... |
| equcomi1 39529 | Proof of ~ equcomi from ~ ... |
| aecom-o 39530 | Commutation law for identi... |
| aecoms-o 39531 | A commutation rule for ide... |
| hbae-o 39532 | All variables are effectiv... |
| dral1-o 39533 | Formula-building lemma for... |
| ax12fromc15 39534 | Rederivation of Axiom ~ ax... |
| ax13fromc9 39535 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 39536 | Axiom to quantify a variab... |
| sps-o 39537 | Generalization of antecede... |
| hbequid 39538 | Bound-variable hypothesis ... |
| nfequid-o 39539 | Bound-variable hypothesis ... |
| axc5c7 39540 | Proof of a single axiom th... |
| axc5c7toc5 39541 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 39542 | Rederivation of ~ ax-c7 fr... |
| axc711 39543 | Proof of a single axiom th... |
| nfa1-o 39544 | ` x ` is not free in ` A. ... |
| axc711toc7 39545 | Rederivation of ~ ax-c7 fr... |
| axc711to11 39546 | Rederivation of ~ ax-11 fr... |
| axc5c711 39547 | Proof of a single axiom th... |
| axc5c711toc5 39548 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 39549 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 39550 | Rederivation of ~ ax-11 fr... |
| equidqe 39551 | ~ equid with existential q... |
| axc5sp1 39552 | A special case of ~ ax-c5 ... |
| equidq 39553 | ~ equid with universal qua... |
| equid1ALT 39554 | Alternate proof of ~ equid... |
| axc11nfromc11 39555 | Rederivation of ~ ax-c11n ... |
| naecoms-o 39556 | A commutation rule for dis... |
| hbnae-o 39557 | All variables are effectiv... |
| dvelimf-o 39558 | Proof of ~ dvelimh that us... |
| dral2-o 39559 | Formula-building lemma for... |
| aev-o 39560 | A "distinctor elimination"... |
| ax5eq 39561 | Theorem to add distinct qu... |
| dveeq2-o 39562 | Quantifier introduction wh... |
| axc16g-o 39563 | A generalization of Axiom ... |
| dveeq1-o 39564 | Quantifier introduction wh... |
| dveeq1-o16 39565 | Version of ~ dveeq1 using ... |
| ax5el 39566 | Theorem to add distinct qu... |
| axc11n-16 39567 | This theorem shows that, g... |
| dveel2ALT 39568 | Alternate proof of ~ dveel... |
| ax12f 39569 | Basis step for constructin... |
| ax12eq 39570 | Basis step for constructin... |
| ax12el 39571 | Basis step for constructin... |
| ax12indn 39572 | Induction step for constru... |
| ax12indi 39573 | Induction step for constru... |
| ax12indalem 39574 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 39575 | Alternate proof of ~ ax12i... |
| ax12inda2 39576 | Induction step for constru... |
| ax12inda 39577 | Induction step for constru... |
| ax12v2-o 39578 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 39579 | Derive ~ ax-c15 from a hyp... |
| axc11-o 39580 | Show that ~ ax-c11 can be ... |
| fsumshftd 39581 | Index shift of a finite su... |
| riotaclbgBAD 39583 | Closure of restricted iota... |
| riotaclbBAD 39584 | Closure of restricted iota... |
| riotasvd 39585 | Deduction version of ~ rio... |
| riotasv2d 39586 | Value of description binde... |
| riotasv2s 39587 | The value of description b... |
| riotasv 39588 | Value of description binde... |
| riotasv3d 39589 | A property ` ch ` holding ... |
| elimhyps 39590 | A version of ~ elimhyp usi... |
| dedths 39591 | A version of weak deductio... |
| renegclALT 39592 | Closure law for negative o... |
| elimhyps2 39593 | Generalization of ~ elimhy... |
| dedths2 39594 | Generalization of ~ dedths... |
| nfcxfrdf 39595 | A utility lemma to transfe... |
| nfded 39596 | A deduction theorem that c... |
| nfded2 39597 | A deduction theorem that c... |
| nfunidALT2 39598 | Deduction version of ~ nfu... |
| nfunidALT 39599 | Deduction version of ~ nfu... |
| nfopdALT 39600 | Deduction version of bound... |
| cnaddcom 39601 | Recover the commutative la... |
| toycom 39602 | Show the commutative law f... |
| lshpset 39607 | The set of all hyperplanes... |
| islshp 39608 | The predicate "is a hyperp... |
| islshpsm 39609 | Hyperplane properties expr... |
| lshplss 39610 | A hyperplane is a subspace... |
| lshpne 39611 | A hyperplane is not equal ... |
| lshpnel 39612 | A hyperplane's generating ... |
| lshpnelb 39613 | The subspace sum of a hype... |
| lshpnel2N 39614 | Condition that determines ... |
| lshpne0 39615 | The member of the span in ... |
| lshpdisj 39616 | A hyperplane and the span ... |
| lshpcmp 39617 | If two hyperplanes are com... |
| lshpinN 39618 | The intersection of two di... |
| lsatset 39619 | The set of all 1-dim subsp... |
| islsat 39620 | The predicate "is a 1-dim ... |
| lsatlspsn2 39621 | The span of a nonzero sing... |
| lsatlspsn 39622 | The span of a nonzero sing... |
| islsati 39623 | A 1-dim subspace (atom) (o... |
| lsateln0 39624 | A 1-dim subspace (atom) (o... |
| lsatlss 39625 | The set of 1-dim subspaces... |
| lsatlssel 39626 | An atom is a subspace. (C... |
| lsatssv 39627 | An atom is a set of vector... |
| lsatn0 39628 | A 1-dim subspace (atom) of... |
| lsatspn0 39629 | The span of a vector is an... |
| lsator0sp 39630 | The span of a vector is ei... |
| lsatssn0 39631 | A subspace (or any class) ... |
| lsatcmp 39632 | If two atoms are comparabl... |
| lsatcmp2 39633 | If an atom is included in ... |
| lsatel 39634 | A nonzero vector in an ato... |
| lsatelbN 39635 | A nonzero vector in an ato... |
| lsat2el 39636 | Two atoms sharing a nonzer... |
| lsmsat 39637 | Convert comparison of atom... |
| lsatfixedN 39638 | Show equality with the spa... |
| lsmsatcv 39639 | Subspace sum has the cover... |
| lssatomic 39640 | The lattice of subspaces i... |
| lssats 39641 | The lattice of subspaces i... |
| lpssat 39642 | Two subspaces in a proper ... |
| lrelat 39643 | Subspaces are relatively a... |
| lssatle 39644 | The ordering of two subspa... |
| lssat 39645 | Two subspaces in a proper ... |
| islshpat 39646 | Hyperplane properties expr... |
| lcvfbr 39649 | The covers relation for a ... |
| lcvbr 39650 | The covers relation for a ... |
| lcvbr2 39651 | The covers relation for a ... |
| lcvbr3 39652 | The covers relation for a ... |
| lcvpss 39653 | The covers relation implie... |
| lcvnbtwn 39654 | The covers relation implie... |
| lcvntr 39655 | The covers relation is not... |
| lcvnbtwn2 39656 | The covers relation implie... |
| lcvnbtwn3 39657 | The covers relation implie... |
| lsmcv2 39658 | Subspace sum has the cover... |
| lcvat 39659 | If a subspace covers anoth... |
| lsatcv0 39660 | An atom covers the zero su... |
| lsatcveq0 39661 | A subspace covered by an a... |
| lsat0cv 39662 | A subspace is an atom iff ... |
| lcvexchlem1 39663 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 39664 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 39665 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 39666 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 39667 | Lemma for ~ lcvexch . (Co... |
| lcvexch 39668 | Subspaces satisfy the exch... |
| lcvp 39669 | Covering property of Defin... |
| lcv1 39670 | Covering property of a sub... |
| lcv2 39671 | Covering property of a sub... |
| lsatexch 39672 | The atom exchange property... |
| lsatnle 39673 | The meet of a subspace and... |
| lsatnem0 39674 | The meet of distinct atoms... |
| lsatexch1 39675 | The atom exch1ange propert... |
| lsatcv0eq 39676 | If the sum of two atoms co... |
| lsatcv1 39677 | Two atoms covering the zer... |
| lsatcvatlem 39678 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 39679 | A nonzero subspace less th... |
| lsatcvat2 39680 | A subspace covered by the ... |
| lsatcvat3 39681 | A condition implying that ... |
| islshpcv 39682 | Hyperplane properties expr... |
| l1cvpat 39683 | A subspace covered by the ... |
| l1cvat 39684 | Create an atom under an el... |
| lshpat 39685 | Create an atom under a hyp... |
| lflset 39688 | The set of linear function... |
| islfl 39689 | The predicate "is a linear... |
| lfli 39690 | Property of a linear funct... |
| islfld 39691 | Properties that determine ... |
| lflf 39692 | A linear functional is a f... |
| lflcl 39693 | A linear functional value ... |
| lfl0 39694 | A linear functional is zer... |
| lfladd 39695 | Property of a linear funct... |
| lflsub 39696 | Property of a linear funct... |
| lflmul 39697 | Property of a linear funct... |
| lfl0f 39698 | The zero function is a fun... |
| lfl1 39699 | A nonzero functional has a... |
| lfladdcl 39700 | Closure of addition of two... |
| lfladdcom 39701 | Commutativity of functiona... |
| lfladdass 39702 | Associativity of functiona... |
| lfladd0l 39703 | Functional addition with t... |
| lflnegcl 39704 | Closure of the negative of... |
| lflnegl 39705 | A functional plus its nega... |
| lflvscl 39706 | Closure of a scalar produc... |
| lflvsdi1 39707 | Distributive law for (righ... |
| lflvsdi2 39708 | Reverse distributive law f... |
| lflvsdi2a 39709 | Reverse distributive law f... |
| lflvsass 39710 | Associative law for (right... |
| lfl0sc 39711 | The (right vector space) s... |
| lflsc0N 39712 | The scalar product with th... |
| lfl1sc 39713 | The (right vector space) s... |
| lkrfval 39716 | The kernel of a functional... |
| lkrval 39717 | Value of the kernel of a f... |
| ellkr 39718 | Membership in the kernel o... |
| lkrval2 39719 | Value of the kernel of a f... |
| ellkr2 39720 | Membership in the kernel o... |
| lkrcl 39721 | A member of the kernel of ... |
| lkrf0 39722 | The value of a functional ... |
| lkr0f 39723 | The kernel of the zero fun... |
| lkrlss 39724 | The kernel of a linear fun... |
| lkrssv 39725 | The kernel of a linear fun... |
| lkrsc 39726 | The kernel of a nonzero sc... |
| lkrscss 39727 | The kernel of a scalar pro... |
| eqlkr 39728 | Two functionals with the s... |
| eqlkr2 39729 | Two functionals with the s... |
| eqlkr3 39730 | Two functionals with the s... |
| lkrlsp 39731 | The subspace sum of a kern... |
| lkrlsp2 39732 | The subspace sum of a kern... |
| lkrlsp3 39733 | The subspace sum of a kern... |
| lkrshp 39734 | The kernel of a nonzero fu... |
| lkrshp3 39735 | The kernels of nonzero fun... |
| lkrshpor 39736 | The kernel of a functional... |
| lkrshp4 39737 | A kernel is a hyperplane i... |
| lshpsmreu 39738 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 39739 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 39740 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 39741 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 39742 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 39743 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 39744 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 39745 | The set ` G ` defined by h... |
| lshpkr 39746 | The kernel of functional `... |
| lshpkrex 39747 | There exists a functional ... |
| lshpset2N 39748 | The set of all hyperplanes... |
| islshpkrN 39749 | The predicate "is a hyperp... |
| lfl1dim 39750 | Equivalent expressions for... |
| lfl1dim2N 39751 | Equivalent expressions for... |
| ldualset 39754 | Define the (left) dual of ... |
| ldualvbase 39755 | The vectors of a dual spac... |
| ldualelvbase 39756 | Utility theorem for conver... |
| ldualfvadd 39757 | Vector addition in the dua... |
| ldualvadd 39758 | Vector addition in the dua... |
| ldualvaddcl 39759 | The value of vector additi... |
| ldualvaddval 39760 | The value of the value of ... |
| ldualsca 39761 | The ring of scalars of the... |
| ldualsbase 39762 | Base set of scalar ring fo... |
| ldualsaddN 39763 | Scalar addition for the du... |
| ldualsmul 39764 | Scalar multiplication for ... |
| ldualfvs 39765 | Scalar product operation f... |
| ldualvs 39766 | Scalar product operation v... |
| ldualvsval 39767 | Value of scalar product op... |
| ldualvscl 39768 | The scalar product operati... |
| ldualvaddcom 39769 | Commutative law for vector... |
| ldualvsass 39770 | Associative law for scalar... |
| ldualvsass2 39771 | Associative law for scalar... |
| ldualvsdi1 39772 | Distributive law for scala... |
| ldualvsdi2 39773 | Reverse distributive law f... |
| ldualgrplem 39774 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 39775 | The dual of a vector space... |
| ldual0 39776 | The zero scalar of the dua... |
| ldual1 39777 | The unit scalar of the dua... |
| ldualneg 39778 | The negative of a scalar o... |
| ldual0v 39779 | The zero vector of the dua... |
| ldual0vcl 39780 | The dual zero vector is a ... |
| lduallmodlem 39781 | Lemma for ~ lduallmod . (... |
| lduallmod 39782 | The dual of a left module ... |
| lduallvec 39783 | The dual of a left vector ... |
| ldualvsub 39784 | The value of vector subtra... |
| ldualvsubcl 39785 | Closure of vector subtract... |
| ldualvsubval 39786 | The value of the value of ... |
| ldualssvscl 39787 | Closure of scalar product ... |
| ldualssvsubcl 39788 | Closure of vector subtract... |
| ldual0vs 39789 | Scalar zero times a functi... |
| lkr0f2 39790 | The kernel of the zero fun... |
| lduallkr3 39791 | The kernels of nonzero fun... |
| lkrpssN 39792 | Proper subset relation bet... |
| lkrin 39793 | Intersection of the kernel... |
| eqlkr4 39794 | Two functionals with the s... |
| ldual1dim 39795 | Equivalent expressions for... |
| ldualkrsc 39796 | The kernel of a nonzero sc... |
| lkrss 39797 | The kernel of a scalar pro... |
| lkrss2N 39798 | Two functionals with kerne... |
| lkreqN 39799 | Proportional functionals h... |
| lkrlspeqN 39800 | Condition for colinear fun... |
| isopos 39809 | The predicate "is an ortho... |
| opposet 39810 | Every orthoposet is a pose... |
| oposlem 39811 | Lemma for orthoposet prope... |
| op01dm 39812 | Conditions necessary for z... |
| op0cl 39813 | An orthoposet has a zero e... |
| op1cl 39814 | An orthoposet has a unity ... |
| op0le 39815 | Orthoposet zero is less th... |
| ople0 39816 | An element less than or eq... |
| opnlen0 39817 | An element not less than a... |
| lub0N 39818 | The least upper bound of t... |
| opltn0 39819 | A lattice element greater ... |
| ople1 39820 | Any element is less than t... |
| op1le 39821 | If the orthoposet unity is... |
| glb0N 39822 | The greatest lower bound o... |
| opoccl 39823 | Closure of orthocomplement... |
| opococ 39824 | Double negative law for or... |
| opcon3b 39825 | Contraposition law for ort... |
| opcon2b 39826 | Orthocomplement contraposi... |
| opcon1b 39827 | Orthocomplement contraposi... |
| oplecon3 39828 | Contraposition law for ort... |
| oplecon3b 39829 | Contraposition law for ort... |
| oplecon1b 39830 | Contraposition law for str... |
| opoc1 39831 | Orthocomplement of orthopo... |
| opoc0 39832 | Orthocomplement of orthopo... |
| opltcon3b 39833 | Contraposition law for str... |
| opltcon1b 39834 | Contraposition law for str... |
| opltcon2b 39835 | Contraposition law for str... |
| opexmid 39836 | Law of excluded middle for... |
| opnoncon 39837 | Law of contradiction for o... |
| riotaocN 39838 | The orthocomplement of the... |
| cmtfvalN 39839 | Value of commutes relation... |
| cmtvalN 39840 | Equivalence for commutes r... |
| isolat 39841 | The predicate "is an ortho... |
| ollat 39842 | An ortholattice is a latti... |
| olop 39843 | An ortholattice is an orth... |
| olposN 39844 | An ortholattice is a poset... |
| isolatiN 39845 | Properties that determine ... |
| oldmm1 39846 | De Morgan's law for meet i... |
| oldmm2 39847 | De Morgan's law for meet i... |
| oldmm3N 39848 | De Morgan's law for meet i... |
| oldmm4 39849 | De Morgan's law for meet i... |
| oldmj1 39850 | De Morgan's law for join i... |
| oldmj2 39851 | De Morgan's law for join i... |
| oldmj3 39852 | De Morgan's law for join i... |
| oldmj4 39853 | De Morgan's law for join i... |
| olj01 39854 | An ortholattice element jo... |
| olj02 39855 | An ortholattice element jo... |
| olm11 39856 | The meet of an ortholattic... |
| olm12 39857 | The meet of an ortholattic... |
| latmassOLD 39858 | Ortholattice meet is assoc... |
| latm12 39859 | A rearrangement of lattice... |
| latm32 39860 | A rearrangement of lattice... |
| latmrot 39861 | Rotate lattice meet of 3 c... |
| latm4 39862 | Rearrangement of lattice m... |
| latmmdiN 39863 | Lattice meet distributes o... |
| latmmdir 39864 | Lattice meet distributes o... |
| olm01 39865 | Meet with lattice zero is ... |
| olm02 39866 | Meet with lattice zero is ... |
| isoml 39867 | The predicate "is an ortho... |
| isomliN 39868 | Properties that determine ... |
| omlol 39869 | An orthomodular lattice is... |
| omlop 39870 | An orthomodular lattice is... |
| omllat 39871 | An orthomodular lattice is... |
| omllaw 39872 | The orthomodular law. (Co... |
| omllaw2N 39873 | Variation of orthomodular ... |
| omllaw3 39874 | Orthomodular law equivalen... |
| omllaw4 39875 | Orthomodular law equivalen... |
| omllaw5N 39876 | The orthomodular law. Rem... |
| cmtcomlemN 39877 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 39878 | Commutation is symmetric. ... |
| cmt2N 39879 | Commutation with orthocomp... |
| cmt3N 39880 | Commutation with orthocomp... |
| cmt4N 39881 | Commutation with orthocomp... |
| cmtbr2N 39882 | Alternate definition of th... |
| cmtbr3N 39883 | Alternate definition for t... |
| cmtbr4N 39884 | Alternate definition for t... |
| lecmtN 39885 | Ordered elements commute. ... |
| cmtidN 39886 | Any element commutes with ... |
| omlfh1N 39887 | Foulis-Holland Theorem, pa... |
| omlfh3N 39888 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 39889 | Analogue of modular law ~ ... |
| omlspjN 39890 | Contraction of a Sasaki pr... |
| cvrfval 39897 | Value of covers relation "... |
| cvrval 39898 | Binary relation expressing... |
| cvrlt 39899 | The covers relation implie... |
| cvrnbtwn 39900 | There is no element betwee... |
| ncvr1 39901 | No element covers the latt... |
| cvrletrN 39902 | Property of an element abo... |
| cvrval2 39903 | Binary relation expressing... |
| cvrnbtwn2 39904 | The covers relation implie... |
| cvrnbtwn3 39905 | The covers relation implie... |
| cvrcon3b 39906 | Contraposition law for the... |
| cvrle 39907 | The covers relation implie... |
| cvrnbtwn4 39908 | The covers relation implie... |
| cvrnle 39909 | The covers relation implie... |
| cvrne 39910 | The covers relation implie... |
| cvrnrefN 39911 | The covers relation is not... |
| cvrcmp 39912 | If two lattice elements th... |
| cvrcmp2 39913 | If two lattice elements co... |
| pats 39914 | The set of atoms in a pose... |
| isat 39915 | The predicate "is an atom"... |
| isat2 39916 | The predicate "is an atom"... |
| atcvr0 39917 | An atom covers zero. ( ~ ... |
| atbase 39918 | An atom is a member of the... |
| atssbase 39919 | The set of atoms is a subs... |
| 0ltat 39920 | An atom is greater than ze... |
| leatb 39921 | A poset element less than ... |
| leat 39922 | A poset element less than ... |
| leat2 39923 | A nonzero poset element le... |
| leat3 39924 | A poset element less than ... |
| meetat 39925 | The meet of any element wi... |
| meetat2 39926 | The meet of any element wi... |
| isatl 39928 | The predicate "is an atomi... |
| atllat 39929 | An atomic lattice is a lat... |
| atlpos 39930 | An atomic lattice is a pos... |
| atl0dm 39931 | Condition necessary for ze... |
| atl0cl 39932 | An atomic lattice has a ze... |
| atl0le 39933 | Orthoposet zero is less th... |
| atlle0 39934 | An element less than or eq... |
| atlltn0 39935 | A lattice element greater ... |
| isat3 39936 | The predicate "is an atom"... |
| atn0 39937 | An atom is not zero. ( ~ ... |
| atnle0 39938 | An atom is not less than o... |
| atlen0 39939 | A lattice element is nonze... |
| atcmp 39940 | If two atoms are comparabl... |
| atncmp 39941 | Frequently-used variation ... |
| atnlt 39942 | Two atoms cannot satisfy t... |
| atcvreq0 39943 | An element covered by an a... |
| atncvrN 39944 | Two atoms cannot satisfy t... |
| atlex 39945 | Every nonzero element of a... |
| atnle 39946 | Two ways of expressing "an... |
| atnem0 39947 | The meet of distinct atoms... |
| atlatmstc 39948 | An atomic, complete, ortho... |
| atlatle 39949 | The ordering of two Hilber... |
| atlrelat1 39950 | An atomistic lattice with ... |
| iscvlat 39952 | The predicate "is an atomi... |
| iscvlat2N 39953 | The predicate "is an atomi... |
| cvlatl 39954 | An atomic lattice with the... |
| cvllat 39955 | An atomic lattice with the... |
| cvlposN 39956 | An atomic lattice with the... |
| cvlexch1 39957 | An atomic covering lattice... |
| cvlexch2 39958 | An atomic covering lattice... |
| cvlexchb1 39959 | An atomic covering lattice... |
| cvlexchb2 39960 | An atomic covering lattice... |
| cvlexch3 39961 | An atomic covering lattice... |
| cvlexch4N 39962 | An atomic covering lattice... |
| cvlatexchb1 39963 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 39964 | A version of ~ cvlexchb2 f... |
| cvlatexch1 39965 | Atom exchange property. (... |
| cvlatexch2 39966 | Atom exchange property. (... |
| cvlatexch3 39967 | Atom exchange property. (... |
| cvlcvr1 39968 | The covering property. Pr... |
| cvlcvrp 39969 | A Hilbert lattice satisfie... |
| cvlatcvr1 39970 | An atom is covered by its ... |
| cvlatcvr2 39971 | An atom is covered by its ... |
| cvlsupr2 39972 | Two equivalent ways of exp... |
| cvlsupr3 39973 | Two equivalent ways of exp... |
| cvlsupr4 39974 | Consequence of superpositi... |
| cvlsupr5 39975 | Consequence of superpositi... |
| cvlsupr6 39976 | Consequence of superpositi... |
| cvlsupr7 39977 | Consequence of superpositi... |
| cvlsupr8 39978 | Consequence of superpositi... |
| ishlat1 39981 | The predicate "is a Hilber... |
| ishlat2 39982 | The predicate "is a Hilber... |
| ishlat3N 39983 | The predicate "is a Hilber... |
| ishlatiN 39984 | Properties that determine ... |
| hlomcmcv 39985 | A Hilbert lattice is ortho... |
| hloml 39986 | A Hilbert lattice is ortho... |
| hlclat 39987 | A Hilbert lattice is compl... |
| hlcvl 39988 | A Hilbert lattice is an at... |
| hlatl 39989 | A Hilbert lattice is atomi... |
| hlol 39990 | A Hilbert lattice is an or... |
| hlop 39991 | A Hilbert lattice is an or... |
| hllat 39992 | A Hilbert lattice is a lat... |
| hllatd 39993 | Deduction form of ~ hllat ... |
| hlomcmat 39994 | A Hilbert lattice is ortho... |
| hlpos 39995 | A Hilbert lattice is a pos... |
| hlatjcl 39996 | Closure of join operation.... |
| hlatjcom 39997 | Commutatitivity of join op... |
| hlatjidm 39998 | Idempotence of join operat... |
| hlatjass 39999 | Lattice join is associativ... |
| hlatj12 40000 | Swap 1st and 2nd members o... |
| hlatj32 40001 | Swap 2nd and 3rd members o... |
| hlatjrot 40002 | Rotate lattice join of 3 c... |
| hlatj4 40003 | Rearrangement of lattice j... |
| hlatlej1 40004 | A join's first argument is... |
| hlatlej2 40005 | A join's second argument i... |
| glbconN 40006 | De Morgan's law for GLB an... |
| glbconxN 40007 | De Morgan's law for GLB an... |
| atnlej1 40008 | If an atom is not less tha... |
| atnlej2 40009 | If an atom is not less tha... |
| hlsuprexch 40010 | A Hilbert lattice has the ... |
| hlexch1 40011 | A Hilbert lattice has the ... |
| hlexch2 40012 | A Hilbert lattice has the ... |
| hlexchb1 40013 | A Hilbert lattice has the ... |
| hlexchb2 40014 | A Hilbert lattice has the ... |
| hlsupr 40015 | A Hilbert lattice has the ... |
| hlsupr2 40016 | A Hilbert lattice has the ... |
| hlhgt4 40017 | A Hilbert lattice has a he... |
| hlhgt2 40018 | A Hilbert lattice has a he... |
| hl0lt1N 40019 | Lattice 0 is less than lat... |
| hlexch3 40020 | A Hilbert lattice has the ... |
| hlexch4N 40021 | A Hilbert lattice has the ... |
| hlatexchb1 40022 | A version of ~ hlexchb1 fo... |
| hlatexchb2 40023 | A version of ~ hlexchb2 fo... |
| hlatexch1 40024 | Atom exchange property. (... |
| hlatexch2 40025 | Atom exchange property. (... |
| hlatmstcOLDN 40026 | An atomic, complete, ortho... |
| hlatle 40027 | The ordering of two Hilber... |
| hlateq 40028 | The equality of two Hilber... |
| hlrelat1 40029 | An atomistic lattice with ... |
| hlrelat5N 40030 | An atomistic lattice with ... |
| hlrelat 40031 | A Hilbert lattice is relat... |
| hlrelat2 40032 | A consequence of relative ... |
| exatleN 40033 | A condition for an atom to... |
| hl2at 40034 | A Hilbert lattice has at l... |
| atex 40035 | At least one atom exists. ... |
| intnatN 40036 | If the intersection with a... |
| 2llnne2N 40037 | Condition implying that tw... |
| 2llnneN 40038 | Condition implying that tw... |
| cvr1 40039 | A Hilbert lattice has the ... |
| cvr2N 40040 | Less-than and covers equiv... |
| hlrelat3 40041 | The Hilbert lattice is rel... |
| cvrval3 40042 | Binary relation expressing... |
| cvrval4N 40043 | Binary relation expressing... |
| cvrval5 40044 | Binary relation expressing... |
| cvrp 40045 | A Hilbert lattice satisfie... |
| atcvr1 40046 | An atom is covered by its ... |
| atcvr2 40047 | An atom is covered by its ... |
| cvrexchlem 40048 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 40049 | A Hilbert lattice satisfie... |
| cvratlem 40050 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 40051 | A nonzero Hilbert lattice ... |
| ltltncvr 40052 | A chained strong ordering ... |
| ltcvrntr 40053 | Non-transitive condition f... |
| cvrntr 40054 | The covers relation is not... |
| atcvr0eq 40055 | The covers relation is not... |
| lnnat 40056 | A line (the join of two di... |
| atcvrj0 40057 | Two atoms covering the zer... |
| cvrat2 40058 | A Hilbert lattice element ... |
| atcvrneN 40059 | Inequality derived from at... |
| atcvrj1 40060 | Condition for an atom to b... |
| atcvrj2b 40061 | Condition for an atom to b... |
| atcvrj2 40062 | Condition for an atom to b... |
| atleneN 40063 | Inequality derived from at... |
| atltcvr 40064 | An equivalence of less-tha... |
| atle 40065 | Any nonzero element has an... |
| atlt 40066 | Two atoms are unequal iff ... |
| atlelt 40067 | Transfer less-than relatio... |
| 2atlt 40068 | Given an atom less than an... |
| atexchcvrN 40069 | Atom exchange property. V... |
| atexchltN 40070 | Atom exchange property. V... |
| cvrat3 40071 | A condition implying that ... |
| cvrat4 40072 | A condition implying exist... |
| cvrat42 40073 | Commuted version of ~ cvra... |
| 2atjm 40074 | The meet of a line (expres... |
| atbtwn 40075 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 40076 | There exists a 3rd atom ` ... |
| atbtwnex 40077 | Given atoms ` P ` in ` X `... |
| 3noncolr2 40078 | Two ways to express 3 non-... |
| 3noncolr1N 40079 | Two ways to express 3 non-... |
| hlatcon3 40080 | Atom exchange combined wit... |
| hlatcon2 40081 | Atom exchange combined wit... |
| 4noncolr3 40082 | A way to express 4 non-col... |
| 4noncolr2 40083 | A way to express 4 non-col... |
| 4noncolr1 40084 | A way to express 4 non-col... |
| athgt 40085 | A Hilbert lattice, whose h... |
| 3dim0 40086 | There exists a 3-dimension... |
| 3dimlem1 40087 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 40088 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 40089 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 40090 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 40091 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 40092 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 40093 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 40094 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 40095 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 40096 | Construct a 3-dimensional ... |
| 3dim2 40097 | Construct 2 new layers on ... |
| 3dim3 40098 | Construct a new layer on t... |
| 2dim 40099 | Generate a height-3 elemen... |
| 1dimN 40100 | An atom is covered by a he... |
| 1cvrco 40101 | The orthocomplement of an ... |
| 1cvratex 40102 | There exists an atom less ... |
| 1cvratlt 40103 | An atom less than or equal... |
| 1cvrjat 40104 | An element covered by the ... |
| 1cvrat 40105 | Create an atom under an el... |
| ps-1 40106 | The join of two atoms ` R ... |
| ps-2 40107 | Lattice analogue for the p... |
| 2atjlej 40108 | Two atoms are different if... |
| hlatexch3N 40109 | Rearrange join of atoms in... |
| hlatexch4 40110 | Exchange 2 atoms. (Contri... |
| ps-2b 40111 | Variation of projective ge... |
| 3atlem1 40112 | Lemma for ~ 3at . (Contri... |
| 3atlem2 40113 | Lemma for ~ 3at . (Contri... |
| 3atlem3 40114 | Lemma for ~ 3at . (Contri... |
| 3atlem4 40115 | Lemma for ~ 3at . (Contri... |
| 3atlem5 40116 | Lemma for ~ 3at . (Contri... |
| 3atlem6 40117 | Lemma for ~ 3at . (Contri... |
| 3atlem7 40118 | Lemma for ~ 3at . (Contri... |
| 3at 40119 | Any three non-colinear ato... |
| llnset 40134 | The set of lattice lines i... |
| islln 40135 | The predicate "is a lattic... |
| islln4 40136 | The predicate "is a lattic... |
| llni 40137 | Condition implying a latti... |
| llnbase 40138 | A lattice line is a lattic... |
| islln3 40139 | The predicate "is a lattic... |
| islln2 40140 | The predicate "is a lattic... |
| llni2 40141 | The join of two different ... |
| llnnleat 40142 | An atom cannot majorize a ... |
| llnneat 40143 | A lattice line is not an a... |
| 2atneat 40144 | The join of two distinct a... |
| llnn0 40145 | A lattice line is nonzero.... |
| islln2a 40146 | The predicate "is a lattic... |
| llnle 40147 | Any element greater than 0... |
| atcvrlln2 40148 | An atom under a line is co... |
| atcvrlln 40149 | An element covering an ato... |
| llnexatN 40150 | Given an atom on a line, t... |
| llncmp 40151 | If two lattice lines are c... |
| llnnlt 40152 | Two lattice lines cannot s... |
| 2llnmat 40153 | Two intersecting lines int... |
| 2at0mat0 40154 | Special case of ~ 2atmat0 ... |
| 2atmat0 40155 | The meet of two unequal li... |
| 2atm 40156 | An atom majorized by two d... |
| ps-2c 40157 | Variation of projective ge... |
| lplnset 40158 | The set of lattice planes ... |
| islpln 40159 | The predicate "is a lattic... |
| islpln4 40160 | The predicate "is a lattic... |
| lplni 40161 | Condition implying a latti... |
| islpln3 40162 | The predicate "is a lattic... |
| lplnbase 40163 | A lattice plane is a latti... |
| islpln5 40164 | The predicate "is a lattic... |
| islpln2 40165 | The predicate "is a lattic... |
| lplni2 40166 | The join of 3 different at... |
| lvolex3N 40167 | There is an atom outside o... |
| llnmlplnN 40168 | The intersection of a line... |
| lplnle 40169 | Any element greater than 0... |
| lplnnle2at 40170 | A lattice line (or atom) c... |
| lplnnleat 40171 | A lattice plane cannot maj... |
| lplnnlelln 40172 | A lattice plane is not les... |
| 2atnelpln 40173 | The join of two atoms is n... |
| lplnneat 40174 | No lattice plane is an ato... |
| lplnnelln 40175 | No lattice plane is a latt... |
| lplnn0N 40176 | A lattice plane is nonzero... |
| islpln2a 40177 | The predicate "is a lattic... |
| islpln2ah 40178 | The predicate "is a lattic... |
| lplnriaN 40179 | Property of a lattice plan... |
| lplnribN 40180 | Property of a lattice plan... |
| lplnric 40181 | Property of a lattice plan... |
| lplnri1 40182 | Property of a lattice plan... |
| lplnri2N 40183 | Property of a lattice plan... |
| lplnri3N 40184 | Property of a lattice plan... |
| lplnllnneN 40185 | Two lattice lines defined ... |
| llncvrlpln2 40186 | A lattice line under a lat... |
| llncvrlpln 40187 | An element covering a latt... |
| 2lplnmN 40188 | If the join of two lattice... |
| 2llnmj 40189 | The meet of two lattice li... |
| 2atmat 40190 | The meet of two intersecti... |
| lplncmp 40191 | If two lattice planes are ... |
| lplnexatN 40192 | Given a lattice line on a ... |
| lplnexllnN 40193 | Given an atom on a lattice... |
| lplnnlt 40194 | Two lattice planes cannot ... |
| 2llnjaN 40195 | The join of two different ... |
| 2llnjN 40196 | The join of two different ... |
| 2llnm2N 40197 | The meet of two different ... |
| 2llnm3N 40198 | Two lattice lines in a lat... |
| 2llnm4 40199 | Two lattice lines that maj... |
| 2llnmeqat 40200 | An atom equals the interse... |
| lvolset 40201 | The set of 3-dim lattice v... |
| islvol 40202 | The predicate "is a 3-dim ... |
| islvol4 40203 | The predicate "is a 3-dim ... |
| lvoli 40204 | Condition implying a 3-dim... |
| islvol3 40205 | The predicate "is a 3-dim ... |
| lvoli3 40206 | Condition implying a 3-dim... |
| lvolbase 40207 | A 3-dim lattice volume is ... |
| islvol5 40208 | The predicate "is a 3-dim ... |
| islvol2 40209 | The predicate "is a 3-dim ... |
| lvoli2 40210 | The join of 4 different at... |
| lvolnle3at 40211 | A lattice plane (or lattic... |
| lvolnleat 40212 | An atom cannot majorize a ... |
| lvolnlelln 40213 | A lattice line cannot majo... |
| lvolnlelpln 40214 | A lattice plane cannot maj... |
| 3atnelvolN 40215 | The join of 3 atoms is not... |
| 2atnelvolN 40216 | The join of two atoms is n... |
| lvolneatN 40217 | No lattice volume is an at... |
| lvolnelln 40218 | No lattice volume is a lat... |
| lvolnelpln 40219 | No lattice volume is a lat... |
| lvoln0N 40220 | A lattice volume is nonzer... |
| islvol2aN 40221 | The predicate "is a lattic... |
| 4atlem0a 40222 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 40223 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 40224 | Lemma for ~ 4at . (Contri... |
| 4atlem3 40225 | Lemma for ~ 4at . Break i... |
| 4atlem3a 40226 | Lemma for ~ 4at . Break i... |
| 4atlem3b 40227 | Lemma for ~ 4at . Break i... |
| 4atlem4a 40228 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 40229 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 40230 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 40231 | Lemma for ~ 4at . Frequen... |
| 4atlem9 40232 | Lemma for ~ 4at . Substit... |
| 4atlem10a 40233 | Lemma for ~ 4at . Substit... |
| 4atlem10b 40234 | Lemma for ~ 4at . Substit... |
| 4atlem10 40235 | Lemma for ~ 4at . Combine... |
| 4atlem11a 40236 | Lemma for ~ 4at . Substit... |
| 4atlem11b 40237 | Lemma for ~ 4at . Substit... |
| 4atlem11 40238 | Lemma for ~ 4at . Combine... |
| 4atlem12a 40239 | Lemma for ~ 4at . Substit... |
| 4atlem12b 40240 | Lemma for ~ 4at . Substit... |
| 4atlem12 40241 | Lemma for ~ 4at . Combine... |
| 4at 40242 | Four atoms determine a lat... |
| 4at2 40243 | Four atoms determine a lat... |
| lplncvrlvol2 40244 | A lattice line under a lat... |
| lplncvrlvol 40245 | An element covering a latt... |
| lvolcmp 40246 | If two lattice planes are ... |
| lvolnltN 40247 | Two lattice volumes cannot... |
| 2lplnja 40248 | The join of two different ... |
| 2lplnj 40249 | The join of two different ... |
| 2lplnm2N 40250 | The meet of two different ... |
| 2lplnmj 40251 | The meet of two lattice pl... |
| dalemkehl 40252 | Lemma for ~ dath . Freque... |
| dalemkelat 40253 | Lemma for ~ dath . Freque... |
| dalemkeop 40254 | Lemma for ~ dath . Freque... |
| dalempea 40255 | Lemma for ~ dath . Freque... |
| dalemqea 40256 | Lemma for ~ dath . Freque... |
| dalemrea 40257 | Lemma for ~ dath . Freque... |
| dalemsea 40258 | Lemma for ~ dath . Freque... |
| dalemtea 40259 | Lemma for ~ dath . Freque... |
| dalemuea 40260 | Lemma for ~ dath . Freque... |
| dalemyeo 40261 | Lemma for ~ dath . Freque... |
| dalemzeo 40262 | Lemma for ~ dath . Freque... |
| dalemclpjs 40263 | Lemma for ~ dath . Freque... |
| dalemclqjt 40264 | Lemma for ~ dath . Freque... |
| dalemclrju 40265 | Lemma for ~ dath . Freque... |
| dalem-clpjq 40266 | Lemma for ~ dath . Freque... |
| dalemceb 40267 | Lemma for ~ dath . Freque... |
| dalempeb 40268 | Lemma for ~ dath . Freque... |
| dalemqeb 40269 | Lemma for ~ dath . Freque... |
| dalemreb 40270 | Lemma for ~ dath . Freque... |
| dalemseb 40271 | Lemma for ~ dath . Freque... |
| dalemteb 40272 | Lemma for ~ dath . Freque... |
| dalemueb 40273 | Lemma for ~ dath . Freque... |
| dalempjqeb 40274 | Lemma for ~ dath . Freque... |
| dalemsjteb 40275 | Lemma for ~ dath . Freque... |
| dalemtjueb 40276 | Lemma for ~ dath . Freque... |
| dalemqrprot 40277 | Lemma for ~ dath . Freque... |
| dalemyeb 40278 | Lemma for ~ dath . Freque... |
| dalemcnes 40279 | Lemma for ~ dath . Freque... |
| dalempnes 40280 | Lemma for ~ dath . Freque... |
| dalemqnet 40281 | Lemma for ~ dath . Freque... |
| dalempjsen 40282 | Lemma for ~ dath . Freque... |
| dalemply 40283 | Lemma for ~ dath . Freque... |
| dalemsly 40284 | Lemma for ~ dath . Freque... |
| dalemswapyz 40285 | Lemma for ~ dath . Swap t... |
| dalemrot 40286 | Lemma for ~ dath . Rotate... |
| dalemrotyz 40287 | Lemma for ~ dath . Rotate... |
| dalem1 40288 | Lemma for ~ dath . Show t... |
| dalemcea 40289 | Lemma for ~ dath . Freque... |
| dalem2 40290 | Lemma for ~ dath . Show t... |
| dalemdea 40291 | Lemma for ~ dath . Freque... |
| dalemeea 40292 | Lemma for ~ dath . Freque... |
| dalem3 40293 | Lemma for ~ dalemdnee . (... |
| dalem4 40294 | Lemma for ~ dalemdnee . (... |
| dalemdnee 40295 | Lemma for ~ dath . Axis o... |
| dalem5 40296 | Lemma for ~ dath . Atom `... |
| dalem6 40297 | Lemma for ~ dath . Analog... |
| dalem7 40298 | Lemma for ~ dath . Analog... |
| dalem8 40299 | Lemma for ~ dath . Plane ... |
| dalem-cly 40300 | Lemma for ~ dalem9 . Cent... |
| dalem9 40301 | Lemma for ~ dath . Since ... |
| dalem10 40302 | Lemma for ~ dath . Atom `... |
| dalem11 40303 | Lemma for ~ dath . Analog... |
| dalem12 40304 | Lemma for ~ dath . Analog... |
| dalem13 40305 | Lemma for ~ dalem14 . (Co... |
| dalem14 40306 | Lemma for ~ dath . Planes... |
| dalem15 40307 | Lemma for ~ dath . The ax... |
| dalem16 40308 | Lemma for ~ dath . The at... |
| dalem17 40309 | Lemma for ~ dath . When p... |
| dalem18 40310 | Lemma for ~ dath . Show t... |
| dalem19 40311 | Lemma for ~ dath . Show t... |
| dalemccea 40312 | Lemma for ~ dath . Freque... |
| dalemddea 40313 | Lemma for ~ dath . Freque... |
| dalem-ccly 40314 | Lemma for ~ dath . Freque... |
| dalem-ddly 40315 | Lemma for ~ dath . Freque... |
| dalemccnedd 40316 | Lemma for ~ dath . Freque... |
| dalemclccjdd 40317 | Lemma for ~ dath . Freque... |
| dalemcceb 40318 | Lemma for ~ dath . Freque... |
| dalemswapyzps 40319 | Lemma for ~ dath . Swap t... |
| dalemrotps 40320 | Lemma for ~ dath . Rotate... |
| dalemcjden 40321 | Lemma for ~ dath . Show t... |
| dalem20 40322 | Lemma for ~ dath . Show t... |
| dalem21 40323 | Lemma for ~ dath . Show t... |
| dalem22 40324 | Lemma for ~ dath . Show t... |
| dalem23 40325 | Lemma for ~ dath . Show t... |
| dalem24 40326 | Lemma for ~ dath . Show t... |
| dalem25 40327 | Lemma for ~ dath . Show t... |
| dalem27 40328 | Lemma for ~ dath . Show t... |
| dalem28 40329 | Lemma for ~ dath . Lemma ... |
| dalem29 40330 | Lemma for ~ dath . Analog... |
| dalem30 40331 | Lemma for ~ dath . Analog... |
| dalem31N 40332 | Lemma for ~ dath . Analog... |
| dalem32 40333 | Lemma for ~ dath . Analog... |
| dalem33 40334 | Lemma for ~ dath . Analog... |
| dalem34 40335 | Lemma for ~ dath . Analog... |
| dalem35 40336 | Lemma for ~ dath . Analog... |
| dalem36 40337 | Lemma for ~ dath . Analog... |
| dalem37 40338 | Lemma for ~ dath . Analog... |
| dalem38 40339 | Lemma for ~ dath . Plane ... |
| dalem39 40340 | Lemma for ~ dath . Auxili... |
| dalem40 40341 | Lemma for ~ dath . Analog... |
| dalem41 40342 | Lemma for ~ dath . (Contr... |
| dalem42 40343 | Lemma for ~ dath . Auxili... |
| dalem43 40344 | Lemma for ~ dath . Planes... |
| dalem44 40345 | Lemma for ~ dath . Dummy ... |
| dalem45 40346 | Lemma for ~ dath . Dummy ... |
| dalem46 40347 | Lemma for ~ dath . Analog... |
| dalem47 40348 | Lemma for ~ dath . Analog... |
| dalem48 40349 | Lemma for ~ dath . Analog... |
| dalem49 40350 | Lemma for ~ dath . Analog... |
| dalem50 40351 | Lemma for ~ dath . Analog... |
| dalem51 40352 | Lemma for ~ dath . Constr... |
| dalem52 40353 | Lemma for ~ dath . Lines ... |
| dalem53 40354 | Lemma for ~ dath . The au... |
| dalem54 40355 | Lemma for ~ dath . Line `... |
| dalem55 40356 | Lemma for ~ dath . Lines ... |
| dalem56 40357 | Lemma for ~ dath . Analog... |
| dalem57 40358 | Lemma for ~ dath . Axis o... |
| dalem58 40359 | Lemma for ~ dath . Analog... |
| dalem59 40360 | Lemma for ~ dath . Analog... |
| dalem60 40361 | Lemma for ~ dath . ` B ` i... |
| dalem61 40362 | Lemma for ~ dath . Show t... |
| dalem62 40363 | Lemma for ~ dath . Elimin... |
| dalem63 40364 | Lemma for ~ dath . Combin... |
| dath 40365 | Desargues's theorem of pro... |
| dath2 40366 | Version of Desargues's the... |
| lineset 40367 | The set of lines in a Hilb... |
| isline 40368 | The predicate "is a line".... |
| islinei 40369 | Condition implying "is a l... |
| pointsetN 40370 | The set of points in a Hil... |
| ispointN 40371 | The predicate "is a point"... |
| atpointN 40372 | The singleton of an atom i... |
| psubspset 40373 | The set of projective subs... |
| ispsubsp 40374 | The predicate "is a projec... |
| ispsubsp2 40375 | The predicate "is a projec... |
| psubspi 40376 | Property of a projective s... |
| psubspi2N 40377 | Property of a projective s... |
| 0psubN 40378 | The empty set is a project... |
| snatpsubN 40379 | The singleton of an atom i... |
| pointpsubN 40380 | A point (singleton of an a... |
| linepsubN 40381 | A line is a projective sub... |
| atpsubN 40382 | The set of all atoms is a ... |
| psubssat 40383 | A projective subspace cons... |
| psubatN 40384 | A member of a projective s... |
| pmapfval 40385 | The projective map of a Hi... |
| pmapval 40386 | Value of the projective ma... |
| elpmap 40387 | Member of a projective map... |
| pmapssat 40388 | The projective map of a Hi... |
| pmapssbaN 40389 | A weakening of ~ pmapssat ... |
| pmaple 40390 | The projective map of a Hi... |
| pmap11 40391 | The projective map of a Hi... |
| pmapat 40392 | The projective map of an a... |
| elpmapat 40393 | Member of the projective m... |
| pmap0 40394 | Value of the projective ma... |
| pmapeq0 40395 | A projective map value is ... |
| pmap1N 40396 | Value of the projective ma... |
| pmapsub 40397 | The projective map of a Hi... |
| pmapglbx 40398 | The projective map of the ... |
| pmapglb 40399 | The projective map of the ... |
| pmapglb2N 40400 | The projective map of the ... |
| pmapglb2xN 40401 | The projective map of the ... |
| pmapmeet 40402 | The projective map of a me... |
| isline2 40403 | Definition of line in term... |
| linepmap 40404 | A line described with a pr... |
| isline3 40405 | Definition of line in term... |
| isline4N 40406 | Definition of line in term... |
| lneq2at 40407 | A line equals the join of ... |
| lnatexN 40408 | There is an atom in a line... |
| lnjatN 40409 | Given an atom in a line, t... |
| lncvrelatN 40410 | A lattice element covered ... |
| lncvrat 40411 | A line covers the atoms it... |
| lncmp 40412 | If two lines are comparabl... |
| 2lnat 40413 | Two intersecting lines int... |
| 2atm2atN 40414 | Two joins with a common at... |
| 2llnma1b 40415 | Generalization of ~ 2llnma... |
| 2llnma1 40416 | Two different intersecting... |
| 2llnma3r 40417 | Two different intersecting... |
| 2llnma2 40418 | Two different intersecting... |
| 2llnma2rN 40419 | Two different intersecting... |
| cdlema1N 40420 | A condition for required f... |
| cdlema2N 40421 | A condition for required f... |
| cdlemblem 40422 | Lemma for ~ cdlemb . (Con... |
| cdlemb 40423 | Given two atoms not less t... |
| paddfval 40426 | Projective subspace sum op... |
| paddval 40427 | Projective subspace sum op... |
| elpadd 40428 | Member of a projective sub... |
| elpaddn0 40429 | Member of projective subsp... |
| paddvaln0N 40430 | Projective subspace sum op... |
| elpaddri 40431 | Condition implying members... |
| elpaddatriN 40432 | Condition implying members... |
| elpaddat 40433 | Membership in a projective... |
| elpaddatiN 40434 | Consequence of membership ... |
| elpadd2at 40435 | Membership in a projective... |
| elpadd2at2 40436 | Membership in a projective... |
| paddunssN 40437 | Projective subspace sum in... |
| elpadd0 40438 | Member of projective subsp... |
| paddval0 40439 | Projective subspace sum wi... |
| padd01 40440 | Projective subspace sum wi... |
| padd02 40441 | Projective subspace sum wi... |
| paddcom 40442 | Projective subspace sum co... |
| paddssat 40443 | A projective subspace sum ... |
| sspadd1 40444 | A projective subspace sum ... |
| sspadd2 40445 | A projective subspace sum ... |
| paddss1 40446 | Subset law for projective ... |
| paddss2 40447 | Subset law for projective ... |
| paddss12 40448 | Subset law for projective ... |
| paddasslem1 40449 | Lemma for ~ paddass . (Co... |
| paddasslem2 40450 | Lemma for ~ paddass . (Co... |
| paddasslem3 40451 | Lemma for ~ paddass . Res... |
| paddasslem4 40452 | Lemma for ~ paddass . Com... |
| paddasslem5 40453 | Lemma for ~ paddass . Sho... |
| paddasslem6 40454 | Lemma for ~ paddass . (Co... |
| paddasslem7 40455 | Lemma for ~ paddass . Com... |
| paddasslem8 40456 | Lemma for ~ paddass . (Co... |
| paddasslem9 40457 | Lemma for ~ paddass . Com... |
| paddasslem10 40458 | Lemma for ~ paddass . Use... |
| paddasslem11 40459 | Lemma for ~ paddass . The... |
| paddasslem12 40460 | Lemma for ~ paddass . The... |
| paddasslem13 40461 | Lemma for ~ paddass . The... |
| paddasslem14 40462 | Lemma for ~ paddass . Rem... |
| paddasslem15 40463 | Lemma for ~ paddass . Use... |
| paddasslem16 40464 | Lemma for ~ paddass . Use... |
| paddasslem17 40465 | Lemma for ~ paddass . The... |
| paddasslem18 40466 | Lemma for ~ paddass . Com... |
| paddass 40467 | Projective subspace sum is... |
| padd12N 40468 | Commutative/associative la... |
| padd4N 40469 | Rearrangement of 4 terms i... |
| paddidm 40470 | Projective subspace sum is... |
| paddclN 40471 | The projective sum of two ... |
| paddssw1 40472 | Subset law for projective ... |
| paddssw2 40473 | Subset law for projective ... |
| paddss 40474 | Subset law for projective ... |
| pmodlem1 40475 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 40476 | Lemma for ~ pmod1i . (Con... |
| pmod1i 40477 | The modular law holds in a... |
| pmod2iN 40478 | Dual of the modular law. ... |
| pmodN 40479 | The modular law for projec... |
| pmodl42N 40480 | Lemma derived from modular... |
| pmapjoin 40481 | The projective map of the ... |
| pmapjat1 40482 | The projective map of the ... |
| pmapjat2 40483 | The projective map of the ... |
| pmapjlln1 40484 | The projective map of the ... |
| hlmod1i 40485 | A version of the modular l... |
| atmod1i1 40486 | Version of modular law ~ p... |
| atmod1i1m 40487 | Version of modular law ~ p... |
| atmod1i2 40488 | Version of modular law ~ p... |
| llnmod1i2 40489 | Version of modular law ~ p... |
| atmod2i1 40490 | Version of modular law ~ p... |
| atmod2i2 40491 | Version of modular law ~ p... |
| llnmod2i2 40492 | Version of modular law ~ p... |
| atmod3i1 40493 | Version of modular law tha... |
| atmod3i2 40494 | Version of modular law tha... |
| atmod4i1 40495 | Version of modular law tha... |
| atmod4i2 40496 | Version of modular law tha... |
| llnexchb2lem 40497 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 40498 | Line exchange property (co... |
| llnexch2N 40499 | Line exchange property (co... |
| dalawlem1 40500 | Lemma for ~ dalaw . Speci... |
| dalawlem2 40501 | Lemma for ~ dalaw . Utili... |
| dalawlem3 40502 | Lemma for ~ dalaw . First... |
| dalawlem4 40503 | Lemma for ~ dalaw . Secon... |
| dalawlem5 40504 | Lemma for ~ dalaw . Speci... |
| dalawlem6 40505 | Lemma for ~ dalaw . First... |
| dalawlem7 40506 | Lemma for ~ dalaw . Secon... |
| dalawlem8 40507 | Lemma for ~ dalaw . Speci... |
| dalawlem9 40508 | Lemma for ~ dalaw . Speci... |
| dalawlem10 40509 | Lemma for ~ dalaw . Combi... |
| dalawlem11 40510 | Lemma for ~ dalaw . First... |
| dalawlem12 40511 | Lemma for ~ dalaw . Secon... |
| dalawlem13 40512 | Lemma for ~ dalaw . Speci... |
| dalawlem14 40513 | Lemma for ~ dalaw . Combi... |
| dalawlem15 40514 | Lemma for ~ dalaw . Swap ... |
| dalaw 40515 | Desargues's law, derived f... |
| pclfvalN 40518 | The projective subspace cl... |
| pclvalN 40519 | Value of the projective su... |
| pclclN 40520 | Closure of the projective ... |
| elpclN 40521 | Membership in the projecti... |
| elpcliN 40522 | Implication of membership ... |
| pclssN 40523 | Ordering is preserved by s... |
| pclssidN 40524 | A set of atoms is included... |
| pclidN 40525 | The projective subspace cl... |
| pclbtwnN 40526 | A projective subspace sand... |
| pclunN 40527 | The projective subspace cl... |
| pclun2N 40528 | The projective subspace cl... |
| pclfinN 40529 | The projective subspace cl... |
| pclcmpatN 40530 | The set of projective subs... |
| polfvalN 40533 | The projective subspace po... |
| polvalN 40534 | Value of the projective su... |
| polval2N 40535 | Alternate expression for v... |
| polsubN 40536 | The polarity of a set of a... |
| polssatN 40537 | The polarity of a set of a... |
| pol0N 40538 | The polarity of the empty ... |
| pol1N 40539 | The polarity of the whole ... |
| 2pol0N 40540 | The closed subspace closur... |
| polpmapN 40541 | The polarity of a projecti... |
| 2polpmapN 40542 | Double polarity of a proje... |
| 2polvalN 40543 | Value of double polarity. ... |
| 2polssN 40544 | A set of atoms is a subset... |
| 3polN 40545 | Triple polarity cancels to... |
| polcon3N 40546 | Contraposition law for pol... |
| 2polcon4bN 40547 | Contraposition law for pol... |
| polcon2N 40548 | Contraposition law for pol... |
| polcon2bN 40549 | Contraposition law for pol... |
| pclss2polN 40550 | The projective subspace cl... |
| pcl0N 40551 | The projective subspace cl... |
| pcl0bN 40552 | The projective subspace cl... |
| pmaplubN 40553 | The LUB of a projective ma... |
| sspmaplubN 40554 | A set of atoms is a subset... |
| 2pmaplubN 40555 | Double projective map of a... |
| paddunN 40556 | The closure of the project... |
| poldmj1N 40557 | De Morgan's law for polari... |
| pmapj2N 40558 | The projective map of the ... |
| pmapocjN 40559 | The projective map of the ... |
| polatN 40560 | The polarity of the single... |
| 2polatN 40561 | Double polarity of the sin... |
| pnonsingN 40562 | The intersection of a set ... |
| psubclsetN 40565 | The set of closed projecti... |
| ispsubclN 40566 | The predicate "is a closed... |
| psubcliN 40567 | Property of a closed proje... |
| psubcli2N 40568 | Property of a closed proje... |
| psubclsubN 40569 | A closed projective subspa... |
| psubclssatN 40570 | A closed projective subspa... |
| pmapidclN 40571 | Projective map of the LUB ... |
| 0psubclN 40572 | The empty set is a closed ... |
| 1psubclN 40573 | The set of all atoms is a ... |
| atpsubclN 40574 | A point (singleton of an a... |
| pmapsubclN 40575 | A projective map value is ... |
| ispsubcl2N 40576 | Alternate predicate for "i... |
| psubclinN 40577 | The intersection of two cl... |
| paddatclN 40578 | The projective sum of a cl... |
| pclfinclN 40579 | The projective subspace cl... |
| linepsubclN 40580 | A line is a closed project... |
| polsubclN 40581 | A polarity is a closed pro... |
| poml4N 40582 | Orthomodular law for proje... |
| poml5N 40583 | Orthomodular law for proje... |
| poml6N 40584 | Orthomodular law for proje... |
| osumcllem1N 40585 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 40586 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 40587 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 40588 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 40589 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 40590 | Lemma for ~ osumclN . Use... |
| osumcllem7N 40591 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 40592 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 40593 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 40594 | Lemma for ~ osumclN . Con... |
| osumcllem11N 40595 | Lemma for ~ osumclN . (Co... |
| osumclN 40596 | Closure of orthogonal sum.... |
| pmapojoinN 40597 | For orthogonal elements, p... |
| pexmidN 40598 | Excluded middle law for cl... |
| pexmidlem1N 40599 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 40600 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 40601 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 40602 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 40603 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 40604 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 40605 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 40606 | Lemma for ~ pexmidN . The... |
| pexmidALTN 40607 | Excluded middle law for cl... |
| pl42lem1N 40608 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 40609 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 40610 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 40611 | Lemma for ~ pl42N . (Cont... |
| pl42N 40612 | Law holding in a Hilbert l... |
| watfvalN 40621 | The W atoms function. (Co... |
| watvalN 40622 | Value of the W atoms funct... |
| iswatN 40623 | The predicate "is a W atom... |
| lhpset 40624 | The set of co-atoms (latti... |
| islhp 40625 | The predicate "is a co-ato... |
| islhp2 40626 | The predicate "is a co-ato... |
| lhpbase 40627 | A co-atom is a member of t... |
| lhp1cvr 40628 | The lattice unity covers a... |
| lhplt 40629 | An atom under a co-atom is... |
| lhp2lt 40630 | The join of two atoms unde... |
| lhpexlt 40631 | There exists an atom less ... |
| lhp0lt 40632 | A co-atom is greater than ... |
| lhpn0 40633 | A co-atom is nonzero. TOD... |
| lhpexle 40634 | There exists an atom under... |
| lhpexnle 40635 | There exists an atom not u... |
| lhpexle1lem 40636 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 40637 | There exists an atom under... |
| lhpexle2lem 40638 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 40639 | There exists atom under a ... |
| lhpexle3lem 40640 | There exists atom under a ... |
| lhpexle3 40641 | There exists atom under a ... |
| lhpex2leN 40642 | There exist at least two d... |
| lhpoc 40643 | The orthocomplement of a c... |
| lhpoc2N 40644 | The orthocomplement of an ... |
| lhpocnle 40645 | The orthocomplement of a c... |
| lhpocat 40646 | The orthocomplement of a c... |
| lhpocnel 40647 | The orthocomplement of a c... |
| lhpocnel2 40648 | The orthocomplement of a c... |
| lhpjat1 40649 | The join of a co-atom (hyp... |
| lhpjat2 40650 | The join of a co-atom (hyp... |
| lhpj1 40651 | The join of a co-atom (hyp... |
| lhpmcvr 40652 | The meet of a lattice hype... |
| lhpmcvr2 40653 | Alternate way to express t... |
| lhpmcvr3 40654 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 40655 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 40656 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 40657 | Specialization of ~ lhpmcv... |
| lhpm0atN 40658 | If the meet of a lattice h... |
| lhpmat 40659 | An element covered by the ... |
| lhpmatb 40660 | An element covered by the ... |
| lhp2at0 40661 | Join and meet with differe... |
| lhp2atnle 40662 | Inequality for 2 different... |
| lhp2atne 40663 | Inequality for joins with ... |
| lhp2at0nle 40664 | Inequality for 2 different... |
| lhp2at0ne 40665 | Inequality for joins with ... |
| lhpelim 40666 | Eliminate an atom not unde... |
| lhpmod2i2 40667 | Modular law for hyperplane... |
| lhpmod6i1 40668 | Modular law for hyperplane... |
| lhprelat3N 40669 | The Hilbert lattice is rel... |
| cdlemb2 40670 | Given two atoms not under ... |
| lhple 40671 | Property of a lattice elem... |
| lhpat 40672 | Create an atom under a co-... |
| lhpat4N 40673 | Property of an atom under ... |
| lhpat2 40674 | Create an atom under a co-... |
| lhpat3 40675 | There is only one atom und... |
| 4atexlemk 40676 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 40677 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 40678 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 40679 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 40680 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 40681 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 40682 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 40683 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 40684 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 40685 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 40686 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 40687 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 40688 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 40689 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 40690 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 40691 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 40692 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 40693 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 40694 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 40695 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 40696 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 40697 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 40698 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 40699 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 40700 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 40701 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 40702 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 40703 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 40704 | Whenever there are at leas... |
| 4atex 40705 | Whenever there are at leas... |
| 4atex2 40706 | More general version of ~ ... |
| 4atex2-0aOLDN 40707 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 40708 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 40709 | Same as ~ 4atex2 except th... |
| 4atex3 40710 | More general version of ~ ... |
| lautset 40711 | The set of lattice automor... |
| islaut 40712 | The predicate "is a lattic... |
| lautle 40713 | Less-than or equal propert... |
| laut1o 40714 | A lattice automorphism is ... |
| laut11 40715 | One-to-one property of a l... |
| lautcl 40716 | A lattice automorphism val... |
| lautcnvclN 40717 | Reverse closure of a latti... |
| lautcnvle 40718 | Less-than or equal propert... |
| lautcnv 40719 | The converse of a lattice ... |
| lautlt 40720 | Less-than property of a la... |
| lautcvr 40721 | Covering property of a lat... |
| lautj 40722 | Meet property of a lattice... |
| lautm 40723 | Meet property of a lattice... |
| lauteq 40724 | A lattice automorphism arg... |
| idlaut 40725 | The identity function is a... |
| lautco 40726 | The composition of two lat... |
| pautsetN 40727 | The set of projective auto... |
| ispautN 40728 | The predicate "is a projec... |
| ldilfset 40737 | The mapping from fiducial ... |
| ldilset 40738 | The set of lattice dilatio... |
| isldil 40739 | The predicate "is a lattic... |
| ldillaut 40740 | A lattice dilation is an a... |
| ldil1o 40741 | A lattice dilation is a on... |
| ldilval 40742 | Value of a lattice dilatio... |
| idldil 40743 | The identity function is a... |
| ldilcnv 40744 | The converse of a lattice ... |
| ldilco 40745 | The composition of two lat... |
| ltrnfset 40746 | The set of all lattice tra... |
| ltrnset 40747 | The set of lattice transla... |
| isltrn 40748 | The predicate "is a lattic... |
| isltrn2N 40749 | The predicate "is a lattic... |
| ltrnu 40750 | Uniqueness property of a l... |
| ltrnldil 40751 | A lattice translation is a... |
| ltrnlaut 40752 | A lattice translation is a... |
| ltrn1o 40753 | A lattice translation is a... |
| ltrncl 40754 | Closure of a lattice trans... |
| ltrn11 40755 | One-to-one property of a l... |
| ltrncnvnid 40756 | If a translation is differ... |
| ltrncoidN 40757 | Two translations are equal... |
| ltrnle 40758 | Less-than or equal propert... |
| ltrncnvleN 40759 | Less-than or equal propert... |
| ltrnm 40760 | Lattice translation of a m... |
| ltrnj 40761 | Lattice translation of a m... |
| ltrncvr 40762 | Covering property of a lat... |
| ltrnval1 40763 | Value of a lattice transla... |
| ltrnid 40764 | A lattice translation is t... |
| ltrnnid 40765 | If a lattice translation i... |
| ltrnatb 40766 | The lattice translation of... |
| ltrncnvatb 40767 | The converse of the lattic... |
| ltrnel 40768 | The lattice translation of... |
| ltrnat 40769 | The lattice translation of... |
| ltrncnvat 40770 | The converse of the lattic... |
| ltrncnvel 40771 | The converse of the lattic... |
| ltrncoelN 40772 | Composition of lattice tra... |
| ltrncoat 40773 | Composition of lattice tra... |
| ltrncoval 40774 | Two ways to express value ... |
| ltrncnv 40775 | The converse of a lattice ... |
| ltrn11at 40776 | Frequently used one-to-one... |
| ltrneq2 40777 | The equality of two transl... |
| ltrneq 40778 | The equality of two transl... |
| idltrn 40779 | The identity function is a... |
| ltrnmw 40780 | Property of lattice transl... |
| dilfsetN 40781 | The mapping from fiducial ... |
| dilsetN 40782 | The set of dilations for a... |
| isdilN 40783 | The predicate "is a dilati... |
| trnfsetN 40784 | The mapping from fiducial ... |
| trnsetN 40785 | The set of translations fo... |
| istrnN 40786 | The predicate "is a transl... |
| trlfset 40789 | The set of all traces of l... |
| trlset 40790 | The set of traces of latti... |
| trlval 40791 | The value of the trace of ... |
| trlval2 40792 | The value of the trace of ... |
| trlcl 40793 | Closure of the trace of a ... |
| trlcnv 40794 | The trace of the converse ... |
| trljat1 40795 | The value of a translation... |
| trljat2 40796 | The value of a translation... |
| trljat3 40797 | The value of a translation... |
| trlat 40798 | If an atom differs from it... |
| trl0 40799 | If an atom not under the f... |
| trlator0 40800 | The trace of a lattice tra... |
| trlatn0 40801 | The trace of a lattice tra... |
| trlnidat 40802 | The trace of a lattice tra... |
| ltrnnidn 40803 | If a lattice translation i... |
| ltrnideq 40804 | Property of the identity l... |
| trlid0 40805 | The trace of the identity ... |
| trlnidatb 40806 | A lattice translation is n... |
| trlid0b 40807 | A lattice translation is t... |
| trlnid 40808 | Different translations wit... |
| ltrn2ateq 40809 | Property of the equality o... |
| ltrnateq 40810 | If any atom (under ` W ` )... |
| ltrnatneq 40811 | If any atom (under ` W ` )... |
| ltrnatlw 40812 | If the value of an atom eq... |
| trlle 40813 | The trace of a lattice tra... |
| trlne 40814 | The trace of a lattice tra... |
| trlnle 40815 | The atom not under the fid... |
| trlval3 40816 | The value of the trace of ... |
| trlval4 40817 | The value of the trace of ... |
| trlval5 40818 | The value of the trace of ... |
| arglem1N 40819 | Lemma for Desargues's law.... |
| cdlemc1 40820 | Part of proof of Lemma C i... |
| cdlemc2 40821 | Part of proof of Lemma C i... |
| cdlemc3 40822 | Part of proof of Lemma C i... |
| cdlemc4 40823 | Part of proof of Lemma C i... |
| cdlemc5 40824 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 40825 | Lemma for ~ cdlemc . (Con... |
| cdlemc 40826 | Lemma C in [Crawley] p. 11... |
| cdlemd1 40827 | Part of proof of Lemma D i... |
| cdlemd2 40828 | Part of proof of Lemma D i... |
| cdlemd3 40829 | Part of proof of Lemma D i... |
| cdlemd4 40830 | Part of proof of Lemma D i... |
| cdlemd5 40831 | Part of proof of Lemma D i... |
| cdlemd6 40832 | Part of proof of Lemma D i... |
| cdlemd7 40833 | Part of proof of Lemma D i... |
| cdlemd8 40834 | Part of proof of Lemma D i... |
| cdlemd9 40835 | Part of proof of Lemma D i... |
| cdlemd 40836 | If two translations agree ... |
| ltrneq3 40837 | Two translations agree at ... |
| cdleme00a 40838 | Part of proof of Lemma E i... |
| cdleme0aa 40839 | Part of proof of Lemma E i... |
| cdleme0a 40840 | Part of proof of Lemma E i... |
| cdleme0b 40841 | Part of proof of Lemma E i... |
| cdleme0c 40842 | Part of proof of Lemma E i... |
| cdleme0cp 40843 | Part of proof of Lemma E i... |
| cdleme0cq 40844 | Part of proof of Lemma E i... |
| cdleme0dN 40845 | Part of proof of Lemma E i... |
| cdleme0e 40846 | Part of proof of Lemma E i... |
| cdleme0fN 40847 | Part of proof of Lemma E i... |
| cdleme0gN 40848 | Part of proof of Lemma E i... |
| cdlemeulpq 40849 | Part of proof of Lemma E i... |
| cdleme01N 40850 | Part of proof of Lemma E i... |
| cdleme02N 40851 | Part of proof of Lemma E i... |
| cdleme0ex1N 40852 | Part of proof of Lemma E i... |
| cdleme0ex2N 40853 | Part of proof of Lemma E i... |
| cdleme0moN 40854 | Part of proof of Lemma E i... |
| cdleme1b 40855 | Part of proof of Lemma E i... |
| cdleme1 40856 | Part of proof of Lemma E i... |
| cdleme2 40857 | Part of proof of Lemma E i... |
| cdleme3b 40858 | Part of proof of Lemma E i... |
| cdleme3c 40859 | Part of proof of Lemma E i... |
| cdleme3d 40860 | Part of proof of Lemma E i... |
| cdleme3e 40861 | Part of proof of Lemma E i... |
| cdleme3fN 40862 | Part of proof of Lemma E i... |
| cdleme3g 40863 | Part of proof of Lemma E i... |
| cdleme3h 40864 | Part of proof of Lemma E i... |
| cdleme3fa 40865 | Part of proof of Lemma E i... |
| cdleme3 40866 | Part of proof of Lemma E i... |
| cdleme4 40867 | Part of proof of Lemma E i... |
| cdleme4a 40868 | Part of proof of Lemma E i... |
| cdleme5 40869 | Part of proof of Lemma E i... |
| cdleme6 40870 | Part of proof of Lemma E i... |
| cdleme7aa 40871 | Part of proof of Lemma E i... |
| cdleme7a 40872 | Part of proof of Lemma E i... |
| cdleme7b 40873 | Part of proof of Lemma E i... |
| cdleme7c 40874 | Part of proof of Lemma E i... |
| cdleme7d 40875 | Part of proof of Lemma E i... |
| cdleme7e 40876 | Part of proof of Lemma E i... |
| cdleme7ga 40877 | Part of proof of Lemma E i... |
| cdleme7 40878 | Part of proof of Lemma E i... |
| cdleme8 40879 | Part of proof of Lemma E i... |
| cdleme9a 40880 | Part of proof of Lemma E i... |
| cdleme9b 40881 | Utility lemma for Lemma E ... |
| cdleme9 40882 | Part of proof of Lemma E i... |
| cdleme10 40883 | Part of proof of Lemma E i... |
| cdleme8tN 40884 | Part of proof of Lemma E i... |
| cdleme9taN 40885 | Part of proof of Lemma E i... |
| cdleme9tN 40886 | Part of proof of Lemma E i... |
| cdleme10tN 40887 | Part of proof of Lemma E i... |
| cdleme16aN 40888 | Part of proof of Lemma E i... |
| cdleme11a 40889 | Part of proof of Lemma E i... |
| cdleme11c 40890 | Part of proof of Lemma E i... |
| cdleme11dN 40891 | Part of proof of Lemma E i... |
| cdleme11e 40892 | Part of proof of Lemma E i... |
| cdleme11fN 40893 | Part of proof of Lemma E i... |
| cdleme11g 40894 | Part of proof of Lemma E i... |
| cdleme11h 40895 | Part of proof of Lemma E i... |
| cdleme11j 40896 | Part of proof of Lemma E i... |
| cdleme11k 40897 | Part of proof of Lemma E i... |
| cdleme11l 40898 | Part of proof of Lemma E i... |
| cdleme11 40899 | Part of proof of Lemma E i... |
| cdleme12 40900 | Part of proof of Lemma E i... |
| cdleme13 40901 | Part of proof of Lemma E i... |
| cdleme14 40902 | Part of proof of Lemma E i... |
| cdleme15a 40903 | Part of proof of Lemma E i... |
| cdleme15b 40904 | Part of proof of Lemma E i... |
| cdleme15c 40905 | Part of proof of Lemma E i... |
| cdleme15d 40906 | Part of proof of Lemma E i... |
| cdleme15 40907 | Part of proof of Lemma E i... |
| cdleme16b 40908 | Part of proof of Lemma E i... |
| cdleme16c 40909 | Part of proof of Lemma E i... |
| cdleme16d 40910 | Part of proof of Lemma E i... |
| cdleme16e 40911 | Part of proof of Lemma E i... |
| cdleme16f 40912 | Part of proof of Lemma E i... |
| cdleme16g 40913 | Part of proof of Lemma E i... |
| cdleme16 40914 | Part of proof of Lemma E i... |
| cdleme17a 40915 | Part of proof of Lemma E i... |
| cdleme17b 40916 | Lemma leading to ~ cdleme1... |
| cdleme17c 40917 | Part of proof of Lemma E i... |
| cdleme17d1 40918 | Part of proof of Lemma E i... |
| cdleme0nex 40919 | Part of proof of Lemma E i... |
| cdleme18a 40920 | Part of proof of Lemma E i... |
| cdleme18b 40921 | Part of proof of Lemma E i... |
| cdleme18c 40922 | Part of proof of Lemma E i... |
| cdleme22gb 40923 | Utility lemma for Lemma E ... |
| cdleme18d 40924 | Part of proof of Lemma E i... |
| cdlemesner 40925 | Part of proof of Lemma E i... |
| cdlemedb 40926 | Part of proof of Lemma E i... |
| cdlemeda 40927 | Part of proof of Lemma E i... |
| cdlemednpq 40928 | Part of proof of Lemma E i... |
| cdlemednuN 40929 | Part of proof of Lemma E i... |
| cdleme20zN 40930 | Part of proof of Lemma E i... |
| cdleme20y 40931 | Part of proof of Lemma E i... |
| cdleme19a 40932 | Part of proof of Lemma E i... |
| cdleme19b 40933 | Part of proof of Lemma E i... |
| cdleme19c 40934 | Part of proof of Lemma E i... |
| cdleme19d 40935 | Part of proof of Lemma E i... |
| cdleme19e 40936 | Part of proof of Lemma E i... |
| cdleme19f 40937 | Part of proof of Lemma E i... |
| cdleme20aN 40938 | Part of proof of Lemma E i... |
| cdleme20bN 40939 | Part of proof of Lemma E i... |
| cdleme20c 40940 | Part of proof of Lemma E i... |
| cdleme20d 40941 | Part of proof of Lemma E i... |
| cdleme20e 40942 | Part of proof of Lemma E i... |
| cdleme20f 40943 | Part of proof of Lemma E i... |
| cdleme20g 40944 | Part of proof of Lemma E i... |
| cdleme20h 40945 | Part of proof of Lemma E i... |
| cdleme20i 40946 | Part of proof of Lemma E i... |
| cdleme20j 40947 | Part of proof of Lemma E i... |
| cdleme20k 40948 | Part of proof of Lemma E i... |
| cdleme20l1 40949 | Part of proof of Lemma E i... |
| cdleme20l2 40950 | Part of proof of Lemma E i... |
| cdleme20l 40951 | Part of proof of Lemma E i... |
| cdleme20m 40952 | Part of proof of Lemma E i... |
| cdleme20 40953 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 40954 | Part of proof of Lemma E i... |
| cdleme21b 40955 | Part of proof of Lemma E i... |
| cdleme21c 40956 | Part of proof of Lemma E i... |
| cdleme21at 40957 | Part of proof of Lemma E i... |
| cdleme21ct 40958 | Part of proof of Lemma E i... |
| cdleme21d 40959 | Part of proof of Lemma E i... |
| cdleme21e 40960 | Part of proof of Lemma E i... |
| cdleme21f 40961 | Part of proof of Lemma E i... |
| cdleme21g 40962 | Part of proof of Lemma E i... |
| cdleme21h 40963 | Part of proof of Lemma E i... |
| cdleme21i 40964 | Part of proof of Lemma E i... |
| cdleme21j 40965 | Combine ~ cdleme20 and ~ c... |
| cdleme21 40966 | Part of proof of Lemma E i... |
| cdleme21k 40967 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 40968 | Part of proof of Lemma E i... |
| cdleme22a 40969 | Part of proof of Lemma E i... |
| cdleme22b 40970 | Part of proof of Lemma E i... |
| cdleme22cN 40971 | Part of proof of Lemma E i... |
| cdleme22d 40972 | Part of proof of Lemma E i... |
| cdleme22e 40973 | Part of proof of Lemma E i... |
| cdleme22eALTN 40974 | Part of proof of Lemma E i... |
| cdleme22f 40975 | Part of proof of Lemma E i... |
| cdleme22f2 40976 | Part of proof of Lemma E i... |
| cdleme22g 40977 | Part of proof of Lemma E i... |
| cdleme23a 40978 | Part of proof of Lemma E i... |
| cdleme23b 40979 | Part of proof of Lemma E i... |
| cdleme23c 40980 | Part of proof of Lemma E i... |
| cdleme24 40981 | Quantified version of ~ cd... |
| cdleme25a 40982 | Lemma for ~ cdleme25b . (... |
| cdleme25b 40983 | Transform ~ cdleme24 . TO... |
| cdleme25c 40984 | Transform ~ cdleme25b . (... |
| cdleme25dN 40985 | Transform ~ cdleme25c . (... |
| cdleme25cl 40986 | Show closure of the unique... |
| cdleme25cv 40987 | Change bound variables in ... |
| cdleme26e 40988 | Part of proof of Lemma E i... |
| cdleme26ee 40989 | Part of proof of Lemma E i... |
| cdleme26eALTN 40990 | Part of proof of Lemma E i... |
| cdleme26fALTN 40991 | Part of proof of Lemma E i... |
| cdleme26f 40992 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 40993 | Part of proof of Lemma E i... |
| cdleme26f2 40994 | Part of proof of Lemma E i... |
| cdleme27cl 40995 | Part of proof of Lemma E i... |
| cdleme27a 40996 | Part of proof of Lemma E i... |
| cdleme27b 40997 | Lemma for ~ cdleme27N . (... |
| cdleme27N 40998 | Part of proof of Lemma E i... |
| cdleme28a 40999 | Lemma for ~ cdleme25b . T... |
| cdleme28b 41000 | Lemma for ~ cdleme25b . T... |
| cdleme28c 41001 | Part of proof of Lemma E i... |
| cdleme28 41002 | Quantified version of ~ cd... |
| cdleme29ex 41003 | Lemma for ~ cdleme29b . (... |
| cdleme29b 41004 | Transform ~ cdleme28 . (C... |
| cdleme29c 41005 | Transform ~ cdleme28b . (... |
| cdleme29cl 41006 | Show closure of the unique... |
| cdleme30a 41007 | Part of proof of Lemma E i... |
| cdleme31so 41008 | Part of proof of Lemma E i... |
| cdleme31sn 41009 | Part of proof of Lemma E i... |
| cdleme31sn1 41010 | Part of proof of Lemma E i... |
| cdleme31se 41011 | Part of proof of Lemma D i... |
| cdleme31se2 41012 | Part of proof of Lemma D i... |
| cdleme31sc 41013 | Part of proof of Lemma E i... |
| cdleme31sde 41014 | Part of proof of Lemma D i... |
| cdleme31snd 41015 | Part of proof of Lemma D i... |
| cdleme31sdnN 41016 | Part of proof of Lemma E i... |
| cdleme31sn1c 41017 | Part of proof of Lemma E i... |
| cdleme31sn2 41018 | Part of proof of Lemma E i... |
| cdleme31fv 41019 | Part of proof of Lemma E i... |
| cdleme31fv1 41020 | Part of proof of Lemma E i... |
| cdleme31fv1s 41021 | Part of proof of Lemma E i... |
| cdleme31fv2 41022 | Part of proof of Lemma E i... |
| cdleme31id 41023 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 41024 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 41025 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 41026 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 41027 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 41028 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 41029 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 41030 | Part of proof of Lemma E i... |
| cdlemefr29exN 41031 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 41032 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 41033 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 41034 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 41035 | TODO fix comment. (Contri... |
| cdlemefr29clN 41036 | Show closure of the unique... |
| cdleme43frv1snN 41037 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 41038 | Part of proof of Lemma E i... |
| cdlemefr32fva1 41039 | Part of proof of Lemma E i... |
| cdlemefr31fv1 41040 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 41041 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 41042 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 41043 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 41044 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 41045 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 41046 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 41047 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 41048 | Show closure of the unique... |
| cdleme43fsv1snlem 41049 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 41050 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 41051 | Part of proof of Lemma E i... |
| cdlemefs32fva1 41052 | Part of proof of Lemma E i... |
| cdlemefs31fv1 41053 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 41054 | Value of f(r) when r is an... |
| cdlemefs44 41055 | Value of f_s(r) when r is ... |
| cdlemefr45 41056 | Value of f(r) when r is an... |
| cdlemefr45e 41057 | Explicit expansion of ~ cd... |
| cdlemefs45 41058 | Value of f_s(r) when r is ... |
| cdlemefs45ee 41059 | Explicit expansion of ~ cd... |
| cdlemefs45eN 41060 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 41061 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 41062 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 41063 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 41064 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 41065 | Show closure of ` [_ R / s... |
| cdleme32fva 41066 | Part of proof of Lemma D i... |
| cdleme32fva1 41067 | Part of proof of Lemma D i... |
| cdleme32fvaw 41068 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 41069 | Part of proof of Lemma D i... |
| cdleme32a 41070 | Part of proof of Lemma D i... |
| cdleme32b 41071 | Part of proof of Lemma D i... |
| cdleme32c 41072 | Part of proof of Lemma D i... |
| cdleme32d 41073 | Part of proof of Lemma D i... |
| cdleme32e 41074 | Part of proof of Lemma D i... |
| cdleme32f 41075 | Part of proof of Lemma D i... |
| cdleme32le 41076 | Part of proof of Lemma D i... |
| cdleme35a 41077 | Part of proof of Lemma E i... |
| cdleme35fnpq 41078 | Part of proof of Lemma E i... |
| cdleme35b 41079 | Part of proof of Lemma E i... |
| cdleme35c 41080 | Part of proof of Lemma E i... |
| cdleme35d 41081 | Part of proof of Lemma E i... |
| cdleme35e 41082 | Part of proof of Lemma E i... |
| cdleme35f 41083 | Part of proof of Lemma E i... |
| cdleme35g 41084 | Part of proof of Lemma E i... |
| cdleme35h 41085 | Part of proof of Lemma E i... |
| cdleme35h2 41086 | Part of proof of Lemma E i... |
| cdleme35sn2aw 41087 | Part of proof of Lemma E i... |
| cdleme35sn3a 41088 | Part of proof of Lemma E i... |
| cdleme36a 41089 | Part of proof of Lemma E i... |
| cdleme36m 41090 | Part of proof of Lemma E i... |
| cdleme37m 41091 | Part of proof of Lemma E i... |
| cdleme38m 41092 | Part of proof of Lemma E i... |
| cdleme38n 41093 | Part of proof of Lemma E i... |
| cdleme39a 41094 | Part of proof of Lemma E i... |
| cdleme39n 41095 | Part of proof of Lemma E i... |
| cdleme40m 41096 | Part of proof of Lemma E i... |
| cdleme40n 41097 | Part of proof of Lemma E i... |
| cdleme40v 41098 | Part of proof of Lemma E i... |
| cdleme40w 41099 | Part of proof of Lemma E i... |
| cdleme42a 41100 | Part of proof of Lemma E i... |
| cdleme42c 41101 | Part of proof of Lemma E i... |
| cdleme42d 41102 | Part of proof of Lemma E i... |
| cdleme41sn3aw 41103 | Part of proof of Lemma E i... |
| cdleme41sn4aw 41104 | Part of proof of Lemma E i... |
| cdleme41snaw 41105 | Part of proof of Lemma E i... |
| cdleme41fva11 41106 | Part of proof of Lemma E i... |
| cdleme42b 41107 | Part of proof of Lemma E i... |
| cdleme42e 41108 | Part of proof of Lemma E i... |
| cdleme42f 41109 | Part of proof of Lemma E i... |
| cdleme42g 41110 | Part of proof of Lemma E i... |
| cdleme42h 41111 | Part of proof of Lemma E i... |
| cdleme42i 41112 | Part of proof of Lemma E i... |
| cdleme42k 41113 | Part of proof of Lemma E i... |
| cdleme42ke 41114 | Part of proof of Lemma E i... |
| cdleme42keg 41115 | Part of proof of Lemma E i... |
| cdleme42mN 41116 | Part of proof of Lemma E i... |
| cdleme42mgN 41117 | Part of proof of Lemma E i... |
| cdleme43aN 41118 | Part of proof of Lemma E i... |
| cdleme43bN 41119 | Lemma for Lemma E in [Craw... |
| cdleme43cN 41120 | Part of proof of Lemma E i... |
| cdleme43dN 41121 | Part of proof of Lemma E i... |
| cdleme46f2g2 41122 | Conversion for ` G ` to re... |
| cdleme46f2g1 41123 | Conversion for ` G ` to re... |
| cdleme17d2 41124 | Part of proof of Lemma E i... |
| cdleme17d3 41125 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 41126 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 41127 | Part of proof of Lemma E i... |
| cdleme48fv 41128 | Part of proof of Lemma D i... |
| cdleme48fvg 41129 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 41130 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 41131 | TODO: fix comment. TODO: ... |
| cdleme48b 41132 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 41133 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 41134 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 41135 | TODO: fix comment. (Contr... |
| cdleme4gfv 41136 | Part of proof of Lemma D i... |
| cdlemeg47b 41137 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 41138 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 41139 | Value of g_s(r) when r is ... |
| cdlemeg49le 41140 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 41141 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 41142 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 41143 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 41144 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 41145 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 41146 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 41147 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 41148 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 41149 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 41150 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 41151 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 41152 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 41153 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 41154 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 41155 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 41156 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 41157 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 41158 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 41159 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 41160 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 41161 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 41162 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 41163 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 41164 | TODO: fix comment. (Contr... |
| cdleme48gfv1 41165 | TODO: fix comment. (Contr... |
| cdleme48gfv 41166 | TODO: fix comment. (Contr... |
| cdleme48fgv 41167 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 41168 | Part of proof of Lemma D i... |
| cdleme50lebi 41169 | Part of proof of Lemma D i... |
| cdleme50eq 41170 | Part of proof of Lemma D i... |
| cdleme50f 41171 | Part of proof of Lemma D i... |
| cdleme50f1 41172 | Part of proof of Lemma D i... |
| cdleme50rnlem 41173 | Part of proof of Lemma D i... |
| cdleme50rn 41174 | Part of proof of Lemma D i... |
| cdleme50f1o 41175 | Part of proof of Lemma D i... |
| cdleme50laut 41176 | Part of proof of Lemma D i... |
| cdleme50ldil 41177 | Part of proof of Lemma D i... |
| cdleme50trn1 41178 | Part of proof that ` F ` i... |
| cdleme50trn2a 41179 | Part of proof that ` F ` i... |
| cdleme50trn2 41180 | Part of proof that ` F ` i... |
| cdleme50trn12 41181 | Part of proof that ` F ` i... |
| cdleme50trn3 41182 | Part of proof that ` F ` i... |
| cdleme50trn123 41183 | Part of proof that ` F ` i... |
| cdleme51finvfvN 41184 | Part of proof of Lemma E i... |
| cdleme51finvN 41185 | Part of proof of Lemma E i... |
| cdleme50ltrn 41186 | Part of proof of Lemma E i... |
| cdleme51finvtrN 41187 | Part of proof of Lemma E i... |
| cdleme50ex 41188 | Part of Lemma E in [Crawle... |
| cdleme 41189 | Lemma E in [Crawley] p. 11... |
| cdlemf1 41190 | Part of Lemma F in [Crawle... |
| cdlemf2 41191 | Part of Lemma F in [Crawle... |
| cdlemf 41192 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 41193 | ~ cdlemf with additional c... |
| cdlemftr3 41194 | Special case of ~ cdlemf s... |
| cdlemftr2 41195 | Special case of ~ cdlemf s... |
| cdlemftr1 41196 | Part of proof of Lemma G o... |
| cdlemftr0 41197 | Special case of ~ cdlemf s... |
| trlord 41198 | The ordering of two Hilber... |
| cdlemg1a 41199 | Shorter expression for ` G... |
| cdlemg1b2 41200 | This theorem can be used t... |
| cdlemg1idlemN 41201 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 41202 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 41203 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 41204 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 41205 | This theorem can be used t... |
| cdlemg1idN 41206 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 41207 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 41208 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 41209 | Version of ~ cdleme51finvt... |
| ltrniotaval 41210 | Value of the unique transl... |
| ltrniotacnvval 41211 | Converse value of the uniq... |
| ltrniotaidvalN 41212 | Value of the unique transl... |
| ltrniotavalbN 41213 | Value of the unique transl... |
| cdlemeiota 41214 | A translation is uniquely ... |
| cdlemg1ci2 41215 | Any function of the form o... |
| cdlemg1cN 41216 | Any translation belongs to... |
| cdlemg1cex 41217 | Any translation is one of ... |
| cdlemg2cN 41218 | Any translation belongs to... |
| cdlemg2dN 41219 | This theorem can be used t... |
| cdlemg2cex 41220 | Any translation is one of ... |
| cdlemg2ce 41221 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 41222 | Part of proof of Lemma E i... |
| cdlemg2fvlem 41223 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 41224 | ~ cdleme42keg with simpler... |
| cdlemg2idN 41225 | Version of ~ cdleme31id wi... |
| cdlemg3a 41226 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 41227 | TODO: Replace this with ~... |
| cdlemg2fv 41228 | Value of a translation in ... |
| cdlemg2fv2 41229 | Value of a translation in ... |
| cdlemg2k 41230 | ~ cdleme42keg with simpler... |
| cdlemg2kq 41231 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 41232 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 41233 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 41234 | TODO: Is there a simpler ... |
| cdlemb3 41235 | Given two atoms not under ... |
| cdlemg7fvbwN 41236 | Properties of a translatio... |
| cdlemg4a 41237 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 41238 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 41239 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 41240 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 41241 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 41242 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 41243 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 41244 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 41245 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 41246 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 41247 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 41248 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 41249 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 41250 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 41251 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 41252 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 41253 | Value of a translation com... |
| cdlemg7aN 41254 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 41255 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 41256 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 41257 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 41258 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 41259 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 41260 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 41261 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 41262 | The triples ` <. P , ( F `... |
| cdlemg9 41263 | The triples ` <. P , ( F `... |
| cdlemg10b 41264 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 41265 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 41266 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 41267 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 41268 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 41269 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 41270 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 41271 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 41272 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 41273 | The triples ` <. P , ( F `... |
| cdlemg12c 41274 | The triples ` <. P , ( F `... |
| cdlemg12d 41275 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 41276 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 41277 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 41278 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 41279 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 41280 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 41281 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 41282 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 41283 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 41284 | Eliminate the ` ( F `` P )... |
| cdlemg15 41285 | Eliminate the ` ( (... |
| cdlemg16 41286 | Part of proof of Lemma G o... |
| cdlemg16ALTN 41287 | This version of ~ cdlemg16... |
| cdlemg16z 41288 | Eliminate ` ( ( F `... |
| cdlemg16zz 41289 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 41290 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 41291 | Part of proof of Lemma G i... |
| cdlemg17dN 41292 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 41293 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 41294 | TODO: fix comment. (Contr... |
| cdlemg17f 41295 | TODO: fix comment. (Contr... |
| cdlemg17g 41296 | TODO: fix comment. (Contr... |
| cdlemg17h 41297 | TODO: fix comment. (Contr... |
| cdlemg17i 41298 | TODO: fix comment. (Contr... |
| cdlemg17ir 41299 | TODO: fix comment. (Contr... |
| cdlemg17j 41300 | TODO: fix comment. (Contr... |
| cdlemg17pq 41301 | Utility theorem for swappi... |
| cdlemg17bq 41302 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 41303 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 41304 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 41305 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 41306 | Part of Lemma G of [Crawle... |
| cdlemg18a 41307 | Show two lines are differe... |
| cdlemg18b 41308 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 41309 | Show two lines intersect a... |
| cdlemg18d 41310 | Show two lines intersect a... |
| cdlemg18 41311 | Show two lines intersect a... |
| cdlemg19a 41312 | Show two lines intersect a... |
| cdlemg19 41313 | Show two lines intersect a... |
| cdlemg20 41314 | Show two lines intersect a... |
| cdlemg21 41315 | Version of cdlemg19 with `... |
| cdlemg22 41316 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 41317 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 41318 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 41319 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 41320 | ~ cdlemg16zz restated for ... |
| cdlemg27a 41321 | For use with case when ` (... |
| cdlemg28a 41322 | Part of proof of Lemma G o... |
| cdlemg31b0N 41323 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 41324 | TODO: Fix comment. (Cont... |
| cdlemg27b 41325 | TODO: Fix comment. (Cont... |
| cdlemg31a 41326 | TODO: fix comment. (Contr... |
| cdlemg31b 41327 | TODO: fix comment. (Contr... |
| cdlemg31c 41328 | Show that when ` N ` is an... |
| cdlemg31d 41329 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 41330 | TODO: Fix comment. (Cont... |
| cdlemg33c0 41331 | TODO: Fix comment. (Cont... |
| cdlemg28b 41332 | Part of proof of Lemma G o... |
| cdlemg28 41333 | Part of proof of Lemma G o... |
| cdlemg29 41334 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 41335 | TODO: Fix comment. (Cont... |
| cdlemg33b 41336 | TODO: Fix comment. (Cont... |
| cdlemg33c 41337 | TODO: Fix comment. (Cont... |
| cdlemg33d 41338 | TODO: Fix comment. (Cont... |
| cdlemg33e 41339 | TODO: Fix comment. (Cont... |
| cdlemg33 41340 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 41341 | Use cdlemg33 to eliminate ... |
| cdlemg35 41342 | TODO: Fix comment. TODO:... |
| cdlemg36 41343 | Use cdlemg35 to eliminate ... |
| cdlemg38 41344 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 41345 | Eliminate ` =/= ` conditio... |
| cdlemg40 41346 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 41347 | Convert ~ cdlemg40 to func... |
| ltrnco 41348 | The composition of two tra... |
| trlcocnv 41349 | Swap the arguments of the ... |
| trlcoabs 41350 | Absorption into a composit... |
| trlcoabs2N 41351 | Absorption of the trace of... |
| trlcoat 41352 | The trace of a composition... |
| trlcocnvat 41353 | Commonly used special case... |
| trlconid 41354 | The composition of two dif... |
| trlcolem 41355 | Lemma for ~ trlco . (Cont... |
| trlco 41356 | The trace of a composition... |
| trlcone 41357 | If two translations have d... |
| cdlemg42 41358 | Part of proof of Lemma G o... |
| cdlemg43 41359 | Part of proof of Lemma G o... |
| cdlemg44a 41360 | Part of proof of Lemma G o... |
| cdlemg44b 41361 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 41362 | Part of proof of Lemma G o... |
| cdlemg47a 41363 | TODO: fix comment. TODO: ... |
| cdlemg46 41364 | Part of proof of Lemma G o... |
| cdlemg47 41365 | Part of proof of Lemma G o... |
| cdlemg48 41366 | Eliminate ` h ` from ~ cdl... |
| ltrncom 41367 | Composition is commutative... |
| ltrnco4 41368 | Rearrange a composition of... |
| trljco 41369 | Trace joined with trace of... |
| trljco2 41370 | Trace joined with trace of... |
| tgrpfset 41373 | The translation group maps... |
| tgrpset 41374 | The translation group for ... |
| tgrpbase 41375 | The base set of the transl... |
| tgrpopr 41376 | The group operation of the... |
| tgrpov 41377 | The group operation value ... |
| tgrpgrplem 41378 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 41379 | The translation group is a... |
| tgrpabl 41380 | The translation group is a... |
| tendofset 41387 | The set of all trace-prese... |
| tendoset 41388 | The set of trace-preservin... |
| istendo 41389 | The predicate "is a trace-... |
| tendotp 41390 | Trace-preserving property ... |
| istendod 41391 | Deduce the predicate "is a... |
| tendof 41392 | Functionality of a trace-p... |
| tendoeq1 41393 | Condition determining equa... |
| tendovalco 41394 | Value of composition of tr... |
| tendocoval 41395 | Value of composition of en... |
| tendocl 41396 | Closure of a trace-preserv... |
| tendoco2 41397 | Distribution of compositio... |
| tendoidcl 41398 | The identity is a trace-pr... |
| tendo1mul 41399 | Multiplicative identity mu... |
| tendo1mulr 41400 | Multiplicative identity mu... |
| tendococl 41401 | The composition of two tra... |
| tendoid 41402 | The identity value of a tr... |
| tendoeq2 41403 | Condition determining equa... |
| tendoplcbv 41404 | Define sum operation for t... |
| tendopl 41405 | Value of endomorphism sum ... |
| tendopl2 41406 | Value of result of endomor... |
| tendoplcl2 41407 | Value of result of endomor... |
| tendoplco2 41408 | Value of result of endomor... |
| tendopltp 41409 | Trace-preserving property ... |
| tendoplcl 41410 | Endomorphism sum is a trac... |
| tendoplcom 41411 | The endomorphism sum opera... |
| tendoplass 41412 | The endomorphism sum opera... |
| tendodi1 41413 | Endomorphism composition d... |
| tendodi2 41414 | Endomorphism composition d... |
| tendo0cbv 41415 | Define additive identity f... |
| tendo02 41416 | Value of additive identity... |
| tendo0co2 41417 | The additive identity trac... |
| tendo0tp 41418 | Trace-preserving property ... |
| tendo0cl 41419 | The additive identity is a... |
| tendo0pl 41420 | Property of the additive i... |
| tendo0plr 41421 | Property of the additive i... |
| tendoicbv 41422 | Define inverse function fo... |
| tendoi 41423 | Value of inverse endomorph... |
| tendoi2 41424 | Value of additive inverse ... |
| tendoicl 41425 | Closure of the additive in... |
| tendoipl 41426 | Property of the additive i... |
| tendoipl2 41427 | Property of the additive i... |
| erngfset 41428 | The division rings on trac... |
| erngset 41429 | The division ring on trace... |
| erngbase 41430 | The base set of the divisi... |
| erngfplus 41431 | Ring addition operation. ... |
| erngplus 41432 | Ring addition operation. ... |
| erngplus2 41433 | Ring addition operation. ... |
| erngfmul 41434 | Ring multiplication operat... |
| erngmul 41435 | Ring addition operation. ... |
| erngfset-rN 41436 | The division rings on trac... |
| erngset-rN 41437 | The division ring on trace... |
| erngbase-rN 41438 | The base set of the divisi... |
| erngfplus-rN 41439 | Ring addition operation. ... |
| erngplus-rN 41440 | Ring addition operation. ... |
| erngplus2-rN 41441 | Ring addition operation. ... |
| erngfmul-rN 41442 | Ring multiplication operat... |
| erngmul-rN 41443 | Ring addition operation. ... |
| cdlemh1 41444 | Part of proof of Lemma H o... |
| cdlemh2 41445 | Part of proof of Lemma H o... |
| cdlemh 41446 | Lemma H of [Crawley] p. 11... |
| cdlemi1 41447 | Part of proof of Lemma I o... |
| cdlemi2 41448 | Part of proof of Lemma I o... |
| cdlemi 41449 | Lemma I of [Crawley] p. 11... |
| cdlemj1 41450 | Part of proof of Lemma J o... |
| cdlemj2 41451 | Part of proof of Lemma J o... |
| cdlemj3 41452 | Part of proof of Lemma J o... |
| tendocan 41453 | Cancellation law: if the v... |
| tendoid0 41454 | A trace-preserving endomor... |
| tendo0mul 41455 | Additive identity multipli... |
| tendo0mulr 41456 | Additive identity multipli... |
| tendo1ne0 41457 | The identity (unity) is no... |
| tendoconid 41458 | The composition (product) ... |
| tendotr 41459 | The trace of the value of ... |
| cdlemk1 41460 | Part of proof of Lemma K o... |
| cdlemk2 41461 | Part of proof of Lemma K o... |
| cdlemk3 41462 | Part of proof of Lemma K o... |
| cdlemk4 41463 | Part of proof of Lemma K o... |
| cdlemk5a 41464 | Part of proof of Lemma K o... |
| cdlemk5 41465 | Part of proof of Lemma K o... |
| cdlemk6 41466 | Part of proof of Lemma K o... |
| cdlemk8 41467 | Part of proof of Lemma K o... |
| cdlemk9 41468 | Part of proof of Lemma K o... |
| cdlemk9bN 41469 | Part of proof of Lemma K o... |
| cdlemki 41470 | Part of proof of Lemma K o... |
| cdlemkvcl 41471 | Part of proof of Lemma K o... |
| cdlemk10 41472 | Part of proof of Lemma K o... |
| cdlemksv 41473 | Part of proof of Lemma K o... |
| cdlemksel 41474 | Part of proof of Lemma K o... |
| cdlemksat 41475 | Part of proof of Lemma K o... |
| cdlemksv2 41476 | Part of proof of Lemma K o... |
| cdlemk7 41477 | Part of proof of Lemma K o... |
| cdlemk11 41478 | Part of proof of Lemma K o... |
| cdlemk12 41479 | Part of proof of Lemma K o... |
| cdlemkoatnle 41480 | Utility lemma. (Contribut... |
| cdlemk13 41481 | Part of proof of Lemma K o... |
| cdlemkole 41482 | Utility lemma. (Contribut... |
| cdlemk14 41483 | Part of proof of Lemma K o... |
| cdlemk15 41484 | Part of proof of Lemma K o... |
| cdlemk16a 41485 | Part of proof of Lemma K o... |
| cdlemk16 41486 | Part of proof of Lemma K o... |
| cdlemk17 41487 | Part of proof of Lemma K o... |
| cdlemk1u 41488 | Part of proof of Lemma K o... |
| cdlemk5auN 41489 | Part of proof of Lemma K o... |
| cdlemk5u 41490 | Part of proof of Lemma K o... |
| cdlemk6u 41491 | Part of proof of Lemma K o... |
| cdlemkj 41492 | Part of proof of Lemma K o... |
| cdlemkuvN 41493 | Part of proof of Lemma K o... |
| cdlemkuel 41494 | Part of proof of Lemma K o... |
| cdlemkuat 41495 | Part of proof of Lemma K o... |
| cdlemkuv2 41496 | Part of proof of Lemma K o... |
| cdlemk18 41497 | Part of proof of Lemma K o... |
| cdlemk19 41498 | Part of proof of Lemma K o... |
| cdlemk7u 41499 | Part of proof of Lemma K o... |
| cdlemk11u 41500 | Part of proof of Lemma K o... |
| cdlemk12u 41501 | Part of proof of Lemma K o... |
| cdlemk21N 41502 | Part of proof of Lemma K o... |
| cdlemk20 41503 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 41504 | Utility lemma. (Contribut... |
| cdlemk13-2N 41505 | Part of proof of Lemma K o... |
| cdlemkole-2N 41506 | Utility lemma. (Contribut... |
| cdlemk14-2N 41507 | Part of proof of Lemma K o... |
| cdlemk15-2N 41508 | Part of proof of Lemma K o... |
| cdlemk16-2N 41509 | Part of proof of Lemma K o... |
| cdlemk17-2N 41510 | Part of proof of Lemma K o... |
| cdlemkj-2N 41511 | Part of proof of Lemma K o... |
| cdlemkuv-2N 41512 | Part of proof of Lemma K o... |
| cdlemkuel-2N 41513 | Part of proof of Lemma K o... |
| cdlemkuv2-2 41514 | Part of proof of Lemma K o... |
| cdlemk18-2N 41515 | Part of proof of Lemma K o... |
| cdlemk19-2N 41516 | Part of proof of Lemma K o... |
| cdlemk7u-2N 41517 | Part of proof of Lemma K o... |
| cdlemk11u-2N 41518 | Part of proof of Lemma K o... |
| cdlemk12u-2N 41519 | Part of proof of Lemma K o... |
| cdlemk21-2N 41520 | Part of proof of Lemma K o... |
| cdlemk20-2N 41521 | Part of proof of Lemma K o... |
| cdlemk22 41522 | Part of proof of Lemma K o... |
| cdlemk30 41523 | Part of proof of Lemma K o... |
| cdlemkuu 41524 | Convert between function a... |
| cdlemk31 41525 | Part of proof of Lemma K o... |
| cdlemk32 41526 | Part of proof of Lemma K o... |
| cdlemkuel-3 41527 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 41528 | Part of proof of Lemma K o... |
| cdlemk18-3N 41529 | Part of proof of Lemma K o... |
| cdlemk22-3 41530 | Part of proof of Lemma K o... |
| cdlemk23-3 41531 | Part of proof of Lemma K o... |
| cdlemk24-3 41532 | Part of proof of Lemma K o... |
| cdlemk25-3 41533 | Part of proof of Lemma K o... |
| cdlemk26b-3 41534 | Part of proof of Lemma K o... |
| cdlemk26-3 41535 | Part of proof of Lemma K o... |
| cdlemk27-3 41536 | Part of proof of Lemma K o... |
| cdlemk28-3 41537 | Part of proof of Lemma K o... |
| cdlemk33N 41538 | Part of proof of Lemma K o... |
| cdlemk34 41539 | Part of proof of Lemma K o... |
| cdlemk29-3 41540 | Part of proof of Lemma K o... |
| cdlemk35 41541 | Part of proof of Lemma K o... |
| cdlemk36 41542 | Part of proof of Lemma K o... |
| cdlemk37 41543 | Part of proof of Lemma K o... |
| cdlemk38 41544 | Part of proof of Lemma K o... |
| cdlemk39 41545 | Part of proof of Lemma K o... |
| cdlemk40 41546 | TODO: fix comment. (Contr... |
| cdlemk40t 41547 | TODO: fix comment. (Contr... |
| cdlemk40f 41548 | TODO: fix comment. (Contr... |
| cdlemk41 41549 | Part of proof of Lemma K o... |
| cdlemkfid1N 41550 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 41551 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 41552 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 41553 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 41554 | TODO: is this useful or sh... |
| cdlemky 41555 | Part of proof of Lemma K o... |
| cdlemkyu 41556 | Convert between function a... |
| cdlemkyuu 41557 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 41558 | Part of proof of Lemma K o... |
| cdlemk19ylem 41559 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 41560 | Part of proof of Lemma K o... |
| cdlemk19y 41561 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 41562 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 41563 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 41564 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 41565 | The value of the tau funct... |
| cdlemk35s 41566 | Substitution version of ~ ... |
| cdlemk35s-id 41567 | Substitution version of ~ ... |
| cdlemk39s 41568 | Substitution version of ~ ... |
| cdlemk39s-id 41569 | Substitution version of ~ ... |
| cdlemk42 41570 | Part of proof of Lemma K o... |
| cdlemk19xlem 41571 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 41572 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 41573 | Part of proof of Lemma K o... |
| cdlemk11tc 41574 | Part of proof of Lemma K o... |
| cdlemk11t 41575 | Part of proof of Lemma K o... |
| cdlemk45 41576 | Part of proof of Lemma K o... |
| cdlemk46 41577 | Part of proof of Lemma K o... |
| cdlemk47 41578 | Part of proof of Lemma K o... |
| cdlemk48 41579 | Part of proof of Lemma K o... |
| cdlemk49 41580 | Part of proof of Lemma K o... |
| cdlemk50 41581 | Part of proof of Lemma K o... |
| cdlemk51 41582 | Part of proof of Lemma K o... |
| cdlemk52 41583 | Part of proof of Lemma K o... |
| cdlemk53a 41584 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 41585 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 41586 | Part of proof of Lemma K o... |
| cdlemk54 41587 | Part of proof of Lemma K o... |
| cdlemk55a 41588 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 41589 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 41590 | Part of proof of Lemma K o... |
| cdlemkyyN 41591 | Part of proof of Lemma K o... |
| cdlemk43N 41592 | Part of proof of Lemma K o... |
| cdlemk35u 41593 | Substitution version of ~ ... |
| cdlemk55u1 41594 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 41595 | Part of proof of Lemma K o... |
| cdlemk39u1 41596 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 41597 | Part of proof of Lemma K o... |
| cdlemk19u1 41598 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 41599 | Part of Lemma K of [Crawle... |
| cdlemk56 41600 | Part of Lemma K of [Crawle... |
| cdlemk19w 41601 | Use a fixed element to eli... |
| cdlemk56w 41602 | Use a fixed element to eli... |
| cdlemk 41603 | Lemma K of [Crawley] p. 11... |
| tendoex 41604 | Generalization of Lemma K ... |
| cdleml1N 41605 | Part of proof of Lemma L o... |
| cdleml2N 41606 | Part of proof of Lemma L o... |
| cdleml3N 41607 | Part of proof of Lemma L o... |
| cdleml4N 41608 | Part of proof of Lemma L o... |
| cdleml5N 41609 | Part of proof of Lemma L o... |
| cdleml6 41610 | Part of proof of Lemma L o... |
| cdleml7 41611 | Part of proof of Lemma L o... |
| cdleml8 41612 | Part of proof of Lemma L o... |
| cdleml9 41613 | Part of proof of Lemma L o... |
| dva1dim 41614 | Two expressions for the 1-... |
| dvhb1dimN 41615 | Two expressions for the 1-... |
| erng1lem 41616 | Value of the endomorphism ... |
| erngdvlem1 41617 | Lemma for ~ eringring . (... |
| erngdvlem2N 41618 | Lemma for ~ eringring . (... |
| erngdvlem3 41619 | Lemma for ~ eringring . (... |
| erngdvlem4 41620 | Lemma for ~ erngdv . (Con... |
| eringring 41621 | An endomorphism ring is a ... |
| erngdv 41622 | An endomorphism ring is a ... |
| erng0g 41623 | The division ring zero of ... |
| erng1r 41624 | The division ring unity of... |
| erngdvlem1-rN 41625 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 41626 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 41627 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 41628 | Lemma for ~ erngdv . (Con... |
| erngring-rN 41629 | An endomorphism ring is a ... |
| erngdv-rN 41630 | An endomorphism ring is a ... |
| dvafset 41633 | The constructed partial ve... |
| dvaset 41634 | The constructed partial ve... |
| dvasca 41635 | The ring base set of the c... |
| dvabase 41636 | The ring base set of the c... |
| dvafplusg 41637 | Ring addition operation fo... |
| dvaplusg 41638 | Ring addition operation fo... |
| dvaplusgv 41639 | Ring addition operation fo... |
| dvafmulr 41640 | Ring multiplication operat... |
| dvamulr 41641 | Ring multiplication operat... |
| dvavbase 41642 | The vectors (vector base s... |
| dvafvadd 41643 | The vector sum operation f... |
| dvavadd 41644 | Ring addition operation fo... |
| dvafvsca 41645 | Ring addition operation fo... |
| dvavsca 41646 | Ring addition operation fo... |
| tendospcl 41647 | Closure of endomorphism sc... |
| tendospass 41648 | Associative law for endomo... |
| tendospdi1 41649 | Forward distributive law f... |
| tendocnv 41650 | Converse of a trace-preser... |
| tendospdi2 41651 | Reverse distributive law f... |
| tendospcanN 41652 | Cancellation law for trace... |
| dvaabl 41653 | The constructed partial ve... |
| dvalveclem 41654 | Lemma for ~ dvalvec . (Co... |
| dvalvec 41655 | The constructed partial ve... |
| dva0g 41656 | The zero vector of partial... |
| diaffval 41659 | The partial isomorphism A ... |
| diafval 41660 | The partial isomorphism A ... |
| diaval 41661 | The partial isomorphism A ... |
| diaelval 41662 | Member of the partial isom... |
| diafn 41663 | Functionality and domain o... |
| diadm 41664 | Domain of the partial isom... |
| diaeldm 41665 | Member of domain of the pa... |
| diadmclN 41666 | A member of domain of the ... |
| diadmleN 41667 | A member of domain of the ... |
| dian0 41668 | The value of the partial i... |
| dia0eldmN 41669 | The lattice zero belongs t... |
| dia1eldmN 41670 | The fiducial hyperplane (t... |
| diass 41671 | The value of the partial i... |
| diael 41672 | A member of the value of t... |
| diatrl 41673 | Trace of a member of the p... |
| diaelrnN 41674 | Any value of the partial i... |
| dialss 41675 | The value of partial isomo... |
| diaord 41676 | The partial isomorphism A ... |
| dia11N 41677 | The partial isomorphism A ... |
| diaf11N 41678 | The partial isomorphism A ... |
| diaclN 41679 | Closure of partial isomorp... |
| diacnvclN 41680 | Closure of partial isomorp... |
| dia0 41681 | The value of the partial i... |
| dia1N 41682 | The value of the partial i... |
| dia1elN 41683 | The largest subspace in th... |
| diaglbN 41684 | Partial isomorphism A of a... |
| diameetN 41685 | Partial isomorphism A of a... |
| diainN 41686 | Inverse partial isomorphis... |
| diaintclN 41687 | The intersection of partia... |
| diasslssN 41688 | The partial isomorphism A ... |
| diassdvaN 41689 | The partial isomorphism A ... |
| dia1dim 41690 | Two expressions for the 1-... |
| dia1dim2 41691 | Two expressions for a 1-di... |
| dia1dimid 41692 | A vector (translation) bel... |
| dia2dimlem1 41693 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 41694 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 41695 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 41696 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 41697 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 41698 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 41699 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 41700 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 41701 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 41702 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 41703 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 41704 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 41705 | Lemma for ~ dia2dim . Eli... |
| dia2dim 41706 | A two-dimensional subspace... |
| dvhfset 41709 | The constructed full vecto... |
| dvhset 41710 | The constructed full vecto... |
| dvhsca 41711 | The ring of scalars of the... |
| dvhbase 41712 | The ring base set of the c... |
| dvhfplusr 41713 | Ring addition operation fo... |
| dvhfmulr 41714 | Ring multiplication operat... |
| dvhmulr 41715 | Ring multiplication operat... |
| dvhvbase 41716 | The vectors (vector base s... |
| dvhelvbasei 41717 | Vector membership in the c... |
| dvhvaddcbv 41718 | Change bound variables to ... |
| dvhvaddval 41719 | The vector sum operation f... |
| dvhfvadd 41720 | The vector sum operation f... |
| dvhvadd 41721 | The vector sum operation f... |
| dvhopvadd 41722 | The vector sum operation f... |
| dvhopvadd2 41723 | The vector sum operation f... |
| dvhvaddcl 41724 | Closure of the vector sum ... |
| dvhvaddcomN 41725 | Commutativity of vector su... |
| dvhvaddass 41726 | Associativity of vector su... |
| dvhvscacbv 41727 | Change bound variables to ... |
| dvhvscaval 41728 | The scalar product operati... |
| dvhfvsca 41729 | Scalar product operation f... |
| dvhvsca 41730 | Scalar product operation f... |
| dvhopvsca 41731 | Scalar product operation f... |
| dvhvscacl 41732 | Closure of the scalar prod... |
| tendoinvcl 41733 | Closure of multiplicative ... |
| tendolinv 41734 | Left multiplicative invers... |
| tendorinv 41735 | Right multiplicative inver... |
| dvhgrp 41736 | The full vector space ` U ... |
| dvhlveclem 41737 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 41738 | The full vector space ` U ... |
| dvhlmod 41739 | The full vector space ` U ... |
| dvh0g 41740 | The zero vector of vector ... |
| dvheveccl 41741 | Properties of a unit vecto... |
| dvhopclN 41742 | Closure of a ` DVecH ` vec... |
| dvhopaddN 41743 | Sum of ` DVecH ` vectors e... |
| dvhopspN 41744 | Scalar product of ` DVecH ... |
| dvhopN 41745 | Decompose a ` DVecH ` vect... |
| dvhopellsm 41746 | Ordered pair membership in... |
| cdlemm10N 41747 | The image of the map ` G `... |
| docaffvalN 41750 | Subspace orthocomplement f... |
| docafvalN 41751 | Subspace orthocomplement f... |
| docavalN 41752 | Subspace orthocomplement f... |
| docaclN 41753 | Closure of subspace orthoc... |
| diaocN 41754 | Value of partial isomorphi... |
| doca2N 41755 | Double orthocomplement of ... |
| doca3N 41756 | Double orthocomplement of ... |
| dvadiaN 41757 | Any closed subspace is a m... |
| diarnN 41758 | Partial isomorphism A maps... |
| diaf1oN 41759 | The partial isomorphism A ... |
| djaffvalN 41762 | Subspace join for ` DVecA ... |
| djafvalN 41763 | Subspace join for ` DVecA ... |
| djavalN 41764 | Subspace join for ` DVecA ... |
| djaclN 41765 | Closure of subspace join f... |
| djajN 41766 | Transfer lattice join to `... |
| dibffval 41769 | The partial isomorphism B ... |
| dibfval 41770 | The partial isomorphism B ... |
| dibval 41771 | The partial isomorphism B ... |
| dibopelvalN 41772 | Member of the partial isom... |
| dibval2 41773 | Value of the partial isomo... |
| dibopelval2 41774 | Member of the partial isom... |
| dibval3N 41775 | Value of the partial isomo... |
| dibelval3 41776 | Member of the partial isom... |
| dibopelval3 41777 | Member of the partial isom... |
| dibelval1st 41778 | Membership in value of the... |
| dibelval1st1 41779 | Membership in value of the... |
| dibelval1st2N 41780 | Membership in value of the... |
| dibelval2nd 41781 | Membership in value of the... |
| dibn0 41782 | The value of the partial i... |
| dibfna 41783 | Functionality and domain o... |
| dibdiadm 41784 | Domain of the partial isom... |
| dibfnN 41785 | Functionality and domain o... |
| dibdmN 41786 | Domain of the partial isom... |
| dibeldmN 41787 | Member of domain of the pa... |
| dibord 41788 | The isomorphism B for a la... |
| dib11N 41789 | The isomorphism B for a la... |
| dibf11N 41790 | The partial isomorphism A ... |
| dibclN 41791 | Closure of partial isomorp... |
| dibvalrel 41792 | The value of partial isomo... |
| dib0 41793 | The value of partial isomo... |
| dib1dim 41794 | Two expressions for the 1-... |
| dibglbN 41795 | Partial isomorphism B of a... |
| dibintclN 41796 | The intersection of partia... |
| dib1dim2 41797 | Two expressions for a 1-di... |
| dibss 41798 | The partial isomorphism B ... |
| diblss 41799 | The value of partial isomo... |
| diblsmopel 41800 | Membership in subspace sum... |
| dicffval 41803 | The partial isomorphism C ... |
| dicfval 41804 | The partial isomorphism C ... |
| dicval 41805 | The partial isomorphism C ... |
| dicopelval 41806 | Membership in value of the... |
| dicelvalN 41807 | Membership in value of the... |
| dicval2 41808 | The partial isomorphism C ... |
| dicelval3 41809 | Member of the partial isom... |
| dicopelval2 41810 | Membership in value of the... |
| dicelval2N 41811 | Membership in value of the... |
| dicfnN 41812 | Functionality and domain o... |
| dicdmN 41813 | Domain of the partial isom... |
| dicvalrelN 41814 | The value of partial isomo... |
| dicssdvh 41815 | The partial isomorphism C ... |
| dicelval1sta 41816 | Membership in value of the... |
| dicelval1stN 41817 | Membership in value of the... |
| dicelval2nd 41818 | Membership in value of the... |
| dicvaddcl 41819 | Membership in value of the... |
| dicvscacl 41820 | Membership in value of the... |
| dicn0 41821 | The value of the partial i... |
| diclss 41822 | The value of partial isomo... |
| diclspsn 41823 | The value of isomorphism C... |
| cdlemn2 41824 | Part of proof of Lemma N o... |
| cdlemn2a 41825 | Part of proof of Lemma N o... |
| cdlemn3 41826 | Part of proof of Lemma N o... |
| cdlemn4 41827 | Part of proof of Lemma N o... |
| cdlemn4a 41828 | Part of proof of Lemma N o... |
| cdlemn5pre 41829 | Part of proof of Lemma N o... |
| cdlemn5 41830 | Part of proof of Lemma N o... |
| cdlemn6 41831 | Part of proof of Lemma N o... |
| cdlemn7 41832 | Part of proof of Lemma N o... |
| cdlemn8 41833 | Part of proof of Lemma N o... |
| cdlemn9 41834 | Part of proof of Lemma N o... |
| cdlemn10 41835 | Part of proof of Lemma N o... |
| cdlemn11a 41836 | Part of proof of Lemma N o... |
| cdlemn11b 41837 | Part of proof of Lemma N o... |
| cdlemn11c 41838 | Part of proof of Lemma N o... |
| cdlemn11pre 41839 | Part of proof of Lemma N o... |
| cdlemn11 41840 | Part of proof of Lemma N o... |
| cdlemn 41841 | Lemma N of [Crawley] p. 12... |
| dihordlem6 41842 | Part of proof of Lemma N o... |
| dihordlem7 41843 | Part of proof of Lemma N o... |
| dihordlem7b 41844 | Part of proof of Lemma N o... |
| dihjustlem 41845 | Part of proof after Lemma ... |
| dihjust 41846 | Part of proof after Lemma ... |
| dihord1 41847 | Part of proof after Lemma ... |
| dihord2a 41848 | Part of proof after Lemma ... |
| dihord2b 41849 | Part of proof after Lemma ... |
| dihord2cN 41850 | Part of proof after Lemma ... |
| dihord11b 41851 | Part of proof after Lemma ... |
| dihord10 41852 | Part of proof after Lemma ... |
| dihord11c 41853 | Part of proof after Lemma ... |
| dihord2pre 41854 | Part of proof after Lemma ... |
| dihord2pre2 41855 | Part of proof after Lemma ... |
| dihord2 41856 | Part of proof after Lemma ... |
| dihffval 41859 | The isomorphism H for a la... |
| dihfval 41860 | Isomorphism H for a lattic... |
| dihval 41861 | Value of isomorphism H for... |
| dihvalc 41862 | Value of isomorphism H for... |
| dihlsscpre 41863 | Closure of isomorphism H f... |
| dihvalcqpre 41864 | Value of isomorphism H for... |
| dihvalcq 41865 | Value of isomorphism H for... |
| dihvalb 41866 | Value of isomorphism H for... |
| dihopelvalbN 41867 | Ordered pair member of the... |
| dihvalcqat 41868 | Value of isomorphism H for... |
| dih1dimb 41869 | Two expressions for a 1-di... |
| dih1dimb2 41870 | Isomorphism H at an atom u... |
| dih1dimc 41871 | Isomorphism H at an atom n... |
| dib2dim 41872 | Extend ~ dia2dim to partia... |
| dih2dimb 41873 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 41874 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 41875 | Ordered pair member of the... |
| dihvalcq2 41876 | Value of isomorphism H for... |
| dihopelvalcpre 41877 | Member of value of isomorp... |
| dihopelvalc 41878 | Member of value of isomorp... |
| dihlss 41879 | The value of isomorphism H... |
| dihss 41880 | The value of isomorphism H... |
| dihssxp 41881 | An isomorphism H value is ... |
| dihopcl 41882 | Closure of an ordered pair... |
| xihopellsmN 41883 | Ordered pair membership in... |
| dihopellsm 41884 | Ordered pair membership in... |
| dihord6apre 41885 | Part of proof that isomorp... |
| dihord3 41886 | The isomorphism H for a la... |
| dihord4 41887 | The isomorphism H for a la... |
| dihord5b 41888 | Part of proof that isomorp... |
| dihord6b 41889 | Part of proof that isomorp... |
| dihord6a 41890 | Part of proof that isomorp... |
| dihord5apre 41891 | Part of proof that isomorp... |
| dihord5a 41892 | Part of proof that isomorp... |
| dihord 41893 | The isomorphism H is order... |
| dih11 41894 | The isomorphism H is one-t... |
| dihf11lem 41895 | Functionality of the isomo... |
| dihf11 41896 | The isomorphism H for a la... |
| dihfn 41897 | Functionality and domain o... |
| dihdm 41898 | Domain of isomorphism H. (... |
| dihcl 41899 | Closure of isomorphism H. ... |
| dihcnvcl 41900 | Closure of isomorphism H c... |
| dihcnvid1 41901 | The converse isomorphism o... |
| dihcnvid2 41902 | The isomorphism of a conve... |
| dihcnvord 41903 | Ordering property for conv... |
| dihcnv11 41904 | The converse of isomorphis... |
| dihsslss 41905 | The isomorphism H maps to ... |
| dihrnlss 41906 | The isomorphism H maps to ... |
| dihrnss 41907 | The isomorphism H maps to ... |
| dihvalrel 41908 | The value of isomorphism H... |
| dih0 41909 | The value of isomorphism H... |
| dih0bN 41910 | A lattice element is zero ... |
| dih0vbN 41911 | A vector is zero iff its s... |
| dih0cnv 41912 | The isomorphism H converse... |
| dih0rn 41913 | The zero subspace belongs ... |
| dih0sb 41914 | A subspace is zero iff the... |
| dih1 41915 | The value of isomorphism H... |
| dih1rn 41916 | The full vector space belo... |
| dih1cnv 41917 | The isomorphism H converse... |
| dihwN 41918 | Value of isomorphism H at ... |
| dihmeetlem1N 41919 | Isomorphism H of a conjunc... |
| dihglblem5apreN 41920 | A conjunction property of ... |
| dihglblem5aN 41921 | A conjunction property of ... |
| dihglblem2aN 41922 | Lemma for isomorphism H of... |
| dihglblem2N 41923 | The GLB of a set of lattic... |
| dihglblem3N 41924 | Isomorphism H of a lattice... |
| dihglblem3aN 41925 | Isomorphism H of a lattice... |
| dihglblem4 41926 | Isomorphism H of a lattice... |
| dihglblem5 41927 | Isomorphism H of a lattice... |
| dihmeetlem2N 41928 | Isomorphism H of a conjunc... |
| dihglbcpreN 41929 | Isomorphism H of a lattice... |
| dihglbcN 41930 | Isomorphism H of a lattice... |
| dihmeetcN 41931 | Isomorphism H of a lattice... |
| dihmeetbN 41932 | Isomorphism H of a lattice... |
| dihmeetbclemN 41933 | Lemma for isomorphism H of... |
| dihmeetlem3N 41934 | Lemma for isomorphism H of... |
| dihmeetlem4preN 41935 | Lemma for isomorphism H of... |
| dihmeetlem4N 41936 | Lemma for isomorphism H of... |
| dihmeetlem5 41937 | Part of proof that isomorp... |
| dihmeetlem6 41938 | Lemma for isomorphism H of... |
| dihmeetlem7N 41939 | Lemma for isomorphism H of... |
| dihjatc1 41940 | Lemma for isomorphism H of... |
| dihjatc2N 41941 | Isomorphism H of join with... |
| dihjatc3 41942 | Isomorphism H of join with... |
| dihmeetlem8N 41943 | Lemma for isomorphism H of... |
| dihmeetlem9N 41944 | Lemma for isomorphism H of... |
| dihmeetlem10N 41945 | Lemma for isomorphism H of... |
| dihmeetlem11N 41946 | Lemma for isomorphism H of... |
| dihmeetlem12N 41947 | Lemma for isomorphism H of... |
| dihmeetlem13N 41948 | Lemma for isomorphism H of... |
| dihmeetlem14N 41949 | Lemma for isomorphism H of... |
| dihmeetlem15N 41950 | Lemma for isomorphism H of... |
| dihmeetlem16N 41951 | Lemma for isomorphism H of... |
| dihmeetlem17N 41952 | Lemma for isomorphism H of... |
| dihmeetlem18N 41953 | Lemma for isomorphism H of... |
| dihmeetlem19N 41954 | Lemma for isomorphism H of... |
| dihmeetlem20N 41955 | Lemma for isomorphism H of... |
| dihmeetALTN 41956 | Isomorphism H of a lattice... |
| dih1dimatlem0 41957 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 41958 | Lemma for ~ dih1dimat . (... |
| dih1dimat 41959 | Any 1-dimensional subspace... |
| dihlsprn 41960 | The span of a vector belon... |
| dihlspsnssN 41961 | A subspace included in a 1... |
| dihlspsnat 41962 | The inverse isomorphism H ... |
| dihatlat 41963 | The isomorphism H of an at... |
| dihat 41964 | There exists at least one ... |
| dihpN 41965 | The value of isomorphism H... |
| dihlatat 41966 | The reverse isomorphism H ... |
| dihatexv 41967 | There is a nonzero vector ... |
| dihatexv2 41968 | There is a nonzero vector ... |
| dihglblem6 41969 | Isomorphism H of a lattice... |
| dihglb 41970 | Isomorphism H of a lattice... |
| dihglb2 41971 | Isomorphism H of a lattice... |
| dihmeet 41972 | Isomorphism H of a lattice... |
| dihintcl 41973 | The intersection of closed... |
| dihmeetcl 41974 | Closure of closed subspace... |
| dihmeet2 41975 | Reverse isomorphism H of a... |
| dochffval 41978 | Subspace orthocomplement f... |
| dochfval 41979 | Subspace orthocomplement f... |
| dochval 41980 | Subspace orthocomplement f... |
| dochval2 41981 | Subspace orthocomplement f... |
| dochcl 41982 | Closure of subspace orthoc... |
| dochlss 41983 | A subspace orthocomplement... |
| dochssv 41984 | A subspace orthocomplement... |
| dochfN 41985 | Domain and codomain of the... |
| dochvalr 41986 | Orthocomplement of a close... |
| doch0 41987 | Orthocomplement of the zer... |
| doch1 41988 | Orthocomplement of the uni... |
| dochoc0 41989 | The zero subspace is close... |
| dochoc1 41990 | The unit subspace (all vec... |
| dochvalr2 41991 | Orthocomplement of a close... |
| dochvalr3 41992 | Orthocomplement of a close... |
| doch2val2 41993 | Double orthocomplement for... |
| dochss 41994 | Subset law for orthocomple... |
| dochocss 41995 | Double negative law for or... |
| dochoc 41996 | Double negative law for or... |
| dochsscl 41997 | If a set of vectors is inc... |
| dochoccl 41998 | A set of vectors is closed... |
| dochord 41999 | Ordering law for orthocomp... |
| dochord2N 42000 | Ordering law for orthocomp... |
| dochord3 42001 | Ordering law for orthocomp... |
| doch11 42002 | Orthocomplement is one-to-... |
| dochsordN 42003 | Strict ordering law for or... |
| dochn0nv 42004 | An orthocomplement is nonz... |
| dihoml4c 42005 | Version of ~ dihoml4 with ... |
| dihoml4 42006 | Orthomodular law for const... |
| dochspss 42007 | The span of a set of vecto... |
| dochocsp 42008 | The span of an orthocomple... |
| dochspocN 42009 | The span of an orthocomple... |
| dochocsn 42010 | The double orthocomplement... |
| dochsncom 42011 | Swap vectors in an orthoco... |
| dochsat 42012 | The double orthocomplement... |
| dochshpncl 42013 | If a hyperplane is not clo... |
| dochlkr 42014 | Equivalent conditions for ... |
| dochkrshp 42015 | The closure of a kernel is... |
| dochkrshp2 42016 | Properties of the closure ... |
| dochkrshp3 42017 | Properties of the closure ... |
| dochkrshp4 42018 | Properties of the closure ... |
| dochdmj1 42019 | De Morgan-like law for sub... |
| dochnoncon 42020 | Law of noncontradiction. ... |
| dochnel2 42021 | A nonzero member of a subs... |
| dochnel 42022 | A nonzero vector doesn't b... |
| djhffval 42025 | Subspace join for ` DVecH ... |
| djhfval 42026 | Subspace join for ` DVecH ... |
| djhval 42027 | Subspace join for ` DVecH ... |
| djhval2 42028 | Value of subspace join for... |
| djhcl 42029 | Closure of subspace join f... |
| djhlj 42030 | Transfer lattice join to `... |
| djhljjN 42031 | Lattice join in terms of `... |
| djhjlj 42032 | ` DVecH ` vector space clo... |
| djhj 42033 | ` DVecH ` vector space clo... |
| djhcom 42034 | Subspace join commutes. (... |
| djhspss 42035 | Subspace span of union is ... |
| djhsumss 42036 | Subspace sum is a subset o... |
| dihsumssj 42037 | The subspace sum of two is... |
| djhunssN 42038 | Subspace union is a subset... |
| dochdmm1 42039 | De Morgan-like law for clo... |
| djhexmid 42040 | Excluded middle property o... |
| djh01 42041 | Closed subspace join with ... |
| djh02 42042 | Closed subspace join with ... |
| djhlsmcl 42043 | A closed subspace sum equa... |
| djhcvat42 42044 | A covering property. ( ~ ... |
| dihjatb 42045 | Isomorphism H of lattice j... |
| dihjatc 42046 | Isomorphism H of lattice j... |
| dihjatcclem1 42047 | Lemma for isomorphism H of... |
| dihjatcclem2 42048 | Lemma for isomorphism H of... |
| dihjatcclem3 42049 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 42050 | Lemma for isomorphism H of... |
| dihjatcc 42051 | Isomorphism H of lattice j... |
| dihjat 42052 | Isomorphism H of lattice j... |
| dihprrnlem1N 42053 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 42054 | Lemma for ~ dihprrn . (Co... |
| dihprrn 42055 | The span of a vector pair ... |
| djhlsmat 42056 | The sum of two subspace at... |
| dihjat1lem 42057 | Subspace sum of a closed s... |
| dihjat1 42058 | Subspace sum of a closed s... |
| dihsmsprn 42059 | Subspace sum of a closed s... |
| dihjat2 42060 | The subspace sum of a clos... |
| dihjat3 42061 | Isomorphism H of lattice j... |
| dihjat4 42062 | Transfer the subspace sum ... |
| dihjat6 42063 | Transfer the subspace sum ... |
| dihsmsnrn 42064 | The subspace sum of two si... |
| dihsmatrn 42065 | The subspace sum of a clos... |
| dihjat5N 42066 | Transfer lattice join with... |
| dvh4dimat 42067 | There is an atom that is o... |
| dvh3dimatN 42068 | There is an atom that is o... |
| dvh2dimatN 42069 | Given an atom, there exist... |
| dvh1dimat 42070 | There exists an atom. (Co... |
| dvh1dim 42071 | There exists a nonzero vec... |
| dvh4dimlem 42072 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 42073 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 42074 | There is a vector that is ... |
| dvh3dim 42075 | There is a vector that is ... |
| dvh4dimN 42076 | There is a vector that is ... |
| dvh3dim2 42077 | There is a vector that is ... |
| dvh3dim3N 42078 | There is a vector that is ... |
| dochsnnz 42079 | The orthocomplement of a s... |
| dochsatshp 42080 | The orthocomplement of a s... |
| dochsatshpb 42081 | The orthocomplement of a s... |
| dochsnshp 42082 | The orthocomplement of a n... |
| dochshpsat 42083 | A hyperplane is closed iff... |
| dochkrsat 42084 | The orthocomplement of a k... |
| dochkrsat2 42085 | The orthocomplement of a k... |
| dochsat0 42086 | The orthocomplement of a k... |
| dochkrsm 42087 | The subspace sum of a clos... |
| dochexmidat 42088 | Special case of excluded m... |
| dochexmidlem1 42089 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 42090 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 42091 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 42092 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 42093 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 42094 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 42095 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 42096 | Lemma for ~ dochexmid . T... |
| dochexmid 42097 | Excluded middle law for cl... |
| dochsnkrlem1 42098 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 42099 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 42100 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 42101 | A (closed) kernel expresse... |
| dochsnkr2 42102 | Kernel of the explicit fun... |
| dochsnkr2cl 42103 | The ` X ` determining func... |
| dochflcl 42104 | Closure of the explicit fu... |
| dochfl1 42105 | The value of the explicit ... |
| dochfln0 42106 | The value of a functional ... |
| dochkr1 42107 | A nonzero functional has a... |
| dochkr1OLDN 42108 | A nonzero functional has a... |
| lpolsetN 42111 | The set of polarities of a... |
| islpolN 42112 | The predicate "is a polari... |
| islpoldN 42113 | Properties that determine ... |
| lpolfN 42114 | Functionality of a polarit... |
| lpolvN 42115 | The polarity of the whole ... |
| lpolconN 42116 | Contraposition property of... |
| lpolsatN 42117 | The polarity of an atomic ... |
| lpolpolsatN 42118 | Property of a polarity. (... |
| dochpolN 42119 | The subspace orthocompleme... |
| lcfl1lem 42120 | Property of a functional w... |
| lcfl1 42121 | Property of a functional w... |
| lcfl2 42122 | Property of a functional w... |
| lcfl3 42123 | Property of a functional w... |
| lcfl4N 42124 | Property of a functional w... |
| lcfl5 42125 | Property of a functional w... |
| lcfl5a 42126 | Property of a functional w... |
| lcfl6lem 42127 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 42128 | Lemma for ~ lcfl7N . If t... |
| lcfl6 42129 | Property of a functional w... |
| lcfl7N 42130 | Property of a functional w... |
| lcfl8 42131 | Property of a functional w... |
| lcfl8a 42132 | Property of a functional w... |
| lcfl8b 42133 | Property of a nonzero func... |
| lcfl9a 42134 | Property implying that a f... |
| lclkrlem1 42135 | The set of functionals hav... |
| lclkrlem2a 42136 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 42137 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 42138 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 42139 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 42140 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 42141 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 42142 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 42143 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 42144 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 42145 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 42146 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 42147 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 42148 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 42149 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 42150 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 42151 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 42152 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 42153 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 42154 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 42155 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 42156 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 42157 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 42158 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 42159 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 42160 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 42161 | The set of functionals hav... |
| lclkr 42162 | The set of functionals wit... |
| lcfls1lem 42163 | Property of a functional w... |
| lcfls1N 42164 | Property of a functional w... |
| lcfls1c 42165 | Property of a functional w... |
| lclkrslem1 42166 | The set of functionals hav... |
| lclkrslem2 42167 | The set of functionals hav... |
| lclkrs 42168 | The set of functionals hav... |
| lclkrs2 42169 | The set of functionals wit... |
| lcfrvalsnN 42170 | Reconstruction from the du... |
| lcfrlem1 42171 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 42172 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 42173 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 42174 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 42175 | Lemma for ~ lcfr . The se... |
| lcfrlem6 42176 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 42177 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 42178 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 42179 | Lemma for ~ lcf1o . (This... |
| lcf1o 42180 | Define a function ` J ` th... |
| lcfrlem10 42181 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 42182 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 42183 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 42184 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 42185 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 42186 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 42187 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 42188 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 42189 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 42190 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 42191 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 42192 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 42193 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 42194 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 42195 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 42196 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 42197 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 42198 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 42199 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 42200 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 42201 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 42202 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 42203 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 42204 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 42205 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 42206 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 42207 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 42208 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 42209 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 42210 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 42211 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 42212 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 42213 | Lemma for ~ lcfr . Elimin... |
| lcfr 42214 | Reconstruction of a subspa... |
| lcdfval 42217 | Dual vector space of funct... |
| lcdval 42218 | Dual vector space of funct... |
| lcdval2 42219 | Dual vector space of funct... |
| lcdlvec 42220 | The dual vector space of f... |
| lcdlmod 42221 | The dual vector space of f... |
| lcdvbase 42222 | Vector base set of a dual ... |
| lcdvbasess 42223 | The vector base set of the... |
| lcdvbaselfl 42224 | A vector in the base set o... |
| lcdvbasecl 42225 | Closure of the value of a ... |
| lcdvadd 42226 | Vector addition for the cl... |
| lcdvaddval 42227 | The value of the value of ... |
| lcdsca 42228 | The ring of scalars of the... |
| lcdsbase 42229 | Base set of scalar ring fo... |
| lcdsadd 42230 | Scalar addition for the cl... |
| lcdsmul 42231 | Scalar multiplication for ... |
| lcdvs 42232 | Scalar product for the clo... |
| lcdvsval 42233 | Value of scalar product op... |
| lcdvscl 42234 | The scalar product operati... |
| lcdlssvscl 42235 | Closure of scalar product ... |
| lcdvsass 42236 | Associative law for scalar... |
| lcd0 42237 | The zero scalar of the clo... |
| lcd1 42238 | The unit scalar of the clo... |
| lcdneg 42239 | The unit scalar of the clo... |
| lcd0v 42240 | The zero functional in the... |
| lcd0v2 42241 | The zero functional in the... |
| lcd0vvalN 42242 | Value of the zero function... |
| lcd0vcl 42243 | Closure of the zero functi... |
| lcd0vs 42244 | A scalar zero times a func... |
| lcdvs0N 42245 | A scalar times the zero fu... |
| lcdvsub 42246 | The value of vector subtra... |
| lcdvsubval 42247 | The value of the value of ... |
| lcdlss 42248 | Subspaces of a dual vector... |
| lcdlss2N 42249 | Subspaces of a dual vector... |
| lcdlsp 42250 | Span in the set of functio... |
| lcdlkreqN 42251 | Colinear functionals have ... |
| lcdlkreq2N 42252 | Colinear functionals have ... |
| mapdffval 42255 | Projectivity from vector s... |
| mapdfval 42256 | Projectivity from vector s... |
| mapdval 42257 | Value of projectivity from... |
| mapdvalc 42258 | Value of projectivity from... |
| mapdval2N 42259 | Value of projectivity from... |
| mapdval3N 42260 | Value of projectivity from... |
| mapdval4N 42261 | Value of projectivity from... |
| mapdval5N 42262 | Value of projectivity from... |
| mapdordlem1a 42263 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 42264 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 42265 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 42266 | Lemma for ~ mapdord . Ord... |
| mapdord 42267 | Ordering property of the m... |
| mapd11 42268 | The map defined by ~ df-ma... |
| mapddlssN 42269 | The mapping of a subspace ... |
| mapdsn 42270 | Value of the map defined b... |
| mapdsn2 42271 | Value of the map defined b... |
| mapdsn3 42272 | Value of the map defined b... |
| mapd1dim2lem1N 42273 | Value of the map defined b... |
| mapdrvallem2 42274 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 42275 | Lemma for ~ mapdrval . (C... |
| mapdrval 42276 | Given a dual subspace ` R ... |
| mapd1o 42277 | The map defined by ~ df-ma... |
| mapdrn 42278 | Range of the map defined b... |
| mapdunirnN 42279 | Union of the range of the ... |
| mapdrn2 42280 | Range of the map defined b... |
| mapdcnvcl 42281 | Closure of the converse of... |
| mapdcl 42282 | Closure the value of the m... |
| mapdcnvid1N 42283 | Converse of the value of t... |
| mapdsord 42284 | Strong ordering property o... |
| mapdcl2 42285 | The mapping of a subspace ... |
| mapdcnvid2 42286 | Value of the converse of t... |
| mapdcnvordN 42287 | Ordering property of the c... |
| mapdcnv11N 42288 | The converse of the map de... |
| mapdcv 42289 | Covering property of the c... |
| mapdincl 42290 | Closure of dual subspace i... |
| mapdin 42291 | Subspace intersection is p... |
| mapdlsmcl 42292 | Closure of dual subspace s... |
| mapdlsm 42293 | Subspace sum is preserved ... |
| mapd0 42294 | Projectivity map of the ze... |
| mapdcnvatN 42295 | Atoms are preserved by the... |
| mapdat 42296 | Atoms are preserved by the... |
| mapdspex 42297 | The map of a span equals t... |
| mapdn0 42298 | Transfer nonzero property ... |
| mapdncol 42299 | Transfer non-colinearity f... |
| mapdindp 42300 | Transfer (part of) vector ... |
| mapdpglem1 42301 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 42302 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 42303 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 42304 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 42305 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 42306 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 42307 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 42308 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 42309 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 42310 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 42311 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 42312 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 42313 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 42314 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 42315 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 42316 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 42317 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 42318 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 42319 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 42320 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 42321 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 42322 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 42323 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 42324 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 42325 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 42326 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 42327 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 42328 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 42329 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 42330 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 42331 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 42332 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 42333 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 42334 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 42335 | Part 1 of proof of the fir... |
| baerlem3lem1 42336 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 42337 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 42338 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 42339 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 42340 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 42341 | Lemma for ~ baerlem5b . (... |
| baerlem3 42342 | An equality that holds whe... |
| baerlem5a 42343 | An equality that holds whe... |
| baerlem5b 42344 | An equality that holds whe... |
| baerlem5amN 42345 | An equality that holds whe... |
| baerlem5bmN 42346 | An equality that holds whe... |
| baerlem5abmN 42347 | An equality that holds whe... |
| mapdindp0 42348 | Vector independence lemma.... |
| mapdindp1 42349 | Vector independence lemma.... |
| mapdindp2 42350 | Vector independence lemma.... |
| mapdindp3 42351 | Vector independence lemma.... |
| mapdindp4 42352 | Vector independence lemma.... |
| mapdhval 42353 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 42354 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 42355 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 42356 | Lemmma for ~~? mapdh . (C... |
| mapdheq 42357 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 42358 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 42359 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 42360 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 42361 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 42362 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 42363 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 42364 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 42365 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 42366 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 42367 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 42368 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 42369 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 42370 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 42371 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 42372 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 42373 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 42374 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 42375 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 42376 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 42377 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 42378 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 42379 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 42380 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 42381 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 42382 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 42383 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 42384 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 42387 | Map from nonzero vectors t... |
| hvmapfval 42388 | Map from nonzero vectors t... |
| hvmapval 42389 | Value of map from nonzero ... |
| hvmapvalvalN 42390 | Value of value of map (i.e... |
| hvmapidN 42391 | The value of the vector to... |
| hvmap1o 42392 | The vector to functional m... |
| hvmapclN 42393 | Closure of the vector to f... |
| hvmap1o2 42394 | The vector to functional m... |
| hvmapcl2 42395 | Closure of the vector to f... |
| hvmaplfl 42396 | The vector to functional m... |
| hvmaplkr 42397 | Kernel of the vector to fu... |
| mapdhvmap 42398 | Relationship between ` map... |
| lspindp5 42399 | Obtain an independent vect... |
| hdmaplem1 42400 | Lemma to convert a frequen... |
| hdmaplem2N 42401 | Lemma to convert a frequen... |
| hdmaplem3 42402 | Lemma to convert a frequen... |
| hdmaplem4 42403 | Lemma to convert a frequen... |
| mapdh8a 42404 | Part of Part (8) in [Baer]... |
| mapdh8aa 42405 | Part of Part (8) in [Baer]... |
| mapdh8ab 42406 | Part of Part (8) in [Baer]... |
| mapdh8ac 42407 | Part of Part (8) in [Baer]... |
| mapdh8ad 42408 | Part of Part (8) in [Baer]... |
| mapdh8b 42409 | Part of Part (8) in [Baer]... |
| mapdh8c 42410 | Part of Part (8) in [Baer]... |
| mapdh8d0N 42411 | Part of Part (8) in [Baer]... |
| mapdh8d 42412 | Part of Part (8) in [Baer]... |
| mapdh8e 42413 | Part of Part (8) in [Baer]... |
| mapdh8g 42414 | Part of Part (8) in [Baer]... |
| mapdh8i 42415 | Part of Part (8) in [Baer]... |
| mapdh8j 42416 | Part of Part (8) in [Baer]... |
| mapdh8 42417 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 42418 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 42419 | Lemma for part (9) in [Bae... |
| hdmap1ffval 42424 | Preliminary map from vecto... |
| hdmap1fval 42425 | Preliminary map from vecto... |
| hdmap1vallem 42426 | Value of preliminary map f... |
| hdmap1val 42427 | Value of preliminary map f... |
| hdmap1val0 42428 | Value of preliminary map f... |
| hdmap1val2 42429 | Value of preliminary map f... |
| hdmap1eq 42430 | The defining equation for ... |
| hdmap1cbv 42431 | Frequently used lemma to c... |
| hdmap1valc 42432 | Connect the value of the p... |
| hdmap1cl 42433 | Convert closure theorem ~ ... |
| hdmap1eq2 42434 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 42435 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 42436 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 42437 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 42438 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 42439 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 42440 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 42441 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 42442 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 42443 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 42444 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 42445 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 42446 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 42447 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 42448 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 42449 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 42450 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 42451 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 42452 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 42453 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 42454 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 42455 | Map from vectors to functi... |
| hdmapfval 42456 | Map from vectors to functi... |
| hdmapval 42457 | Value of map from vectors ... |
| hdmapfnN 42458 | Functionality of map from ... |
| hdmapcl 42459 | Closure of map from vector... |
| hdmapval2lem 42460 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 42461 | Value of map from vectors ... |
| hdmapval0 42462 | Value of map from vectors ... |
| hdmapeveclem 42463 | Lemma for ~ hdmapevec . T... |
| hdmapevec 42464 | Value of map from vectors ... |
| hdmapevec2 42465 | The inner product of the r... |
| hdmapval3lemN 42466 | Value of map from vectors ... |
| hdmapval3N 42467 | Value of map from vectors ... |
| hdmap10lem 42468 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 42469 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 42470 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 42471 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 42472 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 42473 | Part of proof of part 12 i... |
| hdmapnzcl 42474 | Nonzero vector closure of ... |
| hdmapneg 42475 | Part of proof of part 12 i... |
| hdmapsub 42476 | Part of proof of part 12 i... |
| hdmap11 42477 | Part of proof of part 12 i... |
| hdmaprnlem1N 42478 | Part of proof of part 12 i... |
| hdmaprnlem3N 42479 | Part of proof of part 12 i... |
| hdmaprnlem3uN 42480 | Part of proof of part 12 i... |
| hdmaprnlem4tN 42481 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 42482 | Part of proof of part 12 i... |
| hdmaprnlem6N 42483 | Part of proof of part 12 i... |
| hdmaprnlem7N 42484 | Part of proof of part 12 i... |
| hdmaprnlem8N 42485 | Part of proof of part 12 i... |
| hdmaprnlem9N 42486 | Part of proof of part 12 i... |
| hdmaprnlem3eN 42487 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 42488 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 42489 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 42490 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 42491 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 42492 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 42493 | Part of proof of part 12 i... |
| hdmapf1oN 42494 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 42495 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 42496 | Prior to part 14 in [Baer]... |
| hdmap14lem1 42497 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 42498 | Prior to part 14 in [Baer]... |
| hdmap14lem3 42499 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 42500 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 42501 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 42502 | Case where ` F ` is zero. ... |
| hdmap14lem7 42503 | Combine cases of ` F ` . ... |
| hdmap14lem8 42504 | Part of proof of part 14 i... |
| hdmap14lem9 42505 | Part of proof of part 14 i... |
| hdmap14lem10 42506 | Part of proof of part 14 i... |
| hdmap14lem11 42507 | Part of proof of part 14 i... |
| hdmap14lem12 42508 | Lemma for proof of part 14... |
| hdmap14lem13 42509 | Lemma for proof of part 14... |
| hdmap14lem14 42510 | Part of proof of part 14 i... |
| hdmap14lem15 42511 | Part of proof of part 14 i... |
| hgmapffval 42514 | Map from the scalar divisi... |
| hgmapfval 42515 | Map from the scalar divisi... |
| hgmapval 42516 | Value of map from the scal... |
| hgmapfnN 42517 | Functionality of scalar si... |
| hgmapcl 42518 | Closure of scalar sigma ma... |
| hgmapdcl 42519 | Closure of the vector spac... |
| hgmapvs 42520 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 42521 | Value of the scalar sigma ... |
| hgmapval1 42522 | Value of the scalar sigma ... |
| hgmapadd 42523 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 42524 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 42525 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 42526 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 42527 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 42528 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 42529 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 42530 | Part of proof of part 16 i... |
| hgmap11 42531 | The scalar sigma map is on... |
| hgmapf1oN 42532 | The scalar sigma map is a ... |
| hgmapeq0 42533 | The scalar sigma map is ze... |
| hdmapipcl 42534 | The inner product (Hermiti... |
| hdmapln1 42535 | Linearity property that wi... |
| hdmaplna1 42536 | Additive property of first... |
| hdmaplns1 42537 | Subtraction property of fi... |
| hdmaplnm1 42538 | Multiplicative property of... |
| hdmaplna2 42539 | Additive property of secon... |
| hdmapglnm2 42540 | g-linear property of secon... |
| hdmapgln2 42541 | g-linear property that wil... |
| hdmaplkr 42542 | Kernel of the vector to du... |
| hdmapellkr 42543 | Membership in the kernel (... |
| hdmapip0 42544 | Zero property that will be... |
| hdmapip1 42545 | Construct a proportional v... |
| hdmapip0com 42546 | Commutation property of Ba... |
| hdmapinvlem1 42547 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 42548 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 42549 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 42550 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 42551 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 42552 | Involution property of sca... |
| hgmapvvlem2 42553 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 42554 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 42555 | Value of a double involuti... |
| hdmapglem7a 42556 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 42557 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 42558 | Lemma for ~ hdmapg . Line... |
| hdmapg 42559 | Apply the scalar sigma fun... |
| hdmapoc 42560 | Express our constructed or... |
| hlhilset 42563 | The final Hilbert space co... |
| hlhilsca 42564 | The scalar of the final co... |
| hlhilbase 42565 | The base set of the final ... |
| hlhilplus 42566 | The vector addition for th... |
| hlhilslem 42567 | Lemma for ~ hlhilsbase etc... |
| hlhilsbase 42568 | The scalar base set of the... |
| hlhilsplus 42569 | Scalar addition for the fi... |
| hlhilsmul 42570 | Scalar multiplication for ... |
| hlhilsbase2 42571 | The scalar base set of the... |
| hlhilsplus2 42572 | Scalar addition for the fi... |
| hlhilsmul2 42573 | Scalar multiplication for ... |
| hlhils0 42574 | The scalar ring zero for t... |
| hlhils1N 42575 | The scalar ring unity for ... |
| hlhilvsca 42576 | The scalar product for the... |
| hlhilip 42577 | Inner product operation fo... |
| hlhilipval 42578 | Value of inner product ope... |
| hlhilnvl 42579 | The involution operation o... |
| hlhillvec 42580 | The final constructed Hilb... |
| hlhildrng 42581 | The star division ring for... |
| hlhilsrnglem 42582 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 42583 | The star division ring for... |
| hlhil0 42584 | The zero vector for the fi... |
| hlhillsm 42585 | The vector sum operation f... |
| hlhilocv 42586 | The orthocomplement for th... |
| hlhillcs 42587 | The closed subspaces of th... |
| hlhilphllem 42588 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 42589 | Lemma for ~ hlhil . (Cont... |
| hlathil 42590 | Construction of a Hilbert ... |
| iscsrg 42593 | A commutative semiring is ... |
| rhmzrhval 42594 | Evaluation of integers acr... |
| zndvdchrrhm 42595 | Construction of a ring hom... |
| relogbcld 42596 | Closure of the general log... |
| relogbexpd 42597 | Identity law for general l... |
| relogbzexpd 42598 | Power law for the general ... |
| logblebd 42599 | The general logarithm is m... |
| uzindd 42600 | Induction on the upper int... |
| fzadd2d 42601 | Membership of a sum in a f... |
| fzne2d 42602 | Elementhood in a finite se... |
| eqfnfv2d2 42603 | Equality of functions is d... |
| fzsplitnd 42604 | Split a finite interval of... |
| fzsplitnr 42605 | Split a finite interval of... |
| addassnni 42606 | Associative law for additi... |
| addcomnni 42607 | Commutative law for additi... |
| mulassnni 42608 | Associative law for multip... |
| mulcomnni 42609 | Commutative law for multip... |
| gcdcomnni 42610 | Commutative law for gcd. ... |
| gcdnegnni 42611 | Negation invariance for gc... |
| neggcdnni 42612 | Negation invariance for gc... |
| bccl2d 42613 | Closure of the binomial co... |
| recbothd 42614 | Take reciprocal on both si... |
| gcdmultiplei 42615 | The GCD of a multiple of a... |
| gcdaddmzz2nni 42616 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 42617 | Adding a multiple of one o... |
| gcdnncli 42618 | Closure of the gcd operato... |
| muldvds1d 42619 | If a product divides an in... |
| muldvds2d 42620 | If a product divides an in... |
| nndivdvdsd 42621 | A positive integer divides... |
| nnproddivdvdsd 42622 | A product of natural numbe... |
| coprmdvds2d 42623 | If an integer is divisible... |
| imadomfi 42624 | An image of a function und... |
| 12gcd5e1 42625 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 42626 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 42627 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 42628 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 42629 | Calculate the least common... |
| 12lcm5e60 42630 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 42631 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 42632 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 42633 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 42634 | Useful equation to calcula... |
| lcm1un 42635 | Least common multiple of n... |
| lcm2un 42636 | Least common multiple of n... |
| lcm3un 42637 | Least common multiple of n... |
| lcm4un 42638 | Least common multiple of n... |
| lcm5un 42639 | Least common multiple of n... |
| lcm6un 42640 | Least common multiple of n... |
| lcm7un 42641 | Least common multiple of n... |
| lcm8un 42642 | Least common multiple of n... |
| 3factsumint1 42643 | Move constants out of inte... |
| 3factsumint2 42644 | Move constants out of inte... |
| 3factsumint3 42645 | Move constants out of inte... |
| 3factsumint4 42646 | Move constants out of inte... |
| 3factsumint 42647 | Helpful equation for lcm i... |
| resopunitintvd 42648 | Restrict continuous functi... |
| resclunitintvd 42649 | Restrict continuous functi... |
| resdvopclptsd 42650 | Restrict derivative on uni... |
| lcmineqlem1 42651 | Part of lcm inequality lem... |
| lcmineqlem2 42652 | Part of lcm inequality lem... |
| lcmineqlem3 42653 | Part of lcm inequality lem... |
| lcmineqlem4 42654 | Part of lcm inequality lem... |
| lcmineqlem5 42655 | Technical lemma for recipr... |
| lcmineqlem6 42656 | Part of lcm inequality lem... |
| lcmineqlem7 42657 | Derivative of 1-x for chai... |
| lcmineqlem8 42658 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 42659 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 42660 | Induction step of ~ lcmine... |
| lcmineqlem11 42661 | Induction step, continuati... |
| lcmineqlem12 42662 | Base case for induction. ... |
| lcmineqlem13 42663 | Induction proof for lcm in... |
| lcmineqlem14 42664 | Technical lemma for inequa... |
| lcmineqlem15 42665 | F times the least common m... |
| lcmineqlem16 42666 | Technical divisibility lem... |
| lcmineqlem17 42667 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 42668 | Technical lemma to shift f... |
| lcmineqlem19 42669 | Dividing implies inequalit... |
| lcmineqlem20 42670 | Inequality for lcm lemma. ... |
| lcmineqlem21 42671 | The lcm inequality lemma w... |
| lcmineqlem22 42672 | The lcm inequality lemma w... |
| lcmineqlem23 42673 | Penultimate step to the lc... |
| lcmineqlem 42674 | The least common multiple ... |
| 3exp7 42675 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 42676 | First inequality in inequa... |
| 3lexlogpow5ineq2 42677 | Second inequality in inequ... |
| 3lexlogpow5ineq4 42678 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 42679 | Combined inequality chain ... |
| 3lexlogpow2ineq1 42680 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 42681 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 42682 | Result for bound in AKS in... |
| intlewftc 42683 | Inequality inference by in... |
| aks4d1lem1 42684 | Technical lemma to reduce ... |
| aks4d1p1p1 42685 | Exponential law for finite... |
| dvrelog2 42686 | The derivative of the loga... |
| dvrelog3 42687 | The derivative of the loga... |
| dvrelog2b 42688 | Derivative of the binary l... |
| 0nonelalab 42689 | Technical lemma for open i... |
| dvrelogpow2b 42690 | Derivative of the power of... |
| aks4d1p1p3 42691 | Bound of a ceiling of the ... |
| aks4d1p1p2 42692 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 42693 | Technical step for inequal... |
| dvle2 42694 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 42695 | Inequality lift to differe... |
| aks4d1p1p7 42696 | Bound of intermediary of i... |
| aks4d1p1p5 42697 | Show inequality for existe... |
| aks4d1p1 42698 | Show inequality for existe... |
| aks4d1p2 42699 | Technical lemma for existe... |
| aks4d1p3 42700 | There exists a small enoug... |
| aks4d1p4 42701 | There exists a small enoug... |
| aks4d1p5 42702 | Show that ` N ` and ` R ` ... |
| aks4d1p6 42703 | The maximal prime power ex... |
| aks4d1p7d1 42704 | Technical step in AKS lemm... |
| aks4d1p7 42705 | Technical step in AKS lemm... |
| aks4d1p8d1 42706 | If a prime divides one num... |
| aks4d1p8d2 42707 | Any prime power dividing a... |
| aks4d1p8d3 42708 | The remainder of a divisio... |
| aks4d1p8 42709 | Show that ` N ` and ` R ` ... |
| aks4d1p9 42710 | Show that the order is bou... |
| aks4d1 42711 | Lemma 4.1 from ~ https://w... |
| fldhmf1 42712 | A field homomorphism is in... |
| isprimroot 42715 | The value of a primitive r... |
| isprimroot2 42716 | Alternative way of creatin... |
| mndmolinv 42717 | An element of a monoid tha... |
| linvh 42718 | If an element has a unique... |
| primrootsunit1 42719 | Primitive roots have left ... |
| primrootsunit 42720 | Primitive roots have left ... |
| primrootscoprmpow 42721 | Coprime powers of primitiv... |
| posbezout 42722 | Bezout's identity restrict... |
| primrootscoprf 42723 | Coprime powers of primitiv... |
| primrootscoprbij 42724 | A bijection between coprim... |
| primrootscoprbij2 42725 | A bijection between coprim... |
| remexz 42726 | Division with rest. (Cont... |
| primrootlekpowne0 42727 | There is no smaller power ... |
| primrootspoweq0 42728 | The power of a ` R ` -th p... |
| aks6d1c1p1 42729 | Definition of the introspe... |
| aks6d1c1p1rcl 42730 | Reverse closure of the int... |
| aks6d1c1p2 42731 | ` P ` and linear factors a... |
| aks6d1c1p3 42732 | In a field with a Frobeniu... |
| aks6d1c1p4 42733 | The product of polynomials... |
| aks6d1c1p5 42734 | The product of exponents i... |
| aks6d1c1p7 42735 | ` X ` is introspective to ... |
| aks6d1c1p6 42736 | If a polynomials ` F ` is ... |
| aks6d1c1p8 42737 | If a number ` E ` is intro... |
| aks6d1c1 42738 | Claim 1 of Theorem 6.1 ~ h... |
| evl1gprodd 42739 | Polynomial evaluation buil... |
| aks6d1c2p1 42740 | In the AKS-theorem the sub... |
| aks6d1c2p2 42741 | Injective condition for co... |
| hashscontpowcl 42742 | Closure of E for ~ https:/... |
| hashscontpow1 42743 | Helper lemma for to prove ... |
| hashscontpow 42744 | If a set contains all ` N ... |
| aks6d1c3 42745 | Claim 3 of Theorem 6.1 of ... |
| aks6d1c4 42746 | Claim 4 of Theorem 6.1 of ... |
| aks6d1c1rh 42747 | Claim 1 of AKS primality p... |
| aks6d1c2lem3 42748 | Lemma for ~ aks6d1c2 to si... |
| aks6d1c2lem4 42749 | Claim 2 of Theorem 6.1 AKS... |
| hashnexinj 42750 | If the number of elements ... |
| hashnexinjle 42751 | If the number of elements ... |
| aks6d1c2 42752 | Claim 2 of Theorem 6.1 of ... |
| rspcsbnea 42753 | Special case related to ~ ... |
| idomnnzpownz 42754 | A nonzero power in an inte... |
| idomnnzgmulnz 42755 | A finite product of nonzer... |
| ringexp0nn 42756 | Zero to the power of a pos... |
| aks6d1c5lem0 42757 | Lemma for Claim 5 of Theor... |
| aks6d1c5lem1 42758 | Lemma for claim 5, evaluat... |
| aks6d1c5lem3 42759 | Lemma for Claim 5, polynom... |
| aks6d1c5lem2 42760 | Lemma for Claim 5, contrad... |
| aks6d1c5 42761 | Claim 5 of Theorem 6.1 ~ h... |
| deg1gprod 42762 | Degree multiplication is a... |
| deg1pow 42763 | Exact degree of a power of... |
| 5bc2eq10 42764 | The value of 5 choose 2. ... |
| facp2 42765 | The factorial of a success... |
| 2np3bcnp1 42766 | Part of induction step for... |
| 2ap1caineq 42767 | Inequality for Theorem 6.6... |
| sticksstones1 42768 | Different strictly monoton... |
| sticksstones2 42769 | The range function on stri... |
| sticksstones3 42770 | The range function on stri... |
| sticksstones4 42771 | Equinumerosity lemma for s... |
| sticksstones5 42772 | Count the number of strict... |
| sticksstones6 42773 | Function induces an order ... |
| sticksstones7 42774 | Closure property of sticks... |
| sticksstones8 42775 | Establish mapping between ... |
| sticksstones9 42776 | Establish mapping between ... |
| sticksstones10 42777 | Establish mapping between ... |
| sticksstones11 42778 | Establish bijective mappin... |
| sticksstones12a 42779 | Establish bijective mappin... |
| sticksstones12 42780 | Establish bijective mappin... |
| sticksstones13 42781 | Establish bijective mappin... |
| sticksstones14 42782 | Sticks and stones with def... |
| sticksstones15 42783 | Sticks and stones with alm... |
| sticksstones16 42784 | Sticks and stones with col... |
| sticksstones17 42785 | Extend sticks and stones t... |
| sticksstones18 42786 | Extend sticks and stones t... |
| sticksstones19 42787 | Extend sticks and stones t... |
| sticksstones20 42788 | Lift sticks and stones to ... |
| sticksstones21 42789 | Lift sticks and stones to ... |
| sticksstones22 42790 | Non-exhaustive sticks and ... |
| sticksstones23 42791 | Non-exhaustive sticks and ... |
| aks6d1c6lem1 42792 | Lemma for claim 6, deduce ... |
| aks6d1c6lem2 42793 | Every primitive root is ro... |
| aks6d1c6lem3 42794 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6lem4 42795 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6isolem1 42796 | Lemma to construct the map... |
| aks6d1c6isolem2 42797 | Lemma to construct the gro... |
| aks6d1c6isolem3 42798 | The preimage of a map send... |
| aks6d1c6lem5 42799 | Eliminate the size hypothe... |
| bcled 42800 | Inequality for binomial co... |
| bcle2d 42801 | Inequality for binomial co... |
| aks6d1c7lem1 42802 | The last set of inequaliti... |
| aks6d1c7lem2 42803 | Contradiction to Claim 2 a... |
| aks6d1c7lem3 42804 | Remove lots of hypotheses ... |
| aks6d1c7lem4 42805 | In the AKS algorithm there... |
| aks6d1c7 42806 | ` N ` is a prime power if ... |
| rhmqusspan 42807 | Ring homomorphism out of a... |
| aks5lem1 42808 | Section 5 of ~ https://www... |
| aks5lem2 42809 | Lemma for section 5 ~ http... |
| ply1asclzrhval 42810 | Transfer results from alge... |
| aks5lem3a 42811 | Lemma for AKS section 5. ... |
| aks5lem4a 42812 | Lemma for AKS section 5, r... |
| aks5lem5a 42813 | Lemma for AKS, section 5, ... |
| aks5lem6 42814 | Connect results of section... |
| indstrd 42815 | Strong induction, deductio... |
| grpods 42816 | Relate sums of elements of... |
| unitscyglem1 42817 | Lemma for unitscyg . (Con... |
| unitscyglem2 42818 | Lemma for unitscyg . (Con... |
| unitscyglem3 42819 | Lemma for unitscyg . (Con... |
| unitscyglem4 42820 | Lemma for unitscyg . (Con... |
| unitscyglem5 42821 | Lemma for unitscyg . (Con... |
| aks5lem7 42822 | Lemma for aks5. We clean ... |
| aks5lem8 42823 | Lemma for aks5. Clean up ... |
| exfinfldd 42825 | For any prime ` P ` and an... |
| aks5 42826 | The AKS Primality test, gi... |
| jarrii 42827 | Inference associated with ... |
| intnanrt 42828 | Introduction of conjunct i... |
| ioin9i8 42829 | Miscellaneous inference cr... |
| jaodd 42830 | Double deduction form of ~... |
| syl3an12 42831 | A double syllogism inferen... |
| exbiii 42832 | Inference associated with ... |
| sbtd 42833 | A true statement is true u... |
| sbor2 42834 | One direction of ~ sbor , ... |
| sbalexi 42835 | Inference form of ~ sbalex... |
| nfalh 42836 | Version of ~ nfal with an ... |
| nfe2 42837 | An inner existential quant... |
| nfale2 42838 | An inner existential quant... |
| 19.9dev 42839 | ~ 19.9d in the case of an ... |
| 3rspcedvd 42840 | Triple application of ~ rs... |
| sn-axrep5v 42841 | A condensed form of ~ axre... |
| sn-axprlem3 42842 | ~ axprlem3 using only Tars... |
| sn-exelALT 42843 | Alternate proof of ~ exel ... |
| ssabdv 42844 | Deduction of abstraction s... |
| sn-iotalem 42845 | An unused lemma showing th... |
| sn-iotalemcor 42846 | Corollary of ~ sn-iotalem ... |
| abbi1sn 42847 | Originally part of ~ uniab... |
| brif2 42848 | Move a relation inside and... |
| brif12 42849 | Move a relation inside and... |
| pssexg 42850 | The proper subset of a set... |
| pssn0 42851 | A proper superset is nonem... |
| psspwb 42852 | Classes are proper subclas... |
| xppss12 42853 | Proper subset theorem for ... |
| elpwbi 42854 | Membership in a power set,... |
| imaopab 42855 | The image of a class of or... |
| eqresfnbd 42856 | Property of being the rest... |
| fmpocos 42857 | Composition of two functio... |
| ovmpogad 42858 | Value of an operation give... |
| ofun 42859 | A function operation of un... |
| dfqs3 42860 | Alternate definition of qu... |
| qseq12d 42861 | Equality theorem for quoti... |
| qsalrel 42862 | The quotient set is equal ... |
| supinf 42863 | The supremum is the infimu... |
| mapcod 42864 | Compose two mappings. (Co... |
| fisdomnn 42865 | A finite set is dominated ... |
| ltex 42866 | The less-than relation is ... |
| leex 42867 | The less-than-or-equal-to ... |
| subex 42868 | The subtraction operation ... |
| absex 42869 | The absolute value functio... |
| cjex 42870 | The conjugate function is ... |
| fzosumm1 42871 | Separate out the last term... |
| ccatcan2d 42872 | Cancellation law for conca... |
| c0exALT 42873 | Alternate proof of ~ c0ex ... |
| 0cnALT3 42874 | Alternate proof of ~ 0cn u... |
| elre0re 42875 | Specialized version of ~ 0... |
| lttrii 42876 | 'Less than' is transitive.... |
| remulcan2d 42877 | ~ mulcan2d for real number... |
| readdridaddlidd 42878 | Given some real number ` B... |
| 1p3e4 42879 | 1 + 3 = 4. (Contributed b... |
| 5ne0 42880 | The number 5 is nonzero. ... |
| 6ne0 42881 | The number 6 is nonzero. ... |
| 7ne0 42882 | The number 7 is nonzero. ... |
| 8ne0 42883 | The number 8 is nonzero. ... |
| 9ne0 42884 | The number 9 is nonzero. ... |
| sn-1ne2 42885 | A proof of ~ 1ne2 without ... |
| nnn1suc 42886 | A positive integer that is... |
| readdrcl2d 42887 | Reverse closure for additi... |
| mvrrsubd 42888 | Move a subtraction in the ... |
| laddrotrd 42889 | Rotate the variables right... |
| raddswap12d 42890 | Swap the first two variabl... |
| lsubrotld 42891 | Rotate the variables left ... |
| rsubrotld 42892 | Rotate the variables left ... |
| lsubswap23d 42893 | Swap the second and third ... |
| addsubeq4com 42894 | Relation between sums and ... |
| sqsumi 42895 | A sum squared. (Contribut... |
| negn0nposznnd 42896 | Lemma for ~ dffltz . (Con... |
| sqmid3api 42897 | Value of the square of the... |
| decaddcom 42898 | Commute ones place in addi... |
| sqn5i 42899 | The square of a number end... |
| sqn5ii 42900 | The square of a number end... |
| decpmulnc 42901 | Partial products algorithm... |
| decpmul 42902 | Partial products algorithm... |
| sqdeccom12 42903 | The square of a number in ... |
| sq3deccom12 42904 | Variant of ~ sqdeccom12 wi... |
| 4t5e20 42905 | 4 times 5 equals 20. (Con... |
| 3rdpwhole 42906 | A third of a number plus t... |
| sq4 42907 | The square of 4 is 16. (C... |
| sq5 42908 | The square of 5 is 25. (C... |
| sq6 42909 | The square of 6 is 36. (C... |
| sq7 42910 | The square of 7 is 49. (C... |
| sq8 42911 | The square of 8 is 64. (C... |
| sq9 42912 | The square of 9 is 81. (C... |
| rpsscn 42913 | The positive reals are a s... |
| 4rp 42914 | 4 is a positive real. (Co... |
| 6rp 42915 | 6 is a positive real. (Co... |
| 7rp 42916 | 7 is a positive real. (Co... |
| 8rp 42917 | 8 is a positive real. (Co... |
| 9rp 42918 | 9 is a positive real. (Co... |
| 235t711 42919 | Calculate a product by lon... |
| ex-decpmul 42920 | Example usage of ~ decpmul... |
| eluzp1 42921 | Membership in a successor ... |
| sn-eluzp1l 42922 | Shorter proof of ~ eluzp1l... |
| fz1sumconst 42923 | The sum of ` N ` constant ... |
| fz1sump1 42924 | Add one more term to a sum... |
| oddnumth 42925 | The Odd Number Theorem. T... |
| nicomachus 42926 | Nicomachus's Theorem. The... |
| sumcubes 42927 | The sum of the first ` N `... |
| ine1 42928 | ` _i ` is not 1. (Contrib... |
| 0tie0 42929 | 0 times ` _i ` equals 0. ... |
| it1ei 42930 | ` _i ` times 1 equals ` _i... |
| 1tiei 42931 | 1 times ` _i ` equals ` _i... |
| itrere 42932 | ` _i ` times a real is rea... |
| retire 42933 | A real times ` _i ` is rea... |
| iocioodisjd 42934 | Adjacent intervals where t... |
| rpabsid 42935 | A positive real is its own... |
| oexpreposd 42936 | Lemma for ~ dffltz . For ... |
| explt1d 42937 | A nonnegative real number ... |
| expeq1d 42938 | A nonnegative real number ... |
| expeqidd 42939 | A nonnegative real number ... |
| exp11d 42940 | ~ exp11nnd for nonzero int... |
| 0dvds0 42941 | 0 divides 0. (Contributed... |
| absdvdsabsb 42942 | Divisibility is invariant ... |
| gcdnn0id 42943 | The ` gcd ` of a nonnegati... |
| gcdle1d 42944 | The greatest common diviso... |
| gcdle2d 42945 | The greatest common diviso... |
| dvdsexpad 42946 | Deduction associated with ... |
| dvdsexpnn 42947 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 42948 | ~ dvdsexpnn generalized to... |
| dvdsexpb 42949 | ~ dvdssq generalized to po... |
| posqsqznn 42950 | When a positive rational s... |
| zdivgd 42951 | Two ways to express " ` N ... |
| efsubd 42952 | Difference of exponents la... |
| ef11d 42953 | General condition for the ... |
| logccne0d 42954 | The logarithm isn't 0 if i... |
| cxp112d 42955 | General condition for comp... |
| cxp111d 42956 | General condition for comp... |
| cxpi11d 42957 | ` _i ` to the powers of ` ... |
| logne0d 42958 | Deduction form of ~ logne0... |
| rxp112d 42959 | Real exponentiation is one... |
| log11d 42960 | The natural logarithm is o... |
| rplog11d 42961 | The natural logarithm is o... |
| rxp11d 42962 | Real exponentiation is one... |
| tanhalfpim 42963 | The tangent of ` _pi / 2 `... |
| sinpim 42964 | Sine of a number subtracte... |
| cospim 42965 | Cosine of a number subtrac... |
| tan3rdpi 42966 | The tangent of ` _pi / 3 `... |
| sin2t3rdpi 42967 | The sine of ` 2 x. ( _pi /... |
| cos2t3rdpi 42968 | The cosine of ` 2 x. ( _pi... |
| sin4t3rdpi 42969 | The sine of ` 4 x. ( _pi /... |
| cos4t3rdpi 42970 | The cosine of ` 4 x. ( _pi... |
| asin1half 42971 | The arcsine of ` 1 / 2 ` i... |
| acos1half 42972 | The arccosine of ` 1 / 2 `... |
| dvun 42973 | Condition for the union of... |
| redvmptabs 42974 | The derivative of the abso... |
| readvrec2 42975 | The antiderivative of 1/x ... |
| readvrec 42976 | For real numbers, the anti... |
| resuppsinopn 42977 | The support of sin ( ~ df-... |
| readvcot 42978 | Real antiderivative of cot... |
| resubval 42981 | Value of real subtraction,... |
| renegeulemv 42982 | Lemma for ~ renegeu and si... |
| renegeulem 42983 | Lemma for ~ renegeu and si... |
| renegeu 42984 | Existential uniqueness of ... |
| rernegcl 42985 | Closure law for negative r... |
| renegadd 42986 | Relationship between real ... |
| renegid 42987 | Addition of a real number ... |
| reneg0addlid 42988 | Negative zero is a left ad... |
| resubeulem1 42989 | Lemma for ~ resubeu . A v... |
| resubeulem2 42990 | Lemma for ~ resubeu . A v... |
| resubeu 42991 | Existential uniqueness of ... |
| rersubcl 42992 | Closure for real subtracti... |
| resubadd 42993 | Relation between real subt... |
| resubaddd 42994 | Relationship between subtr... |
| resubf 42995 | Real subtraction is an ope... |
| repncan2 42996 | Addition and subtraction o... |
| repncan3 42997 | Addition and subtraction o... |
| readdsub 42998 | Law for addition and subtr... |
| reladdrsub 42999 | Move LHS of a sum into RHS... |
| reltsub1 43000 | Subtraction from both side... |
| reltsubadd2 43001 | 'Less than' relationship b... |
| resubcan2 43002 | Cancellation law for real ... |
| resubsub4 43003 | Law for double subtraction... |
| rennncan2 43004 | Cancellation law for real ... |
| renpncan3 43005 | Cancellation law for real ... |
| repnpcan 43006 | Cancellation law for addit... |
| reppncan 43007 | Cancellation law for mixed... |
| resubidaddlidlem 43008 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 43009 | Any real number subtracted... |
| resubdi 43010 | Distribution of multiplica... |
| re1m1e0m0 43011 | Equality of two left-addit... |
| sn-00idlem1 43012 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 43013 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 43014 | Lemma for ~ sn-00id . (Co... |
| sn-00id 43015 | ~ 00id proven without ~ ax... |
| re0m0e0 43016 | Real number version of ~ 0... |
| readdlid 43017 | Real number version of ~ a... |
| sn-addlid 43018 | ~ addlid without ~ ax-mulc... |
| remul02 43019 | Real number version of ~ m... |
| sn-0ne2 43020 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 43021 | Real number version of ~ m... |
| sn-remul0ord 43022 | A product is zero iff one ... |
| resubid 43023 | Subtraction of a real numb... |
| readdrid 43024 | Real number version of ~ a... |
| resubid1 43025 | Real number version of ~ s... |
| renegneg 43026 | A real number is equal to ... |
| readdcan2 43027 | Commuted version of ~ read... |
| renegid2 43028 | Commuted version of ~ rene... |
| remulneg2d 43029 | Product with negative is n... |
| sn-it0e0 43030 | Proof of ~ it0e0 without ~... |
| sn-negex12 43031 | A combination of ~ cnegex ... |
| sn-negex 43032 | Proof of ~ cnegex without ... |
| sn-negex2 43033 | Proof of ~ cnegex2 without... |
| sn-addcand 43034 | ~ addcand without ~ ax-mul... |
| sn-addrid 43035 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 43036 | ~ addcan2d without ~ ax-mu... |
| reixi 43037 | ~ ixi without ~ ax-mulcom ... |
| rei4 43038 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 43039 | A number that sums to itse... |
| sn-mul01 43040 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 43041 | ~ negeu without ~ ax-mulco... |
| sn-subcl 43042 | ~ subcl without ~ ax-mulco... |
| sn-subf 43043 | ~ subf without ~ ax-mulcom... |
| resubeqsub 43044 | Equivalence between real s... |
| subresre 43045 | Subtraction restricted to ... |
| addinvcom 43046 | A number commutes with its... |
| remulinvcom 43047 | A left multiplicative inve... |
| remullid 43048 | Commuted version of ~ ax-1... |
| sn-1ticom 43049 | Lemma for ~ sn-mullid and ... |
| sn-mullid 43050 | ~ mullid without ~ ax-mulc... |
| sn-it1ei 43051 | ~ it1ei without ~ ax-mulco... |
| ipiiie0 43052 | The multiplicative inverse... |
| remulcand 43053 | Commuted version of ~ remu... |
| redivvald 43056 | Value of real division, wh... |
| rediveud 43057 | Existential uniqueness of ... |
| sn-redivcld 43058 | Closure law for real divis... |
| redivmuld 43059 | Relationship between divis... |
| redivmul2d 43060 | Relationship between divis... |
| redivcan2d 43061 | A cancellation law for div... |
| redivcan3d 43062 | A cancellation law for div... |
| rediveq0d 43063 | A ratio is zero iff the nu... |
| redivne0bd 43064 | The ratio of nonzero numbe... |
| rediveq1d 43065 | Equality in terms of unit ... |
| sn-rediv1d 43066 | A number divided by 1 is i... |
| sn-rediv0d 43067 | Division into zero is zero... |
| sn-redividd 43068 | A number divided by itself... |
| sn-rereccld 43069 | Closure law for reciprocal... |
| rerecne0d 43070 | The reciprocal of a nonzer... |
| rerecidd 43071 | Multiplication of a number... |
| rerecid2d 43072 | Multiplication of a number... |
| rerecrecd 43073 | A number is equal to the r... |
| redivrec2d 43074 | Relationship between divis... |
| rediv23d 43075 | A "commutative"/associativ... |
| redivdird 43076 | Distribution of division o... |
| rediv11d 43077 | One-to-one relationship fo... |
| sn-0tie0 43078 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 43079 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 43080 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 43081 | ~ ltaddneg without ~ ax-mu... |
| reposdif 43082 | Comparison of two numbers ... |
| relt0neg1 43083 | Comparison of a real and i... |
| relt0neg2 43084 | Comparison of a real and i... |
| sn-addlt0d 43085 | The sum of negative number... |
| sn-addgt0d 43086 | The sum of positive number... |
| sn-nnne0 43087 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 43088 | ~ elznn0nn restated using ... |
| nn0addcom 43089 | Addition is commutative fo... |
| zaddcomlem 43090 | Lemma for ~ zaddcom . (Co... |
| zaddcom 43091 | Addition is commutative fo... |
| renegmulnnass 43092 | Move multiplication by a n... |
| nn0mulcom 43093 | Multiplication is commutat... |
| zmulcomlem 43094 | Lemma for ~ zmulcom . (Co... |
| zmulcom 43095 | Multiplication is commutat... |
| mulgt0con1dlem 43096 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 43097 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 43098 | Lemma for ~ mulgt0b1d and ... |
| mulgt0b1d 43099 | Biconditional, deductive f... |
| sn-ltmul2d 43100 | ~ ltmul2d without ~ ax-mul... |
| sn-ltmulgt11d 43101 | ~ ltmulgt11d without ~ ax-... |
| sn-0lt1 43102 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 43103 | ~ ltp1 without ~ ax-mulcom... |
| sn-recgt0d 43104 | The reciprocal of a positi... |
| mulgt0b2d 43105 | Biconditional, deductive f... |
| sn-mulgt1d 43106 | ~ mulgt1d without ~ ax-mul... |
| reneg1lt0 43107 | Negative one is a negative... |
| sn-reclt0d 43108 | The reciprocal of a negati... |
| mulltgt0d 43109 | Negative times positive is... |
| mullt0b1d 43110 | When the first term is neg... |
| mullt0b2d 43111 | When the second term is ne... |
| sn-mullt0d 43112 | The product of two negativ... |
| sn-msqgt0d 43113 | A nonzero square is positi... |
| sn-inelr 43114 | ~ inelr without ~ ax-mulco... |
| sn-itrere 43115 | ` _i ` times a real is rea... |
| sn-retire 43116 | Commuted version of ~ sn-i... |
| cnreeu 43117 | The reals in the expressio... |
| sn-sup2 43118 | ~ sup2 with exactly the sa... |
| sn-sup3d 43119 | ~ sup3 without ~ ax-mulcom... |
| sn-suprcld 43120 | ~ suprcld without ~ ax-mul... |
| sn-suprubd 43121 | ~ suprubd without ~ ax-mul... |
| sn-base0 43122 | Avoid axioms in ~ base0 by... |
| nelsubginvcld 43123 | The inverse of a non-subgr... |
| nelsubgcld 43124 | A non-subgroup-member plus... |
| nelsubgsubcld 43125 | A non-subgroup-member minu... |
| rnasclg 43126 | The set of injected scalar... |
| frlmfielbas 43127 | The vectors of a finite fr... |
| frlmfzwrd 43128 | A vector of a module with ... |
| frlmfzowrd 43129 | A vector of a module with ... |
| frlmfzolen 43130 | The dimension of a vector ... |
| frlmfzowrdb 43131 | The vectors of a module wi... |
| frlmfzoccat 43132 | The concatenation of two v... |
| frlmvscadiccat 43133 | Scalar multiplication dist... |
| grpasscan2d 43134 | An associative cancellatio... |
| grpcominv1 43135 | If two elements commute, t... |
| grpcominv2 43136 | If two elements commute, t... |
| finsubmsubg 43137 | A submonoid of a finite gr... |
| opprmndb 43138 | A class is a monoid if and... |
| opprgrpb 43139 | A class is a group if and ... |
| opprablb 43140 | A class is an Abelian grou... |
| imacrhmcl 43141 | The image of a commutative... |
| rimco 43142 | The composition of ring is... |
| rictr 43143 | Ring isomorphism is transi... |
| riccrng1 43144 | Ring isomorphism preserves... |
| riccrng 43145 | A ring is commutative if a... |
| domnexpgn0cl 43146 | In a domain, a (nonnegativ... |
| drnginvrn0d 43147 | A multiplicative inverse i... |
| drngmullcan 43148 | Cancellation of a nonzero ... |
| drngmulrcan 43149 | Cancellation of a nonzero ... |
| drnginvmuld 43150 | Inverse of a nonzero produ... |
| ricdrng1 43151 | A ring isomorphism maps a ... |
| ricdrng 43152 | A ring is a division ring ... |
| ricfld 43153 | A ring is a field if and o... |
| asclf1 43154 | Two ways of saying the sca... |
| abvexp 43155 | Move exponentiation in and... |
| fimgmcyclem 43156 | Lemma for ~ fimgmcyc . (C... |
| fimgmcyc 43157 | Version of ~ odcl2 for fin... |
| fidomncyc 43158 | Version of ~ odcl2 for mul... |
| fiabv 43159 | In a finite domain (a fini... |
| lvecgrp 43160 | A vector space is a group.... |
| lvecring 43161 | The scalar component of a ... |
| frlm0vald 43162 | All coordinates of the zer... |
| frlmsnic 43163 | Given a free module with a... |
| uvccl 43164 | A unit vector is a vector.... |
| uvcn0 43165 | A unit vector is nonzero. ... |
| psrmnd 43166 | The ring of power series i... |
| mhmcopsr 43167 | The composition of a monoi... |
| mhmcoaddpsr 43168 | Show that the ring homomor... |
| rhmcomulpsr 43169 | Show that the ring homomor... |
| rhmpsr 43170 | Provide a ring homomorphis... |
| rhmpsr1 43171 | Provide a ring homomorphis... |
| evl0 43172 | The zero polynomial evalua... |
| evlsbagval 43173 | Polynomial evaluation buil... |
| evlvvvallem 43174 | Lemma for theorems using ~... |
| evlselvlem 43175 | Lemma for ~ evlselv . Use... |
| evlselv 43176 | Evaluating a selection of ... |
| fsuppind 43177 | Induction on functions ` F... |
| fsuppssindlem1 43178 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 43179 | Lemma for ~ fsuppssind . ... |
| fsuppssind 43180 | Induction on functions ` F... |
| mhpind 43181 | The homogeneous polynomial... |
| evlsmhpvvval 43182 | Give a formula for the eva... |
| mhphflem 43183 | Lemma for ~ mhphf . Add s... |
| mhphf 43184 | A homogeneous polynomial d... |
| mhphf2 43185 | A homogeneous polynomial d... |
| mhphf3 43186 | A homogeneous polynomial d... |
| mhphf4 43187 | A homogeneous polynomial d... |
| prjspval 43190 | Value of the projective sp... |
| prjsprel 43191 | Utility theorem regarding ... |
| prjspertr 43192 | The relation in ` PrjSp ` ... |
| prjsperref 43193 | The relation in ` PrjSp ` ... |
| prjspersym 43194 | The relation in ` PrjSp ` ... |
| prjsper 43195 | The relation used to defin... |
| prjspreln0 43196 | Two nonzero vectors are eq... |
| prjspvs 43197 | A nonzero multiple of a ve... |
| prjsprellsp 43198 | Two vectors are equivalent... |
| prjspeclsp 43199 | The vectors equivalent to ... |
| prjspval2 43200 | Alternate definition of pr... |
| prjspnval 43203 | Value of the n-dimensional... |
| prjspnerlem 43204 | A lemma showing that the e... |
| prjspnval2 43205 | Value of the n-dimensional... |
| prjspner 43206 | The relation used to defin... |
| prjspnvs 43207 | A nonzero multiple of a ve... |
| prjspnssbas 43208 | A projective point spans a... |
| prjspnn0 43209 | A projective point is none... |
| 0prjspnlem 43210 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 43211 | Any vector is equivalent t... |
| prjspner01 43212 | Any vector is equivalent t... |
| prjspner1 43213 | Two vectors whose zeroth c... |
| 0prjspnrel 43214 | In the zero-dimensional pr... |
| 0prjspn 43215 | A zero-dimensional project... |
| prjcrvfval 43218 | Value of the projective cu... |
| prjcrvval 43219 | Value of the projective cu... |
| prjcrv0 43220 | The "curve" (zero set) cor... |
| dffltz 43221 | Fermat's Last Theorem (FLT... |
| fltmul 43222 | A counterexample to FLT st... |
| fltdiv 43223 | A counterexample to FLT st... |
| flt0 43224 | A counterexample for FLT d... |
| fltdvdsabdvdsc 43225 | Any factor of both ` A ` a... |
| fltabcoprmex 43226 | A counterexample to FLT im... |
| fltaccoprm 43227 | A counterexample to FLT wi... |
| fltbccoprm 43228 | A counterexample to FLT wi... |
| fltabcoprm 43229 | A counterexample to FLT wi... |
| infdesc 43230 | Infinite descent. The hyp... |
| fltne 43231 | If a counterexample to FLT... |
| flt4lem 43232 | Raising a number to the fo... |
| flt4lem1 43233 | Satisfy the antecedent use... |
| flt4lem2 43234 | If ` A ` is even, ` B ` is... |
| flt4lem3 43235 | Equivalent to ~ pythagtrip... |
| flt4lem4 43236 | If the product of two copr... |
| flt4lem5 43237 | In the context of the lemm... |
| flt4lem5elem 43238 | Version of ~ fltaccoprm an... |
| flt4lem5a 43239 | Part 1 of Equation 1 of ... |
| flt4lem5b 43240 | Part 2 of Equation 1 of ... |
| flt4lem5c 43241 | Part 2 of Equation 2 of ... |
| flt4lem5d 43242 | Part 3 of Equation 2 of ... |
| flt4lem5e 43243 | Satisfy the hypotheses of ... |
| flt4lem5f 43244 | Final equation of ~... |
| flt4lem6 43245 | Remove shared factors in a... |
| flt4lem7 43246 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 43247 | Strengthening of Fermat's ... |
| fltltc 43248 | ` ( C ^ N ) ` is the large... |
| fltnltalem 43249 | Lemma for ~ fltnlta . A l... |
| fltnlta 43250 | In a Fermat counterexample... |
| iddii 43251 | Version of ~ a1ii with the... |
| bicomdALT 43252 | Alternate proof of ~ bicom... |
| alan 43253 | Alias for ~ 19.26 for easi... |
| exor 43254 | Alias for ~ 19.43 for easi... |
| rexor 43255 | Alias for ~ r19.43 for eas... |
| ruvALT 43256 | Alternate proof of ~ ruv w... |
| sn-wcdeq 43257 | Alternative to ~ wcdeq and... |
| sq45 43258 | 45 squared is 2025. (Cont... |
| sum9cubes 43259 | The sum of the first nine ... |
| sn-isghm 43260 | Longer proof of ~ isghm , ... |
| aprilfools2025 43261 | An abuse of notation. (Co... |
| nfa1w 43262 | Replace ~ ax-10 in ~ nfa1 ... |
| eu6w 43263 | Replace ~ ax-10 , ~ ax-12 ... |
| abbibw 43264 | Replace ~ ax-10 , ~ ax-11 ... |
| absnw 43265 | Replace ~ ax-10 , ~ ax-11 ... |
| euabsn2w 43266 | Replace ~ ax-10 , ~ ax-11 ... |
| cu3addd 43267 | Cube of sum of three numbe... |
| negexpidd 43268 | The sum of a real number t... |
| rexlimdv3d 43269 | An extended version of ~ r... |
| 3cubeslem1 43270 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 43271 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 43272 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 43273 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 43274 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 43275 | Lemma for ~ 3cubes . This... |
| 3cubes 43276 | Every rational number is a... |
| rntrclfvOAI 43277 | The range of the transitiv... |
| moxfr 43278 | Transfer at-most-one betwe... |
| imaiinfv 43279 | Indexed intersection of an... |
| elrfi 43280 | Elementhood in a set of re... |
| elrfirn 43281 | Elementhood in a set of re... |
| elrfirn2 43282 | Elementhood in a set of re... |
| cmpfiiin 43283 | In a compact topology, a s... |
| ismrcd1 43284 | Any function from the subs... |
| ismrcd2 43285 | Second half of ~ ismrcd1 .... |
| istopclsd 43286 | A closure function which s... |
| ismrc 43287 | A function is a Moore clos... |
| isnacs 43290 | Expand definition of Noeth... |
| nacsfg 43291 | In a Noetherian-type closu... |
| isnacs2 43292 | Express Noetherian-type cl... |
| mrefg2 43293 | Slight variation on finite... |
| mrefg3 43294 | Slight variation on finite... |
| nacsacs 43295 | A closure system of Noethe... |
| isnacs3 43296 | A choice-free order equiva... |
| incssnn0 43297 | Transitivity induction of ... |
| nacsfix 43298 | An increasing sequence of ... |
| constmap 43299 | A constant (represented wi... |
| mapco2g 43300 | Renaming indices in a tupl... |
| mapco2 43301 | Post-composition (renaming... |
| mapfzcons 43302 | Extending a one-based mapp... |
| mapfzcons1 43303 | Recover prefix mapping fro... |
| mapfzcons1cl 43304 | A nonempty mapping has a p... |
| mapfzcons2 43305 | Recover added element from... |
| mptfcl 43306 | Interpret range of a maps-... |
| mzpclval 43311 | Substitution lemma for ` m... |
| elmzpcl 43312 | Double substitution lemma ... |
| mzpclall 43313 | The set of all functions w... |
| mzpcln0 43314 | Corollary of ~ mzpclall : ... |
| mzpcl1 43315 | Defining property 1 of a p... |
| mzpcl2 43316 | Defining property 2 of a p... |
| mzpcl34 43317 | Defining properties 3 and ... |
| mzpval 43318 | Value of the ` mzPoly ` fu... |
| dmmzp 43319 | ` mzPoly ` is defined for ... |
| mzpincl 43320 | Polynomial closedness is a... |
| mzpconst 43321 | Constant functions are pol... |
| mzpf 43322 | A polynomial function is a... |
| mzpproj 43323 | A projection function is p... |
| mzpadd 43324 | The pointwise sum of two p... |
| mzpmul 43325 | The pointwise product of t... |
| mzpconstmpt 43326 | A constant function expres... |
| mzpaddmpt 43327 | Sum of polynomial function... |
| mzpmulmpt 43328 | Product of polynomial func... |
| mzpsubmpt 43329 | The difference of two poly... |
| mzpnegmpt 43330 | Negation of a polynomial f... |
| mzpexpmpt 43331 | Raise a polynomial functio... |
| mzpindd 43332 | "Structural" induction to ... |
| mzpmfp 43333 | Relationship between multi... |
| mzpsubst 43334 | Substituting polynomials f... |
| mzprename 43335 | Simplified version of ~ mz... |
| mzpresrename 43336 | A polynomial is a polynomi... |
| mzpcompact2lem 43337 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 43338 | Polynomials are finitary o... |
| coeq0i 43339 | ~ coeq0 but without explic... |
| fzsplit1nn0 43340 | Split a finite 1-based set... |
| eldiophb 43343 | Initial expression of Diop... |
| eldioph 43344 | Condition for a set to be ... |
| diophrw 43345 | Renaming and adding unused... |
| eldioph2lem1 43346 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 43347 | Lemma for ~ eldioph2 . Co... |
| eldioph2 43348 | Construct a Diophantine se... |
| eldioph2b 43349 | While Diophantine sets wer... |
| eldiophelnn0 43350 | Remove antecedent on ` B `... |
| eldioph3b 43351 | Define Diophantine sets in... |
| eldioph3 43352 | Inference version of ~ eld... |
| ellz1 43353 | Membership in a lower set ... |
| lzunuz 43354 | The union of a lower set o... |
| fz1eqin 43355 | Express a one-based finite... |
| lzenom 43356 | Lower integers are countab... |
| elmapresaunres2 43357 | ~ fresaunres2 transposed t... |
| diophin 43358 | If two sets are Diophantin... |
| diophun 43359 | If two sets are Diophantin... |
| eldiophss 43360 | Diophantine sets are sets ... |
| diophrex 43361 | Projecting a Diophantine s... |
| eq0rabdioph 43362 | This is the first of a num... |
| eqrabdioph 43363 | Diophantine set builder fo... |
| 0dioph 43364 | The null set is Diophantin... |
| vdioph 43365 | The "universal" set (as la... |
| anrabdioph 43366 | Diophantine set builder fo... |
| orrabdioph 43367 | Diophantine set builder fo... |
| 3anrabdioph 43368 | Diophantine set builder fo... |
| 3orrabdioph 43369 | Diophantine set builder fo... |
| 2sbcrex 43370 | Exchange an existential qu... |
| sbc2rex 43371 | Exchange a substitution wi... |
| sbc4rex 43372 | Exchange a substitution wi... |
| sbcrot3 43373 | Rotate a sequence of three... |
| sbcrot5 43374 | Rotate a sequence of five ... |
| sbccomieg 43375 | Commute two explicit subst... |
| rexrabdioph 43376 | Diophantine set builder fo... |
| rexfrabdioph 43377 | Diophantine set builder fo... |
| 2rexfrabdioph 43378 | Diophantine set builder fo... |
| 3rexfrabdioph 43379 | Diophantine set builder fo... |
| 4rexfrabdioph 43380 | Diophantine set builder fo... |
| 6rexfrabdioph 43381 | Diophantine set builder fo... |
| 7rexfrabdioph 43382 | Diophantine set builder fo... |
| rabdiophlem1 43383 | Lemma for arithmetic dioph... |
| rabdiophlem2 43384 | Lemma for arithmetic dioph... |
| elnn0rabdioph 43385 | Diophantine set builder fo... |
| rexzrexnn0 43386 | Rewrite an existential qua... |
| lerabdioph 43387 | Diophantine set builder fo... |
| eluzrabdioph 43388 | Diophantine set builder fo... |
| elnnrabdioph 43389 | Diophantine set builder fo... |
| ltrabdioph 43390 | Diophantine set builder fo... |
| nerabdioph 43391 | Diophantine set builder fo... |
| dvdsrabdioph 43392 | Divisibility is a Diophant... |
| eldioph4b 43393 | Membership in ` Dioph ` ex... |
| eldioph4i 43394 | Forward-only version of ~ ... |
| diophren 43395 | Change variables in a Diop... |
| rabrenfdioph 43396 | Change variable numbers in... |
| rabren3dioph 43397 | Change variable numbers in... |
| fphpd 43398 | Pigeonhole principle expre... |
| fphpdo 43399 | Pigeonhole principle for s... |
| ctbnfien 43400 | An infinite subset of a co... |
| fiphp3d 43401 | Infinite pigeonhole princi... |
| rencldnfilem 43402 | Lemma for ~ rencldnfi . (... |
| rencldnfi 43403 | A set of real numbers whic... |
| irrapxlem1 43404 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 43405 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 43406 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 43407 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 43408 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 43409 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 43410 | Dirichlet's approximation ... |
| pellexlem1 43411 | Lemma for ~ pellex . Arit... |
| pellexlem2 43412 | Lemma for ~ pellex . Arit... |
| pellexlem3 43413 | Lemma for ~ pellex . To e... |
| pellexlem4 43414 | Lemma for ~ pellex . Invo... |
| pellexlem5 43415 | Lemma for ~ pellex . Invo... |
| pellexlem6 43416 | Lemma for ~ pellex . Doin... |
| pellex 43417 | Every Pell equation has a ... |
| pell1qrval 43428 | Value of the set of first-... |
| elpell1qr 43429 | Membership in a first-quad... |
| pell14qrval 43430 | Value of the set of positi... |
| elpell14qr 43431 | Membership in the set of p... |
| pell1234qrval 43432 | Value of the set of genera... |
| elpell1234qr 43433 | Membership in the set of g... |
| pell1234qrre 43434 | General Pell solutions are... |
| pell1234qrne0 43435 | No solution to a Pell equa... |
| pell1234qrreccl 43436 | General solutions of the P... |
| pell1234qrmulcl 43437 | General solutions of the P... |
| pell14qrss1234 43438 | A positive Pell solution i... |
| pell14qrre 43439 | A positive Pell solution i... |
| pell14qrne0 43440 | A positive Pell solution i... |
| pell14qrgt0 43441 | A positive Pell solution i... |
| pell14qrrp 43442 | A positive Pell solution i... |
| pell1234qrdich 43443 | A general Pell solution is... |
| elpell14qr2 43444 | A number is a positive Pel... |
| pell14qrmulcl 43445 | Positive Pell solutions ar... |
| pell14qrreccl 43446 | Positive Pell solutions ar... |
| pell14qrdivcl 43447 | Positive Pell solutions ar... |
| pell14qrexpclnn0 43448 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 43449 | Positive Pell solutions ar... |
| pell1qrss14 43450 | First-quadrant Pell soluti... |
| pell14qrdich 43451 | A positive Pell solution i... |
| pell1qrge1 43452 | A Pell solution in the fir... |
| pell1qr1 43453 | 1 is a Pell solution and i... |
| elpell1qr2 43454 | The first quadrant solutio... |
| pell1qrgaplem 43455 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 43456 | First-quadrant Pell soluti... |
| pell14qrgap 43457 | Positive Pell solutions ar... |
| pell14qrgapw 43458 | Positive Pell solutions ar... |
| pellqrexplicit 43459 | Condition for a calculated... |
| infmrgelbi 43460 | Any lower bound of a nonem... |
| pellqrex 43461 | There is a nontrivial solu... |
| pellfundval 43462 | Value of the fundamental s... |
| pellfundre 43463 | The fundamental solution o... |
| pellfundge 43464 | Lower bound on the fundame... |
| pellfundgt1 43465 | Weak lower bound on the Pe... |
| pellfundlb 43466 | A nontrivial first quadran... |
| pellfundglb 43467 | If a real is larger than t... |
| pellfundex 43468 | The fundamental solution a... |
| pellfund14gap 43469 | There are no solutions bet... |
| pellfundrp 43470 | The fundamental Pell solut... |
| pellfundne1 43471 | The fundamental Pell solut... |
| reglogcl 43472 | General logarithm is a rea... |
| reglogltb 43473 | General logarithm preserve... |
| reglogleb 43474 | General logarithm preserve... |
| reglogmul 43475 | Multiplication law for gen... |
| reglogexp 43476 | Power law for general log.... |
| reglogbas 43477 | General log of the base is... |
| reglog1 43478 | General log of 1 is 0. (C... |
| reglogexpbas 43479 | General log of a power of ... |
| pellfund14 43480 | Every positive Pell soluti... |
| pellfund14b 43481 | The positive Pell solution... |
| rmxfval 43486 | Value of the X sequence. ... |
| rmyfval 43487 | Value of the Y sequence. ... |
| rmspecsqrtnq 43488 | The discriminant used to d... |
| rmspecnonsq 43489 | The discriminant used to d... |
| qirropth 43490 | This lemma implements the ... |
| rmspecfund 43491 | The base of exponent used ... |
| rmxyelqirr 43492 | The solutions used to cons... |
| rmxypairf1o 43493 | The function used to extra... |
| rmxyelxp 43494 | Lemma for ~ frmx and ~ frm... |
| frmx 43495 | The X sequence is a nonneg... |
| frmy 43496 | The Y sequence is an integ... |
| rmxyval 43497 | Main definition of the X a... |
| rmspecpos 43498 | The discriminant used to d... |
| rmxycomplete 43499 | The X and Y sequences take... |
| rmxynorm 43500 | The X and Y sequences defi... |
| rmbaserp 43501 | The base of exponentiation... |
| rmxyneg 43502 | Negation law for X and Y s... |
| rmxyadd 43503 | Addition formula for X and... |
| rmxy1 43504 | Value of the X and Y seque... |
| rmxy0 43505 | Value of the X and Y seque... |
| rmxneg 43506 | Negation law (even functio... |
| rmx0 43507 | Value of X sequence at 0. ... |
| rmx1 43508 | Value of X sequence at 1. ... |
| rmxadd 43509 | Addition formula for X seq... |
| rmyneg 43510 | Negation formula for Y seq... |
| rmy0 43511 | Value of Y sequence at 0. ... |
| rmy1 43512 | Value of Y sequence at 1. ... |
| rmyadd 43513 | Addition formula for Y seq... |
| rmxp1 43514 | Special addition-of-1 form... |
| rmyp1 43515 | Special addition of 1 form... |
| rmxm1 43516 | Subtraction of 1 formula f... |
| rmym1 43517 | Subtraction of 1 formula f... |
| rmxluc 43518 | The X sequence is a Lucas ... |
| rmyluc 43519 | The Y sequence is a Lucas ... |
| rmyluc2 43520 | Lucas sequence property of... |
| rmxdbl 43521 | "Double-angle formula" for... |
| rmydbl 43522 | "Double-angle formula" for... |
| monotuz 43523 | A function defined on an u... |
| monotoddzzfi 43524 | A function which is odd an... |
| monotoddzz 43525 | A function (given implicit... |
| oddcomabszz 43526 | An odd function which take... |
| 2nn0ind 43527 | Induction on nonnegative i... |
| zindbi 43528 | Inductively transfer a pro... |
| rmxypos 43529 | For all nonnegative indice... |
| ltrmynn0 43530 | The Y-sequence is strictly... |
| ltrmxnn0 43531 | The X-sequence is strictly... |
| lermxnn0 43532 | The X-sequence is monotoni... |
| rmxnn 43533 | The X-sequence is defined ... |
| ltrmy 43534 | The Y-sequence is strictly... |
| rmyeq0 43535 | Y is zero only at zero. (... |
| rmyeq 43536 | Y is one-to-one. (Contrib... |
| lermy 43537 | Y is monotonic (non-strict... |
| rmynn 43538 | ` rmY ` is positive for po... |
| rmynn0 43539 | ` rmY ` is nonnegative for... |
| rmyabs 43540 | ` rmY ` commutes with ` ab... |
| jm2.24nn 43541 | X(n) is strictly greater t... |
| jm2.17a 43542 | First half of lemma 2.17 o... |
| jm2.17b 43543 | Weak form of the second ha... |
| jm2.17c 43544 | Second half of lemma 2.17 ... |
| jm2.24 43545 | Lemma 2.24 of [JonesMatija... |
| rmygeid 43546 | Y(n) increases faster than... |
| congtr 43547 | A wff of the form ` A || (... |
| congadd 43548 | If two pairs of numbers ar... |
| congmul 43549 | If two pairs of numbers ar... |
| congsym 43550 | Congruence mod ` A ` is a ... |
| congneg 43551 | If two integers are congru... |
| congsub 43552 | If two pairs of numbers ar... |
| congid 43553 | Every integer is congruent... |
| mzpcong 43554 | Polynomials commute with c... |
| congrep 43555 | Every integer is congruent... |
| congabseq 43556 | If two integers are congru... |
| acongid 43557 | A wff like that in this th... |
| acongsym 43558 | Symmetry of alternating co... |
| acongneg2 43559 | Negate right side of alter... |
| acongtr 43560 | Transitivity of alternatin... |
| acongeq12d 43561 | Substitution deduction for... |
| acongrep 43562 | Every integer is alternati... |
| fzmaxdif 43563 | Bound on the difference be... |
| fzneg 43564 | Reflection of a finite ran... |
| acongeq 43565 | Two numbers in the fundame... |
| dvdsacongtr 43566 | Alternating congruence pas... |
| coprmdvdsb 43567 | Multiplication by a coprim... |
| modabsdifz 43568 | Divisibility in terms of m... |
| dvdsabsmod0 43569 | Divisibility in terms of m... |
| jm2.18 43570 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 43571 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 43572 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 43573 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 43574 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 43575 | Lemma 2.19 of [JonesMatija... |
| jm2.21 43576 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 43577 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 43578 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 43579 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 43580 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 43581 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 43582 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 43583 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 43584 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 43585 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 43586 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 43587 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 43588 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 43589 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 43590 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 43591 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 43592 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 43593 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 43594 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 43595 | Lemma for ~ rmydioph . Us... |
| rmydioph 43596 | ~ jm2.27 restated in terms... |
| rmxdiophlem 43597 | X can be expressed in term... |
| rmxdioph 43598 | X is a Diophantine functio... |
| jm3.1lem1 43599 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 43600 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 43601 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 43602 | Diophantine expression for... |
| expdiophlem1 43603 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 43604 | Lemma for ~ expdioph . Ex... |
| expdioph 43605 | The exponential function i... |
| setindtr 43606 | Set induction for sets con... |
| setindtrs 43607 | Set induction scheme witho... |
| dford3lem1 43608 | Lemma for ~ dford3 . (Con... |
| dford3lem2 43609 | Lemma for ~ dford3 . (Con... |
| dford3 43610 | Ordinals are precisely the... |
| dford4 43611 | ~ dford3 expressed in prim... |
| wopprc 43612 | Unrelated: Wiener pairs t... |
| rpnnen3lem 43613 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 43614 | Dedekind cut injection of ... |
| axac10 43615 | Characterization of choice... |
| harinf 43616 | The Hartogs number of an i... |
| wdom2d2 43617 | Deduction for weak dominan... |
| ttac 43618 | Tarski's theorem about cho... |
| pw2f1ocnv 43619 | Define a bijection between... |
| pw2f1o2 43620 | Define a bijection between... |
| pw2f1o2val 43621 | Function value of the ~ pw... |
| pw2f1o2val2 43622 | Membership in a mapped set... |
| limsuc2 43623 | Limit ordinals in the sens... |
| wepwsolem 43624 | Transfer an ordering on ch... |
| wepwso 43625 | A well-ordering induces a ... |
| dnnumch1 43626 | Define an enumeration of a... |
| dnnumch2 43627 | Define an enumeration (wea... |
| dnnumch3lem 43628 | Value of the ordinal injec... |
| dnnumch3 43629 | Define an injection from a... |
| dnwech 43630 | Define a well-ordering fro... |
| fnwe2val 43631 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 43632 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 43633 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 43634 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 43635 | A well-ordering can be con... |
| aomclem1 43636 | Lemma for ~ dfac11 . This... |
| aomclem2 43637 | Lemma for ~ dfac11 . Succ... |
| aomclem3 43638 | Lemma for ~ dfac11 . Succ... |
| aomclem4 43639 | Lemma for ~ dfac11 . Limi... |
| aomclem5 43640 | Lemma for ~ dfac11 . Comb... |
| aomclem6 43641 | Lemma for ~ dfac11 . Tran... |
| aomclem7 43642 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 43643 | Lemma for ~ dfac11 . Perf... |
| dfac11 43644 | The right-hand side of thi... |
| kelac1 43645 | Kelley's choice, basic for... |
| kelac2lem 43646 | Lemma for ~ kelac2 and ~ d... |
| kelac2 43647 | Kelley's choice, most comm... |
| dfac21 43648 | Tychonoff's theorem is a c... |
| islmodfg 43651 | Property of a finitely gen... |
| islssfg 43652 | Property of a finitely gen... |
| islssfg2 43653 | Property of a finitely gen... |
| islssfgi 43654 | Finitely spanned subspaces... |
| fglmod 43655 | Finitely generated left mo... |
| lsmfgcl 43656 | The sum of two finitely ge... |
| islnm 43659 | Property of being a Noethe... |
| islnm2 43660 | Property of being a Noethe... |
| lnmlmod 43661 | A Noetherian left module i... |
| lnmlssfg 43662 | A submodule of Noetherian ... |
| lnmlsslnm 43663 | All submodules of a Noethe... |
| lnmfg 43664 | A Noetherian left module i... |
| kercvrlsm 43665 | The domain of a linear fun... |
| lmhmfgima 43666 | A homomorphism maps finite... |
| lnmepi 43667 | Epimorphic images of Noeth... |
| lmhmfgsplit 43668 | If the kernel and range of... |
| lmhmlnmsplit 43669 | If the kernel and range of... |
| lnmlmic 43670 | Noetherian is an invariant... |
| pwssplit4 43671 | Splitting for structure po... |
| filnm 43672 | Finite left modules are No... |
| pwslnmlem0 43673 | Zeroeth powers are Noether... |
| pwslnmlem1 43674 | First powers are Noetheria... |
| pwslnmlem2 43675 | A sum of powers is Noether... |
| pwslnm 43676 | Finite powers of Noetheria... |
| unxpwdom3 43677 | Weaker version of ~ unxpwd... |
| pwfi2f1o 43678 | The ~ pw2f1o bijection rel... |
| pwfi2en 43679 | Finitely supported indicat... |
| frlmpwfi 43680 | Formal linear combinations... |
| gicabl 43681 | Being Abelian is a group i... |
| imasgim 43682 | A relabeling of the elemen... |
| isnumbasgrplem1 43683 | A set which is equipollent... |
| harn0 43684 | The Hartogs number of a se... |
| numinfctb 43685 | A numerable infinite set c... |
| isnumbasgrplem2 43686 | If the (to be thought of a... |
| isnumbasgrplem3 43687 | Every nonempty numerable s... |
| isnumbasabl 43688 | A set is numerable iff it ... |
| isnumbasgrp 43689 | A set is numerable iff it ... |
| dfacbasgrp 43690 | A choice equivalent in abs... |
| islnr 43693 | Property of a left-Noether... |
| lnrring 43694 | Left-Noetherian rings are ... |
| lnrlnm 43695 | Left-Noetherian rings have... |
| islnr2 43696 | Property of being a left-N... |
| islnr3 43697 | Relate left-Noetherian rin... |
| lnr2i 43698 | Given an ideal in a left-N... |
| lpirlnr 43699 | Left principal ideal rings... |
| lnrfrlm 43700 | Finite-dimensional free mo... |
| lnrfg 43701 | Finitely-generated modules... |
| lnrfgtr 43702 | A submodule of a finitely ... |
| hbtlem1 43705 | Value of the leading coeff... |
| hbtlem2 43706 | Leading coefficient ideals... |
| hbtlem7 43707 | Functionality of leading c... |
| hbtlem4 43708 | The leading ideal function... |
| hbtlem3 43709 | The leading ideal function... |
| hbtlem5 43710 | The leading ideal function... |
| hbtlem6 43711 | There is a finite set of p... |
| hbt 43712 | The Hilbert Basis Theorem ... |
| dgrsub2 43717 | Subtracting two polynomial... |
| elmnc 43718 | Property of a monic polyno... |
| mncply 43719 | A monic polynomial is a po... |
| mnccoe 43720 | A monic polynomial has lea... |
| mncn0 43721 | A monic polynomial is not ... |
| dgraaval 43726 | Value of the degree functi... |
| dgraalem 43727 | Properties of the degree o... |
| dgraacl 43728 | Closure of the degree func... |
| dgraaf 43729 | Degree function on algebra... |
| dgraaub 43730 | Upper bound on degree of a... |
| dgraa0p 43731 | A rational polynomial of d... |
| mpaaeu 43732 | An algebraic number has ex... |
| mpaaval 43733 | Value of the minimal polyn... |
| mpaalem 43734 | Properties of the minimal ... |
| mpaacl 43735 | Minimal polynomial is a po... |
| mpaadgr 43736 | Minimal polynomial has deg... |
| mpaaroot 43737 | The minimal polynomial of ... |
| mpaamn 43738 | Minimal polynomial is moni... |
| itgoval 43743 | Value of the integral-over... |
| aaitgo 43744 | The standard algebraic num... |
| itgoss 43745 | An integral element is int... |
| itgocn 43746 | All integral elements are ... |
| cnsrexpcl 43747 | Exponentiation is closed i... |
| fsumcnsrcl 43748 | Finite sums are closed in ... |
| cnsrplycl 43749 | Polynomials are closed in ... |
| rgspnid 43750 | The span of a subring is i... |
| rngunsnply 43751 | Adjoining one element to a... |
| flcidc 43752 | Finite linear combinations... |
| algstr 43755 | Lemma to shorten proofs of... |
| algbase 43756 | The base set of a construc... |
| algaddg 43757 | The additive operation of ... |
| algmulr 43758 | The multiplicative operati... |
| algsca 43759 | The set of scalars of a co... |
| algvsca 43760 | The scalar product operati... |
| mendval 43761 | Value of the module endomo... |
| mendbas 43762 | Base set of the module end... |
| mendplusgfval 43763 | Addition in the module end... |
| mendplusg 43764 | A specific addition in the... |
| mendmulrfval 43765 | Multiplication in the modu... |
| mendmulr 43766 | A specific multiplication ... |
| mendsca 43767 | The module endomorphism al... |
| mendvscafval 43768 | Scalar multiplication in t... |
| mendvsca 43769 | A specific scalar multipli... |
| mendring 43770 | The module endomorphism al... |
| mendlmod 43771 | The module endomorphism al... |
| mendassa 43772 | The module endomorphism al... |
| idomodle 43773 | Limit on the number of ` N... |
| fiuneneq 43774 | Two finite sets of equal s... |
| idomsubgmo 43775 | The units of an integral d... |
| proot1mul 43776 | Any primitive ` N ` -th ro... |
| proot1hash 43777 | If an integral domain has ... |
| proot1ex 43778 | The complex field has prim... |
| mon1psubm 43781 | Monic polynomials are a mu... |
| deg1mhm 43782 | Homomorphic property of th... |
| cytpfn 43783 | Functionality of the cyclo... |
| cytpval 43784 | Substitutions for the Nth ... |
| fgraphopab 43785 | Express a function as a su... |
| fgraphxp 43786 | Express a function as a su... |
| hausgraph 43787 | The graph of a continuous ... |
| r1sssucd 43792 | Deductive form of ~ r1sssu... |
| iocunico 43793 | Split an open interval int... |
| iocinico 43794 | The intersection of two se... |
| iocmbl 43795 | An open-below, closed-abov... |
| cnioobibld 43796 | A bounded, continuous func... |
| arearect 43797 | The area of a rectangle wh... |
| areaquad 43798 | The area of a quadrilatera... |
| uniel 43799 | Two ways to say a union is... |
| unielss 43800 | Two ways to say the union ... |
| unielid 43801 | Two ways to say the union ... |
| ssunib 43802 | Two ways to say a class is... |
| rp-intrabeq 43803 | Equality theorem for supre... |
| rp-unirabeq 43804 | Equality theorem for infim... |
| onmaxnelsup 43805 | Two ways to say the maximu... |
| onsupneqmaxlim0 43806 | If the supremum of a class... |
| onsupcl2 43807 | The supremum of a set of o... |
| onuniintrab 43808 | The union of a set of ordi... |
| onintunirab 43809 | The intersection of a non-... |
| onsupnmax 43810 | If the union of a class of... |
| onsupuni 43811 | The supremum of a set of o... |
| onsupuni2 43812 | The supremum of a set of o... |
| onsupintrab 43813 | The supremum of a set of o... |
| onsupintrab2 43814 | The supremum of a set of o... |
| onsupcl3 43815 | The supremum of a set of o... |
| onsupex3 43816 | The supremum of a set of o... |
| onuniintrab2 43817 | The union of a set of ordi... |
| oninfint 43818 | The infimum of a non-empty... |
| oninfunirab 43819 | The infimum of a non-empty... |
| oninfcl2 43820 | The infimum of a non-empty... |
| onsupmaxb 43821 | The union of a class of or... |
| onexgt 43822 | For any ordinal, there is ... |
| onexomgt 43823 | For any ordinal, there is ... |
| omlimcl2 43824 | The product of a limit ord... |
| onexlimgt 43825 | For any ordinal, there is ... |
| onexoegt 43826 | For any ordinal, there is ... |
| oninfex2 43827 | The infimum of a non-empty... |
| onsupeqmax 43828 | Condition when the supremu... |
| onsupeqnmax 43829 | Condition when the supremu... |
| onsuplub 43830 | The supremum of a set of o... |
| onsupnub 43831 | An upper bound of a set of... |
| onfisupcl 43832 | Sufficient condition when ... |
| onelord 43833 | Every element of a ordinal... |
| onepsuc 43834 | Every ordinal is less than... |
| epsoon 43835 | The ordinals are strictly ... |
| epirron 43836 | The strict order on the or... |
| oneptr 43837 | The strict order on the or... |
| oneltr 43838 | The elementhood relation o... |
| oneptri 43839 | The strict, complete (line... |
| ordeldif 43840 | Membership in the differen... |
| ordeldifsucon 43841 | Membership in the differen... |
| ordeldif1o 43842 | Membership in the differen... |
| ordne0gt0 43843 | Ordinal zero is less than ... |
| ondif1i 43844 | Ordinal zero is less than ... |
| onsucelab 43845 | The successor of every ord... |
| dflim6 43846 | A limit ordinal is a nonze... |
| limnsuc 43847 | A limit ordinal is not an ... |
| onsucss 43848 | If one ordinal is less tha... |
| ordnexbtwnsuc 43849 | For any distinct pair of o... |
| orddif0suc 43850 | For any distinct pair of o... |
| onsucf1lem 43851 | For ordinals, the successo... |
| onsucf1olem 43852 | The successor operation is... |
| onsucrn 43853 | The successor operation is... |
| onsucf1o 43854 | The successor operation is... |
| dflim7 43855 | A limit ordinal is a nonze... |
| onov0suclim 43856 | Compactly express rules fo... |
| oa0suclim 43857 | Closed form expression of ... |
| om0suclim 43858 | Closed form expression of ... |
| oe0suclim 43859 | Closed form expression of ... |
| oaomoecl 43860 | The operations of addition... |
| onsupsucismax 43861 | If the union of a set of o... |
| onsssupeqcond 43862 | If for every element of a ... |
| limexissup 43863 | An ordinal which is a limi... |
| limiun 43864 | A limit ordinal is the uni... |
| limexissupab 43865 | An ordinal which is a limi... |
| om1om1r 43866 | Ordinal one is both a left... |
| oe0rif 43867 | Ordinal zero raised to any... |
| oasubex 43868 | While subtraction can't be... |
| nnamecl 43869 | Natural numbers are closed... |
| onsucwordi 43870 | The successor operation pr... |
| oalim2cl 43871 | The ordinal sum of any ord... |
| oaltublim 43872 | Given ` C ` is a limit ord... |
| oaordi3 43873 | Ordinal addition of the sa... |
| oaord3 43874 | When the same ordinal is a... |
| 1oaomeqom 43875 | Ordinal one plus omega is ... |
| oaabsb 43876 | The right addend absorbs t... |
| oaordnrex 43877 | When omega is added on the... |
| oaordnr 43878 | When the same ordinal is a... |
| omge1 43879 | Any nonzero ordinal produc... |
| omge2 43880 | Any nonzero ordinal produc... |
| omlim2 43881 | The nonzero product with a... |
| omord2lim 43882 | Given a limit ordinal, the... |
| omord2i 43883 | Ordinal multiplication of ... |
| omord2com 43884 | When the same nonzero ordi... |
| 2omomeqom 43885 | Ordinal two times omega is... |
| omnord1ex 43886 | When omega is multiplied o... |
| omnord1 43887 | When the same nonzero ordi... |
| oege1 43888 | Any nonzero ordinal power ... |
| oege2 43889 | Any power of an ordinal at... |
| rp-oelim2 43890 | The power of an ordinal at... |
| oeord2lim 43891 | Given a limit ordinal, the... |
| oeord2i 43892 | Ordinal exponentiation of ... |
| oeord2com 43893 | When the same base at leas... |
| nnoeomeqom 43894 | Any natural number at leas... |
| df3o2 43895 | Ordinal 3 is the unordered... |
| df3o3 43896 | Ordinal 3, fully expanded.... |
| oenord1ex 43897 | When ordinals two and thre... |
| oenord1 43898 | When two ordinals (both at... |
| oaomoencom 43899 | Ordinal addition, multipli... |
| oenassex 43900 | Ordinal two raised to two ... |
| oenass 43901 | Ordinal exponentiation is ... |
| cantnftermord 43902 | For terms of the form of a... |
| cantnfub 43903 | Given a finite number of t... |
| cantnfub2 43904 | Given a finite number of t... |
| bropabg 43905 | Equivalence for two classe... |
| cantnfresb 43906 | A Cantor normal form which... |
| cantnf2 43907 | For every ordinal, ` A ` ,... |
| oawordex2 43908 | If ` C ` is between ` A ` ... |
| nnawordexg 43909 | If an ordinal, ` B ` , is ... |
| succlg 43910 | Closure law for ordinal su... |
| dflim5 43911 | A limit ordinal is either ... |
| oacl2g 43912 | Closure law for ordinal ad... |
| onmcl 43913 | If an ordinal is less than... |
| omabs2 43914 | Ordinal multiplication by ... |
| omcl2 43915 | Closure law for ordinal mu... |
| omcl3g 43916 | Closure law for ordinal mu... |
| ordsssucb 43917 | An ordinal number is less ... |
| tfsconcatlem 43918 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 43919 | The concatenation of two t... |
| tfsconcatfn 43920 | The concatenation of two t... |
| tfsconcatfv1 43921 | An early value of the conc... |
| tfsconcatfv2 43922 | A latter value of the conc... |
| tfsconcatfv 43923 | The value of the concatena... |
| tfsconcatrn 43924 | The range of the concatena... |
| tfsconcatfo 43925 | The concatenation of two t... |
| tfsconcatb0 43926 | The concatentation with th... |
| tfsconcat0i 43927 | The concatentation with th... |
| tfsconcat0b 43928 | The concatentation with th... |
| tfsconcat00 43929 | The concatentation of two ... |
| tfsconcatrev 43930 | If the domain of a transfi... |
| tfsconcatrnss12 43931 | The range of the concatena... |
| tfsconcatrnss 43932 | The concatenation of trans... |
| tfsconcatrnsson 43933 | The concatenation of trans... |
| tfsnfin 43934 | A transfinite sequence is ... |
| rp-tfslim 43935 | The limit of a sequence of... |
| ofoafg 43936 | Addition operator for func... |
| ofoaf 43937 | Addition operator for func... |
| ofoafo 43938 | Addition operator for func... |
| ofoacl 43939 | Closure law for component ... |
| ofoaid1 43940 | Identity law for component... |
| ofoaid2 43941 | Identity law for component... |
| ofoaass 43942 | Component-wise addition of... |
| ofoacom 43943 | Component-wise addition of... |
| naddcnff 43944 | Addition operator for Cant... |
| naddcnffn 43945 | Addition operator for Cant... |
| naddcnffo 43946 | Addition of Cantor normal ... |
| naddcnfcl 43947 | Closure law for component-... |
| naddcnfcom 43948 | Component-wise ordinal add... |
| naddcnfid1 43949 | Identity law for component... |
| naddcnfid2 43950 | Identity law for component... |
| naddcnfass 43951 | Component-wise addition of... |
| onsucunifi 43952 | The successor to the union... |
| sucunisn 43953 | The successor to the union... |
| onsucunipr 43954 | The successor to the union... |
| onsucunitp 43955 | The successor to the union... |
| oaun3lem1 43956 | The class of all ordinal s... |
| oaun3lem2 43957 | The class of all ordinal s... |
| oaun3lem3 43958 | The class of all ordinal s... |
| oaun3lem4 43959 | The class of all ordinal s... |
| rp-abid 43960 | Two ways to express a clas... |
| oadif1lem 43961 | Express the set difference... |
| oadif1 43962 | Express the set difference... |
| oaun2 43963 | Ordinal addition as a unio... |
| oaun3 43964 | Ordinal addition as a unio... |
| naddov4 43965 | Alternate expression for n... |
| nadd2rabtr 43966 | The set of ordinals which ... |
| nadd2rabord 43967 | The set of ordinals which ... |
| nadd2rabex 43968 | The class of ordinals whic... |
| nadd2rabon 43969 | The set of ordinals which ... |
| nadd1rabtr 43970 | The set of ordinals which ... |
| nadd1rabord 43971 | The set of ordinals which ... |
| nadd1rabex 43972 | The class of ordinals whic... |
| nadd1rabon 43973 | The set of ordinals which ... |
| nadd1suc 43974 | Natural addition with 1 is... |
| naddass1 43975 | Natural addition of ordina... |
| naddgeoa 43976 | Natural addition results i... |
| naddonnn 43977 | Natural addition with a na... |
| naddwordnexlem0 43978 | When ` A ` is the sum of a... |
| naddwordnexlem1 43979 | When ` A ` is the sum of a... |
| naddwordnexlem2 43980 | When ` A ` is the sum of a... |
| naddwordnexlem3 43981 | When ` A ` is the sum of a... |
| oawordex3 43982 | When ` A ` is the sum of a... |
| naddwordnexlem4 43983 | When ` A ` is the sum of a... |
| ordsssucim 43984 | If an ordinal is less than... |
| insucid 43985 | The intersection of a clas... |
| oaltom 43986 | Multiplication eventually ... |
| oe2 43987 | Two ways to square an ordi... |
| omltoe 43988 | Exponentiation eventually ... |
| abeqabi 43989 | Generalized condition for ... |
| abpr 43990 | Condition for a class abst... |
| abtp 43991 | Condition for a class abst... |
| ralopabb 43992 | Restricted universal quant... |
| fpwfvss 43993 | Functions into a powerset ... |
| sdomne0 43994 | A class that strictly domi... |
| sdomne0d 43995 | A class that strictly domi... |
| safesnsupfiss 43996 | If ` B ` is a finite subse... |
| safesnsupfiub 43997 | If ` B ` is a finite subse... |
| safesnsupfidom1o 43998 | If ` B ` is a finite subse... |
| safesnsupfilb 43999 | If ` B ` is a finite subse... |
| isoeq145d 44000 | Equality deduction for iso... |
| resisoeq45d 44001 | Equality deduction for equ... |
| negslem1 44002 | An equivalence between ide... |
| nvocnvb 44003 | Equivalence to saying the ... |
| rp-brsslt 44004 | Binary relation form of a ... |
| nla0002 44005 | Extending a linear order t... |
| nla0003 44006 | Extending a linear order t... |
| nla0001 44007 | Extending a linear order t... |
| faosnf0.11b 44008 | ` B ` is called a non-limi... |
| dfno2 44009 | A surreal number, in the f... |
| onnoxpg 44010 | Every ordinal maps to a su... |
| onnobdayg 44011 | Every ordinal maps to a su... |
| bdaybndex 44012 | Bounds formed from the bir... |
| bdaybndbday 44013 | Bounds formed from the bir... |
| onnoxp 44014 | Every ordinal maps to a su... |
| onnoxpi 44015 | Every ordinal maps to a su... |
| 0fno 44016 | Ordinal zero maps to a sur... |
| 1fno 44017 | Ordinal one maps to a surr... |
| 2fno 44018 | Ordinal two maps to a surr... |
| 3fno 44019 | Ordinal three maps to a su... |
| 4fno 44020 | Ordinal four maps to a sur... |
| fnimafnex 44021 | The functional image of a ... |
| nlimsuc 44022 | A successor is not a limit... |
| nlim1NEW 44023 | 1 is not a limit ordinal. ... |
| nlim2NEW 44024 | 2 is not a limit ordinal. ... |
| nlim3 44025 | 3 is not a limit ordinal. ... |
| nlim4 44026 | 4 is not a limit ordinal. ... |
| oa1un 44027 | Given ` A e. On ` , let ` ... |
| oa1cl 44028 | ` A +o 1o ` is in ` On ` .... |
| 0finon 44029 | 0 is a finite ordinal. Se... |
| 1finon 44030 | 1 is a finite ordinal. Se... |
| 2finon 44031 | 2 is a finite ordinal. Se... |
| 3finon 44032 | 3 is a finite ordinal. Se... |
| 4finon 44033 | 4 is a finite ordinal. Se... |
| finona1cl 44034 | The finite ordinals are cl... |
| finonex 44035 | The finite ordinals are a ... |
| fzunt 44036 | Union of two adjacent fini... |
| fzuntd 44037 | Union of two adjacent fini... |
| fzunt1d 44038 | Union of two overlapping f... |
| fzuntgd 44039 | Union of two adjacent or o... |
| ifpan123g 44040 | Conjunction of conditional... |
| ifpan23 44041 | Conjunction of conditional... |
| ifpdfor2 44042 | Define or in terms of cond... |
| ifporcor 44043 | Corollary of commutation o... |
| ifpdfan2 44044 | Define and with conditiona... |
| ifpancor 44045 | Corollary of commutation o... |
| ifpdfor 44046 | Define or in terms of cond... |
| ifpdfan 44047 | Define and with conditiona... |
| ifpbi2 44048 | Equivalence theorem for co... |
| ifpbi3 44049 | Equivalence theorem for co... |
| ifpim1 44050 | Restate implication as con... |
| ifpnot 44051 | Restate negated wff as con... |
| ifpid2 44052 | Restate wff as conditional... |
| ifpim2 44053 | Restate implication as con... |
| ifpbi23 44054 | Equivalence theorem for co... |
| ifpbiidcor 44055 | Restatement of ~ biid . (... |
| ifpbicor 44056 | Corollary of commutation o... |
| ifpxorcor 44057 | Corollary of commutation o... |
| ifpbi1 44058 | Equivalence theorem for co... |
| ifpnot23 44059 | Negation of conditional lo... |
| ifpnotnotb 44060 | Factor conditional logic o... |
| ifpnorcor 44061 | Corollary of commutation o... |
| ifpnancor 44062 | Corollary of commutation o... |
| ifpnot23b 44063 | Negation of conditional lo... |
| ifpbiidcor2 44064 | Restatement of ~ biid . (... |
| ifpnot23c 44065 | Negation of conditional lo... |
| ifpnot23d 44066 | Negation of conditional lo... |
| ifpdfnan 44067 | Define nand as conditional... |
| ifpdfxor 44068 | Define xor as conditional ... |
| ifpbi12 44069 | Equivalence theorem for co... |
| ifpbi13 44070 | Equivalence theorem for co... |
| ifpbi123 44071 | Equivalence theorem for co... |
| ifpidg 44072 | Restate wff as conditional... |
| ifpid3g 44073 | Restate wff as conditional... |
| ifpid2g 44074 | Restate wff as conditional... |
| ifpid1g 44075 | Restate wff as conditional... |
| ifpim23g 44076 | Restate implication as con... |
| ifpim3 44077 | Restate implication as con... |
| ifpnim1 44078 | Restate negated implicatio... |
| ifpim4 44079 | Restate implication as con... |
| ifpnim2 44080 | Restate negated implicatio... |
| ifpim123g 44081 | Implication of conditional... |
| ifpim1g 44082 | Implication of conditional... |
| ifp1bi 44083 | Substitute the first eleme... |
| ifpbi1b 44084 | When the first variable is... |
| ifpimimb 44085 | Factor conditional logic o... |
| ifpororb 44086 | Factor conditional logic o... |
| ifpananb 44087 | Factor conditional logic o... |
| ifpnannanb 44088 | Factor conditional logic o... |
| ifpor123g 44089 | Disjunction of conditional... |
| ifpimim 44090 | Consequnce of implication.... |
| ifpbibib 44091 | Factor conditional logic o... |
| ifpxorxorb 44092 | Factor conditional logic o... |
| rp-fakeimass 44093 | A special case where impli... |
| rp-fakeanorass 44094 | A special case where a mix... |
| rp-fakeoranass 44095 | A special case where a mix... |
| rp-fakeinunass 44096 | A special case where a mix... |
| rp-fakeuninass 44097 | A special case where a mix... |
| rp-isfinite5 44098 | A set is said to be finite... |
| rp-isfinite6 44099 | A set is said to be finite... |
| intabssd 44100 | When for each element ` y ... |
| eu0 44101 | There is only one empty se... |
| epelon2 44102 | Over the ordinal numbers, ... |
| ontric3g 44103 | For all ` x , y e. On ` , ... |
| dfsucon 44104 | ` A ` is called a successo... |
| snen1g 44105 | A singleton is equinumerou... |
| snen1el 44106 | A singleton is equinumerou... |
| sn1dom 44107 | A singleton is dominated b... |
| pr2dom 44108 | An unordered pair is domin... |
| tr3dom 44109 | An unordered triple is dom... |
| ensucne0 44110 | A class equinumerous to a ... |
| ensucne0OLD 44111 | A class equinumerous to a ... |
| dfom6 44112 | Let ` _om ` be defined to ... |
| infordmin 44113 | ` _om ` is the smallest in... |
| iscard4 44114 | Two ways to express the pr... |
| minregex 44115 | Given any cardinal number ... |
| minregex2 44116 | Given any cardinal number ... |
| iscard5 44117 | Two ways to express the pr... |
| elrncard 44118 | Let us define a cardinal n... |
| harval3 44119 | ` ( har `` A ) ` is the le... |
| harval3on 44120 | For any ordinal number ` A... |
| omssrncard 44121 | All natural numbers are ca... |
| 0iscard 44122 | 0 is a cardinal number. (... |
| 1iscard 44123 | 1 is a cardinal number. (... |
| omiscard 44124 | ` _om ` is a cardinal numb... |
| sucomisnotcard 44125 | ` _om +o 1o ` is not a car... |
| nna1iscard 44126 | For any natural number, th... |
| har2o 44127 | The least cardinal greater... |
| en2pr 44128 | A class is equinumerous to... |
| pr2cv 44129 | If an unordered pair is eq... |
| pr2el1 44130 | If an unordered pair is eq... |
| pr2cv1 44131 | If an unordered pair is eq... |
| pr2el2 44132 | If an unordered pair is eq... |
| pr2cv2 44133 | If an unordered pair is eq... |
| pren2 44134 | An unordered pair is equin... |
| pr2eldif1 44135 | If an unordered pair is eq... |
| pr2eldif2 44136 | If an unordered pair is eq... |
| pren2d 44137 | A pair of two distinct set... |
| aleph1min 44138 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 44139 | ` aleph ` is a strictly or... |
| alephiso3 44140 | ` aleph ` is a strictly or... |
| pwelg 44141 | The powerclass is an eleme... |
| pwinfig 44142 | The powerclass of an infin... |
| pwinfi2 44143 | The powerclass of an infin... |
| pwinfi3 44144 | The powerclass of an infin... |
| pwinfi 44145 | The powerclass of an infin... |
| fipjust 44146 | A definition of the finite... |
| cllem0 44147 | The class of all sets with... |
| superficl 44148 | The class of all supersets... |
| superuncl 44149 | The class of all supersets... |
| ssficl 44150 | The class of all subsets o... |
| ssuncl 44151 | The class of all subsets o... |
| ssdifcl 44152 | The class of all subsets o... |
| sssymdifcl 44153 | The class of all subsets o... |
| fiinfi 44154 | If two classes have the fi... |
| rababg 44155 | Condition when restricted ... |
| elinintab 44156 | Two ways of saying a set i... |
| elmapintrab 44157 | Two ways to say a set is a... |
| elinintrab 44158 | Two ways of saying a set i... |
| inintabss 44159 | Upper bound on intersectio... |
| inintabd 44160 | Value of the intersection ... |
| xpinintabd 44161 | Value of the intersection ... |
| relintabex 44162 | If the intersection of a c... |
| elcnvcnvintab 44163 | Two ways of saying a set i... |
| relintab 44164 | Value of the intersection ... |
| nonrel 44165 | A non-relation is equal to... |
| elnonrel 44166 | Only an ordered pair where... |
| cnvssb 44167 | Subclass theorem for conve... |
| relnonrel 44168 | The non-relation part of a... |
| cnvnonrel 44169 | The converse of the non-re... |
| brnonrel 44170 | A non-relation cannot rela... |
| dmnonrel 44171 | The domain of the non-rela... |
| rnnonrel 44172 | The range of the non-relat... |
| resnonrel 44173 | A restriction of the non-r... |
| imanonrel 44174 | An image under the non-rel... |
| cononrel1 44175 | Composition with the non-r... |
| cononrel2 44176 | Composition with the non-r... |
| elmapintab 44177 | Two ways to say a set is a... |
| fvnonrel 44178 | The function value of any ... |
| elinlem 44179 | Two ways to say a set is a... |
| elcnvcnvlem 44180 | Two ways to say a set is a... |
| cnvcnvintabd 44181 | Value of the relationship ... |
| elcnvlem 44182 | Two ways to say a set is a... |
| elcnvintab 44183 | Two ways of saying a set i... |
| cnvintabd 44184 | Value of the converse of t... |
| undmrnresiss 44185 | Two ways of saying the ide... |
| reflexg 44186 | Two ways of saying a relat... |
| cnvssco 44187 | A condition weaker than re... |
| refimssco 44188 | Reflexive relations are su... |
| cleq2lem 44189 | Equality implies bijection... |
| cbvcllem 44190 | Change of bound variable i... |
| clublem 44191 | If a superset ` Y ` of ` X... |
| clss2lem 44192 | The closure of a property ... |
| dfid7 44193 | Definition of identity rel... |
| mptrcllem 44194 | Show two versions of a clo... |
| cotrintab 44195 | The intersection of a clas... |
| rclexi 44196 | The reflexive closure of a... |
| rtrclexlem 44197 | Existence of relation impl... |
| rtrclex 44198 | The reflexive-transitive c... |
| trclubgNEW 44199 | If a relation exists then ... |
| trclubNEW 44200 | If a relation exists then ... |
| trclexi 44201 | The transitive closure of ... |
| rtrclexi 44202 | The reflexive-transitive c... |
| clrellem 44203 | When the property ` ps ` h... |
| clcnvlem 44204 | When ` A ` , an upper boun... |
| cnvtrucl0 44205 | The converse of the trivia... |
| cnvrcl0 44206 | The converse of the reflex... |
| cnvtrcl0 44207 | The converse of the transi... |
| dmtrcl 44208 | The domain of the transiti... |
| rntrcl 44209 | The range of the transitiv... |
| dfrtrcl5 44210 | Definition of reflexive-tr... |
| trcleq2lemRP 44211 | Equality implies bijection... |
| sqrtcvallem1 44212 | Two ways of saying a compl... |
| reabsifneg 44213 | Alternate expression for t... |
| reabsifnpos 44214 | Alternate expression for t... |
| reabsifpos 44215 | Alternate expression for t... |
| reabsifnneg 44216 | Alternate expression for t... |
| reabssgn 44217 | Alternate expression for t... |
| sqrtcvallem2 44218 | Equivalent to saying that ... |
| sqrtcvallem3 44219 | Equivalent to saying that ... |
| sqrtcvallem4 44220 | Equivalent to saying that ... |
| sqrtcvallem5 44221 | Equivalent to saying that ... |
| sqrtcval 44222 | Explicit formula for the c... |
| sqrtcval2 44223 | Explicit formula for the c... |
| resqrtval 44224 | Real part of the complex s... |
| imsqrtval 44225 | Imaginary part of the comp... |
| resqrtvalex 44226 | Example for ~ resqrtval . ... |
| imsqrtvalex 44227 | Example for ~ imsqrtval . ... |
| al3im 44228 | Version of ~ ax-4 for a ne... |
| intima0 44229 | Two ways of expressing the... |
| elimaint 44230 | Element of image of inters... |
| cnviun 44231 | Converse of indexed union.... |
| imaiun1 44232 | The image of an indexed un... |
| coiun1 44233 | Composition with an indexe... |
| elintima 44234 | Element of intersection of... |
| intimass 44235 | The image under the inters... |
| intimass2 44236 | The image under the inters... |
| intimag 44237 | Requirement for the image ... |
| intimasn 44238 | Two ways to express the im... |
| intimasn2 44239 | Two ways to express the im... |
| ss2iundf 44240 | Subclass theorem for index... |
| ss2iundv 44241 | Subclass theorem for index... |
| cbviuneq12df 44242 | Rule used to change the bo... |
| cbviuneq12dv 44243 | Rule used to change the bo... |
| conrel1d 44244 | Deduction about compositio... |
| conrel2d 44245 | Deduction about compositio... |
| trrelind 44246 | The intersection of transi... |
| xpintrreld 44247 | The intersection of a tran... |
| restrreld 44248 | The restriction of a trans... |
| trrelsuperreldg 44249 | Concrete construction of a... |
| trficl 44250 | The class of all transitiv... |
| cnvtrrel 44251 | The converse of a transiti... |
| trrelsuperrel2dg 44252 | Concrete construction of a... |
| dfrcl2 44255 | Reflexive closure of a rel... |
| dfrcl3 44256 | Reflexive closure of a rel... |
| dfrcl4 44257 | Reflexive closure of a rel... |
| relexp2 44258 | A set operated on by the r... |
| relexpnul 44259 | If the domain and range of... |
| eliunov2 44260 | Membership in the indexed ... |
| eltrclrec 44261 | Membership in the indexed ... |
| elrtrclrec 44262 | Membership in the indexed ... |
| briunov2 44263 | Two classes related by the... |
| brmptiunrelexpd 44264 | If two elements are connec... |
| fvmptiunrelexplb0d 44265 | If the indexed union range... |
| fvmptiunrelexplb0da 44266 | If the indexed union range... |
| fvmptiunrelexplb1d 44267 | If the indexed union range... |
| brfvid 44268 | If two elements are connec... |
| brfvidRP 44269 | If two elements are connec... |
| fvilbd 44270 | A set is a subset of its i... |
| fvilbdRP 44271 | A set is a subset of its i... |
| brfvrcld 44272 | If two elements are connec... |
| brfvrcld2 44273 | If two elements are connec... |
| fvrcllb0d 44274 | A restriction of the ident... |
| fvrcllb0da 44275 | A restriction of the ident... |
| fvrcllb1d 44276 | A set is a subset of its i... |
| brtrclrec 44277 | Two classes related by the... |
| brrtrclrec 44278 | Two classes related by the... |
| briunov2uz 44279 | Two classes related by the... |
| eliunov2uz 44280 | Membership in the indexed ... |
| ov2ssiunov2 44281 | Any particular operator va... |
| relexp0eq 44282 | The zeroth power of relati... |
| iunrelexp0 44283 | Simplification of zeroth p... |
| relexpxpnnidm 44284 | Any positive power of a Ca... |
| relexpiidm 44285 | Any power of any restricti... |
| relexpss1d 44286 | The relational power of a ... |
| comptiunov2i 44287 | The composition two indexe... |
| corclrcl 44288 | The reflexive closure is i... |
| iunrelexpmin1 44289 | The indexed union of relat... |
| relexpmulnn 44290 | With exponents limited to ... |
| relexpmulg 44291 | With ordered exponents, th... |
| trclrelexplem 44292 | The union of relational po... |
| iunrelexpmin2 44293 | The indexed union of relat... |
| relexp01min 44294 | With exponents limited to ... |
| relexp1idm 44295 | Repeated raising a relatio... |
| relexp0idm 44296 | Repeated raising a relatio... |
| relexp0a 44297 | Absorption law for zeroth ... |
| relexpxpmin 44298 | The composition of powers ... |
| relexpaddss 44299 | The composition of two pow... |
| iunrelexpuztr 44300 | The indexed union of relat... |
| dftrcl3 44301 | Transitive closure of a re... |
| brfvtrcld 44302 | If two elements are connec... |
| fvtrcllb1d 44303 | A set is a subset of its i... |
| trclfvcom 44304 | The transitive closure of ... |
| cnvtrclfv 44305 | The converse of the transi... |
| cotrcltrcl 44306 | The transitive closure is ... |
| trclimalb2 44307 | Lower bound for image unde... |
| brtrclfv2 44308 | Two ways to indicate two e... |
| trclfvdecomr 44309 | The transitive closure of ... |
| trclfvdecoml 44310 | The transitive closure of ... |
| dmtrclfvRP 44311 | The domain of the transiti... |
| rntrclfvRP 44312 | The range of the transitiv... |
| rntrclfv 44313 | The range of the transitiv... |
| dfrtrcl3 44314 | Reflexive-transitive closu... |
| brfvrtrcld 44315 | If two elements are connec... |
| fvrtrcllb0d 44316 | A restriction of the ident... |
| fvrtrcllb0da 44317 | A restriction of the ident... |
| fvrtrcllb1d 44318 | A set is a subset of its i... |
| dfrtrcl4 44319 | Reflexive-transitive closu... |
| corcltrcl 44320 | The composition of the ref... |
| cortrcltrcl 44321 | Composition with the refle... |
| corclrtrcl 44322 | Composition with the refle... |
| cotrclrcl 44323 | The composition of the ref... |
| cortrclrcl 44324 | Composition with the refle... |
| cotrclrtrcl 44325 | Composition with the refle... |
| cortrclrtrcl 44326 | The reflexive-transitive c... |
| frege77d 44327 | If the images of both ` { ... |
| frege81d 44328 | If the image of ` U ` is a... |
| frege83d 44329 | If the image of the union ... |
| frege96d 44330 | If ` C ` follows ` A ` in ... |
| frege87d 44331 | If the images of both ` { ... |
| frege91d 44332 | If ` B ` follows ` A ` in ... |
| frege97d 44333 | If ` A ` contains all elem... |
| frege98d 44334 | If ` C ` follows ` A ` and... |
| frege102d 44335 | If either ` A ` and ` C ` ... |
| frege106d 44336 | If ` B ` follows ` A ` in ... |
| frege108d 44337 | If either ` A ` and ` C ` ... |
| frege109d 44338 | If ` A ` contains all elem... |
| frege114d 44339 | If either ` R ` relates ` ... |
| frege111d 44340 | If either ` A ` and ` C ` ... |
| frege122d 44341 | If ` F ` is a function, ` ... |
| frege124d 44342 | If ` F ` is a function, ` ... |
| frege126d 44343 | If ` F ` is a function, ` ... |
| frege129d 44344 | If ` F ` is a function and... |
| frege131d 44345 | If ` F ` is a function and... |
| frege133d 44346 | If ` F ` is a function and... |
| dfxor4 44347 | Express exclusive-or in te... |
| dfxor5 44348 | Express exclusive-or in te... |
| df3or2 44349 | Express triple-or in terms... |
| df3an2 44350 | Express triple-and in term... |
| nev 44351 | Express that not every set... |
| 0pssin 44352 | Express that an intersecti... |
| dfhe2 44355 | The property of relation `... |
| dfhe3 44356 | The property of relation `... |
| heeq12 44357 | Equality law for relations... |
| heeq1 44358 | Equality law for relations... |
| heeq2 44359 | Equality law for relations... |
| sbcheg 44360 | Distribute proper substitu... |
| hess 44361 | Subclass law for relations... |
| xphe 44362 | Any Cartesian product is h... |
| 0he 44363 | The empty relation is here... |
| 0heALT 44364 | The empty relation is here... |
| he0 44365 | Any relation is hereditary... |
| unhe1 44366 | The union of two relations... |
| snhesn 44367 | Any singleton is hereditar... |
| idhe 44368 | The identity relation is h... |
| psshepw 44369 | The relation between sets ... |
| sshepw 44370 | The relation between sets ... |
| rp-simp2-frege 44373 | Simplification of triple c... |
| rp-simp2 44374 | Simplification of triple c... |
| rp-frege3g 44375 | Add antecedent to ~ ax-fre... |
| frege3 44376 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 44377 | Double-use of ~ ax-frege2 ... |
| rp-frege24 44378 | Introducing an embedded an... |
| rp-frege4g 44379 | Deduction related to distr... |
| frege4 44380 | Special case of closed for... |
| frege5 44381 | A closed form of ~ syl . ... |
| rp-7frege 44382 | Distribute antecedent and ... |
| rp-4frege 44383 | Elimination of a nested an... |
| rp-6frege 44384 | Elimination of a nested an... |
| rp-8frege 44385 | Eliminate antecedent when ... |
| rp-frege25 44386 | Closed form for ~ a1dd . ... |
| frege6 44387 | A closed form of ~ imim2d ... |
| axfrege8 44388 | Swap antecedents. Identic... |
| frege7 44389 | A closed form of ~ syl6 . ... |
| frege26 44391 | Identical to ~ idd . Prop... |
| frege27 44392 | We cannot (at the same tim... |
| frege9 44393 | Closed form of ~ syl with ... |
| frege12 44394 | A closed form of ~ com23 .... |
| frege11 44395 | Elimination of a nested an... |
| frege24 44396 | Closed form for ~ a1d . D... |
| frege16 44397 | A closed form of ~ com34 .... |
| frege25 44398 | Closed form for ~ a1dd . ... |
| frege18 44399 | Closed form of a syllogism... |
| frege22 44400 | A closed form of ~ com45 .... |
| frege10 44401 | Result commuting anteceden... |
| frege17 44402 | A closed form of ~ com3l .... |
| frege13 44403 | A closed form of ~ com3r .... |
| frege14 44404 | Closed form of a deduction... |
| frege19 44405 | A closed form of ~ syl6 . ... |
| frege23 44406 | Syllogism followed by rota... |
| frege15 44407 | A closed form of ~ com4r .... |
| frege21 44408 | Replace antecedent in ante... |
| frege20 44409 | A closed form of ~ syl8 . ... |
| axfrege28 44410 | Contraposition. Identical... |
| frege29 44412 | Closed form of ~ con3d . ... |
| frege30 44413 | Commuted, closed form of ~... |
| axfrege31 44414 | Identical to ~ notnotr . ... |
| frege32 44416 | Deduce ~ con1 from ~ con3 ... |
| frege33 44417 | If ` ph ` or ` ps ` takes ... |
| frege34 44418 | If as a consequence of the... |
| frege35 44419 | Commuted, closed form of ~... |
| frege36 44420 | The case in which ` ps ` i... |
| frege37 44421 | If ` ch ` is a necessary c... |
| frege38 44422 | Identical to ~ pm2.21 . P... |
| frege39 44423 | Syllogism between ~ pm2.18... |
| frege40 44424 | Anything implies ~ pm2.18 ... |
| axfrege41 44425 | Identical to ~ notnot . A... |
| frege42 44427 | Not not ~ id . Propositio... |
| frege43 44428 | If there is a choice only ... |
| frege44 44429 | Similar to a commuted ~ pm... |
| frege45 44430 | Deduce ~ pm2.6 from ~ con1... |
| frege46 44431 | If ` ps ` holds when ` ph ... |
| frege47 44432 | Deduce consequence follows... |
| frege48 44433 | Closed form of syllogism w... |
| frege49 44434 | Closed form of deduction w... |
| frege50 44435 | Closed form of ~ jaoi . P... |
| frege51 44436 | Compare with ~ jaod . Pro... |
| axfrege52a 44437 | Justification for ~ ax-fre... |
| frege52aid 44439 | The case when the content ... |
| frege53aid 44440 | Specialization of ~ frege5... |
| frege53a 44441 | Lemma for ~ frege55a . Pr... |
| axfrege54a 44442 | Justification for ~ ax-fre... |
| frege54cor0a 44444 | Synonym for logical equiva... |
| frege54cor1a 44445 | Reflexive equality. (Cont... |
| frege55aid 44446 | Lemma for ~ frege57aid . ... |
| frege55lem1a 44447 | Necessary deduction regard... |
| frege55lem2a 44448 | Core proof of Proposition ... |
| frege55a 44449 | Proposition 55 of [Frege18... |
| frege55cor1a 44450 | Proposition 55 of [Frege18... |
| frege56aid 44451 | Lemma for ~ frege57aid . ... |
| frege56a 44452 | Proposition 56 of [Frege18... |
| frege57aid 44453 | This is the all important ... |
| frege57a 44454 | Analogue of ~ frege57aid .... |
| axfrege58a 44455 | Identical to ~ anifp . Ju... |
| frege58acor 44457 | Lemma for ~ frege59a . (C... |
| frege59a 44458 | A kind of Aristotelian inf... |
| frege60a 44459 | Swap antecedents of ~ ax-f... |
| frege61a 44460 | Lemma for ~ frege65a . Pr... |
| frege62a 44461 | A kind of Aristotelian inf... |
| frege63a 44462 | Proposition 63 of [Frege18... |
| frege64a 44463 | Lemma for ~ frege65a . Pr... |
| frege65a 44464 | A kind of Aristotelian inf... |
| frege66a 44465 | Swap antecedents of ~ freg... |
| frege67a 44466 | Lemma for ~ frege68a . Pr... |
| frege68a 44467 | Combination of applying a ... |
| axfrege52c 44468 | Justification for ~ ax-fre... |
| frege52b 44470 | The case when the content ... |
| frege53b 44471 | Lemma for frege102 (via ~ ... |
| axfrege54c 44472 | Reflexive equality of clas... |
| frege54b 44474 | Reflexive equality of sets... |
| frege54cor1b 44475 | Reflexive equality. (Cont... |
| frege55lem1b 44476 | Necessary deduction regard... |
| frege55lem2b 44477 | Lemma for ~ frege55b . Co... |
| frege55b 44478 | Lemma for ~ frege57b . Pr... |
| frege56b 44479 | Lemma for ~ frege57b . Pr... |
| frege57b 44480 | Analogue of ~ frege57aid .... |
| axfrege58b 44481 | If ` A. x ph ` is affirmed... |
| frege58bid 44483 | If ` A. x ph ` is affirmed... |
| frege58bcor 44484 | Lemma for ~ frege59b . (C... |
| frege59b 44485 | A kind of Aristotelian inf... |
| frege60b 44486 | Swap antecedents of ~ ax-f... |
| frege61b 44487 | Lemma for ~ frege65b . Pr... |
| frege62b 44488 | A kind of Aristotelian inf... |
| frege63b 44489 | Lemma for ~ frege91 . Pro... |
| frege64b 44490 | Lemma for ~ frege65b . Pr... |
| frege65b 44491 | A kind of Aristotelian inf... |
| frege66b 44492 | Swap antecedents of ~ freg... |
| frege67b 44493 | Lemma for ~ frege68b . Pr... |
| frege68b 44494 | Combination of applying a ... |
| frege53c 44495 | Proposition 53 of [Frege18... |
| frege54cor1c 44496 | Reflexive equality. (Cont... |
| frege55lem1c 44497 | Necessary deduction regard... |
| frege55lem2c 44498 | Core proof of Proposition ... |
| frege55c 44499 | Proposition 55 of [Frege18... |
| frege56c 44500 | Lemma for ~ frege57c . Pr... |
| frege57c 44501 | Swap order of implication ... |
| frege58c 44502 | Principle related to ~ sp ... |
| frege59c 44503 | A kind of Aristotelian inf... |
| frege60c 44504 | Swap antecedents of ~ freg... |
| frege61c 44505 | Lemma for ~ frege65c . Pr... |
| frege62c 44506 | A kind of Aristotelian inf... |
| frege63c 44507 | Analogue of ~ frege63b . ... |
| frege64c 44508 | Lemma for ~ frege65c . Pr... |
| frege65c 44509 | A kind of Aristotelian inf... |
| frege66c 44510 | Swap antecedents of ~ freg... |
| frege67c 44511 | Lemma for ~ frege68c . Pr... |
| frege68c 44512 | Combination of applying a ... |
| dffrege69 44513 | If from the proposition th... |
| frege70 44514 | Lemma for ~ frege72 . Pro... |
| frege71 44515 | Lemma for ~ frege72 . Pro... |
| frege72 44516 | If property ` A ` is hered... |
| frege73 44517 | Lemma for ~ frege87 . Pro... |
| frege74 44518 | If ` X ` has a property ` ... |
| frege75 44519 | If from the proposition th... |
| dffrege76 44520 | If from the two propositio... |
| frege77 44521 | If ` Y ` follows ` X ` in ... |
| frege78 44522 | Commuted form of ~ frege77... |
| frege79 44523 | Distributed form of ~ freg... |
| frege80 44524 | Add additional condition t... |
| frege81 44525 | If ` X ` has a property ` ... |
| frege82 44526 | Closed-form deduction base... |
| frege83 44527 | Apply commuted form of ~ f... |
| frege84 44528 | Commuted form of ~ frege81... |
| frege85 44529 | Commuted form of ~ frege77... |
| frege86 44530 | Conclusion about element o... |
| frege87 44531 | If ` Z ` is a result of an... |
| frege88 44532 | Commuted form of ~ frege87... |
| frege89 44533 | One direction of ~ dffrege... |
| frege90 44534 | Add antecedent to ~ frege8... |
| frege91 44535 | Every result of an applica... |
| frege92 44536 | Inference from ~ frege91 .... |
| frege93 44537 | Necessary condition for tw... |
| frege94 44538 | Looking one past a pair re... |
| frege95 44539 | Looking one past a pair re... |
| frege96 44540 | Every result of an applica... |
| frege97 44541 | The property of following ... |
| frege98 44542 | If ` Y ` follows ` X ` and... |
| dffrege99 44543 | If ` Z ` is identical with... |
| frege100 44544 | One direction of ~ dffrege... |
| frege101 44545 | Lemma for ~ frege102 . Pr... |
| frege102 44546 | If ` Z ` belongs to the ` ... |
| frege103 44547 | Proposition 103 of [Frege1... |
| frege104 44548 | Proposition 104 of [Frege1... |
| frege105 44549 | Proposition 105 of [Frege1... |
| frege106 44550 | Whatever follows ` X ` in ... |
| frege107 44551 | Proposition 107 of [Frege1... |
| frege108 44552 | If ` Y ` belongs to the ` ... |
| frege109 44553 | The property of belonging ... |
| frege110 44554 | Proposition 110 of [Frege1... |
| frege111 44555 | If ` Y ` belongs to the ` ... |
| frege112 44556 | Identity implies belonging... |
| frege113 44557 | Proposition 113 of [Frege1... |
| frege114 44558 | If ` X ` belongs to the ` ... |
| dffrege115 44559 | If from the circumstance t... |
| frege116 44560 | One direction of ~ dffrege... |
| frege117 44561 | Lemma for ~ frege118 . Pr... |
| frege118 44562 | Simplified application of ... |
| frege119 44563 | Lemma for ~ frege120 . Pr... |
| frege120 44564 | Simplified application of ... |
| frege121 44565 | Lemma for ~ frege122 . Pr... |
| frege122 44566 | If ` X ` is a result of an... |
| frege123 44567 | Lemma for ~ frege124 . Pr... |
| frege124 44568 | If ` X ` is a result of an... |
| frege125 44569 | Lemma for ~ frege126 . Pr... |
| frege126 44570 | If ` M ` follows ` Y ` in ... |
| frege127 44571 | Communte antecedents of ~ ... |
| frege128 44572 | Lemma for ~ frege129 . Pr... |
| frege129 44573 | If the procedure ` R ` is ... |
| frege130 44574 | Lemma for ~ frege131 . Pr... |
| frege131 44575 | If the procedure ` R ` is ... |
| frege132 44576 | Lemma for ~ frege133 . Pr... |
| frege133 44577 | If the procedure ` R ` is ... |
| enrelmap 44578 | The set of all possible re... |
| enrelmapr 44579 | The set of all possible re... |
| enmappw 44580 | The set of all mappings fr... |
| enmappwid 44581 | The set of all mappings fr... |
| rfovd 44582 | Value of the operator, ` (... |
| rfovfvd 44583 | Value of the operator, ` (... |
| rfovfvfvd 44584 | Value of the operator, ` (... |
| rfovcnvf1od 44585 | Properties of the operator... |
| rfovcnvd 44586 | Value of the converse of t... |
| rfovf1od 44587 | The value of the operator,... |
| rfovcnvfvd 44588 | Value of the converse of t... |
| fsovd 44589 | Value of the operator, ` (... |
| fsovrfovd 44590 | The operator which gives a... |
| fsovfvd 44591 | Value of the operator, ` (... |
| fsovfvfvd 44592 | Value of the operator, ` (... |
| fsovfd 44593 | The operator, ` ( A O B ) ... |
| fsovcnvlem 44594 | The ` O ` operator, which ... |
| fsovcnvd 44595 | The value of the converse ... |
| fsovcnvfvd 44596 | The value of the converse ... |
| fsovf1od 44597 | The value of ` ( A O B ) `... |
| dssmapfvd 44598 | Value of the duality opera... |
| dssmapfv2d 44599 | Value of the duality opera... |
| dssmapfv3d 44600 | Value of the duality opera... |
| dssmapnvod 44601 | For any base set ` B ` the... |
| dssmapf1od 44602 | For any base set ` B ` the... |
| dssmap2d 44603 | For any base set ` B ` the... |
| or3or 44604 | Decompose disjunction into... |
| andi3or 44605 | Distribute over triple dis... |
| uneqsn 44606 | If a union of classes is e... |
| brfvimex 44607 | If a binary relation holds... |
| brovmptimex 44608 | If a binary relation holds... |
| brovmptimex1 44609 | If a binary relation holds... |
| brovmptimex2 44610 | If a binary relation holds... |
| brcoffn 44611 | Conditions allowing the de... |
| brcofffn 44612 | Conditions allowing the de... |
| brco2f1o 44613 | Conditions allowing the de... |
| brco3f1o 44614 | Conditions allowing the de... |
| ntrclsbex 44615 | If (pseudo-)interior and (... |
| ntrclsrcomplex 44616 | The relative complement of... |
| neik0imk0p 44617 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 44618 | If an interior function is... |
| ntrkbimka 44619 | If the interiors of disjoi... |
| ntrk0kbimka 44620 | If the interiors of disjoi... |
| clsk3nimkb 44621 | If the base set is not emp... |
| clsk1indlem0 44622 | The ansatz closure functio... |
| clsk1indlem2 44623 | The ansatz closure functio... |
| clsk1indlem3 44624 | The ansatz closure functio... |
| clsk1indlem4 44625 | The ansatz closure functio... |
| clsk1indlem1 44626 | The ansatz closure functio... |
| clsk1independent 44627 | For generalized closure fu... |
| neik0pk1imk0 44628 | Kuratowski's K0' and K1 ax... |
| isotone1 44629 | Two different ways to say ... |
| isotone2 44630 | Two different ways to say ... |
| ntrk1k3eqk13 44631 | An interior function is bo... |
| ntrclsf1o 44632 | If (pseudo-)interior and (... |
| ntrclsnvobr 44633 | If (pseudo-)interior and (... |
| ntrclsiex 44634 | If (pseudo-)interior and (... |
| ntrclskex 44635 | If (pseudo-)interior and (... |
| ntrclsfv1 44636 | If (pseudo-)interior and (... |
| ntrclsfv2 44637 | If (pseudo-)interior and (... |
| ntrclselnel1 44638 | If (pseudo-)interior and (... |
| ntrclselnel2 44639 | If (pseudo-)interior and (... |
| ntrclsfv 44640 | The value of the interior ... |
| ntrclsfveq1 44641 | If interior and closure fu... |
| ntrclsfveq2 44642 | If interior and closure fu... |
| ntrclsfveq 44643 | If interior and closure fu... |
| ntrclsss 44644 | If interior and closure fu... |
| ntrclsneine0lem 44645 | If (pseudo-)interior and (... |
| ntrclsneine0 44646 | If (pseudo-)interior and (... |
| ntrclscls00 44647 | If (pseudo-)interior and (... |
| ntrclsiso 44648 | If (pseudo-)interior and (... |
| ntrclsk2 44649 | An interior function is co... |
| ntrclskb 44650 | The interiors of disjoint ... |
| ntrclsk3 44651 | The intersection of interi... |
| ntrclsk13 44652 | The interior of the inters... |
| ntrclsk4 44653 | Idempotence of the interio... |
| ntrneibex 44654 | If (pseudo-)interior and (... |
| ntrneircomplex 44655 | The relative complement of... |
| ntrneif1o 44656 | If (pseudo-)interior and (... |
| ntrneiiex 44657 | If (pseudo-)interior and (... |
| ntrneinex 44658 | If (pseudo-)interior and (... |
| ntrneicnv 44659 | If (pseudo-)interior and (... |
| ntrneifv1 44660 | If (pseudo-)interior and (... |
| ntrneifv2 44661 | If (pseudo-)interior and (... |
| ntrneiel 44662 | If (pseudo-)interior and (... |
| ntrneifv3 44663 | The value of the neighbors... |
| ntrneineine0lem 44664 | If (pseudo-)interior and (... |
| ntrneineine1lem 44665 | If (pseudo-)interior and (... |
| ntrneifv4 44666 | The value of the interior ... |
| ntrneiel2 44667 | Membership in iterated int... |
| ntrneineine0 44668 | If (pseudo-)interior and (... |
| ntrneineine1 44669 | If (pseudo-)interior and (... |
| ntrneicls00 44670 | If (pseudo-)interior and (... |
| ntrneicls11 44671 | If (pseudo-)interior and (... |
| ntrneiiso 44672 | If (pseudo-)interior and (... |
| ntrneik2 44673 | An interior function is co... |
| ntrneix2 44674 | An interior (closure) func... |
| ntrneikb 44675 | The interiors of disjoint ... |
| ntrneixb 44676 | The interiors (closures) o... |
| ntrneik3 44677 | The intersection of interi... |
| ntrneix3 44678 | The closure of the union o... |
| ntrneik13 44679 | The interior of the inters... |
| ntrneix13 44680 | The closure of the union o... |
| ntrneik4w 44681 | Idempotence of the interio... |
| ntrneik4 44682 | Idempotence of the interio... |
| clsneibex 44683 | If (pseudo-)closure and (p... |
| clsneircomplex 44684 | The relative complement of... |
| clsneif1o 44685 | If a (pseudo-)closure func... |
| clsneicnv 44686 | If a (pseudo-)closure func... |
| clsneikex 44687 | If closure and neighborhoo... |
| clsneinex 44688 | If closure and neighborhoo... |
| clsneiel1 44689 | If a (pseudo-)closure func... |
| clsneiel2 44690 | If a (pseudo-)closure func... |
| clsneifv3 44691 | Value of the neighborhoods... |
| clsneifv4 44692 | Value of the closure (inte... |
| neicvgbex 44693 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 44694 | The relative complement of... |
| neicvgf1o 44695 | If neighborhood and conver... |
| neicvgnvo 44696 | If neighborhood and conver... |
| neicvgnvor 44697 | If neighborhood and conver... |
| neicvgmex 44698 | If the neighborhoods and c... |
| neicvgnex 44699 | If the neighborhoods and c... |
| neicvgel1 44700 | A subset being an element ... |
| neicvgel2 44701 | The complement of a subset... |
| neicvgfv 44702 | The value of the neighborh... |
| ntrrn 44703 | The range of the interior ... |
| ntrf 44704 | The interior function of a... |
| ntrf2 44705 | The interior function is a... |
| ntrelmap 44706 | The interior function is a... |
| clsf2 44707 | The closure function is a ... |
| clselmap 44708 | The closure function is a ... |
| dssmapntrcls 44709 | The interior and closure o... |
| dssmapclsntr 44710 | The closure and interior o... |
| gneispa 44711 | Each point ` p ` of the ne... |
| gneispb 44712 | Given a neighborhood ` N `... |
| gneispace2 44713 | The predicate that ` F ` i... |
| gneispace3 44714 | The predicate that ` F ` i... |
| gneispace 44715 | The predicate that ` F ` i... |
| gneispacef 44716 | A generic neighborhood spa... |
| gneispacef2 44717 | A generic neighborhood spa... |
| gneispacefun 44718 | A generic neighborhood spa... |
| gneispacern 44719 | A generic neighborhood spa... |
| gneispacern2 44720 | A generic neighborhood spa... |
| gneispace0nelrn 44721 | A generic neighborhood spa... |
| gneispace0nelrn2 44722 | A generic neighborhood spa... |
| gneispace0nelrn3 44723 | A generic neighborhood spa... |
| gneispaceel 44724 | Every neighborhood of a po... |
| gneispaceel2 44725 | Every neighborhood of a po... |
| gneispacess 44726 | All supersets of a neighbo... |
| gneispacess2 44727 | All supersets of a neighbo... |
| k0004lem1 44728 | Application of ~ ssin to r... |
| k0004lem2 44729 | A mapping with a particula... |
| k0004lem3 44730 | When the value of a mappin... |
| k0004val 44731 | The topological simplex of... |
| k0004ss1 44732 | The topological simplex of... |
| k0004ss2 44733 | The topological simplex of... |
| k0004ss3 44734 | The topological simplex of... |
| k0004val0 44735 | The topological simplex of... |
| inductionexd 44736 | Simple induction example. ... |
| wwlemuld 44737 | Natural deduction form of ... |
| leeq1d 44738 | Specialization of ~ breq1d... |
| leeq2d 44739 | Specialization of ~ breq2d... |
| absmulrposd 44740 | Specialization of absmuld ... |
| imadisjld 44741 | Natural dduction form of o... |
| wnefimgd 44742 | The image of a mapping fro... |
| fco2d 44743 | Natural deduction form of ... |
| wfximgfd 44744 | The value of a function on... |
| extoimad 44745 | If |f(x)| <= C for all x t... |
| imo72b2lem0 44746 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 44747 | Natural deduction form of ... |
| imo72b2lem2 44748 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 44749 | Natural deduction form of ... |
| imo72b2lem1 44750 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 44751 | 'Less than or equal to' re... |
| lemuldiv4d 44752 | 'Less than or equal to' re... |
| imo72b2 44753 | IMO 1972 B2. (14th Intern... |
| int-addcomd 44754 | AdditionCommutativity gene... |
| int-addassocd 44755 | AdditionAssociativity gene... |
| int-addsimpd 44756 | AdditionSimplification gen... |
| int-mulcomd 44757 | MultiplicationCommutativit... |
| int-mulassocd 44758 | MultiplicationAssociativit... |
| int-mulsimpd 44759 | MultiplicationSimplificati... |
| int-leftdistd 44760 | AdditionMultiplicationLeft... |
| int-rightdistd 44761 | AdditionMultiplicationRigh... |
| int-sqdefd 44762 | SquareDefinition generator... |
| int-mul11d 44763 | First MultiplicationOne ge... |
| int-mul12d 44764 | Second MultiplicationOne g... |
| int-add01d 44765 | First AdditionZero generat... |
| int-add02d 44766 | Second AdditionZero genera... |
| int-sqgeq0d 44767 | SquareGEQZero generator ru... |
| int-eqprincd 44768 | PrincipleOfEquality genera... |
| int-eqtransd 44769 | EqualityTransitivity gener... |
| int-eqmvtd 44770 | EquMoveTerm generator rule... |
| int-eqineqd 44771 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 44772 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 44773 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 44774 | SecondPrincipleOfInequalit... |
| int-ineqtransd 44775 | InequalityTransitivity gen... |
| unitadd 44776 | Theorem used in conjunctio... |
| gsumws3 44777 | Valuation of a length 3 wo... |
| gsumws4 44778 | Valuation of a length 4 wo... |
| amgm2d 44779 | Arithmetic-geometric mean ... |
| amgm3d 44780 | Arithmetic-geometric mean ... |
| amgm4d 44781 | Arithmetic-geometric mean ... |
| spALT 44782 | ~ sp can be proven from th... |
| elnelneqd 44783 | Two classes are not equal ... |
| elnelneq2d 44784 | Two classes are not equal ... |
| rr-spce 44785 | Prove an existential. (Co... |
| rexlimdvaacbv 44786 | Unpack a restricted existe... |
| rexlimddvcbvw 44787 | Unpack a restricted existe... |
| rexlimddvcbv 44788 | Unpack a restricted existe... |
| rr-elrnmpt3d 44789 | Elementhood in an image se... |
| rr-phpd 44790 | Equivalent of ~ php withou... |
| tfindsd 44791 | Deduction associated with ... |
| mnringvald 44794 | Value of the monoid ring f... |
| mnringnmulrd 44795 | Components of a monoid rin... |
| mnringbased 44796 | The base set of a monoid r... |
| mnringbaserd 44797 | The base set of a monoid r... |
| mnringelbased 44798 | Membership in the base set... |
| mnringbasefd 44799 | Elements of a monoid ring ... |
| mnringbasefsuppd 44800 | Elements of a monoid ring ... |
| mnringaddgd 44801 | The additive operation of ... |
| mnring0gd 44802 | The additive identity of a... |
| mnring0g2d 44803 | The additive identity of a... |
| mnringmulrd 44804 | The ring product of a mono... |
| mnringscad 44805 | The scalar ring of a monoi... |
| mnringvscad 44806 | The scalar product of a mo... |
| mnringlmodd 44807 | Monoid rings are left modu... |
| mnringmulrvald 44808 | Value of multiplication in... |
| mnringmulrcld 44809 | Monoid rings are closed un... |
| gru0eld 44810 | A nonempty Grothendieck un... |
| grusucd 44811 | Grothendieck universes are... |
| r1rankcld 44812 | Any rank of the cumulative... |
| grur1cld 44813 | Grothendieck universes are... |
| grurankcld 44814 | Grothendieck universes are... |
| grurankrcld 44815 | If a Grothendieck universe... |
| scotteqd 44818 | Equality theorem for the S... |
| scotteq 44819 | Closed form of ~ scotteqd ... |
| nfscott 44820 | Bound-variable hypothesis ... |
| scottabf 44821 | Value of the Scott operati... |
| scottab 44822 | Value of the Scott operati... |
| scottabes 44823 | Value of the Scott operati... |
| scottss 44824 | Scott's trick produces a s... |
| elscottab 44825 | An element of the output o... |
| scottex2 44826 | ~ scottex expressed using ... |
| scotteld 44827 | The Scott operation sends ... |
| scottelrankd 44828 | Property of a Scott's tric... |
| scottrankd 44829 | Rank of a nonempty Scott's... |
| gruscottcld 44830 | If a Grothendieck universe... |
| dfcoll2 44833 | Alternate definition of th... |
| colleq12d 44834 | Equality theorem for the c... |
| colleq1 44835 | Equality theorem for the c... |
| colleq2 44836 | Equality theorem for the c... |
| nfcoll 44837 | Bound-variable hypothesis ... |
| collexd 44838 | The output of the collecti... |
| cpcolld 44839 | Property of the collection... |
| cpcoll2d 44840 | ~ cpcolld with an extra ex... |
| grucollcld 44841 | A Grothendieck universe co... |
| ismnu 44842 | The hypothesis of this the... |
| mnuop123d 44843 | Operations of a minimal un... |
| mnussd 44844 | Minimal universes are clos... |
| mnuss2d 44845 | ~ mnussd with arguments pr... |
| mnu0eld 44846 | A nonempty minimal univers... |
| mnuop23d 44847 | Second and third operation... |
| mnupwd 44848 | Minimal universes are clos... |
| mnusnd 44849 | Minimal universes are clos... |
| mnuprssd 44850 | A minimal universe contain... |
| mnuprss2d 44851 | Special case of ~ mnuprssd... |
| mnuop3d 44852 | Third operation of a minim... |
| mnuprdlem1 44853 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 44854 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 44855 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 44856 | Lemma for ~ mnuprd . Gene... |
| mnuprd 44857 | Minimal universes are clos... |
| mnuunid 44858 | Minimal universes are clos... |
| mnuund 44859 | Minimal universes are clos... |
| mnutrcld 44860 | Minimal universes contain ... |
| mnutrd 44861 | Minimal universes are tran... |
| mnurndlem1 44862 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 44863 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 44864 | Minimal universes contain ... |
| mnugrud 44865 | Minimal universes are Grot... |
| grumnudlem 44866 | Lemma for ~ grumnud . (Co... |
| grumnud 44867 | Grothendieck universes are... |
| grumnueq 44868 | The class of Grothendieck ... |
| expandan 44869 | Expand conjunction to prim... |
| expandexn 44870 | Expand an existential quan... |
| expandral 44871 | Expand a restricted univer... |
| expandrexn 44872 | Expand a restricted existe... |
| expandrex 44873 | Expand a restricted existe... |
| expanduniss 44874 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 44875 | Express the predicate on `... |
| rr-grothprimbi 44876 | Express "every set is cont... |
| inagrud 44877 | Inaccessible levels of the... |
| inaex 44878 | Assuming the Tarski-Grothe... |
| gruex 44879 | Assuming the Tarski-Grothe... |
| rr-groth 44880 | An equivalent of ~ ax-grot... |
| rr-grothprim 44881 | An equivalent of ~ ax-grot... |
| ismnushort 44882 | Express the predicate on `... |
| dfuniv2 44883 | Alternative definition of ... |
| rr-grothshortbi 44884 | Express "every set is cont... |
| rr-grothshort 44885 | A shorter equivalent of ~ ... |
| nanorxor 44886 | 'nand' is equivalent to th... |
| undisjrab 44887 | Union of two disjoint rest... |
| iso0 44888 | The empty set is an ` R , ... |
| ssrecnpr 44889 | ` RR ` is a subset of both... |
| seff 44890 | Let set ` S ` be the real ... |
| sblpnf 44891 | The infinity ball in the a... |
| prmunb2 44892 | The primes are unbounded. ... |
| dvgrat 44893 | Ratio test for divergence ... |
| cvgdvgrat 44894 | Ratio test for convergence... |
| radcnvrat 44895 | Let ` L ` be the limit, if... |
| reldvds 44896 | The divides relation is in... |
| nznngen 44897 | All positive integers in t... |
| nzss 44898 | The set of multiples of _m... |
| nzin 44899 | The intersection of the se... |
| nzprmdif 44900 | Subtract one prime's multi... |
| hashnzfz 44901 | Special case of ~ hashdvds... |
| hashnzfz2 44902 | Special case of ~ hashnzfz... |
| hashnzfzclim 44903 | As the upper bound ` K ` o... |
| caofcan 44904 | Transfer a cancellation la... |
| ofsubid 44905 | Function analogue of ~ sub... |
| ofmul12 44906 | Function analogue of ~ mul... |
| ofdivrec 44907 | Function analogue of ~ div... |
| ofdivcan4 44908 | Function analogue of ~ div... |
| ofdivdiv2 44909 | Function analogue of ~ div... |
| lhe4.4ex1a 44910 | Example of the Fundamental... |
| dvsconst 44911 | Derivative of a constant f... |
| dvsid 44912 | Derivative of the identity... |
| dvsef 44913 | Derivative of the exponent... |
| expgrowthi 44914 | Exponential growth and dec... |
| dvconstbi 44915 | The derivative of a functi... |
| expgrowth 44916 | Exponential growth and dec... |
| bccval 44919 | Value of the generalized b... |
| bcccl 44920 | Closure of the generalized... |
| bcc0 44921 | The generalized binomial c... |
| bccp1k 44922 | Generalized binomial coeff... |
| bccm1k 44923 | Generalized binomial coeff... |
| bccn0 44924 | Generalized binomial coeff... |
| bccn1 44925 | Generalized binomial coeff... |
| bccbc 44926 | The binomial coefficient a... |
| uzmptshftfval 44927 | When ` F ` is a maps-to fu... |
| dvradcnv2 44928 | The radius of convergence ... |
| binomcxplemwb 44929 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 44930 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 44931 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 44932 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 44933 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 44934 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 44935 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 44936 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 44937 | Lemma for ~ binomcxp . Wh... |
| binomcxp 44938 | Generalize the binomial th... |
| pm10.12 44939 | Theorem *10.12 in [Whitehe... |
| pm10.14 44940 | Theorem *10.14 in [Whitehe... |
| pm10.251 44941 | Theorem *10.251 in [Whiteh... |
| pm10.252 44942 | Theorem *10.252 in [Whiteh... |
| pm10.253 44943 | Theorem *10.253 in [Whiteh... |
| albitr 44944 | Theorem *10.301 in [Whiteh... |
| pm10.42 44945 | Theorem *10.42 in [Whitehe... |
| pm10.52 44946 | Theorem *10.52 in [Whitehe... |
| pm10.53 44947 | Theorem *10.53 in [Whitehe... |
| pm10.541 44948 | Theorem *10.541 in [Whiteh... |
| pm10.542 44949 | Theorem *10.542 in [Whiteh... |
| pm10.55 44950 | Theorem *10.55 in [Whitehe... |
| pm10.56 44951 | Theorem *10.56 in [Whitehe... |
| pm10.57 44952 | Theorem *10.57 in [Whitehe... |
| 2alanimi 44953 | Removes two universal quan... |
| 2al2imi 44954 | Removes two universal quan... |
| pm11.11 44955 | Theorem *11.11 in [Whitehe... |
| pm11.12 44956 | Theorem *11.12 in [Whitehe... |
| 19.21vv 44957 | Compare Theorem *11.3 in [... |
| 2alim 44958 | Theorem *11.32 in [Whitehe... |
| 2albi 44959 | Theorem *11.33 in [Whitehe... |
| 2exim 44960 | Theorem *11.34 in [Whitehe... |
| 2exbi 44961 | Theorem *11.341 in [Whiteh... |
| spsbce-2 44962 | Theorem *11.36 in [Whitehe... |
| 19.33-2 44963 | Theorem *11.421 in [Whiteh... |
| 19.36vv 44964 | Theorem *11.43 in [Whitehe... |
| 19.31vv 44965 | Theorem *11.44 in [Whitehe... |
| 19.37vv 44966 | Theorem *11.46 in [Whitehe... |
| 19.28vv 44967 | Theorem *11.47 in [Whitehe... |
| pm11.52 44968 | Theorem *11.52 in [Whitehe... |
| aaanv 44969 | Theorem *11.56 in [Whitehe... |
| pm11.57 44970 | Theorem *11.57 in [Whitehe... |
| pm11.58 44971 | Theorem *11.58 in [Whitehe... |
| pm11.59 44972 | Theorem *11.59 in [Whitehe... |
| pm11.6 44973 | Theorem *11.6 in [Whitehea... |
| pm11.61 44974 | Theorem *11.61 in [Whitehe... |
| pm11.62 44975 | Theorem *11.62 in [Whitehe... |
| pm11.63 44976 | Theorem *11.63 in [Whitehe... |
| pm11.7 44977 | Theorem *11.7 in [Whitehea... |
| pm11.71 44978 | Theorem *11.71 in [Whitehe... |
| sbeqal1 44979 | If ` x = y ` always implie... |
| sbeqal1i 44980 | Suppose you know ` x = y `... |
| sbeqal2i 44981 | If ` x = y ` implies ` x =... |
| axc5c4c711 44982 | Proof of a theorem that ca... |
| axc5c4c711toc5 44983 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 44984 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 44985 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 44986 | Rederivation of ~ ax-11 fr... |
| axc11next 44987 | This theorem shows that, g... |
| pm13.13a 44988 | One result of theorem *13.... |
| pm13.13b 44989 | Theorem *13.13 in [Whitehe... |
| pm13.14 44990 | Theorem *13.14 in [Whitehe... |
| pm13.192 44991 | Theorem *13.192 in [Whiteh... |
| pm13.193 44992 | Theorem *13.193 in [Whiteh... |
| pm13.194 44993 | Theorem *13.194 in [Whiteh... |
| pm13.195 44994 | Theorem *13.195 in [Whiteh... |
| pm13.196a 44995 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 44996 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 44997 | Theorem *13.22 in [Whitehe... |
| iotain 44998 | Equivalence between two di... |
| iotaexeu 44999 | The iota class exists. Th... |
| iotasbc 45000 | Definition *14.01 in [Whit... |
| iotasbc2 45001 | Theorem *14.111 in [Whiteh... |
| pm14.12 45002 | Theorem *14.12 in [Whitehe... |
| pm14.122a 45003 | Theorem *14.122 in [Whiteh... |
| pm14.122b 45004 | Theorem *14.122 in [Whiteh... |
| pm14.122c 45005 | Theorem *14.122 in [Whiteh... |
| pm14.123a 45006 | Theorem *14.123 in [Whiteh... |
| pm14.123b 45007 | Theorem *14.123 in [Whiteh... |
| pm14.123c 45008 | Theorem *14.123 in [Whiteh... |
| pm14.18 45009 | Theorem *14.18 in [Whitehe... |
| iotaequ 45010 | Theorem *14.2 in [Whitehea... |
| iotavalb 45011 | Theorem *14.202 in [Whiteh... |
| iotasbc5 45012 | Theorem *14.205 in [Whiteh... |
| pm14.24 45013 | Theorem *14.24 in [Whitehe... |
| iotavalsb 45014 | Theorem *14.242 in [Whiteh... |
| sbiota1 45015 | Theorem *14.25 in [Whitehe... |
| sbaniota 45016 | Theorem *14.26 in [Whitehe... |
| iotasbcq 45017 | Theorem *14.272 in [Whiteh... |
| elnev 45018 | Any set that contains one ... |
| rusbcALT 45019 | A version of Russell's par... |
| compeq 45020 | Equality between two ways ... |
| compne 45021 | The complement of ` A ` is... |
| compab 45022 | Two ways of saying "the co... |
| conss2 45023 | Contrapositive law for sub... |
| conss1 45024 | Contrapositive law for sub... |
| ralbidar 45025 | More general form of ~ ral... |
| rexbidar 45026 | More general form of ~ rex... |
| dropab1 45027 | Theorem to aid use of the ... |
| dropab2 45028 | Theorem to aid use of the ... |
| ipo0 45029 | If the identity relation p... |
| ifr0 45030 | A class that is founded by... |
| ordpss 45031 | ~ ordelpss with an anteced... |
| fvsb 45032 | Explicit substitution of a... |
| fveqsb 45033 | Implicit substitution of a... |
| xpexb 45034 | A Cartesian product exists... |
| trelpss 45035 | An element of a transitive... |
| addcomgi 45036 | Generalization of commutat... |
| addrval 45046 | Value of the operation of ... |
| subrval 45047 | Value of the operation of ... |
| mulvval 45048 | Value of the operation of ... |
| addrfv 45049 | Vector addition at a value... |
| subrfv 45050 | Vector subtraction at a va... |
| mulvfv 45051 | Scalar multiplication at a... |
| addrfn 45052 | Vector addition produces a... |
| subrfn 45053 | Vector subtraction produce... |
| mulvfn 45054 | Scalar multiplication prod... |
| addrcom 45055 | Vector addition is commuta... |
| idiALT 45059 | Placeholder for ~ idi . T... |
| exbir 45060 | Exportation implication al... |
| 3impexpbicom 45061 | Version of ~ 3impexp where... |
| 3impexpbicomi 45062 | Inference associated with ... |
| bi1imp 45063 | Importation inference simi... |
| bi2imp 45064 | Importation inference simi... |
| bi3impb 45065 | Similar to ~ 3impb with im... |
| bi3impa 45066 | Similar to ~ 3impa with im... |
| bi23impib 45067 | ~ 3impib with the inner im... |
| bi13impib 45068 | ~ 3impib with the outer im... |
| bi123impib 45069 | ~ 3impib with the implicat... |
| bi13impia 45070 | ~ 3impia with the outer im... |
| bi123impia 45071 | ~ 3impia with the implicat... |
| bi33imp12 45072 | ~ 3imp with innermost impl... |
| bi13imp23 45073 | ~ 3imp with outermost impl... |
| bi13imp2 45074 | Similar to ~ 3imp except t... |
| bi12imp3 45075 | Similar to ~ 3imp except a... |
| bi23imp1 45076 | Similar to ~ 3imp except a... |
| bi123imp0 45077 | Similar to ~ 3imp except a... |
| 4animp1 45078 | A single hypothesis unific... |
| 4an31 45079 | A rearrangement of conjunc... |
| 4an4132 45080 | A rearrangement of conjunc... |
| expcomdg 45081 | Biconditional form of ~ ex... |
| iidn3 45082 | ~ idn3 without virtual ded... |
| ee222 45083 | ~ e222 without virtual ded... |
| ee3bir 45084 | Right-biconditional form o... |
| ee13 45085 | ~ e13 without virtual dedu... |
| ee121 45086 | ~ e121 without virtual ded... |
| ee122 45087 | ~ e122 without virtual ded... |
| ee333 45088 | ~ e333 without virtual ded... |
| ee323 45089 | ~ e323 without virtual ded... |
| 3ornot23 45090 | If the second and third di... |
| orbi1r 45091 | ~ orbi1 with order of disj... |
| 3orbi123 45092 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 45093 | Closed form of ~ syl5 . D... |
| impexpd 45094 | The following User's Proof... |
| com3rgbi 45095 | The following User's Proof... |
| impexpdcom 45096 | The following User's Proof... |
| ee1111 45097 | Non-virtual deduction form... |
| pm2.43bgbi 45098 | Logical equivalence of a 2... |
| pm2.43cbi 45099 | Logical equivalence of a 3... |
| ee233 45100 | Non-virtual deduction form... |
| imbi13 45101 | Join three logical equival... |
| ee33 45102 | Non-virtual deduction form... |
| con5 45103 | Biconditional contrapositi... |
| con5i 45104 | Inference form of ~ con5 .... |
| exlimexi 45105 | Inference similar to Theor... |
| sb5ALT 45106 | Equivalence for substituti... |
| eexinst01 45107 | ~ exinst01 without virtual... |
| eexinst11 45108 | ~ exinst11 without virtual... |
| vk15.4j 45109 | Excercise 4j of Unit 15 of... |
| notnotrALT 45110 | Converse of double negatio... |
| con3ALT2 45111 | Contraposition. Alternate... |
| ssralv2 45112 | Quantification restricted ... |
| sbc3or 45113 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 45114 | Closed form of ~ alrimi wi... |
| rspsbc2 45115 | ~ rspsbc with two quantify... |
| sbcoreleleq 45116 | Substitution of a setvar v... |
| tratrb 45117 | If a class is transitive a... |
| ordelordALT 45118 | An element of an ordinal c... |
| sbcim2g 45119 | Distribution of class subs... |
| sbcbi 45120 | Implication form of ~ sbcb... |
| trsbc 45121 | Formula-building inference... |
| truniALT 45122 | The union of a class of tr... |
| onfrALTlem5 45123 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 45124 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 45125 | Lemma for ~ onfrALT . (Co... |
| ggen31 45126 | ~ gen31 without virtual de... |
| onfrALTlem2 45127 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 45128 | A theorem pertaining to th... |
| onfrALTlem1 45129 | Lemma for ~ onfrALT . (Co... |
| onfrALT 45130 | The membership relation is... |
| 19.41rg 45131 | Closed form of right-to-le... |
| opelopab4 45132 | Ordered pair membership in... |
| 2pm13.193 45133 | ~ pm13.193 for two variabl... |
| hbntal 45134 | A closed form of ~ hbn . ~... |
| hbimpg 45135 | A closed form of ~ hbim . ... |
| hbalg 45136 | Closed form of ~ hbal . D... |
| hbexg 45137 | Closed form of ~ nfex . D... |
| ax6e2eq 45138 | Alternate form of ~ ax6e f... |
| ax6e2nd 45139 | If at least two sets exist... |
| ax6e2ndeq 45140 | "At least two sets exist" ... |
| 2sb5nd 45141 | Equivalence for double sub... |
| 2uasbanh 45142 | Distribute the unabbreviat... |
| 2uasban 45143 | Distribute the unabbreviat... |
| e2ebind 45144 | Absorption of an existenti... |
| elpwgded 45145 | ~ elpwgdedVD in convention... |
| trelded 45146 | Deduction form of ~ trel .... |
| jaoded 45147 | Deduction form of ~ jao . ... |
| sbtT 45148 | A substitution into a theo... |
| not12an2impnot1 45149 | If a double conjunction is... |
| in1 45152 | Inference form of ~ df-vd1... |
| iin1 45153 | ~ in1 without virtual dedu... |
| dfvd1ir 45154 | Inference form of ~ df-vd1... |
| idn1 45155 | Virtual deduction identity... |
| dfvd1imp 45156 | Left-to-right part of defi... |
| dfvd1impr 45157 | Right-to-left part of defi... |
| dfvd2 45160 | Definition of a 2-hypothes... |
| dfvd2an 45163 | Definition of a 2-hypothes... |
| dfvd2ani 45164 | Inference form of ~ dfvd2a... |
| dfvd2anir 45165 | Right-to-left inference fo... |
| dfvd2i 45166 | Inference form of ~ dfvd2 ... |
| dfvd2ir 45167 | Right-to-left inference fo... |
| dfvd3 45172 | Definition of a 3-hypothes... |
| dfvd3i 45173 | Inference form of ~ dfvd3 ... |
| dfvd3ir 45174 | Right-to-left inference fo... |
| dfvd3an 45175 | Definition of a 3-hypothes... |
| dfvd3ani 45176 | Inference form of ~ dfvd3a... |
| dfvd3anir 45177 | Right-to-left inference fo... |
| vd01 45178 | A virtual hypothesis virtu... |
| vd02 45179 | Two virtual hypotheses vir... |
| vd03 45180 | A theorem is virtually inf... |
| vd12 45181 | A virtual deduction with 1... |
| vd13 45182 | A virtual deduction with 1... |
| vd23 45183 | A virtual deduction with 2... |
| dfvd2imp 45184 | The virtual deduction form... |
| dfvd2impr 45185 | A 2-antecedent nested impl... |
| in2 45186 | The virtual deduction intr... |
| int2 45187 | The virtual deduction intr... |
| iin2 45188 | ~ in2 without virtual dedu... |
| in2an 45189 | The virtual deduction intr... |
| in3 45190 | The virtual deduction intr... |
| iin3 45191 | ~ in3 without virtual dedu... |
| in3an 45192 | The virtual deduction intr... |
| int3 45193 | The virtual deduction intr... |
| idn2 45194 | Virtual deduction identity... |
| iden2 45195 | Virtual deduction identity... |
| idn3 45196 | Virtual deduction identity... |
| gen11 45197 | Virtual deduction generali... |
| gen11nv 45198 | Virtual deduction generali... |
| gen12 45199 | Virtual deduction generali... |
| gen21 45200 | Virtual deduction generali... |
| gen21nv 45201 | Virtual deduction form of ... |
| gen31 45202 | Virtual deduction generali... |
| gen22 45203 | Virtual deduction generali... |
| ggen22 45204 | ~ gen22 without virtual de... |
| exinst 45205 | Existential Instantiation.... |
| exinst01 45206 | Existential Instantiation.... |
| exinst11 45207 | Existential Instantiation.... |
| e1a 45208 | A Virtual deduction elimin... |
| el1 45209 | A Virtual deduction elimin... |
| e1bi 45210 | Biconditional form of ~ e1... |
| e1bir 45211 | Right biconditional form o... |
| e2 45212 | A virtual deduction elimin... |
| e2bi 45213 | Biconditional form of ~ e2... |
| e2bir 45214 | Right biconditional form o... |
| ee223 45215 | ~ e223 without virtual ded... |
| e223 45216 | A virtual deduction elimin... |
| e222 45217 | A virtual deduction elimin... |
| e220 45218 | A virtual deduction elimin... |
| ee220 45219 | ~ e220 without virtual ded... |
| e202 45220 | A virtual deduction elimin... |
| ee202 45221 | ~ e202 without virtual ded... |
| e022 45222 | A virtual deduction elimin... |
| ee022 45223 | ~ e022 without virtual ded... |
| e002 45224 | A virtual deduction elimin... |
| ee002 45225 | ~ e002 without virtual ded... |
| e020 45226 | A virtual deduction elimin... |
| ee020 45227 | ~ e020 without virtual ded... |
| e200 45228 | A virtual deduction elimin... |
| ee200 45229 | ~ e200 without virtual ded... |
| e221 45230 | A virtual deduction elimin... |
| ee221 45231 | ~ e221 without virtual ded... |
| e212 45232 | A virtual deduction elimin... |
| ee212 45233 | ~ e212 without virtual ded... |
| e122 45234 | A virtual deduction elimin... |
| e112 45235 | A virtual deduction elimin... |
| ee112 45236 | ~ e112 without virtual ded... |
| e121 45237 | A virtual deduction elimin... |
| e211 45238 | A virtual deduction elimin... |
| ee211 45239 | ~ e211 without virtual ded... |
| e210 45240 | A virtual deduction elimin... |
| ee210 45241 | ~ e210 without virtual ded... |
| e201 45242 | A virtual deduction elimin... |
| ee201 45243 | ~ e201 without virtual ded... |
| e120 45244 | A virtual deduction elimin... |
| ee120 45245 | Virtual deduction rule ~ e... |
| e021 45246 | A virtual deduction elimin... |
| ee021 45247 | ~ e021 without virtual ded... |
| e012 45248 | A virtual deduction elimin... |
| ee012 45249 | ~ e012 without virtual ded... |
| e102 45250 | A virtual deduction elimin... |
| ee102 45251 | ~ e102 without virtual ded... |
| e22 45252 | A virtual deduction elimin... |
| e22an 45253 | Conjunction form of ~ e22 ... |
| ee22an 45254 | ~ e22an without virtual de... |
| e111 45255 | A virtual deduction elimin... |
| e1111 45256 | A virtual deduction elimin... |
| e110 45257 | A virtual deduction elimin... |
| ee110 45258 | ~ e110 without virtual ded... |
| e101 45259 | A virtual deduction elimin... |
| ee101 45260 | ~ e101 without virtual ded... |
| e011 45261 | A virtual deduction elimin... |
| ee011 45262 | ~ e011 without virtual ded... |
| e100 45263 | A virtual deduction elimin... |
| ee100 45264 | ~ e100 without virtual ded... |
| e010 45265 | A virtual deduction elimin... |
| ee010 45266 | ~ e010 without virtual ded... |
| e001 45267 | A virtual deduction elimin... |
| ee001 45268 | ~ e001 without virtual ded... |
| e11 45269 | A virtual deduction elimin... |
| e11an 45270 | Conjunction form of ~ e11 ... |
| ee11an 45271 | ~ e11an without virtual de... |
| e01 45272 | A virtual deduction elimin... |
| e01an 45273 | Conjunction form of ~ e01 ... |
| ee01an 45274 | ~ e01an without virtual de... |
| e10 45275 | A virtual deduction elimin... |
| e10an 45276 | Conjunction form of ~ e10 ... |
| ee10an 45277 | ~ e10an without virtual de... |
| e02 45278 | A virtual deduction elimin... |
| e02an 45279 | Conjunction form of ~ e02 ... |
| ee02an 45280 | ~ e02an without virtual de... |
| eel021old 45281 | ~ el021old without virtual... |
| el021old 45282 | A virtual deduction elimin... |
| eel000cT 45283 | An elimination deduction. ... |
| eel0TT 45284 | An elimination deduction. ... |
| eelT00 45285 | An elimination deduction. ... |
| eelTTT 45286 | An elimination deduction. ... |
| eelT11 45287 | An elimination deduction. ... |
| eelT1 45288 | Syllogism inference combin... |
| eelT12 45289 | An elimination deduction. ... |
| eelTT1 45290 | An elimination deduction. ... |
| eelT01 45291 | An elimination deduction. ... |
| eel0T1 45292 | An elimination deduction. ... |
| eel12131 45293 | An elimination deduction. ... |
| eel2131 45294 | ~ syl2an with antecedents ... |
| eel3132 45295 | ~ syl2an with antecedents ... |
| eel0321old 45296 | ~ el0321old without virtua... |
| el0321old 45297 | A virtual deduction elimin... |
| eel2122old 45298 | ~ el2122old without virtua... |
| el2122old 45299 | A virtual deduction elimin... |
| eel0000 45300 | Elimination rule similar t... |
| eel00001 45301 | An elimination deduction. ... |
| eel00000 45302 | Elimination rule similar ~... |
| eel11111 45303 | Five-hypothesis eliminatio... |
| e12 45304 | A virtual deduction elimin... |
| e12an 45305 | Conjunction form of ~ e12 ... |
| el12 45306 | Virtual deduction form of ... |
| e20 45307 | A virtual deduction elimin... |
| e20an 45308 | Conjunction form of ~ e20 ... |
| ee20an 45309 | ~ e20an without virtual de... |
| e21 45310 | A virtual deduction elimin... |
| e21an 45311 | Conjunction form of ~ e21 ... |
| ee21an 45312 | ~ e21an without virtual de... |
| e333 45313 | A virtual deduction elimin... |
| e33 45314 | A virtual deduction elimin... |
| e33an 45315 | Conjunction form of ~ e33 ... |
| ee33an 45316 | ~ e33an without virtual de... |
| e3 45317 | Meta-connective form of ~ ... |
| e3bi 45318 | Biconditional form of ~ e3... |
| e3bir 45319 | Right biconditional form o... |
| e03 45320 | A virtual deduction elimin... |
| ee03 45321 | ~ e03 without virtual dedu... |
| e03an 45322 | Conjunction form of ~ e03 ... |
| ee03an 45323 | Conjunction form of ~ ee03... |
| e30 45324 | A virtual deduction elimin... |
| ee30 45325 | ~ e30 without virtual dedu... |
| e30an 45326 | A virtual deduction elimin... |
| ee30an 45327 | Conjunction form of ~ ee30... |
| e13 45328 | A virtual deduction elimin... |
| e13an 45329 | A virtual deduction elimin... |
| ee13an 45330 | ~ e13an without virtual de... |
| e31 45331 | A virtual deduction elimin... |
| ee31 45332 | ~ e31 without virtual dedu... |
| e31an 45333 | A virtual deduction elimin... |
| ee31an 45334 | ~ e31an without virtual de... |
| e23 45335 | A virtual deduction elimin... |
| e23an 45336 | A virtual deduction elimin... |
| ee23an 45337 | ~ e23an without virtual de... |
| e32 45338 | A virtual deduction elimin... |
| ee32 45339 | ~ e32 without virtual dedu... |
| e32an 45340 | A virtual deduction elimin... |
| ee32an 45341 | ~ e33an without virtual de... |
| e123 45342 | A virtual deduction elimin... |
| ee123 45343 | ~ e123 without virtual ded... |
| el123 45344 | A virtual deduction elimin... |
| e233 45345 | A virtual deduction elimin... |
| e323 45346 | A virtual deduction elimin... |
| e000 45347 | A virtual deduction elimin... |
| e00 45348 | Elimination rule identical... |
| e00an 45349 | Elimination rule identical... |
| eel00cT 45350 | An elimination deduction. ... |
| eelTT 45351 | An elimination deduction. ... |
| e0a 45352 | Elimination rule identical... |
| eelT 45353 | An elimination deduction. ... |
| eel0cT 45354 | An elimination deduction. ... |
| eelT0 45355 | An elimination deduction. ... |
| e0bi 45356 | Elimination rule identical... |
| e0bir 45357 | Elimination rule identical... |
| uun0.1 45358 | Convention notation form o... |
| un0.1 45359 | ` T. ` is the constant tru... |
| uunT1 45360 | A deduction unionizing a n... |
| uunT1p1 45361 | A deduction unionizing a n... |
| uunT21 45362 | A deduction unionizing a n... |
| uun121 45363 | A deduction unionizing a n... |
| uun121p1 45364 | A deduction unionizing a n... |
| uun132 45365 | A deduction unionizing a n... |
| uun132p1 45366 | A deduction unionizing a n... |
| anabss7p1 45367 | A deduction unionizing a n... |
| un10 45368 | A unionizing deduction. (... |
| un01 45369 | A unionizing deduction. (... |
| un2122 45370 | A deduction unionizing a n... |
| uun2131 45371 | A deduction unionizing a n... |
| uun2131p1 45372 | A deduction unionizing a n... |
| uunTT1 45373 | A deduction unionizing a n... |
| uunTT1p1 45374 | A deduction unionizing a n... |
| uunTT1p2 45375 | A deduction unionizing a n... |
| uunT11 45376 | A deduction unionizing a n... |
| uunT11p1 45377 | A deduction unionizing a n... |
| uunT11p2 45378 | A deduction unionizing a n... |
| uunT12 45379 | A deduction unionizing a n... |
| uunT12p1 45380 | A deduction unionizing a n... |
| uunT12p2 45381 | A deduction unionizing a n... |
| uunT12p3 45382 | A deduction unionizing a n... |
| uunT12p4 45383 | A deduction unionizing a n... |
| uunT12p5 45384 | A deduction unionizing a n... |
| uun111 45385 | A deduction unionizing a n... |
| 3anidm12p1 45386 | A deduction unionizing a n... |
| 3anidm12p2 45387 | A deduction unionizing a n... |
| uun123 45388 | A deduction unionizing a n... |
| uun123p1 45389 | A deduction unionizing a n... |
| uun123p2 45390 | A deduction unionizing a n... |
| uun123p3 45391 | A deduction unionizing a n... |
| uun123p4 45392 | A deduction unionizing a n... |
| uun2221 45393 | A deduction unionizing a n... |
| uun2221p1 45394 | A deduction unionizing a n... |
| uun2221p2 45395 | A deduction unionizing a n... |
| 3impdirp1 45396 | A deduction unionizing a n... |
| 3impcombi 45397 | A 1-hypothesis proposition... |
| trsspwALT 45398 | Virtual deduction proof of... |
| trsspwALT2 45399 | Virtual deduction proof of... |
| trsspwALT3 45400 | Short predicate calculus p... |
| sspwtr 45401 | Virtual deduction proof of... |
| sspwtrALT 45402 | Virtual deduction proof of... |
| sspwtrALT2 45403 | Short predicate calculus p... |
| pwtrVD 45404 | Virtual deduction proof of... |
| pwtrrVD 45405 | Virtual deduction proof of... |
| suctrALT 45406 | The successor of a transit... |
| snssiALTVD 45407 | Virtual deduction proof of... |
| snssiALT 45408 | If a class is an element o... |
| snsslVD 45409 | Virtual deduction proof of... |
| snssl 45410 | If a singleton is a subcla... |
| snelpwrVD 45411 | Virtual deduction proof of... |
| unipwrVD 45412 | Virtual deduction proof of... |
| unipwr 45413 | A class is a subclass of t... |
| sstrALT2VD 45414 | Virtual deduction proof of... |
| sstrALT2 45415 | Virtual deduction proof of... |
| suctrALT2VD 45416 | Virtual deduction proof of... |
| suctrALT2 45417 | Virtual deduction proof of... |
| elex2VD 45418 | Virtual deduction proof of... |
| elex22VD 45419 | Virtual deduction proof of... |
| eqsbc2VD 45420 | Virtual deduction proof of... |
| zfregs2VD 45421 | Virtual deduction proof of... |
| tpid3gVD 45422 | Virtual deduction proof of... |
| en3lplem1VD 45423 | Virtual deduction proof of... |
| en3lplem2VD 45424 | Virtual deduction proof of... |
| en3lpVD 45425 | Virtual deduction proof of... |
| simplbi2VD 45426 | Virtual deduction proof of... |
| 3ornot23VD 45427 | Virtual deduction proof of... |
| orbi1rVD 45428 | Virtual deduction proof of... |
| bitr3VD 45429 | Virtual deduction proof of... |
| 3orbi123VD 45430 | Virtual deduction proof of... |
| sbc3orgVD 45431 | Virtual deduction proof of... |
| 19.21a3con13vVD 45432 | Virtual deduction proof of... |
| exbirVD 45433 | Virtual deduction proof of... |
| exbiriVD 45434 | Virtual deduction proof of... |
| rspsbc2VD 45435 | Virtual deduction proof of... |
| 3impexpVD 45436 | Virtual deduction proof of... |
| 3impexpbicomVD 45437 | Virtual deduction proof of... |
| 3impexpbicomiVD 45438 | Virtual deduction proof of... |
| sbcoreleleqVD 45439 | Virtual deduction proof of... |
| hbra2VD 45440 | Virtual deduction proof of... |
| tratrbVD 45441 | Virtual deduction proof of... |
| al2imVD 45442 | Virtual deduction proof of... |
| syl5impVD 45443 | Virtual deduction proof of... |
| idiVD 45444 | Virtual deduction proof of... |
| ancomstVD 45445 | Closed form of ~ ancoms . ... |
| ssralv2VD 45446 | Quantification restricted ... |
| ordelordALTVD 45447 | An element of an ordinal c... |
| equncomVD 45448 | If a class equals the unio... |
| equncomiVD 45449 | Inference form of ~ equnco... |
| sucidALTVD 45450 | A set belongs to its succe... |
| sucidALT 45451 | A set belongs to its succe... |
| sucidVD 45452 | A set belongs to its succe... |
| imbi12VD 45453 | Implication form of ~ imbi... |
| imbi13VD 45454 | Join three logical equival... |
| sbcim2gVD 45455 | Distribution of class subs... |
| sbcbiVD 45456 | Implication form of ~ sbcb... |
| trsbcVD 45457 | Formula-building inference... |
| truniALTVD 45458 | The union of a class of tr... |
| ee33VD 45459 | Non-virtual deduction form... |
| trintALTVD 45460 | The intersection of a clas... |
| trintALT 45461 | The intersection of a clas... |
| undif3VD 45462 | The first equality of Exer... |
| sbcssgVD 45463 | Virtual deduction proof of... |
| csbingVD 45464 | Virtual deduction proof of... |
| onfrALTlem5VD 45465 | Virtual deduction proof of... |
| onfrALTlem4VD 45466 | Virtual deduction proof of... |
| onfrALTlem3VD 45467 | Virtual deduction proof of... |
| simplbi2comtVD 45468 | Virtual deduction proof of... |
| onfrALTlem2VD 45469 | Virtual deduction proof of... |
| onfrALTlem1VD 45470 | Virtual deduction proof of... |
| onfrALTVD 45471 | Virtual deduction proof of... |
| csbeq2gVD 45472 | Virtual deduction proof of... |
| csbsngVD 45473 | Virtual deduction proof of... |
| csbxpgVD 45474 | Virtual deduction proof of... |
| csbresgVD 45475 | Virtual deduction proof of... |
| csbrngVD 45476 | Virtual deduction proof of... |
| csbima12gALTVD 45477 | Virtual deduction proof of... |
| csbunigVD 45478 | Virtual deduction proof of... |
| csbfv12gALTVD 45479 | Virtual deduction proof of... |
| con5VD 45480 | Virtual deduction proof of... |
| relopabVD 45481 | Virtual deduction proof of... |
| 19.41rgVD 45482 | Virtual deduction proof of... |
| 2pm13.193VD 45483 | Virtual deduction proof of... |
| hbimpgVD 45484 | Virtual deduction proof of... |
| hbalgVD 45485 | Virtual deduction proof of... |
| hbexgVD 45486 | Virtual deduction proof of... |
| ax6e2eqVD 45487 | The following User's Proof... |
| ax6e2ndVD 45488 | The following User's Proof... |
| ax6e2ndeqVD 45489 | The following User's Proof... |
| 2sb5ndVD 45490 | The following User's Proof... |
| 2uasbanhVD 45491 | The following User's Proof... |
| e2ebindVD 45492 | The following User's Proof... |
| sb5ALTVD 45493 | The following User's Proof... |
| vk15.4jVD 45494 | The following User's Proof... |
| notnotrALTVD 45495 | The following User's Proof... |
| con3ALTVD 45496 | The following User's Proof... |
| elpwgdedVD 45497 | Membership in a power clas... |
| sspwimp 45498 | If a class is a subclass o... |
| sspwimpVD 45499 | The following User's Proof... |
| sspwimpcf 45500 | If a class is a subclass o... |
| sspwimpcfVD 45501 | The following User's Proof... |
| suctrALTcf 45502 | The successor of a transit... |
| suctrALTcfVD 45503 | The following User's Proof... |
| suctrALT3 45504 | The successor of a transit... |
| sspwimpALT 45505 | If a class is a subclass o... |
| unisnALT 45506 | A set equals the union of ... |
| notnotrALT2 45507 | Converse of double negatio... |
| sspwimpALT2 45508 | If a class is a subclass o... |
| e2ebindALT 45509 | Absorption of an existenti... |
| ax6e2ndALT 45510 | If at least two sets exist... |
| ax6e2ndeqALT 45511 | "At least two sets exist" ... |
| 2sb5ndALT 45512 | Equivalence for double sub... |
| chordthmALT 45513 | The intersecting chords th... |
| isosctrlem1ALT 45514 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 45515 | The indexed union of conne... |
| iunconnALT 45516 | The indexed union of conne... |
| sineq0ALT 45517 | A complex number whose sin... |
| rspesbcd 45518 | Restricted quantifier vers... |
| rext0 45519 | Nonempty existential quant... |
| dfbi1ALTa 45520 | Version of ~ dfbi1ALT usin... |
| simprimi 45521 | Inference associated with ... |
| dfbi1ALTb 45522 | Further shorten ~ dfbi1ALT... |
| relpeq1 45525 | Equality theorem for relat... |
| relpeq2 45526 | Equality theorem for relat... |
| relpeq3 45527 | Equality theorem for relat... |
| relpeq4 45528 | Equality theorem for relat... |
| relpeq5 45529 | Equality theorem for relat... |
| nfrelp 45530 | Bound-variable hypothesis ... |
| relpf 45531 | A relation-preserving func... |
| relprel 45532 | A relation-preserving func... |
| relpmin 45533 | A preimage of a minimal el... |
| relpfrlem 45534 | Lemma for ~ relpfr . Prov... |
| relpfr 45535 | If the image of a set unde... |
| orbitex 45536 | Orbits exist. Given a set... |
| orbitinit 45537 | A set is contained in its ... |
| orbitcl 45538 | The orbit under a function... |
| orbitclmpt 45539 | Version of ~ orbitcl using... |
| trwf 45540 | The class of well-founded ... |
| rankrelp 45541 | The rank function preserve... |
| wffr 45542 | The class of well-founded ... |
| trfr 45543 | A transitive class well-fo... |
| tcfr 45544 | A set is well-founded if a... |
| xpwf 45545 | The Cartesian product of t... |
| dmwf 45546 | The domain of a well-found... |
| rnwf 45547 | The range of a well-founde... |
| relwf 45548 | A relation is a well-found... |
| ralabso 45549 | Simplification of restrict... |
| rexabso 45550 | Simplification of restrict... |
| ralabsod 45551 | Deduction form of ~ ralabs... |
| rexabsod 45552 | Deduction form of ~ rexabs... |
| ralabsobidv 45553 | Formula-building lemma for... |
| rexabsobidv 45554 | Formula-building lemma for... |
| ssabso 45555 | The notion " ` x ` is a su... |
| disjabso 45556 | Disjointness is absolute f... |
| n0abso 45557 | Nonemptiness is absolute f... |
| traxext 45558 | A transitive class models ... |
| modelaxreplem1 45559 | Lemma for ~ modelaxrep . ... |
| modelaxreplem2 45560 | Lemma for ~ modelaxrep . ... |
| modelaxreplem3 45561 | Lemma for ~ modelaxrep . ... |
| modelaxrep 45562 | Conditions which guarantee... |
| ssclaxsep 45563 | A class that is closed und... |
| 0elaxnul 45564 | A class that contains the ... |
| pwclaxpow 45565 | Suppose ` M ` is a transit... |
| prclaxpr 45566 | A class that is closed und... |
| uniclaxun 45567 | A class that is closed und... |
| sswfaxreg 45568 | A subclass of the class of... |
| omssaxinf2 45569 | A class that contains all ... |
| omelaxinf2 45570 | A transitive class that co... |
| dfac5prim 45571 | ~ dfac5 expanded into prim... |
| ac8prim 45572 | ~ ac8 expanded into primit... |
| modelac8prim 45573 | If ` M ` is a transitive c... |
| wfaxext 45574 | The class of well-founded ... |
| wfaxrep 45575 | The class of well-founded ... |
| wfaxsep 45576 | The class of well-founded ... |
| wfaxnul 45577 | The class of well-founded ... |
| wfaxpow 45578 | The class of well-founded ... |
| wfaxpr 45579 | The class of well-founded ... |
| wfaxun 45580 | The class of well-founded ... |
| wfaxreg 45581 | The class of well-founded ... |
| wfaxinf2 45582 | The class of well-founded ... |
| wfac8prim 45583 | The class of well-founded ... |
| brpermmodel 45584 | The membership relation in... |
| brpermmodelcnv 45585 | Ordinary membership expres... |
| permaxext 45586 | The Axiom of Extensionalit... |
| permaxrep 45587 | The Axiom of Replacement ~... |
| permaxsep 45588 | The Axiom of Separation ~ ... |
| permaxnul 45589 | The Null Set Axiom ~ ax-nu... |
| permaxpow 45590 | The Axiom of Power Sets ~ ... |
| permaxpr 45591 | The Axiom of Pairing ~ ax-... |
| permaxun 45592 | The Axiom of Union ~ ax-un... |
| permaxinf2lem 45593 | Lemma for ~ permaxinf2 . ... |
| permaxinf2 45594 | The Axiom of Infinity ~ ax... |
| permac8prim 45595 | The Axiom of Choice ~ ac8p... |
| nregmodelf1o 45596 | Define a permutation ` F `... |
| nregmodellem 45597 | Lemma for ~ nregmodel . (... |
| nregmodel 45598 | The Axiom of Regularity ~ ... |
| nregmodelaxext 45599 | The Axiom of Extensionalit... |
| evth2f 45600 | A version of ~ evth2 using... |
| elunif 45601 | A version of ~ eluni using... |
| rzalf 45602 | A version of ~ rzal using ... |
| fvelrnbf 45603 | A version of ~ fvelrnb usi... |
| rfcnpre1 45604 | If F is a continuous funct... |
| ubelsupr 45605 | If U belongs to A and U is... |
| fsumcnf 45606 | A finite sum of functions ... |
| mulltgt0 45607 | The product of a negative ... |
| rspcegf 45608 | A version of ~ rspcev usin... |
| rabexgf 45609 | A version of ~ rabexg usin... |
| fcnre 45610 | A function continuous with... |
| sumsnd 45611 | A sum of a singleton is th... |
| evthf 45612 | A version of ~ evth using ... |
| cnfex 45613 | The class of continuous fu... |
| fnchoice 45614 | For a finite set, a choice... |
| refsumcn 45615 | A finite sum of continuous... |
| rfcnpre2 45616 | If ` F ` is a continuous f... |
| cncmpmax 45617 | When the hypothesis for th... |
| rfcnpre3 45618 | If F is a continuous funct... |
| rfcnpre4 45619 | If F is a continuous funct... |
| sumpair 45620 | Sum of two distinct comple... |
| rfcnnnub 45621 | Given a real continuous fu... |
| refsum2cnlem1 45622 | This is the core Lemma for... |
| refsum2cn 45623 | The sum of two continuus r... |
| adantlllr 45624 | Deduction adding a conjunc... |
| 3adantlr3 45625 | Deduction adding a conjunc... |
| 3adantll2 45626 | Deduction adding a conjunc... |
| 3adantll3 45627 | Deduction adding a conjunc... |
| ssnel 45628 | If not element of a set, t... |
| sncldre 45629 | A singleton is closed w.r.... |
| n0p 45630 | A polynomial with a nonzer... |
| pm2.65ni 45631 | Inference rule for proof b... |
| iuneq2df 45632 | Equality deduction for ind... |
| nnfoctb 45633 | There exists a mapping fro... |
| elpwinss 45634 | An element of the powerset... |
| unidmex 45635 | If ` F ` is a set, then ` ... |
| ndisj2 45636 | A non-disjointness conditi... |
| zenom 45637 | The set of integer numbers... |
| uzwo4 45638 | Well-ordering principle: a... |
| unisn0 45639 | The union of the singleton... |
| ssin0 45640 | If two classes are disjoin... |
| inabs3 45641 | Absorption law for interse... |
| pwpwuni 45642 | Relationship between power... |
| disjiun2 45643 | In a disjoint collection, ... |
| 0pwfi 45644 | The empty set is in any po... |
| ssinss2d 45645 | Intersection preserves sub... |
| zct 45646 | The set of integer numbers... |
| pwfin0 45647 | A finite set always belong... |
| uzct 45648 | An upper integer set is co... |
| iunxsnf 45649 | A singleton index picks ou... |
| fiiuncl 45650 | If a set is closed under t... |
| iunp1 45651 | The addition of the next s... |
| fiunicl 45652 | If a set is closed under t... |
| ixpeq2d 45653 | Equality theorem for infin... |
| disjxp1 45654 | The sets of a cartesian pr... |
| disjsnxp 45655 | The sets in the cartesian ... |
| eliind 45656 | Membership in indexed inte... |
| rspcef 45657 | Restricted existential spe... |
| ixpssmapc 45658 | An infinite Cartesian prod... |
| elintd 45659 | Membership in class inters... |
| ssdf 45660 | A sufficient condition for... |
| brneqtrd 45661 | Substitution of equal clas... |
| ssnct 45662 | A set containing an uncoun... |
| ssuniint 45663 | Sufficient condition for b... |
| elintdv 45664 | Membership in class inters... |
| ssd 45665 | A sufficient condition for... |
| ralimralim 45666 | Introducing any antecedent... |
| snelmap 45667 | Membership of the element ... |
| xrnmnfpnf 45668 | An extended real that is n... |
| iuneq1i 45669 | Equality theorem for index... |
| ssinc 45670 | Inclusion relation for a m... |
| ssdec 45671 | Inclusion relation for a m... |
| elixpconstg 45672 | Membership in an infinite ... |
| iineq1d 45673 | Equality theorem for index... |
| metpsmet 45674 | A metric is a pseudometric... |
| ixpssixp 45675 | Subclass theorem for infin... |
| ballss3 45676 | A sufficient condition for... |
| iunincfi 45677 | Given a sequence of increa... |
| nsstr 45678 | If it's not a subclass, it... |
| rexanuz3 45679 | Combine two different uppe... |
| cbvmpo2 45680 | Rule to change the second ... |
| cbvmpo1 45681 | Rule to change the first b... |
| eliuniin 45682 | Indexed union of indexed i... |
| ssabf 45683 | Subclass of a class abstra... |
| pssnssi 45684 | A proper subclass does not... |
| rabidim2 45685 | Membership in a restricted... |
| eluni2f 45686 | Membership in class union.... |
| eliin2f 45687 | Membership in indexed inte... |
| nssd 45688 | Negation of subclass relat... |
| iineq12dv 45689 | Equality deduction for ind... |
| supxrcld 45690 | The supremum of an arbitra... |
| elrestd 45691 | A sufficient condition for... |
| eliuniincex 45692 | Counterexample to show tha... |
| eliincex 45693 | Counterexample to show tha... |
| eliinid 45694 | Membership in an indexed i... |
| abssf 45695 | Class abstraction in a sub... |
| supxrubd 45696 | A member of a set of exten... |
| ssrabf 45697 | Subclass of a restricted c... |
| ssrabdf 45698 | Subclass of a restricted c... |
| eliin2 45699 | Membership in indexed inte... |
| ssrab2f 45700 | Subclass relation for a re... |
| restuni3 45701 | The underlying set of a su... |
| rabssf 45702 | Restricted class abstracti... |
| eliuniin2 45703 | Indexed union of indexed i... |
| restuni4 45704 | The underlying set of a su... |
| restuni6 45705 | The underlying set of a su... |
| restuni5 45706 | The underlying set of a su... |
| unirestss 45707 | The union of an elementwis... |
| iniin1 45708 | Indexed intersection of in... |
| iniin2 45709 | Indexed intersection of in... |
| cbvrabv2 45710 | A more general version of ... |
| cbvrabv2w 45711 | A more general version of ... |
| iinssiin 45712 | Subset implication for an ... |
| eliind2 45713 | Membership in indexed inte... |
| iinssd 45714 | Subset implication for an ... |
| rabbida2 45715 | Equivalent wff's yield equ... |
| iinexd 45716 | The existence of an indexe... |
| rabexf 45717 | Separation Scheme in terms... |
| rabbida3 45718 | Equivalent wff's yield equ... |
| r19.36vf 45719 | Restricted quantifier vers... |
| raleqd 45720 | Equality deduction for res... |
| iinssf 45721 | Subset implication for an ... |
| iinssdf 45722 | Subset implication for an ... |
| resabs2i 45723 | Absorption law for restric... |
| ssdf2 45724 | A sufficient condition for... |
| rabssd 45725 | Restricted class abstracti... |
| rexnegd 45726 | Minus a real number. (Con... |
| rexlimd3 45727 | * Inference from Theorem 1... |
| nel1nelini 45728 | Membership in an intersect... |
| nel2nelini 45729 | Membership in an intersect... |
| eliunid 45730 | Membership in indexed unio... |
| reximdd 45731 | Deduction from Theorem 19.... |
| inopnd 45732 | The intersection of two op... |
| ss2rabdf 45733 | Deduction of restricted ab... |
| restopn3 45734 | If ` A ` is open, then ` A... |
| restopnssd 45735 | A topology restricted to a... |
| restsubel 45736 | A subset belongs in the sp... |
| toprestsubel 45737 | A subset is open in the to... |
| rabidd 45738 | An "identity" law of concr... |
| iunssdf 45739 | Subset theorem for an inde... |
| iinss2d 45740 | Subset implication for an ... |
| r19.3rzf 45741 | Restricted quantification ... |
| r19.28zf 45742 | Restricted quantifier vers... |
| iindif2f 45743 | Indexed intersection of cl... |
| ralfal 45744 | Two ways of expressing emp... |
| archd 45745 | Archimedean property of re... |
| nimnbi 45746 | If an implication is false... |
| nimnbi2 45747 | If an implication is false... |
| notbicom 45748 | Commutative law for the ne... |
| rexeqif 45749 | Equality inference for res... |
| rspced 45750 | Restricted existential spe... |
| fnresdmss 45751 | A function does not change... |
| fmptsnxp 45752 | Maps-to notation and Carte... |
| fvmpt2bd 45753 | Value of a function given ... |
| rnmptfi 45754 | The range of a function wi... |
| fresin2 45755 | Restriction of a function ... |
| ffi 45756 | A function with finite dom... |
| suprnmpt 45757 | An explicit bound for the ... |
| rnffi 45758 | The range of a function wi... |
| mptelpm 45759 | A function in maps-to nota... |
| rnmptpr 45760 | Range of a function define... |
| resmpti 45761 | Restriction of the mapping... |
| founiiun 45762 | Union expressed as an inde... |
| rnresun 45763 | Distribution law for range... |
| elrnmptf 45764 | The range of a function in... |
| rnmptssrn 45765 | Inclusion relation for two... |
| disjf1 45766 | A 1 to 1 mapping built fro... |
| rnsnf 45767 | The range of a function wh... |
| wessf1ornlem 45768 | Given a function ` F ` on ... |
| wessf1orn 45769 | Given a function ` F ` on ... |
| nelrnres 45770 | If ` A ` is not in the ran... |
| disjrnmpt2 45771 | Disjointness of the range ... |
| elrnmpt1sf 45772 | Elementhood in an image se... |
| founiiun0 45773 | Union expressed as an inde... |
| disjf1o 45774 | A bijection built from dis... |
| disjinfi 45775 | Only a finite number of di... |
| fvovco 45776 | Value of the composition o... |
| ssnnf1octb 45777 | There exists a bijection b... |
| nnf1oxpnn 45778 | There is a bijection betwe... |
| projf1o 45779 | A biijection from a set to... |
| fvmap 45780 | Function value for a membe... |
| fvixp2 45781 | Projection of a factor of ... |
| choicefi 45782 | For a finite set, a choice... |
| mpct 45783 | The exponentiation of a co... |
| cnmetcoval 45784 | Value of the distance func... |
| fcomptss 45785 | Express composition of two... |
| elmapsnd 45786 | Membership in a set expone... |
| mapss2 45787 | Subset inheritance for set... |
| difmap 45788 | Difference of two sets exp... |
| unirnmap 45789 | Given a subset of a set ex... |
| inmap 45790 | Intersection of two sets e... |
| fcoss 45791 | Composition of two mapping... |
| fsneqrn 45792 | Equality condition for two... |
| difmapsn 45793 | Difference of two sets exp... |
| mapssbi 45794 | Subset inheritance for set... |
| unirnmapsn 45795 | Equality theorem for a sub... |
| iunmapss 45796 | The indexed union of set e... |
| ssmapsn 45797 | A subset ` C ` of a set ex... |
| iunmapsn 45798 | The indexed union of set e... |
| absfico 45799 | Mapping domain and codomai... |
| icof 45800 | The set of left-closed rig... |
| elpmrn 45801 | The range of a partial fun... |
| imaexi 45802 | The image of a set is a se... |
| axccdom 45803 | Relax the constraint on ax... |
| dmmptdff 45804 | The domain of the mapping ... |
| dmmptdf 45805 | The domain of the mapping ... |
| elpmi2 45806 | The domain of a partial fu... |
| dmrelrnrel 45807 | A relation preserving func... |
| elrnmpoid 45808 | Membership in the range of... |
| axccd 45809 | An alternative version of ... |
| axccd2 45810 | An alternative version of ... |
| feqresmptf 45811 | Express a restricted funct... |
| dmmptssf 45812 | The domain of a mapping is... |
| dmmptdf2 45813 | The domain of the mapping ... |
| dmuz 45814 | Domain of the upper intege... |
| fmptd2f 45815 | Domain and codomain of the... |
| mpteq1df 45816 | An equality theorem for th... |
| mptexf 45817 | If the domain of a functio... |
| fvmpt4 45818 | Value of a function given ... |
| fmptf 45819 | Functionality of the mappi... |
| resimass 45820 | The image of a restriction... |
| mptssid 45821 | The mapping operation expr... |
| mptfnd 45822 | The maps-to notation defin... |
| rnmptlb 45823 | Boundness below of the ran... |
| rnmptbddlem 45824 | Boundness of the range of ... |
| rnmptbdd 45825 | Boundness of the range of ... |
| funimaeq 45826 | Membership relation for th... |
| rnmptssf 45827 | The range of a function gi... |
| rnmptbd2lem 45828 | Boundness below of the ran... |
| rnmptbd2 45829 | Boundness below of the ran... |
| infnsuprnmpt 45830 | The indexed infimum of rea... |
| suprclrnmpt 45831 | Closure of the indexed sup... |
| suprubrnmpt2 45832 | A member of a nonempty ind... |
| suprubrnmpt 45833 | A member of a nonempty ind... |
| rnmptssdf 45834 | The range of a function gi... |
| rnmptbdlem 45835 | Boundness above of the ran... |
| rnmptbd 45836 | Boundness above of the ran... |
| rnmptss2 45837 | The range of a function gi... |
| elmptima 45838 | The image of a function in... |
| ralrnmpt3 45839 | A restricted quantifier ov... |
| rnmptssbi 45840 | The range of a function gi... |
| imass2d 45841 | Subset theorem for image. ... |
| imassmpt 45842 | Membership relation for th... |
| fpmd 45843 | A total function is a part... |
| fconst7 45844 | An alternative way to expr... |
| fnmptif 45845 | Functionality and domain o... |
| dmmptif 45846 | Domain of the mapping oper... |
| mpteq2dfa 45847 | Slightly more general equa... |
| dmmpt1 45848 | The domain of the mapping ... |
| fmptff 45849 | Functionality of the mappi... |
| fvmptelcdmf 45850 | The value of a function at... |
| fmptdff 45851 | A version of ~ fmptd using... |
| fvmpt2df 45852 | Deduction version of ~ fvm... |
| rn1st 45853 | The range of a function wi... |
| rnmptssff 45854 | The range of a function gi... |
| rnmptssdff 45855 | The range of a function gi... |
| fvmpt4d 45856 | Value of a function given ... |
| sub2times 45857 | Subtracting from a number,... |
| nnxrd 45858 | A natural number is an ext... |
| nnxr 45859 | A natural number is an ext... |
| abssubrp 45860 | The distance of two distin... |
| elfzfzo 45861 | Relationship between membe... |
| oddfl 45862 | Odd number representation ... |
| abscosbd 45863 | Bound for the absolute val... |
| mul13d 45864 | Commutative/associative la... |
| negpilt0 45865 | Negative ` _pi ` is negati... |
| dstregt0 45866 | A complex number ` A ` tha... |
| subadd4b 45867 | Rearrangement of 4 terms i... |
| xrlttri5d 45868 | Not equal and not larger i... |
| zltlesub 45869 | If an integer ` N ` is les... |
| divlt0gt0d 45870 | The ratio of a negative nu... |
| subsub23d 45871 | Swap subtrahend and result... |
| 2timesgt 45872 | Double of a positive real ... |
| reopn 45873 | The reals are open with re... |
| sub31 45874 | Swap the first and third t... |
| nnne1ge2 45875 | A positive integer which i... |
| lefldiveq 45876 | A closed enough, smaller r... |
| negsubdi3d 45877 | Distribution of negative o... |
| ltdiv2dd 45878 | Division of a positive num... |
| abssinbd 45879 | Bound for the absolute val... |
| halffl 45880 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 45881 | Ordering relation for a st... |
| hashssle 45882 | The size of a subset of a ... |
| lttri5d 45883 | Not equal and not larger i... |
| fzisoeu 45884 | A finite ordered set has a... |
| lt3addmuld 45885 | If three real numbers are ... |
| absnpncan2d 45886 | Triangular inequality, com... |
| fperiodmullem 45887 | A function with period ` T... |
| fperiodmul 45888 | A function with period T i... |
| upbdrech 45889 | Choice of an upper bound f... |
| lt4addmuld 45890 | If four real numbers are l... |
| absnpncan3d 45891 | Triangular inequality, com... |
| upbdrech2 45892 | Choice of an upper bound f... |
| ssfiunibd 45893 | A finite union of bounded ... |
| fzdifsuc2 45894 | Remove a successor from th... |
| fzsscn 45895 | A finite sequence of integ... |
| divcan8d 45896 | A cancellation law for div... |
| dmmcand 45897 | Cancellation law for divis... |
| fzssre 45898 | A finite sequence of integ... |
| bccld 45899 | A binomial coefficient, in... |
| fzssnn0 45900 | A finite set of sequential... |
| xreqle 45901 | Equality implies 'less tha... |
| xaddlidd 45902 | ` 0 ` is a left identity f... |
| xadd0ge 45903 | A number is less than or e... |
| xrleneltd 45904 | 'Less than or equal to' an... |
| xaddcomd 45905 | The extended real addition... |
| supxrre3 45906 | The supremum of a nonempty... |
| uzfissfz 45907 | For any finite subset of t... |
| xleadd2d 45908 | Addition of extended reals... |
| suprltrp 45909 | The supremum of a nonempty... |
| xleadd1d 45910 | Addition of extended reals... |
| xreqled 45911 | Equality implies 'less tha... |
| xrgepnfd 45912 | An extended real greater t... |
| xrge0nemnfd 45913 | A nonnegative extended rea... |
| supxrgere 45914 | If a real number can be ap... |
| iuneqfzuzlem 45915 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 45916 | If two unions indexed by u... |
| xle2addd 45917 | Adding both side of two in... |
| supxrgelem 45918 | If an extended real number... |
| supxrge 45919 | If an extended real number... |
| suplesup 45920 | If any element of ` A ` ca... |
| infxrglb 45921 | The infimum of a set of ex... |
| xadd0ge2 45922 | A number is less than or e... |
| nepnfltpnf 45923 | An extended real that is n... |
| ltadd12dd 45924 | Addition to both sides of ... |
| nemnftgtmnft 45925 | An extended real that is n... |
| xrgtso 45926 | 'Greater than' is a strict... |
| rpex 45927 | The positive reals form a ... |
| xrge0ge0 45928 | A nonnegative extended rea... |
| xrssre 45929 | A subset of extended reals... |
| ssuzfz 45930 | A finite subset of the upp... |
| absfun 45931 | The absolute value is a fu... |
| infrpge 45932 | The infimum of a nonempty,... |
| xrlexaddrp 45933 | If an extended real number... |
| supsubc 45934 | The supremum function dist... |
| xralrple2 45935 | Show that ` A ` is less th... |
| nnuzdisj 45936 | The first ` N ` elements o... |
| ltdivgt1 45937 | Divsion by a number greate... |
| xrltned 45938 | 'Less than' implies not eq... |
| nnsplit 45939 | Express the set of positiv... |
| divdiv3d 45940 | Division into a fraction. ... |
| abslt2sqd 45941 | Comparison of the square o... |
| qenom 45942 | The set of rational number... |
| qct 45943 | The set of rational number... |
| lenlteq 45944 | 'less than or equal to' bu... |
| xrred 45945 | An extended real that is n... |
| rr2sscn2 45946 | The cartesian square of ` ... |
| infxr 45947 | The infimum of a set of ex... |
| infxrunb2 45948 | The infimum of an unbounde... |
| infxrbnd2 45949 | The infimum of a bounded-b... |
| infleinflem1 45950 | Lemma for ~ infleinf , cas... |
| infleinflem2 45951 | Lemma for ~ infleinf , whe... |
| infleinf 45952 | If any element of ` B ` ca... |
| xralrple4 45953 | Show that ` A ` is less th... |
| xralrple3 45954 | Show that ` A ` is less th... |
| eluzelzd 45955 | A member of an upper set o... |
| suplesup2 45956 | If any element of ` A ` is... |
| recnnltrp 45957 | ` N ` is a natural number ... |
| nnn0 45958 | The set of positive intege... |
| fzct 45959 | A finite set of sequential... |
| rpgtrecnn 45960 | Any positive real number i... |
| fzossuz 45961 | A half-open integer interv... |
| infxrrefi 45962 | The real and extended real... |
| xrralrecnnle 45963 | Show that ` A ` is less th... |
| fzoct 45964 | A finite set of sequential... |
| frexr 45965 | A function taking real val... |
| nnrecrp 45966 | The reciprocal of a positi... |
| reclt0d 45967 | The reciprocal of a negati... |
| lt0neg1dd 45968 | If a number is negative, i... |
| infxrcld 45969 | The infimum of an arbitrar... |
| xrralrecnnge 45970 | Show that ` A ` is less th... |
| reclt0 45971 | The reciprocal of a negati... |
| ltmulneg 45972 | Multiplying by a negative ... |
| allbutfi 45973 | For all but finitely many.... |
| ltdiv23neg 45974 | Swap denominator with othe... |
| xreqnltd 45975 | A consequence of trichotom... |
| mnfnre2 45976 | Minus infinity is not a re... |
| zssxr 45977 | The integers are a subset ... |
| fisupclrnmpt 45978 | A nonempty finite indexed ... |
| supxrunb3 45979 | The supremum of an unbound... |
| fimaxre4 45980 | A nonempty finite set of r... |
| ren0 45981 | The set of reals is nonemp... |
| eluzelz2 45982 | A member of an upper set o... |
| resabs2d 45983 | Absorption law for restric... |
| uzid2 45984 | Membership of the least me... |
| supxrleubrnmpt 45985 | The supremum of a nonempty... |
| uzssre2 45986 | An upper set of integers i... |
| uzssd 45987 | Subset relationship for tw... |
| eluzd 45988 | Membership in an upper set... |
| infxrlbrnmpt2 45989 | A member of a nonempty ind... |
| xrre4 45990 | An extended real is real i... |
| uz0 45991 | The upper integers functio... |
| eluzelz2d 45992 | A member of an upper set o... |
| infleinf2 45993 | If any element in ` B ` is... |
| unb2ltle 45994 | "Unbounded below" expresse... |
| uzidd2 45995 | Membership of the least me... |
| uzssd2 45996 | Subset relationship for tw... |
| rexabslelem 45997 | An indexed set of absolute... |
| rexabsle 45998 | An indexed set of absolute... |
| allbutfiinf 45999 | Given a "for all but finit... |
| supxrrernmpt 46000 | The real and extended real... |
| suprleubrnmpt 46001 | The supremum of a nonempty... |
| infrnmptle 46002 | An indexed infimum of exte... |
| infxrunb3 46003 | The infimum of an unbounde... |
| uzn0d 46004 | The upper integers are all... |
| uzssd3 46005 | Subset relationship for tw... |
| rexabsle2 46006 | An indexed set of absolute... |
| infxrunb3rnmpt 46007 | The infimum of an unbounde... |
| supxrre3rnmpt 46008 | The indexed supremum of a ... |
| uzublem 46009 | A set of reals, indexed by... |
| uzub 46010 | A set of reals, indexed by... |
| ssrexr 46011 | A subset of the reals is a... |
| supxrmnf2 46012 | Removing minus infinity fr... |
| supxrcli 46013 | The supremum of an arbitra... |
| uzid3 46014 | Membership of the least me... |
| infxrlesupxr 46015 | The supremum of a nonempty... |
| xnegeqd 46016 | Equality of two extended n... |
| xnegrecl 46017 | The extended real negative... |
| xnegnegi 46018 | Extended real version of ~... |
| xnegeqi 46019 | Equality of two extended n... |
| nfxnegd 46020 | Deduction version of ~ nfx... |
| xnegnegd 46021 | Extended real version of ~... |
| uzred 46022 | An upper integer is a real... |
| xnegcli 46023 | Closure of extended real n... |
| supminfrnmpt 46024 | The indexed supremum of a ... |
| infxrpnf 46025 | Adding plus infinity to a ... |
| infxrrnmptcl 46026 | The infimum of an arbitrar... |
| leneg2d 46027 | Negative of one side of 'l... |
| supxrltinfxr 46028 | The supremum of the empty ... |
| max1d 46029 | A number is less than or e... |
| supxrleubrnmptf 46030 | The supremum of a nonempty... |
| nleltd 46031 | 'Not less than or equal to... |
| zxrd 46032 | An integer is an extended ... |
| infxrgelbrnmpt 46033 | The infimum of an indexed ... |
| rphalfltd 46034 | Half of a positive real is... |
| uzssz2 46035 | An upper set of integers i... |
| leneg3d 46036 | Negative of one side of 'l... |
| max2d 46037 | A number is less than or e... |
| uzn0bi 46038 | The upper integers functio... |
| xnegrecl2 46039 | If the extended real negat... |
| nfxneg 46040 | Bound-variable hypothesis ... |
| uzxrd 46041 | An upper integer is an ext... |
| infxrpnf2 46042 | Removing plus infinity fro... |
| supminfxr 46043 | The extended real suprema ... |
| infrpgernmpt 46044 | The infimum of a nonempty,... |
| xnegre 46045 | An extended real is real i... |
| xnegrecl2d 46046 | If the extended real negat... |
| uzxr 46047 | An upper integer is an ext... |
| supminfxr2 46048 | The extended real suprema ... |
| xnegred 46049 | An extended real is real i... |
| supminfxrrnmpt 46050 | The indexed supremum of a ... |
| min1d 46051 | The minimum of two numbers... |
| min2d 46052 | The minimum of two numbers... |
| xrnpnfmnf 46053 | An extended real that is n... |
| uzsscn 46054 | An upper set of integers i... |
| absimnre 46055 | The absolute value of the ... |
| uzsscn2 46056 | An upper set of integers i... |
| xrtgcntopre 46057 | The standard topologies on... |
| absimlere 46058 | The absolute value of the ... |
| rpssxr 46059 | The positive reals are a s... |
| monoordxrv 46060 | Ordering relation for a mo... |
| monoordxr 46061 | Ordering relation for a mo... |
| monoord2xrv 46062 | Ordering relation for a mo... |
| monoord2xr 46063 | Ordering relation for a mo... |
| xrpnf 46064 | An extended real is plus i... |
| xlenegcon1 46065 | Extended real version of ~... |
| xlenegcon2 46066 | Extended real version of ~... |
| pimxrneun 46067 | The preimage of a set of e... |
| caucvgbf 46068 | A function is convergent i... |
| cvgcau 46069 | A convergent function is C... |
| cvgcaule 46070 | A convergent function is C... |
| rexanuz2nf 46071 | A simple counterexample re... |
| gtnelioc 46072 | A real number larger than ... |
| ioossioc 46073 | An open interval is a subs... |
| ioondisj2 46074 | A condition for two open i... |
| ioondisj1 46075 | A condition for two open i... |
| ioogtlb 46076 | An element of a closed int... |
| evthiccabs 46077 | Extreme Value Theorem on y... |
| ltnelicc 46078 | A real number smaller than... |
| eliood 46079 | Membership in an open real... |
| iooabslt 46080 | An upper bound for the dis... |
| gtnelicc 46081 | A real number greater than... |
| iooinlbub 46082 | An open interval has empty... |
| iocgtlb 46083 | An element of a left-open ... |
| iocleub 46084 | An element of a left-open ... |
| eliccd 46085 | Membership in a closed rea... |
| eliccre 46086 | A member of a closed inter... |
| eliooshift 46087 | Element of an open interva... |
| eliocd 46088 | Membership in a left-open ... |
| icoltub 46089 | An element of a left-close... |
| eliocre 46090 | A member of a left-open ri... |
| iooltub 46091 | An element of an open inte... |
| ioontr 46092 | The interior of an interva... |
| snunioo1 46093 | The closure of one end of ... |
| lbioc 46094 | A left-open right-closed i... |
| ioomidp 46095 | The midpoint is an element... |
| iccdifioo 46096 | If the open inverval is re... |
| iccdifprioo 46097 | An open interval is the cl... |
| ioossioobi 46098 | Biconditional form of ~ io... |
| iccshift 46099 | A closed interval shifted ... |
| iccsuble 46100 | An upper bound to the dist... |
| iocopn 46101 | A left-open right-closed i... |
| eliccelioc 46102 | Membership in a closed int... |
| iooshift 46103 | An open interval shifted b... |
| iccintsng 46104 | Intersection of two adiace... |
| icoiccdif 46105 | Left-closed right-open int... |
| icoopn 46106 | A left-closed right-open i... |
| icoub 46107 | A left-closed, right-open ... |
| eliccxrd 46108 | Membership in a closed rea... |
| pnfel0pnf 46109 | ` +oo ` is a nonnegative e... |
| eliccnelico 46110 | An element of a closed int... |
| eliccelicod 46111 | A member of a closed inter... |
| ge0xrre 46112 | A nonnegative extended rea... |
| ge0lere 46113 | A nonnegative extended Rea... |
| elicores 46114 | Membership in a left-close... |
| inficc 46115 | The infimum of a nonempty ... |
| qinioo 46116 | The rational numbers are d... |
| lenelioc 46117 | A real number smaller than... |
| ioonct 46118 | A nonempty open interval i... |
| xrgtnelicc 46119 | A real number greater than... |
| iccdificc 46120 | The difference of two clos... |
| iocnct 46121 | A nonempty left-open, righ... |
| iccnct 46122 | A closed interval, with mo... |
| iooiinicc 46123 | A closed interval expresse... |
| iccgelbd 46124 | An element of a closed int... |
| iooltubd 46125 | An element of an open inte... |
| icoltubd 46126 | An element of a left-close... |
| qelioo 46127 | The rational numbers are d... |
| tgqioo2 46128 | Every open set of reals is... |
| iccleubd 46129 | An element of a closed int... |
| elioored 46130 | A member of an open interv... |
| ioogtlbd 46131 | An element of a closed int... |
| ioofun 46132 | ` (,) ` is a function. (C... |
| icomnfinre 46133 | A left-closed, right-open,... |
| sqrlearg 46134 | The square compared with i... |
| ressiocsup 46135 | If the supremum belongs to... |
| ressioosup 46136 | If the supremum does not b... |
| iooiinioc 46137 | A left-open, right-closed ... |
| ressiooinf 46138 | If the infimum does not be... |
| iocleubd 46139 | An element of a left-open ... |
| uzinico 46140 | An upper interval of integ... |
| preimaiocmnf 46141 | Preimage of a right-closed... |
| uzinico2 46142 | An upper interval of integ... |
| uzinico3 46143 | An upper interval of integ... |
| dmico 46144 | The domain of the closed-b... |
| ndmico 46145 | The closed-below, open-abo... |
| uzubioo 46146 | The upper integers are unb... |
| uzubico 46147 | The upper integers are unb... |
| uzubioo2 46148 | The upper integers are unb... |
| uzubico2 46149 | The upper integers are unb... |
| iocgtlbd 46150 | An element of a left-open ... |
| xrtgioo2 46151 | The topology on the extend... |
| fsummulc1f 46152 | Closure of a finite sum of... |
| fsumnncl 46153 | Closure of a nonempty, fin... |
| fsumge0cl 46154 | The finite sum of nonnegat... |
| fsumf1of 46155 | Re-index a finite sum usin... |
| fsumiunss 46156 | Sum over a disjoint indexe... |
| fsumreclf 46157 | Closure of a finite sum of... |
| fsumlessf 46158 | A shorter sum of nonnegati... |
| fsumsupp0 46159 | Finite sum of function val... |
| fsumsermpt 46160 | A finite sum expressed in ... |
| fmul01 46161 | Multiplying a finite numbe... |
| fmulcl 46162 | If ' Y ' is closed under t... |
| fmuldfeqlem1 46163 | induction step for the pro... |
| fmuldfeq 46164 | X and Z are two equivalent... |
| fmul01lt1lem1 46165 | Given a finite multiplicat... |
| fmul01lt1lem2 46166 | Given a finite multiplicat... |
| fmul01lt1 46167 | Given a finite multiplicat... |
| cncfmptss 46168 | A continuous complex funct... |
| rrpsscn 46169 | The positive reals are a s... |
| mulc1cncfg 46170 | A version of ~ mulc1cncf u... |
| infrglb 46171 | The infimum of a nonempty ... |
| expcnfg 46172 | If ` F ` is a complex cont... |
| prodeq2ad 46173 | Equality deduction for pro... |
| fprodsplit1 46174 | Separate out a term in a f... |
| fprodexp 46175 | Positive integer exponenti... |
| fprodabs2 46176 | The absolute value of a fi... |
| fprod0 46177 | A finite product with a ze... |
| mccllem 46178 | * Induction step for ~ mcc... |
| mccl 46179 | A multinomial coefficient,... |
| fprodcnlem 46180 | A finite product of functi... |
| fprodcn 46181 | A finite product of functi... |
| clim1fr1 46182 | A class of sequences of fr... |
| isumneg 46183 | Negation of a converging s... |
| climrec 46184 | Limit of the reciprocal of... |
| climmulf 46185 | A version of ~ climmul usi... |
| climexp 46186 | The limit of natural power... |
| climinf 46187 | A bounded monotonic noninc... |
| climsuselem1 46188 | The subsequence index ` I ... |
| climsuse 46189 | A subsequence ` G ` of a c... |
| climrecf 46190 | A version of ~ climrec usi... |
| climneg 46191 | Complex limit of the negat... |
| climinff 46192 | A version of ~ climinf usi... |
| climdivf 46193 | Limit of the ratio of two ... |
| climreeq 46194 | If ` F ` is a real functio... |
| ellimciota 46195 | An explicit value for the ... |
| climaddf 46196 | A version of ~ climadd usi... |
| mullimc 46197 | Limit of the product of tw... |
| ellimcabssub0 46198 | An equivalent condition fo... |
| limcdm0 46199 | If a function has empty do... |
| islptre 46200 | An equivalence condition f... |
| limccog 46201 | Limit of the composition o... |
| limciccioolb 46202 | The limit of a function at... |
| climf 46203 | Express the predicate: Th... |
| mullimcf 46204 | Limit of the multiplicatio... |
| constlimc 46205 | Limit of constant function... |
| rexlim2d 46206 | Inference removing two res... |
| idlimc 46207 | Limit of the identity func... |
| divcnvg 46208 | The sequence of reciprocal... |
| limcperiod 46209 | If ` F ` is a periodic fun... |
| limcrecl 46210 | If ` F ` is a real-valued ... |
| sumnnodd 46211 | A series indexed by ` NN `... |
| lptioo2 46212 | The upper bound of an open... |
| lptioo1 46213 | The lower bound of an open... |
| limcmptdm 46214 | The domain of a maps-to fu... |
| clim2f 46215 | Express the predicate: Th... |
| limcicciooub 46216 | The limit of a function at... |
| ltmod 46217 | A sufficient condition for... |
| islpcn 46218 | A characterization for a l... |
| lptre2pt 46219 | If a set in the real line ... |
| limsupre 46220 | If a sequence is bounded, ... |
| limcresiooub 46221 | The left limit doesn't cha... |
| limcresioolb 46222 | The right limit doesn't ch... |
| limcleqr 46223 | If the left and the right ... |
| lptioo2cn 46224 | The upper bound of an open... |
| lptioo1cn 46225 | The lower bound of an open... |
| neglimc 46226 | Limit of the negative func... |
| addlimc 46227 | Sum of two limits. (Contr... |
| 0ellimcdiv 46228 | If the numerator converges... |
| clim2cf 46229 | Express the predicate ` F ... |
| limclner 46230 | For a limit point, both fr... |
| sublimc 46231 | Subtraction of two limits.... |
| reclimc 46232 | Limit of the reciprocal of... |
| clim0cf 46233 | Express the predicate ` F ... |
| limclr 46234 | For a limit point, both fr... |
| divlimc 46235 | Limit of the quotient of t... |
| expfac 46236 | Factorial grows faster tha... |
| climconstmpt 46237 | A constant sequence conver... |
| climresmpt 46238 | A function restricted to u... |
| climsubmpt 46239 | Limit of the difference of... |
| climsubc2mpt 46240 | Limit of the difference of... |
| climsubc1mpt 46241 | Limit of the difference of... |
| fnlimfv 46242 | The value of the limit fun... |
| climreclf 46243 | The limit of a convergent ... |
| climeldmeq 46244 | Two functions that are eve... |
| climf2 46245 | Express the predicate: Th... |
| fnlimcnv 46246 | The sequence of function v... |
| climeldmeqmpt 46247 | Two functions that are eve... |
| climfveq 46248 | Two functions that are eve... |
| clim2f2 46249 | Express the predicate: Th... |
| climfveqmpt 46250 | Two functions that are eve... |
| climd 46251 | Express the predicate: Th... |
| clim2d 46252 | The limit of complex numbe... |
| fnlimfvre 46253 | The limit function of real... |
| allbutfifvre 46254 | Given a sequence of real-v... |
| climleltrp 46255 | The limit of complex numbe... |
| fnlimfvre2 46256 | The limit function of real... |
| fnlimf 46257 | The limit function of real... |
| fnlimabslt 46258 | A sequence of function val... |
| climfveqf 46259 | Two functions that are eve... |
| climmptf 46260 | Exhibit a function ` G ` w... |
| climfveqmpt3 46261 | Two functions that are eve... |
| climeldmeqf 46262 | Two functions that are eve... |
| climreclmpt 46263 | The limit of B convergent ... |
| limsupref 46264 | If a sequence is bounded, ... |
| limsupbnd1f 46265 | If a sequence is eventuall... |
| climbddf 46266 | A converging sequence of c... |
| climeqf 46267 | Two functions that are eve... |
| climeldmeqmpt3 46268 | Two functions that are eve... |
| limsupcld 46269 | Closure of the superior li... |
| climfv 46270 | The limit of a convergent ... |
| limsupval3 46271 | The superior limit of an i... |
| climfveqmpt2 46272 | Two functions that are eve... |
| limsup0 46273 | The superior limit of the ... |
| climeldmeqmpt2 46274 | Two functions that are eve... |
| limsupresre 46275 | The supremum limit of a fu... |
| climeqmpt 46276 | Two functions that are eve... |
| climfvd 46277 | The limit of a convergent ... |
| limsuplesup 46278 | An upper bound for the sup... |
| limsupresico 46279 | The superior limit doesn't... |
| limsuppnfdlem 46280 | If the restriction of a fu... |
| limsuppnfd 46281 | If the restriction of a fu... |
| limsupresuz 46282 | If the real part of the do... |
| limsupub 46283 | If the limsup is not ` +oo... |
| limsupres 46284 | The superior limit of a re... |
| climinf2lem 46285 | A convergent, nonincreasin... |
| climinf2 46286 | A convergent, nonincreasin... |
| limsupvaluz 46287 | The superior limit, when t... |
| limsupresuz2 46288 | If the domain of a functio... |
| limsuppnflem 46289 | If the restriction of a fu... |
| limsuppnf 46290 | If the restriction of a fu... |
| limsupubuzlem 46291 | If the limsup is not ` +oo... |
| limsupubuz 46292 | For a real-valued function... |
| climinf2mpt 46293 | A bounded below, monotonic... |
| climinfmpt 46294 | A bounded below, monotonic... |
| climinf3 46295 | A convergent, nonincreasin... |
| limsupvaluzmpt 46296 | The superior limit, when t... |
| limsupequzmpt2 46297 | Two functions that are eve... |
| limsupubuzmpt 46298 | If the limsup is not ` +oo... |
| limsupmnflem 46299 | The superior limit of a fu... |
| limsupmnf 46300 | The superior limit of a fu... |
| limsupequzlem 46301 | Two functions that are eve... |
| limsupequz 46302 | Two functions that are eve... |
| limsupre2lem 46303 | Given a function on the ex... |
| limsupre2 46304 | Given a function on the ex... |
| limsupmnfuzlem 46305 | The superior limit of a fu... |
| limsupmnfuz 46306 | The superior limit of a fu... |
| limsupequzmptlem 46307 | Two functions that are eve... |
| limsupequzmpt 46308 | Two functions that are eve... |
| limsupre2mpt 46309 | Given a function on the ex... |
| limsupequzmptf 46310 | Two functions that are eve... |
| limsupre3lem 46311 | Given a function on the ex... |
| limsupre3 46312 | Given a function on the ex... |
| limsupre3mpt 46313 | Given a function on the ex... |
| limsupre3uzlem 46314 | Given a function on the ex... |
| limsupre3uz 46315 | Given a function on the ex... |
| limsupreuz 46316 | Given a function on the re... |
| limsupvaluz2 46317 | The superior limit, when t... |
| limsupreuzmpt 46318 | Given a function on the re... |
| supcnvlimsup 46319 | If a function on a set of ... |
| supcnvlimsupmpt 46320 | If a function on a set of ... |
| 0cnv 46321 | If ` (/) ` is a complex nu... |
| climuzlem 46322 | Express the predicate: Th... |
| climuz 46323 | Express the predicate: Th... |
| lmbr3v 46324 | Express the binary relatio... |
| climisp 46325 | If a sequence converges to... |
| lmbr3 46326 | Express the binary relatio... |
| climrescn 46327 | A sequence converging w.r.... |
| climxrrelem 46328 | If a sequence ranging over... |
| climxrre 46329 | If a sequence ranging over... |
| limsuplt2 46332 | The defining property of t... |
| liminfgord 46333 | Ordering property of the i... |
| limsupvald 46334 | The superior limit of a se... |
| limsupresicompt 46335 | The superior limit doesn't... |
| limsupcli 46336 | Closure of the superior li... |
| liminfgf 46337 | Closure of the inferior li... |
| liminfval 46338 | The inferior limit of a se... |
| climlimsup 46339 | A sequence of real numbers... |
| limsupge 46340 | The defining property of t... |
| liminfgval 46341 | Value of the inferior limi... |
| liminfcl 46342 | Closure of the inferior li... |
| liminfvald 46343 | The inferior limit of a se... |
| liminfval5 46344 | The inferior limit of an i... |
| limsupresxr 46345 | The superior limit of a fu... |
| liminfresxr 46346 | The inferior limit of a fu... |
| liminfval2 46347 | The superior limit, relati... |
| climlimsupcex 46348 | Counterexample for ~ climl... |
| liminfcld 46349 | Closure of the inferior li... |
| liminfresico 46350 | The inferior limit doesn't... |
| limsup10exlem 46351 | The range of the given fun... |
| limsup10ex 46352 | The superior limit of a fu... |
| liminf10ex 46353 | The inferior limit of a fu... |
| liminflelimsuplem 46354 | The superior limit is grea... |
| liminflelimsup 46355 | The superior limit is grea... |
| limsupgtlem 46356 | For any positive real, the... |
| limsupgt 46357 | Given a sequence of real n... |
| liminfresre 46358 | The inferior limit of a fu... |
| liminfresicompt 46359 | The inferior limit doesn't... |
| liminfltlimsupex 46360 | An example where the ` lim... |
| liminfgelimsup 46361 | The inferior limit is grea... |
| liminfvalxr 46362 | Alternate definition of ` ... |
| liminfresuz 46363 | If the real part of the do... |
| liminflelimsupuz 46364 | The superior limit is grea... |
| liminfvalxrmpt 46365 | Alternate definition of ` ... |
| liminfresuz2 46366 | If the domain of a functio... |
| liminfgelimsupuz 46367 | The inferior limit is grea... |
| liminfval4 46368 | Alternate definition of ` ... |
| liminfval3 46369 | Alternate definition of ` ... |
| liminfequzmpt2 46370 | Two functions that are eve... |
| liminfvaluz 46371 | Alternate definition of ` ... |
| liminf0 46372 | The inferior limit of the ... |
| limsupval4 46373 | Alternate definition of ` ... |
| liminfvaluz2 46374 | Alternate definition of ` ... |
| liminfvaluz3 46375 | Alternate definition of ` ... |
| liminflelimsupcex 46376 | A counterexample for ~ lim... |
| limsupvaluz3 46377 | Alternate definition of ` ... |
| liminfvaluz4 46378 | Alternate definition of ` ... |
| limsupvaluz4 46379 | Alternate definition of ` ... |
| climliminflimsupd 46380 | If a sequence of real numb... |
| liminfreuzlem 46381 | Given a function on the re... |
| liminfreuz 46382 | Given a function on the re... |
| liminfltlem 46383 | Given a sequence of real n... |
| liminflt 46384 | Given a sequence of real n... |
| climliminf 46385 | A sequence of real numbers... |
| liminflimsupclim 46386 | A sequence of real numbers... |
| climliminflimsup 46387 | A sequence of real numbers... |
| climliminflimsup2 46388 | A sequence of real numbers... |
| climliminflimsup3 46389 | A sequence of real numbers... |
| climliminflimsup4 46390 | A sequence of real numbers... |
| limsupub2 46391 | A extended real valued fun... |
| limsupubuz2 46392 | A sequence with values in ... |
| xlimpnfxnegmnf 46393 | A sequence converges to ` ... |
| liminflbuz2 46394 | A sequence with values in ... |
| liminfpnfuz 46395 | The inferior limit of a fu... |
| liminflimsupxrre 46396 | A sequence with values in ... |
| xlimrel 46399 | The limit on extended real... |
| xlimres 46400 | A function converges iff i... |
| xlimcl 46401 | The limit of a sequence of... |
| rexlimddv2 46402 | Restricted existential eli... |
| xlimclim 46403 | Given a sequence of reals,... |
| xlimconst 46404 | A constant sequence conver... |
| climxlim 46405 | A converging sequence in t... |
| xlimbr 46406 | Express the binary relatio... |
| fuzxrpmcn 46407 | A function mapping from an... |
| cnrefiisplem 46408 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 46409 | A non-real, complex number... |
| xlimxrre 46410 | If a sequence ranging over... |
| xlimmnfvlem1 46411 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 46412 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 46413 | A function converges to mi... |
| xlimconst2 46414 | A sequence that eventually... |
| xlimpnfvlem1 46415 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 46416 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 46417 | A function converges to pl... |
| xlimclim2lem 46418 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 46419 | Given a sequence of extend... |
| xlimmnf 46420 | A function converges to mi... |
| xlimpnf 46421 | A function converges to pl... |
| xlimmnfmpt 46422 | A function converges to pl... |
| xlimpnfmpt 46423 | A function converges to pl... |
| climxlim2lem 46424 | In this lemma for ~ climxl... |
| climxlim2 46425 | A sequence of extended rea... |
| dfxlim2v 46426 | An alternative definition ... |
| dfxlim2 46427 | An alternative definition ... |
| climresd 46428 | A function restricted to u... |
| climresdm 46429 | A real function converges ... |
| dmclimxlim 46430 | A real valued sequence tha... |
| xlimmnflimsup2 46431 | A sequence of extended rea... |
| xlimuni 46432 | An infinite sequence conve... |
| xlimclimdm 46433 | A sequence of extended rea... |
| xlimfun 46434 | The convergence relation o... |
| xlimmnflimsup 46435 | If a sequence of extended ... |
| xlimdm 46436 | Two ways to express that a... |
| xlimpnfxnegmnf2 46437 | A sequence converges to ` ... |
| xlimresdm 46438 | A function converges in th... |
| xlimpnfliminf 46439 | If a sequence of extended ... |
| xlimpnfliminf2 46440 | A sequence of extended rea... |
| xlimliminflimsup 46441 | A sequence of extended rea... |
| xlimlimsupleliminf 46442 | A sequence of extended rea... |
| coseq0 46443 | A complex number whose cos... |
| sinmulcos 46444 | Multiplication formula for... |
| coskpi2 46445 | The cosine of an integer m... |
| cosnegpi 46446 | The cosine of negative ` _... |
| sinaover2ne0 46447 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 46448 | The cosine of an integer m... |
| mulcncff 46449 | The multiplication of two ... |
| cncfmptssg 46450 | A continuous complex funct... |
| constcncfg 46451 | A constant function is a c... |
| idcncfg 46452 | The identity function is a... |
| cncfshift 46453 | A periodic continuous func... |
| resincncf 46454 | ` sin ` restricted to real... |
| addccncf2 46455 | Adding a constant is a con... |
| 0cnf 46456 | The empty set is a continu... |
| fsumcncf 46457 | The finite sum of continuo... |
| cncfperiod 46458 | A periodic continuous func... |
| subcncff 46459 | The subtraction of two con... |
| negcncfg 46460 | The opposite of a continuo... |
| cnfdmsn 46461 | A function with a singleto... |
| cncfcompt 46462 | Composition of continuous ... |
| addcncff 46463 | The sum of two continuous ... |
| ioccncflimc 46464 | Limit at the upper bound o... |
| cncfuni 46465 | A complex function on a su... |
| icccncfext 46466 | A continuous function on a... |
| cncficcgt0 46467 | A the absolute value of a ... |
| icocncflimc 46468 | Limit at the lower bound, ... |
| cncfdmsn 46469 | A complex function with a ... |
| divcncff 46470 | The quotient of two contin... |
| cncfshiftioo 46471 | A periodic continuous func... |
| cncfiooicclem1 46472 | A continuous function ` F ... |
| cncfiooicc 46473 | A continuous function ` F ... |
| cncfiooiccre 46474 | A continuous function ` F ... |
| cncfioobdlem 46475 | ` G ` actually extends ` F... |
| cncfioobd 46476 | A continuous function ` F ... |
| jumpncnp 46477 | Jump discontinuity or disc... |
| cxpcncf2 46478 | The complex power function... |
| fprodcncf 46479 | The finite product of cont... |
| add1cncf 46480 | Addition to a constant is ... |
| add2cncf 46481 | Addition to a constant is ... |
| sub1cncfd 46482 | Subtracting a constant is ... |
| sub2cncfd 46483 | Subtraction from a constan... |
| fprodsub2cncf 46484 | ` F ` is continuous. (Con... |
| fprodadd2cncf 46485 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 46486 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 46487 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 46488 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 46489 | The sequence ` S ` of fini... |
| dvsinexp 46490 | The derivative of sin^N . ... |
| dvcosre 46491 | The real derivative of the... |
| dvsinax 46492 | Derivative exercise: the d... |
| dvsubf 46493 | The subtraction rule for e... |
| dvmptconst 46494 | Function-builder for deriv... |
| dvcnre 46495 | From complex differentiati... |
| dvmptidg 46496 | Function-builder for deriv... |
| dvresntr 46497 | Function-builder for deriv... |
| fperdvper 46498 | The derivative of a period... |
| dvasinbx 46499 | Derivative exercise: the d... |
| dvresioo 46500 | Restriction of a derivativ... |
| dvdivf 46501 | The quotient rule for ever... |
| dvdivbd 46502 | A sufficient condition for... |
| dvsubcncf 46503 | A sufficient condition for... |
| dvmulcncf 46504 | A sufficient condition for... |
| dvcosax 46505 | Derivative exercise: the d... |
| dvdivcncf 46506 | A sufficient condition for... |
| dvbdfbdioolem1 46507 | Given a function with boun... |
| dvbdfbdioolem2 46508 | A function on an open inte... |
| dvbdfbdioo 46509 | A function on an open inte... |
| ioodvbdlimc1lem1 46510 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 46511 | Limit at the lower bound o... |
| ioodvbdlimc1 46512 | A real function with bound... |
| ioodvbdlimc2lem 46513 | Limit at the upper bound o... |
| ioodvbdlimc2 46514 | A real function with bound... |
| dvdmsscn 46515 | ` X ` is a subset of ` CC ... |
| dvmptmulf 46516 | Function-builder for deriv... |
| dvnmptdivc 46517 | Function-builder for itera... |
| dvdsn1add 46518 | If ` K ` divides ` N ` but... |
| dvxpaek 46519 | Derivative of the polynomi... |
| dvnmptconst 46520 | The ` N ` -th derivative o... |
| dvnxpaek 46521 | The ` n ` -th derivative o... |
| dvnmul 46522 | Function-builder for the `... |
| dvmptfprodlem 46523 | Induction step for ~ dvmpt... |
| dvmptfprod 46524 | Function-builder for deriv... |
| dvnprodlem1 46525 | ` D ` is bijective. (Cont... |
| dvnprodlem2 46526 | Induction step for ~ dvnpr... |
| dvnprodlem3 46527 | The multinomial formula fo... |
| dvnprod 46528 | The multinomial formula fo... |
| itgsin0pilem1 46529 | Calculation of the integra... |
| ibliccsinexp 46530 | sin^n on a closed interval... |
| itgsin0pi 46531 | Calculation of the integra... |
| iblioosinexp 46532 | sin^n on an open integral ... |
| itgsinexplem1 46533 | Integration by parts is ap... |
| itgsinexp 46534 | A recursive formula for th... |
| iblconstmpt 46535 | A constant function is int... |
| itgeq1d 46536 | Equality theorem for an in... |
| mbfres2cn 46537 | Measurability of a piecewi... |
| vol0 46538 | The measure of the empty s... |
| ditgeqiooicc 46539 | A function ` F ` on an ope... |
| volge0 46540 | The volume of a set is alw... |
| cnbdibl 46541 | A continuous bounded funct... |
| snmbl 46542 | A singleton is measurable.... |
| ditgeq3d 46543 | Equality theorem for the d... |
| iblempty 46544 | The empty function is inte... |
| iblsplit 46545 | The union of two integrabl... |
| volsn 46546 | A singleton has 0 Lebesgue... |
| itgvol0 46547 | If the domani is negligibl... |
| itgcoscmulx 46548 | Exercise: the integral of ... |
| iblsplitf 46549 | A version of ~ iblsplit us... |
| ibliooicc 46550 | If a function is integrabl... |
| volioc 46551 | The measure of a left-open... |
| iblspltprt 46552 | If a function is integrabl... |
| itgsincmulx 46553 | Exercise: the integral of ... |
| itgsubsticclem 46554 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 46555 | Integration by u-substitut... |
| itgioocnicc 46556 | The integral of a piecewis... |
| iblcncfioo 46557 | A continuous function ` F ... |
| itgspltprt 46558 | The ` S. ` integral splits... |
| itgiccshift 46559 | The integral of a function... |
| itgperiod 46560 | The integral of a periodic... |
| itgsbtaddcnst 46561 | Integral substitution, add... |
| volico 46562 | The measure of left-closed... |
| sublevolico 46563 | The Lebesgue measure of a ... |
| dmvolss 46564 | Lebesgue measurable sets a... |
| ismbl3 46565 | The predicate " ` A ` is L... |
| volioof 46566 | The function that assigns ... |
| ovolsplit 46567 | The Lebesgue outer measure... |
| fvvolioof 46568 | The function value of the ... |
| volioore 46569 | The measure of an open int... |
| fvvolicof 46570 | The function value of the ... |
| voliooico 46571 | An open interval and a lef... |
| ismbl4 46572 | The predicate " ` A ` is L... |
| volioofmpt 46573 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 46574 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 46575 | The Lebesgue measure of op... |
| volicofmpt 46576 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 46577 | The Lebesgue measure of a ... |
| voliccico 46578 | A closed interval and a le... |
| mbfdmssre 46579 | The domain of a measurable... |
| stoweidlem1 46580 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 46581 | lemma for ~ stoweid : here... |
| stoweidlem3 46582 | Lemma for ~ stoweid : if `... |
| stoweidlem4 46583 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 46584 | There exists a δ as ... |
| stoweidlem6 46585 | Lemma for ~ stoweid : two ... |
| stoweidlem7 46586 | This lemma is used to prov... |
| stoweidlem8 46587 | Lemma for ~ stoweid : two ... |
| stoweidlem9 46588 | Lemma for ~ stoweid : here... |
| stoweidlem10 46589 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 46590 | This lemma is used to prov... |
| stoweidlem12 46591 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 46592 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 46593 | There exists a ` k ` as in... |
| stoweidlem15 46594 | This lemma is used to prov... |
| stoweidlem16 46595 | Lemma for ~ stoweid . The... |
| stoweidlem17 46596 | This lemma proves that the... |
| stoweidlem18 46597 | This theorem proves Lemma ... |
| stoweidlem19 46598 | If a set of real functions... |
| stoweidlem20 46599 | If a set A of real functio... |
| stoweidlem21 46600 | Once the Stone Weierstrass... |
| stoweidlem22 46601 | If a set of real functions... |
| stoweidlem23 46602 | This lemma is used to prov... |
| stoweidlem24 46603 | This lemma proves that for... |
| stoweidlem25 46604 | This lemma proves that for... |
| stoweidlem26 46605 | This lemma is used to prov... |
| stoweidlem27 46606 | This lemma is used to prov... |
| stoweidlem28 46607 | There exists a δ as ... |
| stoweidlem29 46608 | When the hypothesis for th... |
| stoweidlem30 46609 | This lemma is used to prov... |
| stoweidlem31 46610 | This lemma is used to prov... |
| stoweidlem32 46611 | If a set A of real functio... |
| stoweidlem33 46612 | If a set of real functions... |
| stoweidlem34 46613 | This lemma proves that for... |
| stoweidlem35 46614 | This lemma is used to prov... |
| stoweidlem36 46615 | This lemma is used to prov... |
| stoweidlem37 46616 | This lemma is used to prov... |
| stoweidlem38 46617 | This lemma is used to prov... |
| stoweidlem39 46618 | This lemma is used to prov... |
| stoweidlem40 46619 | This lemma proves that q_n... |
| stoweidlem41 46620 | This lemma is used to prov... |
| stoweidlem42 46621 | This lemma is used to prov... |
| stoweidlem43 46622 | This lemma is used to prov... |
| stoweidlem44 46623 | This lemma is used to prov... |
| stoweidlem45 46624 | This lemma proves that, gi... |
| stoweidlem46 46625 | This lemma proves that set... |
| stoweidlem47 46626 | Subtracting a constant fro... |
| stoweidlem48 46627 | This lemma is used to prov... |
| stoweidlem49 46628 | There exists a function q_... |
| stoweidlem50 46629 | This lemma proves that set... |
| stoweidlem51 46630 | There exists a function x ... |
| stoweidlem52 46631 | There exists a neighborhoo... |
| stoweidlem53 46632 | This lemma is used to prov... |
| stoweidlem54 46633 | There exists a function ` ... |
| stoweidlem55 46634 | This lemma proves the exis... |
| stoweidlem56 46635 | This theorem proves Lemma ... |
| stoweidlem57 46636 | There exists a function x ... |
| stoweidlem58 46637 | This theorem proves Lemma ... |
| stoweidlem59 46638 | This lemma proves that the... |
| stoweidlem60 46639 | This lemma proves that the... |
| stoweidlem61 46640 | This lemma proves that the... |
| stoweidlem62 46641 | This theorem proves the St... |
| stoweid 46642 | This theorem proves the St... |
| stowei 46643 | This theorem proves the St... |
| wallispilem1 46644 | ` I ` is monotone: increas... |
| wallispilem2 46645 | A first set of properties ... |
| wallispilem3 46646 | I maps to real values. (C... |
| wallispilem4 46647 | ` F ` maps to explicit exp... |
| wallispilem5 46648 | The sequence ` H ` converg... |
| wallispi 46649 | Wallis' formula for π :... |
| wallispi2lem1 46650 | An intermediate step betwe... |
| wallispi2lem2 46651 | Two expressions are proven... |
| wallispi2 46652 | An alternative version of ... |
| stirlinglem1 46653 | A simple limit of fraction... |
| stirlinglem2 46654 | ` A ` maps to positive rea... |
| stirlinglem3 46655 | Long but simple algebraic ... |
| stirlinglem4 46656 | Algebraic manipulation of ... |
| stirlinglem5 46657 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 46658 | A series that converges to... |
| stirlinglem7 46659 | Algebraic manipulation of ... |
| stirlinglem8 46660 | If ` A ` converges to ` C ... |
| stirlinglem9 46661 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 46662 | A bound for any B(N)-B(N +... |
| stirlinglem11 46663 | ` B ` is decreasing. (Con... |
| stirlinglem12 46664 | The sequence ` B ` is boun... |
| stirlinglem13 46665 | ` B ` is decreasing and ha... |
| stirlinglem14 46666 | The sequence ` A ` converg... |
| stirlinglem15 46667 | The Stirling's formula is ... |
| stirling 46668 | Stirling's approximation f... |
| stirlingr 46669 | Stirling's approximation f... |
| dirkerval 46670 | The N_th Dirichlet kernel.... |
| dirker2re 46671 | The Dirichlet kernel value... |
| dirkerdenne0 46672 | The Dirichlet kernel denom... |
| dirkerval2 46673 | The N_th Dirichlet kernel ... |
| dirkerre 46674 | The Dirichlet kernel at an... |
| dirkerper 46675 | the Dirichlet kernel has p... |
| dirkerf 46676 | For any natural number ` N... |
| dirkertrigeqlem1 46677 | Sum of an even number of a... |
| dirkertrigeqlem2 46678 | Trigonometric equality lem... |
| dirkertrigeqlem3 46679 | Trigonometric equality lem... |
| dirkertrigeq 46680 | Trigonometric equality for... |
| dirkeritg 46681 | The definite integral of t... |
| dirkercncflem1 46682 | If ` Y ` is a multiple of ... |
| dirkercncflem2 46683 | Lemma used to prove that t... |
| dirkercncflem3 46684 | The Dirichlet kernel is co... |
| dirkercncflem4 46685 | The Dirichlet kernel is co... |
| dirkercncf 46686 | For any natural number ` N... |
| fourierdlem1 46687 | A partition interval is a ... |
| fourierdlem2 46688 | Membership in a partition.... |
| fourierdlem3 46689 | Membership in a partition.... |
| fourierdlem4 46690 | ` E ` is a function that m... |
| fourierdlem5 46691 | ` S ` is a function. (Con... |
| fourierdlem6 46692 | ` X ` is in the periodic p... |
| fourierdlem7 46693 | The difference between the... |
| fourierdlem8 46694 | A partition interval is a ... |
| fourierdlem9 46695 | ` H ` is a complex functio... |
| fourierdlem10 46696 | Condition on the bounds of... |
| fourierdlem11 46697 | If there is a partition, t... |
| fourierdlem12 46698 | A point of a partition is ... |
| fourierdlem13 46699 | Value of ` V ` in terms of... |
| fourierdlem14 46700 | Given the partition ` V ` ... |
| fourierdlem15 46701 | The range of the partition... |
| fourierdlem16 46702 | The coefficients of the fo... |
| fourierdlem17 46703 | The defined ` L ` is actua... |
| fourierdlem18 46704 | The function ` S ` is cont... |
| fourierdlem19 46705 | If two elements of ` D ` h... |
| fourierdlem20 46706 | Every interval in the part... |
| fourierdlem21 46707 | The coefficients of the fo... |
| fourierdlem22 46708 | The coefficients of the fo... |
| fourierdlem23 46709 | If ` F ` is continuous and... |
| fourierdlem24 46710 | A sufficient condition for... |
| fourierdlem25 46711 | If ` C ` is not in the ran... |
| fourierdlem26 46712 | Periodic image of a point ... |
| fourierdlem27 46713 | A partition open interval ... |
| fourierdlem28 46714 | Derivative of ` ( F `` ( X... |
| fourierdlem29 46715 | Explicit function value fo... |
| fourierdlem30 46716 | Sum of three small pieces ... |
| fourierdlem31 46717 | If ` A ` is finite and for... |
| fourierdlem32 46718 | Limit of a continuous func... |
| fourierdlem33 46719 | Limit of a continuous func... |
| fourierdlem34 46720 | A partition is one to one.... |
| fourierdlem35 46721 | There is a single point in... |
| fourierdlem36 46722 | ` F ` is an isomorphism. ... |
| fourierdlem37 46723 | ` I ` is a function that m... |
| fourierdlem38 46724 | The function ` F ` is cont... |
| fourierdlem39 46725 | Integration by parts of ... |
| fourierdlem40 46726 | ` H ` is a continuous func... |
| fourierdlem41 46727 | Lemma used to prove that e... |
| fourierdlem42 46728 | The set of points in a mov... |
| fourierdlem43 46729 | ` K ` is a real function. ... |
| fourierdlem44 46730 | A condition for having ` (... |
| fourierdlem46 46731 | The function ` F ` has a l... |
| fourierdlem47 46732 | For ` r ` large enough, th... |
| fourierdlem48 46733 | The given periodic functio... |
| fourierdlem49 46734 | The given periodic functio... |
| fourierdlem50 46735 | Continuity of ` O ` and it... |
| fourierdlem51 46736 | ` X ` is in the periodic p... |
| fourierdlem52 46737 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 46738 | The limit of ` F ( s ) ` a... |
| fourierdlem54 46739 | Given a partition ` Q ` an... |
| fourierdlem55 46740 | ` U ` is a real function. ... |
| fourierdlem56 46741 | Derivative of the ` K ` fu... |
| fourierdlem57 46742 | The derivative of ` O ` . ... |
| fourierdlem58 46743 | The derivative of ` K ` is... |
| fourierdlem59 46744 | The derivative of ` H ` is... |
| fourierdlem60 46745 | Given a differentiable fun... |
| fourierdlem61 46746 | Given a differentiable fun... |
| fourierdlem62 46747 | The function ` K ` is cont... |
| fourierdlem63 46748 | The upper bound of interva... |
| fourierdlem64 46749 | The partition ` V ` is fin... |
| fourierdlem65 46750 | The distance of two adjace... |
| fourierdlem66 46751 | Value of the ` G ` functio... |
| fourierdlem67 46752 | ` G ` is a function. (Con... |
| fourierdlem68 46753 | The derivative of ` O ` is... |
| fourierdlem69 46754 | A piecewise continuous fun... |
| fourierdlem70 46755 | A piecewise continuous fun... |
| fourierdlem71 46756 | A periodic piecewise conti... |
| fourierdlem72 46757 | The derivative of ` O ` is... |
| fourierdlem73 46758 | A version of the Riemann L... |
| fourierdlem74 46759 | Given a piecewise smooth f... |
| fourierdlem75 46760 | Given a piecewise smooth f... |
| fourierdlem76 46761 | Continuity of ` O ` and it... |
| fourierdlem77 46762 | If ` H ` is bounded, then ... |
| fourierdlem78 46763 | ` G ` is continuous when r... |
| fourierdlem79 46764 | ` E ` projects every inter... |
| fourierdlem80 46765 | The derivative of ` O ` is... |
| fourierdlem81 46766 | The integral of a piecewis... |
| fourierdlem82 46767 | Integral by substitution, ... |
| fourierdlem83 46768 | The fourier partial sum fo... |
| fourierdlem84 46769 | If ` F ` is piecewise cont... |
| fourierdlem85 46770 | Limit of the function ` G ... |
| fourierdlem86 46771 | Continuity of ` O ` and it... |
| fourierdlem87 46772 | The integral of ` G ` goes... |
| fourierdlem88 46773 | Given a piecewise continuo... |
| fourierdlem89 46774 | Given a piecewise continuo... |
| fourierdlem90 46775 | Given a piecewise continuo... |
| fourierdlem91 46776 | Given a piecewise continuo... |
| fourierdlem92 46777 | The integral of a piecewis... |
| fourierdlem93 46778 | Integral by substitution (... |
| fourierdlem94 46779 | For a piecewise smooth fun... |
| fourierdlem95 46780 | Algebraic manipulation of ... |
| fourierdlem96 46781 | limit for ` F ` at the low... |
| fourierdlem97 46782 | ` F ` is continuous on the... |
| fourierdlem98 46783 | ` F ` is continuous on the... |
| fourierdlem99 46784 | limit for ` F ` at the upp... |
| fourierdlem100 46785 | A piecewise continuous fun... |
| fourierdlem101 46786 | Integral by substitution f... |
| fourierdlem102 46787 | For a piecewise smooth fun... |
| fourierdlem103 46788 | The half lower part of the... |
| fourierdlem104 46789 | The half upper part of the... |
| fourierdlem105 46790 | A piecewise continuous fun... |
| fourierdlem106 46791 | For a piecewise smooth fun... |
| fourierdlem107 46792 | The integral of a piecewis... |
| fourierdlem108 46793 | The integral of a piecewis... |
| fourierdlem109 46794 | The integral of a piecewis... |
| fourierdlem110 46795 | The integral of a piecewis... |
| fourierdlem111 46796 | The fourier partial sum fo... |
| fourierdlem112 46797 | Here abbreviations (local ... |
| fourierdlem113 46798 | Fourier series convergence... |
| fourierdlem114 46799 | Fourier series convergence... |
| fourierdlem115 46800 | Fourier serier convergence... |
| fourierd 46801 | Fourier series convergence... |
| fourierclimd 46802 | Fourier series convergence... |
| fourierclim 46803 | Fourier series convergence... |
| fourier 46804 | Fourier series convergence... |
| fouriercnp 46805 | If ` F ` is continuous at ... |
| fourier2 46806 | Fourier series convergence... |
| sqwvfoura 46807 | Fourier coefficients for t... |
| sqwvfourb 46808 | Fourier series ` B ` coeff... |
| fourierswlem 46809 | The Fourier series for the... |
| fouriersw 46810 | Fourier series convergence... |
| fouriercn 46811 | If the derivative of ` F `... |
| elaa2lem 46812 | Elementhood in the set of ... |
| elaa2 46813 | Elementhood in the set of ... |
| etransclem1 46814 | ` H ` is a function. (Con... |
| etransclem2 46815 | Derivative of ` G ` . (Co... |
| etransclem3 46816 | The given ` if ` term is a... |
| etransclem4 46817 | ` F ` expressed as a finit... |
| etransclem5 46818 | A change of bound variable... |
| etransclem6 46819 | A change of bound variable... |
| etransclem7 46820 | The given product is an in... |
| etransclem8 46821 | ` F ` is a function. (Con... |
| etransclem9 46822 | If ` K ` divides ` N ` but... |
| etransclem10 46823 | The given ` if ` term is a... |
| etransclem11 46824 | A change of bound variable... |
| etransclem12 46825 | ` C ` applied to ` N ` . ... |
| etransclem13 46826 | ` F ` applied to ` Y ` . ... |
| etransclem14 46827 | Value of the term ` T ` , ... |
| etransclem15 46828 | Value of the term ` T ` , ... |
| etransclem16 46829 | Every element in the range... |
| etransclem17 46830 | The ` N ` -th derivative o... |
| etransclem18 46831 | The given function is inte... |
| etransclem19 46832 | The ` N ` -th derivative o... |
| etransclem20 46833 | ` H ` is smooth. (Contrib... |
| etransclem21 46834 | The ` N ` -th derivative o... |
| etransclem22 46835 | The ` N ` -th derivative o... |
| etransclem23 46836 | This is the claim proof in... |
| etransclem24 46837 | ` P ` divides the I -th de... |
| etransclem25 46838 | ` P ` factorial divides th... |
| etransclem26 46839 | Every term in the sum of t... |
| etransclem27 46840 | The ` N ` -th derivative o... |
| etransclem28 46841 | ` ( P - 1 ) ` factorial di... |
| etransclem29 46842 | The ` N ` -th derivative o... |
| etransclem30 46843 | The ` N ` -th derivative o... |
| etransclem31 46844 | The ` N ` -th derivative o... |
| etransclem32 46845 | This is the proof for the ... |
| etransclem33 46846 | ` F ` is smooth. (Contrib... |
| etransclem34 46847 | The ` N ` -th derivative o... |
| etransclem35 46848 | ` P ` does not divide the ... |
| etransclem36 46849 | The ` N ` -th derivative o... |
| etransclem37 46850 | ` ( P - 1 ) ` factorial di... |
| etransclem38 46851 | ` P ` divides the I -th de... |
| etransclem39 46852 | ` G ` is a function. (Con... |
| etransclem40 46853 | The ` N ` -th derivative o... |
| etransclem41 46854 | ` P ` does not divide the ... |
| etransclem42 46855 | The ` N ` -th derivative o... |
| etransclem43 46856 | ` G ` is a continuous func... |
| etransclem44 46857 | The given finite sum is no... |
| etransclem45 46858 | ` K ` is an integer. (Con... |
| etransclem46 46859 | This is the proof for equa... |
| etransclem47 46860 | ` _e ` is transcendental. ... |
| etransclem48 46861 | ` _e ` is transcendental. ... |
| etransc 46862 | ` _e ` is transcendental. ... |
| rrxtopn 46863 | The topology of the genera... |
| rrxngp 46864 | Generalized Euclidean real... |
| rrxtps 46865 | Generalized Euclidean real... |
| rrxtopnfi 46866 | The topology of the n-dime... |
| rrxtopon 46867 | The topology on generalize... |
| rrxtop 46868 | The topology on generalize... |
| rrndistlt 46869 | Given two points in the sp... |
| rrxtoponfi 46870 | The topology on n-dimensio... |
| rrxunitopnfi 46871 | The base set of the standa... |
| rrxtopn0 46872 | The topology of the zero-d... |
| qndenserrnbllem 46873 | n-dimensional rational num... |
| qndenserrnbl 46874 | n-dimensional rational num... |
| rrxtopn0b 46875 | The topology of the zero-d... |
| qndenserrnopnlem 46876 | n-dimensional rational num... |
| qndenserrnopn 46877 | n-dimensional rational num... |
| qndenserrn 46878 | n-dimensional rational num... |
| rrxsnicc 46879 | A multidimensional singlet... |
| rrnprjdstle 46880 | The distance between two p... |
| rrndsmet 46881 | ` D ` is a metric for the ... |
| rrndsxmet 46882 | ` D ` is an extended metri... |
| ioorrnopnlem 46883 | The a point in an indexed ... |
| ioorrnopn 46884 | The indexed product of ope... |
| ioorrnopnxrlem 46885 | Given a point ` F ` that b... |
| ioorrnopnxr 46886 | The indexed product of ope... |
| issal 46893 | Express the predicate " ` ... |
| pwsal 46894 | The power set of a given s... |
| salunicl 46895 | SAlg sigma-algebra is clos... |
| saluncl 46896 | The union of two sets in a... |
| prsal 46897 | The pair of the empty set ... |
| saldifcl 46898 | The complement of an eleme... |
| 0sal 46899 | The empty set belongs to e... |
| salgenval 46900 | The sigma-algebra generate... |
| saliunclf 46901 | SAlg sigma-algebra is clos... |
| saliuncl 46902 | SAlg sigma-algebra is clos... |
| salincl 46903 | The intersection of two se... |
| saluni 46904 | A set is an element of any... |
| saliinclf 46905 | SAlg sigma-algebra is clos... |
| saliincl 46906 | SAlg sigma-algebra is clos... |
| saldifcl2 46907 | The difference of two elem... |
| intsaluni 46908 | The union of an arbitrary ... |
| intsal 46909 | The arbitrary intersection... |
| salgenn0 46910 | The set used in the defini... |
| salgencl 46911 | ` SalGen ` actually genera... |
| issald 46912 | Sufficient condition to pr... |
| salexct 46913 | An example of nontrivial s... |
| sssalgen 46914 | A set is a subset of the s... |
| salgenss 46915 | The sigma-algebra generate... |
| salgenuni 46916 | The base set of the sigma-... |
| issalgend 46917 | One side of ~ dfsalgen2 . ... |
| salexct2 46918 | An example of a subset tha... |
| unisalgen 46919 | The union of a set belongs... |
| dfsalgen2 46920 | Alternate characterization... |
| salexct3 46921 | An example of a sigma-alge... |
| salgencntex 46922 | This counterexample shows ... |
| salgensscntex 46923 | This counterexample shows ... |
| issalnnd 46924 | Sufficient condition to pr... |
| dmvolsal 46925 | Lebesgue measurable sets f... |
| saldifcld 46926 | The complement of an eleme... |
| saluncld 46927 | The union of two sets in a... |
| salgencld 46928 | ` SalGen ` actually genera... |
| 0sald 46929 | The empty set belongs to e... |
| iooborel 46930 | An open interval is a Bore... |
| salincld 46931 | The intersection of two se... |
| salunid 46932 | A set is an element of any... |
| unisalgen2 46933 | The union of a set belongs... |
| bor1sal 46934 | The Borel sigma-algebra on... |
| iocborel 46935 | A left-open, right-closed ... |
| subsaliuncllem 46936 | A subspace sigma-algebra i... |
| subsaliuncl 46937 | A subspace sigma-algebra i... |
| subsalsal 46938 | A subspace sigma-algebra i... |
| subsaluni 46939 | A set belongs to the subsp... |
| salrestss 46940 | A sigma-algebra restricted... |
| sge0rnre 46943 | When ` sum^ ` is applied t... |
| fge0icoicc 46944 | If ` F ` maps to nonnegati... |
| sge0val 46945 | The value of the sum of no... |
| fge0npnf 46946 | If ` F ` maps to nonnegati... |
| sge0rnn0 46947 | The range used in the defi... |
| sge0vald 46948 | The value of the sum of no... |
| fge0iccico 46949 | A range of nonnegative ext... |
| gsumge0cl 46950 | Closure of group sum, for ... |
| sge0reval 46951 | Value of the sum of nonneg... |
| sge0pnfval 46952 | If a term in the sum of no... |
| fge0iccre 46953 | A range of nonnegative ext... |
| sge0z 46954 | Any nonnegative extended s... |
| sge00 46955 | The sum of nonnegative ext... |
| fsumlesge0 46956 | Every finite subsum of non... |
| sge0revalmpt 46957 | Value of the sum of nonneg... |
| sge0sn 46958 | A sum of a nonnegative ext... |
| sge0tsms 46959 | ` sum^ ` applied to a nonn... |
| sge0cl 46960 | The arbitrary sum of nonne... |
| sge0f1o 46961 | Re-index a nonnegative ext... |
| sge0snmpt 46962 | A sum of a nonnegative ext... |
| sge0ge0 46963 | The sum of nonnegative ext... |
| sge0xrcl 46964 | The arbitrary sum of nonne... |
| sge0repnf 46965 | The of nonnegative extende... |
| sge0fsum 46966 | The arbitrary sum of a fin... |
| sge0rern 46967 | If the sum of nonnegative ... |
| sge0supre 46968 | If the arbitrary sum of no... |
| sge0fsummpt 46969 | The arbitrary sum of a fin... |
| sge0sup 46970 | The arbitrary sum of nonne... |
| sge0less 46971 | A shorter sum of nonnegati... |
| sge0rnbnd 46972 | The range used in the defi... |
| sge0pr 46973 | Sum of a pair of nonnegati... |
| sge0gerp 46974 | The arbitrary sum of nonne... |
| sge0pnffigt 46975 | If the sum of nonnegative ... |
| sge0ssre 46976 | If a sum of nonnegative ex... |
| sge0lefi 46977 | A sum of nonnegative exten... |
| sge0lessmpt 46978 | A shorter sum of nonnegati... |
| sge0ltfirp 46979 | If the sum of nonnegative ... |
| sge0prle 46980 | The sum of a pair of nonne... |
| sge0gerpmpt 46981 | The arbitrary sum of nonne... |
| sge0resrnlem 46982 | The sum of nonnegative ext... |
| sge0resrn 46983 | The sum of nonnegative ext... |
| sge0ssrempt 46984 | If a sum of nonnegative ex... |
| sge0resplit 46985 | ` sum^ ` splits into two p... |
| sge0le 46986 | If all of the terms of sum... |
| sge0ltfirpmpt 46987 | If the extended sum of non... |
| sge0split 46988 | Split a sum of nonnegative... |
| sge0lempt 46989 | If all of the terms of sum... |
| sge0splitmpt 46990 | Split a sum of nonnegative... |
| sge0ss 46991 | Change the index set to a ... |
| sge0iunmptlemfi 46992 | Sum of nonnegative extende... |
| sge0p1 46993 | The addition of the next t... |
| sge0iunmptlemre 46994 | Sum of nonnegative extende... |
| sge0fodjrnlem 46995 | Re-index a nonnegative ext... |
| sge0fodjrn 46996 | Re-index a nonnegative ext... |
| sge0iunmpt 46997 | Sum of nonnegative extende... |
| sge0iun 46998 | Sum of nonnegative extende... |
| sge0nemnf 46999 | The generalized sum of non... |
| sge0rpcpnf 47000 | The sum of an infinite num... |
| sge0rernmpt 47001 | If the sum of nonnegative ... |
| sge0lefimpt 47002 | A sum of nonnegative exten... |
| nn0ssge0 47003 | Nonnegative integers are n... |
| sge0clmpt 47004 | The generalized sum of non... |
| sge0ltfirpmpt2 47005 | If the extended sum of non... |
| sge0isum 47006 | If a series of nonnegative... |
| sge0xrclmpt 47007 | The generalized sum of non... |
| sge0xp 47008 | Combine two generalized su... |
| sge0isummpt 47009 | If a series of nonnegative... |
| sge0ad2en 47010 | The value of the infinite ... |
| sge0isummpt2 47011 | If a series of nonnegative... |
| sge0xaddlem1 47012 | The extended addition of t... |
| sge0xaddlem2 47013 | The extended addition of t... |
| sge0xadd 47014 | The extended addition of t... |
| sge0fsummptf 47015 | The generalized sum of a f... |
| sge0snmptf 47016 | A sum of a nonnegative ext... |
| sge0ge0mpt 47017 | The sum of nonnegative ext... |
| sge0repnfmpt 47018 | The of nonnegative extende... |
| sge0pnffigtmpt 47019 | If the generalized sum of ... |
| sge0splitsn 47020 | Separate out a term in a g... |
| sge0pnffsumgt 47021 | If the sum of nonnegative ... |
| sge0gtfsumgt 47022 | If the generalized sum of ... |
| sge0uzfsumgt 47023 | If a real number is smalle... |
| sge0pnfmpt 47024 | If a term in the sum of no... |
| sge0seq 47025 | A series of nonnegative re... |
| sge0reuz 47026 | Value of the generalized s... |
| sge0reuzb 47027 | Value of the generalized s... |
| ismea 47030 | Express the predicate " ` ... |
| dmmeasal 47031 | The domain of a measure is... |
| meaf 47032 | A measure is a function th... |
| mea0 47033 | The measure of the empty s... |
| nnfoctbdjlem 47034 | There exists a mapping fro... |
| nnfoctbdj 47035 | There exists a mapping fro... |
| meadjuni 47036 | The measure of the disjoin... |
| meacl 47037 | The measure of a set is a ... |
| iundjiunlem 47038 | The sets in the sequence `... |
| iundjiun 47039 | Given a sequence ` E ` of ... |
| meaxrcl 47040 | The measure of a set is an... |
| meadjun 47041 | The measure of the union o... |
| meassle 47042 | The measure of a set is gr... |
| meaunle 47043 | The measure of the union o... |
| meadjiunlem 47044 | The sum of nonnegative ext... |
| meadjiun 47045 | The measure of the disjoin... |
| ismeannd 47046 | Sufficient condition to pr... |
| meaiunlelem 47047 | The measure of the union o... |
| meaiunle 47048 | The measure of the union o... |
| psmeasurelem 47049 | ` M ` applied to a disjoin... |
| psmeasure 47050 | Point supported measure, R... |
| voliunsge0lem 47051 | The Lebesgue measure funct... |
| voliunsge0 47052 | The Lebesgue measure funct... |
| volmea 47053 | The Lebesgue measure on th... |
| meage0 47054 | If the measure of a measur... |
| meadjunre 47055 | The measure of the union o... |
| meassre 47056 | If the measure of a measur... |
| meale0eq0 47057 | A measure that is less tha... |
| meadif 47058 | The measure of the differe... |
| meaiuninclem 47059 | Measures are continuous fr... |
| meaiuninc 47060 | Measures are continuous fr... |
| meaiuninc2 47061 | Measures are continuous fr... |
| meaiunincf 47062 | Measures are continuous fr... |
| meaiuninc3v 47063 | Measures are continuous fr... |
| meaiuninc3 47064 | Measures are continuous fr... |
| meaiininclem 47065 | Measures are continuous fr... |
| meaiininc 47066 | Measures are continuous fr... |
| meaiininc2 47067 | Measures are continuous fr... |
| caragenval 47072 | The sigma-algebra generate... |
| isome 47073 | Express the predicate " ` ... |
| caragenel 47074 | Membership in the Caratheo... |
| omef 47075 | An outer measure is a func... |
| ome0 47076 | The outer measure of the e... |
| omessle 47077 | The outer measure of a set... |
| omedm 47078 | The domain of an outer mea... |
| caragensplit 47079 | If ` E ` is in the set gen... |
| caragenelss 47080 | An element of the Caratheo... |
| carageneld 47081 | Membership in the Caratheo... |
| omecl 47082 | The outer measure of a set... |
| caragenss 47083 | The sigma-algebra generate... |
| omeunile 47084 | The outer measure of the u... |
| caragen0 47085 | The empty set belongs to a... |
| omexrcl 47086 | The outer measure of a set... |
| caragenunidm 47087 | The base set of an outer m... |
| caragensspw 47088 | The sigma-algebra generate... |
| omessre 47089 | If the outer measure of a ... |
| caragenuni 47090 | The base set of the sigma-... |
| caragenuncllem 47091 | The Caratheodory's constru... |
| caragenuncl 47092 | The Caratheodory's constru... |
| caragendifcl 47093 | The Caratheodory's constru... |
| caragenfiiuncl 47094 | The Caratheodory's constru... |
| omeunle 47095 | The outer measure of the u... |
| omeiunle 47096 | The outer measure of the i... |
| omelesplit 47097 | The outer measure of a set... |
| omeiunltfirp 47098 | If the outer measure of a ... |
| omeiunlempt 47099 | The outer measure of the i... |
| carageniuncllem1 47100 | The outer measure of ` A i... |
| carageniuncllem2 47101 | The Caratheodory's constru... |
| carageniuncl 47102 | The Caratheodory's constru... |
| caragenunicl 47103 | The Caratheodory's constru... |
| caragensal 47104 | Caratheodory's method gene... |
| caratheodorylem1 47105 | Lemma used to prove that C... |
| caratheodorylem2 47106 | Caratheodory's constructio... |
| caratheodory 47107 | Caratheodory's constructio... |
| 0ome 47108 | The map that assigns 0 to ... |
| isomenndlem 47109 | ` O ` is sub-additive w.r.... |
| isomennd 47110 | Sufficient condition to pr... |
| caragenel2d 47111 | Membership in the Caratheo... |
| omege0 47112 | If the outer measure of a ... |
| omess0 47113 | If the outer measure of a ... |
| caragencmpl 47114 | A measure built with the C... |
| vonval 47119 | Value of the Lebesgue meas... |
| ovnval 47120 | Value of the Lebesgue oute... |
| elhoi 47121 | Membership in a multidimen... |
| icoresmbl 47122 | A closed-below, open-above... |
| hoissre 47123 | The projection of a half-o... |
| ovnval2 47124 | Value of the Lebesgue oute... |
| volicorecl 47125 | The Lebesgue measure of a ... |
| hoiprodcl 47126 | The pre-measure of half-op... |
| hoicvr 47127 | ` I ` is a countable set o... |
| hoissrrn 47128 | A half-open interval is a ... |
| ovn0val 47129 | The Lebesgue outer measure... |
| ovnn0val 47130 | The value of a (multidimen... |
| ovnval2b 47131 | Value of the Lebesgue oute... |
| volicorescl 47132 | The Lebesgue measure of a ... |
| ovnprodcl 47133 | The product used in the de... |
| hoiprodcl2 47134 | The pre-measure of half-op... |
| hoicvrrex 47135 | Any subset of the multidim... |
| ovnsupge0 47136 | The set used in the defini... |
| ovnlecvr 47137 | Given a subset of multidim... |
| ovnpnfelsup 47138 | ` +oo ` is an element of t... |
| ovnsslelem 47139 | The (multidimensional, non... |
| ovnssle 47140 | The (multidimensional) Leb... |
| ovnlerp 47141 | The Lebesgue outer measure... |
| ovnf 47142 | The Lebesgue outer measure... |
| ovncvrrp 47143 | The Lebesgue outer measure... |
| ovn0lem 47144 | For any finite dimension, ... |
| ovn0 47145 | For any finite dimension, ... |
| ovncl 47146 | The Lebesgue outer measure... |
| ovn02 47147 | For the zero-dimensional s... |
| ovnxrcl 47148 | The Lebesgue outer measure... |
| ovnsubaddlem1 47149 | The Lebesgue outer measure... |
| ovnsubaddlem2 47150 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 47151 | ` ( voln* `` X ) ` is suba... |
| ovnome 47152 | ` ( voln* `` X ) ` is an o... |
| vonmea 47153 | ` ( voln `` X ) ` is a mea... |
| volicon0 47154 | The measure of a nonempty ... |
| hsphoif 47155 | ` H ` is a function (that ... |
| hoidmvval 47156 | The dimensional volume of ... |
| hoissrrn2 47157 | A half-open interval is a ... |
| hsphoival 47158 | ` H ` is a function (that ... |
| hoiprodcl3 47159 | The pre-measure of half-op... |
| volicore 47160 | The Lebesgue measure of a ... |
| hoidmvcl 47161 | The dimensional volume of ... |
| hoidmv0val 47162 | The dimensional volume of ... |
| hoidmvn0val 47163 | The dimensional volume of ... |
| hsphoidmvle2 47164 | The dimensional volume of ... |
| hsphoidmvle 47165 | The dimensional volume of ... |
| hoidmvval0 47166 | The dimensional volume of ... |
| hoiprodp1 47167 | The dimensional volume of ... |
| sge0hsphoire 47168 | If the generalized sum of ... |
| hoidmvval0b 47169 | The dimensional volume of ... |
| hoidmv1lelem1 47170 | The supremum of ` U ` belo... |
| hoidmv1lelem2 47171 | This is the contradiction ... |
| hoidmv1lelem3 47172 | The dimensional volume of ... |
| hoidmv1le 47173 | The dimensional volume of ... |
| hoidmvlelem1 47174 | The supremum of ` U ` belo... |
| hoidmvlelem2 47175 | This is the contradiction ... |
| hoidmvlelem3 47176 | This is the contradiction ... |
| hoidmvlelem4 47177 | The dimensional volume of ... |
| hoidmvlelem5 47178 | The dimensional volume of ... |
| hoidmvle 47179 | The dimensional volume of ... |
| ovnhoilem1 47180 | The Lebesgue outer measure... |
| ovnhoilem2 47181 | The Lebesgue outer measure... |
| ovnhoi 47182 | The Lebesgue outer measure... |
| dmovn 47183 | The domain of the Lebesgue... |
| hoicoto2 47184 | The half-open interval exp... |
| dmvon 47185 | Lebesgue measurable n-dime... |
| hoi2toco 47186 | The half-open interval exp... |
| hoidifhspval 47187 | ` D ` is a function that r... |
| hspval 47188 | The value of the half-spac... |
| ovnlecvr2 47189 | Given a subset of multidim... |
| ovncvr2 47190 | ` B ` and ` T ` are the le... |
| dmovnsal 47191 | The domain of the Lebesgue... |
| unidmovn 47192 | Base set of the n-dimensio... |
| rrnmbl 47193 | The set of n-dimensional R... |
| hoidifhspval2 47194 | ` D ` is a function that r... |
| hspdifhsp 47195 | A n-dimensional half-open ... |
| unidmvon 47196 | Base set of the n-dimensio... |
| hoidifhspf 47197 | ` D ` is a function that r... |
| hoidifhspval3 47198 | ` D ` is a function that r... |
| hoidifhspdmvle 47199 | The dimensional volume of ... |
| voncmpl 47200 | The Lebesgue measure is co... |
| hoiqssbllem1 47201 | The center of the n-dimens... |
| hoiqssbllem2 47202 | The center of the n-dimens... |
| hoiqssbllem3 47203 | A n-dimensional ball conta... |
| hoiqssbl 47204 | A n-dimensional ball conta... |
| hspmbllem1 47205 | Any half-space of the n-di... |
| hspmbllem2 47206 | Any half-space of the n-di... |
| hspmbllem3 47207 | Any half-space of the n-di... |
| hspmbl 47208 | Any half-space of the n-di... |
| hoimbllem 47209 | Any n-dimensional half-ope... |
| hoimbl 47210 | Any n-dimensional half-ope... |
| opnvonmbllem1 47211 | The half-open interval exp... |
| opnvonmbllem2 47212 | An open subset of the n-di... |
| opnvonmbl 47213 | An open subset of the n-di... |
| opnssborel 47214 | Open sets of a generalized... |
| borelmbl 47215 | All Borel subsets of the n... |
| volicorege0 47216 | The Lebesgue measure of a ... |
| isvonmbl 47217 | The predicate " ` A ` is m... |
| mblvon 47218 | The n-dimensional Lebesgue... |
| vonmblss 47219 | n-dimensional Lebesgue mea... |
| volico2 47220 | The measure of left-closed... |
| vonmblss2 47221 | n-dimensional Lebesgue mea... |
| ovolval2lem 47222 | The value of the Lebesgue ... |
| ovolval2 47223 | The value of the Lebesgue ... |
| ovnsubadd2lem 47224 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 47225 | ` ( voln* `` X ) ` is suba... |
| ovolval3 47226 | The value of the Lebesgue ... |
| ovnsplit 47227 | The n-dimensional Lebesgue... |
| ovolval4lem1 47228 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 47229 | The value of the Lebesgue ... |
| ovolval4 47230 | The value of the Lebesgue ... |
| ovolval5lem1 47231 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 47232 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 47233 | The value of the Lebesgue ... |
| ovolval5 47234 | The value of the Lebesgue ... |
| ovnovollem1 47235 | if ` F ` is a cover of ` B... |
| ovnovollem2 47236 | if ` I ` is a cover of ` (... |
| ovnovollem3 47237 | The 1-dimensional Lebesgue... |
| ovnovol 47238 | The 1-dimensional Lebesgue... |
| vonvolmbllem 47239 | If a subset ` B ` of real ... |
| vonvolmbl 47240 | A subset of Real numbers i... |
| vonvol 47241 | The 1-dimensional Lebesgue... |
| vonvolmbl2 47242 | A subset ` X ` of the spac... |
| vonvol2 47243 | The 1-dimensional Lebesgue... |
| hoimbl2 47244 | Any n-dimensional half-ope... |
| voncl 47245 | The Lebesgue measure of a ... |
| vonhoi 47246 | The Lebesgue outer measure... |
| vonxrcl 47247 | The Lebesgue measure of a ... |
| ioosshoi 47248 | A n-dimensional open inter... |
| vonn0hoi 47249 | The Lebesgue outer measure... |
| von0val 47250 | The Lebesgue measure (for ... |
| vonhoire 47251 | The Lebesgue measure of a ... |
| iinhoiicclem 47252 | A n-dimensional closed int... |
| iinhoiicc 47253 | A n-dimensional closed int... |
| iunhoiioolem 47254 | A n-dimensional open inter... |
| iunhoiioo 47255 | A n-dimensional open inter... |
| ioovonmbl 47256 | Any n-dimensional open int... |
| iccvonmbllem 47257 | Any n-dimensional closed i... |
| iccvonmbl 47258 | Any n-dimensional closed i... |
| vonioolem1 47259 | The sequence of the measur... |
| vonioolem2 47260 | The n-dimensional Lebesgue... |
| vonioo 47261 | The n-dimensional Lebesgue... |
| vonicclem1 47262 | The sequence of the measur... |
| vonicclem2 47263 | The n-dimensional Lebesgue... |
| vonicc 47264 | The n-dimensional Lebesgue... |
| snvonmbl 47265 | A n-dimensional singleton ... |
| vonn0ioo 47266 | The n-dimensional Lebesgue... |
| vonn0icc 47267 | The n-dimensional Lebesgue... |
| ctvonmbl 47268 | Any n-dimensional countabl... |
| vonn0ioo2 47269 | The n-dimensional Lebesgue... |
| vonsn 47270 | The n-dimensional Lebesgue... |
| vonn0icc2 47271 | The n-dimensional Lebesgue... |
| vonct 47272 | The n-dimensional Lebesgue... |
| vitali2 47273 | There are non-measurable s... |
| pimltmnf2f 47276 | Given a real-valued functi... |
| pimltmnf2 47277 | Given a real-valued functi... |
| preimagelt 47278 | The preimage of a right-op... |
| preimalegt 47279 | The preimage of a left-ope... |
| pimconstlt0 47280 | Given a constant function,... |
| pimconstlt1 47281 | Given a constant function,... |
| pimltpnff 47282 | Given a real-valued functi... |
| pimltpnf 47283 | Given a real-valued functi... |
| pimgtpnf2f 47284 | Given a real-valued functi... |
| pimgtpnf2 47285 | Given a real-valued functi... |
| salpreimagelt 47286 | If all the preimages of le... |
| pimrecltpos 47287 | The preimage of an unbound... |
| salpreimalegt 47288 | If all the preimages of ri... |
| pimiooltgt 47289 | The preimage of an open in... |
| preimaicomnf 47290 | Preimage of an open interv... |
| pimltpnf2f 47291 | Given a real-valued functi... |
| pimltpnf2 47292 | Given a real-valued functi... |
| pimgtmnf2 47293 | Given a real-valued functi... |
| pimdecfgtioc 47294 | Given a nonincreasing func... |
| pimincfltioc 47295 | Given a nondecreasing func... |
| pimdecfgtioo 47296 | Given a nondecreasing func... |
| pimincfltioo 47297 | Given a nondecreasing func... |
| preimaioomnf 47298 | Preimage of an open interv... |
| preimageiingt 47299 | A preimage of a left-close... |
| preimaleiinlt 47300 | A preimage of a left-open,... |
| pimgtmnff 47301 | Given a real-valued functi... |
| pimgtmnf 47302 | Given a real-valued functi... |
| pimrecltneg 47303 | The preimage of an unbound... |
| salpreimagtge 47304 | If all the preimages of le... |
| salpreimaltle 47305 | If all the preimages of ri... |
| issmflem 47306 | The predicate " ` F ` is a... |
| issmf 47307 | The predicate " ` F ` is a... |
| salpreimalelt 47308 | If all the preimages of ri... |
| salpreimagtlt 47309 | If all the preimages of le... |
| smfpreimalt 47310 | Given a function measurabl... |
| smff 47311 | A function measurable w.r.... |
| smfdmss 47312 | The domain of a function m... |
| issmff 47313 | The predicate " ` F ` is a... |
| issmfd 47314 | A sufficient condition for... |
| smfpreimaltf 47315 | Given a function measurabl... |
| issmfdf 47316 | A sufficient condition for... |
| sssmf 47317 | The restriction of a sigma... |
| mbfresmf 47318 | A real-valued measurable f... |
| cnfsmf 47319 | A continuous function is m... |
| incsmflem 47320 | A nondecreasing function i... |
| incsmf 47321 | A real-valued, nondecreasi... |
| smfsssmf 47322 | If a function is measurabl... |
| issmflelem 47323 | The predicate " ` F ` is a... |
| issmfle 47324 | The predicate " ` F ` is a... |
| smfpimltmpt 47325 | Given a function measurabl... |
| smfpimltxr 47326 | Given a function measurabl... |
| issmfdmpt 47327 | A sufficient condition for... |
| smfconst 47328 | Given a sigma-algebra over... |
| sssmfmpt 47329 | The restriction of a sigma... |
| cnfrrnsmf 47330 | A function, continuous fro... |
| smfid 47331 | The identity function is B... |
| bormflebmf 47332 | A Borel measurable functio... |
| smfpreimale 47333 | Given a function measurabl... |
| issmfgtlem 47334 | The predicate " ` F ` is a... |
| issmfgt 47335 | The predicate " ` F ` is a... |
| issmfled 47336 | A sufficient condition for... |
| smfpimltxrmptf 47337 | Given a function measurabl... |
| smfpimltxrmpt 47338 | Given a function measurabl... |
| smfmbfcex 47339 | A constant function, with ... |
| issmfgtd 47340 | A sufficient condition for... |
| smfpreimagt 47341 | Given a function measurabl... |
| smfaddlem1 47342 | Given the sum of two funct... |
| smfaddlem2 47343 | The sum of two sigma-measu... |
| smfadd 47344 | The sum of two sigma-measu... |
| decsmflem 47345 | A nonincreasing function i... |
| decsmf 47346 | A real-valued, nonincreasi... |
| smfpreimagtf 47347 | Given a function measurabl... |
| issmfgelem 47348 | The predicate " ` F ` is a... |
| issmfge 47349 | The predicate " ` F ` is a... |
| smflimlem1 47350 | Lemma for the proof that t... |
| smflimlem2 47351 | Lemma for the proof that t... |
| smflimlem3 47352 | The limit of sigma-measura... |
| smflimlem4 47353 | Lemma for the proof that t... |
| smflimlem5 47354 | Lemma for the proof that t... |
| smflimlem6 47355 | Lemma for the proof that t... |
| smflim 47356 | The limit of sigma-measura... |
| nsssmfmbflem 47357 | The sigma-measurable funct... |
| nsssmfmbf 47358 | The sigma-measurable funct... |
| smfpimgtxr 47359 | Given a function measurabl... |
| smfpimgtmpt 47360 | Given a function measurabl... |
| smfpreimage 47361 | Given a function measurabl... |
| mbfpsssmf 47362 | Real-valued measurable fun... |
| smfpimgtxrmptf 47363 | Given a function measurabl... |
| smfpimgtxrmpt 47364 | Given a function measurabl... |
| smfpimioompt 47365 | Given a function measurabl... |
| smfpimioo 47366 | Given a function measurabl... |
| smfresal 47367 | Given a sigma-measurable f... |
| smfrec 47368 | The reciprocal of a sigma-... |
| smfres 47369 | The restriction of sigma-m... |
| smfmullem1 47370 | The multiplication of two ... |
| smfmullem2 47371 | The multiplication of two ... |
| smfmullem3 47372 | The multiplication of two ... |
| smfmullem4 47373 | The multiplication of two ... |
| smfmul 47374 | The multiplication of two ... |
| smfmulc1 47375 | A sigma-measurable functio... |
| smfdiv 47376 | The fraction of two sigma-... |
| smfpimbor1lem1 47377 | Every open set belongs to ... |
| smfpimbor1lem2 47378 | Given a sigma-measurable f... |
| smfpimbor1 47379 | Given a sigma-measurable f... |
| smf2id 47380 | Twice the identity functio... |
| smfco 47381 | The composition of a Borel... |
| smfneg 47382 | The negative of a sigma-me... |
| smffmptf 47383 | A function measurable w.r.... |
| smffmpt 47384 | A function measurable w.r.... |
| smflim2 47385 | The limit of a sequence of... |
| smfpimcclem 47386 | Lemma for ~ smfpimcc given... |
| smfpimcc 47387 | Given a countable set of s... |
| issmfle2d 47388 | A sufficient condition for... |
| smflimmpt 47389 | The limit of a sequence of... |
| smfsuplem1 47390 | The supremum of a countabl... |
| smfsuplem2 47391 | The supremum of a countabl... |
| smfsuplem3 47392 | The supremum of a countabl... |
| smfsup 47393 | The supremum of a countabl... |
| smfsupmpt 47394 | The supremum of a countabl... |
| smfsupxr 47395 | The supremum of a countabl... |
| smfinflem 47396 | The infimum of a countable... |
| smfinf 47397 | The infimum of a countable... |
| smfinfmpt 47398 | The infimum of a countable... |
| smflimsuplem1 47399 | If ` H ` converges, the ` ... |
| smflimsuplem2 47400 | The superior limit of a se... |
| smflimsuplem3 47401 | The limit of the ` ( H `` ... |
| smflimsuplem4 47402 | If ` H ` converges, the ` ... |
| smflimsuplem5 47403 | ` H ` converges to the sup... |
| smflimsuplem6 47404 | The superior limit of a se... |
| smflimsuplem7 47405 | The superior limit of a se... |
| smflimsuplem8 47406 | The superior limit of a se... |
| smflimsup 47407 | The superior limit of a se... |
| smflimsupmpt 47408 | The superior limit of a se... |
| smfliminflem 47409 | The inferior limit of a co... |
| smfliminf 47410 | The inferior limit of a co... |
| smfliminfmpt 47411 | The inferior limit of a co... |
| adddmmbl 47412 | If two functions have doma... |
| adddmmbl2 47413 | If two functions have doma... |
| muldmmbl 47414 | If two functions have doma... |
| muldmmbl2 47415 | If two functions have doma... |
| smfdmmblpimne 47416 | If a measurable function w... |
| smfdivdmmbl 47417 | If a functions and a sigma... |
| smfpimne 47418 | Given a function measurabl... |
| smfpimne2 47419 | Given a function measurabl... |
| smfdivdmmbl2 47420 | If a functions and a sigma... |
| fsupdm 47421 | The domain of the sup func... |
| fsupdm2 47422 | The domain of the sup func... |
| smfsupdmmbllem 47423 | If a countable set of sigm... |
| smfsupdmmbl 47424 | If a countable set of sigm... |
| finfdm 47425 | The domain of the inf func... |
| finfdm2 47426 | The domain of the inf func... |
| smfinfdmmbllem 47427 | If a countable set of sigm... |
| smfinfdmmbl 47428 | If a countable set of sigm... |
| sigarval 47429 | Define the signed area by ... |
| sigarim 47430 | Signed area takes value in... |
| sigarac 47431 | Signed area is anticommuta... |
| sigaraf 47432 | Signed area is additive by... |
| sigarmf 47433 | Signed area is additive (w... |
| sigaras 47434 | Signed area is additive by... |
| sigarms 47435 | Signed area is additive (w... |
| sigarls 47436 | Signed area is linear by t... |
| sigarid 47437 | Signed area of a flat para... |
| sigarexp 47438 | Expand the signed area for... |
| sigarperm 47439 | Signed area ` ( A - C ) G ... |
| sigardiv 47440 | If signed area between vec... |
| sigarimcd 47441 | Signed area takes value in... |
| sigariz 47442 | If signed area is zero, th... |
| sigarcol 47443 | Given three points ` A ` ,... |
| sharhght 47444 | Let ` A B C ` be a triangl... |
| sigaradd 47445 | Subtracting (double) area ... |
| cevathlem1 47446 | Ceva's theorem first lemma... |
| cevathlem2 47447 | Ceva's theorem second lemm... |
| cevath 47448 | Ceva's theorem. Let ` A B... |
| simpcntrab 47449 | The center of a simple gro... |
| et-ltneverrefl 47450 | Less-than class is never r... |
| et-equeucl 47451 | Alternative proof that equ... |
| et-sqrtnegnre 47452 | The square root of a negat... |
| quantgodel 47453 | There can be no formula as... |
| quantgodelALT 47454 | There can be no formula as... |
| ormklocald 47455 | If elements of a certain s... |
| ormkglobd 47456 | If all adjacent elements o... |
| natlocalincr 47457 | Global monotonicity on hal... |
| natglobalincr 47458 | Local monotonicity on half... |
| chnsubseqword 47459 | A subsequence of a chain i... |
| chnsubseqwl 47460 | A subsequence of a chain h... |
| chnsubseq 47461 | An order-preserving subseq... |
| chnsuslle 47462 | Length of a subsequence is... |
| chnerlem1 47463 | In a chain constructed on ... |
| chnerlem2 47464 | Lemma for ~ chner where th... |
| chnerlem3 47465 | Lemma for ~ chner - tricho... |
| chner 47466 | Any two elements are equiv... |
| nthrucw 47467 | Some number sets form a ch... |
| evenwodadd 47468 | If an integer is multiplie... |
| squeezedltsq 47469 | If a real value is squeeze... |
| sin3t 47470 | Triple-angle formula for s... |
| cos3t 47471 | Triple-angle formula for c... |
| sin5tlem1 47472 | Lemma 1 for quintupled ang... |
| sin5tlem2 47473 | Lemma 2 for quintupled ang... |
| sin5tlem3 47474 | Lemma 3 for quintupled ang... |
| sin5tlem4 47475 | Lemma 4 for quintupled ang... |
| sin5tlem5 47476 | Lemma 5 for quintupled ang... |
| sin5t 47477 | Five-times-angle formula f... |
| cos5t 47478 | Five-times-angle formula f... |
| cos5teq 47479 | Five-times-angle formula f... |
| goldrarr 47480 | The golden ratio is a real... |
| goldrasin 47481 | Alternative trigonometric ... |
| goldrapos 47482 | Golden ratio is positive. ... |
| goldrarp 47483 | The golden ratio is a posi... |
| goldracos5teq 47484 | Lemma 1 for determining th... |
| goldratmolem2 47485 | Lemma 2 for determining th... |
| lambert0 47486 | A value of Lambert W (prod... |
| lamberte 47487 | A value of Lambert W (prod... |
| cjnpoly 47488 | Complex conjugation operat... |
| tannpoly 47489 | The tangent function is no... |
| sinnpoly 47490 | Sine function is not a pol... |
| hirstL-ax3 47491 | The third axiom of a syste... |
| ax3h 47492 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 47493 | A closed form showing (a i... |
| aibandbiaiaiffb 47494 | A closed form showing (a i... |
| notatnand 47495 | Do not use. Use intnanr i... |
| aistia 47496 | Given a is equivalent to `... |
| aisfina 47497 | Given a is equivalent to `... |
| bothtbothsame 47498 | Given both a, b are equiva... |
| bothfbothsame 47499 | Given both a, b are equiva... |
| aiffbbtat 47500 | Given a is equivalent to b... |
| aisbbisfaisf 47501 | Given a is equivalent to b... |
| axorbtnotaiffb 47502 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 47503 | Given a is equivalent to (... |
| axorbciffatcxorb 47504 | Given a is equivalent to (... |
| aibnbna 47505 | Given a implies b, (not b)... |
| aibnbaif 47506 | Given a implies b, not b, ... |
| aiffbtbat 47507 | Given a is equivalent to b... |
| astbstanbst 47508 | Given a is equivalent to T... |
| aistbistaandb 47509 | Given a is equivalent to T... |
| aisbnaxb 47510 | Given a is equivalent to b... |
| atbiffatnnb 47511 | If a implies b, then a imp... |
| bisaiaisb 47512 | Application of bicom1 with... |
| atbiffatnnbalt 47513 | If a implies b, then a imp... |
| abnotbtaxb 47514 | Assuming a, not b, there e... |
| abnotataxb 47515 | Assuming not a, b, there e... |
| conimpf 47516 | Assuming a, not b, and a i... |
| conimpfalt 47517 | Assuming a, not b, and a i... |
| aistbisfiaxb 47518 | Given a is equivalent to T... |
| aisfbistiaxb 47519 | Given a is equivalent to F... |
| aifftbifffaibif 47520 | Given a is equivalent to T... |
| aifftbifffaibifff 47521 | Given a is equivalent to T... |
| atnaiana 47522 | Given a, it is not the cas... |
| ainaiaandna 47523 | Given a, a implies it is n... |
| abcdta 47524 | Given (((a and b) and c) a... |
| abcdtb 47525 | Given (((a and b) and c) a... |
| abcdtc 47526 | Given (((a and b) and c) a... |
| abcdtd 47527 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 47528 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 47529 | Operands in a biconditiona... |
| nabctnabc 47530 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 47531 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 47532 | From one negated implicati... |
| twonotinotbothi 47533 | From these two negated imp... |
| clifte 47534 | show d is the same as an i... |
| cliftet 47535 | show d is the same as an i... |
| clifteta 47536 | show d is the same as an i... |
| cliftetb 47537 | show d is the same as an i... |
| confun 47538 | Given the hypotheses there... |
| confun2 47539 | Confun simplified to two p... |
| confun3 47540 | Confun's more complex form... |
| confun4 47541 | An attempt at derivative. ... |
| confun5 47542 | An attempt at derivative. ... |
| plcofph 47543 | Given, a,b and a "definiti... |
| pldofph 47544 | Given, a,b c, d, "definiti... |
| plvcofph 47545 | Given, a,b,d, and "definit... |
| plvcofphax 47546 | Given, a,b,d, and "definit... |
| plvofpos 47547 | rh is derivable because ON... |
| mdandyv0 47548 | Given the equivalences set... |
| mdandyv1 47549 | Given the equivalences set... |
| mdandyv2 47550 | Given the equivalences set... |
| mdandyv3 47551 | Given the equivalences set... |
| mdandyv4 47552 | Given the equivalences set... |
| mdandyv5 47553 | Given the equivalences set... |
| mdandyv6 47554 | Given the equivalences set... |
| mdandyv7 47555 | Given the equivalences set... |
| mdandyv8 47556 | Given the equivalences set... |
| mdandyv9 47557 | Given the equivalences set... |
| mdandyv10 47558 | Given the equivalences set... |
| mdandyv11 47559 | Given the equivalences set... |
| mdandyv12 47560 | Given the equivalences set... |
| mdandyv13 47561 | Given the equivalences set... |
| mdandyv14 47562 | Given the equivalences set... |
| mdandyv15 47563 | Given the equivalences set... |
| mdandyvr0 47564 | Given the equivalences set... |
| mdandyvr1 47565 | Given the equivalences set... |
| mdandyvr2 47566 | Given the equivalences set... |
| mdandyvr3 47567 | Given the equivalences set... |
| mdandyvr4 47568 | Given the equivalences set... |
| mdandyvr5 47569 | Given the equivalences set... |
| mdandyvr6 47570 | Given the equivalences set... |
| mdandyvr7 47571 | Given the equivalences set... |
| mdandyvr8 47572 | Given the equivalences set... |
| mdandyvr9 47573 | Given the equivalences set... |
| mdandyvr10 47574 | Given the equivalences set... |
| mdandyvr11 47575 | Given the equivalences set... |
| mdandyvr12 47576 | Given the equivalences set... |
| mdandyvr13 47577 | Given the equivalences set... |
| mdandyvr14 47578 | Given the equivalences set... |
| mdandyvr15 47579 | Given the equivalences set... |
| mdandyvrx0 47580 | Given the exclusivities se... |
| mdandyvrx1 47581 | Given the exclusivities se... |
| mdandyvrx2 47582 | Given the exclusivities se... |
| mdandyvrx3 47583 | Given the exclusivities se... |
| mdandyvrx4 47584 | Given the exclusivities se... |
| mdandyvrx5 47585 | Given the exclusivities se... |
| mdandyvrx6 47586 | Given the exclusivities se... |
| mdandyvrx7 47587 | Given the exclusivities se... |
| mdandyvrx8 47588 | Given the exclusivities se... |
| mdandyvrx9 47589 | Given the exclusivities se... |
| mdandyvrx10 47590 | Given the exclusivities se... |
| mdandyvrx11 47591 | Given the exclusivities se... |
| mdandyvrx12 47592 | Given the exclusivities se... |
| mdandyvrx13 47593 | Given the exclusivities se... |
| mdandyvrx14 47594 | Given the exclusivities se... |
| mdandyvrx15 47595 | Given the exclusivities se... |
| H15NH16TH15IH16 47596 | Given 15 hypotheses and a ... |
| dandysum2p2e4 47597 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 47598 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 47599 | Replacement of a nested an... |
| adh-minim 47600 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 47601 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 47602 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 47603 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 47604 | Fourth lemma for the deriv... |
| adh-minim-ax1 47605 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 47606 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 47607 | Sixth lemma for the deriva... |
| adh-minim-ax2c 47608 | Derivation of a commuted f... |
| adh-minim-ax2 47609 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 47610 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 47611 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 47612 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 47613 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 47614 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 47615 | Third lemma for the deriva... |
| adh-minimp-sylsimp 47616 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 47617 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 47618 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 47619 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 47620 | Fourth lemma for the deriv... |
| adh-minimp-ax2 47621 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 47622 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 47623 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 47624 | If a class with one elemen... |
| eusnsn 47625 | There is a unique element ... |
| absnsb 47626 | If the class abstraction `... |
| euabsneu 47627 | Another way to express exi... |
| elprneb 47628 | An element of a proper uno... |
| oppr 47629 | Equality for ordered pairs... |
| opprb 47630 | Equality for unordered pai... |
| or2expropbilem1 47631 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 47632 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 47633 | If two classes are strictl... |
| eubrv 47634 | If there is a unique set w... |
| eubrdm 47635 | If there is a unique set w... |
| eldmressn 47636 | Element of the domain of a... |
| iota0def 47637 | Example for a defined iota... |
| iota0ndef 47638 | Example for an undefined i... |
| fveqvfvv 47639 | If a function's value at a... |
| fnresfnco 47640 | Composition of two functio... |
| funcoressn 47641 | A composition restricted t... |
| funressnfv 47642 | A restriction to a singlet... |
| funressndmfvrn 47643 | The value of a function ` ... |
| funressnvmo 47644 | A function restricted to a... |
| funressnmo 47645 | A function restricted to a... |
| funressneu 47646 | There is exactly one value... |
| fresfo 47647 | Conditions for a restricti... |
| fsetsniunop 47648 | The class of all functions... |
| fsetabsnop 47649 | The class of all functions... |
| fsetsnf 47650 | The mapping of an element ... |
| fsetsnf1 47651 | The mapping of an element ... |
| fsetsnfo 47652 | The mapping of an element ... |
| fsetsnf1o 47653 | The mapping of an element ... |
| fsetsnprcnex 47654 | The class of all functions... |
| cfsetssfset 47655 | The class of constant func... |
| cfsetsnfsetfv 47656 | The function value of the ... |
| cfsetsnfsetf 47657 | The mapping of the class o... |
| cfsetsnfsetf1 47658 | The mapping of the class o... |
| cfsetsnfsetfo 47659 | The mapping of the class o... |
| cfsetsnfsetf1o 47660 | The mapping of the class o... |
| fsetprcnexALT 47661 | First version of proof for... |
| fcoreslem1 47662 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 47663 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 47664 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 47665 | Lemma 4 for ~ fcores . (C... |
| fcores 47666 | Every composite function `... |
| fcoresf1lem 47667 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 47668 | If a composition is inject... |
| fcoresf1b 47669 | A composition is injective... |
| fcoresfo 47670 | If a composition is surjec... |
| fcoresfob 47671 | A composition is surjectiv... |
| fcoresf1ob 47672 | A composition is bijective... |
| f1cof1blem 47673 | Lemma for ~ f1cof1b and ~ ... |
| 3f1oss1 47674 | The composition of three b... |
| 3f1oss2 47675 | The composition of three b... |
| f1cof1b 47676 | If the range of ` F ` equa... |
| funfocofob 47677 | If the domain of a functio... |
| fnfocofob 47678 | If the domain of a functio... |
| focofob 47679 | If the domain of a functio... |
| f1ocof1ob 47680 | If the range of ` F ` equa... |
| f1ocof1ob2 47681 | If the range of ` F ` equa... |
| aiotajust 47683 | Soundness justification th... |
| dfaiota2 47685 | Alternate definition of th... |
| reuabaiotaiota 47686 | The iota and the alternate... |
| reuaiotaiota 47687 | The iota and the alternate... |
| aiotaexb 47688 | The alternate iota over a ... |
| aiotavb 47689 | The alternate iota over a ... |
| aiotaint 47690 | This is to ~ df-aiota what... |
| dfaiota3 47691 | Alternate definition of ` ... |
| iotan0aiotaex 47692 | If the iota over a wff ` p... |
| aiotaexaiotaiota 47693 | The alternate iota over a ... |
| aiotaval 47694 | Theorem 8.19 in [Quine] p.... |
| aiota0def 47695 | Example for a defined alte... |
| aiota0ndef 47696 | Example for an undefined a... |
| r19.32 47697 | Theorem 19.32 of [Margaris... |
| rexsb 47698 | An equivalent expression f... |
| rexrsb 47699 | An equivalent expression f... |
| 2rexsb 47700 | An equivalent expression f... |
| 2rexrsb 47701 | An equivalent expression f... |
| cbvral2 47702 | Change bound variables of ... |
| cbvrex2 47703 | Change bound variables of ... |
| ralndv1 47704 | Example for a theorem abou... |
| ralndv2 47705 | Second example for a theor... |
| reuf1odnf 47706 | There is exactly one eleme... |
| reuf1od 47707 | There is exactly one eleme... |
| euoreqb 47708 | There is a set which is eq... |
| 2reu3 47709 | Double restricted existent... |
| 2reu7 47710 | Two equivalent expressions... |
| 2reu8 47711 | Two equivalent expressions... |
| 2reu8i 47712 | Implication of a double re... |
| 2reuimp0 47713 | Implication of a double re... |
| 2reuimp 47714 | Implication of a double re... |
| ralbinrald 47721 | Elemination of a restricte... |
| nvelim 47722 | If a class is the universa... |
| alneu 47723 | If a statement holds for a... |
| eu2ndop1stv 47724 | If there is a unique secon... |
| dfateq12d 47725 | Equality deduction for "de... |
| nfdfat 47726 | Bound-variable hypothesis ... |
| dfdfat2 47727 | Alternate definition of th... |
| fundmdfat 47728 | A function is defined at a... |
| dfatprc 47729 | A function is not defined ... |
| dfatelrn 47730 | The value of a function ` ... |
| dfafv2 47731 | Alternative definition of ... |
| afveq12d 47732 | Equality deduction for fun... |
| afveq1 47733 | Equality theorem for funct... |
| afveq2 47734 | Equality theorem for funct... |
| nfafv 47735 | Bound-variable hypothesis ... |
| csbafv12g 47736 | Move class substitution in... |
| afvfundmfveq 47737 | If a class is a function r... |
| afvnfundmuv 47738 | If a set is not in the dom... |
| ndmafv 47739 | The value of a class outsi... |
| afvvdm 47740 | If the function value of a... |
| nfunsnafv 47741 | If the restriction of a cl... |
| afvvfunressn 47742 | If the function value of a... |
| afvprc 47743 | A function's value at a pr... |
| afvvv 47744 | If a function's value at a... |
| afvpcfv0 47745 | If the value of the altern... |
| afvnufveq 47746 | The value of the alternati... |
| afvvfveq 47747 | The value of the alternati... |
| afv0fv0 47748 | If the value of the altern... |
| afvfvn0fveq 47749 | If the function's value at... |
| afv0nbfvbi 47750 | The function's value at an... |
| afvfv0bi 47751 | The function's value at an... |
| afveu 47752 | The value of a function at... |
| fnbrafvb 47753 | Equivalence of function va... |
| fnopafvb 47754 | Equivalence of function va... |
| funbrafvb 47755 | Equivalence of function va... |
| funopafvb 47756 | Equivalence of function va... |
| funbrafv 47757 | The second argument of a b... |
| funbrafv2b 47758 | Function value in terms of... |
| dfafn5a 47759 | Representation of a functi... |
| dfafn5b 47760 | Representation of a functi... |
| fnrnafv 47761 | The range of a function ex... |
| afvelrnb 47762 | A member of a function's r... |
| afvelrnb0 47763 | A member of a function's r... |
| dfaimafn 47764 | Alternate definition of th... |
| dfaimafn2 47765 | Alternate definition of th... |
| afvelima 47766 | Function value in an image... |
| afvelrn 47767 | A function's value belongs... |
| fnafvelrn 47768 | A function's value belongs... |
| fafvelcdm 47769 | A function's value belongs... |
| ffnafv 47770 | A function maps to a class... |
| afvres 47771 | The value of a restricted ... |
| tz6.12-afv 47772 | Function value. Theorem 6... |
| tz6.12-1-afv 47773 | Function value (Theorem 6.... |
| dmfcoafv 47774 | Domains of a function comp... |
| afvco2 47775 | Value of a function compos... |
| rlimdmafv 47776 | Two ways to express that a... |
| aoveq123d 47777 | Equality deduction for ope... |
| nfaov 47778 | Bound-variable hypothesis ... |
| csbaovg 47779 | Move class substitution in... |
| aovfundmoveq 47780 | If a class is a function r... |
| aovnfundmuv 47781 | If an ordered pair is not ... |
| ndmaov 47782 | The value of an operation ... |
| ndmaovg 47783 | The value of an operation ... |
| aovvdm 47784 | If the operation value of ... |
| nfunsnaov 47785 | If the restriction of a cl... |
| aovvfunressn 47786 | If the operation value of ... |
| aovprc 47787 | The value of an operation ... |
| aovrcl 47788 | Reverse closure for an ope... |
| aovpcov0 47789 | If the alternative value o... |
| aovnuoveq 47790 | The alternative value of t... |
| aovvoveq 47791 | The alternative value of t... |
| aov0ov0 47792 | If the alternative value o... |
| aovovn0oveq 47793 | If the operation's value a... |
| aov0nbovbi 47794 | The operation's value on a... |
| aovov0bi 47795 | The operation's value on a... |
| rspceaov 47796 | A frequently used special ... |
| fnotaovb 47797 | Equivalence of operation v... |
| ffnaov 47798 | An operation maps to a cla... |
| faovcl 47799 | Closure law for an operati... |
| aovmpt4g 47800 | Value of a function given ... |
| aoprssdm 47801 | Domain of closure of an op... |
| ndmaovcl 47802 | The "closure" of an operat... |
| ndmaovrcl 47803 | Reverse closure law, in co... |
| ndmaovcom 47804 | Any operation is commutati... |
| ndmaovass 47805 | Any operation is associati... |
| ndmaovdistr 47806 | Any operation is distribut... |
| dfatafv2iota 47809 | If a function is defined a... |
| ndfatafv2 47810 | The alternate function val... |
| ndfatafv2undef 47811 | The alternate function val... |
| dfatafv2ex 47812 | The alternate function val... |
| afv2ex 47813 | The alternate function val... |
| afv2eq12d 47814 | Equality deduction for fun... |
| afv2eq1 47815 | Equality theorem for funct... |
| afv2eq2 47816 | Equality theorem for funct... |
| nfafv2 47817 | Bound-variable hypothesis ... |
| csbafv212g 47818 | Move class substitution in... |
| fexafv2ex 47819 | The alternate function val... |
| ndfatafv2nrn 47820 | The alternate function val... |
| ndmafv2nrn 47821 | The value of a class outsi... |
| funressndmafv2rn 47822 | The alternate function val... |
| afv2ndefb 47823 | Two ways to say that an al... |
| nfunsnafv2 47824 | If the restriction of a cl... |
| afv2prc 47825 | A function's value at a pr... |
| dfatafv2rnb 47826 | The alternate function val... |
| afv2orxorb 47827 | If a set is in the range o... |
| dmafv2rnb 47828 | The alternate function val... |
| fundmafv2rnb 47829 | The alternate function val... |
| afv2elrn 47830 | An alternate function valu... |
| afv20defat 47831 | If the alternate function ... |
| fnafv2elrn 47832 | An alternate function valu... |
| fafv2elcdm 47833 | An alternate function valu... |
| fafv2elrnb 47834 | An alternate function valu... |
| fcdmvafv2v 47835 | If the codomain of a funct... |
| tz6.12-2-afv2 47836 | Function value when ` F ` ... |
| afv2eu 47837 | The value of a function at... |
| afv2res 47838 | The value of a restricted ... |
| tz6.12-afv2 47839 | Function value (Theorem 6.... |
| tz6.12-1-afv2 47840 | Function value (Theorem 6.... |
| tz6.12c-afv2 47841 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 47842 | Corollary of Theorem 6.12(... |
| funressnbrafv2 47843 | The second argument of a b... |
| dfatbrafv2b 47844 | Equivalence of function va... |
| dfatopafv2b 47845 | Equivalence of function va... |
| funbrafv2 47846 | The second argument of a b... |
| fnbrafv2b 47847 | Equivalence of function va... |
| fnopafv2b 47848 | Equivalence of function va... |
| funbrafv22b 47849 | Equivalence of function va... |
| funopafv2b 47850 | Equivalence of function va... |
| dfatsnafv2 47851 | Singleton of function valu... |
| dfafv23 47852 | A definition of function v... |
| dfatdmfcoafv2 47853 | Domain of a function compo... |
| dfatcolem 47854 | Lemma for ~ dfatco . (Con... |
| dfatco 47855 | The predicate "defined at"... |
| afv2co2 47856 | Value of a function compos... |
| rlimdmafv2 47857 | Two ways to express that a... |
| dfafv22 47858 | Alternate definition of ` ... |
| afv2ndeffv0 47859 | If the alternate function ... |
| dfatafv2eqfv 47860 | If a function is defined a... |
| afv2rnfveq 47861 | If the alternate function ... |
| afv20fv0 47862 | If the alternate function ... |
| afv2fvn0fveq 47863 | If the function's value at... |
| afv2fv0 47864 | If the function's value at... |
| afv2fv0b 47865 | The function's value at an... |
| afv2fv0xorb 47866 | If a set is in the range o... |
| an4com24 47867 | Rearrangement of 4 conjunc... |
| 3an4ancom24 47868 | Commutative law for a conj... |
| 4an21 47869 | Rearrangement of 4 conjunc... |
| dfnelbr2 47872 | Alternate definition of th... |
| nelbr 47873 | The binary relation of a s... |
| nelbrim 47874 | If a set is related to ano... |
| nelbrnel 47875 | A set is related to anothe... |
| nelbrnelim 47876 | If a set is related to ano... |
| ralralimp 47877 | Selecting one of two alter... |
| otiunsndisjX 47878 | The union of singletons co... |
| fvifeq 47879 | Equality of function value... |
| rnfdmpr 47880 | The range of a one-to-one ... |
| imarnf1pr 47881 | The image of the range of ... |
| funop1 47882 | A function is an ordered p... |
| fun2dmnopgexmpl 47883 | A function with a domain c... |
| opabresex0d 47884 | A collection of ordered pa... |
| opabbrfex0d 47885 | A collection of ordered pa... |
| opabresexd 47886 | A collection of ordered pa... |
| opabbrfexd 47887 | A collection of ordered pa... |
| f1oresf1orab 47888 | Build a bijection by restr... |
| f1oresf1o 47889 | Build a bijection by restr... |
| f1oresf1o2 47890 | Build a bijection by restr... |
| fvmptrab 47891 | Value of a function mappin... |
| fvmptrabdm 47892 | Value of a function mappin... |
| cnambpcma 47893 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 47894 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 47895 | The sum of two complex num... |
| leaddsuble 47896 | Addition and subtraction o... |
| 2leaddle2 47897 | If two real numbers are le... |
| ltnltne 47898 | Variant of trichotomy law ... |
| p1lep2 47899 | A real number increasd by ... |
| ltsubsubaddltsub 47900 | If the result of subtracti... |
| zm1nn 47901 | An integer minus 1 is posi... |
| readdcnnred 47902 | The sum of a real number a... |
| resubcnnred 47903 | The difference of a real n... |
| recnmulnred 47904 | The product of a real numb... |
| cndivrenred 47905 | The quotient of an imagina... |
| sqrtnegnre 47906 | The square root of a negat... |
| nn0resubcl 47907 | Closure law for subtractio... |
| zgeltp1eq 47908 | If an integer is between a... |
| 1t10e1p1e11 47909 | 11 is 1 times 10 to the po... |
| deccarry 47910 | Add 1 to a 2 digit number ... |
| eluzge0nn0 47911 | If an integer is greater t... |
| nltle2tri 47912 | Negated extended trichotom... |
| ssfz12 47913 | Subset relationship for fi... |
| elfz2z 47914 | Membership of an integer i... |
| 2elfz3nn0 47915 | If there are two elements ... |
| fz0addcom 47916 | The addition of two member... |
| 2elfz2melfz 47917 | If the sum of two integers... |
| fz0addge0 47918 | The sum of two integers in... |
| elfzlble 47919 | Membership of an integer i... |
| elfzelfzlble 47920 | Membership of an element o... |
| elfz2nn 47921 | A member of a finite set o... |
| fzopred 47922 | Join a predecessor to the ... |
| fzopredsuc 47923 | Join a predecessor and a s... |
| 1fzopredsuc 47924 | Join 0 and a successor to ... |
| el1fzopredsuc 47925 | An element of an open inte... |
| subsubelfzo0 47926 | Subtracting a difference f... |
| 2ffzoeq 47927 | Two functions over a half-... |
| elfzo2nn 47928 | A member of a half-open ra... |
| nnmul2 47929 | If one factor of a product... |
| nnmul2b 47930 | A factor of a product of i... |
| 2ltceilhalf 47931 | The ceiling of half of an ... |
| ceilhalfgt1 47932 | The ceiling of half of an ... |
| ceilhalfelfzo1 47933 | A positive integer less th... |
| gpgedgvtx1lem 47934 | Lemma for ~ gpgedgvtx1 . ... |
| 2tceilhalfelfzo1 47935 | Two times a positive integ... |
| ceilbi 47936 | A condition equivalent to ... |
| ceilhalf1 47937 | The ceiling of one half is... |
| rehalfge1 47938 | Half of a real number grea... |
| ceilhalfnn 47939 | The ceiling of half of a p... |
| 1elfzo1ceilhalf1 47940 | 1 is in the half-open inte... |
| nnge2recfl0 47941 | The floor of the reciproca... |
| flmrecm1 47942 | The floor of an integer mi... |
| fldivmod 47943 | Expressing the floor of a ... |
| ceildivmod 47944 | Expressing the ceiling of ... |
| ceil5half3 47945 | The ceiling of half of 5 i... |
| submodaddmod 47946 | Subtraction and addition m... |
| difltmodne 47947 | Two nonnegative integers a... |
| zplusmodne 47948 | A nonnegative integer is n... |
| addmodne 47949 | The sum of a nonnegative i... |
| plusmod5ne 47950 | A nonnegative integer is n... |
| zp1modne 47951 | An integer is not itself p... |
| p1modne 47952 | A nonnegative integer is n... |
| m1modne 47953 | A nonnegative integer is n... |
| minusmod5ne 47954 | A nonnegative integer is n... |
| submodlt 47955 | The difference of an eleme... |
| submodneaddmod 47956 | An integer minus ` B ` is ... |
| m1modnep2mod 47957 | A nonnegative integer minu... |
| minusmodnep2tmod 47958 | A nonnegative integer minu... |
| m1mod0mod1 47959 | An integer decreased by 1 ... |
| elmod2 47960 | An integer modulo 2 is eit... |
| mod0mul 47961 | If an integer is 0 modulo ... |
| modn0mul 47962 | If an integer is not 0 mod... |
| m1modmmod 47963 | An integer decreased by 1 ... |
| difmodm1lt 47964 | The difference between an ... |
| 8mod5e3 47965 | 8 modulo 5 is 3. (Contrib... |
| modmkpkne 47966 | If an integer minus a cons... |
| modmknepk 47967 | A nonnegative integer less... |
| modlt0b 47968 | An integer with an absolut... |
| mod2addne 47969 | The sums of a nonnegative ... |
| modm1nep1 47970 | A nonnegative integer less... |
| modm2nep1 47971 | A nonnegative integer less... |
| modp2nep1 47972 | A nonnegative integer less... |
| modm1nep2 47973 | A nonnegative integer less... |
| modm1nem2 47974 | A nonnegative integer less... |
| modm1p1ne 47975 | If an integer minus one eq... |
| smonoord 47976 | Ordering relation for a st... |
| 2timesltsq 47977 | Two times an integer great... |
| 2timesltsqm1 47978 | Two times an integer great... |
| fsummsndifre 47979 | A finite sum with one of i... |
| fsumsplitsndif 47980 | Separate out a term in a f... |
| fsummmodsndifre 47981 | A finite sum of summands m... |
| fsummmodsnunz 47982 | A finite sum of summands m... |
| nndivides2 47983 | Definition of the divides ... |
| facnn0dvdsfac 47984 | The factorial of a nonnega... |
| muldvdsfacgt 47985 | The product of two differe... |
| muldvdsfacm1 47986 | The product of two differe... |
| setsidel 47987 | The injected slot is an el... |
| setsnidel 47988 | The injected slot is an el... |
| setsv 47989 | The value of the structure... |
| preimafvsnel 47990 | The preimage of a function... |
| preimafvn0 47991 | The preimage of a function... |
| uniimafveqt 47992 | The union of the image of ... |
| uniimaprimaeqfv 47993 | The union of the image of ... |
| setpreimafvex 47994 | The class ` P ` of all pre... |
| elsetpreimafvb 47995 | The characterization of an... |
| elsetpreimafv 47996 | An element of the class ` ... |
| elsetpreimafvssdm 47997 | An element of the class ` ... |
| fvelsetpreimafv 47998 | There is an element in a p... |
| preimafvelsetpreimafv 47999 | The preimage of a function... |
| preimafvsspwdm 48000 | The class ` P ` of all pre... |
| 0nelsetpreimafv 48001 | The empty set is not an el... |
| elsetpreimafvbi 48002 | An element of the preimage... |
| elsetpreimafveqfv 48003 | The elements of the preima... |
| eqfvelsetpreimafv 48004 | If an element of the domai... |
| elsetpreimafvrab 48005 | An element of the preimage... |
| imaelsetpreimafv 48006 | The image of an element of... |
| uniimaelsetpreimafv 48007 | The union of the image of ... |
| elsetpreimafveq 48008 | If two preimages of functi... |
| fundcmpsurinjlem1 48009 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 48010 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 48011 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 48012 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 48013 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 48014 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 48015 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 48016 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 48017 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 48018 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 48019 | Every function ` F : A -->... |
| fundcmpsurinj 48020 | Every function ` F : A -->... |
| fundcmpsurbijinj 48021 | Every function ` F : A -->... |
| fundcmpsurinjimaid 48022 | Every function ` F : A -->... |
| fundcmpsurinjALT 48023 | Alternate proof of ~ fundc... |
| iccpval 48026 | Partition consisting of a ... |
| iccpart 48027 | A special partition. Corr... |
| iccpartimp 48028 | Implications for a class b... |
| iccpartres 48029 | The restriction of a parti... |
| iccpartxr 48030 | If there is a partition, t... |
| iccpartgtprec 48031 | If there is a partition, t... |
| iccpartipre 48032 | If there is a partition, t... |
| iccpartiltu 48033 | If there is a partition, t... |
| iccpartigtl 48034 | If there is a partition, t... |
| iccpartlt 48035 | If there is a partition, t... |
| iccpartltu 48036 | If there is a partition, t... |
| iccpartgtl 48037 | If there is a partition, t... |
| iccpartgt 48038 | If there is a partition, t... |
| iccpartleu 48039 | If there is a partition, t... |
| iccpartgel 48040 | If there is a partition, t... |
| iccpartrn 48041 | If there is a partition, t... |
| iccpartf 48042 | The range of the partition... |
| iccpartel 48043 | If there is a partition, t... |
| iccelpart 48044 | An element of any partitio... |
| iccpartiun 48045 | A half-open interval of ex... |
| icceuelpartlem 48046 | Lemma for ~ icceuelpart . ... |
| icceuelpart 48047 | An element of a partitione... |
| iccpartdisj 48048 | The segments of a partitio... |
| iccpartnel 48049 | A point of a partition is ... |
| fargshiftfv 48050 | If a class is a function, ... |
| fargshiftf 48051 | If a class is a function, ... |
| fargshiftf1 48052 | If a function is 1-1, then... |
| fargshiftfo 48053 | If a function is onto, the... |
| fargshiftfva 48054 | The values of a shifted fu... |
| lswn0 48055 | The last symbol of a nonem... |
| nfich1 48058 | The first interchangeable ... |
| nfich2 48059 | The second interchangeable... |
| ichv 48060 | Setvar variables are inter... |
| ichf 48061 | Setvar variables are inter... |
| ichid 48062 | A setvar variable is alway... |
| icht 48063 | A theorem is interchangeab... |
| ichbidv 48064 | Formula building rule for ... |
| ichcircshi 48065 | The setvar variables are i... |
| ichan 48066 | If two setvar variables ar... |
| ichn 48067 | Negation does not affect i... |
| ichim 48068 | Formula building rule for ... |
| dfich2 48069 | Alternate definition of th... |
| ichcom 48070 | The interchangeability of ... |
| ichbi12i 48071 | Equivalence for interchang... |
| icheqid 48072 | In an equality for the sam... |
| icheq 48073 | In an equality of setvar v... |
| ichnfimlem 48074 | Lemma for ~ ichnfim : A s... |
| ichnfim 48075 | If in an interchangeabilit... |
| ichnfb 48076 | If ` x ` and ` y ` are int... |
| ichal 48077 | Move a universal quantifie... |
| ich2al 48078 | Two setvar variables are a... |
| ich2ex 48079 | Two setvar variables are a... |
| ichexmpl1 48080 | Example for interchangeabl... |
| ichexmpl2 48081 | Example for interchangeabl... |
| ich2exprop 48082 | If the setvar variables ar... |
| ichnreuop 48083 | If the setvar variables ar... |
| ichreuopeq 48084 | If the setvar variables ar... |
| sprid 48085 | Two identical representati... |
| elsprel 48086 | An unordered pair is an el... |
| spr0nelg 48087 | The empty set is not an el... |
| sprval 48090 | The set of all unordered p... |
| sprvalpw 48091 | The set of all unordered p... |
| sprssspr 48092 | The set of all unordered p... |
| spr0el 48093 | The empty set is not an un... |
| sprvalpwn0 48094 | The set of all unordered p... |
| sprel 48095 | An element of the set of a... |
| prssspr 48096 | An element of a subset of ... |
| prelspr 48097 | An unordered pair of eleme... |
| prsprel 48098 | The elements of a pair fro... |
| prsssprel 48099 | The elements of a pair fro... |
| sprvalpwle2 48100 | The set of all unordered p... |
| sprsymrelfvlem 48101 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 48102 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 48103 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 48104 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 48105 | The value of the function ... |
| sprsymrelf 48106 | The mapping ` F ` is a fun... |
| sprsymrelf1 48107 | The mapping ` F ` is a one... |
| sprsymrelfo 48108 | The mapping ` F ` is a fun... |
| sprsymrelf1o 48109 | The mapping ` F ` is a bij... |
| sprbisymrel 48110 | There is a bijection betwe... |
| sprsymrelen 48111 | The class ` P ` of subsets... |
| prpair 48112 | Characterization of a prop... |
| prproropf1olem0 48113 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 48114 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 48115 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 48116 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 48117 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 48118 | There is a bijection betwe... |
| prproropen 48119 | The set of proper pairs an... |
| prproropreud 48120 | There is exactly one order... |
| pairreueq 48121 | Two equivalent representat... |
| paireqne 48122 | Two sets are not equal iff... |
| prprval 48125 | The set of all proper unor... |
| prprvalpw 48126 | The set of all proper unor... |
| prprelb 48127 | An element of the set of a... |
| prprelprb 48128 | A set is an element of the... |
| prprspr2 48129 | The set of all proper unor... |
| prprsprreu 48130 | There is a unique proper u... |
| prprreueq 48131 | There is a unique proper u... |
| sbcpr 48132 | The proper substitution of... |
| reupr 48133 | There is a unique unordere... |
| reuprpr 48134 | There is a unique proper u... |
| poprelb 48135 | Equality for unordered pai... |
| 2exopprim 48136 | The existence of an ordere... |
| reuopreuprim 48137 | There is a unique unordere... |
| nprmmul1 48138 | Special factorization of a... |
| nprmmul2 48139 | Special factorization of a... |
| nprmmul3 48140 | Special factorization of a... |
| fmtno 48143 | The ` N ` th Fermat number... |
| fmtnoge3 48144 | Each Fermat number is grea... |
| fmtnonn 48145 | Each Fermat number is a po... |
| fmtnom1nn 48146 | A Fermat number minus one ... |
| fmtnoodd 48147 | Each Fermat number is odd.... |
| fmtnorn 48148 | A Fermat number is a funct... |
| fmtnof1 48149 | The enumeration of the Fer... |
| fmtnoinf 48150 | The set of Fermat numbers ... |
| fmtnorec1 48151 | The first recurrence relat... |
| sqrtpwpw2p 48152 | The floor of the square ro... |
| fmtnosqrt 48153 | The floor of the square ro... |
| fmtno0 48154 | The ` 0 ` th Fermat number... |
| fmtno1 48155 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 48156 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 48157 | The second recurrence rela... |
| fmtnodvds 48158 | Any Fermat number divides ... |
| goldbachthlem1 48159 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 48160 | Lemma 2 for ~ goldbachth .... |
| goldbachth 48161 | Goldbach's theorem: Two d... |
| fmtnorec3 48162 | The third recurrence relat... |
| fmtnorec4 48163 | The fourth recurrence rela... |
| fmtno2 48164 | The ` 2 ` nd Fermat number... |
| fmtno3 48165 | The ` 3 ` rd Fermat number... |
| fmtno4 48166 | The ` 4 ` th Fermat number... |
| fmtno5lem1 48167 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 48168 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 48169 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 48170 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 48171 | The ` 5 ` th Fermat number... |
| fmtno0prm 48172 | The ` 0 ` th Fermat number... |
| fmtno1prm 48173 | The ` 1 ` st Fermat number... |
| fmtno2prm 48174 | The ` 2 ` nd Fermat number... |
| 257prm 48175 | 257 is a prime number (the... |
| fmtno3prm 48176 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 48177 | Any power of two is coprim... |
| fmtnoprmfac1lem 48178 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 48179 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 48180 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 48181 | Divisor of Fermat number (... |
| fmtnofac2lem 48182 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 48183 | Divisor of Fermat number (... |
| fmtnofac1 48184 | Divisor of Fermat number (... |
| fmtno4sqrt 48185 | The floor of the square ro... |
| fmtno4prmfac 48186 | If P was a (prime) factor ... |
| fmtno4prmfac193 48187 | If P was a (prime) factor ... |
| fmtno4nprmfac193 48188 | 193 is not a (prime) facto... |
| fmtno4prm 48189 | The ` 4 `-th Fermat number... |
| 65537prm 48190 | 65537 is a prime number (t... |
| fmtnofz04prm 48191 | The first five Fermat numb... |
| fmtnole4prm 48192 | The first five Fermat numb... |
| fmtno5faclem1 48193 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 48194 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 48195 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 48196 | The factorization of the `... |
| fmtno5nprm 48197 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 48198 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 48199 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 48200 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 48201 | The mapping of a Fermat nu... |
| prminf2 48202 | The set of prime numbers i... |
| 2pwp1prm 48203 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 48204 | Every prime number of the ... |
| m2prm 48205 | The second Mersenne number... |
| m3prm 48206 | The third Mersenne number ... |
| flsqrt 48207 | A condition equivalent to ... |
| flsqrt5 48208 | The floor of the square ro... |
| 3ndvds4 48209 | 3 does not divide 4. (Con... |
| 139prmALT 48210 | 139 is a prime number. In... |
| 31prm 48211 | 31 is a prime number. In ... |
| m5prm 48212 | The fifth Mersenne number ... |
| 127prm 48213 | 127 is a prime number. (C... |
| m7prm 48214 | The seventh Mersenne numbe... |
| m11nprm 48215 | The eleventh Mersenne numb... |
| mod42tp1mod8 48216 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 48217 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 48218 | If ` P ` is a Sophie Germa... |
| lighneallem1 48219 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 48220 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 48221 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 48222 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 48223 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 48224 | Lemma 3 for ~ lighneal . ... |
| lighneal 48225 | If a power of a prime ` P ... |
| modexp2m1d 48226 | The square of an integer w... |
| proththdlem 48227 | Lemma for ~ proththd . (C... |
| proththd 48228 | Proth's theorem (1878). I... |
| 5tcu2e40 48229 | 5 times the cube of 2 is 4... |
| 3exp4mod41 48230 | 3 to the fourth power is -... |
| 41prothprmlem1 48231 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 48232 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 48233 | 41 is a _Proth prime_. (C... |
| nprmdvdsfacm1lem1 48234 | Lemma 1 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem2 48235 | Lemma 2 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem3 48236 | Lemma 3 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem4 48237 | Lemma 4 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1 48238 | A non-prime integer greate... |
| ppivalnnprm 48239 | Value of a term of the pri... |
| ppivalnnnprmge6 48240 | Value of a term of the pri... |
| ppivalnn4 48241 | Value of the term of the p... |
| ppivalnnnprm 48242 | Value of a term of the pri... |
| indprm 48243 | An indicator function for ... |
| indprmfz 48244 | An indicator function for ... |
| ppi1sum 48245 | Value of the prime-countin... |
| ppivalnn 48246 | Value of the prime-countin... |
| quad1 48247 | A condition for a quadrati... |
| requad01 48248 | A condition for a quadrati... |
| requad1 48249 | A condition for a quadrati... |
| requad2 48250 | A condition for a quadrati... |
| iseven 48255 | The predicate "is an even ... |
| isodd 48256 | The predicate "is an odd n... |
| evenz 48257 | An even number is an integ... |
| oddz 48258 | An odd number is an intege... |
| evendiv2z 48259 | The result of dividing an ... |
| oddp1div2z 48260 | The result of dividing an ... |
| oddm1div2z 48261 | The result of dividing an ... |
| isodd2 48262 | The predicate "is an odd n... |
| dfodd2 48263 | Alternate definition for o... |
| dfodd6 48264 | Alternate definition for o... |
| dfeven4 48265 | Alternate definition for e... |
| evenm1odd 48266 | The predecessor of an even... |
| evenp1odd 48267 | The successor of an even n... |
| oddp1eveni 48268 | The successor of an odd nu... |
| oddm1eveni 48269 | The predecessor of an odd ... |
| evennodd 48270 | An even number is not an o... |
| oddneven 48271 | An odd number is not an ev... |
| enege 48272 | The negative of an even nu... |
| onego 48273 | The negative of an odd num... |
| m1expevenALTV 48274 | Exponentiation of -1 by an... |
| m1expoddALTV 48275 | Exponentiation of -1 by an... |
| dfeven2 48276 | Alternate definition for e... |
| dfodd3 48277 | Alternate definition for o... |
| iseven2 48278 | The predicate "is an even ... |
| isodd3 48279 | The predicate "is an odd n... |
| 2dvdseven 48280 | 2 divides an even number. ... |
| m2even 48281 | A multiple of 2 is an even... |
| 2ndvdsodd 48282 | 2 does not divide an odd n... |
| 2dvdsoddp1 48283 | 2 divides an odd number in... |
| 2dvdsoddm1 48284 | 2 divides an odd number de... |
| dfeven3 48285 | Alternate definition for e... |
| dfodd4 48286 | Alternate definition for o... |
| dfodd5 48287 | Alternate definition for o... |
| zefldiv2ALTV 48288 | The floor of an even numbe... |
| zofldiv2ALTV 48289 | The floor of an odd number... |
| oddflALTV 48290 | Odd number representation ... |
| iseven5 48291 | The predicate "is an even ... |
| isodd7 48292 | The predicate "is an odd n... |
| dfeven5 48293 | Alternate definition for e... |
| dfodd7 48294 | Alternate definition for o... |
| gcd2odd1 48295 | The greatest common diviso... |
| zneoALTV 48296 | No even integer equals an ... |
| zeoALTV 48297 | An integer is even or odd.... |
| zeo2ALTV 48298 | An integer is even or odd ... |
| nneoALTV 48299 | A positive integer is even... |
| nneoiALTV 48300 | A positive integer is even... |
| odd2np1ALTV 48301 | An integer is odd iff it i... |
| oddm1evenALTV 48302 | An integer is odd iff its ... |
| oddp1evenALTV 48303 | An integer is odd iff its ... |
| oexpnegALTV 48304 | The exponential of the neg... |
| oexpnegnz 48305 | The exponential of the neg... |
| bits0ALTV 48306 | Value of the zeroth bit. ... |
| bits0eALTV 48307 | The zeroth bit of an even ... |
| bits0oALTV 48308 | The zeroth bit of an odd n... |
| divgcdoddALTV 48309 | Either ` A / ( A gcd B ) `... |
| opoeALTV 48310 | The sum of two odds is eve... |
| opeoALTV 48311 | The sum of an odd and an e... |
| omoeALTV 48312 | The difference of two odds... |
| omeoALTV 48313 | The difference of an odd a... |
| oddprmALTV 48314 | A prime not equal to ` 2 `... |
| 0evenALTV 48315 | 0 is an even number. (Con... |
| 0noddALTV 48316 | 0 is not an odd number. (... |
| 1oddALTV 48317 | 1 is an odd number. (Cont... |
| 1nevenALTV 48318 | 1 is not an even number. ... |
| 2evenALTV 48319 | 2 is an even number. (Con... |
| 2noddALTV 48320 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 48321 | An odd nonnegative integer... |
| nnoALTV 48322 | An alternate characterizat... |
| nn0oALTV 48323 | An alternate characterizat... |
| nn0e 48324 | An alternate characterizat... |
| nneven 48325 | An alternate characterizat... |
| nn0onn0exALTV 48326 | For each odd nonnegative i... |
| nn0enn0exALTV 48327 | For each even nonnegative ... |
| nnennexALTV 48328 | For each even positive int... |
| nnpw2evenALTV 48329 | 2 to the power of a positi... |
| epoo 48330 | The sum of an even and an ... |
| emoo 48331 | The difference of an even ... |
| epee 48332 | The sum of two even number... |
| emee 48333 | The difference of two even... |
| evensumeven 48334 | If a summand is even, the ... |
| 3odd 48335 | 3 is an odd number. (Cont... |
| 4even 48336 | 4 is an even number. (Con... |
| 5odd 48337 | 5 is an odd number. (Cont... |
| 6even 48338 | 6 is an even number. (Con... |
| 7odd 48339 | 7 is an odd number. (Cont... |
| 8even 48340 | 8 is an even number. (Con... |
| evenprm2 48341 | A prime number is even iff... |
| oddprmne2 48342 | Every prime number not bei... |
| oddprmuzge3 48343 | A prime number which is od... |
| evenltle 48344 | If an even number is great... |
| odd2prm2 48345 | If an odd number is the su... |
| even3prm2 48346 | If an even number is the s... |
| mogoldbblem 48347 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 48348 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 48349 | Lemma for ~ perfectALTV . ... |
| perfectALTV 48350 | The Euclid-Euler theorem, ... |
| fppr 48353 | The set of Fermat pseudopr... |
| fpprmod 48354 | The set of Fermat pseudopr... |
| fpprel 48355 | A Fermat pseudoprime to th... |
| fpprbasnn 48356 | The base of a Fermat pseud... |
| fpprnn 48357 | A Fermat pseudoprime to th... |
| fppr2odd 48358 | A Fermat pseudoprime to th... |
| 11t31e341 48359 | 341 is the product of 11 a... |
| 2exp340mod341 48360 | Eight to the eighth power ... |
| 341fppr2 48361 | 341 is the (smallest) _Pou... |
| 4fppr1 48362 | 4 is the (smallest) Fermat... |
| 8exp8mod9 48363 | Eight to the eighth power ... |
| 9fppr8 48364 | 9 is the (smallest) Fermat... |
| dfwppr 48365 | Alternate definition of a ... |
| fpprwppr 48366 | A Fermat pseudoprime to th... |
| fpprwpprb 48367 | An integer ` X ` which is ... |
| fpprel2 48368 | An alternate definition fo... |
| nfermltl8rev 48369 | Fermat's little theorem wi... |
| nfermltl2rev 48370 | Fermat's little theorem wi... |
| nfermltlrev 48371 | Fermat's little theorem re... |
| isgbe 48378 | The predicate "is an even ... |
| isgbow 48379 | The predicate "is a weak o... |
| isgbo 48380 | The predicate "is an odd G... |
| gbeeven 48381 | An even Goldbach number is... |
| gbowodd 48382 | A weak odd Goldbach number... |
| gbogbow 48383 | A (strong) odd Goldbach nu... |
| gboodd 48384 | An odd Goldbach number is ... |
| gbepos 48385 | Any even Goldbach number i... |
| gbowpos 48386 | Any weak odd Goldbach numb... |
| gbopos 48387 | Any odd Goldbach number is... |
| gbegt5 48388 | Any even Goldbach number i... |
| gbowgt5 48389 | Any weak odd Goldbach numb... |
| gbowge7 48390 | Any weak odd Goldbach numb... |
| gboge9 48391 | Any odd Goldbach number is... |
| gbege6 48392 | Any even Goldbach number i... |
| gbpart6 48393 | The Goldbach partition of ... |
| gbpart7 48394 | The (weak) Goldbach partit... |
| gbpart8 48395 | The Goldbach partition of ... |
| gbpart9 48396 | The (strong) Goldbach part... |
| gbpart11 48397 | The (strong) Goldbach part... |
| 6gbe 48398 | 6 is an even Goldbach numb... |
| 7gbow 48399 | 7 is a weak odd Goldbach n... |
| 8gbe 48400 | 8 is an even Goldbach numb... |
| 9gbo 48401 | 9 is an odd Goldbach numbe... |
| 11gbo 48402 | 11 is an odd Goldbach numb... |
| stgoldbwt 48403 | If the strong ternary Gold... |
| sbgoldbwt 48404 | If the strong binary Goldb... |
| sbgoldbst 48405 | If the strong binary Goldb... |
| sbgoldbaltlem1 48406 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 48407 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 48408 | An alternate (related to t... |
| sbgoldbb 48409 | If the strong binary Goldb... |
| sgoldbeven3prm 48410 | If the binary Goldbach con... |
| sbgoldbm 48411 | If the strong binary Goldb... |
| mogoldbb 48412 | If the modern version of t... |
| sbgoldbmb 48413 | The strong binary Goldbach... |
| sbgoldbo 48414 | If the strong binary Goldb... |
| nnsum3primes4 48415 | 4 is the sum of at most 3 ... |
| nnsum4primes4 48416 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 48417 | Every prime is "the sum of... |
| nnsum4primesprm 48418 | Every prime is "the sum of... |
| nnsum3primesgbe 48419 | Any even Goldbach number i... |
| nnsum4primesgbe 48420 | Any even Goldbach number i... |
| nnsum3primesle9 48421 | Every integer greater than... |
| nnsum4primesle9 48422 | Every integer greater than... |
| nnsum4primesodd 48423 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 48424 | If the (strong) ternary Go... |
| evengpop3 48425 | If the (weak) ternary Gold... |
| evengpoap3 48426 | If the (strong) ternary Go... |
| nnsum4primeseven 48427 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 48428 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 48429 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 48430 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 48431 | If the binary Goldbach con... |
| bgoldbtbndlem1 48432 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 48433 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 48434 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 48435 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 48436 | If the binary Goldbach con... |
| tgoldbachgtALTV 48439 | Variant of Thierry Arnoux'... |
| bgoldbachlt 48440 | The binary Goldbach conjec... |
| tgblthelfgott 48442 | The ternary Goldbach conje... |
| tgoldbachlt 48443 | The ternary Goldbach conje... |
| tgoldbach 48444 | The ternary Goldbach conje... |
| clnbgrprc0 48447 | The closed neighborhood is... |
| clnbgrcl 48448 | If a class ` X ` has at le... |
| clnbgrval 48449 | The closed neighborhood of... |
| dfclnbgr2 48450 | Alternate definition of th... |
| dfclnbgr4 48451 | Alternate definition of th... |
| elclnbgrelnbgr 48452 | An element of the closed n... |
| dfclnbgr3 48453 | Alternate definition of th... |
| clnbgrnvtx0 48454 | If a class ` X ` is not a ... |
| clnbgrel 48455 | Characterization of a memb... |
| clnbgrvtxel 48456 | Every vertex ` K ` is a me... |
| clnbgrisvtx 48457 | Every member ` N ` of the ... |
| clnbgrssvtx 48458 | The closed neighborhood of... |
| clnbgrn0 48459 | The closed neighborhood of... |
| clnbupgr 48460 | The closed neighborhood of... |
| clnbupgrel 48461 | A member of the closed nei... |
| clnbupgreli 48462 | A member of the closed nei... |
| clnbgr0vtx 48463 | In a null graph (with no v... |
| clnbgr0edg 48464 | In an empty graph (with no... |
| clnbgrsym 48465 | In a graph, the closed nei... |
| predgclnbgrel 48466 | If a (not necessarily prop... |
| clnbgredg 48467 | A vertex connected by an e... |
| clnbgrssedg 48468 | The vertices connected by ... |
| edgusgrclnbfin 48469 | The size of the closed nei... |
| clnbusgrfi 48470 | The closed neighborhood of... |
| clnbfiusgrfi 48471 | The closed neighborhood of... |
| clnbgrlevtx 48472 | The size of the closed nei... |
| dfsclnbgr2 48473 | Alternate definition of th... |
| sclnbgrel 48474 | Characterization of a memb... |
| sclnbgrelself 48475 | A vertex ` N ` is a member... |
| sclnbgrisvtx 48476 | Every member ` X ` of the ... |
| dfclnbgr5 48477 | Alternate definition of th... |
| dfnbgr5 48478 | Alternate definition of th... |
| dfnbgrss 48479 | Subset chain for different... |
| dfvopnbgr2 48480 | Alternate definition of th... |
| vopnbgrel 48481 | Characterization of a memb... |
| vopnbgrelself 48482 | A vertex ` N ` is a member... |
| dfclnbgr6 48483 | Alternate definition of th... |
| dfnbgr6 48484 | Alternate definition of th... |
| dfsclnbgr6 48485 | Alternate definition of a ... |
| dfnbgrss2 48486 | Subset chain for different... |
| isisubgr 48489 | The subgraph induced by a ... |
| isubgriedg 48490 | The edges of an induced su... |
| isubgrvtxuhgr 48491 | The subgraph induced by th... |
| isubgredgss 48492 | The edges of an induced su... |
| isubgredg 48493 | An edge of an induced subg... |
| isubgrvtx 48494 | The vertices of an induced... |
| isubgruhgr 48495 | An induced subgraph of a h... |
| isubgrsubgr 48496 | An induced subgraph of a h... |
| isubgrupgr 48497 | An induced subgraph of a p... |
| isubgrumgr 48498 | An induced subgraph of a m... |
| isubgrusgr 48499 | An induced subgraph of a s... |
| isubgr0uhgr 48500 | The subgraph induced by an... |
| grimfn 48506 | The graph isomorphism func... |
| grimdmrel 48507 | The domain of the graph is... |
| isgrim 48509 | An isomorphism of graphs i... |
| grimprop 48510 | Properties of an isomorphi... |
| grimf1o 48511 | An isomorphism of graphs i... |
| grimidvtxedg 48512 | The identity relation rest... |
| grimid 48513 | The identity relation rest... |
| grimuhgr 48514 | If there is a graph isomor... |
| grimcnv 48515 | The converse of a graph is... |
| grimco 48516 | The composition of graph i... |
| uhgrimedgi 48517 | An isomorphism between gra... |
| uhgrimedg 48518 | An isomorphism between gra... |
| uhgrimprop 48519 | An isomorphism between hyp... |
| isuspgrim0lem 48520 | An isomorphism of simple p... |
| isuspgrim0 48521 | An isomorphism of simple p... |
| isuspgrimlem 48522 | Lemma for ~ isuspgrim . (... |
| isuspgrim 48523 | A class is an isomorphism ... |
| upgrimwlklem1 48524 | Lemma 1 for ~ upgrimwlk an... |
| upgrimwlklem2 48525 | Lemma 2 for ~ upgrimwlk . ... |
| upgrimwlklem3 48526 | Lemma 3 for ~ upgrimwlk . ... |
| upgrimwlklem4 48527 | Lemma 4 for ~ upgrimwlk . ... |
| upgrimwlklem5 48528 | Lemma 5 for ~ upgrimwlk . ... |
| upgrimwlk 48529 | Graph isomorphisms between... |
| upgrimwlklen 48530 | Graph isomorphisms between... |
| upgrimtrlslem1 48531 | Lemma 1 for ~ upgrimtrls .... |
| upgrimtrlslem2 48532 | Lemma 2 for ~ upgrimtrls .... |
| upgrimtrls 48533 | Graph isomorphisms between... |
| upgrimpthslem1 48534 | Lemma 1 for ~ upgrimpths .... |
| upgrimpthslem2 48535 | Lemma 2 for ~ upgrimpths .... |
| upgrimpths 48536 | Graph isomorphisms between... |
| upgrimspths 48537 | Graph isomorphisms between... |
| upgrimcycls 48538 | Graph isomorphisms between... |
| brgric 48539 | The relation "is isomorphi... |
| brgrici 48540 | Prove that two graphs are ... |
| gricrcl 48541 | Reverse closure of the "is... |
| dfgric2 48542 | Alternate, explicit defini... |
| gricbri 48543 | Implications of two graphs... |
| gricushgr 48544 | The "is isomorphic to" rel... |
| gricuspgr 48545 | The "is isomorphic to" rel... |
| gricrel 48546 | The "is isomorphic to" rel... |
| gricref 48547 | Graph isomorphism is refle... |
| gricsym 48548 | Graph isomorphism is symme... |
| gricsymb 48549 | Graph isomorphism is symme... |
| grictr 48550 | Graph isomorphism is trans... |
| gricer 48551 | Isomorphism is an equivale... |
| gricen 48552 | Isomorphic graphs have equ... |
| opstrgric 48553 | A graph represented as an ... |
| ushggricedg 48554 | A simple hypergraph (with ... |
| cycldlenngric 48555 | Two simple pseudographs ar... |
| isubgrgrim 48556 | Isomorphic subgraphs induc... |
| uhgrimisgrgriclem 48557 | Lemma for ~ uhgrimisgrgric... |
| uhgrimisgrgric 48558 | For isomorphic hypergraphs... |
| clnbgrisubgrgrim 48559 | Isomorphic subgraphs induc... |
| clnbgrgrimlem 48560 | Lemma for ~ clnbgrgrim : ... |
| clnbgrgrim 48561 | Graph isomorphisms between... |
| grimedg 48562 | For two isomorphic graphs,... |
| grimedgi 48563 | Graph isomorphisms map edg... |
| grtriproplem 48566 | Lemma for ~ grtriprop . (... |
| grtri 48567 | The triangles in a graph. ... |
| grtriprop 48568 | The properties of a triang... |
| grtrif1o 48569 | Any bijection onto a trian... |
| isgrtri 48570 | A triangle in a graph. (C... |
| grtrissvtx 48571 | A triangle is a subset of ... |
| grtriclwlk3 48572 | A triangle induces a close... |
| cycl3grtrilem 48573 | Lemma for ~ cycl3grtri . ... |
| cycl3grtri 48574 | The vertices of a cycle of... |
| grtrimap 48575 | Conditions for mapping tri... |
| grimgrtri 48576 | Graph isomorphisms map tri... |
| usgrgrtrirex 48577 | Conditions for a simple gr... |
| stgrfv 48580 | The star graph S_N. (Contr... |
| stgrvtx 48581 | The vertices of the star g... |
| stgriedg 48582 | The indexed edges of the s... |
| stgredg 48583 | The edges of the star grap... |
| stgredgel 48584 | An edge of the star graph ... |
| stgredgiun 48585 | The edges of the star grap... |
| stgrusgra 48586 | The star graph S_N is a si... |
| stgr0 48587 | The star graph S_0 consist... |
| stgr1 48588 | The star graph S_1 consist... |
| stgrvtx0 48589 | The center ("internal node... |
| stgrorder 48590 | The order of a star graph ... |
| stgrnbgr0 48591 | All vertices of a star gra... |
| stgrclnbgr0 48592 | All vertices of a star gra... |
| isubgr3stgrlem1 48593 | Lemma 1 for ~ isubgr3stgr ... |
| isubgr3stgrlem2 48594 | Lemma 2 for ~ isubgr3stgr ... |
| isubgr3stgrlem3 48595 | Lemma 3 for ~ isubgr3stgr ... |
| isubgr3stgrlem4 48596 | Lemma 4 for ~ isubgr3stgr ... |
| isubgr3stgrlem5 48597 | Lemma 5 for ~ isubgr3stgr ... |
| isubgr3stgrlem6 48598 | Lemma 6 for ~ isubgr3stgr ... |
| isubgr3stgrlem7 48599 | Lemma 7 for ~ isubgr3stgr ... |
| isubgr3stgrlem8 48600 | Lemma 8 for ~ isubgr3stgr ... |
| isubgr3stgrlem9 48601 | Lemma 9 for ~ isubgr3stgr ... |
| isubgr3stgr 48602 | If a vertex of a simple gr... |
| grlimfn 48606 | The graph local isomorphis... |
| grlimdmrel 48607 | The domain of the graph lo... |
| isgrlim 48609 | A local isomorphism of gra... |
| isgrlim2 48610 | A local isomorphism of gra... |
| grlimprop 48611 | Properties of a local isom... |
| grlimf1o 48612 | A local isomorphism of gra... |
| grlimprop2 48613 | Properties of a local isom... |
| uhgrimgrlim 48614 | An isomorphism of hypergra... |
| uspgrlimlem1 48615 | Lemma 1 for ~ uspgrlim . ... |
| uspgrlimlem2 48616 | Lemma 2 for ~ uspgrlim . ... |
| uspgrlimlem3 48617 | Lemma 3 for ~ uspgrlim . ... |
| uspgrlimlem4 48618 | Lemma 4 for ~ uspgrlim . ... |
| uspgrlim 48619 | A local isomorphism of sim... |
| usgrlimprop 48620 | Properties of a local isom... |
| clnbgrvtxedg 48621 | An edge ` E ` containing a... |
| grlimedgclnbgr 48622 | For two locally isomorphic... |
| grlimprclnbgr 48623 | For two locally isomorphic... |
| grlimprclnbgredg 48624 | For two locally isomorphic... |
| grlimpredg 48625 | For two locally isomorphic... |
| grlimprclnbgrvtx 48626 | For two locally isomorphic... |
| grlimgredgex 48627 | Local isomorphisms between... |
| grlimgrtrilem1 48628 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtrilem2 48629 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtri 48630 | If one of two locally isom... |
| brgrlic 48631 | The relation "is locally i... |
| brgrilci 48632 | Prove that two graphs are ... |
| grlicrel 48633 | The "is locally isomorphic... |
| grlicrcl 48634 | Reverse closure of the "is... |
| dfgrlic2 48635 | Alternate, explicit defini... |
| grilcbri 48636 | Implications of two graphs... |
| dfgrlic3 48637 | Alternate, explicit defini... |
| grilcbri2 48638 | Implications of two graphs... |
| grlicref 48639 | Graph local isomorphism is... |
| grlicsym 48640 | Graph local isomorphism is... |
| grlicsymb 48641 | Graph local isomorphism is... |
| grlictr 48642 | Graph local isomorphism is... |
| grlicer 48643 | Local isomorphism is an eq... |
| grlicen 48644 | Locally isomorphic graphs ... |
| gricgrlic 48645 | Isomorphic hypergraphs are... |
| clnbgr3stgrgrlim 48646 | If all (closed) neighborho... |
| clnbgr3stgrgrlic 48647 | If all (closed) neighborho... |
| usgrexmpl1lem 48648 | Lemma for ~ usgrexmpl1 . ... |
| usgrexmpl1 48649 | ` G ` is a simple graph of... |
| usgrexmpl1vtx 48650 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl1edg 48651 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl1tri 48652 | ` G ` contains a triangle ... |
| usgrexmpl2lem 48653 | Lemma for ~ usgrexmpl2 . ... |
| usgrexmpl2 48654 | ` G ` is a simple graph of... |
| usgrexmpl2vtx 48655 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl2edg 48656 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl2nblem 48657 | Lemma for ~ usgrexmpl2nb0 ... |
| usgrexmpl2nb0 48658 | The neighborhood of the fi... |
| usgrexmpl2nb1 48659 | The neighborhood of the se... |
| usgrexmpl2nb2 48660 | The neighborhood of the th... |
| usgrexmpl2nb3 48661 | The neighborhood of the fo... |
| usgrexmpl2nb4 48662 | The neighborhood of the fi... |
| usgrexmpl2nb5 48663 | The neighborhood of the si... |
| usgrexmpl2trifr 48664 | ` G ` is triangle-free. (... |
| usgrexmpl12ngric 48665 | The graphs ` H ` and ` G `... |
| usgrexmpl12ngrlic 48666 | The graphs ` H ` and ` G `... |
| gpgov 48669 | The generalized Petersen g... |
| gpgvtx 48670 | The vertices of the genera... |
| gpgiedg 48671 | The indexed edges of the g... |
| gpgedg 48672 | The edges of the generaliz... |
| gpgiedgdmellem 48673 | Lemma for ~ gpgiedgdmel an... |
| gpgvtxel 48674 | A vertex in a generalized ... |
| gpgvtxel2 48675 | The second component of a ... |
| gpgiedgdmel 48676 | An index of edges of the g... |
| gpgedgel 48677 | An edge in a generalized P... |
| gpgprismgriedgdmel 48678 | An index of edges of the g... |
| gpgprismgriedgdmss 48679 | A subset of the index of e... |
| gpgvtx0 48680 | The outside vertices in a ... |
| gpgvtx1 48681 | The inside vertices in a g... |
| opgpgvtx 48682 | A vertex in a generalized ... |
| gpgusgralem 48683 | Lemma for ~ gpgusgra . (C... |
| gpgusgra 48684 | The generalized Petersen g... |
| gpgprismgrusgra 48685 | The generalized Petersen g... |
| gpgorder 48686 | The order of the generaliz... |
| gpg5order 48687 | The order of a generalized... |
| gpgedgvtx0 48688 | The edges starting at an o... |
| gpgedgvtx1 48689 | The edges starting at an i... |
| gpgvtxedg0 48690 | The edges starting at an o... |
| gpgvtxedg1 48691 | The edges starting at an i... |
| gpgedgiov 48692 | The edges of the generaliz... |
| gpgedg2ov 48693 | The edges of the generaliz... |
| gpgedg2iv 48694 | The edges of the generaliz... |
| gpg5nbgrvtx03starlem1 48695 | Lemma 1 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem2 48696 | Lemma 2 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem3 48697 | Lemma 3 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx13starlem1 48698 | Lemma 1 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem2 48699 | Lemma 2 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem3 48700 | Lemma 3 for ~ gpg5nbgr3sta... |
| gpgnbgrvtx0 48701 | The (open) neighborhood of... |
| gpgnbgrvtx1 48702 | The (open) neighborhood of... |
| gpg3nbgrvtx0 48703 | In a generalized Petersen ... |
| gpg3nbgrvtx0ALT 48704 | In a generalized Petersen ... |
| gpg3nbgrvtx1 48705 | In a generalized Petersen ... |
| gpgcubic 48706 | Every generalized Petersen... |
| gpg5nbgrvtx03star 48707 | In a generalized Petersen ... |
| gpg5nbgr3star 48708 | In a generalized Petersen ... |
| gpgvtxdg3 48709 | Every vertex in a generali... |
| gpg3kgrtriexlem1 48710 | Lemma 1 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem2 48711 | Lemma 2 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem3 48712 | Lemma 3 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem4 48713 | Lemma 4 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem5 48714 | Lemma 5 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem6 48715 | Lemma 6 for ~ gpg3kgrtriex... |
| gpg3kgrtriex 48716 | All generalized Petersen g... |
| gpg5gricstgr3 48717 | Each closed neighborhood i... |
| pglem 48718 | Lemma for theorems about P... |
| pgjsgr 48719 | A Petersen graph is a simp... |
| gpg5grlim 48720 | A local isomorphism betwee... |
| gpg5grlic 48721 | The two generalized Peters... |
| gpgprismgr4cycllem1 48722 | Lemma 1 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem2 48723 | Lemma 2 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem3 48724 | Lemma 3 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem4 48725 | Lemma 4 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem5 48726 | Lemma 5 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem6 48727 | Lemma 6 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem7 48728 | Lemma 7 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem8 48729 | Lemma 8 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem9 48730 | Lemma 9 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem10 48731 | Lemma 10 for ~ gpgprismgr4... |
| gpgprismgr4cycllem11 48732 | Lemma 11 for ~ gpgprismgr4... |
| gpgprismgr4cycl0 48733 | The generalized Petersen g... |
| gpgprismgr4cyclex 48734 | The generalized Petersen g... |
| pgnioedg1 48735 | An inside and an outside v... |
| pgnioedg2 48736 | An inside and an outside v... |
| pgnioedg3 48737 | An inside and an outside v... |
| pgnioedg4 48738 | An inside and an outside v... |
| pgnioedg5 48739 | An inside and an outside v... |
| pgnbgreunbgrlem1 48740 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem1 48741 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem2 48742 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem3 48743 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2 48744 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem3 48745 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem4 48746 | Lemma 4 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem1 48747 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem2 48748 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem3 48749 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5 48750 | Lemma 5 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem6 48751 | Lemma 6 for ~ pgnbgreunbgr... |
| pgnbgreunbgr 48752 | In a Petersen graph, two d... |
| pgn4cyclex 48753 | A cycle in a Petersen grap... |
| pg4cyclnex 48754 | In the Petersen graph G(5,... |
| gpg5ngric 48755 | The two generalized Peters... |
| lgricngricex 48756 | There are two different lo... |
| gpg5edgnedg 48757 | Two consecutive (according... |
| grlimedgnedg 48758 | In general, the image of a... |
| 1hegrlfgr 48759 | A graph ` G ` with one hyp... |
| upwlksfval 48762 | The set of simple walks (i... |
| isupwlk 48763 | Properties of a pair of fu... |
| isupwlkg 48764 | Generalization of ~ isupwl... |
| upwlkbprop 48765 | Basic properties of a simp... |
| upwlkwlk 48766 | A simple walk is a walk. ... |
| upgrwlkupwlk 48767 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 48768 | In a pseudograph, the defi... |
| upgrisupwlkALT 48769 | Alternate proof of ~ upgri... |
| upgredgssspr 48770 | The set of edges of a pseu... |
| uspgropssxp 48771 | The set ` G ` of "simple p... |
| uspgrsprfv 48772 | The value of the function ... |
| uspgrsprf 48773 | The mapping ` F ` is a fun... |
| uspgrsprf1 48774 | The mapping ` F ` is a one... |
| uspgrsprfo 48775 | The mapping ` F ` is a fun... |
| uspgrsprf1o 48776 | The mapping ` F ` is a bij... |
| uspgrex 48777 | The class ` G ` of all "si... |
| uspgrbispr 48778 | There is a bijection betwe... |
| uspgrspren 48779 | The set ` G ` of the "simp... |
| uspgrymrelen 48780 | The set ` G ` of the "simp... |
| uspgrbisymrel 48781 | There is a bijection betwe... |
| uspgrbisymrelALT 48782 | Alternate proof of ~ uspgr... |
| ovn0dmfun 48783 | If a class operation value... |
| xpsnopab 48784 | A Cartesian product with a... |
| xpiun 48785 | A Cartesian product expres... |
| ovn0ssdmfun 48786 | If a class' operation valu... |
| fnxpdmdm 48787 | The domain of the domain o... |
| cnfldsrngbas 48788 | The base set of a subring ... |
| cnfldsrngadd 48789 | The group addition operati... |
| cnfldsrngmul 48790 | The ring multiplication op... |
| plusfreseq 48791 | If the empty set is not co... |
| mgmplusfreseq 48792 | If the empty set is not co... |
| 0mgm 48793 | A set with an empty base s... |
| opmpoismgm 48794 | A structure with a group a... |
| copissgrp 48795 | A structure with a constan... |
| copisnmnd 48796 | A structure with a constan... |
| 0nodd 48797 | 0 is not an odd integer. ... |
| 1odd 48798 | 1 is an odd integer. (Con... |
| 2nodd 48799 | 2 is not an odd integer. ... |
| oddibas 48800 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 48801 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 48802 | The structure of all odd i... |
| nnsgrpmgm 48803 | The structure of positive ... |
| nnsgrp 48804 | The structure of positive ... |
| nnsgrpnmnd 48805 | The structure of positive ... |
| nn0mnd 48806 | The set of nonnegative int... |
| gsumsplit2f 48807 | Split a group sum into two... |
| gsumdifsndf 48808 | Extract a summand from a f... |
| gsumfsupp 48809 | A group sum of a family ca... |
| iscllaw 48816 | The predicate "is a closed... |
| iscomlaw 48817 | The predicate "is a commut... |
| clcllaw 48818 | Closure of a closed operat... |
| isasslaw 48819 | The predicate "is an assoc... |
| asslawass 48820 | Associativity of an associ... |
| mgmplusgiopALT 48821 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 48822 | Slot 2 (group operation) o... |
| intopval 48829 | The internal (binary) oper... |
| intop 48830 | An internal (binary) opera... |
| clintopval 48831 | The closed (internal binar... |
| assintopval 48832 | The associative (closed in... |
| assintopmap 48833 | The associative (closed in... |
| isclintop 48834 | The predicate "is a closed... |
| clintop 48835 | A closed (internal binary)... |
| assintop 48836 | An associative (closed int... |
| isassintop 48837 | The predicate "is an assoc... |
| clintopcllaw 48838 | The closure law holds for ... |
| assintopcllaw 48839 | The closure low holds for ... |
| assintopasslaw 48840 | The associative low holds ... |
| assintopass 48841 | An associative (closed int... |
| ismgmALT 48850 | The predicate "is a magma"... |
| iscmgmALT 48851 | The predicate "is a commut... |
| issgrpALT 48852 | The predicate "is a semigr... |
| iscsgrpALT 48853 | The predicate "is a commut... |
| mgm2mgm 48854 | Equivalence of the two def... |
| sgrp2sgrp 48855 | Equivalence of the two def... |
| lmod0rng 48856 | If the scalar ring of a mo... |
| nzrneg1ne0 48857 | The additive inverse of th... |
| lidldomn1 48858 | If a (left) ideal (which i... |
| lidlabl 48859 | A (left) ideal of a ring i... |
| lidlrng 48860 | A (left) ideal of a ring i... |
| zlidlring 48861 | The zero (left) ideal of a... |
| uzlidlring 48862 | Only the zero (left) ideal... |
| lidldomnnring 48863 | A (left) ideal of a domain... |
| 0even 48864 | 0 is an even integer. (Co... |
| 1neven 48865 | 1 is not an even integer. ... |
| 2even 48866 | 2 is an even integer. (Co... |
| 2zlidl 48867 | The even integers are a (l... |
| 2zrng 48868 | The ring of integers restr... |
| 2zrngbas 48869 | The base set of R is the s... |
| 2zrngadd 48870 | The group addition operati... |
| 2zrng0 48871 | The additive identity of R... |
| 2zrngamgm 48872 | R is an (additive) magma. ... |
| 2zrngasgrp 48873 | R is an (additive) semigro... |
| 2zrngamnd 48874 | R is an (additive) monoid.... |
| 2zrngacmnd 48875 | R is a commutative (additi... |
| 2zrngagrp 48876 | R is an (additive) group. ... |
| 2zrngaabl 48877 | R is an (additive) abelian... |
| 2zrngmul 48878 | The ring multiplication op... |
| 2zrngmmgm 48879 | R is a (multiplicative) ma... |
| 2zrngmsgrp 48880 | R is a (multiplicative) se... |
| 2zrngALT 48881 | The ring of integers restr... |
| 2zrngnmlid 48882 | R has no multiplicative (l... |
| 2zrngnmrid 48883 | R has no multiplicative (r... |
| 2zrngnmlid2 48884 | R has no multiplicative (l... |
| 2zrngnring 48885 | R is not a unital ring. (... |
| cznrnglem 48886 | Lemma for ~ cznrng : The ... |
| cznabel 48887 | The ring constructed from ... |
| cznrng 48888 | The ring constructed from ... |
| cznnring 48889 | The ring constructed from ... |
| rngcvalALTV 48892 | Value of the category of n... |
| rngcbasALTV 48893 | Set of objects of the cate... |
| rngchomfvalALTV 48894 | Set of arrows of the categ... |
| rngchomALTV 48895 | Set of arrows of the categ... |
| elrngchomALTV 48896 | A morphism of non-unital r... |
| rngccofvalALTV 48897 | Composition in the categor... |
| rngccoALTV 48898 | Composition in the categor... |
| rngccatidALTV 48899 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 48900 | The category of non-unital... |
| rngcidALTV 48901 | The identity arrow in the ... |
| rngcsectALTV 48902 | A section in the category ... |
| rngcinvALTV 48903 | An inverse in the category... |
| rngcisoALTV 48904 | An isomorphism in the cate... |
| rngchomffvalALTV 48905 | The value of the functiona... |
| rngchomrnghmresALTV 48906 | The value of the functiona... |
| rngcrescrhmALTV 48907 | The category of non-unital... |
| rhmsubcALTVlem1 48908 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 48909 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 48910 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 48911 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 48912 | According to ~ df-subc , t... |
| rhmsubcALTVcat 48913 | The restriction of the cat... |
| ringcvalALTV 48916 | Value of the category of r... |
| funcringcsetcALTV2lem1 48917 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 48918 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 48919 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 48920 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 48921 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 48922 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 48923 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 48924 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 48925 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 48926 | The "natural forgetful fun... |
| ringcbasALTV 48927 | Set of objects of the cate... |
| ringchomfvalALTV 48928 | Set of arrows of the categ... |
| ringchomALTV 48929 | Set of arrows of the categ... |
| elringchomALTV 48930 | A morphism of rings is a f... |
| ringccofvalALTV 48931 | Composition in the categor... |
| ringccoALTV 48932 | Composition in the categor... |
| ringccatidALTV 48933 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 48934 | The category of rings is a... |
| ringcidALTV 48935 | The identity arrow in the ... |
| ringcsectALTV 48936 | A section in the category ... |
| ringcinvALTV 48937 | An inverse in the category... |
| ringcisoALTV 48938 | An isomorphism in the cate... |
| ringcbasbasALTV 48939 | An element of the base set... |
| funcringcsetclem1ALTV 48940 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 48941 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 48942 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 48943 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 48944 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 48945 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 48946 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 48947 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 48948 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 48949 | The "natural forgetful fun... |
| srhmsubcALTVlem1 48950 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 48951 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 48952 | According to ~ df-subc , t... |
| sringcatALTV 48953 | The restriction of the cat... |
| crhmsubcALTV 48954 | According to ~ df-subc , t... |
| cringcatALTV 48955 | The restriction of the cat... |
| drhmsubcALTV 48956 | According to ~ df-subc , t... |
| drngcatALTV 48957 | The restriction of the cat... |
| fldcatALTV 48958 | The restriction of the cat... |
| fldcALTV 48959 | The restriction of the cat... |
| fldhmsubcALTV 48960 | According to ~ df-subc , t... |
| eliunxp2 48961 | Membership in a union of C... |
| mpomptx2 48962 | Express a two-argument fun... |
| cbvmpox2 48963 | Rule to change the bound v... |
| dmmpossx2 48964 | The domain of a mapping is... |
| mpoexxg2 48965 | Existence of an operation ... |
| ovmpordxf 48966 | Value of an operation give... |
| ovmpordx 48967 | Value of an operation give... |
| ovmpox2 48968 | The value of an operation ... |
| fdmdifeqresdif 48969 | The restriction of a condi... |
| ofaddmndmap 48970 | The function operation app... |
| mapsnop 48971 | A singleton of an ordered ... |
| fprmappr 48972 | A function with a domain o... |
| mapprop 48973 | An unordered pair containi... |
| ztprmneprm 48974 | A prime is not an integer ... |
| 2t6m3t4e0 48975 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 48976 | For any finite subset of `... |
| nn0sumltlt 48977 | If the sum of two nonnegat... |
| bcpascm1 48978 | Pascal's rule for the bino... |
| altgsumbc 48979 | The sum of binomial coeffi... |
| altgsumbcALT 48980 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 48981 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 48982 | An element of the (base se... |
| zlmodzxz0 48983 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 48984 | The scalar multiplication ... |
| zlmodzxzadd 48985 | The addition of the ` ZZ `... |
| zlmodzxzsubm 48986 | The subtraction of the ` Z... |
| zlmodzxzsub 48987 | The subtraction of the ` Z... |
| mgpsumunsn 48988 | Extract a summand/factor f... |
| mgpsumz 48989 | If the group sum for the m... |
| mgpsumn 48990 | If the group sum for the m... |
| exple2lt6 48991 | A nonnegative integer to t... |
| pgrple2abl 48992 | Every symmetric group on a... |
| pgrpgt2nabl 48993 | Every symmetric group on a... |
| invginvrid 48994 | Identity for a multiplicat... |
| rmsupp0 48995 | The support of a mapping o... |
| domnmsuppn0 48996 | The support of a mapping o... |
| rmsuppss 48997 | The support of a mapping o... |
| scmsuppss 48998 | The support of a mapping o... |
| rmsuppfi 48999 | The support of a mapping o... |
| rmfsupp 49000 | A mapping of a multiplicat... |
| scmsuppfi 49001 | The support of a mapping o... |
| scmfsupp 49002 | A mapping of a scalar mult... |
| suppmptcfin 49003 | The support of a mapping w... |
| mptcfsupp 49004 | A mapping with value 0 exc... |
| fsuppmptdmf 49005 | A mapping with a finite do... |
| lmodvsmdi 49006 | Multiple distributive law ... |
| gsumlsscl 49007 | Closure of a group sum in ... |
| assaascl0 49008 | The scalar 0 embedded into... |
| assaascl1 49009 | The scalar 1 embedded into... |
| ply1vr1smo 49010 | The variable in a polynomi... |
| ply1sclrmsm 49011 | The ring multiplication of... |
| coe1sclmulval 49012 | The value of the coefficie... |
| ply1mulgsumlem1 49013 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 49014 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 49015 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 49016 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 49017 | The product of two polynom... |
| evl1at0 49018 | Polynomial evaluation for ... |
| evl1at1 49019 | Polynomial evaluation for ... |
| linply1 49020 | A term of the form ` x - C... |
| lineval 49021 | A term of the form ` x - C... |
| linevalexample 49022 | The polynomial ` x - 3 ` o... |
| dmatALTval 49027 | The algebra of ` N ` x ` N... |
| dmatALTbas 49028 | The base set of the algebr... |
| dmatALTbasel 49029 | An element of the base set... |
| dmatbas 49030 | The set of all ` N ` x ` N... |
| lincop 49035 | A linear combination as op... |
| lincval 49036 | The value of a linear comb... |
| dflinc2 49037 | Alternative definition of ... |
| lcoop 49038 | A linear combination as op... |
| lcoval 49039 | The value of a linear comb... |
| lincfsuppcl 49040 | A linear combination of ve... |
| linccl 49041 | A linear combination of ve... |
| lincval0 49042 | The value of an empty line... |
| lincvalsng 49043 | The linear combination ove... |
| lincvalsn 49044 | The linear combination ove... |
| lincvalpr 49045 | The linear combination ove... |
| lincval1 49046 | The linear combination ove... |
| lcosn0 49047 | Properties of a linear com... |
| lincvalsc0 49048 | The linear combination whe... |
| lcoc0 49049 | Properties of a linear com... |
| linc0scn0 49050 | If a set contains the zero... |
| lincdifsn 49051 | A vector is a linear combi... |
| linc1 49052 | A vector is a linear combi... |
| lincellss 49053 | A linear combination of a ... |
| lco0 49054 | The set of empty linear co... |
| lcoel0 49055 | The zero vector is always ... |
| lincsum 49056 | The sum of two linear comb... |
| lincscm 49057 | A linear combinations mult... |
| lincsumcl 49058 | The sum of two linear comb... |
| lincscmcl 49059 | The multiplication of a li... |
| lincsumscmcl 49060 | The sum of a linear combin... |
| lincolss 49061 | According to the statement... |
| ellcoellss 49062 | Every linear combination o... |
| lcoss 49063 | A set of vectors of a modu... |
| lspsslco 49064 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 49065 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 49066 | Equivalence of a _span_ of... |
| rellininds 49070 | The class defining the rel... |
| linindsv 49072 | The classes of the module ... |
| islininds 49073 | The property of being a li... |
| linindsi 49074 | The implications of being ... |
| linindslinci 49075 | The implications of being ... |
| islinindfis 49076 | The property of being a li... |
| islinindfiss 49077 | The property of being a li... |
| linindscl 49078 | A linearly independent set... |
| lindepsnlininds 49079 | A linearly dependent subse... |
| islindeps 49080 | The property of being a li... |
| lincext1 49081 | Property 1 of an extension... |
| lincext2 49082 | Property 2 of an extension... |
| lincext3 49083 | Property 3 of an extension... |
| lindslinindsimp1 49084 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 49085 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 49086 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 49087 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 49088 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 49089 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 49090 | Implication 2 for ~ lindsl... |
| lindslininds 49091 | Equivalence of definitions... |
| linds0 49092 | The empty set is always a ... |
| el0ldep 49093 | A set containing the zero ... |
| el0ldepsnzr 49094 | A set containing the zero ... |
| lindsrng01 49095 | Any subset of a module is ... |
| lindszr 49096 | Any subset of a module ove... |
| snlindsntorlem 49097 | Lemma for ~ snlindsntor . ... |
| snlindsntor 49098 | A singleton is linearly in... |
| ldepsprlem 49099 | Lemma for ~ ldepspr . (Co... |
| ldepspr 49100 | If a vector is a scalar mu... |
| lincresunit3lem3 49101 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 49102 | Lemma 1 for properties of ... |
| lincresunitlem2 49103 | Lemma for properties of a ... |
| lincresunit1 49104 | Property 1 of a specially ... |
| lincresunit2 49105 | Property 2 of a specially ... |
| lincresunit3lem1 49106 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 49107 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 49108 | Property 3 of a specially ... |
| lincreslvec3 49109 | Property 3 of a specially ... |
| islindeps2 49110 | Conditions for being a lin... |
| islininds2 49111 | Implication of being a lin... |
| isldepslvec2 49112 | Alternative definition of ... |
| lindssnlvec 49113 | A singleton not containing... |
| lmod1lem1 49114 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 49115 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 49116 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 49117 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 49118 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 49119 | The (smallest) structure r... |
| lmod1zr 49120 | The (smallest) structure r... |
| lmod1zrnlvec 49121 | There is a (left) module (... |
| lmodn0 49122 | Left modules exist. (Cont... |
| zlmodzxzequa 49123 | Example of an equation wit... |
| zlmodzxznm 49124 | Example of a linearly depe... |
| zlmodzxzldeplem 49125 | A and B are not equal. (C... |
| zlmodzxzequap 49126 | Example of an equation wit... |
| zlmodzxzldeplem1 49127 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 49128 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 49129 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 49130 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 49131 | { A , B } is a linearly de... |
| ldepsnlinclem1 49132 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 49133 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 49134 | The class of all (left) ve... |
| ldepsnlinc 49135 | The reverse implication of... |
| ldepslinc 49136 | For (left) vector spaces, ... |
| suppdm 49137 | If the range of a function... |
| eluz2cnn0n1 49138 | An integer greater than 1 ... |
| divge1b 49139 | The ratio of a real number... |
| divgt1b 49140 | The ratio of a real number... |
| ltsubaddb 49141 | Equivalence for the "less ... |
| ltsubsubb 49142 | Equivalence for the "less ... |
| ltsubadd2b 49143 | Equivalence for the "less ... |
| divsub1dir 49144 | Distribution of division o... |
| expnegico01 49145 | An integer greater than 1 ... |
| elfzolborelfzop1 49146 | An element of a half-open ... |
| pw2m1lepw2m1 49147 | 2 to the power of a positi... |
| zgtp1leeq 49148 | If an integer is between a... |
| flsubz 49149 | An integer can be moved in... |
| nn0onn0ex 49150 | For each odd nonnegative i... |
| nn0enn0ex 49151 | For each even nonnegative ... |
| nnennex 49152 | For each even positive int... |
| nneop 49153 | A positive integer is even... |
| nneom 49154 | A positive integer is even... |
| nn0eo 49155 | A nonnegative integer is e... |
| nnpw2even 49156 | 2 to the power of a positi... |
| zefldiv2 49157 | The floor of an even integ... |
| zofldiv2 49158 | The floor of an odd intege... |
| nn0ofldiv2 49159 | The floor of an odd nonneg... |
| flnn0div2ge 49160 | The floor of a positive in... |
| flnn0ohalf 49161 | The floor of the half of a... |
| logcxp0 49162 | Logarithm of a complex pow... |
| regt1loggt0 49163 | The natural logarithm for ... |
| fdivval 49166 | The quotient of two functi... |
| fdivmpt 49167 | The quotient of two functi... |
| fdivmptf 49168 | The quotient of two functi... |
| refdivmptf 49169 | The quotient of two functi... |
| fdivpm 49170 | The quotient of two functi... |
| refdivpm 49171 | The quotient of two functi... |
| fdivmptfv 49172 | The function value of a qu... |
| refdivmptfv 49173 | The function value of a qu... |
| bigoval 49176 | Set of functions of order ... |
| elbigofrcl 49177 | Reverse closure of the "bi... |
| elbigo 49178 | Properties of a function o... |
| elbigo2 49179 | Properties of a function o... |
| elbigo2r 49180 | Sufficient condition for a... |
| elbigof 49181 | A function of order G(x) i... |
| elbigodm 49182 | The domain of a function o... |
| elbigoimp 49183 | The defining property of a... |
| elbigolo1 49184 | A function (into the posit... |
| rege1logbrege0 49185 | The general logarithm, wit... |
| rege1logbzge0 49186 | The general logarithm, wit... |
| fllogbd 49187 | A real number is between t... |
| relogbmulbexp 49188 | The logarithm of the produ... |
| relogbdivb 49189 | The logarithm of the quoti... |
| logbge0b 49190 | The logarithm of a number ... |
| logblt1b 49191 | The logarithm of a number ... |
| fldivexpfllog2 49192 | The floor of a positive re... |
| nnlog2ge0lt1 49193 | A positive integer is 1 if... |
| logbpw2m1 49194 | The floor of the binary lo... |
| fllog2 49195 | The floor of the binary lo... |
| blenval 49198 | The binary length of an in... |
| blen0 49199 | The binary length of 0. (... |
| blenn0 49200 | The binary length of a "nu... |
| blenre 49201 | The binary length of a pos... |
| blennn 49202 | The binary length of a pos... |
| blennnelnn 49203 | The binary length of a pos... |
| blennn0elnn 49204 | The binary length of a non... |
| blenpw2 49205 | The binary length of a pow... |
| blenpw2m1 49206 | The binary length of a pow... |
| nnpw2blen 49207 | A positive integer is betw... |
| nnpw2blenfzo 49208 | A positive integer is betw... |
| nnpw2blenfzo2 49209 | A positive integer is eith... |
| nnpw2pmod 49210 | Every positive integer can... |
| blen1 49211 | The binary length of 1. (... |
| blen2 49212 | The binary length of 2. (... |
| nnpw2p 49213 | Every positive integer can... |
| nnpw2pb 49214 | A number is a positive int... |
| blen1b 49215 | The binary length of a non... |
| blennnt2 49216 | The binary length of a pos... |
| nnolog2flm1 49217 | The floor of the binary lo... |
| blennn0em1 49218 | The binary length of the h... |
| blennngt2o2 49219 | The binary length of an od... |
| blengt1fldiv2p1 49220 | The binary length of an in... |
| blennn0e2 49221 | The binary length of an ev... |
| digfval 49224 | Operation to obtain the ` ... |
| digval 49225 | The ` K ` th digit of a no... |
| digvalnn0 49226 | The ` K ` th digit of a no... |
| nn0digval 49227 | The ` K ` th digit of a no... |
| dignn0fr 49228 | The digits of the fraction... |
| dignn0ldlem 49229 | Lemma for ~ dignnld . (Co... |
| dignnld 49230 | The leading digits of a po... |
| dig2nn0ld 49231 | The leading digits of a po... |
| dig2nn1st 49232 | The first (relevant) digit... |
| dig0 49233 | All digits of 0 are 0. (C... |
| digexp 49234 | The ` K ` th digit of a po... |
| dig1 49235 | All but one digits of 1 ar... |
| 0dig1 49236 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 49237 | The integers 0 and 1 corre... |
| dig2nn0 49238 | A digit of a nonnegative i... |
| 0dig2nn0e 49239 | The last bit of an even in... |
| 0dig2nn0o 49240 | The last bit of an odd int... |
| dig2bits 49241 | The ` K ` th digit of a no... |
| dignn0flhalflem1 49242 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 49243 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 49244 | The digits of the half of ... |
| dignn0flhalf 49245 | The digits of the rounded ... |
| nn0sumshdiglemA 49246 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 49247 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 49248 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 49249 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 49250 | A nonnegative integer can ... |
| nn0mulfsum 49251 | Trivial algorithm to calcu... |
| nn0mullong 49252 | Standard algorithm (also k... |
| naryfval 49255 | The set of the n-ary (endo... |
| naryfvalixp 49256 | The set of the n-ary (endo... |
| naryfvalel 49257 | An n-ary (endo)function on... |
| naryrcl 49258 | Reverse closure for n-ary ... |
| naryfvalelfv 49259 | The value of an n-ary (end... |
| naryfvalelwrdf 49260 | An n-ary (endo)function on... |
| 0aryfvalel 49261 | A nullary (endo)function o... |
| 0aryfvalelfv 49262 | The value of a nullary (en... |
| 1aryfvalel 49263 | A unary (endo)function on ... |
| fv1arycl 49264 | Closure of a unary (endo)f... |
| 1arympt1 49265 | A unary (endo)function in ... |
| 1arympt1fv 49266 | The value of a unary (endo... |
| 1arymaptfv 49267 | The value of the mapping o... |
| 1arymaptf 49268 | The mapping of unary (endo... |
| 1arymaptf1 49269 | The mapping of unary (endo... |
| 1arymaptfo 49270 | The mapping of unary (endo... |
| 1arymaptf1o 49271 | The mapping of unary (endo... |
| 1aryenef 49272 | The set of unary (endo)fun... |
| 1aryenefmnd 49273 | The set of unary (endo)fun... |
| 2aryfvalel 49274 | A binary (endo)function on... |
| fv2arycl 49275 | Closure of a binary (endo)... |
| 2arympt 49276 | A binary (endo)function in... |
| 2arymptfv 49277 | The value of a binary (end... |
| 2arymaptfv 49278 | The value of the mapping o... |
| 2arymaptf 49279 | The mapping of binary (end... |
| 2arymaptf1 49280 | The mapping of binary (end... |
| 2arymaptfo 49281 | The mapping of binary (end... |
| 2arymaptf1o 49282 | The mapping of binary (end... |
| 2aryenef 49283 | The set of binary (endo)fu... |
| itcoval 49288 | The value of the function ... |
| itcoval0 49289 | A function iterated zero t... |
| itcoval1 49290 | A function iterated once. ... |
| itcoval2 49291 | A function iterated twice.... |
| itcoval3 49292 | A function iterated three ... |
| itcoval0mpt 49293 | A mapping iterated zero ti... |
| itcovalsuc 49294 | The value of the function ... |
| itcovalsucov 49295 | The value of the function ... |
| itcovalendof 49296 | The n-th iterate of an end... |
| itcovalpclem1 49297 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 49298 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 49299 | The value of the function ... |
| itcovalt2lem2lem1 49300 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 49301 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 49302 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 49303 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 49304 | The value of the function ... |
| ackvalsuc1mpt 49305 | The Ackermann function at ... |
| ackvalsuc1 49306 | The Ackermann function at ... |
| ackval0 49307 | The Ackermann function at ... |
| ackval1 49308 | The Ackermann function at ... |
| ackval2 49309 | The Ackermann function at ... |
| ackval3 49310 | The Ackermann function at ... |
| ackendofnn0 49311 | The Ackermann function at ... |
| ackfnnn0 49312 | The Ackermann function at ... |
| ackval0val 49313 | The Ackermann function at ... |
| ackvalsuc0val 49314 | The Ackermann function at ... |
| ackvalsucsucval 49315 | The Ackermann function at ... |
| ackval0012 49316 | The Ackermann function at ... |
| ackval1012 49317 | The Ackermann function at ... |
| ackval2012 49318 | The Ackermann function at ... |
| ackval3012 49319 | The Ackermann function at ... |
| ackval40 49320 | The Ackermann function at ... |
| ackval41a 49321 | The Ackermann function at ... |
| ackval41 49322 | The Ackermann function at ... |
| ackval42 49323 | The Ackermann function at ... |
| ackval42a 49324 | The Ackermann function at ... |
| ackval50 49325 | The Ackermann function at ... |
| fv1prop 49326 | The function value of unor... |
| fv2prop 49327 | The function value of unor... |
| submuladdmuld 49328 | Transformation of a sum of... |
| affinecomb1 49329 | Combination of two real af... |
| affinecomb2 49330 | Combination of two real af... |
| affineid 49331 | Identity of an affine comb... |
| 1subrec1sub 49332 | Subtract the reciprocal of... |
| resum2sqcl 49333 | The sum of two squares of ... |
| resum2sqgt0 49334 | The sum of the square of a... |
| resum2sqrp 49335 | The sum of the square of a... |
| resum2sqorgt0 49336 | The sum of the square of t... |
| reorelicc 49337 | Membership in and outside ... |
| rrx2pxel 49338 | The x-coordinate of a poin... |
| rrx2pyel 49339 | The y-coordinate of a poin... |
| prelrrx2 49340 | An unordered pair of order... |
| prelrrx2b 49341 | An unordered pair of order... |
| rrx2pnecoorneor 49342 | If two different points ` ... |
| rrx2pnedifcoorneor 49343 | If two different points ` ... |
| rrx2pnedifcoorneorr 49344 | If two different points ` ... |
| rrx2xpref1o 49345 | There is a bijection betwe... |
| rrx2xpreen 49346 | The set of points in the t... |
| rrx2plord 49347 | The lexicographical orderi... |
| rrx2plord1 49348 | The lexicographical orderi... |
| rrx2plord2 49349 | The lexicographical orderi... |
| rrx2plordisom 49350 | The set of points in the t... |
| rrx2plordso 49351 | The lexicographical orderi... |
| ehl2eudisval0 49352 | The Euclidean distance of ... |
| ehl2eudis0lt 49353 | An upper bound of the Eucl... |
| lines 49358 | The lines passing through ... |
| line 49359 | The line passing through t... |
| rrxlines 49360 | Definition of lines passin... |
| rrxline 49361 | The line passing through t... |
| rrxlinesc 49362 | Definition of lines passin... |
| rrxlinec 49363 | The line passing through t... |
| eenglngeehlnmlem1 49364 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 49365 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 49366 | The line definition in the... |
| rrx2line 49367 | The line passing through t... |
| rrx2vlinest 49368 | The vertical line passing ... |
| rrx2linest 49369 | The line passing through t... |
| rrx2linesl 49370 | The line passing through t... |
| rrx2linest2 49371 | The line passing through t... |
| elrrx2linest2 49372 | The line passing through t... |
| spheres 49373 | The spheres for given cent... |
| sphere 49374 | A sphere with center ` X `... |
| rrxsphere 49375 | The sphere with center ` M... |
| 2sphere 49376 | The sphere with center ` M... |
| 2sphere0 49377 | The sphere around the orig... |
| line2ylem 49378 | Lemma for ~ line2y . This... |
| line2 49379 | Example for a line ` G ` p... |
| line2xlem 49380 | Lemma for ~ line2x . This... |
| line2x 49381 | Example for a horizontal l... |
| line2y 49382 | Example for a vertical lin... |
| itsclc0lem1 49383 | Lemma for theorems about i... |
| itsclc0lem2 49384 | Lemma for theorems about i... |
| itsclc0lem3 49385 | Lemma for theorems about i... |
| itscnhlc0yqe 49386 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 49387 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 49388 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 49389 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 49390 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 49391 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 49392 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 49393 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 49394 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 49395 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 49396 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 49397 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 49398 | The intersection points of... |
| itsclc0b 49399 | The intersection points of... |
| itsclinecirc0 49400 | The intersection points of... |
| itsclinecirc0b 49401 | The intersection points of... |
| itsclinecirc0in 49402 | The intersection points of... |
| itsclquadb 49403 | Quadratic equation for the... |
| itsclquadeu 49404 | Quadratic equation for the... |
| 2itscplem1 49405 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 49406 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 49407 | Lemma D for ~ 2itscp . (C... |
| 2itscp 49408 | A condition for a quadrati... |
| itscnhlinecirc02plem1 49409 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 49410 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 49411 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 49412 | Intersection of a nonhoriz... |
| inlinecirc02plem 49413 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 49414 | Intersection of a line wit... |
| inlinecirc02preu 49415 | Intersection of a line wit... |
| pm4.71da 49416 | Deduction converting a bic... |
| logic1 49417 | Distribution of implicatio... |
| logic1a 49418 | Variant of ~ logic1 . (Co... |
| logic2 49419 | Variant of ~ logic1 . (Co... |
| pm5.32dav 49420 | Distribution of implicatio... |
| pm5.32dra 49421 | Reverse distribution of im... |
| exp12bd 49422 | The import-export theorem ... |
| mpbiran3d 49423 | Equivalence with a conjunc... |
| mpbiran4d 49424 | Equivalence with a conjunc... |
| dtrucor3 49425 | An example of how ~ ax-5 w... |
| ralbidb 49426 | Formula-building rule for ... |
| ralbidc 49427 | Formula-building rule for ... |
| r19.41dv 49428 | A complex deduction form o... |
| rmotru 49429 | Two ways of expressing "at... |
| reutru 49430 | Two ways of expressing "ex... |
| reutruALT 49431 | Alternate proof of ~ reutr... |
| reueqbidva 49432 | Formula-building rule for ... |
| reuxfr1dd 49433 | Transfer existential uniqu... |
| ssdisjd 49434 | Subset preserves disjointn... |
| ssdisjdr 49435 | Subset preserves disjointn... |
| disjdifb 49436 | Relative complement is ant... |
| predisj 49437 | Preimages of disjoint sets... |
| vsn 49438 | The singleton of the unive... |
| mosn 49439 | "At most one" element in a... |
| mo0 49440 | "At most one" element in a... |
| mosssn 49441 | "At most one" element in a... |
| mo0sn 49442 | Two ways of expressing "at... |
| mosssn2 49443 | Two ways of expressing "at... |
| unilbss 49444 | Superclass of the greatest... |
| iuneq0 49445 | An indexed union is empty ... |
| iineq0 49446 | An indexed intersection is... |
| iunlub 49447 | The indexed union is the t... |
| iinglb 49448 | The indexed intersection i... |
| iuneqconst2 49449 | Indexed union of identical... |
| iineqconst2 49450 | Indexed intersection of id... |
| inpw 49451 | Two ways of expressing a c... |
| opth1neg 49452 | Two ordered pairs are not ... |
| opth2neg 49453 | Two ordered pairs are not ... |
| brab2dd 49454 | Expressing that two sets a... |
| brab2ddw 49455 | Expressing that two sets a... |
| brab2ddw2 49456 | Expressing that two sets a... |
| iinxp 49457 | Indexed intersection of Ca... |
| intxp 49458 | Intersection of Cartesian ... |
| coxp 49459 | Composition with a Cartesi... |
| cosn 49460 | Composition with an ordere... |
| cosni 49461 | Composition with an ordere... |
| inisegn0a 49462 | The inverse image of a sin... |
| dmrnxp 49463 | A Cartesian product is the... |
| mof0 49464 | There is at most one funct... |
| mof02 49465 | A variant of ~ mof0 . (Co... |
| mof0ALT 49466 | Alternate proof of ~ mof0 ... |
| eufsnlem 49467 | There is exactly one funct... |
| eufsn 49468 | There is exactly one funct... |
| eufsn2 49469 | There is exactly one funct... |
| mofsn 49470 | There is at most one funct... |
| mofsn2 49471 | There is at most one funct... |
| mofsssn 49472 | There is at most one funct... |
| mofmo 49473 | There is at most one funct... |
| mofeu 49474 | The uniqueness of a functi... |
| elfvne0 49475 | If a function value has a ... |
| fdomne0 49476 | A function with non-empty ... |
| f1sn2g 49477 | A function that maps a sin... |
| f102g 49478 | A function that maps the e... |
| f1mo 49479 | A function that maps a set... |
| f002 49480 | A function with an empty c... |
| map0cor 49481 | A function exists iff an e... |
| ffvbr 49482 | Relation with function val... |
| xpco2 49483 | Composition of a Cartesian... |
| ovsng 49484 | The operation value of a s... |
| ovsng2 49485 | The operation value of a s... |
| ovsn 49486 | The operation value of a s... |
| ovsn2 49487 | The operation value of a s... |
| fvconstr 49488 | Two ways of expressing ` A... |
| fvconstrn0 49489 | Two ways of expressing ` A... |
| fvconstr2 49490 | Two ways of expressing ` A... |
| ovmpt4d 49491 | Deduction version of ~ ovm... |
| eqfnovd 49492 | Deduction for equality of ... |
| fonex 49493 | The domain of a surjection... |
| eloprab1st2nd 49494 | Reconstruction of a nested... |
| fmpodg 49495 | Domain and codomain of the... |
| fmpod 49496 | Domain and codomain of the... |
| resinsnlem 49497 | Lemma for ~ resinsnALT . ... |
| resinsn 49498 | Restriction to the interse... |
| resinsnALT 49499 | Restriction to the interse... |
| dftpos5 49500 | Alternate definition of ` ... |
| dftpos6 49501 | Alternate definition of ` ... |
| dmtposss 49502 | The domain of ` tpos F ` i... |
| tposres0 49503 | The transposition of a set... |
| tposresg 49504 | The transposition restrict... |
| tposrescnv 49505 | The transposition restrict... |
| tposres2 49506 | The transposition restrict... |
| tposres3 49507 | The transposition restrict... |
| tposres 49508 | The transposition restrict... |
| tposresxp 49509 | The transposition restrict... |
| tposf1o 49510 | Condition of a bijective t... |
| tposid 49511 | Swap an ordered pair. (Co... |
| tposidres 49512 | Swap an ordered pair. (Co... |
| tposidf1o 49513 | The swap function, or the ... |
| tposideq 49514 | Two ways of expressing the... |
| tposideq2 49515 | Two ways of expressing the... |
| ixpv 49516 | Infinite Cartesian product... |
| fvconst0ci 49517 | A constant function's valu... |
| fvconstdomi 49518 | A constant function's valu... |
| f1omo 49519 | There is at most one eleme... |
| f1omoOLD 49520 | Obsolete version of ~ f1om... |
| f1omoALT 49521 | There is at most one eleme... |
| iccin 49522 | Intersection of two closed... |
| iccdisj2 49523 | If the upper bound of one ... |
| iccdisj 49524 | If the upper bound of one ... |
| slotresfo 49525 | The condition of a structu... |
| mreuniss 49526 | The union of a collection ... |
| clduni 49527 | The union of closed sets i... |
| opncldeqv 49528 | Conditions on open sets ar... |
| opndisj 49529 | Two ways of saying that tw... |
| clddisj 49530 | Two ways of saying that tw... |
| neircl 49531 | Reverse closure of the nei... |
| opnneilem 49532 | Lemma factoring out common... |
| opnneir 49533 | If something is true for a... |
| opnneirv 49534 | A variant of ~ opnneir wit... |
| opnneilv 49535 | The converse of ~ opnneir ... |
| opnneil 49536 | A variant of ~ opnneilv . ... |
| opnneieqv 49537 | The equivalence between ne... |
| opnneieqvv 49538 | The equivalence between ne... |
| restcls2lem 49539 | A closed set in a subspace... |
| restcls2 49540 | A closed set in a subspace... |
| restclsseplem 49541 | Lemma for ~ restclssep . ... |
| restclssep 49542 | Two disjoint closed sets i... |
| cnneiima 49543 | Given a continuous functio... |
| iooii 49544 | Open intervals are open se... |
| icccldii 49545 | Closed intervals are close... |
| i0oii 49546 | ` ( 0 [,) A ) ` is open in... |
| io1ii 49547 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 49548 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 49549 | Open neighborhood and neig... |
| sepnsepo 49550 | Open neighborhood and neig... |
| sepdisj 49551 | Separated sets are disjoin... |
| seposep 49552 | If two sets are separated ... |
| sepcsepo 49553 | If two sets are separated ... |
| sepfsepc 49554 | If two sets are separated ... |
| seppsepf 49555 | If two sets are precisely ... |
| seppcld 49556 | If two sets are precisely ... |
| isnrm4 49557 | A topological space is nor... |
| dfnrm2 49558 | A topological space is nor... |
| dfnrm3 49559 | A topological space is nor... |
| iscnrm3lem1 49560 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 49561 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem4 49562 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 49563 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 49564 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 49565 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 49566 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 49567 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 49568 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 49569 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 49570 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 49571 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 49572 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 49573 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 49574 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 49575 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 49576 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 49577 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 49578 | A completely normal topolo... |
| iscnrm3v 49579 | A topology is completely n... |
| iscnrm4 49580 | A completely normal topolo... |
| isprsd 49581 | Property of being a preord... |
| lubeldm2 49582 | Member of the domain of th... |
| glbeldm2 49583 | Member of the domain of th... |
| lubeldm2d 49584 | Member of the domain of th... |
| glbeldm2d 49585 | Member of the domain of th... |
| lubsscl 49586 | If a subset of ` S ` conta... |
| glbsscl 49587 | If a subset of ` S ` conta... |
| lubprlem 49588 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 49589 | The set of two comparable ... |
| lubpr 49590 | The LUB of the set of two ... |
| glbprlem 49591 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 49592 | The set of two comparable ... |
| glbpr 49593 | The GLB of the set of two ... |
| joindm2 49594 | The join of any two elemen... |
| joindm3 49595 | The join of any two elemen... |
| meetdm2 49596 | The meet of any two elemen... |
| meetdm3 49597 | The meet of any two elemen... |
| posjidm 49598 | Poset join is idempotent. ... |
| posmidm 49599 | Poset meet is idempotent. ... |
| resiposbas 49600 | Construct a poset ( ~ resi... |
| resipos 49601 | A set equipped with an ord... |
| exbaspos 49602 | There exists a poset for a... |
| exbasprs 49603 | There exists a preordered ... |
| basresposfo 49604 | The base function restrict... |
| basresprsfo 49605 | The base function restrict... |
| posnex 49606 | The class of posets is a p... |
| prsnex 49607 | The class of preordered se... |
| toslat 49608 | A toset is a lattice. (Co... |
| isclatd 49609 | The predicate "is a comple... |
| intubeu 49610 | Existential uniqueness of ... |
| unilbeu 49611 | Existential uniqueness of ... |
| ipolublem 49612 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 49613 | The domain of the LUB of t... |
| ipolub 49614 | The LUB of the inclusion p... |
| ipoglblem 49615 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 49616 | The domain of the GLB of t... |
| ipoglb 49617 | The GLB of the inclusion p... |
| ipolub0 49618 | The LUB of the empty set i... |
| ipolub00 49619 | The LUB of the empty set i... |
| ipoglb0 49620 | The GLB of the empty set i... |
| mrelatlubALT 49621 | Least upper bounds in a Mo... |
| mrelatglbALT 49622 | Greatest lower bounds in a... |
| mreclat 49623 | A Moore space is a complet... |
| topclat 49624 | A topology is a complete l... |
| toplatglb0 49625 | The empty intersection in ... |
| toplatlub 49626 | Least upper bounds in a to... |
| toplatglb 49627 | Greatest lower bounds in a... |
| toplatjoin 49628 | Joins in a topology are re... |
| toplatmeet 49629 | Meets in a topology are re... |
| topdlat 49630 | A topology is a distributi... |
| elmgpcntrd 49631 | The center of a ring. (Co... |
| asclelbasALT 49632 | Alternate proof of ~ ascle... |
| asclcntr 49633 | The algebra scalar lifting... |
| asclcom 49634 | Scalars are commutative af... |
| homf0 49635 | The base is empty iff the ... |
| catprslem 49636 | Lemma for ~ catprs . (Con... |
| catprs 49637 | A preorder can be extracte... |
| catprs2 49638 | A category equipped with t... |
| catprsc 49639 | A construction of the preo... |
| catprsc2 49640 | An alternate construction ... |
| endmndlem 49641 | A diagonal hom-set in a ca... |
| oppccatb 49642 | An opposite category is a ... |
| oppcmndclem 49643 | Lemma for ~ oppcmndc . Ev... |
| oppcendc 49644 | The opposite category of a... |
| oppcmndc 49645 | The opposite category of a... |
| idmon 49646 | An identity arrow, or an i... |
| idepi 49647 | An identity arrow, or an i... |
| sectrcl 49648 | Reverse closure for sectio... |
| sectrcl2 49649 | Reverse closure for sectio... |
| invrcl 49650 | Reverse closure for invers... |
| invrcl2 49651 | Reverse closure for invers... |
| isinv2 49652 | The property " ` F ` is an... |
| isisod 49653 | The predicate "is an isomo... |
| upeu2lem 49654 | Lemma for ~ upeu2 . There... |
| sectfn 49655 | The function value of the ... |
| invfn 49656 | The function value of the ... |
| isofnALT 49657 | The function value of the ... |
| isofval2 49658 | Function value of the func... |
| isorcl 49659 | Reverse closure for isomor... |
| isorcl2 49660 | Reverse closure for isomor... |
| isoval2 49661 | The isomorphisms are the d... |
| sectpropdlem 49662 | Lemma for ~ sectpropd . (... |
| sectpropd 49663 | Two structures with the sa... |
| invpropdlem 49664 | Lemma for ~ invpropd . (C... |
| invpropd 49665 | Two structures with the sa... |
| isopropdlem 49666 | Lemma for ~ isopropd . (C... |
| isopropd 49667 | Two structures with the sa... |
| cicfn 49668 | ` ~=c ` is a function on `... |
| cicrcl2 49669 | Isomorphism implies the st... |
| oppccic 49670 | Isomorphic objects are iso... |
| relcic 49671 | The set of isomorphic obje... |
| cicerALT 49672 | Isomorphism is an equivale... |
| cic1st2nd 49673 | Reconstruction of a pair o... |
| cic1st2ndbr 49674 | Rewrite the predicate of i... |
| cicpropdlem 49675 | Lemma for ~ cicpropd . (C... |
| cicpropd 49676 | Two structures with the sa... |
| oppccicb 49677 | Isomorphic objects are iso... |
| oppcciceq 49678 | The opposite category has ... |
| dmdm 49679 | The double domain of a fun... |
| iinfssclem1 49680 | Lemma for ~ iinfssc . (Co... |
| iinfssclem2 49681 | Lemma for ~ iinfssc . (Co... |
| iinfssclem3 49682 | Lemma for ~ iinfssc . (Co... |
| iinfssc 49683 | Indexed intersection of su... |
| iinfsubc 49684 | Indexed intersection of su... |
| iinfprg 49685 | Indexed intersection of fu... |
| infsubc 49686 | The intersection of two su... |
| infsubc2 49687 | The intersection of two su... |
| infsubc2d 49688 | The intersection of two su... |
| discsubclem 49689 | Lemma for ~ discsubc . (C... |
| discsubc 49690 | A discrete category, whose... |
| iinfconstbaslem 49691 | Lemma for ~ iinfconstbas .... |
| iinfconstbas 49692 | The discrete category is t... |
| nelsubclem 49693 | Lemma for ~ nelsubc . (Co... |
| nelsubc 49694 | An empty "hom-set" for non... |
| nelsubc2 49695 | An empty "hom-set" for non... |
| nelsubc3lem 49696 | Lemma for ~ nelsubc3 . (C... |
| nelsubc3 49697 | Remark 4.2(2) of [Adamek] ... |
| ssccatid 49698 | A category ` C ` restricte... |
| resccatlem 49699 | Lemma for ~ resccat . (Co... |
| resccat 49700 | A class ` C ` restricted b... |
| reldmfunc 49701 | The domain of ` Func ` is ... |
| func1st2nd 49702 | Rewrite the functor predic... |
| func1st 49703 | Extract the first member o... |
| func2nd 49704 | Extract the second member ... |
| funcrcl2 49705 | Reverse closure for a func... |
| funcrcl3 49706 | Reverse closure for a func... |
| funcf2lem 49707 | A utility theorem for prov... |
| funcf2lem2 49708 | A utility theorem for prov... |
| 0funcglem 49709 | Lemma for ~ 0funcg . (Con... |
| 0funcg2 49710 | The functor from the empty... |
| 0funcg 49711 | The functor from the empty... |
| 0funclem 49712 | Lemma for ~ 0funcALT . (C... |
| 0func 49713 | The functor from the empty... |
| 0funcALT 49714 | Alternate proof of ~ 0func... |
| func0g 49715 | The source category of a f... |
| func0g2 49716 | The source category of a f... |
| initc 49717 | Sets with empty base are t... |
| cofu1st2nd 49718 | Rewrite the functor compos... |
| rescofuf 49719 | The restriction of functor... |
| cofu1a 49720 | Value of the object part o... |
| cofu2a 49721 | Value of the morphism part... |
| cofucla 49722 | The composition of two fun... |
| funchomf 49723 | Source categories of a fun... |
| idfurcl 49724 | Reverse closure for an ide... |
| idfu1stf1o 49725 | The identity functor/inclu... |
| idfu1stalem 49726 | Lemma for ~ idfu1sta . (C... |
| idfu1sta 49727 | Value of the object part o... |
| idfu1a 49728 | Value of the object part o... |
| idfu2nda 49729 | Value of the morphism part... |
| imasubclem1 49730 | Lemma for ~ imasubc . (Co... |
| imasubclem2 49731 | Lemma for ~ imasubc . (Co... |
| imasubclem3 49732 | Lemma for ~ imasubc . (Co... |
| imaf1homlem 49733 | Lemma for ~ imaf1hom and o... |
| imaf1hom 49734 | The hom-set of an image of... |
| imaidfu2lem 49735 | Lemma for ~ imaidfu2 . (C... |
| imaidfu 49736 | The image of the identity ... |
| imaidfu2 49737 | The image of the identity ... |
| cofid1a 49738 | Express the object part of... |
| cofid2a 49739 | Express the morphism part ... |
| cofid1 49740 | Express the object part of... |
| cofid2 49741 | Express the morphism part ... |
| cofidvala 49742 | The property " ` F ` is a ... |
| cofidf2a 49743 | If " ` F ` is a section of... |
| cofidf1a 49744 | If " ` F ` is a section of... |
| cofidval 49745 | The property " ` <. F , G ... |
| cofidf2 49746 | If " ` F ` is a section of... |
| cofidf1 49747 | If " ` <. F , G >. ` is a ... |
| oppffn 49750 | ` oppFunc ` is a function ... |
| reldmoppf 49751 | The domain of ` oppFunc ` ... |
| oppfvalg 49752 | Value of the opposite func... |
| oppfrcllem 49753 | Lemma for ~ oppfrcl . (Co... |
| oppfrcl 49754 | If an opposite functor of ... |
| oppfrcl2 49755 | If an opposite functor of ... |
| oppfrcl3 49756 | If an opposite functor of ... |
| oppf1st2nd 49757 | Rewrite the opposite funct... |
| 2oppf 49758 | The double opposite functo... |
| eloppf 49759 | The pre-image of a non-emp... |
| eloppf2 49760 | Both components of a pre-i... |
| oppfvallem 49761 | Lemma for ~ oppfval . (Co... |
| oppfval 49762 | Value of the opposite func... |
| oppfval2 49763 | Value of the opposite func... |
| oppfval3 49764 | Value of the opposite func... |
| oppf1 49765 | Value of the object part o... |
| oppf2 49766 | Value of the morphism part... |
| oppfoppc 49767 | The opposite functor is a ... |
| oppfoppc2 49768 | The opposite functor is a ... |
| funcoppc2 49769 | A functor on opposite cate... |
| funcoppc4 49770 | A functor on opposite cate... |
| funcoppc5 49771 | A functor on opposite cate... |
| 2oppffunc 49772 | The opposite functor of an... |
| funcoppc3 49773 | A functor on opposite cate... |
| oppff1 49774 | The operation generating o... |
| oppff1o 49775 | The operation generating o... |
| cofuoppf 49776 | Composition of opposite fu... |
| imasubc 49777 | An image of a full functor... |
| imasubc2 49778 | An image of a full functor... |
| imassc 49779 | An image of a functor sati... |
| imaid 49780 | An image of a functor pres... |
| imaf1co 49781 | An image of a functor whos... |
| imasubc3 49782 | An image of a functor inje... |
| fthcomf 49783 | Source categories of a fai... |
| idfth 49784 | The inclusion functor is a... |
| idemb 49785 | The inclusion functor is a... |
| idsubc 49786 | The source category of an ... |
| idfullsubc 49787 | The source category of an ... |
| cofidfth 49788 | If " ` F ` is a section of... |
| fulloppf 49789 | The opposite functor of a ... |
| fthoppf 49790 | The opposite functor of a ... |
| ffthoppf 49791 | The opposite functor of a ... |
| upciclem1 49792 | Lemma for ~ upcic , ~ upeu... |
| upciclem2 49793 | Lemma for ~ upciclem3 and ... |
| upciclem3 49794 | Lemma for ~ upciclem4 . (... |
| upciclem4 49795 | Lemma for ~ upcic and ~ up... |
| upcic 49796 | A universal property defin... |
| upeu 49797 | A universal property defin... |
| upeu2 49798 | Generate new universal mor... |
| reldmup 49801 | The domain of ` UP ` is a ... |
| upfval 49802 | Function value of the clas... |
| upfval2 49803 | Function value of the clas... |
| upfval3 49804 | Function value of the clas... |
| isuplem 49805 | Lemma for ~ isup and other... |
| isup 49806 | The predicate "is a univer... |
| uppropd 49807 | If two categories have the... |
| reldmup2 49808 | The domain of ` ( D UP E )... |
| relup 49809 | The set of universal pairs... |
| uprcl 49810 | Reverse closure for the cl... |
| up1st2nd 49811 | Rewrite the universal prop... |
| up1st2ndr 49812 | Combine separated parts in... |
| up1st2ndb 49813 | Combine/separate parts in ... |
| up1st2nd2 49814 | Rewrite the universal prop... |
| uprcl2 49815 | Reverse closure for the cl... |
| uprcl3 49816 | Reverse closure for the cl... |
| uprcl4 49817 | Reverse closure for the cl... |
| uprcl5 49818 | Reverse closure for the cl... |
| uobrcl 49819 | Reverse closure for univer... |
| isup2 49820 | The universal property of ... |
| upeu3 49821 | The universal pair ` <. X ... |
| upeu4 49822 | Generate a new universal m... |
| uptposlem 49823 | Lemma for ~ uptpos . (Con... |
| uptpos 49824 | Rewrite the predicate of u... |
| oppcuprcl4 49825 | Reverse closure for the cl... |
| oppcuprcl3 49826 | Reverse closure for the cl... |
| oppcuprcl5 49827 | Reverse closure for the cl... |
| oppcuprcl2 49828 | Reverse closure for the cl... |
| uprcl2a 49829 | Reverse closure for the cl... |
| oppfuprcl 49830 | Reverse closure for the cl... |
| oppfuprcl2 49831 | Reverse closure for the cl... |
| oppcup3lem 49832 | Lemma for ~ oppcup3 . (Co... |
| oppcup 49833 | The universal pair ` <. X ... |
| oppcup2 49834 | The universal property for... |
| oppcup3 49835 | The universal property for... |
| uptrlem1 49836 | Lemma for ~ uptr . (Contr... |
| uptrlem2 49837 | Lemma for ~ uptr . (Contr... |
| uptrlem3 49838 | Lemma for ~ uptr . (Contr... |
| uptr 49839 | Universal property and ful... |
| uptri 49840 | Universal property and ful... |
| uptra 49841 | Universal property and ful... |
| uptrar 49842 | Universal property and ful... |
| uptrai 49843 | Universal property and ful... |
| uobffth 49844 | A fully faithful functor g... |
| uobeqw 49845 | If a full functor (in fact... |
| uobeq 49846 | If a full functor (in fact... |
| uptr2 49847 | Universal property and ful... |
| uptr2a 49848 | Universal property and ful... |
| isnatd 49849 | Property of being a natura... |
| natrcl2 49850 | Reverse closure for a natu... |
| natrcl3 49851 | Reverse closure for a natu... |
| catbas 49852 | The base of the category s... |
| cathomfval 49853 | The hom-sets of the catego... |
| catcofval 49854 | Composition of the categor... |
| natoppf 49855 | A natural transformation i... |
| natoppf2 49856 | A natural transformation i... |
| natoppfb 49857 | A natural transformation i... |
| initoo2 49858 | An initial object is an ob... |
| termoo2 49859 | A terminal object is an ob... |
| zeroo2 49860 | A zero object is an object... |
| oppcinito 49861 | Initial objects are termin... |
| oppctermo 49862 | Terminal objects are initi... |
| oppczeroo 49863 | Zero objects are zero in t... |
| termoeu2 49864 | Terminal objects are essen... |
| initopropdlemlem 49865 | Lemma for ~ initopropdlem ... |
| initopropdlem 49866 | Lemma for ~ initopropd . ... |
| termopropdlem 49867 | Lemma for ~ termopropd . ... |
| zeroopropdlem 49868 | Lemma for ~ zeroopropd . ... |
| initopropd 49869 | Two structures with the sa... |
| termopropd 49870 | Two structures with the sa... |
| zeroopropd 49871 | Two structures with the sa... |
| reldmxpc 49872 | The binary product of cate... |
| reldmxpcALT 49873 | Alternate proof of ~ reldm... |
| elxpcbasex1 49874 | A non-empty base set of th... |
| elxpcbasex1ALT 49875 | Alternate proof of ~ elxpc... |
| elxpcbasex2 49876 | A non-empty base set of th... |
| elxpcbasex2ALT 49877 | Alternate proof of ~ elxpc... |
| xpcfucbas 49878 | The base set of the produc... |
| xpcfuchomfval 49879 | Set of morphisms of the bi... |
| xpcfuchom 49880 | Set of morphisms of the bi... |
| xpcfuchom2 49881 | Value of the set of morphi... |
| xpcfucco2 49882 | Value of composition in th... |
| xpcfuccocl 49883 | The composition of two nat... |
| xpcfucco3 49884 | Value of composition in th... |
| dfswapf2 49887 | Alternate definition of ` ... |
| swapfval 49888 | Value of the swap functor.... |
| swapfelvv 49889 | A swap functor is an order... |
| swapf2fvala 49890 | The morphism part of the s... |
| swapf2fval 49891 | The morphism part of the s... |
| swapf1vala 49892 | The object part of the swa... |
| swapf1val 49893 | The object part of the swa... |
| swapf2fn 49894 | The morphism part of the s... |
| swapf1a 49895 | The object part of the swa... |
| swapf2vala 49896 | The morphism part of the s... |
| swapf2a 49897 | The morphism part of the s... |
| swapf1 49898 | The object part of the swa... |
| swapf2val 49899 | The morphism part of the s... |
| swapf2 49900 | The morphism part of the s... |
| swapf1f1o 49901 | The object part of the swa... |
| swapf2f1o 49902 | The morphism part of the s... |
| swapf2f1oa 49903 | The morphism part of the s... |
| swapf2f1oaALT 49904 | Alternate proof of ~ swapf... |
| swapfid 49905 | Each identity morphism in ... |
| swapfida 49906 | Each identity morphism in ... |
| swapfcoa 49907 | Composition in the source ... |
| swapffunc 49908 | The swap functor is a func... |
| swapfffth 49909 | The swap functor is a full... |
| swapffunca 49910 | The swap functor is a func... |
| swapfiso 49911 | The swap functor is an iso... |
| swapciso 49912 | The product category is ca... |
| oppc1stflem 49913 | A utility theorem for prov... |
| oppc1stf 49914 | The opposite functor of th... |
| oppc2ndf 49915 | The opposite functor of th... |
| 1stfpropd 49916 | If two categories have the... |
| 2ndfpropd 49917 | If two categories have the... |
| diagpropd 49918 | If two categories have the... |
| cofuswapfcl 49919 | The bifunctor pre-composed... |
| cofuswapf1 49920 | The object part of a bifun... |
| cofuswapf2 49921 | The morphism part of a bif... |
| tposcurf1cl 49922 | The partially evaluated tr... |
| tposcurf11 49923 | Value of the double evalua... |
| tposcurf12 49924 | The partially evaluated tr... |
| tposcurf1 49925 | Value of the object part o... |
| tposcurf2 49926 | Value of the transposed cu... |
| tposcurf2val 49927 | Value of a component of th... |
| tposcurf2cl 49928 | The transposed curry funct... |
| tposcurfcl 49929 | The transposed curry funct... |
| diag1 49930 | The constant functor of ` ... |
| diag1a 49931 | The constant functor of ` ... |
| diag1f1lem 49932 | The object part of the dia... |
| diag1f1 49933 | The object part of the dia... |
| diag2f1lem 49934 | Lemma for ~ diag2f1 . The... |
| diag2f1 49935 | If ` B ` is non-empty, the... |
| fucofulem1 49936 | Lemma for proving functor ... |
| fucofulem2 49937 | Lemma for proving functor ... |
| fuco2el 49938 | Equivalence of product fun... |
| fuco2eld 49939 | Equivalence of product fun... |
| fuco2eld2 49940 | Equivalence of product fun... |
| fuco2eld3 49941 | Equivalence of product fun... |
| fucofvalg 49944 | Value of the function givi... |
| fucofval 49945 | Value of the function givi... |
| fucoelvv 49946 | A functor composition bifu... |
| fuco1 49947 | The object part of the fun... |
| fucof1 49948 | The object part of the fun... |
| fuco2 49949 | The morphism part of the f... |
| fucofn2 49950 | The morphism part of the f... |
| fucofvalne 49951 | Value of the function givi... |
| fuco11 49952 | The object part of the fun... |
| fuco11cl 49953 | The object part of the fun... |
| fuco11a 49954 | The object part of the fun... |
| fuco112 49955 | The object part of the fun... |
| fuco111 49956 | The object part of the fun... |
| fuco111x 49957 | The object part of the fun... |
| fuco112x 49958 | The object part of the fun... |
| fuco112xa 49959 | The object part of the fun... |
| fuco11id 49960 | The identity morphism of t... |
| fuco11idx 49961 | The identity morphism of t... |
| fuco21 49962 | The morphism part of the f... |
| fuco11b 49963 | The object part of the fun... |
| fuco11bALT 49964 | Alternate proof of ~ fuco1... |
| fuco22 49965 | The morphism part of the f... |
| fucofn22 49966 | The morphism part of the f... |
| fuco23 49967 | The morphism part of the f... |
| fuco22natlem1 49968 | Lemma for ~ fuco22nat . T... |
| fuco22natlem2 49969 | Lemma for ~ fuco22nat . T... |
| fuco22natlem3 49970 | Combine ~ fuco22natlem2 wi... |
| fuco22natlem 49971 | The composed natural trans... |
| fuco22nat 49972 | The composed natural trans... |
| fucof21 49973 | The morphism part of the f... |
| fucoid 49974 | Each identity morphism in ... |
| fucoid2 49975 | Each identity morphism in ... |
| fuco22a 49976 | The morphism part of the f... |
| fuco23alem 49977 | The naturality property ( ... |
| fuco23a 49978 | The morphism part of the f... |
| fucocolem1 49979 | Lemma for ~ fucoco . Asso... |
| fucocolem2 49980 | Lemma for ~ fucoco . The ... |
| fucocolem3 49981 | Lemma for ~ fucoco . The ... |
| fucocolem4 49982 | Lemma for ~ fucoco . The ... |
| fucoco 49983 | Composition in the source ... |
| fucoco2 49984 | Composition in the source ... |
| fucofunc 49985 | The functor composition bi... |
| fucofunca 49986 | The functor composition bi... |
| fucolid 49987 | Post-compose a natural tra... |
| fucorid 49988 | Pre-composing a natural tr... |
| fucorid2 49989 | Pre-composing a natural tr... |
| postcofval 49990 | Value of the post-composit... |
| postcofcl 49991 | The post-composition funct... |
| precofvallem 49992 | Lemma for ~ precofval to e... |
| precofval 49993 | Value of the pre-compositi... |
| precofvalALT 49994 | Alternate proof of ~ preco... |
| precofval2 49995 | Value of the pre-compositi... |
| precofcl 49996 | The pre-composition functo... |
| precofval3 49997 | Value of the pre-compositi... |
| precoffunc 49998 | The pre-composition functo... |
| reldmprcof 50001 | The domain of ` -o.F ` is ... |
| prcofvalg 50002 | Value of the pre-compositi... |
| prcofvala 50003 | Value of the pre-compositi... |
| prcofval 50004 | Value of the pre-compositi... |
| prcofpropd 50005 | If the categories have the... |
| prcofelvv 50006 | The pre-composition functo... |
| reldmprcof1 50007 | The domain of the object p... |
| reldmprcof2 50008 | The domain of the morphism... |
| prcoftposcurfuco 50009 | The pre-composition functo... |
| prcoftposcurfucoa 50010 | The pre-composition functo... |
| prcoffunc 50011 | The pre-composition functo... |
| prcoffunca 50012 | The pre-composition functo... |
| prcoffunca2 50013 | The pre-composition functo... |
| prcof1 50014 | The object part of the pre... |
| prcof2a 50015 | The morphism part of the p... |
| prcof2 50016 | The morphism part of the p... |
| prcof21a 50017 | The morphism part of the p... |
| prcof22a 50018 | The morphism part of the p... |
| prcofdiag1 50019 | A constant functor pre-com... |
| prcofdiag 50020 | A diagonal functor post-co... |
| catcrcl 50021 | Reverse closure for the ca... |
| catcrcl2 50022 | Reverse closure for the ca... |
| elcatchom 50023 | A morphism of the category... |
| catcsect 50024 | The property " ` F ` is a ... |
| catcinv 50025 | The property " ` F ` is an... |
| catcisoi 50026 | A functor is an isomorphis... |
| uobeq2 50027 | If a full functor (in fact... |
| uobeq3 50028 | An isomorphism between cat... |
| opf11 50029 | The object part of the op ... |
| opf12 50030 | The object part of the op ... |
| opf2fval 50031 | The morphism part of the o... |
| opf2 50032 | The morphism part of the o... |
| fucoppclem 50033 | Lemma for ~ fucoppc . (Co... |
| fucoppcid 50034 | The opposite category of f... |
| fucoppcco 50035 | The opposite category of f... |
| fucoppc 50036 | The isomorphism from the o... |
| fucoppcffth 50037 | A fully faithful functor f... |
| fucoppcfunc 50038 | A functor from the opposit... |
| fucoppccic 50039 | The opposite category of f... |
| oppfdiag1 50040 | A constant functor for opp... |
| oppfdiag1a 50041 | A constant functor for opp... |
| oppfdiag 50042 | A diagonal functor for opp... |
| isthinc 50045 | The predicate "is a thin c... |
| isthinc2 50046 | A thin category is a categ... |
| isthinc3 50047 | A thin category is a categ... |
| thincc 50048 | A thin category is a categ... |
| thinccd 50049 | A thin category is a categ... |
| thincssc 50050 | A thin category is a categ... |
| isthincd2lem1 50051 | Lemma for ~ isthincd2 and ... |
| thincmo2 50052 | Morphisms in the same hom-... |
| thinchom 50053 | A non-empty hom-set of a t... |
| thincmo 50054 | There is at most one morph... |
| thincmoALT 50055 | Alternate proof of ~ thinc... |
| thincmod 50056 | At most one morphism in ea... |
| thincn0eu 50057 | In a thin category, a hom-... |
| thincid 50058 | In a thin category, a morp... |
| thincmon 50059 | In a thin category, all mo... |
| thincepi 50060 | In a thin category, all mo... |
| isthincd2lem2 50061 | Lemma for ~ isthincd2 . (... |
| isthincd 50062 | The predicate "is a thin c... |
| isthincd2 50063 | The predicate " ` C ` is a... |
| oppcthin 50064 | The opposite category of a... |
| oppcthinco 50065 | If the opposite category o... |
| oppcthinendc 50066 | The opposite category of a... |
| oppcthinendcALT 50067 | Alternate proof of ~ oppct... |
| thincpropd 50068 | Two structures with the sa... |
| subthinc 50069 | A subcategory of a thin ca... |
| functhinclem1 50070 | Lemma for ~ functhinc . G... |
| functhinclem2 50071 | Lemma for ~ functhinc . (... |
| functhinclem3 50072 | Lemma for ~ functhinc . T... |
| functhinclem4 50073 | Lemma for ~ functhinc . O... |
| functhinc 50074 | A functor to a thin catego... |
| functhincfun 50075 | A functor to a thin catego... |
| fullthinc 50076 | A functor to a thin catego... |
| fullthinc2 50077 | A full functor to a thin c... |
| thincfth 50078 | A functor from a thin cate... |
| thincciso 50079 | Two thin categories are is... |
| thinccisod 50080 | Two thin categories are is... |
| thincciso2 50081 | Categories isomorphic to a... |
| thincciso3 50082 | Categories isomorphic to a... |
| thincciso4 50083 | Two isomorphic categories ... |
| 0thincg 50084 | Any structure with an empt... |
| 0thinc 50085 | The empty category (see ~ ... |
| indcthing 50086 | An indiscrete category, i.... |
| discthing 50087 | A discrete category, i.e.,... |
| indthinc 50088 | An indiscrete category in ... |
| indthincALT 50089 | An alternate proof of ~ in... |
| prsthinc 50090 | Preordered sets as categor... |
| setcthin 50091 | A category of sets all of ... |
| setc2othin 50092 | The category ` ( SetCat ``... |
| thincsect 50093 | In a thin category, one mo... |
| thincsect2 50094 | In a thin category, ` F ` ... |
| thincinv 50095 | In a thin category, ` F ` ... |
| thinciso 50096 | In a thin category, ` F : ... |
| thinccic 50097 | In a thin category, two ob... |
| istermc 50100 | The predicate "is a termin... |
| istermc2 50101 | The predicate "is a termin... |
| istermc3 50102 | The predicate "is a termin... |
| termcthin 50103 | A terminal category is a t... |
| termcthind 50104 | A terminal category is a t... |
| termccd 50105 | A terminal category is a c... |
| termcbas 50106 | The base of a terminal cat... |
| termco 50107 | The object of a terminal c... |
| termcbas2 50108 | The base of a terminal cat... |
| termcbasmo 50109 | Two objects in a terminal ... |
| termchomn0 50110 | All hom-sets of a terminal... |
| termchommo 50111 | All morphisms of a termina... |
| termcid 50112 | The morphism of a terminal... |
| termcid2 50113 | The morphism of a terminal... |
| termchom 50114 | The hom-set of a terminal ... |
| termchom2 50115 | The hom-set of a terminal ... |
| setcsnterm 50116 | The category of one set, e... |
| setc1oterm 50117 | The category ` ( SetCat ``... |
| setc1obas 50118 | The base of the trivial ca... |
| setc1ohomfval 50119 | Set of morphisms of the tr... |
| setc1ocofval 50120 | Composition in the trivial... |
| setc1oid 50121 | The identity morphism of t... |
| funcsetc1ocl 50122 | The functor to the trivial... |
| funcsetc1o 50123 | Value of the functor to th... |
| isinito2lem 50124 | The predicate "is an initi... |
| isinito2 50125 | The predicate "is an initi... |
| isinito3 50126 | The predicate "is an initi... |
| dfinito4 50127 | An alternate definition of... |
| dftermo4 50128 | An alternate definition of... |
| termcpropd 50129 | Two structures with the sa... |
| oppctermhom 50130 | The opposite category of a... |
| oppctermco 50131 | The opposite category of a... |
| oppcterm 50132 | The opposite category of a... |
| functermclem 50133 | Lemma for ~ functermc . (... |
| functermc 50134 | Functor to a terminal cate... |
| functermc2 50135 | Functor to a terminal cate... |
| functermceu 50136 | There exists a unique func... |
| fulltermc 50137 | A functor to a terminal ca... |
| fulltermc2 50138 | Given a full functor to a ... |
| termcterm 50139 | A terminal category is a t... |
| termcterm2 50140 | A terminal object of the c... |
| termcterm3 50141 | In the category of small c... |
| termcciso 50142 | A category is isomorphic t... |
| termccisoeu 50143 | The isomorphism between te... |
| termc2 50144 | If there exists a unique f... |
| termc 50145 | Alternate definition of ` ... |
| dftermc2 50146 | Alternate definition of ` ... |
| eufunclem 50147 | If there exists a unique f... |
| eufunc 50148 | If there exists a unique f... |
| idfudiag1lem 50149 | Lemma for ~ idfudiag1bas a... |
| idfudiag1bas 50150 | If the identity functor of... |
| idfudiag1 50151 | If the identity functor of... |
| euendfunc 50152 | If there exists a unique e... |
| euendfunc2 50153 | If there exists a unique e... |
| termcarweu 50154 | There exists a unique disj... |
| arweuthinc 50155 | If a structure has a uniqu... |
| arweutermc 50156 | If a structure has a uniqu... |
| dftermc3 50157 | Alternate definition of ` ... |
| termcfuncval 50158 | The value of a functor fro... |
| diag1f1olem 50159 | To any functor from a term... |
| diag1f1o 50160 | The object part of the dia... |
| termcnatval 50161 | Value of natural transform... |
| diag2f1olem 50162 | Lemma for ~ diag2f1o . (C... |
| diag2f1o 50163 | If ` D ` is terminal, the ... |
| diagffth 50164 | The diagonal functor is a ... |
| diagciso 50165 | The diagonal functor is an... |
| diagcic 50166 | Any category ` C ` is isom... |
| funcsn 50167 | The category of one functo... |
| fucterm 50168 | The category of functors t... |
| 0fucterm 50169 | The category of functors f... |
| termfucterm 50170 | All functors between two t... |
| cofuterm 50171 | Post-compose with a functo... |
| uobeqterm 50172 | Universal objects and term... |
| isinito4 50173 | The predicate "is an initi... |
| isinito4a 50174 | The predicate "is an initi... |
| prstcval 50177 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 50178 | Lemma for ~ prstcnid and ~... |
| prstcnid 50179 | Components other than ` Ho... |
| prstcbas 50180 | The base set is unchanged.... |
| prstcleval 50181 | Value of the less-than-or-... |
| prstcle 50182 | Value of the less-than-or-... |
| prstcocval 50183 | Orthocomplementation is un... |
| prstcoc 50184 | Orthocomplementation is un... |
| prstchomval 50185 | Hom-sets of the constructe... |
| prstcprs 50186 | The category is a preorder... |
| prstcthin 50187 | The preordered set is equi... |
| prstchom 50188 | Hom-sets of the constructe... |
| prstchom2 50189 | Hom-sets of the constructe... |
| prstchom2ALT 50190 | Hom-sets of the constructe... |
| oduoppcbas 50191 | The dual of a preordered s... |
| oduoppcciso 50192 | The dual of a preordered s... |
| postcpos 50193 | The converted category is ... |
| postcposALT 50194 | Alternate proof of ~ postc... |
| postc 50195 | The converted category is ... |
| discsntermlem 50196 | A singlegon is an element ... |
| basrestermcfolem 50197 | An element of the class of... |
| discbas 50198 | A discrete category (a cat... |
| discthin 50199 | A discrete category (a cat... |
| discsnterm 50200 | A discrete category (a cat... |
| basrestermcfo 50201 | The base function restrict... |
| termcnex 50202 | The class of all terminal ... |
| mndtcval 50205 | Value of the category buil... |
| mndtcbasval 50206 | The base set of the catego... |
| mndtcbas 50207 | The category built from a ... |
| mndtcob 50208 | Lemma for ~ mndtchom and ~... |
| mndtcbas2 50209 | Two objects in a category ... |
| mndtchom 50210 | The only hom-set of the ca... |
| mndtcco 50211 | The composition of the cat... |
| mndtcco2 50212 | The composition of the cat... |
| mndtccatid 50213 | Lemma for ~ mndtccat and ~... |
| mndtccat 50214 | The function value is a ca... |
| mndtcid 50215 | The identity morphism, or ... |
| oppgoppchom 50216 | The converted opposite mon... |
| oppgoppcco 50217 | The converted opposite mon... |
| oppgoppcid 50218 | The converted opposite mon... |
| grptcmon 50219 | All morphisms in a categor... |
| grptcepi 50220 | All morphisms in a categor... |
| 2arwcatlem1 50221 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem2 50222 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem3 50223 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem4 50224 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem5 50225 | Lemma for ~ 2arwcat . (Co... |
| 2arwcat 50226 | The condition for a struct... |
| incat 50227 | Constructing a category wi... |
| setc1onsubc 50228 | Construct a category with ... |
| cnelsubclem 50229 | Lemma for ~ cnelsubc . (C... |
| cnelsubc 50230 | Remark 4.2(2) of [Adamek] ... |
| lanfn 50235 | ` Lan ` is a function on `... |
| ranfn 50236 | ` Ran ` is a function on `... |
| reldmlan 50237 | The domain of ` Lan ` is a... |
| reldmran 50238 | The domain of ` Ran ` is a... |
| lanfval 50239 | Value of the function gene... |
| ranfval 50240 | Value of the function gene... |
| lanpropd 50241 | If the categories have the... |
| ranpropd 50242 | If the categories have the... |
| reldmlan2 50243 | The domain of ` ( P Lan E ... |
| reldmran2 50244 | The domain of ` ( P Ran E ... |
| lanval 50245 | Value of the set of left K... |
| ranval 50246 | Value of the set of right ... |
| lanrcl 50247 | Reverse closure for left K... |
| ranrcl 50248 | Reverse closure for right ... |
| rellan 50249 | The set of left Kan extens... |
| relran 50250 | The set of right Kan exten... |
| islan 50251 | A left Kan extension is a ... |
| islan2 50252 | A left Kan extension is a ... |
| lanval2 50253 | The set of left Kan extens... |
| isran 50254 | A right Kan extension is a... |
| isran2 50255 | A right Kan extension is a... |
| ranval2 50256 | The set of right Kan exten... |
| ranval3 50257 | The set of right Kan exten... |
| lanrcl2 50258 | Reverse closure for left K... |
| lanrcl3 50259 | Reverse closure for left K... |
| lanrcl4 50260 | The first component of a l... |
| lanrcl5 50261 | The second component of a ... |
| ranrcl2 50262 | Reverse closure for right ... |
| ranrcl3 50263 | Reverse closure for right ... |
| ranrcl4lem 50264 | Lemma for ~ ranrcl4 and ~ ... |
| ranrcl4 50265 | The first component of a r... |
| ranrcl5 50266 | The second component of a ... |
| lanup 50267 | The universal property of ... |
| ranup 50268 | The universal property of ... |
| reldmlmd 50273 | The domain of ` Limit ` is... |
| reldmcmd 50274 | The domain of ` Colimit ` ... |
| lmdfval 50275 | Function value of ` Limit ... |
| cmdfval 50276 | Function value of ` Colimi... |
| lmdrcl 50277 | Reverse closure for a limi... |
| cmdrcl 50278 | Reverse closure for a coli... |
| reldmlmd2 50279 | The domain of ` ( C Limit ... |
| reldmcmd2 50280 | The domain of ` ( C Colimi... |
| lmdfval2 50281 | The set of limits of a dia... |
| cmdfval2 50282 | The set of colimits of a d... |
| lmdpropd 50283 | If the categories have the... |
| cmdpropd 50284 | If the categories have the... |
| rellmd 50285 | The set of limits of a dia... |
| relcmd 50286 | The set of colimits of a d... |
| concl 50287 | A natural transformation f... |
| coccl 50288 | A natural transformation t... |
| concom 50289 | A cone to a diagram commut... |
| coccom 50290 | A co-cone to a diagram com... |
| islmd 50291 | The universal property of ... |
| iscmd 50292 | The universal property of ... |
| lmddu 50293 | The duality of limits and ... |
| cmddu 50294 | The duality of limits and ... |
| initocmd 50295 | Initial objects are the ob... |
| termolmd 50296 | Terminal objects are the o... |
| lmdran 50297 | To each limit of a diagram... |
| cmdlan 50298 | To each colimit of a diagr... |
| nfintd 50299 | Bound-variable hypothesis ... |
| nfiund 50300 | Bound-variable hypothesis ... |
| nfiundg 50301 | Bound-variable hypothesis ... |
| iunord 50302 | The indexed union of a col... |
| iunordi 50303 | The indexed union of a col... |
| spd 50304 | Specialization deduction, ... |
| spcdvw 50305 | A version of ~ spcdv where... |
| tfis2d 50306 | Transfinite Induction Sche... |
| bnd2d 50307 | Deduction form of ~ bnd2 .... |
| dffun3f 50308 | Alternate definition of fu... |
| setrecseq 50311 | Equality theorem for set r... |
| nfsetrecs 50312 | Bound-variable hypothesis ... |
| setrec1lem1 50313 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 50314 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 50315 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 50316 | Lemma for ~ setrec1 . If ... |
| setrec1 50317 | This is the first of two f... |
| setrec2fun 50318 | This is the second of two ... |
| setrec2lem1 50319 | Lemma for ~ setrec2 . The... |
| setrec2lem2 50320 | Lemma for ~ setrec2 . The... |
| setrec2 50321 | This is the second of two ... |
| setrec2v 50322 | Version of ~ setrec2 with ... |
| setrec2mpt 50323 | Version of ~ setrec2 where... |
| setis 50324 | Version of ~ setrec2 expre... |
| elsetrecslem 50325 | Lemma for ~ elsetrecs . A... |
| elsetrecs 50326 | A set ` A ` is an element ... |
| setrecsss 50327 | The ` setrecs ` operator r... |
| setrecsres 50328 | A recursively generated cl... |
| vsetrec 50329 | Construct ` _V ` using set... |
| 0setrec 50330 | If a function sends the em... |
| onsetreclem1 50331 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 50332 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 50333 | Lemma for ~ onsetrec . (C... |
| onsetrec 50334 | Construct ` On ` using set... |
| elpglem1 50337 | Lemma for ~ elpg . (Contr... |
| elpglem2 50338 | Lemma for ~ elpg . (Contr... |
| elpglem3 50339 | Lemma for ~ elpg . (Contr... |
| elpg 50340 | Membership in the class of... |
| pgindlem 50341 | Lemma for ~ pgind . (Cont... |
| pgindnf 50342 | Version of ~ pgind with ex... |
| pgind 50343 | Induction on partizan game... |
| sbidd 50344 | An identity theorem for su... |
| sbidd-misc 50345 | An identity theorem for su... |
| gte-lte 50350 | Simple relationship betwee... |
| gt-lt 50351 | Simple relationship betwee... |
| gte-lteh 50352 | Relationship between ` <_ ... |
| gt-lth 50353 | Relationship between ` < `... |
| ex-gt 50354 | Simple example of ` > ` , ... |
| ex-gte 50355 | Simple example of ` >_ ` ,... |
| sinhval-named 50362 | Value of the named sinh fu... |
| coshval-named 50363 | Value of the named cosh fu... |
| tanhval-named 50364 | Value of the named tanh fu... |
| sinh-conventional 50365 | Conventional definition of... |
| sinhpcosh 50366 | Prove that ` ( sinh `` A )... |
| secval 50373 | Value of the secant functi... |
| cscval 50374 | Value of the cosecant func... |
| cotval 50375 | Value of the cotangent fun... |
| seccl 50376 | The closure of the secant ... |
| csccl 50377 | The closure of the cosecan... |
| cotcl 50378 | The closure of the cotange... |
| reseccl 50379 | The closure of the secant ... |
| recsccl 50380 | The closure of the cosecan... |
| recotcl 50381 | The closure of the cotange... |
| recsec 50382 | The reciprocal of secant i... |
| reccsc 50383 | The reciprocal of cosecant... |
| reccot 50384 | The reciprocal of cotangen... |
| rectan 50385 | The reciprocal of tangent ... |
| sec0 50386 | The value of the secant fu... |
| onetansqsecsq 50387 | Prove the tangent squared ... |
| cotsqcscsq 50388 | Prove the tangent squared ... |
| ifnmfalse 50389 | If A is not a member of B,... |
| logb2aval 50390 | Define the value of the ` ... |
| mvlraddi 50397 | Move the right term in a s... |
| assraddsubi 50398 | Associate RHS addition-sub... |
| joinlmuladdmuli 50399 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 50400 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 50401 | Join AB-CB into (A-C) on L... |
| mvlrmuld 50402 | Move the right term in a p... |
| mvlrmuli 50403 | Move the right term in a p... |
| i2linesi 50404 | Solve for the intersection... |
| i2linesd 50405 | Solve for the intersection... |
| alimp-surprise 50406 | Demonstrate that when usin... |
| alimp-no-surprise 50407 | There is no "surprise" in ... |
| empty-surprise 50408 | Demonstrate that when usin... |
| empty-surprise2 50409 | "Prove" that false is true... |
| eximp-surprise 50410 | Show what implication insi... |
| eximp-surprise2 50411 | Show that "there exists" w... |
| alsconv 50416 | There is an equivalence be... |
| alsi1d 50417 | Deduction rule: Given "al... |
| alsi2d 50418 | Deduction rule: Given "al... |
| alsc1d 50419 | Deduction rule: Given "al... |
| alsc2d 50420 | Deduction rule: Given "al... |
| alscn0d 50421 | Deduction rule: Given "al... |
| alsi-no-surprise 50422 | Demonstrate that there is ... |
| 5m4e1 50423 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 50424 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 50425 | Resolution rule. This is ... |
| testable 50426 | In classical logic all wff... |
| aacllem 50427 | Lemma for other theorems a... |
| amgmwlem 50428 | Weighted version of ~ amgm... |
| amgmlemALT 50429 | Alternate proof of ~ amgml... |
| amgmw2d 50430 | Weighted arithmetic-geomet... |
| young2d 50431 | Young's inequality for ` n... |
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