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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.... |
| impt 178 | Importation theorem ~ pm3.... |
| pm2.61d 179 | Deduction eliminating an a... |
| pm2.61d1 180 | Inference eliminating an a... |
| pm2.61d2 181 | Inference eliminating an a... |
| pm2.61i 182 | Inference eliminating an a... |
| pm2.61ii 183 | Inference eliminating two ... |
| pm2.61nii 184 | Inference eliminating two ... |
| pm2.61iii 185 | Inference eliminating thre... |
| ja 186 | Inference joining the ante... |
| jad 187 | Deduction form of ~ ja . ... |
| pm2.01 188 | Weak Clavius law. If a fo... |
| pm2.01d 189 | Deduction based on reducti... |
| pm2.6 190 | Theorem *2.6 of [Whitehead... |
| pm2.61 191 | Theorem *2.61 of [Whitehea... |
| pm2.65 192 | Theorem *2.65 of [Whitehea... |
| pm2.65i 193 | Inference for proof by con... |
| pm2.21dd 194 | A contradiction implies an... |
| pm2.65d 195 | Deduction for proof by con... |
| mto 196 | The rule of modus tollens.... |
| mtod 197 | Modus tollens deduction. ... |
| mtoi 198 | Modus tollens inference. ... |
| mt2 199 | A rule similar to modus to... |
| mt3 200 | A rule similar to modus to... |
| peirce 201 | Peirce's axiom. A non-int... |
| looinv 202 | The Inversion Axiom of the... |
| bijust0 203 | A self-implication (see ~ ... |
| bijust 204 | Theorem used to justify th... |
| impbi 207 | Property of the biconditio... |
| impbii 208 | Infer an equivalence from ... |
| impbidd 209 | Deduce an equivalence from... |
| impbid21d 210 | Deduce an equivalence from... |
| impbid 211 | Deduce an equivalence from... |
| dfbi1 212 | Relate the biconditional c... |
| dfbi1ALT 213 | Alternate proof of ~ dfbi1... |
| biimp 214 | Property of the biconditio... |
| biimpi 215 | Infer an implication from ... |
| sylbi 216 | A mixed syllogism inferenc... |
| sylib 217 | A mixed syllogism inferenc... |
| sylbb 218 | A mixed syllogism inferenc... |
| biimpr 219 | Property of the biconditio... |
| bicom1 220 | Commutative law for the bi... |
| bicom 221 | Commutative law for the bi... |
| bicomd 222 | Commute two sides of a bic... |
| bicomi 223 | Inference from commutative... |
| impbid1 224 | Infer an equivalence from ... |
| impbid2 225 | Infer an equivalence from ... |
| impcon4bid 226 | A variation on ~ impbid wi... |
| biimpri 227 | Infer a converse implicati... |
| biimpd 228 | Deduce an implication from... |
| mpbi 229 | An inference from a bicond... |
| mpbir 230 | An inference from a bicond... |
| mpbid 231 | A deduction from a bicondi... |
| mpbii 232 | An inference from a nested... |
| sylibr 233 | A mixed syllogism inferenc... |
| sylbir 234 | A mixed syllogism inferenc... |
| sylbbr 235 | A mixed syllogism inferenc... |
| sylbb1 236 | A mixed syllogism inferenc... |
| sylbb2 237 | A mixed syllogism inferenc... |
| sylibd 238 | A syllogism deduction. (C... |
| sylbid 239 | A syllogism deduction. (C... |
| mpbidi 240 | A deduction from a bicondi... |
| biimtrid 241 | A mixed syllogism inferenc... |
| biimtrrid 242 | A mixed syllogism inferenc... |
| imbitrid 243 | A mixed syllogism inferenc... |
| syl5ibcom 244 | A mixed syllogism inferenc... |
| imbitrrid 245 | A mixed syllogism inferenc... |
| syl5ibrcom 246 | A mixed syllogism inferenc... |
| biimprd 247 | Deduce a converse implicat... |
| biimpcd 248 | Deduce a commuted implicat... |
| biimprcd 249 | Deduce a converse commuted... |
| imbitrdi 250 | A mixed syllogism inferenc... |
| syl6ib 251 | A mixed syllogism inferenc... |
| syl6ibr 252 | A mixed syllogism inferenc... |
| syl6bi 253 | A mixed syllogism inferenc... |
| syl6bir 254 | A mixed syllogism inferenc... |
| syl7bi 255 | A mixed syllogism inferenc... |
| syl8ib 256 | A syllogism rule of infere... |
| mpbird 257 | A deduction from a bicondi... |
| mpbiri 258 | An inference from a nested... |
| sylibrd 259 | A syllogism deduction. (C... |
| sylbird 260 | A syllogism deduction. (C... |
| biid 261 | Principle of identity for ... |
| biidd 262 | Principle of identity with... |
| pm5.1im 263 | Two propositions are equiv... |
| 2th 264 | Two truths are equivalent.... |
| 2thd 265 | Two truths are equivalent.... |
| monothetic 266 | Two self-implications (see... |
| ibi 267 | Inference that converts a ... |
| ibir 268 | Inference that converts a ... |
| ibd 269 | Deduction that converts a ... |
| pm5.74 270 | Distribution of implicatio... |
| pm5.74i 271 | Distribution of implicatio... |
| pm5.74ri 272 | Distribution of implicatio... |
| pm5.74d 273 | Distribution of implicatio... |
| pm5.74rd 274 | Distribution of implicatio... |
| bitri 275 | An inference from transiti... |
| bitr2i 276 | An inference from transiti... |
| bitr3i 277 | An inference from transiti... |
| bitr4i 278 | An inference from transiti... |
| bitrd 279 | Deduction form of ~ bitri ... |
| bitr2d 280 | Deduction form of ~ bitr2i... |
| bitr3d 281 | Deduction form of ~ bitr3i... |
| bitr4d 282 | Deduction form of ~ bitr4i... |
| bitrid 283 | A syllogism inference from... |
| bitr2id 284 | A syllogism inference from... |
| bitr3id 285 | A syllogism inference from... |
| bitr3di 286 | A syllogism inference from... |
| bitrdi 287 | A syllogism inference from... |
| bitr2di 288 | A syllogism inference from... |
| bitr4di 289 | A syllogism inference from... |
| bitr4id 290 | A syllogism inference from... |
| 3imtr3i 291 | A mixed syllogism inferenc... |
| 3imtr4i 292 | A mixed syllogism inferenc... |
| 3imtr3d 293 | More general version of ~ ... |
| 3imtr4d 294 | More general version of ~ ... |
| 3imtr3g 295 | More general version of ~ ... |
| 3imtr4g 296 | More general version of ~ ... |
| 3bitri 297 | A chained inference from t... |
| 3bitrri 298 | A chained inference from t... |
| 3bitr2i 299 | A chained inference from t... |
| 3bitr2ri 300 | A chained inference from t... |
| 3bitr3i 301 | A chained inference from t... |
| 3bitr3ri 302 | A chained inference from t... |
| 3bitr4i 303 | A chained inference from t... |
| 3bitr4ri 304 | A chained inference from t... |
| 3bitrd 305 | Deduction from transitivit... |
| 3bitrrd 306 | Deduction from transitivit... |
| 3bitr2d 307 | Deduction from transitivit... |
| 3bitr2rd 308 | Deduction from transitivit... |
| 3bitr3d 309 | Deduction from transitivit... |
| 3bitr3rd 310 | Deduction from transitivit... |
| 3bitr4d 311 | Deduction from transitivit... |
| 3bitr4rd 312 | Deduction from transitivit... |
| 3bitr3g 313 | More general version of ~ ... |
| 3bitr4g 314 | More general version of ~ ... |
| notnotb 315 | Double negation. Theorem ... |
| con34b 316 | A biconditional form of co... |
| con4bid 317 | A contraposition deduction... |
| notbid 318 | Deduction negating both si... |
| notbi 319 | Contraposition. Theorem *... |
| notbii 320 | Negate both sides of a log... |
| con4bii 321 | A contraposition inference... |
| mtbi 322 | An inference from a bicond... |
| mtbir 323 | An inference from a bicond... |
| mtbid 324 | A deduction from a bicondi... |
| mtbird 325 | A deduction from a bicondi... |
| mtbii 326 | An inference from a bicond... |
| mtbiri 327 | An inference from a bicond... |
| sylnib 328 | A mixed syllogism inferenc... |
| sylnibr 329 | A mixed syllogism inferenc... |
| sylnbi 330 | A mixed syllogism inferenc... |
| sylnbir 331 | A mixed syllogism inferenc... |
| xchnxbi 332 | Replacement of a subexpres... |
| xchnxbir 333 | Replacement of a subexpres... |
| xchbinx 334 | Replacement of a subexpres... |
| xchbinxr 335 | Replacement of a subexpres... |
| imbi2i 336 | Introduce an antecedent to... |
| jcndOLD 337 | Obsolete version of ~ jcnd... |
| bibi2i 338 | Inference adding a bicondi... |
| bibi1i 339 | Inference adding a bicondi... |
| bibi12i 340 | The equivalence of two equ... |
| imbi2d 341 | Deduction adding an antece... |
| imbi1d 342 | Deduction adding a consequ... |
| bibi2d 343 | Deduction adding a bicondi... |
| bibi1d 344 | Deduction adding a bicondi... |
| imbi12d 345 | Deduction joining two equi... |
| bibi12d 346 | Deduction joining two equi... |
| imbi12 347 | Closed form of ~ imbi12i .... |
| imbi1 348 | Theorem *4.84 of [Whitehea... |
| imbi2 349 | Theorem *4.85 of [Whitehea... |
| imbi1i 350 | Introduce a consequent to ... |
| imbi12i 351 | Join two logical equivalen... |
| bibi1 352 | Theorem *4.86 of [Whitehea... |
| bitr3 353 | Closed nested implication ... |
| con2bi 354 | Contraposition. Theorem *... |
| con2bid 355 | A contraposition deduction... |
| con1bid 356 | A contraposition deduction... |
| con1bii 357 | A contraposition inference... |
| con2bii 358 | A contraposition inference... |
| con1b 359 | Contraposition. Bidirecti... |
| con2b 360 | Contraposition. Bidirecti... |
| biimt 361 | A wff is equivalent to its... |
| pm5.5 362 | Theorem *5.5 of [Whitehead... |
| a1bi 363 | Inference introducing a th... |
| mt2bi 364 | A false consequent falsifi... |
| mtt 365 | Modus-tollens-like theorem... |
| imnot 366 | If a proposition is false,... |
| pm5.501 367 | Theorem *5.501 of [Whitehe... |
| ibib 368 | Implication in terms of im... |
| ibibr 369 | Implication in terms of im... |
| tbt 370 | A wff is equivalent to its... |
| nbn2 371 | The negation of a wff is e... |
| bibif 372 | Transfer negation via an e... |
| nbn 373 | The negation of a wff is e... |
| nbn3 374 | Transfer falsehood via equ... |
| pm5.21im 375 | Two propositions are equiv... |
| 2false 376 | Two falsehoods are equival... |
| 2falsed 377 | Two falsehoods are equival... |
| 2falsedOLD 378 | Obsolete version of ~ 2fal... |
| pm5.21ni 379 | Two propositions implying ... |
| pm5.21nii 380 | Eliminate an antecedent im... |
| pm5.21ndd 381 | Eliminate an antecedent im... |
| bija 382 | Combine antecedents into a... |
| pm5.18 383 | Theorem *5.18 of [Whitehea... |
| xor3 384 | Two ways to express "exclu... |
| nbbn 385 | Move negation outside of b... |
| biass 386 | Associative law for the bi... |
| biluk 387 | Lukasiewicz's shortest axi... |
| pm5.19 388 | Theorem *5.19 of [Whitehea... |
| bi2.04 389 | Logical equivalence of com... |
| pm5.4 390 | Antecedent absorption impl... |
| imdi 391 | Distributive law for impli... |
| pm5.41 392 | Theorem *5.41 of [Whitehea... |
| imbibi 393 | The antecedent of one side... |
| pm4.8 394 | Theorem *4.8 of [Whitehead... |
| pm4.81 395 | A formula is equivalent to... |
| imim21b 396 | Simplify an implication be... |
| pm4.63 399 | Theorem *4.63 of [Whitehea... |
| pm4.67 400 | Theorem *4.67 of [Whitehea... |
| imnan 401 | Express an implication in ... |
| imnani 402 | Infer an implication from ... |
| iman 403 | Implication in terms of co... |
| pm3.24 404 | Law of noncontradiction. ... |
| annim 405 | Express a conjunction in t... |
| pm4.61 406 | Theorem *4.61 of [Whitehea... |
| pm4.65 407 | Theorem *4.65 of [Whitehea... |
| imp 408 | Importation inference. (C... |
| impcom 409 | Importation inference with... |
| con3dimp 410 | Variant of ~ con3d with im... |
| mpnanrd 411 | Eliminate the right side o... |
| impd 412 | Importation deduction. (C... |
| impcomd 413 | Importation deduction with... |
| ex 414 | Exportation inference. (T... |
| expcom 415 | Exportation inference with... |
| expdcom 416 | Commuted form of ~ expd . ... |
| expd 417 | Exportation deduction. (C... |
| expcomd 418 | Deduction form of ~ expcom... |
| imp31 419 | An importation inference. ... |
| imp32 420 | An importation inference. ... |
| exp31 421 | An exportation inference. ... |
| exp32 422 | An exportation inference. ... |
| imp4b 423 | An importation inference. ... |
| imp4a 424 | An importation inference. ... |
| imp4c 425 | An importation inference. ... |
| imp4d 426 | An importation inference. ... |
| imp41 427 | An importation inference. ... |
| imp42 428 | An importation inference. ... |
| imp43 429 | An importation inference. ... |
| imp44 430 | An importation inference. ... |
| imp45 431 | An importation inference. ... |
| exp4b 432 | An exportation inference. ... |
| exp4a 433 | An exportation inference. ... |
| exp4c 434 | An exportation inference. ... |
| exp4d 435 | An exportation inference. ... |
| exp41 436 | An exportation inference. ... |
| exp42 437 | An exportation inference. ... |
| exp43 438 | An exportation inference. ... |
| exp44 439 | An exportation inference. ... |
| exp45 440 | An exportation inference. ... |
| imp5d 441 | An importation inference. ... |
| imp5a 442 | An importation inference. ... |
| imp5g 443 | An importation inference. ... |
| imp55 444 | An importation inference. ... |
| imp511 445 | An importation inference. ... |
| exp5c 446 | An exportation inference. ... |
| exp5j 447 | An exportation inference. ... |
| exp5l 448 | An exportation inference. ... |
| exp53 449 | An exportation inference. ... |
| pm3.3 450 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 451 | Theorem *3.31 (Imp) of [Wh... |
| impexp 452 | Import-export theorem. Pa... |
| impancom 453 | Mixed importation/commutat... |
| expdimp 454 | A deduction version of exp... |
| expimpd 455 | Exportation followed by a ... |
| impr 456 | Import a wff into a right ... |
| impl 457 | Export a wff from a left c... |
| expr 458 | Export a wff from a right ... |
| expl 459 | Export a wff from a left c... |
| ancoms 460 | Inference commuting conjun... |
| pm3.22 461 | Theorem *3.22 of [Whitehea... |
| ancom 462 | Commutative law for conjun... |
| ancomd 463 | Commutation of conjuncts i... |
| biancomi 464 | Commuting conjunction in a... |
| biancomd 465 | Commuting conjunction in a... |
| ancomst 466 | Closed form of ~ ancoms . ... |
| ancomsd 467 | Deduction commuting conjun... |
| anasss 468 | Associative law for conjun... |
| anassrs 469 | Associative law for conjun... |
| anass 470 | Associative law for conjun... |
| pm3.2 471 | Join antecedents with conj... |
| pm3.2i 472 | Infer conjunction of premi... |
| pm3.21 473 | Join antecedents with conj... |
| pm3.43i 474 | Nested conjunction of ante... |
| pm3.43 475 | Theorem *3.43 (Comp) of [W... |
| dfbi2 476 | A theorem similar to the s... |
| dfbi 477 | Definition ~ df-bi rewritt... |
| biimpa 478 | Importation inference from... |
| biimpar 479 | Importation inference from... |
| biimpac 480 | Importation inference from... |
| biimparc 481 | Importation inference from... |
| adantr 482 | Inference adding a conjunc... |
| adantl 483 | Inference adding a conjunc... |
| simpl 484 | Elimination of a conjunct.... |
| simpli 485 | Inference eliminating a co... |
| simpr 486 | Elimination of a conjunct.... |
| simpri 487 | Inference eliminating a co... |
| intnan 488 | Introduction of conjunct i... |
| intnanr 489 | Introduction of conjunct i... |
| intnand 490 | Introduction of conjunct i... |
| intnanrd 491 | Introduction of conjunct i... |
| adantld 492 | Deduction adding a conjunc... |
| adantrd 493 | Deduction adding a conjunc... |
| pm3.41 494 | Theorem *3.41 of [Whitehea... |
| pm3.42 495 | Theorem *3.42 of [Whitehea... |
| simpld 496 | Deduction eliminating a co... |
| simprd 497 | Deduction eliminating a co... |
| simprbi 498 | Deduction eliminating a co... |
| simplbi 499 | Deduction eliminating a co... |
| simprbda 500 | Deduction eliminating a co... |
| simplbda 501 | Deduction eliminating a co... |
| simplbi2 502 | Deduction eliminating a co... |
| simplbi2comt 503 | Closed form of ~ simplbi2c... |
| simplbi2com 504 | A deduction eliminating a ... |
| simpl2im 505 | Implication from an elimin... |
| simplbiim 506 | Implication from an elimin... |
| impel 507 | An inference for implicati... |
| mpan9 508 | Modus ponens conjoining di... |
| sylan9 509 | Nested syllogism inference... |
| sylan9r 510 | Nested syllogism inference... |
| sylan9bb 511 | Nested syllogism inference... |
| sylan9bbr 512 | Nested syllogism inference... |
| jca 513 | Deduce conjunction of the ... |
| jcad 514 | Deduction conjoining the c... |
| jca2 515 | Inference conjoining the c... |
| jca31 516 | Join three consequents. (... |
| jca32 517 | Join three consequents. (... |
| jcai 518 | Deduction replacing implic... |
| jcab 519 | Distributive law for impli... |
| pm4.76 520 | Theorem *4.76 of [Whitehea... |
| jctil 521 | Inference conjoining a the... |
| jctir 522 | Inference conjoining a the... |
| jccir 523 | Inference conjoining a con... |
| jccil 524 | Inference conjoining a con... |
| jctl 525 | Inference conjoining a the... |
| jctr 526 | Inference conjoining a the... |
| jctild 527 | Deduction conjoining a the... |
| jctird 528 | Deduction conjoining a the... |
| iba 529 | Introduction of antecedent... |
| ibar 530 | Introduction of antecedent... |
| biantru 531 | A wff is equivalent to its... |
| biantrur 532 | A wff is equivalent to its... |
| biantrud 533 | A wff is equivalent to its... |
| biantrurd 534 | A wff is equivalent to its... |
| bianfi 535 | A wff conjoined with false... |
| bianfd 536 | A wff conjoined with false... |
| baib 537 | Move conjunction outside o... |
| baibr 538 | Move conjunction outside o... |
| rbaibr 539 | Move conjunction outside o... |
| rbaib 540 | Move conjunction outside o... |
| baibd 541 | Move conjunction outside o... |
| rbaibd 542 | Move conjunction outside o... |
| bianabs 543 | Absorb a hypothesis into t... |
| pm5.44 544 | Theorem *5.44 of [Whitehea... |
| pm5.42 545 | Theorem *5.42 of [Whitehea... |
| ancl 546 | Conjoin antecedent to left... |
| anclb 547 | Conjoin antecedent to left... |
| ancr 548 | Conjoin antecedent to righ... |
| ancrb 549 | Conjoin antecedent to righ... |
| ancli 550 | Deduction conjoining antec... |
| ancri 551 | Deduction conjoining antec... |
| ancld 552 | Deduction conjoining antec... |
| ancrd 553 | Deduction conjoining antec... |
| impac 554 | Importation with conjuncti... |
| anc2l 555 | Conjoin antecedent to left... |
| anc2r 556 | Conjoin antecedent to righ... |
| anc2li 557 | Deduction conjoining antec... |
| anc2ri 558 | Deduction conjoining antec... |
| pm4.71 559 | Implication in terms of bi... |
| pm4.71r 560 | Implication in terms of bi... |
| pm4.71i 561 | Inference converting an im... |
| pm4.71ri 562 | Inference converting an im... |
| pm4.71d 563 | Deduction converting an im... |
| pm4.71rd 564 | Deduction converting an im... |
| pm4.24 565 | Theorem *4.24 of [Whitehea... |
| anidm 566 | Idempotent law for conjunc... |
| anidmdbi 567 | Conjunction idempotence wi... |
| anidms 568 | Inference from idempotent ... |
| imdistan 569 | Distribution of implicatio... |
| imdistani 570 | Distribution of implicatio... |
| imdistanri 571 | Distribution of implicatio... |
| imdistand 572 | Distribution of implicatio... |
| imdistanda 573 | Distribution of implicatio... |
| pm5.3 574 | Theorem *5.3 of [Whitehead... |
| pm5.32 575 | Distribution of implicatio... |
| pm5.32i 576 | Distribution of implicatio... |
| pm5.32ri 577 | Distribution of implicatio... |
| pm5.32d 578 | Distribution of implicatio... |
| pm5.32rd 579 | Distribution of implicatio... |
| pm5.32da 580 | Distribution of implicatio... |
| sylan 581 | A syllogism inference. (C... |
| sylanb 582 | A syllogism inference. (C... |
| sylanbr 583 | A syllogism inference. (C... |
| sylanbrc 584 | Syllogism inference. (Con... |
| syl2anc 585 | Syllogism inference combin... |
| syl2anc2 586 | Double syllogism inference... |
| sylancl 587 | Syllogism inference combin... |
| sylancr 588 | Syllogism inference combin... |
| sylancom 589 | Syllogism inference with c... |
| sylanblc 590 | Syllogism inference combin... |
| sylanblrc 591 | Syllogism inference combin... |
| syldan 592 | A syllogism deduction with... |
| sylbida 593 | A syllogism deduction. (C... |
| sylan2 594 | A syllogism inference. (C... |
| sylan2b 595 | A syllogism inference. (C... |
| sylan2br 596 | A syllogism inference. (C... |
| syl2an 597 | A double syllogism inferen... |
| syl2anr 598 | A double syllogism inferen... |
| syl2anb 599 | A double syllogism inferen... |
| syl2anbr 600 | A double syllogism inferen... |
| sylancb 601 | A syllogism inference comb... |
| sylancbr 602 | A syllogism inference comb... |
| syldanl 603 | A syllogism deduction with... |
| syland 604 | A syllogism deduction. (C... |
| sylani 605 | A syllogism inference. (C... |
| sylan2d 606 | A syllogism deduction. (C... |
| sylan2i 607 | A syllogism inference. (C... |
| syl2ani 608 | A syllogism inference. (C... |
| syl2and 609 | A syllogism deduction. (C... |
| anim12d 610 | Conjoin antecedents and co... |
| anim12d1 611 | Variant of ~ anim12d where... |
| anim1d 612 | Add a conjunct to right of... |
| anim2d 613 | Add a conjunct to left of ... |
| anim12i 614 | Conjoin antecedents and co... |
| anim12ci 615 | Variant of ~ anim12i with ... |
| anim1i 616 | Introduce conjunct to both... |
| anim1ci 617 | Introduce conjunct to both... |
| anim2i 618 | Introduce conjunct to both... |
| anim12ii 619 | Conjoin antecedents and co... |
| anim12dan 620 | Conjoin antecedents and co... |
| im2anan9 621 | Deduction joining nested i... |
| im2anan9r 622 | Deduction joining nested i... |
| pm3.45 623 | Theorem *3.45 (Fact) of [W... |
| anbi2i 624 | Introduce a left conjunct ... |
| anbi1i 625 | Introduce a right conjunct... |
| anbi2ci 626 | Variant of ~ anbi2i with c... |
| anbi1ci 627 | Variant of ~ anbi1i with c... |
| anbi12i 628 | Conjoin both sides of two ... |
| anbi12ci 629 | Variant of ~ anbi12i with ... |
| anbi2d 630 | Deduction adding a left co... |
| anbi1d 631 | Deduction adding a right c... |
| anbi12d 632 | Deduction joining two equi... |
| anbi1 633 | Introduce a right conjunct... |
| anbi2 634 | Introduce a left conjunct ... |
| anbi1cd 635 | Introduce a proposition as... |
| an2anr 636 | Double commutation in conj... |
| pm4.38 637 | Theorem *4.38 of [Whitehea... |
| bi2anan9 638 | Deduction joining two equi... |
| bi2anan9r 639 | Deduction joining two equi... |
| bi2bian9 640 | Deduction joining two bico... |
| bianass 641 | An inference to merge two ... |
| bianassc 642 | An inference to merge two ... |
| an21 643 | Swap two conjuncts. (Cont... |
| an12 644 | Swap two conjuncts. Note ... |
| an32 645 | A rearrangement of conjunc... |
| an13 646 | A rearrangement of conjunc... |
| an31 647 | A rearrangement of conjunc... |
| an12s 648 | Swap two conjuncts in ante... |
| ancom2s 649 | Inference commuting a nest... |
| an13s 650 | Swap two conjuncts in ante... |
| an32s 651 | Swap two conjuncts in ante... |
| ancom1s 652 | Inference commuting a nest... |
| an31s 653 | Swap two conjuncts in ante... |
| anass1rs 654 | Commutative-associative la... |
| an4 655 | Rearrangement of 4 conjunc... |
| an42 656 | Rearrangement of 4 conjunc... |
| an43 657 | Rearrangement of 4 conjunc... |
| an3 658 | A rearrangement of conjunc... |
| an4s 659 | Inference rearranging 4 co... |
| an42s 660 | Inference rearranging 4 co... |
| anabs1 661 | Absorption into embedded c... |
| anabs5 662 | Absorption into embedded c... |
| anabs7 663 | Absorption into embedded c... |
| anabsan 664 | Absorption of antecedent w... |
| anabss1 665 | Absorption of antecedent i... |
| anabss4 666 | Absorption of antecedent i... |
| anabss5 667 | Absorption of antecedent i... |
| anabsi5 668 | Absorption of antecedent i... |
| anabsi6 669 | Absorption of antecedent i... |
| anabsi7 670 | Absorption of antecedent i... |
| anabsi8 671 | Absorption of antecedent i... |
| anabss7 672 | Absorption of antecedent i... |
| anabsan2 673 | Absorption of antecedent w... |
| anabss3 674 | Absorption of antecedent i... |
| anandi 675 | Distribution of conjunctio... |
| anandir 676 | Distribution of conjunctio... |
| anandis 677 | Inference that undistribut... |
| anandirs 678 | Inference that undistribut... |
| sylanl1 679 | A syllogism inference. (C... |
| sylanl2 680 | A syllogism inference. (C... |
| sylanr1 681 | A syllogism inference. (C... |
| sylanr2 682 | A syllogism inference. (C... |
| syl6an 683 | A syllogism deduction comb... |
| syl2an2r 684 | ~ syl2anr with antecedents... |
| syl2an2 685 | ~ syl2an with antecedents ... |
| mpdan 686 | An inference based on modu... |
| mpancom 687 | An inference based on modu... |
| mpidan 688 | A deduction which "stacks"... |
| mpan 689 | An inference based on modu... |
| mpan2 690 | An inference based on modu... |
| mp2an 691 | An inference based on modu... |
| mp4an 692 | An inference based on modu... |
| mpan2d 693 | A deduction based on modus... |
| mpand 694 | A deduction based on modus... |
| mpani 695 | An inference based on modu... |
| mpan2i 696 | An inference based on modu... |
| mp2ani 697 | An inference based on modu... |
| mp2and 698 | A deduction based on modus... |
| mpanl1 699 | An inference based on modu... |
| mpanl2 700 | An inference based on modu... |
| mpanl12 701 | An inference based on modu... |
| mpanr1 702 | An inference based on modu... |
| mpanr2 703 | An inference based on modu... |
| mpanr12 704 | An inference based on modu... |
| mpanlr1 705 | An inference based on modu... |
| mpbirand 706 | Detach truth from conjunct... |
| mpbiran2d 707 | Detach truth from conjunct... |
| mpbiran 708 | Detach truth from conjunct... |
| mpbiran2 709 | Detach truth from conjunct... |
| mpbir2an 710 | Detach a conjunction of tr... |
| mpbi2and 711 | Detach a conjunction of tr... |
| mpbir2and 712 | Detach a conjunction of tr... |
| adantll 713 | Deduction adding a conjunc... |
| adantlr 714 | Deduction adding a conjunc... |
| adantrl 715 | Deduction adding a conjunc... |
| adantrr 716 | Deduction adding a conjunc... |
| adantlll 717 | Deduction adding a conjunc... |
| adantllr 718 | Deduction adding a conjunc... |
| adantlrl 719 | Deduction adding a conjunc... |
| adantlrr 720 | Deduction adding a conjunc... |
| adantrll 721 | Deduction adding a conjunc... |
| adantrlr 722 | Deduction adding a conjunc... |
| adantrrl 723 | Deduction adding a conjunc... |
| adantrrr 724 | Deduction adding a conjunc... |
| ad2antrr 725 | Deduction adding two conju... |
| ad2antlr 726 | Deduction adding two conju... |
| ad2antrl 727 | Deduction adding two conju... |
| ad2antll 728 | Deduction adding conjuncts... |
| ad3antrrr 729 | Deduction adding three con... |
| ad3antlr 730 | Deduction adding three con... |
| ad4antr 731 | Deduction adding 4 conjunc... |
| ad4antlr 732 | Deduction adding 4 conjunc... |
| ad5antr 733 | Deduction adding 5 conjunc... |
| ad5antlr 734 | Deduction adding 5 conjunc... |
| ad6antr 735 | Deduction adding 6 conjunc... |
| ad6antlr 736 | Deduction adding 6 conjunc... |
| ad7antr 737 | Deduction adding 7 conjunc... |
| ad7antlr 738 | Deduction adding 7 conjunc... |
| ad8antr 739 | Deduction adding 8 conjunc... |
| ad8antlr 740 | Deduction adding 8 conjunc... |
| ad9antr 741 | Deduction adding 9 conjunc... |
| ad9antlr 742 | Deduction adding 9 conjunc... |
| ad10antr 743 | Deduction adding 10 conjun... |
| ad10antlr 744 | Deduction adding 10 conjun... |
| ad2ant2l 745 | Deduction adding two conju... |
| ad2ant2r 746 | Deduction adding two conju... |
| ad2ant2lr 747 | Deduction adding two conju... |
| ad2ant2rl 748 | Deduction adding two conju... |
| adantl3r 749 | Deduction adding 1 conjunc... |
| ad4ant13 750 | Deduction adding conjuncts... |
| ad4ant14 751 | Deduction adding conjuncts... |
| ad4ant23 752 | Deduction adding conjuncts... |
| ad4ant24 753 | Deduction adding conjuncts... |
| adantl4r 754 | Deduction adding 1 conjunc... |
| ad5ant12 755 | Deduction adding conjuncts... |
| ad5ant13 756 | Deduction adding conjuncts... |
| ad5ant14 757 | Deduction adding conjuncts... |
| ad5ant15 758 | Deduction adding conjuncts... |
| ad5ant23 759 | Deduction adding conjuncts... |
| ad5ant24 760 | Deduction adding conjuncts... |
| ad5ant25 761 | Deduction adding conjuncts... |
| adantl5r 762 | Deduction adding 1 conjunc... |
| adantl6r 763 | Deduction adding 1 conjunc... |
| pm3.33 764 | Theorem *3.33 (Syll) of [W... |
| pm3.34 765 | Theorem *3.34 (Syll) of [W... |
| simpll 766 | Simplification of a conjun... |
| simplld 767 | Deduction form of ~ simpll... |
| simplr 768 | Simplification of a conjun... |
| simplrd 769 | Deduction eliminating a do... |
| simprl 770 | Simplification of a conjun... |
| simprld 771 | Deduction eliminating a do... |
| simprr 772 | Simplification of a conjun... |
| simprrd 773 | Deduction form of ~ simprr... |
| simplll 774 | Simplification of a conjun... |
| simpllr 775 | Simplification of a conjun... |
| simplrl 776 | Simplification of a conjun... |
| simplrr 777 | Simplification of a conjun... |
| simprll 778 | Simplification of a conjun... |
| simprlr 779 | Simplification of a conjun... |
| simprrl 780 | Simplification of a conjun... |
| simprrr 781 | Simplification of a conjun... |
| simp-4l 782 | Simplification of a conjun... |
| simp-4r 783 | Simplification of a conjun... |
| simp-5l 784 | Simplification of a conjun... |
| simp-5r 785 | Simplification of a conjun... |
| simp-6l 786 | Simplification of a conjun... |
| simp-6r 787 | Simplification of a conjun... |
| simp-7l 788 | Simplification of a conjun... |
| simp-7r 789 | Simplification of a conjun... |
| simp-8l 790 | Simplification of a conjun... |
| simp-8r 791 | Simplification of a conjun... |
| simp-9l 792 | Simplification of a conjun... |
| simp-9r 793 | Simplification of a conjun... |
| simp-10l 794 | Simplification of a conjun... |
| simp-10r 795 | Simplification of a conjun... |
| simp-11l 796 | Simplification of a conjun... |
| simp-11r 797 | Simplification of a conjun... |
| pm2.01da 798 | Deduction based on reducti... |
| pm2.18da 799 | Deduction based on reducti... |
| impbida 800 | Deduce an equivalence from... |
| pm5.21nd 801 | Eliminate an antecedent im... |
| pm3.35 802 | Conjunctive detachment. T... |
| pm5.74da 803 | Distribution of implicatio... |
| bitr 804 | Theorem *4.22 of [Whitehea... |
| biantr 805 | A transitive law of equiva... |
| pm4.14 806 | Theorem *4.14 of [Whitehea... |
| pm3.37 807 | Theorem *3.37 (Transp) of ... |
| anim12 808 | Conjoin antecedents and co... |
| pm3.4 809 | Conjunction implies implic... |
| exbiri 810 | Inference form of ~ exbir ... |
| pm2.61ian 811 | Elimination of an antecede... |
| pm2.61dan 812 | Elimination of an antecede... |
| pm2.61ddan 813 | Elimination of two anteced... |
| pm2.61dda 814 | Elimination of two anteced... |
| mtand 815 | A modus tollens deduction.... |
| pm2.65da 816 | Deduction for proof by con... |
| condan 817 | Proof by contradiction. (... |
| biadan 818 | An implication is equivale... |
| biadani 819 | Inference associated with ... |
| biadaniALT 820 | Alternate proof of ~ biada... |
| biadanii 821 | Inference associated with ... |
| biadanid 822 | Deduction associated with ... |
| pm5.1 823 | Two propositions are equiv... |
| pm5.21 824 | Two propositions are equiv... |
| pm5.35 825 | Theorem *5.35 of [Whitehea... |
| abai 826 | Introduce one conjunct as ... |
| pm4.45im 827 | Conjunction with implicati... |
| impimprbi 828 | An implication and its rev... |
| nan 829 | Theorem to move a conjunct... |
| pm5.31 830 | Theorem *5.31 of [Whitehea... |
| pm5.31r 831 | Variant of ~ pm5.31 . (Co... |
| pm4.15 832 | Theorem *4.15 of [Whitehea... |
| pm5.36 833 | Theorem *5.36 of [Whitehea... |
| annotanannot 834 | A conjunction with a negat... |
| pm5.33 835 | Theorem *5.33 of [Whitehea... |
| syl12anc 836 | Syllogism combined with co... |
| syl21anc 837 | Syllogism combined with co... |
| syl22anc 838 | Syllogism combined with co... |
| syl1111anc 839 | Four-hypothesis eliminatio... |
| syldbl2 840 | Stacked hypotheseis implie... |
| mpsyl4anc 841 | An elimination deduction. ... |
| pm4.87 842 | Theorem *4.87 of [Whitehea... |
| bimsc1 843 | Removal of conjunct from o... |
| a2and 844 | Deduction distributing a c... |
| animpimp2impd 845 | Deduction deriving nested ... |
| pm4.64 848 | Theorem *4.64 of [Whitehea... |
| pm4.66 849 | Theorem *4.66 of [Whitehea... |
| pm2.53 850 | Theorem *2.53 of [Whitehea... |
| pm2.54 851 | Theorem *2.54 of [Whitehea... |
| imor 852 | Implication in terms of di... |
| imori 853 | Infer disjunction from imp... |
| imorri 854 | Infer implication from dis... |
| pm4.62 855 | Theorem *4.62 of [Whitehea... |
| jaoi 856 | Inference disjoining the a... |
| jao1i 857 | Add a disjunct in the ante... |
| jaod 858 | Deduction disjoining the a... |
| mpjaod 859 | Eliminate a disjunction in... |
| ori 860 | Infer implication from dis... |
| orri 861 | Infer disjunction from imp... |
| orrd 862 | Deduce disjunction from im... |
| ord 863 | Deduce implication from di... |
| orci 864 | Deduction introducing a di... |
| olci 865 | Deduction introducing a di... |
| orc 866 | Introduction of a disjunct... |
| olc 867 | Introduction of a disjunct... |
| pm1.4 868 | Axiom *1.4 of [WhiteheadRu... |
| orcom 869 | Commutative law for disjun... |
| orcomd 870 | Commutation of disjuncts i... |
| orcoms 871 | Commutation of disjuncts i... |
| orcd 872 | Deduction introducing a di... |
| olcd 873 | Deduction introducing a di... |
| orcs 874 | Deduction eliminating disj... |
| olcs 875 | Deduction eliminating disj... |
| olcnd 876 | A lemma for Conjunctive No... |
| unitreslOLD 877 | Obsolete version of ~ olcn... |
| orcnd 878 | A lemma for Conjunctive No... |
| mtord 879 | A modus tollens deduction ... |
| pm3.2ni 880 | Infer negated disjunction ... |
| pm2.45 881 | Theorem *2.45 of [Whitehea... |
| pm2.46 882 | Theorem *2.46 of [Whitehea... |
| pm2.47 883 | Theorem *2.47 of [Whitehea... |
| pm2.48 884 | Theorem *2.48 of [Whitehea... |
| pm2.49 885 | Theorem *2.49 of [Whitehea... |
| norbi 886 | If neither of two proposit... |
| nbior 887 | If two propositions are no... |
| orel1 888 | Elimination of disjunction... |
| pm2.25 889 | Theorem *2.25 of [Whitehea... |
| orel2 890 | Elimination of disjunction... |
| pm2.67-2 891 | Slight generalization of T... |
| pm2.67 892 | Theorem *2.67 of [Whitehea... |
| curryax 893 | A non-intuitionistic posit... |
| exmid 894 | Law of excluded middle, al... |
| exmidd 895 | Law of excluded middle in ... |
| pm2.1 896 | Theorem *2.1 of [Whitehead... |
| pm2.13 897 | Theorem *2.13 of [Whitehea... |
| pm2.621 898 | Theorem *2.621 of [Whitehe... |
| pm2.62 899 | Theorem *2.62 of [Whitehea... |
| pm2.68 900 | Theorem *2.68 of [Whitehea... |
| dfor2 901 | Logical 'or' expressed in ... |
| pm2.07 902 | Theorem *2.07 of [Whitehea... |
| pm1.2 903 | Axiom *1.2 of [WhiteheadRu... |
| oridm 904 | Idempotent law for disjunc... |
| pm4.25 905 | Theorem *4.25 of [Whitehea... |
| pm2.4 906 | Theorem *2.4 of [Whitehead... |
| pm2.41 907 | Theorem *2.41 of [Whitehea... |
| orim12i 908 | Disjoin antecedents and co... |
| orim1i 909 | Introduce disjunct to both... |
| orim2i 910 | Introduce disjunct to both... |
| orim12dALT 911 | Alternate proof of ~ orim1... |
| orbi2i 912 | Inference adding a left di... |
| orbi1i 913 | Inference adding a right d... |
| orbi12i 914 | Infer the disjunction of t... |
| orbi2d 915 | Deduction adding a left di... |
| orbi1d 916 | Deduction adding a right d... |
| orbi1 917 | Theorem *4.37 of [Whitehea... |
| orbi12d 918 | Deduction joining two equi... |
| pm1.5 919 | Axiom *1.5 (Assoc) of [Whi... |
| or12 920 | Swap two disjuncts. (Cont... |
| orass 921 | Associative law for disjun... |
| pm2.31 922 | Theorem *2.31 of [Whitehea... |
| pm2.32 923 | Theorem *2.32 of [Whitehea... |
| pm2.3 924 | Theorem *2.3 of [Whitehead... |
| or32 925 | A rearrangement of disjunc... |
| or4 926 | Rearrangement of 4 disjunc... |
| or42 927 | Rearrangement of 4 disjunc... |
| orordi 928 | Distribution of disjunctio... |
| orordir 929 | Distribution of disjunctio... |
| orimdi 930 | Disjunction distributes ov... |
| pm2.76 931 | Theorem *2.76 of [Whitehea... |
| pm2.85 932 | Theorem *2.85 of [Whitehea... |
| pm2.75 933 | Theorem *2.75 of [Whitehea... |
| pm4.78 934 | Implication distributes ov... |
| biort 935 | A disjunction with a true ... |
| biorf 936 | A wff is equivalent to its... |
| biortn 937 | A wff is equivalent to its... |
| biorfi 938 | A wff is equivalent to its... |
| pm2.26 939 | Theorem *2.26 of [Whitehea... |
| pm2.63 940 | Theorem *2.63 of [Whitehea... |
| pm2.64 941 | Theorem *2.64 of [Whitehea... |
| pm2.42 942 | Theorem *2.42 of [Whitehea... |
| pm5.11g 943 | A general instance of Theo... |
| pm5.11 944 | Theorem *5.11 of [Whitehea... |
| pm5.12 945 | Theorem *5.12 of [Whitehea... |
| pm5.14 946 | Theorem *5.14 of [Whitehea... |
| pm5.13 947 | Theorem *5.13 of [Whitehea... |
| pm5.55 948 | Theorem *5.55 of [Whitehea... |
| pm4.72 949 | Implication in terms of bi... |
| imimorb 950 | Simplify an implication be... |
| oibabs 951 | Absorption of disjunction ... |
| orbidi 952 | Disjunction distributes ov... |
| pm5.7 953 | Disjunction distributes ov... |
| jaao 954 | Inference conjoining and d... |
| jaoa 955 | Inference disjoining and c... |
| jaoian 956 | Inference disjoining the a... |
| jaodan 957 | Deduction disjoining the a... |
| mpjaodan 958 | Eliminate a disjunction in... |
| pm3.44 959 | Theorem *3.44 of [Whitehea... |
| jao 960 | Disjunction of antecedents... |
| jaob 961 | Disjunction of antecedents... |
| pm4.77 962 | Theorem *4.77 of [Whitehea... |
| pm3.48 963 | Theorem *3.48 of [Whitehea... |
| orim12d 964 | Disjoin antecedents and co... |
| orim1d 965 | Disjoin antecedents and co... |
| orim2d 966 | Disjoin antecedents and co... |
| orim2 967 | Axiom *1.6 (Sum) of [White... |
| pm2.38 968 | Theorem *2.38 of [Whitehea... |
| pm2.36 969 | Theorem *2.36 of [Whitehea... |
| pm2.37 970 | Theorem *2.37 of [Whitehea... |
| pm2.81 971 | Theorem *2.81 of [Whitehea... |
| pm2.8 972 | Theorem *2.8 of [Whitehead... |
| pm2.73 973 | Theorem *2.73 of [Whitehea... |
| pm2.74 974 | Theorem *2.74 of [Whitehea... |
| pm2.82 975 | Theorem *2.82 of [Whitehea... |
| pm4.39 976 | Theorem *4.39 of [Whitehea... |
| animorl 977 | Conjunction implies disjun... |
| animorr 978 | Conjunction implies disjun... |
| animorlr 979 | Conjunction implies disjun... |
| animorrl 980 | Conjunction implies disjun... |
| ianor 981 | Negated conjunction in ter... |
| anor 982 | Conjunction in terms of di... |
| ioran 983 | Negated disjunction in ter... |
| pm4.52 984 | Theorem *4.52 of [Whitehea... |
| pm4.53 985 | Theorem *4.53 of [Whitehea... |
| pm4.54 986 | Theorem *4.54 of [Whitehea... |
| pm4.55 987 | Theorem *4.55 of [Whitehea... |
| pm4.56 988 | Theorem *4.56 of [Whitehea... |
| oran 989 | Disjunction in terms of co... |
| pm4.57 990 | Theorem *4.57 of [Whitehea... |
| pm3.1 991 | Theorem *3.1 of [Whitehead... |
| pm3.11 992 | Theorem *3.11 of [Whitehea... |
| pm3.12 993 | Theorem *3.12 of [Whitehea... |
| pm3.13 994 | Theorem *3.13 of [Whitehea... |
| pm3.14 995 | Theorem *3.14 of [Whitehea... |
| pm4.44 996 | Theorem *4.44 of [Whitehea... |
| pm4.45 997 | Theorem *4.45 of [Whitehea... |
| orabs 998 | Absorption of redundant in... |
| oranabs 999 | Absorb a disjunct into a c... |
| pm5.61 1000 | Theorem *5.61 of [Whitehea... |
| pm5.6 1001 | Conjunction in antecedent ... |
| orcanai 1002 | Change disjunction in cons... |
| pm4.79 1003 | Theorem *4.79 of [Whitehea... |
| pm5.53 1004 | Theorem *5.53 of [Whitehea... |
| ordi 1005 | Distributive law for disju... |
| ordir 1006 | Distributive law for disju... |
| andi 1007 | Distributive law for conju... |
| andir 1008 | Distributive law for conju... |
| orddi 1009 | Double distributive law fo... |
| anddi 1010 | Double distributive law fo... |
| pm5.17 1011 | Theorem *5.17 of [Whitehea... |
| pm5.15 1012 | Theorem *5.15 of [Whitehea... |
| pm5.16 1013 | Theorem *5.16 of [Whitehea... |
| xor 1014 | Two ways to express exclus... |
| nbi2 1015 | Two ways to express "exclu... |
| xordi 1016 | Conjunction distributes ov... |
| pm5.54 1017 | Theorem *5.54 of [Whitehea... |
| pm5.62 1018 | Theorem *5.62 of [Whitehea... |
| pm5.63 1019 | Theorem *5.63 of [Whitehea... |
| niabn 1020 | Miscellaneous inference re... |
| ninba 1021 | Miscellaneous inference re... |
| pm4.43 1022 | Theorem *4.43 of [Whitehea... |
| pm4.82 1023 | Theorem *4.82 of [Whitehea... |
| pm4.83 1024 | Theorem *4.83 of [Whitehea... |
| pclem6 1025 | Negation inferred from emb... |
| bigolden 1026 | Dijkstra-Scholten's Golden... |
| pm5.71 1027 | Theorem *5.71 of [Whitehea... |
| pm5.75 1028 | Theorem *5.75 of [Whitehea... |
| ecase2d 1029 | Deduction for elimination ... |
| ecase2dOLD 1030 | Obsolete version of ~ ecas... |
| ecase3 1031 | Inference for elimination ... |
| ecase 1032 | Inference for elimination ... |
| ecase3d 1033 | Deduction for elimination ... |
| ecased 1034 | Deduction for elimination ... |
| ecase3ad 1035 | Deduction for elimination ... |
| ecase3adOLD 1036 | Obsolete version of ~ ecas... |
| ccase 1037 | Inference for combining ca... |
| ccased 1038 | Deduction for combining ca... |
| ccase2 1039 | Inference for combining ca... |
| 4cases 1040 | Inference eliminating two ... |
| 4casesdan 1041 | Deduction eliminating two ... |
| cases 1042 | Case disjunction according... |
| dedlem0a 1043 | Lemma for an alternate ver... |
| dedlem0b 1044 | Lemma for an alternate ver... |
| dedlema 1045 | Lemma for weak deduction t... |
| dedlemb 1046 | Lemma for weak deduction t... |
| cases2 1047 | Case disjunction according... |
| cases2ALT 1048 | Alternate proof of ~ cases... |
| dfbi3 1049 | An alternate definition of... |
| pm5.24 1050 | Theorem *5.24 of [Whitehea... |
| 4exmid 1051 | The disjunction of the fou... |
| consensus 1052 | The consensus theorem. Th... |
| pm4.42 1053 | Theorem *4.42 of [Whitehea... |
| prlem1 1054 | A specialized lemma for se... |
| prlem2 1055 | A specialized lemma for se... |
| oplem1 1056 | A specialized lemma for se... |
| dn1 1057 | A single axiom for Boolean... |
| bianir 1058 | A closed form of ~ mpbir ,... |
| jaoi2 1059 | Inference removing a negat... |
| jaoi3 1060 | Inference separating a dis... |
| ornld 1061 | Selecting one statement fr... |
| dfifp2 1064 | Alternate definition of th... |
| dfifp3 1065 | Alternate definition of th... |
| dfifp4 1066 | Alternate definition of th... |
| dfifp5 1067 | Alternate definition of th... |
| dfifp6 1068 | Alternate definition of th... |
| dfifp7 1069 | Alternate definition of th... |
| ifpdfbi 1070 | Define the biconditional a... |
| anifp 1071 | The conditional operator i... |
| ifpor 1072 | The conditional operator i... |
| ifpn 1073 | Conditional operator for t... |
| ifpnOLD 1074 | Obsolete version of ~ ifpn... |
| ifptru 1075 | Value of the conditional o... |
| ifpfal 1076 | Value of the conditional o... |
| ifpid 1077 | Value of the conditional o... |
| casesifp 1078 | Version of ~ cases express... |
| ifpbi123d 1079 | Equivalence deduction for ... |
| ifpbi123dOLD 1080 | Obsolete version of ~ ifpb... |
| ifpbi23d 1081 | Equivalence deduction for ... |
| ifpimpda 1082 | Separation of the values o... |
| 1fpid3 1083 | The value of the condition... |
| elimh 1084 | Hypothesis builder for the... |
| dedt 1085 | The weak deduction theorem... |
| con3ALT 1086 | Proof of ~ con3 from its a... |
| 3orass 1091 | Associative law for triple... |
| 3orel1 1092 | Partial elimination of a t... |
| 3orrot 1093 | Rotation law for triple di... |
| 3orcoma 1094 | Commutation law for triple... |
| 3orcomb 1095 | Commutation law for triple... |
| 3anass 1096 | Associative law for triple... |
| 3anan12 1097 | Convert triple conjunction... |
| 3anan32 1098 | Convert triple conjunction... |
| 3ancoma 1099 | Commutation law for triple... |
| 3ancomb 1100 | Commutation law for triple... |
| 3anrot 1101 | Rotation law for triple co... |
| 3anrev 1102 | Reversal law for triple co... |
| anandi3 1103 | Distribution of triple con... |
| anandi3r 1104 | Distribution of triple con... |
| 3anidm 1105 | Idempotent law for conjunc... |
| 3an4anass 1106 | Associative law for four c... |
| 3ioran 1107 | Negated triple disjunction... |
| 3ianor 1108 | Negated triple conjunction... |
| 3anor 1109 | Triple conjunction express... |
| 3oran 1110 | Triple disjunction in term... |
| 3impa 1111 | Importation from double to... |
| 3imp 1112 | Importation inference. (C... |
| 3imp31 1113 | The importation inference ... |
| 3imp231 1114 | Importation inference. (C... |
| 3imp21 1115 | The importation inference ... |
| 3impb 1116 | Importation from double to... |
| 3impib 1117 | Importation to triple conj... |
| 3impia 1118 | Importation to triple conj... |
| 3expa 1119 | Exportation from triple to... |
| 3exp 1120 | Exportation inference. (C... |
| 3expb 1121 | Exportation from triple to... |
| 3expia 1122 | Exportation from triple co... |
| 3expib 1123 | Exportation from triple co... |
| 3com12 1124 | Commutation in antecedent.... |
| 3com13 1125 | Commutation in antecedent.... |
| 3comr 1126 | Commutation in antecedent.... |
| 3com23 1127 | Commutation in antecedent.... |
| 3coml 1128 | Commutation in antecedent.... |
| 3jca 1129 | Join consequents with conj... |
| 3jcad 1130 | Deduction conjoining the c... |
| 3adant1 1131 | Deduction adding a conjunc... |
| 3adant2 1132 | Deduction adding a conjunc... |
| 3adant3 1133 | Deduction adding a conjunc... |
| 3ad2ant1 1134 | Deduction adding conjuncts... |
| 3ad2ant2 1135 | Deduction adding conjuncts... |
| 3ad2ant3 1136 | Deduction adding conjuncts... |
| simp1 1137 | Simplification of triple c... |
| simp2 1138 | Simplification of triple c... |
| simp3 1139 | Simplification of triple c... |
| simp1i 1140 | Infer a conjunct from a tr... |
| simp2i 1141 | Infer a conjunct from a tr... |
| simp3i 1142 | Infer a conjunct from a tr... |
| simp1d 1143 | Deduce a conjunct from a t... |
| simp2d 1144 | Deduce a conjunct from a t... |
| simp3d 1145 | Deduce a conjunct from a t... |
| simp1bi 1146 | Deduce a conjunct from a t... |
| simp2bi 1147 | Deduce a conjunct from a t... |
| simp3bi 1148 | Deduce a conjunct from a t... |
| 3simpa 1149 | Simplification of triple c... |
| 3simpb 1150 | Simplification of triple c... |
| 3simpc 1151 | Simplification of triple c... |
| 3anim123i 1152 | Join antecedents and conse... |
| 3anim1i 1153 | Add two conjuncts to antec... |
| 3anim2i 1154 | Add two conjuncts to antec... |
| 3anim3i 1155 | Add two conjuncts to antec... |
| 3anbi123i 1156 | Join 3 biconditionals with... |
| 3orbi123i 1157 | Join 3 biconditionals with... |
| 3anbi1i 1158 | Inference adding two conju... |
| 3anbi2i 1159 | Inference adding two conju... |
| 3anbi3i 1160 | Inference adding two conju... |
| syl3an 1161 | A triple syllogism inferen... |
| syl3anb 1162 | A triple syllogism inferen... |
| syl3anbr 1163 | A triple syllogism inferen... |
| syl3an1 1164 | A syllogism inference. (C... |
| syl3an2 1165 | A syllogism inference. (C... |
| syl3an3 1166 | A syllogism inference. (C... |
| 3adantl1 1167 | Deduction adding a conjunc... |
| 3adantl2 1168 | Deduction adding a conjunc... |
| 3adantl3 1169 | Deduction adding a conjunc... |
| 3adantr1 1170 | Deduction adding a conjunc... |
| 3adantr2 1171 | Deduction adding a conjunc... |
| 3adantr3 1172 | Deduction adding a conjunc... |
| ad4ant123 1173 | Deduction adding conjuncts... |
| ad4ant124 1174 | Deduction adding conjuncts... |
| ad4ant134 1175 | Deduction adding conjuncts... |
| ad4ant234 1176 | Deduction adding conjuncts... |
| 3adant1l 1177 | Deduction adding a conjunc... |
| 3adant1r 1178 | Deduction adding a conjunc... |
| 3adant2l 1179 | Deduction adding a conjunc... |
| 3adant2r 1180 | Deduction adding a conjunc... |
| 3adant3l 1181 | Deduction adding a conjunc... |
| 3adant3r 1182 | Deduction adding a conjunc... |
| 3adant3r1 1183 | Deduction adding a conjunc... |
| 3adant3r2 1184 | Deduction adding a conjunc... |
| 3adant3r3 1185 | Deduction adding a conjunc... |
| 3ad2antl1 1186 | Deduction adding conjuncts... |
| 3ad2antl2 1187 | Deduction adding conjuncts... |
| 3ad2antl3 1188 | Deduction adding conjuncts... |
| 3ad2antr1 1189 | Deduction adding conjuncts... |
| 3ad2antr2 1190 | Deduction adding conjuncts... |
| 3ad2antr3 1191 | Deduction adding conjuncts... |
| simpl1 1192 | Simplification of conjunct... |
| simpl2 1193 | Simplification of conjunct... |
| simpl3 1194 | Simplification of conjunct... |
| simpr1 1195 | Simplification of conjunct... |
| simpr2 1196 | Simplification of conjunct... |
| simpr3 1197 | Simplification of conjunct... |
| simp1l 1198 | Simplification of triple c... |
| simp1r 1199 | Simplification of triple c... |
| simp2l 1200 | Simplification of triple c... |
| simp2r 1201 | Simplification of triple c... |
| simp3l 1202 | Simplification of triple c... |
| simp3r 1203 | Simplification of triple c... |
| simp11 1204 | Simplification of doubly t... |
| simp12 1205 | Simplification of doubly t... |
| simp13 1206 | Simplification of doubly t... |
| simp21 1207 | Simplification of doubly t... |
| simp22 1208 | Simplification of doubly t... |
| simp23 1209 | Simplification of doubly t... |
| simp31 1210 | Simplification of doubly t... |
| simp32 1211 | Simplification of doubly t... |
| simp33 1212 | Simplification of doubly t... |
| simpll1 1213 | Simplification of conjunct... |
| simpll2 1214 | Simplification of conjunct... |
| simpll3 1215 | Simplification of conjunct... |
| simplr1 1216 | Simplification of conjunct... |
| simplr2 1217 | Simplification of conjunct... |
| simplr3 1218 | Simplification of conjunct... |
| simprl1 1219 | Simplification of conjunct... |
| simprl2 1220 | Simplification of conjunct... |
| simprl3 1221 | Simplification of conjunct... |
| simprr1 1222 | Simplification of conjunct... |
| simprr2 1223 | Simplification of conjunct... |
| simprr3 1224 | Simplification of conjunct... |
| simpl1l 1225 | Simplification of conjunct... |
| simpl1r 1226 | Simplification of conjunct... |
| simpl2l 1227 | Simplification of conjunct... |
| simpl2r 1228 | Simplification of conjunct... |
| simpl3l 1229 | Simplification of conjunct... |
| simpl3r 1230 | Simplification of conjunct... |
| simpr1l 1231 | Simplification of conjunct... |
| simpr1r 1232 | Simplification of conjunct... |
| simpr2l 1233 | Simplification of conjunct... |
| simpr2r 1234 | Simplification of conjunct... |
| simpr3l 1235 | Simplification of conjunct... |
| simpr3r 1236 | Simplification of conjunct... |
| simp1ll 1237 | Simplification of conjunct... |
| simp1lr 1238 | Simplification of conjunct... |
| simp1rl 1239 | Simplification of conjunct... |
| simp1rr 1240 | Simplification of conjunct... |
| simp2ll 1241 | Simplification of conjunct... |
| simp2lr 1242 | Simplification of conjunct... |
| simp2rl 1243 | Simplification of conjunct... |
| simp2rr 1244 | Simplification of conjunct... |
| simp3ll 1245 | Simplification of conjunct... |
| simp3lr 1246 | Simplification of conjunct... |
| simp3rl 1247 | Simplification of conjunct... |
| simp3rr 1248 | Simplification of conjunct... |
| simpl11 1249 | Simplification of conjunct... |
| simpl12 1250 | Simplification of conjunct... |
| simpl13 1251 | Simplification of conjunct... |
| simpl21 1252 | Simplification of conjunct... |
| simpl22 1253 | Simplification of conjunct... |
| simpl23 1254 | Simplification of conjunct... |
| simpl31 1255 | Simplification of conjunct... |
| simpl32 1256 | Simplification of conjunct... |
| simpl33 1257 | Simplification of conjunct... |
| simpr11 1258 | Simplification of conjunct... |
| simpr12 1259 | Simplification of conjunct... |
| simpr13 1260 | Simplification of conjunct... |
| simpr21 1261 | Simplification of conjunct... |
| simpr22 1262 | Simplification of conjunct... |
| simpr23 1263 | Simplification of conjunct... |
| simpr31 1264 | Simplification of conjunct... |
| simpr32 1265 | Simplification of conjunct... |
| simpr33 1266 | Simplification of conjunct... |
| simp1l1 1267 | Simplification of conjunct... |
| simp1l2 1268 | Simplification of conjunct... |
| simp1l3 1269 | Simplification of conjunct... |
| simp1r1 1270 | Simplification of conjunct... |
| simp1r2 1271 | Simplification of conjunct... |
| simp1r3 1272 | Simplification of conjunct... |
| simp2l1 1273 | Simplification of conjunct... |
| simp2l2 1274 | Simplification of conjunct... |
| simp2l3 1275 | Simplification of conjunct... |
| simp2r1 1276 | Simplification of conjunct... |
| simp2r2 1277 | Simplification of conjunct... |
| simp2r3 1278 | Simplification of conjunct... |
| simp3l1 1279 | Simplification of conjunct... |
| simp3l2 1280 | Simplification of conjunct... |
| simp3l3 1281 | Simplification of conjunct... |
| simp3r1 1282 | Simplification of conjunct... |
| simp3r2 1283 | Simplification of conjunct... |
| simp3r3 1284 | Simplification of conjunct... |
| simp11l 1285 | Simplification of conjunct... |
| simp11r 1286 | Simplification of conjunct... |
| simp12l 1287 | Simplification of conjunct... |
| simp12r 1288 | Simplification of conjunct... |
| simp13l 1289 | Simplification of conjunct... |
| simp13r 1290 | Simplification of conjunct... |
| simp21l 1291 | Simplification of conjunct... |
| simp21r 1292 | Simplification of conjunct... |
| simp22l 1293 | Simplification of conjunct... |
| simp22r 1294 | Simplification of conjunct... |
| simp23l 1295 | Simplification of conjunct... |
| simp23r 1296 | Simplification of conjunct... |
| simp31l 1297 | Simplification of conjunct... |
| simp31r 1298 | Simplification of conjunct... |
| simp32l 1299 | Simplification of conjunct... |
| simp32r 1300 | Simplification of conjunct... |
| simp33l 1301 | Simplification of conjunct... |
| simp33r 1302 | Simplification of conjunct... |
| simp111 1303 | Simplification of conjunct... |
| simp112 1304 | Simplification of conjunct... |
| simp113 1305 | Simplification of conjunct... |
| simp121 1306 | Simplification of conjunct... |
| simp122 1307 | Simplification of conjunct... |
| simp123 1308 | Simplification of conjunct... |
| simp131 1309 | Simplification of conjunct... |
| simp132 1310 | Simplification of conjunct... |
| simp133 1311 | Simplification of conjunct... |
| simp211 1312 | Simplification of conjunct... |
| simp212 1313 | Simplification of conjunct... |
| simp213 1314 | Simplification of conjunct... |
| simp221 1315 | Simplification of conjunct... |
| simp222 1316 | Simplification of conjunct... |
| simp223 1317 | Simplification of conjunct... |
| simp231 1318 | Simplification of conjunct... |
| simp232 1319 | Simplification of conjunct... |
| simp233 1320 | Simplification of conjunct... |
| simp311 1321 | Simplification of conjunct... |
| simp312 1322 | Simplification of conjunct... |
| simp313 1323 | Simplification of conjunct... |
| simp321 1324 | Simplification of conjunct... |
| simp322 1325 | Simplification of conjunct... |
| simp323 1326 | Simplification of conjunct... |
| simp331 1327 | Simplification of conjunct... |
| simp332 1328 | Simplification of conjunct... |
| simp333 1329 | Simplification of conjunct... |
| 3anibar 1330 | Remove a hypothesis from t... |
| 3mix1 1331 | Introduction in triple dis... |
| 3mix2 1332 | Introduction in triple dis... |
| 3mix3 1333 | Introduction in triple dis... |
| 3mix1i 1334 | Introduction in triple dis... |
| 3mix2i 1335 | Introduction in triple dis... |
| 3mix3i 1336 | Introduction in triple dis... |
| 3mix1d 1337 | Deduction introducing trip... |
| 3mix2d 1338 | Deduction introducing trip... |
| 3mix3d 1339 | Deduction introducing trip... |
| 3pm3.2i 1340 | Infer conjunction of premi... |
| pm3.2an3 1341 | Version of ~ pm3.2 for a t... |
| mpbir3an 1342 | Detach a conjunction of tr... |
| mpbir3and 1343 | Detach a conjunction of tr... |
| syl3anbrc 1344 | Syllogism inference. (Con... |
| syl21anbrc 1345 | Syllogism inference. (Con... |
| 3imp3i2an 1346 | An elimination deduction. ... |
| ex3 1347 | Apply ~ ex to a hypothesis... |
| 3imp1 1348 | Importation to left triple... |
| 3impd 1349 | Importation deduction for ... |
| 3imp2 1350 | Importation to right tripl... |
| 3impdi 1351 | Importation inference (und... |
| 3impdir 1352 | Importation inference (und... |
| 3exp1 1353 | Exportation from left trip... |
| 3expd 1354 | Exportation deduction for ... |
| 3exp2 1355 | Exportation from right tri... |
| exp5o 1356 | A triple exportation infer... |
| exp516 1357 | A triple exportation infer... |
| exp520 1358 | A triple exportation infer... |
| 3impexp 1359 | Version of ~ impexp for a ... |
| 3an1rs 1360 | Swap conjuncts. (Contribu... |
| 3anassrs 1361 | Associative law for conjun... |
| ad5ant245 1362 | Deduction adding conjuncts... |
| ad5ant234 1363 | Deduction adding conjuncts... |
| ad5ant235 1364 | Deduction adding conjuncts... |
| ad5ant123 1365 | Deduction adding conjuncts... |
| ad5ant124 1366 | Deduction adding conjuncts... |
| ad5ant125 1367 | Deduction adding conjuncts... |
| ad5ant134 1368 | Deduction adding conjuncts... |
| ad5ant135 1369 | Deduction adding conjuncts... |
| ad5ant145 1370 | Deduction adding conjuncts... |
| ad5ant2345 1371 | Deduction adding conjuncts... |
| syl3anc 1372 | Syllogism combined with co... |
| syl13anc 1373 | Syllogism combined with co... |
| syl31anc 1374 | Syllogism combined with co... |
| syl112anc 1375 | Syllogism combined with co... |
| syl121anc 1376 | Syllogism combined with co... |
| syl211anc 1377 | Syllogism combined with co... |
| syl23anc 1378 | Syllogism combined with co... |
| syl32anc 1379 | Syllogism combined with co... |
| syl122anc 1380 | Syllogism combined with co... |
| syl212anc 1381 | Syllogism combined with co... |
| syl221anc 1382 | Syllogism combined with co... |
| syl113anc 1383 | Syllogism combined with co... |
| syl131anc 1384 | Syllogism combined with co... |
| syl311anc 1385 | Syllogism combined with co... |
| syl33anc 1386 | Syllogism combined with co... |
| syl222anc 1387 | Syllogism combined with co... |
| syl123anc 1388 | Syllogism combined with co... |
| syl132anc 1389 | Syllogism combined with co... |
| syl213anc 1390 | Syllogism combined with co... |
| syl231anc 1391 | Syllogism combined with co... |
| syl312anc 1392 | Syllogism combined with co... |
| syl321anc 1393 | Syllogism combined with co... |
| syl133anc 1394 | Syllogism combined with co... |
| syl313anc 1395 | Syllogism combined with co... |
| syl331anc 1396 | Syllogism combined with co... |
| syl223anc 1397 | Syllogism combined with co... |
| syl232anc 1398 | Syllogism combined with co... |
| syl322anc 1399 | Syllogism combined with co... |
| syl233anc 1400 | Syllogism combined with co... |
| syl323anc 1401 | Syllogism combined with co... |
| syl332anc 1402 | Syllogism combined with co... |
| syl333anc 1403 | A syllogism inference comb... |
| syl3an1b 1404 | A syllogism inference. (C... |
| syl3an2b 1405 | A syllogism inference. (C... |
| syl3an3b 1406 | A syllogism inference. (C... |
| syl3an1br 1407 | A syllogism inference. (C... |
| syl3an2br 1408 | A syllogism inference. (C... |
| syl3an3br 1409 | A syllogism inference. (C... |
| syld3an3 1410 | A syllogism inference. (C... |
| syld3an1 1411 | A syllogism inference. (C... |
| syld3an2 1412 | A syllogism inference. (C... |
| syl3anl1 1413 | A syllogism inference. (C... |
| syl3anl2 1414 | A syllogism inference. (C... |
| syl3anl3 1415 | A syllogism inference. (C... |
| syl3anl 1416 | A triple syllogism inferen... |
| syl3anr1 1417 | A syllogism inference. (C... |
| syl3anr2 1418 | A syllogism inference. (C... |
| syl3anr3 1419 | A syllogism inference. (C... |
| 3anidm12 1420 | Inference from idempotent ... |
| 3anidm13 1421 | Inference from idempotent ... |
| 3anidm23 1422 | Inference from idempotent ... |
| syl2an3an 1423 | ~ syl3an with antecedents ... |
| syl2an23an 1424 | Deduction related to ~ syl... |
| 3ori 1425 | Infer implication from tri... |
| 3jao 1426 | Disjunction of three antec... |
| 3jaob 1427 | Disjunction of three antec... |
| 3jaoi 1428 | Disjunction of three antec... |
| 3jaod 1429 | Disjunction of three antec... |
| 3jaoian 1430 | Disjunction of three antec... |
| 3jaodan 1431 | Disjunction of three antec... |
| mpjao3dan 1432 | Eliminate a three-way disj... |
| mpjao3danOLD 1433 | Obsolete version of ~ mpja... |
| 3jaao 1434 | Inference conjoining and d... |
| syl3an9b 1435 | Nested syllogism inference... |
| 3orbi123d 1436 | Deduction joining 3 equiva... |
| 3anbi123d 1437 | Deduction joining 3 equiva... |
| 3anbi12d 1438 | Deduction conjoining and a... |
| 3anbi13d 1439 | Deduction conjoining and a... |
| 3anbi23d 1440 | Deduction conjoining and a... |
| 3anbi1d 1441 | Deduction adding conjuncts... |
| 3anbi2d 1442 | Deduction adding conjuncts... |
| 3anbi3d 1443 | Deduction adding conjuncts... |
| 3anim123d 1444 | Deduction joining 3 implic... |
| 3orim123d 1445 | Deduction joining 3 implic... |
| an6 1446 | Rearrangement of 6 conjunc... |
| 3an6 1447 | Analogue of ~ an4 for trip... |
| 3or6 1448 | Analogue of ~ or4 for trip... |
| mp3an1 1449 | An inference based on modu... |
| mp3an2 1450 | An inference based on modu... |
| mp3an3 1451 | An inference based on modu... |
| mp3an12 1452 | An inference based on modu... |
| mp3an13 1453 | An inference based on modu... |
| mp3an23 1454 | An inference based on modu... |
| mp3an1i 1455 | An inference based on modu... |
| mp3anl1 1456 | An inference based on modu... |
| mp3anl2 1457 | An inference based on modu... |
| mp3anl3 1458 | An inference based on modu... |
| mp3anr1 1459 | An inference based on modu... |
| mp3anr2 1460 | An inference based on modu... |
| mp3anr3 1461 | An inference based on modu... |
| mp3an 1462 | An inference based on modu... |
| mpd3an3 1463 | An inference based on modu... |
| mpd3an23 1464 | An inference based on modu... |
| mp3and 1465 | A deduction based on modus... |
| mp3an12i 1466 | ~ mp3an with antecedents i... |
| mp3an2i 1467 | ~ mp3an with antecedents i... |
| mp3an3an 1468 | ~ mp3an with antecedents i... |
| mp3an2ani 1469 | An elimination deduction. ... |
| biimp3a 1470 | Infer implication from a l... |
| biimp3ar 1471 | Infer implication from a l... |
| 3anandis 1472 | Inference that undistribut... |
| 3anandirs 1473 | Inference that undistribut... |
| ecase23d 1474 | Deduction for elimination ... |
| 3ecase 1475 | Inference for elimination ... |
| 3bior1fd 1476 | A disjunction is equivalen... |
| 3bior1fand 1477 | A disjunction is equivalen... |
| 3bior2fd 1478 | A wff is equivalent to its... |
| 3biant1d 1479 | A conjunction is equivalen... |
| intn3an1d 1480 | Introduction of a triple c... |
| intn3an2d 1481 | Introduction of a triple c... |
| intn3an3d 1482 | Introduction of a triple c... |
| an3andi 1483 | Distribution of conjunctio... |
| an33rean 1484 | Rearrange a 9-fold conjunc... |
| an33reanOLD 1485 | Obsolete version of ~ an33... |
| 3orel2 1486 | Partial elimination of a t... |
| 3orel3 1487 | Partial elimination of a t... |
| 3orel13 1488 | Elimination of two disjunc... |
| 3pm3.2ni 1489 | Triple negated disjunction... |
| nanan 1492 | Conjunction in terms of al... |
| dfnan2 1493 | Alternative denial in term... |
| nanor 1494 | Alternative denial in term... |
| nancom 1495 | Alternative denial is comm... |
| nannan 1496 | Nested alternative denials... |
| nanim 1497 | Implication in terms of al... |
| nannot 1498 | Negation in terms of alter... |
| nanbi 1499 | Biconditional in terms of ... |
| nanbi1 1500 | Introduce a right anti-con... |
| nanbi2 1501 | Introduce a left anti-conj... |
| nanbi12 1502 | Join two logical equivalen... |
| nanbi1i 1503 | Introduce a right anti-con... |
| nanbi2i 1504 | Introduce a left anti-conj... |
| nanbi12i 1505 | Join two logical equivalen... |
| nanbi1d 1506 | Introduce a right anti-con... |
| nanbi2d 1507 | Introduce a left anti-conj... |
| nanbi12d 1508 | Join two logical equivalen... |
| nanass 1509 | A characterization of when... |
| xnor 1512 | Two ways to write XNOR (ex... |
| xorcom 1513 | The connector ` \/_ ` is c... |
| xorcomOLD 1514 | Obsolete version of ~ xorc... |
| xorass 1515 | The connector ` \/_ ` is a... |
| excxor 1516 | This tautology shows that ... |
| xor2 1517 | Two ways to express "exclu... |
| xoror 1518 | Exclusive disjunction impl... |
| xornan 1519 | Exclusive disjunction impl... |
| xornan2 1520 | XOR implies NAND (written ... |
| xorneg2 1521 | The connector ` \/_ ` is n... |
| xorneg1 1522 | The connector ` \/_ ` is n... |
| xorneg 1523 | The connector ` \/_ ` is u... |
| xorbi12i 1524 | Equality property for excl... |
| xorbi12iOLD 1525 | Obsolete version of ~ xorb... |
| xorbi12d 1526 | Equality property for excl... |
| anxordi 1527 | Conjunction distributes ov... |
| xorexmid 1528 | Exclusive-or variant of th... |
| norcom 1531 | The connector ` -\/ ` is c... |
| norcomOLD 1532 | Obsolete version of ~ norc... |
| nornot 1533 | ` -. ` is expressible via ... |
| noran 1534 | ` /\ ` is expressible via ... |
| noror 1535 | ` \/ ` is expressible via ... |
| norasslem1 1536 | This lemma shows the equiv... |
| norasslem2 1537 | This lemma specializes ~ b... |
| norasslem3 1538 | This lemma specializes ~ b... |
| norass 1539 | A characterization of when... |
| trujust 1544 | Soundness justification th... |
| tru 1546 | The truth value ` T. ` is ... |
| dftru2 1547 | An alternate definition of... |
| trut 1548 | A proposition is equivalen... |
| mptru 1549 | Eliminate ` T. ` as an ant... |
| tbtru 1550 | A proposition is equivalen... |
| bitru 1551 | A theorem is equivalent to... |
| trud 1552 | Anything implies ` T. ` . ... |
| truan 1553 | True can be removed from a... |
| fal 1556 | The truth value ` F. ` is ... |
| nbfal 1557 | The negation of a proposit... |
| bifal 1558 | A contradiction is equival... |
| falim 1559 | The truth value ` F. ` imp... |
| falimd 1560 | The truth value ` F. ` imp... |
| dfnot 1561 | Given falsum ` F. ` , we c... |
| inegd 1562 | Negation introduction rule... |
| efald 1563 | Deduction based on reducti... |
| pm2.21fal 1564 | If a wff and its negation ... |
| truimtru 1565 | A ` -> ` identity. (Contr... |
| truimfal 1566 | A ` -> ` identity. (Contr... |
| falimtru 1567 | A ` -> ` identity. (Contr... |
| falimfal 1568 | A ` -> ` identity. (Contr... |
| nottru 1569 | A ` -. ` identity. (Contr... |
| notfal 1570 | A ` -. ` identity. (Contr... |
| trubitru 1571 | A ` <-> ` identity. (Cont... |
| falbitru 1572 | A ` <-> ` identity. (Cont... |
| trubifal 1573 | A ` <-> ` identity. (Cont... |
| falbifal 1574 | A ` <-> ` identity. (Cont... |
| truantru 1575 | A ` /\ ` identity. (Contr... |
| truanfal 1576 | A ` /\ ` identity. (Contr... |
| falantru 1577 | A ` /\ ` identity. (Contr... |
| falanfal 1578 | A ` /\ ` identity. (Contr... |
| truortru 1579 | A ` \/ ` identity. (Contr... |
| truorfal 1580 | A ` \/ ` identity. (Contr... |
| falortru 1581 | A ` \/ ` identity. (Contr... |
| falorfal 1582 | A ` \/ ` identity. (Contr... |
| trunantru 1583 | A ` -/\ ` identity. (Cont... |
| trunanfal 1584 | A ` -/\ ` identity. (Cont... |
| falnantru 1585 | A ` -/\ ` identity. (Cont... |
| falnanfal 1586 | A ` -/\ ` identity. (Cont... |
| truxortru 1587 | A ` \/_ ` identity. (Cont... |
| truxorfal 1588 | A ` \/_ ` identity. (Cont... |
| falxortru 1589 | A ` \/_ ` identity. (Cont... |
| falxorfal 1590 | A ` \/_ ` identity. (Cont... |
| trunortru 1591 | A ` -\/ ` identity. (Cont... |
| trunorfal 1592 | A ` -\/ ` identity. (Cont... |
| falnortru 1593 | A ` -\/ ` identity. (Cont... |
| falnorfal 1594 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1597 | Equality theorem for the a... |
| hadbi123i 1598 | Equality theorem for the a... |
| hadass 1599 | Associative law for the ad... |
| hadbi 1600 | The adder sum is the same ... |
| hadcoma 1601 | Commutative law for the ad... |
| hadcomb 1602 | Commutative law for the ad... |
| hadrot 1603 | Rotation law for the adder... |
| hadnot 1604 | The adder sum distributes ... |
| had1 1605 | If the first input is true... |
| had0 1606 | If the first input is fals... |
| hadifp 1607 | The value of the adder sum... |
| cador 1610 | The adder carry in disjunc... |
| cadan 1611 | The adder carry in conjunc... |
| cadbi123d 1612 | Equality theorem for the a... |
| cadbi123i 1613 | Equality theorem for the a... |
| cadcoma 1614 | Commutative law for the ad... |
| cadcomb 1615 | Commutative law for the ad... |
| cadrot 1616 | Rotation law for the adder... |
| cadnot 1617 | The adder carry distribute... |
| cad11 1618 | If (at least) two inputs a... |
| cad1 1619 | If one input is true, then... |
| cad0 1620 | If one input is false, the... |
| cad0OLD 1621 | Obsolete version of ~ cad0... |
| cadifp 1622 | The value of the carry is,... |
| cadtru 1623 | The adder carry is true as... |
| minimp 1624 | A single axiom for minimal... |
| minimp-syllsimp 1625 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1626 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1627 | Derivation of a commuted f... |
| minimp-ax2 1628 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1629 | Derivation of ~ pm2.43 (al... |
| impsingle 1630 | The shortest single axiom ... |
| impsingle-step4 1631 | Derivation of impsingle-st... |
| impsingle-step8 1632 | Derivation of impsingle-st... |
| impsingle-ax1 1633 | Derivation of impsingle-ax... |
| impsingle-step15 1634 | Derivation of impsingle-st... |
| impsingle-step18 1635 | Derivation of impsingle-st... |
| impsingle-step19 1636 | Derivation of impsingle-st... |
| impsingle-step20 1637 | Derivation of impsingle-st... |
| impsingle-step21 1638 | Derivation of impsingle-st... |
| impsingle-step22 1639 | Derivation of impsingle-st... |
| impsingle-step25 1640 | Derivation of impsingle-st... |
| impsingle-imim1 1641 | Derivation of impsingle-im... |
| impsingle-peirce 1642 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1643 | Derivation of ~ ax-2 from ... |
| meredith 1644 | Carew Meredith's sole axio... |
| merlem1 1645 | Step 3 of Meredith's proof... |
| merlem2 1646 | Step 4 of Meredith's proof... |
| merlem3 1647 | Step 7 of Meredith's proof... |
| merlem4 1648 | Step 8 of Meredith's proof... |
| merlem5 1649 | Step 11 of Meredith's proo... |
| merlem6 1650 | Step 12 of Meredith's proo... |
| merlem7 1651 | Between steps 14 and 15 of... |
| merlem8 1652 | Step 15 of Meredith's proo... |
| merlem9 1653 | Step 18 of Meredith's proo... |
| merlem10 1654 | Step 19 of Meredith's proo... |
| merlem11 1655 | Step 20 of Meredith's proo... |
| merlem12 1656 | Step 28 of Meredith's proo... |
| merlem13 1657 | Step 35 of Meredith's proo... |
| luk-1 1658 | 1 of 3 axioms for proposit... |
| luk-2 1659 | 2 of 3 axioms for proposit... |
| luk-3 1660 | 3 of 3 axioms for proposit... |
| luklem1 1661 | Used to rederive standard ... |
| luklem2 1662 | Used to rederive standard ... |
| luklem3 1663 | Used to rederive standard ... |
| luklem4 1664 | Used to rederive standard ... |
| luklem5 1665 | Used to rederive standard ... |
| luklem6 1666 | Used to rederive standard ... |
| luklem7 1667 | Used to rederive standard ... |
| luklem8 1668 | Used to rederive standard ... |
| ax1 1669 | Standard propositional axi... |
| ax2 1670 | Standard propositional axi... |
| ax3 1671 | Standard propositional axi... |
| nic-dfim 1672 | This theorem "defines" imp... |
| nic-dfneg 1673 | This theorem "defines" neg... |
| nic-mp 1674 | Derive Nicod's rule of mod... |
| nic-mpALT 1675 | A direct proof of ~ nic-mp... |
| nic-ax 1676 | Nicod's axiom derived from... |
| nic-axALT 1677 | A direct proof of ~ nic-ax... |
| nic-imp 1678 | Inference for ~ nic-mp usi... |
| nic-idlem1 1679 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1680 | Lemma for ~ nic-id . Infe... |
| nic-id 1681 | Theorem ~ id expressed wit... |
| nic-swap 1682 | The connector ` -/\ ` is s... |
| nic-isw1 1683 | Inference version of ~ nic... |
| nic-isw2 1684 | Inference for swapping nes... |
| nic-iimp1 1685 | Inference version of ~ nic... |
| nic-iimp2 1686 | Inference version of ~ nic... |
| nic-idel 1687 | Inference to remove the tr... |
| nic-ich 1688 | Chained inference. (Contr... |
| nic-idbl 1689 | Double the terms. Since d... |
| nic-bijust 1690 | Biconditional justificatio... |
| nic-bi1 1691 | Inference to extract one s... |
| nic-bi2 1692 | Inference to extract the o... |
| nic-stdmp 1693 | Derive the standard modus ... |
| nic-luk1 1694 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1695 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1696 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1697 | This alternative axiom for... |
| lukshefth1 1698 | Lemma for ~ renicax . (Co... |
| lukshefth2 1699 | Lemma for ~ renicax . (Co... |
| renicax 1700 | A rederivation of ~ nic-ax... |
| tbw-bijust 1701 | Justification for ~ tbw-ne... |
| tbw-negdf 1702 | The definition of negation... |
| tbw-ax1 1703 | The first of four axioms i... |
| tbw-ax2 1704 | The second of four axioms ... |
| tbw-ax3 1705 | The third of four axioms i... |
| tbw-ax4 1706 | The fourth of four axioms ... |
| tbwsyl 1707 | Used to rederive the Lukas... |
| tbwlem1 1708 | Used to rederive the Lukas... |
| tbwlem2 1709 | Used to rederive the Lukas... |
| tbwlem3 1710 | Used to rederive the Lukas... |
| tbwlem4 1711 | Used to rederive the Lukas... |
| tbwlem5 1712 | Used to rederive the Lukas... |
| re1luk1 1713 | ~ luk-1 derived from the T... |
| re1luk2 1714 | ~ luk-2 derived from the T... |
| re1luk3 1715 | ~ luk-3 derived from the T... |
| merco1 1716 | A single axiom for proposi... |
| merco1lem1 1717 | Used to rederive the Tarsk... |
| retbwax4 1718 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1719 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1720 | Used to rederive the Tarsk... |
| merco1lem3 1721 | Used to rederive the Tarsk... |
| merco1lem4 1722 | Used to rederive the Tarsk... |
| merco1lem5 1723 | Used to rederive the Tarsk... |
| merco1lem6 1724 | Used to rederive the Tarsk... |
| merco1lem7 1725 | Used to rederive the Tarsk... |
| retbwax3 1726 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1727 | Used to rederive the Tarsk... |
| merco1lem9 1728 | Used to rederive the Tarsk... |
| merco1lem10 1729 | Used to rederive the Tarsk... |
| merco1lem11 1730 | Used to rederive the Tarsk... |
| merco1lem12 1731 | Used to rederive the Tarsk... |
| merco1lem13 1732 | Used to rederive the Tarsk... |
| merco1lem14 1733 | Used to rederive the Tarsk... |
| merco1lem15 1734 | Used to rederive the Tarsk... |
| merco1lem16 1735 | Used to rederive the Tarsk... |
| merco1lem17 1736 | Used to rederive the Tarsk... |
| merco1lem18 1737 | Used to rederive the Tarsk... |
| retbwax1 1738 | ~ tbw-ax1 rederived from ~... |
| merco2 1739 | A single axiom for proposi... |
| mercolem1 1740 | Used to rederive the Tarsk... |
| mercolem2 1741 | Used to rederive the Tarsk... |
| mercolem3 1742 | Used to rederive the Tarsk... |
| mercolem4 1743 | Used to rederive the Tarsk... |
| mercolem5 1744 | Used to rederive the Tarsk... |
| mercolem6 1745 | Used to rederive the Tarsk... |
| mercolem7 1746 | Used to rederive the Tarsk... |
| mercolem8 1747 | Used to rederive the Tarsk... |
| re1tbw1 1748 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1749 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1750 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1751 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1752 | Justification for ~ rb-imd... |
| rb-imdf 1753 | The definition of implicat... |
| anmp 1754 | Modus ponens for ` { \/ , ... |
| rb-ax1 1755 | The first of four axioms i... |
| rb-ax2 1756 | The second of four axioms ... |
| rb-ax3 1757 | The third of four axioms i... |
| rb-ax4 1758 | The fourth of four axioms ... |
| rbsyl 1759 | Used to rederive the Lukas... |
| rblem1 1760 | Used to rederive the Lukas... |
| rblem2 1761 | Used to rederive the Lukas... |
| rblem3 1762 | Used to rederive the Lukas... |
| rblem4 1763 | Used to rederive the Lukas... |
| rblem5 1764 | Used to rederive the Lukas... |
| rblem6 1765 | Used to rederive the Lukas... |
| rblem7 1766 | Used to rederive the Lukas... |
| re1axmp 1767 | ~ ax-mp derived from Russe... |
| re2luk1 1768 | ~ luk-1 derived from Russe... |
| re2luk2 1769 | ~ luk-2 derived from Russe... |
| re2luk3 1770 | ~ luk-3 derived from Russe... |
| mptnan 1771 | Modus ponendo tollens 1, o... |
| mptxor 1772 | Modus ponendo tollens 2, o... |
| mtpor 1773 | Modus tollendo ponens (inc... |
| mtpxor 1774 | Modus tollendo ponens (ori... |
| stoic1a 1775 | Stoic logic Thema 1 (part ... |
| stoic1b 1776 | Stoic logic Thema 1 (part ... |
| stoic2a 1777 | Stoic logic Thema 2 versio... |
| stoic2b 1778 | Stoic logic Thema 2 versio... |
| stoic3 1779 | Stoic logic Thema 3. Stat... |
| stoic4a 1780 | Stoic logic Thema 4 versio... |
| stoic4b 1781 | Stoic logic Thema 4 versio... |
| alnex 1784 | Universal quantification o... |
| eximal 1785 | An equivalence between an ... |
| nf2 1788 | Alternate definition of no... |
| nf3 1789 | Alternate definition of no... |
| nf4 1790 | Alternate definition of no... |
| nfi 1791 | Deduce that ` x ` is not f... |
| nfri 1792 | Consequence of the definit... |
| nfd 1793 | Deduce that ` x ` is not f... |
| nfrd 1794 | Consequence of the definit... |
| nftht 1795 | Closed form of ~ nfth . (... |
| nfntht 1796 | Closed form of ~ nfnth . ... |
| nfntht2 1797 | Closed form of ~ nfnth . ... |
| gen2 1799 | Generalization applied twi... |
| mpg 1800 | Modus ponens combined with... |
| mpgbi 1801 | Modus ponens on biconditio... |
| mpgbir 1802 | Modus ponens on biconditio... |
| nex 1803 | Generalization rule for ne... |
| nfth 1804 | No variable is (effectivel... |
| nfnth 1805 | No variable is (effectivel... |
| hbth 1806 | No variable is (effectivel... |
| nftru 1807 | The true constant has no f... |
| nffal 1808 | The false constant has no ... |
| sptruw 1809 | Version of ~ sp when ` ph ... |
| altru 1810 | For all sets, ` T. ` is tr... |
| alfal 1811 | For all sets, ` -. F. ` is... |
| alim 1813 | Restatement of Axiom ~ ax-... |
| alimi 1814 | Inference quantifying both... |
| 2alimi 1815 | Inference doubly quantifyi... |
| ala1 1816 | Add an antecedent in a uni... |
| al2im 1817 | Closed form of ~ al2imi . ... |
| al2imi 1818 | Inference quantifying ante... |
| alanimi 1819 | Variant of ~ al2imi with c... |
| alimdh 1820 | Deduction form of Theorem ... |
| albi 1821 | Theorem 19.15 of [Margaris... |
| albii 1822 | Inference adding universal... |
| 2albii 1823 | Inference adding two unive... |
| 3albii 1824 | Inference adding three uni... |
| sylgt 1825 | Closed form of ~ sylg . (... |
| sylg 1826 | A syllogism combined with ... |
| alrimih 1827 | Inference form of Theorem ... |
| hbxfrbi 1828 | A utility lemma to transfe... |
| alex 1829 | Universal quantifier in te... |
| exnal 1830 | Existential quantification... |
| 2nalexn 1831 | Part of theorem *11.5 in [... |
| 2exnaln 1832 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1833 | Theorem *11.25 in [Whitehe... |
| alimex 1834 | An equivalence between an ... |
| aleximi 1835 | A variant of ~ al2imi : in... |
| alexbii 1836 | Biconditional form of ~ al... |
| exim 1837 | Theorem 19.22 of [Margaris... |
| eximi 1838 | Inference adding existenti... |
| 2eximi 1839 | Inference adding two exist... |
| eximii 1840 | Inference associated with ... |
| exa1 1841 | Add an antecedent in an ex... |
| 19.38 1842 | Theorem 19.38 of [Margaris... |
| 19.38a 1843 | Under a nonfreeness hypoth... |
| 19.38b 1844 | Under a nonfreeness hypoth... |
| imnang 1845 | Quantified implication in ... |
| alinexa 1846 | A transformation of quanti... |
| exnalimn 1847 | Existential quantification... |
| alexn 1848 | A relationship between two... |
| 2exnexn 1849 | Theorem *11.51 in [Whitehe... |
| exbi 1850 | Theorem 19.18 of [Margaris... |
| exbii 1851 | Inference adding existenti... |
| 2exbii 1852 | Inference adding two exist... |
| 3exbii 1853 | Inference adding three exi... |
| nfbiit 1854 | Equivalence theorem for th... |
| nfbii 1855 | Equality theorem for the n... |
| nfxfr 1856 | A utility lemma to transfe... |
| nfxfrd 1857 | A utility lemma to transfe... |
| nfnbi 1858 | A variable is nonfree in a... |
| nfnbiOLD 1859 | Obsolete version of ~ nfnb... |
| nfnt 1860 | If a variable is nonfree i... |
| nfn 1861 | Inference associated with ... |
| nfnd 1862 | Deduction associated with ... |
| exanali 1863 | A transformation of quanti... |
| 2exanali 1864 | Theorem *11.521 in [Whiteh... |
| exancom 1865 | Commutation of conjunction... |
| exan 1866 | Place a conjunct in the sc... |
| alrimdh 1867 | Deduction form of Theorem ... |
| eximdh 1868 | Deduction from Theorem 19.... |
| nexdh 1869 | Deduction for generalizati... |
| albidh 1870 | Formula-building rule for ... |
| exbidh 1871 | Formula-building rule for ... |
| exsimpl 1872 | Simplification of an exist... |
| exsimpr 1873 | Simplification of an exist... |
| 19.26 1874 | Theorem 19.26 of [Margaris... |
| 19.26-2 1875 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1876 | Theorem ~ 19.26 with tripl... |
| 19.29 1877 | Theorem 19.29 of [Margaris... |
| 19.29r 1878 | Variation of ~ 19.29 . (C... |
| 19.29r2 1879 | Variation of ~ 19.29r with... |
| 19.29x 1880 | Variation of ~ 19.29 with ... |
| 19.35 1881 | Theorem 19.35 of [Margaris... |
| 19.35i 1882 | Inference associated with ... |
| 19.35ri 1883 | Inference associated with ... |
| 19.25 1884 | Theorem 19.25 of [Margaris... |
| 19.30 1885 | Theorem 19.30 of [Margaris... |
| 19.43 1886 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1887 | Obsolete proof of ~ 19.43 ... |
| 19.33 1888 | Theorem 19.33 of [Margaris... |
| 19.33b 1889 | The antecedent provides a ... |
| 19.40 1890 | Theorem 19.40 of [Margaris... |
| 19.40-2 1891 | Theorem *11.42 in [Whitehe... |
| 19.40b 1892 | The antecedent provides a ... |
| albiim 1893 | Split a biconditional and ... |
| 2albiim 1894 | Split a biconditional and ... |
| exintrbi 1895 | Add/remove a conjunct in t... |
| exintr 1896 | Introduce a conjunct in th... |
| alsyl 1897 | Universally quantified and... |
| nfimd 1898 | If in a context ` x ` is n... |
| nfimt 1899 | Closed form of ~ nfim and ... |
| nfim 1900 | If ` x ` is not free in ` ... |
| nfand 1901 | If in a context ` x ` is n... |
| nf3and 1902 | Deduction form of bound-va... |
| nfan 1903 | If ` x ` is not free in ` ... |
| nfnan 1904 | If ` x ` is not free in ` ... |
| nf3an 1905 | If ` x ` is not free in ` ... |
| nfbid 1906 | If in a context ` x ` is n... |
| nfbi 1907 | If ` x ` is not free in ` ... |
| nfor 1908 | If ` x ` is not free in ` ... |
| nf3or 1909 | If ` x ` is not free in ` ... |
| empty 1910 | Two characterizations of t... |
| emptyex 1911 | On the empty domain, any e... |
| emptyal 1912 | On the empty domain, any u... |
| emptynf 1913 | On the empty domain, any v... |
| ax5d 1915 | Version of ~ ax-5 with ant... |
| ax5e 1916 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1917 | If a formula holds for som... |
| nfv 1918 | If ` x ` is not present in... |
| nfvd 1919 | ~ nfv with antecedent. Us... |
| alimdv 1920 | Deduction form of Theorem ... |
| eximdv 1921 | Deduction form of Theorem ... |
| 2alimdv 1922 | Deduction form of Theorem ... |
| 2eximdv 1923 | Deduction form of Theorem ... |
| albidv 1924 | Formula-building rule for ... |
| exbidv 1925 | Formula-building rule for ... |
| nfbidv 1926 | An equality theorem for no... |
| 2albidv 1927 | Formula-building rule for ... |
| 2exbidv 1928 | Formula-building rule for ... |
| 3exbidv 1929 | Formula-building rule for ... |
| 4exbidv 1930 | Formula-building rule for ... |
| alrimiv 1931 | Inference form of Theorem ... |
| alrimivv 1932 | Inference form of Theorem ... |
| alrimdv 1933 | Deduction form of Theorem ... |
| exlimiv 1934 | Inference form of Theorem ... |
| exlimiiv 1935 | Inference (Rule C) associa... |
| exlimivv 1936 | Inference form of Theorem ... |
| exlimdv 1937 | Deduction form of Theorem ... |
| exlimdvv 1938 | Deduction form of Theorem ... |
| exlimddv 1939 | Existential elimination ru... |
| nexdv 1940 | Deduction for generalizati... |
| 2ax5 1941 | Quantification of two vari... |
| stdpc5v 1942 | Version of ~ stdpc5 with a... |
| 19.21v 1943 | Version of ~ 19.21 with a ... |
| 19.32v 1944 | Version of ~ 19.32 with a ... |
| 19.31v 1945 | Version of ~ 19.31 with a ... |
| 19.23v 1946 | Version of ~ 19.23 with a ... |
| 19.23vv 1947 | Theorem ~ 19.23v extended ... |
| pm11.53v 1948 | Version of ~ pm11.53 with ... |
| 19.36imv 1949 | One direction of ~ 19.36v ... |
| 19.36imvOLD 1950 | Obsolete version of ~ 19.3... |
| 19.36iv 1951 | Inference associated with ... |
| 19.37imv 1952 | One direction of ~ 19.37v ... |
| 19.37iv 1953 | Inference associated with ... |
| 19.41v 1954 | Version of ~ 19.41 with a ... |
| 19.41vv 1955 | Version of ~ 19.41 with tw... |
| 19.41vvv 1956 | Version of ~ 19.41 with th... |
| 19.41vvvv 1957 | Version of ~ 19.41 with fo... |
| 19.42v 1958 | Version of ~ 19.42 with a ... |
| exdistr 1959 | Distribution of existentia... |
| exdistrv 1960 | Distribute a pair of exist... |
| 4exdistrv 1961 | Distribute two pairs of ex... |
| 19.42vv 1962 | Version of ~ 19.42 with tw... |
| exdistr2 1963 | Distribution of existentia... |
| 19.42vvv 1964 | Version of ~ 19.42 with th... |
| 3exdistr 1965 | Distribution of existentia... |
| 4exdistr 1966 | Distribution of existentia... |
| weq 1967 | Extend wff definition to i... |
| speimfw 1968 | Specialization, with addit... |
| speimfwALT 1969 | Alternate proof of ~ speim... |
| spimfw 1970 | Specialization, with addit... |
| ax12i 1971 | Inference that has ~ ax-12... |
| ax6v 1973 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1974 | At least one individual ex... |
| spimw 1975 | Specialization. Lemma 8 o... |
| spimew 1976 | Existential introduction, ... |
| speiv 1977 | Inference from existential... |
| speivw 1978 | Version of ~ spei with a d... |
| exgen 1979 | Rule of existential genera... |
| extru 1980 | There exists a variable su... |
| 19.2 1981 | Theorem 19.2 of [Margaris]... |
| 19.2d 1982 | Deduction associated with ... |
| 19.8w 1983 | Weak version of ~ 19.8a an... |
| spnfw 1984 | Weak version of ~ sp . Us... |
| spvw 1985 | Version of ~ sp when ` x `... |
| 19.3v 1986 | Version of ~ 19.3 with a d... |
| 19.8v 1987 | Version of ~ 19.8a with a ... |
| 19.9v 1988 | Version of ~ 19.9 with a d... |
| 19.39 1989 | Theorem 19.39 of [Margaris... |
| 19.24 1990 | Theorem 19.24 of [Margaris... |
| 19.34 1991 | Theorem 19.34 of [Margaris... |
| 19.36v 1992 | Version of ~ 19.36 with a ... |
| 19.12vvv 1993 | Version of ~ 19.12vv with ... |
| 19.27v 1994 | Version of ~ 19.27 with a ... |
| 19.28v 1995 | Version of ~ 19.28 with a ... |
| 19.37v 1996 | Version of ~ 19.37 with a ... |
| 19.44v 1997 | Version of ~ 19.44 with a ... |
| 19.45v 1998 | Version of ~ 19.45 with a ... |
| spimevw 1999 | Existential introduction, ... |
| spimvw 2000 | A weak form of specializat... |
| spvv 2001 | Specialization, using impl... |
| spfalw 2002 | Version of ~ sp when ` ph ... |
| chvarvv 2003 | Implicit substitution of `... |
| equs4v 2004 | Version of ~ equs4 with a ... |
| alequexv 2005 | Version of ~ equs4v with i... |
| exsbim 2006 | One direction of the equiv... |
| equsv 2007 | If a formula does not cont... |
| equsalvw 2008 | Version of ~ equsalv with ... |
| equsexvw 2009 | Version of ~ equsexv with ... |
| cbvaliw 2010 | Change bound variable. Us... |
| cbvalivw 2011 | Change bound variable. Us... |
| ax7v 2013 | Weakened version of ~ ax-7... |
| ax7v1 2014 | First of two weakened vers... |
| ax7v2 2015 | Second of two weakened ver... |
| equid 2016 | Identity law for equality.... |
| nfequid 2017 | Bound-variable hypothesis ... |
| equcomiv 2018 | Weaker form of ~ equcomi w... |
| ax6evr 2019 | A commuted form of ~ ax6ev... |
| ax7 2020 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2021 | Commutative law for equali... |
| equcom 2022 | Commutative law for equali... |
| equcomd 2023 | Deduction form of ~ equcom... |
| equcoms 2024 | An inference commuting equ... |
| equtr 2025 | A transitive law for equal... |
| equtrr 2026 | A transitive law for equal... |
| equeuclr 2027 | Commuted version of ~ eque... |
| equeucl 2028 | Equality is a left-Euclide... |
| equequ1 2029 | An equivalence law for equ... |
| equequ2 2030 | An equivalence law for equ... |
| equtr2 2031 | Equality is a left-Euclide... |
| stdpc6 2032 | One of the two equality ax... |
| equvinv 2033 | A variable introduction la... |
| equvinva 2034 | A modified version of the ... |
| equvelv 2035 | A biconditional form of ~ ... |
| ax13b 2036 | An equivalence between two... |
| spfw 2037 | Weak version of ~ sp . Us... |
| spw 2038 | Weak version of the specia... |
| cbvalw 2039 | Change bound variable. Us... |
| cbvalvw 2040 | Change bound variable. Us... |
| cbvexvw 2041 | Change bound variable. Us... |
| cbvaldvaw 2042 | Rule used to change the bo... |
| cbvexdvaw 2043 | Rule used to change the bo... |
| cbval2vw 2044 | Rule used to change bound ... |
| cbvex2vw 2045 | Rule used to change bound ... |
| cbvex4vw 2046 | Rule used to change bound ... |
| alcomiw 2047 | Weak version of ~ ax-11 . ... |
| alcomw 2048 | Weak version of ~ alcom an... |
| hbn1fw 2049 | Weak version of ~ ax-10 fr... |
| hbn1w 2050 | Weak version of ~ hbn1 . ... |
| hba1w 2051 | Weak version of ~ hba1 . ... |
| hbe1w 2052 | Weak version of ~ hbe1 . ... |
| hbalw 2053 | Weak version of ~ hbal . ... |
| 19.8aw 2054 | If a formula is true, then... |
| exexw 2055 | Existential quantification... |
| spaev 2056 | A special instance of ~ sp... |
| cbvaev 2057 | Change bound variable in a... |
| aevlem0 2058 | Lemma for ~ aevlem . Inst... |
| aevlem 2059 | Lemma for ~ aev and ~ axc1... |
| aeveq 2060 | The antecedent ` A. x x = ... |
| aev 2061 | A "distinctor elimination"... |
| aev2 2062 | A version of ~ aev with tw... |
| hbaev 2063 | All variables are effectiv... |
| naev 2064 | If some set variables can ... |
| naev2 2065 | Generalization of ~ hbnaev... |
| hbnaev 2066 | Any variable is free in ` ... |
| sbjust 2067 | Justification theorem for ... |
| sbt 2070 | A substitution into a theo... |
| sbtru 2071 | The result of substituting... |
| stdpc4 2072 | The specialization axiom o... |
| sbtALT 2073 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2074 | A double specialization us... |
| sbi1 2075 | Distribute substitution ov... |
| spsbim 2076 | Distribute substitution ov... |
| spsbbi 2077 | Biconditional property for... |
| sbimi 2078 | Distribute substitution ov... |
| sb2imi 2079 | Distribute substitution ov... |
| sbbii 2080 | Infer substitution into bo... |
| 2sbbii 2081 | Infer double substitution ... |
| sbimdv 2082 | Deduction substituting bot... |
| sbbidv 2083 | Deduction substituting bot... |
| sban 2084 | Conjunction inside and out... |
| sb3an 2085 | Threefold conjunction insi... |
| spsbe 2086 | Existential generalization... |
| sbequ 2087 | Equality property for subs... |
| sbequi 2088 | An equality theorem for su... |
| sb6 2089 | Alternate definition of su... |
| 2sb6 2090 | Equivalence for double sub... |
| sb1v 2091 | One direction of ~ sb5 , p... |
| sbv 2092 | Substitution for a variabl... |
| sbcom4 2093 | Commutativity law for subs... |
| pm11.07 2094 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2095 | Substitution in an implica... |
| sbievw 2096 | Conversion of implicit sub... |
| sbiedvw 2097 | Conversion of implicit sub... |
| 2sbievw 2098 | Conversion of double impli... |
| sbcom3vv 2099 | Substituting ` y ` for ` x... |
| sbievw2 2100 | ~ sbievw applied twice, av... |
| sbco2vv 2101 | A composition law for subs... |
| equsb3 2102 | Substitution in an equalit... |
| equsb3r 2103 | Substitution applied to th... |
| equsb1v 2104 | Substitution applied to an... |
| nsb 2105 | Any substitution in an alw... |
| sbn1 2106 | One direction of ~ sbn , u... |
| wel 2108 | Extend wff definition to i... |
| ax8v 2110 | Weakened version of ~ ax-8... |
| ax8v1 2111 | First of two weakened vers... |
| ax8v2 2112 | Second of two weakened ver... |
| ax8 2113 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2114 | An identity law for the no... |
| elsb1 2115 | Substitution for the first... |
| cleljust 2116 | When the class variables i... |
| ax9v 2118 | Weakened version of ~ ax-9... |
| ax9v1 2119 | First of two weakened vers... |
| ax9v2 2120 | Second of two weakened ver... |
| ax9 2121 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2122 | An identity law for the no... |
| elequ2g 2123 | A form of ~ elequ2 with a ... |
| elsb2 2124 | Substitution for the secon... |
| ax6dgen 2125 | Tarski's system uses the w... |
| ax10w 2126 | Weak version of ~ ax-10 fr... |
| ax11w 2127 | Weak version of ~ ax-11 fr... |
| ax11dgen 2128 | Degenerate instance of ~ a... |
| ax12wlem 2129 | Lemma for weak version of ... |
| ax12w 2130 | Weak version of ~ ax-12 fr... |
| ax12dgen 2131 | Degenerate instance of ~ a... |
| ax12wdemo 2132 | Example of an application ... |
| ax13w 2133 | Weak version (principal in... |
| ax13dgen1 2134 | Degenerate instance of ~ a... |
| ax13dgen2 2135 | Degenerate instance of ~ a... |
| ax13dgen3 2136 | Degenerate instance of ~ a... |
| ax13dgen4 2137 | Degenerate instance of ~ a... |
| hbn1 2139 | Alias for ~ ax-10 to be us... |
| hbe1 2140 | The setvar ` x ` is not fr... |
| hbe1a 2141 | Dual statement of ~ hbe1 .... |
| nf5-1 2142 | One direction of ~ nf5 can... |
| nf5i 2143 | Deduce that ` x ` is not f... |
| nf5dh 2144 | Deduce that ` x ` is not f... |
| nf5dv 2145 | Apply the definition of no... |
| nfnaew 2146 | All variables are effectiv... |
| nfnaewOLD 2147 | Obsolete version of ~ nfna... |
| nfe1 2148 | The setvar ` x ` is not fr... |
| nfa1 2149 | The setvar ` x ` is not fr... |
| nfna1 2150 | A convenience theorem part... |
| nfia1 2151 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2152 | The setvar ` x ` is not fr... |
| modal5 2153 | The analogue in our predic... |
| nfs1v 2154 | The setvar ` x ` is not fr... |
| alcoms 2156 | Swap quantifiers in an ant... |
| alcom 2157 | Theorem 19.5 of [Margaris]... |
| alrot3 2158 | Theorem *11.21 in [Whitehe... |
| alrot4 2159 | Rotate four universal quan... |
| sbal 2160 | Move universal quantifier ... |
| sbalv 2161 | Quantify with new variable... |
| sbcom2 2162 | Commutativity law for subs... |
| excom 2163 | Theorem 19.11 of [Margaris... |
| excomim 2164 | One direction of Theorem 1... |
| excom13 2165 | Swap 1st and 3rd existenti... |
| exrot3 2166 | Rotate existential quantif... |
| exrot4 2167 | Rotate existential quantif... |
| hbal 2168 | If ` x ` is not free in ` ... |
| hbald 2169 | Deduction form of bound-va... |
| hbsbw 2170 | If ` z ` is not free in ` ... |
| nfa2 2171 | Lemma 24 of [Monk2] p. 114... |
| ax12v 2173 | This is essentially Axiom ... |
| ax12v2 2174 | It is possible to remove a... |
| 19.8a 2175 | If a wff is true, it is tr... |
| 19.8ad 2176 | If a wff is true, it is tr... |
| sp 2177 | Specialization. A univers... |
| spi 2178 | Inference rule of universa... |
| sps 2179 | Generalization of antecede... |
| 2sp 2180 | A double specialization (s... |
| spsd 2181 | Deduction generalizing ant... |
| 19.2g 2182 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2183 | Inference form of ~ 19.21 ... |
| 19.21bbi 2184 | Inference removing two uni... |
| 19.23bi 2185 | Inference form of Theorem ... |
| nexr 2186 | Inference associated with ... |
| qexmid 2187 | Quantified excluded middle... |
| nf5r 2188 | Consequence of the definit... |
| nf5ri 2189 | Consequence of the definit... |
| nf5rd 2190 | Consequence of the definit... |
| spimedv 2191 | Deduction version of ~ spi... |
| spimefv 2192 | Version of ~ spime with a ... |
| nfim1 2193 | A closed form of ~ nfim . ... |
| nfan1 2194 | A closed form of ~ nfan . ... |
| 19.3t 2195 | Closed form of ~ 19.3 and ... |
| 19.3 2196 | A wff may be quantified wi... |
| 19.9d 2197 | A deduction version of one... |
| 19.9t 2198 | Closed form of ~ 19.9 and ... |
| 19.9 2199 | A wff may be existentially... |
| 19.21t 2200 | Closed form of Theorem 19.... |
| 19.21 2201 | Theorem 19.21 of [Margaris... |
| stdpc5 2202 | An axiom scheme of standar... |
| 19.21-2 2203 | Version of ~ 19.21 with tw... |
| 19.23t 2204 | Closed form of Theorem 19.... |
| 19.23 2205 | Theorem 19.23 of [Margaris... |
| alimd 2206 | Deduction form of Theorem ... |
| alrimi 2207 | Inference form of Theorem ... |
| alrimdd 2208 | Deduction form of Theorem ... |
| alrimd 2209 | Deduction form of Theorem ... |
| eximd 2210 | Deduction form of Theorem ... |
| exlimi 2211 | Inference associated with ... |
| exlimd 2212 | Deduction form of Theorem ... |
| exlimimdd 2213 | Existential elimination ru... |
| exlimdd 2214 | Existential elimination ru... |
| nexd 2215 | Deduction for generalizati... |
| albid 2216 | Formula-building rule for ... |
| exbid 2217 | Formula-building rule for ... |
| nfbidf 2218 | An equality theorem for ef... |
| 19.16 2219 | Theorem 19.16 of [Margaris... |
| 19.17 2220 | Theorem 19.17 of [Margaris... |
| 19.27 2221 | Theorem 19.27 of [Margaris... |
| 19.28 2222 | Theorem 19.28 of [Margaris... |
| 19.19 2223 | Theorem 19.19 of [Margaris... |
| 19.36 2224 | Theorem 19.36 of [Margaris... |
| 19.36i 2225 | Inference associated with ... |
| 19.37 2226 | Theorem 19.37 of [Margaris... |
| 19.32 2227 | Theorem 19.32 of [Margaris... |
| 19.31 2228 | Theorem 19.31 of [Margaris... |
| 19.41 2229 | Theorem 19.41 of [Margaris... |
| 19.42 2230 | Theorem 19.42 of [Margaris... |
| 19.44 2231 | Theorem 19.44 of [Margaris... |
| 19.45 2232 | Theorem 19.45 of [Margaris... |
| spimfv 2233 | Specialization, using impl... |
| chvarfv 2234 | Implicit substitution of `... |
| cbv3v2 2235 | Version of ~ cbv3 with two... |
| sbalex 2236 | Equivalence of two ways to... |
| sb4av 2237 | Version of ~ sb4a with a d... |
| sbimd 2238 | Deduction substituting bot... |
| sbbid 2239 | Deduction substituting bot... |
| 2sbbid 2240 | Deduction doubly substitut... |
| sbequ1 2241 | An equality theorem for su... |
| sbequ2 2242 | An equality theorem for su... |
| stdpc7 2243 | One of the two equality ax... |
| sbequ12 2244 | An equality theorem for su... |
| sbequ12r 2245 | An equality theorem for su... |
| sbelx 2246 | Elimination of substitutio... |
| sbequ12a 2247 | An equality theorem for su... |
| sbid 2248 | An identity theorem for su... |
| sbcov 2249 | A composition law for subs... |
| sb6a 2250 | Equivalence for substituti... |
| sbid2vw 2251 | Reverting substitution yie... |
| axc16g 2252 | Generalization of ~ axc16 ... |
| axc16 2253 | Proof of older axiom ~ ax-... |
| axc16gb 2254 | Biconditional strengthenin... |
| axc16nf 2255 | If ~ dtru is false, then t... |
| axc11v 2256 | Version of ~ axc11 with a ... |
| axc11rv 2257 | Version of ~ axc11r with a... |
| drsb2 2258 | Formula-building lemma for... |
| equsalv 2259 | An equivalence related to ... |
| equsexv 2260 | An equivalence related to ... |
| equsexvOLD 2261 | Obsolete version of ~ equs... |
| sbft 2262 | Substitution has no effect... |
| sbf 2263 | Substitution for a variabl... |
| sbf2 2264 | Substitution has no effect... |
| sbh 2265 | Substitution for a variabl... |
| hbs1 2266 | The setvar ` x ` is not fr... |
| nfs1f 2267 | If ` x ` is not free in ` ... |
| sb5 2268 | Alternate definition of su... |
| sb5OLD 2269 | Obsolete version of ~ sb5 ... |
| sb56OLD 2270 | Obsolete version of ~ sbal... |
| equs5av 2271 | A property related to subs... |
| 2sb5 2272 | Equivalence for double sub... |
| sbco4lem 2273 | Lemma for ~ sbco4 . It re... |
| sbco4lemOLD 2274 | Obsolete version of ~ sbco... |
| sbco4 2275 | Two ways of exchanging two... |
| dfsb7 2276 | An alternate definition of... |
| sbn 2277 | Negation inside and outsid... |
| sbex 2278 | Move existential quantifie... |
| nf5 2279 | Alternate definition of ~ ... |
| nf6 2280 | An alternate definition of... |
| nf5d 2281 | Deduce that ` x ` is not f... |
| nf5di 2282 | Since the converse holds b... |
| 19.9h 2283 | A wff may be existentially... |
| 19.21h 2284 | Theorem 19.21 of [Margaris... |
| 19.23h 2285 | Theorem 19.23 of [Margaris... |
| exlimih 2286 | Inference associated with ... |
| exlimdh 2287 | Deduction form of Theorem ... |
| equsalhw 2288 | Version of ~ equsalh with ... |
| equsexhv 2289 | An equivalence related to ... |
| hba1 2290 | The setvar ` x ` is not fr... |
| hbnt 2291 | Closed theorem version of ... |
| hbn 2292 | If ` x ` is not free in ` ... |
| hbnd 2293 | Deduction form of bound-va... |
| hbim1 2294 | A closed form of ~ hbim . ... |
| hbimd 2295 | Deduction form of bound-va... |
| hbim 2296 | If ` x ` is not free in ` ... |
| hban 2297 | If ` x ` is not free in ` ... |
| hb3an 2298 | If ` x ` is not free in ` ... |
| sbi2 2299 | Introduction of implicatio... |
| sbim 2300 | Implication inside and out... |
| sbrim 2301 | Substitution in an implica... |
| sbrimOLD 2302 | Obsolete version of ~ sbri... |
| sblim 2303 | Substitution in an implica... |
| sbor 2304 | Disjunction inside and out... |
| sbbi 2305 | Equivalence inside and out... |
| sblbis 2306 | Introduce left bicondition... |
| sbrbis 2307 | Introduce right biconditio... |
| sbrbif 2308 | Introduce right biconditio... |
| sbiev 2309 | Conversion of implicit sub... |
| sbiedw 2310 | Conversion of implicit sub... |
| axc7 2311 | Show that the original axi... |
| axc7e 2312 | Abbreviated version of ~ a... |
| modal-b 2313 | The analogue in our predic... |
| 19.9ht 2314 | A closed version of ~ 19.9... |
| axc4 2315 | Show that the original axi... |
| axc4i 2316 | Inference version of ~ axc... |
| nfal 2317 | If ` x ` is not free in ` ... |
| nfex 2318 | If ` x ` is not free in ` ... |
| hbex 2319 | If ` x ` is not free in ` ... |
| nfnf 2320 | If ` x ` is not free in ` ... |
| 19.12 2321 | Theorem 19.12 of [Margaris... |
| nfald 2322 | Deduction form of ~ nfal .... |
| nfexd 2323 | If ` x ` is not free in ` ... |
| nfsbv 2324 | If ` z ` is not free in ` ... |
| nfsbvOLD 2325 | Obsolete version of ~ nfsb... |
| hbsbwOLD 2326 | Obsolete version of ~ hbsb... |
| sbco2v 2327 | A composition law for subs... |
| aaan 2328 | Distribute universal quant... |
| aaanOLD 2329 | Obsolete version of ~ aaan... |
| eeor 2330 | Distribute existential qua... |
| eeorOLD 2331 | Obsolete version of ~ eeor... |
| cbv3v 2332 | Rule used to change bound ... |
| cbv1v 2333 | Rule used to change bound ... |
| cbv2w 2334 | Rule used to change bound ... |
| cbvaldw 2335 | Deduction used to change b... |
| cbvexdw 2336 | Deduction used to change b... |
| cbv3hv 2337 | Rule used to change bound ... |
| cbvalv1 2338 | Rule used to change bound ... |
| cbvexv1 2339 | Rule used to change bound ... |
| cbval2v 2340 | Rule used to change bound ... |
| cbvex2v 2341 | Rule used to change bound ... |
| dvelimhw 2342 | Proof of ~ dvelimh without... |
| pm11.53 2343 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2344 | Special case of ~ 19.12 wh... |
| eean 2345 | Distribute existential qua... |
| eeanv 2346 | Distribute a pair of exist... |
| eeeanv 2347 | Distribute three existenti... |
| ee4anv 2348 | Distribute two pairs of ex... |
| sb8v 2349 | Substitution of variable i... |
| sb8f 2350 | Substitution of variable i... |
| sb8fOLD 2351 | Obsolete version of ~ sb8f... |
| sb8ef 2352 | Substitution of variable i... |
| 2sb8ef 2353 | An equivalent expression f... |
| sb6rfv 2354 | Reversed substitution. Ve... |
| sbnf2 2355 | Two ways of expressing " `... |
| exsb 2356 | An equivalent expression f... |
| 2exsb 2357 | An equivalent expression f... |
| sbbib 2358 | Reversal of substitution. ... |
| sbbibvv 2359 | Reversal of substitution. ... |
| cbvsbv 2360 | Change the bound variable ... |
| cbvsbvf 2361 | Change the bound variable ... |
| cleljustALT 2362 | Alternate proof of ~ clelj... |
| cleljustALT2 2363 | Alternate proof of ~ clelj... |
| equs5aALT 2364 | Alternate proof of ~ equs5... |
| equs5eALT 2365 | Alternate proof of ~ equs5... |
| axc11r 2366 | Same as ~ axc11 but with r... |
| dral1v 2367 | Formula-building lemma for... |
| dral1vOLD 2368 | Obsolete version of ~ dral... |
| drex1v 2369 | Formula-building lemma for... |
| drnf1v 2370 | Formula-building lemma for... |
| drnf1vOLD 2371 | Obsolete version of ~ drnf... |
| ax13v 2373 | A weaker version of ~ ax-1... |
| ax13lem1 2374 | A version of ~ ax13v with ... |
| ax13 2375 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2376 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2377 | An equation between setvar... |
| dveeq2 2378 | Quantifier introduction wh... |
| nfeqf1 2379 | An equation between setvar... |
| dveeq1 2380 | Quantifier introduction wh... |
| nfeqf 2381 | A variable is effectively ... |
| axc9 2382 | Derive set.mm's original ~... |
| ax6e 2383 | At least one individual ex... |
| ax6 2384 | Theorem showing that ~ ax-... |
| axc10 2385 | Show that the original axi... |
| spimt 2386 | Closed theorem form of ~ s... |
| spim 2387 | Specialization, using impl... |
| spimed 2388 | Deduction version of ~ spi... |
| spime 2389 | Existential introduction, ... |
| spimv 2390 | A version of ~ spim with a... |
| spimvALT 2391 | Alternate proof of ~ spimv... |
| spimev 2392 | Distinct-variable version ... |
| spv 2393 | Specialization, using impl... |
| spei 2394 | Inference from existential... |
| chvar 2395 | Implicit substitution of `... |
| chvarv 2396 | Implicit substitution of `... |
| cbv3 2397 | Rule used to change bound ... |
| cbval 2398 | Rule used to change bound ... |
| cbvex 2399 | Rule used to change bound ... |
| cbvalv 2400 | Rule used to change bound ... |
| cbvexv 2401 | Rule used to change bound ... |
| cbv1 2402 | Rule used to change bound ... |
| cbv2 2403 | Rule used to change bound ... |
| cbv3h 2404 | Rule used to change bound ... |
| cbv1h 2405 | Rule used to change bound ... |
| cbv2h 2406 | Rule used to change bound ... |
| cbvald 2407 | Deduction used to change b... |
| cbvexd 2408 | Deduction used to change b... |
| cbvaldva 2409 | Rule used to change the bo... |
| cbvexdva 2410 | Rule used to change the bo... |
| cbval2 2411 | Rule used to change bound ... |
| cbvex2 2412 | Rule used to change bound ... |
| cbval2vv 2413 | Rule used to change bound ... |
| cbvex2vv 2414 | Rule used to change bound ... |
| cbvex4v 2415 | Rule used to change bound ... |
| equs4 2416 | Lemma used in proofs of im... |
| equsal 2417 | An equivalence related to ... |
| equsex 2418 | An equivalence related to ... |
| equsexALT 2419 | Alternate proof of ~ equse... |
| equsalh 2420 | An equivalence related to ... |
| equsexh 2421 | An equivalence related to ... |
| axc15 2422 | Derivation of set.mm's ori... |
| ax12 2423 | Rederivation of Axiom ~ ax... |
| ax12b 2424 | A bidirectional version of... |
| ax13ALT 2425 | Alternate proof of ~ ax13 ... |
| axc11n 2426 | Derive set.mm's original ~... |
| aecom 2427 | Commutation law for identi... |
| aecoms 2428 | A commutation rule for ide... |
| naecoms 2429 | A commutation rule for dis... |
| axc11 2430 | Show that ~ ax-c11 can be ... |
| hbae 2431 | All variables are effectiv... |
| hbnae 2432 | All variables are effectiv... |
| nfae 2433 | All variables are effectiv... |
| nfnae 2434 | All variables are effectiv... |
| hbnaes 2435 | Rule that applies ~ hbnae ... |
| axc16i 2436 | Inference with ~ axc16 as ... |
| axc16nfALT 2437 | Alternate proof of ~ axc16... |
| dral2 2438 | Formula-building lemma for... |
| dral1 2439 | Formula-building lemma for... |
| dral1ALT 2440 | Alternate proof of ~ dral1... |
| drex1 2441 | Formula-building lemma for... |
| drex2 2442 | Formula-building lemma for... |
| drnf1 2443 | Formula-building lemma for... |
| drnf2 2444 | Formula-building lemma for... |
| nfald2 2445 | Variation on ~ nfald which... |
| nfexd2 2446 | Variation on ~ nfexd which... |
| exdistrf 2447 | Distribution of existentia... |
| dvelimf 2448 | Version of ~ dvelimv witho... |
| dvelimdf 2449 | Deduction form of ~ dvelim... |
| dvelimh 2450 | Version of ~ dvelim withou... |
| dvelim 2451 | This theorem can be used t... |
| dvelimv 2452 | Similar to ~ dvelim with f... |
| dvelimnf 2453 | Version of ~ dvelim using ... |
| dveeq2ALT 2454 | Alternate proof of ~ dveeq... |
| equvini 2455 | A variable introduction la... |
| equvel 2456 | A variable elimination law... |
| equs5a 2457 | A property related to subs... |
| equs5e 2458 | A property related to subs... |
| equs45f 2459 | Two ways of expressing sub... |
| equs5 2460 | Lemma used in proofs of su... |
| dveel1 2461 | Quantifier introduction wh... |
| dveel2 2462 | Quantifier introduction wh... |
| axc14 2463 | Axiom ~ ax-c14 is redundan... |
| sb6x 2464 | Equivalence involving subs... |
| sbequ5 2465 | Substitution does not chan... |
| sbequ6 2466 | Substitution does not chan... |
| sb5rf 2467 | Reversed substitution. Us... |
| sb6rf 2468 | Reversed substitution. Fo... |
| ax12vALT 2469 | Alternate proof of ~ ax12v... |
| 2ax6elem 2470 | We can always find values ... |
| 2ax6e 2471 | We can always find values ... |
| 2sb5rf 2472 | Reversed double substituti... |
| 2sb6rf 2473 | Reversed double substituti... |
| sbel2x 2474 | Elimination of double subs... |
| sb4b 2475 | Simplified definition of s... |
| sb3b 2476 | Simplified definition of s... |
| sb3 2477 | One direction of a simplif... |
| sb1 2478 | One direction of a simplif... |
| sb2 2479 | One direction of a simplif... |
| sb4a 2480 | A version of one implicati... |
| dfsb1 2481 | Alternate definition of su... |
| hbsb2 2482 | Bound-variable hypothesis ... |
| nfsb2 2483 | Bound-variable hypothesis ... |
| hbsb2a 2484 | Special case of a bound-va... |
| sb4e 2485 | One direction of a simplif... |
| hbsb2e 2486 | Special case of a bound-va... |
| hbsb3 2487 | If ` y ` is not free in ` ... |
| nfs1 2488 | If ` y ` is not free in ` ... |
| axc16ALT 2489 | Alternate proof of ~ axc16... |
| axc16gALT 2490 | Alternate proof of ~ axc16... |
| equsb1 2491 | Substitution applied to an... |
| equsb2 2492 | Substitution applied to an... |
| dfsb2 2493 | An alternate definition of... |
| dfsb3 2494 | An alternate definition of... |
| drsb1 2495 | Formula-building lemma for... |
| sb2ae 2496 | In the case of two success... |
| sb6f 2497 | Equivalence for substituti... |
| sb5f 2498 | Equivalence for substituti... |
| nfsb4t 2499 | A variable not free in a p... |
| nfsb4 2500 | A variable not free in a p... |
| sbequ8 2501 | Elimination of equality fr... |
| sbie 2502 | Conversion of implicit sub... |
| sbied 2503 | Conversion of implicit sub... |
| sbiedv 2504 | Conversion of implicit sub... |
| 2sbiev 2505 | Conversion of double impli... |
| sbcom3 2506 | Substituting ` y ` for ` x... |
| sbco 2507 | A composition law for subs... |
| sbid2 2508 | An identity law for substi... |
| sbid2v 2509 | An identity law for substi... |
| sbidm 2510 | An idempotent law for subs... |
| sbco2 2511 | A composition law for subs... |
| sbco2d 2512 | A composition law for subs... |
| sbco3 2513 | A composition law for subs... |
| sbcom 2514 | A commutativity law for su... |
| sbtrt 2515 | Partially closed form of ~... |
| sbtr 2516 | A partial converse to ~ sb... |
| sb8 2517 | Substitution of variable i... |
| sb8e 2518 | Substitution of variable i... |
| sb9 2519 | Commutation of quantificat... |
| sb9i 2520 | Commutation of quantificat... |
| sbhb 2521 | Two ways of expressing " `... |
| nfsbd 2522 | Deduction version of ~ nfs... |
| nfsb 2523 | If ` z ` is not free in ` ... |
| hbsb 2524 | If ` z ` is not free in ` ... |
| sb7f 2525 | This version of ~ dfsb7 do... |
| sb7h 2526 | This version of ~ dfsb7 do... |
| sb10f 2527 | Hao Wang's identity axiom ... |
| sbal1 2528 | Check out ~ sbal for a ver... |
| sbal2 2529 | Move quantifier in and out... |
| 2sb8e 2530 | An equivalent expression f... |
| dfmoeu 2531 | An elementary proof of ~ m... |
| dfeumo 2532 | An elementary proof showin... |
| mojust 2534 | Soundness justification th... |
| nexmo 2536 | Nonexistence implies uniqu... |
| exmo 2537 | Any proposition holds for ... |
| moabs 2538 | Absorption of existence co... |
| moim 2539 | The at-most-one quantifier... |
| moimi 2540 | The at-most-one quantifier... |
| moimdv 2541 | The at-most-one quantifier... |
| mobi 2542 | Equivalence theorem for th... |
| mobii 2543 | Formula-building rule for ... |
| mobidv 2544 | Formula-building rule for ... |
| mobid 2545 | Formula-building rule for ... |
| moa1 2546 | If an implication holds fo... |
| moan 2547 | "At most one" is still the... |
| moani 2548 | "At most one" is still tru... |
| moor 2549 | "At most one" is still the... |
| mooran1 2550 | "At most one" imports disj... |
| mooran2 2551 | "At most one" exports disj... |
| nfmo1 2552 | Bound-variable hypothesis ... |
| nfmod2 2553 | Bound-variable hypothesis ... |
| nfmodv 2554 | Bound-variable hypothesis ... |
| nfmov 2555 | Bound-variable hypothesis ... |
| nfmod 2556 | Bound-variable hypothesis ... |
| nfmo 2557 | Bound-variable hypothesis ... |
| mof 2558 | Version of ~ df-mo with di... |
| mo3 2559 | Alternate definition of th... |
| mo 2560 | Equivalent definitions of ... |
| mo4 2561 | At-most-one quantifier exp... |
| mo4f 2562 | At-most-one quantifier exp... |
| eu3v 2565 | An alternate way to expres... |
| eujust 2566 | Soundness justification th... |
| eujustALT 2567 | Alternate proof of ~ eujus... |
| eu6lem 2568 | Lemma of ~ eu6im . A diss... |
| eu6 2569 | Alternate definition of th... |
| eu6im 2570 | One direction of ~ eu6 nee... |
| euf 2571 | Version of ~ eu6 with disj... |
| euex 2572 | Existential uniqueness imp... |
| eumo 2573 | Existential uniqueness imp... |
| eumoi 2574 | Uniqueness inferred from e... |
| exmoeub 2575 | Existence implies that uni... |
| exmoeu 2576 | Existence is equivalent to... |
| moeuex 2577 | Uniqueness implies that ex... |
| moeu 2578 | Uniqueness is equivalent t... |
| eubi 2579 | Equivalence theorem for th... |
| eubii 2580 | Introduce unique existenti... |
| eubidv 2581 | Formula-building rule for ... |
| eubid 2582 | Formula-building rule for ... |
| nfeu1 2583 | Bound-variable hypothesis ... |
| nfeu1ALT 2584 | Alternate proof of ~ nfeu1... |
| nfeud2 2585 | Bound-variable hypothesis ... |
| nfeudw 2586 | Bound-variable hypothesis ... |
| nfeud 2587 | Bound-variable hypothesis ... |
| nfeuw 2588 | Bound-variable hypothesis ... |
| nfeu 2589 | Bound-variable hypothesis ... |
| dfeu 2590 | Rederive ~ df-eu from the ... |
| dfmo 2591 | Rederive ~ df-mo from the ... |
| euequ 2592 | There exists a unique set ... |
| sb8eulem 2593 | Lemma. Factor out the com... |
| sb8euv 2594 | Variable substitution in u... |
| sb8eu 2595 | Variable substitution in u... |
| sb8mo 2596 | Variable substitution for ... |
| cbvmovw 2597 | Change bound variable. Us... |
| cbvmow 2598 | Rule used to change bound ... |
| cbvmowOLD 2599 | Obsolete version of ~ cbvm... |
| cbvmo 2600 | Rule used to change bound ... |
| cbveuvw 2601 | Change bound variable. Us... |
| cbveuw 2602 | Version of ~ cbveu with a ... |
| cbveuwOLD 2603 | Obsolete version of ~ cbve... |
| cbveu 2604 | Rule used to change bound ... |
| cbveuALT 2605 | Alternative proof of ~ cbv... |
| eu2 2606 | An alternate way of defini... |
| eu1 2607 | An alternate way to expres... |
| euor 2608 | Introduce a disjunct into ... |
| euorv 2609 | Introduce a disjunct into ... |
| euor2 2610 | Introduce or eliminate a d... |
| sbmo 2611 | Substitution into an at-mo... |
| eu4 2612 | Uniqueness using implicit ... |
| euimmo 2613 | Existential uniqueness imp... |
| euim 2614 | Add unique existential qua... |
| moanimlem 2615 | Factor out the common proo... |
| moanimv 2616 | Introduction of a conjunct... |
| moanim 2617 | Introduction of a conjunct... |
| euan 2618 | Introduction of a conjunct... |
| moanmo 2619 | Nested at-most-one quantif... |
| moaneu 2620 | Nested at-most-one and uni... |
| euanv 2621 | Introduction of a conjunct... |
| mopick 2622 | "At most one" picks a vari... |
| moexexlem 2623 | Factor out the proof skele... |
| 2moexv 2624 | Double quantification with... |
| moexexvw 2625 | "At most one" double quant... |
| 2moswapv 2626 | A condition allowing to sw... |
| 2euswapv 2627 | A condition allowing to sw... |
| 2euexv 2628 | Double quantification with... |
| 2exeuv 2629 | Double existential uniquen... |
| eupick 2630 | Existential uniqueness "pi... |
| eupicka 2631 | Version of ~ eupick with c... |
| eupickb 2632 | Existential uniqueness "pi... |
| eupickbi 2633 | Theorem *14.26 in [Whitehe... |
| mopick2 2634 | "At most one" can show the... |
| moexex 2635 | "At most one" double quant... |
| moexexv 2636 | "At most one" double quant... |
| 2moex 2637 | Double quantification with... |
| 2euex 2638 | Double quantification with... |
| 2eumo 2639 | Nested unique existential ... |
| 2eu2ex 2640 | Double existential uniquen... |
| 2moswap 2641 | A condition allowing to sw... |
| 2euswap 2642 | A condition allowing to sw... |
| 2exeu 2643 | Double existential uniquen... |
| 2mo2 2644 | Two ways of expressing "th... |
| 2mo 2645 | Two ways of expressing "th... |
| 2mos 2646 | Double "there exists at mo... |
| 2eu1 2647 | Double existential uniquen... |
| 2eu1v 2648 | Double existential uniquen... |
| 2eu2 2649 | Double existential uniquen... |
| 2eu3 2650 | Double existential uniquen... |
| 2eu4 2651 | This theorem provides us w... |
| 2eu5 2652 | An alternate definition of... |
| 2eu6 2653 | Two equivalent expressions... |
| 2eu7 2654 | Two equivalent expressions... |
| 2eu8 2655 | Two equivalent expressions... |
| euae 2656 | Two ways to express "exact... |
| exists1 2657 | Two ways to express "exact... |
| exists2 2658 | A condition implying that ... |
| barbara 2659 | "Barbara", one of the fund... |
| celarent 2660 | "Celarent", one of the syl... |
| darii 2661 | "Darii", one of the syllog... |
| dariiALT 2662 | Alternate proof of ~ darii... |
| ferio 2663 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2664 | Lemma for ~ barbari and th... |
| barbari 2665 | "Barbari", one of the syll... |
| barbariALT 2666 | Alternate proof of ~ barba... |
| celaront 2667 | "Celaront", one of the syl... |
| cesare 2668 | "Cesare", one of the syllo... |
| camestres 2669 | "Camestres", one of the sy... |
| festino 2670 | "Festino", one of the syll... |
| festinoALT 2671 | Alternate proof of ~ festi... |
| baroco 2672 | "Baroco", one of the syllo... |
| barocoALT 2673 | Alternate proof of ~ festi... |
| cesaro 2674 | "Cesaro", one of the syllo... |
| camestros 2675 | "Camestros", one of the sy... |
| datisi 2676 | "Datisi", one of the syllo... |
| disamis 2677 | "Disamis", one of the syll... |
| ferison 2678 | "Ferison", one of the syll... |
| bocardo 2679 | "Bocardo", one of the syll... |
| darapti 2680 | "Darapti", one of the syll... |
| daraptiALT 2681 | Alternate proof of ~ darap... |
| felapton 2682 | "Felapton", one of the syl... |
| calemes 2683 | "Calemes", one of the syll... |
| dimatis 2684 | "Dimatis", one of the syll... |
| fresison 2685 | "Fresison", one of the syl... |
| calemos 2686 | "Calemos", one of the syll... |
| fesapo 2687 | "Fesapo", one of the syllo... |
| bamalip 2688 | "Bamalip", one of the syll... |
| axia1 2689 | Left 'and' elimination (in... |
| axia2 2690 | Right 'and' elimination (i... |
| axia3 2691 | 'And' introduction (intuit... |
| axin1 2692 | 'Not' introduction (intuit... |
| axin2 2693 | 'Not' elimination (intuiti... |
| axio 2694 | Definition of 'or' (intuit... |
| axi4 2695 | Specialization (intuitioni... |
| axi5r 2696 | Converse of ~ axc4 (intuit... |
| axial 2697 | The setvar ` x ` is not fr... |
| axie1 2698 | The setvar ` x ` is not fr... |
| axie2 2699 | A key property of existent... |
| axi9 2700 | Axiom of existence (intuit... |
| axi10 2701 | Axiom of Quantifier Substi... |
| axi12 2702 | Axiom of Quantifier Introd... |
| axbnd 2703 | Axiom of Bundling (intuiti... |
| axexte 2705 | The axiom of extensionalit... |
| axextg 2706 | A generalization of the ax... |
| axextb 2707 | A bidirectional version of... |
| axextmo 2708 | There exists at most one s... |
| nulmo 2709 | There exists at most one e... |
| eleq1ab 2712 | Extension (in the sense of... |
| cleljustab 2713 | Extension of ~ cleljust fr... |
| abid 2714 | Simplification of class ab... |
| vexwt 2715 | A standard theorem of pred... |
| vexw 2716 | If ` ph ` is a theorem, th... |
| vextru 2717 | Every setvar is a member o... |
| nfsab1 2718 | Bound-variable hypothesis ... |
| hbab1 2719 | Bound-variable hypothesis ... |
| hbab1OLD 2720 | Obsolete version of ~ hbab... |
| hbab 2721 | Bound-variable hypothesis ... |
| hbabg 2722 | Bound-variable hypothesis ... |
| nfsab 2723 | Bound-variable hypothesis ... |
| nfsabg 2724 | Bound-variable hypothesis ... |
| dfcleq 2726 | The defining characterizat... |
| cvjust 2727 | Every set is a class. Pro... |
| ax9ALT 2728 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2729 | A weaker version of ~ eleq... |
| eqriv 2730 | Infer equality of classes ... |
| eqrdv 2731 | Deduce equality of classes... |
| eqrdav 2732 | Deduce equality of classes... |
| eqid 2733 | Law of identity (reflexivi... |
| eqidd 2734 | Class identity law with an... |
| eqeq1d 2735 | Deduction from equality to... |
| eqeq1dALT 2736 | Alternate proof of ~ eqeq1... |
| eqeq1 2737 | Equality implies equivalen... |
| eqeq1i 2738 | Inference from equality to... |
| eqcomd 2739 | Deduction from commutative... |
| eqcom 2740 | Commutative law for class ... |
| eqcoms 2741 | Inference applying commuta... |
| eqcomi 2742 | Inference from commutative... |
| neqcomd 2743 | Commute an inequality. (C... |
| eqeq2d 2744 | Deduction from equality to... |
| eqeq2 2745 | Equality implies equivalen... |
| eqeq2i 2746 | Inference from equality to... |
| eqeqan12d 2747 | A useful inference for sub... |
| eqeqan12rd 2748 | A useful inference for sub... |
| eqeq12d 2749 | A useful inference for sub... |
| eqeq12 2750 | Equality relationship amon... |
| eqeq12i 2751 | A useful inference for sub... |
| eqeq12OLD 2752 | Obsolete version of ~ eqeq... |
| eqeq12dOLD 2753 | Obsolete version of ~ eqeq... |
| eqeqan12dOLD 2754 | Obsolete version of ~ eqeq... |
| eqeqan12dALT 2755 | Alternate proof of ~ eqeqa... |
| eqtr 2756 | Transitive law for class e... |
| eqtr2 2757 | A transitive law for class... |
| eqtr2OLD 2758 | Obsolete version of eqtr2 ... |
| eqtr3 2759 | A transitive law for class... |
| eqtr3OLD 2760 | Obsolete version of ~ eqtr... |
| eqtri 2761 | An equality transitivity i... |
| eqtr2i 2762 | An equality transitivity i... |
| eqtr3i 2763 | An equality transitivity i... |
| eqtr4i 2764 | An equality transitivity i... |
| 3eqtri 2765 | An inference from three ch... |
| 3eqtrri 2766 | An inference from three ch... |
| 3eqtr2i 2767 | An inference from three ch... |
| 3eqtr2ri 2768 | An inference from three ch... |
| 3eqtr3i 2769 | An inference from three ch... |
| 3eqtr3ri 2770 | An inference from three ch... |
| 3eqtr4i 2771 | An inference from three ch... |
| 3eqtr4ri 2772 | An inference from three ch... |
| eqtrd 2773 | An equality transitivity d... |
| eqtr2d 2774 | An equality transitivity d... |
| eqtr3d 2775 | An equality transitivity e... |
| eqtr4d 2776 | An equality transitivity e... |
| 3eqtrd 2777 | A deduction from three cha... |
| 3eqtrrd 2778 | A deduction from three cha... |
| 3eqtr2d 2779 | A deduction from three cha... |
| 3eqtr2rd 2780 | A deduction from three cha... |
| 3eqtr3d 2781 | A deduction from three cha... |
| 3eqtr3rd 2782 | A deduction from three cha... |
| 3eqtr4d 2783 | A deduction from three cha... |
| 3eqtr4rd 2784 | A deduction from three cha... |
| eqtrid 2785 | An equality transitivity d... |
| eqtr2id 2786 | An equality transitivity d... |
| eqtr3id 2787 | An equality transitivity d... |
| eqtr3di 2788 | An equality transitivity d... |
| eqtrdi 2789 | An equality transitivity d... |
| eqtr2di 2790 | An equality transitivity d... |
| eqtr4di 2791 | An equality transitivity d... |
| eqtr4id 2792 | An equality transitivity d... |
| sylan9eq 2793 | An equality transitivity d... |
| sylan9req 2794 | An equality transitivity d... |
| sylan9eqr 2795 | An equality transitivity d... |
| 3eqtr3g 2796 | A chained equality inferen... |
| 3eqtr3a 2797 | A chained equality inferen... |
| 3eqtr4g 2798 | A chained equality inferen... |
| 3eqtr4a 2799 | A chained equality inferen... |
| eq2tri 2800 | A compound transitive infe... |
| abbi 2801 | Equivalent formulas yield ... |
| abbidv 2802 | Equivalent wff's yield equ... |
| abbii 2803 | Equivalent wff's yield equ... |
| abbid 2804 | Equivalent wff's yield equ... |
| abbib 2805 | Equal class abstractions r... |
| cbvabv 2806 | Rule used to change bound ... |
| cbvabw 2807 | Rule used to change bound ... |
| cbvabwOLD 2808 | Obsolete version of ~ cbva... |
| cbvab 2809 | Rule used to change bound ... |
| eqabbw 2810 | Version of ~ eqabb using i... |
| dfclel 2812 | Characterization of the el... |
| elex2 2813 | If a class contains anothe... |
| issetlem 2814 | Lemma for ~ elisset and ~ ... |
| elissetv 2815 | An element of a class exis... |
| elisset 2816 | An element of a class exis... |
| eleq1w 2817 | Weaker version of ~ eleq1 ... |
| eleq2w 2818 | Weaker version of ~ eleq2 ... |
| eleq1d 2819 | Deduction from equality to... |
| eleq2d 2820 | Deduction from equality to... |
| eleq2dALT 2821 | Alternate proof of ~ eleq2... |
| eleq1 2822 | Equality implies equivalen... |
| eleq2 2823 | Equality implies equivalen... |
| eleq12 2824 | Equality implies equivalen... |
| eleq1i 2825 | Inference from equality to... |
| eleq2i 2826 | Inference from equality to... |
| eleq12i 2827 | Inference from equality to... |
| eleq12d 2828 | Deduction from equality to... |
| eleq1a 2829 | A transitive-type law rela... |
| eqeltri 2830 | Substitution of equal clas... |
| eqeltrri 2831 | Substitution of equal clas... |
| eleqtri 2832 | Substitution of equal clas... |
| eleqtrri 2833 | Substitution of equal clas... |
| eqeltrd 2834 | Substitution of equal clas... |
| eqeltrrd 2835 | Deduction that substitutes... |
| eleqtrd 2836 | Deduction that substitutes... |
| eleqtrrd 2837 | Deduction that substitutes... |
| eqeltrid 2838 | A membership and equality ... |
| eqeltrrid 2839 | A membership and equality ... |
| eleqtrid 2840 | A membership and equality ... |
| eleqtrrid 2841 | A membership and equality ... |
| eqeltrdi 2842 | A membership and equality ... |
| eqeltrrdi 2843 | A membership and equality ... |
| eleqtrdi 2844 | A membership and equality ... |
| eleqtrrdi 2845 | A membership and equality ... |
| 3eltr3i 2846 | Substitution of equal clas... |
| 3eltr4i 2847 | Substitution of equal clas... |
| 3eltr3d 2848 | Substitution of equal clas... |
| 3eltr4d 2849 | Substitution of equal clas... |
| 3eltr3g 2850 | Substitution of equal clas... |
| 3eltr4g 2851 | Substitution of equal clas... |
| eleq2s 2852 | Substitution of equal clas... |
| eqneltri 2853 | If a class is not an eleme... |
| eqneltrd 2854 | If a class is not an eleme... |
| eqneltrrd 2855 | If a class is not an eleme... |
| neleqtrd 2856 | If a class is not an eleme... |
| neleqtrrd 2857 | If a class is not an eleme... |
| nelneq 2858 | A way of showing two class... |
| nelneq2 2859 | A way of showing two class... |
| eqsb1 2860 | Substitution for the left-... |
| clelsb1 2861 | Substitution for the first... |
| clelsb2 2862 | Substitution for the secon... |
| clelsb2OLD 2863 | Obsolete version of ~ clel... |
| cleqh 2864 | Establish equality between... |
| hbxfreq 2865 | A utility lemma to transfe... |
| hblem 2866 | Change the free variable o... |
| hblemg 2867 | Change the free variable o... |
| eqabdv 2868 | Deduction from a wff to a ... |
| eqabcdv 2869 | Deduction from a wff to a ... |
| eqabi 2870 | Equality of a class variab... |
| abid1 2871 | Every class is equal to a ... |
| abid2 2872 | A simplification of class ... |
| eqab 2873 | One direction of ~ eqabb i... |
| eqabb 2874 | Equality of a class variab... |
| eqabbOLD 2875 | Obsolete version of ~ eqab... |
| eqabcb 2876 | Equality of a class variab... |
| eqabrd 2877 | Equality of a class variab... |
| eqabri 2878 | Equality of a class variab... |
| eqabcri 2879 | Equality of a class variab... |
| clelab 2880 | Membership of a class vari... |
| clelabOLD 2881 | Obsolete version of ~ clel... |
| clabel 2882 | Membership of a class abst... |
| sbab 2883 | The right-hand side of the... |
| nfcjust 2885 | Justification theorem for ... |
| nfci 2887 | Deduce that a class ` A ` ... |
| nfcii 2888 | Deduce that a class ` A ` ... |
| nfcr 2889 | Consequence of the not-fre... |
| nfcrALT 2890 | Alternate version of ~ nfc... |
| nfcri 2891 | Consequence of the not-fre... |
| nfcd 2892 | Deduce that a class ` A ` ... |
| nfcrd 2893 | Consequence of the not-fre... |
| nfcriOLD 2894 | Obsolete version of ~ nfcr... |
| nfcriOLDOLD 2895 | Obsolete version of ~ nfcr... |
| nfcrii 2896 | Consequence of the not-fre... |
| nfcriiOLD 2897 | Obsolete version of ~ nfcr... |
| nfcriOLDOLDOLD 2898 | Obsolete version of ~ nfcr... |
| nfceqdf 2899 | An equality theorem for ef... |
| nfceqdfOLD 2900 | Obsolete version of ~ nfce... |
| nfceqi 2901 | Equality theorem for class... |
| nfcxfr 2902 | A utility lemma to transfe... |
| nfcxfrd 2903 | A utility lemma to transfe... |
| nfcv 2904 | If ` x ` is disjoint from ... |
| nfcvd 2905 | If ` x ` is disjoint from ... |
| nfab1 2906 | Bound-variable hypothesis ... |
| nfnfc1 2907 | The setvar ` x ` is bound ... |
| clelsb1fw 2908 | Substitution for the first... |
| clelsb1f 2909 | Substitution for the first... |
| nfab 2910 | Bound-variable hypothesis ... |
| nfabg 2911 | Bound-variable hypothesis ... |
| nfaba1 2912 | Bound-variable hypothesis ... |
| nfaba1g 2913 | Bound-variable hypothesis ... |
| nfeqd 2914 | Hypothesis builder for equ... |
| nfeld 2915 | Hypothesis builder for ele... |
| nfnfc 2916 | Hypothesis builder for ` F... |
| nfeq 2917 | Hypothesis builder for equ... |
| nfel 2918 | Hypothesis builder for ele... |
| nfeq1 2919 | Hypothesis builder for equ... |
| nfel1 2920 | Hypothesis builder for ele... |
| nfeq2 2921 | Hypothesis builder for equ... |
| nfel2 2922 | Hypothesis builder for ele... |
| drnfc1 2923 | Formula-building lemma for... |
| drnfc1OLD 2924 | Obsolete version of ~ drnf... |
| drnfc2 2925 | Formula-building lemma for... |
| drnfc2OLD 2926 | Obsolete version of ~ drnf... |
| nfabdw 2927 | Bound-variable hypothesis ... |
| nfabdwOLD 2928 | Obsolete version of ~ nfab... |
| nfabd 2929 | Bound-variable hypothesis ... |
| nfabd2 2930 | Bound-variable hypothesis ... |
| dvelimdc 2931 | Deduction form of ~ dvelim... |
| dvelimc 2932 | Version of ~ dvelim for cl... |
| nfcvf 2933 | If ` x ` and ` y ` are dis... |
| nfcvf2 2934 | If ` x ` and ` y ` are dis... |
| cleqf 2935 | Establish equality between... |
| eqabf 2936 | Equality of a class variab... |
| abid2f 2937 | A simplification of class ... |
| abid2fOLD 2938 | Obsolete version of ~ abid... |
| sbabel 2939 | Theorem to move a substitu... |
| sbabelOLD 2940 | Obsolete version of ~ sbab... |
| neii 2943 | Inference associated with ... |
| neir 2944 | Inference associated with ... |
| nne 2945 | Negation of inequality. (... |
| neneqd 2946 | Deduction eliminating ineq... |
| neneq 2947 | From inequality to non-equ... |
| neqned 2948 | If it is not the case that... |
| neqne 2949 | From non-equality to inequ... |
| neirr 2950 | No class is unequal to its... |
| exmidne 2951 | Excluded middle with equal... |
| eqneqall 2952 | A contradiction concerning... |
| nonconne 2953 | Law of noncontradiction wi... |
| necon3ad 2954 | Contrapositive law deducti... |
| necon3bd 2955 | Contrapositive law deducti... |
| necon2ad 2956 | Contrapositive inference f... |
| necon2bd 2957 | Contrapositive inference f... |
| necon1ad 2958 | Contrapositive deduction f... |
| necon1bd 2959 | Contrapositive deduction f... |
| necon4ad 2960 | Contrapositive inference f... |
| necon4bd 2961 | Contrapositive inference f... |
| necon3d 2962 | Contrapositive law deducti... |
| necon1d 2963 | Contrapositive law deducti... |
| necon2d 2964 | Contrapositive inference f... |
| necon4d 2965 | Contrapositive inference f... |
| necon3ai 2966 | Contrapositive inference f... |
| necon3aiOLD 2967 | Obsolete version of ~ neco... |
| necon3bi 2968 | Contrapositive inference f... |
| necon1ai 2969 | Contrapositive inference f... |
| necon1bi 2970 | Contrapositive inference f... |
| necon2ai 2971 | Contrapositive inference f... |
| necon2bi 2972 | Contrapositive inference f... |
| necon4ai 2973 | Contrapositive inference f... |
| necon3i 2974 | Contrapositive inference f... |
| necon1i 2975 | Contrapositive inference f... |
| necon2i 2976 | Contrapositive inference f... |
| necon4i 2977 | Contrapositive inference f... |
| necon3abid 2978 | Deduction from equality to... |
| necon3bbid 2979 | Deduction from equality to... |
| necon1abid 2980 | Contrapositive deduction f... |
| necon1bbid 2981 | Contrapositive inference f... |
| necon4abid 2982 | Contrapositive law deducti... |
| necon4bbid 2983 | Contrapositive law deducti... |
| necon2abid 2984 | Contrapositive deduction f... |
| necon2bbid 2985 | Contrapositive deduction f... |
| necon3bid 2986 | Deduction from equality to... |
| necon4bid 2987 | Contrapositive law deducti... |
| necon3abii 2988 | Deduction from equality to... |
| necon3bbii 2989 | Deduction from equality to... |
| necon1abii 2990 | Contrapositive inference f... |
| necon1bbii 2991 | Contrapositive inference f... |
| necon2abii 2992 | Contrapositive inference f... |
| necon2bbii 2993 | Contrapositive inference f... |
| necon3bii 2994 | Inference from equality to... |
| necom 2995 | Commutation of inequality.... |
| necomi 2996 | Inference from commutative... |
| necomd 2997 | Deduction from commutative... |
| nesym 2998 | Characterization of inequa... |
| nesymi 2999 | Inference associated with ... |
| nesymir 3000 | Inference associated with ... |
| neeq1d 3001 | Deduction for inequality. ... |
| neeq2d 3002 | Deduction for inequality. ... |
| neeq12d 3003 | Deduction for inequality. ... |
| neeq1 3004 | Equality theorem for inequ... |
| neeq2 3005 | Equality theorem for inequ... |
| neeq1i 3006 | Inference for inequality. ... |
| neeq2i 3007 | Inference for inequality. ... |
| neeq12i 3008 | Inference for inequality. ... |
| eqnetrd 3009 | Substitution of equal clas... |
| eqnetrrd 3010 | Substitution of equal clas... |
| neeqtrd 3011 | Substitution of equal clas... |
| eqnetri 3012 | Substitution of equal clas... |
| eqnetrri 3013 | Substitution of equal clas... |
| neeqtri 3014 | Substitution of equal clas... |
| neeqtrri 3015 | Substitution of equal clas... |
| neeqtrrd 3016 | Substitution of equal clas... |
| eqnetrrid 3017 | A chained equality inferen... |
| 3netr3d 3018 | Substitution of equality i... |
| 3netr4d 3019 | Substitution of equality i... |
| 3netr3g 3020 | Substitution of equality i... |
| 3netr4g 3021 | Substitution of equality i... |
| nebi 3022 | Contraposition law for ine... |
| pm13.18 3023 | Theorem *13.18 in [Whitehe... |
| pm13.181 3024 | Theorem *13.181 in [Whiteh... |
| pm13.181OLD 3025 | Obsolete version of ~ pm13... |
| pm2.61ine 3026 | Inference eliminating an i... |
| pm2.21ddne 3027 | A contradiction implies an... |
| pm2.61ne 3028 | Deduction eliminating an i... |
| pm2.61dne 3029 | Deduction eliminating an i... |
| pm2.61dane 3030 | Deduction eliminating an i... |
| pm2.61da2ne 3031 | Deduction eliminating two ... |
| pm2.61da3ne 3032 | Deduction eliminating thre... |
| pm2.61iine 3033 | Equality version of ~ pm2.... |
| mteqand 3034 | A modus tollens deduction ... |
| neor 3035 | Logical OR with an equalit... |
| neanior 3036 | A De Morgan's law for ineq... |
| ne3anior 3037 | A De Morgan's law for ineq... |
| neorian 3038 | A De Morgan's law for ineq... |
| nemtbir 3039 | An inference from an inequ... |
| nelne1 3040 | Two classes are different ... |
| nelne2 3041 | Two classes are different ... |
| nelelne 3042 | Two classes are different ... |
| neneor 3043 | If two classes are differe... |
| nfne 3044 | Bound-variable hypothesis ... |
| nfned 3045 | Bound-variable hypothesis ... |
| nabbib 3046 | Not equivalent wff's corre... |
| neli 3049 | Inference associated with ... |
| nelir 3050 | Inference associated with ... |
| nelcon3d 3051 | Contrapositive law deducti... |
| neleq12d 3052 | Equality theorem for negat... |
| neleq1 3053 | Equality theorem for negat... |
| neleq2 3054 | Equality theorem for negat... |
| nfnel 3055 | Bound-variable hypothesis ... |
| nfneld 3056 | Bound-variable hypothesis ... |
| nnel 3057 | Negation of negated member... |
| elnelne1 3058 | Two classes are different ... |
| elnelne2 3059 | Two classes are different ... |
| pm2.24nel 3060 | A contradiction concerning... |
| pm2.61danel 3061 | Deduction eliminating an e... |
| rgen 3064 | Generalization rule for re... |
| ralel 3065 | All elements of a class ar... |
| rgenw 3066 | Generalization rule for re... |
| rgen2w 3067 | Generalization rule for re... |
| mprg 3068 | Modus ponens combined with... |
| mprgbir 3069 | Modus ponens on biconditio... |
| raln 3070 | Restricted universally qua... |
| ralnex 3073 | Relationship between restr... |
| dfrex2 3074 | Relationship between restr... |
| nrex 3075 | Inference adding restricte... |
| alral 3076 | Universal quantification i... |
| rexex 3077 | Restricted existence impli... |
| rextru 3078 | Two ways of expressing tha... |
| ralimi2 3079 | Inference quantifying both... |
| reximi2 3080 | Inference quantifying both... |
| ralimia 3081 | Inference quantifying both... |
| reximia 3082 | Inference quantifying both... |
| ralimiaa 3083 | Inference quantifying both... |
| ralimi 3084 | Inference quantifying both... |
| reximi 3085 | Inference quantifying both... |
| ral2imi 3086 | Inference quantifying ante... |
| ralim 3087 | Distribution of restricted... |
| rexim 3088 | Theorem 19.22 of [Margaris... |
| reximiaOLD 3089 | Obsolete version of ~ rexi... |
| ralbii2 3090 | Inference adding different... |
| rexbii2 3091 | Inference adding different... |
| ralbiia 3092 | Inference adding restricte... |
| rexbiia 3093 | Inference adding restricte... |
| ralbii 3094 | Inference adding restricte... |
| rexbii 3095 | Inference adding restricte... |
| ralanid 3096 | Cancellation law for restr... |
| rexanid 3097 | Cancellation law for restr... |
| ralcom3 3098 | A commutation law for rest... |
| ralcom3OLD 3099 | Obsolete version of ~ ralc... |
| dfral2 3100 | Relationship between restr... |
| rexnal 3101 | Relationship between restr... |
| ralinexa 3102 | A transformation of restri... |
| rexanali 3103 | A transformation of restri... |
| ralbi 3104 | Distribute a restricted un... |
| rexbi 3105 | Distribute restricted quan... |
| rexbiOLD 3106 | Obsolete version of ~ rexb... |
| ralrexbid 3107 | Formula-building rule for ... |
| ralrexbidOLD 3108 | Obsolete version of ~ ralr... |
| r19.35 3109 | Restricted quantifier vers... |
| r19.35OLD 3110 | Obsolete version of ~ 19.3... |
| r19.26m 3111 | Version of ~ 19.26 and ~ r... |
| r19.26 3112 | Restricted quantifier vers... |
| r19.26-3 3113 | Version of ~ r19.26 with t... |
| ralbiim 3114 | Split a biconditional and ... |
| r19.29 3115 | Restricted quantifier vers... |
| r19.29OLD 3116 | Obsolete version of ~ r19.... |
| r19.29r 3117 | Restricted quantifier vers... |
| r19.29rOLD 3118 | Obsolete version of ~ r19.... |
| r19.29imd 3119 | Theorem 19.29 of [Margaris... |
| r19.40 3120 | Restricted quantifier vers... |
| r19.30 3121 | Restricted quantifier vers... |
| r19.30OLD 3122 | Obsolete version of ~ 19.3... |
| r19.43 3123 | Restricted quantifier vers... |
| 2ralimi 3124 | Inference quantifying both... |
| 3ralimi 3125 | Inference quantifying both... |
| 4ralimi 3126 | Inference quantifying both... |
| 5ralimi 3127 | Inference quantifying both... |
| 6ralimi 3128 | Inference quantifying both... |
| 2ralbii 3129 | Inference adding two restr... |
| 2rexbii 3130 | Inference adding two restr... |
| 3ralbii 3131 | Inference adding three res... |
| 4ralbii 3132 | Inference adding four rest... |
| 2ralbiim 3133 | Split a biconditional and ... |
| ralnex2 3134 | Relationship between two r... |
| ralnex3 3135 | Relationship between three... |
| rexnal2 3136 | Relationship between two r... |
| rexnal3 3137 | Relationship between three... |
| nrexralim 3138 | Negation of a complex pred... |
| r19.26-2 3139 | Restricted quantifier vers... |
| 2r19.29 3140 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3141 | Theorem 19.29 of [Margaris... |
| r19.29d2rOLD 3142 | Obsolete version of ~ r19.... |
| r2allem 3143 | Lemma factoring out common... |
| r2exlem 3144 | Lemma factoring out common... |
| hbralrimi 3145 | Inference from Theorem 19.... |
| ralrimiv 3146 | Inference from Theorem 19.... |
| ralrimiva 3147 | Inference from Theorem 19.... |
| rexlimiva 3148 | Inference from Theorem 19.... |
| rexlimiv 3149 | Inference from Theorem 19.... |
| nrexdv 3150 | Deduction adding restricte... |
| ralrimivw 3151 | Inference from Theorem 19.... |
| rexlimivw 3152 | Weaker version of ~ rexlim... |
| ralrimdv 3153 | Inference from Theorem 19.... |
| rexlimdv 3154 | Inference from Theorem 19.... |
| ralrimdva 3155 | Inference from Theorem 19.... |
| rexlimdva 3156 | Inference from Theorem 19.... |
| rexlimdvaa 3157 | Inference from Theorem 19.... |
| rexlimdva2 3158 | Inference from Theorem 19.... |
| r19.29an 3159 | A commonly used pattern in... |
| rexlimdv3a 3160 | Inference from Theorem 19.... |
| rexlimdvw 3161 | Inference from Theorem 19.... |
| rexlimddv 3162 | Restricted existential eli... |
| r19.29a 3163 | A commonly used pattern in... |
| ralimdv2 3164 | Inference quantifying both... |
| reximdv2 3165 | Deduction quantifying both... |
| reximdvai 3166 | Deduction quantifying both... |
| reximdvaiOLD 3167 | Obsolete version of ~ rexi... |
| ralimdva 3168 | Deduction quantifying both... |
| reximdva 3169 | Deduction quantifying both... |
| ralimdv 3170 | Deduction quantifying both... |
| reximdv 3171 | Deduction from Theorem 19.... |
| reximddv 3172 | Deduction from Theorem 19.... |
| reximssdv 3173 | Derivation of a restricted... |
| ralbidv2 3174 | Formula-building rule for ... |
| rexbidv2 3175 | Formula-building rule for ... |
| ralbidva 3176 | Formula-building rule for ... |
| rexbidva 3177 | Formula-building rule for ... |
| ralbidv 3178 | Formula-building rule for ... |
| rexbidv 3179 | Formula-building rule for ... |
| r19.21v 3180 | Restricted quantifier vers... |
| r19.21vOLD 3181 | Obsolete version of ~ r19.... |
| r19.37v 3182 | Restricted quantifier vers... |
| r19.23v 3183 | Restricted quantifier vers... |
| r19.36v 3184 | Restricted quantifier vers... |
| rexlimivOLD 3185 | Obsolete version of ~ rexl... |
| rexlimivaOLD 3186 | Obsolete version of ~ rexl... |
| rexlimivwOLD 3187 | Obsolete version of ~ rexl... |
| r19.27v 3188 | Restricted quantitifer ver... |
| r19.41v 3189 | Restricted quantifier vers... |
| r19.28v 3190 | Restricted quantifier vers... |
| r19.42v 3191 | Restricted quantifier vers... |
| r19.32v 3192 | Restricted quantifier vers... |
| r19.45v 3193 | Restricted quantifier vers... |
| r19.44v 3194 | One direction of a restric... |
| r2al 3195 | Double restricted universa... |
| r2ex 3196 | Double restricted existent... |
| r3al 3197 | Triple restricted universa... |
| rgen2 3198 | Generalization rule for re... |
| ralrimivv 3199 | Inference from Theorem 19.... |
| rexlimivv 3200 | Inference from Theorem 19.... |
| ralrimivva 3201 | Inference from Theorem 19.... |
| ralrimdvv 3202 | Inference from Theorem 19.... |
| rgen3 3203 | Generalization rule for re... |
| ralrimivvva 3204 | Inference from Theorem 19.... |
| ralimdvva 3205 | Deduction doubly quantifyi... |
| reximdvva 3206 | Deduction doubly quantifyi... |
| ralimdvv 3207 | Deduction doubly quantifyi... |
| ralimd4v 3208 | Deduction quadrupally quan... |
| ralimd6v 3209 | Deduction sextupally quant... |
| ralrimdvva 3210 | Inference from Theorem 19.... |
| rexlimdvv 3211 | Inference from Theorem 19.... |
| rexlimdvva 3212 | Inference from Theorem 19.... |
| reximddv2 3213 | Double deduction from Theo... |
| r19.29vva 3214 | A commonly used pattern ba... |
| r19.29vvaOLD 3215 | Obsolete version of ~ r19.... |
| 2rexbiia 3216 | Inference adding two restr... |
| 2ralbidva 3217 | Formula-building rule for ... |
| 2rexbidva 3218 | Formula-building rule for ... |
| 2ralbidv 3219 | Formula-building rule for ... |
| 2rexbidv 3220 | Formula-building rule for ... |
| rexralbidv 3221 | Formula-building rule for ... |
| 3ralbidv 3222 | Formula-building rule for ... |
| 4ralbidv 3223 | Formula-building rule for ... |
| 6ralbidv 3224 | Formula-building rule for ... |
| r19.41vv 3225 | Version of ~ r19.41v with ... |
| reeanlem 3226 | Lemma factoring out common... |
| reeanv 3227 | Rearrange restricted exist... |
| 3reeanv 3228 | Rearrange three restricted... |
| 2ralor 3229 | Distribute restricted univ... |
| 2ralorOLD 3230 | Obsolete version of ~ 2ral... |
| risset 3231 | Two ways to say " ` A ` be... |
| nelb 3232 | A definition of ` -. A e. ... |
| nelbOLD 3233 | Obsolete version of ~ nelb... |
| rspw 3234 | Restricted specialization.... |
| cbvralvw 3235 | Change the bound variable ... |
| cbvrexvw 3236 | Change the bound variable ... |
| cbvraldva 3237 | Rule used to change the bo... |
| cbvrexdva 3238 | Rule used to change the bo... |
| cbvral2vw 3239 | Change bound variables of ... |
| cbvrex2vw 3240 | Change bound variables of ... |
| cbvral3vw 3241 | Change bound variables of ... |
| cbvral4vw 3242 | Change bound variables of ... |
| cbvral6vw 3243 | Change bound variables of ... |
| cbvral8vw 3244 | Change bound variables of ... |
| rsp 3245 | Restricted specialization.... |
| rspa 3246 | Restricted specialization.... |
| rspe 3247 | Restricted specialization.... |
| rspec 3248 | Specialization rule for re... |
| r19.21bi 3249 | Inference from Theorem 19.... |
| r19.21be 3250 | Inference from Theorem 19.... |
| r19.21t 3251 | Restricted quantifier vers... |
| r19.21 3252 | Restricted quantifier vers... |
| r19.23t 3253 | Closed theorem form of ~ r... |
| r19.23 3254 | Restricted quantifier vers... |
| ralrimi 3255 | Inference from Theorem 19.... |
| ralrimia 3256 | Inference from Theorem 19.... |
| rexlimi 3257 | Restricted quantifier vers... |
| ralimdaa 3258 | Deduction quantifying both... |
| reximdai 3259 | Deduction from Theorem 19.... |
| r19.37 3260 | Restricted quantifier vers... |
| r19.41 3261 | Restricted quantifier vers... |
| ralrimd 3262 | Inference from Theorem 19.... |
| rexlimd2 3263 | Version of ~ rexlimd with ... |
| rexlimd 3264 | Deduction form of ~ rexlim... |
| r19.29af2 3265 | A commonly used pattern ba... |
| r19.29af 3266 | A commonly used pattern ba... |
| reximd2a 3267 | Deduction quantifying both... |
| ralbida 3268 | Formula-building rule for ... |
| ralbidaOLD 3269 | Obsolete version of ~ ralb... |
| rexbida 3270 | Formula-building rule for ... |
| ralbid 3271 | Formula-building rule for ... |
| rexbid 3272 | Formula-building rule for ... |
| rexbidvALT 3273 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3274 | Alternate proof of ~ rexbi... |
| rsp2 3275 | Restricted specialization,... |
| rsp2e 3276 | Restricted specialization.... |
| rspec2 3277 | Specialization rule for re... |
| rspec3 3278 | Specialization rule for re... |
| r2alf 3279 | Double restricted universa... |
| r2exf 3280 | Double restricted existent... |
| 2ralbida 3281 | Formula-building rule for ... |
| nfra1 3282 | The setvar ` x ` is not fr... |
| nfre1 3283 | The setvar ` x ` is not fr... |
| ralcom4 3284 | Commutation of restricted ... |
| ralcom4OLD 3285 | Obsolete version of ~ ralc... |
| rexcom4 3286 | Commutation of restricted ... |
| ralcom 3287 | Commutation of restricted ... |
| rexcom 3288 | Commutation of restricted ... |
| rexcomOLD 3289 | Obsolete version of ~ rexc... |
| rexcom4a 3290 | Specialized existential co... |
| ralrot3 3291 | Rotate three restricted un... |
| ralcom13 3292 | Swap first and third restr... |
| ralcom13OLD 3293 | Obsolete version of ~ ralc... |
| rexcom13 3294 | Swap first and third restr... |
| rexrot4 3295 | Rotate four restricted exi... |
| 2ex2rexrot 3296 | Rotate two existential qua... |
| nfra2w 3297 | Similar to Lemma 24 of [Mo... |
| nfra2wOLD 3298 | Obsolete version of ~ nfra... |
| hbra1 3299 | The setvar ` x ` is not fr... |
| ralcomf 3300 | Commutation of restricted ... |
| rexcomf 3301 | Commutation of restricted ... |
| cbvralfw 3302 | Rule used to change bound ... |
| cbvrexfw 3303 | Rule used to change bound ... |
| cbvralw 3304 | Rule used to change bound ... |
| cbvrexw 3305 | Rule used to change bound ... |
| hbral 3306 | Bound-variable hypothesis ... |
| nfraldw 3307 | Deduction version of ~ nfr... |
| nfrexdw 3308 | Deduction version of ~ nfr... |
| nfralw 3309 | Bound-variable hypothesis ... |
| nfralwOLD 3310 | Obsolete version of ~ nfra... |
| nfrexw 3311 | Bound-variable hypothesis ... |
| r19.12 3312 | Restricted quantifier vers... |
| r19.12OLD 3313 | Obsolete version of ~ 19.1... |
| 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... |
| cbvrexsvwOLD 3318 | Obsolete version of ~ cbvr... |
| nfraldwOLD 3319 | Obsolete version of ~ nfra... |
| nfra2wOLDOLD 3320 | Obsolete version of ~ nfra... |
| cbvralfwOLD 3321 | Obsolete version of ~ cbvr... |
| rexeq 3322 | Equality theorem for restr... |
| raleq 3323 | Equality theorem for restr... |
| raleqi 3324 | Equality inference for res... |
| rexeqi 3325 | Equality inference for res... |
| raleqdv 3326 | Equality deduction for res... |
| rexeqdv 3327 | Equality deduction for res... |
| raleqbidva 3328 | Equality deduction for res... |
| rexeqbidva 3329 | Equality deduction for res... |
| raleqbidvv 3330 | Version of ~ raleqbidv wit... |
| raleqbidvvOLD 3331 | Version of ~ raleqbidv wit... |
| rexeqbidvv 3332 | Version of ~ rexeqbidv wit... |
| rexeqbidvvOLD 3333 | Version of ~ rexeqbidv wit... |
| raleqbi1dv 3334 | Equality deduction for res... |
| rexeqbi1dv 3335 | Equality deduction for res... |
| raleqOLD 3336 | Equality theorem for restr... |
| rexeqOLD 3337 | Equality theorem for restr... |
| raleleq 3338 | All elements of a class ar... |
| raleqbii 3339 | Equality deduction for res... |
| rexeqbii 3340 | Equality deduction for res... |
| raleleqOLD 3341 | Obsolete version of ~ rale... |
| raleleqALT 3342 | Alternate proof of ~ ralel... |
| raleqbidv 3343 | Equality deduction for res... |
| rexeqbidv 3344 | Equality deduction for res... |
| cbvraldva2 3345 | Rule used to change the bo... |
| cbvrexdva2 3346 | Rule used to change the bo... |
| cbvrexdva2OLD 3347 | Obsolete version of ~ cbvr... |
| cbvraldvaOLD 3348 | Obsolete version of ~ cbvr... |
| cbvrexdvaOLD 3349 | Obsolete version of ~ cbvr... |
| raleqf 3350 | Equality theorem for restr... |
| rexeqf 3351 | Equality theorem for restr... |
| rexeqfOLD 3352 | Obsolete version of ~ rexe... |
| raleqbid 3353 | Equality deduction for res... |
| rexeqbid 3354 | Equality deduction for res... |
| sbralie 3355 | Implicit to explicit subst... |
| sbralieALT 3356 | Alternative shorter proof ... |
| cbvralf 3357 | Rule used to change bound ... |
| cbvrexf 3358 | Rule used to change bound ... |
| cbvral 3359 | Rule used to change bound ... |
| cbvrex 3360 | Rule used to change bound ... |
| cbvralv 3361 | Change the bound variable ... |
| cbvrexv 3362 | Change the bound variable ... |
| cbvralsv 3363 | Change bound variable by u... |
| cbvrexsv 3364 | Change bound variable by u... |
| cbvral2v 3365 | Change bound variables of ... |
| cbvrex2v 3366 | Change bound variables of ... |
| cbvral3v 3367 | Change bound variables of ... |
| rgen2a 3368 | Generalization rule for re... |
| nfrald 3369 | Deduction version of ~ nfr... |
| nfrexd 3370 | Deduction version of ~ nfr... |
| nfral 3371 | Bound-variable hypothesis ... |
| nfrex 3372 | Bound-variable hypothesis ... |
| nfra2 3373 | Similar to Lemma 24 of [Mo... |
| ralcom2 3374 | Commutation of restricted ... |
| reu5 3379 | Restricted uniqueness in t... |
| reurmo 3380 | Restricted existential uni... |
| reurex 3381 | Restricted unique existenc... |
| mormo 3382 | Unrestricted "at most one"... |
| rmobiia 3383 | Formula-building rule for ... |
| reubiia 3384 | Formula-building rule for ... |
| rmobii 3385 | Formula-building rule for ... |
| reubii 3386 | Formula-building rule for ... |
| rmoanid 3387 | Cancellation law for restr... |
| reuanid 3388 | Cancellation law for restr... |
| rmoanidOLD 3389 | Obsolete version of ~ rmoa... |
| reuanidOLD 3390 | Obsolete version of ~ reua... |
| 2reu2rex 3391 | Double restricted existent... |
| rmobidva 3392 | Formula-building rule for ... |
| reubidva 3393 | Formula-building rule for ... |
| rmobidv 3394 | Formula-building rule for ... |
| reubidv 3395 | Formula-building rule for ... |
| reueubd 3396 | Restricted existential uni... |
| rmo5 3397 | Restricted "at most one" i... |
| nrexrmo 3398 | Nonexistence implies restr... |
| moel 3399 | "At most one" element in a... |
| cbvrmovw 3400 | Change the bound variable ... |
| cbvreuvw 3401 | Change the bound variable ... |
| moelOLD 3402 | Obsolete version of ~ moel... |
| rmobida 3403 | Formula-building rule for ... |
| reubida 3404 | Formula-building rule for ... |
| rmobidvaOLD 3405 | Obsolete version of ~ rmob... |
| cbvrmow 3406 | Change the bound variable ... |
| cbvreuw 3407 | Change the bound variable ... |
| nfrmo1 3408 | The setvar ` x ` is not fr... |
| nfreu1 3409 | The setvar ` x ` is not fr... |
| nfrmow 3410 | Bound-variable hypothesis ... |
| nfreuw 3411 | Bound-variable hypothesis ... |
| cbvrmowOLD 3412 | Obsolete version of ~ cbvr... |
| cbvreuwOLD 3413 | Obsolete version of ~ cbvr... |
| cbvreuvwOLD 3414 | Obsolete version of ~ cbvr... |
| rmoeq1 3415 | Equality theorem for restr... |
| reueq1 3416 | Equality theorem for restr... |
| rmoeq1OLD 3417 | Obsolete version of ~ rmoe... |
| reueq1OLD 3418 | Obsolete version of ~ reue... |
| rmoeqd 3419 | Equality deduction for res... |
| reueqd 3420 | Equality deduction for res... |
| rmoeq1f 3421 | Equality theorem for restr... |
| reueq1f 3422 | Equality theorem for restr... |
| nfreuwOLD 3423 | Obsolete version of ~ nfre... |
| nfrmowOLD 3424 | Obsolete version of ~ nfrm... |
| cbvreu 3425 | Change the bound variable ... |
| cbvrmo 3426 | Change the bound variable ... |
| cbvrmov 3427 | Change the bound variable ... |
| cbvreuv 3428 | Change the bound variable ... |
| nfrmod 3429 | Deduction version of ~ nfr... |
| nfreud 3430 | Deduction version of ~ nfr... |
| nfrmo 3431 | Bound-variable hypothesis ... |
| nfreu 3432 | Bound-variable hypothesis ... |
| rabbidva2 3435 | Equivalent wff's yield equ... |
| rabbia2 3436 | Equivalent wff's yield equ... |
| rabbiia 3437 | Equivalent formulas yield ... |
| rabbiiaOLD 3438 | Obsolete version of ~ rabb... |
| rabbii 3439 | Equivalent wff's correspon... |
| rabbidva 3440 | Equivalent wff's yield equ... |
| rabbidv 3441 | Equivalent wff's yield equ... |
| rabswap 3442 | Swap with a membership rel... |
| cbvrabv 3443 | Rule to change the bound v... |
| rabeqcda 3444 | When ` ps ` is always true... |
| rabeqc 3445 | A restricted class abstrac... |
| rabeqi 3446 | Equality theorem for restr... |
| rabeq 3447 | Equality theorem for restr... |
| rabeqdv 3448 | Equality of restricted cla... |
| rabeqbidva 3449 | Equality of restricted cla... |
| rabeqbidv 3450 | Equality of restricted cla... |
| rabrabi 3451 | Abstract builder restricte... |
| nfrab1 3452 | The abstraction variable i... |
| rabid 3453 | An "identity" law of concr... |
| rabidim1 3454 | Membership in a restricted... |
| reqabi 3455 | Inference from equality of... |
| rabrab 3456 | Abstract builder restricte... |
| rabrabiOLD 3457 | Obsolete version of ~ rabr... |
| rabbida4 3458 | Version of ~ rabbidva2 wit... |
| rabbida 3459 | Equivalent wff's yield equ... |
| rabbid 3460 | Version of ~ rabbidv with ... |
| rabeqd 3461 | Deduction form of ~ rabeq ... |
| rabeqbida 3462 | Version of ~ rabeqbidva wi... |
| rabbi 3463 | Equivalent wff's correspon... |
| rabid2f 3464 | An "identity" law for rest... |
| rabid2 3465 | An "identity" law for rest... |
| rabid2OLD 3466 | Obsolete version of ~ rabi... |
| rabeqf 3467 | Equality theorem for restr... |
| cbvrabw 3468 | Rule to change the bound v... |
| nfrabw 3469 | A variable not free in a w... |
| nfrabwOLD 3470 | Obsolete version of ~ nfra... |
| rabbidaOLD 3471 | Obsolete version of ~ rabb... |
| rabeqiOLD 3472 | Obsolete version of ~ rabe... |
| nfrab 3473 | A variable not free in a w... |
| cbvrab 3474 | Rule to change the bound v... |
| vjust 3476 | Justification theorem for ... |
| dfv2 3478 | Alternate definition of th... |
| vex 3479 | All setvar variables are s... |
| vexOLD 3480 | Obsolete version of ~ vex ... |
| elv 3481 | If a proposition is implie... |
| elvd 3482 | If a proposition is implie... |
| el2v 3483 | If a proposition is implie... |
| eqv 3484 | The universe contains ever... |
| eqvf 3485 | The universe contains ever... |
| abv 3486 | The class of sets verifyin... |
| abvALT 3487 | Alternate proof of ~ abv ,... |
| isset 3488 | Two ways to express that "... |
| issetf 3489 | A version of ~ isset that ... |
| isseti 3490 | A way to say " ` A ` is a ... |
| issetri 3491 | A way to say " ` A ` is a ... |
| eqvisset 3492 | A class equal to a variabl... |
| elex 3493 | If a class is a member of ... |
| elexi 3494 | If a class is a member of ... |
| elexd 3495 | If a class is a member of ... |
| elex2OLD 3496 | Obsolete version of ~ elex... |
| elex22 3497 | If two classes each contai... |
| prcnel 3498 | A proper class doesn't bel... |
| ralv 3499 | A universal quantifier res... |
| rexv 3500 | An existential quantifier ... |
| reuv 3501 | A unique existential quant... |
| rmov 3502 | An at-most-one quantifier ... |
| rabab 3503 | A class abstraction restri... |
| rexcom4b 3504 | Specialized existential co... |
| ceqsalt 3505 | Closed theorem version of ... |
| ceqsralt 3506 | Restricted quantifier vers... |
| ceqsalg 3507 | A representation of explic... |
| ceqsalgALT 3508 | Alternate proof of ~ ceqsa... |
| ceqsal 3509 | A representation of explic... |
| ceqsalALT 3510 | A representation of explic... |
| ceqsalv 3511 | A representation of explic... |
| ceqsalvOLD 3512 | Obsolete version of ~ ceqs... |
| ceqsralv 3513 | Restricted quantifier vers... |
| ceqsralvOLD 3514 | Obsolete version of ~ ceqs... |
| gencl 3515 | Implicit substitution for ... |
| 2gencl 3516 | Implicit substitution for ... |
| 3gencl 3517 | Implicit substitution for ... |
| cgsexg 3518 | Implicit substitution infe... |
| cgsex2g 3519 | Implicit substitution infe... |
| cgsex4g 3520 | An implicit substitution i... |
| cgsex4gOLD 3521 | Obsolete version of ~ cgse... |
| cgsex4gOLDOLD 3522 | Obsolete version of ~ cgse... |
| ceqsex 3523 | Elimination of an existent... |
| ceqsexOLD 3524 | Obsolete version of ~ ceqs... |
| ceqsexv 3525 | Elimination of an existent... |
| ceqsexvOLD 3526 | Obsolete version of ~ ceqs... |
| ceqsexvOLDOLD 3527 | Obsolete version of ~ ceqs... |
| ceqsexv2d 3528 | Elimination of an existent... |
| ceqsex2 3529 | Elimination of two existen... |
| ceqsex2v 3530 | Elimination of two existen... |
| ceqsex3v 3531 | Elimination of three exist... |
| ceqsex4v 3532 | Elimination of four existe... |
| ceqsex6v 3533 | Elimination of six existen... |
| ceqsex8v 3534 | Elimination of eight exist... |
| gencbvex 3535 | Change of bound variable u... |
| gencbvex2 3536 | Restatement of ~ gencbvex ... |
| gencbval 3537 | Change of bound variable u... |
| sbhypf 3538 | Introduce an explicit subs... |
| sbhypfOLD 3539 | Obsolete version of ~ sbhy... |
| vtoclgft 3540 | Closed theorem form of ~ v... |
| vtocldf 3541 | Implicit substitution of a... |
| vtocld 3542 | Implicit substitution of a... |
| vtocldOLD 3543 | Obsolete version of ~ vtoc... |
| vtocl2d 3544 | Implicit substitution of t... |
| vtocleg 3545 | Implicit substitution of a... |
| vtoclef 3546 | Implicit substitution of a... |
| vtoclf 3547 | Implicit substitution of a... |
| vtoclfOLD 3548 | Obsolete version of ~ vtoc... |
| vtocl 3549 | Implicit substitution of a... |
| vtoclALT 3550 | Alternate proof of ~ vtocl... |
| vtocl2 3551 | Implicit substitution of c... |
| vtocl3 3552 | Implicit substitution of c... |
| vtoclb 3553 | Implicit substitution of a... |
| vtoclgf 3554 | Implicit substitution of a... |
| vtoclg1f 3555 | Version of ~ vtoclgf with ... |
| vtoclg 3556 | Implicit substitution of a... |
| vtoclgOLD 3557 | Obsolete version of ~ vtoc... |
| vtoclgOLDOLD 3558 | Obsolete version of ~ vtoc... |
| vtoclbg 3559 | Implicit substitution of a... |
| vtocl2gf 3560 | Implicit substitution of a... |
| vtocl3gf 3561 | Implicit substitution of a... |
| vtocl2g 3562 | Implicit substitution of 2... |
| vtocl3g 3563 | Implicit substitution of a... |
| vtoclgaf 3564 | Implicit substitution of a... |
| vtoclga 3565 | Implicit substitution of a... |
| vtocl2ga 3566 | Implicit substitution of 2... |
| vtocl2gaf 3567 | Implicit substitution of 2... |
| vtocl3gaf 3568 | Implicit substitution of 3... |
| vtocl3ga 3569 | Implicit substitution of 3... |
| vtocl3gaOLD 3570 | Obsolete version of ~ vtoc... |
| vtocl4g 3571 | Implicit substitution of 4... |
| vtocl4ga 3572 | Implicit substitution of 4... |
| vtoclegft 3573 | Implicit substitution of a... |
| vtoclegftOLD 3574 | Obsolete version of ~ vtoc... |
| vtocle 3575 | Implicit substitution of a... |
| vtoclri 3576 | Implicit substitution of a... |
| spcimgft 3577 | A closed version of ~ spci... |
| spcgft 3578 | A closed version of ~ spcg... |
| spcimgf 3579 | Rule of specialization, us... |
| spcimegf 3580 | Existential specialization... |
| spcgf 3581 | Rule of specialization, us... |
| spcegf 3582 | Existential specialization... |
| spcimdv 3583 | Restricted specialization,... |
| spcdv 3584 | Rule of specialization, us... |
| spcimedv 3585 | Restricted existential spe... |
| spcgv 3586 | Rule of specialization, us... |
| spcegv 3587 | Existential specialization... |
| spcedv 3588 | Existential specialization... |
| spc2egv 3589 | Existential specialization... |
| spc2gv 3590 | Specialization with two qu... |
| spc2ed 3591 | Existential specialization... |
| spc2d 3592 | Specialization with 2 quan... |
| spc3egv 3593 | Existential specialization... |
| spc3gv 3594 | Specialization with three ... |
| spcv 3595 | Rule of specialization, us... |
| spcev 3596 | Existential specialization... |
| spc2ev 3597 | Existential specialization... |
| rspct 3598 | A closed version of ~ rspc... |
| rspcdf 3599 | Restricted specialization,... |
| rspc 3600 | Restricted specialization,... |
| rspce 3601 | Restricted existential spe... |
| rspcimdv 3602 | Restricted specialization,... |
| rspcimedv 3603 | Restricted existential spe... |
| rspcdv 3604 | Restricted specialization,... |
| rspcedv 3605 | Restricted existential spe... |
| rspcebdv 3606 | Restricted existential spe... |
| rspcdv2 3607 | Restricted specialization,... |
| rspcv 3608 | Restricted specialization,... |
| rspccv 3609 | Restricted specialization,... |
| rspcva 3610 | Restricted specialization,... |
| rspccva 3611 | Restricted specialization,... |
| rspcev 3612 | Restricted existential spe... |
| rspcdva 3613 | Restricted specialization,... |
| rspcedvd 3614 | Restricted existential spe... |
| rspcime 3615 | Prove a restricted existen... |
| rspceaimv 3616 | Restricted existential spe... |
| rspcedeq1vd 3617 | Restricted existential spe... |
| rspcedeq2vd 3618 | Restricted existential spe... |
| rspc2 3619 | Restricted specialization ... |
| rspc2gv 3620 | Restricted specialization ... |
| rspc2v 3621 | 2-variable restricted spec... |
| rspc2va 3622 | 2-variable restricted spec... |
| rspc2ev 3623 | 2-variable restricted exis... |
| 2rspcedvdw 3624 | Double application of ~ rs... |
| rspc2dv 3625 | 2-variable restricted spec... |
| rspc3v 3626 | 3-variable restricted spec... |
| rspc3ev 3627 | 3-variable restricted exis... |
| rspc3dv 3628 | 3-variable restricted spec... |
| rspc4v 3629 | 4-variable restricted spec... |
| rspc6v 3630 | 6-variable restricted spec... |
| rspc8v 3631 | 8-variable restricted spec... |
| rspceeqv 3632 | Restricted existential spe... |
| ralxpxfr2d 3633 | Transfer a universal quant... |
| rexraleqim 3634 | Statement following from e... |
| eqvincg 3635 | A variable introduction la... |
| eqvinc 3636 | A variable introduction la... |
| eqvincf 3637 | A variable introduction la... |
| alexeqg 3638 | Two ways to express substi... |
| ceqex 3639 | Equality implies equivalen... |
| ceqsexg 3640 | A representation of explic... |
| ceqsexgv 3641 | Elimination of an existent... |
| ceqsrexv 3642 | Elimination of a restricte... |
| ceqsrexbv 3643 | Elimination of a restricte... |
| ceqsralbv 3644 | Elimination of a restricte... |
| ceqsrex2v 3645 | Elimination of a restricte... |
| clel2g 3646 | Alternate definition of me... |
| clel2gOLD 3647 | Obsolete version of ~ clel... |
| clel2 3648 | Alternate definition of me... |
| clel3g 3649 | Alternate definition of me... |
| clel3 3650 | Alternate definition of me... |
| clel4g 3651 | Alternate definition of me... |
| clel4 3652 | Alternate definition of me... |
| clel4OLD 3653 | Obsolete version of ~ clel... |
| clel5 3654 | Alternate definition of cl... |
| pm13.183 3655 | Compare theorem *13.183 in... |
| rr19.3v 3656 | Restricted quantifier vers... |
| rr19.28v 3657 | Restricted quantifier vers... |
| elab6g 3658 | Membership in a class abst... |
| elabd2 3659 | Membership in a class abst... |
| elabd3 3660 | Membership in a class abst... |
| elabgt 3661 | Membership in a class abst... |
| elabgtOLD 3662 | Obsolete version of ~ elab... |
| elabgf 3663 | Membership in a class abst... |
| elabf 3664 | Membership in a class abst... |
| elabg 3665 | Membership in a class abst... |
| elabgOLD 3666 | Obsolete version of ~ elab... |
| elab 3667 | Membership in a class abst... |
| elabOLD 3668 | Obsolete version of ~ elab... |
| elab2g 3669 | Membership in a class abst... |
| elabd 3670 | Explicit demonstration the... |
| elab2 3671 | Membership in a class abst... |
| elab4g 3672 | Membership in a class abst... |
| elab3gf 3673 | Membership in a class abst... |
| elab3g 3674 | Membership in a class abst... |
| elab3 3675 | Membership in a class abst... |
| elrabi 3676 | Implication for the member... |
| elrabiOLD 3677 | Obsolete version of ~ elra... |
| elrabf 3678 | Membership in a restricted... |
| rabtru 3679 | Abstract builder using the... |
| rabeqcOLD 3680 | Obsolete version of ~ rabe... |
| elrab3t 3681 | Membership in a restricted... |
| elrab 3682 | Membership in a restricted... |
| elrab3 3683 | Membership in a restricted... |
| elrabd 3684 | Membership in a restricted... |
| elrab2 3685 | Membership in a restricted... |
| ralab 3686 | Universal quantification o... |
| ralabOLD 3687 | Obsolete version of ~ rala... |
| ralrab 3688 | Universal quantification o... |
| rexab 3689 | Existential quantification... |
| rexabOLD 3690 | Obsolete version of ~ rexa... |
| rexrab 3691 | Existential quantification... |
| ralab2 3692 | Universal quantification o... |
| ralrab2 3693 | Universal quantification o... |
| rexab2 3694 | Existential quantification... |
| rexrab2 3695 | Existential quantification... |
| reurab 3696 | Restricted existential uni... |
| abidnf 3697 | Identity used to create cl... |
| dedhb 3698 | A deduction theorem for co... |
| class2seteq 3699 | Writing a set as a class a... |
| nelrdva 3700 | Deduce negative membership... |
| eqeu 3701 | A condition which implies ... |
| moeq 3702 | There exists at most one s... |
| eueq 3703 | A class is a set if and on... |
| eueqi 3704 | There exists a unique set ... |
| eueq2 3705 | Equality has existential u... |
| eueq3 3706 | Equality has existential u... |
| moeq3 3707 | "At most one" property of ... |
| mosub 3708 | "At most one" remains true... |
| mo2icl 3709 | Theorem for inferring "at ... |
| mob2 3710 | Consequence of "at most on... |
| moi2 3711 | Consequence of "at most on... |
| mob 3712 | Equality implied by "at mo... |
| moi 3713 | Equality implied by "at mo... |
| morex 3714 | Derive membership from uni... |
| euxfr2w 3715 | Transfer existential uniqu... |
| euxfrw 3716 | Transfer existential uniqu... |
| euxfr2 3717 | Transfer existential uniqu... |
| euxfr 3718 | Transfer existential uniqu... |
| euind 3719 | Existential uniqueness via... |
| reu2 3720 | A way to express restricte... |
| reu6 3721 | A way to express restricte... |
| reu3 3722 | A way to express restricte... |
| reu6i 3723 | A condition which implies ... |
| eqreu 3724 | A condition which implies ... |
| rmo4 3725 | Restricted "at most one" u... |
| reu4 3726 | Restricted uniqueness usin... |
| reu7 3727 | Restricted uniqueness usin... |
| reu8 3728 | Restricted uniqueness usin... |
| rmo3f 3729 | Restricted "at most one" u... |
| rmo4f 3730 | Restricted "at most one" u... |
| reu2eqd 3731 | Deduce equality from restr... |
| reueq 3732 | Equality has existential u... |
| rmoeq 3733 | Equality's restricted exis... |
| rmoan 3734 | Restricted "at most one" s... |
| rmoim 3735 | Restricted "at most one" i... |
| rmoimia 3736 | Restricted "at most one" i... |
| rmoimi 3737 | Restricted "at most one" i... |
| rmoimi2 3738 | Restricted "at most one" i... |
| 2reu5a 3739 | Double restricted existent... |
| reuimrmo 3740 | Restricted uniqueness impl... |
| 2reuswap 3741 | A condition allowing swap ... |
| 2reuswap2 3742 | A condition allowing swap ... |
| reuxfrd 3743 | Transfer existential uniqu... |
| reuxfr 3744 | Transfer existential uniqu... |
| reuxfr1d 3745 | Transfer existential uniqu... |
| reuxfr1ds 3746 | Transfer existential uniqu... |
| reuxfr1 3747 | Transfer existential uniqu... |
| reuind 3748 | Existential uniqueness via... |
| 2rmorex 3749 | Double restricted quantifi... |
| 2reu5lem1 3750 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3751 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3752 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3753 | Double restricted existent... |
| 2reurmo 3754 | Double restricted quantifi... |
| 2reurex 3755 | Double restricted quantifi... |
| 2rmoswap 3756 | A condition allowing to sw... |
| 2rexreu 3757 | Double restricted existent... |
| cdeqi 3760 | Deduce conditional equalit... |
| cdeqri 3761 | Property of conditional eq... |
| cdeqth 3762 | Deduce conditional equalit... |
| cdeqnot 3763 | Distribute conditional equ... |
| cdeqal 3764 | Distribute conditional equ... |
| cdeqab 3765 | Distribute conditional equ... |
| cdeqal1 3766 | Distribute conditional equ... |
| cdeqab1 3767 | Distribute conditional equ... |
| cdeqim 3768 | Distribute conditional equ... |
| cdeqcv 3769 | Conditional equality for s... |
| cdeqeq 3770 | Distribute conditional equ... |
| cdeqel 3771 | Distribute conditional equ... |
| nfcdeq 3772 | If we have a conditional e... |
| nfccdeq 3773 | Variation of ~ nfcdeq for ... |
| rru 3774 | Relative version of Russel... |
| ru 3775 | Russell's Paradox. Propos... |
| dfsbcq 3778 | Proper substitution of a c... |
| dfsbcq2 3779 | This theorem, which is sim... |
| sbsbc 3780 | Show that ~ df-sb and ~ df... |
| sbceq1d 3781 | Equality theorem for class... |
| sbceq1dd 3782 | Equality theorem for class... |
| sbceqbid 3783 | Equality theorem for class... |
| sbc8g 3784 | This is the closest we can... |
| sbc2or 3785 | The disjunction of two equ... |
| sbcex 3786 | By our definition of prope... |
| sbceq1a 3787 | Equality theorem for class... |
| sbceq2a 3788 | Equality theorem for class... |
| spsbc 3789 | Specialization: if a formu... |
| spsbcd 3790 | Specialization: if a formu... |
| sbcth 3791 | A substitution into a theo... |
| sbcthdv 3792 | Deduction version of ~ sbc... |
| sbcid 3793 | An identity theorem for su... |
| nfsbc1d 3794 | Deduction version of ~ nfs... |
| nfsbc1 3795 | Bound-variable hypothesis ... |
| nfsbc1v 3796 | Bound-variable hypothesis ... |
| nfsbcdw 3797 | Deduction version of ~ nfs... |
| nfsbcw 3798 | Bound-variable hypothesis ... |
| sbccow 3799 | A composition law for clas... |
| nfsbcd 3800 | Deduction version of ~ nfs... |
| nfsbc 3801 | Bound-variable hypothesis ... |
| sbcco 3802 | A composition law for clas... |
| sbcco2 3803 | A composition law for clas... |
| sbc5 3804 | An equivalence for class s... |
| sbc5ALT 3805 | Alternate proof of ~ sbc5 ... |
| sbc6g 3806 | An equivalence for class s... |
| sbc6gOLD 3807 | Obsolete version of ~ sbc6... |
| sbc6 3808 | An equivalence for class s... |
| sbc7 3809 | An equivalence for class s... |
| cbvsbcw 3810 | Change bound variables in ... |
| cbvsbcvw 3811 | Change the bound variable ... |
| cbvsbc 3812 | Change bound variables in ... |
| cbvsbcv 3813 | Change the bound variable ... |
| sbciegft 3814 | Conversion of implicit sub... |
| sbciegf 3815 | Conversion of implicit sub... |
| sbcieg 3816 | Conversion of implicit sub... |
| sbciegOLD 3817 | Obsolete version of ~ sbci... |
| sbcie2g 3818 | Conversion of implicit sub... |
| sbcie 3819 | Conversion of implicit sub... |
| sbciedf 3820 | Conversion of implicit sub... |
| sbcied 3821 | Conversion of implicit sub... |
| sbciedOLD 3822 | Obsolete version of ~ sbci... |
| sbcied2 3823 | Conversion of implicit sub... |
| elrabsf 3824 | Membership in a restricted... |
| eqsbc1 3825 | Substitution for the left-... |
| sbcng 3826 | Move negation in and out o... |
| sbcimg 3827 | Distribution of class subs... |
| sbcan 3828 | Distribution of class subs... |
| sbcor 3829 | Distribution of class subs... |
| sbcbig 3830 | Distribution of class subs... |
| sbcn1 3831 | Move negation in and out o... |
| sbcim1 3832 | Distribution of class subs... |
| sbcim1OLD 3833 | Obsolete version of ~ sbci... |
| sbcbid 3834 | Formula-building deduction... |
| sbcbidv 3835 | Formula-building deduction... |
| sbcbii 3836 | Formula-building inference... |
| sbcbi1 3837 | Distribution of class subs... |
| sbcbi2 3838 | Substituting into equivale... |
| sbcbi2OLD 3839 | Obsolete proof of ~ sbcbi2... |
| sbcal 3840 | Move universal quantifier ... |
| sbcex2 3841 | Move existential quantifie... |
| sbceqal 3842 | Class version of one impli... |
| sbceqalOLD 3843 | Obsolete version of ~ sbce... |
| sbeqalb 3844 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3845 | Substitution for the right... |
| sbc3an 3846 | Distribution of class subs... |
| sbcel1v 3847 | Class substitution into a ... |
| sbcel2gv 3848 | Class substitution into a ... |
| sbcel21v 3849 | Class substitution into a ... |
| sbcimdv 3850 | Substitution analogue of T... |
| sbcimdvOLD 3851 | Obsolete version of ~ sbci... |
| sbctt 3852 | Substitution for a variabl... |
| sbcgf 3853 | Substitution for a variabl... |
| sbc19.21g 3854 | Substitution for a variabl... |
| sbcg 3855 | Substitution for a variabl... |
| sbcgOLD 3856 | Obsolete version of ~ sbcg... |
| sbcgfi 3857 | Substitution for a variabl... |
| sbc2iegf 3858 | Conversion of implicit sub... |
| sbc2ie 3859 | Conversion of implicit sub... |
| sbc2ieOLD 3860 | Obsolete version of ~ sbc2... |
| sbc2iedv 3861 | Conversion of implicit sub... |
| sbc3ie 3862 | Conversion of implicit sub... |
| sbccomlem 3863 | Lemma for ~ sbccom . (Con... |
| sbccom 3864 | Commutative law for double... |
| sbcralt 3865 | Interchange class substitu... |
| sbcrext 3866 | Interchange class substitu... |
| sbcralg 3867 | Interchange class substitu... |
| sbcrex 3868 | Interchange class substitu... |
| sbcreu 3869 | Interchange class substitu... |
| reu8nf 3870 | Restricted uniqueness usin... |
| sbcabel 3871 | Interchange class substitu... |
| rspsbc 3872 | Restricted quantifier vers... |
| rspsbca 3873 | Restricted quantifier vers... |
| rspesbca 3874 | Existence form of ~ rspsbc... |
| spesbc 3875 | Existence form of ~ spsbc ... |
| spesbcd 3876 | form of ~ spsbc . (Contri... |
| sbcth2 3877 | A substitution into a theo... |
| ra4v 3878 | Version of ~ ra4 with a di... |
| ra4 3879 | Restricted quantifier vers... |
| rmo2 3880 | Alternate definition of re... |
| rmo2i 3881 | Condition implying restric... |
| rmo3 3882 | Restricted "at most one" u... |
| rmob 3883 | Consequence of "at most on... |
| rmoi 3884 | Consequence of "at most on... |
| rmob2 3885 | Consequence of "restricted... |
| rmoi2 3886 | Consequence of "restricted... |
| rmoanim 3887 | Introduction of a conjunct... |
| rmoanimALT 3888 | Alternate proof of ~ rmoan... |
| reuan 3889 | Introduction of a conjunct... |
| 2reu1 3890 | Double restricted existent... |
| 2reu2 3891 | Double restricted existent... |
| csb2 3894 | Alternate expression for t... |
| csbeq1 3895 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3896 | Equality deduction for pro... |
| csbeq2 3897 | Substituting into equivale... |
| csbeq2d 3898 | Formula-building deduction... |
| csbeq2dv 3899 | Formula-building deduction... |
| csbeq2i 3900 | Formula-building inference... |
| csbeq12dv 3901 | Formula-building inference... |
| cbvcsbw 3902 | Change bound variables in ... |
| cbvcsb 3903 | Change bound variables in ... |
| cbvcsbv 3904 | Change the bound variable ... |
| csbid 3905 | Analogue of ~ sbid for pro... |
| csbeq1a 3906 | Equality theorem for prope... |
| csbcow 3907 | Composition law for chaine... |
| csbco 3908 | Composition law for chaine... |
| csbtt 3909 | Substitution doesn't affec... |
| csbconstgf 3910 | Substitution doesn't affec... |
| csbconstg 3911 | Substitution doesn't affec... |
| csbconstgOLD 3912 | Obsolete version of ~ csbc... |
| csbgfi 3913 | Substitution for a variabl... |
| csbconstgi 3914 | The proper substitution of... |
| nfcsb1d 3915 | Bound-variable hypothesis ... |
| nfcsb1 3916 | Bound-variable hypothesis ... |
| nfcsb1v 3917 | Bound-variable hypothesis ... |
| nfcsbd 3918 | Deduction version of ~ nfc... |
| nfcsbw 3919 | Bound-variable hypothesis ... |
| nfcsb 3920 | Bound-variable hypothesis ... |
| csbhypf 3921 | Introduce an explicit subs... |
| csbiebt 3922 | Conversion of implicit sub... |
| csbiedf 3923 | Conversion of implicit sub... |
| csbieb 3924 | Bidirectional conversion b... |
| csbiebg 3925 | Bidirectional conversion b... |
| csbiegf 3926 | Conversion of implicit sub... |
| csbief 3927 | Conversion of implicit sub... |
| csbie 3928 | Conversion of implicit sub... |
| csbieOLD 3929 | Obsolete version of ~ csbi... |
| csbied 3930 | Conversion of implicit sub... |
| csbiedOLD 3931 | Obsolete version of ~ csbi... |
| csbied2 3932 | Conversion of implicit sub... |
| csbie2t 3933 | Conversion of implicit sub... |
| csbie2 3934 | Conversion of implicit sub... |
| csbie2g 3935 | Conversion of implicit sub... |
| cbvrabcsfw 3936 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3937 | A more general version of ... |
| cbvrexcsf 3938 | A more general version of ... |
| cbvreucsf 3939 | A more general version of ... |
| cbvrabcsf 3940 | A more general version of ... |
| cbvralv2 3941 | Rule used to change the bo... |
| cbvrexv2 3942 | Rule used to change the bo... |
| rspc2vd 3943 | Deduction version of 2-var... |
| difjust 3949 | Soundness justification th... |
| unjust 3951 | Soundness justification th... |
| injust 3953 | Soundness justification th... |
| dfin5 3955 | Alternate definition for t... |
| dfdif2 3956 | Alternate definition of cl... |
| eldif 3957 | Expansion of membership in... |
| eldifd 3958 | If a class is in one class... |
| eldifad 3959 | If a class is in the diffe... |
| eldifbd 3960 | If a class is in the diffe... |
| elneeldif 3961 | The elements of a set diff... |
| velcomp 3962 | Characterization of setvar... |
| elin 3963 | Expansion of membership in... |
| dfss 3965 | Variant of subclass defini... |
| dfss2 3967 | Alternate definition of th... |
| dfss2OLD 3968 | Obsolete version of ~ dfss... |
| dfss3 3969 | Alternate definition of su... |
| dfss6 3970 | Alternate definition of su... |
| dfss2f 3971 | Equivalence for subclass r... |
| dfss3f 3972 | Equivalence for subclass r... |
| nfss 3973 | If ` x ` is not free in ` ... |
| ssel 3974 | Membership relationships f... |
| sselOLD 3975 | Obsolete version of ~ ssel... |
| ssel2 3976 | Membership relationships f... |
| sseli 3977 | Membership implication fro... |
| sselii 3978 | Membership inference from ... |
| sselid 3979 | Membership inference from ... |
| sseld 3980 | Membership deduction from ... |
| sselda 3981 | Membership deduction from ... |
| sseldd 3982 | Membership inference from ... |
| ssneld 3983 | If a class is not in anoth... |
| ssneldd 3984 | If an element is not in a ... |
| ssriv 3985 | Inference based on subclas... |
| ssrd 3986 | Deduction based on subclas... |
| ssrdv 3987 | Deduction based on subclas... |
| sstr2 3988 | Transitivity of subclass r... |
| sstr 3989 | Transitivity of subclass r... |
| sstri 3990 | Subclass transitivity infe... |
| sstrd 3991 | Subclass transitivity dedu... |
| sstrid 3992 | Subclass transitivity dedu... |
| sstrdi 3993 | Subclass transitivity dedu... |
| sylan9ss 3994 | A subclass transitivity de... |
| sylan9ssr 3995 | A subclass transitivity de... |
| eqss 3996 | The subclass relationship ... |
| eqssi 3997 | Infer equality from two su... |
| eqssd 3998 | Equality deduction from tw... |
| sssseq 3999 | If a class is a subclass o... |
| eqrd 4000 | Deduce equality of classes... |
| eqri 4001 | Infer equality of classes ... |
| eqelssd 4002 | Equality deduction from su... |
| ssid 4003 | Any class is a subclass of... |
| ssidd 4004 | Weakening of ~ ssid . (Co... |
| ssv 4005 | Any class is a subclass of... |
| sseq1 4006 | Equality theorem for subcl... |
| sseq2 4007 | Equality theorem for the s... |
| sseq12 4008 | Equality theorem for the s... |
| sseq1i 4009 | An equality inference for ... |
| sseq2i 4010 | An equality inference for ... |
| sseq12i 4011 | An equality inference for ... |
| sseq1d 4012 | An equality deduction for ... |
| sseq2d 4013 | An equality deduction for ... |
| sseq12d 4014 | An equality deduction for ... |
| eqsstri 4015 | Substitution of equality i... |
| eqsstrri 4016 | Substitution of equality i... |
| sseqtri 4017 | Substitution of equality i... |
| sseqtrri 4018 | Substitution of equality i... |
| eqsstrd 4019 | Substitution of equality i... |
| eqsstrrd 4020 | Substitution of equality i... |
| sseqtrd 4021 | Substitution of equality i... |
| sseqtrrd 4022 | Substitution of equality i... |
| 3sstr3i 4023 | Substitution of equality i... |
| 3sstr4i 4024 | Substitution of equality i... |
| 3sstr3g 4025 | Substitution of equality i... |
| 3sstr4g 4026 | Substitution of equality i... |
| 3sstr3d 4027 | Substitution of equality i... |
| 3sstr4d 4028 | Substitution of equality i... |
| eqsstrid 4029 | A chained subclass and equ... |
| eqsstrrid 4030 | A chained subclass and equ... |
| sseqtrdi 4031 | A chained subclass and equ... |
| sseqtrrdi 4032 | A chained subclass and equ... |
| sseqtrid 4033 | Subclass transitivity dedu... |
| sseqtrrid 4034 | Subclass transitivity dedu... |
| eqsstrdi 4035 | A chained subclass and equ... |
| eqsstrrdi 4036 | A chained subclass and equ... |
| eqimssd 4037 | Equality implies inclusion... |
| eqimsscd 4038 | Equality implies inclusion... |
| eqimss 4039 | Equality implies inclusion... |
| eqimss2 4040 | Equality implies inclusion... |
| eqimssi 4041 | Infer subclass relationshi... |
| eqimss2i 4042 | Infer subclass relationshi... |
| nssne1 4043 | Two classes are different ... |
| nssne2 4044 | Two classes are different ... |
| nss 4045 | Negation of subclass relat... |
| nelss 4046 | Demonstrate by witnesses t... |
| ssrexf 4047 | Restricted existential qua... |
| ssrmof 4048 | "At most one" existential ... |
| ssralv 4049 | Quantification restricted ... |
| ssrexv 4050 | Existential quantification... |
| ss2ralv 4051 | Two quantifications restri... |
| ss2rexv 4052 | Two existential quantifica... |
| ralss 4053 | Restricted universal quant... |
| rexss 4054 | Restricted existential qua... |
| ss2ab 4055 | Class abstractions in a su... |
| abss 4056 | Class abstraction in a sub... |
| ssab 4057 | Subclass of a class abstra... |
| ssabral 4058 | The relation for a subclas... |
| ss2abdv 4059 | Deduction of abstraction s... |
| ss2abdvALT 4060 | Alternate proof of ~ ss2ab... |
| ss2abdvOLD 4061 | Obsolete version of ~ ss2a... |
| ss2abi 4062 | Inference of abstraction s... |
| ss2abiOLD 4063 | Obsolete version of ~ ss2a... |
| abssdv 4064 | Deduction of abstraction s... |
| abssdvOLD 4065 | Obsolete version of ~ abss... |
| abssi 4066 | Inference of abstraction s... |
| ss2rab 4067 | Restricted abstraction cla... |
| rabss 4068 | Restricted class abstracti... |
| ssrab 4069 | Subclass of a restricted c... |
| ssrabdv 4070 | Subclass of a restricted c... |
| rabssdv 4071 | Subclass of a restricted c... |
| ss2rabdv 4072 | Deduction of restricted ab... |
| ss2rabi 4073 | Inference of restricted ab... |
| rabss2 4074 | Subclass law for restricte... |
| ssab2 4075 | Subclass relation for the ... |
| ssrab2 4076 | Subclass relation for a re... |
| ssrab2OLD 4077 | Obsolete version of ~ ssra... |
| rabss3d 4078 | Subclass law for restricte... |
| ssrab3 4079 | Subclass relation for a re... |
| rabssrabd 4080 | Subclass of a restricted c... |
| ssrabeq 4081 | If the restricting class o... |
| rabssab 4082 | A restricted class is a su... |
| uniiunlem 4083 | A subset relationship usef... |
| dfpss2 4084 | Alternate definition of pr... |
| dfpss3 4085 | Alternate definition of pr... |
| psseq1 4086 | Equality theorem for prope... |
| psseq2 4087 | Equality theorem for prope... |
| psseq1i 4088 | An equality inference for ... |
| psseq2i 4089 | An equality inference for ... |
| psseq12i 4090 | An equality inference for ... |
| psseq1d 4091 | An equality deduction for ... |
| psseq2d 4092 | An equality deduction for ... |
| psseq12d 4093 | An equality deduction for ... |
| pssss 4094 | A proper subclass is a sub... |
| pssne 4095 | Two classes in a proper su... |
| pssssd 4096 | Deduce subclass from prope... |
| pssned 4097 | Proper subclasses are uneq... |
| sspss 4098 | Subclass in terms of prope... |
| pssirr 4099 | Proper subclass is irrefle... |
| pssn2lp 4100 | Proper subclass has no 2-c... |
| sspsstri 4101 | Two ways of stating tricho... |
| ssnpss 4102 | Partial trichotomy law for... |
| psstr 4103 | Transitive law for proper ... |
| sspsstr 4104 | Transitive law for subclas... |
| psssstr 4105 | Transitive law for subclas... |
| psstrd 4106 | Proper subclass inclusion ... |
| sspsstrd 4107 | Transitivity involving sub... |
| psssstrd 4108 | Transitivity involving sub... |
| npss 4109 | A class is not a proper su... |
| ssnelpss 4110 | A subclass missing a membe... |
| ssnelpssd 4111 | Subclass inclusion with on... |
| ssexnelpss 4112 | If there is an element of ... |
| dfdif3 4113 | Alternate definition of cl... |
| difeq1 4114 | Equality theorem for class... |
| difeq2 4115 | Equality theorem for class... |
| difeq12 4116 | Equality theorem for class... |
| difeq1i 4117 | Inference adding differenc... |
| difeq2i 4118 | Inference adding differenc... |
| difeq12i 4119 | Equality inference for cla... |
| difeq1d 4120 | Deduction adding differenc... |
| difeq2d 4121 | Deduction adding differenc... |
| difeq12d 4122 | Equality deduction for cla... |
| difeqri 4123 | Inference from membership ... |
| nfdif 4124 | Bound-variable hypothesis ... |
| eldifi 4125 | Implication of membership ... |
| eldifn 4126 | Implication of membership ... |
| elndif 4127 | A set does not belong to a... |
| neldif 4128 | Implication of membership ... |
| difdif 4129 | Double class difference. ... |
| difss 4130 | Subclass relationship for ... |
| difssd 4131 | A difference of two classe... |
| difss2 4132 | If a class is contained in... |
| difss2d 4133 | If a class is contained in... |
| ssdifss 4134 | Preservation of a subclass... |
| ddif 4135 | Double complement under un... |
| ssconb 4136 | Contraposition law for sub... |
| sscon 4137 | Contraposition law for sub... |
| ssdif 4138 | Difference law for subsets... |
| ssdifd 4139 | If ` A ` is contained in `... |
| sscond 4140 | If ` A ` is contained in `... |
| ssdifssd 4141 | If ` A ` is contained in `... |
| ssdif2d 4142 | If ` A ` is contained in `... |
| raldifb 4143 | Restricted universal quant... |
| rexdifi 4144 | Restricted existential qua... |
| complss 4145 | Complementation reverses i... |
| compleq 4146 | Two classes are equal if a... |
| elun 4147 | Expansion of membership in... |
| elunnel1 4148 | A member of a union that i... |
| elunnel2 4149 | A member of a union that i... |
| uneqri 4150 | Inference from membership ... |
| unidm 4151 | Idempotent law for union o... |
| uncom 4152 | Commutative law for union ... |
| equncom 4153 | If a class equals the unio... |
| equncomi 4154 | Inference form of ~ equnco... |
| uneq1 4155 | Equality theorem for the u... |
| uneq2 4156 | Equality theorem for the u... |
| uneq12 4157 | Equality theorem for the u... |
| uneq1i 4158 | Inference adding union to ... |
| uneq2i 4159 | Inference adding union to ... |
| uneq12i 4160 | Equality inference for the... |
| uneq1d 4161 | Deduction adding union to ... |
| uneq2d 4162 | Deduction adding union to ... |
| uneq12d 4163 | Equality deduction for the... |
| nfun 4164 | Bound-variable hypothesis ... |
| unass 4165 | Associative law for union ... |
| un12 4166 | A rearrangement of union. ... |
| un23 4167 | A rearrangement of union. ... |
| un4 4168 | A rearrangement of the uni... |
| unundi 4169 | Union distributes over its... |
| unundir 4170 | Union distributes over its... |
| ssun1 4171 | Subclass relationship for ... |
| ssun2 4172 | Subclass relationship for ... |
| ssun3 4173 | Subclass law for union of ... |
| ssun4 4174 | Subclass law for union of ... |
| elun1 4175 | Membership law for union o... |
| elun2 4176 | Membership law for union o... |
| elunant 4177 | A statement is true for ev... |
| unss1 4178 | Subclass law for union of ... |
| ssequn1 4179 | A relationship between sub... |
| unss2 4180 | Subclass law for union of ... |
| unss12 4181 | Subclass law for union of ... |
| ssequn2 4182 | A relationship between sub... |
| unss 4183 | The union of two subclasse... |
| unssi 4184 | An inference showing the u... |
| unssd 4185 | A deduction showing the un... |
| unssad 4186 | If ` ( A u. B ) ` is conta... |
| unssbd 4187 | If ` ( A u. B ) ` is conta... |
| ssun 4188 | A condition that implies i... |
| rexun 4189 | Restricted existential qua... |
| ralunb 4190 | Restricted quantification ... |
| ralun 4191 | Restricted quantification ... |
| elini 4192 | Membership in an intersect... |
| elind 4193 | Deduce membership in an in... |
| elinel1 4194 | Membership in an intersect... |
| elinel2 4195 | Membership in an intersect... |
| elin2 4196 | Membership in a class defi... |
| elin1d 4197 | Elementhood in the first s... |
| elin2d 4198 | Elementhood in the first s... |
| elin3 4199 | Membership in a class defi... |
| incom 4200 | Commutative law for inters... |
| ineqcom 4201 | Two ways of expressing tha... |
| ineqcomi 4202 | Two ways of expressing tha... |
| ineqri 4203 | Inference from membership ... |
| ineq1 4204 | Equality theorem for inter... |
| ineq2 4205 | Equality theorem for inter... |
| ineq12 4206 | Equality theorem for inter... |
| ineq1i 4207 | Equality inference for int... |
| ineq2i 4208 | Equality inference for int... |
| ineq12i 4209 | Equality inference for int... |
| ineq1d 4210 | Equality deduction for int... |
| ineq2d 4211 | Equality deduction for int... |
| ineq12d 4212 | Equality deduction for int... |
| ineqan12d 4213 | Equality deduction for int... |
| sseqin2 4214 | A relationship between sub... |
| nfin 4215 | Bound-variable hypothesis ... |
| rabbi2dva 4216 | Deduction from a wff to a ... |
| inidm 4217 | Idempotent law for interse... |
| inass 4218 | Associative law for inters... |
| in12 4219 | A rearrangement of interse... |
| in32 4220 | A rearrangement of interse... |
| in13 4221 | A rearrangement of interse... |
| in31 4222 | A rearrangement of interse... |
| inrot 4223 | Rotate the intersection of... |
| in4 4224 | Rearrangement of intersect... |
| inindi 4225 | Intersection distributes o... |
| inindir 4226 | Intersection distributes o... |
| inss1 4227 | The intersection of two cl... |
| inss2 4228 | The intersection of two cl... |
| ssin 4229 | Subclass of intersection. ... |
| ssini 4230 | An inference showing that ... |
| ssind 4231 | A deduction showing that a... |
| ssrin 4232 | Add right intersection to ... |
| sslin 4233 | Add left intersection to s... |
| ssrind 4234 | Add right intersection to ... |
| ss2in 4235 | Intersection of subclasses... |
| ssinss1 4236 | Intersection preserves sub... |
| inss 4237 | Inclusion of an intersecti... |
| rexin 4238 | Restricted existential qua... |
| dfss7 4239 | Alternate definition of su... |
| symdifcom 4242 | Symmetric difference commu... |
| symdifeq1 4243 | Equality theorem for symme... |
| symdifeq2 4244 | Equality theorem for symme... |
| nfsymdif 4245 | Hypothesis builder for sym... |
| elsymdif 4246 | Membership in a symmetric ... |
| dfsymdif4 4247 | Alternate definition of th... |
| elsymdifxor 4248 | Membership in a symmetric ... |
| dfsymdif2 4249 | Alternate definition of th... |
| symdifass 4250 | Symmetric difference is as... |
| difsssymdif 4251 | The symmetric difference c... |
| difsymssdifssd 4252 | If the symmetric differenc... |
| unabs 4253 | Absorption law for union. ... |
| inabs 4254 | Absorption law for interse... |
| nssinpss 4255 | Negation of subclass expre... |
| nsspssun 4256 | Negation of subclass expre... |
| dfss4 4257 | Subclass defined in terms ... |
| dfun2 4258 | An alternate definition of... |
| dfin2 4259 | An alternate definition of... |
| difin 4260 | Difference with intersecti... |
| ssdifim 4261 | Implication of a class dif... |
| ssdifsym 4262 | Symmetric class difference... |
| dfss5 4263 | Alternate definition of su... |
| dfun3 4264 | Union defined in terms of ... |
| dfin3 4265 | Intersection defined in te... |
| dfin4 4266 | Alternate definition of th... |
| invdif 4267 | Intersection with universa... |
| indif 4268 | Intersection with class di... |
| indif2 4269 | Bring an intersection in a... |
| indif1 4270 | Bring an intersection in a... |
| indifcom 4271 | Commutation law for inters... |
| indi 4272 | Distributive law for inter... |
| undi 4273 | Distributive law for union... |
| indir 4274 | Distributive law for inter... |
| undir 4275 | Distributive law for union... |
| unineq 4276 | Infer equality from equali... |
| uneqin 4277 | Equality of union and inte... |
| difundi 4278 | Distributive law for class... |
| difundir 4279 | Distributive law for class... |
| difindi 4280 | Distributive law for class... |
| difindir 4281 | Distributive law for class... |
| indifdi 4282 | Distribute intersection ov... |
| indifdir 4283 | Distribute intersection ov... |
| indifdirOLD 4284 | Obsolete version of ~ indi... |
| difdif2 4285 | Class difference by a clas... |
| undm 4286 | De Morgan's law for union.... |
| indm 4287 | De Morgan's law for inters... |
| difun1 4288 | A relationship involving d... |
| undif3 4289 | An equality involving clas... |
| difin2 4290 | Represent a class differen... |
| dif32 4291 | Swap second and third argu... |
| difabs 4292 | Absorption-like law for cl... |
| sscon34b 4293 | Relative complementation r... |
| rcompleq 4294 | Two subclasses are equal i... |
| dfsymdif3 4295 | Alternate definition of th... |
| unabw 4296 | Union of two class abstrac... |
| unab 4297 | Union of two class abstrac... |
| inab 4298 | Intersection of two class ... |
| difab 4299 | Difference of two class ab... |
| abanssl 4300 | A class abstraction with a... |
| abanssr 4301 | A class abstraction with a... |
| notabw 4302 | A class abstraction define... |
| notab 4303 | A class abstraction define... |
| unrab 4304 | Union of two restricted cl... |
| inrab 4305 | Intersection of two restri... |
| inrab2 4306 | Intersection with a restri... |
| difrab 4307 | Difference of two restrict... |
| dfrab3 4308 | Alternate definition of re... |
| dfrab2 4309 | Alternate definition of re... |
| notrab 4310 | Complementation of restric... |
| dfrab3ss 4311 | Restricted class abstracti... |
| rabun2 4312 | Abstraction restricted to ... |
| reuun2 4313 | Transfer uniqueness to a s... |
| reuss2 4314 | Transfer uniqueness to a s... |
| reuss 4315 | Transfer uniqueness to a s... |
| reuun1 4316 | Transfer uniqueness to a s... |
| reupick 4317 | Restricted uniqueness "pic... |
| reupick3 4318 | Restricted uniqueness "pic... |
| reupick2 4319 | Restricted uniqueness "pic... |
| euelss 4320 | Transfer uniqueness of an ... |
| dfnul4 4323 | Alternate definition of th... |
| dfnul2 4324 | Alternate definition of th... |
| dfnul3 4325 | Alternate definition of th... |
| dfnul2OLD 4326 | Obsolete version of ~ dfnu... |
| dfnul3OLD 4327 | Obsolete version of ~ dfnu... |
| dfnul4OLD 4328 | Obsolete version of ~ dfnu... |
| noel 4329 | The empty set has no eleme... |
| noelOLD 4330 | Obsolete version of ~ noel... |
| nel02 4331 | The empty set has no eleme... |
| n0i 4332 | If a class has elements, t... |
| ne0i 4333 | If a class has elements, t... |
| ne0d 4334 | Deduction form of ~ ne0i .... |
| n0ii 4335 | If a class has elements, t... |
| ne0ii 4336 | If a class has elements, t... |
| vn0 4337 | The universal class is not... |
| vn0ALT 4338 | Alternate proof of ~ vn0 .... |
| eq0f 4339 | A class is equal to the em... |
| neq0f 4340 | A class is not empty if an... |
| n0f 4341 | A class is nonempty if and... |
| eq0 4342 | A class is equal to the em... |
| eq0ALT 4343 | Alternate proof of ~ eq0 .... |
| neq0 4344 | A class is not empty if an... |
| n0 4345 | A class is nonempty if and... |
| eq0OLDOLD 4346 | Obsolete version of ~ eq0 ... |
| neq0OLD 4347 | Obsolete version of ~ neq0... |
| n0OLD 4348 | Obsolete version of ~ n0 a... |
| nel0 4349 | From the general negation ... |
| reximdva0 4350 | Restricted existence deduc... |
| rspn0 4351 | Specialization for restric... |
| rspn0OLD 4352 | Obsolete version of ~ rspn... |
| n0rex 4353 | There is an element in a n... |
| ssn0rex 4354 | There is an element in a c... |
| n0moeu 4355 | A case of equivalence of "... |
| rex0 4356 | Vacuous restricted existen... |
| reu0 4357 | Vacuous restricted uniquen... |
| rmo0 4358 | Vacuous restricted at-most... |
| 0el 4359 | Membership of the empty se... |
| n0el 4360 | Negated membership of the ... |
| eqeuel 4361 | A condition which implies ... |
| ssdif0 4362 | Subclass expressed in term... |
| difn0 4363 | If the difference of two s... |
| pssdifn0 4364 | A proper subclass has a no... |
| pssdif 4365 | A proper subclass has a no... |
| ndisj 4366 | Express that an intersecti... |
| difin0ss 4367 | Difference, intersection, ... |
| inssdif0 4368 | Intersection, subclass, an... |
| difid 4369 | The difference between a c... |
| difidALT 4370 | Alternate proof of ~ difid... |
| dif0 4371 | The difference between a c... |
| ab0w 4372 | The class of sets verifyin... |
| ab0 4373 | The class of sets verifyin... |
| ab0OLD 4374 | Obsolete version of ~ ab0 ... |
| ab0ALT 4375 | Alternate proof of ~ ab0 ,... |
| dfnf5 4376 | Characterization of nonfre... |
| ab0orv 4377 | The class abstraction defi... |
| ab0orvALT 4378 | Alternate proof of ~ ab0or... |
| abn0 4379 | Nonempty class abstraction... |
| abn0OLD 4380 | Obsolete version of ~ abn0... |
| rab0 4381 | Any restricted class abstr... |
| rabeq0w 4382 | Condition for a restricted... |
| rabeq0 4383 | Condition for a restricted... |
| rabn0 4384 | Nonempty restricted class ... |
| rabxm 4385 | Law of excluded middle, in... |
| rabnc 4386 | Law of noncontradiction, i... |
| elneldisj 4387 | The set of elements ` s ` ... |
| elnelun 4388 | The union of the set of el... |
| un0 4389 | The union of a class with ... |
| in0 4390 | The intersection of a clas... |
| 0un 4391 | The union of the empty set... |
| 0in 4392 | The intersection of the em... |
| inv1 4393 | The intersection of a clas... |
| unv 4394 | The union of a class with ... |
| 0ss 4395 | The null set is a subset o... |
| ss0b 4396 | Any subset of the empty se... |
| ss0 4397 | Any subset of the empty se... |
| sseq0 4398 | A subclass of an empty cla... |
| ssn0 4399 | A class with a nonempty su... |
| 0dif 4400 | The difference between the... |
| abf 4401 | A class abstraction determ... |
| abfOLD 4402 | Obsolete version of ~ abf ... |
| eq0rdv 4403 | Deduction for equality to ... |
| eq0rdvALT 4404 | Alternate proof of ~ eq0rd... |
| csbprc 4405 | The proper substitution of... |
| csb0 4406 | The proper substitution of... |
| sbcel12 4407 | Distribute proper substitu... |
| sbceqg 4408 | Distribute proper substitu... |
| sbceqi 4409 | Distribution of class subs... |
| sbcnel12g 4410 | Distribute proper substitu... |
| sbcne12 4411 | Distribute proper substitu... |
| sbcel1g 4412 | Move proper substitution i... |
| sbceq1g 4413 | Move proper substitution t... |
| sbcel2 4414 | Move proper substitution i... |
| sbceq2g 4415 | Move proper substitution t... |
| csbcom 4416 | Commutative law for double... |
| sbcnestgfw 4417 | Nest the composition of tw... |
| csbnestgfw 4418 | Nest the composition of tw... |
| sbcnestgw 4419 | Nest the composition of tw... |
| csbnestgw 4420 | Nest the composition of tw... |
| sbcco3gw 4421 | Composition of two substit... |
| sbcnestgf 4422 | Nest the composition of tw... |
| csbnestgf 4423 | Nest the composition of tw... |
| sbcnestg 4424 | Nest the composition of tw... |
| csbnestg 4425 | Nest the composition of tw... |
| sbcco3g 4426 | Composition of two substit... |
| csbco3g 4427 | Composition of two class s... |
| csbnest1g 4428 | Nest the composition of tw... |
| csbidm 4429 | Idempotent law for class s... |
| csbvarg 4430 | The proper substitution of... |
| csbvargi 4431 | The proper substitution of... |
| sbccsb 4432 | Substitution into a wff ex... |
| sbccsb2 4433 | Substitution into a wff ex... |
| rspcsbela 4434 | Special case related to ~ ... |
| sbnfc2 4435 | Two ways of expressing " `... |
| csbab 4436 | Move substitution into a c... |
| csbun 4437 | Distribution of class subs... |
| csbin 4438 | Distribute proper substitu... |
| csbie2df 4439 | Conversion of implicit sub... |
| 2nreu 4440 | If there are two different... |
| un00 4441 | Two classes are empty iff ... |
| vss 4442 | Only the universal class h... |
| 0pss 4443 | The null set is a proper s... |
| npss0 4444 | No set is a proper subset ... |
| pssv 4445 | Any non-universal class is... |
| disj 4446 | Two ways of saying that tw... |
| disjOLD 4447 | Obsolete version of ~ disj... |
| disjr 4448 | Two ways of saying that tw... |
| disj1 4449 | Two ways of saying that tw... |
| reldisj 4450 | Two ways of saying that tw... |
| reldisjOLD 4451 | Obsolete version of ~ reld... |
| disj3 4452 | Two ways of saying that tw... |
| disjne 4453 | Members of disjoint sets a... |
| disjeq0 4454 | Two disjoint sets are equa... |
| disjel 4455 | A set can't belong to both... |
| disj2 4456 | Two ways of saying that tw... |
| disj4 4457 | Two ways of saying that tw... |
| ssdisj 4458 | Intersection with a subcla... |
| disjpss 4459 | A class is a proper subset... |
| undisj1 4460 | The union of disjoint clas... |
| undisj2 4461 | The union of disjoint clas... |
| ssindif0 4462 | Subclass expressed in term... |
| inelcm 4463 | The intersection of classe... |
| minel 4464 | A minimum element of a cla... |
| undif4 4465 | Distribute union over diff... |
| disjssun 4466 | Subset relation for disjoi... |
| vdif0 4467 | Universal class equality i... |
| difrab0eq 4468 | If the difference between ... |
| pssnel 4469 | A proper subclass has a me... |
| disjdif 4470 | A class and its relative c... |
| disjdifr 4471 | A class and its relative c... |
| difin0 4472 | The difference of a class ... |
| unvdif 4473 | The union of a class and i... |
| undif1 4474 | Absorption of difference b... |
| undif2 4475 | Absorption of difference b... |
| undifabs 4476 | Absorption of difference b... |
| inundif 4477 | The intersection and class... |
| disjdif2 4478 | The difference of a class ... |
| difun2 4479 | Absorption of union by dif... |
| undif 4480 | Union of complementary par... |
| undifr 4481 | Union of complementary par... |
| undifrOLD 4482 | Obsolete version of ~ undi... |
| undif5 4483 | An equality involving clas... |
| ssdifin0 4484 | A subset of a difference d... |
| ssdifeq0 4485 | A class is a subclass of i... |
| ssundif 4486 | A condition equivalent to ... |
| difcom 4487 | Swap the arguments of a cl... |
| pssdifcom1 4488 | Two ways to express overla... |
| pssdifcom2 4489 | Two ways to express non-co... |
| difdifdir 4490 | Distributive law for class... |
| uneqdifeq 4491 | Two ways to say that ` A `... |
| raldifeq 4492 | Equality theorem for restr... |
| r19.2z 4493 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4494 | A response to the notion t... |
| r19.3rz 4495 | Restricted quantification ... |
| r19.28z 4496 | Restricted quantifier vers... |
| r19.3rzv 4497 | Restricted quantification ... |
| r19.9rzv 4498 | Restricted quantification ... |
| r19.28zv 4499 | Restricted quantifier vers... |
| r19.37zv 4500 | Restricted quantifier vers... |
| r19.45zv 4501 | Restricted version of Theo... |
| r19.44zv 4502 | Restricted version of Theo... |
| r19.27z 4503 | Restricted quantifier vers... |
| r19.27zv 4504 | Restricted quantifier vers... |
| r19.36zv 4505 | Restricted quantifier vers... |
| ralidmw 4506 | Idempotent law for restric... |
| rzal 4507 | Vacuous quantification is ... |
| rzalALT 4508 | Alternate proof of ~ rzal ... |
| rexn0 4509 | Restricted existential qua... |
| ralidm 4510 | Idempotent law for restric... |
| ral0 4511 | Vacuous universal quantifi... |
| ralf0 4512 | The quantification of a fa... |
| rexn0OLD 4513 | Obsolete version of ~ rexn... |
| ralidmOLD 4514 | Obsolete version of ~ rali... |
| ral0OLD 4515 | Obsolete version of ~ ral0... |
| ralf0OLD 4516 | Obsolete version of ~ ralf... |
| ralnralall 4517 | A contradiction concerning... |
| falseral0 4518 | A false statement can only... |
| raaan 4519 | Rearrange restricted quant... |
| raaanv 4520 | Rearrange restricted quant... |
| sbss 4521 | Set substitution into the ... |
| sbcssg 4522 | Distribute proper substitu... |
| raaan2 4523 | Rearrange restricted quant... |
| 2reu4lem 4524 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4525 | Definition of double restr... |
| csbdif 4526 | Distribution of class subs... |
| dfif2 4529 | An alternate definition of... |
| dfif6 4530 | An alternate definition of... |
| ifeq1 4531 | Equality theorem for condi... |
| ifeq2 4532 | Equality theorem for condi... |
| iftrue 4533 | Value of the conditional o... |
| iftruei 4534 | Inference associated with ... |
| iftrued 4535 | Value of the conditional o... |
| iffalse 4536 | Value of the conditional o... |
| iffalsei 4537 | Inference associated with ... |
| iffalsed 4538 | Value of the conditional o... |
| ifnefalse 4539 | When values are unequal, b... |
| ifsb 4540 | Distribute a function over... |
| dfif3 4541 | Alternate definition of th... |
| dfif4 4542 | Alternate definition of th... |
| dfif5 4543 | Alternate definition of th... |
| ifssun 4544 | A conditional class is inc... |
| ifeq12 4545 | Equality theorem for condi... |
| ifeq1d 4546 | Equality deduction for con... |
| ifeq2d 4547 | Equality deduction for con... |
| ifeq12d 4548 | Equality deduction for con... |
| ifbi 4549 | Equivalence theorem for co... |
| ifbid 4550 | Equivalence deduction for ... |
| ifbieq1d 4551 | Equivalence/equality deduc... |
| ifbieq2i 4552 | Equivalence/equality infer... |
| ifbieq2d 4553 | Equivalence/equality deduc... |
| ifbieq12i 4554 | Equivalence deduction for ... |
| ifbieq12d 4555 | Equivalence deduction for ... |
| nfifd 4556 | Deduction form of ~ nfif .... |
| nfif 4557 | Bound-variable hypothesis ... |
| ifeq1da 4558 | Conditional equality. (Co... |
| ifeq2da 4559 | Conditional equality. (Co... |
| ifeq12da 4560 | Equivalence deduction for ... |
| ifbieq12d2 4561 | Equivalence deduction for ... |
| ifclda 4562 | Conditional closure. (Con... |
| ifeqda 4563 | Separation of the values o... |
| elimif 4564 | Elimination of a condition... |
| ifbothda 4565 | A wff ` th ` containing a ... |
| ifboth 4566 | A wff ` th ` containing a ... |
| ifid 4567 | Identical true and false a... |
| eqif 4568 | Expansion of an equality w... |
| ifval 4569 | Another expression of the ... |
| elif 4570 | Membership in a conditiona... |
| ifel 4571 | Membership of a conditiona... |
| ifcl 4572 | Membership (closure) of a ... |
| ifcld 4573 | Membership (closure) of a ... |
| ifcli 4574 | Inference associated with ... |
| ifexd 4575 | Existence of the condition... |
| ifexg 4576 | Existence of the condition... |
| ifex 4577 | Existence of the condition... |
| ifeqor 4578 | The possible values of a c... |
| ifnot 4579 | Negating the first argumen... |
| ifan 4580 | Rewrite a conjunction in a... |
| ifor 4581 | Rewrite a disjunction in a... |
| 2if2 4582 | Resolve two nested conditi... |
| ifcomnan 4583 | Commute the conditions in ... |
| csbif 4584 | Distribute proper substitu... |
| dedth 4585 | Weak deduction theorem tha... |
| dedth2h 4586 | Weak deduction theorem eli... |
| dedth3h 4587 | Weak deduction theorem eli... |
| dedth4h 4588 | Weak deduction theorem eli... |
| dedth2v 4589 | Weak deduction theorem for... |
| dedth3v 4590 | Weak deduction theorem for... |
| dedth4v 4591 | Weak deduction theorem for... |
| elimhyp 4592 | Eliminate a hypothesis con... |
| elimhyp2v 4593 | Eliminate a hypothesis con... |
| elimhyp3v 4594 | Eliminate a hypothesis con... |
| elimhyp4v 4595 | Eliminate a hypothesis con... |
| elimel 4596 | Eliminate a membership hyp... |
| elimdhyp 4597 | Version of ~ elimhyp where... |
| keephyp 4598 | Transform a hypothesis ` p... |
| keephyp2v 4599 | Keep a hypothesis containi... |
| keephyp3v 4600 | Keep a hypothesis containi... |
| pwjust 4602 | Soundness justification th... |
| elpwg 4604 | Membership in a power clas... |
| elpw 4605 | Membership in a power clas... |
| velpw 4606 | Setvar variable membership... |
| elpwd 4607 | Membership in a power clas... |
| elpwi 4608 | Subset relation implied by... |
| elpwb 4609 | Characterization of the el... |
| elpwid 4610 | An element of a power clas... |
| elelpwi 4611 | If ` A ` belongs to a part... |
| sspw 4612 | The powerclass preserves i... |
| sspwi 4613 | The powerclass preserves i... |
| sspwd 4614 | The powerclass preserves i... |
| pweq 4615 | Equality theorem for power... |
| pweqALT 4616 | Alternate proof of ~ pweq ... |
| pweqi 4617 | Equality inference for pow... |
| pweqd 4618 | Equality deduction for pow... |
| pwunss 4619 | The power class of the uni... |
| nfpw 4620 | Bound-variable hypothesis ... |
| pwidg 4621 | A set is an element of its... |
| pwidb 4622 | A class is an element of i... |
| pwid 4623 | A set is a member of its p... |
| pwss 4624 | Subclass relationship for ... |
| pwundif 4625 | Break up the power class o... |
| snjust 4626 | Soundness justification th... |
| sneq 4637 | Equality theorem for singl... |
| sneqi 4638 | Equality inference for sin... |
| sneqd 4639 | Equality deduction for sin... |
| dfsn2 4640 | Alternate definition of si... |
| elsng 4641 | There is exactly one eleme... |
| elsn 4642 | There is exactly one eleme... |
| velsn 4643 | There is only one element ... |
| elsni 4644 | There is at most one eleme... |
| absn 4645 | Condition for a class abst... |
| dfpr2 4646 | Alternate definition of a ... |
| dfsn2ALT 4647 | Alternate definition of si... |
| elprg 4648 | A member of a pair of clas... |
| elpri 4649 | If a class is an element o... |
| elpr 4650 | A member of a pair of clas... |
| elpr2g 4651 | A member of a pair of sets... |
| elpr2 4652 | A member of a pair of sets... |
| elpr2OLD 4653 | Obsolete version of ~ elpr... |
| nelpr2 4654 | If a class is not an eleme... |
| nelpr1 4655 | If a class is not an eleme... |
| nelpri 4656 | If an element doesn't matc... |
| prneli 4657 | If an element doesn't matc... |
| nelprd 4658 | If an element doesn't matc... |
| eldifpr 4659 | Membership in a set with t... |
| rexdifpr 4660 | Restricted existential qua... |
| snidg 4661 | A set is a member of its s... |
| snidb 4662 | A class is a set iff it is... |
| snid 4663 | A set is a member of its s... |
| vsnid 4664 | A setvar variable is a mem... |
| elsn2g 4665 | There is exactly one eleme... |
| elsn2 4666 | There is exactly one eleme... |
| nelsn 4667 | If a class is not equal to... |
| rabeqsn 4668 | Conditions for a restricte... |
| rabsssn 4669 | Conditions for a restricte... |
| rabeqsnd 4670 | Conditions for a restricte... |
| ralsnsg 4671 | Substitution expressed in ... |
| rexsns 4672 | Restricted existential qua... |
| rexsngf 4673 | Restricted existential qua... |
| ralsngf 4674 | Restricted universal quant... |
| reusngf 4675 | Restricted existential uni... |
| ralsng 4676 | Substitution expressed in ... |
| rexsng 4677 | Restricted existential qua... |
| reusng 4678 | Restricted existential uni... |
| 2ralsng 4679 | Substitution expressed in ... |
| ralsngOLD 4680 | Obsolete version of ~ rals... |
| rexsngOLD 4681 | Obsolete version of ~ rexs... |
| rexreusng 4682 | Restricted existential uni... |
| exsnrex 4683 | There is a set being the e... |
| ralsn 4684 | Convert a universal quanti... |
| rexsn 4685 | Convert an existential qua... |
| elpwunsn 4686 | Membership in an extension... |
| eqoreldif 4687 | An element of a set is eit... |
| eltpg 4688 | Members of an unordered tr... |
| eldiftp 4689 | Membership in a set with t... |
| eltpi 4690 | A member of an unordered t... |
| eltp 4691 | A member of an unordered t... |
| dftp2 4692 | Alternate definition of un... |
| nfpr 4693 | Bound-variable hypothesis ... |
| ifpr 4694 | Membership of a conditiona... |
| ralprgf 4695 | Convert a restricted unive... |
| rexprgf 4696 | Convert a restricted exist... |
| ralprg 4697 | Convert a restricted unive... |
| ralprgOLD 4698 | Obsolete version of ~ ralp... |
| rexprg 4699 | Convert a restricted exist... |
| rexprgOLD 4700 | Obsolete version of ~ rexp... |
| raltpg 4701 | Convert a restricted unive... |
| rextpg 4702 | Convert a restricted exist... |
| ralpr 4703 | Convert a restricted unive... |
| rexpr 4704 | Convert a restricted exist... |
| reuprg0 4705 | Convert a restricted exist... |
| reuprg 4706 | Convert a restricted exist... |
| reurexprg 4707 | Convert a restricted exist... |
| raltp 4708 | Convert a universal quanti... |
| rextp 4709 | Convert an existential qua... |
| nfsn 4710 | Bound-variable hypothesis ... |
| csbsng 4711 | Distribute proper substitu... |
| csbprg 4712 | Distribute proper substitu... |
| elinsn 4713 | If the intersection of two... |
| disjsn 4714 | Intersection with the sing... |
| disjsn2 4715 | Two distinct singletons ar... |
| disjpr2 4716 | Two completely distinct un... |
| disjprsn 4717 | The disjoint intersection ... |
| disjtpsn 4718 | The disjoint intersection ... |
| disjtp2 4719 | Two completely distinct un... |
| snprc 4720 | The singleton of a proper ... |
| snnzb 4721 | A singleton is nonempty if... |
| rmosn 4722 | A restricted at-most-one q... |
| r19.12sn 4723 | Special case of ~ r19.12 w... |
| rabsn 4724 | Condition where a restrict... |
| rabsnifsb 4725 | A restricted class abstrac... |
| rabsnif 4726 | A restricted class abstrac... |
| rabrsn 4727 | A restricted class abstrac... |
| euabsn2 4728 | Another way to express exi... |
| euabsn 4729 | Another way to express exi... |
| reusn 4730 | A way to express restricte... |
| absneu 4731 | Restricted existential uni... |
| rabsneu 4732 | Restricted existential uni... |
| eusn 4733 | Two ways to express " ` A ... |
| rabsnt 4734 | Truth implied by equality ... |
| prcom 4735 | Commutative law for unorde... |
| preq1 4736 | Equality theorem for unord... |
| preq2 4737 | Equality theorem for unord... |
| preq12 4738 | Equality theorem for unord... |
| preq1i 4739 | Equality inference for uno... |
| preq2i 4740 | Equality inference for uno... |
| preq12i 4741 | Equality inference for uno... |
| preq1d 4742 | Equality deduction for uno... |
| preq2d 4743 | Equality deduction for uno... |
| preq12d 4744 | Equality deduction for uno... |
| tpeq1 4745 | Equality theorem for unord... |
| tpeq2 4746 | Equality theorem for unord... |
| tpeq3 4747 | Equality theorem for unord... |
| tpeq1d 4748 | Equality theorem for unord... |
| tpeq2d 4749 | Equality theorem for unord... |
| tpeq3d 4750 | Equality theorem for unord... |
| tpeq123d 4751 | Equality theorem for unord... |
| tprot 4752 | Rotation of the elements o... |
| tpcoma 4753 | Swap 1st and 2nd members o... |
| tpcomb 4754 | Swap 2nd and 3rd members o... |
| tpass 4755 | Split off the first elemen... |
| qdass 4756 | Two ways to write an unord... |
| qdassr 4757 | Two ways to write an unord... |
| tpidm12 4758 | Unordered triple ` { A , A... |
| tpidm13 4759 | Unordered triple ` { A , B... |
| tpidm23 4760 | Unordered triple ` { A , B... |
| tpidm 4761 | Unordered triple ` { A , A... |
| tppreq3 4762 | An unordered triple is an ... |
| prid1g 4763 | An unordered pair contains... |
| prid2g 4764 | An unordered pair contains... |
| prid1 4765 | An unordered pair contains... |
| prid2 4766 | An unordered pair contains... |
| ifpprsnss 4767 | An unordered pair is a sin... |
| prprc1 4768 | A proper class vanishes in... |
| prprc2 4769 | A proper class vanishes in... |
| prprc 4770 | An unordered pair containi... |
| tpid1 4771 | One of the three elements ... |
| tpid1g 4772 | Closed theorem form of ~ t... |
| tpid2 4773 | One of the three elements ... |
| tpid2g 4774 | Closed theorem form of ~ t... |
| tpid3g 4775 | Closed theorem form of ~ t... |
| tpid3 4776 | One of the three elements ... |
| snnzg 4777 | The singleton of a set is ... |
| snn0d 4778 | The singleton of a set is ... |
| snnz 4779 | The singleton of a set is ... |
| prnz 4780 | A pair containing a set is... |
| prnzg 4781 | A pair containing a set is... |
| tpnz 4782 | An unordered triple contai... |
| tpnzd 4783 | An unordered triple contai... |
| raltpd 4784 | Convert a universal quanti... |
| snssb 4785 | Characterization of the in... |
| snssg 4786 | The singleton formed on a ... |
| snssgOLD 4787 | Obsolete version of ~ snss... |
| snss 4788 | The singleton of an elemen... |
| eldifsn 4789 | Membership in a set with a... |
| ssdifsn 4790 | Subset of a set with an el... |
| elpwdifsn 4791 | A subset of a set is an el... |
| eldifsni 4792 | Membership in a set with a... |
| eldifsnneq 4793 | An element of a difference... |
| neldifsn 4794 | The class ` A ` is not in ... |
| neldifsnd 4795 | The class ` A ` is not in ... |
| rexdifsn 4796 | Restricted existential qua... |
| raldifsni 4797 | Rearrangement of a propert... |
| raldifsnb 4798 | Restricted universal quant... |
| eldifvsn 4799 | A set is an element of the... |
| difsn 4800 | An element not in a set ca... |
| difprsnss 4801 | Removal of a singleton fro... |
| difprsn1 4802 | Removal of a singleton fro... |
| difprsn2 4803 | Removal of a singleton fro... |
| diftpsn3 4804 | Removal of a singleton fro... |
| difpr 4805 | Removing two elements as p... |
| tpprceq3 4806 | An unordered triple is an ... |
| tppreqb 4807 | An unordered triple is an ... |
| difsnb 4808 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4809 | ` ( B \ { A } ) ` is a pro... |
| snssi 4810 | The singleton of an elemen... |
| snssd 4811 | The singleton of an elemen... |
| difsnid 4812 | If we remove a single elem... |
| eldifeldifsn 4813 | An element of a difference... |
| pw0 4814 | Compute the power set of t... |
| pwpw0 4815 | Compute the power set of t... |
| snsspr1 4816 | A singleton is a subset of... |
| snsspr2 4817 | A singleton is a subset of... |
| snsstp1 4818 | A singleton is a subset of... |
| snsstp2 4819 | A singleton is a subset of... |
| snsstp3 4820 | A singleton is a subset of... |
| prssg 4821 | A pair of elements of a cl... |
| prss 4822 | A pair of elements of a cl... |
| prssi 4823 | A pair of elements of a cl... |
| prssd 4824 | Deduction version of ~ prs... |
| prsspwg 4825 | An unordered pair belongs ... |
| ssprss 4826 | A pair as subset of a pair... |
| ssprsseq 4827 | A proper pair is a subset ... |
| sssn 4828 | The subsets of a singleton... |
| ssunsn2 4829 | The property of being sand... |
| ssunsn 4830 | Possible values for a set ... |
| eqsn 4831 | Two ways to express that a... |
| issn 4832 | A sufficient condition for... |
| n0snor2el 4833 | A nonempty set is either a... |
| ssunpr 4834 | Possible values for a set ... |
| sspr 4835 | The subsets of a pair. (C... |
| sstp 4836 | The subsets of an unordere... |
| tpss 4837 | An unordered triple of ele... |
| tpssi 4838 | An unordered triple of ele... |
| sneqrg 4839 | Closed form of ~ sneqr . ... |
| sneqr 4840 | If the singletons of two s... |
| snsssn 4841 | If a singleton is a subset... |
| mosneq 4842 | There exists at most one s... |
| sneqbg 4843 | Two singletons of sets are... |
| snsspw 4844 | The singleton of a class i... |
| prsspw 4845 | An unordered pair belongs ... |
| preq1b 4846 | Biconditional equality lem... |
| preq2b 4847 | Biconditional equality lem... |
| preqr1 4848 | Reverse equality lemma for... |
| preqr2 4849 | Reverse equality lemma for... |
| preq12b 4850 | Equality relationship for ... |
| opthpr 4851 | An unordered pair has the ... |
| preqr1g 4852 | Reverse equality lemma for... |
| preq12bg 4853 | Closed form of ~ preq12b .... |
| prneimg 4854 | Two pairs are not equal if... |
| prnebg 4855 | A (proper) pair is not equ... |
| pr1eqbg 4856 | A (proper) pair is equal t... |
| pr1nebg 4857 | A (proper) pair is not equ... |
| preqsnd 4858 | Equivalence for a pair equ... |
| prnesn 4859 | A proper unordered pair is... |
| prneprprc 4860 | A proper unordered pair is... |
| preqsn 4861 | Equivalence for a pair equ... |
| preq12nebg 4862 | Equality relationship for ... |
| prel12g 4863 | Equality of two unordered ... |
| opthprneg 4864 | An unordered pair has the ... |
| elpreqprlem 4865 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4866 | Equality and membership ru... |
| elpreqprb 4867 | A set is an element of an ... |
| elpr2elpr 4868 | For an element ` A ` of an... |
| dfopif 4869 | Rewrite ~ df-op using ` if... |
| dfopg 4870 | Value of the ordered pair ... |
| dfop 4871 | Value of an ordered pair w... |
| opeq1 4872 | Equality theorem for order... |
| opeq2 4873 | Equality theorem for order... |
| opeq12 4874 | Equality theorem for order... |
| opeq1i 4875 | Equality inference for ord... |
| opeq2i 4876 | Equality inference for ord... |
| opeq12i 4877 | Equality inference for ord... |
| opeq1d 4878 | Equality deduction for ord... |
| opeq2d 4879 | Equality deduction for ord... |
| opeq12d 4880 | Equality deduction for ord... |
| oteq1 4881 | Equality theorem for order... |
| oteq2 4882 | Equality theorem for order... |
| oteq3 4883 | Equality theorem for order... |
| oteq1d 4884 | Equality deduction for ord... |
| oteq2d 4885 | Equality deduction for ord... |
| oteq3d 4886 | Equality deduction for ord... |
| oteq123d 4887 | Equality deduction for ord... |
| nfop 4888 | Bound-variable hypothesis ... |
| nfopd 4889 | Deduction version of bound... |
| csbopg 4890 | Distribution of class subs... |
| opidg 4891 | The ordered pair ` <. A , ... |
| opid 4892 | The ordered pair ` <. A , ... |
| ralunsn 4893 | Restricted quantification ... |
| 2ralunsn 4894 | Double restricted quantifi... |
| opprc 4895 | Expansion of an ordered pa... |
| opprc1 4896 | Expansion of an ordered pa... |
| opprc2 4897 | Expansion of an ordered pa... |
| oprcl 4898 | If an ordered pair has an ... |
| pwsn 4899 | The power set of a singlet... |
| pwsnOLD 4900 | Obsolete version of ~ pwsn... |
| pwpr 4901 | The power set of an unorde... |
| pwtp 4902 | The power set of an unorde... |
| pwpwpw0 4903 | Compute the power set of t... |
| pwv 4904 | The power class of the uni... |
| prproe 4905 | For an element of a proper... |
| 3elpr2eq 4906 | If there are three element... |
| dfuni2 4909 | Alternate definition of cl... |
| eluni 4910 | Membership in class union.... |
| eluni2 4911 | Membership in class union.... |
| elunii 4912 | Membership in class union.... |
| nfunid 4913 | Deduction version of ~ nfu... |
| nfuni 4914 | Bound-variable hypothesis ... |
| uniss 4915 | Subclass relationship for ... |
| unissi 4916 | Subclass relationship for ... |
| unissd 4917 | Subclass relationship for ... |
| unieq 4918 | Equality theorem for class... |
| unieqOLD 4919 | Obsolete version of ~ unie... |
| unieqi 4920 | Inference of equality of t... |
| unieqd 4921 | Deduction of equality of t... |
| eluniab 4922 | Membership in union of a c... |
| elunirab 4923 | Membership in union of a c... |
| uniprg 4924 | The union of a pair is the... |
| unipr 4925 | The union of a pair is the... |
| uniprOLD 4926 | Obsolete version of ~ unip... |
| uniprgOLD 4927 | Obsolete version of ~ unip... |
| unisng 4928 | A set equals the union of ... |
| unisn 4929 | A set equals the union of ... |
| unisnv 4930 | A set equals the union of ... |
| unisn3 4931 | Union of a singleton in th... |
| dfnfc2 4932 | An alternative statement o... |
| uniun 4933 | The class union of the uni... |
| uniin 4934 | The class union of the int... |
| ssuni 4935 | Subclass relationship for ... |
| uni0b 4936 | The union of a set is empt... |
| uni0c 4937 | The union of a set is empt... |
| uni0 4938 | The union of the empty set... |
| csbuni 4939 | Distribute proper substitu... |
| elssuni 4940 | An element of a class is a... |
| unissel 4941 | Condition turning a subcla... |
| unissb 4942 | Relationship involving mem... |
| unissbOLD 4943 | Obsolete version of ~ unis... |
| uniss2 4944 | A subclass condition on th... |
| unidif 4945 | If the difference ` A \ B ... |
| ssunieq 4946 | Relationship implying unio... |
| unimax 4947 | Any member of a class is t... |
| pwuni 4948 | A class is a subclass of t... |
| dfint2 4951 | Alternate definition of cl... |
| inteq 4952 | Equality law for intersect... |
| inteqi 4953 | Equality inference for cla... |
| inteqd 4954 | Equality deduction for cla... |
| elint 4955 | Membership in class inters... |
| elint2 4956 | Membership in class inters... |
| elintg 4957 | Membership in class inters... |
| elinti 4958 | Membership in class inters... |
| nfint 4959 | Bound-variable hypothesis ... |
| elintabg 4960 | Two ways of saying a set i... |
| elintab 4961 | Membership in the intersec... |
| elintabOLD 4962 | Obsolete version of ~ elin... |
| elintrab 4963 | Membership in the intersec... |
| elintrabg 4964 | Membership in the intersec... |
| int0 4965 | The intersection of the em... |
| intss1 4966 | An element of a class incl... |
| ssint 4967 | Subclass of a class inters... |
| ssintab 4968 | Subclass of the intersecti... |
| ssintub 4969 | Subclass of the least uppe... |
| ssmin 4970 | Subclass of the minimum va... |
| intmin 4971 | Any member of a class is t... |
| intss 4972 | Intersection of subclasses... |
| intssuni 4973 | The intersection of a none... |
| ssintrab 4974 | Subclass of the intersecti... |
| unissint 4975 | If the union of a class is... |
| intssuni2 4976 | Subclass relationship for ... |
| intminss 4977 | Under subset ordering, the... |
| intmin2 4978 | Any set is the smallest of... |
| intmin3 4979 | Under subset ordering, the... |
| intmin4 4980 | Elimination of a conjunct ... |
| intab 4981 | The intersection of a spec... |
| int0el 4982 | The intersection of a clas... |
| intun 4983 | The class intersection of ... |
| intprg 4984 | The intersection of a pair... |
| intpr 4985 | The intersection of a pair... |
| intprOLD 4986 | Obsolete version of ~ intp... |
| intprgOLD 4987 | Obsolete version of ~ intp... |
| intsng 4988 | Intersection of a singleto... |
| intsn 4989 | The intersection of a sing... |
| uniintsn 4990 | Two ways to express " ` A ... |
| uniintab 4991 | The union and the intersec... |
| intunsn 4992 | Theorem joining a singleto... |
| rint0 4993 | Relative intersection of a... |
| elrint 4994 | Membership in a restricted... |
| elrint2 4995 | Membership in a restricted... |
| eliun 5000 | Membership in indexed unio... |
| eliin 5001 | Membership in indexed inte... |
| eliuni 5002 | Membership in an indexed u... |
| iuncom 5003 | Commutation of indexed uni... |
| iuncom4 5004 | Commutation of union with ... |
| iunconst 5005 | Indexed union of a constan... |
| iinconst 5006 | Indexed intersection of a ... |
| iuneqconst 5007 | Indexed union of identical... |
| iuniin 5008 | Law combining indexed unio... |
| iinssiun 5009 | An indexed intersection is... |
| iunss1 5010 | Subclass theorem for index... |
| iinss1 5011 | Subclass theorem for index... |
| iuneq1 5012 | Equality theorem for index... |
| iineq1 5013 | Equality theorem for index... |
| ss2iun 5014 | Subclass theorem for index... |
| iuneq2 5015 | Equality theorem for index... |
| iineq2 5016 | Equality theorem for index... |
| iuneq2i 5017 | Equality inference for ind... |
| iineq2i 5018 | Equality inference for ind... |
| iineq2d 5019 | Equality deduction for ind... |
| iuneq2dv 5020 | Equality deduction for ind... |
| iineq2dv 5021 | Equality deduction for ind... |
| iuneq12df 5022 | Equality deduction for ind... |
| iuneq1d 5023 | Equality theorem for index... |
| iuneq12d 5024 | Equality deduction for ind... |
| iuneq2d 5025 | Equality deduction for ind... |
| nfiun 5026 | Bound-variable hypothesis ... |
| nfiin 5027 | Bound-variable hypothesis ... |
| nfiung 5028 | Bound-variable hypothesis ... |
| nfiing 5029 | Bound-variable hypothesis ... |
| nfiu1 5030 | Bound-variable hypothesis ... |
| nfii1 5031 | Bound-variable hypothesis ... |
| dfiun2g 5032 | Alternate definition of in... |
| dfiun2gOLD 5033 | Obsolete version of ~ dfiu... |
| dfiin2g 5034 | Alternate definition of in... |
| dfiun2 5035 | Alternate definition of in... |
| dfiin2 5036 | Alternate definition of in... |
| dfiunv2 5037 | Define double indexed unio... |
| cbviun 5038 | Rule used to change the bo... |
| cbviin 5039 | Change bound variables in ... |
| cbviung 5040 | Rule used to change the bo... |
| cbviing 5041 | Change bound variables in ... |
| cbviunv 5042 | Rule used to change the bo... |
| cbviinv 5043 | Change bound variables in ... |
| cbviunvg 5044 | Rule used to change the bo... |
| cbviinvg 5045 | Change bound variables in ... |
| iunssf 5046 | Subset theorem for an inde... |
| iunss 5047 | Subset theorem for an inde... |
| ssiun 5048 | Subset implication for an ... |
| ssiun2 5049 | Identity law for subset of... |
| ssiun2s 5050 | Subset relationship for an... |
| iunss2 5051 | A subclass condition on th... |
| iunssd 5052 | Subset theorem for an inde... |
| iunab 5053 | The indexed union of a cla... |
| iunrab 5054 | The indexed union of a res... |
| iunxdif2 5055 | Indexed union with a class... |
| ssiinf 5056 | Subset theorem for an inde... |
| ssiin 5057 | Subset theorem for an inde... |
| iinss 5058 | Subset implication for an ... |
| iinss2 5059 | An indexed intersection is... |
| uniiun 5060 | Class union in terms of in... |
| intiin 5061 | Class intersection in term... |
| iunid 5062 | An indexed union of single... |
| iunidOLD 5063 | Obsolete version of ~ iuni... |
| iun0 5064 | An indexed union of the em... |
| 0iun 5065 | An empty indexed union is ... |
| 0iin 5066 | An empty indexed intersect... |
| viin 5067 | Indexed intersection with ... |
| iunsn 5068 | Indexed union of a singlet... |
| iunn0 5069 | There is a nonempty class ... |
| iinab 5070 | Indexed intersection of a ... |
| iinrab 5071 | Indexed intersection of a ... |
| iinrab2 5072 | Indexed intersection of a ... |
| iunin2 5073 | Indexed union of intersect... |
| iunin1 5074 | Indexed union of intersect... |
| iinun2 5075 | Indexed intersection of un... |
| iundif2 5076 | Indexed union of class dif... |
| iindif1 5077 | Indexed intersection of cl... |
| 2iunin 5078 | Rearrange indexed unions o... |
| iindif2 5079 | Indexed intersection of cl... |
| iinin2 5080 | Indexed intersection of in... |
| iinin1 5081 | Indexed intersection of in... |
| iinvdif 5082 | The indexed intersection o... |
| elriin 5083 | Elementhood in a relative ... |
| riin0 5084 | Relative intersection of a... |
| riinn0 5085 | Relative intersection of a... |
| riinrab 5086 | Relative intersection of a... |
| symdif0 5087 | Symmetric difference with ... |
| symdifv 5088 | The symmetric difference w... |
| symdifid 5089 | The symmetric difference o... |
| iinxsng 5090 | A singleton index picks ou... |
| iinxprg 5091 | Indexed intersection with ... |
| iunxsng 5092 | A singleton index picks ou... |
| iunxsn 5093 | A singleton index picks ou... |
| iunxsngf 5094 | A singleton index picks ou... |
| iunun 5095 | Separate a union in an ind... |
| iunxun 5096 | Separate a union in the in... |
| iunxdif3 5097 | An indexed union where som... |
| iunxprg 5098 | A pair index picks out two... |
| iunxiun 5099 | Separate an indexed union ... |
| iinuni 5100 | A relationship involving u... |
| iununi 5101 | A relationship involving u... |
| sspwuni 5102 | Subclass relationship for ... |
| pwssb 5103 | Two ways to express a coll... |
| elpwpw 5104 | Characterization of the el... |
| pwpwab 5105 | The double power class wri... |
| pwpwssunieq 5106 | The class of sets whose un... |
| elpwuni 5107 | Relationship for power cla... |
| iinpw 5108 | The power class of an inte... |
| iunpwss 5109 | Inclusion of an indexed un... |
| intss2 5110 | A nonempty intersection of... |
| rintn0 5111 | Relative intersection of a... |
| dfdisj2 5114 | Alternate definition for d... |
| disjss2 5115 | If each element of a colle... |
| disjeq2 5116 | Equality theorem for disjo... |
| disjeq2dv 5117 | Equality deduction for dis... |
| disjss1 5118 | A subset of a disjoint col... |
| disjeq1 5119 | Equality theorem for disjo... |
| disjeq1d 5120 | Equality theorem for disjo... |
| disjeq12d 5121 | Equality theorem for disjo... |
| cbvdisj 5122 | Change bound variables in ... |
| cbvdisjv 5123 | Change bound variables in ... |
| nfdisjw 5124 | Bound-variable hypothesis ... |
| nfdisj 5125 | Bound-variable hypothesis ... |
| nfdisj1 5126 | Bound-variable hypothesis ... |
| disjor 5127 | Two ways to say that a col... |
| disjors 5128 | Two ways to say that a col... |
| disji2 5129 | Property of a disjoint col... |
| disji 5130 | Property of a disjoint col... |
| invdisj 5131 | If there is a function ` C... |
| invdisjrabw 5132 | Version of ~ invdisjrab wi... |
| invdisjrab 5133 | The restricted class abstr... |
| disjiun 5134 | A disjoint collection yiel... |
| disjord 5135 | Conditions for a collectio... |
| disjiunb 5136 | Two ways to say that a col... |
| disjiund 5137 | Conditions for a collectio... |
| sndisj 5138 | Any collection of singleto... |
| 0disj 5139 | Any collection of empty se... |
| disjxsn 5140 | A singleton collection is ... |
| disjx0 5141 | An empty collection is dis... |
| disjprgw 5142 | Version of ~ disjprg with ... |
| disjprg 5143 | A pair collection is disjo... |
| disjxiun 5144 | An indexed union of a disj... |
| disjxun 5145 | The union of two disjoint ... |
| disjss3 5146 | Expand a disjoint collecti... |
| breq 5149 | Equality theorem for binar... |
| breq1 5150 | Equality theorem for a bin... |
| breq2 5151 | Equality theorem for a bin... |
| breq12 5152 | Equality theorem for a bin... |
| breqi 5153 | Equality inference for bin... |
| breq1i 5154 | Equality inference for a b... |
| breq2i 5155 | Equality inference for a b... |
| breq12i 5156 | Equality inference for a b... |
| breq1d 5157 | Equality deduction for a b... |
| breqd 5158 | Equality deduction for a b... |
| breq2d 5159 | Equality deduction for a b... |
| breq12d 5160 | Equality deduction for a b... |
| breq123d 5161 | Equality deduction for a b... |
| breqdi 5162 | Equality deduction for a b... |
| breqan12d 5163 | Equality deduction for a b... |
| breqan12rd 5164 | Equality deduction for a b... |
| eqnbrtrd 5165 | Substitution of equal clas... |
| nbrne1 5166 | Two classes are different ... |
| nbrne2 5167 | Two classes are different ... |
| eqbrtri 5168 | Substitution of equal clas... |
| eqbrtrd 5169 | Substitution of equal clas... |
| eqbrtrri 5170 | Substitution of equal clas... |
| eqbrtrrd 5171 | Substitution of equal clas... |
| breqtri 5172 | Substitution of equal clas... |
| breqtrd 5173 | Substitution of equal clas... |
| breqtrri 5174 | Substitution of equal clas... |
| breqtrrd 5175 | Substitution of equal clas... |
| 3brtr3i 5176 | Substitution of equality i... |
| 3brtr4i 5177 | Substitution of equality i... |
| 3brtr3d 5178 | Substitution of equality i... |
| 3brtr4d 5179 | Substitution of equality i... |
| 3brtr3g 5180 | Substitution of equality i... |
| 3brtr4g 5181 | Substitution of equality i... |
| eqbrtrid 5182 | A chained equality inferen... |
| eqbrtrrid 5183 | A chained equality inferen... |
| breqtrid 5184 | A chained equality inferen... |
| breqtrrid 5185 | A chained equality inferen... |
| eqbrtrdi 5186 | A chained equality inferen... |
| eqbrtrrdi 5187 | A chained equality inferen... |
| breqtrdi 5188 | A chained equality inferen... |
| breqtrrdi 5189 | A chained equality inferen... |
| ssbrd 5190 | Deduction from a subclass ... |
| ssbr 5191 | Implication from a subclas... |
| ssbri 5192 | Inference from a subclass ... |
| nfbrd 5193 | Deduction version of bound... |
| nfbr 5194 | Bound-variable hypothesis ... |
| brab1 5195 | Relationship between a bin... |
| br0 5196 | The empty binary relation ... |
| brne0 5197 | If two sets are in a binar... |
| brun 5198 | The union of two binary re... |
| brin 5199 | The intersection of two re... |
| brdif 5200 | The difference of two bina... |
| sbcbr123 5201 | Move substitution in and o... |
| sbcbr 5202 | Move substitution in and o... |
| sbcbr12g 5203 | Move substitution in and o... |
| sbcbr1g 5204 | Move substitution in and o... |
| sbcbr2g 5205 | Move substitution in and o... |
| brsymdif 5206 | Characterization of the sy... |
| brralrspcev 5207 | Restricted existential spe... |
| brimralrspcev 5208 | Restricted existential spe... |
| opabss 5211 | The collection of ordered ... |
| opabbid 5212 | Equivalent wff's yield equ... |
| opabbidv 5213 | Equivalent wff's yield equ... |
| opabbii 5214 | Equivalent wff's yield equ... |
| nfopabd 5215 | Bound-variable hypothesis ... |
| nfopab 5216 | Bound-variable hypothesis ... |
| nfopab1 5217 | The first abstraction vari... |
| nfopab2 5218 | The second abstraction var... |
| cbvopab 5219 | Rule used to change bound ... |
| cbvopabv 5220 | Rule used to change bound ... |
| cbvopabvOLD 5221 | Obsolete version of ~ cbvo... |
| cbvopab1 5222 | Change first bound variabl... |
| cbvopab1g 5223 | Change first bound variabl... |
| cbvopab2 5224 | Change second bound variab... |
| cbvopab1s 5225 | Change first bound variabl... |
| cbvopab1v 5226 | Rule used to change the fi... |
| cbvopab1vOLD 5227 | Obsolete version of ~ cbvo... |
| cbvopab2v 5228 | Rule used to change the se... |
| unopab 5229 | Union of two ordered pair ... |
| mpteq12da 5232 | An equality inference for ... |
| mpteq12df 5233 | An equality inference for ... |
| mpteq12dfOLD 5234 | Obsolete version of ~ mpte... |
| mpteq12f 5235 | An equality theorem for th... |
| mpteq12dva 5236 | An equality inference for ... |
| mpteq12dvaOLD 5237 | Obsolete version of ~ mpte... |
| mpteq12dv 5238 | An equality inference for ... |
| mpteq12 5239 | An equality theorem for th... |
| mpteq1 5240 | An equality theorem for th... |
| mpteq1OLD 5241 | Obsolete version of ~ mpte... |
| mpteq1d 5242 | An equality theorem for th... |
| mpteq1i 5243 | An equality theorem for th... |
| mpteq1iOLD 5244 | An equality theorem for th... |
| mpteq2da 5245 | Slightly more general equa... |
| mpteq2daOLD 5246 | Obsolete version of ~ mpte... |
| mpteq2dva 5247 | Slightly more general equa... |
| mpteq2dvaOLD 5248 | Obsolete version of ~ mpte... |
| mpteq2dv 5249 | An equality inference for ... |
| mpteq2ia 5250 | An equality inference for ... |
| mpteq2iaOLD 5251 | Obsolete version of ~ mpte... |
| mpteq2i 5252 | An equality inference for ... |
| mpteq12i 5253 | An equality inference for ... |
| nfmpt 5254 | Bound-variable hypothesis ... |
| nfmpt1 5255 | Bound-variable hypothesis ... |
| cbvmptf 5256 | Rule to change the bound v... |
| cbvmptfg 5257 | Rule to change the bound v... |
| cbvmpt 5258 | Rule to change the bound v... |
| cbvmptg 5259 | Rule to change the bound v... |
| cbvmptv 5260 | Rule to change the bound v... |
| cbvmptvOLD 5261 | Obsolete version of ~ cbvm... |
| cbvmptvg 5262 | Rule to change the bound v... |
| mptv 5263 | Function with universal do... |
| dftr2 5266 | An alternate way of defini... |
| dftr2c 5267 | Variant of ~ dftr2 with co... |
| dftr5 5268 | An alternate way of defini... |
| dftr5OLD 5269 | Obsolete version of ~ dftr... |
| dftr3 5270 | An alternate way of defini... |
| dftr4 5271 | An alternate way of defini... |
| treq 5272 | Equality theorem for the t... |
| trel 5273 | In a transitive class, the... |
| trel3 5274 | In a transitive class, the... |
| trss 5275 | An element of a transitive... |
| trin 5276 | The intersection of transi... |
| tr0 5277 | The empty set is transitiv... |
| trv 5278 | The universe is transitive... |
| triun 5279 | An indexed union of a clas... |
| truni 5280 | The union of a class of tr... |
| triin 5281 | An indexed intersection of... |
| trint 5282 | The intersection of a clas... |
| trintss 5283 | Any nonempty transitive cl... |
| axrep1 5285 | The version of the Axiom o... |
| axreplem 5286 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5287 | Axiom of Replacement expre... |
| axrep3 5288 | Axiom of Replacement sligh... |
| axrep4 5289 | A more traditional version... |
| axrep5 5290 | Axiom of Replacement (simi... |
| axrep6 5291 | A condensed form of ~ ax-r... |
| axrep6g 5292 | ~ axrep6 in class notation... |
| zfrepclf 5293 | An inference based on the ... |
| zfrep3cl 5294 | An inference based on the ... |
| zfrep4 5295 | A version of Replacement u... |
| axsepgfromrep 5296 | A more general version ~ a... |
| axsep 5297 | Axiom scheme of separation... |
| axsepg 5299 | A more general version of ... |
| zfauscl 5300 | Separation Scheme (Aussond... |
| bm1.3ii 5301 | Convert implication to equ... |
| ax6vsep 5302 | Derive ~ ax6v (a weakened ... |
| axnulALT 5303 | Alternate proof of ~ axnul... |
| axnul 5304 | The Null Set Axiom of ZF s... |
| 0ex 5306 | The Null Set Axiom of ZF s... |
| al0ssb 5307 | The empty set is the uniqu... |
| sseliALT 5308 | Alternate proof of ~ sseli... |
| csbexg 5309 | The existence of proper su... |
| csbex 5310 | The existence of proper su... |
| unisn2 5311 | A version of ~ unisn witho... |
| nalset 5312 | No set contains all sets. ... |
| vnex 5313 | The universal class does n... |
| vprc 5314 | The universal class is not... |
| nvel 5315 | The universal class does n... |
| inex1 5316 | Separation Scheme (Aussond... |
| inex2 5317 | Separation Scheme (Aussond... |
| inex1g 5318 | Closed-form, generalized S... |
| inex2g 5319 | Sufficient condition for a... |
| ssex 5320 | The subset of a set is als... |
| ssexi 5321 | The subset of a set is als... |
| ssexg 5322 | The subset of a set is als... |
| ssexd 5323 | A subclass of a set is a s... |
| prcssprc 5324 | The superclass of a proper... |
| sselpwd 5325 | Elementhood to a power set... |
| difexg 5326 | Existence of a difference.... |
| difexi 5327 | Existence of a difference,... |
| difexd 5328 | Existence of a difference.... |
| zfausab 5329 | Separation Scheme (Aussond... |
| rabexg 5330 | Separation Scheme in terms... |
| rabex 5331 | Separation Scheme in terms... |
| rabexd 5332 | Separation Scheme in terms... |
| rabex2 5333 | Separation Scheme in terms... |
| rab2ex 5334 | A class abstraction based ... |
| elssabg 5335 | Membership in a class abst... |
| intex 5336 | The intersection of a none... |
| intnex 5337 | If a class intersection is... |
| intexab 5338 | The intersection of a none... |
| intexrab 5339 | The intersection of a none... |
| iinexg 5340 | The existence of a class i... |
| intabs 5341 | Absorption of a redundant ... |
| inuni 5342 | The intersection of a unio... |
| elpw2g 5343 | Membership in a power clas... |
| elpw2 5344 | Membership in a power clas... |
| elpwi2 5345 | Membership in a power clas... |
| elpwi2OLD 5346 | Obsolete version of ~ elpw... |
| axpweq 5347 | Two equivalent ways to exp... |
| pwnss 5348 | The power set of a set is ... |
| pwne 5349 | No set equals its power se... |
| difelpw 5350 | A difference is an element... |
| rabelpw 5351 | A restricted class abstrac... |
| class2set 5352 | The class of elements of `... |
| 0elpw 5353 | Every power class contains... |
| pwne0 5354 | A power class is never emp... |
| 0nep0 5355 | The empty set and its powe... |
| 0inp0 5356 | Something cannot be equal ... |
| unidif0 5357 | The removal of the empty s... |
| eqsnuniex 5358 | If a class is equal to the... |
| iin0 5359 | An indexed intersection of... |
| notzfaus 5360 | In the Separation Scheme ~... |
| intv 5361 | The intersection of the un... |
| zfpow 5363 | Axiom of Power Sets expres... |
| axpow2 5364 | A variant of the Axiom of ... |
| axpow3 5365 | A variant of the Axiom of ... |
| elALT2 5366 | Alternate proof of ~ el us... |
| dtruALT2 5367 | Alternate proof of ~ dtru ... |
| dtrucor 5368 | Corollary of ~ dtru . Thi... |
| dtrucor2 5369 | The theorem form of the de... |
| dvdemo1 5370 | Demonstration of a theorem... |
| dvdemo2 5371 | Demonstration of a theorem... |
| nfnid 5372 | A setvar variable is not f... |
| nfcvb 5373 | The "distinctor" expressio... |
| vpwex 5374 | Power set axiom: the power... |
| pwexg 5375 | Power set axiom expressed ... |
| pwexd 5376 | Deduction version of the p... |
| pwex 5377 | Power set axiom expressed ... |
| pwel 5378 | Quantitative version of ~ ... |
| abssexg 5379 | Existence of a class of su... |
| snexALT 5380 | Alternate proof of ~ snex ... |
| p0ex 5381 | The power set of the empty... |
| p0exALT 5382 | Alternate proof of ~ p0ex ... |
| pp0ex 5383 | The power set of the power... |
| ord3ex 5384 | The ordinal number 3 is a ... |
| dtruALT 5385 | Alternate proof of ~ dtru ... |
| axc16b 5386 | This theorem shows that Ax... |
| eunex 5387 | Existential uniqueness imp... |
| eusv1 5388 | Two ways to express single... |
| eusvnf 5389 | Even if ` x ` is free in `... |
| eusvnfb 5390 | Two ways to say that ` A (... |
| eusv2i 5391 | Two ways to express single... |
| eusv2nf 5392 | Two ways to express single... |
| eusv2 5393 | Two ways to express single... |
| reusv1 5394 | Two ways to express single... |
| reusv2lem1 5395 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5396 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5397 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5398 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5399 | Lemma for ~ reusv2 . (Con... |
| reusv2 5400 | Two ways to express single... |
| reusv3i 5401 | Two ways of expressing exi... |
| reusv3 5402 | Two ways to express single... |
| eusv4 5403 | Two ways to express single... |
| alxfr 5404 | Transfer universal quantif... |
| ralxfrd 5405 | Transfer universal quantif... |
| rexxfrd 5406 | Transfer universal quantif... |
| ralxfr2d 5407 | Transfer universal quantif... |
| rexxfr2d 5408 | Transfer universal quantif... |
| ralxfrd2 5409 | Transfer universal quantif... |
| rexxfrd2 5410 | Transfer existence from a ... |
| ralxfr 5411 | Transfer universal quantif... |
| ralxfrALT 5412 | Alternate proof of ~ ralxf... |
| rexxfr 5413 | Transfer existence from a ... |
| rabxfrd 5414 | Membership in a restricted... |
| rabxfr 5415 | Membership in a restricted... |
| reuhypd 5416 | A theorem useful for elimi... |
| reuhyp 5417 | A theorem useful for elimi... |
| zfpair 5418 | The Axiom of Pairing of Ze... |
| axprALT 5419 | Alternate proof of ~ axpr ... |
| axprlem1 5420 | Lemma for ~ axpr . There ... |
| axprlem2 5421 | Lemma for ~ axpr . There ... |
| axprlem3 5422 | Lemma for ~ axpr . Elimin... |
| axprlem4 5423 | Lemma for ~ axpr . The fi... |
| axprlem5 5424 | Lemma for ~ axpr . The se... |
| axpr 5425 | Unabbreviated version of t... |
| zfpair2 5427 | Derive the abbreviated ver... |
| vsnex 5428 | A singleton built on a set... |
| snexg 5429 | A singleton built on a set... |
| snex 5430 | A singleton is a set. The... |
| prex 5431 | The Axiom of Pairing using... |
| exel 5432 | There exist two sets, one ... |
| exexneq 5433 | There exist two different ... |
| exneq 5434 | Given any set (the " ` y `... |
| dtru 5435 | Given any set (the " ` y `... |
| el 5436 | Any set is an element of s... |
| sels 5437 | If a class is a set, then ... |
| selsALT 5438 | Alternate proof of ~ sels ... |
| elALT 5439 | Alternate proof of ~ el , ... |
| dtruOLD 5440 | Obsolete proof of ~ dtru a... |
| snelpwg 5441 | A singleton of a set is a ... |
| snelpwi 5442 | If a set is a member of a ... |
| snelpwiOLD 5443 | Obsolete version of ~ snel... |
| snelpw 5444 | A singleton of a set is a ... |
| prelpw 5445 | An unordered pair of two s... |
| prelpwi 5446 | If two sets are members of... |
| rext 5447 | A theorem similar to exten... |
| sspwb 5448 | The powerclass constructio... |
| unipw 5449 | A class equals the union o... |
| univ 5450 | The union of the universe ... |
| pwtr 5451 | A class is transitive iff ... |
| ssextss 5452 | An extensionality-like pri... |
| ssext 5453 | An extensionality-like pri... |
| nssss 5454 | Negation of subclass relat... |
| pweqb 5455 | Classes are equal if and o... |
| intidg 5456 | The intersection of all se... |
| intidOLD 5457 | Obsolete version of ~ inti... |
| moabex 5458 | "At most one" existence im... |
| rmorabex 5459 | Restricted "at most one" e... |
| euabex 5460 | The abstraction of a wff w... |
| nnullss 5461 | A nonempty class (even if ... |
| exss 5462 | Restricted existence in a ... |
| opex 5463 | An ordered pair of classes... |
| otex 5464 | An ordered triple of class... |
| elopg 5465 | Characterization of the el... |
| elop 5466 | Characterization of the el... |
| opi1 5467 | One of the two elements in... |
| opi2 5468 | One of the two elements of... |
| opeluu 5469 | Each member of an ordered ... |
| op1stb 5470 | Extract the first member o... |
| brv 5471 | Two classes are always in ... |
| opnz 5472 | An ordered pair is nonempt... |
| opnzi 5473 | An ordered pair is nonempt... |
| opth1 5474 | Equality of the first memb... |
| opth 5475 | The ordered pair theorem. ... |
| opthg 5476 | Ordered pair theorem. ` C ... |
| opth1g 5477 | Equality of the first memb... |
| opthg2 5478 | Ordered pair theorem. (Co... |
| opth2 5479 | Ordered pair theorem. (Co... |
| opthneg 5480 | Two ordered pairs are not ... |
| opthne 5481 | Two ordered pairs are not ... |
| otth2 5482 | Ordered triple theorem, wi... |
| otth 5483 | Ordered triple theorem. (... |
| otthg 5484 | Ordered triple theorem, cl... |
| otthne 5485 | Contrapositive of the orde... |
| eqvinop 5486 | A variable introduction la... |
| sbcop1 5487 | The proper substitution of... |
| sbcop 5488 | The proper substitution of... |
| copsexgw 5489 | Version of ~ copsexg with ... |
| copsexg 5490 | Substitution of class ` A ... |
| copsex2t 5491 | Closed theorem form of ~ c... |
| copsex2g 5492 | Implicit substitution infe... |
| copsex2gOLD 5493 | Obsolete version of ~ cops... |
| copsex4g 5494 | An implicit substitution i... |
| 0nelop 5495 | A property of ordered pair... |
| opwo0id 5496 | An ordered pair is equal t... |
| opeqex 5497 | Equivalence of existence i... |
| oteqex2 5498 | Equivalence of existence i... |
| oteqex 5499 | Equivalence of existence i... |
| opcom 5500 | An ordered pair commutes i... |
| moop2 5501 | "At most one" property of ... |
| opeqsng 5502 | Equivalence for an ordered... |
| opeqsn 5503 | Equivalence for an ordered... |
| opeqpr 5504 | Equivalence for an ordered... |
| snopeqop 5505 | Equivalence for an ordered... |
| propeqop 5506 | Equivalence for an ordered... |
| propssopi 5507 | If a pair of ordered pairs... |
| snopeqopsnid 5508 | Equivalence for an ordered... |
| mosubopt 5509 | "At most one" remains true... |
| mosubop 5510 | "At most one" remains true... |
| euop2 5511 | Transfer existential uniqu... |
| euotd 5512 | Prove existential uniquene... |
| opthwiener 5513 | Justification theorem for ... |
| uniop 5514 | The union of an ordered pa... |
| uniopel 5515 | Ordered pair membership is... |
| opthhausdorff 5516 | Justification theorem for ... |
| opthhausdorff0 5517 | Justification theorem for ... |
| otsndisj 5518 | The singletons consisting ... |
| otiunsndisj 5519 | The union of singletons co... |
| iunopeqop 5520 | Implication of an ordered ... |
| brsnop 5521 | Binary relation for an ord... |
| brtp 5522 | A necessary and sufficient... |
| opabidw 5523 | The law of concretion. Sp... |
| opabid 5524 | The law of concretion. Sp... |
| elopabw 5525 | Membership in a class abst... |
| elopab 5526 | Membership in a class abst... |
| rexopabb 5527 | Restricted existential qua... |
| vopelopabsb 5528 | The law of concretion in t... |
| opelopabsb 5529 | The law of concretion in t... |
| brabsb 5530 | The law of concretion in t... |
| opelopabt 5531 | Closed theorem form of ~ o... |
| opelopabga 5532 | The law of concretion. Th... |
| brabga 5533 | The law of concretion for ... |
| opelopab2a 5534 | Ordered pair membership in... |
| opelopaba 5535 | The law of concretion. Th... |
| braba 5536 | The law of concretion for ... |
| opelopabg 5537 | The law of concretion. Th... |
| brabg 5538 | The law of concretion for ... |
| opelopabgf 5539 | The law of concretion. Th... |
| opelopab2 5540 | Ordered pair membership in... |
| opelopab 5541 | The law of concretion. Th... |
| brab 5542 | The law of concretion for ... |
| opelopabaf 5543 | The law of concretion. Th... |
| opelopabf 5544 | The law of concretion. Th... |
| ssopab2 5545 | Equivalence of ordered pai... |
| ssopab2bw 5546 | Equivalence of ordered pai... |
| eqopab2bw 5547 | Equivalence of ordered pai... |
| ssopab2b 5548 | Equivalence of ordered pai... |
| ssopab2i 5549 | Inference of ordered pair ... |
| ssopab2dv 5550 | Inference of ordered pair ... |
| eqopab2b 5551 | Equivalence of ordered pai... |
| opabn0 5552 | Nonempty ordered pair clas... |
| opab0 5553 | Empty ordered pair class a... |
| csbopab 5554 | Move substitution into a c... |
| csbopabgALT 5555 | Move substitution into a c... |
| csbmpt12 5556 | Move substitution into a m... |
| csbmpt2 5557 | Move substitution into the... |
| iunopab 5558 | Move indexed union inside ... |
| iunopabOLD 5559 | Obsolete version of ~ iuno... |
| elopabr 5560 | Membership in an ordered-p... |
| elopabran 5561 | Membership in an ordered-p... |
| elopabrOLD 5562 | Obsolete version of ~ elop... |
| rbropapd 5563 | Properties of a pair in an... |
| rbropap 5564 | Properties of a pair in a ... |
| 2rbropap 5565 | Properties of a pair in a ... |
| 0nelopab 5566 | The empty set is never an ... |
| 0nelopabOLD 5567 | Obsolete version of ~ 0nel... |
| brabv 5568 | If two classes are in a re... |
| pwin 5569 | The power class of the int... |
| pwssun 5570 | The power class of the uni... |
| pwun 5571 | The power class of the uni... |
| dfid4 5574 | The identity function expr... |
| dfid2 5575 | Alternate definition of th... |
| dfid3 5576 | A stronger version of ~ df... |
| dfid2OLD 5577 | Obsolete version of ~ dfid... |
| epelg 5580 | The membership relation an... |
| epeli 5581 | The membership relation an... |
| epel 5582 | The membership relation an... |
| 0sn0ep 5583 | An example for the members... |
| epn0 5584 | The membership relation is... |
| poss 5589 | Subset theorem for the par... |
| poeq1 5590 | Equality theorem for parti... |
| poeq2 5591 | Equality theorem for parti... |
| nfpo 5592 | Bound-variable hypothesis ... |
| nfso 5593 | Bound-variable hypothesis ... |
| pocl 5594 | Characteristic properties ... |
| poclOLD 5595 | Obsolete version of ~ pocl... |
| ispod 5596 | Sufficient conditions for ... |
| swopolem 5597 | Perform the substitutions ... |
| swopo 5598 | A strict weak order is a p... |
| poirr 5599 | A partial order is irrefle... |
| potr 5600 | A partial order is a trans... |
| po2nr 5601 | A partial order has no 2-c... |
| po3nr 5602 | A partial order has no 3-c... |
| po2ne 5603 | Two sets related by a part... |
| po0 5604 | Any relation is a partial ... |
| pofun 5605 | The inverse image of a par... |
| sopo 5606 | A strict linear order is a... |
| soss 5607 | Subset theorem for the str... |
| soeq1 5608 | Equality theorem for the s... |
| soeq2 5609 | Equality theorem for the s... |
| sonr 5610 | A strict order relation is... |
| sotr 5611 | A strict order relation is... |
| solin 5612 | A strict order relation is... |
| so2nr 5613 | A strict order relation ha... |
| so3nr 5614 | A strict order relation ha... |
| sotric 5615 | A strict order relation sa... |
| sotrieq 5616 | Trichotomy law for strict ... |
| sotrieq2 5617 | Trichotomy law for strict ... |
| soasym 5618 | Asymmetry law for strict o... |
| sotr2 5619 | A transitivity relation. ... |
| issod 5620 | An irreflexive, transitive... |
| issoi 5621 | An irreflexive, transitive... |
| isso2i 5622 | Deduce strict ordering fro... |
| so0 5623 | Any relation is a strict o... |
| somo 5624 | A totally ordered set has ... |
| sotrine 5625 | Trichotomy law for strict ... |
| sotr3 5626 | Transitivity law for stric... |
| dffr6 5633 | Alternate definition of ~ ... |
| frd 5634 | A nonempty subset of an ` ... |
| fri 5635 | A nonempty subset of an ` ... |
| friOLD 5636 | Obsolete version of ~ fri ... |
| seex 5637 | The ` R ` -preimage of an ... |
| exse 5638 | Any relation on a set is s... |
| dffr2 5639 | Alternate definition of we... |
| dffr2ALT 5640 | Alternate proof of ~ dffr2... |
| frc 5641 | Property of well-founded r... |
| frss 5642 | Subset theorem for the wel... |
| sess1 5643 | Subset theorem for the set... |
| sess2 5644 | Subset theorem for the set... |
| freq1 5645 | Equality theorem for the w... |
| freq2 5646 | Equality theorem for the w... |
| seeq1 5647 | Equality theorem for the s... |
| seeq2 5648 | Equality theorem for the s... |
| nffr 5649 | Bound-variable hypothesis ... |
| nfse 5650 | Bound-variable hypothesis ... |
| nfwe 5651 | Bound-variable hypothesis ... |
| frirr 5652 | A well-founded relation is... |
| fr2nr 5653 | A well-founded relation ha... |
| fr0 5654 | Any relation is well-found... |
| frminex 5655 | If an element of a well-fo... |
| efrirr 5656 | A well-founded class does ... |
| efrn2lp 5657 | A well-founded class conta... |
| epse 5658 | The membership relation is... |
| tz7.2 5659 | Similar to Theorem 7.2 of ... |
| dfepfr 5660 | An alternate way of saying... |
| epfrc 5661 | A subset of a well-founded... |
| wess 5662 | Subset theorem for the wel... |
| weeq1 5663 | Equality theorem for the w... |
| weeq2 5664 | Equality theorem for the w... |
| wefr 5665 | A well-ordering is well-fo... |
| weso 5666 | A well-ordering is a stric... |
| wecmpep 5667 | The elements of a class we... |
| wetrep 5668 | On a class well-ordered by... |
| wefrc 5669 | A nonempty subclass of a c... |
| we0 5670 | Any relation is a well-ord... |
| wereu 5671 | A nonempty subset of an ` ... |
| wereu2 5672 | A nonempty subclass of an ... |
| xpeq1 5689 | Equality theorem for Carte... |
| xpss12 5690 | Subset theorem for Cartesi... |
| xpss 5691 | A Cartesian product is inc... |
| inxpssres 5692 | Intersection with a Cartes... |
| relxp 5693 | A Cartesian product is a r... |
| xpss1 5694 | Subset relation for Cartes... |
| xpss2 5695 | Subset relation for Cartes... |
| xpeq2 5696 | Equality theorem for Carte... |
| elxpi 5697 | Membership in a Cartesian ... |
| elxp 5698 | Membership in a Cartesian ... |
| elxp2 5699 | Membership in a Cartesian ... |
| xpeq12 5700 | Equality theorem for Carte... |
| xpeq1i 5701 | Equality inference for Car... |
| xpeq2i 5702 | Equality inference for Car... |
| xpeq12i 5703 | Equality inference for Car... |
| xpeq1d 5704 | Equality deduction for Car... |
| xpeq2d 5705 | Equality deduction for Car... |
| xpeq12d 5706 | Equality deduction for Car... |
| sqxpeqd 5707 | Equality deduction for a C... |
| nfxp 5708 | Bound-variable hypothesis ... |
| 0nelxp 5709 | The empty set is not a mem... |
| 0nelelxp 5710 | A member of a Cartesian pr... |
| opelxp 5711 | Ordered pair membership in... |
| opelxpi 5712 | Ordered pair membership in... |
| opelxpd 5713 | Ordered pair membership in... |
| opelvv 5714 | Ordered pair membership in... |
| opelvvg 5715 | Ordered pair membership in... |
| opelxp1 5716 | The first member of an ord... |
| opelxp2 5717 | The second member of an or... |
| otelxp 5718 | Ordered triple membership ... |
| otelxp1 5719 | The first member of an ord... |
| otel3xp 5720 | An ordered triple is an el... |
| opabssxpd 5721 | An ordered-pair class abst... |
| rabxp 5722 | Class abstraction restrict... |
| brxp 5723 | Binary relation on a Carte... |
| pwvrel 5724 | A set is a binary relation... |
| pwvabrel 5725 | The powerclass of the cart... |
| brrelex12 5726 | Two classes related by a b... |
| brrelex1 5727 | If two classes are related... |
| brrelex2 5728 | If two classes are related... |
| brrelex12i 5729 | Two classes that are relat... |
| brrelex1i 5730 | The first argument of a bi... |
| brrelex2i 5731 | The second argument of a b... |
| nprrel12 5732 | Proper classes are not rel... |
| nprrel 5733 | No proper class is related... |
| 0nelrel0 5734 | A binary relation does not... |
| 0nelrel 5735 | A binary relation does not... |
| fconstmpt 5736 | Representation of a consta... |
| vtoclr 5737 | Variable to class conversi... |
| opthprc 5738 | Justification theorem for ... |
| brel 5739 | Two things in a binary rel... |
| elxp3 5740 | Membership in a Cartesian ... |
| opeliunxp 5741 | Membership in a union of C... |
| xpundi 5742 | Distributive law for Carte... |
| xpundir 5743 | Distributive law for Carte... |
| xpiundi 5744 | Distributive law for Carte... |
| xpiundir 5745 | Distributive law for Carte... |
| iunxpconst 5746 | Membership in a union of C... |
| xpun 5747 | The Cartesian product of t... |
| elvv 5748 | Membership in universal cl... |
| elvvv 5749 | Membership in universal cl... |
| elvvuni 5750 | An ordered pair contains i... |
| brinxp2 5751 | Intersection of binary rel... |
| brinxp 5752 | Intersection of binary rel... |
| opelinxp 5753 | Ordered pair element in an... |
| poinxp 5754 | Intersection of partial or... |
| soinxp 5755 | Intersection of total orde... |
| frinxp 5756 | Intersection of well-found... |
| seinxp 5757 | Intersection of set-like r... |
| weinxp 5758 | Intersection of well-order... |
| posn 5759 | Partial ordering of a sing... |
| sosn 5760 | Strict ordering on a singl... |
| frsn 5761 | Founded relation on a sing... |
| wesn 5762 | Well-ordering of a singlet... |
| elopaelxp 5763 | Membership in an ordered-p... |
| elopaelxpOLD 5764 | Obsolete version of ~ elop... |
| bropaex12 5765 | Two classes related by an ... |
| opabssxp 5766 | An abstraction relation is... |
| brab2a 5767 | The law of concretion for ... |
| optocl 5768 | Implicit substitution of c... |
| 2optocl 5769 | Implicit substitution of c... |
| 3optocl 5770 | Implicit substitution of c... |
| opbrop 5771 | Ordered pair membership in... |
| 0xp 5772 | The Cartesian product with... |
| csbxp 5773 | Distribute proper substitu... |
| releq 5774 | Equality theorem for the r... |
| releqi 5775 | Equality inference for the... |
| releqd 5776 | Equality deduction for the... |
| nfrel 5777 | Bound-variable hypothesis ... |
| sbcrel 5778 | Distribute proper substitu... |
| relss 5779 | Subclass theorem for relat... |
| ssrel 5780 | A subclass relationship de... |
| ssrelOLD 5781 | Obsolete version of ~ ssre... |
| eqrel 5782 | Extensionality principle f... |
| ssrel2 5783 | A subclass relationship de... |
| ssrel3 5784 | Subclass relation in anoth... |
| relssi 5785 | Inference from subclass pr... |
| relssdv 5786 | Deduction from subclass pr... |
| eqrelriv 5787 | Inference from extensional... |
| eqrelriiv 5788 | Inference from extensional... |
| eqbrriv 5789 | Inference from extensional... |
| eqrelrdv 5790 | Deduce equality of relatio... |
| eqbrrdv 5791 | Deduction from extensional... |
| eqbrrdiv 5792 | Deduction from extensional... |
| eqrelrdv2 5793 | A version of ~ eqrelrdv . ... |
| ssrelrel 5794 | A subclass relationship de... |
| eqrelrel 5795 | Extensionality principle f... |
| elrel 5796 | A member of a relation is ... |
| rel0 5797 | The empty set is a relatio... |
| nrelv 5798 | The universal class is not... |
| relsng 5799 | A singleton is a relation ... |
| relsnb 5800 | An at-most-singleton is a ... |
| relsnopg 5801 | A singleton of an ordered ... |
| relsn 5802 | A singleton is a relation ... |
| relsnop 5803 | A singleton of an ordered ... |
| copsex2gb 5804 | Implicit substitution infe... |
| copsex2ga 5805 | Implicit substitution infe... |
| elopaba 5806 | Membership in an ordered-p... |
| xpsspw 5807 | A Cartesian product is inc... |
| unixpss 5808 | The double class union of ... |
| relun 5809 | The union of two relations... |
| relin1 5810 | The intersection with a re... |
| relin2 5811 | The intersection with a re... |
| relinxp 5812 | Intersection with a Cartes... |
| reldif 5813 | A difference cutting down ... |
| reliun 5814 | An indexed union is a rela... |
| reliin 5815 | An indexed intersection is... |
| reluni 5816 | The union of a class is a ... |
| relint 5817 | The intersection of a clas... |
| relopabiv 5818 | A class of ordered pairs i... |
| relopabv 5819 | A class of ordered pairs i... |
| relopabi 5820 | A class of ordered pairs i... |
| relopabiALT 5821 | Alternate proof of ~ relop... |
| relopab 5822 | A class of ordered pairs i... |
| mptrel 5823 | The maps-to notation alway... |
| reli 5824 | The identity relation is a... |
| rele 5825 | The membership relation is... |
| opabid2 5826 | A relation expressed as an... |
| inopab 5827 | Intersection of two ordere... |
| difopab 5828 | Difference of two ordered-... |
| difopabOLD 5829 | Obsolete version of ~ difo... |
| inxp 5830 | Intersection of two Cartes... |
| xpindi 5831 | Distributive law for Carte... |
| xpindir 5832 | Distributive law for Carte... |
| xpiindi 5833 | Distributive law for Carte... |
| xpriindi 5834 | Distributive law for Carte... |
| eliunxp 5835 | Membership in a union of C... |
| opeliunxp2 5836 | Membership in a union of C... |
| raliunxp 5837 | Write a double restricted ... |
| rexiunxp 5838 | Write a double restricted ... |
| ralxp 5839 | Universal quantification r... |
| rexxp 5840 | Existential quantification... |
| exopxfr 5841 | Transfer ordered-pair exis... |
| exopxfr2 5842 | Transfer ordered-pair exis... |
| djussxp 5843 | Disjoint union is a subset... |
| ralxpf 5844 | Version of ~ ralxp with bo... |
| rexxpf 5845 | Version of ~ rexxp with bo... |
| iunxpf 5846 | Indexed union on a Cartesi... |
| opabbi2dv 5847 | Deduce equality of a relat... |
| relop 5848 | A necessary and sufficient... |
| ideqg 5849 | For sets, the identity rel... |
| ideq 5850 | For sets, the identity rel... |
| ididg 5851 | A set is identical to itse... |
| issetid 5852 | Two ways of expressing set... |
| coss1 5853 | Subclass theorem for compo... |
| coss2 5854 | Subclass theorem for compo... |
| coeq1 5855 | Equality theorem for compo... |
| coeq2 5856 | Equality theorem for compo... |
| coeq1i 5857 | Equality inference for com... |
| coeq2i 5858 | Equality inference for com... |
| coeq1d 5859 | Equality deduction for com... |
| coeq2d 5860 | Equality deduction for com... |
| coeq12i 5861 | Equality inference for com... |
| coeq12d 5862 | Equality deduction for com... |
| nfco 5863 | Bound-variable hypothesis ... |
| brcog 5864 | Ordered pair membership in... |
| opelco2g 5865 | Ordered pair membership in... |
| brcogw 5866 | Ordered pair membership in... |
| eqbrrdva 5867 | Deduction from extensional... |
| brco 5868 | Binary relation on a compo... |
| opelco 5869 | Ordered pair membership in... |
| cnvss 5870 | Subset theorem for convers... |
| cnveq 5871 | Equality theorem for conve... |
| cnveqi 5872 | Equality inference for con... |
| cnveqd 5873 | Equality deduction for con... |
| elcnv 5874 | Membership in a converse r... |
| elcnv2 5875 | Membership in a converse r... |
| nfcnv 5876 | Bound-variable hypothesis ... |
| brcnvg 5877 | The converse of a binary r... |
| opelcnvg 5878 | Ordered-pair membership in... |
| opelcnv 5879 | Ordered-pair membership in... |
| brcnv 5880 | The converse of a binary r... |
| csbcnv 5881 | Move class substitution in... |
| csbcnvgALT 5882 | Move class substitution in... |
| cnvco 5883 | Distributive law of conver... |
| cnvuni 5884 | The converse of a class un... |
| dfdm3 5885 | Alternate definition of do... |
| dfrn2 5886 | Alternate definition of ra... |
| dfrn3 5887 | Alternate definition of ra... |
| elrn2g 5888 | Membership in a range. (C... |
| elrng 5889 | Membership in a range. (C... |
| elrn2 5890 | Membership in a range. (C... |
| elrn 5891 | Membership in a range. (C... |
| ssrelrn 5892 | If a relation is a subset ... |
| dfdm4 5893 | Alternate definition of do... |
| dfdmf 5894 | Definition of domain, usin... |
| csbdm 5895 | Distribute proper substitu... |
| eldmg 5896 | Domain membership. Theore... |
| eldm2g 5897 | Domain membership. Theore... |
| eldm 5898 | Membership in a domain. T... |
| eldm2 5899 | Membership in a domain. T... |
| dmss 5900 | Subset theorem for domain.... |
| dmeq 5901 | Equality theorem for domai... |
| dmeqi 5902 | Equality inference for dom... |
| dmeqd 5903 | Equality deduction for dom... |
| opeldmd 5904 | Membership of first of an ... |
| opeldm 5905 | Membership of first of an ... |
| breldm 5906 | Membership of first of a b... |
| breldmg 5907 | Membership of first of a b... |
| dmun 5908 | The domain of a union is t... |
| dmin 5909 | The domain of an intersect... |
| breldmd 5910 | Membership of first of a b... |
| dmiun 5911 | The domain of an indexed u... |
| dmuni 5912 | The domain of a union. Pa... |
| dmopab 5913 | The domain of a class of o... |
| dmopabelb 5914 | A set is an element of the... |
| dmopab2rex 5915 | The domain of an ordered p... |
| dmopabss 5916 | Upper bound for the domain... |
| dmopab3 5917 | The domain of a restricted... |
| dm0 5918 | The domain of the empty se... |
| dmi 5919 | The domain of the identity... |
| dmv 5920 | The domain of the universe... |
| dmep 5921 | The domain of the membersh... |
| dm0rn0 5922 | An empty domain is equival... |
| rn0 5923 | The range of the empty set... |
| rnep 5924 | The range of the membershi... |
| reldm0 5925 | A relation is empty iff it... |
| dmxp 5926 | The domain of a Cartesian ... |
| dmxpid 5927 | The domain of a Cartesian ... |
| dmxpin 5928 | The domain of the intersec... |
| xpid11 5929 | The Cartesian square is a ... |
| dmcnvcnv 5930 | The domain of the double c... |
| rncnvcnv 5931 | The range of the double co... |
| elreldm 5932 | The first member of an ord... |
| rneq 5933 | Equality theorem for range... |
| rneqi 5934 | Equality inference for ran... |
| rneqd 5935 | Equality deduction for ran... |
| rnss 5936 | Subset theorem for range. ... |
| rnssi 5937 | Subclass inference for ran... |
| brelrng 5938 | The second argument of a b... |
| brelrn 5939 | The second argument of a b... |
| opelrn 5940 | Membership of second membe... |
| releldm 5941 | The first argument of a bi... |
| relelrn 5942 | The second argument of a b... |
| releldmb 5943 | Membership in a domain. (... |
| relelrnb 5944 | Membership in a range. (C... |
| releldmi 5945 | The first argument of a bi... |
| relelrni 5946 | The second argument of a b... |
| dfrnf 5947 | Definition of range, using... |
| nfdm 5948 | Bound-variable hypothesis ... |
| nfrn 5949 | Bound-variable hypothesis ... |
| dmiin 5950 | Domain of an intersection.... |
| rnopab 5951 | The range of a class of or... |
| rnmpt 5952 | The range of a function in... |
| elrnmpt 5953 | The range of a function in... |
| elrnmpt1s 5954 | Elementhood in an image se... |
| elrnmpt1 5955 | Elementhood in an image se... |
| elrnmptg 5956 | Membership in the range of... |
| elrnmpti 5957 | Membership in the range of... |
| elrnmptd 5958 | The range of a function in... |
| elrnmptdv 5959 | Elementhood in the range o... |
| elrnmpt2d 5960 | Elementhood in the range o... |
| dfiun3g 5961 | Alternate definition of in... |
| dfiin3g 5962 | Alternate definition of in... |
| dfiun3 5963 | Alternate definition of in... |
| dfiin3 5964 | Alternate definition of in... |
| riinint 5965 | Express a relative indexed... |
| relrn0 5966 | A relation is empty iff it... |
| dmrnssfld 5967 | The domain and range of a ... |
| dmcoss 5968 | Domain of a composition. ... |
| rncoss 5969 | Range of a composition. (... |
| dmcosseq 5970 | Domain of a composition. ... |
| dmcoeq 5971 | Domain of a composition. ... |
| rncoeq 5972 | Range of a composition. (... |
| reseq1 5973 | Equality theorem for restr... |
| reseq2 5974 | Equality theorem for restr... |
| reseq1i 5975 | Equality inference for res... |
| reseq2i 5976 | Equality inference for res... |
| reseq12i 5977 | Equality inference for res... |
| reseq1d 5978 | Equality deduction for res... |
| reseq2d 5979 | Equality deduction for res... |
| reseq12d 5980 | Equality deduction for res... |
| nfres 5981 | Bound-variable hypothesis ... |
| csbres 5982 | Distribute proper substitu... |
| res0 5983 | A restriction to the empty... |
| dfres3 5984 | Alternate definition of re... |
| opelres 5985 | Ordered pair elementhood i... |
| brres 5986 | Binary relation on a restr... |
| opelresi 5987 | Ordered pair membership in... |
| brresi 5988 | Binary relation on a restr... |
| opres 5989 | Ordered pair membership in... |
| resieq 5990 | A restricted identity rela... |
| opelidres 5991 | ` <. A , A >. ` belongs to... |
| resres 5992 | The restriction of a restr... |
| resundi 5993 | Distributive law for restr... |
| resundir 5994 | Distributive law for restr... |
| resindi 5995 | Class restriction distribu... |
| resindir 5996 | Class restriction distribu... |
| inres 5997 | Move intersection into cla... |
| resdifcom 5998 | Commutative law for restri... |
| resiun1 5999 | Distribution of restrictio... |
| resiun2 6000 | Distribution of restrictio... |
| dmres 6001 | The domain of a restrictio... |
| ssdmres 6002 | A domain restricted to a s... |
| dmresexg 6003 | The domain of a restrictio... |
| resss 6004 | A class includes its restr... |
| rescom 6005 | Commutative law for restri... |
| ssres 6006 | Subclass theorem for restr... |
| ssres2 6007 | Subclass theorem for restr... |
| relres 6008 | A restriction is a relatio... |
| resabs1 6009 | Absorption law for restric... |
| resabs1d 6010 | Absorption law for restric... |
| resabs2 6011 | Absorption law for restric... |
| residm 6012 | Idempotent law for restric... |
| resima 6013 | A restriction to an image.... |
| resima2 6014 | Image under a restricted c... |
| rnresss 6015 | The range of a restriction... |
| xpssres 6016 | Restriction of a constant ... |
| elinxp 6017 | Membership in an intersect... |
| elres 6018 | Membership in a restrictio... |
| elsnres 6019 | Membership in restriction ... |
| relssres 6020 | Simplification law for res... |
| dmressnsn 6021 | The domain of a restrictio... |
| eldmressnsn 6022 | The element of the domain ... |
| eldmeldmressn 6023 | An element of the domain (... |
| resdm 6024 | A relation restricted to i... |
| resexg 6025 | The restriction of a set i... |
| resexd 6026 | The restriction of a set i... |
| resex 6027 | The restriction of a set i... |
| resindm 6028 | When restricting a relatio... |
| resdmdfsn 6029 | Restricting a relation to ... |
| reldisjun 6030 | Split a relation into two ... |
| relresdm1 6031 | Restriction of a disjoint ... |
| resopab 6032 | Restriction of a class abs... |
| iss 6033 | A subclass of the identity... |
| resopab2 6034 | Restriction of a class abs... |
| resmpt 6035 | Restriction of the mapping... |
| resmpt3 6036 | Unconditional restriction ... |
| resmptf 6037 | Restriction of the mapping... |
| resmptd 6038 | Restriction of the mapping... |
| dfres2 6039 | Alternate definition of th... |
| mptss 6040 | Sufficient condition for i... |
| elidinxp 6041 | Characterization of the el... |
| elidinxpid 6042 | Characterization of the el... |
| elrid 6043 | Characterization of the el... |
| idinxpres 6044 | The intersection of the id... |
| idinxpresid 6045 | The intersection of the id... |
| idssxp 6046 | A diagonal set as a subset... |
| opabresid 6047 | The restricted identity re... |
| mptresid 6048 | The restricted identity re... |
| dmresi 6049 | The domain of a restricted... |
| restidsing 6050 | Restriction of the identit... |
| iresn0n0 6051 | The identity function rest... |
| imaeq1 6052 | Equality theorem for image... |
| imaeq2 6053 | Equality theorem for image... |
| imaeq1i 6054 | Equality theorem for image... |
| imaeq2i 6055 | Equality theorem for image... |
| imaeq1d 6056 | Equality theorem for image... |
| imaeq2d 6057 | Equality theorem for image... |
| imaeq12d 6058 | Equality theorem for image... |
| dfima2 6059 | Alternate definition of im... |
| dfima3 6060 | Alternate definition of im... |
| elimag 6061 | Membership in an image. T... |
| elima 6062 | Membership in an image. T... |
| elima2 6063 | Membership in an image. T... |
| elima3 6064 | Membership in an image. T... |
| nfima 6065 | Bound-variable hypothesis ... |
| nfimad 6066 | Deduction version of bound... |
| imadmrn 6067 | The image of the domain of... |
| imassrn 6068 | The image of a class is a ... |
| mptima 6069 | Image of a function in map... |
| imai 6070 | Image under the identity r... |
| rnresi 6071 | The range of the restricte... |
| resiima 6072 | The image of a restriction... |
| ima0 6073 | Image of the empty set. T... |
| 0ima 6074 | Image under the empty rela... |
| csbima12 6075 | Move class substitution in... |
| imadisj 6076 | A class whose image under ... |
| cnvimass 6077 | A preimage under any class... |
| cnvimarndm 6078 | The preimage of the range ... |
| imasng 6079 | The image of a singleton. ... |
| relimasn 6080 | The image of a singleton. ... |
| elrelimasn 6081 | Elementhood in the image o... |
| elimasng1 6082 | Membership in an image of ... |
| elimasn1 6083 | Membership in an image of ... |
| elimasng 6084 | Membership in an image of ... |
| elimasn 6085 | Membership in an image of ... |
| elimasngOLD 6086 | Obsolete version of ~ elim... |
| elimasni 6087 | Membership in an image of ... |
| args 6088 | Two ways to express the cl... |
| elinisegg 6089 | Membership in the inverse ... |
| eliniseg 6090 | Membership in the inverse ... |
| epin 6091 | Any set is equal to its pr... |
| epini 6092 | Any set is equal to its pr... |
| iniseg 6093 | An idiom that signifies an... |
| inisegn0 6094 | Nonemptiness of an initial... |
| dffr3 6095 | Alternate definition of we... |
| dfse2 6096 | Alternate definition of se... |
| imass1 6097 | Subset theorem for image. ... |
| imass2 6098 | Subset theorem for image. ... |
| ndmima 6099 | The image of a singleton o... |
| relcnv 6100 | A converse is a relation. ... |
| relbrcnvg 6101 | When ` R ` is a relation, ... |
| eliniseg2 6102 | Eliminate the class existe... |
| relbrcnv 6103 | When ` R ` is a relation, ... |
| relco 6104 | A composition is a relatio... |
| cotrg 6105 | Two ways of saying that th... |
| cotrgOLD 6106 | Obsolete version of ~ cotr... |
| cotrgOLDOLD 6107 | Obsolete version of ~ cotr... |
| cotr 6108 | Two ways of saying a relat... |
| idrefALT 6109 | Alternate proof of ~ idref... |
| cnvsym 6110 | Two ways of saying a relat... |
| cnvsymOLD 6111 | Obsolete proof of ~ cnvsym... |
| cnvsymOLDOLD 6112 | Obsolete proof of ~ cnvsym... |
| intasym 6113 | Two ways of saying a relat... |
| asymref 6114 | Two ways of saying a relat... |
| asymref2 6115 | Two ways of saying a relat... |
| intirr 6116 | Two ways of saying a relat... |
| brcodir 6117 | Two ways of saying that tw... |
| codir 6118 | Two ways of saying a relat... |
| qfto 6119 | A quantifier-free way of e... |
| xpidtr 6120 | A Cartesian square is a tr... |
| trin2 6121 | The intersection of two tr... |
| poirr2 6122 | A partial order is irrefle... |
| trinxp 6123 | The relation induced by a ... |
| soirri 6124 | A strict order relation is... |
| sotri 6125 | A strict order relation is... |
| son2lpi 6126 | A strict order relation ha... |
| sotri2 6127 | A transitivity relation. ... |
| sotri3 6128 | A transitivity relation. ... |
| poleloe 6129 | Express "less than or equa... |
| poltletr 6130 | Transitive law for general... |
| somin1 6131 | Property of a minimum in a... |
| somincom 6132 | Commutativity of minimum i... |
| somin2 6133 | Property of a minimum in a... |
| soltmin 6134 | Being less than a minimum,... |
| cnvopab 6135 | The converse of a class ab... |
| mptcnv 6136 | The converse of a mapping ... |
| cnv0 6137 | The converse of the empty ... |
| cnvi 6138 | The converse of the identi... |
| cnvun 6139 | The converse of a union is... |
| cnvdif 6140 | Distributive law for conve... |
| cnvin 6141 | Distributive law for conve... |
| rnun 6142 | Distributive law for range... |
| rnin 6143 | The range of an intersecti... |
| rniun 6144 | The range of an indexed un... |
| rnuni 6145 | The range of a union. Par... |
| imaundi 6146 | Distributive law for image... |
| imaundir 6147 | The image of a union. (Co... |
| cnvimassrndm 6148 | The preimage of a superset... |
| dminss 6149 | An upper bound for interse... |
| imainss 6150 | An upper bound for interse... |
| inimass 6151 | The image of an intersecti... |
| inimasn 6152 | The intersection of the im... |
| cnvxp 6153 | The converse of a Cartesia... |
| xp0 6154 | The Cartesian product with... |
| xpnz 6155 | The Cartesian product of n... |
| xpeq0 6156 | At least one member of an ... |
| xpdisj1 6157 | Cartesian products with di... |
| xpdisj2 6158 | Cartesian products with di... |
| xpsndisj 6159 | Cartesian products with tw... |
| difxp 6160 | Difference of Cartesian pr... |
| difxp1 6161 | Difference law for Cartesi... |
| difxp2 6162 | Difference law for Cartesi... |
| djudisj 6163 | Disjoint unions with disjo... |
| xpdifid 6164 | The set of distinct couple... |
| resdisj 6165 | A double restriction to di... |
| rnxp 6166 | The range of a Cartesian p... |
| dmxpss 6167 | The domain of a Cartesian ... |
| rnxpss 6168 | The range of a Cartesian p... |
| rnxpid 6169 | The range of a Cartesian s... |
| ssxpb 6170 | A Cartesian product subcla... |
| xp11 6171 | The Cartesian product of n... |
| xpcan 6172 | Cancellation law for Carte... |
| xpcan2 6173 | Cancellation law for Carte... |
| ssrnres 6174 | Two ways to express surjec... |
| rninxp 6175 | Two ways to express surjec... |
| dminxp 6176 | Two ways to express totali... |
| imainrect 6177 | Image by a restricted and ... |
| xpima 6178 | Direct image by a Cartesia... |
| xpima1 6179 | Direct image by a Cartesia... |
| xpima2 6180 | Direct image by a Cartesia... |
| xpimasn 6181 | Direct image of a singleto... |
| sossfld 6182 | The base set of a strict o... |
| sofld 6183 | The base set of a nonempty... |
| cnvcnv3 6184 | The set of all ordered pai... |
| dfrel2 6185 | Alternate definition of re... |
| dfrel4v 6186 | A relation can be expresse... |
| dfrel4 6187 | A relation can be expresse... |
| cnvcnv 6188 | The double converse of a c... |
| cnvcnv2 6189 | The double converse of a c... |
| cnvcnvss 6190 | The double converse of a c... |
| cnvrescnv 6191 | Two ways to express the co... |
| cnveqb 6192 | Equality theorem for conve... |
| cnveq0 6193 | A relation empty iff its c... |
| dfrel3 6194 | Alternate definition of re... |
| elid 6195 | Characterization of the el... |
| dmresv 6196 | The domain of a universal ... |
| rnresv 6197 | The range of a universal r... |
| dfrn4 6198 | Range defined in terms of ... |
| csbrn 6199 | Distribute proper substitu... |
| rescnvcnv 6200 | The restriction of the dou... |
| cnvcnvres 6201 | The double converse of the... |
| imacnvcnv 6202 | The image of the double co... |
| dmsnn0 6203 | The domain of a singleton ... |
| rnsnn0 6204 | The range of a singleton i... |
| dmsn0 6205 | The domain of the singleto... |
| cnvsn0 6206 | The converse of the single... |
| dmsn0el 6207 | The domain of a singleton ... |
| relsn2 6208 | A singleton is a relation ... |
| dmsnopg 6209 | The domain of a singleton ... |
| dmsnopss 6210 | The domain of a singleton ... |
| dmpropg 6211 | The domain of an unordered... |
| dmsnop 6212 | The domain of a singleton ... |
| dmprop 6213 | The domain of an unordered... |
| dmtpop 6214 | The domain of an unordered... |
| cnvcnvsn 6215 | Double converse of a singl... |
| dmsnsnsn 6216 | The domain of the singleto... |
| rnsnopg 6217 | The range of a singleton o... |
| rnpropg 6218 | The range of a pair of ord... |
| cnvsng 6219 | Converse of a singleton of... |
| rnsnop 6220 | The range of a singleton o... |
| op1sta 6221 | Extract the first member o... |
| cnvsn 6222 | Converse of a singleton of... |
| op2ndb 6223 | Extract the second member ... |
| op2nda 6224 | Extract the second member ... |
| opswap 6225 | Swap the members of an ord... |
| cnvresima 6226 | An image under the convers... |
| resdm2 6227 | A class restricted to its ... |
| resdmres 6228 | Restriction to the domain ... |
| resresdm 6229 | A restriction by an arbitr... |
| imadmres 6230 | The image of the domain of... |
| resdmss 6231 | Subset relationship for th... |
| resdifdi 6232 | Distributive law for restr... |
| resdifdir 6233 | Distributive law for restr... |
| mptpreima 6234 | The preimage of a function... |
| mptiniseg 6235 | Converse singleton image o... |
| dmmpt 6236 | The domain of the mapping ... |
| dmmptss 6237 | The domain of a mapping is... |
| dmmptg 6238 | The domain of the mapping ... |
| rnmpt0f 6239 | The range of a function in... |
| rnmptn0 6240 | The range of a function in... |
| dfco2 6241 | Alternate definition of a ... |
| dfco2a 6242 | Generalization of ~ dfco2 ... |
| coundi 6243 | Class composition distribu... |
| coundir 6244 | Class composition distribu... |
| cores 6245 | Restricted first member of... |
| resco 6246 | Associative law for the re... |
| imaco 6247 | Image of the composition o... |
| rnco 6248 | The range of the compositi... |
| rnco2 6249 | The range of the compositi... |
| dmco 6250 | The domain of a compositio... |
| coeq0 6251 | A composition of two relat... |
| coiun 6252 | Composition with an indexe... |
| cocnvcnv1 6253 | A composition is not affec... |
| cocnvcnv2 6254 | A composition is not affec... |
| cores2 6255 | Absorption of a reverse (p... |
| co02 6256 | Composition with the empty... |
| co01 6257 | Composition with the empty... |
| coi1 6258 | Composition with the ident... |
| coi2 6259 | Composition with the ident... |
| coires1 6260 | Composition with a restric... |
| coass 6261 | Associative law for class ... |
| relcnvtrg 6262 | General form of ~ relcnvtr... |
| relcnvtr 6263 | A relation is transitive i... |
| relssdmrn 6264 | A relation is included in ... |
| relssdmrnOLD 6265 | Obsolete version of ~ rels... |
| resssxp 6266 | If the ` R ` -image of a c... |
| cnvssrndm 6267 | The converse is a subset o... |
| cossxp 6268 | Composition as a subset of... |
| relrelss 6269 | Two ways to describe the s... |
| unielrel 6270 | The membership relation fo... |
| relfld 6271 | The double union of a rela... |
| relresfld 6272 | Restriction of a relation ... |
| relcoi2 6273 | Composition with the ident... |
| relcoi1 6274 | Composition with the ident... |
| unidmrn 6275 | The double union of the co... |
| relcnvfld 6276 | if ` R ` is a relation, it... |
| dfdm2 6277 | Alternate definition of do... |
| unixp 6278 | The double class union of ... |
| unixp0 6279 | A Cartesian product is emp... |
| unixpid 6280 | Field of a Cartesian squar... |
| ressn 6281 | Restriction of a class to ... |
| cnviin 6282 | The converse of an interse... |
| cnvpo 6283 | The converse of a partial ... |
| cnvso 6284 | The converse of a strict o... |
| xpco 6285 | Composition of two Cartesi... |
| xpcoid 6286 | Composition of two Cartesi... |
| elsnxp 6287 | Membership in a Cartesian ... |
| reu3op 6288 | There is a unique ordered ... |
| reuop 6289 | There is a unique ordered ... |
| opreu2reurex 6290 | There is a unique ordered ... |
| opreu2reu 6291 | If there is a unique order... |
| dfpo2 6292 | Quantifier-free definition... |
| csbcog 6293 | Distribute proper substitu... |
| snres0 6294 | Condition for restriction ... |
| imaindm 6295 | The image is unaffected by... |
| predeq123 6298 | Equality theorem for the p... |
| predeq1 6299 | Equality theorem for the p... |
| predeq2 6300 | Equality theorem for the p... |
| predeq3 6301 | Equality theorem for the p... |
| nfpred 6302 | Bound-variable hypothesis ... |
| csbpredg 6303 | Move class substitution in... |
| predpredss 6304 | If ` A ` is a subset of ` ... |
| predss 6305 | The predecessor class of `... |
| sspred 6306 | Another subset/predecessor... |
| dfpred2 6307 | An alternate definition of... |
| dfpred3 6308 | An alternate definition of... |
| dfpred3g 6309 | An alternate definition of... |
| elpredgg 6310 | Membership in a predecesso... |
| elpredg 6311 | Membership in a predecesso... |
| elpredimg 6312 | Membership in a predecesso... |
| elpredim 6313 | Membership in a predecesso... |
| elpred 6314 | Membership in a predecesso... |
| predexg 6315 | The predecessor class exis... |
| predasetexOLD 6316 | Obsolete form of ~ predexg... |
| dffr4 6317 | Alternate definition of we... |
| predel 6318 | Membership in the predeces... |
| predbrg 6319 | Closed form of ~ elpredim ... |
| predtrss 6320 | If ` R ` is transitive ove... |
| predpo 6321 | Property of the predecesso... |
| predso 6322 | Property of the predecesso... |
| setlikespec 6323 | If ` R ` is set-like in ` ... |
| predidm 6324 | Idempotent law for the pre... |
| predin 6325 | Intersection law for prede... |
| predun 6326 | Union law for predecessor ... |
| preddif 6327 | Difference law for predece... |
| predep 6328 | The predecessor under the ... |
| trpred 6329 | The class of predecessors ... |
| preddowncl 6330 | A property of classes that... |
| predpoirr 6331 | Given a partial ordering, ... |
| predfrirr 6332 | Given a well-founded relat... |
| pred0 6333 | The predecessor class over... |
| dfse3 6334 | Alternate definition of se... |
| predrelss 6335 | Subset carries from relati... |
| predprc 6336 | The predecessor of a prope... |
| predres 6337 | Predecessor class is unaff... |
| frpomin 6338 | Every nonempty (possibly p... |
| frpomin2 6339 | Every nonempty (possibly p... |
| frpoind 6340 | The principle of well-foun... |
| frpoinsg 6341 | Well-Founded Induction Sch... |
| frpoins2fg 6342 | Well-Founded Induction sch... |
| frpoins2g 6343 | Well-Founded Induction sch... |
| frpoins3g 6344 | Well-Founded Induction sch... |
| tz6.26 6345 | All nonempty subclasses of... |
| tz6.26OLD 6346 | Obsolete proof of ~ tz6.26... |
| tz6.26i 6347 | All nonempty subclasses of... |
| wfi 6348 | The Principle of Well-Orde... |
| wfiOLD 6349 | Obsolete proof of ~ wfi as... |
| wfii 6350 | The Principle of Well-Orde... |
| wfisg 6351 | Well-Ordered Induction Sch... |
| wfisgOLD 6352 | Obsolete proof of ~ wfisg ... |
| wfis 6353 | Well-Ordered Induction Sch... |
| wfis2fg 6354 | Well-Ordered Induction Sch... |
| wfis2fgOLD 6355 | Obsolete proof of ~ wfis2f... |
| wfis2f 6356 | Well-Ordered Induction sch... |
| wfis2g 6357 | Well-Ordered Induction Sch... |
| wfis2 6358 | Well-Ordered Induction sch... |
| wfis3 6359 | Well-Ordered Induction sch... |
| ordeq 6368 | Equality theorem for the o... |
| elong 6369 | An ordinal number is an or... |
| elon 6370 | An ordinal number is an or... |
| eloni 6371 | An ordinal number has the ... |
| elon2 6372 | An ordinal number is an or... |
| limeq 6373 | Equality theorem for the l... |
| ordwe 6374 | Membership well-orders eve... |
| ordtr 6375 | An ordinal class is transi... |
| ordfr 6376 | Membership is well-founded... |
| ordelss 6377 | An element of an ordinal c... |
| trssord 6378 | A transitive subclass of a... |
| ordirr 6379 | No ordinal class is a memb... |
| nordeq 6380 | A member of an ordinal cla... |
| ordn2lp 6381 | An ordinal class cannot be... |
| tz7.5 6382 | A nonempty subclass of an ... |
| ordelord 6383 | An element of an ordinal c... |
| tron 6384 | The class of all ordinal n... |
| ordelon 6385 | An element of an ordinal c... |
| onelon 6386 | An element of an ordinal n... |
| tz7.7 6387 | A transitive class belongs... |
| ordelssne 6388 | For ordinal classes, membe... |
| ordelpss 6389 | For ordinal classes, membe... |
| ordsseleq 6390 | For ordinal classes, inclu... |
| ordin 6391 | The intersection of two or... |
| onin 6392 | The intersection of two or... |
| ordtri3or 6393 | A trichotomy law for ordin... |
| ordtri1 6394 | A trichotomy law for ordin... |
| ontri1 6395 | A trichotomy law for ordin... |
| ordtri2 6396 | A trichotomy law for ordin... |
| ordtri3 6397 | A trichotomy law for ordin... |
| ordtri4 6398 | A trichotomy law for ordin... |
| orddisj 6399 | An ordinal class and its s... |
| onfr 6400 | The ordinal class is well-... |
| onelpss 6401 | Relationship between membe... |
| onsseleq 6402 | Relationship between subse... |
| onelss 6403 | An element of an ordinal n... |
| ordtr1 6404 | Transitive law for ordinal... |
| ordtr2 6405 | Transitive law for ordinal... |
| ordtr3 6406 | Transitive law for ordinal... |
| ontr1 6407 | Transitive law for ordinal... |
| ontr2 6408 | Transitive law for ordinal... |
| onelssex 6409 | Ordinal less than is equiv... |
| ordunidif 6410 | The union of an ordinal st... |
| ordintdif 6411 | If ` B ` is smaller than `... |
| onintss 6412 | If a property is true for ... |
| oneqmini 6413 | A way to show that an ordi... |
| ord0 6414 | The empty set is an ordina... |
| 0elon 6415 | The empty set is an ordina... |
| ord0eln0 6416 | A nonempty ordinal contain... |
| on0eln0 6417 | An ordinal number contains... |
| dflim2 6418 | An alternate definition of... |
| inton 6419 | The intersection of the cl... |
| nlim0 6420 | The empty set is not a lim... |
| limord 6421 | A limit ordinal is ordinal... |
| limuni 6422 | A limit ordinal is its own... |
| limuni2 6423 | The union of a limit ordin... |
| 0ellim 6424 | A limit ordinal contains t... |
| limelon 6425 | A limit ordinal class that... |
| onn0 6426 | The class of all ordinal n... |
| suceq 6427 | Equality of successors. (... |
| elsuci 6428 | Membership in a successor.... |
| elsucg 6429 | Membership in a successor.... |
| elsuc2g 6430 | Variant of membership in a... |
| elsuc 6431 | Membership in a successor.... |
| elsuc2 6432 | Membership in a successor.... |
| nfsuc 6433 | Bound-variable hypothesis ... |
| elelsuc 6434 | Membership in a successor.... |
| sucel 6435 | Membership of a successor ... |
| suc0 6436 | The successor of the empty... |
| sucprc 6437 | A proper class is its own ... |
| unisucs 6438 | The union of the successor... |
| unisucg 6439 | A transitive class is equa... |
| unisuc 6440 | A transitive class is equa... |
| sssucid 6441 | A class is included in its... |
| sucidg 6442 | Part of Proposition 7.23 o... |
| sucid 6443 | A set belongs to its succe... |
| nsuceq0 6444 | No successor is empty. (C... |
| eqelsuc 6445 | A set belongs to the succe... |
| iunsuc 6446 | Inductive definition for t... |
| suctr 6447 | The successor of a transit... |
| trsuc 6448 | A set whose successor belo... |
| trsucss 6449 | A member of the successor ... |
| ordsssuc 6450 | An ordinal is a subset of ... |
| onsssuc 6451 | A subset of an ordinal num... |
| ordsssuc2 6452 | An ordinal subset of an or... |
| onmindif 6453 | When its successor is subt... |
| ordnbtwn 6454 | There is no set between an... |
| onnbtwn 6455 | There is no set between an... |
| sucssel 6456 | A set whose successor is a... |
| orddif 6457 | Ordinal derived from its s... |
| orduniss 6458 | An ordinal class includes ... |
| ordtri2or 6459 | A trichotomy law for ordin... |
| ordtri2or2 6460 | A trichotomy law for ordin... |
| ordtri2or3 6461 | A consequence of total ord... |
| ordelinel 6462 | The intersection of two or... |
| ordssun 6463 | Property of a subclass of ... |
| ordequn 6464 | The maximum (i.e. union) o... |
| ordun 6465 | The maximum (i.e., union) ... |
| onunel 6466 | The union of two ordinals ... |
| ordunisssuc 6467 | A subclass relationship fo... |
| suc11 6468 | The successor operation be... |
| onun2 6469 | The union of two ordinals ... |
| ontr 6470 | An ordinal number is a tra... |
| onunisuc 6471 | An ordinal number is equal... |
| onordi 6472 | An ordinal number is an or... |
| ontrciOLD 6473 | Obsolete version of ~ ontr... |
| onirri 6474 | An ordinal number is not a... |
| oneli 6475 | A member of an ordinal num... |
| onelssi 6476 | A member of an ordinal num... |
| onssneli 6477 | An ordering law for ordina... |
| onssnel2i 6478 | An ordering law for ordina... |
| onelini 6479 | An element of an ordinal n... |
| oneluni 6480 | An ordinal number equals i... |
| onunisuci 6481 | An ordinal number is equal... |
| onsseli 6482 | Subset is equivalent to me... |
| onun2i 6483 | The union of two ordinal n... |
| unizlim 6484 | An ordinal equal to its ow... |
| on0eqel 6485 | An ordinal number either e... |
| snsn0non 6486 | The singleton of the singl... |
| onxpdisj 6487 | Ordinal numbers and ordere... |
| onnev 6488 | The class of ordinal numbe... |
| onnevOLD 6489 | Obsolete version of ~ onne... |
| iotajust 6491 | Soundness justification th... |
| dfiota2 6493 | Alternate definition for d... |
| nfiota1 6494 | Bound-variable hypothesis ... |
| nfiotadw 6495 | Deduction version of ~ nfi... |
| nfiotaw 6496 | Bound-variable hypothesis ... |
| nfiotad 6497 | Deduction version of ~ nfi... |
| nfiota 6498 | Bound-variable hypothesis ... |
| cbviotaw 6499 | Change bound variables in ... |
| cbviotavw 6500 | Change bound variables in ... |
| cbviotavwOLD 6501 | Obsolete version of ~ cbvi... |
| cbviota 6502 | Change bound variables in ... |
| cbviotav 6503 | Change bound variables in ... |
| sb8iota 6504 | Variable substitution in d... |
| iotaeq 6505 | Equality theorem for descr... |
| iotabi 6506 | Equivalence theorem for de... |
| uniabio 6507 | Part of Theorem 8.17 in [Q... |
| iotaval2 6508 | Version of ~ iotaval using... |
| iotauni2 6509 | Version of ~ iotauni using... |
| iotanul2 6510 | Version of ~ iotanul using... |
| iotaval 6511 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6512 | The ` iota ` class is a su... |
| iotaex 6513 | Theorem 8.23 in [Quine] p.... |
| iotavalOLD 6514 | Obsolete version of ~ iota... |
| iotauni 6515 | Equivalence between two di... |
| iotaint 6516 | Equivalence between two di... |
| iota1 6517 | Property of iota. (Contri... |
| iotanul 6518 | Theorem 8.22 in [Quine] p.... |
| iotassuniOLD 6519 | Obsolete version of ~ iota... |
| iotaexOLD 6520 | Obsolete version of ~ iota... |
| iota4 6521 | Theorem *14.22 in [Whitehe... |
| iota4an 6522 | Theorem *14.23 in [Whitehe... |
| iota5 6523 | A method for computing iot... |
| iotabidv 6524 | Formula-building deduction... |
| iotabii 6525 | Formula-building deduction... |
| iotacl 6526 | Membership law for descrip... |
| iota2df 6527 | A condition that allows to... |
| iota2d 6528 | A condition that allows to... |
| iota2 6529 | The unique element such th... |
| iotan0 6530 | Representation of "the uni... |
| sniota 6531 | A class abstraction with a... |
| dfiota4 6532 | The ` iota ` operation usi... |
| csbiota 6533 | Class substitution within ... |
| dffun2 6550 | Alternate definition of a ... |
| dffun2OLD 6551 | Obsolete version of ~ dffu... |
| dffun2OLDOLD 6552 | Obsolete version of ~ dffu... |
| dffun6 6553 | Alternate definition of a ... |
| dffun3 6554 | Alternate definition of fu... |
| dffun3OLD 6555 | Obsolete version of ~ dffu... |
| dffun4 6556 | Alternate definition of a ... |
| dffun5 6557 | Alternate definition of fu... |
| dffun6f 6558 | Definition of function, us... |
| dffun6OLD 6559 | Obsolete version of ~ dffu... |
| funmo 6560 | A function has at most one... |
| funmoOLD 6561 | Obsolete version of ~ funm... |
| funrel 6562 | A function is a relation. ... |
| 0nelfun 6563 | A function does not contai... |
| funss 6564 | Subclass theorem for funct... |
| funeq 6565 | Equality theorem for funct... |
| funeqi 6566 | Equality inference for the... |
| funeqd 6567 | Equality deduction for the... |
| nffun 6568 | Bound-variable hypothesis ... |
| sbcfung 6569 | Distribute proper substitu... |
| funeu 6570 | There is exactly one value... |
| funeu2 6571 | There is exactly one value... |
| dffun7 6572 | Alternate definition of a ... |
| dffun8 6573 | Alternate definition of a ... |
| dffun9 6574 | Alternate definition of a ... |
| funfn 6575 | A class is a function if a... |
| funfnd 6576 | A function is a function o... |
| funi 6577 | The identity relation is a... |
| nfunv 6578 | The universal class is not... |
| funopg 6579 | A Kuratowski ordered pair ... |
| funopab 6580 | A class of ordered pairs i... |
| funopabeq 6581 | A class of ordered pairs o... |
| funopab4 6582 | A class of ordered pairs o... |
| funmpt 6583 | A function in maps-to nota... |
| funmpt2 6584 | Functionality of a class g... |
| funco 6585 | The composition of two fun... |
| funresfunco 6586 | Composition of two functio... |
| funres 6587 | A restriction of a functio... |
| funresd 6588 | A restriction of a functio... |
| funssres 6589 | The restriction of a funct... |
| fun2ssres 6590 | Equality of restrictions o... |
| funun 6591 | The union of functions wit... |
| fununmo 6592 | If the union of classes is... |
| fununfun 6593 | If the union of classes is... |
| fundif 6594 | A function with removed el... |
| funcnvsn 6595 | The converse singleton of ... |
| funsng 6596 | A singleton of an ordered ... |
| fnsng 6597 | Functionality and domain o... |
| funsn 6598 | A singleton of an ordered ... |
| funprg 6599 | A set of two pairs is a fu... |
| funtpg 6600 | A set of three pairs is a ... |
| funpr 6601 | A function with a domain o... |
| funtp 6602 | A function with a domain o... |
| fnsn 6603 | Functionality and domain o... |
| fnprg 6604 | Function with a domain of ... |
| fntpg 6605 | Function with a domain of ... |
| fntp 6606 | A function with a domain o... |
| funcnvpr 6607 | The converse pair of order... |
| funcnvtp 6608 | The converse triple of ord... |
| funcnvqp 6609 | The converse quadruple of ... |
| fun0 6610 | The empty set is a functio... |
| funcnv0 6611 | The converse of the empty ... |
| funcnvcnv 6612 | The double converse of a f... |
| funcnv2 6613 | A simpler equivalence for ... |
| funcnv 6614 | The converse of a class is... |
| funcnv3 6615 | A condition showing a clas... |
| fun2cnv 6616 | The double converse of a c... |
| svrelfun 6617 | A single-valued relation i... |
| fncnv 6618 | Single-rootedness (see ~ f... |
| fun11 6619 | Two ways of stating that `... |
| fununi 6620 | The union of a chain (with... |
| funin 6621 | The intersection with a fu... |
| funres11 6622 | The restriction of a one-t... |
| funcnvres 6623 | The converse of a restrict... |
| cnvresid 6624 | Converse of a restricted i... |
| funcnvres2 6625 | The converse of a restrict... |
| funimacnv 6626 | The image of the preimage ... |
| funimass1 6627 | A kind of contraposition l... |
| funimass2 6628 | A kind of contraposition l... |
| imadif 6629 | The image of a difference ... |
| imain 6630 | The image of an intersecti... |
| funimaexg 6631 | Axiom of Replacement using... |
| funimaexgOLD 6632 | Obsolete version of ~ funi... |
| funimaex 6633 | The image of a set under a... |
| isarep1 6634 | Part of a study of the Axi... |
| isarep1OLD 6635 | Obsolete version of ~ isar... |
| isarep2 6636 | Part of a study of the Axi... |
| fneq1 6637 | Equality theorem for funct... |
| fneq2 6638 | Equality theorem for funct... |
| fneq1d 6639 | Equality deduction for fun... |
| fneq2d 6640 | Equality deduction for fun... |
| fneq12d 6641 | Equality deduction for fun... |
| fneq12 6642 | Equality theorem for funct... |
| fneq1i 6643 | Equality inference for fun... |
| fneq2i 6644 | Equality inference for fun... |
| nffn 6645 | Bound-variable hypothesis ... |
| fnfun 6646 | A function with domain is ... |
| fnfund 6647 | A function with domain is ... |
| fnrel 6648 | A function with domain is ... |
| fndm 6649 | The domain of a function. ... |
| fndmi 6650 | The domain of a function. ... |
| fndmd 6651 | The domain of a function. ... |
| funfni 6652 | Inference to convert a fun... |
| fndmu 6653 | A function has a unique do... |
| fnbr 6654 | The first argument of bina... |
| fnop 6655 | The first argument of an o... |
| fneu 6656 | There is exactly one value... |
| fneu2 6657 | There is exactly one value... |
| fnunres1 6658 | Restriction of a disjoint ... |
| fnunres2 6659 | Restriction of a disjoint ... |
| fnun 6660 | The union of two functions... |
| fnund 6661 | The union of two functions... |
| fnunop 6662 | Extension of a function wi... |
| fncofn 6663 | Composition of a function ... |
| fnco 6664 | Composition of two functio... |
| fncoOLD 6665 | Obsolete version of ~ fnco... |
| fnresdm 6666 | A function does not change... |
| fnresdisj 6667 | A function restricted to a... |
| 2elresin 6668 | Membership in two function... |
| fnssresb 6669 | Restriction of a function ... |
| fnssres 6670 | Restriction of a function ... |
| fnssresd 6671 | Restriction of a function ... |
| fnresin1 6672 | Restriction of a function'... |
| fnresin2 6673 | Restriction of a function'... |
| fnres 6674 | An equivalence for functio... |
| idfn 6675 | The identity relation is a... |
| fnresi 6676 | The restricted identity re... |
| fnima 6677 | The image of a function's ... |
| fn0 6678 | A function with empty doma... |
| fnimadisj 6679 | A class that is disjoint w... |
| fnimaeq0 6680 | Images under a function ne... |
| dfmpt3 6681 | Alternate definition for t... |
| mptfnf 6682 | The maps-to notation defin... |
| fnmptf 6683 | The maps-to notation defin... |
| fnopabg 6684 | Functionality and domain o... |
| fnopab 6685 | Functionality and domain o... |
| mptfng 6686 | The maps-to notation defin... |
| fnmpt 6687 | The maps-to notation defin... |
| fnmptd 6688 | The maps-to notation defin... |
| mpt0 6689 | A mapping operation with e... |
| fnmpti 6690 | Functionality and domain o... |
| dmmpti 6691 | Domain of the mapping oper... |
| dmmptd 6692 | The domain of the mapping ... |
| mptun 6693 | Union of mappings which ar... |
| partfun 6694 | Rewrite a function defined... |
| feq1 6695 | Equality theorem for funct... |
| feq2 6696 | Equality theorem for funct... |
| feq3 6697 | Equality theorem for funct... |
| feq23 6698 | Equality theorem for funct... |
| feq1d 6699 | Equality deduction for fun... |
| feq2d 6700 | Equality deduction for fun... |
| feq3d 6701 | Equality deduction for fun... |
| feq12d 6702 | Equality deduction for fun... |
| feq123d 6703 | Equality deduction for fun... |
| feq123 6704 | Equality theorem for funct... |
| feq1i 6705 | Equality inference for fun... |
| feq2i 6706 | Equality inference for fun... |
| feq12i 6707 | Equality inference for fun... |
| feq23i 6708 | Equality inference for fun... |
| feq23d 6709 | Equality deduction for fun... |
| nff 6710 | Bound-variable hypothesis ... |
| sbcfng 6711 | Distribute proper substitu... |
| sbcfg 6712 | Distribute proper substitu... |
| elimf 6713 | Eliminate a mapping hypoth... |
| ffn 6714 | A mapping is a function wi... |
| ffnd 6715 | A mapping is a function wi... |
| dffn2 6716 | Any function is a mapping ... |
| ffun 6717 | A mapping is a function. ... |
| ffund 6718 | A mapping is a function, d... |
| frel 6719 | A mapping is a relation. ... |
| freld 6720 | A mapping is a relation. ... |
| frn 6721 | The range of a mapping. (... |
| frnd 6722 | Deduction form of ~ frn . ... |
| fdm 6723 | The domain of a mapping. ... |
| fdmOLD 6724 | Obsolete version of ~ fdm ... |
| fdmd 6725 | Deduction form of ~ fdm . ... |
| fdmi 6726 | Inference associated with ... |
| dffn3 6727 | A function maps to its ran... |
| ffrn 6728 | A function maps to its ran... |
| ffrnb 6729 | Characterization of a func... |
| ffrnbd 6730 | A function maps to its ran... |
| fss 6731 | Expanding the codomain of ... |
| fssd 6732 | Expanding the codomain of ... |
| fssdmd 6733 | Expressing that a class is... |
| fssdm 6734 | Expressing that a class is... |
| fimass 6735 | The image of a class under... |
| fimacnv 6736 | The preimage of the codoma... |
| fcof 6737 | Composition of a function ... |
| fco 6738 | Composition of two functio... |
| fcoOLD 6739 | Obsolete version of ~ fco ... |
| fcod 6740 | Composition of two mapping... |
| fco2 6741 | Functionality of a composi... |
| fssxp 6742 | A mapping is a class of or... |
| funssxp 6743 | Two ways of specifying a p... |
| ffdm 6744 | A mapping is a partial fun... |
| ffdmd 6745 | The domain of a function. ... |
| fdmrn 6746 | A different way to write `... |
| funcofd 6747 | Composition of two functio... |
| fco3OLD 6748 | Obsolete version of ~ func... |
| opelf 6749 | The members of an ordered ... |
| fun 6750 | The union of two functions... |
| fun2 6751 | The union of two functions... |
| fun2d 6752 | The union of functions wit... |
| fnfco 6753 | Composition of two functio... |
| fssres 6754 | Restriction of a function ... |
| fssresd 6755 | Restriction of a function ... |
| fssres2 6756 | Restriction of a restricte... |
| fresin 6757 | An identity for the mappin... |
| resasplit 6758 | If two functions agree on ... |
| fresaun 6759 | The union of two functions... |
| fresaunres2 6760 | From the union of two func... |
| fresaunres1 6761 | From the union of two func... |
| fcoi1 6762 | Composition of a mapping a... |
| fcoi2 6763 | Composition of restricted ... |
| feu 6764 | There is exactly one value... |
| fcnvres 6765 | The converse of a restrict... |
| fimacnvdisj 6766 | The preimage of a class di... |
| fint 6767 | Function into an intersect... |
| fin 6768 | Mapping into an intersecti... |
| f0 6769 | The empty function. (Cont... |
| f00 6770 | A class is a function with... |
| f0bi 6771 | A function with empty doma... |
| f0dom0 6772 | A function is empty iff it... |
| f0rn0 6773 | If there is no element in ... |
| fconst 6774 | A Cartesian product with a... |
| fconstg 6775 | A Cartesian product with a... |
| fnconstg 6776 | A Cartesian product with a... |
| fconst6g 6777 | Constant function with loo... |
| fconst6 6778 | A constant function as a m... |
| f1eq1 6779 | Equality theorem for one-t... |
| f1eq2 6780 | Equality theorem for one-t... |
| f1eq3 6781 | Equality theorem for one-t... |
| nff1 6782 | Bound-variable hypothesis ... |
| dff12 6783 | Alternate definition of a ... |
| f1f 6784 | A one-to-one mapping is a ... |
| f1fn 6785 | A one-to-one mapping is a ... |
| f1fun 6786 | A one-to-one mapping is a ... |
| f1rel 6787 | A one-to-one onto mapping ... |
| f1dm 6788 | The domain of a one-to-one... |
| f1dmOLD 6789 | Obsolete version of ~ f1dm... |
| f1ss 6790 | A function that is one-to-... |
| f1ssr 6791 | A function that is one-to-... |
| f1ssres 6792 | A function that is one-to-... |
| f1resf1 6793 | The restriction of an inje... |
| f1cnvcnv 6794 | Two ways to express that a... |
| f1cof1 6795 | Composition of two one-to-... |
| f1co 6796 | Composition of one-to-one ... |
| f1coOLD 6797 | Obsolete version of ~ f1co... |
| foeq1 6798 | Equality theorem for onto ... |
| foeq2 6799 | Equality theorem for onto ... |
| foeq3 6800 | Equality theorem for onto ... |
| nffo 6801 | Bound-variable hypothesis ... |
| fof 6802 | An onto mapping is a mappi... |
| fofun 6803 | An onto mapping is a funct... |
| fofn 6804 | An onto mapping is a funct... |
| forn 6805 | The codomain of an onto fu... |
| dffo2 6806 | Alternate definition of an... |
| foima 6807 | The image of the domain of... |
| dffn4 6808 | A function maps onto its r... |
| funforn 6809 | A function maps its domain... |
| fodmrnu 6810 | An onto function has uniqu... |
| fimadmfo 6811 | A function is a function o... |
| fores 6812 | Restriction of an onto fun... |
| fimadmfoALT 6813 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6814 | The preimage of the codoma... |
| focofo 6815 | Composition of onto functi... |
| foco 6816 | Composition of onto functi... |
| foconst 6817 | A nonzero constant functio... |
| f1oeq1 6818 | Equality theorem for one-t... |
| f1oeq2 6819 | Equality theorem for one-t... |
| f1oeq3 6820 | Equality theorem for one-t... |
| f1oeq23 6821 | Equality theorem for one-t... |
| f1eq123d 6822 | Equality deduction for one... |
| foeq123d 6823 | Equality deduction for ont... |
| f1oeq123d 6824 | Equality deduction for one... |
| f1oeq1d 6825 | Equality deduction for one... |
| f1oeq2d 6826 | Equality deduction for one... |
| f1oeq3d 6827 | Equality deduction for one... |
| nff1o 6828 | Bound-variable hypothesis ... |
| f1of1 6829 | A one-to-one onto mapping ... |
| f1of 6830 | A one-to-one onto mapping ... |
| f1ofn 6831 | A one-to-one onto mapping ... |
| f1ofun 6832 | A one-to-one onto mapping ... |
| f1orel 6833 | A one-to-one onto mapping ... |
| f1odm 6834 | The domain of a one-to-one... |
| dff1o2 6835 | Alternate definition of on... |
| dff1o3 6836 | Alternate definition of on... |
| f1ofo 6837 | A one-to-one onto function... |
| dff1o4 6838 | Alternate definition of on... |
| dff1o5 6839 | Alternate definition of on... |
| f1orn 6840 | A one-to-one function maps... |
| f1f1orn 6841 | A one-to-one function maps... |
| f1ocnv 6842 | The converse of a one-to-o... |
| f1ocnvb 6843 | A relation is a one-to-one... |
| f1ores 6844 | The restriction of a one-t... |
| f1orescnv 6845 | The converse of a one-to-o... |
| f1imacnv 6846 | Preimage of an image. (Co... |
| foimacnv 6847 | A reverse version of ~ f1i... |
| foun 6848 | The union of two onto func... |
| f1oun 6849 | The union of two one-to-on... |
| f1un 6850 | The union of two one-to-on... |
| resdif 6851 | The restriction of a one-t... |
| resin 6852 | The restriction of a one-t... |
| f1oco 6853 | Composition of one-to-one ... |
| f1cnv 6854 | The converse of an injecti... |
| funcocnv2 6855 | Composition with the conve... |
| fococnv2 6856 | The composition of an onto... |
| f1ococnv2 6857 | The composition of a one-t... |
| f1cocnv2 6858 | Composition of an injectiv... |
| f1ococnv1 6859 | The composition of a one-t... |
| f1cocnv1 6860 | Composition of an injectiv... |
| funcoeqres 6861 | Express a constraint on a ... |
| f1ssf1 6862 | A subset of an injective f... |
| f10 6863 | The empty set maps one-to-... |
| f10d 6864 | The empty set maps one-to-... |
| f1o00 6865 | One-to-one onto mapping of... |
| fo00 6866 | Onto mapping of the empty ... |
| f1o0 6867 | One-to-one onto mapping of... |
| f1oi 6868 | A restriction of the ident... |
| f1ovi 6869 | The identity relation is a... |
| f1osn 6870 | A singleton of an ordered ... |
| f1osng 6871 | A singleton of an ordered ... |
| f1sng 6872 | A singleton of an ordered ... |
| fsnd 6873 | A singleton of an ordered ... |
| f1oprswap 6874 | A two-element swap is a bi... |
| f1oprg 6875 | An unordered pair of order... |
| tz6.12-2 6876 | Function value when ` F ` ... |
| fveu 6877 | The value of a function at... |
| brprcneu 6878 | If ` A ` is a proper class... |
| brprcneuALT 6879 | Alternate proof of ~ brprc... |
| fvprc 6880 | A function's value at a pr... |
| fvprcALT 6881 | Alternate proof of ~ fvprc... |
| rnfvprc 6882 | The range of a function va... |
| fv2 6883 | Alternate definition of fu... |
| dffv3 6884 | A definition of function v... |
| dffv4 6885 | The previous definition of... |
| elfv 6886 | Membership in a function v... |
| fveq1 6887 | Equality theorem for funct... |
| fveq2 6888 | Equality theorem for funct... |
| fveq1i 6889 | Equality inference for fun... |
| fveq1d 6890 | Equality deduction for fun... |
| fveq2i 6891 | Equality inference for fun... |
| fveq2d 6892 | Equality deduction for fun... |
| 2fveq3 6893 | Equality theorem for neste... |
| fveq12i 6894 | Equality deduction for fun... |
| fveq12d 6895 | Equality deduction for fun... |
| fveqeq2d 6896 | Equality deduction for fun... |
| fveqeq2 6897 | Equality deduction for fun... |
| nffv 6898 | Bound-variable hypothesis ... |
| nffvmpt1 6899 | Bound-variable hypothesis ... |
| nffvd 6900 | Deduction version of bound... |
| fvex 6901 | The value of a class exist... |
| fvexi 6902 | The value of a class exist... |
| fvexd 6903 | The value of a class exist... |
| fvif 6904 | Move a conditional outside... |
| iffv 6905 | Move a conditional outside... |
| fv3 6906 | Alternate definition of th... |
| fvres 6907 | The value of a restricted ... |
| fvresd 6908 | The value of a restricted ... |
| funssfv 6909 | The value of a member of t... |
| tz6.12c 6910 | Corollary of Theorem 6.12(... |
| tz6.12-1 6911 | Function value. Theorem 6... |
| tz6.12-1OLD 6912 | Obsolete version of ~ tz6.... |
| tz6.12 6913 | Function value. Theorem 6... |
| tz6.12f 6914 | Function value, using boun... |
| tz6.12cOLD 6915 | Obsolete version of ~ tz6.... |
| tz6.12i 6916 | Corollary of Theorem 6.12(... |
| fvbr0 6917 | Two possibilities for the ... |
| fvrn0 6918 | A function value is a memb... |
| fvn0fvelrn 6919 | If the value of a function... |
| elfvunirn 6920 | A function value is a subs... |
| fvssunirn 6921 | The result of a function v... |
| fvssunirnOLD 6922 | Obsolete version of ~ fvss... |
| ndmfv 6923 | The value of a class outsi... |
| ndmfvrcl 6924 | Reverse closure law for fu... |
| elfvdm 6925 | If a function value has a ... |
| elfvex 6926 | If a function value has a ... |
| elfvexd 6927 | If a function value has a ... |
| eliman0 6928 | A nonempty function value ... |
| nfvres 6929 | The value of a non-member ... |
| nfunsn 6930 | If the restriction of a cl... |
| fvfundmfvn0 6931 | If the "value of a class" ... |
| 0fv 6932 | Function value of the empt... |
| fv2prc 6933 | A function value of a func... |
| elfv2ex 6934 | If a function value of a f... |
| fveqres 6935 | Equal values imply equal v... |
| csbfv12 6936 | Move class substitution in... |
| csbfv2g 6937 | Move class substitution in... |
| csbfv 6938 | Substitution for a functio... |
| funbrfv 6939 | The second argument of a b... |
| funopfv 6940 | The second element in an o... |
| fnbrfvb 6941 | Equivalence of function va... |
| fnopfvb 6942 | Equivalence of function va... |
| funbrfvb 6943 | Equivalence of function va... |
| funopfvb 6944 | Equivalence of function va... |
| fnbrfvb2 6945 | Version of ~ fnbrfvb for f... |
| funbrfv2b 6946 | Function value in terms of... |
| dffn5 6947 | Representation of a functi... |
| fnrnfv 6948 | The range of a function ex... |
| fvelrnb 6949 | A member of a function's r... |
| foelcdmi 6950 | A member of a surjective f... |
| dfimafn 6951 | Alternate definition of th... |
| dfimafn2 6952 | Alternate definition of th... |
| funimass4 6953 | Membership relation for th... |
| fvelima 6954 | Function value in an image... |
| fvelimad 6955 | Function value in an image... |
| feqmptd 6956 | Deduction form of ~ dffn5 ... |
| feqresmpt 6957 | Express a restricted funct... |
| feqmptdf 6958 | Deduction form of ~ dffn5f... |
| dffn5f 6959 | Representation of a functi... |
| fvelimab 6960 | Function value in an image... |
| fvelimabd 6961 | Deduction form of ~ fvelim... |
| unima 6962 | Image of a union. (Contri... |
| fvi 6963 | The value of the identity ... |
| fviss 6964 | The value of the identity ... |
| fniinfv 6965 | The indexed intersection o... |
| fnsnfv 6966 | Singleton of function valu... |
| fnsnfvOLD 6967 | Obsolete version of ~ fnsn... |
| opabiotafun 6968 | Define a function whose va... |
| opabiotadm 6969 | Define a function whose va... |
| opabiota 6970 | Define a function whose va... |
| fnimapr 6971 | The image of a pair under ... |
| ssimaex 6972 | The existence of a subimag... |
| ssimaexg 6973 | The existence of a subimag... |
| funfv 6974 | A simplified expression fo... |
| funfv2 6975 | The value of a function. ... |
| funfv2f 6976 | The value of a function. ... |
| fvun 6977 | Value of the union of two ... |
| fvun1 6978 | The value of a union when ... |
| fvun2 6979 | The value of a union when ... |
| fvun1d 6980 | The value of a union when ... |
| fvun2d 6981 | The value of a union when ... |
| dffv2 6982 | Alternate definition of fu... |
| dmfco 6983 | Domains of a function comp... |
| fvco2 6984 | Value of a function compos... |
| fvco 6985 | Value of a function compos... |
| fvco3 6986 | Value of a function compos... |
| fvco3d 6987 | Value of a function compos... |
| fvco4i 6988 | Conditions for a compositi... |
| fvopab3g 6989 | Value of a function given ... |
| fvopab3ig 6990 | Value of a function given ... |
| brfvopabrbr 6991 | The binary relation of a f... |
| fvmptg 6992 | Value of a function given ... |
| fvmpti 6993 | Value of a function given ... |
| fvmpt 6994 | Value of a function given ... |
| fvmpt2f 6995 | Value of a function given ... |
| fvtresfn 6996 | Functionality of a tuple-r... |
| fvmpts 6997 | Value of a function given ... |
| fvmpt3 6998 | Value of a function given ... |
| fvmpt3i 6999 | Value of a function given ... |
| fvmptdf 7000 | Deduction version of ~ fvm... |
| fvmptd 7001 | Deduction version of ~ fvm... |
| fvmptd2 7002 | Deduction version of ~ fvm... |
| mptrcl 7003 | Reverse closure for a mapp... |
| fvmpt2i 7004 | Value of a function given ... |
| fvmpt2 7005 | Value of a function given ... |
| fvmptss 7006 | If all the values of the m... |
| fvmpt2d 7007 | Deduction version of ~ fvm... |
| fvmptex 7008 | Express a function ` F ` w... |
| fvmptd3f 7009 | Alternate deduction versio... |
| fvmptd2f 7010 | Alternate deduction versio... |
| fvmptdv 7011 | Alternate deduction versio... |
| fvmptdv2 7012 | Alternate deduction versio... |
| mpteqb 7013 | Bidirectional equality the... |
| fvmptt 7014 | Closed theorem form of ~ f... |
| fvmptf 7015 | Value of a function given ... |
| fvmptnf 7016 | The value of a function gi... |
| fvmptd3 7017 | Deduction version of ~ fvm... |
| fvmptn 7018 | This somewhat non-intuitiv... |
| fvmptss2 7019 | A mapping always evaluates... |
| elfvmptrab1w 7020 | Implications for the value... |
| elfvmptrab1 7021 | Implications for the value... |
| elfvmptrab 7022 | Implications for the value... |
| fvopab4ndm 7023 | Value of a function given ... |
| fvmptndm 7024 | Value of a function given ... |
| fvmptrabfv 7025 | Value of a function mappin... |
| fvopab5 7026 | The value of a function th... |
| fvopab6 7027 | Value of a function given ... |
| eqfnfv 7028 | Equality of functions is d... |
| eqfnfv2 7029 | Equality of functions is d... |
| eqfnfv3 7030 | Derive equality of functio... |
| eqfnfvd 7031 | Deduction for equality of ... |
| eqfnfv2f 7032 | Equality of functions is d... |
| eqfunfv 7033 | Equality of functions is d... |
| eqfnun 7034 | Two functions on ` A u. B ... |
| fvreseq0 7035 | Equality of restricted fun... |
| fvreseq1 7036 | Equality of a function res... |
| fvreseq 7037 | Equality of restricted fun... |
| fnmptfvd 7038 | A function with a given do... |
| fndmdif 7039 | Two ways to express the lo... |
| fndmdifcom 7040 | The difference set between... |
| fndmdifeq0 7041 | The difference set of two ... |
| fndmin 7042 | Two ways to express the lo... |
| fneqeql 7043 | Two functions are equal if... |
| fneqeql2 7044 | Two functions are equal if... |
| fnreseql 7045 | Two functions are equal on... |
| chfnrn 7046 | The range of a choice func... |
| funfvop 7047 | Ordered pair with function... |
| funfvbrb 7048 | Two ways to say that ` A `... |
| fvimacnvi 7049 | A member of a preimage is ... |
| fvimacnv 7050 | The argument of a function... |
| funimass3 7051 | A kind of contraposition l... |
| funimass5 7052 | A subclass of a preimage i... |
| funconstss 7053 | Two ways of specifying tha... |
| fvimacnvALT 7054 | Alternate proof of ~ fvima... |
| elpreima 7055 | Membership in the preimage... |
| elpreimad 7056 | Membership in the preimage... |
| fniniseg 7057 | Membership in the preimage... |
| fncnvima2 7058 | Inverse images under funct... |
| fniniseg2 7059 | Inverse point images under... |
| unpreima 7060 | Preimage of a union. (Con... |
| inpreima 7061 | Preimage of an intersectio... |
| difpreima 7062 | Preimage of a difference. ... |
| respreima 7063 | The preimage of a restrict... |
| cnvimainrn 7064 | The preimage of the inters... |
| sspreima 7065 | The preimage of a subset i... |
| iinpreima 7066 | Preimage of an intersectio... |
| intpreima 7067 | Preimage of an intersectio... |
| fimacnvOLD 7068 | Obsolete version of ~ fima... |
| fimacnvinrn 7069 | Taking the converse image ... |
| fimacnvinrn2 7070 | Taking the converse image ... |
| rescnvimafod 7071 | The restriction of a funct... |
| fvn0ssdmfun 7072 | If a class' function value... |
| fnopfv 7073 | Ordered pair with function... |
| fvelrn 7074 | A function's value belongs... |
| nelrnfvne 7075 | A function value cannot be... |
| fveqdmss 7076 | If the empty set is not co... |
| fveqressseq 7077 | If the empty set is not co... |
| fnfvelrn 7078 | A function's value belongs... |
| ffvelcdm 7079 | A function's value belongs... |
| fnfvelrnd 7080 | A function's value belongs... |
| ffvelcdmi 7081 | A function's value belongs... |
| ffvelcdmda 7082 | A function's value belongs... |
| ffvelcdmd 7083 | A function's value belongs... |
| rexrn 7084 | Restricted existential qua... |
| ralrn 7085 | Restricted universal quant... |
| elrnrexdm 7086 | For any element in the ran... |
| elrnrexdmb 7087 | For any element in the ran... |
| eldmrexrn 7088 | For any element in the dom... |
| eldmrexrnb 7089 | For any element in the dom... |
| fvcofneq 7090 | The values of two function... |
| ralrnmptw 7091 | A restricted quantifier ov... |
| rexrnmptw 7092 | A restricted quantifier ov... |
| ralrnmpt 7093 | A restricted quantifier ov... |
| rexrnmpt 7094 | A restricted quantifier ov... |
| f0cli 7095 | Unconditional closure of a... |
| dff2 7096 | Alternate definition of a ... |
| dff3 7097 | Alternate definition of a ... |
| dff4 7098 | Alternate definition of a ... |
| dffo3 7099 | An onto mapping expressed ... |
| dffo4 7100 | Alternate definition of an... |
| dffo5 7101 | Alternate definition of an... |
| exfo 7102 | A relation equivalent to t... |
| foelrn 7103 | Property of a surjective f... |
| foco2 7104 | If a composition of two fu... |
| fmpt 7105 | Functionality of the mappi... |
| f1ompt 7106 | Express bijection for a ma... |
| fmpti 7107 | Functionality of the mappi... |
| fvmptelcdm 7108 | The value of a function at... |
| fmptd 7109 | Domain and codomain of the... |
| fmpttd 7110 | Version of ~ fmptd with in... |
| fmpt3d 7111 | Domain and codomain of the... |
| fmptdf 7112 | A version of ~ fmptd using... |
| ffnfv 7113 | A function maps to a class... |
| ffnfvf 7114 | A function maps to a class... |
| fnfvrnss 7115 | An upper bound for range d... |
| fcdmssb 7116 | A function is a function i... |
| rnmptss 7117 | The range of an operation ... |
| fmpt2d 7118 | Domain and codomain of the... |
| ffvresb 7119 | A necessary and sufficient... |
| f1oresrab 7120 | Build a bijection between ... |
| f1ossf1o 7121 | Restricting a bijection, w... |
| fmptco 7122 | Composition of two functio... |
| fmptcof 7123 | Version of ~ fmptco where ... |
| fmptcos 7124 | Composition of two functio... |
| cofmpt 7125 | Express composition of a m... |
| fcompt 7126 | Express composition of two... |
| fcoconst 7127 | Composition with a constan... |
| fsn 7128 | A function maps a singleto... |
| fsn2 7129 | A function that maps a sin... |
| fsng 7130 | A function maps a singleto... |
| fsn2g 7131 | A function that maps a sin... |
| xpsng 7132 | The Cartesian product of t... |
| xpprsng 7133 | The Cartesian product of a... |
| xpsn 7134 | The Cartesian product of t... |
| f1o2sn 7135 | A singleton consisting in ... |
| residpr 7136 | Restriction of the identit... |
| dfmpt 7137 | Alternate definition for t... |
| fnasrn 7138 | A function expressed as th... |
| idref 7139 | Two ways to state that a r... |
| funiun 7140 | A function is a union of s... |
| funopsn 7141 | If a function is an ordere... |
| funop 7142 | An ordered pair is a funct... |
| funopdmsn 7143 | The domain of a function w... |
| funsndifnop 7144 | A singleton of an ordered ... |
| funsneqopb 7145 | A singleton of an ordered ... |
| ressnop0 7146 | If ` A ` is not in ` C ` ,... |
| fpr 7147 | A function with a domain o... |
| fprg 7148 | A function with a domain o... |
| ftpg 7149 | A function with a domain o... |
| ftp 7150 | A function with a domain o... |
| fnressn 7151 | A function restricted to a... |
| funressn 7152 | A function restricted to a... |
| fressnfv 7153 | The value of a function re... |
| fvrnressn 7154 | If the value of a function... |
| fvressn 7155 | The value of a function re... |
| fvn0fvelrnOLD 7156 | Obsolete version of ~ fvn0... |
| fvconst 7157 | The value of a constant fu... |
| fnsnr 7158 | If a class belongs to a fu... |
| fnsnb 7159 | A function whose domain is... |
| fmptsn 7160 | Express a singleton functi... |
| fmptsng 7161 | Express a singleton functi... |
| fmptsnd 7162 | Express a singleton functi... |
| fmptap 7163 | Append an additional value... |
| fmptapd 7164 | Append an additional value... |
| fmptpr 7165 | Express a pair function in... |
| fvresi 7166 | The value of a restricted ... |
| fninfp 7167 | Express the class of fixed... |
| fnelfp 7168 | Property of a fixed point ... |
| fndifnfp 7169 | Express the class of non-f... |
| fnelnfp 7170 | Property of a non-fixed po... |
| fnnfpeq0 7171 | A function is the identity... |
| fvunsn 7172 | Remove an ordered pair not... |
| fvsng 7173 | The value of a singleton o... |
| fvsn 7174 | The value of a singleton o... |
| fvsnun1 7175 | The value of a function wi... |
| fvsnun2 7176 | The value of a function wi... |
| fnsnsplit 7177 | Split a function into a si... |
| fsnunf 7178 | Adjoining a point to a fun... |
| fsnunf2 7179 | Adjoining a point to a pun... |
| fsnunfv 7180 | Recover the added point fr... |
| fsnunres 7181 | Recover the original funct... |
| funresdfunsn 7182 | Restricting a function to ... |
| fvpr1g 7183 | The value of a function wi... |
| fvpr2g 7184 | The value of a function wi... |
| fvpr2gOLD 7185 | Obsolete version of ~ fvpr... |
| fvpr1 7186 | The value of a function wi... |
| fvpr1OLD 7187 | Obsolete version of ~ fvpr... |
| fvpr2 7188 | The value of a function wi... |
| fvpr2OLD 7189 | Obsolete version of ~ fvpr... |
| fprb 7190 | A condition for functionho... |
| fvtp1 7191 | The first value of a funct... |
| fvtp2 7192 | The second value of a func... |
| fvtp3 7193 | The third value of a funct... |
| fvtp1g 7194 | The value of a function wi... |
| fvtp2g 7195 | The value of a function wi... |
| fvtp3g 7196 | The value of a function wi... |
| tpres 7197 | An unordered triple of ord... |
| fvconst2g 7198 | The value of a constant fu... |
| fconst2g 7199 | A constant function expres... |
| fvconst2 7200 | The value of a constant fu... |
| fconst2 7201 | A constant function expres... |
| fconst5 7202 | Two ways to express that a... |
| rnmptc 7203 | Range of a constant functi... |
| rnmptcOLD 7204 | Obsolete version of ~ rnmp... |
| fnprb 7205 | A function whose domain ha... |
| fntpb 7206 | A function whose domain ha... |
| fnpr2g 7207 | A function whose domain ha... |
| fpr2g 7208 | A function that maps a pai... |
| fconstfv 7209 | A constant function expres... |
| fconst3 7210 | Two ways to express a cons... |
| fconst4 7211 | Two ways to express a cons... |
| resfunexg 7212 | The restriction of a funct... |
| resiexd 7213 | The restriction of the ide... |
| fnex 7214 | If the domain of a functio... |
| fnexd 7215 | If the domain of a functio... |
| funex 7216 | If the domain of a functio... |
| opabex 7217 | Existence of a function ex... |
| mptexg 7218 | If the domain of a functio... |
| mptexgf 7219 | If the domain of a functio... |
| mptex 7220 | If the domain of a functio... |
| mptexd 7221 | If the domain of a functio... |
| mptrabex 7222 | If the domain of a functio... |
| fex 7223 | If the domain of a mapping... |
| fexd 7224 | If the domain of a mapping... |
| mptfvmpt 7225 | A function in maps-to nota... |
| eufnfv 7226 | A function is uniquely det... |
| funfvima 7227 | A function's value in a pr... |
| funfvima2 7228 | A function's value in an i... |
| funfvima2d 7229 | A function's value in a pr... |
| fnfvima 7230 | The function value of an o... |
| fnfvimad 7231 | A function's value belongs... |
| resfvresima 7232 | The value of the function ... |
| funfvima3 7233 | A class including a functi... |
| rexima 7234 | Existential quantification... |
| ralima 7235 | Universal quantification u... |
| fvclss 7236 | Upper bound for the class ... |
| elabrex 7237 | Elementhood in an image se... |
| abrexco 7238 | Composition of two image m... |
| imaiun 7239 | The image of an indexed un... |
| imauni 7240 | The image of a union is th... |
| fniunfv 7241 | The indexed union of a fun... |
| funiunfv 7242 | The indexed union of a fun... |
| funiunfvf 7243 | The indexed union of a fun... |
| eluniima 7244 | Membership in the union of... |
| elunirn 7245 | Membership in the union of... |
| elunirnALT 7246 | Alternate proof of ~ eluni... |
| elunirn2OLD 7247 | Obsolete version of ~ elfv... |
| fnunirn 7248 | Membership in a union of s... |
| dff13 7249 | A one-to-one function in t... |
| dff13f 7250 | A one-to-one function in t... |
| f1veqaeq 7251 | If the values of a one-to-... |
| f1cofveqaeq 7252 | If the values of a composi... |
| f1cofveqaeqALT 7253 | Alternate proof of ~ f1cof... |
| 2f1fvneq 7254 | If two one-to-one function... |
| f1mpt 7255 | Express injection for a ma... |
| f1fveq 7256 | Equality of function value... |
| f1elima 7257 | Membership in the image of... |
| f1imass 7258 | Taking images under a one-... |
| f1imaeq 7259 | Taking images under a one-... |
| f1imapss 7260 | Taking images under a one-... |
| fpropnf1 7261 | A function, given by an un... |
| f1dom3fv3dif 7262 | The function values for a ... |
| f1dom3el3dif 7263 | The codomain of a 1-1 func... |
| dff14a 7264 | A one-to-one function in t... |
| dff14b 7265 | A one-to-one function in t... |
| f12dfv 7266 | A one-to-one function with... |
| f13dfv 7267 | A one-to-one function with... |
| dff1o6 7268 | A one-to-one onto function... |
| f1ocnvfv1 7269 | The converse value of the ... |
| f1ocnvfv2 7270 | The value of the converse ... |
| f1ocnvfv 7271 | Relationship between the v... |
| f1ocnvfvb 7272 | Relationship between the v... |
| nvof1o 7273 | An involution is a bijecti... |
| nvocnv 7274 | The converse of an involut... |
| f1cdmsn 7275 | If a one-to-one function w... |
| fsnex 7276 | Relate a function with a s... |
| f1prex 7277 | Relate a one-to-one functi... |
| f1ocnvdm 7278 | The value of the converse ... |
| f1ocnvfvrneq 7279 | If the values of a one-to-... |
| fcof1 7280 | An application is injectiv... |
| fcofo 7281 | An application is surjecti... |
| cbvfo 7282 | Change bound variable betw... |
| cbvexfo 7283 | Change bound variable betw... |
| cocan1 7284 | An injection is left-cance... |
| cocan2 7285 | A surjection is right-canc... |
| fcof1oinvd 7286 | Show that a function is th... |
| fcof1od 7287 | A function is bijective if... |
| 2fcoidinvd 7288 | Show that a function is th... |
| fcof1o 7289 | Show that two functions ar... |
| 2fvcoidd 7290 | Show that the composition ... |
| 2fvidf1od 7291 | A function is bijective if... |
| 2fvidinvd 7292 | Show that two functions ar... |
| foeqcnvco 7293 | Condition for function equ... |
| f1eqcocnv 7294 | Condition for function equ... |
| f1eqcocnvOLD 7295 | Obsolete version of ~ f1eq... |
| fveqf1o 7296 | Given a bijection ` F ` , ... |
| nf1const 7297 | A constant function from a... |
| nf1oconst 7298 | A constant function from a... |
| f1ofvswap 7299 | Swapping two values in a b... |
| fliftrel 7300 | ` F ` , a function lift, i... |
| fliftel 7301 | Elementhood in the relatio... |
| fliftel1 7302 | Elementhood in the relatio... |
| fliftcnv 7303 | Converse of the relation `... |
| fliftfun 7304 | The function ` F ` is the ... |
| fliftfund 7305 | The function ` F ` is the ... |
| fliftfuns 7306 | The function ` F ` is the ... |
| fliftf 7307 | The domain and range of th... |
| fliftval 7308 | The value of the function ... |
| isoeq1 7309 | Equality theorem for isomo... |
| isoeq2 7310 | Equality theorem for isomo... |
| isoeq3 7311 | Equality theorem for isomo... |
| isoeq4 7312 | Equality theorem for isomo... |
| isoeq5 7313 | Equality theorem for isomo... |
| nfiso 7314 | Bound-variable hypothesis ... |
| isof1o 7315 | An isomorphism is a one-to... |
| isof1oidb 7316 | A function is a bijection ... |
| isof1oopb 7317 | A function is a bijection ... |
| isorel 7318 | An isomorphism connects bi... |
| soisores 7319 | Express the condition of i... |
| soisoi 7320 | Infer isomorphism from one... |
| isoid 7321 | Identity law for isomorphi... |
| isocnv 7322 | Converse law for isomorphi... |
| isocnv2 7323 | Converse law for isomorphi... |
| isocnv3 7324 | Complementation law for is... |
| isores2 7325 | An isomorphism from one we... |
| isores1 7326 | An isomorphism from one we... |
| isores3 7327 | Induced isomorphism on a s... |
| isotr 7328 | Composition (transitive) l... |
| isomin 7329 | Isomorphisms preserve mini... |
| isoini 7330 | Isomorphisms preserve init... |
| isoini2 7331 | Isomorphisms are isomorphi... |
| isofrlem 7332 | Lemma for ~ isofr . (Cont... |
| isoselem 7333 | Lemma for ~ isose . (Cont... |
| isofr 7334 | An isomorphism preserves w... |
| isose 7335 | An isomorphism preserves s... |
| isofr2 7336 | A weak form of ~ isofr tha... |
| isopolem 7337 | Lemma for ~ isopo . (Cont... |
| isopo 7338 | An isomorphism preserves t... |
| isosolem 7339 | Lemma for ~ isoso . (Cont... |
| isoso 7340 | An isomorphism preserves t... |
| isowe 7341 | An isomorphism preserves t... |
| isowe2 7342 | A weak form of ~ isowe tha... |
| f1oiso 7343 | Any one-to-one onto functi... |
| f1oiso2 7344 | Any one-to-one onto functi... |
| f1owe 7345 | Well-ordering of isomorphi... |
| weniso 7346 | A set-like well-ordering h... |
| weisoeq 7347 | Thus, there is at most one... |
| weisoeq2 7348 | Thus, there is at most one... |
| knatar 7349 | The Knaster-Tarski theorem... |
| fvresval 7350 | The value of a restricted ... |
| funeldmb 7351 | If ` (/) ` is not part of ... |
| eqfunresadj 7352 | Law for adjoining an eleme... |
| eqfunressuc 7353 | Law for equality of restri... |
| fnssintima 7354 | Condition for subset of an... |
| imaeqsexv 7355 | Substitute a function valu... |
| imaeqsalv 7356 | Substitute a function valu... |
| canth 7357 | No set ` A ` is equinumero... |
| ncanth 7358 | Cantor's theorem fails for... |
| riotaeqdv 7361 | Formula-building deduction... |
| riotabidv 7362 | Formula-building deduction... |
| riotaeqbidv 7363 | Equality deduction for res... |
| riotaex 7364 | Restricted iota is a set. ... |
| riotav 7365 | An iota restricted to the ... |
| riotauni 7366 | Restricted iota in terms o... |
| nfriota1 7367 | The abstraction variable i... |
| nfriotadw 7368 | Deduction version of ~ nfr... |
| cbvriotaw 7369 | Change bound variable in a... |
| cbvriotavw 7370 | Change bound variable in a... |
| cbvriotavwOLD 7371 | Obsolete version of ~ cbvr... |
| nfriotad 7372 | Deduction version of ~ nfr... |
| nfriota 7373 | A variable not free in a w... |
| cbvriota 7374 | Change bound variable in a... |
| cbvriotav 7375 | Change bound variable in a... |
| csbriota 7376 | Interchange class substitu... |
| riotacl2 7377 | Membership law for "the un... |
| riotacl 7378 | Closure of restricted iota... |
| riotasbc 7379 | Substitution law for descr... |
| riotabidva 7380 | Equivalent wff's yield equ... |
| riotabiia 7381 | Equivalent wff's yield equ... |
| riota1 7382 | Property of restricted iot... |
| riota1a 7383 | Property of iota. (Contri... |
| riota2df 7384 | A deduction version of ~ r... |
| riota2f 7385 | This theorem shows a condi... |
| riota2 7386 | This theorem shows a condi... |
| riotaeqimp 7387 | If two restricted iota des... |
| riotaprop 7388 | Properties of a restricted... |
| riota5f 7389 | A method for computing res... |
| riota5 7390 | A method for computing res... |
| riotass2 7391 | Restriction of a unique el... |
| riotass 7392 | Restriction of a unique el... |
| moriotass 7393 | Restriction of a unique el... |
| snriota 7394 | A restricted class abstrac... |
| riotaxfrd 7395 | Change the variable ` x ` ... |
| eusvobj2 7396 | Specify the same property ... |
| eusvobj1 7397 | Specify the same object in... |
| f1ofveu 7398 | There is one domain elemen... |
| f1ocnvfv3 7399 | Value of the converse of a... |
| riotaund 7400 | Restricted iota equals the... |
| riotassuni 7401 | The restricted iota class ... |
| riotaclb 7402 | Bidirectional closure of r... |
| riotarab 7403 | Restricted iota of a restr... |
| oveq 7410 | Equality theorem for opera... |
| oveq1 7411 | Equality theorem for opera... |
| oveq2 7412 | Equality theorem for opera... |
| oveq12 7413 | Equality theorem for opera... |
| oveq1i 7414 | Equality inference for ope... |
| oveq2i 7415 | Equality inference for ope... |
| oveq12i 7416 | Equality inference for ope... |
| oveqi 7417 | Equality inference for ope... |
| oveq123i 7418 | Equality inference for ope... |
| oveq1d 7419 | Equality deduction for ope... |
| oveq2d 7420 | Equality deduction for ope... |
| oveqd 7421 | Equality deduction for ope... |
| oveq12d 7422 | Equality deduction for ope... |
| oveqan12d 7423 | Equality deduction for ope... |
| oveqan12rd 7424 | Equality deduction for ope... |
| oveq123d 7425 | Equality deduction for ope... |
| fvoveq1d 7426 | Equality deduction for nes... |
| fvoveq1 7427 | Equality theorem for neste... |
| ovanraleqv 7428 | Equality theorem for a con... |
| imbrov2fvoveq 7429 | Equality theorem for neste... |
| ovrspc2v 7430 | If an operation value is e... |
| oveqrspc2v 7431 | Restricted specialization ... |
| oveqdr 7432 | Equality of two operations... |
| nfovd 7433 | Deduction version of bound... |
| nfov 7434 | Bound-variable hypothesis ... |
| oprabidw 7435 | The law of concretion. Sp... |
| oprabid 7436 | The law of concretion. Sp... |
| ovex 7437 | The result of an operation... |
| ovexi 7438 | The result of an operation... |
| ovexd 7439 | The result of an operation... |
| ovssunirn 7440 | The result of an operation... |
| 0ov 7441 | Operation value of the emp... |
| ovprc 7442 | The value of an operation ... |
| ovprc1 7443 | The value of an operation ... |
| ovprc2 7444 | The value of an operation ... |
| ovrcl 7445 | Reverse closure for an ope... |
| csbov123 7446 | Move class substitution in... |
| csbov 7447 | Move class substitution in... |
| csbov12g 7448 | Move class substitution in... |
| csbov1g 7449 | Move class substitution in... |
| csbov2g 7450 | Move class substitution in... |
| rspceov 7451 | A frequently used special ... |
| elovimad 7452 | Elementhood of the image s... |
| fnbrovb 7453 | Value of a binary operatio... |
| fnotovb 7454 | Equivalence of operation v... |
| opabbrex 7455 | A collection of ordered pa... |
| opabresex2 7456 | Restrictions of a collecti... |
| opabresex2d 7457 | Obsolete version of ~ opab... |
| fvmptopab 7458 | The function value of a ma... |
| fvmptopabOLD 7459 | Obsolete version of ~ fvmp... |
| f1opr 7460 | Condition for an operation... |
| brfvopab 7461 | The classes involved in a ... |
| dfoprab2 7462 | Class abstraction for oper... |
| reloprab 7463 | An operation class abstrac... |
| oprabv 7464 | If a pair and a class are ... |
| nfoprab1 7465 | The abstraction variables ... |
| nfoprab2 7466 | The abstraction variables ... |
| nfoprab3 7467 | The abstraction variables ... |
| nfoprab 7468 | Bound-variable hypothesis ... |
| oprabbid 7469 | Equivalent wff's yield equ... |
| oprabbidv 7470 | Equivalent wff's yield equ... |
| oprabbii 7471 | Equivalent wff's yield equ... |
| ssoprab2 7472 | Equivalence of ordered pai... |
| ssoprab2b 7473 | Equivalence of ordered pai... |
| eqoprab2bw 7474 | Equivalence of ordered pai... |
| eqoprab2b 7475 | Equivalence of ordered pai... |
| mpoeq123 7476 | An equality theorem for th... |
| mpoeq12 7477 | An equality theorem for th... |
| mpoeq123dva 7478 | An equality deduction for ... |
| mpoeq123dv 7479 | An equality deduction for ... |
| mpoeq123i 7480 | An equality inference for ... |
| mpoeq3dva 7481 | Slightly more general equa... |
| mpoeq3ia 7482 | An equality inference for ... |
| mpoeq3dv 7483 | An equality deduction for ... |
| nfmpo1 7484 | Bound-variable hypothesis ... |
| nfmpo2 7485 | Bound-variable hypothesis ... |
| nfmpo 7486 | Bound-variable hypothesis ... |
| 0mpo0 7487 | A mapping operation with e... |
| mpo0v 7488 | A mapping operation with e... |
| mpo0 7489 | A mapping operation with e... |
| oprab4 7490 | Two ways to state the doma... |
| cbvoprab1 7491 | Rule used to change first ... |
| cbvoprab2 7492 | Change the second bound va... |
| cbvoprab12 7493 | Rule used to change first ... |
| cbvoprab12v 7494 | Rule used to change first ... |
| cbvoprab3 7495 | Rule used to change the th... |
| cbvoprab3v 7496 | Rule used to change the th... |
| cbvmpox 7497 | Rule to change the bound v... |
| cbvmpo 7498 | Rule to change the bound v... |
| cbvmpov 7499 | Rule to change the bound v... |
| elimdelov 7500 | Eliminate a hypothesis whi... |
| ovif 7501 | Move a conditional outside... |
| ovif2 7502 | Move a conditional outside... |
| ovif12 7503 | Move a conditional outside... |
| ifov 7504 | Move a conditional outside... |
| dmoprab 7505 | The domain of an operation... |
| dmoprabss 7506 | The domain of an operation... |
| rnoprab 7507 | The range of an operation ... |
| rnoprab2 7508 | The range of a restricted ... |
| reldmoprab 7509 | The domain of an operation... |
| oprabss 7510 | Structure of an operation ... |
| eloprabga 7511 | The law of concretion for ... |
| eloprabgaOLD 7512 | Obsolete version of ~ elop... |
| eloprabg 7513 | The law of concretion for ... |
| ssoprab2i 7514 | Inference of operation cla... |
| mpov 7515 | Operation with universal d... |
| mpomptx 7516 | Express a two-argument fun... |
| mpompt 7517 | Express a two-argument fun... |
| mpodifsnif 7518 | A mapping with two argumen... |
| mposnif 7519 | A mapping with two argumen... |
| fconstmpo 7520 | Representation of a consta... |
| resoprab 7521 | Restriction of an operatio... |
| resoprab2 7522 | Restriction of an operator... |
| resmpo 7523 | Restriction of the mapping... |
| funoprabg 7524 | "At most one" is a suffici... |
| funoprab 7525 | "At most one" is a suffici... |
| fnoprabg 7526 | Functionality and domain o... |
| mpofun 7527 | The maps-to notation for a... |
| mpofunOLD 7528 | Obsolete version of ~ mpof... |
| fnoprab 7529 | Functionality and domain o... |
| ffnov 7530 | An operation maps to a cla... |
| fovcld 7531 | Closure law for an operati... |
| fovcl 7532 | Closure law for an operati... |
| eqfnov 7533 | Equality of two operations... |
| eqfnov2 7534 | Two operators with the sam... |
| fnov 7535 | Representation of a functi... |
| mpo2eqb 7536 | Bidirectional equality the... |
| rnmpo 7537 | The range of an operation ... |
| reldmmpo 7538 | The domain of an operation... |
| elrnmpog 7539 | Membership in the range of... |
| elrnmpo 7540 | Membership in the range of... |
| elrnmpores 7541 | Membership in the range of... |
| ralrnmpo 7542 | A restricted quantifier ov... |
| rexrnmpo 7543 | A restricted quantifier ov... |
| ovid 7544 | The value of an operation ... |
| ovidig 7545 | The value of an operation ... |
| ovidi 7546 | The value of an operation ... |
| ov 7547 | The value of an operation ... |
| ovigg 7548 | The value of an operation ... |
| ovig 7549 | The value of an operation ... |
| ovmpt4g 7550 | Value of a function given ... |
| ovmpos 7551 | Value of a function given ... |
| ov2gf 7552 | The value of an operation ... |
| ovmpodxf 7553 | Value of an operation give... |
| ovmpodx 7554 | Value of an operation give... |
| ovmpod 7555 | Value of an operation give... |
| ovmpox 7556 | The value of an operation ... |
| ovmpoga 7557 | Value of an operation give... |
| ovmpoa 7558 | Value of an operation give... |
| ovmpodf 7559 | Alternate deduction versio... |
| ovmpodv 7560 | Alternate deduction versio... |
| ovmpodv2 7561 | Alternate deduction versio... |
| ovmpog 7562 | Value of an operation give... |
| ovmpo 7563 | Value of an operation give... |
| fvmpopr2d 7564 | Value of an operation give... |
| ov3 7565 | The value of an operation ... |
| ov6g 7566 | The value of an operation ... |
| ovg 7567 | The value of an operation ... |
| ovres 7568 | The value of a restricted ... |
| ovresd 7569 | Lemma for converting metri... |
| oprres 7570 | The restriction of an oper... |
| oprssov 7571 | The value of a member of t... |
| fovcdm 7572 | An operation's value belon... |
| fovcdmda 7573 | An operation's value belon... |
| fovcdmd 7574 | An operation's value belon... |
| fnrnov 7575 | The range of an operation ... |
| foov 7576 | An onto mapping of an oper... |
| fnovrn 7577 | An operation's value belon... |
| ovelrn 7578 | A member of an operation's... |
| funimassov 7579 | Membership relation for th... |
| ovelimab 7580 | Operation value in an imag... |
| ovima0 7581 | An operation value is a me... |
| ovconst2 7582 | The value of a constant op... |
| oprssdm 7583 | Domain of closure of an op... |
| nssdmovg 7584 | The value of an operation ... |
| ndmovg 7585 | The value of an operation ... |
| ndmov 7586 | The value of an operation ... |
| ndmovcl 7587 | The closure of an operatio... |
| ndmovrcl 7588 | Reverse closure law, when ... |
| ndmovcom 7589 | Any operation is commutati... |
| ndmovass 7590 | Any operation is associati... |
| ndmovdistr 7591 | Any operation is distribut... |
| ndmovord 7592 | Elimination of redundant a... |
| ndmovordi 7593 | Elimination of redundant a... |
| caovclg 7594 | Convert an operation closu... |
| caovcld 7595 | Convert an operation closu... |
| caovcl 7596 | Convert an operation closu... |
| caovcomg 7597 | Convert an operation commu... |
| caovcomd 7598 | Convert an operation commu... |
| caovcom 7599 | Convert an operation commu... |
| caovassg 7600 | Convert an operation assoc... |
| caovassd 7601 | Convert an operation assoc... |
| caovass 7602 | Convert an operation assoc... |
| caovcang 7603 | Convert an operation cance... |
| caovcand 7604 | Convert an operation cance... |
| caovcanrd 7605 | Commute the arguments of a... |
| caovcan 7606 | Convert an operation cance... |
| caovordig 7607 | Convert an operation order... |
| caovordid 7608 | Convert an operation order... |
| caovordg 7609 | Convert an operation order... |
| caovordd 7610 | Convert an operation order... |
| caovord2d 7611 | Operation ordering law wit... |
| caovord3d 7612 | Ordering law. (Contribute... |
| caovord 7613 | Convert an operation order... |
| caovord2 7614 | Operation ordering law wit... |
| caovord3 7615 | Ordering law. (Contribute... |
| caovdig 7616 | Convert an operation distr... |
| caovdid 7617 | Convert an operation distr... |
| caovdir2d 7618 | Convert an operation distr... |
| caovdirg 7619 | Convert an operation rever... |
| caovdird 7620 | Convert an operation distr... |
| caovdi 7621 | Convert an operation distr... |
| caov32d 7622 | Rearrange arguments in a c... |
| caov12d 7623 | Rearrange arguments in a c... |
| caov31d 7624 | Rearrange arguments in a c... |
| caov13d 7625 | Rearrange arguments in a c... |
| caov4d 7626 | Rearrange arguments in a c... |
| caov411d 7627 | Rearrange arguments in a c... |
| caov42d 7628 | Rearrange arguments in a c... |
| caov32 7629 | Rearrange arguments in a c... |
| caov12 7630 | Rearrange arguments in a c... |
| caov31 7631 | Rearrange arguments in a c... |
| caov13 7632 | Rearrange arguments in a c... |
| caov4 7633 | Rearrange arguments in a c... |
| caov411 7634 | Rearrange arguments in a c... |
| caov42 7635 | Rearrange arguments in a c... |
| caovdir 7636 | Reverse distributive law. ... |
| caovdilem 7637 | Lemma used by real number ... |
| caovlem2 7638 | Lemma used in real number ... |
| caovmo 7639 | Uniqueness of inverse elem... |
| imaeqexov 7640 | Substitute an operation va... |
| imaeqalov 7641 | Substitute an operation va... |
| mpondm0 7642 | The value of an operation ... |
| elmpocl 7643 | If a two-parameter class i... |
| elmpocl1 7644 | If a two-parameter class i... |
| elmpocl2 7645 | If a two-parameter class i... |
| elovmpo 7646 | Utility lemma for two-para... |
| elovmporab 7647 | Implications for the value... |
| elovmporab1w 7648 | Implications for the value... |
| elovmporab1 7649 | Implications for the value... |
| 2mpo0 7650 | If the operation value of ... |
| relmptopab 7651 | Any function to sets of or... |
| f1ocnvd 7652 | Describe an implicit one-t... |
| f1od 7653 | Describe an implicit one-t... |
| f1ocnv2d 7654 | Describe an implicit one-t... |
| f1o2d 7655 | Describe an implicit one-t... |
| f1opw2 7656 | A one-to-one mapping induc... |
| f1opw 7657 | A one-to-one mapping induc... |
| elovmpt3imp 7658 | If the value of a function... |
| ovmpt3rab1 7659 | The value of an operation ... |
| ovmpt3rabdm 7660 | If the value of a function... |
| elovmpt3rab1 7661 | Implications for the value... |
| elovmpt3rab 7662 | Implications for the value... |
| ofeqd 7667 | Equality theorem for funct... |
| ofeq 7668 | Equality theorem for funct... |
| ofreq 7669 | Equality theorem for funct... |
| ofexg 7670 | A function operation restr... |
| nfof 7671 | Hypothesis builder for fun... |
| nfofr 7672 | Hypothesis builder for fun... |
| ofrfvalg 7673 | Value of a relation applie... |
| offval 7674 | Value of an operation appl... |
| ofrfval 7675 | Value of a relation applie... |
| ofval 7676 | Evaluate a function operat... |
| ofrval 7677 | Exhibit a function relatio... |
| offn 7678 | The function operation pro... |
| offun 7679 | The function operation pro... |
| offval2f 7680 | The function operation exp... |
| ofmresval 7681 | Value of a restriction of ... |
| fnfvof 7682 | Function value of a pointw... |
| off 7683 | The function operation pro... |
| ofres 7684 | Restrict the operands of a... |
| offval2 7685 | The function operation exp... |
| ofrfval2 7686 | The function relation acti... |
| ofmpteq 7687 | Value of a pointwise opera... |
| ofco 7688 | The composition of a funct... |
| offveq 7689 | Convert an identity of the... |
| offveqb 7690 | Equivalent expressions for... |
| ofc1 7691 | Left operation by a consta... |
| ofc2 7692 | Right operation by a const... |
| ofc12 7693 | Function operation on two ... |
| caofref 7694 | Transfer a reflexive law t... |
| caofinvl 7695 | Transfer a left inverse la... |
| caofid0l 7696 | Transfer a left identity l... |
| caofid0r 7697 | Transfer a right identity ... |
| caofid1 7698 | Transfer a right absorptio... |
| caofid2 7699 | Transfer a right absorptio... |
| caofcom 7700 | Transfer a commutative law... |
| 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... |
| unex 7728 | The union of two sets is a... |
| tpex 7729 | An unordered triple of cla... |
| unexb 7730 | Existence of union is equi... |
| unexg 7731 | A union of two sets is a s... |
| xpexg 7732 | The Cartesian product of t... |
| xpexd 7733 | The Cartesian product of t... |
| 3xpexg 7734 | The Cartesian product of t... |
| xpex 7735 | The Cartesian product of t... |
| unexd 7736 | The union of two sets is a... |
| sqxpexg 7737 | The Cartesian square of a ... |
| abnexg 7738 | Sufficient condition for a... |
| abnex 7739 | Sufficient condition for a... |
| snnex 7740 | The class of all singleton... |
| pwnex 7741 | The class of all power set... |
| difex2 7742 | If the subtrahend of a cla... |
| difsnexi 7743 | If the difference of a cla... |
| uniuni 7744 | Expression for double unio... |
| uniexr 7745 | Converse of the Axiom of U... |
| uniexb 7746 | The Axiom of Union and its... |
| pwexr 7747 | Converse of the Axiom of P... |
| pwexb 7748 | The Axiom of Power Sets an... |
| elpwpwel 7749 | A class belongs to a doubl... |
| eldifpw 7750 | Membership in a power clas... |
| elpwun 7751 | Membership in the power cl... |
| pwuncl 7752 | Power classes are closed u... |
| iunpw 7753 | An indexed union of a powe... |
| fr3nr 7754 | A well-founded relation ha... |
| epne3 7755 | A well-founded class conta... |
| dfwe2 7756 | Alternate definition of we... |
| epweon 7757 | The membership relation we... |
| epweonALT 7758 | Alternate proof of ~ epweo... |
| ordon 7759 | The class of all ordinal n... |
| onprc 7760 | No set contains all ordina... |
| ssorduni 7761 | The union of a class of or... |
| ssonuni 7762 | The union of a set of ordi... |
| ssonunii 7763 | The union of a set of ordi... |
| ordeleqon 7764 | A way to express the ordin... |
| ordsson 7765 | Any ordinal class is a sub... |
| dford5 7766 | A class is ordinal iff it ... |
| onss 7767 | An ordinal number is a sub... |
| predon 7768 | The predecessor of an ordi... |
| predonOLD 7769 | Obsolete version of ~ pred... |
| ssonprc 7770 | Two ways of saying a class... |
| onuni 7771 | The union of an ordinal nu... |
| orduni 7772 | The union of an ordinal cl... |
| onint 7773 | The intersection (infimum)... |
| onint0 7774 | The intersection of a clas... |
| onssmin 7775 | A nonempty class of ordina... |
| onminesb 7776 | If a property is true for ... |
| onminsb 7777 | If a property is true for ... |
| oninton 7778 | The intersection of a none... |
| onintrab 7779 | The intersection of a clas... |
| onintrab2 7780 | An existence condition equ... |
| onnmin 7781 | No member of a set of ordi... |
| onnminsb 7782 | An ordinal number smaller ... |
| oneqmin 7783 | A way to show that an ordi... |
| uniordint 7784 | The union of a set of ordi... |
| onminex 7785 | If a wff is true for an or... |
| sucon 7786 | The class of all ordinal n... |
| sucexb 7787 | A successor exists iff its... |
| sucexg 7788 | The successor of a set is ... |
| sucex 7789 | The successor of a set is ... |
| onmindif2 7790 | The minimum of a class of ... |
| ordsuci 7791 | The successor of an ordina... |
| sucexeloni 7792 | If the successor of an ord... |
| sucexeloniOLD 7793 | Obsolete version of ~ suce... |
| onsuc 7794 | The successor of an ordina... |
| suceloniOLD 7795 | Obsolete version of ~ onsu... |
| ordsuc 7796 | A class is ordinal if and ... |
| ordsucOLD 7797 | Obsolete version of ~ ords... |
| ordpwsuc 7798 | The collection of ordinals... |
| onpwsuc 7799 | The collection of ordinal ... |
| onsucb 7800 | A class is an ordinal numb... |
| ordsucss 7801 | The successor of an elemen... |
| onpsssuc 7802 | An ordinal number is a pro... |
| ordelsuc 7803 | A set belongs to an ordina... |
| onsucmin 7804 | The successor of an ordina... |
| ordsucelsuc 7805 | Membership is inherited by... |
| ordsucsssuc 7806 | The subclass relationship ... |
| ordsucuniel 7807 | Given an element ` A ` of ... |
| ordsucun 7808 | The successor of the maxim... |
| ordunpr 7809 | The maximum of two ordinal... |
| ordunel 7810 | The maximum of two ordinal... |
| onsucuni 7811 | A class of ordinal numbers... |
| ordsucuni 7812 | An ordinal class is a subc... |
| orduniorsuc 7813 | An ordinal class is either... |
| unon 7814 | The class of all ordinal n... |
| ordunisuc 7815 | An ordinal class is equal ... |
| orduniss2 7816 | The union of the ordinal s... |
| onsucuni2 7817 | A successor ordinal is the... |
| 0elsuc 7818 | The successor of an ordina... |
| limon 7819 | The class of ordinal numbe... |
| onuniorsuc 7820 | An ordinal number is eithe... |
| onssi 7821 | An ordinal number is a sub... |
| onsuci 7822 | The successor of an ordina... |
| onuniorsuciOLD 7823 | Obsolete version of ~ onun... |
| onuninsuci 7824 | An ordinal is equal to its... |
| onsucssi 7825 | A set belongs to an ordina... |
| nlimsucg 7826 | A successor is not a limit... |
| orduninsuc 7827 | An ordinal class is equal ... |
| ordunisuc2 7828 | An ordinal equal to its un... |
| ordzsl 7829 | An ordinal is zero, a succ... |
| onzsl 7830 | An ordinal number is zero,... |
| dflim3 7831 | An alternate definition of... |
| dflim4 7832 | An alternate definition of... |
| limsuc 7833 | The successor of a member ... |
| limsssuc 7834 | A class includes a limit o... |
| nlimon 7835 | Two ways to express the cl... |
| limuni3 7836 | The union of a nonempty cl... |
| tfi 7837 | The Principle of Transfini... |
| tfisg 7838 | A closed form of ~ tfis . ... |
| tfis 7839 | Transfinite Induction Sche... |
| tfis2f 7840 | Transfinite Induction Sche... |
| tfis2 7841 | Transfinite Induction Sche... |
| tfis3 7842 | Transfinite Induction Sche... |
| tfisi 7843 | A transfinite induction sc... |
| tfinds 7844 | Principle of Transfinite I... |
| tfindsg 7845 | Transfinite Induction (inf... |
| tfindsg2 7846 | Transfinite Induction (inf... |
| tfindes 7847 | Transfinite Induction with... |
| tfinds2 7848 | Transfinite Induction (inf... |
| tfinds3 7849 | Principle of Transfinite I... |
| dfom2 7852 | An alternate definition of... |
| elom 7853 | Membership in omega. The ... |
| omsson 7854 | Omega is a subset of ` On ... |
| limomss 7855 | The class of natural numbe... |
| nnon 7856 | A natural number is an ord... |
| nnoni 7857 | A natural number is an ord... |
| nnord 7858 | A natural number is ordina... |
| trom 7859 | The class of finite ordina... |
| ordom 7860 | The class of finite ordina... |
| elnn 7861 | A member of a natural numb... |
| omon 7862 | The class of natural numbe... |
| omelon2 7863 | Omega is an ordinal number... |
| nnlim 7864 | A natural number is not a ... |
| omssnlim 7865 | The class of natural numbe... |
| limom 7866 | Omega is a limit ordinal. ... |
| peano2b 7867 | A class belongs to omega i... |
| nnsuc 7868 | A nonzero natural number i... |
| omsucne 7869 | A natural number is not th... |
| ssnlim 7870 | An ordinal subclass of non... |
| omsinds 7871 | Strong (or "total") induct... |
| omsindsOLD 7872 | Obsolete version of ~ omsi... |
| omun 7873 | The union of two finite or... |
| peano1 7874 | Zero is a natural number. ... |
| peano1OLD 7875 | Obsolete version of ~ pean... |
| peano2 7876 | The successor of any natur... |
| peano3 7877 | The successor of any natur... |
| peano4 7878 | Two natural numbers are eq... |
| peano5 7879 | The induction postulate: a... |
| peano5OLD 7880 | Obsolete version of ~ pean... |
| nn0suc 7881 | A natural number is either... |
| find 7882 | The Principle of Finite In... |
| findOLD 7883 | Obsolete version of ~ find... |
| finds 7884 | Principle of Finite Induct... |
| findsg 7885 | Principle of Finite Induct... |
| finds2 7886 | Principle of Finite Induct... |
| finds1 7887 | Principle of Finite Induct... |
| findes 7888 | Finite induction with expl... |
| dmexg 7889 | The domain of a set is a s... |
| rnexg 7890 | The range of a set is a se... |
| dmexd 7891 | The domain of a set is a s... |
| fndmexd 7892 | If a function is a set, it... |
| dmfex 7893 | If a mapping is a set, its... |
| fndmexb 7894 | The domain of a function i... |
| fdmexb 7895 | The domain of a function i... |
| dmfexALT 7896 | Alternate proof of ~ dmfex... |
| dmex 7897 | The domain of a set is a s... |
| rnex 7898 | The range of a set is a se... |
| iprc 7899 | The identity function is a... |
| resiexg 7900 | The existence of a restric... |
| imaexg 7901 | The image of a set is a se... |
| imaex 7902 | The image of a set is a se... |
| exse2 7903 | Any set relation is set-li... |
| xpexr 7904 | If a Cartesian product is ... |
| xpexr2 7905 | If a nonempty Cartesian pr... |
| xpexcnv 7906 | A condition where the conv... |
| soex 7907 | If the relation in a stric... |
| elxp4 7908 | Membership in a Cartesian ... |
| elxp5 7909 | Membership in a Cartesian ... |
| cnvexg 7910 | The converse of a set is a... |
| cnvex 7911 | The converse of a set is a... |
| relcnvexb 7912 | A relation is a set iff it... |
| f1oexrnex 7913 | If the range of a 1-1 onto... |
| f1oexbi 7914 | There is a one-to-one onto... |
| coexg 7915 | The composition of two set... |
| coex 7916 | The composition of two set... |
| funcnvuni 7917 | The union of a chain (with... |
| fun11uni 7918 | The union of a chain (with... |
| fex2 7919 | A function with bounded do... |
| fabexg 7920 | Existence of a set of func... |
| fabex 7921 | Existence of a set of func... |
| f1oabexg 7922 | The class of all 1-1-onto ... |
| fiunlem 7923 | Lemma for ~ fiun and ~ f1i... |
| fiun 7924 | The union of a chain (with... |
| f1iun 7925 | The union of a chain (with... |
| fviunfun 7926 | The function value of an i... |
| ffoss 7927 | Relationship between a map... |
| f11o 7928 | Relationship between one-t... |
| resfunexgALT 7929 | Alternate proof of ~ resfu... |
| cofunexg 7930 | Existence of a composition... |
| cofunex2g 7931 | Existence of a composition... |
| fnexALT 7932 | Alternate proof of ~ fnex ... |
| funexw 7933 | Weak version of ~ funex th... |
| mptexw 7934 | Weak version of ~ mptex th... |
| funrnex 7935 | If the domain of a functio... |
| zfrep6 7936 | A version of the Axiom of ... |
| focdmex 7937 | If the domain of an onto f... |
| f1dmex 7938 | If the codomain of a one-t... |
| f1ovv 7939 | The codomain/range of a 1-... |
| fvclex 7940 | Existence of the class of ... |
| fvresex 7941 | Existence of the class of ... |
| abrexexg 7942 | Existence of a class abstr... |
| abrexexgOLD 7943 | Obsolete version of ~ abre... |
| abrexex 7944 | Existence of a class abstr... |
| iunexg 7945 | The existence of an indexe... |
| abrexex2g 7946 | Existence of an existentia... |
| opabex3d 7947 | Existence of an ordered pa... |
| opabex3rd 7948 | Existence of an ordered pa... |
| opabex3 7949 | Existence of an ordered pa... |
| iunex 7950 | The existence of an indexe... |
| abrexex2 7951 | Existence of an existentia... |
| abexssex 7952 | Existence of a class abstr... |
| abexex 7953 | A condition where a class ... |
| f1oweALT 7954 | Alternate proof of ~ f1owe... |
| wemoiso 7955 | Thus, there is at most one... |
| wemoiso2 7956 | Thus, there is at most one... |
| oprabexd 7957 | Existence of an operator a... |
| oprabex 7958 | Existence of an operation ... |
| oprabex3 7959 | Existence of an operation ... |
| oprabrexex2 7960 | Existence of an existentia... |
| ab2rexex 7961 | Existence of a class abstr... |
| ab2rexex2 7962 | Existence of an existentia... |
| xpexgALT 7963 | Alternate proof of ~ xpexg... |
| offval3 7964 | General value of ` ( F oF ... |
| offres 7965 | Pointwise combination comm... |
| ofmres 7966 | Equivalent expressions for... |
| ofmresex 7967 | Existence of a restriction... |
| 1stval 7972 | The value of the function ... |
| 2ndval 7973 | The value of the function ... |
| 1stnpr 7974 | Value of the first-member ... |
| 2ndnpr 7975 | Value of the second-member... |
| 1st0 7976 | The value of the first-mem... |
| 2nd0 7977 | The value of the second-me... |
| op1st 7978 | Extract the first member o... |
| op2nd 7979 | Extract the second member ... |
| op1std 7980 | Extract the first member o... |
| op2ndd 7981 | Extract the second member ... |
| op1stg 7982 | Extract the first member o... |
| op2ndg 7983 | Extract the second member ... |
| ot1stg 7984 | Extract the first member o... |
| ot2ndg 7985 | Extract the second member ... |
| ot3rdg 7986 | Extract the third member o... |
| 1stval2 7987 | Alternate value of the fun... |
| 2ndval2 7988 | Alternate value of the fun... |
| oteqimp 7989 | The components of an order... |
| fo1st 7990 | The ` 1st ` function maps ... |
| fo2nd 7991 | The ` 2nd ` function maps ... |
| br1steqg 7992 | Uniqueness condition for t... |
| br2ndeqg 7993 | Uniqueness condition for t... |
| f1stres 7994 | Mapping of a restriction o... |
| f2ndres 7995 | Mapping of a restriction o... |
| fo1stres 7996 | Onto mapping of a restrict... |
| fo2ndres 7997 | Onto mapping of a restrict... |
| 1st2val 7998 | Value of an alternate defi... |
| 2nd2val 7999 | Value of an alternate defi... |
| 1stcof 8000 | Composition of the first m... |
| 2ndcof 8001 | Composition of the second ... |
| xp1st 8002 | Location of the first elem... |
| xp2nd 8003 | Location of the second ele... |
| elxp6 8004 | Membership in a Cartesian ... |
| elxp7 8005 | Membership in a Cartesian ... |
| eqopi 8006 | Equality with an ordered p... |
| xp2 8007 | Representation of Cartesia... |
| unielxp 8008 | The membership relation fo... |
| 1st2nd2 8009 | Reconstruction of a member... |
| 1st2ndb 8010 | Reconstruction of an order... |
| xpopth 8011 | An ordered pair theorem fo... |
| eqop 8012 | Two ways to express equali... |
| eqop2 8013 | Two ways to express equali... |
| op1steq 8014 | Two ways of expressing tha... |
| opreuopreu 8015 | There is a unique ordered ... |
| el2xptp 8016 | A member of a nested Carte... |
| el2xptp0 8017 | A member of a nested Carte... |
| el2xpss 8018 | Version of ~ elrel for tri... |
| 2nd1st 8019 | Swap the members of an ord... |
| 1st2nd 8020 | Reconstruction of a member... |
| 1stdm 8021 | The first ordered pair com... |
| 2ndrn 8022 | The second ordered pair co... |
| 1st2ndbr 8023 | Express an element of a re... |
| releldm2 8024 | Two ways of expressing mem... |
| reldm 8025 | An expression for the doma... |
| releldmdifi 8026 | One way of expressing memb... |
| funfv1st2nd 8027 | The function value for the... |
| funelss 8028 | If the first component of ... |
| funeldmdif 8029 | Two ways of expressing mem... |
| sbcopeq1a 8030 | Equality theorem for subst... |
| csbopeq1a 8031 | Equality theorem for subst... |
| sbcoteq1a 8032 | Equality theorem for subst... |
| dfopab2 8033 | A way to define an ordered... |
| dfoprab3s 8034 | A way to define an operati... |
| dfoprab3 8035 | Operation class abstractio... |
| dfoprab4 8036 | Operation class abstractio... |
| dfoprab4f 8037 | Operation class abstractio... |
| opabex2 8038 | Condition for an operation... |
| opabn1stprc 8039 | An ordered-pair class abst... |
| opiota 8040 | The property of a uniquely... |
| cnvoprab 8041 | The converse of a class ab... |
| dfxp3 8042 | Define the Cartesian produ... |
| elopabi 8043 | A consequence of membershi... |
| eloprabi 8044 | A consequence of membershi... |
| mpomptsx 8045 | Express a two-argument fun... |
| mpompts 8046 | Express a two-argument fun... |
| dmmpossx 8047 | The domain of a mapping is... |
| fmpox 8048 | Functionality, domain and ... |
| fmpo 8049 | Functionality, domain and ... |
| fnmpo 8050 | Functionality and domain o... |
| fnmpoi 8051 | Functionality and domain o... |
| dmmpo 8052 | Domain of a class given by... |
| ovmpoelrn 8053 | An operation's value belon... |
| dmmpoga 8054 | Domain of an operation giv... |
| dmmpogaOLD 8055 | Obsolete version of ~ dmmp... |
| dmmpog 8056 | Domain of an operation giv... |
| mpoexxg 8057 | Existence of an operation ... |
| mpoexg 8058 | Existence of an operation ... |
| mpoexga 8059 | If the domain of an operat... |
| mpoexw 8060 | Weak version of ~ mpoex th... |
| mpoex 8061 | If the domain of an operat... |
| mptmpoopabbrd 8062 | The operation value of a f... |
| mptmpoopabovd 8063 | The operation value of a f... |
| mptmpoopabbrdOLD 8064 | Obsolete version of ~ mptm... |
| mptmpoopabovdOLD 8065 | Obsolete version of ~ mptm... |
| 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 ... |
| frxp 8107 | A lexicographical ordering... |
| xporderlem 8108 | Lemma for lexicographical ... |
| poxp 8109 | A lexicographical ordering... |
| soxp 8110 | A lexicographical ordering... |
| wexp 8111 | A lexicographical ordering... |
| fnwelem 8112 | Lemma for ~ fnwe . (Contr... |
| fnwe 8113 | A variant on lexicographic... |
| fnse 8114 | Condition for the well-ord... |
| fvproj 8115 | Value of a function on ord... |
| fimaproj 8116 | Image of a cartesian produ... |
| ralxpes 8117 | A version of ~ ralxp with ... |
| ralxp3f 8118 | Restricted for all over a ... |
| ralxp3 8119 | Restricted for all over a ... |
| ralxp3es 8120 | Restricted for-all over a ... |
| frpoins3xpg 8121 | Special case of founded pa... |
| frpoins3xp3g 8122 | Special case of founded pa... |
| xpord2lem 8123 | Lemma for Cartesian produc... |
| poxp2 8124 | Another way of partially o... |
| frxp2 8125 | Another way of giving a we... |
| xpord2pred 8126 | Calculate the predecessor ... |
| sexp2 8127 | Condition for the relation... |
| xpord2indlem 8128 | Induction over the Cartesi... |
| xpord2ind 8129 | Induction over the Cartesi... |
| xpord3lem 8130 | Lemma for triple ordering.... |
| poxp3 8131 | Triple Cartesian product p... |
| frxp3 8132 | Give well-foundedness over... |
| xpord3pred 8133 | Calculate the predecsessor... |
| sexp3 8134 | Show that the triple order... |
| xpord3inddlem 8135 | Induction over the triple ... |
| xpord3indd 8136 | Induction over the triple ... |
| xpord3ind 8137 | Induction over the triple ... |
| orderseqlem 8138 | Lemma for ~ poseq and ~ so... |
| poseq 8139 | A partial ordering of ordi... |
| soseq 8140 | A linear ordering of ordin... |
| suppval 8143 | The value of the operation... |
| supp0prc 8144 | The support of a class is ... |
| suppvalbr 8145 | The value of the operation... |
| supp0 8146 | The support of the empty s... |
| suppval1 8147 | The value of the operation... |
| suppvalfng 8148 | The value of the operation... |
| suppvalfn 8149 | The value of the operation... |
| elsuppfng 8150 | An element of the support ... |
| elsuppfn 8151 | An element of the support ... |
| cnvimadfsn 8152 | The support of functions "... |
| suppimacnvss 8153 | The support of functions "... |
| suppimacnv 8154 | Support sets of functions ... |
| fsuppeq 8155 | Two ways of writing the su... |
| fsuppeqg 8156 | Version of ~ fsuppeq avoid... |
| suppssdm 8157 | The support of a function ... |
| suppsnop 8158 | The support of a singleton... |
| snopsuppss 8159 | The support of a singleton... |
| fvn0elsupp 8160 | If the function value for ... |
| fvn0elsuppb 8161 | The function value for a g... |
| rexsupp 8162 | Existential quantification... |
| ressuppss 8163 | The support of the restric... |
| suppun 8164 | The support of a class/fun... |
| ressuppssdif 8165 | The support of the restric... |
| mptsuppdifd 8166 | The support of a function ... |
| mptsuppd 8167 | The support of a function ... |
| extmptsuppeq 8168 | The support of an extended... |
| suppfnss 8169 | The support of a function ... |
| funsssuppss 8170 | The support of a function ... |
| fnsuppres 8171 | Two ways to express restri... |
| fnsuppeq0 8172 | The support of a function ... |
| fczsupp0 8173 | The support of a constant ... |
| suppss 8174 | Show that the support of a... |
| suppssOLD 8175 | Obsolete version of ~ supp... |
| suppssr 8176 | A function is zero outside... |
| suppssrg 8177 | A function is zero outside... |
| suppssov1 8178 | Formula building theorem f... |
| suppssof1 8179 | Formula building theorem f... |
| suppss2 8180 | Show that the support of a... |
| suppsssn 8181 | Show that the support of a... |
| suppssfv 8182 | Formula building theorem f... |
| suppofssd 8183 | Condition for the support ... |
| suppofss1d 8184 | Condition for the support ... |
| suppofss2d 8185 | Condition for the support ... |
| suppco 8186 | The support of the composi... |
| suppcoss 8187 | The support of the composi... |
| supp0cosupp0 8188 | The support of the composi... |
| imacosupp 8189 | The image of the support o... |
| opeliunxp2f 8190 | Membership in a union of C... |
| mpoxeldm 8191 | If there is an element of ... |
| mpoxneldm 8192 | If the first argument of a... |
| mpoxopn0yelv 8193 | If there is an element of ... |
| mpoxopynvov0g 8194 | If the second argument of ... |
| mpoxopxnop0 8195 | If the first argument of a... |
| mpoxopx0ov0 8196 | If the first argument of a... |
| mpoxopxprcov0 8197 | If the components of the f... |
| mpoxopynvov0 8198 | If the second argument of ... |
| mpoxopoveq 8199 | Value of an operation give... |
| mpoxopovel 8200 | Element of the value of an... |
| mpoxopoveqd 8201 | Value of an operation give... |
| brovex 8202 | A binary relation of the v... |
| brovmpoex 8203 | A binary relation of the v... |
| sprmpod 8204 | The extension of a binary ... |
| tposss 8207 | Subset theorem for transpo... |
| tposeq 8208 | Equality theorem for trans... |
| tposeqd 8209 | Equality theorem for trans... |
| tposssxp 8210 | The transposition is a sub... |
| reltpos 8211 | The transposition is a rel... |
| brtpos2 8212 | Value of the transposition... |
| brtpos0 8213 | The behavior of ` tpos ` w... |
| reldmtpos 8214 | Necessary and sufficient c... |
| brtpos 8215 | The transposition swaps ar... |
| ottpos 8216 | The transposition swaps th... |
| relbrtpos 8217 | The transposition swaps ar... |
| dmtpos 8218 | The domain of ` tpos F ` w... |
| rntpos 8219 | The range of ` tpos F ` wh... |
| tposexg 8220 | The transposition of a set... |
| ovtpos 8221 | The transposition swaps th... |
| tposfun 8222 | The transposition of a fun... |
| dftpos2 8223 | Alternate definition of ` ... |
| dftpos3 8224 | Alternate definition of ` ... |
| dftpos4 8225 | Alternate definition of ` ... |
| tpostpos 8226 | Value of the double transp... |
| tpostpos2 8227 | Value of the double transp... |
| tposfn2 8228 | The domain of a transposit... |
| tposfo2 8229 | Condition for a surjective... |
| tposf2 8230 | The domain and codomain of... |
| tposf12 8231 | Condition for an injective... |
| tposf1o2 8232 | Condition of a bijective t... |
| tposfo 8233 | The domain and codomain/ra... |
| tposf 8234 | The domain and codomain of... |
| tposfn 8235 | Functionality of a transpo... |
| tpos0 8236 | Transposition of the empty... |
| tposco 8237 | Transposition of a composi... |
| tpossym 8238 | Two ways to say a function... |
| tposeqi 8239 | Equality theorem for trans... |
| tposex 8240 | A transposition is a set. ... |
| nftpos 8241 | Hypothesis builder for tra... |
| tposoprab 8242 | Transposition of a class o... |
| tposmpo 8243 | Transposition of a two-arg... |
| tposconst 8244 | The transposition of a con... |
| mpocurryd 8249 | The currying of an operati... |
| mpocurryvald 8250 | The value of a curried ope... |
| fvmpocurryd 8251 | The value of the value of ... |
| pwuninel2 8254 | Direct proof of ~ pwuninel... |
| pwuninel 8255 | The power set of the union... |
| undefval 8256 | Value of the undefined val... |
| undefnel2 8257 | The undefined value genera... |
| undefnel 8258 | The undefined value genera... |
| undefne0 8259 | The undefined value genera... |
| frecseq123 8262 | Equality theorem for the w... |
| nffrecs 8263 | Bound-variable hypothesis ... |
| csbfrecsg 8264 | Move class substitution in... |
| fpr3g 8265 | Functions defined by well-... |
| frrlem1 8266 | Lemma for well-founded rec... |
| frrlem2 8267 | Lemma for well-founded rec... |
| frrlem3 8268 | Lemma for well-founded rec... |
| frrlem4 8269 | Lemma for well-founded rec... |
| frrlem5 8270 | Lemma for well-founded rec... |
| frrlem6 8271 | Lemma for well-founded rec... |
| frrlem7 8272 | Lemma for well-founded rec... |
| frrlem8 8273 | Lemma for well-founded rec... |
| frrlem9 8274 | Lemma for well-founded rec... |
| frrlem10 8275 | Lemma for well-founded rec... |
| frrlem11 8276 | Lemma for well-founded rec... |
| frrlem12 8277 | Lemma for well-founded rec... |
| frrlem13 8278 | Lemma for well-founded rec... |
| frrlem14 8279 | Lemma for well-founded rec... |
| fprlem1 8280 | Lemma for well-founded rec... |
| fprlem2 8281 | Lemma for well-founded rec... |
| fpr2a 8282 | Weak version of ~ fpr2 whi... |
| fpr1 8283 | Law of well-founded recurs... |
| fpr2 8284 | Law of well-founded recurs... |
| fpr3 8285 | Law of well-founded recurs... |
| frrrel 8286 | Show without using the axi... |
| frrdmss 8287 | Show without using the axi... |
| frrdmcl 8288 | Show without using the axi... |
| fprfung 8289 | A "function" defined by we... |
| fprresex 8290 | The restriction of a funct... |
| dfwrecsOLD 8293 | Obsolete definition of the... |
| wrecseq123 8294 | General equality theorem f... |
| wrecseq123OLD 8295 | Obsolete proof of ~ wrecse... |
| nfwrecs 8296 | Bound-variable hypothesis ... |
| nfwrecsOLD 8297 | Obsolete proof of ~ nfwrec... |
| 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-... |
| wfrlem1OLD 8303 | Lemma for well-ordered rec... |
| wfrlem2OLD 8304 | Lemma for well-ordered rec... |
| wfrlem3OLD 8305 | Lemma for well-ordered rec... |
| wfrlem3OLDa 8306 | Lemma for well-ordered rec... |
| wfrlem4OLD 8307 | Lemma for well-ordered rec... |
| wfrlem5OLD 8308 | Lemma for well-ordered rec... |
| wfrrelOLD 8309 | Obsolete proof of ~ wfrrel... |
| wfrdmssOLD 8310 | Obsolete proof of ~ wfrdms... |
| wfrlem8OLD 8311 | Lemma for well-ordered rec... |
| wfrdmclOLD 8312 | Obsolete proof of ~ wfrdmc... |
| wfrlem10OLD 8313 | Lemma for well-ordered rec... |
| wfrfunOLD 8314 | Obsolete proof of ~ wfrfun... |
| wfrlem12OLD 8315 | Lemma for well-ordered rec... |
| wfrlem13OLD 8316 | Lemma for well-ordered rec... |
| wfrlem14OLD 8317 | Lemma for well-ordered rec... |
| wfrlem15OLD 8318 | Lemma for well-ordered rec... |
| wfrlem16OLD 8319 | Lemma for well-ordered rec... |
| wfrlem17OLD 8320 | Without using ~ ax-rep , s... |
| wfr2aOLD 8321 | Obsolete proof of ~ wfr2a ... |
| wfr1OLD 8322 | Obsolete proof of ~ wfr1 a... |
| wfr2OLD 8323 | Obsolete proof of ~ wfr2 a... |
| wfrrel 8324 | The well-ordered recursion... |
| wfrdmss 8325 | The domain of the well-ord... |
| wfrdmcl 8326 | The predecessor class of a... |
| wfrfun 8327 | The "function" generated b... |
| wfrresex 8328 | Show without using the axi... |
| wfr2a 8329 | A weak version of ~ wfr2 w... |
| wfr1 8330 | The Principle of Well-Orde... |
| wfr2 8331 | The Principle of Well-Orde... |
| wfr3 8332 | The principle of Well-Orde... |
| wfr3OLD 8333 | Obsolete form of ~ wfr3 as... |
| iunon 8334 | The indexed union of a set... |
| iinon 8335 | The nonempty indexed inter... |
| onfununi 8336 | A property of functions on... |
| onovuni 8337 | A variant of ~ onfununi fo... |
| onoviun 8338 | A variant of ~ onovuni wit... |
| onnseq 8339 | There are no length ` _om ... |
| dfsmo2 8342 | Alternate definition of a ... |
| issmo 8343 | Conditions for which ` A `... |
| issmo2 8344 | Alternate definition of a ... |
| smoeq 8345 | Equality theorem for stric... |
| smodm 8346 | The domain of a strictly m... |
| smores 8347 | A strictly monotone functi... |
| smores3 8348 | A strictly monotone functi... |
| smores2 8349 | A strictly monotone ordina... |
| smodm2 8350 | The domain of a strictly m... |
| smofvon2 8351 | The function values of a s... |
| iordsmo 8352 | The identity relation rest... |
| smo0 8353 | The null set is a strictly... |
| smofvon 8354 | If ` B ` is a strictly mon... |
| smoel 8355 | If ` x ` is less than ` y ... |
| smoiun 8356 | The value of a strictly mo... |
| smoiso 8357 | If ` F ` is an isomorphism... |
| smoel2 8358 | A strictly monotone ordina... |
| smo11 8359 | A strictly monotone ordina... |
| smoord 8360 | A strictly monotone ordina... |
| smoword 8361 | A strictly monotone ordina... |
| smogt 8362 | A strictly monotone ordina... |
| smocdmdom 8363 | The codomain of a strictly... |
| smoiso2 8364 | The strictly monotone ordi... |
| dfrecs3 8367 | The old definition of tran... |
| dfrecs3OLD 8368 | Obsolete proof of ~ dfrecs... |
| recseq 8369 | Equality theorem for ` rec... |
| nfrecs 8370 | Bound-variable hypothesis ... |
| tfrlem1 8371 | A technical lemma for tran... |
| tfrlem3a 8372 | Lemma for transfinite recu... |
| tfrlem3 8373 | Lemma for transfinite recu... |
| tfrlem4 8374 | Lemma for transfinite recu... |
| tfrlem5 8375 | Lemma for transfinite recu... |
| recsfval 8376 | Lemma for transfinite recu... |
| tfrlem6 8377 | Lemma for transfinite recu... |
| tfrlem7 8378 | Lemma for transfinite recu... |
| tfrlem8 8379 | Lemma for transfinite recu... |
| tfrlem9 8380 | Lemma for transfinite recu... |
| tfrlem9a 8381 | Lemma for transfinite recu... |
| tfrlem10 8382 | Lemma for transfinite recu... |
| tfrlem11 8383 | Lemma for transfinite recu... |
| tfrlem12 8384 | Lemma for transfinite recu... |
| tfrlem13 8385 | Lemma for transfinite recu... |
| tfrlem14 8386 | Lemma for transfinite recu... |
| tfrlem15 8387 | Lemma for transfinite recu... |
| tfrlem16 8388 | Lemma for finite recursion... |
| tfr1a 8389 | A weak version of ~ tfr1 w... |
| tfr2a 8390 | A weak version of ~ tfr2 w... |
| tfr2b 8391 | Without assuming ~ ax-rep ... |
| tfr1 8392 | Principle of Transfinite R... |
| tfr2 8393 | Principle of Transfinite R... |
| tfr3 8394 | Principle of Transfinite R... |
| tfr1ALT 8395 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8396 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8397 | Alternate proof of ~ tfr3 ... |
| recsfnon 8398 | Strong transfinite recursi... |
| recsval 8399 | Strong transfinite recursi... |
| tz7.44lem1 8400 | The ordered pair abstracti... |
| tz7.44-1 8401 | The value of ` F ` at ` (/... |
| tz7.44-2 8402 | The value of ` F ` at a su... |
| tz7.44-3 8403 | The value of ` F ` at a li... |
| rdgeq1 8406 | Equality theorem for the r... |
| rdgeq2 8407 | Equality theorem for the r... |
| rdgeq12 8408 | Equality theorem for the r... |
| nfrdg 8409 | Bound-variable hypothesis ... |
| rdglem1 8410 | Lemma used with the recurs... |
| rdgfun 8411 | The recursive definition g... |
| rdgdmlim 8412 | The domain of the recursiv... |
| rdgfnon 8413 | The recursive definition g... |
| rdgvalg 8414 | Value of the recursive def... |
| rdgval 8415 | Value of the recursive def... |
| rdg0 8416 | The initial value of the r... |
| rdgseg 8417 | The initial segments of th... |
| rdgsucg 8418 | The value of the recursive... |
| rdgsuc 8419 | The value of the recursive... |
| rdglimg 8420 | The value of the recursive... |
| rdglim 8421 | The value of the recursive... |
| rdg0g 8422 | The initial value of the r... |
| rdgsucmptf 8423 | The value of the recursive... |
| rdgsucmptnf 8424 | The value of the recursive... |
| rdgsucmpt2 8425 | This version of ~ rdgsucmp... |
| rdgsucmpt 8426 | The value of the recursive... |
| rdglim2 8427 | The value of the recursive... |
| rdglim2a 8428 | The value of the recursive... |
| rdg0n 8429 | If ` A ` is a proper class... |
| frfnom 8430 | The function generated by ... |
| fr0g 8431 | The initial value resultin... |
| frsuc 8432 | The successor value result... |
| frsucmpt 8433 | The successor value result... |
| frsucmptn 8434 | The value of the finite re... |
| frsucmpt2 8435 | The successor value result... |
| tz7.48lem 8436 | A way of showing an ordina... |
| tz7.48-2 8437 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8438 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8439 | Proposition 7.48(3) of [Ta... |
| tz7.49 8440 | Proposition 7.49 of [Takeu... |
| tz7.49c 8441 | Corollary of Proposition 7... |
| seqomlem0 8444 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8445 | Lemma for ` seqom ` . The... |
| seqomlem2 8446 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8447 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8448 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8449 | Equality theorem for ` seq... |
| fnseqom 8450 | An index-aware recursive d... |
| seqom0g 8451 | Value of an index-aware re... |
| seqomsuc 8452 | Value of an index-aware re... |
| omsucelsucb 8453 | Membership is inherited by... |
| df1o2 8468 | Expanded value of the ordi... |
| df2o3 8469 | Expanded value of the ordi... |
| df2o2 8470 | Expanded value of the ordi... |
| 1oex 8471 | Ordinal 1 is a set. (Cont... |
| 2oex 8472 | ` 2o ` is a set. (Contrib... |
| 1on 8473 | Ordinal 1 is an ordinal nu... |
| 1onOLD 8474 | Obsolete version of ~ 1on ... |
| 2on 8475 | Ordinal 2 is an ordinal nu... |
| 2onOLD 8476 | Obsolete version of ~ 2on ... |
| 2on0 8477 | Ordinal two is not zero. ... |
| ord3 8478 | Ordinal 3 is an ordinal cl... |
| 3on 8479 | Ordinal 3 is an ordinal nu... |
| 4on 8480 | Ordinal 4 is an ordinal nu... |
| 1oexOLD 8481 | Obsolete version of ~ 1oex... |
| 2oexOLD 8482 | Obsolete version of ~ 2oex... |
| 1n0 8483 | Ordinal one is not equal t... |
| nlim1 8484 | 1 is not a limit ordinal. ... |
| nlim2 8485 | 2 is not a limit ordinal. ... |
| xp01disj 8486 | Cartesian products with th... |
| xp01disjl 8487 | Cartesian products with th... |
| ordgt0ge1 8488 | Two ways to express that a... |
| ordge1n0 8489 | An ordinal greater than or... |
| el1o 8490 | Membership in ordinal one.... |
| ord1eln01 8491 | An ordinal that is not 0 o... |
| ord2eln012 8492 | An ordinal that is not 0, ... |
| 1ellim 8493 | A limit ordinal contains 1... |
| 2ellim 8494 | A limit ordinal contains 2... |
| dif1o 8495 | Two ways to say that ` A `... |
| ondif1 8496 | Two ways to say that ` A `... |
| ondif2 8497 | Two ways to say that ` A `... |
| 2oconcl 8498 | Closure of the pair swappi... |
| 0lt1o 8499 | Ordinal zero is less than ... |
| dif20el 8500 | An ordinal greater than on... |
| 0we1 8501 | The empty set is a well-or... |
| brwitnlem 8502 | Lemma for relations which ... |
| fnoa 8503 | Functionality and domain o... |
| fnom 8504 | Functionality and domain o... |
| fnoe 8505 | Functionality and domain o... |
| oav 8506 | Value of ordinal addition.... |
| omv 8507 | Value of ordinal multiplic... |
| oe0lem 8508 | A helper lemma for ~ oe0 a... |
| oev 8509 | Value of ordinal exponenti... |
| oevn0 8510 | Value of ordinal exponenti... |
| oa0 8511 | Addition with zero. Propo... |
| om0 8512 | Ordinal multiplication wit... |
| oe0m 8513 | Value of zero raised to an... |
| om0x 8514 | Ordinal multiplication wit... |
| oe0m0 8515 | Ordinal exponentiation wit... |
| oe0m1 8516 | Ordinal exponentiation wit... |
| oe0 8517 | Ordinal exponentiation wit... |
| oev2 8518 | Alternate value of ordinal... |
| oasuc 8519 | Addition with successor. ... |
| oesuclem 8520 | Lemma for ~ oesuc . (Cont... |
| omsuc 8521 | Multiplication with succes... |
| oesuc 8522 | Ordinal exponentiation wit... |
| onasuc 8523 | Addition with successor. ... |
| onmsuc 8524 | Multiplication with succes... |
| onesuc 8525 | Exponentiation with a succ... |
| oa1suc 8526 | Addition with 1 is same as... |
| oalim 8527 | Ordinal addition with a li... |
| omlim 8528 | Ordinal multiplication wit... |
| oelim 8529 | Ordinal exponentiation wit... |
| oacl 8530 | Closure law for ordinal ad... |
| omcl 8531 | Closure law for ordinal mu... |
| oecl 8532 | Closure law for ordinal ex... |
| oa0r 8533 | Ordinal addition with zero... |
| om0r 8534 | Ordinal multiplication wit... |
| o1p1e2 8535 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8536 | 2 + 2 = 4 for ordinal numb... |
| o2p2e4OLD 8537 | Obsolete version of ~ o2p2... |
| om1 8538 | Ordinal multiplication wit... |
| om1r 8539 | Ordinal multiplication wit... |
| oe1 8540 | Ordinal exponentiation wit... |
| oe1m 8541 | Ordinal exponentiation wit... |
| oaordi 8542 | Ordering property of ordin... |
| oaord 8543 | Ordering property of ordin... |
| oacan 8544 | Left cancellation law for ... |
| oaword 8545 | Weak ordering property of ... |
| oawordri 8546 | Weak ordering property of ... |
| oaord1 8547 | An ordinal is less than it... |
| oaword1 8548 | An ordinal is less than or... |
| oaword2 8549 | An ordinal is less than or... |
| oawordeulem 8550 | Lemma for ~ oawordex . (C... |
| oawordeu 8551 | Existence theorem for weak... |
| oawordexr 8552 | Existence theorem for weak... |
| oawordex 8553 | Existence theorem for weak... |
| oaordex 8554 | Existence theorem for orde... |
| oa00 8555 | An ordinal sum is zero iff... |
| oalimcl 8556 | The ordinal sum with a lim... |
| oaass 8557 | Ordinal addition is associ... |
| oarec 8558 | Recursive definition of or... |
| oaf1o 8559 | Left addition by a constan... |
| oacomf1olem 8560 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8561 | Define a bijection from ` ... |
| omordi 8562 | Ordering property of ordin... |
| omord2 8563 | Ordering property of ordin... |
| omord 8564 | Ordering property of ordin... |
| omcan 8565 | Left cancellation law for ... |
| omword 8566 | Weak ordering property of ... |
| omwordi 8567 | Weak ordering property of ... |
| omwordri 8568 | Weak ordering property of ... |
| omword1 8569 | An ordinal is less than or... |
| omword2 8570 | An ordinal is less than or... |
| om00 8571 | The product of two ordinal... |
| om00el 8572 | The product of two nonzero... |
| omordlim 8573 | Ordering involving the pro... |
| omlimcl 8574 | The product of any nonzero... |
| odi 8575 | Distributive law for ordin... |
| omass 8576 | Multiplication of ordinal ... |
| oneo 8577 | If an ordinal number is ev... |
| omeulem1 8578 | Lemma for ~ omeu : existen... |
| omeulem2 8579 | Lemma for ~ omeu : uniquen... |
| omopth2 8580 | An ordered pair-like theor... |
| omeu 8581 | The division algorithm for... |
| oen0 8582 | Ordinal exponentiation wit... |
| oeordi 8583 | Ordering law for ordinal e... |
| oeord 8584 | Ordering property of ordin... |
| oecan 8585 | Left cancellation law for ... |
| oeword 8586 | Weak ordering property of ... |
| oewordi 8587 | Weak ordering property of ... |
| oewordri 8588 | Weak ordering property of ... |
| oeworde 8589 | Ordinal exponentiation com... |
| oeordsuc 8590 | Ordering property of ordin... |
| oelim2 8591 | Ordinal exponentiation wit... |
| oeoalem 8592 | Lemma for ~ oeoa . (Contr... |
| oeoa 8593 | Sum of exponents law for o... |
| oeoelem 8594 | Lemma for ~ oeoe . (Contr... |
| oeoe 8595 | Product of exponents law f... |
| oelimcl 8596 | The ordinal exponential wi... |
| oeeulem 8597 | Lemma for ~ oeeu . (Contr... |
| oeeui 8598 | The division algorithm for... |
| oeeu 8599 | The division algorithm for... |
| nna0 8600 | Addition with zero. Theor... |
| nnm0 8601 | Multiplication with zero. ... |
| nnasuc 8602 | Addition with successor. ... |
| nnmsuc 8603 | Multiplication with succes... |
| nnesuc 8604 | Exponentiation with a succ... |
| nna0r 8605 | Addition to zero. Remark ... |
| nnm0r 8606 | Multiplication with zero. ... |
| nnacl 8607 | Closure of addition of nat... |
| nnmcl 8608 | Closure of multiplication ... |
| nnecl 8609 | Closure of exponentiation ... |
| nnacli 8610 | ` _om ` is closed under ad... |
| nnmcli 8611 | ` _om ` is closed under mu... |
| nnarcl 8612 | Reverse closure law for ad... |
| nnacom 8613 | Addition of natural number... |
| nnaordi 8614 | Ordering property of addit... |
| nnaord 8615 | Ordering property of addit... |
| nnaordr 8616 | Ordering property of addit... |
| nnawordi 8617 | Adding to both sides of an... |
| nnaass 8618 | Addition of natural number... |
| nndi 8619 | Distributive law for natur... |
| nnmass 8620 | Multiplication of natural ... |
| nnmsucr 8621 | Multiplication with succes... |
| nnmcom 8622 | Multiplication of natural ... |
| nnaword 8623 | Weak ordering property of ... |
| nnacan 8624 | Cancellation law for addit... |
| nnaword1 8625 | Weak ordering property of ... |
| nnaword2 8626 | Weak ordering property of ... |
| nnmordi 8627 | Ordering property of multi... |
| nnmord 8628 | Ordering property of multi... |
| nnmword 8629 | Weak ordering property of ... |
| nnmcan 8630 | Cancellation law for multi... |
| nnmwordi 8631 | Weak ordering property of ... |
| nnmwordri 8632 | Weak ordering property of ... |
| nnawordex 8633 | Equivalence for weak order... |
| nnaordex 8634 | Equivalence for ordering. ... |
| 1onn 8635 | The ordinal 1 is a natural... |
| 1onnALT 8636 | Shorter proof of ~ 1onn us... |
| 2onn 8637 | The ordinal 2 is a natural... |
| 2onnALT 8638 | Shorter proof of ~ 2onn us... |
| 3onn 8639 | The ordinal 3 is a natural... |
| 4onn 8640 | The ordinal 4 is a natural... |
| 1one2o 8641 | Ordinal one is not ordinal... |
| oaabslem 8642 | Lemma for ~ oaabs . (Cont... |
| oaabs 8643 | Ordinal addition absorbs a... |
| oaabs2 8644 | The absorption law ~ oaabs... |
| omabslem 8645 | Lemma for ~ omabs . (Cont... |
| omabs 8646 | Ordinal multiplication is ... |
| nnm1 8647 | Multiply an element of ` _... |
| nnm2 8648 | Multiply an element of ` _... |
| nn2m 8649 | Multiply an element of ` _... |
| nnneo 8650 | If a natural number is eve... |
| nneob 8651 | A natural number is even i... |
| omsmolem 8652 | Lemma for ~ omsmo . (Cont... |
| omsmo 8653 | A strictly monotonic ordin... |
| omopthlem1 8654 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8655 | Lemma for ~ omopthi . (Co... |
| omopthi 8656 | An ordered pair theorem fo... |
| omopth 8657 | An ordered pair theorem fo... |
| nnasmo 8658 | There is at most one left ... |
| eldifsucnn 8659 | Condition for membership i... |
| on2recsfn 8662 | Show that double recursion... |
| on2recsov 8663 | Calculate the value of the... |
| on2ind 8664 | Double induction over ordi... |
| on3ind 8665 | Triple induction over ordi... |
| coflton 8666 | Cofinality theorem for ord... |
| cofon1 8667 | Cofinality theorem for ord... |
| cofon2 8668 | Cofinality theorem for ord... |
| cofonr 8669 | Inverse cofinality law for... |
| naddfn 8670 | Natural addition is a func... |
| naddcllem 8671 | Lemma for ordinal addition... |
| naddcl 8672 | Closure law for natural ad... |
| naddov 8673 | The value of natural addit... |
| naddov2 8674 | Alternate expression for n... |
| naddov3 8675 | Alternate expression for n... |
| naddf 8676 | Function statement for nat... |
| naddcom 8677 | Natural addition commutes.... |
| naddrid 8678 | Ordinal zero is the additi... |
| naddlid 8679 | Ordinal zero is the additi... |
| naddssim 8680 | Ordinal less-than-or-equal... |
| naddelim 8681 | Ordinal less-than is prese... |
| naddel1 8682 | Ordinal less-than is not a... |
| naddel2 8683 | Ordinal less-than is not a... |
| naddss1 8684 | Ordinal less-than-or-equal... |
| naddss2 8685 | Ordinal less-than-or-equal... |
| naddword1 8686 | Weak-ordering principle fo... |
| naddword2 8687 | Weak-ordering principle fo... |
| naddunif 8688 | Uniformity theorem for nat... |
| naddasslem1 8689 | Lemma for ~ naddass . Exp... |
| naddasslem2 8690 | Lemma for ~ naddass . Exp... |
| naddass 8691 | Natural ordinal addition i... |
| nadd32 8692 | Commutative/associative la... |
| nadd4 8693 | Rearragement of terms in a... |
| nadd42 8694 | Rearragement of terms in a... |
| naddel12 8695 | Natural addition to both s... |
| dfer2 8700 | Alternate definition of eq... |
| dfec2 8702 | Alternate definition of ` ... |
| ecexg 8703 | An equivalence class modul... |
| ecexr 8704 | A nonempty equivalence cla... |
| ereq1 8706 | Equality theorem for equiv... |
| ereq2 8707 | Equality theorem for equiv... |
| errel 8708 | An equivalence relation is... |
| erdm 8709 | The domain of an equivalen... |
| ercl 8710 | Elementhood in the field o... |
| ersym 8711 | An equivalence relation is... |
| ercl2 8712 | Elementhood in the field o... |
| ersymb 8713 | An equivalence relation is... |
| ertr 8714 | An equivalence relation is... |
| ertrd 8715 | A transitivity relation fo... |
| ertr2d 8716 | A transitivity relation fo... |
| ertr3d 8717 | A transitivity relation fo... |
| ertr4d 8718 | A transitivity relation fo... |
| erref 8719 | An equivalence relation is... |
| ercnv 8720 | The converse of an equival... |
| errn 8721 | The range and domain of an... |
| erssxp 8722 | An equivalence relation is... |
| erex 8723 | An equivalence relation is... |
| erexb 8724 | An equivalence relation is... |
| iserd 8725 | A reflexive, symmetric, tr... |
| iseri 8726 | A reflexive, symmetric, tr... |
| iseriALT 8727 | Alternate proof of ~ iseri... |
| brdifun 8728 | Evaluate the incomparabili... |
| swoer 8729 | Incomparability under a st... |
| swoord1 8730 | The incomparability equiva... |
| swoord2 8731 | The incomparability equiva... |
| swoso 8732 | If the incomparability rel... |
| eqerlem 8733 | Lemma for ~ eqer . (Contr... |
| eqer 8734 | Equivalence relation invol... |
| ider 8735 | The identity relation is a... |
| 0er 8736 | The empty set is an equiva... |
| eceq1 8737 | Equality theorem for equiv... |
| eceq1d 8738 | Equality theorem for equiv... |
| eceq2 8739 | Equality theorem for equiv... |
| eceq2i 8740 | Equality theorem for the `... |
| eceq2d 8741 | Equality theorem for the `... |
| elecg 8742 | Membership in an equivalen... |
| elec 8743 | Membership in an equivalen... |
| relelec 8744 | Membership in an equivalen... |
| ecss 8745 | An equivalence class is a ... |
| ecdmn0 8746 | A representative of a none... |
| ereldm 8747 | Equality of equivalence cl... |
| erth 8748 | Basic property of equivale... |
| erth2 8749 | Basic property of equivale... |
| erthi 8750 | Basic property of equivale... |
| erdisj 8751 | Equivalence classes do not... |
| ecidsn 8752 | An equivalence class modul... |
| qseq1 8753 | Equality theorem for quoti... |
| qseq2 8754 | Equality theorem for quoti... |
| qseq2i 8755 | Equality theorem for quoti... |
| qseq2d 8756 | Equality theorem for quoti... |
| qseq12 8757 | Equality theorem for quoti... |
| elqsg 8758 | Closed form of ~ elqs . (... |
| elqs 8759 | Membership in a quotient s... |
| elqsi 8760 | Membership in a quotient s... |
| elqsecl 8761 | Membership in a quotient s... |
| ecelqsg 8762 | Membership of an equivalen... |
| ecelqsi 8763 | Membership of an equivalen... |
| ecopqsi 8764 | "Closure" law for equivale... |
| qsexg 8765 | A quotient set exists. (C... |
| qsex 8766 | A quotient set exists. (C... |
| uniqs 8767 | The union of a quotient se... |
| qsss 8768 | A quotient set is a set of... |
| uniqs2 8769 | The union of a quotient se... |
| snec 8770 | The singleton of an equiva... |
| ecqs 8771 | Equivalence class in terms... |
| ecid 8772 | A set is equal to its cose... |
| qsid 8773 | A set is equal to its quot... |
| ectocld 8774 | Implicit substitution of c... |
| ectocl 8775 | Implicit substitution of c... |
| elqsn0 8776 | A quotient set does not co... |
| ecelqsdm 8777 | Membership of an equivalen... |
| xpider 8778 | A Cartesian square is an e... |
| iiner 8779 | The intersection of a none... |
| riiner 8780 | The relative intersection ... |
| erinxp 8781 | A restricted equivalence r... |
| ecinxp 8782 | Restrict the relation in a... |
| qsinxp 8783 | Restrict the equivalence r... |
| qsdisj 8784 | Members of a quotient set ... |
| qsdisj2 8785 | A quotient set is a disjoi... |
| qsel 8786 | If an element of a quotien... |
| uniinqs 8787 | Class union distributes ov... |
| qliftlem 8788 | Lemma for theorems about a... |
| qliftrel 8789 | ` F ` , a function lift, i... |
| qliftel 8790 | Elementhood in the relatio... |
| qliftel1 8791 | Elementhood in the relatio... |
| qliftfun 8792 | The function ` F ` is the ... |
| qliftfund 8793 | The function ` F ` is the ... |
| qliftfuns 8794 | The function ` F ` is the ... |
| qliftf 8795 | The domain and codomain of... |
| qliftval 8796 | The value of the function ... |
| ecoptocl 8797 | Implicit substitution of c... |
| 2ecoptocl 8798 | Implicit substitution of c... |
| 3ecoptocl 8799 | Implicit substitution of c... |
| brecop 8800 | Binary relation on a quoti... |
| brecop2 8801 | Binary relation on a quoti... |
| eroveu 8802 | Lemma for ~ erov and ~ ero... |
| erovlem 8803 | Lemma for ~ erov and ~ ero... |
| erov 8804 | The value of an operation ... |
| eroprf 8805 | Functionality of an operat... |
| erov2 8806 | The value of an operation ... |
| eroprf2 8807 | Functionality of an operat... |
| ecopoveq 8808 | This is the first of sever... |
| ecopovsym 8809 | Assuming the operation ` F... |
| ecopovtrn 8810 | Assuming that operation ` ... |
| ecopover 8811 | Assuming that operation ` ... |
| eceqoveq 8812 | Equality of equivalence re... |
| ecovcom 8813 | Lemma used to transfer a c... |
| ecovass 8814 | Lemma used to transfer an ... |
| ecovdi 8815 | Lemma used to transfer a d... |
| mapprc 8820 | When ` A ` is a proper cla... |
| pmex 8821 | The class of all partial f... |
| mapex 8822 | The class of all functions... |
| fnmap 8823 | Set exponentiation has a u... |
| fnpm 8824 | Partial function exponenti... |
| reldmmap 8825 | Set exponentiation is a we... |
| mapvalg 8826 | The value of set exponenti... |
| pmvalg 8827 | The value of the partial m... |
| mapval 8828 | The value of set exponenti... |
| elmapg 8829 | Membership relation for se... |
| elmapd 8830 | Deduction form of ~ elmapg... |
| elmapdd 8831 | Deduction associated with ... |
| mapdm0 8832 | The empty set is the only ... |
| elpmg 8833 | The predicate "is a partia... |
| elpm2g 8834 | The predicate "is a partia... |
| elpm2r 8835 | Sufficient condition for b... |
| elpmi 8836 | A partial function is a fu... |
| pmfun 8837 | A partial function is a fu... |
| elmapex 8838 | Eliminate antecedent for m... |
| elmapi 8839 | A mapping is a function, f... |
| mapfset 8840 | If ` B ` is a set, the val... |
| mapssfset 8841 | The value of the set expon... |
| mapfoss 8842 | The value of the set expon... |
| fsetsspwxp 8843 | The class of all functions... |
| fset0 8844 | The set of functions from ... |
| fsetdmprc0 8845 | The set of functions with ... |
| fsetex 8846 | The set of functions betwe... |
| f1setex 8847 | The set of injections betw... |
| fosetex 8848 | The set of surjections bet... |
| f1osetex 8849 | The set of bijections betw... |
| fsetfcdm 8850 | The class of functions wit... |
| fsetfocdm 8851 | The class of functions wit... |
| fsetprcnex 8852 | The class of all functions... |
| fsetcdmex 8853 | The class of all functions... |
| fsetexb 8854 | The class of all functions... |
| elmapfn 8855 | A mapping is a function wi... |
| elmapfun 8856 | A mapping is always a func... |
| elmapssres 8857 | A restricted mapping is a ... |
| fpmg 8858 | A total function is a part... |
| pmss12g 8859 | Subset relation for the se... |
| pmresg 8860 | Elementhood of a restricte... |
| elmap 8861 | Membership relation for se... |
| mapval2 8862 | Alternate expression for t... |
| elpm 8863 | The predicate "is a partia... |
| elpm2 8864 | The predicate "is a partia... |
| fpm 8865 | A total function is a part... |
| mapsspm 8866 | Set exponentiation is a su... |
| pmsspw 8867 | Partial maps are a subset ... |
| mapsspw 8868 | Set exponentiation is a su... |
| mapfvd 8869 | The value of a function th... |
| elmapresaun 8870 | ~ fresaun transposed to ma... |
| fvmptmap 8871 | Special case of ~ fvmpt fo... |
| map0e 8872 | Set exponentiation with an... |
| map0b 8873 | Set exponentiation with an... |
| map0g 8874 | Set exponentiation is empt... |
| 0map0sn0 8875 | The set of mappings of the... |
| mapsnd 8876 | The value of set exponenti... |
| map0 8877 | Set exponentiation is empt... |
| mapsn 8878 | The value of set exponenti... |
| mapss 8879 | Subset inheritance for set... |
| fdiagfn 8880 | Functionality of the diago... |
| fvdiagfn 8881 | Functionality of the diago... |
| mapsnconst 8882 | Every singleton map is a c... |
| mapsncnv 8883 | Expression for the inverse... |
| mapsnf1o2 8884 | Explicit bijection between... |
| mapsnf1o3 8885 | Explicit bijection in the ... |
| ralxpmap 8886 | Quantification over functi... |
| dfixp 8889 | Eliminate the expression `... |
| ixpsnval 8890 | The value of an infinite C... |
| elixp2 8891 | Membership in an infinite ... |
| fvixp 8892 | Projection of a factor of ... |
| ixpfn 8893 | A nuple is a function. (C... |
| elixp 8894 | Membership in an infinite ... |
| elixpconst 8895 | Membership in an infinite ... |
| ixpconstg 8896 | Infinite Cartesian product... |
| ixpconst 8897 | Infinite Cartesian product... |
| ixpeq1 8898 | Equality theorem for infin... |
| ixpeq1d 8899 | Equality theorem for infin... |
| ss2ixp 8900 | Subclass theorem for infin... |
| ixpeq2 8901 | Equality theorem for infin... |
| ixpeq2dva 8902 | Equality theorem for infin... |
| ixpeq2dv 8903 | Equality theorem for infin... |
| cbvixp 8904 | Change bound variable in a... |
| cbvixpv 8905 | Change bound variable in a... |
| nfixpw 8906 | Bound-variable hypothesis ... |
| nfixp 8907 | Bound-variable hypothesis ... |
| nfixp1 8908 | The index variable in an i... |
| ixpprc 8909 | A cartesian product of pro... |
| ixpf 8910 | A member of an infinite Ca... |
| uniixp 8911 | The union of an infinite C... |
| ixpexg 8912 | The existence of an infini... |
| ixpin 8913 | The intersection of two in... |
| ixpiin 8914 | The indexed intersection o... |
| ixpint 8915 | The intersection of a coll... |
| ixp0x 8916 | An infinite Cartesian prod... |
| ixpssmap2g 8917 | An infinite Cartesian prod... |
| ixpssmapg 8918 | An infinite Cartesian prod... |
| 0elixp 8919 | Membership of the empty se... |
| ixpn0 8920 | The infinite Cartesian pro... |
| ixp0 8921 | The infinite Cartesian pro... |
| ixpssmap 8922 | An infinite Cartesian prod... |
| resixp 8923 | Restriction of an element ... |
| undifixp 8924 | Union of two projections o... |
| mptelixpg 8925 | Condition for an explicit ... |
| resixpfo 8926 | Restriction of elements of... |
| elixpsn 8927 | Membership in a class of s... |
| ixpsnf1o 8928 | A bijection between a clas... |
| mapsnf1o 8929 | A bijection between a set ... |
| boxriin 8930 | A rectangular subset of a ... |
| boxcutc 8931 | The relative complement of... |
| relen 8940 | Equinumerosity is a relati... |
| reldom 8941 | Dominance is a relation. ... |
| relsdom 8942 | Strict dominance is a rela... |
| encv 8943 | If two classes are equinum... |
| breng 8944 | Equinumerosity relation. ... |
| bren 8945 | Equinumerosity relation. ... |
| brenOLD 8946 | Obsolete version of ~ bren... |
| brdom2g 8947 | Dominance relation. This ... |
| brdomg 8948 | Dominance relation. (Cont... |
| brdomgOLD 8949 | Obsolete version of ~ brdo... |
| brdomi 8950 | Dominance relation. (Cont... |
| brdomiOLD 8951 | Obsolete version of ~ brdo... |
| brdom 8952 | Dominance relation. (Cont... |
| domen 8953 | Dominance in terms of equi... |
| domeng 8954 | Dominance in terms of equi... |
| ctex 8955 | A countable set is a set. ... |
| f1oen4g 8956 | The domain and range of a ... |
| f1dom4g 8957 | The domain of a one-to-one... |
| f1oen3g 8958 | The domain and range of a ... |
| f1dom3g 8959 | The domain of a one-to-one... |
| f1oen2g 8960 | The domain and range of a ... |
| f1dom2g 8961 | The domain of a one-to-one... |
| f1dom2gOLD 8962 | Obsolete version of ~ f1do... |
| f1oeng 8963 | The domain and range of a ... |
| f1domg 8964 | The domain of a one-to-one... |
| f1oen 8965 | The domain and range of a ... |
| f1dom 8966 | The domain of a one-to-one... |
| brsdom 8967 | Strict dominance relation,... |
| isfi 8968 | Express " ` A ` is finite"... |
| enssdom 8969 | Equinumerosity implies dom... |
| dfdom2 8970 | Alternate definition of do... |
| endom 8971 | Equinumerosity implies dom... |
| sdomdom 8972 | Strict dominance implies d... |
| sdomnen 8973 | Strict dominance implies n... |
| brdom2 8974 | Dominance in terms of stri... |
| bren2 8975 | Equinumerosity expressed i... |
| enrefg 8976 | Equinumerosity is reflexiv... |
| enref 8977 | Equinumerosity is reflexiv... |
| eqeng 8978 | Equality implies equinumer... |
| domrefg 8979 | Dominance is reflexive. (... |
| en2d 8980 | Equinumerosity inference f... |
| en3d 8981 | Equinumerosity inference f... |
| en2i 8982 | Equinumerosity inference f... |
| en3i 8983 | Equinumerosity inference f... |
| dom2lem 8984 | A mapping (first hypothesi... |
| dom2d 8985 | A mapping (first hypothesi... |
| dom3d 8986 | A mapping (first hypothesi... |
| dom2 8987 | A mapping (first hypothesi... |
| dom3 8988 | A mapping (first hypothesi... |
| idssen 8989 | Equality implies equinumer... |
| domssl 8990 | If ` A ` is a subset of ` ... |
| domssr 8991 | If ` C ` is a superset of ... |
| ssdomg 8992 | A set dominates its subset... |
| ener 8993 | Equinumerosity is an equiv... |
| ensymb 8994 | Symmetry of equinumerosity... |
| ensym 8995 | Symmetry of equinumerosity... |
| ensymi 8996 | Symmetry of equinumerosity... |
| ensymd 8997 | Symmetry of equinumerosity... |
| entr 8998 | Transitivity of equinumero... |
| domtr 8999 | Transitivity of dominance ... |
| entri 9000 | A chained equinumerosity i... |
| entr2i 9001 | A chained equinumerosity i... |
| entr3i 9002 | A chained equinumerosity i... |
| entr4i 9003 | A chained equinumerosity i... |
| endomtr 9004 | Transitivity of equinumero... |
| domentr 9005 | Transitivity of dominance ... |
| f1imaeng 9006 | If a function is one-to-on... |
| f1imaen2g 9007 | If a function is one-to-on... |
| f1imaen 9008 | If a function is one-to-on... |
| en0 9009 | The empty set is equinumer... |
| en0OLD 9010 | Obsolete version of ~ en0 ... |
| en0ALT 9011 | Shorter proof of ~ en0 , d... |
| en0r 9012 | The empty set is equinumer... |
| ensn1 9013 | A singleton is equinumerou... |
| ensn1OLD 9014 | Obsolete version of ~ ensn... |
| ensn1g 9015 | A singleton is equinumerou... |
| enpr1g 9016 | ` { A , A } ` has only one... |
| en1 9017 | A set is equinumerous to o... |
| en1OLD 9018 | Obsolete version of ~ en1 ... |
| en1b 9019 | A set is equinumerous to o... |
| en1bOLD 9020 | Obsolete version of ~ en1b... |
| reuen1 9021 | Two ways to express "exact... |
| euen1 9022 | Two ways to express "exact... |
| euen1b 9023 | Two ways to express " ` A ... |
| en1uniel 9024 | A singleton contains its s... |
| en1unielOLD 9025 | Obsolete version of ~ en1u... |
| 2dom 9026 | A set that dominates ordin... |
| fundmen 9027 | A function is equinumerous... |
| fundmeng 9028 | A function is equinumerous... |
| cnven 9029 | A relational set is equinu... |
| cnvct 9030 | If a set is countable, so ... |
| fndmeng 9031 | A function is equinumerate... |
| mapsnend 9032 | Set exponentiation to a si... |
| mapsnen 9033 | Set exponentiation to a si... |
| snmapen 9034 | Set exponentiation: a sing... |
| snmapen1 9035 | Set exponentiation: a sing... |
| map1 9036 | Set exponentiation: ordina... |
| en2sn 9037 | Two singletons are equinum... |
| en2snOLD 9038 | Obsolete version of ~ en2s... |
| en2snOLDOLD 9039 | Obsolete version of ~ en2s... |
| snfi 9040 | A singleton is finite. (C... |
| fiprc 9041 | The class of finite sets i... |
| unen 9042 | Equinumerosity of union of... |
| enrefnn 9043 | Equinumerosity is reflexiv... |
| en2prd 9044 | Two unordered pairs are eq... |
| enpr2d 9045 | A pair with distinct eleme... |
| enpr2dOLD 9046 | Obsolete version of ~ enpr... |
| ssct 9047 | Any subset of a countable ... |
| ssctOLD 9048 | Obsolete version of ~ ssct... |
| difsnen 9049 | All decrements of a set ar... |
| domdifsn 9050 | Dominance over a set with ... |
| xpsnen 9051 | A set is equinumerous to i... |
| xpsneng 9052 | A set is equinumerous to i... |
| xp1en 9053 | One times a cardinal numbe... |
| endisj 9054 | Any two sets are equinumer... |
| undom 9055 | Dominance law for union. ... |
| undomOLD 9056 | Obsolete version of ~ undo... |
| xpcomf1o 9057 | The canonical bijection fr... |
| xpcomco 9058 | Composition with the bijec... |
| xpcomen 9059 | Commutative law for equinu... |
| xpcomeng 9060 | Commutative law for equinu... |
| xpsnen2g 9061 | A set is equinumerous to i... |
| xpassen 9062 | Associative law for equinu... |
| xpdom2 9063 | Dominance law for Cartesia... |
| xpdom2g 9064 | Dominance law for Cartesia... |
| xpdom1g 9065 | Dominance law for Cartesia... |
| xpdom3 9066 | A set is dominated by its ... |
| xpdom1 9067 | Dominance law for Cartesia... |
| domunsncan 9068 | A singleton cancellation l... |
| omxpenlem 9069 | Lemma for ~ omxpen . (Con... |
| omxpen 9070 | The cardinal and ordinal p... |
| omf1o 9071 | Construct an explicit bije... |
| pw2f1olem 9072 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9073 | The power set of a set is ... |
| pw2eng 9074 | The power set of a set is ... |
| pw2en 9075 | The power set of a set is ... |
| fopwdom 9076 | Covering implies injection... |
| enfixsn 9077 | Given two equipollent sets... |
| sucdom2OLD 9078 | Obsolete version of ~ sucd... |
| sbthlem1 9079 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9080 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9081 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9082 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9083 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9084 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9085 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9086 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9087 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9088 | Lemma for ~ sbth . (Contr... |
| sbth 9089 | Schroeder-Bernstein Theore... |
| sbthb 9090 | Schroeder-Bernstein Theore... |
| sbthcl 9091 | Schroeder-Bernstein Theore... |
| dfsdom2 9092 | Alternate definition of st... |
| brsdom2 9093 | Alternate definition of st... |
| sdomnsym 9094 | Strict dominance is asymme... |
| domnsym 9095 | Theorem 22(i) of [Suppes] ... |
| 0domg 9096 | Any set dominates the empt... |
| 0domgOLD 9097 | Obsolete version of ~ 0dom... |
| dom0 9098 | A set dominated by the emp... |
| dom0OLD 9099 | Obsolete version of ~ dom0... |
| 0sdomg 9100 | A set strictly dominates t... |
| 0sdomgOLD 9101 | Obsolete version of ~ 0sdo... |
| 0dom 9102 | Any set dominates the empt... |
| 0sdom 9103 | A set strictly dominates t... |
| sdom0 9104 | The empty set does not str... |
| sdom0OLD 9105 | Obsolete version of ~ sdom... |
| sdomdomtr 9106 | Transitivity of strict dom... |
| sdomentr 9107 | Transitivity of strict dom... |
| domsdomtr 9108 | Transitivity of dominance ... |
| ensdomtr 9109 | Transitivity of equinumero... |
| sdomirr 9110 | Strict dominance is irrefl... |
| sdomtr 9111 | Strict dominance is transi... |
| sdomn2lp 9112 | Strict dominance has no 2-... |
| enen1 9113 | Equality-like theorem for ... |
| enen2 9114 | Equality-like theorem for ... |
| domen1 9115 | Equality-like theorem for ... |
| domen2 9116 | Equality-like theorem for ... |
| sdomen1 9117 | Equality-like theorem for ... |
| sdomen2 9118 | Equality-like theorem for ... |
| domtriord 9119 | Dominance is trichotomous ... |
| sdomel 9120 | For ordinals, strict domin... |
| sdomdif 9121 | The difference of a set fr... |
| onsdominel 9122 | An ordinal with more eleme... |
| domunsn 9123 | Dominance over a set with ... |
| fodomr 9124 | There exists a mapping fro... |
| pwdom 9125 | Injection of sets implies ... |
| canth2 9126 | Cantor's Theorem. No set ... |
| canth2g 9127 | Cantor's theorem with the ... |
| 2pwuninel 9128 | The power set of the power... |
| 2pwne 9129 | No set equals the power se... |
| disjen 9130 | A stronger form of ~ pwuni... |
| disjenex 9131 | Existence version of ~ dis... |
| domss2 9132 | A corollary of ~ disjenex ... |
| domssex2 9133 | A corollary of ~ disjenex ... |
| domssex 9134 | Weakening of ~ domssex2 to... |
| xpf1o 9135 | Construct a bijection on a... |
| xpen 9136 | Equinumerosity law for Car... |
| mapen 9137 | Two set exponentiations ar... |
| mapdom1 9138 | Order-preserving property ... |
| mapxpen 9139 | Equinumerosity law for dou... |
| xpmapenlem 9140 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9141 | Equinumerosity law for set... |
| mapunen 9142 | Equinumerosity law for set... |
| map2xp 9143 | A cardinal power with expo... |
| mapdom2 9144 | Order-preserving property ... |
| mapdom3 9145 | Set exponentiation dominat... |
| pwen 9146 | If two sets are equinumero... |
| ssenen 9147 | Equinumerosity of equinume... |
| limenpsi 9148 | A limit ordinal is equinum... |
| limensuci 9149 | A limit ordinal is equinum... |
| limensuc 9150 | A limit ordinal is equinum... |
| infensuc 9151 | Any infinite ordinal is eq... |
| dif1enlem 9152 | Lemma for ~ rexdif1en and ... |
| dif1enlemOLD 9153 | Obsolete version of ~ dif1... |
| rexdif1en 9154 | If a set is equinumerous t... |
| rexdif1enOLD 9155 | Obsolete version of ~ rexd... |
| dif1en 9156 | If a set ` A ` is equinume... |
| dif1ennn 9157 | If a set ` A ` is equinume... |
| dif1enOLD 9158 | Obsolete version of ~ dif1... |
| findcard 9159 | Schema for induction on th... |
| findcard2 9160 | Schema for induction on th... |
| findcard2s 9161 | Variation of ~ findcard2 r... |
| findcard2d 9162 | Deduction version of ~ fin... |
| nnfi 9163 | Natural numbers are finite... |
| pssnn 9164 | A proper subset of a natur... |
| ssnnfi 9165 | A subset of a natural numb... |
| ssnnfiOLD 9166 | Obsolete version of ~ ssnn... |
| 0fin 9167 | The empty set is finite. ... |
| unfi 9168 | The union of two finite se... |
| ssfi 9169 | A subset of a finite set i... |
| ssfiALT 9170 | Shorter proof of ~ ssfi us... |
| imafi 9171 | Images of finite sets are ... |
| pwfir 9172 | If the power set of a set ... |
| pwfilem 9173 | Lemma for ~ pwfi . (Contr... |
| pwfi 9174 | The power set of a finite ... |
| diffi 9175 | If ` A ` is finite, ` ( A ... |
| cnvfi 9176 | If a set is finite, its co... |
| fnfi 9177 | A version of ~ fnex for fi... |
| f1oenfi 9178 | If the domain of a one-to-... |
| f1oenfirn 9179 | If the range of a one-to-o... |
| f1domfi 9180 | If the codomain of a one-t... |
| f1domfi2 9181 | If the domain of a one-to-... |
| enreffi 9182 | Equinumerosity is reflexiv... |
| ensymfib 9183 | Symmetry of equinumerosity... |
| entrfil 9184 | Transitivity of equinumero... |
| enfii 9185 | A set equinumerous to a fi... |
| enfi 9186 | Equinumerous sets have the... |
| enfiALT 9187 | Shorter proof of ~ enfi us... |
| domfi 9188 | A set dominated by a finit... |
| entrfi 9189 | Transitivity of equinumero... |
| entrfir 9190 | Transitivity of equinumero... |
| domtrfil 9191 | Transitivity of dominance ... |
| domtrfi 9192 | Transitivity of dominance ... |
| domtrfir 9193 | Transitivity of dominance ... |
| f1imaenfi 9194 | If a function is one-to-on... |
| ssdomfi 9195 | A finite set dominates its... |
| ssdomfi2 9196 | A set dominates its finite... |
| sbthfilem 9197 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9198 | Schroeder-Bernstein Theore... |
| domnsymfi 9199 | If a set dominates a finit... |
| sdomdomtrfi 9200 | Transitivity of strict dom... |
| domsdomtrfi 9201 | Transitivity of dominance ... |
| sucdom2 9202 | Strict dominance of a set ... |
| phplem1 9203 | Lemma for Pigeonhole Princ... |
| phplem2 9204 | Lemma for Pigeonhole Princ... |
| nneneq 9205 | Two equinumerous natural n... |
| php 9206 | Pigeonhole Principle. A n... |
| php2 9207 | Corollary of Pigeonhole Pr... |
| php3 9208 | Corollary of Pigeonhole Pr... |
| php4 9209 | Corollary of the Pigeonhol... |
| php5 9210 | Corollary of the Pigeonhol... |
| phpeqd 9211 | Corollary of the Pigeonhol... |
| nndomog 9212 | Cardinal ordering agrees w... |
| phplem1OLD 9213 | Obsolete lemma for ~ php .... |
| phplem2OLD 9214 | Obsolete lemma for ~ php .... |
| phplem3OLD 9215 | Obsolete version of ~ phpl... |
| phplem4OLD 9216 | Obsolete version of ~ phpl... |
| nneneqOLD 9217 | Obsolete version of ~ nnen... |
| phpOLD 9218 | Obsolete version of ~ php ... |
| php2OLD 9219 | Obsolete version of ~ php2... |
| php3OLD 9220 | Obsolete version of ~ php3... |
| phpeqdOLD 9221 | Obsolete version of ~ phpe... |
| nndomogOLD 9222 | Obsolete version of ~ nndo... |
| snnen2oOLD 9223 | Obsolete version of ~ snne... |
| onomeneq 9224 | An ordinal number equinume... |
| onomeneqOLD 9225 | Obsolete version of ~ onom... |
| onfin 9226 | An ordinal number is finit... |
| onfin2 9227 | A set is a natural number ... |
| nnfiOLD 9228 | Obsolete version of ~ nnfi... |
| nndomo 9229 | Cardinal ordering agrees w... |
| nnsdomo 9230 | Cardinal ordering agrees w... |
| sucdom 9231 | Strict dominance of a set ... |
| sucdomOLD 9232 | Obsolete version of ~ sucd... |
| snnen2o 9233 | A singleton ` { A } ` is n... |
| 0sdom1dom 9234 | Strict dominance over 0 is... |
| 0sdom1domALT 9235 | Alternate proof of ~ 0sdom... |
| 1sdom2 9236 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9237 | Alternate proof of ~ 1sdom... |
| sdom1 9238 | A set has less than one me... |
| sdom1OLD 9239 | Obsolete version of ~ sdom... |
| modom 9240 | Two ways to express "at mo... |
| modom2 9241 | Two ways to express "at mo... |
| rex2dom 9242 | A set that has at least 2 ... |
| 1sdom2dom 9243 | Strict dominance over 1 is... |
| 1sdom 9244 | A set that strictly domina... |
| 1sdomOLD 9245 | Obsolete version of ~ 1sdo... |
| unxpdomlem1 9246 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9247 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9248 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9249 | Cartesian product dominate... |
| unxpdom2 9250 | Corollary of ~ unxpdom . ... |
| sucxpdom 9251 | Cartesian product dominate... |
| pssinf 9252 | A set equinumerous to a pr... |
| fisseneq 9253 | A finite set is equal to i... |
| ominf 9254 | The set of natural numbers... |
| ominfOLD 9255 | Obsolete version of ~ omin... |
| isinf 9256 | Any set that is not finite... |
| isinfOLD 9257 | Obsolete version of ~ isin... |
| fineqvlem 9258 | Lemma for ~ fineqv . (Con... |
| fineqv 9259 | If the Axiom of Infinity i... |
| enfiiOLD 9260 | Obsolete version of ~ enfi... |
| pssnnOLD 9261 | Obsolete version of ~ pssn... |
| xpfir 9262 | The components of a nonemp... |
| ssfid 9263 | A subset of a finite set i... |
| infi 9264 | The intersection of two se... |
| rabfi 9265 | A restricted class built f... |
| finresfin 9266 | The restriction of a finit... |
| f1finf1o 9267 | Any injection from one fin... |
| f1finf1oOLD 9268 | Obsolete version of ~ f1fi... |
| nfielex 9269 | If a class is not finite, ... |
| en1eqsn 9270 | A set with one element is ... |
| en1eqsnOLD 9271 | Obsolete version of ~ en1e... |
| en1eqsnbi 9272 | A set containing an elemen... |
| dif1ennnALT 9273 | Alternate proof of ~ dif1e... |
| enp1ilem 9274 | Lemma for uses of ~ enp1i ... |
| enp1i 9275 | Proof induction for ~ en2 ... |
| enp1iOLD 9276 | Obsolete version of ~ enp1... |
| en2 9277 | A set equinumerous to ordi... |
| en3 9278 | A set equinumerous to ordi... |
| en4 9279 | A set equinumerous to ordi... |
| findcard2OLD 9280 | Obsolete version of ~ find... |
| findcard3 9281 | Schema for strong inductio... |
| findcard3OLD 9282 | Obsolete version of ~ find... |
| ac6sfi 9283 | A version of ~ ac6s for fi... |
| frfi 9284 | A partial order is well-fo... |
| fimax2g 9285 | A finite set has a maximum... |
| fimaxg 9286 | A finite set has a maximum... |
| fisupg 9287 | Lemma showing existence an... |
| wofi 9288 | A total order on a finite ... |
| ordunifi 9289 | The maximum of a finite co... |
| nnunifi 9290 | The union (supremum) of a ... |
| unblem1 9291 | Lemma for ~ unbnn . After... |
| unblem2 9292 | Lemma for ~ unbnn . The v... |
| unblem3 9293 | Lemma for ~ unbnn . The v... |
| unblem4 9294 | Lemma for ~ unbnn . The f... |
| unbnn 9295 | Any unbounded subset of na... |
| unbnn2 9296 | Version of ~ unbnn that do... |
| isfinite2 9297 | Any set strictly dominated... |
| nnsdomg 9298 | Omega strictly dominates a... |
| nnsdomgOLD 9299 | Obsolete version of ~ nnsd... |
| isfiniteg 9300 | A set is finite iff it is ... |
| infsdomnn 9301 | An infinite set strictly d... |
| infsdomnnOLD 9302 | Obsolete version of ~ infs... |
| infn0 9303 | An infinite set is not emp... |
| infn0ALT 9304 | Shorter proof of ~ infn0 u... |
| fin2inf 9305 | This (useless) theorem, wh... |
| unfilem1 9306 | Lemma for proving that the... |
| unfilem2 9307 | Lemma for proving that the... |
| unfilem3 9308 | Lemma for proving that the... |
| unfiOLD 9309 | Obsolete version of ~ unfi... |
| unfir 9310 | If a union is finite, the ... |
| unfi2 9311 | The union of two finite se... |
| difinf 9312 | An infinite set ` A ` minu... |
| xpfi 9313 | The Cartesian product of t... |
| xpfiOLD 9314 | Obsolete version of ~ xpfi... |
| 3xpfi 9315 | The Cartesian product of t... |
| domunfican 9316 | A finite set union cancell... |
| infcntss 9317 | Every infinite set has a d... |
| prfi 9318 | An unordered pair is finit... |
| tpfi 9319 | An unordered triple is fin... |
| fiint 9320 | Equivalent ways of stating... |
| fodomfi 9321 | An onto function implies d... |
| fodomfib 9322 | Equivalence of an onto map... |
| fofinf1o 9323 | Any surjection from one fi... |
| rneqdmfinf1o 9324 | Any function from a finite... |
| fidomdm 9325 | Any finite set dominates i... |
| dmfi 9326 | The domain of a finite set... |
| fundmfibi 9327 | A function is finite if an... |
| resfnfinfin 9328 | The restriction of a funct... |
| residfi 9329 | A restricted identity func... |
| cnvfiALT 9330 | Shorter proof of ~ cnvfi u... |
| rnfi 9331 | The range of a finite set ... |
| f1dmvrnfibi 9332 | A one-to-one function whos... |
| f1vrnfibi 9333 | A one-to-one function whic... |
| fofi 9334 | If an onto function has a ... |
| f1fi 9335 | If a 1-to-1 function has a... |
| iunfi 9336 | The finite union of finite... |
| unifi 9337 | The finite union of finite... |
| unifi2 9338 | The finite union of finite... |
| infssuni 9339 | If an infinite set ` A ` i... |
| unirnffid 9340 | The union of the range of ... |
| imafiALT 9341 | Shorter proof of ~ imafi u... |
| pwfilemOLD 9342 | Obsolete version of ~ pwfi... |
| pwfiOLD 9343 | Obsolete version of ~ pwfi... |
| mapfi 9344 | Set exponentiation of fini... |
| ixpfi 9345 | A Cartesian product of fin... |
| ixpfi2 9346 | A Cartesian product of fin... |
| mptfi 9347 | A finite mapping set is fi... |
| abrexfi 9348 | An image set from a finite... |
| cnvimamptfin 9349 | A preimage of a mapping wi... |
| elfpw 9350 | Membership in a class of f... |
| unifpw 9351 | A set is the union of its ... |
| f1opwfi 9352 | A one-to-one mapping induc... |
| fissuni 9353 | A finite subset of a union... |
| fipreima 9354 | Given a finite subset ` A ... |
| finsschain 9355 | A finite subset of the uni... |
| indexfi 9356 | If for every element of a ... |
| relfsupp 9359 | The property of a function... |
| relprcnfsupp 9360 | A proper class is never fi... |
| isfsupp 9361 | The property of a class to... |
| isfsuppd 9362 | Deduction form of ~ isfsup... |
| funisfsupp 9363 | The property of a function... |
| fsuppimp 9364 | Implications of a class be... |
| fsuppimpd 9365 | A finitely supported funct... |
| fisuppfi 9366 | A function on a finite set... |
| fidmfisupp 9367 | A function with a finite d... |
| fdmfisuppfi 9368 | The support of a function ... |
| fdmfifsupp 9369 | A function with a finite d... |
| fsuppmptdm 9370 | A mapping with a finite do... |
| fndmfisuppfi 9371 | The support of a function ... |
| fndmfifsupp 9372 | A function with a finite d... |
| suppeqfsuppbi 9373 | If two functions have the ... |
| suppssfifsupp 9374 | If the support of a functi... |
| fsuppsssupp 9375 | If the support of a functi... |
| fsuppxpfi 9376 | The cartesian product of t... |
| fczfsuppd 9377 | A constant function with v... |
| fsuppun 9378 | The union of two finitely ... |
| fsuppunfi 9379 | The union of the support o... |
| fsuppunbi 9380 | If the union of two classe... |
| 0fsupp 9381 | The empty set is a finitel... |
| snopfsupp 9382 | A singleton containing an ... |
| funsnfsupp 9383 | Finite support for a funct... |
| fsuppres 9384 | The restriction of a finit... |
| fmptssfisupp 9385 | The restriction of a mappi... |
| ressuppfi 9386 | If the support of the rest... |
| resfsupp 9387 | If the restriction of a fu... |
| resfifsupp 9388 | The restriction of a funct... |
| ffsuppbi 9389 | Two ways of saying that a ... |
| fsuppmptif 9390 | A function mapping an argu... |
| sniffsupp 9391 | A function mapping all but... |
| fsuppcolem 9392 | Lemma for ~ fsuppco . For... |
| fsuppco 9393 | The composition of a 1-1 f... |
| fsuppco2 9394 | The composition of a funct... |
| fsuppcor 9395 | The composition of a funct... |
| mapfienlem1 9396 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9397 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9398 | Lemma 3 for ~ mapfien . (... |
| mapfien 9399 | A bijection of the base se... |
| mapfien2 9400 | Equinumerousity relation f... |
| fival 9403 | The set of all the finite ... |
| elfi 9404 | Specific properties of an ... |
| elfi2 9405 | The empty intersection nee... |
| elfir 9406 | Sufficient condition for a... |
| intrnfi 9407 | Sufficient condition for t... |
| iinfi 9408 | An indexed intersection of... |
| inelfi 9409 | The intersection of two se... |
| ssfii 9410 | Any element of a set ` A `... |
| fi0 9411 | The set of finite intersec... |
| fieq0 9412 | A set is empty iff the cla... |
| fiin 9413 | The elements of ` ( fi `` ... |
| dffi2 9414 | The set of finite intersec... |
| fiss 9415 | Subset relationship for fu... |
| inficl 9416 | A set which is closed unde... |
| fipwuni 9417 | The set of finite intersec... |
| fisn 9418 | A singleton is closed unde... |
| fiuni 9419 | The union of the finite in... |
| fipwss 9420 | If a set is a family of su... |
| elfiun 9421 | A finite intersection of e... |
| dffi3 9422 | The set of finite intersec... |
| fifo 9423 | Describe a surjection from... |
| marypha1lem 9424 | Core induction for Philip ... |
| marypha1 9425 | (Philip) Hall's marriage t... |
| marypha2lem1 9426 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9427 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9428 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9429 | Lemma for ~ marypha2 . Pr... |
| marypha2 9430 | Version of ~ marypha1 usin... |
| dfsup2 9435 | Quantifier-free definition... |
| supeq1 9436 | Equality theorem for supre... |
| supeq1d 9437 | Equality deduction for sup... |
| supeq1i 9438 | Equality inference for sup... |
| supeq2 9439 | Equality theorem for supre... |
| supeq3 9440 | Equality theorem for supre... |
| supeq123d 9441 | Equality deduction for sup... |
| nfsup 9442 | Hypothesis builder for sup... |
| supmo 9443 | Any class ` B ` has at mos... |
| supexd 9444 | A supremum is a set. (Con... |
| supeu 9445 | A supremum is unique. Sim... |
| supval2 9446 | Alternate expression for t... |
| eqsup 9447 | Sufficient condition for a... |
| eqsupd 9448 | Sufficient condition for a... |
| supcl 9449 | A supremum belongs to its ... |
| supub 9450 | A supremum is an upper bou... |
| suplub 9451 | A supremum is the least up... |
| suplub2 9452 | Bidirectional form of ~ su... |
| supnub 9453 | An upper bound is not less... |
| supex 9454 | A supremum is a set. (Con... |
| sup00 9455 | The supremum under an empt... |
| sup0riota 9456 | The supremum of an empty s... |
| sup0 9457 | The supremum of an empty s... |
| supmax 9458 | The greatest element of a ... |
| fisup2g 9459 | A finite set satisfies the... |
| fisupcl 9460 | A nonempty finite set cont... |
| supgtoreq 9461 | The supremum of a finite s... |
| suppr 9462 | The supremum of a pair. (... |
| supsn 9463 | The supremum of a singleto... |
| supisolem 9464 | Lemma for ~ supiso . (Con... |
| supisoex 9465 | Lemma for ~ supiso . (Con... |
| supiso 9466 | Image of a supremum under ... |
| infeq1 9467 | Equality theorem for infim... |
| infeq1d 9468 | Equality deduction for inf... |
| infeq1i 9469 | Equality inference for inf... |
| infeq2 9470 | Equality theorem for infim... |
| infeq3 9471 | Equality theorem for infim... |
| infeq123d 9472 | Equality deduction for inf... |
| nfinf 9473 | Hypothesis builder for inf... |
| infexd 9474 | An infimum is a set. (Con... |
| eqinf 9475 | Sufficient condition for a... |
| eqinfd 9476 | Sufficient condition for a... |
| infval 9477 | Alternate expression for t... |
| infcllem 9478 | Lemma for ~ infcl , ~ infl... |
| infcl 9479 | An infimum belongs to its ... |
| inflb 9480 | An infimum is a lower boun... |
| infglb 9481 | An infimum is the greatest... |
| infglbb 9482 | Bidirectional form of ~ in... |
| infnlb 9483 | A lower bound is not great... |
| infex 9484 | An infimum is a set. (Con... |
| infmin 9485 | The smallest element of a ... |
| infmo 9486 | Any class ` B ` has at mos... |
| infeu 9487 | An infimum is unique. (Co... |
| fimin2g 9488 | A finite set has a minimum... |
| fiming 9489 | A finite set has a minimum... |
| fiinfg 9490 | Lemma showing existence an... |
| fiinf2g 9491 | A finite set satisfies the... |
| fiinfcl 9492 | A nonempty finite set cont... |
| infltoreq 9493 | The infimum of a finite se... |
| infpr 9494 | The infimum of a pair. (C... |
| infsupprpr 9495 | The infimum of a proper pa... |
| infsn 9496 | The infimum of a singleton... |
| inf00 9497 | The infimum regarding an e... |
| infempty 9498 | The infimum of an empty se... |
| infiso 9499 | Image of an infimum under ... |
| dfoi 9502 | Rewrite ~ df-oi with abbre... |
| oieq1 9503 | Equality theorem for ordin... |
| oieq2 9504 | Equality theorem for ordin... |
| nfoi 9505 | Hypothesis builder for ord... |
| ordiso2 9506 | Generalize ~ ordiso to pro... |
| ordiso 9507 | Order-isomorphic ordinal n... |
| ordtypecbv 9508 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9509 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9510 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9511 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9512 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9513 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9514 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9515 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9516 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9517 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9518 | Lemma for ~ ordtype . Usi... |
| oi0 9519 | Definition of the ordinal ... |
| oicl 9520 | The order type of the well... |
| oif 9521 | The order isomorphism of t... |
| oiiso2 9522 | The order isomorphism of t... |
| ordtype 9523 | For any set-like well-orde... |
| oiiniseg 9524 | ` ran F ` is an initial se... |
| ordtype2 9525 | For any set-like well-orde... |
| oiexg 9526 | The order isomorphism on a... |
| oion 9527 | The order type of the well... |
| oiiso 9528 | The order isomorphism of t... |
| oien 9529 | The order type of a well-o... |
| oieu 9530 | Uniqueness of the unique o... |
| oismo 9531 | When ` A ` is a subclass o... |
| oiid 9532 | The order type of an ordin... |
| hartogslem1 9533 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9534 | Lemma for ~ hartogs . (Co... |
| hartogs 9535 | The class of ordinals domi... |
| wofib 9536 | The only sets which are we... |
| wemaplem1 9537 | Value of the lexicographic... |
| wemaplem2 9538 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9539 | Lemma for ~ wemapso . Tra... |
| wemappo 9540 | Construct lexicographic or... |
| wemapsolem 9541 | Lemma for ~ wemapso . (Co... |
| wemapso 9542 | Construct lexicographic or... |
| wemapso2lem 9543 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9544 | An alternative to having a... |
| card2on 9545 | The alternate definition o... |
| card2inf 9546 | The alternate definition o... |
| harf 9549 | Functionality of the Harto... |
| harcl 9550 | Values of the Hartogs func... |
| harval 9551 | Function value of the Hart... |
| elharval 9552 | The Hartogs number of a se... |
| harndom 9553 | The Hartogs number of a se... |
| harword 9554 | Weak ordering property of ... |
| relwdom 9557 | Weak dominance is a relati... |
| brwdom 9558 | Property of weak dominance... |
| brwdomi 9559 | Property of weak dominance... |
| brwdomn0 9560 | Weak dominance over nonemp... |
| 0wdom 9561 | Any set weakly dominates t... |
| fowdom 9562 | An onto function implies w... |
| wdomref 9563 | Reflexivity of weak domina... |
| brwdom2 9564 | Alternate characterization... |
| domwdom 9565 | Weak dominance is implied ... |
| wdomtr 9566 | Transitivity of weak domin... |
| wdomen1 9567 | Equality-like theorem for ... |
| wdomen2 9568 | Equality-like theorem for ... |
| wdompwdom 9569 | Weak dominance strengthens... |
| canthwdom 9570 | Cantor's Theorem, stated u... |
| wdom2d 9571 | Deduce weak dominance from... |
| wdomd 9572 | Deduce weak dominance from... |
| brwdom3 9573 | Condition for weak dominan... |
| brwdom3i 9574 | Weak dominance implies exi... |
| unwdomg 9575 | Weak dominance of a (disjo... |
| xpwdomg 9576 | Weak dominance of a Cartes... |
| wdomima2g 9577 | A set is weakly dominant o... |
| wdomimag 9578 | A set is weakly dominant o... |
| unxpwdom2 9579 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9580 | If a Cartesian product is ... |
| ixpiunwdom 9581 | Describe an onto function ... |
| harwdom 9582 | The value of the Hartogs f... |
| axreg2 9584 | Axiom of Regularity expres... |
| zfregcl 9585 | The Axiom of Regularity wi... |
| zfreg 9586 | The Axiom of Regularity us... |
| elirrv 9587 | The membership relation is... |
| elirr 9588 | No class is a member of it... |
| elneq 9589 | A class is not equal to an... |
| nelaneq 9590 | A class is not an element ... |
| epinid0 9591 | The membership relation an... |
| sucprcreg 9592 | A class is equal to its su... |
| ruv 9593 | The Russell class is equal... |
| ruALT 9594 | Alternate proof of ~ ru , ... |
| disjcsn 9595 | A class is disjoint from i... |
| zfregfr 9596 | The membership relation is... |
| en2lp 9597 | No class has 2-cycle membe... |
| elnanel 9598 | Two classes are not elemen... |
| cnvepnep 9599 | The membership (epsilon) r... |
| epnsym 9600 | The membership (epsilon) r... |
| elnotel 9601 | A class cannot be an eleme... |
| elnel 9602 | A class cannot be an eleme... |
| en3lplem1 9603 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9604 | Lemma for ~ en3lp . (Cont... |
| en3lp 9605 | No class has 3-cycle membe... |
| preleqg 9606 | Equality of two unordered ... |
| preleq 9607 | Equality of two unordered ... |
| preleqALT 9608 | Alternate proof of ~ prele... |
| opthreg 9609 | Theorem for alternate repr... |
| suc11reg 9610 | The successor operation be... |
| dford2 9611 | Assuming ~ ax-reg , an ord... |
| inf0 9612 | Existence of ` _om ` impli... |
| inf1 9613 | Variation of Axiom of Infi... |
| inf2 9614 | Variation of Axiom of Infi... |
| inf3lema 9615 | Lemma for our Axiom of Inf... |
| inf3lemb 9616 | Lemma for our Axiom of Inf... |
| inf3lemc 9617 | Lemma for our Axiom of Inf... |
| inf3lemd 9618 | Lemma for our Axiom of Inf... |
| inf3lem1 9619 | Lemma for our Axiom of Inf... |
| inf3lem2 9620 | Lemma for our Axiom of Inf... |
| inf3lem3 9621 | Lemma for our Axiom of Inf... |
| inf3lem4 9622 | Lemma for our Axiom of Inf... |
| inf3lem5 9623 | Lemma for our Axiom of Inf... |
| inf3lem6 9624 | Lemma for our Axiom of Inf... |
| inf3lem7 9625 | Lemma for our Axiom of Inf... |
| inf3 9626 | Our Axiom of Infinity ~ ax... |
| infeq5i 9627 | Half of ~ infeq5 . (Contr... |
| infeq5 9628 | The statement "there exist... |
| zfinf 9630 | Axiom of Infinity expresse... |
| axinf2 9631 | A standard version of Axio... |
| zfinf2 9633 | A standard version of the ... |
| omex 9634 | The existence of omega (th... |
| axinf 9635 | The first version of the A... |
| inf5 9636 | The statement "there exist... |
| omelon 9637 | Omega is an ordinal number... |
| dfom3 9638 | The class of natural numbe... |
| elom3 9639 | A simplification of ~ elom... |
| dfom4 9640 | A simplification of ~ df-o... |
| dfom5 9641 | ` _om ` is the smallest li... |
| oancom 9642 | Ordinal addition is not co... |
| isfinite 9643 | A set is finite iff it is ... |
| fict 9644 | A finite set is countable ... |
| nnsdom 9645 | A natural number is strict... |
| omenps 9646 | Omega is equinumerous to a... |
| omensuc 9647 | The set of natural numbers... |
| infdifsn 9648 | Removing a singleton from ... |
| infdiffi 9649 | Removing a finite set from... |
| unbnn3 9650 | Any unbounded subset of na... |
| noinfep 9651 | Using the Axiom of Regular... |
| cantnffval 9654 | The value of the Cantor no... |
| cantnfdm 9655 | The domain of the Cantor n... |
| cantnfvalf 9656 | Lemma for ~ cantnf . The ... |
| cantnfs 9657 | Elementhood in the set of ... |
| cantnfcl 9658 | Basic properties of the or... |
| cantnfval 9659 | The value of the Cantor no... |
| cantnfval2 9660 | Alternate expression for t... |
| cantnfsuc 9661 | The value of the recursive... |
| cantnfle 9662 | A lower bound on the ` CNF... |
| cantnflt 9663 | An upper bound on the part... |
| cantnflt2 9664 | An upper bound on the ` CN... |
| cantnff 9665 | The ` CNF ` function is a ... |
| cantnf0 9666 | The value of the zero func... |
| cantnfrescl 9667 | A function is finitely sup... |
| cantnfres 9668 | The ` CNF ` function respe... |
| cantnfp1lem1 9669 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9670 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9671 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9672 | If ` F ` is created by add... |
| oemapso 9673 | The relation ` T ` is a st... |
| oemapval 9674 | Value of the relation ` T ... |
| oemapvali 9675 | If ` F < G ` , then there ... |
| cantnflem1a 9676 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9677 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9678 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9679 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9680 | Lemma for ~ cantnf . This... |
| cantnflem2 9681 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9682 | Lemma for ~ cantnf . Here... |
| cantnflem4 9683 | Lemma for ~ cantnf . Comp... |
| cantnf 9684 | The Cantor Normal Form the... |
| oemapwe 9685 | The lexicographic order on... |
| cantnffval2 9686 | An alternate definition of... |
| cantnff1o 9687 | Simplify the isomorphism o... |
| wemapwe 9688 | Construct lexicographic or... |
| oef1o 9689 | A bijection of the base se... |
| cnfcomlem 9690 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9691 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9692 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9693 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9694 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9695 | Any infinite ordinal ` B `... |
| cnfcom3clem 9696 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9697 | Wrap the construction of ~... |
| ttrcleq 9700 | Equality theorem for trans... |
| nfttrcld 9701 | Bound variable hypothesis ... |
| nfttrcl 9702 | Bound variable hypothesis ... |
| relttrcl 9703 | The transitive closure of ... |
| brttrcl 9704 | Characterization of elemen... |
| brttrcl2 9705 | Characterization of elemen... |
| ssttrcl 9706 | If ` R ` is a relation, th... |
| ttrcltr 9707 | The transitive closure of ... |
| ttrclresv 9708 | The transitive closure of ... |
| ttrclco 9709 | Composition law for the tr... |
| cottrcl 9710 | Composition law for the tr... |
| ttrclss 9711 | If ` R ` is a subclass of ... |
| dmttrcl 9712 | The domain of a transitive... |
| rnttrcl 9713 | The range of a transitive ... |
| ttrclexg 9714 | If ` R ` is a set, then so... |
| dfttrcl2 9715 | When ` R ` is a set and a ... |
| ttrclselem1 9716 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9717 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9718 | If ` R ` is set-like over ... |
| trcl 9719 | For any set ` A ` , show t... |
| tz9.1 9720 | Every set has a transitive... |
| tz9.1c 9721 | Alternate expression for t... |
| epfrs 9722 | The strong form of the Axi... |
| zfregs 9723 | The strong form of the Axi... |
| zfregs2 9724 | Alternate strong form of t... |
| setind 9725 | Set (epsilon) induction. ... |
| setind2 9726 | Set (epsilon) induction, s... |
| tcvalg 9729 | Value of the transitive cl... |
| tcid 9730 | Defining property of the t... |
| tctr 9731 | Defining property of the t... |
| tcmin 9732 | Defining property of the t... |
| tc2 9733 | A variant of the definitio... |
| tcsni 9734 | The transitive closure of ... |
| tcss 9735 | The transitive closure fun... |
| tcel 9736 | The transitive closure fun... |
| tcidm 9737 | The transitive closure fun... |
| tc0 9738 | The transitive closure of ... |
| tc00 9739 | The transitive closure is ... |
| frmin 9740 | Every (possibly proper) su... |
| frind 9741 | A subclass of a well-found... |
| frinsg 9742 | Well-Founded Induction Sch... |
| frins 9743 | Well-Founded Induction Sch... |
| frins2f 9744 | Well-Founded Induction sch... |
| frins2 9745 | Well-Founded Induction sch... |
| frins3 9746 | Well-Founded Induction sch... |
| frr3g 9747 | Functions defined by well-... |
| frrlem15 9748 | Lemma for general well-fou... |
| frrlem16 9749 | Lemma for general well-fou... |
| frr1 9750 | Law of general well-founde... |
| frr2 9751 | Law of general well-founde... |
| frr3 9752 | Law of general well-founde... |
| r1funlim 9757 | The cumulative hierarchy o... |
| r1fnon 9758 | The cumulative hierarchy o... |
| r10 9759 | Value of the cumulative hi... |
| r1sucg 9760 | Value of the cumulative hi... |
| r1suc 9761 | Value of the cumulative hi... |
| r1limg 9762 | Value of the cumulative hi... |
| r1lim 9763 | Value of the cumulative hi... |
| r1fin 9764 | The first ` _om ` levels o... |
| r1sdom 9765 | Each stage in the cumulati... |
| r111 9766 | The cumulative hierarchy i... |
| r1tr 9767 | The cumulative hierarchy o... |
| r1tr2 9768 | The union of a cumulative ... |
| r1ordg 9769 | Ordering relation for the ... |
| r1ord3g 9770 | Ordering relation for the ... |
| r1ord 9771 | Ordering relation for the ... |
| r1ord2 9772 | Ordering relation for the ... |
| r1ord3 9773 | Ordering relation for the ... |
| r1sssuc 9774 | The value of the cumulativ... |
| r1pwss 9775 | Each set of the cumulative... |
| r1sscl 9776 | Each set of the cumulative... |
| r1val1 9777 | The value of the cumulativ... |
| tz9.12lem1 9778 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9779 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9780 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9781 | A set is well-founded if a... |
| tz9.13 9782 | Every set is well-founded,... |
| tz9.13g 9783 | Every set is well-founded,... |
| rankwflemb 9784 | Two ways of saying a set i... |
| rankf 9785 | The domain and codomain of... |
| rankon 9786 | The rank of a set is an or... |
| r1elwf 9787 | Any member of the cumulati... |
| rankvalb 9788 | Value of the rank function... |
| rankr1ai 9789 | One direction of ~ rankr1a... |
| rankvaln 9790 | Value of the rank function... |
| rankidb 9791 | Identity law for the rank ... |
| rankdmr1 9792 | A rank is a member of the ... |
| rankr1ag 9793 | A version of ~ rankr1a tha... |
| rankr1bg 9794 | A relationship between ran... |
| r1rankidb 9795 | Any set is a subset of the... |
| r1elssi 9796 | The range of the ` R1 ` fu... |
| r1elss 9797 | The range of the ` R1 ` fu... |
| pwwf 9798 | A power set is well-founde... |
| sswf 9799 | A subset of a well-founded... |
| snwf 9800 | A singleton is well-founde... |
| unwf 9801 | A binary union is well-fou... |
| prwf 9802 | An unordered pair is well-... |
| opwf 9803 | An ordered pair is well-fo... |
| unir1 9804 | The cumulative hierarchy o... |
| jech9.3 9805 | Every set belongs to some ... |
| rankwflem 9806 | Every set is well-founded,... |
| rankval 9807 | Value of the rank function... |
| rankvalg 9808 | Value of the rank function... |
| rankval2 9809 | Value of an alternate defi... |
| uniwf 9810 | A union is well-founded if... |
| rankr1clem 9811 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9812 | A relationship between the... |
| rankidn 9813 | A relationship between the... |
| rankpwi 9814 | The rank of a power set. ... |
| rankelb 9815 | The membership relation is... |
| wfelirr 9816 | A well-founded set is not ... |
| rankval3b 9817 | The value of the rank func... |
| ranksnb 9818 | The rank of a singleton. ... |
| rankonidlem 9819 | Lemma for ~ rankonid . (C... |
| rankonid 9820 | The rank of an ordinal num... |
| onwf 9821 | The ordinals are all well-... |
| onssr1 9822 | Initial segments of the or... |
| rankr1g 9823 | A relationship between the... |
| rankid 9824 | Identity law for the rank ... |
| rankr1 9825 | A relationship between the... |
| ssrankr1 9826 | A relationship between an ... |
| rankr1a 9827 | A relationship between ran... |
| r1val2 9828 | The value of the cumulativ... |
| r1val3 9829 | The value of the cumulativ... |
| rankel 9830 | The membership relation is... |
| rankval3 9831 | The value of the rank func... |
| bndrank 9832 | Any class whose elements h... |
| unbndrank 9833 | The elements of a proper c... |
| rankpw 9834 | The rank of a power set. ... |
| ranklim 9835 | The rank of a set belongs ... |
| r1pw 9836 | A stronger property of ` R... |
| r1pwALT 9837 | Alternate shorter proof of... |
| r1pwcl 9838 | The cumulative hierarchy o... |
| rankssb 9839 | The subset relation is inh... |
| rankss 9840 | The subset relation is inh... |
| rankunb 9841 | The rank of the union of t... |
| rankprb 9842 | The rank of an unordered p... |
| rankopb 9843 | The rank of an ordered pai... |
| rankuni2b 9844 | The value of the rank func... |
| ranksn 9845 | The rank of a singleton. ... |
| rankuni2 9846 | The rank of a union. Part... |
| rankun 9847 | The rank of the union of t... |
| rankpr 9848 | The rank of an unordered p... |
| rankop 9849 | The rank of an ordered pai... |
| r1rankid 9850 | Any set is a subset of the... |
| rankeq0b 9851 | A set is empty iff its ran... |
| rankeq0 9852 | A set is empty iff its ran... |
| rankr1id 9853 | The rank of the hierarchy ... |
| rankuni 9854 | The rank of a union. Part... |
| rankr1b 9855 | A relationship between ran... |
| ranksuc 9856 | The rank of a successor. ... |
| rankuniss 9857 | Upper bound of the rank of... |
| rankval4 9858 | The rank of a set is the s... |
| rankbnd 9859 | The rank of a set is bound... |
| rankbnd2 9860 | The rank of a set is bound... |
| rankc1 9861 | A relationship that can be... |
| rankc2 9862 | A relationship that can be... |
| rankelun 9863 | Rank membership is inherit... |
| rankelpr 9864 | Rank membership is inherit... |
| rankelop 9865 | Rank membership is inherit... |
| rankxpl 9866 | A lower bound on the rank ... |
| rankxpu 9867 | An upper bound on the rank... |
| rankfu 9868 | An upper bound on the rank... |
| rankmapu 9869 | An upper bound on the rank... |
| rankxplim 9870 | The rank of a Cartesian pr... |
| rankxplim2 9871 | If the rank of a Cartesian... |
| rankxplim3 9872 | The rank of a Cartesian pr... |
| rankxpsuc 9873 | The rank of a Cartesian pr... |
| tcwf 9874 | The transitive closure fun... |
| tcrank 9875 | This theorem expresses two... |
| scottex 9876 | Scott's trick collects all... |
| scott0 9877 | Scott's trick collects all... |
| scottexs 9878 | Theorem scheme version of ... |
| scott0s 9879 | Theorem scheme version of ... |
| cplem1 9880 | Lemma for the Collection P... |
| cplem2 9881 | Lemma for the Collection P... |
| cp 9882 | Collection Principle. Thi... |
| bnd 9883 | A very strong generalizati... |
| bnd2 9884 | A variant of the Boundedne... |
| kardex 9885 | The collection of all sets... |
| karden 9886 | If we allow the Axiom of R... |
| htalem 9887 | Lemma for defining an emul... |
| hta 9888 | A ZFC emulation of Hilbert... |
| djueq12 9895 | Equality theorem for disjo... |
| djueq1 9896 | Equality theorem for disjo... |
| djueq2 9897 | Equality theorem for disjo... |
| nfdju 9898 | Bound-variable hypothesis ... |
| djuex 9899 | The disjoint union of sets... |
| djuexb 9900 | The disjoint union of two ... |
| djulcl 9901 | Left closure of disjoint u... |
| djurcl 9902 | Right closure of disjoint ... |
| djulf1o 9903 | The left injection functio... |
| djurf1o 9904 | The right injection functi... |
| inlresf 9905 | The left injection restric... |
| inlresf1 9906 | The left injection restric... |
| inrresf 9907 | The right injection restri... |
| inrresf1 9908 | The right injection restri... |
| djuin 9909 | The images of any classes ... |
| djur 9910 | A member of a disjoint uni... |
| djuss 9911 | A disjoint union is a subc... |
| djuunxp 9912 | The union of a disjoint un... |
| djuexALT 9913 | Alternate proof of ~ djuex... |
| eldju1st 9914 | The first component of an ... |
| eldju2ndl 9915 | The second component of an... |
| eldju2ndr 9916 | The second component of an... |
| djuun 9917 | The disjoint union of two ... |
| 1stinl 9918 | The first component of the... |
| 2ndinl 9919 | The second component of th... |
| 1stinr 9920 | The first component of the... |
| 2ndinr 9921 | The second component of th... |
| updjudhf 9922 | The mapping of an element ... |
| updjudhcoinlf 9923 | The composition of the map... |
| updjudhcoinrg 9924 | The composition of the map... |
| updjud 9925 | Universal property of the ... |
| cardf2 9934 | The cardinality function i... |
| cardon 9935 | The cardinal number of a s... |
| isnum2 9936 | A way to express well-orde... |
| isnumi 9937 | A set equinumerous to an o... |
| ennum 9938 | Equinumerous sets are equi... |
| finnum 9939 | Every finite set is numera... |
| onenon 9940 | Every ordinal number is nu... |
| tskwe 9941 | A Tarski set is well-order... |
| xpnum 9942 | The cartesian product of n... |
| cardval3 9943 | An alternate definition of... |
| cardid2 9944 | Any numerable set is equin... |
| isnum3 9945 | A set is numerable iff it ... |
| oncardval 9946 | The value of the cardinal ... |
| oncardid 9947 | Any ordinal number is equi... |
| cardonle 9948 | The cardinal of an ordinal... |
| card0 9949 | The cardinality of the emp... |
| cardidm 9950 | The cardinality function i... |
| oncard 9951 | A set is a cardinal number... |
| ficardom 9952 | The cardinal number of a f... |
| ficardid 9953 | A finite set is equinumero... |
| cardnn 9954 | The cardinality of a natur... |
| cardnueq0 9955 | The empty set is the only ... |
| cardne 9956 | No member of a cardinal nu... |
| carden2a 9957 | If two sets have equal non... |
| carden2b 9958 | If two sets are equinumero... |
| card1 9959 | A set has cardinality one ... |
| cardsn 9960 | A singleton has cardinalit... |
| carddomi2 9961 | Two sets have the dominanc... |
| sdomsdomcardi 9962 | A set strictly dominates i... |
| cardlim 9963 | An infinite cardinal is a ... |
| cardsdomelir 9964 | A cardinal strictly domina... |
| cardsdomel 9965 | A cardinal strictly domina... |
| iscard 9966 | Two ways to express the pr... |
| iscard2 9967 | Two ways to express the pr... |
| carddom2 9968 | Two numerable sets have th... |
| harcard 9969 | The class of ordinal numbe... |
| cardprclem 9970 | Lemma for ~ cardprc . (Co... |
| cardprc 9971 | The class of all cardinal ... |
| carduni 9972 | The union of a set of card... |
| cardiun 9973 | The indexed union of a set... |
| cardennn 9974 | If ` A ` is equinumerous t... |
| cardsucinf 9975 | The cardinality of the suc... |
| cardsucnn 9976 | The cardinality of the suc... |
| cardom 9977 | The set of natural numbers... |
| carden2 9978 | Two numerable sets are equ... |
| cardsdom2 9979 | A numerable set is strictl... |
| domtri2 9980 | Trichotomy of dominance fo... |
| nnsdomel 9981 | Strict dominance and eleme... |
| cardval2 9982 | An alternate version of th... |
| isinffi 9983 | An infinite set contains s... |
| fidomtri 9984 | Trichotomy of dominance wi... |
| fidomtri2 9985 | Trichotomy of dominance wi... |
| harsdom 9986 | The Hartogs number of a we... |
| onsdom 9987 | Any well-orderable set is ... |
| harval2 9988 | An alternate expression fo... |
| harsucnn 9989 | The next cardinal after a ... |
| cardmin2 9990 | The smallest ordinal that ... |
| pm54.43lem 9991 | In Theorem *54.43 of [Whit... |
| pm54.43 9992 | Theorem *54.43 of [Whitehe... |
| enpr2 9993 | An unordered pair with dis... |
| pr2nelemOLD 9994 | Obsolete version of ~ enpr... |
| pr2ne 9995 | If an unordered pair has t... |
| pr2neOLD 9996 | Obsolete version of ~ pr2n... |
| prdom2 9997 | An unordered pair has at m... |
| en2eqpr 9998 | Building a set with two el... |
| en2eleq 9999 | Express a set of pair card... |
| en2other2 10000 | Taking the other element t... |
| dif1card 10001 | The cardinality of a nonem... |
| leweon 10002 | Lexicographical order is a... |
| r0weon 10003 | A set-like well-ordering o... |
| infxpenlem 10004 | Lemma for ~ infxpen . (Co... |
| infxpen 10005 | Every infinite ordinal is ... |
| xpomen 10006 | The Cartesian product of o... |
| xpct 10007 | The cartesian product of t... |
| infxpidm2 10008 | Every infinite well-ordera... |
| infxpenc 10009 | A canonical version of ~ i... |
| infxpenc2lem1 10010 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 10011 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 10012 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 10013 | Existence form of ~ infxpe... |
| iunmapdisj 10014 | The union ` U_ n e. C ( A ... |
| fseqenlem1 10015 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 10016 | Lemma for ~ fseqen . (Con... |
| fseqdom 10017 | One half of ~ fseqen . (C... |
| fseqen 10018 | A set that is equinumerous... |
| infpwfidom 10019 | The collection of finite s... |
| dfac8alem 10020 | Lemma for ~ dfac8a . If t... |
| dfac8a 10021 | Numeration theorem: every ... |
| dfac8b 10022 | The well-ordering theorem:... |
| dfac8clem 10023 | Lemma for ~ dfac8c . (Con... |
| dfac8c 10024 | If the union of a set is w... |
| ac10ct 10025 | A proof of the well-orderi... |
| ween 10026 | A set is numerable iff it ... |
| ac5num 10027 | A version of ~ ac5b with t... |
| ondomen 10028 | If a set is dominated by a... |
| numdom 10029 | A set dominated by a numer... |
| ssnum 10030 | A subset of a numerable se... |
| onssnum 10031 | All subsets of the ordinal... |
| indcardi 10032 | Indirect strong induction ... |
| acnrcl 10033 | Reverse closure for the ch... |
| acneq 10034 | Equality theorem for the c... |
| isacn 10035 | The property of being a ch... |
| acni 10036 | The property of being a ch... |
| acni2 10037 | The property of being a ch... |
| acni3 10038 | The property of being a ch... |
| acnlem 10039 | Construct a mapping satisf... |
| numacn 10040 | A well-orderable set has c... |
| finacn 10041 | Every set has finite choic... |
| acndom 10042 | A set with long choice seq... |
| acnnum 10043 | A set ` X ` which has choi... |
| acnen 10044 | The class of choice sets o... |
| acndom2 10045 | A set smaller than one wit... |
| acnen2 10046 | The class of sets with cho... |
| fodomacn 10047 | A version of ~ fodom that ... |
| fodomnum 10048 | A version of ~ fodom that ... |
| fonum 10049 | A surjection maps numerabl... |
| numwdom 10050 | A surjection maps numerabl... |
| fodomfi2 10051 | Onto functions define domi... |
| wdomfil 10052 | Weak dominance agrees with... |
| infpwfien 10053 | Any infinite well-orderabl... |
| inffien 10054 | The set of finite intersec... |
| wdomnumr 10055 | Weak dominance agrees with... |
| alephfnon 10056 | The aleph function is a fu... |
| aleph0 10057 | The first infinite cardina... |
| alephlim 10058 | Value of the aleph functio... |
| alephsuc 10059 | Value of the aleph functio... |
| alephon 10060 | An aleph is an ordinal num... |
| alephcard 10061 | Every aleph is a cardinal ... |
| alephnbtwn 10062 | No cardinal can be sandwic... |
| alephnbtwn2 10063 | No set has equinumerosity ... |
| alephordilem1 10064 | Lemma for ~ alephordi . (... |
| alephordi 10065 | Strict ordering property o... |
| alephord 10066 | Ordering property of the a... |
| alephord2 10067 | Ordering property of the a... |
| alephord2i 10068 | Ordering property of the a... |
| alephord3 10069 | Ordering property of the a... |
| alephsucdom 10070 | A set dominated by an alep... |
| alephsuc2 10071 | An alternate representatio... |
| alephdom 10072 | Relationship between inclu... |
| alephgeom 10073 | Every aleph is greater tha... |
| alephislim 10074 | Every aleph is a limit ord... |
| aleph11 10075 | The aleph function is one-... |
| alephf1 10076 | The aleph function is a on... |
| alephsdom 10077 | If an ordinal is smaller t... |
| alephdom2 10078 | A dominated initial ordina... |
| alephle 10079 | The argument of the aleph ... |
| cardaleph 10080 | Given any transfinite card... |
| cardalephex 10081 | Every transfinite cardinal... |
| infenaleph 10082 | An infinite numerable set ... |
| isinfcard 10083 | Two ways to express the pr... |
| iscard3 10084 | Two ways to express the pr... |
| cardnum 10085 | Two ways to express the cl... |
| alephinit 10086 | An infinite initial ordina... |
| carduniima 10087 | The union of the image of ... |
| cardinfima 10088 | If a mapping to cardinals ... |
| alephiso 10089 | Aleph is an order isomorph... |
| alephprc 10090 | The class of all transfini... |
| alephsson 10091 | The class of transfinite c... |
| unialeph 10092 | The union of the class of ... |
| alephsmo 10093 | The aleph function is stri... |
| alephf1ALT 10094 | Alternate proof of ~ aleph... |
| alephfplem1 10095 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10096 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10097 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10098 | Lemma for ~ alephfp . (Co... |
| alephfp 10099 | The aleph function has a f... |
| alephfp2 10100 | The aleph function has at ... |
| alephval3 10101 | An alternate way to expres... |
| alephsucpw2 10102 | The power set of an aleph ... |
| mappwen 10103 | Power rule for cardinal ar... |
| finnisoeu 10104 | A finite totally ordered s... |
| iunfictbso 10105 | Countability of a countabl... |
| aceq1 10108 | Equivalence of two version... |
| aceq0 10109 | Equivalence of two version... |
| aceq2 10110 | Equivalence of two version... |
| aceq3lem 10111 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10112 | Equivalence of two version... |
| dfac4 10113 | Equivalence of two version... |
| dfac5lem1 10114 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10115 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10116 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10117 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10118 | Lemma for ~ dfac5 . (Cont... |
| dfac5 10119 | Equivalence of two version... |
| dfac2a 10120 | Our Axiom of Choice (in th... |
| dfac2b 10121 | Axiom of Choice (first for... |
| dfac2 10122 | Axiom of Choice (first for... |
| dfac7 10123 | Equivalence of the Axiom o... |
| dfac0 10124 | Equivalence of two version... |
| dfac1 10125 | Equivalence of two version... |
| dfac8 10126 | A proof of the equivalency... |
| dfac9 10127 | Equivalence of the axiom o... |
| dfac10 10128 | Axiom of Choice equivalent... |
| dfac10c 10129 | Axiom of Choice equivalent... |
| dfac10b 10130 | Axiom of Choice equivalent... |
| acacni 10131 | A choice equivalent: every... |
| dfacacn 10132 | A choice equivalent: every... |
| dfac13 10133 | The axiom of choice holds ... |
| dfac12lem1 10134 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10135 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10136 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10137 | The axiom of choice holds ... |
| dfac12k 10138 | Equivalence of ~ dfac12 an... |
| dfac12a 10139 | The axiom of choice holds ... |
| dfac12 10140 | The axiom of choice holds ... |
| kmlem1 10141 | Lemma for 5-quantifier AC ... |
| kmlem2 10142 | Lemma for 5-quantifier AC ... |
| kmlem3 10143 | Lemma for 5-quantifier AC ... |
| kmlem4 10144 | Lemma for 5-quantifier AC ... |
| kmlem5 10145 | Lemma for 5-quantifier AC ... |
| kmlem6 10146 | Lemma for 5-quantifier AC ... |
| kmlem7 10147 | Lemma for 5-quantifier AC ... |
| kmlem8 10148 | Lemma for 5-quantifier AC ... |
| kmlem9 10149 | Lemma for 5-quantifier AC ... |
| kmlem10 10150 | Lemma for 5-quantifier AC ... |
| kmlem11 10151 | Lemma for 5-quantifier AC ... |
| kmlem12 10152 | Lemma for 5-quantifier AC ... |
| kmlem13 10153 | Lemma for 5-quantifier AC ... |
| kmlem14 10154 | Lemma for 5-quantifier AC ... |
| kmlem15 10155 | Lemma for 5-quantifier AC ... |
| kmlem16 10156 | Lemma for 5-quantifier AC ... |
| dfackm 10157 | Equivalence of the Axiom o... |
| undjudom 10158 | Cardinal addition dominate... |
| endjudisj 10159 | Equinumerosity of a disjoi... |
| djuen 10160 | Disjoint unions of equinum... |
| djuenun 10161 | Disjoint union is equinume... |
| dju1en 10162 | Cardinal addition with car... |
| dju1dif 10163 | Adding and subtracting one... |
| dju1p1e2 10164 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10165 | Alternate proof of ~ dju1p... |
| dju0en 10166 | Cardinal addition with car... |
| xp2dju 10167 | Two times a cardinal numbe... |
| djucomen 10168 | Commutative law for cardin... |
| djuassen 10169 | Associative law for cardin... |
| xpdjuen 10170 | Cardinal multiplication di... |
| mapdjuen 10171 | Sum of exponents law for c... |
| pwdjuen 10172 | Sum of exponents law for c... |
| djudom1 10173 | Ordering law for cardinal ... |
| djudom2 10174 | Ordering law for cardinal ... |
| djudoml 10175 | A set is dominated by its ... |
| djuxpdom 10176 | Cartesian product dominate... |
| djufi 10177 | The disjoint union of two ... |
| cdainflem 10178 | Any partition of omega int... |
| djuinf 10179 | A set is infinite iff the ... |
| infdju1 10180 | An infinite set is equinum... |
| pwdju1 10181 | The sum of a powerset with... |
| pwdjuidm 10182 | If the natural numbers inj... |
| djulepw 10183 | If ` A ` is idempotent und... |
| onadju 10184 | The cardinal and ordinal s... |
| cardadju 10185 | The cardinal sum is equinu... |
| djunum 10186 | The disjoint union of two ... |
| unnum 10187 | The union of two numerable... |
| nnadju 10188 | The cardinal and ordinal s... |
| nnadjuALT 10189 | Shorter proof of ~ nnadju ... |
| ficardadju 10190 | The disjoint union of fini... |
| ficardun 10191 | The cardinality of the uni... |
| ficardunOLD 10192 | Obsolete version of ~ fica... |
| ficardun2 10193 | The cardinality of the uni... |
| ficardun2OLD 10194 | Obsolete version of ~ fica... |
| pwsdompw 10195 | Lemma for ~ domtriom . Th... |
| unctb 10196 | The union of two countable... |
| infdjuabs 10197 | Absorption law for additio... |
| infunabs 10198 | An infinite set is equinum... |
| infdju 10199 | The sum of two cardinal nu... |
| infdif 10200 | The cardinality of an infi... |
| infdif2 10201 | Cardinality ordering for a... |
| infxpdom 10202 | Dominance law for multipli... |
| infxpabs 10203 | Absorption law for multipl... |
| infunsdom1 10204 | The union of two sets that... |
| infunsdom 10205 | The union of two sets that... |
| infxp 10206 | Absorption law for multipl... |
| pwdjudom 10207 | A property of dominance ov... |
| infpss 10208 | Every infinite set has an ... |
| infmap2 10209 | An exponentiation law for ... |
| ackbij2lem1 10210 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10211 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10212 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10213 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10214 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10215 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10216 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10217 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10218 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10219 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10220 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10221 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10222 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10223 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10224 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10225 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10226 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10227 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10228 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10229 | The Ackermann bijection, p... |
| ackbij1b 10230 | The Ackermann bijection, p... |
| ackbij2lem2 10231 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10232 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10233 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10234 | The Ackermann bijection, p... |
| r1om 10235 | The set of hereditarily fi... |
| fictb 10236 | A set is countable iff its... |
| cflem 10237 | A lemma used to simplify c... |
| cfval 10238 | Value of the cofinality fu... |
| cff 10239 | Cofinality is a function o... |
| cfub 10240 | An upper bound on cofinali... |
| cflm 10241 | Value of the cofinality fu... |
| cf0 10242 | Value of the cofinality fu... |
| cardcf 10243 | Cofinality is a cardinal n... |
| cflecard 10244 | Cofinality is bounded by t... |
| cfle 10245 | Cofinality is bounded by i... |
| cfon 10246 | The cofinality of any set ... |
| cfeq0 10247 | Only the ordinal zero has ... |
| cfsuc 10248 | Value of the cofinality fu... |
| cff1 10249 | There is always a map from... |
| cfflb 10250 | If there is a cofinal map ... |
| cfval2 10251 | Another expression for the... |
| coflim 10252 | A simpler expression for t... |
| cflim3 10253 | Another expression for the... |
| cflim2 10254 | The cofinality function is... |
| cfom 10255 | Value of the cofinality fu... |
| cfss 10256 | There is a cofinal subset ... |
| cfslb 10257 | Any cofinal subset of ` A ... |
| cfslbn 10258 | Any subset of ` A ` smalle... |
| cfslb2n 10259 | Any small collection of sm... |
| cofsmo 10260 | Any cofinal map implies th... |
| cfsmolem 10261 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10262 | The map in ~ cff1 can be a... |
| cfcoflem 10263 | Lemma for ~ cfcof , showin... |
| coftr 10264 | If there is a cofinal map ... |
| cfcof 10265 | If there is a cofinal map ... |
| cfidm 10266 | The cofinality function is... |
| alephsing 10267 | The cofinality of a limit ... |
| sornom 10268 | The range of a single-step... |
| isfin1a 10283 | Definition of a Ia-finite ... |
| fin1ai 10284 | Property of a Ia-finite se... |
| isfin2 10285 | Definition of a II-finite ... |
| fin2i 10286 | Property of a II-finite se... |
| isfin3 10287 | Definition of a III-finite... |
| isfin4 10288 | Definition of a IV-finite ... |
| fin4i 10289 | Infer that a set is IV-inf... |
| isfin5 10290 | Definition of a V-finite s... |
| isfin6 10291 | Definition of a VI-finite ... |
| isfin7 10292 | Definition of a VII-finite... |
| sdom2en01 10293 | A set with less than two e... |
| infpssrlem1 10294 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10295 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10296 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10297 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10298 | Lemma for ~ infpssr . (Co... |
| infpssr 10299 | Dedekind infinity implies ... |
| fin4en1 10300 | Dedekind finite is a cardi... |
| ssfin4 10301 | Dedekind finite sets have ... |
| domfin4 10302 | A set dominated by a Dedek... |
| ominf4 10303 | ` _om ` is Dedekind infini... |
| infpssALT 10304 | Alternate proof of ~ infps... |
| isfin4-2 10305 | Alternate definition of IV... |
| isfin4p1 10306 | Alternate definition of IV... |
| fin23lem7 10307 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10308 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10309 | A II-finite set contains m... |
| isfin2-2 10310 | ` Fin2 ` expressed in term... |
| ssfin2 10311 | A subset of a II-finite se... |
| enfin2i 10312 | II-finiteness is a cardina... |
| fin23lem24 10313 | Lemma for ~ fin23 . In a ... |
| fincssdom 10314 | In a chain of finite sets,... |
| fin23lem25 10315 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10316 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10317 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10318 | Lemma for ~ fin23 but coul... |
| fin23lem27 10319 | The mapping constructed in... |
| isfin3ds 10320 | Property of a III-finite s... |
| ssfin3ds 10321 | A subset of a III-finite s... |
| fin23lem12 10322 | The beginning of the proof... |
| fin23lem13 10323 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10324 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10325 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10326 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10327 | Lemma for ~ fin23 . The f... |
| fin23lem20 10328 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10329 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10330 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10331 | Lemma for ~ fin23 . The r... |
| fin23lem29 10332 | Lemma for ~ fin23 . The r... |
| fin23lem30 10333 | Lemma for ~ fin23 . The r... |
| fin23lem31 10334 | Lemma for ~ fin23 . The r... |
| fin23lem32 10335 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10336 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10337 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10338 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10339 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10340 | Lemma for ~ fin23 . The c... |
| fin23lem39 10341 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10342 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10343 | Lemma for ~ fin23 . A set... |
| isf32lem1 10344 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10345 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10346 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10347 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10348 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10349 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10350 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10351 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10352 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10353 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10354 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10355 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10356 | One half of ~ isfin3-2 . ... |
| isf33lem 10357 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10358 | Weakly Dedekind-infinite s... |
| isfin3-3 10359 | Weakly Dedekind-infinite s... |
| fin33i 10360 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10361 | Lemma for ~ compsscnv . (... |
| compsscnv 10362 | Complementation on a power... |
| isf34lem1 10363 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10364 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10365 | Complementation is an anti... |
| isf34lem3 10366 | Lemma for ~ isfin3-4 . (C... |
| compss 10367 | Express image under of the... |
| isf34lem4 10368 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10369 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10370 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10371 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10372 | Inference from ~ isfin3-4 ... |
| isfin3-4 10373 | Weakly Dedekind-infinite s... |
| fin11a 10374 | Every I-finite set is Ia-f... |
| enfin1ai 10375 | Ia-finiteness is a cardina... |
| isfin1-2 10376 | A set is finite in the usu... |
| isfin1-3 10377 | A set is I-finite iff ever... |
| isfin1-4 10378 | A set is I-finite iff ever... |
| dffin1-5 10379 | Compact quantifier-free ve... |
| fin23 10380 | Every II-finite set (every... |
| fin34 10381 | Every III-finite set is IV... |
| isfin5-2 10382 | Alternate definition of V-... |
| fin45 10383 | Every IV-finite set is V-f... |
| fin56 10384 | Every V-finite set is VI-f... |
| fin17 10385 | Every I-finite set is VII-... |
| fin67 10386 | Every VI-finite set is VII... |
| isfin7-2 10387 | A set is VII-finite iff it... |
| fin71num 10388 | A well-orderable set is VI... |
| dffin7-2 10389 | Class form of ~ isfin7-2 .... |
| dfacfin7 10390 | Axiom of Choice equivalent... |
| fin1a2lem1 10391 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10392 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10393 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10394 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10395 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10396 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10397 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10398 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10399 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10400 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10401 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10402 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10403 | Lemma for ~ fin1a2 . (Con... |
| fin12 10404 | Weak theorem which skips I... |
| fin1a2s 10405 | An II-infinite set can hav... |
| fin1a2 10406 | Every Ia-finite set is II-... |
| itunifval 10407 | Function value of iterated... |
| itunifn 10408 | Functionality of the itera... |
| ituni0 10409 | A zero-fold iterated union... |
| itunisuc 10410 | Successor iterated union. ... |
| itunitc1 10411 | Each union iterate is a me... |
| itunitc 10412 | The union of all union ite... |
| ituniiun 10413 | Unwrap an iterated union f... |
| hsmexlem7 10414 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10415 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10416 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10417 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10418 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10419 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10420 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10421 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10422 | Lemma for ~ hsmex . (Cont... |
| hsmex 10423 | The collection of heredita... |
| hsmex2 10424 | The set of hereditary size... |
| hsmex3 10425 | The set of hereditary size... |
| axcc2lem 10427 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10428 | A possibly more useful ver... |
| axcc3 10429 | A possibly more useful ver... |
| axcc4 10430 | A version of ~ axcc3 that ... |
| acncc 10431 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10432 | Relax the constraint on ~ ... |
| domtriomlem 10433 | Lemma for ~ domtriom . (C... |
| domtriom 10434 | Trichotomy of equinumerosi... |
| fin41 10435 | Under countable choice, th... |
| dominf 10436 | A nonempty set that is a s... |
| dcomex 10438 | The Axiom of Dependent Cho... |
| axdc2lem 10439 | Lemma for ~ axdc2 . We co... |
| axdc2 10440 | An apparent strengthening ... |
| axdc3lem 10441 | The class ` S ` of finite ... |
| axdc3lem2 10442 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10443 | Simple substitution lemma ... |
| axdc3lem4 10444 | Lemma for ~ axdc3 . We ha... |
| axdc3 10445 | Dependent Choice. Axiom D... |
| axdc4lem 10446 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10447 | A more general version of ... |
| axcclem 10448 | Lemma for ~ axcc . (Contr... |
| axcc 10449 | Although CC can be proven ... |
| zfac 10451 | Axiom of Choice expressed ... |
| ac2 10452 | Axiom of Choice equivalent... |
| ac3 10453 | Axiom of Choice using abbr... |
| axac3 10455 | This theorem asserts that ... |
| ackm 10456 | A remarkable equivalent to... |
| axac2 10457 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10458 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10459 | Apply a choice equivalent.... |
| cardeqv 10460 | All sets are well-orderabl... |
| numth3 10461 | All sets are well-orderabl... |
| numth2 10462 | Numeration theorem: any se... |
| numth 10463 | Numeration theorem: every ... |
| ac7 10464 | An Axiom of Choice equival... |
| ac7g 10465 | An Axiom of Choice equival... |
| ac4 10466 | Equivalent of Axiom of Cho... |
| ac4c 10467 | Equivalent of Axiom of Cho... |
| ac5 10468 | An Axiom of Choice equival... |
| ac5b 10469 | Equivalent of Axiom of Cho... |
| ac6num 10470 | A version of ~ ac6 which t... |
| ac6 10471 | Equivalent of Axiom of Cho... |
| ac6c4 10472 | Equivalent of Axiom of Cho... |
| ac6c5 10473 | Equivalent of Axiom of Cho... |
| ac9 10474 | An Axiom of Choice equival... |
| ac6s 10475 | Equivalent of Axiom of Cho... |
| ac6n 10476 | Equivalent of Axiom of Cho... |
| ac6s2 10477 | Generalization of the Axio... |
| ac6s3 10478 | Generalization of the Axio... |
| ac6sg 10479 | ~ ac6s with sethood as ant... |
| ac6sf 10480 | Version of ~ ac6 with boun... |
| ac6s4 10481 | Generalization of the Axio... |
| ac6s5 10482 | Generalization of the Axio... |
| ac8 10483 | An Axiom of Choice equival... |
| ac9s 10484 | An Axiom of Choice equival... |
| numthcor 10485 | Any set is strictly domina... |
| weth 10486 | Well-ordering theorem: any... |
| zorn2lem1 10487 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10488 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10489 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10490 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10491 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10492 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10493 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10494 | Zorn's Lemma of [Monk1] p.... |
| zorng 10495 | Zorn's Lemma. If the unio... |
| zornn0g 10496 | Variant of Zorn's lemma ~ ... |
| zorn2 10497 | Zorn's Lemma of [Monk1] p.... |
| zorn 10498 | Zorn's Lemma. If the unio... |
| zornn0 10499 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10500 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10501 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10502 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10503 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10504 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10505 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10506 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10507 | The Teichmüller-Tukey... |
| ttukeyg 10508 | The Teichmüller-Tukey... |
| ttukey 10509 | The Teichmüller-Tukey... |
| axdclem 10510 | Lemma for ~ axdc . (Contr... |
| axdclem2 10511 | Lemma for ~ axdc . Using ... |
| axdc 10512 | This theorem derives ~ ax-... |
| fodomg 10513 | An onto function implies d... |
| fodom 10514 | An onto function implies d... |
| dmct 10515 | The domain of a countable ... |
| rnct 10516 | The range of a countable s... |
| fodomb 10517 | Equivalence of an onto map... |
| wdomac 10518 | When assuming AC, weak and... |
| brdom3 10519 | Equivalence to a dominance... |
| brdom5 10520 | An equivalence to a domina... |
| brdom4 10521 | An equivalence to a domina... |
| brdom7disj 10522 | An equivalence to a domina... |
| brdom6disj 10523 | An equivalence to a domina... |
| fin71ac 10524 | Once we allow AC, the "str... |
| imadomg 10525 | An image of a function und... |
| fimact 10526 | The image by a function of... |
| fnrndomg 10527 | The range of a function is... |
| fnct 10528 | If the domain of a functio... |
| mptct 10529 | A countable mapping set is... |
| iunfo 10530 | Existence of an onto funct... |
| iundom2g 10531 | An upper bound for the car... |
| iundomg 10532 | An upper bound for the car... |
| iundom 10533 | An upper bound for the car... |
| unidom 10534 | An upper bound for the car... |
| uniimadom 10535 | An upper bound for the car... |
| uniimadomf 10536 | An upper bound for the car... |
| cardval 10537 | The value of the cardinal ... |
| cardid 10538 | Any set is equinumerous to... |
| cardidg 10539 | Any set is equinumerous to... |
| cardidd 10540 | Any set is equinumerous to... |
| cardf 10541 | The cardinality function i... |
| carden 10542 | Two sets are equinumerous ... |
| cardeq0 10543 | Only the empty set has car... |
| unsnen 10544 | Equinumerosity of a set wi... |
| carddom 10545 | Two sets have the dominanc... |
| cardsdom 10546 | Two sets have the strict d... |
| domtri 10547 | Trichotomy law for dominan... |
| entric 10548 | Trichotomy of equinumerosi... |
| entri2 10549 | Trichotomy of dominance an... |
| entri3 10550 | Trichotomy of dominance. ... |
| sdomsdomcard 10551 | A set strictly dominates i... |
| canth3 10552 | Cantor's theorem in terms ... |
| infxpidm 10553 | Every infinite class is eq... |
| ondomon 10554 | The class of ordinals domi... |
| cardmin 10555 | The smallest ordinal that ... |
| ficard 10556 | A set is finite iff its ca... |
| infinf 10557 | Equivalence between two in... |
| unirnfdomd 10558 | The union of the range of ... |
| konigthlem 10559 | Lemma for ~ konigth . (Co... |
| konigth 10560 | Konig's Theorem. If ` m (... |
| alephsucpw 10561 | The power set of an aleph ... |
| aleph1 10562 | The set exponentiation of ... |
| alephval2 10563 | An alternate way to expres... |
| dominfac 10564 | A nonempty set that is a s... |
| iunctb 10565 | The countable union of cou... |
| unictb 10566 | The countable union of cou... |
| infmap 10567 | An exponentiation law for ... |
| alephadd 10568 | The sum of two alephs is t... |
| alephmul 10569 | The product of two alephs ... |
| alephexp1 10570 | An exponentiation law for ... |
| alephsuc3 10571 | An alternate representatio... |
| alephexp2 10572 | An expression equinumerous... |
| alephreg 10573 | A successor aleph is regul... |
| pwcfsdom 10574 | A corollary of Konig's The... |
| cfpwsdom 10575 | A corollary of Konig's The... |
| alephom 10576 | From ~ canth2 , we know th... |
| smobeth 10577 | The beth function is stric... |
| nd1 10578 | A lemma for proving condit... |
| nd2 10579 | A lemma for proving condit... |
| nd3 10580 | A lemma for proving condit... |
| nd4 10581 | A lemma for proving condit... |
| axextnd 10582 | A version of the Axiom of ... |
| axrepndlem1 10583 | Lemma for the Axiom of Rep... |
| axrepndlem2 10584 | Lemma for the Axiom of Rep... |
| axrepnd 10585 | A version of the Axiom of ... |
| axunndlem1 10586 | Lemma for the Axiom of Uni... |
| axunnd 10587 | A version of the Axiom of ... |
| axpowndlem1 10588 | Lemma for the Axiom of Pow... |
| axpowndlem2 10589 | Lemma for the Axiom of Pow... |
| axpowndlem3 10590 | Lemma for the Axiom of Pow... |
| axpowndlem4 10591 | Lemma for the Axiom of Pow... |
| axpownd 10592 | A version of the Axiom of ... |
| axregndlem1 10593 | Lemma for the Axiom of Reg... |
| axregndlem2 10594 | Lemma for the Axiom of Reg... |
| axregnd 10595 | A version of the Axiom of ... |
| axinfndlem1 10596 | Lemma for the Axiom of Inf... |
| axinfnd 10597 | A version of the Axiom of ... |
| axacndlem1 10598 | Lemma for the Axiom of Cho... |
| axacndlem2 10599 | Lemma for the Axiom of Cho... |
| axacndlem3 10600 | Lemma for the Axiom of Cho... |
| axacndlem4 10601 | Lemma for the Axiom of Cho... |
| axacndlem5 10602 | Lemma for the Axiom of Cho... |
| axacnd 10603 | A version of the Axiom of ... |
| zfcndext 10604 | Axiom of Extensionality ~ ... |
| zfcndrep 10605 | Axiom of Replacement ~ ax-... |
| zfcndun 10606 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10607 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10608 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10609 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10610 | Axiom of Choice ~ ax-ac , ... |
| elgch 10613 | Elementhood in the collect... |
| fingch 10614 | A finite set is a GCH-set.... |
| gchi 10615 | The only GCH-sets which ha... |
| gchen1 10616 | If ` A <_ B < ~P A ` , and... |
| gchen2 10617 | If ` A < B <_ ~P A ` , and... |
| gchor 10618 | If ` A <_ B <_ ~P A ` , an... |
| engch 10619 | The property of being a GC... |
| gchdomtri 10620 | Under certain conditions, ... |
| fpwwe2cbv 10621 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10622 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10623 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10624 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10625 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10626 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10627 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10628 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10629 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10630 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10631 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10632 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10633 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10634 | Given any function ` F ` f... |
| fpwwecbv 10635 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10636 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10637 | Given any function ` F ` f... |
| canth4 10638 | An "effective" form of Can... |
| canthnumlem 10639 | Lemma for ~ canthnum . (C... |
| canthnum 10640 | The set of well-orderable ... |
| canthwelem 10641 | Lemma for ~ canthwe . (Co... |
| canthwe 10642 | The set of well-orders of ... |
| canthp1lem1 10643 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10644 | Lemma for ~ canthp1 . (Co... |
| canthp1 10645 | A slightly stronger form o... |
| finngch 10646 | The exclusion of finite se... |
| gchdju1 10647 | An infinite GCH-set is ide... |
| gchinf 10648 | An infinite GCH-set is Ded... |
| pwfseqlem1 10649 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10650 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10651 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10652 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10653 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10654 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10655 | The powerset of a Dedekind... |
| pwxpndom2 10656 | The powerset of a Dedekind... |
| pwxpndom 10657 | The powerset of a Dedekind... |
| pwdjundom 10658 | The powerset of a Dedekind... |
| gchdjuidm 10659 | An infinite GCH-set is ide... |
| gchxpidm 10660 | An infinite GCH-set is ide... |
| gchpwdom 10661 | A relationship between dom... |
| gchaleph 10662 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10663 | If ` ( aleph `` A ) ` and ... |
| hargch 10664 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10665 | If ` ( aleph `` suc A ) ` ... |
| gch2 10666 | It is sufficient to requir... |
| gch3 10667 | An equivalent formulation ... |
| gch-kn 10668 | The equivalence of two ver... |
| gchaclem 10669 | Lemma for ~ gchac (obsolet... |
| gchhar 10670 | A "local" form of ~ gchac ... |
| gchacg 10671 | A "local" form of ~ gchac ... |
| gchac 10672 | The Generalized Continuum ... |
| elwina 10677 | Conditions of weak inacces... |
| elina 10678 | Conditions of strong inacc... |
| winaon 10679 | A weakly inaccessible card... |
| inawinalem 10680 | Lemma for ~ inawina . (Co... |
| inawina 10681 | Every strongly inaccessibl... |
| omina 10682 | ` _om ` is a strongly inac... |
| winacard 10683 | A weakly inaccessible card... |
| winainflem 10684 | A weakly inaccessible card... |
| winainf 10685 | A weakly inaccessible card... |
| winalim 10686 | A weakly inaccessible card... |
| winalim2 10687 | A nontrivial weakly inacce... |
| winafp 10688 | A nontrivial weakly inacce... |
| winafpi 10689 | This theorem, which states... |
| gchina 10690 | Assuming the GCH, weakly a... |
| iswun 10695 | Properties of a weak unive... |
| wuntr 10696 | A weak universe is transit... |
| wununi 10697 | A weak universe is closed ... |
| wunpw 10698 | A weak universe is closed ... |
| wunelss 10699 | The elements of a weak uni... |
| wunpr 10700 | A weak universe is closed ... |
| wunun 10701 | A weak universe is closed ... |
| wuntp 10702 | A weak universe is closed ... |
| wunss 10703 | A weak universe is closed ... |
| wunin 10704 | A weak universe is closed ... |
| wundif 10705 | A weak universe is closed ... |
| wunint 10706 | A weak universe is closed ... |
| wunsn 10707 | A weak universe is closed ... |
| wunsuc 10708 | A weak universe is closed ... |
| wun0 10709 | A weak universe contains t... |
| wunr1om 10710 | A weak universe is infinit... |
| wunom 10711 | A weak universe contains a... |
| wunfi 10712 | A weak universe contains a... |
| wunop 10713 | A weak universe is closed ... |
| wunot 10714 | A weak universe is closed ... |
| wunxp 10715 | A weak universe is closed ... |
| wunpm 10716 | A weak universe is closed ... |
| wunmap 10717 | A weak universe is closed ... |
| wunf 10718 | A weak universe is closed ... |
| wundm 10719 | A weak universe is closed ... |
| wunrn 10720 | A weak universe is closed ... |
| wuncnv 10721 | A weak universe is closed ... |
| wunres 10722 | A weak universe is closed ... |
| wunfv 10723 | A weak universe is closed ... |
| wunco 10724 | A weak universe is closed ... |
| wuntpos 10725 | A weak universe is closed ... |
| intwun 10726 | The intersection of a coll... |
| r1limwun 10727 | Each limit stage in the cu... |
| r1wunlim 10728 | The weak universes in the ... |
| wunex2 10729 | Construct a weak universe ... |
| wunex 10730 | Construct a weak universe ... |
| uniwun 10731 | Every set is contained in ... |
| wunex3 10732 | Construct a weak universe ... |
| wuncval 10733 | Value of the weak universe... |
| wuncid 10734 | The weak universe closure ... |
| wunccl 10735 | The weak universe closure ... |
| wuncss 10736 | The weak universe closure ... |
| wuncidm 10737 | The weak universe closure ... |
| wuncval2 10738 | Our earlier expression for... |
| eltskg 10741 | Properties of a Tarski cla... |
| eltsk2g 10742 | Properties of a Tarski cla... |
| tskpwss 10743 | First axiom of a Tarski cl... |
| tskpw 10744 | Second axiom of a Tarski c... |
| tsken 10745 | Third axiom of a Tarski cl... |
| 0tsk 10746 | The empty set is a (transi... |
| tsksdom 10747 | An element of a Tarski cla... |
| tskssel 10748 | A part of a Tarski class s... |
| tskss 10749 | The subsets of an element ... |
| tskin 10750 | The intersection of two el... |
| tsksn 10751 | A singleton of an element ... |
| tsktrss 10752 | A transitive element of a ... |
| tsksuc 10753 | If an element of a Tarski ... |
| tsk0 10754 | A nonempty Tarski class co... |
| tsk1 10755 | One is an element of a non... |
| tsk2 10756 | Two is an element of a non... |
| 2domtsk 10757 | If a Tarski class is not e... |
| tskr1om 10758 | A nonempty Tarski class is... |
| tskr1om2 10759 | A nonempty Tarski class co... |
| tskinf 10760 | A nonempty Tarski class is... |
| tskpr 10761 | If ` A ` and ` B ` are mem... |
| tskop 10762 | If ` A ` and ` B ` are mem... |
| tskxpss 10763 | A Cartesian product of two... |
| tskwe2 10764 | A Tarski class is well-ord... |
| inttsk 10765 | The intersection of a coll... |
| inar1 10766 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10767 | Alternate proof of ~ r1om ... |
| rankcf 10768 | Any set must be at least a... |
| inatsk 10769 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10770 | The set of hereditarily fi... |
| tskord 10771 | A Tarski class contains al... |
| tskcard 10772 | An even more direct relati... |
| r1tskina 10773 | There is a direct relation... |
| tskuni 10774 | The union of an element of... |
| tskwun 10775 | A nonempty transitive Tars... |
| tskint 10776 | The intersection of an ele... |
| tskun 10777 | The union of two elements ... |
| tskxp 10778 | The Cartesian product of t... |
| tskmap 10779 | Set exponentiation is an e... |
| tskurn 10780 | A transitive Tarski class ... |
| elgrug 10783 | Properties of a Grothendie... |
| grutr 10784 | A Grothendieck universe is... |
| gruelss 10785 | A Grothendieck universe is... |
| grupw 10786 | A Grothendieck universe co... |
| gruss 10787 | Any subset of an element o... |
| grupr 10788 | A Grothendieck universe co... |
| gruurn 10789 | A Grothendieck universe co... |
| gruiun 10790 | If ` B ( x ) ` is a family... |
| gruuni 10791 | A Grothendieck universe co... |
| grurn 10792 | A Grothendieck universe co... |
| gruima 10793 | A Grothendieck universe co... |
| gruel 10794 | Any element of an element ... |
| grusn 10795 | A Grothendieck universe co... |
| gruop 10796 | A Grothendieck universe co... |
| gruun 10797 | A Grothendieck universe co... |
| gruxp 10798 | A Grothendieck universe co... |
| grumap 10799 | A Grothendieck universe co... |
| gruixp 10800 | A Grothendieck universe co... |
| gruiin 10801 | A Grothendieck universe co... |
| gruf 10802 | A Grothendieck universe co... |
| gruen 10803 | A Grothendieck universe co... |
| gruwun 10804 | A nonempty Grothendieck un... |
| intgru 10805 | The intersection of a fami... |
| ingru 10806 | The intersection of a univ... |
| wfgru 10807 | The wellfounded part of a ... |
| grudomon 10808 | Each ordinal that is compa... |
| gruina 10809 | If a Grothendieck universe... |
| grur1a 10810 | A characterization of Grot... |
| grur1 10811 | A characterization of Grot... |
| grutsk1 10812 | Grothendieck universes are... |
| grutsk 10813 | Grothendieck universes are... |
| axgroth5 10815 | The Tarski-Grothendieck ax... |
| axgroth2 10816 | Alternate version of the T... |
| grothpw 10817 | Derive the Axiom of Power ... |
| grothpwex 10818 | Derive the Axiom of Power ... |
| axgroth6 10819 | The Tarski-Grothendieck ax... |
| grothomex 10820 | The Tarski-Grothendieck Ax... |
| grothac 10821 | The Tarski-Grothendieck Ax... |
| axgroth3 10822 | Alternate version of the T... |
| axgroth4 10823 | Alternate version of the T... |
| grothprimlem 10824 | Lemma for ~ grothprim . E... |
| grothprim 10825 | The Tarski-Grothendieck Ax... |
| grothtsk 10826 | The Tarski-Grothendieck Ax... |
| inaprc 10827 | An equivalent to the Tarsk... |
| tskmval 10830 | Value of our tarski map. ... |
| tskmid 10831 | The set ` A ` is an elemen... |
| tskmcl 10832 | A Tarski class that contai... |
| sstskm 10833 | Being a part of ` ( tarski... |
| eltskm 10834 | Belonging to ` ( tarskiMap... |
| elni 10867 | Membership in the class of... |
| elni2 10868 | Membership in the class of... |
| pinn 10869 | A positive integer is a na... |
| pion 10870 | A positive integer is an o... |
| piord 10871 | A positive integer is ordi... |
| niex 10872 | The class of positive inte... |
| 0npi 10873 | The empty set is not a pos... |
| 1pi 10874 | Ordinal 'one' is a positiv... |
| addpiord 10875 | Positive integer addition ... |
| mulpiord 10876 | Positive integer multiplic... |
| mulidpi 10877 | 1 is an identity element f... |
| ltpiord 10878 | Positive integer 'less tha... |
| ltsopi 10879 | Positive integer 'less tha... |
| ltrelpi 10880 | Positive integer 'less tha... |
| dmaddpi 10881 | Domain of addition on posi... |
| dmmulpi 10882 | Domain of multiplication o... |
| addclpi 10883 | Closure of addition of pos... |
| mulclpi 10884 | Closure of multiplication ... |
| addcompi 10885 | Addition of positive integ... |
| addasspi 10886 | Addition of positive integ... |
| mulcompi 10887 | Multiplication of positive... |
| mulasspi 10888 | Multiplication of positive... |
| distrpi 10889 | Multiplication of positive... |
| addcanpi 10890 | Addition cancellation law ... |
| mulcanpi 10891 | Multiplication cancellatio... |
| addnidpi 10892 | There is no identity eleme... |
| ltexpi 10893 | Ordering on positive integ... |
| ltapi 10894 | Ordering property of addit... |
| ltmpi 10895 | Ordering property of multi... |
| 1lt2pi 10896 | One is less than two (one ... |
| nlt1pi 10897 | No positive integer is les... |
| indpi 10898 | Principle of Finite Induct... |
| enqbreq 10910 | Equivalence relation for p... |
| enqbreq2 10911 | Equivalence relation for p... |
| enqer 10912 | The equivalence relation f... |
| enqex 10913 | The equivalence relation f... |
| nqex 10914 | The class of positive frac... |
| 0nnq 10915 | The empty set is not a pos... |
| elpqn 10916 | Each positive fraction is ... |
| ltrelnq 10917 | Positive fraction 'less th... |
| pinq 10918 | The representatives of pos... |
| 1nq 10919 | The positive fraction 'one... |
| nqereu 10920 | There is a unique element ... |
| nqerf 10921 | Corollary of ~ nqereu : th... |
| nqercl 10922 | Corollary of ~ nqereu : cl... |
| nqerrel 10923 | Any member of ` ( N. X. N.... |
| nqerid 10924 | Corollary of ~ nqereu : th... |
| enqeq 10925 | Corollary of ~ nqereu : if... |
| nqereq 10926 | The function ` /Q ` acts a... |
| addpipq2 10927 | Addition of positive fract... |
| addpipq 10928 | Addition of positive fract... |
| addpqnq 10929 | Addition of positive fract... |
| mulpipq2 10930 | Multiplication of positive... |
| mulpipq 10931 | Multiplication of positive... |
| mulpqnq 10932 | Multiplication of positive... |
| ordpipq 10933 | Ordering of positive fract... |
| ordpinq 10934 | Ordering of positive fract... |
| addpqf 10935 | Closure of addition on pos... |
| addclnq 10936 | Closure of addition on pos... |
| mulpqf 10937 | Closure of multiplication ... |
| mulclnq 10938 | Closure of multiplication ... |
| addnqf 10939 | Domain of addition on posi... |
| mulnqf 10940 | Domain of multiplication o... |
| addcompq 10941 | Addition of positive fract... |
| addcomnq 10942 | Addition of positive fract... |
| mulcompq 10943 | Multiplication of positive... |
| mulcomnq 10944 | Multiplication of positive... |
| adderpqlem 10945 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10946 | Lemma for ~ mulerpq . (Co... |
| adderpq 10947 | Addition is compatible wit... |
| mulerpq 10948 | Multiplication is compatib... |
| addassnq 10949 | Addition of positive fract... |
| mulassnq 10950 | Multiplication of positive... |
| mulcanenq 10951 | Lemma for distributive law... |
| distrnq 10952 | Multiplication of positive... |
| 1nqenq 10953 | The equivalence class of r... |
| mulidnq 10954 | Multiplication identity el... |
| recmulnq 10955 | Relationship between recip... |
| recidnq 10956 | A positive fraction times ... |
| recclnq 10957 | Closure law for positive f... |
| recrecnq 10958 | Reciprocal of reciprocal o... |
| dmrecnq 10959 | Domain of reciprocal on po... |
| ltsonq 10960 | 'Less than' is a strict or... |
| lterpq 10961 | Compatibility of ordering ... |
| ltanq 10962 | Ordering property of addit... |
| ltmnq 10963 | Ordering property of multi... |
| 1lt2nq 10964 | One is less than two (one ... |
| ltaddnq 10965 | The sum of two fractions i... |
| ltexnq 10966 | Ordering on positive fract... |
| halfnq 10967 | One-half of any positive f... |
| nsmallnq 10968 | The is no smallest positiv... |
| ltbtwnnq 10969 | There exists a number betw... |
| ltrnq 10970 | Ordering property of recip... |
| archnq 10971 | For any fraction, there is... |
| npex 10977 | The class of positive real... |
| elnp 10978 | Membership in positive rea... |
| elnpi 10979 | Membership in positive rea... |
| prn0 10980 | A positive real is not emp... |
| prpssnq 10981 | A positive real is a subse... |
| elprnq 10982 | A positive real is a set o... |
| 0npr 10983 | The empty set is not a pos... |
| prcdnq 10984 | A positive real is closed ... |
| prub 10985 | A positive fraction not in... |
| prnmax 10986 | A positive real has no lar... |
| npomex 10987 | A simplifying observation,... |
| prnmadd 10988 | A positive real has no lar... |
| ltrelpr 10989 | Positive real 'less than' ... |
| genpv 10990 | Value of general operation... |
| genpelv 10991 | Membership in value of gen... |
| genpprecl 10992 | Pre-closure law for genera... |
| genpdm 10993 | Domain of general operatio... |
| genpn0 10994 | The result of an operation... |
| genpss 10995 | The result of an operation... |
| genpnnp 10996 | The result of an operation... |
| genpcd 10997 | Downward closure of an ope... |
| genpnmax 10998 | An operation on positive r... |
| genpcl 10999 | Closure of an operation on... |
| genpass 11000 | Associativity of an operat... |
| plpv 11001 | Value of addition on posit... |
| mpv 11002 | Value of multiplication on... |
| dmplp 11003 | Domain of addition on posi... |
| dmmp 11004 | Domain of multiplication o... |
| nqpr 11005 | The canonical embedding of... |
| 1pr 11006 | The positive real number '... |
| addclprlem1 11007 | Lemma to prove downward cl... |
| addclprlem2 11008 | Lemma to prove downward cl... |
| addclpr 11009 | Closure of addition on pos... |
| mulclprlem 11010 | Lemma to prove downward cl... |
| mulclpr 11011 | Closure of multiplication ... |
| addcompr 11012 | Addition of positive reals... |
| addasspr 11013 | Addition of positive reals... |
| mulcompr 11014 | Multiplication of positive... |
| mulasspr 11015 | Multiplication of positive... |
| distrlem1pr 11016 | Lemma for distributive law... |
| distrlem4pr 11017 | Lemma for distributive law... |
| distrlem5pr 11018 | Lemma for distributive law... |
| distrpr 11019 | Multiplication of positive... |
| 1idpr 11020 | 1 is an identity element f... |
| ltprord 11021 | Positive real 'less than' ... |
| psslinpr 11022 | Proper subset is a linear ... |
| ltsopr 11023 | Positive real 'less than' ... |
| prlem934 11024 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 11025 | The sum of two positive re... |
| ltaddpr2 11026 | The sum of two positive re... |
| ltexprlem1 11027 | Lemma for Proposition 9-3.... |
| ltexprlem2 11028 | Lemma for Proposition 9-3.... |
| ltexprlem3 11029 | Lemma for Proposition 9-3.... |
| ltexprlem4 11030 | Lemma for Proposition 9-3.... |
| ltexprlem5 11031 | Lemma for Proposition 9-3.... |
| ltexprlem6 11032 | Lemma for Proposition 9-3.... |
| ltexprlem7 11033 | Lemma for Proposition 9-3.... |
| ltexpri 11034 | Proposition 9-3.5(iv) of [... |
| ltaprlem 11035 | Lemma for Proposition 9-3.... |
| ltapr 11036 | Ordering property of addit... |
| addcanpr 11037 | Addition cancellation law ... |
| prlem936 11038 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 11039 | Lemma for Proposition 9-3.... |
| reclem3pr 11040 | Lemma for Proposition 9-3.... |
| reclem4pr 11041 | Lemma for Proposition 9-3.... |
| recexpr 11042 | The reciprocal of a positi... |
| suplem1pr 11043 | The union of a nonempty, b... |
| suplem2pr 11044 | The union of a set of posi... |
| supexpr 11045 | The union of a nonempty, b... |
| enrer 11054 | The equivalence relation f... |
| nrex1 11055 | The class of signed reals ... |
| enrbreq 11056 | Equivalence relation for s... |
| enreceq 11057 | Equivalence class equality... |
| enrex 11058 | The equivalence relation f... |
| ltrelsr 11059 | Signed real 'less than' is... |
| addcmpblnr 11060 | Lemma showing compatibilit... |
| mulcmpblnrlem 11061 | Lemma used in lemma showin... |
| mulcmpblnr 11062 | Lemma showing compatibilit... |
| prsrlem1 11063 | Decomposing signed reals i... |
| addsrmo 11064 | There is at most one resul... |
| mulsrmo 11065 | There is at most one resul... |
| addsrpr 11066 | Addition of signed reals i... |
| mulsrpr 11067 | Multiplication of signed r... |
| ltsrpr 11068 | Ordering of signed reals i... |
| gt0srpr 11069 | Greater than zero in terms... |
| 0nsr 11070 | The empty set is not a sig... |
| 0r 11071 | The constant ` 0R ` is a s... |
| 1sr 11072 | The constant ` 1R ` is a s... |
| m1r 11073 | The constant ` -1R ` is a ... |
| addclsr 11074 | Closure of addition on sig... |
| mulclsr 11075 | Closure of multiplication ... |
| dmaddsr 11076 | Domain of addition on sign... |
| dmmulsr 11077 | Domain of multiplication o... |
| addcomsr 11078 | Addition of signed reals i... |
| addasssr 11079 | Addition of signed reals i... |
| mulcomsr 11080 | Multiplication of signed r... |
| mulasssr 11081 | Multiplication of signed r... |
| distrsr 11082 | Multiplication of signed r... |
| m1p1sr 11083 | Minus one plus one is zero... |
| m1m1sr 11084 | Minus one times minus one ... |
| ltsosr 11085 | Signed real 'less than' is... |
| 0lt1sr 11086 | 0 is less than 1 for signe... |
| 1ne0sr 11087 | 1 and 0 are distinct for s... |
| 0idsr 11088 | The signed real number 0 i... |
| 1idsr 11089 | 1 is an identity element f... |
| 00sr 11090 | A signed real times 0 is 0... |
| ltasr 11091 | Ordering property of addit... |
| pn0sr 11092 | A signed real plus its neg... |
| negexsr 11093 | Existence of negative sign... |
| recexsrlem 11094 | The reciprocal of a positi... |
| addgt0sr 11095 | The sum of two positive si... |
| mulgt0sr 11096 | The product of two positiv... |
| sqgt0sr 11097 | The square of a nonzero si... |
| recexsr 11098 | The reciprocal of a nonzer... |
| mappsrpr 11099 | Mapping from positive sign... |
| ltpsrpr 11100 | Mapping of order from posi... |
| map2psrpr 11101 | Equivalence for positive s... |
| supsrlem 11102 | Lemma for supremum theorem... |
| supsr 11103 | A nonempty, bounded set of... |
| opelcn 11120 | Ordered pair membership in... |
| opelreal 11121 | Ordered pair membership in... |
| elreal 11122 | Membership in class of rea... |
| elreal2 11123 | Ordered pair membership in... |
| 0ncn 11124 | The empty set is not a com... |
| ltrelre 11125 | 'Less than' is a relation ... |
| addcnsr 11126 | Addition of complex number... |
| mulcnsr 11127 | Multiplication of complex ... |
| eqresr 11128 | Equality of real numbers i... |
| addresr 11129 | Addition of real numbers i... |
| mulresr 11130 | Multiplication of real num... |
| ltresr 11131 | Ordering of real subset of... |
| ltresr2 11132 | Ordering of real subset of... |
| dfcnqs 11133 | Technical trick to permit ... |
| addcnsrec 11134 | Technical trick to permit ... |
| mulcnsrec 11135 | Technical trick to permit ... |
| axaddf 11136 | Addition is an operation o... |
| axmulf 11137 | Multiplication is an opera... |
| axcnex 11138 | The complex numbers form a... |
| axresscn 11139 | The real numbers are a sub... |
| ax1cn 11140 | 1 is a complex number. Ax... |
| axicn 11141 | ` _i ` is a complex number... |
| axaddcl 11142 | Closure law for addition o... |
| axaddrcl 11143 | Closure law for addition i... |
| axmulcl 11144 | Closure law for multiplica... |
| axmulrcl 11145 | Closure law for multiplica... |
| axmulcom 11146 | Multiplication of complex ... |
| axaddass 11147 | Addition of complex number... |
| axmulass 11148 | Multiplication of complex ... |
| axdistr 11149 | Distributive law for compl... |
| axi2m1 11150 | i-squared equals -1 (expre... |
| ax1ne0 11151 | 1 and 0 are distinct. Axi... |
| ax1rid 11152 | ` 1 ` is an identity eleme... |
| axrnegex 11153 | Existence of negative of r... |
| axrrecex 11154 | Existence of reciprocal of... |
| axcnre 11155 | A complex number can be ex... |
| axpre-lttri 11156 | Ordering on reals satisfie... |
| axpre-lttrn 11157 | Ordering on reals is trans... |
| axpre-ltadd 11158 | Ordering property of addit... |
| axpre-mulgt0 11159 | The product of two positiv... |
| axpre-sup 11160 | A nonempty, bounded-above ... |
| wuncn 11161 | A weak universe containing... |
| cnex 11187 | Alias for ~ ax-cnex . See... |
| addcl 11188 | Alias for ~ ax-addcl , for... |
| readdcl 11189 | Alias for ~ ax-addrcl , fo... |
| mulcl 11190 | Alias for ~ ax-mulcl , for... |
| remulcl 11191 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11192 | Alias for ~ ax-mulcom , fo... |
| addass 11193 | Alias for ~ ax-addass , fo... |
| mulass 11194 | Alias for ~ ax-mulass , fo... |
| adddi 11195 | Alias for ~ ax-distr , for... |
| recn 11196 | A real number is a complex... |
| reex 11197 | The real numbers form a se... |
| reelprrecn 11198 | Reals are a subset of the ... |
| cnelprrecn 11199 | Complex numbers are a subs... |
| elimne0 11200 | Hypothesis for weak deduct... |
| adddir 11201 | Distributive law for compl... |
| 0cn 11202 | Zero is a complex number. ... |
| 0cnd 11203 | Zero is a complex number, ... |
| c0ex 11204 | Zero is a set. (Contribut... |
| 1cnd 11205 | One is a complex number, d... |
| 1ex 11206 | One is a set. (Contribute... |
| cnre 11207 | Alias for ~ ax-cnre , for ... |
| mulrid 11208 | The number 1 is an identit... |
| mullid 11209 | Identity law for multiplic... |
| 1re 11210 | The number 1 is real. Thi... |
| 1red 11211 | The number 1 is real, dedu... |
| 0re 11212 | The number 0 is real. Rem... |
| 0red 11213 | The number 0 is real, dedu... |
| mulridi 11214 | Identity law for multiplic... |
| mullidi 11215 | Identity law for multiplic... |
| addcli 11216 | Closure law for addition. ... |
| mulcli 11217 | Closure law for multiplica... |
| mulcomi 11218 | Commutative law for multip... |
| mulcomli 11219 | Commutative law for multip... |
| addassi 11220 | Associative law for additi... |
| mulassi 11221 | Associative law for multip... |
| adddii 11222 | Distributive law (left-dis... |
| adddiri 11223 | Distributive law (right-di... |
| recni 11224 | A real number is a complex... |
| readdcli 11225 | Closure law for addition o... |
| remulcli 11226 | Closure law for multiplica... |
| mulridd 11227 | Identity law for multiplic... |
| mullidd 11228 | Identity law for multiplic... |
| addcld 11229 | Closure law for addition. ... |
| mulcld 11230 | Closure law for multiplica... |
| mulcomd 11231 | Commutative law for multip... |
| addassd 11232 | Associative law for additi... |
| mulassd 11233 | Associative law for multip... |
| adddid 11234 | Distributive law (left-dis... |
| adddird 11235 | Distributive law (right-di... |
| adddirp1d 11236 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11237 | Join AB+CB into (A+C) on L... |
| recnd 11238 | Deduction from real number... |
| readdcld 11239 | Closure law for addition o... |
| remulcld 11240 | Closure law for multiplica... |
| pnfnre 11251 | Plus infinity is not a rea... |
| pnfnre2 11252 | Plus infinity is not a rea... |
| mnfnre 11253 | Minus infinity is not a re... |
| ressxr 11254 | The standard reals are a s... |
| rexpssxrxp 11255 | The Cartesian product of s... |
| rexr 11256 | A standard real is an exte... |
| 0xr 11257 | Zero is an extended real. ... |
| renepnf 11258 | No (finite) real equals pl... |
| renemnf 11259 | No real equals minus infin... |
| rexrd 11260 | A standard real is an exte... |
| renepnfd 11261 | No (finite) real equals pl... |
| renemnfd 11262 | No real equals minus infin... |
| pnfex 11263 | Plus infinity exists. (Co... |
| pnfxr 11264 | Plus infinity belongs to t... |
| pnfnemnf 11265 | Plus and minus infinity ar... |
| mnfnepnf 11266 | Minus and plus infinity ar... |
| mnfxr 11267 | Minus infinity belongs to ... |
| rexri 11268 | A standard real is an exte... |
| 1xr 11269 | ` 1 ` is an extended real ... |
| renfdisj 11270 | The reals and the infiniti... |
| ltrelxr 11271 | "Less than" is a relation ... |
| ltrel 11272 | "Less than" is a relation.... |
| lerelxr 11273 | "Less than or equal to" is... |
| lerel 11274 | "Less than or equal to" is... |
| xrlenlt 11275 | "Less than or equal to" ex... |
| xrlenltd 11276 | "Less than or equal to" ex... |
| xrltnle 11277 | "Less than" expressed in t... |
| xrnltled 11278 | "Not less than" implies "l... |
| ssxr 11279 | The three (non-exclusive) ... |
| ltxrlt 11280 | The standard less-than ` <... |
| axlttri 11281 | Ordering on reals satisfie... |
| axlttrn 11282 | Ordering on reals is trans... |
| axltadd 11283 | Ordering property of addit... |
| axmulgt0 11284 | The product of two positiv... |
| axsup 11285 | A nonempty, bounded-above ... |
| lttr 11286 | Alias for ~ axlttrn , for ... |
| mulgt0 11287 | The product of two positiv... |
| lenlt 11288 | 'Less than or equal to' ex... |
| ltnle 11289 | 'Less than' expressed in t... |
| ltso 11290 | 'Less than' is a strict or... |
| gtso 11291 | 'Greater than' is a strict... |
| lttri2 11292 | Consequence of trichotomy.... |
| lttri3 11293 | Trichotomy law for 'less t... |
| lttri4 11294 | Trichotomy law for 'less t... |
| letri3 11295 | Trichotomy law. (Contribu... |
| leloe 11296 | 'Less than or equal to' ex... |
| eqlelt 11297 | Equality in terms of 'less... |
| ltle 11298 | 'Less than' implies 'less ... |
| leltne 11299 | 'Less than or equal to' im... |
| lelttr 11300 | Transitive law. (Contribu... |
| leltletr 11301 | Transitive law, weaker for... |
| ltletr 11302 | Transitive law. (Contribu... |
| ltleletr 11303 | Transitive law, weaker for... |
| letr 11304 | Transitive law. (Contribu... |
| ltnr 11305 | 'Less than' is irreflexive... |
| leid 11306 | 'Less than or equal to' is... |
| ltne 11307 | 'Less than' implies not eq... |
| ltnsym 11308 | 'Less than' is not symmetr... |
| ltnsym2 11309 | 'Less than' is antisymmetr... |
| letric 11310 | Trichotomy law. (Contribu... |
| ltlen 11311 | 'Less than' expressed in t... |
| eqle 11312 | Equality implies 'less tha... |
| eqled 11313 | Equality implies 'less tha... |
| ltadd2 11314 | Addition to both sides of ... |
| ne0gt0 11315 | A nonzero nonnegative numb... |
| lecasei 11316 | Ordering elimination by ca... |
| lelttric 11317 | Trichotomy law. (Contribu... |
| ltlecasei 11318 | Ordering elimination by ca... |
| ltnri 11319 | 'Less than' is irreflexive... |
| eqlei 11320 | Equality implies 'less tha... |
| eqlei2 11321 | Equality implies 'less tha... |
| gtneii 11322 | 'Less than' implies not eq... |
| ltneii 11323 | 'Greater than' implies not... |
| lttri2i 11324 | Consequence of trichotomy.... |
| lttri3i 11325 | Consequence of trichotomy.... |
| letri3i 11326 | Consequence of trichotomy.... |
| leloei 11327 | 'Less than or equal to' in... |
| ltleni 11328 | 'Less than' expressed in t... |
| ltnsymi 11329 | 'Less than' is not symmetr... |
| lenlti 11330 | 'Less than or equal to' in... |
| ltnlei 11331 | 'Less than' in terms of 'l... |
| ltlei 11332 | 'Less than' implies 'less ... |
| ltleii 11333 | 'Less than' implies 'less ... |
| ltnei 11334 | 'Less than' implies not eq... |
| letrii 11335 | Trichotomy law for 'less t... |
| lttri 11336 | 'Less than' is transitive.... |
| lelttri 11337 | 'Less than or equal to', '... |
| ltletri 11338 | 'Less than', 'less than or... |
| letri 11339 | 'Less than or equal to' is... |
| le2tri3i 11340 | Extended trichotomy law fo... |
| ltadd2i 11341 | Addition to both sides of ... |
| mulgt0i 11342 | The product of two positiv... |
| mulgt0ii 11343 | The product of two positiv... |
| ltnrd 11344 | 'Less than' is irreflexive... |
| gtned 11345 | 'Less than' implies not eq... |
| ltned 11346 | 'Greater than' implies not... |
| ne0gt0d 11347 | A nonzero nonnegative numb... |
| lttrid 11348 | Ordering on reals satisfie... |
| lttri2d 11349 | Consequence of trichotomy.... |
| lttri3d 11350 | Consequence of trichotomy.... |
| lttri4d 11351 | Trichotomy law for 'less t... |
| letri3d 11352 | Consequence of trichotomy.... |
| leloed 11353 | 'Less than or equal to' in... |
| eqleltd 11354 | Equality in terms of 'less... |
| ltlend 11355 | 'Less than' expressed in t... |
| lenltd 11356 | 'Less than or equal to' in... |
| ltnled 11357 | 'Less than' in terms of 'l... |
| ltled 11358 | 'Less than' implies 'less ... |
| ltnsymd 11359 | 'Less than' implies 'less ... |
| nltled 11360 | 'Not less than ' implies '... |
| lensymd 11361 | 'Less than or equal to' im... |
| letrid 11362 | Trichotomy law for 'less t... |
| leltned 11363 | 'Less than or equal to' im... |
| leneltd 11364 | 'Less than or equal to' an... |
| mulgt0d 11365 | The product of two positiv... |
| ltadd2d 11366 | Addition to both sides of ... |
| letrd 11367 | Transitive law deduction f... |
| lelttrd 11368 | Transitive law deduction f... |
| ltadd2dd 11369 | Addition to both sides of ... |
| ltletrd 11370 | Transitive law deduction f... |
| lttrd 11371 | Transitive law deduction f... |
| lelttrdi 11372 | If a number is less than a... |
| dedekind 11373 | The Dedekind cut theorem. ... |
| dedekindle 11374 | The Dedekind cut theorem, ... |
| mul12 11375 | Commutative/associative la... |
| mul32 11376 | Commutative/associative la... |
| mul31 11377 | Commutative/associative la... |
| mul4 11378 | Rearrangement of 4 factors... |
| mul4r 11379 | Rearrangement of 4 factors... |
| muladd11 11380 | A simple product of sums e... |
| 1p1times 11381 | Two times a number. (Cont... |
| peano2cn 11382 | A theorem for complex numb... |
| peano2re 11383 | A theorem for reals analog... |
| readdcan 11384 | Cancellation law for addit... |
| 00id 11385 | ` 0 ` is its own additive ... |
| mul02lem1 11386 | Lemma for ~ mul02 . If an... |
| mul02lem2 11387 | Lemma for ~ mul02 . Zero ... |
| mul02 11388 | Multiplication by ` 0 ` . ... |
| mul01 11389 | Multiplication by ` 0 ` . ... |
| addrid 11390 | ` 0 ` is an additive ident... |
| cnegex 11391 | Existence of the negative ... |
| cnegex2 11392 | Existence of a left invers... |
| addlid 11393 | ` 0 ` is a left identity f... |
| addcan 11394 | Cancellation law for addit... |
| addcan2 11395 | Cancellation law for addit... |
| addcom 11396 | Addition commutes. This u... |
| addridi 11397 | ` 0 ` is an additive ident... |
| addlidi 11398 | ` 0 ` is a left identity f... |
| mul02i 11399 | Multiplication by 0. Theo... |
| mul01i 11400 | Multiplication by ` 0 ` . ... |
| addcomi 11401 | Addition commutes. Based ... |
| addcomli 11402 | Addition commutes. (Contr... |
| addcani 11403 | Cancellation law for addit... |
| addcan2i 11404 | Cancellation law for addit... |
| mul12i 11405 | Commutative/associative la... |
| mul32i 11406 | Commutative/associative la... |
| mul4i 11407 | Rearrangement of 4 factors... |
| mul02d 11408 | Multiplication by 0. Theo... |
| mul01d 11409 | Multiplication by ` 0 ` . ... |
| addridd 11410 | ` 0 ` is an additive ident... |
| addlidd 11411 | ` 0 ` is a left identity f... |
| addcomd 11412 | Addition commutes. Based ... |
| addcand 11413 | Cancellation law for addit... |
| addcan2d 11414 | Cancellation law for addit... |
| addcanad 11415 | Cancelling a term on the l... |
| addcan2ad 11416 | Cancelling a term on the r... |
| addneintrd 11417 | Introducing a term on the ... |
| addneintr2d 11418 | Introducing a term on the ... |
| mul12d 11419 | Commutative/associative la... |
| mul32d 11420 | Commutative/associative la... |
| mul31d 11421 | Commutative/associative la... |
| mul4d 11422 | Rearrangement of 4 factors... |
| muladd11r 11423 | A simple product of sums e... |
| comraddd 11424 | Commute RHS addition, in d... |
| ltaddneg 11425 | Adding a negative number t... |
| ltaddnegr 11426 | Adding a negative number t... |
| add12 11427 | Commutative/associative la... |
| add32 11428 | Commutative/associative la... |
| add32r 11429 | Commutative/associative la... |
| add4 11430 | Rearrangement of 4 terms i... |
| add42 11431 | Rearrangement of 4 terms i... |
| add12i 11432 | Commutative/associative la... |
| add32i 11433 | Commutative/associative la... |
| add4i 11434 | Rearrangement of 4 terms i... |
| add42i 11435 | Rearrangement of 4 terms i... |
| add12d 11436 | Commutative/associative la... |
| add32d 11437 | Commutative/associative la... |
| add4d 11438 | Rearrangement of 4 terms i... |
| add42d 11439 | Rearrangement of 4 terms i... |
| 0cnALT 11444 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11445 | Alternate proof of ~ 0cnAL... |
| negeu 11446 | Existential uniqueness of ... |
| subval 11447 | Value of subtraction, whic... |
| negeq 11448 | Equality theorem for negat... |
| negeqi 11449 | Equality inference for neg... |
| negeqd 11450 | Equality deduction for neg... |
| nfnegd 11451 | Deduction version of ~ nfn... |
| nfneg 11452 | Bound-variable hypothesis ... |
| csbnegg 11453 | Move class substitution in... |
| negex 11454 | A negative is a set. (Con... |
| subcl 11455 | Closure law for subtractio... |
| negcl 11456 | Closure law for negative. ... |
| negicn 11457 | ` -u _i ` is a complex num... |
| subf 11458 | Subtraction is an operatio... |
| subadd 11459 | Relationship between subtr... |
| subadd2 11460 | Relationship between subtr... |
| subsub23 11461 | Swap subtrahend and result... |
| pncan 11462 | Cancellation law for subtr... |
| pncan2 11463 | Cancellation law for subtr... |
| pncan3 11464 | Subtraction and addition o... |
| npcan 11465 | Cancellation law for subtr... |
| addsubass 11466 | Associative-type law for a... |
| addsub 11467 | Law for addition and subtr... |
| subadd23 11468 | Commutative/associative la... |
| addsub12 11469 | Commutative/associative la... |
| 2addsub 11470 | Law for subtraction and ad... |
| addsubeq4 11471 | Relation between sums and ... |
| pncan3oi 11472 | Subtraction and addition o... |
| mvrraddi 11473 | Move the right term in a s... |
| mvlladdi 11474 | Move the left term in a su... |
| subid 11475 | Subtraction of a number fr... |
| subid1 11476 | Identity law for subtracti... |
| npncan 11477 | Cancellation law for subtr... |
| nppcan 11478 | Cancellation law for subtr... |
| nnpcan 11479 | Cancellation law for subtr... |
| nppcan3 11480 | Cancellation law for subtr... |
| subcan2 11481 | Cancellation law for subtr... |
| subeq0 11482 | If the difference between ... |
| npncan2 11483 | Cancellation law for subtr... |
| subsub2 11484 | Law for double subtraction... |
| nncan 11485 | Cancellation law for subtr... |
| subsub 11486 | Law for double subtraction... |
| nppcan2 11487 | Cancellation law for subtr... |
| subsub3 11488 | Law for double subtraction... |
| subsub4 11489 | Law for double subtraction... |
| sub32 11490 | Swap the second and third ... |
| nnncan 11491 | Cancellation law for subtr... |
| nnncan1 11492 | Cancellation law for subtr... |
| nnncan2 11493 | Cancellation law for subtr... |
| npncan3 11494 | Cancellation law for subtr... |
| pnpcan 11495 | Cancellation law for mixed... |
| pnpcan2 11496 | Cancellation law for mixed... |
| pnncan 11497 | Cancellation law for mixed... |
| ppncan 11498 | Cancellation law for mixed... |
| addsub4 11499 | Rearrangement of 4 terms i... |
| subadd4 11500 | Rearrangement of 4 terms i... |
| sub4 11501 | Rearrangement of 4 terms i... |
| neg0 11502 | Minus 0 equals 0. (Contri... |
| negid 11503 | Addition of a number and i... |
| negsub 11504 | Relationship between subtr... |
| subneg 11505 | Relationship between subtr... |
| negneg 11506 | A number is equal to the n... |
| neg11 11507 | Negative is one-to-one. (... |
| negcon1 11508 | Negative contraposition la... |
| negcon2 11509 | Negative contraposition la... |
| negeq0 11510 | A number is zero iff its n... |
| subcan 11511 | Cancellation law for subtr... |
| negsubdi 11512 | Distribution of negative o... |
| negdi 11513 | Distribution of negative o... |
| negdi2 11514 | Distribution of negative o... |
| negsubdi2 11515 | Distribution of negative o... |
| neg2sub 11516 | Relationship between subtr... |
| renegcli 11517 | Closure law for negative o... |
| resubcli 11518 | Closure law for subtractio... |
| renegcl 11519 | Closure law for negative o... |
| resubcl 11520 | Closure law for subtractio... |
| negreb 11521 | The negative of a real is ... |
| peano2cnm 11522 | "Reverse" second Peano pos... |
| peano2rem 11523 | "Reverse" second Peano pos... |
| negcli 11524 | Closure law for negative. ... |
| negidi 11525 | Addition of a number and i... |
| negnegi 11526 | A number is equal to the n... |
| subidi 11527 | Subtraction of a number fr... |
| subid1i 11528 | Identity law for subtracti... |
| negne0bi 11529 | A number is nonzero iff it... |
| negrebi 11530 | The negative of a real is ... |
| negne0i 11531 | The negative of a nonzero ... |
| subcli 11532 | Closure law for subtractio... |
| pncan3i 11533 | Subtraction and addition o... |
| negsubi 11534 | Relationship between subtr... |
| subnegi 11535 | Relationship between subtr... |
| subeq0i 11536 | If the difference between ... |
| neg11i 11537 | Negative is one-to-one. (... |
| negcon1i 11538 | Negative contraposition la... |
| negcon2i 11539 | Negative contraposition la... |
| negdii 11540 | Distribution of negative o... |
| negsubdii 11541 | Distribution of negative o... |
| negsubdi2i 11542 | Distribution of negative o... |
| subaddi 11543 | Relationship between subtr... |
| subadd2i 11544 | Relationship between subtr... |
| subaddrii 11545 | Relationship between subtr... |
| subsub23i 11546 | Swap subtrahend and result... |
| addsubassi 11547 | Associative-type law for s... |
| addsubi 11548 | Law for subtraction and ad... |
| subcani 11549 | Cancellation law for subtr... |
| subcan2i 11550 | Cancellation law for subtr... |
| pnncani 11551 | Cancellation law for mixed... |
| addsub4i 11552 | Rearrangement of 4 terms i... |
| 0reALT 11553 | Alternate proof of ~ 0re .... |
| negcld 11554 | Closure law for negative. ... |
| subidd 11555 | Subtraction of a number fr... |
| subid1d 11556 | Identity law for subtracti... |
| negidd 11557 | Addition of a number and i... |
| negnegd 11558 | A number is equal to the n... |
| negeq0d 11559 | A number is zero iff its n... |
| negne0bd 11560 | A number is nonzero iff it... |
| negcon1d 11561 | Contraposition law for una... |
| negcon1ad 11562 | Contraposition law for una... |
| neg11ad 11563 | The negatives of two compl... |
| negned 11564 | If two complex numbers are... |
| negne0d 11565 | The negative of a nonzero ... |
| negrebd 11566 | The negative of a real is ... |
| subcld 11567 | Closure law for subtractio... |
| pncand 11568 | Cancellation law for subtr... |
| pncan2d 11569 | Cancellation law for subtr... |
| pncan3d 11570 | Subtraction and addition o... |
| npcand 11571 | Cancellation law for subtr... |
| nncand 11572 | Cancellation law for subtr... |
| negsubd 11573 | Relationship between subtr... |
| subnegd 11574 | Relationship between subtr... |
| subeq0d 11575 | If the difference between ... |
| subne0d 11576 | Two unequal numbers have n... |
| subeq0ad 11577 | The difference of two comp... |
| subne0ad 11578 | If the difference of two c... |
| neg11d 11579 | If the difference between ... |
| negdid 11580 | Distribution of negative o... |
| negdi2d 11581 | Distribution of negative o... |
| negsubdid 11582 | Distribution of negative o... |
| negsubdi2d 11583 | Distribution of negative o... |
| neg2subd 11584 | Relationship between subtr... |
| subaddd 11585 | Relationship between subtr... |
| subadd2d 11586 | Relationship between subtr... |
| addsubassd 11587 | Associative-type law for s... |
| addsubd 11588 | Law for subtraction and ad... |
| subadd23d 11589 | Commutative/associative la... |
| addsub12d 11590 | Commutative/associative la... |
| npncand 11591 | Cancellation law for subtr... |
| nppcand 11592 | Cancellation law for subtr... |
| nppcan2d 11593 | Cancellation law for subtr... |
| nppcan3d 11594 | Cancellation law for subtr... |
| subsubd 11595 | Law for double subtraction... |
| subsub2d 11596 | Law for double subtraction... |
| subsub3d 11597 | Law for double subtraction... |
| subsub4d 11598 | Law for double subtraction... |
| sub32d 11599 | Swap the second and third ... |
| nnncand 11600 | Cancellation law for subtr... |
| nnncan1d 11601 | Cancellation law for subtr... |
| nnncan2d 11602 | Cancellation law for subtr... |
| npncan3d 11603 | Cancellation law for subtr... |
| pnpcand 11604 | Cancellation law for mixed... |
| pnpcan2d 11605 | Cancellation law for mixed... |
| pnncand 11606 | Cancellation law for mixed... |
| ppncand 11607 | Cancellation law for mixed... |
| subcand 11608 | Cancellation law for subtr... |
| subcan2d 11609 | Cancellation law for subtr... |
| subcanad 11610 | Cancellation law for subtr... |
| subneintrd 11611 | Introducing subtraction on... |
| subcan2ad 11612 | Cancellation law for subtr... |
| subneintr2d 11613 | Introducing subtraction on... |
| addsub4d 11614 | Rearrangement of 4 terms i... |
| subadd4d 11615 | Rearrangement of 4 terms i... |
| sub4d 11616 | Rearrangement of 4 terms i... |
| 2addsubd 11617 | Law for subtraction and ad... |
| addsubeq4d 11618 | Relation between sums and ... |
| subeqxfrd 11619 | Transfer two terms of a su... |
| mvlraddd 11620 | Move the right term in a s... |
| mvlladdd 11621 | Move the left term in a su... |
| mvrraddd 11622 | Move the right term in a s... |
| mvrladdd 11623 | Move the left term in a su... |
| assraddsubd 11624 | Associate RHS addition-sub... |
| subaddeqd 11625 | Transfer two terms of a su... |
| addlsub 11626 | Left-subtraction: Subtrac... |
| addrsub 11627 | Right-subtraction: Subtra... |
| subexsub 11628 | A subtraction law: Exchan... |
| addid0 11629 | If adding a number to a an... |
| addn0nid 11630 | Adding a nonzero number to... |
| pnpncand 11631 | Addition/subtraction cance... |
| subeqrev 11632 | Reverse the order of subtr... |
| addeq0 11633 | Two complex numbers add up... |
| pncan1 11634 | Cancellation law for addit... |
| npcan1 11635 | Cancellation law for subtr... |
| subeq0bd 11636 | If two complex numbers are... |
| renegcld 11637 | Closure law for negative o... |
| resubcld 11638 | Closure law for subtractio... |
| negn0 11639 | The image under negation o... |
| negf1o 11640 | Negation is an isomorphism... |
| kcnktkm1cn 11641 | k times k minus 1 is a com... |
| muladd 11642 | Product of two sums. (Con... |
| subdi 11643 | Distribution of multiplica... |
| subdir 11644 | Distribution of multiplica... |
| ine0 11645 | The imaginary unit ` _i ` ... |
| mulneg1 11646 | Product with negative is n... |
| mulneg2 11647 | The product with a negativ... |
| mulneg12 11648 | Swap the negative sign in ... |
| mul2neg 11649 | Product of two negatives. ... |
| submul2 11650 | Convert a subtraction to a... |
| mulm1 11651 | Product with minus one is ... |
| addneg1mul 11652 | Addition with product with... |
| mulsub 11653 | Product of two differences... |
| mulsub2 11654 | Swap the order of subtract... |
| mulm1i 11655 | Product with minus one is ... |
| mulneg1i 11656 | Product with negative is n... |
| mulneg2i 11657 | Product with negative is n... |
| mul2negi 11658 | Product of two negatives. ... |
| subdii 11659 | Distribution of multiplica... |
| subdiri 11660 | Distribution of multiplica... |
| muladdi 11661 | Product of two sums. (Con... |
| mulm1d 11662 | Product with minus one is ... |
| mulneg1d 11663 | Product with negative is n... |
| mulneg2d 11664 | Product with negative is n... |
| mul2negd 11665 | Product of two negatives. ... |
| subdid 11666 | Distribution of multiplica... |
| subdird 11667 | Distribution of multiplica... |
| muladdd 11668 | Product of two sums. (Con... |
| mulsubd 11669 | Product of two differences... |
| muls1d 11670 | Multiplication by one minu... |
| mulsubfacd 11671 | Multiplication followed by... |
| addmulsub 11672 | The product of a sum and a... |
| subaddmulsub 11673 | The difference with a prod... |
| mulsubaddmulsub 11674 | A special difference of a ... |
| gt0ne0 11675 | Positive implies nonzero. ... |
| lt0ne0 11676 | A number which is less tha... |
| ltadd1 11677 | Addition to both sides of ... |
| leadd1 11678 | Addition to both sides of ... |
| leadd2 11679 | Addition to both sides of ... |
| ltsubadd 11680 | 'Less than' relationship b... |
| ltsubadd2 11681 | 'Less than' relationship b... |
| lesubadd 11682 | 'Less than or equal to' re... |
| lesubadd2 11683 | 'Less than or equal to' re... |
| ltaddsub 11684 | 'Less than' relationship b... |
| ltaddsub2 11685 | 'Less than' relationship b... |
| leaddsub 11686 | 'Less than or equal to' re... |
| leaddsub2 11687 | 'Less than or equal to' re... |
| suble 11688 | Swap subtrahends in an ine... |
| lesub 11689 | Swap subtrahends in an ine... |
| ltsub23 11690 | 'Less than' relationship b... |
| ltsub13 11691 | 'Less than' relationship b... |
| le2add 11692 | Adding both sides of two '... |
| ltleadd 11693 | Adding both sides of two o... |
| leltadd 11694 | Adding both sides of two o... |
| lt2add 11695 | Adding both sides of two '... |
| addgt0 11696 | The sum of 2 positive numb... |
| addgegt0 11697 | The sum of nonnegative and... |
| addgtge0 11698 | The sum of nonnegative and... |
| addge0 11699 | The sum of 2 nonnegative n... |
| ltaddpos 11700 | Adding a positive number t... |
| ltaddpos2 11701 | Adding a positive number t... |
| ltsubpos 11702 | Subtracting a positive num... |
| posdif 11703 | Comparison of two numbers ... |
| lesub1 11704 | Subtraction from both side... |
| lesub2 11705 | Subtraction of both sides ... |
| ltsub1 11706 | Subtraction from both side... |
| ltsub2 11707 | Subtraction of both sides ... |
| lt2sub 11708 | Subtracting both sides of ... |
| le2sub 11709 | Subtracting both sides of ... |
| ltneg 11710 | Negative of both sides of ... |
| ltnegcon1 11711 | Contraposition of negative... |
| ltnegcon2 11712 | Contraposition of negative... |
| leneg 11713 | Negative of both sides of ... |
| lenegcon1 11714 | Contraposition of negative... |
| lenegcon2 11715 | Contraposition of negative... |
| lt0neg1 11716 | Comparison of a number and... |
| lt0neg2 11717 | Comparison of a number and... |
| le0neg1 11718 | Comparison of a number and... |
| le0neg2 11719 | Comparison of a number and... |
| addge01 11720 | A number is less than or e... |
| addge02 11721 | A number is less than or e... |
| add20 11722 | Two nonnegative numbers ar... |
| subge0 11723 | Nonnegative subtraction. ... |
| suble0 11724 | Nonpositive subtraction. ... |
| leaddle0 11725 | The sum of a real number a... |
| subge02 11726 | Nonnegative subtraction. ... |
| lesub0 11727 | Lemma to show a nonnegativ... |
| mulge0 11728 | The product of two nonnega... |
| mullt0 11729 | The product of two negativ... |
| msqgt0 11730 | A nonzero square is positi... |
| msqge0 11731 | A square is nonnegative. ... |
| 0lt1 11732 | 0 is less than 1. Theorem... |
| 0le1 11733 | 0 is less than or equal to... |
| relin01 11734 | An interval law for less t... |
| ltordlem 11735 | Lemma for ~ ltord1 . (Con... |
| ltord1 11736 | Infer an ordering relation... |
| leord1 11737 | Infer an ordering relation... |
| eqord1 11738 | A strictly increasing real... |
| ltord2 11739 | Infer an ordering relation... |
| leord2 11740 | Infer an ordering relation... |
| eqord2 11741 | A strictly decreasing real... |
| wloglei 11742 | Form of ~ wlogle where bot... |
| wlogle 11743 | If the predicate ` ch ( x ... |
| leidi 11744 | 'Less than or equal to' is... |
| gt0ne0i 11745 | Positive means nonzero (us... |
| gt0ne0ii 11746 | Positive implies nonzero. ... |
| msqgt0i 11747 | A nonzero square is positi... |
| msqge0i 11748 | A square is nonnegative. ... |
| addgt0i 11749 | Addition of 2 positive num... |
| addge0i 11750 | Addition of 2 nonnegative ... |
| addgegt0i 11751 | Addition of nonnegative an... |
| addgt0ii 11752 | Addition of 2 positive num... |
| add20i 11753 | Two nonnegative numbers ar... |
| ltnegi 11754 | Negative of both sides of ... |
| lenegi 11755 | Negative of both sides of ... |
| ltnegcon2i 11756 | Contraposition of negative... |
| mulge0i 11757 | The product of two nonnega... |
| lesub0i 11758 | Lemma to show a nonnegativ... |
| ltaddposi 11759 | Adding a positive number t... |
| posdifi 11760 | Comparison of two numbers ... |
| ltnegcon1i 11761 | Contraposition of negative... |
| lenegcon1i 11762 | Contraposition of negative... |
| subge0i 11763 | Nonnegative subtraction. ... |
| ltadd1i 11764 | Addition to both sides of ... |
| leadd1i 11765 | Addition to both sides of ... |
| leadd2i 11766 | Addition to both sides of ... |
| ltsubaddi 11767 | 'Less than' relationship b... |
| lesubaddi 11768 | 'Less than or equal to' re... |
| ltsubadd2i 11769 | 'Less than' relationship b... |
| lesubadd2i 11770 | 'Less than or equal to' re... |
| ltaddsubi 11771 | 'Less than' relationship b... |
| lt2addi 11772 | Adding both side of two in... |
| le2addi 11773 | Adding both side of two in... |
| gt0ne0d 11774 | Positive implies nonzero. ... |
| lt0ne0d 11775 | Something less than zero i... |
| leidd 11776 | 'Less than or equal to' is... |
| msqgt0d 11777 | A nonzero square is positi... |
| msqge0d 11778 | A square is nonnegative. ... |
| lt0neg1d 11779 | Comparison of a number and... |
| lt0neg2d 11780 | Comparison of a number and... |
| le0neg1d 11781 | Comparison of a number and... |
| le0neg2d 11782 | Comparison of a number and... |
| addgegt0d 11783 | Addition of nonnegative an... |
| addgtge0d 11784 | Addition of positive and n... |
| addgt0d 11785 | Addition of 2 positive num... |
| addge0d 11786 | Addition of 2 nonnegative ... |
| mulge0d 11787 | The product of two nonnega... |
| ltnegd 11788 | Negative of both sides of ... |
| lenegd 11789 | Negative of both sides of ... |
| ltnegcon1d 11790 | Contraposition of negative... |
| ltnegcon2d 11791 | Contraposition of negative... |
| lenegcon1d 11792 | Contraposition of negative... |
| lenegcon2d 11793 | Contraposition of negative... |
| ltaddposd 11794 | Adding a positive number t... |
| ltaddpos2d 11795 | Adding a positive number t... |
| ltsubposd 11796 | Subtracting a positive num... |
| posdifd 11797 | Comparison of two numbers ... |
| addge01d 11798 | A number is less than or e... |
| addge02d 11799 | A number is less than or e... |
| subge0d 11800 | Nonnegative subtraction. ... |
| suble0d 11801 | Nonpositive subtraction. ... |
| subge02d 11802 | Nonnegative subtraction. ... |
| ltadd1d 11803 | Addition to both sides of ... |
| leadd1d 11804 | Addition to both sides of ... |
| leadd2d 11805 | Addition to both sides of ... |
| ltsubaddd 11806 | 'Less than' relationship b... |
| lesubaddd 11807 | 'Less than or equal to' re... |
| ltsubadd2d 11808 | 'Less than' relationship b... |
| lesubadd2d 11809 | 'Less than or equal to' re... |
| ltaddsubd 11810 | 'Less than' relationship b... |
| ltaddsub2d 11811 | 'Less than' relationship b... |
| leaddsub2d 11812 | 'Less than or equal to' re... |
| subled 11813 | Swap subtrahends in an ine... |
| lesubd 11814 | Swap subtrahends in an ine... |
| ltsub23d 11815 | 'Less than' relationship b... |
| ltsub13d 11816 | 'Less than' relationship b... |
| lesub1d 11817 | Subtraction from both side... |
| lesub2d 11818 | Subtraction of both sides ... |
| ltsub1d 11819 | Subtraction from both side... |
| ltsub2d 11820 | Subtraction of both sides ... |
| ltadd1dd 11821 | Addition to both sides of ... |
| ltsub1dd 11822 | Subtraction from both side... |
| ltsub2dd 11823 | Subtraction of both sides ... |
| leadd1dd 11824 | Addition to both sides of ... |
| leadd2dd 11825 | Addition to both sides of ... |
| lesub1dd 11826 | Subtraction from both side... |
| lesub2dd 11827 | Subtraction of both sides ... |
| lesub3d 11828 | The result of subtracting ... |
| le2addd 11829 | Adding both side of two in... |
| le2subd 11830 | Subtracting both sides of ... |
| ltleaddd 11831 | Adding both sides of two o... |
| leltaddd 11832 | Adding both sides of two o... |
| lt2addd 11833 | Adding both side of two in... |
| lt2subd 11834 | Subtracting both sides of ... |
| possumd 11835 | Condition for a positive s... |
| sublt0d 11836 | When a subtraction gives a... |
| ltaddsublt 11837 | Addition and subtraction o... |
| 1le1 11838 | One is less than or equal ... |
| ixi 11839 | ` _i ` times itself is min... |
| recextlem1 11840 | Lemma for ~ recex . (Cont... |
| recextlem2 11841 | Lemma for ~ recex . (Cont... |
| recex 11842 | Existence of reciprocal of... |
| mulcand 11843 | Cancellation law for multi... |
| mulcan2d 11844 | Cancellation law for multi... |
| mulcanad 11845 | Cancellation of a nonzero ... |
| mulcan2ad 11846 | Cancellation of a nonzero ... |
| mulcan 11847 | Cancellation law for multi... |
| mulcan2 11848 | Cancellation law for multi... |
| mulcani 11849 | Cancellation law for multi... |
| mul0or 11850 | If a product is zero, one ... |
| mulne0b 11851 | The product of two nonzero... |
| mulne0 11852 | The product of two nonzero... |
| mulne0i 11853 | The product of two nonzero... |
| muleqadd 11854 | Property of numbers whose ... |
| receu 11855 | Existential uniqueness of ... |
| mulnzcnopr 11856 | Multiplication maps nonzer... |
| msq0i 11857 | A number is zero iff its s... |
| mul0ori 11858 | If a product is zero, one ... |
| msq0d 11859 | A number is zero iff its s... |
| mul0ord 11860 | If a product is zero, one ... |
| mulne0bd 11861 | The product of two nonzero... |
| mulne0d 11862 | The product of two nonzero... |
| mulcan1g 11863 | A generalized form of the ... |
| mulcan2g 11864 | A generalized form of the ... |
| mulne0bad 11865 | A factor of a nonzero comp... |
| mulne0bbd 11866 | A factor of a nonzero comp... |
| 1div0 11869 | You can't divide by zero, ... |
| divval 11870 | Value of division: if ` A ... |
| divmul 11871 | Relationship between divis... |
| divmul2 11872 | Relationship between divis... |
| divmul3 11873 | Relationship between divis... |
| divcl 11874 | Closure law for division. ... |
| reccl 11875 | Closure law for reciprocal... |
| divcan2 11876 | A cancellation law for div... |
| divcan1 11877 | A cancellation law for div... |
| diveq0 11878 | A ratio is zero iff the nu... |
| divne0b 11879 | The ratio of nonzero numbe... |
| divne0 11880 | The ratio of nonzero numbe... |
| recne0 11881 | The reciprocal of a nonzer... |
| recid 11882 | Multiplication of a number... |
| recid2 11883 | Multiplication of a number... |
| divrec 11884 | Relationship between divis... |
| divrec2 11885 | Relationship between divis... |
| divass 11886 | An associative law for div... |
| div23 11887 | A commutative/associative ... |
| div32 11888 | A commutative/associative ... |
| div13 11889 | A commutative/associative ... |
| div12 11890 | A commutative/associative ... |
| divmulass 11891 | An associative law for div... |
| divmulasscom 11892 | An associative/commutative... |
| divdir 11893 | Distribution of division o... |
| divcan3 11894 | A cancellation law for div... |
| divcan4 11895 | A cancellation law for div... |
| div11 11896 | One-to-one relationship fo... |
| divid 11897 | A number divided by itself... |
| div0 11898 | Division into zero is zero... |
| div1 11899 | A number divided by 1 is i... |
| 1div1e1 11900 | 1 divided by 1 is 1. (Con... |
| diveq1 11901 | Equality in terms of unit ... |
| divneg 11902 | Move negative sign inside ... |
| muldivdir 11903 | Distribution of division o... |
| divsubdir 11904 | Distribution of division o... |
| subdivcomb1 11905 | Bring a term in a subtract... |
| subdivcomb2 11906 | Bring a term in a subtract... |
| recrec 11907 | A number is equal to the r... |
| rec11 11908 | Reciprocal is one-to-one. ... |
| rec11r 11909 | Mutual reciprocals. (Cont... |
| divmuldiv 11910 | Multiplication of two rati... |
| divdivdiv 11911 | Division of two ratios. T... |
| divcan5 11912 | Cancellation of common fac... |
| divmul13 11913 | Swap the denominators in t... |
| divmul24 11914 | Swap the numerators in the... |
| divmuleq 11915 | Cross-multiply in an equal... |
| recdiv 11916 | The reciprocal of a ratio.... |
| divcan6 11917 | Cancellation of inverted f... |
| divdiv32 11918 | Swap denominators in a div... |
| divcan7 11919 | Cancel equal divisors in a... |
| dmdcan 11920 | Cancellation law for divis... |
| divdiv1 11921 | Division into a fraction. ... |
| divdiv2 11922 | Division by a fraction. (... |
| recdiv2 11923 | Division into a reciprocal... |
| ddcan 11924 | Cancellation in a double d... |
| divadddiv 11925 | Addition of two ratios. T... |
| divsubdiv 11926 | Subtraction of two ratios.... |
| conjmul 11927 | Two numbers whose reciproc... |
| rereccl 11928 | Closure law for reciprocal... |
| redivcl 11929 | Closure law for division o... |
| eqneg 11930 | A number equal to its nega... |
| eqnegd 11931 | A complex number equals it... |
| eqnegad 11932 | If a complex number equals... |
| div2neg 11933 | Quotient of two negatives.... |
| divneg2 11934 | Move negative sign inside ... |
| recclzi 11935 | Closure law for reciprocal... |
| recne0zi 11936 | The reciprocal of a nonzer... |
| recidzi 11937 | Multiplication of a number... |
| div1i 11938 | A number divided by 1 is i... |
| eqnegi 11939 | A number equal to its nega... |
| reccli 11940 | Closure law for reciprocal... |
| recidi 11941 | Multiplication of a number... |
| recreci 11942 | A number is equal to the r... |
| dividi 11943 | A number divided by itself... |
| div0i 11944 | Division into zero is zero... |
| divclzi 11945 | Closure law for division. ... |
| divcan1zi 11946 | A cancellation law for div... |
| divcan2zi 11947 | A cancellation law for div... |
| divreczi 11948 | Relationship between divis... |
| divcan3zi 11949 | A cancellation law for div... |
| divcan4zi 11950 | A cancellation law for div... |
| rec11i 11951 | Reciprocal is one-to-one. ... |
| divcli 11952 | Closure law for division. ... |
| divcan2i 11953 | A cancellation law for div... |
| divcan1i 11954 | A cancellation law for div... |
| divreci 11955 | Relationship between divis... |
| divcan3i 11956 | A cancellation law for div... |
| divcan4i 11957 | A cancellation law for div... |
| divne0i 11958 | The ratio of nonzero numbe... |
| rec11ii 11959 | Reciprocal is one-to-one. ... |
| divasszi 11960 | An associative law for div... |
| divmulzi 11961 | Relationship between divis... |
| divdirzi 11962 | Distribution of division o... |
| divdiv23zi 11963 | Swap denominators in a div... |
| divmuli 11964 | Relationship between divis... |
| divdiv32i 11965 | Swap denominators in a div... |
| divassi 11966 | An associative law for div... |
| divdiri 11967 | Distribution of division o... |
| div23i 11968 | A commutative/associative ... |
| div11i 11969 | One-to-one relationship fo... |
| divmuldivi 11970 | Multiplication of two rati... |
| divmul13i 11971 | Swap denominators of two r... |
| divadddivi 11972 | Addition of two ratios. T... |
| divdivdivi 11973 | Division of two ratios. T... |
| rerecclzi 11974 | Closure law for reciprocal... |
| rereccli 11975 | Closure law for reciprocal... |
| redivclzi 11976 | Closure law for division o... |
| redivcli 11977 | Closure law for division o... |
| div1d 11978 | A number divided by 1 is i... |
| reccld 11979 | Closure law for reciprocal... |
| recne0d 11980 | The reciprocal of a nonzer... |
| recidd 11981 | Multiplication of a number... |
| recid2d 11982 | Multiplication of a number... |
| recrecd 11983 | A number is equal to the r... |
| dividd 11984 | A number divided by itself... |
| div0d 11985 | Division into zero is zero... |
| divcld 11986 | Closure law for division. ... |
| divcan1d 11987 | A cancellation law for div... |
| divcan2d 11988 | A cancellation law for div... |
| divrecd 11989 | Relationship between divis... |
| divrec2d 11990 | Relationship between divis... |
| divcan3d 11991 | A cancellation law for div... |
| divcan4d 11992 | A cancellation law for div... |
| diveq0d 11993 | A ratio is zero iff the nu... |
| diveq1d 11994 | Equality in terms of unit ... |
| diveq1ad 11995 | The quotient of two comple... |
| diveq0ad 11996 | A fraction of complex numb... |
| divne1d 11997 | If two complex numbers are... |
| divne0bd 11998 | A ratio is zero iff the nu... |
| divnegd 11999 | Move negative sign inside ... |
| divneg2d 12000 | Move negative sign inside ... |
| div2negd 12001 | Quotient of two negatives.... |
| divne0d 12002 | The ratio of nonzero numbe... |
| recdivd 12003 | The reciprocal of a ratio.... |
| recdiv2d 12004 | Division into a reciprocal... |
| divcan6d 12005 | Cancellation of inverted f... |
| ddcand 12006 | Cancellation in a double d... |
| rec11d 12007 | Reciprocal is one-to-one. ... |
| divmuld 12008 | Relationship between divis... |
| div32d 12009 | A commutative/associative ... |
| div13d 12010 | A commutative/associative ... |
| divdiv32d 12011 | Swap denominators in a div... |
| divcan5d 12012 | Cancellation of common fac... |
| divcan5rd 12013 | Cancellation of common fac... |
| divcan7d 12014 | Cancel equal divisors in a... |
| dmdcand 12015 | Cancellation law for divis... |
| dmdcan2d 12016 | Cancellation law for divis... |
| divdiv1d 12017 | Division into a fraction. ... |
| divdiv2d 12018 | Division by a fraction. (... |
| divmul2d 12019 | Relationship between divis... |
| divmul3d 12020 | Relationship between divis... |
| divassd 12021 | An associative law for div... |
| div12d 12022 | A commutative/associative ... |
| div23d 12023 | A commutative/associative ... |
| divdird 12024 | Distribution of division o... |
| divsubdird 12025 | Distribution of division o... |
| div11d 12026 | One-to-one relationship fo... |
| divmuldivd 12027 | Multiplication of two rati... |
| divmul13d 12028 | Swap denominators of two r... |
| divmul24d 12029 | Swap the numerators in the... |
| divadddivd 12030 | Addition of two ratios. T... |
| divsubdivd 12031 | Subtraction of two ratios.... |
| divmuleqd 12032 | Cross-multiply in an equal... |
| divdivdivd 12033 | Division of two ratios. T... |
| diveq1bd 12034 | If two complex numbers are... |
| div2sub 12035 | Swap the order of subtract... |
| div2subd 12036 | Swap subtrahend and minuen... |
| rereccld 12037 | Closure law for reciprocal... |
| redivcld 12038 | Closure law for division o... |
| subrec 12039 | Subtraction of reciprocals... |
| subreci 12040 | Subtraction of reciprocals... |
| subrecd 12041 | Subtraction of reciprocals... |
| mvllmuld 12042 | Move the left term in a pr... |
| mvllmuli 12043 | Move the left term in a pr... |
| ldiv 12044 | Left-division. (Contribut... |
| rdiv 12045 | Right-division. (Contribu... |
| mdiv 12046 | A division law. (Contribu... |
| lineq 12047 | Solution of a (scalar) lin... |
| elimgt0 12048 | Hypothesis for weak deduct... |
| elimge0 12049 | Hypothesis for weak deduct... |
| ltp1 12050 | A number is less than itse... |
| lep1 12051 | A number is less than or e... |
| ltm1 12052 | A number minus 1 is less t... |
| lem1 12053 | A number minus 1 is less t... |
| letrp1 12054 | A transitive property of '... |
| p1le 12055 | A transitive property of p... |
| recgt0 12056 | The reciprocal of a positi... |
| prodgt0 12057 | Infer that a multiplicand ... |
| prodgt02 12058 | Infer that a multiplier is... |
| ltmul1a 12059 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12060 | Multiplication of both sid... |
| ltmul2 12061 | Multiplication of both sid... |
| lemul1 12062 | Multiplication of both sid... |
| lemul2 12063 | Multiplication of both sid... |
| lemul1a 12064 | Multiplication of both sid... |
| lemul2a 12065 | Multiplication of both sid... |
| ltmul12a 12066 | Comparison of product of t... |
| lemul12b 12067 | Comparison of product of t... |
| lemul12a 12068 | Comparison of product of t... |
| mulgt1 12069 | The product of two numbers... |
| ltmulgt11 12070 | Multiplication by a number... |
| ltmulgt12 12071 | Multiplication by a number... |
| lemulge11 12072 | Multiplication by a number... |
| lemulge12 12073 | Multiplication by a number... |
| ltdiv1 12074 | Division of both sides of ... |
| lediv1 12075 | Division of both sides of ... |
| gt0div 12076 | Division of a positive num... |
| ge0div 12077 | Division of a nonnegative ... |
| divgt0 12078 | The ratio of two positive ... |
| divge0 12079 | The ratio of nonnegative a... |
| mulge0b 12080 | A condition for multiplica... |
| mulle0b 12081 | A condition for multiplica... |
| mulsuble0b 12082 | A condition for multiplica... |
| ltmuldiv 12083 | 'Less than' relationship b... |
| ltmuldiv2 12084 | 'Less than' relationship b... |
| ltdivmul 12085 | 'Less than' relationship b... |
| ledivmul 12086 | 'Less than or equal to' re... |
| ltdivmul2 12087 | 'Less than' relationship b... |
| lt2mul2div 12088 | 'Less than' relationship b... |
| ledivmul2 12089 | 'Less than or equal to' re... |
| lemuldiv 12090 | 'Less than or equal' relat... |
| lemuldiv2 12091 | 'Less than or equal' relat... |
| ltrec 12092 | The reciprocal of both sid... |
| lerec 12093 | The reciprocal of both sid... |
| lt2msq1 12094 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12095 | Two nonnegative numbers co... |
| ltdiv2 12096 | Division of a positive num... |
| ltrec1 12097 | Reciprocal swap in a 'less... |
| lerec2 12098 | Reciprocal swap in a 'less... |
| ledivdiv 12099 | Invert ratios of positive ... |
| lediv2 12100 | Division of a positive num... |
| ltdiv23 12101 | Swap denominator with othe... |
| lediv23 12102 | Swap denominator with othe... |
| lediv12a 12103 | Comparison of ratio of two... |
| lediv2a 12104 | Division of both sides of ... |
| reclt1 12105 | The reciprocal of a positi... |
| recgt1 12106 | The reciprocal of a positi... |
| recgt1i 12107 | The reciprocal of a number... |
| recp1lt1 12108 | Construct a number less th... |
| recreclt 12109 | Given a positive number ` ... |
| le2msq 12110 | The square function on non... |
| msq11 12111 | The square of a nonnegativ... |
| ledivp1 12112 | "Less than or equal to" an... |
| squeeze0 12113 | If a nonnegative number is... |
| ltp1i 12114 | A number is less than itse... |
| recgt0i 12115 | The reciprocal of a positi... |
| recgt0ii 12116 | The reciprocal of a positi... |
| prodgt0i 12117 | Infer that a multiplicand ... |
| divgt0i 12118 | The ratio of two positive ... |
| divge0i 12119 | The ratio of nonnegative a... |
| ltreci 12120 | The reciprocal of both sid... |
| lereci 12121 | The reciprocal of both sid... |
| lt2msqi 12122 | The square function on non... |
| le2msqi 12123 | The square function on non... |
| msq11i 12124 | The square of a nonnegativ... |
| divgt0i2i 12125 | The ratio of two positive ... |
| ltrecii 12126 | The reciprocal of both sid... |
| divgt0ii 12127 | The ratio of two positive ... |
| ltmul1i 12128 | Multiplication of both sid... |
| ltdiv1i 12129 | Division of both sides of ... |
| ltmuldivi 12130 | 'Less than' relationship b... |
| ltmul2i 12131 | Multiplication of both sid... |
| lemul1i 12132 | Multiplication of both sid... |
| lemul2i 12133 | Multiplication of both sid... |
| ltdiv23i 12134 | Swap denominator with othe... |
| ledivp1i 12135 | "Less than or equal to" an... |
| ltdivp1i 12136 | Less-than and division rel... |
| ltdiv23ii 12137 | Swap denominator with othe... |
| ltmul1ii 12138 | Multiplication of both sid... |
| ltdiv1ii 12139 | Division of both sides of ... |
| ltp1d 12140 | A number is less than itse... |
| lep1d 12141 | A number is less than or e... |
| ltm1d 12142 | A number minus 1 is less t... |
| lem1d 12143 | A number minus 1 is less t... |
| recgt0d 12144 | The reciprocal of a positi... |
| divgt0d 12145 | The ratio of two positive ... |
| mulgt1d 12146 | The product of two numbers... |
| lemulge11d 12147 | Multiplication by a number... |
| lemulge12d 12148 | Multiplication by a number... |
| lemul1ad 12149 | Multiplication of both sid... |
| lemul2ad 12150 | Multiplication of both sid... |
| ltmul12ad 12151 | Comparison of product of t... |
| lemul12ad 12152 | Comparison of product of t... |
| lemul12bd 12153 | Comparison of product of t... |
| fimaxre 12154 | A finite set of real numbe... |
| fimaxre2 12155 | A nonempty finite set of r... |
| fimaxre3 12156 | A nonempty finite set of r... |
| fiminre 12157 | A nonempty finite set of r... |
| fiminre2 12158 | A nonempty finite set of r... |
| negfi 12159 | The negation of a finite s... |
| lbreu 12160 | If a set of reals contains... |
| lbcl 12161 | If a set of reals contains... |
| lble 12162 | If a set of reals contains... |
| lbinf 12163 | If a set of reals contains... |
| lbinfcl 12164 | If a set of reals contains... |
| lbinfle 12165 | If a set of reals contains... |
| sup2 12166 | A nonempty, bounded-above ... |
| sup3 12167 | A version of the completen... |
| infm3lem 12168 | Lemma for ~ infm3 . (Cont... |
| infm3 12169 | The completeness axiom for... |
| suprcl 12170 | Closure of supremum of a n... |
| suprub 12171 | A member of a nonempty bou... |
| suprubd 12172 | Natural deduction form of ... |
| suprcld 12173 | Natural deduction form of ... |
| suprlub 12174 | The supremum of a nonempty... |
| suprnub 12175 | An upper bound is not less... |
| suprleub 12176 | The supremum of a nonempty... |
| supaddc 12177 | The supremum function dist... |
| supadd 12178 | The supremum function dist... |
| supmul1 12179 | The supremum function dist... |
| supmullem1 12180 | Lemma for ~ supmul . (Con... |
| supmullem2 12181 | Lemma for ~ supmul . (Con... |
| supmul 12182 | The supremum function dist... |
| sup3ii 12183 | A version of the completen... |
| suprclii 12184 | Closure of supremum of a n... |
| suprubii 12185 | A member of a nonempty bou... |
| suprlubii 12186 | The supremum of a nonempty... |
| suprnubii 12187 | An upper bound is not less... |
| suprleubii 12188 | The supremum of a nonempty... |
| riotaneg 12189 | The negative of the unique... |
| negiso 12190 | Negation is an order anti-... |
| dfinfre 12191 | The infimum of a set of re... |
| infrecl 12192 | Closure of infimum of a no... |
| infrenegsup 12193 | The infimum of a set of re... |
| infregelb 12194 | Any lower bound of a nonem... |
| infrelb 12195 | If a nonempty set of real ... |
| infrefilb 12196 | The infimum of a finite se... |
| supfirege 12197 | The supremum of a finite s... |
| inelr 12198 | The imaginary unit ` _i ` ... |
| rimul 12199 | A real number times the im... |
| cru 12200 | The representation of comp... |
| crne0 12201 | The real representation of... |
| creur 12202 | The real part of a complex... |
| creui 12203 | The imaginary part of a co... |
| cju 12204 | The complex conjugate of a... |
| ofsubeq0 12205 | Function analogue of ~ sub... |
| ofnegsub 12206 | Function analogue of ~ neg... |
| ofsubge0 12207 | Function analogue of ~ sub... |
| nnexALT 12210 | Alternate proof of ~ nnex ... |
| peano5nni 12211 | Peano's inductive postulat... |
| nnssre 12212 | The positive integers are ... |
| nnsscn 12213 | The positive integers are ... |
| nnex 12214 | The set of positive intege... |
| nnre 12215 | A positive integer is a re... |
| nncn 12216 | A positive integer is a co... |
| nnrei 12217 | A positive integer is a re... |
| nncni 12218 | A positive integer is a co... |
| 1nn 12219 | Peano postulate: 1 is a po... |
| peano2nn 12220 | Peano postulate: a success... |
| dfnn2 12221 | Alternate definition of th... |
| dfnn3 12222 | Alternate definition of th... |
| nnred 12223 | A positive integer is a re... |
| nncnd 12224 | A positive integer is a co... |
| peano2nnd 12225 | Peano postulate: a success... |
| nnind 12226 | Principle of Mathematical ... |
| nnindALT 12227 | Principle of Mathematical ... |
| nnindd 12228 | Principle of Mathematical ... |
| nn1m1nn 12229 | Every positive integer is ... |
| nn1suc 12230 | If a statement holds for 1... |
| nnaddcl 12231 | Closure of addition of pos... |
| nnmulcl 12232 | Closure of multiplication ... |
| nnmulcli 12233 | Closure of multiplication ... |
| nnmtmip 12234 | "Minus times minus is plus... |
| nn2ge 12235 | There exists a positive in... |
| nnge1 12236 | A positive integer is one ... |
| nngt1ne1 12237 | A positive integer is grea... |
| nnle1eq1 12238 | A positive integer is less... |
| nngt0 12239 | A positive integer is posi... |
| nnnlt1 12240 | A positive integer is not ... |
| nnnle0 12241 | A positive integer is not ... |
| nnne0 12242 | A positive integer is nonz... |
| nnneneg 12243 | No positive integer is equ... |
| 0nnn 12244 | Zero is not a positive int... |
| 0nnnALT 12245 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12246 | Alternate version of ~ nnn... |
| nngt0i 12247 | A positive integer is posi... |
| nnne0i 12248 | A positive integer is nonz... |
| nndivre 12249 | The quotient of a real and... |
| nnrecre 12250 | The reciprocal of a positi... |
| nnrecgt0 12251 | The reciprocal of a positi... |
| nnsub 12252 | Subtraction of positive in... |
| nnsubi 12253 | Subtraction of positive in... |
| nndiv 12254 | Two ways to express " ` A ... |
| nndivtr 12255 | Transitive property of div... |
| nnge1d 12256 | A positive integer is one ... |
| nngt0d 12257 | A positive integer is posi... |
| nnne0d 12258 | A positive integer is nonz... |
| nnrecred 12259 | The reciprocal of a positi... |
| nnaddcld 12260 | Closure of addition of pos... |
| nnmulcld 12261 | Closure of multiplication ... |
| nndivred 12262 | A positive integer is one ... |
| 0ne1 12279 | Zero is different from one... |
| 1m1e0 12280 | One minus one equals zero.... |
| 2nn 12281 | 2 is a positive integer. ... |
| 2re 12282 | The number 2 is real. (Co... |
| 2cn 12283 | The number 2 is a complex ... |
| 2cnALT 12284 | Alternate proof of ~ 2cn .... |
| 2ex 12285 | The number 2 is a set. (C... |
| 2cnd 12286 | The number 2 is a complex ... |
| 3nn 12287 | 3 is a positive integer. ... |
| 3re 12288 | The number 3 is real. (Co... |
| 3cn 12289 | The number 3 is a complex ... |
| 3ex 12290 | The number 3 is a set. (C... |
| 4nn 12291 | 4 is a positive integer. ... |
| 4re 12292 | The number 4 is real. (Co... |
| 4cn 12293 | The number 4 is a complex ... |
| 5nn 12294 | 5 is a positive integer. ... |
| 5re 12295 | The number 5 is real. (Co... |
| 5cn 12296 | The number 5 is a complex ... |
| 6nn 12297 | 6 is a positive integer. ... |
| 6re 12298 | The number 6 is real. (Co... |
| 6cn 12299 | The number 6 is a complex ... |
| 7nn 12300 | 7 is a positive integer. ... |
| 7re 12301 | The number 7 is real. (Co... |
| 7cn 12302 | The number 7 is a complex ... |
| 8nn 12303 | 8 is a positive integer. ... |
| 8re 12304 | The number 8 is real. (Co... |
| 8cn 12305 | The number 8 is a complex ... |
| 9nn 12306 | 9 is a positive integer. ... |
| 9re 12307 | The number 9 is real. (Co... |
| 9cn 12308 | The number 9 is a complex ... |
| 0le0 12309 | Zero is nonnegative. (Con... |
| 0le2 12310 | The number 0 is less than ... |
| 2pos 12311 | The number 2 is positive. ... |
| 2ne0 12312 | The number 2 is nonzero. ... |
| 3pos 12313 | The number 3 is positive. ... |
| 3ne0 12314 | The number 3 is nonzero. ... |
| 4pos 12315 | The number 4 is positive. ... |
| 4ne0 12316 | The number 4 is nonzero. ... |
| 5pos 12317 | The number 5 is positive. ... |
| 6pos 12318 | The number 6 is positive. ... |
| 7pos 12319 | The number 7 is positive. ... |
| 8pos 12320 | The number 8 is positive. ... |
| 9pos 12321 | The number 9 is positive. ... |
| neg1cn 12322 | -1 is a complex number. (... |
| neg1rr 12323 | -1 is a real number. (Con... |
| neg1ne0 12324 | -1 is nonzero. (Contribut... |
| neg1lt0 12325 | -1 is less than 0. (Contr... |
| negneg1e1 12326 | ` -u -u 1 ` is 1. (Contri... |
| 1pneg1e0 12327 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12328 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12329 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12330 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12331 | Function value at ` N + 1 ... |
| 1p0e1 12332 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12333 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12334 | 2 - 1 = 1. The result is ... |
| 1e2m1 12335 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12336 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12337 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12338 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12339 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12340 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12341 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12342 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12343 | Two plus two equals four. ... |
| 2times 12344 | Two times a number. (Cont... |
| times2 12345 | A number times 2. (Contri... |
| 2timesi 12346 | Two times a number. (Cont... |
| times2i 12347 | A number times 2. (Contri... |
| 2txmxeqx 12348 | Two times a complex number... |
| 2div2e1 12349 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12350 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12351 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12352 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12353 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12354 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12355 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12356 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12357 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12358 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12359 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12360 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12361 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12362 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12363 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12364 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12365 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12366 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12367 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12368 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12369 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12370 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12371 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12372 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12373 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12374 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12375 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12376 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12377 | 2 times 0 equals 0. (Cont... |
| 4d2e2 12378 | One half of four is two. ... |
| 1lt2 12379 | 1 is less than 2. (Contri... |
| 2lt3 12380 | 2 is less than 3. (Contri... |
| 1lt3 12381 | 1 is less than 3. (Contri... |
| 3lt4 12382 | 3 is less than 4. (Contri... |
| 2lt4 12383 | 2 is less than 4. (Contri... |
| 1lt4 12384 | 1 is less than 4. (Contri... |
| 4lt5 12385 | 4 is less than 5. (Contri... |
| 3lt5 12386 | 3 is less than 5. (Contri... |
| 2lt5 12387 | 2 is less than 5. (Contri... |
| 1lt5 12388 | 1 is less than 5. (Contri... |
| 5lt6 12389 | 5 is less than 6. (Contri... |
| 4lt6 12390 | 4 is less than 6. (Contri... |
| 3lt6 12391 | 3 is less than 6. (Contri... |
| 2lt6 12392 | 2 is less than 6. (Contri... |
| 1lt6 12393 | 1 is less than 6. (Contri... |
| 6lt7 12394 | 6 is less than 7. (Contri... |
| 5lt7 12395 | 5 is less than 7. (Contri... |
| 4lt7 12396 | 4 is less than 7. (Contri... |
| 3lt7 12397 | 3 is less than 7. (Contri... |
| 2lt7 12398 | 2 is less than 7. (Contri... |
| 1lt7 12399 | 1 is less than 7. (Contri... |
| 7lt8 12400 | 7 is less than 8. (Contri... |
| 6lt8 12401 | 6 is less than 8. (Contri... |
| 5lt8 12402 | 5 is less than 8. (Contri... |
| 4lt8 12403 | 4 is less than 8. (Contri... |
| 3lt8 12404 | 3 is less than 8. (Contri... |
| 2lt8 12405 | 2 is less than 8. (Contri... |
| 1lt8 12406 | 1 is less than 8. (Contri... |
| 8lt9 12407 | 8 is less than 9. (Contri... |
| 7lt9 12408 | 7 is less than 9. (Contri... |
| 6lt9 12409 | 6 is less than 9. (Contri... |
| 5lt9 12410 | 5 is less than 9. (Contri... |
| 4lt9 12411 | 4 is less than 9. (Contri... |
| 3lt9 12412 | 3 is less than 9. (Contri... |
| 2lt9 12413 | 2 is less than 9. (Contri... |
| 1lt9 12414 | 1 is less than 9. (Contri... |
| 0ne2 12415 | 0 is not equal to 2. (Con... |
| 1ne2 12416 | 1 is not equal to 2. (Con... |
| 1le2 12417 | 1 is less than or equal to... |
| 2cnne0 12418 | 2 is a nonzero complex num... |
| 2rene0 12419 | 2 is a nonzero real number... |
| 1le3 12420 | 1 is less than or equal to... |
| neg1mulneg1e1 12421 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12422 | One-half is real. (Contri... |
| halfcn 12423 | One-half is a complex numb... |
| halfgt0 12424 | One-half is greater than z... |
| halfge0 12425 | One-half is not negative. ... |
| halflt1 12426 | One-half is less than one.... |
| 1mhlfehlf 12427 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12428 | An eighth of four thirds i... |
| halfpm6th 12429 | One half plus or minus one... |
| it0e0 12430 | i times 0 equals 0. (Cont... |
| 2mulicn 12431 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12432 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12433 | Closure of half of a numbe... |
| rehalfcl 12434 | Real closure of half. (Co... |
| half0 12435 | Half of a number is zero i... |
| 2halves 12436 | Two halves make a whole. ... |
| halfpos2 12437 | A number is positive iff i... |
| halfpos 12438 | A positive number is great... |
| halfnneg2 12439 | A number is nonnegative if... |
| halfaddsubcl 12440 | Closure of half-sum and ha... |
| halfaddsub 12441 | Sum and difference of half... |
| subhalfhalf 12442 | Subtracting the half of a ... |
| lt2halves 12443 | A sum is less than the who... |
| addltmul 12444 | Sum is less than product f... |
| nominpos 12445 | There is no smallest posit... |
| avglt1 12446 | Ordering property for aver... |
| avglt2 12447 | Ordering property for aver... |
| avgle1 12448 | Ordering property for aver... |
| avgle2 12449 | Ordering property for aver... |
| avgle 12450 | The average of two numbers... |
| 2timesd 12451 | Two times a number. (Cont... |
| times2d 12452 | A number times 2. (Contri... |
| halfcld 12453 | Closure of half of a numbe... |
| 2halvesd 12454 | Two halves make a whole. ... |
| rehalfcld 12455 | Real closure of half. (Co... |
| lt2halvesd 12456 | A sum is less than the who... |
| rehalfcli 12457 | Half a real number is real... |
| lt2addmuld 12458 | If two real numbers are le... |
| add1p1 12459 | Adding two times 1 to a nu... |
| sub1m1 12460 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12461 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12462 | A complex number increased... |
| div4p1lem1div2 12463 | An integer greater than 5,... |
| nnunb 12464 | The set of positive intege... |
| arch 12465 | Archimedean property of re... |
| nnrecl 12466 | There exists a positive in... |
| bndndx 12467 | A bounded real sequence ` ... |
| elnn0 12470 | Nonnegative integers expre... |
| nnssnn0 12471 | Positive naturals are a su... |
| nn0ssre 12472 | Nonnegative integers are a... |
| nn0sscn 12473 | Nonnegative integers are a... |
| nn0ex 12474 | The set of nonnegative int... |
| nnnn0 12475 | A positive integer is a no... |
| nnnn0i 12476 | A positive integer is a no... |
| nn0re 12477 | A nonnegative integer is a... |
| nn0cn 12478 | A nonnegative integer is a... |
| nn0rei 12479 | A nonnegative integer is a... |
| nn0cni 12480 | A nonnegative integer is a... |
| dfn2 12481 | The set of positive intege... |
| elnnne0 12482 | The positive integer prope... |
| 0nn0 12483 | 0 is a nonnegative integer... |
| 1nn0 12484 | 1 is a nonnegative integer... |
| 2nn0 12485 | 2 is a nonnegative integer... |
| 3nn0 12486 | 3 is a nonnegative integer... |
| 4nn0 12487 | 4 is a nonnegative integer... |
| 5nn0 12488 | 5 is a nonnegative integer... |
| 6nn0 12489 | 6 is a nonnegative integer... |
| 7nn0 12490 | 7 is a nonnegative integer... |
| 8nn0 12491 | 8 is a nonnegative integer... |
| 9nn0 12492 | 9 is a nonnegative integer... |
| nn0ge0 12493 | A nonnegative integer is g... |
| nn0nlt0 12494 | A nonnegative integer is n... |
| nn0ge0i 12495 | Nonnegative integers are n... |
| nn0le0eq0 12496 | A nonnegative integer is l... |
| nn0p1gt0 12497 | A nonnegative integer incr... |
| nnnn0addcl 12498 | A positive integer plus a ... |
| nn0nnaddcl 12499 | A nonnegative integer plus... |
| 0mnnnnn0 12500 | The result of subtracting ... |
| un0addcl 12501 | If ` S ` is closed under a... |
| un0mulcl 12502 | If ` S ` is closed under m... |
| nn0addcl 12503 | Closure of addition of non... |
| nn0mulcl 12504 | Closure of multiplication ... |
| nn0addcli 12505 | Closure of addition of non... |
| nn0mulcli 12506 | Closure of multiplication ... |
| nn0p1nn 12507 | A nonnegative integer plus... |
| peano2nn0 12508 | Second Peano postulate for... |
| nnm1nn0 12509 | A positive integer minus 1... |
| elnn0nn 12510 | The nonnegative integer pr... |
| elnnnn0 12511 | The positive integer prope... |
| elnnnn0b 12512 | The positive integer prope... |
| elnnnn0c 12513 | The positive integer prope... |
| nn0addge1 12514 | A number is less than or e... |
| nn0addge2 12515 | A number is less than or e... |
| nn0addge1i 12516 | A number is less than or e... |
| nn0addge2i 12517 | A number is less than or e... |
| nn0sub 12518 | Subtraction of nonnegative... |
| ltsubnn0 12519 | Subtracting a nonnegative ... |
| nn0negleid 12520 | A nonnegative integer is g... |
| difgtsumgt 12521 | If the difference of a rea... |
| nn0le2xi 12522 | A nonnegative integer is l... |
| nn0lele2xi 12523 | 'Less than or equal to' im... |
| fcdmnn0supp 12524 | Two ways to write the supp... |
| fcdmnn0fsupp 12525 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12526 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12527 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12528 | A positive integer is a no... |
| nn0red 12529 | A nonnegative integer is a... |
| nn0cnd 12530 | A nonnegative integer is a... |
| nn0ge0d 12531 | A nonnegative integer is g... |
| nn0addcld 12532 | Closure of addition of non... |
| nn0mulcld 12533 | Closure of multiplication ... |
| nn0readdcl 12534 | Closure law for addition o... |
| nn0n0n1ge2 12535 | A nonnegative integer whic... |
| nn0n0n1ge2b 12536 | A nonnegative integer is n... |
| nn0ge2m1nn 12537 | If a nonnegative integer i... |
| nn0ge2m1nn0 12538 | If a nonnegative integer i... |
| nn0nndivcl 12539 | Closure law for dividing o... |
| elxnn0 12542 | An extended nonnegative in... |
| nn0ssxnn0 12543 | The standard nonnegative i... |
| nn0xnn0 12544 | A standard nonnegative int... |
| xnn0xr 12545 | An extended nonnegative in... |
| 0xnn0 12546 | Zero is an extended nonneg... |
| pnf0xnn0 12547 | Positive infinity is an ex... |
| nn0nepnf 12548 | No standard nonnegative in... |
| nn0xnn0d 12549 | A standard nonnegative int... |
| nn0nepnfd 12550 | No standard nonnegative in... |
| xnn0nemnf 12551 | No extended nonnegative in... |
| xnn0xrnemnf 12552 | The extended nonnegative i... |
| xnn0nnn0pnf 12553 | An extended nonnegative in... |
| elz 12556 | Membership in the set of i... |
| nnnegz 12557 | The negative of a positive... |
| zre 12558 | An integer is a real. (Co... |
| zcn 12559 | An integer is a complex nu... |
| zrei 12560 | An integer is a real numbe... |
| zssre 12561 | The integers are a subset ... |
| zsscn 12562 | The integers are a subset ... |
| zex 12563 | The set of integers exists... |
| elnnz 12564 | Positive integer property ... |
| 0z 12565 | Zero is an integer. (Cont... |
| 0zd 12566 | Zero is an integer, deduct... |
| elnn0z 12567 | Nonnegative integer proper... |
| elznn0nn 12568 | Integer property expressed... |
| elznn0 12569 | Integer property expressed... |
| elznn 12570 | Integer property expressed... |
| zle0orge1 12571 | There is no integer in the... |
| elz2 12572 | Membership in the set of i... |
| dfz2 12573 | Alternative definition of ... |
| zexALT 12574 | Alternate proof of ~ zex .... |
| nnz 12575 | A positive integer is an i... |
| nnssz 12576 | Positive integers are a su... |
| nn0ssz 12577 | Nonnegative integers are a... |
| nnzOLD 12578 | Obsolete version of ~ nnz ... |
| nn0z 12579 | A nonnegative integer is a... |
| nn0zd 12580 | A nonnegative integer is a... |
| nnzd 12581 | A positive integer is an i... |
| nnzi 12582 | A positive integer is an i... |
| nn0zi 12583 | A nonnegative integer is a... |
| elnnz1 12584 | Positive integer property ... |
| znnnlt1 12585 | An integer is not a positi... |
| nnzrab 12586 | Positive integers expresse... |
| nn0zrab 12587 | Nonnegative integers expre... |
| 1z 12588 | One is an integer. (Contr... |
| 1zzd 12589 | One is an integer, deducti... |
| 2z 12590 | 2 is an integer. (Contrib... |
| 3z 12591 | 3 is an integer. (Contrib... |
| 4z 12592 | 4 is an integer. (Contrib... |
| znegcl 12593 | Closure law for negative i... |
| neg1z 12594 | -1 is an integer. (Contri... |
| znegclb 12595 | A complex number is an int... |
| nn0negz 12596 | The negative of a nonnegat... |
| nn0negzi 12597 | The negative of a nonnegat... |
| zaddcl 12598 | Closure of addition of int... |
| peano2z 12599 | Second Peano postulate gen... |
| zsubcl 12600 | Closure of subtraction of ... |
| peano2zm 12601 | "Reverse" second Peano pos... |
| zletr 12602 | Transitive law of ordering... |
| zrevaddcl 12603 | Reverse closure law for ad... |
| znnsub 12604 | The positive difference of... |
| znn0sub 12605 | The nonnegative difference... |
| nzadd 12606 | The sum of a real number n... |
| zmulcl 12607 | Closure of multiplication ... |
| zltp1le 12608 | Integer ordering relation.... |
| zleltp1 12609 | Integer ordering relation.... |
| zlem1lt 12610 | Integer ordering relation.... |
| zltlem1 12611 | Integer ordering relation.... |
| zgt0ge1 12612 | An integer greater than ` ... |
| nnleltp1 12613 | Positive integer ordering ... |
| nnltp1le 12614 | Positive integer ordering ... |
| nnaddm1cl 12615 | Closure of addition of pos... |
| nn0ltp1le 12616 | Nonnegative integer orderi... |
| nn0leltp1 12617 | Nonnegative integer orderi... |
| nn0ltlem1 12618 | Nonnegative integer orderi... |
| nn0sub2 12619 | Subtraction of nonnegative... |
| nn0lt10b 12620 | A nonnegative integer less... |
| nn0lt2 12621 | A nonnegative integer less... |
| nn0le2is012 12622 | A nonnegative integer whic... |
| nn0lem1lt 12623 | Nonnegative integer orderi... |
| nnlem1lt 12624 | Positive integer ordering ... |
| nnltlem1 12625 | Positive integer ordering ... |
| nnm1ge0 12626 | A positive integer decreas... |
| nn0ge0div 12627 | Division of a nonnegative ... |
| zdiv 12628 | Two ways to express " ` M ... |
| zdivadd 12629 | Property of divisibility: ... |
| zdivmul 12630 | Property of divisibility: ... |
| zextle 12631 | An extensionality-like pro... |
| zextlt 12632 | An extensionality-like pro... |
| recnz 12633 | The reciprocal of a number... |
| btwnnz 12634 | A number between an intege... |
| gtndiv 12635 | A larger number does not d... |
| halfnz 12636 | One-half is not an integer... |
| 3halfnz 12637 | Three halves is not an int... |
| suprzcl 12638 | The supremum of a bounded-... |
| prime 12639 | Two ways to express " ` A ... |
| msqznn 12640 | The square of a nonzero in... |
| zneo 12641 | No even integer equals an ... |
| nneo 12642 | A positive integer is even... |
| nneoi 12643 | A positive integer is even... |
| zeo 12644 | An integer is even or odd.... |
| zeo2 12645 | An integer is even or odd ... |
| peano2uz2 12646 | Second Peano postulate for... |
| peano5uzi 12647 | Peano's inductive postulat... |
| peano5uzti 12648 | Peano's inductive postulat... |
| dfuzi 12649 | An expression for the uppe... |
| uzind 12650 | Induction on the upper int... |
| uzind2 12651 | Induction on the upper int... |
| uzind3 12652 | Induction on the upper int... |
| nn0ind 12653 | Principle of Mathematical ... |
| nn0indALT 12654 | Principle of Mathematical ... |
| nn0indd 12655 | Principle of Mathematical ... |
| fzind 12656 | Induction on the integers ... |
| fnn0ind 12657 | Induction on the integers ... |
| nn0ind-raph 12658 | Principle of Mathematical ... |
| zindd 12659 | Principle of Mathematical ... |
| fzindd 12660 | Induction on the integers ... |
| btwnz 12661 | Any real number can be san... |
| zred 12662 | An integer is a real numbe... |
| zcnd 12663 | An integer is a complex nu... |
| znegcld 12664 | Closure law for negative i... |
| peano2zd 12665 | Deduction from second Pean... |
| zaddcld 12666 | Closure of addition of int... |
| zsubcld 12667 | Closure of subtraction of ... |
| zmulcld 12668 | Closure of multiplication ... |
| znnn0nn 12669 | The negative of a negative... |
| zadd2cl 12670 | Increasing an integer by 2... |
| zriotaneg 12671 | The negative of the unique... |
| suprfinzcl 12672 | The supremum of a nonempty... |
| 9p1e10 12675 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12676 | Version of the definition ... |
| decex 12677 | A decimal number is a set.... |
| deceq1 12678 | Equality theorem for the d... |
| deceq2 12679 | Equality theorem for the d... |
| deceq1i 12680 | Equality theorem for the d... |
| deceq2i 12681 | Equality theorem for the d... |
| deceq12i 12682 | Equality theorem for the d... |
| numnncl 12683 | Closure for a numeral (wit... |
| num0u 12684 | Add a zero in the units pl... |
| num0h 12685 | Add a zero in the higher p... |
| numcl 12686 | Closure for a decimal inte... |
| numsuc 12687 | The successor of a decimal... |
| deccl 12688 | Closure for a numeral. (C... |
| 10nn 12689 | 10 is a positive integer. ... |
| 10pos 12690 | The number 10 is positive.... |
| 10nn0 12691 | 10 is a nonnegative intege... |
| 10re 12692 | The number 10 is real. (C... |
| decnncl 12693 | Closure for a numeral. (C... |
| dec0u 12694 | Add a zero in the units pl... |
| dec0h 12695 | Add a zero in the higher p... |
| numnncl2 12696 | Closure for a decimal inte... |
| decnncl2 12697 | Closure for a decimal inte... |
| numlt 12698 | Comparing two decimal inte... |
| numltc 12699 | Comparing two decimal inte... |
| le9lt10 12700 | A "decimal digit" (i.e. a ... |
| declt 12701 | Comparing two decimal inte... |
| decltc 12702 | Comparing two decimal inte... |
| declth 12703 | Comparing two decimal inte... |
| decsuc 12704 | The successor of a decimal... |
| 3declth 12705 | Comparing two decimal inte... |
| 3decltc 12706 | Comparing two decimal inte... |
| decle 12707 | Comparing two decimal inte... |
| decleh 12708 | Comparing two decimal inte... |
| declei 12709 | Comparing a digit to a dec... |
| numlti 12710 | Comparing a digit to a dec... |
| declti 12711 | Comparing a digit to a dec... |
| decltdi 12712 | Comparing a digit to a dec... |
| numsucc 12713 | The successor of a decimal... |
| decsucc 12714 | The successor of a decimal... |
| 1e0p1 12715 | The successor of zero. (C... |
| dec10p 12716 | Ten plus an integer. (Con... |
| numma 12717 | Perform a multiply-add of ... |
| nummac 12718 | Perform a multiply-add of ... |
| numma2c 12719 | Perform a multiply-add of ... |
| numadd 12720 | Add two decimal integers `... |
| numaddc 12721 | Add two decimal integers `... |
| nummul1c 12722 | The product of a decimal i... |
| nummul2c 12723 | The product of a decimal i... |
| decma 12724 | Perform a multiply-add of ... |
| decmac 12725 | Perform a multiply-add of ... |
| decma2c 12726 | Perform a multiply-add of ... |
| decadd 12727 | Add two numerals ` M ` and... |
| decaddc 12728 | Add two numerals ` M ` and... |
| decaddc2 12729 | Add two numerals ` M ` and... |
| decrmanc 12730 | Perform a multiply-add of ... |
| decrmac 12731 | Perform a multiply-add of ... |
| decaddm10 12732 | The sum of two multiples o... |
| decaddi 12733 | Add two numerals ` M ` and... |
| decaddci 12734 | Add two numerals ` M ` and... |
| decaddci2 12735 | Add two numerals ` M ` and... |
| decsubi 12736 | Difference between a numer... |
| decmul1 12737 | The product of a numeral w... |
| decmul1c 12738 | The product of a numeral w... |
| decmul2c 12739 | The product of a numeral w... |
| decmulnc 12740 | The product of a numeral w... |
| 11multnc 12741 | The product of 11 (as nume... |
| decmul10add 12742 | A multiplication of a numb... |
| 6p5lem 12743 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12744 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12745 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12746 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12747 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12748 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12749 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12750 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12751 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12752 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12753 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12754 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12755 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12756 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12757 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12758 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12759 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12760 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12761 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12762 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12763 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12764 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12765 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12766 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12767 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12768 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12769 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12770 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12771 | 4 times 3 equals 12. (Con... |
| 4t4e16 12772 | 4 times 4 equals 16. (Con... |
| 5t2e10 12773 | 5 times 2 equals 10. (Con... |
| 5t3e15 12774 | 5 times 3 equals 15. (Con... |
| 5t4e20 12775 | 5 times 4 equals 20. (Con... |
| 5t5e25 12776 | 5 times 5 equals 25. (Con... |
| 6t2e12 12777 | 6 times 2 equals 12. (Con... |
| 6t3e18 12778 | 6 times 3 equals 18. (Con... |
| 6t4e24 12779 | 6 times 4 equals 24. (Con... |
| 6t5e30 12780 | 6 times 5 equals 30. (Con... |
| 6t6e36 12781 | 6 times 6 equals 36. (Con... |
| 7t2e14 12782 | 7 times 2 equals 14. (Con... |
| 7t3e21 12783 | 7 times 3 equals 21. (Con... |
| 7t4e28 12784 | 7 times 4 equals 28. (Con... |
| 7t5e35 12785 | 7 times 5 equals 35. (Con... |
| 7t6e42 12786 | 7 times 6 equals 42. (Con... |
| 7t7e49 12787 | 7 times 7 equals 49. (Con... |
| 8t2e16 12788 | 8 times 2 equals 16. (Con... |
| 8t3e24 12789 | 8 times 3 equals 24. (Con... |
| 8t4e32 12790 | 8 times 4 equals 32. (Con... |
| 8t5e40 12791 | 8 times 5 equals 40. (Con... |
| 8t6e48 12792 | 8 times 6 equals 48. (Con... |
| 8t7e56 12793 | 8 times 7 equals 56. (Con... |
| 8t8e64 12794 | 8 times 8 equals 64. (Con... |
| 9t2e18 12795 | 9 times 2 equals 18. (Con... |
| 9t3e27 12796 | 9 times 3 equals 27. (Con... |
| 9t4e36 12797 | 9 times 4 equals 36. (Con... |
| 9t5e45 12798 | 9 times 5 equals 45. (Con... |
| 9t6e54 12799 | 9 times 6 equals 54. (Con... |
| 9t7e63 12800 | 9 times 7 equals 63. (Con... |
| 9t8e72 12801 | 9 times 8 equals 72. (Con... |
| 9t9e81 12802 | 9 times 9 equals 81. (Con... |
| 9t11e99 12803 | 9 times 11 equals 99. (Co... |
| 9lt10 12804 | 9 is less than 10. (Contr... |
| 8lt10 12805 | 8 is less than 10. (Contr... |
| 7lt10 12806 | 7 is less than 10. (Contr... |
| 6lt10 12807 | 6 is less than 10. (Contr... |
| 5lt10 12808 | 5 is less than 10. (Contr... |
| 4lt10 12809 | 4 is less than 10. (Contr... |
| 3lt10 12810 | 3 is less than 10. (Contr... |
| 2lt10 12811 | 2 is less than 10. (Contr... |
| 1lt10 12812 | 1 is less than 10. (Contr... |
| decbin0 12813 | Decompose base 4 into base... |
| decbin2 12814 | Decompose base 4 into base... |
| decbin3 12815 | Decompose base 4 into base... |
| halfthird 12816 | Half minus a third. (Cont... |
| 5recm6rec 12817 | One fifth minus one sixth.... |
| uzval 12820 | The value of the upper int... |
| uzf 12821 | The domain and codomain of... |
| eluz1 12822 | Membership in the upper se... |
| eluzel2 12823 | Implication of membership ... |
| eluz2 12824 | Membership in an upper set... |
| eluzmn 12825 | Membership in an earlier u... |
| eluz1i 12826 | Membership in an upper set... |
| eluzuzle 12827 | An integer in an upper set... |
| eluzelz 12828 | A member of an upper set o... |
| eluzelre 12829 | A member of an upper set o... |
| eluzelcn 12830 | A member of an upper set o... |
| eluzle 12831 | Implication of membership ... |
| eluz 12832 | Membership in an upper set... |
| uzid 12833 | Membership of the least me... |
| uzidd 12834 | Membership of the least me... |
| uzn0 12835 | The upper integers are all... |
| uztrn 12836 | Transitive law for sets of... |
| uztrn2 12837 | Transitive law for sets of... |
| uzneg 12838 | Contraposition law for upp... |
| uzssz 12839 | An upper set of integers i... |
| uzssre 12840 | An upper set of integers i... |
| uzss 12841 | Subset relationship for tw... |
| uztric 12842 | Totality of the ordering r... |
| uz11 12843 | The upper integers functio... |
| eluzp1m1 12844 | Membership in the next upp... |
| eluzp1l 12845 | Strict ordering implied by... |
| eluzp1p1 12846 | Membership in the next upp... |
| eluzadd 12847 | Membership in a later uppe... |
| eluzsub 12848 | Membership in an earlier u... |
| eluzaddi 12849 | Membership in a later uppe... |
| eluzaddiOLD 12850 | Obsolete version of ~ eluz... |
| eluzsubi 12851 | Membership in an earlier u... |
| eluzsubiOLD 12852 | Obsolete version of ~ eluz... |
| eluzaddOLD 12853 | Obsolete version of ~ eluz... |
| eluzsubOLD 12854 | Obsolete version of ~ eluz... |
| subeluzsub 12855 | Membership of a difference... |
| uzm1 12856 | Choices for an element of ... |
| uznn0sub 12857 | The nonnegative difference... |
| uzin 12858 | Intersection of two upper ... |
| uzp1 12859 | Choices for an element of ... |
| nn0uz 12860 | Nonnegative integers expre... |
| nnuz 12861 | Positive integers expresse... |
| elnnuz 12862 | A positive integer express... |
| elnn0uz 12863 | A nonnegative integer expr... |
| eluz2nn 12864 | An integer greater than or... |
| eluz4eluz2 12865 | An integer greater than or... |
| eluz4nn 12866 | An integer greater than or... |
| eluzge2nn0 12867 | If an integer is greater t... |
| eluz2n0 12868 | An integer greater than or... |
| uzuzle23 12869 | An integer in the upper se... |
| eluzge3nn 12870 | If an integer is greater t... |
| uz3m2nn 12871 | An integer greater than or... |
| 1eluzge0 12872 | 1 is an integer greater th... |
| 2eluzge0 12873 | 2 is an integer greater th... |
| 2eluzge1 12874 | 2 is an integer greater th... |
| uznnssnn 12875 | The upper integers startin... |
| raluz 12876 | Restricted universal quant... |
| raluz2 12877 | Restricted universal quant... |
| rexuz 12878 | Restricted existential qua... |
| rexuz2 12879 | Restricted existential qua... |
| 2rexuz 12880 | Double existential quantif... |
| peano2uz 12881 | Second Peano postulate for... |
| peano2uzs 12882 | Second Peano postulate for... |
| peano2uzr 12883 | Reversed second Peano axio... |
| uzaddcl 12884 | Addition closure law for a... |
| nn0pzuz 12885 | The sum of a nonnegative i... |
| uzind4 12886 | Induction on the upper set... |
| uzind4ALT 12887 | Induction on the upper set... |
| uzind4s 12888 | Induction on the upper set... |
| uzind4s2 12889 | Induction on the upper set... |
| uzind4i 12890 | Induction on the upper int... |
| uzwo 12891 | Well-ordering principle: a... |
| uzwo2 12892 | Well-ordering principle: a... |
| nnwo 12893 | Well-ordering principle: a... |
| nnwof 12894 | Well-ordering principle: a... |
| nnwos 12895 | Well-ordering principle: a... |
| indstr 12896 | Strong Mathematical Induct... |
| eluznn0 12897 | Membership in a nonnegativ... |
| eluznn 12898 | Membership in a positive u... |
| eluz2b1 12899 | Two ways to say "an intege... |
| eluz2gt1 12900 | An integer greater than or... |
| eluz2b2 12901 | Two ways to say "an intege... |
| eluz2b3 12902 | Two ways to say "an intege... |
| uz2m1nn 12903 | One less than an integer g... |
| 1nuz2 12904 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12905 | A positive integer is eith... |
| uz2mulcl 12906 | Closure of multiplication ... |
| indstr2 12907 | Strong Mathematical Induct... |
| uzinfi 12908 | Extract the lower bound of... |
| nninf 12909 | The infimum of the set of ... |
| nn0inf 12910 | The infimum of the set of ... |
| infssuzle 12911 | The infimum of a subset of... |
| infssuzcl 12912 | The infimum of a subset of... |
| ublbneg 12913 | The image under negation o... |
| eqreznegel 12914 | Two ways to express the im... |
| supminf 12915 | The supremum of a bounded-... |
| lbzbi 12916 | If a set of reals is bound... |
| zsupss 12917 | Any nonempty bounded subse... |
| suprzcl2 12918 | The supremum of a bounded-... |
| suprzub 12919 | The supremum of a bounded-... |
| uzsupss 12920 | Any bounded subset of an u... |
| nn01to3 12921 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12922 | Alternate proof of ~ nn0ge... |
| uzwo3 12923 | Well-ordering principle: a... |
| zmin 12924 | There is a unique smallest... |
| zmax 12925 | There is a unique largest ... |
| zbtwnre 12926 | There is a unique integer ... |
| rebtwnz 12927 | There is a unique greatest... |
| elq 12930 | Membership in the set of r... |
| qmulz 12931 | If ` A ` is rational, then... |
| znq 12932 | The ratio of an integer an... |
| qre 12933 | A rational number is a rea... |
| zq 12934 | An integer is a rational n... |
| qred 12935 | A rational number is a rea... |
| zssq 12936 | The integers are a subset ... |
| nn0ssq 12937 | The nonnegative integers a... |
| nnssq 12938 | The positive integers are ... |
| qssre 12939 | The rationals are a subset... |
| qsscn 12940 | The rationals are a subset... |
| qex 12941 | The set of rational number... |
| nnq 12942 | A positive integer is rati... |
| qcn 12943 | A rational number is a com... |
| qexALT 12944 | Alternate proof of ~ qex .... |
| qaddcl 12945 | Closure of addition of rat... |
| qnegcl 12946 | Closure law for the negati... |
| qmulcl 12947 | Closure of multiplication ... |
| qsubcl 12948 | Closure of subtraction of ... |
| qreccl 12949 | Closure of reciprocal of r... |
| qdivcl 12950 | Closure of division of rat... |
| qrevaddcl 12951 | Reverse closure law for ad... |
| nnrecq 12952 | The reciprocal of a positi... |
| irradd 12953 | The sum of an irrational n... |
| irrmul 12954 | The product of an irration... |
| elpq 12955 | A positive rational is the... |
| elpqb 12956 | A class is a positive rati... |
| rpnnen1lem2 12957 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12958 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12959 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12960 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12961 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12962 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12963 | One half of ~ rpnnen , whe... |
| reexALT 12964 | Alternate proof of ~ reex ... |
| cnref1o 12965 | There is a natural one-to-... |
| cnexALT 12966 | The set of complex numbers... |
| xrex 12967 | The set of extended reals ... |
| addex 12968 | The addition operation is ... |
| mulex 12969 | The multiplication operati... |
| elrp 12972 | Membership in the set of p... |
| elrpii 12973 | Membership in the set of p... |
| 1rp 12974 | 1 is a positive real. (Co... |
| 2rp 12975 | 2 is a positive real. (Co... |
| 3rp 12976 | 3 is a positive real. (Co... |
| rpssre 12977 | The positive reals are a s... |
| rpre 12978 | A positive real is a real.... |
| rpxr 12979 | A positive real is an exte... |
| rpcn 12980 | A positive real is a compl... |
| nnrp 12981 | A positive integer is a po... |
| rpgt0 12982 | A positive real is greater... |
| rpge0 12983 | A positive real is greater... |
| rpregt0 12984 | A positive real is a posit... |
| rprege0 12985 | A positive real is a nonne... |
| rpne0 12986 | A positive real is nonzero... |
| rprene0 12987 | A positive real is a nonze... |
| rpcnne0 12988 | A positive real is a nonze... |
| rpcndif0 12989 | A positive real number is ... |
| ralrp 12990 | Quantification over positi... |
| rexrp 12991 | Quantification over positi... |
| rpaddcl 12992 | Closure law for addition o... |
| rpmulcl 12993 | Closure law for multiplica... |
| rpmtmip 12994 | "Minus times minus is plus... |
| rpdivcl 12995 | Closure law for division o... |
| rpreccl 12996 | Closure law for reciprocat... |
| rphalfcl 12997 | Closure law for half of a ... |
| rpgecl 12998 | A number greater than or e... |
| rphalflt 12999 | Half of a positive real is... |
| rerpdivcl 13000 | Closure law for division o... |
| ge0p1rp 13001 | A nonnegative number plus ... |
| rpneg 13002 | Either a nonzero real or i... |
| negelrp 13003 | Elementhood of a negation ... |
| negelrpd 13004 | The negation of a negative... |
| 0nrp 13005 | Zero is not a positive rea... |
| ltsubrp 13006 | Subtracting a positive rea... |
| ltaddrp 13007 | Adding a positive number t... |
| difrp 13008 | Two ways to say one number... |
| elrpd 13009 | Membership in the set of p... |
| nnrpd 13010 | A positive integer is a po... |
| zgt1rpn0n1 13011 | An integer greater than 1 ... |
| rpred 13012 | A positive real is a real.... |
| rpxrd 13013 | A positive real is an exte... |
| rpcnd 13014 | A positive real is a compl... |
| rpgt0d 13015 | A positive real is greater... |
| rpge0d 13016 | A positive real is greater... |
| rpne0d 13017 | A positive real is nonzero... |
| rpregt0d 13018 | A positive real is real an... |
| rprege0d 13019 | A positive real is real an... |
| rprene0d 13020 | A positive real is a nonze... |
| rpcnne0d 13021 | A positive real is a nonze... |
| rpreccld 13022 | Closure law for reciprocat... |
| rprecred 13023 | Closure law for reciprocat... |
| rphalfcld 13024 | Closure law for half of a ... |
| reclt1d 13025 | The reciprocal of a positi... |
| recgt1d 13026 | The reciprocal of a positi... |
| rpaddcld 13027 | Closure law for addition o... |
| rpmulcld 13028 | Closure law for multiplica... |
| rpdivcld 13029 | Closure law for division o... |
| ltrecd 13030 | The reciprocal of both sid... |
| lerecd 13031 | The reciprocal of both sid... |
| ltrec1d 13032 | Reciprocal swap in a 'less... |
| lerec2d 13033 | Reciprocal swap in a 'less... |
| lediv2ad 13034 | Division of both sides of ... |
| ltdiv2d 13035 | Division of a positive num... |
| lediv2d 13036 | Division of a positive num... |
| ledivdivd 13037 | Invert ratios of positive ... |
| divge1 13038 | The ratio of a number over... |
| divlt1lt 13039 | A real number divided by a... |
| divle1le 13040 | A real number divided by a... |
| ledivge1le 13041 | If a number is less than o... |
| ge0p1rpd 13042 | A nonnegative number plus ... |
| rerpdivcld 13043 | Closure law for division o... |
| ltsubrpd 13044 | Subtracting a positive rea... |
| ltaddrpd 13045 | Adding a positive number t... |
| ltaddrp2d 13046 | Adding a positive number t... |
| ltmulgt11d 13047 | Multiplication by a number... |
| ltmulgt12d 13048 | Multiplication by a number... |
| gt0divd 13049 | Division of a positive num... |
| ge0divd 13050 | Division of a nonnegative ... |
| rpgecld 13051 | A number greater than or e... |
| divge0d 13052 | The ratio of nonnegative a... |
| ltmul1d 13053 | The ratio of nonnegative a... |
| ltmul2d 13054 | Multiplication of both sid... |
| lemul1d 13055 | Multiplication of both sid... |
| lemul2d 13056 | Multiplication of both sid... |
| ltdiv1d 13057 | Division of both sides of ... |
| lediv1d 13058 | Division of both sides of ... |
| ltmuldivd 13059 | 'Less than' relationship b... |
| ltmuldiv2d 13060 | 'Less than' relationship b... |
| lemuldivd 13061 | 'Less than or equal to' re... |
| lemuldiv2d 13062 | 'Less than or equal to' re... |
| ltdivmuld 13063 | 'Less than' relationship b... |
| ltdivmul2d 13064 | 'Less than' relationship b... |
| ledivmuld 13065 | 'Less than or equal to' re... |
| ledivmul2d 13066 | 'Less than or equal to' re... |
| ltmul1dd 13067 | The ratio of nonnegative a... |
| ltmul2dd 13068 | Multiplication of both sid... |
| ltdiv1dd 13069 | Division of both sides of ... |
| lediv1dd 13070 | Division of both sides of ... |
| lediv12ad 13071 | Comparison of ratio of two... |
| mul2lt0rlt0 13072 | If the result of a multipl... |
| mul2lt0rgt0 13073 | If the result of a multipl... |
| mul2lt0llt0 13074 | If the result of a multipl... |
| mul2lt0lgt0 13075 | If the result of a multipl... |
| mul2lt0bi 13076 | If the result of a multipl... |
| prodge0rd 13077 | Infer that a multiplicand ... |
| prodge0ld 13078 | Infer that a multiplier is... |
| ltdiv23d 13079 | Swap denominator with othe... |
| lediv23d 13080 | Swap denominator with othe... |
| lt2mul2divd 13081 | The ratio of nonnegative a... |
| nnledivrp 13082 | Division of a positive int... |
| nn0ledivnn 13083 | Division of a nonnegative ... |
| addlelt 13084 | If the sum of a real numbe... |
| ltxr 13091 | The 'less than' binary rel... |
| elxr 13092 | Membership in the set of e... |
| xrnemnf 13093 | An extended real other tha... |
| xrnepnf 13094 | An extended real other tha... |
| xrltnr 13095 | The extended real 'less th... |
| ltpnf 13096 | Any (finite) real is less ... |
| ltpnfd 13097 | Any (finite) real is less ... |
| 0ltpnf 13098 | Zero is less than plus inf... |
| mnflt 13099 | Minus infinity is less tha... |
| mnfltd 13100 | Minus infinity is less tha... |
| mnflt0 13101 | Minus infinity is less tha... |
| mnfltpnf 13102 | Minus infinity is less tha... |
| mnfltxr 13103 | Minus infinity is less tha... |
| pnfnlt 13104 | No extended real is greate... |
| nltmnf 13105 | No extended real is less t... |
| pnfge 13106 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13107 | An extended nonnegative in... |
| 0lepnf 13108 | 0 less than or equal to po... |
| xnn0ge0 13109 | An extended nonnegative in... |
| mnfle 13110 | Minus infinity is less tha... |
| mnfled 13111 | Minus infinity is less tha... |
| xrltnsym 13112 | Ordering on the extended r... |
| xrltnsym2 13113 | 'Less than' is antisymmetr... |
| xrlttri 13114 | Ordering on the extended r... |
| xrlttr 13115 | Ordering on the extended r... |
| xrltso 13116 | 'Less than' is a strict or... |
| xrlttri2 13117 | Trichotomy law for 'less t... |
| xrlttri3 13118 | Trichotomy law for 'less t... |
| xrleloe 13119 | 'Less than or equal' expre... |
| xrleltne 13120 | 'Less than or equal to' im... |
| xrltlen 13121 | 'Less than' expressed in t... |
| dfle2 13122 | Alternative definition of ... |
| dflt2 13123 | Alternative definition of ... |
| xrltle 13124 | 'Less than' implies 'less ... |
| xrltled 13125 | 'Less than' implies 'less ... |
| xrleid 13126 | 'Less than or equal to' is... |
| xrleidd 13127 | 'Less than or equal to' is... |
| xrletri 13128 | Trichotomy law for extende... |
| xrletri3 13129 | Trichotomy law for extende... |
| xrletrid 13130 | Trichotomy law for extende... |
| xrlelttr 13131 | Transitive law for orderin... |
| xrltletr 13132 | Transitive law for orderin... |
| xrletr 13133 | Transitive law for orderin... |
| xrlttrd 13134 | Transitive law for orderin... |
| xrlelttrd 13135 | Transitive law for orderin... |
| xrltletrd 13136 | Transitive law for orderin... |
| xrletrd 13137 | Transitive law for orderin... |
| xrltne 13138 | 'Less than' implies not eq... |
| nltpnft 13139 | An extended real is not le... |
| xgepnf 13140 | An extended real which is ... |
| ngtmnft 13141 | An extended real is not gr... |
| xlemnf 13142 | An extended real which is ... |
| xrrebnd 13143 | An extended real is real i... |
| xrre 13144 | A way of proving that an e... |
| xrre2 13145 | An extended real between t... |
| xrre3 13146 | A way of proving that an e... |
| ge0gtmnf 13147 | A nonnegative extended rea... |
| ge0nemnf 13148 | A nonnegative extended rea... |
| xrrege0 13149 | A nonnegative extended rea... |
| xrmax1 13150 | An extended real is less t... |
| xrmax2 13151 | An extended real is less t... |
| xrmin1 13152 | The minimum of two extende... |
| xrmin2 13153 | The minimum of two extende... |
| xrmaxeq 13154 | The maximum of two extende... |
| xrmineq 13155 | The minimum of two extende... |
| xrmaxlt 13156 | Two ways of saying the max... |
| xrltmin 13157 | Two ways of saying an exte... |
| xrmaxle 13158 | Two ways of saying the max... |
| xrlemin 13159 | Two ways of saying a numbe... |
| max1 13160 | A number is less than or e... |
| max1ALT 13161 | A number is less than or e... |
| max2 13162 | A number is less than or e... |
| 2resupmax 13163 | The supremum of two real n... |
| min1 13164 | The minimum of two numbers... |
| min2 13165 | The minimum of two numbers... |
| maxle 13166 | Two ways of saying the max... |
| lemin 13167 | Two ways of saying a numbe... |
| maxlt 13168 | Two ways of saying the max... |
| ltmin 13169 | Two ways of saying a numbe... |
| lemaxle 13170 | A real number which is les... |
| max0sub 13171 | Decompose a real number in... |
| ifle 13172 | An if statement transforms... |
| z2ge 13173 | There exists an integer gr... |
| qbtwnre 13174 | The rational numbers are d... |
| qbtwnxr 13175 | The rational numbers are d... |
| qsqueeze 13176 | If a nonnegative real is l... |
| qextltlem 13177 | Lemma for ~ qextlt and qex... |
| qextlt 13178 | An extensionality-like pro... |
| qextle 13179 | An extensionality-like pro... |
| xralrple 13180 | Show that ` A ` is less th... |
| alrple 13181 | Show that ` A ` is less th... |
| xnegeq 13182 | Equality of two extended n... |
| xnegex 13183 | A negative extended real e... |
| xnegpnf 13184 | Minus ` +oo ` . Remark of... |
| xnegmnf 13185 | Minus ` -oo ` . Remark of... |
| rexneg 13186 | Minus a real number. Rema... |
| xneg0 13187 | The negative of zero. (Co... |
| xnegcl 13188 | Closure of extended real n... |
| xnegneg 13189 | Extended real version of ~... |
| xneg11 13190 | Extended real version of ~... |
| xltnegi 13191 | Forward direction of ~ xlt... |
| xltneg 13192 | Extended real version of ~... |
| xleneg 13193 | Extended real version of ~... |
| xlt0neg1 13194 | Extended real version of ~... |
| xlt0neg2 13195 | Extended real version of ~... |
| xle0neg1 13196 | Extended real version of ~... |
| xle0neg2 13197 | Extended real version of ~... |
| xaddval 13198 | Value of the extended real... |
| xaddf 13199 | The extended real addition... |
| xmulval 13200 | Value of the extended real... |
| xaddpnf1 13201 | Addition of positive infin... |
| xaddpnf2 13202 | Addition of positive infin... |
| xaddmnf1 13203 | Addition of negative infin... |
| xaddmnf2 13204 | Addition of negative infin... |
| pnfaddmnf 13205 | Addition of positive and n... |
| mnfaddpnf 13206 | Addition of negative and p... |
| rexadd 13207 | The extended real addition... |
| rexsub 13208 | Extended real subtraction ... |
| rexaddd 13209 | The extended real addition... |
| xnn0xaddcl 13210 | The extended nonnegative i... |
| xaddnemnf 13211 | Closure of extended real a... |
| xaddnepnf 13212 | Closure of extended real a... |
| xnegid 13213 | Extended real version of ~... |
| xaddcl 13214 | The extended real addition... |
| xaddcom 13215 | The extended real addition... |
| xaddrid 13216 | Extended real version of ~... |
| xaddlid 13217 | Extended real version of ~... |
| xaddridd 13218 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13219 | Extended nonnegative integ... |
| xnn0lenn0nn0 13220 | An extended nonnegative in... |
| xnn0le2is012 13221 | An extended nonnegative in... |
| xnn0xadd0 13222 | The sum of two extended no... |
| xnegdi 13223 | Extended real version of ~... |
| xaddass 13224 | Associativity of extended ... |
| xaddass2 13225 | Associativity of extended ... |
| xpncan 13226 | Extended real version of ~... |
| xnpcan 13227 | Extended real version of ~... |
| xleadd1a 13228 | Extended real version of ~... |
| xleadd2a 13229 | Commuted form of ~ xleadd1... |
| xleadd1 13230 | Weakened version of ~ xlea... |
| xltadd1 13231 | Extended real version of ~... |
| xltadd2 13232 | Extended real version of ~... |
| xaddge0 13233 | The sum of nonnegative ext... |
| xle2add 13234 | Extended real version of ~... |
| xlt2add 13235 | Extended real version of ~... |
| xsubge0 13236 | Extended real version of ~... |
| xposdif 13237 | Extended real version of ~... |
| xlesubadd 13238 | Under certain conditions, ... |
| xmullem 13239 | Lemma for ~ rexmul . (Con... |
| xmullem2 13240 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13241 | Extended real multiplicati... |
| xmul01 13242 | Extended real version of ~... |
| xmul02 13243 | Extended real version of ~... |
| xmulneg1 13244 | Extended real version of ~... |
| xmulneg2 13245 | Extended real version of ~... |
| rexmul 13246 | The extended real multipli... |
| xmulf 13247 | The extended real multipli... |
| xmulcl 13248 | Closure of extended real m... |
| xmulpnf1 13249 | Multiplication by plus inf... |
| xmulpnf2 13250 | Multiplication by plus inf... |
| xmulmnf1 13251 | Multiplication by minus in... |
| xmulmnf2 13252 | Multiplication by minus in... |
| xmulpnf1n 13253 | Multiplication by plus inf... |
| xmulrid 13254 | Extended real version of ~... |
| xmullid 13255 | Extended real version of ~... |
| xmulm1 13256 | Extended real version of ~... |
| xmulasslem2 13257 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13258 | Extended real version of ~... |
| xmulge0 13259 | Extended real version of ~... |
| xmulasslem 13260 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13261 | Lemma for ~ xmulass . (Co... |
| xmulass 13262 | Associativity of the exten... |
| xlemul1a 13263 | Extended real version of ~... |
| xlemul2a 13264 | Extended real version of ~... |
| xlemul1 13265 | Extended real version of ~... |
| xlemul2 13266 | Extended real version of ~... |
| xltmul1 13267 | Extended real version of ~... |
| xltmul2 13268 | Extended real version of ~... |
| xadddilem 13269 | Lemma for ~ xadddi . (Con... |
| xadddi 13270 | Distributive property for ... |
| xadddir 13271 | Commuted version of ~ xadd... |
| xadddi2 13272 | The assumption that the mu... |
| xadddi2r 13273 | Commuted version of ~ xadd... |
| x2times 13274 | Extended real version of ~... |
| xnegcld 13275 | Closure of extended real n... |
| xaddcld 13276 | The extended real addition... |
| xmulcld 13277 | Closure of extended real m... |
| xadd4d 13278 | Rearrangement of 4 terms i... |
| xnn0add4d 13279 | Rearrangement of 4 terms i... |
| xrsupexmnf 13280 | Adding minus infinity to a... |
| xrinfmexpnf 13281 | Adding plus infinity to a ... |
| xrsupsslem 13282 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13283 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13284 | Any subset of extended rea... |
| xrinfmss 13285 | Any subset of extended rea... |
| xrinfmss2 13286 | Any subset of extended rea... |
| xrub 13287 | By quantifying only over r... |
| supxr 13288 | The supremum of a set of e... |
| supxr2 13289 | The supremum of a set of e... |
| supxrcl 13290 | The supremum of an arbitra... |
| supxrun 13291 | The supremum of the union ... |
| supxrmnf 13292 | Adding minus infinity to a... |
| supxrpnf 13293 | The supremum of a set of e... |
| supxrunb1 13294 | The supremum of an unbound... |
| supxrunb2 13295 | The supremum of an unbound... |
| supxrbnd1 13296 | The supremum of a bounded-... |
| supxrbnd2 13297 | The supremum of a bounded-... |
| xrsup0 13298 | The supremum of an empty s... |
| supxrub 13299 | A member of a set of exten... |
| supxrlub 13300 | The supremum of a set of e... |
| supxrleub 13301 | The supremum of a set of e... |
| supxrre 13302 | The real and extended real... |
| supxrbnd 13303 | The supremum of a bounded-... |
| supxrgtmnf 13304 | The supremum of a nonempty... |
| supxrre1 13305 | The supremum of a nonempty... |
| supxrre2 13306 | The supremum of a nonempty... |
| supxrss 13307 | Smaller sets of extended r... |
| infxrcl 13308 | The infimum of an arbitrar... |
| infxrlb 13309 | A member of a set of exten... |
| infxrgelb 13310 | The infimum of a set of ex... |
| infxrre 13311 | The real and extended real... |
| infxrmnf 13312 | The infinimum of a set of ... |
| xrinf0 13313 | The infimum of the empty s... |
| infxrss 13314 | Larger sets of extended re... |
| reltre 13315 | For all real numbers there... |
| rpltrp 13316 | For all positive real numb... |
| reltxrnmnf 13317 | For all extended real numb... |
| infmremnf 13318 | The infimum of the reals i... |
| infmrp1 13319 | The infimum of the positiv... |
| ixxval 13328 | Value of the interval func... |
| elixx1 13329 | Membership in an interval ... |
| ixxf 13330 | The set of intervals of ex... |
| ixxex 13331 | The set of intervals of ex... |
| ixxssxr 13332 | The set of intervals of ex... |
| elixx3g 13333 | Membership in a set of ope... |
| ixxssixx 13334 | An interval is a subset of... |
| ixxdisj 13335 | Split an interval into dis... |
| ixxun 13336 | Split an interval into two... |
| ixxin 13337 | Intersection of two interv... |
| ixxss1 13338 | Subset relationship for in... |
| ixxss2 13339 | Subset relationship for in... |
| ixxss12 13340 | Subset relationship for in... |
| ixxub 13341 | Extract the upper bound of... |
| ixxlb 13342 | Extract the lower bound of... |
| iooex 13343 | The set of open intervals ... |
| iooval 13344 | Value of the open interval... |
| ioo0 13345 | An empty open interval of ... |
| ioon0 13346 | An open interval of extend... |
| ndmioo 13347 | The open interval function... |
| iooid 13348 | An open interval with iden... |
| elioo3g 13349 | Membership in a set of ope... |
| elioore 13350 | A member of an open interv... |
| lbioo 13351 | An open interval does not ... |
| ubioo 13352 | An open interval does not ... |
| iooval2 13353 | Value of the open interval... |
| iooin 13354 | Intersection of two open i... |
| iooss1 13355 | Subset relationship for op... |
| iooss2 13356 | Subset relationship for op... |
| iocval 13357 | Value of the open-below, c... |
| icoval 13358 | Value of the closed-below,... |
| iccval 13359 | Value of the closed interv... |
| elioo1 13360 | Membership in an open inte... |
| elioo2 13361 | Membership in an open inte... |
| elioc1 13362 | Membership in an open-belo... |
| elico1 13363 | Membership in a closed-bel... |
| elicc1 13364 | Membership in a closed int... |
| iccid 13365 | A closed interval with ide... |
| ico0 13366 | An empty open interval of ... |
| ioc0 13367 | An empty open interval of ... |
| icc0 13368 | An empty closed interval o... |
| dfrp2 13369 | Alternate definition of th... |
| elicod 13370 | Membership in a left-close... |
| icogelb 13371 | An element of a left-close... |
| elicore 13372 | A member of a left-closed ... |
| ubioc1 13373 | The upper bound belongs to... |
| lbico1 13374 | The lower bound belongs to... |
| iccleub 13375 | An element of a closed int... |
| iccgelb 13376 | An element of a closed int... |
| elioo5 13377 | Membership in an open inte... |
| eliooxr 13378 | A nonempty open interval s... |
| eliooord 13379 | Ordering implied by a memb... |
| elioo4g 13380 | Membership in an open inte... |
| ioossre 13381 | An open interval is a set ... |
| ioosscn 13382 | An open interval is a set ... |
| elioc2 13383 | Membership in an open-belo... |
| elico2 13384 | Membership in a closed-bel... |
| elicc2 13385 | Membership in a closed rea... |
| elicc2i 13386 | Inference for membership i... |
| elicc4 13387 | Membership in a closed rea... |
| iccss 13388 | Condition for a closed int... |
| iccssioo 13389 | Condition for a closed int... |
| icossico 13390 | Condition for a closed-bel... |
| iccss2 13391 | Condition for a closed int... |
| iccssico 13392 | Condition for a closed int... |
| iccssioo2 13393 | Condition for a closed int... |
| iccssico2 13394 | Condition for a closed int... |
| ioomax 13395 | The open interval from min... |
| iccmax 13396 | The closed interval from m... |
| ioopos 13397 | The set of positive reals ... |
| ioorp 13398 | The set of positive reals ... |
| iooshf 13399 | Shift the arguments of the... |
| iocssre 13400 | A closed-above interval wi... |
| icossre 13401 | A closed-below interval wi... |
| iccssre 13402 | A closed real interval is ... |
| iccssxr 13403 | A closed interval is a set... |
| iocssxr 13404 | An open-below, closed-abov... |
| icossxr 13405 | A closed-below, open-above... |
| ioossicc 13406 | An open interval is a subs... |
| iccssred 13407 | A closed real interval is ... |
| eliccxr 13408 | A member of a closed inter... |
| icossicc 13409 | A closed-below, open-above... |
| iocssicc 13410 | A closed-above, open-below... |
| ioossico 13411 | An open interval is a subs... |
| iocssioo 13412 | Condition for a closed int... |
| icossioo 13413 | Condition for a closed int... |
| ioossioo 13414 | Condition for an open inte... |
| iccsupr 13415 | A nonempty subset of a clo... |
| elioopnf 13416 | Membership in an unbounded... |
| elioomnf 13417 | Membership in an unbounded... |
| elicopnf 13418 | Membership in a closed unb... |
| repos 13419 | Two ways of saying that a ... |
| ioof 13420 | The set of open intervals ... |
| iccf 13421 | The set of closed interval... |
| unirnioo 13422 | The union of the range of ... |
| dfioo2 13423 | Alternate definition of th... |
| ioorebas 13424 | Open intervals are element... |
| xrge0neqmnf 13425 | A nonnegative extended rea... |
| xrge0nre 13426 | An extended real which is ... |
| elrege0 13427 | The predicate "is a nonneg... |
| nn0rp0 13428 | A nonnegative integer is a... |
| rge0ssre 13429 | Nonnegative real numbers a... |
| elxrge0 13430 | Elementhood in the set of ... |
| 0e0icopnf 13431 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13432 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13433 | The nonnegative reals are ... |
| ge0mulcl 13434 | The nonnegative reals are ... |
| ge0xaddcl 13435 | The nonnegative reals are ... |
| ge0xmulcl 13436 | The nonnegative extended r... |
| lbicc2 13437 | The lower bound of a close... |
| ubicc2 13438 | The upper bound of a close... |
| elicc01 13439 | Membership in the closed r... |
| elunitrn 13440 | The closed unit interval i... |
| elunitcn 13441 | The closed unit interval i... |
| 0elunit 13442 | Zero is an element of the ... |
| 1elunit 13443 | One is an element of the c... |
| iooneg 13444 | Membership in a negated op... |
| iccneg 13445 | Membership in a negated cl... |
| icoshft 13446 | A shifted real is a member... |
| icoshftf1o 13447 | Shifting a closed-below, o... |
| icoun 13448 | The union of two adjacent ... |
| icodisj 13449 | Adjacent left-closed right... |
| ioounsn 13450 | The union of an open inter... |
| snunioo 13451 | The closure of one end of ... |
| snunico 13452 | The closure of the open en... |
| snunioc 13453 | The closure of the open en... |
| prunioo 13454 | The closure of an open rea... |
| ioodisj 13455 | If the upper bound of one ... |
| ioojoin 13456 | Join two open intervals to... |
| difreicc 13457 | The class difference of ` ... |
| iccsplit 13458 | Split a closed interval in... |
| iccshftr 13459 | Membership in a shifted in... |
| iccshftri 13460 | Membership in a shifted in... |
| iccshftl 13461 | Membership in a shifted in... |
| iccshftli 13462 | Membership in a shifted in... |
| iccdil 13463 | Membership in a dilated in... |
| iccdili 13464 | Membership in a dilated in... |
| icccntr 13465 | Membership in a contracted... |
| icccntri 13466 | Membership in a contracted... |
| divelunit 13467 | A condition for a ratio to... |
| lincmb01cmp 13468 | A linear combination of tw... |
| iccf1o 13469 | Describe a bijection from ... |
| iccen 13470 | Any nontrivial closed inte... |
| xov1plusxeqvd 13471 | A complex number ` X ` is ... |
| unitssre 13472 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13473 | The closed unit interval i... |
| supicc 13474 | Supremum of a bounded set ... |
| supiccub 13475 | The supremum of a bounded ... |
| supicclub 13476 | The supremum of a bounded ... |
| supicclub2 13477 | The supremum of a bounded ... |
| zltaddlt1le 13478 | The sum of an integer and ... |
| xnn0xrge0 13479 | An extended nonnegative in... |
| fzval 13482 | The value of a finite set ... |
| fzval2 13483 | An alternative way of expr... |
| fzf 13484 | Establish the domain and c... |
| elfz1 13485 | Membership in a finite set... |
| elfz 13486 | Membership in a finite set... |
| elfz2 13487 | Membership in a finite set... |
| elfzd 13488 | Membership in a finite set... |
| elfz5 13489 | Membership in a finite set... |
| elfz4 13490 | Membership in a finite set... |
| elfzuzb 13491 | Membership in a finite set... |
| eluzfz 13492 | Membership in a finite set... |
| elfzuz 13493 | A member of a finite set o... |
| elfzuz3 13494 | Membership in a finite set... |
| elfzel2 13495 | Membership in a finite set... |
| elfzel1 13496 | Membership in a finite set... |
| elfzelz 13497 | A member of a finite set o... |
| elfzelzd 13498 | A member of a finite set o... |
| fzssz 13499 | A finite sequence of integ... |
| elfzle1 13500 | A member of a finite set o... |
| elfzle2 13501 | A member of a finite set o... |
| elfzuz2 13502 | Implication of membership ... |
| elfzle3 13503 | Membership in a finite set... |
| eluzfz1 13504 | Membership in a finite set... |
| eluzfz2 13505 | Membership in a finite set... |
| eluzfz2b 13506 | Membership in a finite set... |
| elfz3 13507 | Membership in a finite set... |
| elfz1eq 13508 | Membership in a finite set... |
| elfzubelfz 13509 | If there is a member in a ... |
| peano2fzr 13510 | A Peano-postulate-like the... |
| fzn0 13511 | Properties of a finite int... |
| fz0 13512 | A finite set of sequential... |
| fzn 13513 | A finite set of sequential... |
| fzen 13514 | A shifted finite set of se... |
| fz1n 13515 | A 1-based finite set of se... |
| 0nelfz1 13516 | 0 is not an element of a f... |
| 0fz1 13517 | Two ways to say a finite 1... |
| fz10 13518 | There are no integers betw... |
| uzsubsubfz 13519 | Membership of an integer g... |
| uzsubsubfz1 13520 | Membership of an integer g... |
| ige3m2fz 13521 | Membership of an integer g... |
| fzsplit2 13522 | Split a finite interval of... |
| fzsplit 13523 | Split a finite interval of... |
| fzdisj 13524 | Condition for two finite i... |
| fz01en 13525 | 0-based and 1-based finite... |
| elfznn 13526 | A member of a finite set o... |
| elfz1end 13527 | A nonempty finite range of... |
| fz1ssnn 13528 | A finite set of positive i... |
| fznn0sub 13529 | Subtraction closure for a ... |
| fzmmmeqm 13530 | Subtracting the difference... |
| fzaddel 13531 | Membership of a sum in a f... |
| fzadd2 13532 | Membership of a sum in a f... |
| fzsubel 13533 | Membership of a difference... |
| fzopth 13534 | A finite set of sequential... |
| fzass4 13535 | Two ways to express a nond... |
| fzss1 13536 | Subset relationship for fi... |
| fzss2 13537 | Subset relationship for fi... |
| fzssuz 13538 | A finite set of sequential... |
| fzsn 13539 | A finite interval of integ... |
| fzssp1 13540 | Subset relationship for fi... |
| fzssnn 13541 | Finite sets of sequential ... |
| ssfzunsnext 13542 | A subset of a finite seque... |
| ssfzunsn 13543 | A subset of a finite seque... |
| fzsuc 13544 | Join a successor to the en... |
| fzpred 13545 | Join a predecessor to the ... |
| fzpreddisj 13546 | A finite set of sequential... |
| elfzp1 13547 | Append an element to a fin... |
| fzp1ss 13548 | Subset relationship for fi... |
| fzelp1 13549 | Membership in a set of seq... |
| fzp1elp1 13550 | Add one to an element of a... |
| fznatpl1 13551 | Shift membership in a fini... |
| fzpr 13552 | A finite interval of integ... |
| fztp 13553 | A finite interval of integ... |
| fz12pr 13554 | An integer range between 1... |
| fzsuc2 13555 | Join a successor to the en... |
| fzp1disj 13556 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13557 | Remove a successor from th... |
| fzprval 13558 | Two ways of defining the f... |
| fztpval 13559 | Two ways of defining the f... |
| fzrev 13560 | Reversal of start and end ... |
| fzrev2 13561 | Reversal of start and end ... |
| fzrev2i 13562 | Reversal of start and end ... |
| fzrev3 13563 | The "complement" of a memb... |
| fzrev3i 13564 | The "complement" of a memb... |
| fznn 13565 | Finite set of sequential i... |
| elfz1b 13566 | Membership in a 1-based fi... |
| elfz1uz 13567 | Membership in a 1-based fi... |
| elfzm11 13568 | Membership in a finite set... |
| uzsplit 13569 | Express an upper integer s... |
| uzdisj 13570 | The first ` N ` elements o... |
| fseq1p1m1 13571 | Add/remove an item to/from... |
| fseq1m1p1 13572 | Add/remove an item to/from... |
| fz1sbc 13573 | Quantification over a one-... |
| elfzp1b 13574 | An integer is a member of ... |
| elfzm1b 13575 | An integer is a member of ... |
| elfzp12 13576 | Options for membership in ... |
| fzm1 13577 | Choices for an element of ... |
| fzneuz 13578 | No finite set of sequentia... |
| fznuz 13579 | Disjointness of the upper ... |
| uznfz 13580 | Disjointness of the upper ... |
| fzp1nel 13581 | One plus the upper bound o... |
| fzrevral 13582 | Reversal of scanning order... |
| fzrevral2 13583 | Reversal of scanning order... |
| fzrevral3 13584 | Reversal of scanning order... |
| fzshftral 13585 | Shift the scanning order i... |
| ige2m1fz1 13586 | Membership of an integer g... |
| ige2m1fz 13587 | Membership in a 0-based fi... |
| elfz2nn0 13588 | Membership in a finite set... |
| fznn0 13589 | Characterization of a fini... |
| elfznn0 13590 | A member of a finite set o... |
| elfz3nn0 13591 | The upper bound of a nonem... |
| fz0ssnn0 13592 | Finite sets of sequential ... |
| fz1ssfz0 13593 | Subset relationship for fi... |
| 0elfz 13594 | 0 is an element of a finit... |
| nn0fz0 13595 | A nonnegative integer is a... |
| elfz0add 13596 | An element of a finite set... |
| fz0sn 13597 | An integer range from 0 to... |
| fz0tp 13598 | An integer range from 0 to... |
| fz0to3un2pr 13599 | An integer range from 0 to... |
| fz0to4untppr 13600 | An integer range from 0 to... |
| elfz0ubfz0 13601 | An element of a finite set... |
| elfz0fzfz0 13602 | A member of a finite set o... |
| fz0fzelfz0 13603 | If a member of a finite se... |
| fznn0sub2 13604 | Subtraction closure for a ... |
| uzsubfz0 13605 | Membership of an integer g... |
| fz0fzdiffz0 13606 | The difference of an integ... |
| elfzmlbm 13607 | Subtracting the lower boun... |
| elfzmlbp 13608 | Subtracting the lower boun... |
| fzctr 13609 | Lemma for theorems about t... |
| difelfzle 13610 | The difference of two inte... |
| difelfznle 13611 | The difference of two inte... |
| nn0split 13612 | Express the set of nonnega... |
| nn0disj 13613 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13614 | A finite set of sequential... |
| fvffz0 13615 | The function value of a fu... |
| 1fv 13616 | A function on a singleton.... |
| 4fvwrd4 13617 | The first four function va... |
| 2ffzeq 13618 | Two functions over 0-based... |
| preduz 13619 | The value of the predecess... |
| prednn 13620 | The value of the predecess... |
| prednn0 13621 | The value of the predecess... |
| predfz 13622 | Calculate the predecessor ... |
| fzof 13625 | Functionality of the half-... |
| elfzoel1 13626 | Reverse closure for half-o... |
| elfzoel2 13627 | Reverse closure for half-o... |
| elfzoelz 13628 | Reverse closure for half-o... |
| fzoval 13629 | Value of the half-open int... |
| elfzo 13630 | Membership in a half-open ... |
| elfzo2 13631 | Membership in a half-open ... |
| elfzouz 13632 | Membership in a half-open ... |
| nelfzo 13633 | An integer not being a mem... |
| fzolb 13634 | The left endpoint of a hal... |
| fzolb2 13635 | The left endpoint of a hal... |
| elfzole1 13636 | A member in a half-open in... |
| elfzolt2 13637 | A member in a half-open in... |
| elfzolt3 13638 | Membership in a half-open ... |
| elfzolt2b 13639 | A member in a half-open in... |
| elfzolt3b 13640 | Membership in a half-open ... |
| elfzop1le2 13641 | A member in a half-open in... |
| fzonel 13642 | A half-open range does not... |
| elfzouz2 13643 | The upper bound of a half-... |
| elfzofz 13644 | A half-open range is conta... |
| elfzo3 13645 | Express membership in a ha... |
| fzon0 13646 | A half-open integer interv... |
| fzossfz 13647 | A half-open range is conta... |
| fzossz 13648 | A half-open integer interv... |
| fzon 13649 | A half-open set of sequent... |
| fzo0n 13650 | A half-open range of nonne... |
| fzonlt0 13651 | A half-open integer range ... |
| fzo0 13652 | Half-open sets with equal ... |
| fzonnsub 13653 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13654 | If ` M < N ` then ` N - M ... |
| fzoss1 13655 | Subset relationship for ha... |
| fzoss2 13656 | Subset relationship for ha... |
| fzossrbm1 13657 | Subset of a half-open rang... |
| fzo0ss1 13658 | Subset relationship for ha... |
| fzossnn0 13659 | A half-open integer range ... |
| fzospliti 13660 | One direction of splitting... |
| fzosplit 13661 | Split a half-open integer ... |
| fzodisj 13662 | Abutting half-open integer... |
| fzouzsplit 13663 | Split an upper integer set... |
| fzouzdisj 13664 | A half-open integer range ... |
| fzoun 13665 | A half-open integer range ... |
| fzodisjsn 13666 | A half-open integer range ... |
| prinfzo0 13667 | The intersection of a half... |
| lbfzo0 13668 | An integer is strictly gre... |
| elfzo0 13669 | Membership in a half-open ... |
| elfzo0z 13670 | Membership in a half-open ... |
| nn0p1elfzo 13671 | A nonnegative integer incr... |
| elfzo0le 13672 | A member in a half-open ra... |
| elfzonn0 13673 | A member of a half-open ra... |
| fzonmapblen 13674 | The result of subtracting ... |
| fzofzim 13675 | If a nonnegative integer i... |
| fz1fzo0m1 13676 | Translation of one between... |
| fzossnn 13677 | Half-open integer ranges s... |
| elfzo1 13678 | Membership in a half-open ... |
| fzo1fzo0n0 13679 | An integer between 1 and a... |
| fzo0n0 13680 | A half-open integer range ... |
| fzoaddel 13681 | Translate membership in a ... |
| fzo0addel 13682 | Translate membership in a ... |
| fzo0addelr 13683 | Translate membership in a ... |
| fzoaddel2 13684 | Translate membership in a ... |
| elfzoext 13685 | Membership of an integer i... |
| elincfzoext 13686 | Membership of an increased... |
| fzosubel 13687 | Translate membership in a ... |
| fzosubel2 13688 | Membership in a translated... |
| fzosubel3 13689 | Membership in a translated... |
| eluzgtdifelfzo 13690 | Membership of the differen... |
| ige2m2fzo 13691 | Membership of an integer g... |
| fzocatel 13692 | Translate membership in a ... |
| ubmelfzo 13693 | If an integer in a 1-based... |
| elfzodifsumelfzo 13694 | If an integer is in a half... |
| elfzom1elp1fzo 13695 | Membership of an integer i... |
| elfzom1elfzo 13696 | Membership in a half-open ... |
| fzval3 13697 | Expressing a closed intege... |
| fz0add1fz1 13698 | Translate membership in a ... |
| fzosn 13699 | Expressing a singleton as ... |
| elfzomin 13700 | Membership of an integer i... |
| zpnn0elfzo 13701 | Membership of an integer i... |
| zpnn0elfzo1 13702 | Membership of an integer i... |
| fzosplitsnm1 13703 | Removing a singleton from ... |
| elfzonlteqm1 13704 | If an element of a half-op... |
| fzonn0p1 13705 | A nonnegative integer is e... |
| fzossfzop1 13706 | A half-open range of nonne... |
| fzonn0p1p1 13707 | If a nonnegative integer i... |
| elfzom1p1elfzo 13708 | Increasing an element of a... |
| fzo0ssnn0 13709 | Half-open integer ranges s... |
| fzo01 13710 | Expressing the singleton o... |
| fzo12sn 13711 | A 1-based half-open intege... |
| fzo13pr 13712 | A 1-based half-open intege... |
| fzo0to2pr 13713 | A half-open integer range ... |
| fzo0to3tp 13714 | A half-open integer range ... |
| fzo0to42pr 13715 | A half-open integer range ... |
| fzo1to4tp 13716 | A half-open integer range ... |
| fzo0sn0fzo1 13717 | A half-open range of nonne... |
| elfzo0l 13718 | A member of a half-open ra... |
| fzoend 13719 | The endpoint of a half-ope... |
| fzo0end 13720 | The endpoint of a zero-bas... |
| ssfzo12 13721 | Subset relationship for ha... |
| ssfzoulel 13722 | If a half-open integer ran... |
| ssfzo12bi 13723 | Subset relationship for ha... |
| ubmelm1fzo 13724 | The result of subtracting ... |
| fzofzp1 13725 | If a point is in a half-op... |
| fzofzp1b 13726 | If a point is in a half-op... |
| elfzom1b 13727 | An integer is a member of ... |
| elfzom1elp1fzo1 13728 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13729 | Membership of a positive i... |
| elfzonelfzo 13730 | If an element of a half-op... |
| fzonfzoufzol 13731 | If an element of a half-op... |
| elfzomelpfzo 13732 | An integer increased by an... |
| elfznelfzo 13733 | A value in a finite set of... |
| elfznelfzob 13734 | A value in a finite set of... |
| peano2fzor 13735 | A Peano-postulate-like the... |
| fzosplitsn 13736 | Extending a half-open rang... |
| fzosplitpr 13737 | Extending a half-open inte... |
| fzosplitprm1 13738 | Extending a half-open inte... |
| fzosplitsni 13739 | Membership in a half-open ... |
| fzisfzounsn 13740 | A finite interval of integ... |
| elfzr 13741 | A member of a finite inter... |
| elfzlmr 13742 | A member of a finite inter... |
| elfz0lmr 13743 | A member of a finite inter... |
| fzostep1 13744 | Two possibilities for a nu... |
| fzoshftral 13745 | Shift the scanning order i... |
| fzind2 13746 | Induction on the integers ... |
| fvinim0ffz 13747 | The function values for th... |
| injresinjlem 13748 | Lemma for ~ injresinj . (... |
| injresinj 13749 | A function whose restricti... |
| subfzo0 13750 | The difference between two... |
| flval 13755 | Value of the floor (greate... |
| flcl 13756 | The floor (greatest intege... |
| reflcl 13757 | The floor (greatest intege... |
| fllelt 13758 | A basic property of the fl... |
| flcld 13759 | The floor (greatest intege... |
| flle 13760 | A basic property of the fl... |
| flltp1 13761 | A basic property of the fl... |
| fllep1 13762 | A basic property of the fl... |
| fraclt1 13763 | The fractional part of a r... |
| fracle1 13764 | The fractional part of a r... |
| fracge0 13765 | The fractional part of a r... |
| flge 13766 | The floor function value i... |
| fllt 13767 | The floor function value i... |
| flflp1 13768 | Move floor function betwee... |
| flid 13769 | An integer is its own floo... |
| flidm 13770 | The floor function is idem... |
| flidz 13771 | A real number equals its f... |
| flltnz 13772 | The floor of a non-integer... |
| flwordi 13773 | Ordering relation for the ... |
| flword2 13774 | Ordering relation for the ... |
| flval2 13775 | An alternate way to define... |
| flval3 13776 | An alternate way to define... |
| flbi 13777 | A condition equivalent to ... |
| flbi2 13778 | A condition equivalent to ... |
| adddivflid 13779 | The floor of a sum of an i... |
| ico01fl0 13780 | The floor of a real number... |
| flge0nn0 13781 | The floor of a number grea... |
| flge1nn 13782 | The floor of a number grea... |
| fldivnn0 13783 | The floor function of a di... |
| refldivcl 13784 | The floor function of a di... |
| divfl0 13785 | The floor of a fraction is... |
| fladdz 13786 | An integer can be moved in... |
| flzadd 13787 | An integer can be moved in... |
| flmulnn0 13788 | Move a nonnegative integer... |
| btwnzge0 13789 | A real bounded between an ... |
| 2tnp1ge0ge0 13790 | Two times an integer plus ... |
| flhalf 13791 | Ordering relation for the ... |
| fldivle 13792 | The floor function of a di... |
| fldivnn0le 13793 | The floor function of a di... |
| flltdivnn0lt 13794 | The floor function of a di... |
| ltdifltdiv 13795 | If the dividend of a divis... |
| fldiv4p1lem1div2 13796 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13797 | The floor of an integer gr... |
| fldiv4lem1div2 13798 | The floor of a positive in... |
| ceilval 13799 | The value of the ceiling f... |
| dfceil2 13800 | Alternative definition of ... |
| ceilval2 13801 | The value of the ceiling f... |
| ceicl 13802 | The ceiling function retur... |
| ceilcl 13803 | Closure of the ceiling fun... |
| ceilcld 13804 | Closure of the ceiling fun... |
| ceige 13805 | The ceiling of a real numb... |
| ceilge 13806 | The ceiling of a real numb... |
| ceilged 13807 | The ceiling of a real numb... |
| ceim1l 13808 | One less than the ceiling ... |
| ceilm1lt 13809 | One less than the ceiling ... |
| ceile 13810 | The ceiling of a real numb... |
| ceille 13811 | The ceiling of a real numb... |
| ceilid 13812 | An integer is its own ceil... |
| ceilidz 13813 | A real number equals its c... |
| flleceil 13814 | The floor of a real number... |
| fleqceilz 13815 | A real number is an intege... |
| quoremz 13816 | Quotient and remainder of ... |
| quoremnn0 13817 | Quotient and remainder of ... |
| quoremnn0ALT 13818 | Alternate proof of ~ quore... |
| intfrac2 13819 | Decompose a real into inte... |
| intfracq 13820 | Decompose a rational numbe... |
| fldiv 13821 | Cancellation of the embedd... |
| fldiv2 13822 | Cancellation of an embedde... |
| fznnfl 13823 | Finite set of sequential i... |
| uzsup 13824 | An upper set of integers i... |
| ioopnfsup 13825 | An upper set of reals is u... |
| icopnfsup 13826 | An upper set of reals is u... |
| rpsup 13827 | The positive reals are unb... |
| resup 13828 | The real numbers are unbou... |
| xrsup 13829 | The extended real numbers ... |
| modval 13832 | The value of the modulo op... |
| modvalr 13833 | The value of the modulo op... |
| modcl 13834 | Closure law for the modulo... |
| flpmodeq 13835 | Partition of a division in... |
| modcld 13836 | Closure law for the modulo... |
| mod0 13837 | ` A mod B ` is zero iff ` ... |
| mulmod0 13838 | The product of an integer ... |
| negmod0 13839 | ` A ` is divisible by ` B ... |
| modge0 13840 | The modulo operation is no... |
| modlt 13841 | The modulo operation is le... |
| modelico 13842 | Modular reduction produces... |
| moddiffl 13843 | Value of the modulo operat... |
| moddifz 13844 | The modulo operation diffe... |
| modfrac 13845 | The fractional part of a n... |
| flmod 13846 | The floor function express... |
| intfrac 13847 | Break a number into its in... |
| zmod10 13848 | An integer modulo 1 is 0. ... |
| zmod1congr 13849 | Two arbitrary integers are... |
| modmulnn 13850 | Move a positive integer in... |
| modvalp1 13851 | The value of the modulo op... |
| zmodcl 13852 | Closure law for the modulo... |
| zmodcld 13853 | Closure law for the modulo... |
| zmodfz 13854 | An integer mod ` B ` lies ... |
| zmodfzo 13855 | An integer mod ` B ` lies ... |
| zmodfzp1 13856 | An integer mod ` B ` lies ... |
| modid 13857 | Identity law for modulo. ... |
| modid0 13858 | A positive real number mod... |
| modid2 13859 | Identity law for modulo. ... |
| zmodid2 13860 | Identity law for modulo re... |
| zmodidfzo 13861 | Identity law for modulo re... |
| zmodidfzoimp 13862 | Identity law for modulo re... |
| 0mod 13863 | Special case: 0 modulo a p... |
| 1mod 13864 | Special case: 1 modulo a r... |
| modabs 13865 | Absorption law for modulo.... |
| modabs2 13866 | Absorption law for modulo.... |
| modcyc 13867 | The modulo operation is pe... |
| modcyc2 13868 | The modulo operation is pe... |
| modadd1 13869 | Addition property of the m... |
| modaddabs 13870 | Absorption law for modulo.... |
| modaddmod 13871 | The sum of a real number m... |
| muladdmodid 13872 | The sum of a positive real... |
| mulp1mod1 13873 | The product of an integer ... |
| modmuladd 13874 | Decomposition of an intege... |
| modmuladdim 13875 | Implication of a decomposi... |
| modmuladdnn0 13876 | Implication of a decomposi... |
| negmod 13877 | The negation of a number m... |
| m1modnnsub1 13878 | Minus one modulo a positiv... |
| m1modge3gt1 13879 | Minus one modulo an intege... |
| addmodid 13880 | The sum of a positive inte... |
| addmodidr 13881 | The sum of a positive inte... |
| modadd2mod 13882 | The sum of a real number m... |
| modm1p1mod0 13883 | If a real number modulo a ... |
| modltm1p1mod 13884 | If a real number modulo a ... |
| modmul1 13885 | Multiplication property of... |
| modmul12d 13886 | Multiplication property of... |
| modnegd 13887 | Negation property of the m... |
| modadd12d 13888 | Additive property of the m... |
| modsub12d 13889 | Subtraction property of th... |
| modsubmod 13890 | The difference of a real n... |
| modsubmodmod 13891 | The difference of a real n... |
| 2txmodxeq0 13892 | Two times a positive real ... |
| 2submod 13893 | If a real number is betwee... |
| modifeq2int 13894 | If a nonnegative integer i... |
| modaddmodup 13895 | The sum of an integer modu... |
| modaddmodlo 13896 | The sum of an integer modu... |
| modmulmod 13897 | The product of a real numb... |
| modmulmodr 13898 | The product of an integer ... |
| modaddmulmod 13899 | The sum of a real number a... |
| moddi 13900 | Distribute multiplication ... |
| modsubdir 13901 | Distribute the modulo oper... |
| modeqmodmin 13902 | A real number equals the d... |
| modirr 13903 | A number modulo an irratio... |
| modfzo0difsn 13904 | For a number within a half... |
| modsumfzodifsn 13905 | The sum of a number within... |
| modlteq 13906 | Two nonnegative integers l... |
| addmodlteq 13907 | Two nonnegative integers l... |
| om2uz0i 13908 | The mapping ` G ` is a one... |
| om2uzsuci 13909 | The value of ` G ` (see ~ ... |
| om2uzuzi 13910 | The value ` G ` (see ~ om2... |
| om2uzlti 13911 | Less-than relation for ` G... |
| om2uzlt2i 13912 | The mapping ` G ` (see ~ o... |
| om2uzrani 13913 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13914 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13915 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13916 | An alternative definition ... |
| om2uzrdg 13917 | A helper lemma for the val... |
| uzrdglem 13918 | A helper lemma for the val... |
| uzrdgfni 13919 | The recursive definition g... |
| uzrdg0i 13920 | Initial value of a recursi... |
| uzrdgsuci 13921 | Successor value of a recur... |
| ltweuz 13922 | ` < ` is a well-founded re... |
| ltwenn 13923 | Less than well-orders the ... |
| ltwefz 13924 | Less than well-orders a se... |
| uzenom 13925 | An upper integer set is de... |
| uzinf 13926 | An upper integer set is in... |
| nnnfi 13927 | The set of positive intege... |
| uzrdgxfr 13928 | Transfer the value of the ... |
| fzennn 13929 | The cardinality of a finit... |
| fzen2 13930 | The cardinality of a finit... |
| cardfz 13931 | The cardinality of a finit... |
| hashgf1o 13932 | ` G ` maps ` _om ` one-to-... |
| fzfi 13933 | A finite interval of integ... |
| fzfid 13934 | Commonly used special case... |
| fzofi 13935 | Half-open integer sets are... |
| fsequb 13936 | The values of a finite rea... |
| fsequb2 13937 | The values of a finite rea... |
| fseqsupcl 13938 | The values of a finite rea... |
| fseqsupubi 13939 | The values of a finite rea... |
| nn0ennn 13940 | The nonnegative integers a... |
| nnenom 13941 | The set of positive intege... |
| nnct 13942 | ` NN ` is countable. (Con... |
| uzindi 13943 | Indirect strong induction ... |
| axdc4uzlem 13944 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13945 | A version of ~ axdc4 that ... |
| ssnn0fi 13946 | A subset of the nonnegativ... |
| rabssnn0fi 13947 | A subset of the nonnegativ... |
| uzsinds 13948 | Strong (or "total") induct... |
| nnsinds 13949 | Strong (or "total") induct... |
| nn0sinds 13950 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 13951 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 13952 | If all functions of a fini... |
| fsuppmapnn0fiubex 13953 | If all functions of a fini... |
| fsuppmapnn0fiub0 13954 | If all functions of a fini... |
| suppssfz 13955 | Condition for a function o... |
| fsuppmapnn0ub 13956 | If a function over the non... |
| fsuppmapnn0fz 13957 | If a function over the non... |
| mptnn0fsupp 13958 | A mapping from the nonnega... |
| mptnn0fsuppd 13959 | A mapping from the nonnega... |
| mptnn0fsuppr 13960 | A finitely supported mappi... |
| f13idfv 13961 | A one-to-one function with... |
| seqex 13964 | Existence of the sequence ... |
| seqeq1 13965 | Equality theorem for the s... |
| seqeq2 13966 | Equality theorem for the s... |
| seqeq3 13967 | Equality theorem for the s... |
| seqeq1d 13968 | Equality deduction for the... |
| seqeq2d 13969 | Equality deduction for the... |
| seqeq3d 13970 | Equality deduction for the... |
| seqeq123d 13971 | Equality deduction for the... |
| nfseq 13972 | Hypothesis builder for the... |
| seqval 13973 | Value of the sequence buil... |
| seqfn 13974 | The sequence builder funct... |
| seq1 13975 | Value of the sequence buil... |
| seq1i 13976 | Value of the sequence buil... |
| seqp1 13977 | Value of the sequence buil... |
| seqexw 13978 | Weak version of ~ seqex th... |
| seqp1d 13979 | Value of the sequence buil... |
| seqp1iOLD 13980 | Obsolete version of ~ seqp... |
| seqm1 13981 | Value of the sequence buil... |
| seqcl2 13982 | Closure properties of the ... |
| seqf2 13983 | Range of the recursive seq... |
| seqcl 13984 | Closure properties of the ... |
| seqf 13985 | Range of the recursive seq... |
| seqfveq2 13986 | Equality of sequences. (C... |
| seqfeq2 13987 | Equality of sequences. (C... |
| seqfveq 13988 | Equality of sequences. (C... |
| seqfeq 13989 | Equality of sequences. (C... |
| seqshft2 13990 | Shifting the index set of ... |
| seqres 13991 | Restricting its characteri... |
| serf 13992 | An infinite series of comp... |
| serfre 13993 | An infinite series of real... |
| monoord 13994 | Ordering relation for a mo... |
| monoord2 13995 | Ordering relation for a mo... |
| sermono 13996 | The partial sums in an inf... |
| seqsplit 13997 | Split a sequence into two ... |
| seq1p 13998 | Removing the first term fr... |
| seqcaopr3 13999 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 14000 | The sum of two infinite se... |
| seqcaopr 14001 | The sum of two infinite se... |
| seqf1olem2a 14002 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 14003 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 14004 | Lemma for ~ seqf1o . (Con... |
| seqf1o 14005 | Rearrange a sum via an arb... |
| seradd 14006 | The sum of two infinite se... |
| sersub 14007 | The difference of two infi... |
| seqid3 14008 | A sequence that consists e... |
| seqid 14009 | Discarding the first few t... |
| seqid2 14010 | The last few partial sums ... |
| seqhomo 14011 | Apply a homomorphism to a ... |
| seqz 14012 | If the operation ` .+ ` ha... |
| seqfeq4 14013 | Equality of series under d... |
| seqfeq3 14014 | Equality of series under d... |
| seqdistr 14015 | The distributive property ... |
| ser0 14016 | The value of the partial s... |
| ser0f 14017 | A zero-valued infinite ser... |
| serge0 14018 | A finite sum of nonnegativ... |
| serle 14019 | Comparison of partial sums... |
| ser1const 14020 | Value of the partial serie... |
| seqof 14021 | Distribute function operat... |
| seqof2 14022 | Distribute function operat... |
| expval 14025 | Value of exponentiation to... |
| expnnval 14026 | Value of exponentiation to... |
| exp0 14027 | Value of a complex number ... |
| 0exp0e1 14028 | The zeroth power of zero e... |
| exp1 14029 | Value of a complex number ... |
| expp1 14030 | Value of a complex number ... |
| expneg 14031 | Value of a complex number ... |
| expneg2 14032 | Value of a complex number ... |
| expn1 14033 | A complex number raised to... |
| expcllem 14034 | Lemma for proving nonnegat... |
| expcl2lem 14035 | Lemma for proving integer ... |
| nnexpcl 14036 | Closure of exponentiation ... |
| nn0expcl 14037 | Closure of exponentiation ... |
| zexpcl 14038 | Closure of exponentiation ... |
| qexpcl 14039 | Closure of exponentiation ... |
| reexpcl 14040 | Closure of exponentiation ... |
| expcl 14041 | Closure law for nonnegativ... |
| rpexpcl 14042 | Closure law for integer ex... |
| qexpclz 14043 | Closure of integer exponen... |
| reexpclz 14044 | Closure of integer exponen... |
| expclzlem 14045 | Lemma for ~ expclz . (Con... |
| expclz 14046 | Closure law for integer ex... |
| m1expcl2 14047 | Closure of integer exponen... |
| m1expcl 14048 | Closure of exponentiation ... |
| zexpcld 14049 | Closure of exponentiation ... |
| nn0expcli 14050 | Closure of exponentiation ... |
| nn0sqcl 14051 | The square of a nonnegativ... |
| expm1t 14052 | Exponentiation in terms of... |
| 1exp 14053 | Value of 1 raised to an in... |
| expeq0 14054 | A positive integer power i... |
| expne0 14055 | A positive integer power i... |
| expne0i 14056 | An integer power is nonzer... |
| expgt0 14057 | A positive real raised to ... |
| expnegz 14058 | Value of a nonzero complex... |
| 0exp 14059 | Value of zero raised to a ... |
| expge0 14060 | A nonnegative real raised ... |
| expge1 14061 | A real greater than or equ... |
| expgt1 14062 | A real greater than 1 rais... |
| mulexp 14063 | Nonnegative integer expone... |
| mulexpz 14064 | Integer exponentiation of ... |
| exprec 14065 | Integer exponentiation of ... |
| expadd 14066 | Sum of exponents law for n... |
| expaddzlem 14067 | Lemma for ~ expaddz . (Co... |
| expaddz 14068 | Sum of exponents law for i... |
| expmul 14069 | Product of exponents law f... |
| expmulz 14070 | Product of exponents law f... |
| m1expeven 14071 | Exponentiation of negative... |
| expsub 14072 | Exponent subtraction law f... |
| expp1z 14073 | Value of a nonzero complex... |
| expm1 14074 | Value of a nonzero complex... |
| expdiv 14075 | Nonnegative integer expone... |
| sqval 14076 | Value of the square of a c... |
| sqneg 14077 | The square of the negative... |
| sqsubswap 14078 | Swap the order of subtract... |
| sqcl 14079 | Closure of square. (Contr... |
| sqmul 14080 | Distribution of squaring o... |
| sqeq0 14081 | A complex number is zero i... |
| sqdiv 14082 | Distribution of squaring o... |
| sqdivid 14083 | The square of a nonzero co... |
| sqne0 14084 | A complex number is nonzer... |
| resqcl 14085 | Closure of squaring in rea... |
| resqcld 14086 | Closure of squaring in rea... |
| sqgt0 14087 | The square of a nonzero re... |
| sqn0rp 14088 | The square of a nonzero re... |
| nnsqcl 14089 | The positive naturals are ... |
| zsqcl 14090 | Integers are closed under ... |
| qsqcl 14091 | The square of a rational i... |
| sq11 14092 | The square function is one... |
| nn0sq11 14093 | The square function is one... |
| lt2sq 14094 | The square function is inc... |
| le2sq 14095 | The square function is non... |
| le2sq2 14096 | The square function is non... |
| sqge0 14097 | The square of a real is no... |
| sqge0d 14098 | The square of a real is no... |
| zsqcl2 14099 | The square of an integer i... |
| 0expd 14100 | Value of zero raised to a ... |
| exp0d 14101 | Value of a complex number ... |
| exp1d 14102 | Value of a complex number ... |
| expeq0d 14103 | If a positive integer powe... |
| sqvald 14104 | Value of square. Inferenc... |
| sqcld 14105 | Closure of square. (Contr... |
| sqeq0d 14106 | A number is zero iff its s... |
| expcld 14107 | Closure law for nonnegativ... |
| expp1d 14108 | Value of a complex number ... |
| expaddd 14109 | Sum of exponents law for n... |
| expmuld 14110 | Product of exponents law f... |
| sqrecd 14111 | Square of reciprocal is re... |
| expclzd 14112 | Closure law for integer ex... |
| expne0d 14113 | A nonnegative integer powe... |
| expnegd 14114 | Value of a nonzero complex... |
| exprecd 14115 | An integer power of a reci... |
| expp1zd 14116 | Value of a nonzero complex... |
| expm1d 14117 | Value of a nonzero complex... |
| expsubd 14118 | Exponent subtraction law f... |
| sqmuld 14119 | Distribution of squaring o... |
| sqdivd 14120 | Distribution of squaring o... |
| expdivd 14121 | Nonnegative integer expone... |
| mulexpd 14122 | Nonnegative integer expone... |
| znsqcld 14123 | The square of a nonzero in... |
| reexpcld 14124 | Closure of exponentiation ... |
| expge0d 14125 | A nonnegative real raised ... |
| expge1d 14126 | A real greater than or equ... |
| ltexp2a 14127 | Exponent ordering relation... |
| expmordi 14128 | Base ordering relationship... |
| rpexpmord 14129 | Base ordering relationship... |
| expcan 14130 | Cancellation law for integ... |
| ltexp2 14131 | Strict ordering law for ex... |
| leexp2 14132 | Ordering law for exponenti... |
| leexp2a 14133 | Weak ordering relationship... |
| ltexp2r 14134 | The integer powers of a fi... |
| leexp2r 14135 | Weak ordering relationship... |
| leexp1a 14136 | Weak base ordering relatio... |
| exple1 14137 | A real between 0 and 1 inc... |
| expubnd 14138 | An upper bound on ` A ^ N ... |
| sumsqeq0 14139 | The sum of two squres of r... |
| sqvali 14140 | Value of square. Inferenc... |
| sqcli 14141 | Closure of square. (Contr... |
| sqeq0i 14142 | A complex number is zero i... |
| sqrecii 14143 | The square of a reciprocal... |
| sqmuli 14144 | Distribution of squaring o... |
| sqdivi 14145 | Distribution of squaring o... |
| resqcli 14146 | Closure of square in reals... |
| sqgt0i 14147 | The square of a nonzero re... |
| sqge0i 14148 | The square of a real is no... |
| lt2sqi 14149 | The square function on non... |
| le2sqi 14150 | The square function on non... |
| sq11i 14151 | The square function is one... |
| sq0 14152 | The square of 0 is 0. (Co... |
| sq0i 14153 | If a number is zero, then ... |
| sq0id 14154 | If a number is zero, then ... |
| sq1 14155 | The square of 1 is 1. (Co... |
| neg1sqe1 14156 | The square of ` -u 1 ` is ... |
| sq2 14157 | The square of 2 is 4. (Co... |
| sq3 14158 | The square of 3 is 9. (Co... |
| sq4e2t8 14159 | The square of 4 is 2 times... |
| cu2 14160 | The cube of 2 is 8. (Cont... |
| irec 14161 | The reciprocal of ` _i ` .... |
| i2 14162 | ` _i ` squared. (Contribu... |
| i3 14163 | ` _i ` cubed. (Contribute... |
| i4 14164 | ` _i ` to the fourth power... |
| nnlesq 14165 | A positive integer is less... |
| zzlesq 14166 | An integer is less than or... |
| iexpcyc 14167 | Taking ` _i ` to the ` K `... |
| expnass 14168 | A counterexample showing t... |
| sqlecan 14169 | Cancel one factor of a squ... |
| subsq 14170 | Factor the difference of t... |
| subsq2 14171 | Express the difference of ... |
| binom2i 14172 | The square of a binomial. ... |
| subsqi 14173 | Factor the difference of t... |
| sqeqori 14174 | The squares of two complex... |
| subsq0i 14175 | The two solutions to the d... |
| sqeqor 14176 | The squares of two complex... |
| binom2 14177 | The square of a binomial. ... |
| binom21 14178 | Special case of ~ binom2 w... |
| binom2sub 14179 | Expand the square of a sub... |
| binom2sub1 14180 | Special case of ~ binom2su... |
| binom2subi 14181 | Expand the square of a sub... |
| mulbinom2 14182 | The square of a binomial w... |
| binom3 14183 | The cube of a binomial. (... |
| sq01 14184 | If a complex number equals... |
| zesq 14185 | An integer is even iff its... |
| nnesq 14186 | A positive integer is even... |
| crreczi 14187 | Reciprocal of a complex nu... |
| bernneq 14188 | Bernoulli's inequality, du... |
| bernneq2 14189 | Variation of Bernoulli's i... |
| bernneq3 14190 | A corollary of ~ bernneq .... |
| expnbnd 14191 | Exponentiation with a base... |
| expnlbnd 14192 | The reciprocal of exponent... |
| expnlbnd2 14193 | The reciprocal of exponent... |
| expmulnbnd 14194 | Exponentiation with a base... |
| digit2 14195 | Two ways to express the ` ... |
| digit1 14196 | Two ways to express the ` ... |
| modexp 14197 | Exponentiation property of... |
| discr1 14198 | A nonnegative quadratic fo... |
| discr 14199 | If a quadratic polynomial ... |
| expnngt1 14200 | If an integer power with a... |
| expnngt1b 14201 | An integer power with an i... |
| sqoddm1div8 14202 | A squared odd number minus... |
| nnsqcld 14203 | The naturals are closed un... |
| nnexpcld 14204 | Closure of exponentiation ... |
| nn0expcld 14205 | Closure of exponentiation ... |
| rpexpcld 14206 | Closure law for exponentia... |
| ltexp2rd 14207 | The power of a positive nu... |
| reexpclzd 14208 | Closure of exponentiation ... |
| sqgt0d 14209 | The square of a nonzero re... |
| ltexp2d 14210 | Ordering relationship for ... |
| leexp2d 14211 | Ordering law for exponenti... |
| expcand 14212 | Ordering relationship for ... |
| leexp2ad 14213 | Ordering relationship for ... |
| leexp2rd 14214 | Ordering relationship for ... |
| lt2sqd 14215 | The square function on non... |
| le2sqd 14216 | The square function on non... |
| sq11d 14217 | The square function is one... |
| mulsubdivbinom2 14218 | The square of a binomial w... |
| muldivbinom2 14219 | The square of a binomial w... |
| sq10 14220 | The square of 10 is 100. ... |
| sq10e99m1 14221 | The square of 10 is 99 plu... |
| 3dec 14222 | A "decimal constructor" wh... |
| nn0le2msqi 14223 | The square function on non... |
| nn0opthlem1 14224 | A rather pretty lemma for ... |
| nn0opthlem2 14225 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14226 | An ordered pair theorem fo... |
| nn0opth2i 14227 | An ordered pair theorem fo... |
| nn0opth2 14228 | An ordered pair theorem fo... |
| facnn 14231 | Value of the factorial fun... |
| fac0 14232 | The factorial of 0. (Cont... |
| fac1 14233 | The factorial of 1. (Cont... |
| facp1 14234 | The factorial of a success... |
| fac2 14235 | The factorial of 2. (Cont... |
| fac3 14236 | The factorial of 3. (Cont... |
| fac4 14237 | The factorial of 4. (Cont... |
| facnn2 14238 | Value of the factorial fun... |
| faccl 14239 | Closure of the factorial f... |
| faccld 14240 | Closure of the factorial f... |
| facmapnn 14241 | The factorial function res... |
| facne0 14242 | The factorial function is ... |
| facdiv 14243 | A positive integer divides... |
| facndiv 14244 | No positive integer (great... |
| facwordi 14245 | Ordering property of facto... |
| faclbnd 14246 | A lower bound for the fact... |
| faclbnd2 14247 | A lower bound for the fact... |
| faclbnd3 14248 | A lower bound for the fact... |
| faclbnd4lem1 14249 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14250 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14251 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14252 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14253 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14254 | The factorial function gro... |
| faclbnd6 14255 | Geometric lower bound for ... |
| facubnd 14256 | An upper bound for the fac... |
| facavg 14257 | The product of two factori... |
| bcval 14260 | Value of the binomial coef... |
| bcval2 14261 | Value of the binomial coef... |
| bcval3 14262 | Value of the binomial coef... |
| bcval4 14263 | Value of the binomial coef... |
| bcrpcl 14264 | Closure of the binomial co... |
| bccmpl 14265 | "Complementing" its second... |
| bcn0 14266 | ` N ` choose 0 is 1. Rema... |
| bc0k 14267 | The binomial coefficient "... |
| bcnn 14268 | ` N ` choose ` N ` is 1. ... |
| bcn1 14269 | Binomial coefficient: ` N ... |
| bcnp1n 14270 | Binomial coefficient: ` N ... |
| bcm1k 14271 | The proportion of one bino... |
| bcp1n 14272 | The proportion of one bino... |
| bcp1nk 14273 | The proportion of one bino... |
| bcval5 14274 | Write out the top and bott... |
| bcn2 14275 | Binomial coefficient: ` N ... |
| bcp1m1 14276 | Compute the binomial coeff... |
| bcpasc 14277 | Pascal's rule for the bino... |
| bccl 14278 | A binomial coefficient, in... |
| bccl2 14279 | A binomial coefficient, in... |
| bcn2m1 14280 | Compute the binomial coeff... |
| bcn2p1 14281 | Compute the binomial coeff... |
| permnn 14282 | The number of permutations... |
| bcnm1 14283 | The binomial coefficent of... |
| 4bc3eq4 14284 | The value of four choose t... |
| 4bc2eq6 14285 | The value of four choose t... |
| hashkf 14288 | The finite part of the siz... |
| hashgval 14289 | The value of the ` # ` fun... |
| hashginv 14290 | The converse of ` G ` maps... |
| hashinf 14291 | The value of the ` # ` fun... |
| hashbnd 14292 | If ` A ` has size bounded ... |
| hashfxnn0 14293 | The size function is a fun... |
| hashf 14294 | The size function maps all... |
| hashxnn0 14295 | The value of the hash func... |
| hashresfn 14296 | Restriction of the domain ... |
| dmhashres 14297 | Restriction of the domain ... |
| hashnn0pnf 14298 | The value of the hash func... |
| hashnnn0genn0 14299 | If the size of a set is no... |
| hashnemnf 14300 | The size of a set is never... |
| hashv01gt1 14301 | The size of a set is eithe... |
| hashfz1 14302 | The set ` ( 1 ... N ) ` ha... |
| hashen 14303 | Two finite sets have the s... |
| hasheni 14304 | Equinumerous sets have the... |
| hasheqf1o 14305 | The size of two finite set... |
| fiinfnf1o 14306 | There is no bijection betw... |
| hasheqf1oi 14307 | The size of two sets is eq... |
| hashf1rn 14308 | The size of a finite set w... |
| hasheqf1od 14309 | The size of two sets is eq... |
| fz1eqb 14310 | Two possibly-empty 1-based... |
| hashcard 14311 | The size function of the c... |
| hashcl 14312 | Closure of the ` # ` funct... |
| hashxrcl 14313 | Extended real closure of t... |
| hashclb 14314 | Reverse closure of the ` #... |
| nfile 14315 | The size of any infinite s... |
| hashvnfin 14316 | A set of finite size is a ... |
| hashnfinnn0 14317 | The size of an infinite se... |
| isfinite4 14318 | A finite set is equinumero... |
| hasheq0 14319 | Two ways of saying a set i... |
| hashneq0 14320 | Two ways of saying a set i... |
| hashgt0n0 14321 | If the size of a set is gr... |
| hashnncl 14322 | Positive natural closure o... |
| hash0 14323 | The empty set has size zer... |
| hashelne0d 14324 | A set with an element has ... |
| hashsng 14325 | The size of a singleton. ... |
| hashen1 14326 | A set has size 1 if and on... |
| hash1elsn 14327 | A set of size 1 with a kno... |
| hashrabrsn 14328 | The size of a restricted c... |
| hashrabsn01 14329 | The size of a restricted c... |
| hashrabsn1 14330 | If the size of a restricte... |
| hashfn 14331 | A function is equinumerous... |
| fseq1hash 14332 | The value of the size func... |
| hashgadd 14333 | ` G ` maps ordinal additio... |
| hashgval2 14334 | A short expression for the... |
| hashdom 14335 | Dominance relation for the... |
| hashdomi 14336 | Non-strict order relation ... |
| hashsdom 14337 | Strict dominance relation ... |
| hashun 14338 | The size of the union of d... |
| hashun2 14339 | The size of the union of f... |
| hashun3 14340 | The size of the union of f... |
| hashinfxadd 14341 | The extended real addition... |
| hashunx 14342 | The size of the union of d... |
| hashge0 14343 | The cardinality of a set i... |
| hashgt0 14344 | The cardinality of a nonem... |
| hashge1 14345 | The cardinality of a nonem... |
| 1elfz0hash 14346 | 1 is an element of the fin... |
| hashnn0n0nn 14347 | If a nonnegative integer i... |
| hashunsng 14348 | The size of the union of a... |
| hashunsngx 14349 | The size of the union of a... |
| hashunsnggt 14350 | The size of a set is great... |
| hashprg 14351 | The size of an unordered p... |
| elprchashprn2 14352 | If one element of an unord... |
| hashprb 14353 | The size of an unordered p... |
| hashprdifel 14354 | The elements of an unorder... |
| prhash2ex 14355 | There is (at least) one se... |
| hashle00 14356 | If the size of a set is le... |
| hashgt0elex 14357 | If the size of a set is gr... |
| hashgt0elexb 14358 | The size of a set is great... |
| hashp1i 14359 | Size of a finite ordinal. ... |
| hash1 14360 | Size of a finite ordinal. ... |
| hash2 14361 | Size of a finite ordinal. ... |
| hash3 14362 | Size of a finite ordinal. ... |
| hash4 14363 | Size of a finite ordinal. ... |
| pr0hash2ex 14364 | There is (at least) one se... |
| hashss 14365 | The size of a subset is le... |
| prsshashgt1 14366 | The size of a superset of ... |
| hashin 14367 | The size of the intersecti... |
| hashssdif 14368 | The size of the difference... |
| hashdif 14369 | The size of the difference... |
| hashdifsn 14370 | The size of the difference... |
| hashdifpr 14371 | The size of the difference... |
| hashsn01 14372 | The size of a singleton is... |
| hashsnle1 14373 | The size of a singleton is... |
| hashsnlei 14374 | Get an upper bound on a co... |
| hash1snb 14375 | The size of a set is 1 if ... |
| euhash1 14376 | The size of a set is 1 in ... |
| hash1n0 14377 | If the size of a set is 1 ... |
| hashgt12el 14378 | In a set with more than on... |
| hashgt12el2 14379 | In a set with more than on... |
| hashgt23el 14380 | A set with more than two e... |
| hashunlei 14381 | Get an upper bound on a co... |
| hashsslei 14382 | Get an upper bound on a co... |
| hashfz 14383 | Value of the numeric cardi... |
| fzsdom2 14384 | Condition for finite range... |
| hashfzo 14385 | Cardinality of a half-open... |
| hashfzo0 14386 | Cardinality of a half-open... |
| hashfzp1 14387 | Value of the numeric cardi... |
| hashfz0 14388 | Value of the numeric cardi... |
| hashxplem 14389 | Lemma for ~ hashxp . (Con... |
| hashxp 14390 | The size of the Cartesian ... |
| hashmap 14391 | The size of the set expone... |
| hashpw 14392 | The size of the power set ... |
| hashfun 14393 | A finite set is a function... |
| hashres 14394 | The number of elements of ... |
| hashreshashfun 14395 | The number of elements of ... |
| hashimarn 14396 | The size of the image of a... |
| hashimarni 14397 | If the size of the image o... |
| hashfundm 14398 | The size of a set function... |
| hashf1dmrn 14399 | The size of the domain of ... |
| resunimafz0 14400 | TODO-AV: Revise using ` F... |
| fnfz0hash 14401 | The size of a function on ... |
| ffz0hash 14402 | The size of a function on ... |
| fnfz0hashnn0 14403 | The size of a function on ... |
| ffzo0hash 14404 | The size of a function on ... |
| fnfzo0hash 14405 | The size of a function on ... |
| fnfzo0hashnn0 14406 | The value of the size func... |
| hashbclem 14407 | Lemma for ~ hashbc : induc... |
| hashbc 14408 | The binomial coefficient c... |
| hashfacen 14409 | The number of bijections b... |
| hashfacenOLD 14410 | Obsolete version of ~ hash... |
| hashf1lem1 14411 | Lemma for ~ hashf1 . (Con... |
| hashf1lem1OLD 14412 | Obsolete version of ~ hash... |
| hashf1lem2 14413 | Lemma for ~ hashf1 . (Con... |
| hashf1 14414 | The permutation number ` |... |
| hashfac 14415 | A factorial counts the num... |
| leiso 14416 | Two ways to write a strict... |
| leisorel 14417 | Version of ~ isorel for st... |
| fz1isolem 14418 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14419 | Any finite ordered set has... |
| ishashinf 14420 | Any set that is not finite... |
| seqcoll 14421 | The function ` F ` contain... |
| seqcoll2 14422 | The function ` F ` contain... |
| phphashd 14423 | Corollary of the Pigeonhol... |
| phphashrd 14424 | Corollary of the Pigeonhol... |
| hashprlei 14425 | An unordered pair has at m... |
| hash2pr 14426 | A set of size two is an un... |
| hash2prde 14427 | A set of size two is an un... |
| hash2exprb 14428 | A set of size two is an un... |
| hash2prb 14429 | A set of size two is a pro... |
| prprrab 14430 | The set of proper pairs of... |
| nehash2 14431 | The cardinality of a set w... |
| hash2prd 14432 | A set of size two is an un... |
| hash2pwpr 14433 | If the size of a subset of... |
| hashle2pr 14434 | A nonempty set of size les... |
| hashle2prv 14435 | A nonempty subset of a pow... |
| pr2pwpr 14436 | The set of subsets of a pa... |
| hashge2el2dif 14437 | A set with size at least 2... |
| hashge2el2difr 14438 | A set with at least 2 diff... |
| hashge2el2difb 14439 | A set has size at least 2 ... |
| hashdmpropge2 14440 | The size of the domain of ... |
| hashtplei 14441 | An unordered triple has at... |
| hashtpg 14442 | The size of an unordered t... |
| hashge3el3dif 14443 | A set with size at least 3... |
| elss2prb 14444 | An element of the set of s... |
| hash2sspr 14445 | A subset of size two is an... |
| exprelprel 14446 | If there is an element of ... |
| hash3tr 14447 | A set of size three is an ... |
| hash1to3 14448 | If the size of a set is be... |
| fundmge2nop0 14449 | A function with a domain c... |
| fundmge2nop 14450 | A function with a domain c... |
| fun2dmnop0 14451 | A function with a domain c... |
| fun2dmnop 14452 | A function with a domain c... |
| hashdifsnp1 14453 | If the size of a set is a ... |
| fi1uzind 14454 | Properties of an ordered p... |
| brfi1uzind 14455 | Properties of a binary rel... |
| brfi1ind 14456 | Properties of a binary rel... |
| brfi1indALT 14457 | Alternate proof of ~ brfi1... |
| opfi1uzind 14458 | Properties of an ordered p... |
| opfi1ind 14459 | Properties of an ordered p... |
| iswrd 14462 | Property of being a word o... |
| wrdval 14463 | Value of the set of words ... |
| iswrdi 14464 | A zero-based sequence is a... |
| wrdf 14465 | A word is a zero-based seq... |
| iswrdb 14466 | A word over an alphabet is... |
| wrddm 14467 | The indices of a word (i.e... |
| sswrd 14468 | The set of words respects ... |
| snopiswrd 14469 | A singleton of an ordered ... |
| wrdexg 14470 | The set of words over a se... |
| wrdexb 14471 | The set of words over a se... |
| wrdexi 14472 | The set of words over a se... |
| wrdsymbcl 14473 | A symbol within a word ove... |
| wrdfn 14474 | A word is a function with ... |
| wrdv 14475 | A word over an alphabet is... |
| wrdlndm 14476 | The length of a word is no... |
| iswrdsymb 14477 | An arbitrary word is a wor... |
| wrdfin 14478 | A word is a finite set. (... |
| lencl 14479 | The length of a word is a ... |
| lennncl 14480 | The length of a nonempty w... |
| wrdffz 14481 | A word is a function from ... |
| wrdeq 14482 | Equality theorem for the s... |
| wrdeqi 14483 | Equality theorem for the s... |
| iswrddm0 14484 | A function with empty doma... |
| wrd0 14485 | The empty set is a word (t... |
| 0wrd0 14486 | The empty word is the only... |
| ffz0iswrd 14487 | A sequence with zero-based... |
| wrdsymb 14488 | A word is a word over the ... |
| nfwrd 14489 | Hypothesis builder for ` W... |
| csbwrdg 14490 | Class substitution for the... |
| wrdnval 14491 | Words of a fixed length ar... |
| wrdmap 14492 | Words as a mapping. (Cont... |
| hashwrdn 14493 | If there is only a finite ... |
| wrdnfi 14494 | If there is only a finite ... |
| wrdsymb0 14495 | A symbol at a position "ou... |
| wrdlenge1n0 14496 | A word with length at leas... |
| len0nnbi 14497 | The length of a word is a ... |
| wrdlenge2n0 14498 | A word with length at leas... |
| wrdsymb1 14499 | The first symbol of a none... |
| wrdlen1 14500 | A word of length 1 starts ... |
| fstwrdne 14501 | The first symbol of a none... |
| fstwrdne0 14502 | The first symbol of a none... |
| eqwrd 14503 | Two words are equal iff th... |
| elovmpowrd 14504 | Implications for the value... |
| elovmptnn0wrd 14505 | Implications for the value... |
| wrdred1 14506 | A word truncated by a symb... |
| wrdred1hash 14507 | The length of a word trunc... |
| lsw 14510 | Extract the last symbol of... |
| lsw0 14511 | The last symbol of an empt... |
| lsw0g 14512 | The last symbol of an empt... |
| lsw1 14513 | The last symbol of a word ... |
| lswcl 14514 | Closure of the last symbol... |
| lswlgt0cl 14515 | The last symbol of a nonem... |
| ccatfn 14518 | The concatenation operator... |
| ccatfval 14519 | Value of the concatenation... |
| ccatcl 14520 | The concatenation of two w... |
| ccatlen 14521 | The length of a concatenat... |
| ccat0 14522 | The concatenation of two w... |
| ccatval1 14523 | Value of a symbol in the l... |
| ccatval2 14524 | Value of a symbol in the r... |
| ccatval3 14525 | Value of a symbol in the r... |
| elfzelfzccat 14526 | An element of a finite set... |
| ccatvalfn 14527 | The concatenation of two w... |
| ccatsymb 14528 | The symbol at a given posi... |
| ccatfv0 14529 | The first symbol of a conc... |
| ccatval1lsw 14530 | The last symbol of the lef... |
| ccatval21sw 14531 | The first symbol of the ri... |
| ccatlid 14532 | Concatenation of a word by... |
| ccatrid 14533 | Concatenation of a word by... |
| ccatass 14534 | Associative law for concat... |
| ccatrn 14535 | The range of a concatenate... |
| ccatidid 14536 | Concatenation of the empty... |
| lswccatn0lsw 14537 | The last symbol of a word ... |
| lswccat0lsw 14538 | The last symbol of a word ... |
| ccatalpha 14539 | A concatenation of two arb... |
| ccatrcl1 14540 | Reverse closure of a conca... |
| ids1 14543 | Identity function protecti... |
| s1val 14544 | Value of a singleton word.... |
| s1rn 14545 | The range of a singleton w... |
| s1eq 14546 | Equality theorem for a sin... |
| s1eqd 14547 | Equality theorem for a sin... |
| s1cl 14548 | A singleton word is a word... |
| s1cld 14549 | A singleton word is a word... |
| s1prc 14550 | Value of a singleton word ... |
| s1cli 14551 | A singleton word is a word... |
| s1len 14552 | Length of a singleton word... |
| s1nz 14553 | A singleton word is not th... |
| s1dm 14554 | The domain of a singleton ... |
| s1dmALT 14555 | Alternate version of ~ s1d... |
| s1fv 14556 | Sole symbol of a singleton... |
| lsws1 14557 | The last symbol of a singl... |
| eqs1 14558 | A word of length 1 is a si... |
| wrdl1exs1 14559 | A word of length 1 is a si... |
| wrdl1s1 14560 | A word of length 1 is a si... |
| s111 14561 | The singleton word functio... |
| ccatws1cl 14562 | The concatenation of a wor... |
| ccatws1clv 14563 | The concatenation of a wor... |
| ccat2s1cl 14564 | The concatenation of two s... |
| ccats1alpha 14565 | A concatenation of a word ... |
| ccatws1len 14566 | The length of the concaten... |
| ccatws1lenp1b 14567 | The length of a word is ` ... |
| wrdlenccats1lenm1 14568 | The length of a word is th... |
| ccat2s1len 14569 | The length of the concaten... |
| ccatw2s1cl 14570 | The concatenation of a wor... |
| ccatw2s1len 14571 | The length of the concaten... |
| ccats1val1 14572 | Value of a symbol in the l... |
| ccats1val2 14573 | Value of the symbol concat... |
| ccat1st1st 14574 | The first symbol of a word... |
| ccat2s1p1 14575 | Extract the first of two c... |
| ccat2s1p2 14576 | Extract the second of two ... |
| ccatw2s1ass 14577 | Associative law for a conc... |
| ccatws1n0 14578 | The concatenation of a wor... |
| ccatws1ls 14579 | The last symbol of the con... |
| lswccats1 14580 | The last symbol of a word ... |
| lswccats1fst 14581 | The last symbol of a nonem... |
| ccatw2s1p1 14582 | Extract the symbol of the ... |
| ccatw2s1p2 14583 | Extract the second of two ... |
| ccat2s1fvw 14584 | Extract a symbol of a word... |
| ccat2s1fst 14585 | The first symbol of the co... |
| swrdnznd 14588 | The value of a subword ope... |
| swrdval 14589 | Value of a subword. (Cont... |
| swrd00 14590 | A zero length substring. ... |
| swrdcl 14591 | Closure of the subword ext... |
| swrdval2 14592 | Value of the subword extra... |
| swrdlen 14593 | Length of an extracted sub... |
| swrdfv 14594 | A symbol in an extracted s... |
| swrdfv0 14595 | The first symbol in an ext... |
| swrdf 14596 | A subword of a word is a f... |
| swrdvalfn 14597 | Value of the subword extra... |
| swrdrn 14598 | The range of a subword of ... |
| swrdlend 14599 | The value of the subword e... |
| swrdnd 14600 | The value of the subword e... |
| swrdnd2 14601 | Value of the subword extra... |
| swrdnnn0nd 14602 | The value of a subword ope... |
| swrdnd0 14603 | The value of a subword ope... |
| swrd0 14604 | A subword of an empty set ... |
| swrdrlen 14605 | Length of a right-anchored... |
| swrdlen2 14606 | Length of an extracted sub... |
| swrdfv2 14607 | A symbol in an extracted s... |
| swrdwrdsymb 14608 | A subword is a word over t... |
| swrdsb0eq 14609 | Two subwords with the same... |
| swrdsbslen 14610 | Two subwords with the same... |
| swrdspsleq 14611 | Two words have a common su... |
| swrds1 14612 | Extract a single symbol fr... |
| swrdlsw 14613 | Extract the last single sy... |
| ccatswrd 14614 | Joining two adjacent subwo... |
| swrdccat2 14615 | Recover the right half of ... |
| pfxnndmnd 14618 | The value of a prefix oper... |
| pfxval 14619 | Value of a prefix operatio... |
| pfx00 14620 | The zero length prefix is ... |
| pfx0 14621 | A prefix of an empty set i... |
| pfxval0 14622 | Value of a prefix operatio... |
| pfxcl 14623 | Closure of the prefix extr... |
| pfxmpt 14624 | Value of the prefix extrac... |
| pfxres 14625 | Value of the subword extra... |
| pfxf 14626 | A prefix of a word is a fu... |
| pfxfn 14627 | Value of the prefix extrac... |
| pfxfv 14628 | A symbol in a prefix of a ... |
| pfxlen 14629 | Length of a prefix. (Cont... |
| pfxid 14630 | A word is a prefix of itse... |
| pfxrn 14631 | The range of a prefix of a... |
| pfxn0 14632 | A prefix consisting of at ... |
| pfxnd 14633 | The value of a prefix oper... |
| pfxnd0 14634 | The value of a prefix oper... |
| pfxwrdsymb 14635 | A prefix of a word is a wo... |
| addlenrevpfx 14636 | The sum of the lengths of ... |
| addlenpfx 14637 | The sum of the lengths of ... |
| pfxfv0 14638 | The first symbol of a pref... |
| pfxtrcfv 14639 | A symbol in a word truncat... |
| pfxtrcfv0 14640 | The first symbol in a word... |
| pfxfvlsw 14641 | The last symbol in a nonem... |
| pfxeq 14642 | The prefixes of two words ... |
| pfxtrcfvl 14643 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14644 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14645 | Two (nonempty) words are e... |
| disjwrdpfx 14646 | Sets of words are disjoint... |
| ccatpfx 14647 | Concatenating a prefix wit... |
| pfxccat1 14648 | Recover the left half of a... |
| pfx1 14649 | The prefix of length one o... |
| swrdswrdlem 14650 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14651 | A subword of a subword is ... |
| pfxswrd 14652 | A prefix of a subword is a... |
| swrdpfx 14653 | A subword of a prefix is a... |
| pfxpfx 14654 | A prefix of a prefix is a ... |
| pfxpfxid 14655 | A prefix of a prefix with ... |
| pfxcctswrd 14656 | The concatenation of the p... |
| lenpfxcctswrd 14657 | The length of the concaten... |
| lenrevpfxcctswrd 14658 | The length of the concaten... |
| pfxlswccat 14659 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14660 | The last symbol of a word ... |
| ccats1pfxeqrex 14661 | There exists a symbol such... |
| ccatopth 14662 | An ~ opth -like theorem fo... |
| ccatopth2 14663 | An ~ opth -like theorem fo... |
| ccatlcan 14664 | Concatenation of words is ... |
| ccatrcan 14665 | Concatenation of words is ... |
| wrdeqs1cat 14666 | Decompose a nonempty word ... |
| cats1un 14667 | Express a word with an ext... |
| wrdind 14668 | Perform induction over the... |
| wrd2ind 14669 | Perform induction over the... |
| swrdccatfn 14670 | The subword of a concatena... |
| swrdccatin1 14671 | The subword of a concatena... |
| pfxccatin12lem4 14672 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14673 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14674 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14675 | The subword of a concatena... |
| pfxccatin12lem2c 14676 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14677 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14678 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14679 | The subword of a concatena... |
| pfxccat3 14680 | The subword of a concatena... |
| swrdccat 14681 | The subword of a concatena... |
| pfxccatpfx1 14682 | A prefix of a concatenatio... |
| pfxccatpfx2 14683 | A prefix of a concatenatio... |
| pfxccat3a 14684 | A prefix of a concatenatio... |
| swrdccat3blem 14685 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14686 | A suffix of a concatenatio... |
| pfxccatid 14687 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14688 | A word is a prefix of a wo... |
| swrdccatin1d 14689 | The subword of a concatena... |
| swrdccatin2d 14690 | The subword of a concatena... |
| pfxccatin12d 14691 | The subword of a concatena... |
| reuccatpfxs1lem 14692 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14693 | There is a unique word hav... |
| reuccatpfxs1v 14694 | There is a unique word hav... |
| splval 14697 | Value of the substring rep... |
| splcl 14698 | Closure of the substring r... |
| splid 14699 | Splicing a subword for the... |
| spllen 14700 | The length of a splice. (... |
| splfv1 14701 | Symbols to the left of a s... |
| splfv2a 14702 | Symbols within the replace... |
| splval2 14703 | Value of a splice, assumin... |
| revval 14706 | Value of the word reversin... |
| revcl 14707 | The reverse of a word is a... |
| revlen 14708 | The reverse of a word has ... |
| revfv 14709 | Reverse of a word at a poi... |
| rev0 14710 | The empty word is its own ... |
| revs1 14711 | Singleton words are their ... |
| revccat 14712 | Antiautomorphic property o... |
| revrev 14713 | Reversal is an involution ... |
| reps 14716 | Construct a function mappi... |
| repsundef 14717 | A function mapping a half-... |
| repsconst 14718 | Construct a function mappi... |
| repsf 14719 | The constructed function m... |
| repswsymb 14720 | The symbols of a "repeated... |
| repsw 14721 | A function mapping a half-... |
| repswlen 14722 | The length of a "repeated ... |
| repsw0 14723 | The "repeated symbol word"... |
| repsdf2 14724 | Alternative definition of ... |
| repswsymball 14725 | All the symbols of a "repe... |
| repswsymballbi 14726 | A word is a "repeated symb... |
| repswfsts 14727 | The first symbol of a none... |
| repswlsw 14728 | The last symbol of a nonem... |
| repsw1 14729 | The "repeated symbol word"... |
| repswswrd 14730 | A subword of a "repeated s... |
| repswpfx 14731 | A prefix of a repeated sym... |
| repswccat 14732 | The concatenation of two "... |
| repswrevw 14733 | The reverse of a "repeated... |
| cshfn 14736 | Perform a cyclical shift f... |
| cshword 14737 | Perform a cyclical shift f... |
| cshnz 14738 | A cyclical shift is the em... |
| 0csh0 14739 | Cyclically shifting an emp... |
| cshw0 14740 | A word cyclically shifted ... |
| cshwmodn 14741 | Cyclically shifting a word... |
| cshwsublen 14742 | Cyclically shifting a word... |
| cshwn 14743 | A word cyclically shifted ... |
| cshwcl 14744 | A cyclically shifted word ... |
| cshwlen 14745 | The length of a cyclically... |
| cshwf 14746 | A cyclically shifted word ... |
| cshwfn 14747 | A cyclically shifted word ... |
| cshwrn 14748 | The range of a cyclically ... |
| cshwidxmod 14749 | The symbol at a given inde... |
| cshwidxmodr 14750 | The symbol at a given inde... |
| cshwidx0mod 14751 | The symbol at index 0 of a... |
| cshwidx0 14752 | The symbol at index 0 of a... |
| cshwidxm1 14753 | The symbol at index ((n-N)... |
| cshwidxm 14754 | The symbol at index (n-N) ... |
| cshwidxn 14755 | The symbol at index (n-1) ... |
| cshf1 14756 | Cyclically shifting a word... |
| cshinj 14757 | If a word is injectiv (reg... |
| repswcshw 14758 | A cyclically shifted "repe... |
| 2cshw 14759 | Cyclically shifting a word... |
| 2cshwid 14760 | Cyclically shifting a word... |
| lswcshw 14761 | The last symbol of a word ... |
| 2cshwcom 14762 | Cyclically shifting a word... |
| cshwleneq 14763 | If the results of cyclical... |
| 3cshw 14764 | Cyclically shifting a word... |
| cshweqdif2 14765 | If cyclically shifting two... |
| cshweqdifid 14766 | If cyclically shifting a w... |
| cshweqrep 14767 | If cyclically shifting a w... |
| cshw1 14768 | If cyclically shifting a w... |
| cshw1repsw 14769 | If cyclically shifting a w... |
| cshwsexa 14770 | The class of (different!) ... |
| cshwsexaOLD 14771 | Obsolete version of ~ cshw... |
| 2cshwcshw 14772 | If a word is a cyclically ... |
| scshwfzeqfzo 14773 | For a nonempty word the se... |
| cshwcshid 14774 | A cyclically shifted word ... |
| cshwcsh2id 14775 | A cyclically shifted word ... |
| cshimadifsn 14776 | The image of a cyclically ... |
| cshimadifsn0 14777 | The image of a cyclically ... |
| wrdco 14778 | Mapping a word by a functi... |
| lenco 14779 | Length of a mapped word is... |
| s1co 14780 | Mapping of a singleton wor... |
| revco 14781 | Mapping of words (i.e., a ... |
| ccatco 14782 | Mapping of words commutes ... |
| cshco 14783 | Mapping of words commutes ... |
| swrdco 14784 | Mapping of words commutes ... |
| pfxco 14785 | Mapping of words commutes ... |
| lswco 14786 | Mapping of (nonempty) word... |
| repsco 14787 | Mapping of words commutes ... |
| cats1cld 14802 | Closure of concatenation w... |
| cats1co 14803 | Closure of concatenation w... |
| cats1cli 14804 | Closure of concatenation w... |
| cats1fvn 14805 | The last symbol of a conca... |
| cats1fv 14806 | A symbol other than the la... |
| cats1len 14807 | The length of concatenatio... |
| cats1cat 14808 | Closure of concatenation w... |
| cats2cat 14809 | Closure of concatenation o... |
| s2eqd 14810 | Equality theorem for a dou... |
| s3eqd 14811 | Equality theorem for a len... |
| s4eqd 14812 | Equality theorem for a len... |
| s5eqd 14813 | Equality theorem for a len... |
| s6eqd 14814 | Equality theorem for a len... |
| s7eqd 14815 | Equality theorem for a len... |
| s8eqd 14816 | Equality theorem for a len... |
| s3eq2 14817 | Equality theorem for a len... |
| s2cld 14818 | A doubleton word is a word... |
| s3cld 14819 | A length 3 string is a wor... |
| s4cld 14820 | A length 4 string is a wor... |
| s5cld 14821 | A length 5 string is a wor... |
| s6cld 14822 | A length 6 string is a wor... |
| s7cld 14823 | A length 7 string is a wor... |
| s8cld 14824 | A length 7 string is a wor... |
| s2cl 14825 | A doubleton word is a word... |
| s3cl 14826 | A length 3 string is a wor... |
| s2cli 14827 | A doubleton word is a word... |
| s3cli 14828 | A length 3 string is a wor... |
| s4cli 14829 | A length 4 string is a wor... |
| s5cli 14830 | A length 5 string is a wor... |
| s6cli 14831 | A length 6 string is a wor... |
| s7cli 14832 | A length 7 string is a wor... |
| s8cli 14833 | A length 8 string is a wor... |
| s2fv0 14834 | Extract the first symbol f... |
| s2fv1 14835 | Extract the second symbol ... |
| s2len 14836 | The length of a doubleton ... |
| s2dm 14837 | The domain of a doubleton ... |
| s3fv0 14838 | Extract the first symbol f... |
| s3fv1 14839 | Extract the second symbol ... |
| s3fv2 14840 | Extract the third symbol f... |
| s3len 14841 | The length of a length 3 s... |
| s4fv0 14842 | Extract the first symbol f... |
| s4fv1 14843 | Extract the second symbol ... |
| s4fv2 14844 | Extract the third symbol f... |
| s4fv3 14845 | Extract the fourth symbol ... |
| s4len 14846 | The length of a length 4 s... |
| s5len 14847 | The length of a length 5 s... |
| s6len 14848 | The length of a length 6 s... |
| s7len 14849 | The length of a length 7 s... |
| s8len 14850 | The length of a length 8 s... |
| lsws2 14851 | The last symbol of a doubl... |
| lsws3 14852 | The last symbol of a 3 let... |
| lsws4 14853 | The last symbol of a 4 let... |
| s2prop 14854 | A length 2 word is an unor... |
| s2dmALT 14855 | Alternate version of ~ s2d... |
| s3tpop 14856 | A length 3 word is an unor... |
| s4prop 14857 | A length 4 word is a union... |
| s3fn 14858 | A length 3 word is a funct... |
| funcnvs1 14859 | The converse of a singleto... |
| funcnvs2 14860 | The converse of a length 2... |
| funcnvs3 14861 | The converse of a length 3... |
| funcnvs4 14862 | The converse of a length 4... |
| s2f1o 14863 | A length 2 word with mutua... |
| f1oun2prg 14864 | A union of unordered pairs... |
| s4f1o 14865 | A length 4 word with mutua... |
| s4dom 14866 | The domain of a length 4 w... |
| s2co 14867 | Mapping a doubleton word b... |
| s3co 14868 | Mapping a length 3 string ... |
| s0s1 14869 | Concatenation of fixed len... |
| s1s2 14870 | Concatenation of fixed len... |
| s1s3 14871 | Concatenation of fixed len... |
| s1s4 14872 | Concatenation of fixed len... |
| s1s5 14873 | Concatenation of fixed len... |
| s1s6 14874 | Concatenation of fixed len... |
| s1s7 14875 | Concatenation of fixed len... |
| s2s2 14876 | Concatenation of fixed len... |
| s4s2 14877 | Concatenation of fixed len... |
| s4s3 14878 | Concatenation of fixed len... |
| s4s4 14879 | Concatenation of fixed len... |
| s3s4 14880 | Concatenation of fixed len... |
| s2s5 14881 | Concatenation of fixed len... |
| s5s2 14882 | Concatenation of fixed len... |
| s2eq2s1eq 14883 | Two length 2 words are equ... |
| s2eq2seq 14884 | Two length 2 words are equ... |
| s3eqs2s1eq 14885 | Two length 3 words are equ... |
| s3eq3seq 14886 | Two length 3 words are equ... |
| swrds2 14887 | Extract two adjacent symbo... |
| swrds2m 14888 | Extract two adjacent symbo... |
| wrdlen2i 14889 | Implications of a word of ... |
| wrd2pr2op 14890 | A word of length two repre... |
| wrdlen2 14891 | A word of length two. (Co... |
| wrdlen2s2 14892 | A word of length two as do... |
| wrdl2exs2 14893 | A word of length two is a ... |
| pfx2 14894 | A prefix of length two. (... |
| wrd3tpop 14895 | A word of length three rep... |
| wrdlen3s3 14896 | A word of length three as ... |
| repsw2 14897 | The "repeated symbol word"... |
| repsw3 14898 | The "repeated symbol word"... |
| swrd2lsw 14899 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14900 | Two words of length at lea... |
| ccatw2s1ccatws2 14901 | The concatenation of a wor... |
| ccat2s1fvwALT 14902 | Alternate proof of ~ ccat2... |
| wwlktovf 14903 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14904 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14905 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14906 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14907 | There is a bijection betwe... |
| eqwrds3 14908 | A word is equal with a len... |
| wrdl3s3 14909 | A word of length 3 is a le... |
| s3sndisj 14910 | The singletons consisting ... |
| s3iunsndisj 14911 | The union of singletons co... |
| ofccat 14912 | Letterwise operations on w... |
| ofs1 14913 | Letterwise operations on a... |
| ofs2 14914 | Letterwise operations on a... |
| coss12d 14915 | Subset deduction for compo... |
| trrelssd 14916 | The composition of subclas... |
| xpcogend 14917 | The most interesting case ... |
| xpcoidgend 14918 | If two classes are not dis... |
| cotr2g 14919 | Two ways of saying that th... |
| cotr2 14920 | Two ways of saying a relat... |
| cotr3 14921 | Two ways of saying a relat... |
| coemptyd 14922 | Deduction about compositio... |
| xptrrel 14923 | The cross product is alway... |
| 0trrel 14924 | The empty class is a trans... |
| cleq1lem 14925 | Equality implies bijection... |
| cleq1 14926 | Equality of relations impl... |
| clsslem 14927 | The closure of a subclass ... |
| trcleq1 14932 | Equality of relations impl... |
| trclsslem 14933 | The transitive closure (as... |
| trcleq2lem 14934 | Equality implies bijection... |
| cvbtrcl 14935 | Change of bound variable i... |
| trcleq12lem 14936 | Equality implies bijection... |
| trclexlem 14937 | Existence of relation impl... |
| trclublem 14938 | If a relation exists then ... |
| trclubi 14939 | The Cartesian product of t... |
| trclubgi 14940 | The union with the Cartesi... |
| trclub 14941 | The Cartesian product of t... |
| trclubg 14942 | The union with the Cartesi... |
| trclfv 14943 | The transitive closure of ... |
| brintclab 14944 | Two ways to express a bina... |
| brtrclfv 14945 | Two ways of expressing the... |
| brcnvtrclfv 14946 | Two ways of expressing the... |
| brtrclfvcnv 14947 | Two ways of expressing the... |
| brcnvtrclfvcnv 14948 | Two ways of expressing the... |
| trclfvss 14949 | The transitive closure (as... |
| trclfvub 14950 | The transitive closure of ... |
| trclfvlb 14951 | The transitive closure of ... |
| trclfvcotr 14952 | The transitive closure of ... |
| trclfvlb2 14953 | The transitive closure of ... |
| trclfvlb3 14954 | The transitive closure of ... |
| cotrtrclfv 14955 | The transitive closure of ... |
| trclidm 14956 | The transitive closure of ... |
| trclun 14957 | Transitive closure of a un... |
| trclfvg 14958 | The value of the transitiv... |
| trclfvcotrg 14959 | The value of the transitiv... |
| reltrclfv 14960 | The transitive closure of ... |
| dmtrclfv 14961 | The domain of the transiti... |
| reldmrelexp 14964 | The domain of the repeated... |
| relexp0g 14965 | A relation composed zero t... |
| relexp0 14966 | A relation composed zero t... |
| relexp0d 14967 | A relation composed zero t... |
| relexpsucnnr 14968 | A reduction for relation e... |
| relexp1g 14969 | A relation composed once i... |
| dfid5 14970 | Identity relation is equal... |
| dfid6 14971 | Identity relation expresse... |
| relexp1d 14972 | A relation composed once i... |
| relexpsucnnl 14973 | A reduction for relation e... |
| relexpsucl 14974 | A reduction for relation e... |
| relexpsucr 14975 | A reduction for relation e... |
| relexpsucrd 14976 | A reduction for relation e... |
| relexpsucld 14977 | A reduction for relation e... |
| relexpcnv 14978 | Commutation of converse an... |
| relexpcnvd 14979 | Commutation of converse an... |
| relexp0rel 14980 | The exponentiation of a cl... |
| relexprelg 14981 | The exponentiation of a cl... |
| relexprel 14982 | The exponentiation of a re... |
| relexpreld 14983 | The exponentiation of a re... |
| relexpnndm 14984 | The domain of an exponenti... |
| relexpdmg 14985 | The domain of an exponenti... |
| relexpdm 14986 | The domain of an exponenti... |
| relexpdmd 14987 | The domain of an exponenti... |
| relexpnnrn 14988 | The range of an exponentia... |
| relexprng 14989 | The range of an exponentia... |
| relexprn 14990 | The range of an exponentia... |
| relexprnd 14991 | The range of an exponentia... |
| relexpfld 14992 | The field of an exponentia... |
| relexpfldd 14993 | The field of an exponentia... |
| relexpaddnn 14994 | Relation composition becom... |
| relexpuzrel 14995 | The exponentiation of a cl... |
| relexpaddg 14996 | Relation composition becom... |
| relexpaddd 14997 | Relation composition becom... |
| rtrclreclem1 15000 | The reflexive, transitive ... |
| dfrtrclrec2 15001 | If two elements are connec... |
| rtrclreclem2 15002 | The reflexive, transitive ... |
| rtrclreclem3 15003 | The reflexive, transitive ... |
| rtrclreclem4 15004 | The reflexive, transitive ... |
| dfrtrcl2 15005 | The two definitions ` t* `... |
| relexpindlem 15006 | Principle of transitive in... |
| relexpind 15007 | Principle of transitive in... |
| rtrclind 15008 | Principle of transitive in... |
| shftlem 15011 | Two ways to write a shifte... |
| shftuz 15012 | A shift of the upper integ... |
| shftfval 15013 | The value of the sequence ... |
| shftdm 15014 | Domain of a relation shift... |
| shftfib 15015 | Value of a fiber of the re... |
| shftfn 15016 | Functionality and domain o... |
| shftval 15017 | Value of a sequence shifte... |
| shftval2 15018 | Value of a sequence shifte... |
| shftval3 15019 | Value of a sequence shifte... |
| shftval4 15020 | Value of a sequence shifte... |
| shftval5 15021 | Value of a shifted sequenc... |
| shftf 15022 | Functionality of a shifted... |
| 2shfti 15023 | Composite shift operations... |
| shftidt2 15024 | Identity law for the shift... |
| shftidt 15025 | Identity law for the shift... |
| shftcan1 15026 | Cancellation law for the s... |
| shftcan2 15027 | Cancellation law for the s... |
| seqshft 15028 | Shifting the index set of ... |
| sgnval 15031 | Value of the signum functi... |
| sgn0 15032 | The signum of 0 is 0. (Co... |
| sgnp 15033 | The signum of a positive e... |
| sgnrrp 15034 | The signum of a positive r... |
| sgn1 15035 | The signum of 1 is 1. (Co... |
| sgnpnf 15036 | The signum of ` +oo ` is 1... |
| sgnn 15037 | The signum of a negative e... |
| sgnmnf 15038 | The signum of ` -oo ` is -... |
| cjval 15045 | The value of the conjugate... |
| cjth 15046 | The defining property of t... |
| cjf 15047 | Domain and codomain of the... |
| cjcl 15048 | The conjugate of a complex... |
| reval 15049 | The value of the real part... |
| imval 15050 | The value of the imaginary... |
| imre 15051 | The imaginary part of a co... |
| reim 15052 | The real part of a complex... |
| recl 15053 | The real part of a complex... |
| imcl 15054 | The imaginary part of a co... |
| ref 15055 | Domain and codomain of the... |
| imf 15056 | Domain and codomain of the... |
| crre 15057 | The real part of a complex... |
| crim 15058 | The real part of a complex... |
| replim 15059 | Reconstruct a complex numb... |
| remim 15060 | Value of the conjugate of ... |
| reim0 15061 | The imaginary part of a re... |
| reim0b 15062 | A number is real iff its i... |
| rereb 15063 | A number is real iff it eq... |
| mulre 15064 | A product with a nonzero r... |
| rere 15065 | A real number equals its r... |
| cjreb 15066 | A number is real iff it eq... |
| recj 15067 | Real part of a complex con... |
| reneg 15068 | Real part of negative. (C... |
| readd 15069 | Real part distributes over... |
| resub 15070 | Real part distributes over... |
| remullem 15071 | Lemma for ~ remul , ~ immu... |
| remul 15072 | Real part of a product. (... |
| remul2 15073 | Real part of a product. (... |
| rediv 15074 | Real part of a division. ... |
| imcj 15075 | Imaginary part of a comple... |
| imneg 15076 | The imaginary part of a ne... |
| imadd 15077 | Imaginary part distributes... |
| imsub 15078 | Imaginary part distributes... |
| immul 15079 | Imaginary part of a produc... |
| immul2 15080 | Imaginary part of a produc... |
| imdiv 15081 | Imaginary part of a divisi... |
| cjre 15082 | A real number equals its c... |
| cjcj 15083 | The conjugate of the conju... |
| cjadd 15084 | Complex conjugate distribu... |
| cjmul 15085 | Complex conjugate distribu... |
| ipcnval 15086 | Standard inner product on ... |
| cjmulrcl 15087 | A complex number times its... |
| cjmulval 15088 | A complex number times its... |
| cjmulge0 15089 | A complex number times its... |
| cjneg 15090 | Complex conjugate of negat... |
| addcj 15091 | A number plus its conjugat... |
| cjsub 15092 | Complex conjugate distribu... |
| cjexp 15093 | Complex conjugate of posit... |
| imval2 15094 | The imaginary part of a nu... |
| re0 15095 | The real part of zero. (C... |
| im0 15096 | The imaginary part of zero... |
| re1 15097 | The real part of one. (Co... |
| im1 15098 | The imaginary part of one.... |
| rei 15099 | The real part of ` _i ` . ... |
| imi 15100 | The imaginary part of ` _i... |
| cj0 15101 | The conjugate of zero. (C... |
| cji 15102 | The complex conjugate of t... |
| cjreim 15103 | The conjugate of a represe... |
| cjreim2 15104 | The conjugate of the repre... |
| cj11 15105 | Complex conjugate is a one... |
| cjne0 15106 | A number is nonzero iff it... |
| cjdiv 15107 | Complex conjugate distribu... |
| cnrecnv 15108 | The inverse to the canonic... |
| sqeqd 15109 | A deduction for showing tw... |
| recli 15110 | The real part of a complex... |
| imcli 15111 | The imaginary part of a co... |
| cjcli 15112 | Closure law for complex co... |
| replimi 15113 | Construct a complex number... |
| cjcji 15114 | The conjugate of the conju... |
| reim0bi 15115 | A number is real iff its i... |
| rerebi 15116 | A real number equals its r... |
| cjrebi 15117 | A number is real iff it eq... |
| recji 15118 | Real part of a complex con... |
| imcji 15119 | Imaginary part of a comple... |
| cjmulrcli 15120 | A complex number times its... |
| cjmulvali 15121 | A complex number times its... |
| cjmulge0i 15122 | A complex number times its... |
| renegi 15123 | Real part of negative. (C... |
| imnegi 15124 | Imaginary part of negative... |
| cjnegi 15125 | Complex conjugate of negat... |
| addcji 15126 | A number plus its conjugat... |
| readdi 15127 | Real part distributes over... |
| imaddi 15128 | Imaginary part distributes... |
| remuli 15129 | Real part of a product. (... |
| immuli 15130 | Imaginary part of a produc... |
| cjaddi 15131 | Complex conjugate distribu... |
| cjmuli 15132 | Complex conjugate distribu... |
| ipcni 15133 | Standard inner product on ... |
| cjdivi 15134 | Complex conjugate distribu... |
| crrei 15135 | The real part of a complex... |
| crimi 15136 | The imaginary part of a co... |
| recld 15137 | The real part of a complex... |
| imcld 15138 | The imaginary part of a co... |
| cjcld 15139 | Closure law for complex co... |
| replimd 15140 | Construct a complex number... |
| remimd 15141 | Value of the conjugate of ... |
| cjcjd 15142 | The conjugate of the conju... |
| reim0bd 15143 | A number is real iff its i... |
| rerebd 15144 | A real number equals its r... |
| cjrebd 15145 | A number is real iff it eq... |
| cjne0d 15146 | A number is nonzero iff it... |
| recjd 15147 | Real part of a complex con... |
| imcjd 15148 | Imaginary part of a comple... |
| cjmulrcld 15149 | A complex number times its... |
| cjmulvald 15150 | A complex number times its... |
| cjmulge0d 15151 | A complex number times its... |
| renegd 15152 | Real part of negative. (C... |
| imnegd 15153 | Imaginary part of negative... |
| cjnegd 15154 | Complex conjugate of negat... |
| addcjd 15155 | A number plus its conjugat... |
| cjexpd 15156 | Complex conjugate of posit... |
| readdd 15157 | Real part distributes over... |
| imaddd 15158 | Imaginary part distributes... |
| resubd 15159 | Real part distributes over... |
| imsubd 15160 | Imaginary part distributes... |
| remuld 15161 | Real part of a product. (... |
| immuld 15162 | Imaginary part of a produc... |
| cjaddd 15163 | Complex conjugate distribu... |
| cjmuld 15164 | Complex conjugate distribu... |
| ipcnd 15165 | Standard inner product on ... |
| cjdivd 15166 | Complex conjugate distribu... |
| rered 15167 | A real number equals its r... |
| reim0d 15168 | The imaginary part of a re... |
| cjred 15169 | A real number equals its c... |
| remul2d 15170 | Real part of a product. (... |
| immul2d 15171 | Imaginary part of a produc... |
| redivd 15172 | Real part of a division. ... |
| imdivd 15173 | Imaginary part of a divisi... |
| crred 15174 | The real part of a complex... |
| crimd 15175 | The imaginary part of a co... |
| sqrtval 15180 | Value of square root funct... |
| absval 15181 | The absolute value (modulu... |
| rennim 15182 | A real number does not lie... |
| cnpart 15183 | The specification of restr... |
| sqrt0 15184 | The square root of zero is... |
| 01sqrexlem1 15185 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15186 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15187 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15188 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15189 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15190 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15191 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15192 | Existence of a square root... |
| resqrex 15193 | Existence of a square root... |
| sqrmo 15194 | Uniqueness for the square ... |
| resqreu 15195 | Existence and uniqueness f... |
| resqrtcl 15196 | Closure of the square root... |
| resqrtthlem 15197 | Lemma for ~ resqrtth . (C... |
| resqrtth 15198 | Square root theorem over t... |
| remsqsqrt 15199 | Square of square root. (C... |
| sqrtge0 15200 | The square root function i... |
| sqrtgt0 15201 | The square root function i... |
| sqrtmul 15202 | Square root distributes ov... |
| sqrtle 15203 | Square root is monotonic. ... |
| sqrtlt 15204 | Square root is strictly mo... |
| sqrt11 15205 | The square root function i... |
| sqrt00 15206 | A square root is zero iff ... |
| rpsqrtcl 15207 | The square root of a posit... |
| sqrtdiv 15208 | Square root distributes ov... |
| sqrtneglem 15209 | The square root of a negat... |
| sqrtneg 15210 | The square root of a negat... |
| sqrtsq2 15211 | Relationship between squar... |
| sqrtsq 15212 | Square root of square. (C... |
| sqrtmsq 15213 | Square root of square. (C... |
| sqrt1 15214 | The square root of 1 is 1.... |
| sqrt4 15215 | The square root of 4 is 2.... |
| sqrt9 15216 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15217 | The square root of 2 is bo... |
| sqrtm1 15218 | The imaginary unit is the ... |
| nn0sqeq1 15219 | A natural number with squa... |
| absneg 15220 | Absolute value of the oppo... |
| abscl 15221 | Real closure of absolute v... |
| abscj 15222 | The absolute value of a nu... |
| absvalsq 15223 | Square of value of absolut... |
| absvalsq2 15224 | Square of value of absolut... |
| sqabsadd 15225 | Square of absolute value o... |
| sqabssub 15226 | Square of absolute value o... |
| absval2 15227 | Value of absolute value fu... |
| abs0 15228 | The absolute value of 0. ... |
| absi 15229 | The absolute value of the ... |
| absge0 15230 | Absolute value is nonnegat... |
| absrpcl 15231 | The absolute value of a no... |
| abs00 15232 | The absolute value of a nu... |
| abs00ad 15233 | A complex number is zero i... |
| abs00bd 15234 | If a complex number is zer... |
| absreimsq 15235 | Square of the absolute val... |
| absreim 15236 | Absolute value of a number... |
| absmul 15237 | Absolute value distributes... |
| absdiv 15238 | Absolute value distributes... |
| absid 15239 | A nonnegative number is it... |
| abs1 15240 | The absolute value of one ... |
| absnid 15241 | A negative number is the n... |
| leabs 15242 | A real number is less than... |
| absor 15243 | The absolute value of a re... |
| absre 15244 | Absolute value of a real n... |
| absresq 15245 | Square of the absolute val... |
| absmod0 15246 | ` A ` is divisible by ` B ... |
| absexp 15247 | Absolute value of positive... |
| absexpz 15248 | Absolute value of integer ... |
| abssq 15249 | Square can be moved in and... |
| sqabs 15250 | The squares of two reals a... |
| absrele 15251 | The absolute value of a co... |
| absimle 15252 | The absolute value of a co... |
| max0add 15253 | The sum of the positive an... |
| absz 15254 | A real number is an intege... |
| nn0abscl 15255 | The absolute value of an i... |
| zabscl 15256 | The absolute value of an i... |
| abslt 15257 | Absolute value and 'less t... |
| absle 15258 | Absolute value and 'less t... |
| abssubne0 15259 | If the absolute value of a... |
| absdiflt 15260 | The absolute value of a di... |
| absdifle 15261 | The absolute value of a di... |
| elicc4abs 15262 | Membership in a symmetric ... |
| lenegsq 15263 | Comparison to a nonnegativ... |
| releabs 15264 | The real part of a number ... |
| recval 15265 | Reciprocal expressed with ... |
| absidm 15266 | The absolute value functio... |
| absgt0 15267 | The absolute value of a no... |
| nnabscl 15268 | The absolute value of a no... |
| abssub 15269 | Swapping order of subtract... |
| abssubge0 15270 | Absolute value of a nonneg... |
| abssuble0 15271 | Absolute value of a nonpos... |
| absmax 15272 | The maximum of two numbers... |
| abstri 15273 | Triangle inequality for ab... |
| abs3dif 15274 | Absolute value of differen... |
| abs2dif 15275 | Difference of absolute val... |
| abs2dif2 15276 | Difference of absolute val... |
| abs2difabs 15277 | Absolute value of differen... |
| abs1m 15278 | For any complex number, th... |
| recan 15279 | Cancellation law involving... |
| absf 15280 | Mapping domain and codomai... |
| abs3lem 15281 | Lemma involving absolute v... |
| abslem2 15282 | Lemma involving absolute v... |
| rddif 15283 | The difference between a r... |
| absrdbnd 15284 | Bound on the absolute valu... |
| fzomaxdiflem 15285 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15286 | A bound on the separation ... |
| uzin2 15287 | The upper integers are clo... |
| rexanuz 15288 | Combine two different uppe... |
| rexanre 15289 | Combine two different uppe... |
| rexfiuz 15290 | Combine finitely many diff... |
| rexuz3 15291 | Restrict the base of the u... |
| rexanuz2 15292 | Combine two different uppe... |
| r19.29uz 15293 | A version of ~ 19.29 for u... |
| r19.2uz 15294 | A version of ~ r19.2z for ... |
| rexuzre 15295 | Convert an upper real quan... |
| rexico 15296 | Restrict the base of an up... |
| cau3lem 15297 | Lemma for ~ cau3 . (Contr... |
| cau3 15298 | Convert between three-quan... |
| cau4 15299 | Change the base of a Cauch... |
| caubnd2 15300 | A Cauchy sequence of compl... |
| caubnd 15301 | A Cauchy sequence of compl... |
| sqreulem 15302 | Lemma for ~ sqreu : write ... |
| sqreu 15303 | Existence and uniqueness f... |
| sqrtcl 15304 | Closure of the square root... |
| sqrtthlem 15305 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15306 | Mapping domain and codomai... |
| sqrtth 15307 | Square root theorem over t... |
| sqrtrege0 15308 | The square root function m... |
| eqsqrtor 15309 | Solve an equation containi... |
| eqsqrtd 15310 | A deduction for showing th... |
| eqsqrt2d 15311 | A deduction for showing th... |
| amgm2 15312 | Arithmetic-geometric mean ... |
| sqrtthi 15313 | Square root theorem. Theo... |
| sqrtcli 15314 | The square root of a nonne... |
| sqrtgt0i 15315 | The square root of a posit... |
| sqrtmsqi 15316 | Square root of square. (C... |
| sqrtsqi 15317 | Square root of square. (C... |
| sqsqrti 15318 | Square of square root. (C... |
| sqrtge0i 15319 | The square root of a nonne... |
| absidi 15320 | A nonnegative number is it... |
| absnidi 15321 | A negative number is the n... |
| leabsi 15322 | A real number is less than... |
| absori 15323 | The absolute value of a re... |
| absrei 15324 | Absolute value of a real n... |
| sqrtpclii 15325 | The square root of a posit... |
| sqrtgt0ii 15326 | The square root of a posit... |
| sqrt11i 15327 | The square root function i... |
| sqrtmuli 15328 | Square root distributes ov... |
| sqrtmulii 15329 | Square root distributes ov... |
| sqrtmsq2i 15330 | Relationship between squar... |
| sqrtlei 15331 | Square root is monotonic. ... |
| sqrtlti 15332 | Square root is strictly mo... |
| abslti 15333 | Absolute value and 'less t... |
| abslei 15334 | Absolute value and 'less t... |
| cnsqrt00 15335 | A square root of a complex... |
| absvalsqi 15336 | Square of value of absolut... |
| absvalsq2i 15337 | Square of value of absolut... |
| abscli 15338 | Real closure of absolute v... |
| absge0i 15339 | Absolute value is nonnegat... |
| absval2i 15340 | Value of absolute value fu... |
| abs00i 15341 | The absolute value of a nu... |
| absgt0i 15342 | The absolute value of a no... |
| absnegi 15343 | Absolute value of negative... |
| abscji 15344 | The absolute value of a nu... |
| releabsi 15345 | The real part of a number ... |
| abssubi 15346 | Swapping order of subtract... |
| absmuli 15347 | Absolute value distributes... |
| sqabsaddi 15348 | Square of absolute value o... |
| sqabssubi 15349 | Square of absolute value o... |
| absdivzi 15350 | Absolute value distributes... |
| abstrii 15351 | Triangle inequality for ab... |
| abs3difi 15352 | Absolute value of differen... |
| abs3lemi 15353 | Lemma involving absolute v... |
| rpsqrtcld 15354 | The square root of a posit... |
| sqrtgt0d 15355 | The square root of a posit... |
| absnidd 15356 | A negative number is the n... |
| leabsd 15357 | A real number is less than... |
| absord 15358 | The absolute value of a re... |
| absred 15359 | Absolute value of a real n... |
| resqrtcld 15360 | The square root of a nonne... |
| sqrtmsqd 15361 | Square root of square. (C... |
| sqrtsqd 15362 | Square root of square. (C... |
| sqrtge0d 15363 | The square root of a nonne... |
| sqrtnegd 15364 | The square root of a negat... |
| absidd 15365 | A nonnegative number is it... |
| sqrtdivd 15366 | Square root distributes ov... |
| sqrtmuld 15367 | Square root distributes ov... |
| sqrtsq2d 15368 | Relationship between squar... |
| sqrtled 15369 | Square root is monotonic. ... |
| sqrtltd 15370 | Square root is strictly mo... |
| sqr11d 15371 | The square root function i... |
| absltd 15372 | Absolute value and 'less t... |
| absled 15373 | Absolute value and 'less t... |
| abssubge0d 15374 | Absolute value of a nonneg... |
| abssuble0d 15375 | Absolute value of a nonpos... |
| absdifltd 15376 | The absolute value of a di... |
| absdifled 15377 | The absolute value of a di... |
| icodiamlt 15378 | Two elements in a half-ope... |
| abscld 15379 | Real closure of absolute v... |
| sqrtcld 15380 | Closure of the square root... |
| sqrtrege0d 15381 | The real part of the squar... |
| sqsqrtd 15382 | Square root theorem. Theo... |
| msqsqrtd 15383 | Square root theorem. Theo... |
| sqr00d 15384 | A square root is zero iff ... |
| absvalsqd 15385 | Square of value of absolut... |
| absvalsq2d 15386 | Square of value of absolut... |
| absge0d 15387 | Absolute value is nonnegat... |
| absval2d 15388 | Value of absolute value fu... |
| abs00d 15389 | The absolute value of a nu... |
| absne0d 15390 | The absolute value of a nu... |
| absrpcld 15391 | The absolute value of a no... |
| absnegd 15392 | Absolute value of negative... |
| abscjd 15393 | The absolute value of a nu... |
| releabsd 15394 | The real part of a number ... |
| absexpd 15395 | Absolute value of positive... |
| abssubd 15396 | Swapping order of subtract... |
| absmuld 15397 | Absolute value distributes... |
| absdivd 15398 | Absolute value distributes... |
| abstrid 15399 | Triangle inequality for ab... |
| abs2difd 15400 | Difference of absolute val... |
| abs2dif2d 15401 | Difference of absolute val... |
| abs2difabsd 15402 | Absolute value of differen... |
| abs3difd 15403 | Absolute value of differen... |
| abs3lemd 15404 | Lemma involving absolute v... |
| reusq0 15405 | A complex number is the sq... |
| bhmafibid1cn 15406 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15407 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15408 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15409 | The Brahmagupta-Fibonacci ... |
| limsupgord 15412 | Ordering property of the s... |
| limsupcl 15413 | Closure of the superior li... |
| limsupval 15414 | The superior limit of an i... |
| limsupgf 15415 | Closure of the superior li... |
| limsupgval 15416 | Value of the superior limi... |
| limsupgle 15417 | The defining property of t... |
| limsuple 15418 | The defining property of t... |
| limsuplt 15419 | The defining property of t... |
| limsupval2 15420 | The superior limit, relati... |
| limsupgre 15421 | If a sequence of real numb... |
| limsupbnd1 15422 | If a sequence is eventuall... |
| limsupbnd2 15423 | If a sequence is eventuall... |
| climrel 15432 | The limit relation is a re... |
| rlimrel 15433 | The limit relation is a re... |
| clim 15434 | Express the predicate: Th... |
| rlim 15435 | Express the predicate: Th... |
| rlim2 15436 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15437 | Use strictly less-than in ... |
| rlim3 15438 | Restrict the range of the ... |
| climcl 15439 | Closure of the limit of a ... |
| rlimpm 15440 | Closure of a function with... |
| rlimf 15441 | Closure of a function with... |
| rlimss 15442 | Domain closure of a functi... |
| rlimcl 15443 | Closure of the limit of a ... |
| clim2 15444 | Express the predicate: Th... |
| clim2c 15445 | Express the predicate ` F ... |
| clim0 15446 | Express the predicate ` F ... |
| clim0c 15447 | Express the predicate ` F ... |
| rlim0 15448 | Express the predicate ` B ... |
| rlim0lt 15449 | Use strictly less-than in ... |
| climi 15450 | Convergence of a sequence ... |
| climi2 15451 | Convergence of a sequence ... |
| climi0 15452 | Convergence of a sequence ... |
| rlimi 15453 | Convergence at infinity of... |
| rlimi2 15454 | Convergence at infinity of... |
| ello1 15455 | Elementhood in the set of ... |
| ello12 15456 | Elementhood in the set of ... |
| ello12r 15457 | Sufficient condition for e... |
| lo1f 15458 | An eventually upper bounde... |
| lo1dm 15459 | An eventually upper bounde... |
| lo1bdd 15460 | The defining property of a... |
| ello1mpt 15461 | Elementhood in the set of ... |
| ello1mpt2 15462 | Elementhood in the set of ... |
| ello1d 15463 | Sufficient condition for e... |
| lo1bdd2 15464 | If an eventually bounded f... |
| lo1bddrp 15465 | Refine ~ o1bdd2 to give a ... |
| elo1 15466 | Elementhood in the set of ... |
| elo12 15467 | Elementhood in the set of ... |
| elo12r 15468 | Sufficient condition for e... |
| o1f 15469 | An eventually bounded func... |
| o1dm 15470 | An eventually bounded func... |
| o1bdd 15471 | The defining property of a... |
| lo1o1 15472 | A function is eventually b... |
| lo1o12 15473 | A function is eventually b... |
| elo1mpt 15474 | Elementhood in the set of ... |
| elo1mpt2 15475 | Elementhood in the set of ... |
| elo1d 15476 | Sufficient condition for e... |
| o1lo1 15477 | A real function is eventua... |
| o1lo12 15478 | A lower bounded real funct... |
| o1lo1d 15479 | A real eventually bounded ... |
| icco1 15480 | Derive eventual boundednes... |
| o1bdd2 15481 | If an eventually bounded f... |
| o1bddrp 15482 | Refine ~ o1bdd2 to give a ... |
| climconst 15483 | An (eventually) constant s... |
| rlimconst 15484 | A constant sequence conver... |
| rlimclim1 15485 | Forward direction of ~ rli... |
| rlimclim 15486 | A sequence on an upper int... |
| climrlim2 15487 | Produce a real limit from ... |
| climconst2 15488 | A constant sequence conver... |
| climz 15489 | The zero sequence converge... |
| rlimuni 15490 | A real function whose doma... |
| rlimdm 15491 | Two ways to express that a... |
| climuni 15492 | An infinite sequence of co... |
| fclim 15493 | The limit relation is func... |
| climdm 15494 | Two ways to express that a... |
| climeu 15495 | An infinite sequence of co... |
| climreu 15496 | An infinite sequence of co... |
| climmo 15497 | An infinite sequence of co... |
| rlimres 15498 | The restriction of a funct... |
| lo1res 15499 | The restriction of an even... |
| o1res 15500 | The restriction of an even... |
| rlimres2 15501 | The restriction of a funct... |
| lo1res2 15502 | The restriction of a funct... |
| o1res2 15503 | The restriction of a funct... |
| lo1resb 15504 | The restriction of a funct... |
| rlimresb 15505 | The restriction of a funct... |
| o1resb 15506 | The restriction of a funct... |
| climeq 15507 | Two functions that are eve... |
| lo1eq 15508 | Two functions that are eve... |
| rlimeq 15509 | Two functions that are eve... |
| o1eq 15510 | Two functions that are eve... |
| climmpt 15511 | Exhibit a function ` G ` w... |
| 2clim 15512 | If two sequences converge ... |
| climmpt2 15513 | Relate an integer limit on... |
| climshftlem 15514 | A shifted function converg... |
| climres 15515 | A function restricted to u... |
| climshft 15516 | A shifted function converg... |
| serclim0 15517 | The zero series converges ... |
| rlimcld2 15518 | If ` D ` is a closed set i... |
| rlimrege0 15519 | The limit of a sequence of... |
| rlimrecl 15520 | The limit of a real sequen... |
| rlimge0 15521 | The limit of a sequence of... |
| climshft2 15522 | A shifted function converg... |
| climrecl 15523 | The limit of a convergent ... |
| climge0 15524 | A nonnegative sequence con... |
| climabs0 15525 | Convergence to zero of the... |
| o1co 15526 | Sufficient condition for t... |
| o1compt 15527 | Sufficient condition for t... |
| rlimcn1 15528 | Image of a limit under a c... |
| rlimcn1b 15529 | Image of a limit under a c... |
| rlimcn3 15530 | Image of a limit under a c... |
| rlimcn2 15531 | Image of a limit under a c... |
| climcn1 15532 | Image of a limit under a c... |
| climcn2 15533 | Image of a limit under a c... |
| addcn2 15534 | Complex number addition is... |
| subcn2 15535 | Complex number subtraction... |
| mulcn2 15536 | Complex number multiplicat... |
| reccn2 15537 | The reciprocal function is... |
| cn1lem 15538 | A sufficient condition for... |
| abscn2 15539 | The absolute value functio... |
| cjcn2 15540 | The complex conjugate func... |
| recn2 15541 | The real part function is ... |
| imcn2 15542 | The imaginary part functio... |
| climcn1lem 15543 | The limit of a continuous ... |
| climabs 15544 | Limit of the absolute valu... |
| climcj 15545 | Limit of the complex conju... |
| climre 15546 | Limit of the real part of ... |
| climim 15547 | Limit of the imaginary par... |
| rlimmptrcl 15548 | Reverse closure for a real... |
| rlimabs 15549 | Limit of the absolute valu... |
| rlimcj 15550 | Limit of the complex conju... |
| rlimre 15551 | Limit of the real part of ... |
| rlimim 15552 | Limit of the imaginary par... |
| o1of2 15553 | Show that a binary operati... |
| o1add 15554 | The sum of two eventually ... |
| o1mul 15555 | The product of two eventua... |
| o1sub 15556 | The difference of two even... |
| rlimo1 15557 | Any function with a finite... |
| rlimdmo1 15558 | A convergent function is e... |
| o1rlimmul 15559 | The product of an eventual... |
| o1const 15560 | A constant function is eve... |
| lo1const 15561 | A constant function is eve... |
| lo1mptrcl 15562 | Reverse closure for an eve... |
| o1mptrcl 15563 | Reverse closure for an eve... |
| o1add2 15564 | The sum of two eventually ... |
| o1mul2 15565 | The product of two eventua... |
| o1sub2 15566 | The product of two eventua... |
| lo1add 15567 | The sum of two eventually ... |
| lo1mul 15568 | The product of an eventual... |
| lo1mul2 15569 | The product of an eventual... |
| o1dif 15570 | If the difference of two f... |
| lo1sub 15571 | The difference of an event... |
| climadd 15572 | Limit of the sum of two co... |
| climmul 15573 | Limit of the product of tw... |
| climsub 15574 | Limit of the difference of... |
| climaddc1 15575 | Limit of a constant ` C ` ... |
| climaddc2 15576 | Limit of a constant ` C ` ... |
| climmulc2 15577 | Limit of a sequence multip... |
| climsubc1 15578 | Limit of a constant ` C ` ... |
| climsubc2 15579 | Limit of a constant ` C ` ... |
| climle 15580 | Comparison of the limits o... |
| climsqz 15581 | Convergence of a sequence ... |
| climsqz2 15582 | Convergence of a sequence ... |
| rlimadd 15583 | Limit of the sum of two co... |
| rlimaddOLD 15584 | Obsolete version of ~ rlim... |
| rlimsub 15585 | Limit of the difference of... |
| rlimmul 15586 | Limit of the product of tw... |
| rlimmulOLD 15587 | Obsolete version of ~ rlim... |
| rlimdiv 15588 | Limit of the quotient of t... |
| rlimneg 15589 | Limit of the negative of a... |
| rlimle 15590 | Comparison of the limits o... |
| rlimsqzlem 15591 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15592 | Convergence of a sequence ... |
| rlimsqz2 15593 | Convergence of a sequence ... |
| lo1le 15594 | Transfer eventual upper bo... |
| o1le 15595 | Transfer eventual boundedn... |
| rlimno1 15596 | A function whose inverse c... |
| clim2ser 15597 | The limit of an infinite s... |
| clim2ser2 15598 | The limit of an infinite s... |
| iserex 15599 | An infinite series converg... |
| isermulc2 15600 | Multiplication of an infin... |
| climlec2 15601 | Comparison of a constant t... |
| iserle 15602 | Comparison of the limits o... |
| iserge0 15603 | The limit of an infinite s... |
| climub 15604 | The limit of a monotonic s... |
| climserle 15605 | The partial sums of a conv... |
| isershft 15606 | Index shift of the limit o... |
| isercolllem1 15607 | Lemma for ~ isercoll . (C... |
| isercolllem2 15608 | Lemma for ~ isercoll . (C... |
| isercolllem3 15609 | Lemma for ~ isercoll . (C... |
| isercoll 15610 | Rearrange an infinite seri... |
| isercoll2 15611 | Generalize ~ isercoll so t... |
| climsup 15612 | A bounded monotonic sequen... |
| climcau 15613 | A converging sequence of c... |
| climbdd 15614 | A converging sequence of c... |
| caucvgrlem 15615 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15616 | A Cauchy sequence of real ... |
| caucvgrlem2 15617 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15618 | A Cauchy sequence of compl... |
| caurcvg 15619 | A Cauchy sequence of real ... |
| caurcvg2 15620 | A Cauchy sequence of real ... |
| caucvg 15621 | A Cauchy sequence of compl... |
| caucvgb 15622 | A function is convergent i... |
| serf0 15623 | If an infinite series conv... |
| iseraltlem1 15624 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15625 | Lemma for ~ iseralt . The... |
| iseraltlem3 15626 | Lemma for ~ iseralt . Fro... |
| iseralt 15627 | The alternating series tes... |
| sumex 15630 | A sum is a set. (Contribu... |
| sumeq1 15631 | Equality theorem for a sum... |
| nfsum1 15632 | Bound-variable hypothesis ... |
| nfsum 15633 | Bound-variable hypothesis ... |
| sumeq2w 15634 | Equality theorem for sum, ... |
| sumeq2ii 15635 | Equality theorem for sum, ... |
| sumeq2 15636 | Equality theorem for sum. ... |
| cbvsum 15637 | Change bound variable in a... |
| cbvsumv 15638 | Change bound variable in a... |
| cbvsumi 15639 | Change bound variable in a... |
| sumeq1i 15640 | Equality inference for sum... |
| sumeq2i 15641 | Equality inference for sum... |
| sumeq12i 15642 | Equality inference for sum... |
| sumeq1d 15643 | Equality deduction for sum... |
| sumeq2d 15644 | Equality deduction for sum... |
| sumeq2dv 15645 | Equality deduction for sum... |
| sumeq2sdv 15646 | Equality deduction for sum... |
| 2sumeq2dv 15647 | Equality deduction for dou... |
| sumeq12dv 15648 | Equality deduction for sum... |
| sumeq12rdv 15649 | Equality deduction for sum... |
| sum2id 15650 | The second class argument ... |
| sumfc 15651 | A lemma to facilitate conv... |
| fz1f1o 15652 | A lemma for working with f... |
| sumrblem 15653 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15654 | The sequence of partial su... |
| sumrb 15655 | Rebase the starting point ... |
| summolem3 15656 | Lemma for ~ summo . (Cont... |
| summolem2a 15657 | Lemma for ~ summo . (Cont... |
| summolem2 15658 | Lemma for ~ summo . (Cont... |
| summo 15659 | A sum has at most one limi... |
| zsum 15660 | Series sum with index set ... |
| isum 15661 | Series sum with an upper i... |
| fsum 15662 | The value of a sum over a ... |
| sum0 15663 | Any sum over the empty set... |
| sumz 15664 | Any sum of zero over a sum... |
| fsumf1o 15665 | Re-index a finite sum usin... |
| sumss 15666 | Change the index set to a ... |
| fsumss 15667 | Change the index set to a ... |
| sumss2 15668 | Change the index set of a ... |
| fsumcvg2 15669 | The sequence of partial su... |
| fsumsers 15670 | Special case of series sum... |
| fsumcvg3 15671 | A finite sum is convergent... |
| fsumser 15672 | A finite sum expressed in ... |
| fsumcl2lem 15673 | - Lemma for finite sum clo... |
| fsumcllem 15674 | - Lemma for finite sum clo... |
| fsumcl 15675 | Closure of a finite sum of... |
| fsumrecl 15676 | Closure of a finite sum of... |
| fsumzcl 15677 | Closure of a finite sum of... |
| fsumnn0cl 15678 | Closure of a finite sum of... |
| fsumrpcl 15679 | Closure of a finite sum of... |
| fsumclf 15680 | Closure of a finite sum of... |
| fsumzcl2 15681 | A finite sum with integer ... |
| fsumadd 15682 | The sum of two finite sums... |
| fsumsplit 15683 | Split a sum into two parts... |
| fsumsplitf 15684 | Split a sum into two parts... |
| sumsnf 15685 | A sum of a singleton is th... |
| fsumsplitsn 15686 | Separate out a term in a f... |
| fsumsplit1 15687 | Separate out a term in a f... |
| sumsn 15688 | A sum of a singleton is th... |
| fsum1 15689 | The finite sum of ` A ( k ... |
| sumpr 15690 | A sum over a pair is the s... |
| sumtp 15691 | A sum over a triple is the... |
| sumsns 15692 | A sum of a singleton is th... |
| fsumm1 15693 | Separate out the last term... |
| fzosump1 15694 | Separate out the last term... |
| fsum1p 15695 | Separate out the first ter... |
| fsummsnunz 15696 | A finite sum all of whose ... |
| fsumsplitsnun 15697 | Separate out a term in a f... |
| fsump1 15698 | The addition of the next t... |
| isumclim 15699 | An infinite sum equals the... |
| isumclim2 15700 | A converging series conver... |
| isumclim3 15701 | The sequence of partial fi... |
| sumnul 15702 | The sum of a non-convergen... |
| isumcl 15703 | The sum of a converging in... |
| isummulc2 15704 | An infinite sum multiplied... |
| isummulc1 15705 | An infinite sum multiplied... |
| isumdivc 15706 | An infinite sum divided by... |
| isumrecl 15707 | The sum of a converging in... |
| isumge0 15708 | An infinite sum of nonnega... |
| isumadd 15709 | Addition of infinite sums.... |
| sumsplit 15710 | Split a sum into two parts... |
| fsump1i 15711 | Optimized version of ~ fsu... |
| fsum2dlem 15712 | Lemma for ~ fsum2d - induc... |
| fsum2d 15713 | Write a double sum as a su... |
| fsumxp 15714 | Combine two sums into a si... |
| fsumcnv 15715 | Transform a region of summ... |
| fsumcom2 15716 | Interchange order of summa... |
| fsumcom 15717 | Interchange order of summa... |
| fsum0diaglem 15718 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15719 | Two ways to express "the s... |
| mptfzshft 15720 | 1-1 onto function in maps-... |
| fsumrev 15721 | Reversal of a finite sum. ... |
| fsumshft 15722 | Index shift of a finite su... |
| fsumshftm 15723 | Negative index shift of a ... |
| fsumrev2 15724 | Reversal of a finite sum. ... |
| fsum0diag2 15725 | Two ways to express "the s... |
| fsummulc2 15726 | A finite sum multiplied by... |
| fsummulc1 15727 | A finite sum multiplied by... |
| fsumdivc 15728 | A finite sum divided by a ... |
| fsumneg 15729 | Negation of a finite sum. ... |
| fsumsub 15730 | Split a finite sum over a ... |
| fsum2mul 15731 | Separate the nested sum of... |
| fsumconst 15732 | The sum of constant terms ... |
| fsumdifsnconst 15733 | The sum of constant terms ... |
| modfsummodslem1 15734 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15735 | Induction step for ~ modfs... |
| modfsummod 15736 | A finite sum modulo a posi... |
| fsumge0 15737 | If all of the terms of a f... |
| fsumless 15738 | A shorter sum of nonnegati... |
| fsumge1 15739 | A sum of nonnegative numbe... |
| fsum00 15740 | A sum of nonnegative numbe... |
| fsumle 15741 | If all of the terms of fin... |
| fsumlt 15742 | If every term in one finit... |
| fsumabs 15743 | Generalized triangle inequ... |
| telfsumo 15744 | Sum of a telescoping serie... |
| telfsumo2 15745 | Sum of a telescoping serie... |
| telfsum 15746 | Sum of a telescoping serie... |
| telfsum2 15747 | Sum of a telescoping serie... |
| fsumparts 15748 | Summation by parts. (Cont... |
| fsumrelem 15749 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15750 | The real part of a sum. (... |
| fsumim 15751 | The imaginary part of a su... |
| fsumcj 15752 | The complex conjugate of a... |
| fsumrlim 15753 | Limit of a finite sum of c... |
| fsumo1 15754 | The finite sum of eventual... |
| o1fsum 15755 | If ` A ( k ) ` is O(1), th... |
| seqabs 15756 | Generalized triangle inequ... |
| iserabs 15757 | Generalized triangle inequ... |
| cvgcmp 15758 | A comparison test for conv... |
| cvgcmpub 15759 | An upper bound for the lim... |
| cvgcmpce 15760 | A comparison test for conv... |
| abscvgcvg 15761 | An absolutely convergent s... |
| climfsum 15762 | Limit of a finite sum of c... |
| fsumiun 15763 | Sum over a disjoint indexe... |
| hashiun 15764 | The cardinality of a disjo... |
| hash2iun 15765 | The cardinality of a neste... |
| hash2iun1dif1 15766 | The cardinality of a neste... |
| hashrabrex 15767 | The number of elements in ... |
| hashuni 15768 | The cardinality of a disjo... |
| qshash 15769 | The cardinality of a set w... |
| ackbijnn 15770 | Translate the Ackermann bi... |
| binomlem 15771 | Lemma for ~ binom (binomia... |
| binom 15772 | The binomial theorem: ` ( ... |
| binom1p 15773 | Special case of the binomi... |
| binom11 15774 | Special case of the binomi... |
| binom1dif 15775 | A summation for the differ... |
| bcxmaslem1 15776 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15777 | Parallel summation (Christ... |
| incexclem 15778 | Lemma for ~ incexc . (Con... |
| incexc 15779 | The inclusion/exclusion pr... |
| incexc2 15780 | The inclusion/exclusion pr... |
| isumshft 15781 | Index shift of an infinite... |
| isumsplit 15782 | Split off the first ` N ` ... |
| isum1p 15783 | The infinite sum of a conv... |
| isumnn0nn 15784 | Sum from 0 to infinity in ... |
| isumrpcl 15785 | The infinite sum of positi... |
| isumle 15786 | Comparison of two infinite... |
| isumless 15787 | A finite sum of nonnegativ... |
| isumsup2 15788 | An infinite sum of nonnega... |
| isumsup 15789 | An infinite sum of nonnega... |
| isumltss 15790 | A partial sum of a series ... |
| climcndslem1 15791 | Lemma for ~ climcnds : bou... |
| climcndslem2 15792 | Lemma for ~ climcnds : bou... |
| climcnds 15793 | The Cauchy condensation te... |
| divrcnv 15794 | The sequence of reciprocal... |
| divcnv 15795 | The sequence of reciprocal... |
| flo1 15796 | The floor function satisfi... |
| divcnvshft 15797 | Limit of a ratio function.... |
| supcvg 15798 | Extract a sequence ` f ` i... |
| infcvgaux1i 15799 | Auxiliary theorem for appl... |
| infcvgaux2i 15800 | Auxiliary theorem for appl... |
| harmonic 15801 | The harmonic series ` H ` ... |
| arisum 15802 | Arithmetic series sum of t... |
| arisum2 15803 | Arithmetic series sum of t... |
| trireciplem 15804 | Lemma for ~ trirecip . Sh... |
| trirecip 15805 | The sum of the reciprocals... |
| expcnv 15806 | A sequence of powers of a ... |
| explecnv 15807 | A sequence of terms conver... |
| geoserg 15808 | The value of the finite ge... |
| geoser 15809 | The value of the finite ge... |
| pwdif 15810 | The difference of two numb... |
| pwm1geoser 15811 | The n-th power of a number... |
| geolim 15812 | The partial sums in the in... |
| geolim2 15813 | The partial sums in the ge... |
| georeclim 15814 | The limit of a geometric s... |
| geo2sum 15815 | The value of the finite ge... |
| geo2sum2 15816 | The value of the finite ge... |
| geo2lim 15817 | The value of the infinite ... |
| geomulcvg 15818 | The geometric series conve... |
| geoisum 15819 | The infinite sum of ` 1 + ... |
| geoisumr 15820 | The infinite sum of recipr... |
| geoisum1 15821 | The infinite sum of ` A ^ ... |
| geoisum1c 15822 | The infinite sum of ` A x.... |
| 0.999... 15823 | The recurring decimal 0.99... |
| geoihalfsum 15824 | Prove that the infinite ge... |
| cvgrat 15825 | Ratio test for convergence... |
| mertenslem1 15826 | Lemma for ~ mertens . (Co... |
| mertenslem2 15827 | Lemma for ~ mertens . (Co... |
| mertens 15828 | Mertens' theorem. If ` A ... |
| prodf 15829 | An infinite product of com... |
| clim2prod 15830 | The limit of an infinite p... |
| clim2div 15831 | The limit of an infinite p... |
| prodfmul 15832 | The product of two infinit... |
| prodf1 15833 | The value of the partial p... |
| prodf1f 15834 | A one-valued infinite prod... |
| prodfclim1 15835 | The constant one product c... |
| prodfn0 15836 | No term of a nonzero infin... |
| prodfrec 15837 | The reciprocal of an infin... |
| prodfdiv 15838 | The quotient of two infini... |
| ntrivcvg 15839 | A non-trivially converging... |
| ntrivcvgn0 15840 | A product that converges t... |
| ntrivcvgfvn0 15841 | Any value of a product seq... |
| ntrivcvgtail 15842 | A tail of a non-trivially ... |
| ntrivcvgmullem 15843 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15844 | The product of two non-tri... |
| prodex 15847 | A product is a set. (Cont... |
| prodeq1f 15848 | Equality theorem for a pro... |
| prodeq1 15849 | Equality theorem for a pro... |
| nfcprod1 15850 | Bound-variable hypothesis ... |
| nfcprod 15851 | Bound-variable hypothesis ... |
| prodeq2w 15852 | Equality theorem for produ... |
| prodeq2ii 15853 | Equality theorem for produ... |
| prodeq2 15854 | Equality theorem for produ... |
| cbvprod 15855 | Change bound variable in a... |
| cbvprodv 15856 | Change bound variable in a... |
| cbvprodi 15857 | Change bound variable in a... |
| prodeq1i 15858 | Equality inference for pro... |
| prodeq2i 15859 | Equality inference for pro... |
| prodeq12i 15860 | Equality inference for pro... |
| prodeq1d 15861 | Equality deduction for pro... |
| prodeq2d 15862 | Equality deduction for pro... |
| prodeq2dv 15863 | Equality deduction for pro... |
| prodeq2sdv 15864 | Equality deduction for pro... |
| 2cprodeq2dv 15865 | Equality deduction for dou... |
| prodeq12dv 15866 | Equality deduction for pro... |
| prodeq12rdv 15867 | Equality deduction for pro... |
| prod2id 15868 | The second class argument ... |
| prodrblem 15869 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15870 | The sequence of partial pr... |
| prodrblem2 15871 | Lemma for ~ prodrb . (Con... |
| prodrb 15872 | Rebase the starting point ... |
| prodmolem3 15873 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15874 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15875 | Lemma for ~ prodmo . (Con... |
| prodmo 15876 | A product has at most one ... |
| zprod 15877 | Series product with index ... |
| iprod 15878 | Series product with an upp... |
| zprodn0 15879 | Nonzero series product wit... |
| iprodn0 15880 | Nonzero series product wit... |
| fprod 15881 | The value of a product ove... |
| fprodntriv 15882 | A non-triviality lemma for... |
| prod0 15883 | A product over the empty s... |
| prod1 15884 | Any product of one over a ... |
| prodfc 15885 | A lemma to facilitate conv... |
| fprodf1o 15886 | Re-index a finite product ... |
| prodss 15887 | Change the index set to a ... |
| fprodss 15888 | Change the index set to a ... |
| fprodser 15889 | A finite product expressed... |
| fprodcl2lem 15890 | Finite product closure lem... |
| fprodcllem 15891 | Finite product closure lem... |
| fprodcl 15892 | Closure of a finite produc... |
| fprodrecl 15893 | Closure of a finite produc... |
| fprodzcl 15894 | Closure of a finite produc... |
| fprodnncl 15895 | Closure of a finite produc... |
| fprodrpcl 15896 | Closure of a finite produc... |
| fprodnn0cl 15897 | Closure of a finite produc... |
| fprodcllemf 15898 | Finite product closure lem... |
| fprodreclf 15899 | Closure of a finite produc... |
| fprodmul 15900 | The product of two finite ... |
| fproddiv 15901 | The quotient of two finite... |
| prodsn 15902 | A product of a singleton i... |
| fprod1 15903 | A finite product of only o... |
| prodsnf 15904 | A product of a singleton i... |
| climprod1 15905 | The limit of a product ove... |
| fprodsplit 15906 | Split a finite product int... |
| fprodm1 15907 | Separate out the last term... |
| fprod1p 15908 | Separate out the first ter... |
| fprodp1 15909 | Multiply in the last term ... |
| fprodm1s 15910 | Separate out the last term... |
| fprodp1s 15911 | Multiply in the last term ... |
| prodsns 15912 | A product of the singleton... |
| fprodfac 15913 | Factorial using product no... |
| fprodabs 15914 | The absolute value of a fi... |
| fprodeq0 15915 | Any finite product contain... |
| fprodshft 15916 | Shift the index of a finit... |
| fprodrev 15917 | Reversal of a finite produ... |
| fprodconst 15918 | The product of constant te... |
| fprodn0 15919 | A finite product of nonzer... |
| fprod2dlem 15920 | Lemma for ~ fprod2d - indu... |
| fprod2d 15921 | Write a double product as ... |
| fprodxp 15922 | Combine two products into ... |
| fprodcnv 15923 | Transform a product region... |
| fprodcom2 15924 | Interchange order of multi... |
| fprodcom 15925 | Interchange product order.... |
| fprod0diag 15926 | Two ways to express "the p... |
| fproddivf 15927 | The quotient of two finite... |
| fprodsplitf 15928 | Split a finite product int... |
| fprodsplitsn 15929 | Separate out a term in a f... |
| fprodsplit1f 15930 | Separate out a term in a f... |
| fprodn0f 15931 | A finite product of nonzer... |
| fprodclf 15932 | Closure of a finite produc... |
| fprodge0 15933 | If all the terms of a fini... |
| fprodeq0g 15934 | Any finite product contain... |
| fprodge1 15935 | If all of the terms of a f... |
| fprodle 15936 | If all the terms of two fi... |
| fprodmodd 15937 | If all factors of two fini... |
| iprodclim 15938 | An infinite product equals... |
| iprodclim2 15939 | A converging product conve... |
| iprodclim3 15940 | The sequence of partial fi... |
| iprodcl 15941 | The product of a non-trivi... |
| iprodrecl 15942 | The product of a non-trivi... |
| iprodmul 15943 | Multiplication of infinite... |
| risefacval 15948 | The value of the rising fa... |
| fallfacval 15949 | The value of the falling f... |
| risefacval2 15950 | One-based value of rising ... |
| fallfacval2 15951 | One-based value of falling... |
| fallfacval3 15952 | A product representation o... |
| risefaccllem 15953 | Lemma for rising factorial... |
| fallfaccllem 15954 | Lemma for falling factoria... |
| risefaccl 15955 | Closure law for rising fac... |
| fallfaccl 15956 | Closure law for falling fa... |
| rerisefaccl 15957 | Closure law for rising fac... |
| refallfaccl 15958 | Closure law for falling fa... |
| nnrisefaccl 15959 | Closure law for rising fac... |
| zrisefaccl 15960 | Closure law for rising fac... |
| zfallfaccl 15961 | Closure law for falling fa... |
| nn0risefaccl 15962 | Closure law for rising fac... |
| rprisefaccl 15963 | Closure law for rising fac... |
| risefallfac 15964 | A relationship between ris... |
| fallrisefac 15965 | A relationship between fal... |
| risefall0lem 15966 | Lemma for ~ risefac0 and ~... |
| risefac0 15967 | The value of the rising fa... |
| fallfac0 15968 | The value of the falling f... |
| risefacp1 15969 | The value of the rising fa... |
| fallfacp1 15970 | The value of the falling f... |
| risefacp1d 15971 | The value of the rising fa... |
| fallfacp1d 15972 | The value of the falling f... |
| risefac1 15973 | The value of rising factor... |
| fallfac1 15974 | The value of falling facto... |
| risefacfac 15975 | Relate rising factorial to... |
| fallfacfwd 15976 | The forward difference of ... |
| 0fallfac 15977 | The value of the zero fall... |
| 0risefac 15978 | The value of the zero risi... |
| binomfallfaclem1 15979 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 15980 | Lemma for ~ binomfallfac .... |
| binomfallfac 15981 | A version of the binomial ... |
| binomrisefac 15982 | A version of the binomial ... |
| fallfacval4 15983 | Represent the falling fact... |
| bcfallfac 15984 | Binomial coefficient in te... |
| fallfacfac 15985 | Relate falling factorial t... |
| bpolylem 15988 | Lemma for ~ bpolyval . (C... |
| bpolyval 15989 | The value of the Bernoulli... |
| bpoly0 15990 | The value of the Bernoulli... |
| bpoly1 15991 | The value of the Bernoulli... |
| bpolycl 15992 | Closure law for Bernoulli ... |
| bpolysum 15993 | A sum for Bernoulli polyno... |
| bpolydiflem 15994 | Lemma for ~ bpolydif . (C... |
| bpolydif 15995 | Calculate the difference b... |
| fsumkthpow 15996 | A closed-form expression f... |
| bpoly2 15997 | The Bernoulli polynomials ... |
| bpoly3 15998 | The Bernoulli polynomials ... |
| bpoly4 15999 | The Bernoulli polynomials ... |
| fsumcube 16000 | Express the sum of cubes i... |
| eftcl 16013 | Closure of a term in the s... |
| reeftcl 16014 | The terms of the series ex... |
| eftabs 16015 | The absolute value of a te... |
| eftval 16016 | The value of a term in the... |
| efcllem 16017 | Lemma for ~ efcl . The se... |
| ef0lem 16018 | The series defining the ex... |
| efval 16019 | Value of the exponential f... |
| esum 16020 | Value of Euler's constant ... |
| eff 16021 | Domain and codomain of the... |
| efcl 16022 | Closure law for the expone... |
| efval2 16023 | Value of the exponential f... |
| efcvg 16024 | The series that defines th... |
| efcvgfsum 16025 | Exponential function conve... |
| reefcl 16026 | The exponential function i... |
| reefcld 16027 | The exponential function i... |
| ere 16028 | Euler's constant ` _e ` = ... |
| ege2le3 16029 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16030 | Value of the exponential f... |
| efcj 16031 | The exponential of a compl... |
| efaddlem 16032 | Lemma for ~ efadd (exponen... |
| efadd 16033 | Sum of exponents law for e... |
| fprodefsum 16034 | Move the exponential funct... |
| efcan 16035 | Cancellation law for expon... |
| efne0 16036 | The exponential of a compl... |
| efneg 16037 | The exponential of the opp... |
| eff2 16038 | The exponential function m... |
| efsub 16039 | Difference of exponents la... |
| efexp 16040 | The exponential of an inte... |
| efzval 16041 | Value of the exponential f... |
| efgt0 16042 | The exponential of a real ... |
| rpefcl 16043 | The exponential of a real ... |
| rpefcld 16044 | The exponential of a real ... |
| eftlcvg 16045 | The tail series of the exp... |
| eftlcl 16046 | Closure of the sum of an i... |
| reeftlcl 16047 | Closure of the sum of an i... |
| eftlub 16048 | An upper bound on the abso... |
| efsep 16049 | Separate out the next term... |
| effsumlt 16050 | The partial sums of the se... |
| eft0val 16051 | The value of the first ter... |
| ef4p 16052 | Separate out the first fou... |
| efgt1p2 16053 | The exponential of a posit... |
| efgt1p 16054 | The exponential of a posit... |
| efgt1 16055 | The exponential of a posit... |
| eflt 16056 | The exponential function o... |
| efle 16057 | The exponential function o... |
| reef11 16058 | The exponential function o... |
| reeff1 16059 | The exponential function m... |
| eflegeo 16060 | The exponential function o... |
| sinval 16061 | Value of the sine function... |
| cosval 16062 | Value of the cosine functi... |
| sinf 16063 | Domain and codomain of the... |
| cosf 16064 | Domain and codomain of the... |
| sincl 16065 | Closure of the sine functi... |
| coscl 16066 | Closure of the cosine func... |
| tanval 16067 | Value of the tangent funct... |
| tancl 16068 | The closure of the tangent... |
| sincld 16069 | Closure of the sine functi... |
| coscld 16070 | Closure of the cosine func... |
| tancld 16071 | Closure of the tangent fun... |
| tanval2 16072 | Express the tangent functi... |
| tanval3 16073 | Express the tangent functi... |
| resinval 16074 | The sine of a real number ... |
| recosval 16075 | The cosine of a real numbe... |
| efi4p 16076 | Separate out the first fou... |
| resin4p 16077 | Separate out the first fou... |
| recos4p 16078 | Separate out the first fou... |
| resincl 16079 | The sine of a real number ... |
| recoscl 16080 | The cosine of a real numbe... |
| retancl 16081 | The closure of the tangent... |
| resincld 16082 | Closure of the sine functi... |
| recoscld 16083 | Closure of the cosine func... |
| retancld 16084 | Closure of the tangent fun... |
| sinneg 16085 | The sine of a negative is ... |
| cosneg 16086 | The cosines of a number an... |
| tanneg 16087 | The tangent of a negative ... |
| sin0 16088 | Value of the sine function... |
| cos0 16089 | Value of the cosine functi... |
| tan0 16090 | The value of the tangent f... |
| efival 16091 | The exponential function i... |
| efmival 16092 | The exponential function i... |
| sinhval 16093 | Value of the hyperbolic si... |
| coshval 16094 | Value of the hyperbolic co... |
| resinhcl 16095 | The hyperbolic sine of a r... |
| rpcoshcl 16096 | The hyperbolic cosine of a... |
| recoshcl 16097 | The hyperbolic cosine of a... |
| retanhcl 16098 | The hyperbolic tangent of ... |
| tanhlt1 16099 | The hyperbolic tangent of ... |
| tanhbnd 16100 | The hyperbolic tangent of ... |
| efeul 16101 | Eulerian representation of... |
| efieq 16102 | The exponentials of two im... |
| sinadd 16103 | Addition formula for sine.... |
| cosadd 16104 | Addition formula for cosin... |
| tanaddlem 16105 | A useful intermediate step... |
| tanadd 16106 | Addition formula for tange... |
| sinsub 16107 | Sine of difference. (Cont... |
| cossub 16108 | Cosine of difference. (Co... |
| addsin 16109 | Sum of sines. (Contribute... |
| subsin 16110 | Difference of sines. (Con... |
| sinmul 16111 | Product of sines can be re... |
| cosmul 16112 | Product of cosines can be ... |
| addcos 16113 | Sum of cosines. (Contribu... |
| subcos 16114 | Difference of cosines. (C... |
| sincossq 16115 | Sine squared plus cosine s... |
| sin2t 16116 | Double-angle formula for s... |
| cos2t 16117 | Double-angle formula for c... |
| cos2tsin 16118 | Double-angle formula for c... |
| sinbnd 16119 | The sine of a real number ... |
| cosbnd 16120 | The cosine of a real numbe... |
| sinbnd2 16121 | The sine of a real number ... |
| cosbnd2 16122 | The cosine of a real numbe... |
| ef01bndlem 16123 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16124 | Bounds on the sine of a po... |
| cos01bnd 16125 | Bounds on the cosine of a ... |
| cos1bnd 16126 | Bounds on the cosine of 1.... |
| cos2bnd 16127 | Bounds on the cosine of 2.... |
| sinltx 16128 | The sine of a positive rea... |
| sin01gt0 16129 | The sine of a positive rea... |
| cos01gt0 16130 | The cosine of a positive r... |
| sin02gt0 16131 | The sine of a positive rea... |
| sincos1sgn 16132 | The signs of the sine and ... |
| sincos2sgn 16133 | The signs of the sine and ... |
| sin4lt0 16134 | The sine of 4 is negative.... |
| absefi 16135 | The absolute value of the ... |
| absef 16136 | The absolute value of the ... |
| absefib 16137 | A complex number is real i... |
| efieq1re 16138 | A number whose imaginary e... |
| demoivre 16139 | De Moivre's Formula. Proo... |
| demoivreALT 16140 | Alternate proof of ~ demoi... |
| eirrlem 16143 | Lemma for ~ eirr . (Contr... |
| eirr 16144 | ` _e ` is irrational. (Co... |
| egt2lt3 16145 | Euler's constant ` _e ` = ... |
| epos 16146 | Euler's constant ` _e ` is... |
| epr 16147 | Euler's constant ` _e ` is... |
| ene0 16148 | ` _e ` is not 0. (Contrib... |
| ene1 16149 | ` _e ` is not 1. (Contrib... |
| xpnnen 16150 | The Cartesian product of t... |
| znnen 16151 | The set of integers and th... |
| qnnen 16152 | The rational numbers are c... |
| rpnnen2lem1 16153 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16154 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16155 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16156 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16157 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16158 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16159 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16160 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16161 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16162 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16163 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16164 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16165 | The other half of ~ rpnnen... |
| rpnnen 16166 | The cardinality of the con... |
| rexpen 16167 | The real numbers are equin... |
| cpnnen 16168 | The complex numbers are eq... |
| rucALT 16169 | Alternate proof of ~ ruc .... |
| ruclem1 16170 | Lemma for ~ ruc (the reals... |
| ruclem2 16171 | Lemma for ~ ruc . Orderin... |
| ruclem3 16172 | Lemma for ~ ruc . The con... |
| ruclem4 16173 | Lemma for ~ ruc . Initial... |
| ruclem6 16174 | Lemma for ~ ruc . Domain ... |
| ruclem7 16175 | Lemma for ~ ruc . Success... |
| ruclem8 16176 | Lemma for ~ ruc . The int... |
| ruclem9 16177 | Lemma for ~ ruc . The fir... |
| ruclem10 16178 | Lemma for ~ ruc . Every f... |
| ruclem11 16179 | Lemma for ~ ruc . Closure... |
| ruclem12 16180 | Lemma for ~ ruc . The sup... |
| ruclem13 16181 | Lemma for ~ ruc . There i... |
| ruc 16182 | The set of positive intege... |
| resdomq 16183 | The set of rationals is st... |
| aleph1re 16184 | There are at least aleph-o... |
| aleph1irr 16185 | There are at least aleph-o... |
| cnso 16186 | The complex numbers can be... |
| sqrt2irrlem 16187 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16188 | The square root of 2 is ir... |
| sqrt2re 16189 | The square root of 2 exist... |
| sqrt2irr0 16190 | The square root of 2 is an... |
| nthruc 16191 | The sequence ` NN ` , ` ZZ... |
| nthruz 16192 | The sequence ` NN ` , ` NN... |
| divides 16195 | Define the divides relatio... |
| dvdsval2 16196 | One nonzero integer divide... |
| dvdsval3 16197 | One nonzero integer divide... |
| dvdszrcl 16198 | Reverse closure for the di... |
| dvdsmod0 16199 | If a positive integer divi... |
| p1modz1 16200 | If a number greater than 1... |
| dvdsmodexp 16201 | If a positive integer divi... |
| nndivdvds 16202 | Strong form of ~ dvdsval2 ... |
| nndivides 16203 | Definition of the divides ... |
| moddvds 16204 | Two ways to say ` A == B `... |
| modm1div 16205 | An integer greater than on... |
| dvds0lem 16206 | A lemma to assist theorems... |
| dvds1lem 16207 | A lemma to assist theorems... |
| dvds2lem 16208 | A lemma to assist theorems... |
| iddvds 16209 | An integer divides itself.... |
| 1dvds 16210 | 1 divides any integer. Th... |
| dvds0 16211 | Any integer divides 0. Th... |
| negdvdsb 16212 | An integer divides another... |
| dvdsnegb 16213 | An integer divides another... |
| absdvdsb 16214 | An integer divides another... |
| dvdsabsb 16215 | An integer divides another... |
| 0dvds 16216 | Only 0 is divisible by 0. ... |
| dvdsmul1 16217 | An integer divides a multi... |
| dvdsmul2 16218 | An integer divides a multi... |
| iddvdsexp 16219 | An integer divides a posit... |
| muldvds1 16220 | If a product divides an in... |
| muldvds2 16221 | If a product divides an in... |
| dvdscmul 16222 | Multiplication by a consta... |
| dvdsmulc 16223 | Multiplication by a consta... |
| dvdscmulr 16224 | Cancellation law for the d... |
| dvdsmulcr 16225 | Cancellation law for the d... |
| summodnegmod 16226 | The sum of two integers mo... |
| modmulconst 16227 | Constant multiplication in... |
| dvds2ln 16228 | If an integer divides each... |
| dvds2add 16229 | If an integer divides each... |
| dvds2sub 16230 | If an integer divides each... |
| dvds2addd 16231 | Deduction form of ~ dvds2a... |
| dvds2subd 16232 | Deduction form of ~ dvds2s... |
| dvdstr 16233 | The divides relation is tr... |
| dvdstrd 16234 | The divides relation is tr... |
| dvdsmultr1 16235 | If an integer divides anot... |
| dvdsmultr1d 16236 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16237 | If an integer divides anot... |
| dvdsmultr2d 16238 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16239 | If an integer divides eith... |
| dvdssub2 16240 | If an integer divides a di... |
| dvdsadd 16241 | An integer divides another... |
| dvdsaddr 16242 | An integer divides another... |
| dvdssub 16243 | An integer divides another... |
| dvdssubr 16244 | An integer divides another... |
| dvdsadd2b 16245 | Adding a multiple of the b... |
| dvdsaddre2b 16246 | Adding a multiple of the b... |
| fsumdvds 16247 | If every term in a sum is ... |
| dvdslelem 16248 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16249 | The divisors of a positive... |
| dvdsleabs 16250 | The divisors of a nonzero ... |
| dvdsleabs2 16251 | Transfer divisibility to a... |
| dvdsabseq 16252 | If two integers divide eac... |
| dvdseq 16253 | If two nonnegative integer... |
| divconjdvds 16254 | If a nonzero integer ` M `... |
| dvdsdivcl 16255 | The complement of a diviso... |
| dvdsflip 16256 | An involution of the divis... |
| dvdsssfz1 16257 | The set of divisors of a n... |
| dvds1 16258 | The only nonnegative integ... |
| alzdvds 16259 | Only 0 is divisible by all... |
| dvdsext 16260 | Poset extensionality for d... |
| fzm1ndvds 16261 | No number between ` 1 ` an... |
| fzo0dvdseq 16262 | Zero is the only one of th... |
| fzocongeq 16263 | Two different elements of ... |
| addmodlteqALT 16264 | Two nonnegative integers l... |
| dvdsfac 16265 | A positive integer divides... |
| dvdsexp2im 16266 | If an integer divides anot... |
| dvdsexp 16267 | A power divides a power wi... |
| dvdsmod 16268 | Any number ` K ` whose mod... |
| mulmoddvds 16269 | If an integer is divisible... |
| 3dvds 16270 | A rule for divisibility by... |
| 3dvdsdec 16271 | A decimal number is divisi... |
| 3dvds2dec 16272 | A decimal number is divisi... |
| fprodfvdvdsd 16273 | A finite product of intege... |
| fproddvdsd 16274 | A finite product of intege... |
| evenelz 16275 | An even number is an integ... |
| zeo3 16276 | An integer is even or odd.... |
| zeo4 16277 | An integer is even or odd ... |
| zeneo 16278 | No even integer equals an ... |
| odd2np1lem 16279 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16280 | An integer is odd iff it i... |
| even2n 16281 | An integer is even iff it ... |
| oddm1even 16282 | An integer is odd iff its ... |
| oddp1even 16283 | An integer is odd iff its ... |
| oexpneg 16284 | The exponential of the neg... |
| mod2eq0even 16285 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16286 | An integer is 1 modulo 2 i... |
| oddnn02np1 16287 | A nonnegative integer is o... |
| oddge22np1 16288 | An integer greater than on... |
| evennn02n 16289 | A nonnegative integer is e... |
| evennn2n 16290 | A positive integer is even... |
| 2tp1odd 16291 | A number which is twice an... |
| mulsucdiv2z 16292 | An integer multiplied with... |
| sqoddm1div8z 16293 | A squared odd number minus... |
| 2teven 16294 | A number which is twice an... |
| zeo5 16295 | An integer is either even ... |
| evend2 16296 | An integer is even iff its... |
| oddp1d2 16297 | An integer is odd iff its ... |
| zob 16298 | Alternate characterization... |
| oddm1d2 16299 | An integer is odd iff its ... |
| ltoddhalfle 16300 | An integer is less than ha... |
| halfleoddlt 16301 | An integer is greater than... |
| opoe 16302 | The sum of two odds is eve... |
| omoe 16303 | The difference of two odds... |
| opeo 16304 | The sum of an odd and an e... |
| omeo 16305 | The difference of an odd a... |
| z0even 16306 | 2 divides 0. That means 0... |
| n2dvds1 16307 | 2 does not divide 1. That... |
| n2dvdsm1 16308 | 2 does not divide -1. Tha... |
| z2even 16309 | 2 divides 2. That means 2... |
| n2dvds3 16310 | 2 does not divide 3. That... |
| z4even 16311 | 2 divides 4. That means 4... |
| 4dvdseven 16312 | An integer which is divisi... |
| m1expe 16313 | Exponentiation of -1 by an... |
| m1expo 16314 | Exponentiation of -1 by an... |
| m1exp1 16315 | Exponentiation of negative... |
| nn0enne 16316 | A positive integer is an e... |
| nn0ehalf 16317 | The half of an even nonneg... |
| nnehalf 16318 | The half of an even positi... |
| nn0onn 16319 | An odd nonnegative integer... |
| nn0o1gt2 16320 | An odd nonnegative integer... |
| nno 16321 | An alternate characterizat... |
| nn0o 16322 | An alternate characterizat... |
| nn0ob 16323 | Alternate characterization... |
| nn0oddm1d2 16324 | A positive integer is odd ... |
| nnoddm1d2 16325 | A positive integer is odd ... |
| sumeven 16326 | If every term in a sum is ... |
| sumodd 16327 | If every term in a sum is ... |
| evensumodd 16328 | If every term in a sum wit... |
| oddsumodd 16329 | If every term in a sum wit... |
| pwp1fsum 16330 | The n-th power of a number... |
| oddpwp1fsum 16331 | An odd power of a number i... |
| divalglem0 16332 | Lemma for ~ divalg . (Con... |
| divalglem1 16333 | Lemma for ~ divalg . (Con... |
| divalglem2 16334 | Lemma for ~ divalg . (Con... |
| divalglem4 16335 | Lemma for ~ divalg . (Con... |
| divalglem5 16336 | Lemma for ~ divalg . (Con... |
| divalglem6 16337 | Lemma for ~ divalg . (Con... |
| divalglem7 16338 | Lemma for ~ divalg . (Con... |
| divalglem8 16339 | Lemma for ~ divalg . (Con... |
| divalglem9 16340 | Lemma for ~ divalg . (Con... |
| divalglem10 16341 | Lemma for ~ divalg . (Con... |
| divalg 16342 | The division algorithm (th... |
| divalgb 16343 | Express the division algor... |
| divalg2 16344 | The division algorithm (th... |
| divalgmod 16345 | The result of the ` mod ` ... |
| divalgmodcl 16346 | The result of the ` mod ` ... |
| modremain 16347 | The result of the modulo o... |
| ndvdssub 16348 | Corollary of the division ... |
| ndvdsadd 16349 | Corollary of the division ... |
| ndvdsp1 16350 | Special case of ~ ndvdsadd... |
| ndvdsi 16351 | A quick test for non-divis... |
| flodddiv4 16352 | The floor of an odd intege... |
| fldivndvdslt 16353 | The floor of an integer di... |
| flodddiv4lt 16354 | The floor of an odd number... |
| flodddiv4t2lthalf 16355 | The floor of an odd number... |
| bitsfval 16360 | Expand the definition of t... |
| bitsval 16361 | Expand the definition of t... |
| bitsval2 16362 | Expand the definition of t... |
| bitsss 16363 | The set of bits of an inte... |
| bitsf 16364 | The ` bits ` function is a... |
| bits0 16365 | Value of the zeroth bit. ... |
| bits0e 16366 | The zeroth bit of an even ... |
| bits0o 16367 | The zeroth bit of an odd n... |
| bitsp1 16368 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16369 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16370 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16371 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16372 | The bits of a number are a... |
| bitsmod 16373 | Truncating the bit sequenc... |
| bitsfi 16374 | Every number is associated... |
| bitscmp 16375 | The bit complement of ` N ... |
| 0bits 16376 | The bits of zero. (Contri... |
| m1bits 16377 | The bits of negative one. ... |
| bitsinv1lem 16378 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16379 | There is an explicit inver... |
| bitsinv2 16380 | There is an explicit inver... |
| bitsf1ocnv 16381 | The ` bits ` function rest... |
| bitsf1o 16382 | The ` bits ` function rest... |
| bitsf1 16383 | The ` bits ` function is a... |
| 2ebits 16384 | The bits of a power of two... |
| bitsinv 16385 | The inverse of the ` bits ... |
| bitsinvp1 16386 | Recursive definition of th... |
| sadadd2lem2 16387 | The core of the proof of ~... |
| sadfval 16389 | Define the addition of two... |
| sadcf 16390 | The carry sequence is a se... |
| sadc0 16391 | The initial element of the... |
| sadcp1 16392 | The carry sequence (which ... |
| sadval 16393 | The full adder sequence is... |
| sadcaddlem 16394 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16395 | Non-recursive definition o... |
| sadadd2lem 16396 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16397 | Sum of initial segments of... |
| sadadd3 16398 | Sum of initial segments of... |
| sadcl 16399 | The sum of two sequences i... |
| sadcom 16400 | The adder sequence functio... |
| saddisjlem 16401 | Lemma for ~ sadadd . (Con... |
| saddisj 16402 | The sum of disjoint sequen... |
| sadaddlem 16403 | Lemma for ~ sadadd . (Con... |
| sadadd 16404 | For sequences that corresp... |
| sadid1 16405 | The adder sequence functio... |
| sadid2 16406 | The adder sequence functio... |
| sadasslem 16407 | Lemma for ~ sadass . (Con... |
| sadass 16408 | Sequence addition is assoc... |
| sadeq 16409 | Any element of a sequence ... |
| bitsres 16410 | Restrict the bits of a num... |
| bitsuz 16411 | The bits of a number are a... |
| bitsshft 16412 | Shifting a bit sequence to... |
| smufval 16414 | The multiplication of two ... |
| smupf 16415 | The sequence of partial su... |
| smup0 16416 | The initial element of the... |
| smupp1 16417 | The initial element of the... |
| smuval 16418 | Define the addition of two... |
| smuval2 16419 | The partial sum sequence s... |
| smupvallem 16420 | If ` A ` only has elements... |
| smucl 16421 | The product of two sequenc... |
| smu01lem 16422 | Lemma for ~ smu01 and ~ sm... |
| smu01 16423 | Multiplication of a sequen... |
| smu02 16424 | Multiplication of a sequen... |
| smupval 16425 | Rewrite the elements of th... |
| smup1 16426 | Rewrite ~ smupp1 using onl... |
| smueqlem 16427 | Any element of a sequence ... |
| smueq 16428 | Any element of a sequence ... |
| smumullem 16429 | Lemma for ~ smumul . (Con... |
| smumul 16430 | For sequences that corresp... |
| gcdval 16433 | The value of the ` gcd ` o... |
| gcd0val 16434 | The value, by convention, ... |
| gcdn0val 16435 | The value of the ` gcd ` o... |
| gcdcllem1 16436 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16437 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16438 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16439 | Closure of the ` gcd ` ope... |
| gcddvds 16440 | The gcd of two integers di... |
| dvdslegcd 16441 | An integer which divides b... |
| nndvdslegcd 16442 | A positive integer which d... |
| gcdcl 16443 | Closure of the ` gcd ` ope... |
| gcdnncl 16444 | Closure of the ` gcd ` ope... |
| gcdcld 16445 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16446 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16447 | An integer is the product ... |
| divgcdz 16448 | An integer divided by the ... |
| gcdf 16449 | Domain and codomain of the... |
| gcdcom 16450 | The ` gcd ` operator is co... |
| gcdcomd 16451 | The ` gcd ` operator is co... |
| divgcdnn 16452 | A positive integer divided... |
| divgcdnnr 16453 | A positive integer divided... |
| gcdeq0 16454 | The gcd of two integers is... |
| gcdn0gt0 16455 | The gcd of two integers is... |
| gcd0id 16456 | The gcd of 0 and an intege... |
| gcdid0 16457 | The gcd of an integer and ... |
| nn0gcdid0 16458 | The gcd of a nonnegative i... |
| gcdneg 16459 | Negating one operand of th... |
| neggcd 16460 | Negating one operand of th... |
| gcdaddmlem 16461 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16462 | Adding a multiple of one o... |
| gcdadd 16463 | The GCD of two numbers is ... |
| gcdid 16464 | The gcd of a number and it... |
| gcd1 16465 | The gcd of a number with 1... |
| gcdabs1 16466 | ` gcd ` of the absolute va... |
| gcdabs2 16467 | ` gcd ` of the absolute va... |
| gcdabs 16468 | The gcd of two integers is... |
| gcdabsOLD 16469 | Obsolete version of ~ gcda... |
| modgcd 16470 | The gcd remains unchanged ... |
| 1gcd 16471 | The GCD of one and an inte... |
| gcdmultipled 16472 | The greatest common diviso... |
| gcdmultiplez 16473 | The GCD of a multiple of a... |
| gcdmultiple 16474 | The GCD of a multiple of a... |
| dvdsgcdidd 16475 | The greatest common diviso... |
| 6gcd4e2 16476 | The greatest common diviso... |
| bezoutlem1 16477 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16478 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16479 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16480 | Lemma for ~ bezout . (Con... |
| bezout 16481 | Bézout's identity: ... |
| dvdsgcd 16482 | An integer which divides e... |
| dvdsgcdb 16483 | Biconditional form of ~ dv... |
| dfgcd2 16484 | Alternate definition of th... |
| gcdass 16485 | Associative law for ` gcd ... |
| mulgcd 16486 | Distribute multiplication ... |
| absmulgcd 16487 | Distribute absolute value ... |
| mulgcdr 16488 | Reverse distribution law f... |
| gcddiv 16489 | Division law for GCD. (Con... |
| gcdzeq 16490 | A positive integer ` A ` i... |
| gcdeq 16491 | ` A ` is equal to its gcd ... |
| dvdssqim 16492 | Unidirectional form of ~ d... |
| dvdsmulgcd 16493 | A divisibility equivalent ... |
| rpmulgcd 16494 | If ` K ` and ` M ` are rel... |
| rplpwr 16495 | If ` A ` and ` B ` are rel... |
| rprpwr 16496 | If ` A ` and ` B ` are rel... |
| rppwr 16497 | If ` A ` and ` B ` are rel... |
| sqgcd 16498 | Square distributes over gc... |
| dvdssqlem 16499 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16500 | Two numbers are divisible ... |
| bezoutr 16501 | Partial converse to ~ bezo... |
| bezoutr1 16502 | Converse of ~ bezout for w... |
| nn0seqcvgd 16503 | A strictly-decreasing nonn... |
| seq1st 16504 | A sequence whose iteration... |
| algr0 16505 | The value of the algorithm... |
| algrf 16506 | An algorithm is a step fun... |
| algrp1 16507 | The value of the algorithm... |
| alginv 16508 | If ` I ` is an invariant o... |
| algcvg 16509 | One way to prove that an a... |
| algcvgblem 16510 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16511 | Two ways of expressing tha... |
| algcvga 16512 | The countdown function ` C... |
| algfx 16513 | If ` F ` reaches a fixed p... |
| eucalgval2 16514 | The value of the step func... |
| eucalgval 16515 | Euclid's Algorithm ~ eucal... |
| eucalgf 16516 | Domain and codomain of the... |
| eucalginv 16517 | The invariant of the step ... |
| eucalglt 16518 | The second member of the s... |
| eucalgcvga 16519 | Once Euclid's Algorithm ha... |
| eucalg 16520 | Euclid's Algorithm compute... |
| lcmval 16525 | Value of the ` lcm ` opera... |
| lcmcom 16526 | The ` lcm ` operator is co... |
| lcm0val 16527 | The value, by convention, ... |
| lcmn0val 16528 | The value of the ` lcm ` o... |
| lcmcllem 16529 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16530 | Closure of the ` lcm ` ope... |
| dvdslcm 16531 | The lcm of two integers is... |
| lcmledvds 16532 | A positive integer which b... |
| lcmeq0 16533 | The lcm of two integers is... |
| lcmcl 16534 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16535 | The greatest common diviso... |
| lcmneg 16536 | Negating one operand of th... |
| neglcm 16537 | Negating one operand of th... |
| lcmabs 16538 | The lcm of two integers is... |
| lcmgcdlem 16539 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16540 | The product of two numbers... |
| lcmdvds 16541 | The lcm of two integers di... |
| lcmid 16542 | The lcm of an integer and ... |
| lcm1 16543 | The lcm of an integer and ... |
| lcmgcdnn 16544 | The product of two positiv... |
| lcmgcdeq 16545 | Two integers' absolute val... |
| lcmdvdsb 16546 | Biconditional form of ~ lc... |
| lcmass 16547 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16548 | The least common multiple ... |
| 6lcm4e12 16549 | The least common multiple ... |
| absproddvds 16550 | The absolute value of the ... |
| absprodnn 16551 | The absolute value of the ... |
| fissn0dvds 16552 | For each finite subset of ... |
| fissn0dvdsn0 16553 | For each finite subset of ... |
| lcmfval 16554 | Value of the ` _lcm ` func... |
| lcmf0val 16555 | The value, by convention, ... |
| lcmfn0val 16556 | The value of the ` _lcm ` ... |
| lcmfnnval 16557 | The value of the ` _lcm ` ... |
| lcmfcllem 16558 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16559 | Closure of the ` _lcm ` fu... |
| lcmfpr 16560 | The value of the ` _lcm ` ... |
| lcmfcl 16561 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16562 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16563 | The least common multiple ... |
| dvdslcmf 16564 | The least common multiple ... |
| lcmfledvds 16565 | A positive integer which i... |
| lcmf 16566 | Characterization of the le... |
| lcmf0 16567 | The least common multiple ... |
| lcmfsn 16568 | The least common multiple ... |
| lcmftp 16569 | The least common multiple ... |
| lcmfunsnlem1 16570 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16571 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16572 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16573 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16574 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16575 | The least common multiple ... |
| lcmfdvdsb 16576 | Biconditional form of ~ lc... |
| lcmfunsn 16577 | The ` _lcm ` function for ... |
| lcmfun 16578 | The ` _lcm ` function for ... |
| lcmfass 16579 | Associative law for the ` ... |
| lcmf2a3a4e12 16580 | The least common multiple ... |
| lcmflefac 16581 | The least common multiple ... |
| coprmgcdb 16582 | Two positive integers are ... |
| ncoprmgcdne1b 16583 | Two positive integers are ... |
| ncoprmgcdgt1b 16584 | Two positive integers are ... |
| coprmdvds1 16585 | If two positive integers a... |
| coprmdvds 16586 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16587 | If an integer is divisible... |
| mulgcddvds 16588 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16589 | If ` M ` is relatively pri... |
| qredeq 16590 | Two equal reduced fraction... |
| qredeu 16591 | Every rational number has ... |
| rpmul 16592 | If ` K ` is relatively pri... |
| rpdvds 16593 | If ` K ` is relatively pri... |
| coprmprod 16594 | The product of the element... |
| coprmproddvdslem 16595 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16596 | If a positive integer is d... |
| congr 16597 | Definition of congruence b... |
| divgcdcoprm0 16598 | Integers divided by gcd ar... |
| divgcdcoprmex 16599 | Integers divided by gcd ar... |
| cncongr1 16600 | One direction of the bicon... |
| cncongr2 16601 | The other direction of the... |
| cncongr 16602 | Cancellability of Congruen... |
| cncongrcoprm 16603 | Corollary 1 of Cancellabil... |
| isprm 16606 | The predicate "is a prime ... |
| prmnn 16607 | A prime number is a positi... |
| prmz 16608 | A prime number is an integ... |
| prmssnn 16609 | The prime numbers are a su... |
| prmex 16610 | The set of prime numbers e... |
| 0nprm 16611 | 0 is not a prime number. ... |
| 1nprm 16612 | 1 is not a prime number. ... |
| 1idssfct 16613 | The positive divisors of a... |
| isprm2lem 16614 | Lemma for ~ isprm2 . (Con... |
| isprm2 16615 | The predicate "is a prime ... |
| isprm3 16616 | The predicate "is a prime ... |
| isprm4 16617 | The predicate "is a prime ... |
| prmind2 16618 | A variation on ~ prmind as... |
| prmind 16619 | Perform induction over the... |
| dvdsprime 16620 | If ` M ` divides a prime, ... |
| nprm 16621 | A product of two integers ... |
| nprmi 16622 | An inference for composite... |
| dvdsnprmd 16623 | If a number is divisible b... |
| prm2orodd 16624 | A prime number is either 2... |
| 2prm 16625 | 2 is a prime number. (Con... |
| 2mulprm 16626 | A multiple of two is prime... |
| 3prm 16627 | 3 is a prime number. (Con... |
| 4nprm 16628 | 4 is not a prime number. ... |
| prmuz2 16629 | A prime number is an integ... |
| prmgt1 16630 | A prime number is an integ... |
| prmm2nn0 16631 | Subtracting 2 from a prime... |
| oddprmgt2 16632 | An odd prime is greater th... |
| oddprmge3 16633 | An odd prime is greater th... |
| ge2nprmge4 16634 | A composite integer greate... |
| sqnprm 16635 | A square is never prime. ... |
| dvdsprm 16636 | An integer greater than or... |
| exprmfct 16637 | Every integer greater than... |
| prmdvdsfz 16638 | Each integer greater than ... |
| nprmdvds1 16639 | No prime number divides 1.... |
| isprm5 16640 | One need only check prime ... |
| isprm7 16641 | One need only check prime ... |
| maxprmfct 16642 | The set of prime factors o... |
| divgcdodd 16643 | Either ` A / ( A gcd B ) `... |
| coprm 16644 | A prime number either divi... |
| prmrp 16645 | Unequal prime numbers are ... |
| euclemma 16646 | Euclid's lemma. A prime n... |
| isprm6 16647 | A number is prime iff it s... |
| prmdvdsexp 16648 | A prime divides a positive... |
| prmdvdsexpb 16649 | A prime divides a positive... |
| prmdvdsexpr 16650 | If a prime divides a nonne... |
| prmdvdssq 16651 | Condition for a prime divi... |
| prmdvdssqOLD 16652 | Obsolete version of ~ prmd... |
| prmexpb 16653 | Two positive prime powers ... |
| prmfac1 16654 | The factorial of a number ... |
| rpexp 16655 | If two numbers ` A ` and `... |
| rpexp1i 16656 | Relative primality passes ... |
| rpexp12i 16657 | Relative primality passes ... |
| prmndvdsfaclt 16658 | A prime number does not di... |
| prmdvdsncoprmbd 16659 | Two positive integers are ... |
| ncoprmlnprm 16660 | If two positive integers a... |
| cncongrprm 16661 | Corollary 2 of Cancellabil... |
| isevengcd2 16662 | The predicate "is an even ... |
| isoddgcd1 16663 | The predicate "is an odd n... |
| 3lcm2e6 16664 | The least common multiple ... |
| qnumval 16669 | Value of the canonical num... |
| qdenval 16670 | Value of the canonical den... |
| qnumdencl 16671 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16672 | The canonical numerator of... |
| qdencl 16673 | The canonical denominator ... |
| fnum 16674 | Canonical numerator define... |
| fden 16675 | Canonical denominator defi... |
| qnumdenbi 16676 | Two numbers are the canoni... |
| qnumdencoprm 16677 | The canonical representati... |
| qeqnumdivden 16678 | Recover a rational number ... |
| qmuldeneqnum 16679 | Multiplying a rational by ... |
| divnumden 16680 | Calculate the reduced form... |
| divdenle 16681 | Reducing a quotient never ... |
| qnumgt0 16682 | A rational is positive iff... |
| qgt0numnn 16683 | A rational is positive iff... |
| nn0gcdsq 16684 | Squaring commutes with GCD... |
| zgcdsq 16685 | ~ nn0gcdsq extended to int... |
| numdensq 16686 | Squaring a rational square... |
| numsq 16687 | Square commutes with canon... |
| densq 16688 | Square commutes with canon... |
| qden1elz 16689 | A rational is an integer i... |
| zsqrtelqelz 16690 | If an integer has a ration... |
| nonsq 16691 | Any integer strictly betwe... |
| phival 16696 | Value of the Euler ` phi `... |
| phicl2 16697 | Bounds and closure for the... |
| phicl 16698 | Closure for the value of t... |
| phibndlem 16699 | Lemma for ~ phibnd . (Con... |
| phibnd 16700 | A slightly tighter bound o... |
| phicld 16701 | Closure for the value of t... |
| phi1 16702 | Value of the Euler ` phi `... |
| dfphi2 16703 | Alternate definition of th... |
| hashdvds 16704 | The number of numbers in a... |
| phiprmpw 16705 | Value of the Euler ` phi `... |
| phiprm 16706 | Value of the Euler ` phi `... |
| crth 16707 | The Chinese Remainder Theo... |
| phimullem 16708 | Lemma for ~ phimul . (Con... |
| phimul 16709 | The Euler ` phi ` function... |
| eulerthlem1 16710 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16711 | Lemma for ~ eulerth . (Co... |
| eulerth 16712 | Euler's theorem, a general... |
| fermltl 16713 | Fermat's little theorem. ... |
| prmdiv 16714 | Show an explicit expressio... |
| prmdiveq 16715 | The modular inverse of ` A... |
| prmdivdiv 16716 | The (modular) inverse of t... |
| hashgcdlem 16717 | A correspondence between e... |
| hashgcdeq 16718 | Number of initial positive... |
| phisum 16719 | The divisor sum identity o... |
| odzval 16720 | Value of the order functio... |
| odzcllem 16721 | - Lemma for ~ odzcl , show... |
| odzcl 16722 | The order of a group eleme... |
| odzid 16723 | Any element raised to the ... |
| odzdvds 16724 | The only powers of ` A ` t... |
| odzphi 16725 | The order of any group ele... |
| modprm1div 16726 | A prime number divides an ... |
| m1dvdsndvds 16727 | If an integer minus 1 is d... |
| modprminv 16728 | Show an explicit expressio... |
| modprminveq 16729 | The modular inverse of ` A... |
| vfermltl 16730 | Variant of Fermat's little... |
| vfermltlALT 16731 | Alternate proof of ~ vferm... |
| powm2modprm 16732 | If an integer minus 1 is d... |
| reumodprminv 16733 | For any prime number and f... |
| modprm0 16734 | For two positive integers ... |
| nnnn0modprm0 16735 | For a positive integer and... |
| modprmn0modprm0 16736 | For an integer not being 0... |
| coprimeprodsq 16737 | If three numbers are copri... |
| coprimeprodsq2 16738 | If three numbers are copri... |
| oddprm 16739 | A prime not equal to ` 2 `... |
| nnoddn2prm 16740 | A prime not equal to ` 2 `... |
| oddn2prm 16741 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16742 | A number is a prime number... |
| prm23lt5 16743 | A prime less than 5 is eit... |
| prm23ge5 16744 | A prime is either 2 or 3 o... |
| pythagtriplem1 16745 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16746 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16747 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16748 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16749 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16750 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16751 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16752 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16753 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16754 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16755 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16756 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16757 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16758 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16759 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16760 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16761 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16762 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16763 | Parameterize the Pythagore... |
| iserodd 16764 | Collect the odd terms in a... |
| pclem 16767 | - Lemma for the prime powe... |
| pcprecl 16768 | Closure of the prime power... |
| pcprendvds 16769 | Non-divisibility property ... |
| pcprendvds2 16770 | Non-divisibility property ... |
| pcpre1 16771 | Value of the prime power p... |
| pcpremul 16772 | Multiplicative property of... |
| pcval 16773 | The value of the prime pow... |
| pceulem 16774 | Lemma for ~ pceu . (Contr... |
| pceu 16775 | Uniqueness for the prime p... |
| pczpre 16776 | Connect the prime count pr... |
| pczcl 16777 | Closure of the prime power... |
| pccl 16778 | Closure of the prime power... |
| pccld 16779 | Closure of the prime power... |
| pcmul 16780 | Multiplication property of... |
| pcdiv 16781 | Division property of the p... |
| pcqmul 16782 | Multiplication property of... |
| pc0 16783 | The value of the prime pow... |
| pc1 16784 | Value of the prime count f... |
| pcqcl 16785 | Closure of the general pri... |
| pcqdiv 16786 | Division property of the p... |
| pcrec 16787 | Prime power of a reciproca... |
| pcexp 16788 | Prime power of an exponent... |
| pcxnn0cl 16789 | Extended nonnegative integ... |
| pcxcl 16790 | Extended real closure of t... |
| pcge0 16791 | The prime count of an inte... |
| pczdvds 16792 | Defining property of the p... |
| pcdvds 16793 | Defining property of the p... |
| pczndvds 16794 | Defining property of the p... |
| pcndvds 16795 | Defining property of the p... |
| pczndvds2 16796 | The remainder after dividi... |
| pcndvds2 16797 | The remainder after dividi... |
| pcdvdsb 16798 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16799 | There are a positive numbe... |
| pceq0 16800 | There are zero powers of a... |
| pcidlem 16801 | The prime count of a prime... |
| pcid 16802 | The prime count of a prime... |
| pcneg 16803 | The prime count of a negat... |
| pcabs 16804 | The prime count of an abso... |
| pcdvdstr 16805 | The prime count increases ... |
| pcgcd1 16806 | The prime count of a GCD i... |
| pcgcd 16807 | The prime count of a GCD i... |
| pc2dvds 16808 | A characterization of divi... |
| pc11 16809 | The prime count function, ... |
| pcz 16810 | The prime count function c... |
| pcprmpw2 16811 | Self-referential expressio... |
| pcprmpw 16812 | Self-referential expressio... |
| dvdsprmpweq 16813 | If a positive integer divi... |
| dvdsprmpweqnn 16814 | If an integer greater than... |
| dvdsprmpweqle 16815 | If a positive integer divi... |
| difsqpwdvds 16816 | If the difference of two s... |
| pcaddlem 16817 | Lemma for ~ pcadd . The o... |
| pcadd 16818 | An inequality for the prim... |
| pcadd2 16819 | The inequality of ~ pcadd ... |
| pcmptcl 16820 | Closure for the prime powe... |
| pcmpt 16821 | Construct a function with ... |
| pcmpt2 16822 | Dividing two prime count m... |
| pcmptdvds 16823 | The partial products of th... |
| pcprod 16824 | The product of the primes ... |
| sumhash 16825 | The sum of 1 over a set is... |
| fldivp1 16826 | The difference between the... |
| pcfaclem 16827 | Lemma for ~ pcfac . (Cont... |
| pcfac 16828 | Calculate the prime count ... |
| pcbc 16829 | Calculate the prime count ... |
| qexpz 16830 | If a power of a rational n... |
| expnprm 16831 | A second or higher power o... |
| oddprmdvds 16832 | Every positive integer whi... |
| prmpwdvds 16833 | A relation involving divis... |
| pockthlem 16834 | Lemma for ~ pockthg . (Co... |
| pockthg 16835 | The generalized Pocklingto... |
| pockthi 16836 | Pocklington's theorem, whi... |
| unbenlem 16837 | Lemma for ~ unben . (Cont... |
| unben 16838 | An unbounded set of positi... |
| infpnlem1 16839 | Lemma for ~ infpn . The s... |
| infpnlem2 16840 | Lemma for ~ infpn . For a... |
| infpn 16841 | There exist infinitely man... |
| infpn2 16842 | There exist infinitely man... |
| prmunb 16843 | The primes are unbounded. ... |
| prminf 16844 | There are an infinite numb... |
| prmreclem1 16845 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16846 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16847 | Lemma for ~ prmrec . The ... |
| prmreclem4 16848 | Lemma for ~ prmrec . Show... |
| prmreclem5 16849 | Lemma for ~ prmrec . Here... |
| prmreclem6 16850 | Lemma for ~ prmrec . If t... |
| prmrec 16851 | The sum of the reciprocals... |
| 1arithlem1 16852 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16853 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16854 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16855 | Lemma for ~ 1arith . (Con... |
| 1arith 16856 | Fundamental theorem of ari... |
| 1arith2 16857 | Fundamental theorem of ari... |
| elgz 16860 | Elementhood in the gaussia... |
| gzcn 16861 | A gaussian integer is a co... |
| zgz 16862 | An integer is a gaussian i... |
| igz 16863 | ` _i ` is a gaussian integ... |
| gznegcl 16864 | The gaussian integers are ... |
| gzcjcl 16865 | The gaussian integers are ... |
| gzaddcl 16866 | The gaussian integers are ... |
| gzmulcl 16867 | The gaussian integers are ... |
| gzreim 16868 | Construct a gaussian integ... |
| gzsubcl 16869 | The gaussian integers are ... |
| gzabssqcl 16870 | The squared norm of a gaus... |
| 4sqlem5 16871 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16872 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16873 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16874 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16875 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16876 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16877 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16878 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16879 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16880 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16881 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16882 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16883 | Euler's four-square identi... |
| 4sqlem11 16884 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16885 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16886 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16887 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16888 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16889 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16890 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16891 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16892 | Lemma for ~ 4sq . The pro... |
| 4sq 16893 | Lagrange's four-square the... |
| vdwapfval 16900 | Define the arithmetic prog... |
| vdwapf 16901 | The arithmetic progression... |
| vdwapval 16902 | Value of the arithmetic pr... |
| vdwapun 16903 | Remove the first element o... |
| vdwapid1 16904 | The first element of an ar... |
| vdwap0 16905 | Value of a length-1 arithm... |
| vdwap1 16906 | Value of a length-1 arithm... |
| vdwmc 16907 | The predicate " The ` <. R... |
| vdwmc2 16908 | Expand out the definition ... |
| vdwpc 16909 | The predicate " The colori... |
| vdwlem1 16910 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16911 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16912 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16913 | Lemma for ~ vdw . (Contri... |
| vdwlem5 16914 | Lemma for ~ vdw . (Contri... |
| vdwlem6 16915 | Lemma for ~ vdw . (Contri... |
| vdwlem7 16916 | Lemma for ~ vdw . (Contri... |
| vdwlem8 16917 | Lemma for ~ vdw . (Contri... |
| vdwlem9 16918 | Lemma for ~ vdw . (Contri... |
| vdwlem10 16919 | Lemma for ~ vdw . Set up ... |
| vdwlem11 16920 | Lemma for ~ vdw . (Contri... |
| vdwlem12 16921 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 16922 | Lemma for ~ vdw . Main in... |
| vdw 16923 | Van der Waerden's theorem.... |
| vdwnnlem1 16924 | Corollary of ~ vdw , and l... |
| vdwnnlem2 16925 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 16926 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 16927 | Van der Waerden's theorem,... |
| ramtlecl 16929 | The set ` T ` of numbers w... |
| hashbcval 16931 | Value of the "binomial set... |
| hashbccl 16932 | The binomial set is a fini... |
| hashbcss 16933 | Subset relation for the bi... |
| hashbc0 16934 | The set of subsets of size... |
| hashbc2 16935 | The size of the binomial s... |
| 0hashbc 16936 | There are no subsets of th... |
| ramval 16937 | The value of the Ramsey nu... |
| ramcl2lem 16938 | Lemma for extended real cl... |
| ramtcl 16939 | The Ramsey number has the ... |
| ramtcl2 16940 | The Ramsey number is an in... |
| ramtub 16941 | The Ramsey number is a low... |
| ramub 16942 | The Ramsey number is a low... |
| ramub2 16943 | It is sufficient to check ... |
| rami 16944 | The defining property of a... |
| ramcl2 16945 | The Ramsey number is eithe... |
| ramxrcl 16946 | The Ramsey number is an ex... |
| ramubcl 16947 | If the Ramsey number is up... |
| ramlb 16948 | Establish a lower bound on... |
| 0ram 16949 | The Ramsey number when ` M... |
| 0ram2 16950 | The Ramsey number when ` M... |
| ram0 16951 | The Ramsey number when ` R... |
| 0ramcl 16952 | Lemma for ~ ramcl : Exist... |
| ramz2 16953 | The Ramsey number when ` F... |
| ramz 16954 | The Ramsey number when ` F... |
| ramub1lem1 16955 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 16956 | Lemma for ~ ramub1 . (Con... |
| ramub1 16957 | Inductive step for Ramsey'... |
| ramcl 16958 | Ramsey's theorem: the Rams... |
| ramsey 16959 | Ramsey's theorem with the ... |
| prmoval 16962 | Value of the primorial fun... |
| prmocl 16963 | Closure of the primorial f... |
| prmone0 16964 | The primorial function is ... |
| prmo0 16965 | The primorial of 0. (Cont... |
| prmo1 16966 | The primorial of 1. (Cont... |
| prmop1 16967 | The primorial of a success... |
| prmonn2 16968 | Value of the primorial fun... |
| prmo2 16969 | The primorial of 2. (Cont... |
| prmo3 16970 | The primorial of 3. (Cont... |
| prmdvdsprmo 16971 | The primorial of a number ... |
| prmdvdsprmop 16972 | The primorial of a number ... |
| fvprmselelfz 16973 | The value of the prime sel... |
| fvprmselgcd1 16974 | The greatest common diviso... |
| prmolefac 16975 | The primorial of a positiv... |
| prmodvdslcmf 16976 | The primorial of a nonnega... |
| prmolelcmf 16977 | The primorial of a positiv... |
| prmgaplem1 16978 | Lemma for ~ prmgap : The ... |
| prmgaplem2 16979 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 16980 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 16981 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 16982 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 16983 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 16984 | Lemma for ~ prmgap : for e... |
| prmgaplem6 16985 | Lemma for ~ prmgap : for e... |
| prmgaplem7 16986 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 16987 | Lemma for ~ prmgap . (Con... |
| prmgap 16988 | The prime gap theorem: for... |
| prmgaplcm 16989 | Alternate proof of ~ prmga... |
| prmgapprmolem 16990 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 16991 | Alternate proof of ~ prmga... |
| dec2dvds 16992 | Divisibility by two is obv... |
| dec5dvds 16993 | Divisibility by five is ob... |
| dec5dvds2 16994 | Divisibility by five is ob... |
| dec5nprm 16995 | Divisibility by five is ob... |
| dec2nprm 16996 | Divisibility by two is obv... |
| modxai 16997 | Add exponents in a power m... |
| mod2xi 16998 | Double exponents in a powe... |
| modxp1i 16999 | Add one to an exponent in ... |
| mod2xnegi 17000 | Version of ~ mod2xi with a... |
| modsubi 17001 | Subtract from within a mod... |
| gcdi 17002 | Calculate a GCD via Euclid... |
| gcdmodi 17003 | Calculate a GCD via Euclid... |
| decexp2 17004 | Calculate a power of two. ... |
| numexp0 17005 | Calculate an integer power... |
| numexp1 17006 | Calculate an integer power... |
| numexpp1 17007 | Calculate an integer power... |
| numexp2x 17008 | Double an integer power. ... |
| decsplit0b 17009 | Split a decimal number int... |
| decsplit0 17010 | Split a decimal number int... |
| decsplit1 17011 | Split a decimal number int... |
| decsplit 17012 | Split a decimal number int... |
| karatsuba 17013 | The Karatsuba multiplicati... |
| 2exp4 17014 | Two to the fourth power is... |
| 2exp5 17015 | Two to the fifth power is ... |
| 2exp6 17016 | Two to the sixth power is ... |
| 2exp7 17017 | Two to the seventh power i... |
| 2exp8 17018 | Two to the eighth power is... |
| 2exp11 17019 | Two to the eleventh power ... |
| 2exp16 17020 | Two to the sixteenth power... |
| 3exp3 17021 | Three to the third power i... |
| 2expltfac 17022 | The factorial grows faster... |
| cshwsidrepsw 17023 | If cyclically shifting a w... |
| cshwsidrepswmod0 17024 | If cyclically shifting a w... |
| cshwshashlem1 17025 | If cyclically shifting a w... |
| cshwshashlem2 17026 | If cyclically shifting a w... |
| cshwshashlem3 17027 | If cyclically shifting a w... |
| cshwsdisj 17028 | The singletons resulting b... |
| cshwsiun 17029 | The set of (different!) wo... |
| cshwsex 17030 | The class of (different!) ... |
| cshws0 17031 | The size of the set of (di... |
| cshwrepswhash1 17032 | The size of the set of (di... |
| cshwshashnsame 17033 | If a word (not consisting ... |
| cshwshash 17034 | If a word has a length bei... |
| prmlem0 17035 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 17036 | A quick proof skeleton to ... |
| prmlem1 17037 | A quick proof skeleton to ... |
| 5prm 17038 | 5 is a prime number. (Con... |
| 6nprm 17039 | 6 is not a prime number. ... |
| 7prm 17040 | 7 is a prime number. (Con... |
| 8nprm 17041 | 8 is not a prime number. ... |
| 9nprm 17042 | 9 is not a prime number. ... |
| 10nprm 17043 | 10 is not a prime number. ... |
| 11prm 17044 | 11 is a prime number. (Co... |
| 13prm 17045 | 13 is a prime number. (Co... |
| 17prm 17046 | 17 is a prime number. (Co... |
| 19prm 17047 | 19 is a prime number. (Co... |
| 23prm 17048 | 23 is a prime number. (Co... |
| prmlem2 17049 | Our last proving session g... |
| 37prm 17050 | 37 is a prime number. (Co... |
| 43prm 17051 | 43 is a prime number. (Co... |
| 83prm 17052 | 83 is a prime number. (Co... |
| 139prm 17053 | 139 is a prime number. (C... |
| 163prm 17054 | 163 is a prime number. (C... |
| 317prm 17055 | 317 is a prime number. (C... |
| 631prm 17056 | 631 is a prime number. (C... |
| prmo4 17057 | The primorial of 4. (Cont... |
| prmo5 17058 | The primorial of 5. (Cont... |
| prmo6 17059 | The primorial of 6. (Cont... |
| 1259lem1 17060 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17061 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17062 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17063 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17064 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17065 | 1259 is a prime number. (... |
| 2503lem1 17066 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17067 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17068 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17069 | 2503 is a prime number. (... |
| 4001lem1 17070 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17071 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17072 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17073 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17074 | 4001 is a prime number. (... |
| brstruct 17077 | The structure relation is ... |
| isstruct2 17078 | The property of being a st... |
| structex 17079 | A structure is a set. (Co... |
| structn0fun 17080 | A structure without the em... |
| isstruct 17081 | The property of being a st... |
| structcnvcnv 17082 | Two ways to express the re... |
| structfung 17083 | The converse of the conver... |
| structfun 17084 | Convert between two kinds ... |
| structfn 17085 | Convert between two kinds ... |
| strleun 17086 | Combine two structures int... |
| strle1 17087 | Make a structure from a si... |
| strle2 17088 | Make a structure from a pa... |
| strle3 17089 | Make a structure from a tr... |
| sbcie2s 17090 | A special version of class... |
| sbcie3s 17091 | A special version of class... |
| reldmsets 17094 | The structure override ope... |
| setsvalg 17095 | Value of the structure rep... |
| setsval 17096 | Value of the structure rep... |
| fvsetsid 17097 | The value of the structure... |
| fsets 17098 | The structure replacement ... |
| setsdm 17099 | The domain of a structure ... |
| setsfun 17100 | A structure with replaceme... |
| setsfun0 17101 | A structure with replaceme... |
| setsn0fun 17102 | The value of the structure... |
| setsstruct2 17103 | An extensible structure wi... |
| setsexstruct2 17104 | An extensible structure wi... |
| setsstruct 17105 | An extensible structure wi... |
| wunsets 17106 | Closure of structure repla... |
| setsres 17107 | The structure replacement ... |
| setsabs 17108 | Replacing the same compone... |
| setscom 17109 | Different components can b... |
| sloteq 17112 | Equality theorem for the `... |
| slotfn 17113 | A slot is a function on se... |
| strfvnd 17114 | Deduction version of ~ str... |
| strfvn 17115 | Value of a structure compo... |
| strfvss 17116 | A structure component extr... |
| wunstr 17117 | Closure of a structure ind... |
| str0 17118 | All components of the empt... |
| strfvi 17119 | Structure slot extractors ... |
| fveqprc 17120 | Lemma for showing the equa... |
| oveqprc 17121 | Lemma for showing the equa... |
| wunndx 17124 | Closure of the index extra... |
| ndxarg 17125 | Get the numeric argument f... |
| ndxid 17126 | A structure component extr... |
| strndxid 17127 | The value of a structure c... |
| setsidvald 17128 | Value of the structure rep... |
| setsidvaldOLD 17129 | Obsolete version of ~ sets... |
| strfvd 17130 | Deduction version of ~ str... |
| strfv2d 17131 | Deduction version of ~ str... |
| strfv2 17132 | A variation on ~ strfv to ... |
| strfv 17133 | Extract a structure compon... |
| strfv3 17134 | Variant on ~ strfv for lar... |
| strssd 17135 | Deduction version of ~ str... |
| strss 17136 | Propagate component extrac... |
| setsid 17137 | Value of the structure rep... |
| setsnid 17138 | Value of the structure rep... |
| setsnidOLD 17139 | Obsolete proof of ~ setsni... |
| baseval 17142 | Value of the base set extr... |
| baseid 17143 | Utility theorem: index-ind... |
| basfn 17144 | The base set extractor is ... |
| base0 17145 | The base set of the empty ... |
| elbasfv 17146 | Utility theorem: reverse c... |
| elbasov 17147 | Utility theorem: reverse c... |
| strov2rcl 17148 | Partial reverse closure fo... |
| basendx 17149 | Index value of the base se... |
| basendxnn 17150 | The index value of the bas... |
| basendxnnOLD 17151 | Obsolete proof of ~ basend... |
| basndxelwund 17152 | The index of the base set ... |
| basprssdmsets 17153 | The pair of the base index... |
| opelstrbas 17154 | The base set of a structur... |
| 1strstr 17155 | A constructed one-slot str... |
| 1strstr1 17156 | A constructed one-slot str... |
| 1strbas 17157 | The base set of a construc... |
| 1strbasOLD 17158 | Obsolete proof of ~ 1strba... |
| 1strwunbndx 17159 | A constructed one-slot str... |
| 1strwun 17160 | A constructed one-slot str... |
| 1strwunOLD 17161 | Obsolete version of ~ 1str... |
| 2strstr 17162 | A constructed two-slot str... |
| 2strbas 17163 | The base set of a construc... |
| 2strop 17164 | The other slot of a constr... |
| 2strstr1 17165 | A constructed two-slot str... |
| 2strstr1OLD 17166 | Obsolete version of ~ 2str... |
| 2strbas1 17167 | The base set of a construc... |
| 2strop1 17168 | The other slot of a constr... |
| reldmress 17171 | The structure restriction ... |
| ressval 17172 | Value of structure restric... |
| ressid2 17173 | General behavior of trivia... |
| ressval2 17174 | Value of nontrivial struct... |
| ressbas 17175 | Base set of a structure re... |
| ressbasOLD 17176 | Obsolete proof of ~ ressba... |
| ressbasssg 17177 | The base set of a restrict... |
| ressbas2 17178 | Base set of a structure re... |
| ressbasss 17179 | The base set of a restrict... |
| ressbasssOLD 17180 | Obsolete proof of ~ ressba... |
| ressbasss2 17181 | The base set of a restrict... |
| resseqnbas 17182 | The components of an exten... |
| resslemOLD 17183 | Obsolete version of ~ ress... |
| ress0 17184 | All restrictions of the nu... |
| ressid 17185 | Behavior of trivial restri... |
| ressinbas 17186 | Restriction only cares abo... |
| ressval3d 17187 | Value of structure restric... |
| ressval3dOLD 17188 | Obsolete version of ~ ress... |
| ressress 17189 | Restriction composition la... |
| ressabs 17190 | Restriction absorption law... |
| wunress 17191 | Closure of structure restr... |
| wunressOLD 17192 | Obsolete proof of ~ wunres... |
| plusgndx 17219 | Index value of the ~ df-pl... |
| plusgid 17220 | Utility theorem: index-ind... |
| plusgndxnn 17221 | The index of the slot for ... |
| basendxltplusgndx 17222 | The index of the slot for ... |
| basendxnplusgndx 17223 | The slot for the base set ... |
| basendxnplusgndxOLD 17224 | Obsolete version of ~ base... |
| grpstr 17225 | A constructed group is a s... |
| grpstrndx 17226 | A constructed group is a s... |
| grpbase 17227 | The base set of a construc... |
| grpbaseOLD 17228 | Obsolete version of ~ grpb... |
| grpplusg 17229 | The operation of a constru... |
| grpplusgOLD 17230 | Obsolete version of ~ grpp... |
| ressplusg 17231 | ` +g ` is unaffected by re... |
| grpbasex 17232 | The base of an explicitly ... |
| grpplusgx 17233 | The operation of an explic... |
| mulrndx 17234 | Index value of the ~ df-mu... |
| mulridx 17235 | Utility theorem: index-ind... |
| basendxnmulrndx 17236 | The slot for the base set ... |
| basendxnmulrndxOLD 17237 | Obsolete proof of ~ basend... |
| plusgndxnmulrndx 17238 | The slot for the group (ad... |
| rngstr 17239 | A constructed ring is a st... |
| rngbase 17240 | The base set of a construc... |
| rngplusg 17241 | The additive operation of ... |
| rngmulr 17242 | The multiplicative operati... |
| starvndx 17243 | Index value of the ~ df-st... |
| starvid 17244 | Utility theorem: index-ind... |
| starvndxnbasendx 17245 | The slot for the involutio... |
| starvndxnplusgndx 17246 | The slot for the involutio... |
| starvndxnmulrndx 17247 | The slot for the involutio... |
| ressmulr 17248 | ` .r ` is unaffected by re... |
| ressstarv 17249 | ` *r ` is unaffected by re... |
| srngstr 17250 | A constructed star ring is... |
| srngbase 17251 | The base set of a construc... |
| srngplusg 17252 | The addition operation of ... |
| srngmulr 17253 | The multiplication operati... |
| srnginvl 17254 | The involution function of... |
| scandx 17255 | Index value of the ~ df-sc... |
| scaid 17256 | Utility theorem: index-ind... |
| scandxnbasendx 17257 | The slot for the scalar is... |
| scandxnplusgndx 17258 | The slot for the scalar fi... |
| scandxnmulrndx 17259 | The slot for the scalar fi... |
| vscandx 17260 | Index value of the ~ df-vs... |
| vscaid 17261 | Utility theorem: index-ind... |
| vscandxnbasendx 17262 | The slot for the scalar pr... |
| vscandxnplusgndx 17263 | The slot for the scalar pr... |
| vscandxnmulrndx 17264 | The slot for the scalar pr... |
| vscandxnscandx 17265 | The slot for the scalar pr... |
| lmodstr 17266 | A constructed left module ... |
| lmodbase 17267 | The base set of a construc... |
| lmodplusg 17268 | The additive operation of ... |
| lmodsca 17269 | The set of scalars of a co... |
| lmodvsca 17270 | The scalar product operati... |
| ipndx 17271 | Index value of the ~ df-ip... |
| ipid 17272 | Utility theorem: index-ind... |
| ipndxnbasendx 17273 | The slot for the inner pro... |
| ipndxnplusgndx 17274 | The slot for the inner pro... |
| ipndxnmulrndx 17275 | The slot for the inner pro... |
| slotsdifipndx 17276 | The slot for the scalar is... |
| ipsstr 17277 | Lemma to shorten proofs of... |
| ipsbase 17278 | The base set of a construc... |
| ipsaddg 17279 | The additive operation of ... |
| ipsmulr 17280 | The multiplicative operati... |
| ipssca 17281 | The set of scalars of a co... |
| ipsvsca 17282 | The scalar product operati... |
| ipsip 17283 | The multiplicative operati... |
| resssca 17284 | ` Scalar ` is unaffected b... |
| ressvsca 17285 | ` .s ` is unaffected by re... |
| ressip 17286 | The inner product is unaff... |
| phlstr 17287 | A constructed pre-Hilbert ... |
| phlbase 17288 | The base set of a construc... |
| phlplusg 17289 | The additive operation of ... |
| phlsca 17290 | The ring of scalars of a c... |
| phlvsca 17291 | The scalar product operati... |
| phlip 17292 | The inner product (Hermiti... |
| tsetndx 17293 | Index value of the ~ df-ts... |
| tsetid 17294 | Utility theorem: index-ind... |
| tsetndxnn 17295 | The index of the slot for ... |
| basendxlttsetndx 17296 | The index of the slot for ... |
| tsetndxnbasendx 17297 | The slot for the topology ... |
| tsetndxnplusgndx 17298 | The slot for the topology ... |
| tsetndxnmulrndx 17299 | The slot for the topology ... |
| tsetndxnstarvndx 17300 | The slot for the topology ... |
| slotstnscsi 17301 | The slots ` Scalar ` , ` .... |
| topgrpstr 17302 | A constructed topological ... |
| topgrpbas 17303 | The base set of a construc... |
| topgrpplusg 17304 | The additive operation of ... |
| topgrptset 17305 | The topology of a construc... |
| resstset 17306 | ` TopSet ` is unaffected b... |
| plendx 17307 | Index value of the ~ df-pl... |
| pleid 17308 | Utility theorem: self-refe... |
| plendxnn 17309 | The index value of the ord... |
| basendxltplendx 17310 | The index value of the ` B... |
| plendxnbasendx 17311 | The slot for the order is ... |
| plendxnplusgndx 17312 | The slot for the "less tha... |
| plendxnmulrndx 17313 | The slot for the "less tha... |
| plendxnscandx 17314 | The slot for the "less tha... |
| plendxnvscandx 17315 | The slot for the "less tha... |
| slotsdifplendx 17316 | The index of the slot for ... |
| otpsstr 17317 | Functionality of a topolog... |
| otpsbas 17318 | The base set of a topologi... |
| otpstset 17319 | The open sets of a topolog... |
| otpsle 17320 | The order of a topological... |
| ressle 17321 | ` le ` is unaffected by re... |
| ocndx 17322 | Index value of the ~ df-oc... |
| ocid 17323 | Utility theorem: index-ind... |
| basendxnocndx 17324 | The slot for the orthocomp... |
| plendxnocndx 17325 | The slot for the orthocomp... |
| dsndx 17326 | Index value of the ~ df-ds... |
| dsid 17327 | Utility theorem: index-ind... |
| dsndxnn 17328 | The index of the slot for ... |
| basendxltdsndx 17329 | The index of the slot for ... |
| dsndxnbasendx 17330 | The slot for the distance ... |
| dsndxnplusgndx 17331 | The slot for the distance ... |
| dsndxnmulrndx 17332 | The slot for the distance ... |
| slotsdnscsi 17333 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17334 | The slot for the distance ... |
| slotsdifdsndx 17335 | The index of the slot for ... |
| unifndx 17336 | Index value of the ~ df-un... |
| unifid 17337 | Utility theorem: index-ind... |
| unifndxnn 17338 | The index of the slot for ... |
| basendxltunifndx 17339 | The index of the slot for ... |
| unifndxnbasendx 17340 | The slot for the uniform s... |
| unifndxntsetndx 17341 | The slot for the uniform s... |
| slotsdifunifndx 17342 | The index of the slot for ... |
| ressunif 17343 | ` UnifSet ` is unaffected ... |
| odrngstr 17344 | Functionality of an ordere... |
| odrngbas 17345 | The base set of an ordered... |
| odrngplusg 17346 | The addition operation of ... |
| odrngmulr 17347 | The multiplication operati... |
| odrngtset 17348 | The open sets of an ordere... |
| odrngle 17349 | The order of an ordered me... |
| odrngds 17350 | The metric of an ordered m... |
| ressds 17351 | ` dist ` is unaffected by ... |
| homndx 17352 | Index value of the ~ df-ho... |
| homid 17353 | Utility theorem: index-ind... |
| ccondx 17354 | Index value of the ~ df-cc... |
| ccoid 17355 | Utility theorem: index-ind... |
| slotsbhcdif 17356 | The slots ` Base ` , ` Hom... |
| slotsbhcdifOLD 17357 | Obsolete proof of ~ slotsb... |
| slotsdifplendx2 17358 | The index of the slot for ... |
| slotsdifocndx 17359 | The index of the slot for ... |
| resshom 17360 | ` Hom ` is unaffected by r... |
| ressco 17361 | ` comp ` is unaffected by ... |
| restfn 17366 | The subspace topology oper... |
| topnfn 17367 | The topology extractor fun... |
| restval 17368 | The subspace topology indu... |
| elrest 17369 | The predicate "is an open ... |
| elrestr 17370 | Sufficient condition for b... |
| 0rest 17371 | Value of the structure res... |
| restid2 17372 | The subspace topology over... |
| restsspw 17373 | The subspace topology is a... |
| firest 17374 | The finite intersections o... |
| restid 17375 | The subspace topology of t... |
| topnval 17376 | Value of the topology extr... |
| topnid 17377 | Value of the topology extr... |
| topnpropd 17378 | The topology extractor fun... |
| reldmprds 17390 | The structure product is a... |
| prdsbasex 17392 | Lemma for structure produc... |
| imasvalstr 17393 | An image structure value i... |
| prdsvalstr 17394 | Structure product value is... |
| prdsbaslem 17395 | Lemma for ~ prdsbas and si... |
| prdsvallem 17396 | Lemma for ~ prdsval . (Co... |
| prdsval 17397 | Value of the structure pro... |
| prdssca 17398 | Scalar ring of a structure... |
| prdsbas 17399 | Base set of a structure pr... |
| prdsplusg 17400 | Addition in a structure pr... |
| prdsmulr 17401 | Multiplication in a struct... |
| prdsvsca 17402 | Scalar multiplication in a... |
| prdsip 17403 | Inner product in a structu... |
| prdsle 17404 | Structure product weak ord... |
| prdsless 17405 | Closure of the order relat... |
| prdsds 17406 | Structure product distance... |
| prdsdsfn 17407 | Structure product distance... |
| prdstset 17408 | Structure product topology... |
| prdshom 17409 | Structure product hom-sets... |
| prdsco 17410 | Structure product composit... |
| prdsbas2 17411 | The base set of a structur... |
| prdsbasmpt 17412 | A constructed tuple is a p... |
| prdsbasfn 17413 | Points in the structure pr... |
| prdsbasprj 17414 | Each point in a structure ... |
| prdsplusgval 17415 | Value of a componentwise s... |
| prdsplusgfval 17416 | Value of a structure produ... |
| prdsmulrval 17417 | Value of a componentwise r... |
| prdsmulrfval 17418 | Value of a structure produ... |
| prdsleval 17419 | Value of the product order... |
| prdsdsval 17420 | Value of the metric in a s... |
| prdsvscaval 17421 | Scalar multiplication in a... |
| prdsvscafval 17422 | Scalar multiplication of a... |
| prdsbas3 17423 | The base set of an indexed... |
| prdsbasmpt2 17424 | A constructed tuple is a p... |
| prdsbascl 17425 | An element of the base has... |
| prdsdsval2 17426 | Value of the metric in a s... |
| prdsdsval3 17427 | Value of the metric in a s... |
| pwsval 17428 | Value of a structure power... |
| pwsbas 17429 | Base set of a structure po... |
| pwselbasb 17430 | Membership in the base set... |
| pwselbas 17431 | An element of a structure ... |
| pwsplusgval 17432 | Value of addition in a str... |
| pwsmulrval 17433 | Value of multiplication in... |
| pwsle 17434 | Ordering in a structure po... |
| pwsleval 17435 | Ordering in a structure po... |
| pwsvscafval 17436 | Scalar multiplication in a... |
| pwsvscaval 17437 | Scalar multiplication of a... |
| pwssca 17438 | The ring of scalars of a s... |
| pwsdiagel 17439 | Membership of diagonal ele... |
| pwssnf1o 17440 | Triviality of singleton po... |
| imasval 17453 | Value of an image structur... |
| imasbas 17454 | The base set of an image s... |
| imasds 17455 | The distance function of a... |
| imasdsfn 17456 | The distance function is a... |
| imasdsval 17457 | The distance function of a... |
| imasdsval2 17458 | The distance function of a... |
| imasplusg 17459 | The group operation in an ... |
| imasmulr 17460 | The ring multiplication in... |
| imassca 17461 | The scalar field of an ima... |
| imasvsca 17462 | The scalar multiplication ... |
| imasip 17463 | The inner product of an im... |
| imastset 17464 | The topology of an image s... |
| imasle 17465 | The ordering of an image s... |
| f1ocpbllem 17466 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17467 | An injection is compatible... |
| f1ovscpbl 17468 | An injection is compatible... |
| f1olecpbl 17469 | An injection is compatible... |
| imasaddfnlem 17470 | The image structure operat... |
| imasaddvallem 17471 | The operation of an image ... |
| imasaddflem 17472 | The image set operations a... |
| imasaddfn 17473 | The image structure's grou... |
| imasaddval 17474 | The value of an image stru... |
| imasaddf 17475 | The image structure's grou... |
| imasmulfn 17476 | The image structure's ring... |
| imasmulval 17477 | The value of an image stru... |
| imasmulf 17478 | The image structure's ring... |
| imasvscafn 17479 | The image structure's scal... |
| imasvscaval 17480 | The value of an image stru... |
| imasvscaf 17481 | The image structure's scal... |
| imasless 17482 | The order relation defined... |
| imasleval 17483 | The value of the image str... |
| qusval 17484 | Value of a quotient struct... |
| quslem 17485 | The function in ~ qusval i... |
| qusin 17486 | Restrict the equivalence r... |
| qusbas 17487 | Base set of a quotient str... |
| quss 17488 | The scalar field of a quot... |
| divsfval 17489 | Value of the function in ~... |
| ercpbllem 17490 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17491 | Translate the function com... |
| erlecpbl 17492 | Translate the relation com... |
| qusaddvallem 17493 | Value of an operation defi... |
| qusaddflem 17494 | The operation of a quotien... |
| qusaddval 17495 | The base set of an image s... |
| qusaddf 17496 | The base set of an image s... |
| qusmulval 17497 | The base set of an image s... |
| qusmulf 17498 | The base set of an image s... |
| fnpr2o 17499 | Function with a domain of ... |
| fnpr2ob 17500 | Biconditional version of ~... |
| fvpr0o 17501 | The value of a function wi... |
| fvpr1o 17502 | The value of a function wi... |
| fvprif 17503 | The value of the pair func... |
| xpsfrnel 17504 | Elementhood in the target ... |
| xpsfeq 17505 | A function on ` 2o ` is de... |
| xpsfrnel2 17506 | Elementhood in the target ... |
| xpscf 17507 | Equivalent condition for t... |
| xpsfval 17508 | The value of the function ... |
| xpsff1o 17509 | The function appearing in ... |
| xpsfrn 17510 | A short expression for the... |
| xpsff1o2 17511 | The function appearing in ... |
| xpsval 17512 | Value of the binary struct... |
| xpsrnbas 17513 | The indexed structure prod... |
| xpsbas 17514 | The base set of the binary... |
| xpsaddlem 17515 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17516 | Value of the addition oper... |
| xpsmul 17517 | Value of the multiplicatio... |
| xpssca 17518 | Value of the scalar field ... |
| xpsvsca 17519 | Value of the scalar multip... |
| xpsless 17520 | Closure of the ordering in... |
| xpsle 17521 | Value of the ordering in a... |
| ismre 17530 | Property of being a Moore ... |
| fnmre 17531 | The Moore collection gener... |
| mresspw 17532 | A Moore collection is a su... |
| mress 17533 | A Moore-closed subset is a... |
| mre1cl 17534 | In any Moore collection th... |
| mreintcl 17535 | A nonempty collection of c... |
| mreiincl 17536 | A nonempty indexed interse... |
| mrerintcl 17537 | The relative intersection ... |
| mreriincl 17538 | The relative intersection ... |
| mreincl 17539 | Two closed sets have a clo... |
| mreuni 17540 | Since the entire base set ... |
| mreunirn 17541 | Two ways to express the no... |
| ismred 17542 | Properties that determine ... |
| ismred2 17543 | Properties that determine ... |
| mremre 17544 | The Moore collections of s... |
| submre 17545 | The subcollection of a clo... |
| mrcflem 17546 | The domain and codomain of... |
| fnmrc 17547 | Moore-closure is a well-be... |
| mrcfval 17548 | Value of the function expr... |
| mrcf 17549 | The Moore closure is a fun... |
| mrcval 17550 | Evaluation of the Moore cl... |
| mrccl 17551 | The Moore closure of a set... |
| mrcsncl 17552 | The Moore closure of a sin... |
| mrcid 17553 | The closure of a closed se... |
| mrcssv 17554 | The closure of a set is a ... |
| mrcidb 17555 | A set is closed iff it is ... |
| mrcss 17556 | Closure preserves subset o... |
| mrcssid 17557 | The closure of a set is a ... |
| mrcidb2 17558 | A set is closed iff it con... |
| mrcidm 17559 | The closure operation is i... |
| mrcsscl 17560 | The closure is the minimal... |
| mrcuni 17561 | Idempotence of closure und... |
| mrcun 17562 | Idempotence of closure und... |
| mrcssvd 17563 | The Moore closure of a set... |
| mrcssd 17564 | Moore closure preserves su... |
| mrcssidd 17565 | A set is contained in its ... |
| mrcidmd 17566 | Moore closure is idempoten... |
| mressmrcd 17567 | In a Moore system, if a se... |
| submrc 17568 | In a closure system which ... |
| mrieqvlemd 17569 | In a Moore system, if ` Y ... |
| mrisval 17570 | Value of the set of indepe... |
| ismri 17571 | Criterion for a set to be ... |
| ismri2 17572 | Criterion for a subset of ... |
| ismri2d 17573 | Criterion for a subset of ... |
| ismri2dd 17574 | Definition of independence... |
| mriss 17575 | An independent set of a Mo... |
| mrissd 17576 | An independent set of a Mo... |
| ismri2dad 17577 | Consequence of a set in a ... |
| mrieqvd 17578 | In a Moore system, a set i... |
| mrieqv2d 17579 | In a Moore system, a set i... |
| mrissmrcd 17580 | In a Moore system, if an i... |
| mrissmrid 17581 | In a Moore system, subsets... |
| mreexd 17582 | In a Moore system, the clo... |
| mreexmrid 17583 | In a Moore system whose cl... |
| mreexexlemd 17584 | This lemma is used to gene... |
| mreexexlem2d 17585 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17586 | Base case of the induction... |
| mreexexlem4d 17587 | Induction step of the indu... |
| mreexexd 17588 | Exchange-type theorem. In... |
| mreexdomd 17589 | In a Moore system whose cl... |
| mreexfidimd 17590 | In a Moore system whose cl... |
| isacs 17591 | A set is an algebraic clos... |
| acsmre 17592 | Algebraic closure systems ... |
| isacs2 17593 | In the definition of an al... |
| acsfiel 17594 | A set is closed in an alge... |
| acsfiel2 17595 | A set is closed in an alge... |
| acsmred 17596 | An algebraic closure syste... |
| isacs1i 17597 | A closure system determine... |
| mreacs 17598 | Algebraicity is a composab... |
| acsfn 17599 | Algebraicity of a conditio... |
| acsfn0 17600 | Algebraicity of a point cl... |
| acsfn1 17601 | Algebraicity of a one-argu... |
| acsfn1c 17602 | Algebraicity of a one-argu... |
| acsfn2 17603 | Algebraicity of a two-argu... |
| iscat 17612 | The predicate "is a catego... |
| iscatd 17613 | Properties that determine ... |
| catidex 17614 | Each object in a category ... |
| catideu 17615 | Each object in a category ... |
| cidfval 17616 | Each object in a category ... |
| cidval 17617 | Each object in a category ... |
| cidffn 17618 | The identity arrow constru... |
| cidfn 17619 | The identity arrow operato... |
| catidd 17620 | Deduce the identity arrow ... |
| iscatd2 17621 | Version of ~ iscatd with a... |
| catidcl 17622 | Each object in a category ... |
| catlid 17623 | Left identity property of ... |
| catrid 17624 | Right identity property of... |
| catcocl 17625 | Closure of a composition a... |
| catass 17626 | Associativity of compositi... |
| catcone0 17627 | Composition of non-empty h... |
| 0catg 17628 | Any structure with an empt... |
| 0cat 17629 | The empty set is a categor... |
| homffval 17630 | Value of the functionalize... |
| fnhomeqhomf 17631 | If the Hom-set operation i... |
| homfval 17632 | Value of the functionalize... |
| homffn 17633 | The functionalized Hom-set... |
| homfeq 17634 | Condition for two categori... |
| homfeqd 17635 | If two structures have the... |
| homfeqbas 17636 | Deduce equality of base se... |
| homfeqval 17637 | Value of the functionalize... |
| comfffval 17638 | Value of the functionalize... |
| comffval 17639 | Value of the functionalize... |
| comfval 17640 | Value of the functionalize... |
| comfffval2 17641 | Value of the functionalize... |
| comffval2 17642 | Value of the functionalize... |
| comfval2 17643 | Value of the functionalize... |
| comfffn 17644 | The functionalized composi... |
| comffn 17645 | The functionalized composi... |
| comfeq 17646 | Condition for two categori... |
| comfeqd 17647 | Condition for two categori... |
| comfeqval 17648 | Equality of two compositio... |
| catpropd 17649 | Two structures with the sa... |
| cidpropd 17650 | Two structures with the sa... |
| oppcval 17653 | Value of the opposite cate... |
| oppchomfval 17654 | Hom-sets of the opposite c... |
| oppchomfvalOLD 17655 | Obsolete proof of ~ oppcho... |
| oppchom 17656 | Hom-sets of the opposite c... |
| oppccofval 17657 | Composition in the opposit... |
| oppcco 17658 | Composition in the opposit... |
| oppcbas 17659 | Base set of an opposite ca... |
| oppcbasOLD 17660 | Obsolete version of ~ oppc... |
| oppccatid 17661 | Lemma for ~ oppccat . (Co... |
| oppchomf 17662 | Hom-sets of the opposite c... |
| oppcid 17663 | Identity function of an op... |
| oppccat 17664 | An opposite category is a ... |
| 2oppcbas 17665 | The double opposite catego... |
| 2oppchomf 17666 | The double opposite catego... |
| 2oppccomf 17667 | The double opposite catego... |
| oppchomfpropd 17668 | If two categories have the... |
| oppccomfpropd 17669 | If two categories have the... |
| oppccatf 17670 | ` oppCat ` restricted to `... |
| monfval 17675 | Definition of a monomorphi... |
| ismon 17676 | Definition of a monomorphi... |
| ismon2 17677 | Write out the monomorphism... |
| monhom 17678 | A monomorphism is a morphi... |
| moni 17679 | Property of a monomorphism... |
| monpropd 17680 | If two categories have the... |
| oppcmon 17681 | A monomorphism in the oppo... |
| oppcepi 17682 | An epimorphism in the oppo... |
| isepi 17683 | Definition of an epimorphi... |
| isepi2 17684 | Write out the epimorphism ... |
| epihom 17685 | An epimorphism is a morphi... |
| epii 17686 | Property of an epimorphism... |
| sectffval 17693 | Value of the section opera... |
| sectfval 17694 | Value of the section relat... |
| sectss 17695 | The section relation is a ... |
| issect 17696 | The property " ` F ` is a ... |
| issect2 17697 | Property of being a sectio... |
| sectcan 17698 | If ` G ` is a section of `... |
| sectco 17699 | Composition of two section... |
| isofval 17700 | Function value of the func... |
| invffval 17701 | Value of the inverse relat... |
| invfval 17702 | Value of the inverse relat... |
| isinv 17703 | Value of the inverse relat... |
| invss 17704 | The inverse relation is a ... |
| invsym 17705 | The inverse relation is sy... |
| invsym2 17706 | The inverse relation is sy... |
| invfun 17707 | The inverse relation is a ... |
| isoval 17708 | The isomorphisms are the d... |
| inviso1 17709 | If ` G ` is an inverse to ... |
| inviso2 17710 | If ` G ` is an inverse to ... |
| invf 17711 | The inverse relation is a ... |
| invf1o 17712 | The inverse relation is a ... |
| invinv 17713 | The inverse of the inverse... |
| invco 17714 | The composition of two iso... |
| dfiso2 17715 | Alternate definition of an... |
| dfiso3 17716 | Alternate definition of an... |
| inveq 17717 | If there are two inverses ... |
| isofn 17718 | The function value of the ... |
| isohom 17719 | An isomorphism is a homomo... |
| isoco 17720 | The composition of two iso... |
| oppcsect 17721 | A section in the opposite ... |
| oppcsect2 17722 | A section in the opposite ... |
| oppcinv 17723 | An inverse in the opposite... |
| oppciso 17724 | An isomorphism in the oppo... |
| sectmon 17725 | If ` F ` is a section of `... |
| monsect 17726 | If ` F ` is a monomorphism... |
| sectepi 17727 | If ` F ` is a section of `... |
| episect 17728 | If ` F ` is an epimorphism... |
| sectid 17729 | The identity is a section ... |
| invid 17730 | The inverse of the identit... |
| idiso 17731 | The identity is an isomorp... |
| idinv 17732 | The inverse of the identit... |
| invisoinvl 17733 | The inverse of an isomorph... |
| invisoinvr 17734 | The inverse of an isomorph... |
| invcoisoid 17735 | The inverse of an isomorph... |
| isocoinvid 17736 | The inverse of an isomorph... |
| rcaninv 17737 | Right cancellation of an i... |
| cicfval 17740 | The set of isomorphic obje... |
| brcic 17741 | The relation "is isomorphi... |
| cic 17742 | Objects ` X ` and ` Y ` in... |
| brcici 17743 | Prove that two objects are... |
| cicref 17744 | Isomorphism is reflexive. ... |
| ciclcl 17745 | Isomorphism implies the le... |
| cicrcl 17746 | Isomorphism implies the ri... |
| cicsym 17747 | Isomorphism is symmetric. ... |
| cictr 17748 | Isomorphism is transitive.... |
| cicer 17749 | Isomorphism is an equivale... |
| sscrel 17756 | The subcategory subset rel... |
| brssc 17757 | The subcategory subset rel... |
| sscpwex 17758 | An analogue of ~ pwex for ... |
| subcrcl 17759 | Reverse closure for the su... |
| sscfn1 17760 | The subcategory subset rel... |
| sscfn2 17761 | The subcategory subset rel... |
| ssclem 17762 | Lemma for ~ ssc1 and simil... |
| isssc 17763 | Value of the subcategory s... |
| ssc1 17764 | Infer subset relation on o... |
| ssc2 17765 | Infer subset relation on m... |
| sscres 17766 | Any function restricted to... |
| sscid 17767 | The subcategory subset rel... |
| ssctr 17768 | The subcategory subset rel... |
| ssceq 17769 | The subcategory subset rel... |
| rescval 17770 | Value of the category rest... |
| rescval2 17771 | Value of the category rest... |
| rescbas 17772 | Base set of the category r... |
| rescbasOLD 17773 | Obsolete version of ~ resc... |
| reschom 17774 | Hom-sets of the category r... |
| reschomf 17775 | Hom-sets of the category r... |
| rescco 17776 | Composition in the categor... |
| resccoOLD 17777 | Obsolete proof of ~ rescco... |
| rescabs 17778 | Restriction absorption law... |
| rescabsOLD 17779 | Obsolete proof of ~ seqp1d... |
| rescabs2 17780 | Restriction absorption law... |
| issubc 17781 | Elementhood in the set of ... |
| issubc2 17782 | Elementhood in the set of ... |
| 0ssc 17783 | For any category ` C ` , t... |
| 0subcat 17784 | For any category ` C ` , t... |
| catsubcat 17785 | For any category ` C ` , `... |
| subcssc 17786 | An element in the set of s... |
| subcfn 17787 | An element in the set of s... |
| subcss1 17788 | The objects of a subcatego... |
| subcss2 17789 | The morphisms of a subcate... |
| subcidcl 17790 | The identity of the origin... |
| subccocl 17791 | A subcategory is closed un... |
| subccatid 17792 | A subcategory is a categor... |
| subcid 17793 | The identity in a subcateg... |
| subccat 17794 | A subcategory is a categor... |
| issubc3 17795 | Alternate definition of a ... |
| fullsubc 17796 | The full subcategory gener... |
| fullresc 17797 | The category formed by str... |
| resscat 17798 | A category restricted to a... |
| subsubc 17799 | A subcategory of a subcate... |
| relfunc 17808 | The set of functors is a r... |
| funcrcl 17809 | Reverse closure for a func... |
| isfunc 17810 | Value of the set of functo... |
| isfuncd 17811 | Deduce that an operation i... |
| funcf1 17812 | The object part of a funct... |
| funcixp 17813 | The morphism part of a fun... |
| funcf2 17814 | The morphism part of a fun... |
| funcfn2 17815 | The morphism part of a fun... |
| funcid 17816 | A functor maps each identi... |
| funcco 17817 | A functor maps composition... |
| funcsect 17818 | The image of a section und... |
| funcinv 17819 | The image of an inverse un... |
| funciso 17820 | The image of an isomorphis... |
| funcoppc 17821 | A functor on categories yi... |
| idfuval 17822 | Value of the identity func... |
| idfu2nd 17823 | Value of the morphism part... |
| idfu2 17824 | Value of the morphism part... |
| idfu1st 17825 | Value of the object part o... |
| idfu1 17826 | Value of the object part o... |
| idfucl 17827 | The identity functor is a ... |
| cofuval 17828 | Value of the composition o... |
| cofu1st 17829 | Value of the object part o... |
| cofu1 17830 | Value of the object part o... |
| cofu2nd 17831 | Value of the morphism part... |
| cofu2 17832 | Value of the morphism part... |
| cofuval2 17833 | Value of the composition o... |
| cofucl 17834 | The composition of two fun... |
| cofuass 17835 | Functor composition is ass... |
| cofulid 17836 | The identity functor is a ... |
| cofurid 17837 | The identity functor is a ... |
| resfval 17838 | Value of the functor restr... |
| resfval2 17839 | Value of the functor restr... |
| resf1st 17840 | Value of the functor restr... |
| resf2nd 17841 | Value of the functor restr... |
| funcres 17842 | A functor restricted to a ... |
| funcres2b 17843 | Condition for a functor to... |
| funcres2 17844 | A functor into a restricte... |
| wunfunc 17845 | A weak universe is closed ... |
| wunfuncOLD 17846 | Obsolete proof of ~ wunfun... |
| funcpropd 17847 | If two categories have the... |
| funcres2c 17848 | Condition for a functor to... |
| fullfunc 17853 | A full functor is a functo... |
| fthfunc 17854 | A faithful functor is a fu... |
| relfull 17855 | The set of full functors i... |
| relfth 17856 | The set of faithful functo... |
| isfull 17857 | Value of the set of full f... |
| isfull2 17858 | Equivalent condition for a... |
| fullfo 17859 | The morphism map of a full... |
| fulli 17860 | The morphism map of a full... |
| isfth 17861 | Value of the set of faithf... |
| isfth2 17862 | Equivalent condition for a... |
| isffth2 17863 | A fully faithful functor i... |
| fthf1 17864 | The morphism map of a fait... |
| fthi 17865 | The morphism map of a fait... |
| ffthf1o 17866 | The morphism map of a full... |
| fullpropd 17867 | If two categories have the... |
| fthpropd 17868 | If two categories have the... |
| fulloppc 17869 | The opposite functor of a ... |
| fthoppc 17870 | The opposite functor of a ... |
| ffthoppc 17871 | The opposite functor of a ... |
| fthsect 17872 | A faithful functor reflect... |
| fthinv 17873 | A faithful functor reflect... |
| fthmon 17874 | A faithful functor reflect... |
| fthepi 17875 | A faithful functor reflect... |
| ffthiso 17876 | A fully faithful functor r... |
| fthres2b 17877 | Condition for a faithful f... |
| fthres2c 17878 | Condition for a faithful f... |
| fthres2 17879 | A faithful functor into a ... |
| idffth 17880 | The identity functor is a ... |
| cofull 17881 | The composition of two ful... |
| cofth 17882 | The composition of two fai... |
| coffth 17883 | The composition of two ful... |
| rescfth 17884 | The inclusion functor from... |
| ressffth 17885 | The inclusion functor from... |
| fullres2c 17886 | Condition for a full funct... |
| ffthres2c 17887 | Condition for a fully fait... |
| fnfuc 17892 | The ` FuncCat ` operation ... |
| natfval 17893 | Value of the function givi... |
| isnat 17894 | Property of being a natura... |
| isnat2 17895 | Property of being a natura... |
| natffn 17896 | The natural transformation... |
| natrcl 17897 | Reverse closure for a natu... |
| nat1st2nd 17898 | Rewrite the natural transf... |
| natixp 17899 | A natural transformation i... |
| natcl 17900 | A component of a natural t... |
| natfn 17901 | A natural transformation i... |
| nati 17902 | Naturality property of a n... |
| wunnat 17903 | A weak universe is closed ... |
| wunnatOLD 17904 | Obsolete proof of ~ wunnat... |
| catstr 17905 | A category structure is a ... |
| fucval 17906 | Value of the functor categ... |
| fuccofval 17907 | Value of the functor categ... |
| fucbas 17908 | The objects of the functor... |
| fuchom 17909 | The morphisms in the funct... |
| fuchomOLD 17910 | Obsolete proof of ~ fuchom... |
| fucco 17911 | Value of the composition o... |
| fuccoval 17912 | Value of the functor categ... |
| fuccocl 17913 | The composition of two nat... |
| fucidcl 17914 | The identity natural trans... |
| fuclid 17915 | Left identity of natural t... |
| fucrid 17916 | Right identity of natural ... |
| fucass 17917 | Associativity of natural t... |
| fuccatid 17918 | The functor category is a ... |
| fuccat 17919 | The functor category is a ... |
| fucid 17920 | The identity morphism in t... |
| fucsect 17921 | Two natural transformation... |
| fucinv 17922 | Two natural transformation... |
| invfuc 17923 | If ` V ( x ) ` is an inver... |
| fuciso 17924 | A natural transformation i... |
| natpropd 17925 | If two categories have the... |
| fucpropd 17926 | If two categories have the... |
| initofn 17933 | ` InitO ` is a function on... |
| termofn 17934 | ` TermO ` is a function on... |
| zeroofn 17935 | ` ZeroO ` is a function on... |
| initorcl 17936 | Reverse closure for an ini... |
| termorcl 17937 | Reverse closure for a term... |
| zeroorcl 17938 | Reverse closure for a zero... |
| initoval 17939 | The value of the initial o... |
| termoval 17940 | The value of the terminal ... |
| zerooval 17941 | The value of the zero obje... |
| isinito 17942 | The predicate "is an initi... |
| istermo 17943 | The predicate "is a termin... |
| iszeroo 17944 | The predicate "is a zero o... |
| isinitoi 17945 | Implication of a class bei... |
| istermoi 17946 | Implication of a class bei... |
| initoid 17947 | For an initial object, the... |
| termoid 17948 | For a terminal object, the... |
| dfinito2 17949 | An initial object is a ter... |
| dftermo2 17950 | A terminal object is an in... |
| dfinito3 17951 | An alternate definition of... |
| dftermo3 17952 | An alternate definition of... |
| initoo 17953 | An initial object is an ob... |
| termoo 17954 | A terminal object is an ob... |
| iszeroi 17955 | Implication of a class bei... |
| 2initoinv 17956 | Morphisms between two init... |
| initoeu1 17957 | Initial objects are essent... |
| initoeu1w 17958 | Initial objects are essent... |
| initoeu2lem0 17959 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 17960 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 17961 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 17962 | Initial objects are essent... |
| 2termoinv 17963 | Morphisms between two term... |
| termoeu1 17964 | Terminal objects are essen... |
| termoeu1w 17965 | Terminal objects are essen... |
| homarcl 17974 | Reverse closure for an arr... |
| homafval 17975 | Value of the disjointified... |
| homaf 17976 | Functionality of the disjo... |
| homaval 17977 | Value of the disjointified... |
| elhoma 17978 | Value of the disjointified... |
| elhomai 17979 | Produce an arrow from a mo... |
| elhomai2 17980 | Produce an arrow from a mo... |
| homarcl2 17981 | Reverse closure for the do... |
| homarel 17982 | An arrow is an ordered pai... |
| homa1 17983 | The first component of an ... |
| homahom2 17984 | The second component of an... |
| homahom 17985 | The second component of an... |
| homadm 17986 | The domain of an arrow wit... |
| homacd 17987 | The codomain of an arrow w... |
| homadmcd 17988 | Decompose an arrow into do... |
| arwval 17989 | The set of arrows is the u... |
| arwrcl 17990 | The first component of an ... |
| arwhoma 17991 | An arrow is contained in t... |
| homarw 17992 | A hom-set is a subset of t... |
| arwdm 17993 | The domain of an arrow is ... |
| arwcd 17994 | The codomain of an arrow i... |
| dmaf 17995 | The domain function is a f... |
| cdaf 17996 | The codomain function is a... |
| arwhom 17997 | The second component of an... |
| arwdmcd 17998 | Decompose an arrow into do... |
| idafval 18003 | Value of the identity arro... |
| idaval 18004 | Value of the identity arro... |
| ida2 18005 | Morphism part of the ident... |
| idahom 18006 | Domain and codomain of the... |
| idadm 18007 | Domain of the identity arr... |
| idacd 18008 | Codomain of the identity a... |
| idaf 18009 | The identity arrow functio... |
| coafval 18010 | The value of the compositi... |
| eldmcoa 18011 | A pair ` <. G , F >. ` is ... |
| dmcoass 18012 | The domain of composition ... |
| homdmcoa 18013 | If ` F : X --> Y ` and ` G... |
| coaval 18014 | Value of composition for c... |
| coa2 18015 | The morphism part of arrow... |
| coahom 18016 | The composition of two com... |
| coapm 18017 | Composition of arrows is a... |
| arwlid 18018 | Left identity of a categor... |
| arwrid 18019 | Right identity of a catego... |
| arwass 18020 | Associativity of compositi... |
| setcval 18023 | Value of the category of s... |
| setcbas 18024 | Set of objects of the cate... |
| setchomfval 18025 | Set of arrows of the categ... |
| setchom 18026 | Set of arrows of the categ... |
| elsetchom 18027 | A morphism of sets is a fu... |
| setccofval 18028 | Composition in the categor... |
| setcco 18029 | Composition in the categor... |
| setccatid 18030 | Lemma for ~ setccat . (Co... |
| setccat 18031 | The category of sets is a ... |
| setcid 18032 | The identity arrow in the ... |
| setcmon 18033 | A monomorphism of sets is ... |
| setcepi 18034 | An epimorphism of sets is ... |
| setcsect 18035 | A section in the category ... |
| setcinv 18036 | An inverse in the category... |
| setciso 18037 | An isomorphism in the cate... |
| resssetc 18038 | The restriction of the cat... |
| funcsetcres2 18039 | A functor into a smaller c... |
| setc2obas 18040 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18041 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18042 | The category of sets in a ... |
| cat1 18043 | The definition of category... |
| catcval 18046 | Value of the category of c... |
| catcbas 18047 | Set of objects of the cate... |
| catchomfval 18048 | Set of arrows of the categ... |
| catchom 18049 | Set of arrows of the categ... |
| catccofval 18050 | Composition in the categor... |
| catcco 18051 | Composition in the categor... |
| catccatid 18052 | Lemma for ~ catccat . (Co... |
| catcid 18053 | The identity arrow in the ... |
| catccat 18054 | The category of categories... |
| resscatc 18055 | The restriction of the cat... |
| catcisolem 18056 | Lemma for ~ catciso . (Co... |
| catciso 18057 | A functor is an isomorphis... |
| catcbascl 18058 | An element of the base set... |
| catcslotelcl 18059 | A slot entry of an element... |
| catcbaselcl 18060 | The base set of an element... |
| catchomcl 18061 | The Hom-set of an element ... |
| catcccocl 18062 | The composition operation ... |
| catcoppccl 18063 | The category of categories... |
| catcoppcclOLD 18064 | Obsolete proof of ~ catcop... |
| catcfuccl 18065 | The category of categories... |
| catcfucclOLD 18066 | Obsolete proof of ~ catcfu... |
| fncnvimaeqv 18067 | The inverse images of the ... |
| bascnvimaeqv 18068 | The inverse image of the u... |
| estrcval 18071 | Value of the category of e... |
| estrcbas 18072 | Set of objects of the cate... |
| estrchomfval 18073 | Set of morphisms ("arrows"... |
| estrchom 18074 | The morphisms between exte... |
| elestrchom 18075 | A morphism between extensi... |
| estrccofval 18076 | Composition in the categor... |
| estrcco 18077 | Composition in the categor... |
| estrcbasbas 18078 | An element of the base set... |
| estrccatid 18079 | Lemma for ~ estrccat . (C... |
| estrccat 18080 | The category of extensible... |
| estrcid 18081 | The identity arrow in the ... |
| estrchomfn 18082 | The Hom-set operation in t... |
| estrchomfeqhom 18083 | The functionalized Hom-set... |
| estrreslem1 18084 | Lemma 1 for ~ estrres . (... |
| estrreslem1OLD 18085 | Obsolete version of ~ estr... |
| estrreslem2 18086 | Lemma 2 for ~ estrres . (... |
| estrres 18087 | Any restriction of a categ... |
| funcestrcsetclem1 18088 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18089 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18090 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18091 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18092 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18093 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18094 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18095 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18096 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18097 | The "natural forgetful fun... |
| fthestrcsetc 18098 | The "natural forgetful fun... |
| fullestrcsetc 18099 | The "natural forgetful fun... |
| equivestrcsetc 18100 | The "natural forgetful fun... |
| setc1strwun 18101 | A constructed one-slot str... |
| funcsetcestrclem1 18102 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18103 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18104 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18105 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18106 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18107 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18108 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18109 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18110 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18111 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18112 | The "embedding functor" fr... |
| fthsetcestrc 18113 | The "embedding functor" fr... |
| fullsetcestrc 18114 | The "embedding functor" fr... |
| embedsetcestrc 18115 | The "embedding functor" fr... |
| fnxpc 18124 | The binary product of cate... |
| xpcval 18125 | Value of the binary produc... |
| xpcbas 18126 | Set of objects of the bina... |
| xpchomfval 18127 | Set of morphisms of the bi... |
| xpchom 18128 | Set of morphisms of the bi... |
| relxpchom 18129 | A hom-set in the binary pr... |
| xpccofval 18130 | Value of composition in th... |
| xpcco 18131 | Value of composition in th... |
| xpcco1st 18132 | Value of composition in th... |
| xpcco2nd 18133 | Value of composition in th... |
| xpchom2 18134 | Value of the set of morphi... |
| xpcco2 18135 | Value of composition in th... |
| xpccatid 18136 | The product of two categor... |
| xpcid 18137 | The identity morphism in t... |
| xpccat 18138 | The product of two categor... |
| 1stfval 18139 | Value of the first project... |
| 1stf1 18140 | Value of the first project... |
| 1stf2 18141 | Value of the first project... |
| 2ndfval 18142 | Value of the first project... |
| 2ndf1 18143 | Value of the first project... |
| 2ndf2 18144 | Value of the first project... |
| 1stfcl 18145 | The first projection funct... |
| 2ndfcl 18146 | The second projection func... |
| prfval 18147 | Value of the pairing funct... |
| prf1 18148 | Value of the pairing funct... |
| prf2fval 18149 | Value of the pairing funct... |
| prf2 18150 | Value of the pairing funct... |
| prfcl 18151 | The pairing of functors ` ... |
| prf1st 18152 | Cancellation of pairing wi... |
| prf2nd 18153 | Cancellation of pairing wi... |
| 1st2ndprf 18154 | Break a functor into a pro... |
| catcxpccl 18155 | The category of categories... |
| catcxpcclOLD 18156 | Obsolete proof of ~ catcxp... |
| xpcpropd 18157 | If two categories have the... |
| evlfval 18166 | Value of the evaluation fu... |
| evlf2 18167 | Value of the evaluation fu... |
| evlf2val 18168 | Value of the evaluation na... |
| evlf1 18169 | Value of the evaluation fu... |
| evlfcllem 18170 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18171 | The evaluation functor is ... |
| curfval 18172 | Value of the curry functor... |
| curf1fval 18173 | Value of the object part o... |
| curf1 18174 | Value of the object part o... |
| curf11 18175 | Value of the double evalua... |
| curf12 18176 | The partially evaluated cu... |
| curf1cl 18177 | The partially evaluated cu... |
| curf2 18178 | Value of the curry functor... |
| curf2val 18179 | Value of a component of th... |
| curf2cl 18180 | The curry functor at a mor... |
| curfcl 18181 | The curry functor of a fun... |
| curfpropd 18182 | If two categories have the... |
| uncfval 18183 | Value of the uncurry funct... |
| uncfcl 18184 | The uncurry operation take... |
| uncf1 18185 | Value of the uncurry funct... |
| uncf2 18186 | Value of the uncurry funct... |
| curfuncf 18187 | Cancellation of curry with... |
| uncfcurf 18188 | Cancellation of uncurry wi... |
| diagval 18189 | Define the diagonal functo... |
| diagcl 18190 | The diagonal functor is a ... |
| diag1cl 18191 | The constant functor of ` ... |
| diag11 18192 | Value of the constant func... |
| diag12 18193 | Value of the constant func... |
| diag2 18194 | Value of the diagonal func... |
| diag2cl 18195 | The diagonal functor at a ... |
| curf2ndf 18196 | As shown in ~ diagval , th... |
| hofval 18201 | Value of the Hom functor, ... |
| hof1fval 18202 | The object part of the Hom... |
| hof1 18203 | The object part of the Hom... |
| hof2fval 18204 | The morphism part of the H... |
| hof2val 18205 | The morphism part of the H... |
| hof2 18206 | The morphism part of the H... |
| hofcllem 18207 | Lemma for ~ hofcl . (Cont... |
| hofcl 18208 | Closure of the Hom functor... |
| oppchofcl 18209 | Closure of the opposite Ho... |
| yonval 18210 | Value of the Yoneda embedd... |
| yoncl 18211 | The Yoneda embedding is a ... |
| yon1cl 18212 | The Yoneda embedding at an... |
| yon11 18213 | Value of the Yoneda embedd... |
| yon12 18214 | Value of the Yoneda embedd... |
| yon2 18215 | Value of the Yoneda embedd... |
| hofpropd 18216 | If two categories have the... |
| yonpropd 18217 | If two categories have the... |
| oppcyon 18218 | Value of the opposite Yone... |
| oyoncl 18219 | The opposite Yoneda embedd... |
| oyon1cl 18220 | The opposite Yoneda embedd... |
| yonedalem1 18221 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18222 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18223 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18224 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18225 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18226 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18227 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18228 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18229 | Lemma for ~ yoneda . (Con... |
| yonedainv 18230 | The Yoneda Lemma with expl... |
| yonffthlem 18231 | Lemma for ~ yonffth . (Co... |
| yoneda 18232 | The Yoneda Lemma. There i... |
| yonffth 18233 | The Yoneda Lemma. The Yon... |
| yoniso 18234 | If the codomain is recover... |
| oduval 18237 | Value of an order dual str... |
| oduleval 18238 | Value of the less-equal re... |
| oduleg 18239 | Truth of the less-equal re... |
| odubas 18240 | Base set of an order dual ... |
| odubasOLD 18241 | Obsolete proof of ~ odubas... |
| isprs 18246 | Property of being a preord... |
| prslem 18247 | Lemma for ~ prsref and ~ p... |
| prsref 18248 | "Less than or equal to" is... |
| prstr 18249 | "Less than or equal to" is... |
| isdrs 18250 | Property of being a direct... |
| drsdir 18251 | Direction of a directed se... |
| drsprs 18252 | A directed set is a proset... |
| drsbn0 18253 | The base of a directed set... |
| drsdirfi 18254 | Any _finite_ number of ele... |
| isdrs2 18255 | Directed sets may be defin... |
| ispos 18263 | The predicate "is a poset"... |
| ispos2 18264 | A poset is an antisymmetri... |
| posprs 18265 | A poset is a proset. (Con... |
| posi 18266 | Lemma for poset properties... |
| posref 18267 | A poset ordering is reflex... |
| posasymb 18268 | A poset ordering is asymme... |
| postr 18269 | A poset ordering is transi... |
| 0pos 18270 | Technical lemma to simplif... |
| 0posOLD 18271 | Obsolete proof of ~ 0pos a... |
| isposd 18272 | Properties that determine ... |
| isposi 18273 | Properties that determine ... |
| isposix 18274 | Properties that determine ... |
| isposixOLD 18275 | Obsolete proof of ~ isposi... |
| pospropd 18276 | Posethood is determined on... |
| odupos 18277 | Being a poset is a self-du... |
| oduposb 18278 | Being a poset is a self-du... |
| pltfval 18280 | Value of the less-than rel... |
| pltval 18281 | Less-than relation. ( ~ d... |
| pltle 18282 | "Less than" implies "less ... |
| pltne 18283 | The "less than" relation i... |
| pltirr 18284 | The "less than" relation i... |
| pleval2i 18285 | One direction of ~ pleval2... |
| pleval2 18286 | "Less than or equal to" in... |
| pltnle 18287 | "Less than" implies not co... |
| pltval3 18288 | Alternate expression for t... |
| pltnlt 18289 | The less-than relation imp... |
| pltn2lp 18290 | The less-than relation has... |
| plttr 18291 | The less-than relation is ... |
| pltletr 18292 | Transitive law for chained... |
| plelttr 18293 | Transitive law for chained... |
| pospo 18294 | Write a poset structure in... |
| lubfval 18299 | Value of the least upper b... |
| lubdm 18300 | Domain of the least upper ... |
| lubfun 18301 | The LUB is a function. (C... |
| lubeldm 18302 | Member of the domain of th... |
| lubelss 18303 | A member of the domain of ... |
| lubeu 18304 | Unique existence proper of... |
| lubval 18305 | Value of the least upper b... |
| lubcl 18306 | The least upper bound func... |
| lubprop 18307 | Properties of greatest low... |
| luble 18308 | The greatest lower bound i... |
| lublecllem 18309 | Lemma for ~ lublecl and ~ ... |
| lublecl 18310 | The set of all elements le... |
| lubid 18311 | The LUB of elements less t... |
| glbfval 18312 | Value of the greatest lowe... |
| glbdm 18313 | Domain of the greatest low... |
| glbfun 18314 | The GLB is a function. (C... |
| glbeldm 18315 | Member of the domain of th... |
| glbelss 18316 | A member of the domain of ... |
| glbeu 18317 | Unique existence proper of... |
| glbval 18318 | Value of the greatest lowe... |
| glbcl 18319 | The least upper bound func... |
| glbprop 18320 | Properties of greatest low... |
| glble 18321 | The greatest lower bound i... |
| joinfval 18322 | Value of join function for... |
| joinfval2 18323 | Value of join function for... |
| joindm 18324 | Domain of join function fo... |
| joindef 18325 | Two ways to say that a joi... |
| joinval 18326 | Join value. Since both si... |
| joincl 18327 | Closure of join of element... |
| joindmss 18328 | Subset property of domain ... |
| joinval2lem 18329 | Lemma for ~ joinval2 and ~... |
| joinval2 18330 | Value of join for a poset ... |
| joineu 18331 | Uniqueness of join of elem... |
| joinlem 18332 | Lemma for join properties.... |
| lejoin1 18333 | A join's first argument is... |
| lejoin2 18334 | A join's second argument i... |
| joinle 18335 | A join is less than or equ... |
| meetfval 18336 | Value of meet function for... |
| meetfval2 18337 | Value of meet function for... |
| meetdm 18338 | Domain of meet function fo... |
| meetdef 18339 | Two ways to say that a mee... |
| meetval 18340 | Meet value. Since both si... |
| meetcl 18341 | Closure of meet of element... |
| meetdmss 18342 | Subset property of domain ... |
| meetval2lem 18343 | Lemma for ~ meetval2 and ~... |
| meetval2 18344 | Value of meet for a poset ... |
| meeteu 18345 | Uniqueness of meet of elem... |
| meetlem 18346 | Lemma for meet properties.... |
| lemeet1 18347 | A meet's first argument is... |
| lemeet2 18348 | A meet's second argument i... |
| meetle 18349 | A meet is less than or equ... |
| joincomALT 18350 | The join of a poset is com... |
| joincom 18351 | The join of a poset is com... |
| meetcomALT 18352 | The meet of a poset is com... |
| meetcom 18353 | The meet of a poset is com... |
| join0 18354 | Lemma for ~ odumeet . (Co... |
| meet0 18355 | Lemma for ~ odujoin . (Co... |
| odulub 18356 | Least upper bounds in a du... |
| odujoin 18357 | Joins in a dual order are ... |
| oduglb 18358 | Greatest lower bounds in a... |
| odumeet 18359 | Meets in a dual order are ... |
| poslubmo 18360 | Least upper bounds in a po... |
| posglbmo 18361 | Greatest lower bounds in a... |
| poslubd 18362 | Properties which determine... |
| poslubdg 18363 | Properties which determine... |
| posglbdg 18364 | Properties which determine... |
| istos 18367 | The predicate "is a toset"... |
| tosso 18368 | Write the totally ordered ... |
| tospos 18369 | A Toset is a Poset. (Cont... |
| tleile 18370 | In a Toset, any two elemen... |
| tltnle 18371 | In a Toset, "less than" is... |
| p0val 18376 | Value of poset zero. (Con... |
| p1val 18377 | Value of poset zero. (Con... |
| p0le 18378 | Any element is less than o... |
| ple1 18379 | Any element is less than o... |
| islat 18382 | The predicate "is a lattic... |
| odulatb 18383 | Being a lattice is self-du... |
| odulat 18384 | Being a lattice is self-du... |
| latcl2 18385 | The join and meet of any t... |
| latlem 18386 | Lemma for lattice properti... |
| latpos 18387 | A lattice is a poset. (Co... |
| latjcl 18388 | Closure of join operation ... |
| latmcl 18389 | Closure of meet operation ... |
| latref 18390 | A lattice ordering is refl... |
| latasymb 18391 | A lattice ordering is asym... |
| latasym 18392 | A lattice ordering is asym... |
| lattr 18393 | A lattice ordering is tran... |
| latasymd 18394 | Deduce equality from latti... |
| lattrd 18395 | A lattice ordering is tran... |
| latjcom 18396 | The join of a lattice comm... |
| latlej1 18397 | A join's first argument is... |
| latlej2 18398 | A join's second argument i... |
| latjle12 18399 | A join is less than or equ... |
| latleeqj1 18400 | "Less than or equal to" in... |
| latleeqj2 18401 | "Less than or equal to" in... |
| latjlej1 18402 | Add join to both sides of ... |
| latjlej2 18403 | Add join to both sides of ... |
| latjlej12 18404 | Add join to both sides of ... |
| latnlej 18405 | An idiom to express that a... |
| latnlej1l 18406 | An idiom to express that a... |
| latnlej1r 18407 | An idiom to express that a... |
| latnlej2 18408 | An idiom to express that a... |
| latnlej2l 18409 | An idiom to express that a... |
| latnlej2r 18410 | An idiom to express that a... |
| latjidm 18411 | Lattice join is idempotent... |
| latmcom 18412 | The join of a lattice comm... |
| latmle1 18413 | A meet is less than or equ... |
| latmle2 18414 | A meet is less than or equ... |
| latlem12 18415 | An element is less than or... |
| latleeqm1 18416 | "Less than or equal to" in... |
| latleeqm2 18417 | "Less than or equal to" in... |
| latmlem1 18418 | Add meet to both sides of ... |
| latmlem2 18419 | Add meet to both sides of ... |
| latmlem12 18420 | Add join to both sides of ... |
| latnlemlt 18421 | Negation of "less than or ... |
| latnle 18422 | Equivalent expressions for... |
| latmidm 18423 | Lattice meet is idempotent... |
| latabs1 18424 | Lattice absorption law. F... |
| latabs2 18425 | Lattice absorption law. F... |
| latledi 18426 | An ortholattice is distrib... |
| latmlej11 18427 | Ordering of a meet and joi... |
| latmlej12 18428 | Ordering of a meet and joi... |
| latmlej21 18429 | Ordering of a meet and joi... |
| latmlej22 18430 | Ordering of a meet and joi... |
| lubsn 18431 | The least upper bound of a... |
| latjass 18432 | Lattice join is associativ... |
| latj12 18433 | Swap 1st and 2nd members o... |
| latj32 18434 | Swap 2nd and 3rd members o... |
| latj13 18435 | Swap 1st and 3rd members o... |
| latj31 18436 | Swap 2nd and 3rd members o... |
| latjrot 18437 | Rotate lattice join of 3 c... |
| latj4 18438 | Rearrangement of lattice j... |
| latj4rot 18439 | Rotate lattice join of 4 c... |
| latjjdi 18440 | Lattice join distributes o... |
| latjjdir 18441 | Lattice join distributes o... |
| mod1ile 18442 | The weak direction of the ... |
| mod2ile 18443 | The weak direction of the ... |
| latmass 18444 | Lattice meet is associativ... |
| latdisdlem 18445 | Lemma for ~ latdisd . (Co... |
| latdisd 18446 | In a lattice, joins distri... |
| isclat 18449 | The predicate "is a comple... |
| clatpos 18450 | A complete lattice is a po... |
| clatlem 18451 | Lemma for properties of a ... |
| clatlubcl 18452 | Any subset of the base set... |
| clatlubcl2 18453 | Any subset of the base set... |
| clatglbcl 18454 | Any subset of the base set... |
| clatglbcl2 18455 | Any subset of the base set... |
| oduclatb 18456 | Being a complete lattice i... |
| clatl 18457 | A complete lattice is a la... |
| isglbd 18458 | Properties that determine ... |
| lublem 18459 | Lemma for the least upper ... |
| lubub 18460 | The LUB of a complete latt... |
| lubl 18461 | The LUB of a complete latt... |
| lubss 18462 | Subset law for least upper... |
| lubel 18463 | An element of a set is les... |
| lubun 18464 | The LUB of a union. (Cont... |
| clatglb 18465 | Properties of greatest low... |
| clatglble 18466 | The greatest lower bound i... |
| clatleglb 18467 | Two ways of expressing "le... |
| clatglbss 18468 | Subset law for greatest lo... |
| isdlat 18471 | Property of being a distri... |
| dlatmjdi 18472 | In a distributive lattice,... |
| dlatl 18473 | A distributive lattice is ... |
| odudlatb 18474 | The dual of a distributive... |
| dlatjmdi 18475 | In a distributive lattice,... |
| ipostr 18478 | The structure of ~ df-ipo ... |
| ipoval 18479 | Value of the inclusion pos... |
| ipobas 18480 | Base set of the inclusion ... |
| ipolerval 18481 | Relation of the inclusion ... |
| ipotset 18482 | Topology of the inclusion ... |
| ipole 18483 | Weak order condition of th... |
| ipolt 18484 | Strict order condition of ... |
| ipopos 18485 | The inclusion poset on a f... |
| isipodrs 18486 | Condition for a family of ... |
| ipodrscl 18487 | Direction by inclusion as ... |
| ipodrsfi 18488 | Finite upper bound propert... |
| fpwipodrs 18489 | The finite subsets of any ... |
| ipodrsima 18490 | The monotone image of a di... |
| isacs3lem 18491 | An algebraic closure syste... |
| acsdrsel 18492 | An algebraic closure syste... |
| isacs4lem 18493 | In a closure system in whi... |
| isacs5lem 18494 | If closure commutes with d... |
| acsdrscl 18495 | In an algebraic closure sy... |
| acsficl 18496 | A closure in an algebraic ... |
| isacs5 18497 | A closure system is algebr... |
| isacs4 18498 | A closure system is algebr... |
| isacs3 18499 | A closure system is algebr... |
| acsficld 18500 | In an algebraic closure sy... |
| acsficl2d 18501 | In an algebraic closure sy... |
| acsfiindd 18502 | In an algebraic closure sy... |
| acsmapd 18503 | In an algebraic closure sy... |
| acsmap2d 18504 | In an algebraic closure sy... |
| acsinfd 18505 | In an algebraic closure sy... |
| acsdomd 18506 | In an algebraic closure sy... |
| acsinfdimd 18507 | In an algebraic closure sy... |
| acsexdimd 18508 | In an algebraic closure sy... |
| mrelatglb 18509 | Greatest lower bounds in a... |
| mrelatglb0 18510 | The empty intersection in ... |
| mrelatlub 18511 | Least upper bounds in a Mo... |
| mreclatBAD 18512 | A Moore space is a complet... |
| isps 18517 | The predicate "is a poset"... |
| psrel 18518 | A poset is a relation. (C... |
| psref2 18519 | A poset is antisymmetric a... |
| pstr2 18520 | A poset is transitive. (C... |
| pslem 18521 | Lemma for ~ psref and othe... |
| psdmrn 18522 | The domain and range of a ... |
| psref 18523 | A poset is reflexive. (Co... |
| psrn 18524 | The range of a poset equal... |
| psasym 18525 | A poset is antisymmetric. ... |
| pstr 18526 | A poset is transitive. (C... |
| cnvps 18527 | The converse of a poset is... |
| cnvpsb 18528 | The converse of a poset is... |
| psss 18529 | Any subset of a partially ... |
| psssdm2 18530 | Field of a subposet. (Con... |
| psssdm 18531 | Field of a subposet. (Con... |
| istsr 18532 | The predicate is a toset. ... |
| istsr2 18533 | The predicate is a toset. ... |
| tsrlin 18534 | A toset is a linear order.... |
| tsrlemax 18535 | Two ways of saying a numbe... |
| tsrps 18536 | A toset is a poset. (Cont... |
| cnvtsr 18537 | The converse of a toset is... |
| tsrss 18538 | Any subset of a totally or... |
| ledm 18539 | The domain of ` <_ ` is ` ... |
| lern 18540 | The range of ` <_ ` is ` R... |
| lefld 18541 | The field of the 'less or ... |
| letsr 18542 | The "less than or equal to... |
| isdir 18547 | A condition for a relation... |
| reldir 18548 | A direction is a relation.... |
| dirdm 18549 | A direction's domain is eq... |
| dirref 18550 | A direction is reflexive. ... |
| dirtr 18551 | A direction is transitive.... |
| dirge 18552 | For any two elements of a ... |
| tsrdir 18553 | A totally ordered set is a... |
| ismgm 18558 | The predicate "is a magma"... |
| ismgmn0 18559 | The predicate "is a magma"... |
| mgmcl 18560 | Closure of the operation o... |
| isnmgm 18561 | A condition for a structur... |
| mgmsscl 18562 | If the base set of a magma... |
| plusffval 18563 | The group addition operati... |
| plusfval 18564 | The group addition operati... |
| plusfeq 18565 | If the addition operation ... |
| plusffn 18566 | The group addition operati... |
| mgmplusf 18567 | The group addition functio... |
| issstrmgm 18568 | Characterize a substructur... |
| intopsn 18569 | The internal operation for... |
| mgmb1mgm1 18570 | The only magma with a base... |
| mgm0 18571 | Any set with an empty base... |
| mgm0b 18572 | The structure with an empt... |
| mgm1 18573 | The structure with one ele... |
| opifismgm 18574 | A structure with a group a... |
| mgmidmo 18575 | A two-sided identity eleme... |
| grpidval 18576 | The value of the identity ... |
| grpidpropd 18577 | If two structures have the... |
| fn0g 18578 | The group zero extractor i... |
| 0g0 18579 | The identity element funct... |
| ismgmid 18580 | The identity element of a ... |
| mgmidcl 18581 | The identity element of a ... |
| mgmlrid 18582 | The identity element of a ... |
| ismgmid2 18583 | Show that a given element ... |
| lidrideqd 18584 | If there is a left and rig... |
| lidrididd 18585 | If there is a left and rig... |
| grpidd 18586 | Deduce the identity elemen... |
| mgmidsssn0 18587 | Property of the set of ide... |
| grprinvlem 18588 | Lemma for ~ grpinva . (Co... |
| grpinva 18589 | Deduce right inverse from ... |
| grprida 18590 | Deduce right identity from... |
| gsumvalx 18591 | Expand out the substitutio... |
| gsumval 18592 | Expand out the substitutio... |
| gsumpropd 18593 | The group sum depends only... |
| gsumpropd2lem 18594 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18595 | A stronger version of ~ gs... |
| gsummgmpropd 18596 | A stronger version of ~ gs... |
| gsumress 18597 | The group sum in a substru... |
| gsumval1 18598 | Value of the group sum ope... |
| gsum0 18599 | Value of the empty group s... |
| gsumval2a 18600 | Value of the group sum ope... |
| gsumval2 18601 | Value of the group sum ope... |
| gsumsplit1r 18602 | Splitting off the rightmos... |
| gsumprval 18603 | Value of the group sum ope... |
| gsumpr12val 18604 | Value of the group sum ope... |
| issgrp 18607 | The predicate "is a semigr... |
| issgrpv 18608 | The predicate "is a semigr... |
| issgrpn0 18609 | The predicate "is a semigr... |
| isnsgrp 18610 | A condition for a structur... |
| sgrpmgm 18611 | A semigroup is a magma. (... |
| sgrpass 18612 | A semigroup operation is a... |
| sgrpcl 18613 | Closure of the operation o... |
| sgrp0 18614 | Any set with an empty base... |
| sgrp0b 18615 | The structure with an empt... |
| sgrp1 18616 | The structure with one ele... |
| issgrpd 18617 | Deduce a semigroup from it... |
| sgrppropd 18618 | If two structures are sets... |
| prdsplusgsgrpcl 18619 | Structure product pointwis... |
| prdssgrpd 18620 | The product of a family of... |
| ismnddef 18623 | The predicate "is a monoid... |
| ismnd 18624 | The predicate "is a monoid... |
| isnmnd 18625 | A condition for a structur... |
| sgrpidmnd 18626 | A semigroup with an identi... |
| mndsgrp 18627 | A monoid is a semigroup. ... |
| mndmgm 18628 | A monoid is a magma. (Con... |
| mndcl 18629 | Closure of the operation o... |
| mndass 18630 | A monoid operation is asso... |
| mndid 18631 | A monoid has a two-sided i... |
| mndideu 18632 | The two-sided identity ele... |
| mnd32g 18633 | Commutative/associative la... |
| mnd12g 18634 | Commutative/associative la... |
| mnd4g 18635 | Commutative/associative la... |
| mndidcl 18636 | The identity element of a ... |
| mndbn0 18637 | The base set of a monoid i... |
| hashfinmndnn 18638 | A finite monoid has positi... |
| mndplusf 18639 | The group addition operati... |
| mndlrid 18640 | A monoid's identity elemen... |
| mndlid 18641 | The identity element of a ... |
| mndrid 18642 | The identity element of a ... |
| ismndd 18643 | Deduce a monoid from its p... |
| mndpfo 18644 | The addition operation of ... |
| mndfo 18645 | The addition operation of ... |
| mndpropd 18646 | If two structures have the... |
| mndprop 18647 | If two structures have the... |
| issubmnd 18648 | Characterize a submonoid b... |
| ress0g 18649 | ` 0g ` is unaffected by re... |
| submnd0 18650 | The zero of a submonoid is... |
| mndinvmod 18651 | Uniqueness of an inverse e... |
| prdsplusgcl 18652 | Structure product pointwis... |
| prdsidlem 18653 | Characterization of identi... |
| prdsmndd 18654 | The product of a family of... |
| prds0g 18655 | Zero in a product of monoi... |
| pwsmnd 18656 | The structure power of a m... |
| pws0g 18657 | Zero in a structure power ... |
| imasmnd2 18658 | The image structure of a m... |
| imasmnd 18659 | The image structure of a m... |
| imasmndf1 18660 | The image of a monoid unde... |
| xpsmnd 18661 | The binary product of mono... |
| xpsmnd0 18662 | The identity element of a ... |
| mnd1 18663 | The (smallest) structure r... |
| mnd1id 18664 | The singleton element of a... |
| ismhm 18669 | Property of a monoid homom... |
| ismhmd 18670 | Deduction version of ~ ism... |
| mhmrcl1 18671 | Reverse closure of a monoi... |
| mhmrcl2 18672 | Reverse closure of a monoi... |
| mhmf 18673 | A monoid homomorphism is a... |
| mhmpropd 18674 | Monoid homomorphism depend... |
| mhmlin 18675 | A monoid homomorphism comm... |
| mhm0 18676 | A monoid homomorphism pres... |
| idmhm 18677 | The identity homomorphism ... |
| mhmf1o 18678 | A monoid homomorphism is b... |
| submrcl 18679 | Reverse closure for submon... |
| issubm 18680 | Expand definition of a sub... |
| issubm2 18681 | Submonoids are subsets tha... |
| issubmndb 18682 | The submonoid predicate. ... |
| issubmd 18683 | Deduction for proving a su... |
| mndissubm 18684 | If the base set of a monoi... |
| resmndismnd 18685 | If the base set of a monoi... |
| submss 18686 | Submonoids are subsets of ... |
| submid 18687 | Every monoid is trivially ... |
| subm0cl 18688 | Submonoids contain zero. ... |
| submcl 18689 | Submonoids are closed unde... |
| submmnd 18690 | Submonoids are themselves ... |
| submbas 18691 | The base set of a submonoi... |
| subm0 18692 | Submonoids have the same i... |
| subsubm 18693 | A submonoid of a submonoid... |
| 0subm 18694 | The zero submonoid of an a... |
| insubm 18695 | The intersection of two su... |
| 0mhm 18696 | The constant zero linear f... |
| resmhm 18697 | Restriction of a monoid ho... |
| resmhm2 18698 | One direction of ~ resmhm2... |
| resmhm2b 18699 | Restriction of the codomai... |
| mhmco 18700 | The composition of monoid ... |
| mhmimalem 18701 | Lemma for ~ mhmima and sim... |
| mhmima 18702 | The homomorphic image of a... |
| mhmeql 18703 | The equalizer of two monoi... |
| submacs 18704 | Submonoids are an algebrai... |
| mndind 18705 | Induction in a monoid. In... |
| prdspjmhm 18706 | A projection from a produc... |
| pwspjmhm 18707 | A projection from a struct... |
| pwsdiagmhm 18708 | Diagonal monoid homomorphi... |
| pwsco1mhm 18709 | Right composition with a f... |
| pwsco2mhm 18710 | Left composition with a mo... |
| gsumvallem2 18711 | Lemma for properties of th... |
| gsumsubm 18712 | Evaluate a group sum in a ... |
| gsumz 18713 | Value of a group sum over ... |
| gsumwsubmcl 18714 | Closure of the composite i... |
| gsumws1 18715 | A singleton composite reco... |
| gsumwcl 18716 | Closure of the composite o... |
| gsumsgrpccat 18717 | Homomorphic property of no... |
| gsumccat 18718 | Homomorphic property of co... |
| gsumws2 18719 | Valuation of a pair in a m... |
| gsumccatsn 18720 | Homomorphic property of co... |
| gsumspl 18721 | The primary purpose of the... |
| gsumwmhm 18722 | Behavior of homomorphisms ... |
| gsumwspan 18723 | The submonoid generated by... |
| frmdval 18728 | Value of the free monoid c... |
| frmdbas 18729 | The base set of a free mon... |
| frmdelbas 18730 | An element of the base set... |
| frmdplusg 18731 | The monoid operation of a ... |
| frmdadd 18732 | Value of the monoid operat... |
| vrmdfval 18733 | The canonical injection fr... |
| vrmdval 18734 | The value of the generatin... |
| vrmdf 18735 | The mapping from the index... |
| frmdmnd 18736 | A free monoid is a monoid.... |
| frmd0 18737 | The identity of the free m... |
| frmdsssubm 18738 | The set of words taking va... |
| frmdgsum 18739 | Any word in a free monoid ... |
| frmdss2 18740 | A subset of generators is ... |
| frmdup1 18741 | Any assignment of the gene... |
| frmdup2 18742 | The evaluation map has the... |
| frmdup3lem 18743 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18744 | Universal property of the ... |
| efmnd 18747 | The monoid of endofunction... |
| efmndbas 18748 | The base set of the monoid... |
| efmndbasabf 18749 | The base set of the monoid... |
| elefmndbas 18750 | Two ways of saying a funct... |
| elefmndbas2 18751 | Two ways of saying a funct... |
| efmndbasf 18752 | Elements in the monoid of ... |
| efmndhash 18753 | The monoid of endofunction... |
| efmndbasfi 18754 | The monoid of endofunction... |
| efmndfv 18755 | The function value of an e... |
| efmndtset 18756 | The topology of the monoid... |
| efmndplusg 18757 | The group operation of a m... |
| efmndov 18758 | The value of the group ope... |
| efmndcl 18759 | The group operation of the... |
| efmndtopn 18760 | The topology of the monoid... |
| symggrplem 18761 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18762 | The monoid of endofunction... |
| efmndsgrp 18763 | The monoid of endofunction... |
| ielefmnd 18764 | The identity function rest... |
| efmndid 18765 | The identity function rest... |
| efmndmnd 18766 | The monoid of endofunction... |
| efmnd0nmnd 18767 | Even the monoid of endofun... |
| efmndbas0 18768 | The base set of the monoid... |
| efmnd1hash 18769 | The monoid of endofunction... |
| efmnd1bas 18770 | The monoid of endofunction... |
| efmnd2hash 18771 | The monoid of endofunction... |
| submefmnd 18772 | If the base set of a monoi... |
| sursubmefmnd 18773 | The set of surjective endo... |
| injsubmefmnd 18774 | The set of injective endof... |
| idressubmefmnd 18775 | The singleton containing o... |
| idresefmnd 18776 | The structure with the sin... |
| smndex1ibas 18777 | The modulo function ` I ` ... |
| smndex1iidm 18778 | The modulo function ` I ` ... |
| smndex1gbas 18779 | The constant functions ` (... |
| smndex1gid 18780 | The composition of a const... |
| smndex1igid 18781 | The composition of the mod... |
| smndex1basss 18782 | The modulo function ` I ` ... |
| smndex1bas 18783 | The base set of the monoid... |
| smndex1mgm 18784 | The monoid of endofunction... |
| smndex1sgrp 18785 | The monoid of endofunction... |
| smndex1mndlem 18786 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18787 | The monoid of endofunction... |
| smndex1id 18788 | The modulo function ` I ` ... |
| smndex1n0mnd 18789 | The identity of the monoid... |
| nsmndex1 18790 | The base set ` B ` of the ... |
| smndex2dbas 18791 | The doubling function ` D ... |
| smndex2dnrinv 18792 | The doubling function ` D ... |
| smndex2hbas 18793 | The halving functions ` H ... |
| smndex2dlinvh 18794 | The halving functions ` H ... |
| mgm2nsgrplem1 18795 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18796 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18797 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18798 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18799 | A small magma (with two el... |
| sgrp2nmndlem1 18800 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18801 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18802 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18803 | A small semigroup (with tw... |
| sgrp2rid2ex 18804 | A small semigroup (with tw... |
| sgrp2nmndlem4 18805 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18806 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18807 | A small semigroup (with tw... |
| mgmnsgrpex 18808 | There is a magma which is ... |
| sgrpnmndex 18809 | There is a semigroup which... |
| sgrpssmgm 18810 | The class of all semigroup... |
| mndsssgrp 18811 | The class of all monoids i... |
| pwmndgplus 18812 | The operation of the monoi... |
| pwmndid 18813 | The identity of the monoid... |
| pwmnd 18814 | The power set of a class `... |
| isgrp 18821 | The predicate "is a group"... |
| grpmnd 18822 | A group is a monoid. (Con... |
| grpcl 18823 | Closure of the operation o... |
| grpass 18824 | A group operation is assoc... |
| grpinvex 18825 | Every member of a group ha... |
| grpideu 18826 | The two-sided identity ele... |
| grpassd 18827 | A group operation is assoc... |
| grpmndd 18828 | A group is a monoid. (Con... |
| grpcld 18829 | Closure of the operation o... |
| grpplusf 18830 | The group addition operati... |
| grpplusfo 18831 | The group addition operati... |
| resgrpplusfrn 18832 | The underlying set of a gr... |
| grppropd 18833 | If two structures have the... |
| grpprop 18834 | If two structures have the... |
| grppropstr 18835 | Generalize a specific 2-el... |
| grpss 18836 | Show that a structure exte... |
| isgrpd2e 18837 | Deduce a group from its pr... |
| isgrpd2 18838 | Deduce a group from its pr... |
| isgrpde 18839 | Deduce a group from its pr... |
| isgrpd 18840 | Deduce a group from its pr... |
| isgrpi 18841 | Properties that determine ... |
| grpsgrp 18842 | A group is a semigroup. (... |
| dfgrp2 18843 | Alternate definition of a ... |
| dfgrp2e 18844 | Alternate definition of a ... |
| isgrpix 18845 | Properties that determine ... |
| grpidcl 18846 | The identity element of a ... |
| grpbn0 18847 | The base set of a group is... |
| grplid 18848 | The identity element of a ... |
| grprid 18849 | The identity element of a ... |
| grplidd 18850 | The identity element of a ... |
| grpridd 18851 | The identity element of a ... |
| grpn0 18852 | A group is not empty. (Co... |
| hashfingrpnn 18853 | A finite group has positiv... |
| grprcan 18854 | Right cancellation law for... |
| grpinveu 18855 | The left inverse element o... |
| grpid 18856 | Two ways of saying that an... |
| isgrpid2 18857 | Properties showing that an... |
| grpidd2 18858 | Deduce the identity elemen... |
| grpinvfval 18859 | The inverse function of a ... |
| grpinvfvalALT 18860 | Shorter proof of ~ grpinvf... |
| grpinvval 18861 | The inverse of a group ele... |
| grpinvfn 18862 | Functionality of the group... |
| grpinvfvi 18863 | The group inverse function... |
| grpsubfval 18864 | Group subtraction (divisio... |
| grpsubfvalALT 18865 | Shorter proof of ~ grpsubf... |
| grpsubval 18866 | Group subtraction (divisio... |
| grpinvf 18867 | The group inversion operat... |
| grpinvcl 18868 | A group element's inverse ... |
| grpinvcld 18869 | A group element's inverse ... |
| grplinv 18870 | The left inverse of a grou... |
| grprinv 18871 | The right inverse of a gro... |
| grpinvid1 18872 | The inverse of a group ele... |
| grpinvid2 18873 | The inverse of a group ele... |
| isgrpinv 18874 | Properties showing that a ... |
| grplinvd 18875 | The left inverse of a grou... |
| grprinvd 18876 | The right inverse of a gro... |
| grplrinv 18877 | In a group, every member h... |
| grpidinv2 18878 | A group's properties using... |
| grpidinv 18879 | A group has a left and rig... |
| grpinvid 18880 | The inverse of the identit... |
| grplcan 18881 | Left cancellation law for ... |
| grpasscan1 18882 | An associative cancellatio... |
| grpasscan2 18883 | An associative cancellatio... |
| grpidrcan 18884 | If right adding an element... |
| grpidlcan 18885 | If left adding an element ... |
| grpinvinv 18886 | Double inverse law for gro... |
| grpinvcnv 18887 | The group inverse is its o... |
| grpinv11 18888 | The group inverse is one-t... |
| grpinvf1o 18889 | The group inverse is a one... |
| grpinvnz 18890 | The inverse of a nonzero g... |
| grpinvnzcl 18891 | The inverse of a nonzero g... |
| grpsubinv 18892 | Subtraction of an inverse.... |
| grplmulf1o 18893 | Left multiplication by a g... |
| grpinvpropd 18894 | If two structures have the... |
| grpidssd 18895 | If the base set of a group... |
| grpinvssd 18896 | If the base set of a group... |
| grpinvadd 18897 | The inverse of the group o... |
| grpsubf 18898 | Functionality of group sub... |
| grpsubcl 18899 | Closure of group subtracti... |
| grpsubrcan 18900 | Right cancellation law for... |
| grpinvsub 18901 | Inverse of a group subtrac... |
| grpinvval2 18902 | A ~ df-neg -like equation ... |
| grpsubid 18903 | Subtraction of a group ele... |
| grpsubid1 18904 | Subtraction of the identit... |
| grpsubeq0 18905 | If the difference between ... |
| grpsubadd0sub 18906 | Subtraction expressed as a... |
| grpsubadd 18907 | Relationship between group... |
| grpsubsub 18908 | Double group subtraction. ... |
| grpaddsubass 18909 | Associative-type law for g... |
| grppncan 18910 | Cancellation law for subtr... |
| grpnpcan 18911 | Cancellation law for subtr... |
| grpsubsub4 18912 | Double group subtraction (... |
| grppnpcan2 18913 | Cancellation law for mixed... |
| grpnpncan 18914 | Cancellation law for group... |
| grpnpncan0 18915 | Cancellation law for group... |
| grpnnncan2 18916 | Cancellation law for group... |
| dfgrp3lem 18917 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 18918 | Alternate definition of a ... |
| dfgrp3e 18919 | Alternate definition of a ... |
| grplactfval 18920 | The left group action of e... |
| grplactval 18921 | The value of the left grou... |
| grplactcnv 18922 | The left group action of e... |
| grplactf1o 18923 | The left group action of e... |
| grpsubpropd 18924 | Weak property deduction fo... |
| grpsubpropd2 18925 | Strong property deduction ... |
| grp1 18926 | The (smallest) structure r... |
| grp1inv 18927 | The inverse function of th... |
| prdsinvlem 18928 | Characterization of invers... |
| prdsgrpd 18929 | The product of a family of... |
| prdsinvgd 18930 | Negation in a product of g... |
| pwsgrp 18931 | A structure power of a gro... |
| pwsinvg 18932 | Negation in a group power.... |
| pwssub 18933 | Subtraction in a group pow... |
| imasgrp2 18934 | The image structure of a g... |
| imasgrp 18935 | The image structure of a g... |
| imasgrpf1 18936 | The image of a group under... |
| qusgrp2 18937 | Prove that a quotient stru... |
| xpsgrp 18938 | The binary product of grou... |
| mhmlem 18939 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 18940 | A surjective monoid morphi... |
| mhmmnd 18941 | The image of a monoid ` G ... |
| mhmfmhm 18942 | The function fulfilling th... |
| ghmgrp 18943 | The image of a group ` G `... |
| mulgfval 18946 | Group multiple (exponentia... |
| mulgfvalALT 18947 | Shorter proof of ~ mulgfva... |
| mulgval 18948 | Value of the group multipl... |
| mulgfn 18949 | Functionality of the group... |
| mulgfvi 18950 | The group multiple operati... |
| mulg0 18951 | Group multiple (exponentia... |
| mulgnn 18952 | Group multiple (exponentia... |
| mulgnngsum 18953 | Group multiple (exponentia... |
| mulgnn0gsum 18954 | Group multiple (exponentia... |
| mulg1 18955 | Group multiple (exponentia... |
| mulgnnp1 18956 | Group multiple (exponentia... |
| mulg2 18957 | Group multiple (exponentia... |
| mulgnegnn 18958 | Group multiple (exponentia... |
| mulgnn0p1 18959 | Group multiple (exponentia... |
| mulgnnsubcl 18960 | Closure of the group multi... |
| mulgnn0subcl 18961 | Closure of the group multi... |
| mulgsubcl 18962 | Closure of the group multi... |
| mulgnncl 18963 | Closure of the group multi... |
| mulgnn0cl 18964 | Closure of the group multi... |
| mulgcl 18965 | Closure of the group multi... |
| mulgneg 18966 | Group multiple (exponentia... |
| mulgnegneg 18967 | The inverse of a negative ... |
| mulgm1 18968 | Group multiple (exponentia... |
| mulgnn0cld 18969 | Closure of the group multi... |
| mulgcld 18970 | Deduction associated with ... |
| mulgaddcomlem 18971 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 18972 | The group multiple operato... |
| mulginvcom 18973 | The group multiple operato... |
| mulginvinv 18974 | The group multiple operato... |
| mulgnn0z 18975 | A group multiple of the id... |
| mulgz 18976 | A group multiple of the id... |
| mulgnndir 18977 | Sum of group multiples, fo... |
| mulgnn0dir 18978 | Sum of group multiples, ge... |
| mulgdirlem 18979 | Lemma for ~ mulgdir . (Co... |
| mulgdir 18980 | Sum of group multiples, ge... |
| mulgp1 18981 | Group multiple (exponentia... |
| mulgneg2 18982 | Group multiple (exponentia... |
| mulgnnass 18983 | Product of group multiples... |
| mulgnn0ass 18984 | Product of group multiples... |
| mulgass 18985 | Product of group multiples... |
| mulgassr 18986 | Reversed product of group ... |
| mulgmodid 18987 | Casting out multiples of t... |
| mulgsubdir 18988 | Distribution of group mult... |
| mhmmulg 18989 | A homomorphism of monoids ... |
| mulgpropd 18990 | Two structures with the sa... |
| submmulgcl 18991 | Closure of the group multi... |
| submmulg 18992 | A group multiple is the sa... |
| pwsmulg 18993 | Value of a group multiple ... |
| issubg 19000 | The subgroup predicate. (... |
| subgss 19001 | A subgroup is a subset. (... |
| subgid 19002 | A group is a subgroup of i... |
| subggrp 19003 | A subgroup is a group. (C... |
| subgbas 19004 | The base of the restricted... |
| subgrcl 19005 | Reverse closure for the su... |
| subg0 19006 | A subgroup of a group must... |
| subginv 19007 | The inverse of an element ... |
| subg0cl 19008 | The group identity is an e... |
| subginvcl 19009 | The inverse of an element ... |
| subgcl 19010 | A subgroup is closed under... |
| subgsubcl 19011 | A subgroup is closed under... |
| subgsub 19012 | The subtraction of element... |
| subgmulgcl 19013 | Closure of the group multi... |
| subgmulg 19014 | A group multiple is the sa... |
| issubg2 19015 | Characterize the subgroups... |
| issubgrpd2 19016 | Prove a subgroup by closur... |
| issubgrpd 19017 | Prove a subgroup by closur... |
| issubg3 19018 | A subgroup is a symmetric ... |
| issubg4 19019 | A subgroup is a nonempty s... |
| grpissubg 19020 | If the base set of a group... |
| resgrpisgrp 19021 | If the base set of a group... |
| subgsubm 19022 | A subgroup is a submonoid.... |
| subsubg 19023 | A subgroup of a subgroup i... |
| subgint 19024 | The intersection of a none... |
| 0subg 19025 | The zero subgroup of an ar... |
| 0subgOLD 19026 | Obsolete version of ~ 0sub... |
| trivsubgd 19027 | The only subgroup of a tri... |
| trivsubgsnd 19028 | The only subgroup of a tri... |
| isnsg 19029 | Property of being a normal... |
| isnsg2 19030 | Weaken the condition of ~ ... |
| nsgbi 19031 | Defining property of a nor... |
| nsgsubg 19032 | A normal subgroup is a sub... |
| nsgconj 19033 | The conjugation of an elem... |
| isnsg3 19034 | A subgroup is normal iff t... |
| subgacs 19035 | Subgroups are an algebraic... |
| nsgacs 19036 | Normal subgroups form an a... |
| elnmz 19037 | Elementhood in the normali... |
| nmzbi 19038 | Defining property of the n... |
| nmzsubg 19039 | The normalizer N_G(S) of a... |
| ssnmz 19040 | A subgroup is a subset of ... |
| isnsg4 19041 | A subgroup is normal iff i... |
| nmznsg 19042 | Any subgroup is a normal s... |
| 0nsg 19043 | The zero subgroup is norma... |
| nsgid 19044 | The whole group is a norma... |
| 0idnsgd 19045 | The whole group and the ze... |
| trivnsgd 19046 | The only normal subgroup o... |
| triv1nsgd 19047 | A trivial group has exactl... |
| 1nsgtrivd 19048 | A group with exactly one n... |
| releqg 19049 | The left coset equivalence... |
| eqgfval 19050 | Value of the subgroup left... |
| eqgval 19051 | Value of the subgroup left... |
| eqger 19052 | The subgroup coset equival... |
| eqglact 19053 | A left coset can be expres... |
| eqgid 19054 | The left coset containing ... |
| eqgen 19055 | Each coset is equipotent t... |
| eqgcpbl 19056 | The subgroup coset equival... |
| quselbas 19057 | Membership in the base set... |
| quseccl0 19058 | Closure of the quotient ma... |
| qusgrp 19059 | If ` Y ` is a normal subgr... |
| quseccl 19060 | Closure of the quotient ma... |
| qusadd 19061 | Value of the group operati... |
| qus0 19062 | Value of the group identit... |
| qusinv 19063 | Value of the group inverse... |
| qussub 19064 | Value of the group subtrac... |
| lagsubg2 19065 | Lagrange's theorem for fin... |
| lagsubg 19066 | Lagrange's theorem for Gro... |
| eqg0subg 19067 | The coset equivalence rela... |
| eqg0subgecsn 19068 | The equivalence classes mo... |
| qus0subgbas 19069 | The base set of a quotient... |
| qus0subgadd 19070 | The addition in a quotient... |
| cycsubmel 19071 | Characterization of an ele... |
| cycsubmcl 19072 | The set of nonnegative int... |
| cycsubm 19073 | The set of nonnegative int... |
| cyccom 19074 | Condition for an operation... |
| cycsubmcom 19075 | The operation of a monoid ... |
| cycsubggend 19076 | The cyclic subgroup genera... |
| cycsubgcl 19077 | The set of integer powers ... |
| cycsubgss 19078 | The cyclic subgroup genera... |
| cycsubg 19079 | The cyclic group generated... |
| cycsubgcld 19080 | The cyclic subgroup genera... |
| cycsubg2 19081 | The subgroup generated by ... |
| cycsubg2cl 19082 | Any multiple of an element... |
| reldmghm 19085 | Lemma for group homomorphi... |
| isghm 19086 | Property of being a homomo... |
| isghm3 19087 | Property of a group homomo... |
| ghmgrp1 19088 | A group homomorphism is on... |
| ghmgrp2 19089 | A group homomorphism is on... |
| ghmf 19090 | A group homomorphism is a ... |
| ghmlin 19091 | A homomorphism of groups i... |
| ghmid 19092 | A homomorphism of groups p... |
| ghminv 19093 | A homomorphism of groups p... |
| ghmsub 19094 | Linearity of subtraction t... |
| isghmd 19095 | Deduction for a group homo... |
| ghmmhm 19096 | A group homomorphism is a ... |
| ghmmhmb 19097 | Group homomorphisms and mo... |
| ghmmulg 19098 | A homomorphism of monoids ... |
| ghmrn 19099 | The range of a homomorphis... |
| 0ghm 19100 | The constant zero linear f... |
| idghm 19101 | The identity homomorphism ... |
| resghm 19102 | Restriction of a homomorph... |
| resghm2 19103 | One direction of ~ resghm2... |
| resghm2b 19104 | Restriction of the codomai... |
| ghmghmrn 19105 | A group homomorphism from ... |
| ghmco 19106 | The composition of group h... |
| ghmima 19107 | The image of a subgroup un... |
| ghmpreima 19108 | The inverse image of a sub... |
| ghmeql 19109 | The equalizer of two group... |
| ghmnsgima 19110 | The image of a normal subg... |
| ghmnsgpreima 19111 | The inverse image of a nor... |
| ghmker 19112 | The kernel of a homomorphi... |
| ghmeqker 19113 | Two source points map to t... |
| pwsdiagghm 19114 | Diagonal homomorphism into... |
| ghmf1 19115 | Two ways of saying a group... |
| ghmf1o 19116 | A bijective group homomorp... |
| conjghm 19117 | Conjugation is an automorp... |
| conjsubg 19118 | A conjugated subgroup is a... |
| conjsubgen 19119 | A conjugated subgroup is e... |
| conjnmz 19120 | A subgroup is unchanged un... |
| conjnmzb 19121 | Alternative condition for ... |
| conjnsg 19122 | A normal subgroup is uncha... |
| qusghm 19123 | If ` Y ` is a normal subgr... |
| ghmpropd 19124 | Group homomorphism depends... |
| gimfn 19129 | The group isomorphism func... |
| isgim 19130 | An isomorphism of groups i... |
| gimf1o 19131 | An isomorphism of groups i... |
| gimghm 19132 | An isomorphism of groups i... |
| isgim2 19133 | A group isomorphism is a h... |
| subggim 19134 | Behavior of subgroups unde... |
| gimcnv 19135 | The converse of a bijectiv... |
| gimco 19136 | The composition of group i... |
| brgic 19137 | The relation "is isomorphi... |
| brgici 19138 | Prove isomorphic by an exp... |
| gicref 19139 | Isomorphism is reflexive. ... |
| giclcl 19140 | Isomorphism implies the le... |
| gicrcl 19141 | Isomorphism implies the ri... |
| gicsym 19142 | Isomorphism is symmetric. ... |
| gictr 19143 | Isomorphism is transitive.... |
| gicer 19144 | Isomorphism is an equivale... |
| gicen 19145 | Isomorphic groups have equ... |
| gicsubgen 19146 | A less trivial example of ... |
| isga 19149 | The predicate "is a (left)... |
| gagrp 19150 | The left argument of a gro... |
| gaset 19151 | The right argument of a gr... |
| gagrpid 19152 | The identity of the group ... |
| gaf 19153 | The mapping of the group a... |
| gafo 19154 | A group action is onto its... |
| gaass 19155 | An "associative" property ... |
| ga0 19156 | The action of a group on t... |
| gaid 19157 | The trivial action of a gr... |
| subgga 19158 | A subgroup acts on its par... |
| gass 19159 | A subset of a group action... |
| gasubg 19160 | The restriction of a group... |
| gaid2 19161 | A group operation is a lef... |
| galcan 19162 | The action of a particular... |
| gacan 19163 | Group inverses cancel in a... |
| gapm 19164 | The action of a particular... |
| gaorb 19165 | The orbit equivalence rela... |
| gaorber 19166 | The orbit equivalence rela... |
| gastacl 19167 | The stabilizer subgroup in... |
| gastacos 19168 | Write the coset relation f... |
| orbstafun 19169 | Existence and uniqueness f... |
| orbstaval 19170 | Value of the function at a... |
| orbsta 19171 | The Orbit-Stabilizer theor... |
| orbsta2 19172 | Relation between the size ... |
| cntrval 19177 | Substitute definition of t... |
| cntzfval 19178 | First level substitution f... |
| cntzval 19179 | Definition substitution fo... |
| elcntz 19180 | Elementhood in the central... |
| cntzel 19181 | Membership in a centralize... |
| cntzsnval 19182 | Special substitution for t... |
| elcntzsn 19183 | Value of the centralizer o... |
| sscntz 19184 | A centralizer expression f... |
| cntzrcl 19185 | Reverse closure for elemen... |
| cntzssv 19186 | The centralizer is uncondi... |
| cntzi 19187 | Membership in a centralize... |
| cntrss 19188 | The center is a subset of ... |
| cntri 19189 | Defining property of the c... |
| resscntz 19190 | Centralizer in a substruct... |
| cntzsgrpcl 19191 | Centralizers are closed un... |
| cntz2ss 19192 | Centralizers reverse the s... |
| cntzrec 19193 | Reciprocity relationship f... |
| cntziinsn 19194 | Express any centralizer as... |
| cntzsubm 19195 | Centralizers in a monoid a... |
| cntzsubg 19196 | Centralizers in a group ar... |
| cntzidss 19197 | If the elements of ` S ` c... |
| cntzmhm 19198 | Centralizers in a monoid a... |
| cntzmhm2 19199 | Centralizers in a monoid a... |
| cntrsubgnsg 19200 | A central subgroup is norm... |
| cntrnsg 19201 | The center of a group is a... |
| oppgval 19204 | Value of the opposite grou... |
| oppgplusfval 19205 | Value of the addition oper... |
| oppgplus 19206 | Value of the addition oper... |
| setsplusg 19207 | The other components of an... |
| oppglemOLD 19208 | Obsolete version of ~ sets... |
| oppgbas 19209 | Base set of an opposite gr... |
| oppgbasOLD 19210 | Obsolete version of ~ oppg... |
| oppgtset 19211 | Topology of an opposite gr... |
| oppgtsetOLD 19212 | Obsolete version of ~ oppg... |
| oppgtopn 19213 | Topology of an opposite gr... |
| oppgmnd 19214 | The opposite of a monoid i... |
| oppgmndb 19215 | Bidirectional form of ~ op... |
| oppgid 19216 | Zero in a monoid is a symm... |
| oppggrp 19217 | The opposite of a group is... |
| oppggrpb 19218 | Bidirectional form of ~ op... |
| oppginv 19219 | Inverses in a group are a ... |
| invoppggim 19220 | The inverse is an antiauto... |
| oppggic 19221 | Every group is (naturally)... |
| oppgsubm 19222 | Being a submonoid is a sym... |
| oppgsubg 19223 | Being a subgroup is a symm... |
| oppgcntz 19224 | A centralizer in a group i... |
| oppgcntr 19225 | The center of a group is t... |
| gsumwrev 19226 | A sum in an opposite monoi... |
| symgval 19229 | The value of the symmetric... |
| permsetexOLD 19230 | Obsolete version of ~ f1os... |
| symgbas 19231 | The base set of the symmet... |
| symgbasexOLD 19232 | Obsolete as of 8-Aug-2024.... |
| elsymgbas2 19233 | Two ways of saying a funct... |
| elsymgbas 19234 | Two ways of saying a funct... |
| symgbasf1o 19235 | Elements in the symmetric ... |
| symgbasf 19236 | A permutation (element of ... |
| symgbasmap 19237 | A permutation (element of ... |
| symghash 19238 | The symmetric group on ` n... |
| symgbasfi 19239 | The symmetric group on a f... |
| symgfv 19240 | The function value of a pe... |
| symgfvne 19241 | The function values of a p... |
| symgressbas 19242 | The symmetric group on ` A... |
| symgplusg 19243 | The group operation of a s... |
| symgov 19244 | The value of the group ope... |
| symgcl 19245 | The group operation of the... |
| idresperm 19246 | The identity function rest... |
| symgmov1 19247 | For a permutation of a set... |
| symgmov2 19248 | For a permutation of a set... |
| symgbas0 19249 | The base set of the symmet... |
| symg1hash 19250 | The symmetric group on a s... |
| symg1bas 19251 | The symmetric group on a s... |
| symg2hash 19252 | The symmetric group on a (... |
| symg2bas 19253 | The symmetric group on a p... |
| 0symgefmndeq 19254 | The symmetric group on the... |
| snsymgefmndeq 19255 | The symmetric group on a s... |
| symgpssefmnd 19256 | For a set ` A ` with more ... |
| symgvalstruct 19257 | The value of the symmetric... |
| symgvalstructOLD 19258 | Obsolete proof of ~ symgva... |
| symgsubmefmnd 19259 | The symmetric group on a s... |
| symgtset 19260 | The topology of the symmet... |
| symggrp 19261 | The symmetric group on a s... |
| symgid 19262 | The group identity element... |
| symginv 19263 | The group inverse in the s... |
| symgsubmefmndALT 19264 | The symmetric group on a s... |
| galactghm 19265 | The currying of a group ac... |
| lactghmga 19266 | The converse of ~ galactgh... |
| symgtopn 19267 | The topology of the symmet... |
| symgga 19268 | The symmetric group induce... |
| pgrpsubgsymgbi 19269 | Every permutation group is... |
| pgrpsubgsymg 19270 | Every permutation group is... |
| idressubgsymg 19271 | The singleton containing o... |
| idrespermg 19272 | The structure with the sin... |
| cayleylem1 19273 | Lemma for ~ cayley . (Con... |
| cayleylem2 19274 | Lemma for ~ cayley . (Con... |
| cayley 19275 | Cayley's Theorem (construc... |
| cayleyth 19276 | Cayley's Theorem (existenc... |
| symgfix2 19277 | If a permutation does not ... |
| symgextf 19278 | The extension of a permuta... |
| symgextfv 19279 | The function value of the ... |
| symgextfve 19280 | The function value of the ... |
| symgextf1lem 19281 | Lemma for ~ symgextf1 . (... |
| symgextf1 19282 | The extension of a permuta... |
| symgextfo 19283 | The extension of a permuta... |
| symgextf1o 19284 | The extension of a permuta... |
| symgextsymg 19285 | The extension of a permuta... |
| symgextres 19286 | The restriction of the ext... |
| gsumccatsymgsn 19287 | Homomorphic property of co... |
| gsmsymgrfixlem1 19288 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19289 | The composition of permuta... |
| fvcosymgeq 19290 | The values of two composit... |
| gsmsymgreqlem1 19291 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19292 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19293 | Two combination of permuta... |
| symgfixelq 19294 | A permutation of a set fix... |
| symgfixels 19295 | The restriction of a permu... |
| symgfixelsi 19296 | The restriction of a permu... |
| symgfixf 19297 | The mapping of a permutati... |
| symgfixf1 19298 | The mapping of a permutati... |
| symgfixfolem1 19299 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19300 | The mapping of a permutati... |
| symgfixf1o 19301 | The mapping of a permutati... |
| f1omvdmvd 19304 | A permutation of any class... |
| f1omvdcnv 19305 | A permutation and its inve... |
| mvdco 19306 | Composing two permutations... |
| f1omvdconj 19307 | Conjugation of a permutati... |
| f1otrspeq 19308 | A transposition is charact... |
| f1omvdco2 19309 | If exactly one of two perm... |
| f1omvdco3 19310 | If a point is moved by exa... |
| pmtrfval 19311 | The function generating tr... |
| pmtrval 19312 | A generated transposition,... |
| pmtrfv 19313 | General value of mapping a... |
| pmtrprfv 19314 | In a transposition of two ... |
| pmtrprfv3 19315 | In a transposition of two ... |
| pmtrf 19316 | Functionality of a transpo... |
| pmtrmvd 19317 | A transposition moves prec... |
| pmtrrn 19318 | Transposing two points giv... |
| pmtrfrn 19319 | A transposition (as a kind... |
| pmtrffv 19320 | Mapping of a point under a... |
| pmtrrn2 19321 | For any transposition ther... |
| pmtrfinv 19322 | A transposition function i... |
| pmtrfmvdn0 19323 | A transposition moves at l... |
| pmtrff1o 19324 | A transposition function i... |
| pmtrfcnv 19325 | A transposition function i... |
| pmtrfb 19326 | An intrinsic characterizat... |
| pmtrfconj 19327 | Any conjugate of a transpo... |
| symgsssg 19328 | The symmetric group has su... |
| symgfisg 19329 | The symmetric group has a ... |
| symgtrf 19330 | Transpositions are element... |
| symggen 19331 | The span of the transposit... |
| symggen2 19332 | A finite permutation group... |
| symgtrinv 19333 | To invert a permutation re... |
| pmtr3ncomlem1 19334 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19335 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19336 | Transpositions over sets w... |
| pmtrdifellem1 19337 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19338 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19339 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19340 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19341 | A transposition of element... |
| pmtrdifwrdellem1 19342 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19343 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19344 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19345 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19346 | A sequence of transpositio... |
| pmtrdifwrdel2 19347 | A sequence of transpositio... |
| pmtrprfval 19348 | The transpositions on a pa... |
| pmtrprfvalrn 19349 | The range of the transposi... |
| psgnunilem1 19354 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19355 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19356 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19357 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19358 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19359 | Addition and subtraction o... |
| psgnuni 19360 | If the same permutation ca... |
| psgnfval 19361 | Function definition of the... |
| psgnfn 19362 | Functionality and domain o... |
| psgndmsubg 19363 | The finitary permutations ... |
| psgneldm 19364 | Property of being a finita... |
| psgneldm2 19365 | The finitary permutations ... |
| psgneldm2i 19366 | A sequence of transpositio... |
| psgneu 19367 | A finitary permutation has... |
| psgnval 19368 | Value of the permutation s... |
| psgnvali 19369 | A finitary permutation has... |
| psgnvalii 19370 | Any representation of a pe... |
| psgnpmtr 19371 | All transpositions are odd... |
| psgn0fv0 19372 | The permutation sign funct... |
| sygbasnfpfi 19373 | The class of non-fixed poi... |
| psgnfvalfi 19374 | Function definition of the... |
| psgnvalfi 19375 | Value of the permutation s... |
| psgnran 19376 | The range of the permutati... |
| gsmtrcl 19377 | The group sum of transposi... |
| psgnfitr 19378 | A permutation of a finite ... |
| psgnfieu 19379 | A permutation of a finite ... |
| pmtrsn 19380 | The value of the transposi... |
| psgnsn 19381 | The permutation sign funct... |
| psgnprfval 19382 | The permutation sign funct... |
| psgnprfval1 19383 | The permutation sign of th... |
| psgnprfval2 19384 | The permutation sign of th... |
| odfval 19393 | Value of the order functio... |
| odfvalALT 19394 | Shorter proof of ~ odfval ... |
| odval 19395 | Second substitution for th... |
| odlem1 19396 | The group element order is... |
| odcl 19397 | The order of a group eleme... |
| odf 19398 | Functionality of the group... |
| odid 19399 | Any element to the power o... |
| odlem2 19400 | Any positive annihilator o... |
| odmodnn0 19401 | Reduce the argument of a g... |
| mndodconglem 19402 | Lemma for ~ mndodcong . (... |
| mndodcong 19403 | If two multipliers are con... |
| mndodcongi 19404 | If two multipliers are con... |
| oddvdsnn0 19405 | The only multiples of ` A ... |
| odnncl 19406 | If a nonzero multiple of a... |
| odmod 19407 | Reduce the argument of a g... |
| oddvds 19408 | The only multiples of ` A ... |
| oddvdsi 19409 | Any group element is annih... |
| odcong 19410 | If two multipliers are con... |
| odeq 19411 | The ~ oddvds property uniq... |
| odval2 19412 | A non-conditional definiti... |
| odcld 19413 | The order of a group eleme... |
| odm1inv 19414 | The (order-1)th multiple o... |
| odmulgid 19415 | A relationship between the... |
| odmulg2 19416 | The order of a multiple di... |
| odmulg 19417 | Relationship between the o... |
| odmulgeq 19418 | A multiple of a point of f... |
| odbezout 19419 | If ` N ` is coprime to the... |
| od1 19420 | The order of the group ide... |
| odeq1 19421 | The group identity is the ... |
| odinv 19422 | The order of the inverse o... |
| odf1 19423 | The multiples of an elemen... |
| odinf 19424 | The multiples of an elemen... |
| dfod2 19425 | An alternative definition ... |
| odcl2 19426 | The order of an element of... |
| oddvds2 19427 | The order of an element of... |
| finodsubmsubg 19428 | A submonoid whose elements... |
| 0subgALT 19429 | A shorter proof of ~ 0subg... |
| submod 19430 | The order of an element is... |
| subgod 19431 | The order of an element is... |
| odsubdvds 19432 | The order of an element of... |
| odf1o1 19433 | An element with zero order... |
| odf1o2 19434 | An element with nonzero or... |
| odhash 19435 | An element of zero order g... |
| odhash2 19436 | If an element has nonzero ... |
| odhash3 19437 | An element which generates... |
| odngen 19438 | A cyclic subgroup of size ... |
| gexval 19439 | Value of the exponent of a... |
| gexlem1 19440 | The group element order is... |
| gexcl 19441 | The exponent of a group is... |
| gexid 19442 | Any element to the power o... |
| gexlem2 19443 | Any positive annihilator o... |
| gexdvdsi 19444 | Any group element is annih... |
| gexdvds 19445 | The only ` N ` that annihi... |
| gexdvds2 19446 | An integer divides the gro... |
| gexod 19447 | Any group element is annih... |
| gexcl3 19448 | If the order of every grou... |
| gexnnod 19449 | Every group element has fi... |
| gexcl2 19450 | The exponent of a finite g... |
| gexdvds3 19451 | The exponent of a finite g... |
| gex1 19452 | A group or monoid has expo... |
| ispgp 19453 | A group is a ` P ` -group ... |
| pgpprm 19454 | Reverse closure for the fi... |
| pgpgrp 19455 | Reverse closure for the se... |
| pgpfi1 19456 | A finite group with order ... |
| pgp0 19457 | The identity subgroup is a... |
| subgpgp 19458 | A subgroup of a p-group is... |
| sylow1lem1 19459 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19460 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19461 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19462 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19463 | Lemma for ~ sylow1 . Usin... |
| sylow1 19464 | Sylow's first theorem. If... |
| odcau 19465 | Cauchy's theorem for the o... |
| pgpfi 19466 | The converse to ~ pgpfi1 .... |
| pgpfi2 19467 | Alternate version of ~ pgp... |
| pgphash 19468 | The order of a p-group. (... |
| isslw 19469 | The property of being a Sy... |
| slwprm 19470 | Reverse closure for the fi... |
| slwsubg 19471 | A Sylow ` P ` -subgroup is... |
| slwispgp 19472 | Defining property of a Syl... |
| slwpss 19473 | A proper superset of a Syl... |
| slwpgp 19474 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19475 | Every ` P ` -subgroup is c... |
| slwn0 19476 | Every finite group contain... |
| subgslw 19477 | A Sylow subgroup that is c... |
| sylow2alem1 19478 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19479 | Lemma for ~ sylow2a . All... |
| sylow2a 19480 | A named lemma of Sylow's s... |
| sylow2blem1 19481 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19482 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19483 | Sylow's second theorem. P... |
| sylow2b 19484 | Sylow's second theorem. A... |
| slwhash 19485 | A sylow subgroup has cardi... |
| fislw 19486 | The sylow subgroups of a f... |
| sylow2 19487 | Sylow's second theorem. S... |
| sylow3lem1 19488 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19489 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19490 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19491 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19492 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19493 | Lemma for ~ sylow3 , secon... |
| sylow3 19494 | Sylow's third theorem. Th... |
| lsmfval 19499 | The subgroup sum function ... |
| lsmvalx 19500 | Subspace sum value (for a ... |
| lsmelvalx 19501 | Subspace sum membership (f... |
| lsmelvalix 19502 | Subspace sum membership (f... |
| oppglsm 19503 | The subspace sum operation... |
| lsmssv 19504 | Subgroup sum is a subset o... |
| lsmless1x 19505 | Subset implies subgroup su... |
| lsmless2x 19506 | Subset implies subgroup su... |
| lsmub1x 19507 | Subgroup sum is an upper b... |
| lsmub2x 19508 | Subgroup sum is an upper b... |
| lsmval 19509 | Subgroup sum value (for a ... |
| lsmelval 19510 | Subgroup sum membership (f... |
| lsmelvali 19511 | Subgroup sum membership (f... |
| lsmelvalm 19512 | Subgroup sum membership an... |
| lsmelvalmi 19513 | Membership of vector subtr... |
| lsmsubm 19514 | The sum of two commuting s... |
| lsmsubg 19515 | The sum of two commuting s... |
| lsmcom2 19516 | Subgroup sum commutes. (C... |
| smndlsmidm 19517 | The direct product is idem... |
| lsmub1 19518 | Subgroup sum is an upper b... |
| lsmub2 19519 | Subgroup sum is an upper b... |
| lsmunss 19520 | Union of subgroups is a su... |
| lsmless1 19521 | Subset implies subgroup su... |
| lsmless2 19522 | Subset implies subgroup su... |
| lsmless12 19523 | Subset implies subgroup su... |
| lsmidm 19524 | Subgroup sum is idempotent... |
| lsmlub 19525 | The least upper bound prop... |
| lsmss1 19526 | Subgroup sum with a subset... |
| lsmss1b 19527 | Subgroup sum with a subset... |
| lsmss2 19528 | Subgroup sum with a subset... |
| lsmss2b 19529 | Subgroup sum with a subset... |
| lsmass 19530 | Subgroup sum is associativ... |
| mndlsmidm 19531 | Subgroup sum is idempotent... |
| lsm01 19532 | Subgroup sum with the zero... |
| lsm02 19533 | Subgroup sum with the zero... |
| subglsm 19534 | The subgroup sum evaluated... |
| lssnle 19535 | Equivalent expressions for... |
| lsmmod 19536 | The modular law holds for ... |
| lsmmod2 19537 | Modular law dual for subgr... |
| lsmpropd 19538 | If two structures have the... |
| cntzrecd 19539 | Commute the "subgroups com... |
| lsmcntz 19540 | The "subgroups commute" pr... |
| lsmcntzr 19541 | The "subgroups commute" pr... |
| lsmdisj 19542 | Disjointness from a subgro... |
| lsmdisj2 19543 | Association of the disjoin... |
| lsmdisj3 19544 | Association of the disjoin... |
| lsmdisjr 19545 | Disjointness from a subgro... |
| lsmdisj2r 19546 | Association of the disjoin... |
| lsmdisj3r 19547 | Association of the disjoin... |
| lsmdisj2a 19548 | Association of the disjoin... |
| lsmdisj2b 19549 | Association of the disjoin... |
| lsmdisj3a 19550 | Association of the disjoin... |
| lsmdisj3b 19551 | Association of the disjoin... |
| subgdisj1 19552 | Vectors belonging to disjo... |
| subgdisj2 19553 | Vectors belonging to disjo... |
| subgdisjb 19554 | Vectors belonging to disjo... |
| pj1fval 19555 | The left projection functi... |
| pj1val 19556 | The left projection functi... |
| pj1eu 19557 | Uniqueness of a left proje... |
| pj1f 19558 | The left projection functi... |
| pj2f 19559 | The right projection funct... |
| pj1id 19560 | Any element of a direct su... |
| pj1eq 19561 | Any element of a direct su... |
| pj1lid 19562 | The left projection functi... |
| pj1rid 19563 | The left projection functi... |
| pj1ghm 19564 | The left projection functi... |
| pj1ghm2 19565 | The left projection functi... |
| lsmhash 19566 | The order of the direct pr... |
| efgmval 19573 | Value of the formal invers... |
| efgmf 19574 | The formal inverse operati... |
| efgmnvl 19575 | The inversion function on ... |
| efgrcl 19576 | Lemma for ~ efgval . (Con... |
| efglem 19577 | Lemma for ~ efgval . (Con... |
| efgval 19578 | Value of the free group co... |
| efger 19579 | Value of the free group co... |
| efgi 19580 | Value of the free group co... |
| efgi0 19581 | Value of the free group co... |
| efgi1 19582 | Value of the free group co... |
| efgtf 19583 | Value of the free group co... |
| efgtval 19584 | Value of the extension fun... |
| efgval2 19585 | Value of the free group co... |
| efgi2 19586 | Value of the free group co... |
| efgtlen 19587 | Value of the free group co... |
| efginvrel2 19588 | The inverse of the reverse... |
| efginvrel1 19589 | The inverse of the reverse... |
| efgsf 19590 | Value of the auxiliary fun... |
| efgsdm 19591 | Elementhood in the domain ... |
| efgsval 19592 | Value of the auxiliary fun... |
| efgsdmi 19593 | Property of the last link ... |
| efgsval2 19594 | Value of the auxiliary fun... |
| efgsrel 19595 | The start and end of any e... |
| efgs1 19596 | A singleton of an irreduci... |
| efgs1b 19597 | Every extension sequence e... |
| efgsp1 19598 | If ` F ` is an extension s... |
| efgsres 19599 | An initial segment of an e... |
| efgsfo 19600 | For any word, there is a s... |
| efgredlema 19601 | The reduced word that form... |
| efgredlemf 19602 | Lemma for ~ efgredleme . ... |
| efgredlemg 19603 | Lemma for ~ efgred . (Con... |
| efgredleme 19604 | Lemma for ~ efgred . (Con... |
| efgredlemd 19605 | The reduced word that form... |
| efgredlemc 19606 | The reduced word that form... |
| efgredlemb 19607 | The reduced word that form... |
| efgredlem 19608 | The reduced word that form... |
| efgred 19609 | The reduced word that form... |
| efgrelexlema 19610 | If two words ` A , B ` are... |
| efgrelexlemb 19611 | If two words ` A , B ` are... |
| efgrelex 19612 | If two words ` A , B ` are... |
| efgredeu 19613 | There is a unique reduced ... |
| efgred2 19614 | Two extension sequences ha... |
| efgcpbllema 19615 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19616 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19617 | Two extension sequences ha... |
| efgcpbl2 19618 | Two extension sequences ha... |
| frgpval 19619 | Value of the free group co... |
| frgpcpbl 19620 | Compatibility of the group... |
| frgp0 19621 | The free group is a group.... |
| frgpeccl 19622 | Closure of the quotient ma... |
| frgpgrp 19623 | The free group is a group.... |
| frgpadd 19624 | Addition in the free group... |
| frgpinv 19625 | The inverse of an element ... |
| frgpmhm 19626 | The "natural map" from wor... |
| vrgpfval 19627 | The canonical injection fr... |
| vrgpval 19628 | The value of the generatin... |
| vrgpf 19629 | The mapping from the index... |
| vrgpinv 19630 | The inverse of a generatin... |
| frgpuptf 19631 | Any assignment of the gene... |
| frgpuptinv 19632 | Any assignment of the gene... |
| frgpuplem 19633 | Any assignment of the gene... |
| frgpupf 19634 | Any assignment of the gene... |
| frgpupval 19635 | Any assignment of the gene... |
| frgpup1 19636 | Any assignment of the gene... |
| frgpup2 19637 | The evaluation map has the... |
| frgpup3lem 19638 | The evaluation map has the... |
| frgpup3 19639 | Universal property of the ... |
| 0frgp 19640 | The free group on zero gen... |
| isabl 19645 | The predicate "is an Abeli... |
| ablgrp 19646 | An Abelian group is a grou... |
| ablgrpd 19647 | An Abelian group is a grou... |
| ablcmn 19648 | An Abelian group is a comm... |
| ablcmnd 19649 | An Abelian group is a comm... |
| iscmn 19650 | The predicate "is a commut... |
| isabl2 19651 | The predicate "is an Abeli... |
| cmnpropd 19652 | If two structures have the... |
| ablpropd 19653 | If two structures have the... |
| ablprop 19654 | If two structures have the... |
| iscmnd 19655 | Properties that determine ... |
| isabld 19656 | Properties that determine ... |
| isabli 19657 | Properties that determine ... |
| cmnmnd 19658 | A commutative monoid is a ... |
| cmncom 19659 | A commutative monoid is co... |
| ablcom 19660 | An Abelian group operation... |
| cmn32 19661 | Commutative/associative la... |
| cmn4 19662 | Commutative/associative la... |
| cmn12 19663 | Commutative/associative la... |
| abl32 19664 | Commutative/associative la... |
| cmnmndd 19665 | A commutative monoid is a ... |
| rinvmod 19666 | Uniqueness of a right inve... |
| ablinvadd 19667 | The inverse of an Abelian ... |
| ablsub2inv 19668 | Abelian group subtraction ... |
| ablsubadd 19669 | Relationship between Abeli... |
| ablsub4 19670 | Commutative/associative su... |
| abladdsub4 19671 | Abelian group addition/sub... |
| abladdsub 19672 | Associative-type law for g... |
| ablsubadd23 19673 | Commutative/associative la... |
| ablsubaddsub 19674 | Double subtraction and add... |
| ablpncan2 19675 | Cancellation law for subtr... |
| ablpncan3 19676 | A cancellation law for Abe... |
| ablsubsub 19677 | Law for double subtraction... |
| ablsubsub4 19678 | Law for double subtraction... |
| ablpnpcan 19679 | Cancellation law for mixed... |
| ablnncan 19680 | Cancellation law for group... |
| ablsub32 19681 | Swap the second and third ... |
| ablnnncan 19682 | Cancellation law for group... |
| ablnnncan1 19683 | Cancellation law for group... |
| ablsubsub23 19684 | Swap subtrahend and result... |
| mulgnn0di 19685 | Group multiple of a sum, f... |
| mulgdi 19686 | Group multiple of a sum. ... |
| mulgmhm 19687 | The map from ` x ` to ` n ... |
| mulgghm 19688 | The map from ` x ` to ` n ... |
| mulgsubdi 19689 | Group multiple of a differ... |
| ghmfghm 19690 | The function fulfilling th... |
| ghmcmn 19691 | The image of a commutative... |
| ghmabl 19692 | The image of an abelian gr... |
| invghm 19693 | The inversion map is a gro... |
| eqgabl 19694 | Value of the subgroup cose... |
| qusecsub 19695 | Two subgroup cosets are eq... |
| subgabl 19696 | A subgroup of an abelian g... |
| subcmn 19697 | A submonoid of a commutati... |
| submcmn 19698 | A submonoid of a commutati... |
| submcmn2 19699 | A submonoid is commutative... |
| cntzcmn 19700 | The centralizer of any sub... |
| cntzcmnss 19701 | Any subset in a commutativ... |
| cntrcmnd 19702 | The center of a monoid is ... |
| cntrabl 19703 | The center of a group is a... |
| cntzspan 19704 | If the generators commute,... |
| cntzcmnf 19705 | Discharge the centralizer ... |
| ghmplusg 19706 | The pointwise sum of two l... |
| ablnsg 19707 | Every subgroup of an abeli... |
| odadd1 19708 | The order of a product in ... |
| odadd2 19709 | The order of a product in ... |
| odadd 19710 | The order of a product is ... |
| gex2abl 19711 | A group with exponent 2 (o... |
| gexexlem 19712 | Lemma for ~ gexex . (Cont... |
| gexex 19713 | In an abelian group with f... |
| torsubg 19714 | The set of all elements of... |
| oddvdssubg 19715 | The set of all elements wh... |
| lsmcomx 19716 | Subgroup sum commutes (ext... |
| ablcntzd 19717 | All subgroups in an abelia... |
| lsmcom 19718 | Subgroup sum commutes. (C... |
| lsmsubg2 19719 | The sum of two subgroups i... |
| lsm4 19720 | Commutative/associative la... |
| prdscmnd 19721 | The product of a family of... |
| prdsabld 19722 | The product of a family of... |
| pwscmn 19723 | The structure power on a c... |
| pwsabl 19724 | The structure power on an ... |
| qusabl 19725 | If ` Y ` is a subgroup of ... |
| abl1 19726 | The (smallest) structure r... |
| abln0 19727 | Abelian groups (and theref... |
| cnaddablx 19728 | The complex numbers are an... |
| cnaddabl 19729 | The complex numbers are an... |
| cnaddid 19730 | The group identity element... |
| cnaddinv 19731 | Value of the group inverse... |
| zaddablx 19732 | The integers are an Abelia... |
| frgpnabllem1 19733 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19734 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19735 | The free group on two or m... |
| imasabl 19736 | The image structure of an ... |
| iscyg 19739 | Definition of a cyclic gro... |
| iscyggen 19740 | The property of being a cy... |
| iscyggen2 19741 | The property of being a cy... |
| iscyg2 19742 | A cyclic group is a group ... |
| cyggeninv 19743 | The inverse of a cyclic ge... |
| cyggenod 19744 | An element is the generato... |
| cyggenod2 19745 | In an infinite cyclic grou... |
| iscyg3 19746 | Definition of a cyclic gro... |
| iscygd 19747 | Definition of a cyclic gro... |
| iscygodd 19748 | Show that a group with an ... |
| cycsubmcmn 19749 | The set of nonnegative int... |
| cyggrp 19750 | A cyclic group is a group.... |
| cygabl 19751 | A cyclic group is abelian.... |
| cygctb 19752 | A cyclic group is countabl... |
| 0cyg 19753 | The trivial group is cycli... |
| prmcyg 19754 | A group with prime order i... |
| lt6abl 19755 | A group with fewer than ` ... |
| ghmcyg 19756 | The image of a cyclic grou... |
| cyggex2 19757 | The exponent of a cyclic g... |
| cyggex 19758 | The exponent of a finite c... |
| cyggexb 19759 | A finite abelian group is ... |
| giccyg 19760 | Cyclicity is a group prope... |
| cycsubgcyg 19761 | The cyclic subgroup genera... |
| cycsubgcyg2 19762 | The cyclic subgroup genera... |
| gsumval3a 19763 | Value of the group sum ope... |
| gsumval3eu 19764 | The group sum as defined i... |
| gsumval3lem1 19765 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19766 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19767 | Value of the group sum ope... |
| gsumcllem 19768 | Lemma for ~ gsumcl and rel... |
| gsumzres 19769 | Extend a finite group sum ... |
| gsumzcl2 19770 | Closure of a finite group ... |
| gsumzcl 19771 | Closure of a finite group ... |
| gsumzf1o 19772 | Re-index a finite group su... |
| gsumres 19773 | Extend a finite group sum ... |
| gsumcl2 19774 | Closure of a finite group ... |
| gsumcl 19775 | Closure of a finite group ... |
| gsumf1o 19776 | Re-index a finite group su... |
| gsumreidx 19777 | Re-index a finite group su... |
| gsumzsubmcl 19778 | Closure of a group sum in ... |
| gsumsubmcl 19779 | Closure of a group sum in ... |
| gsumsubgcl 19780 | Closure of a group sum in ... |
| gsumzaddlem 19781 | The sum of two group sums.... |
| gsumzadd 19782 | The sum of two group sums.... |
| gsumadd 19783 | The sum of two group sums.... |
| gsummptfsadd 19784 | The sum of two group sums ... |
| gsummptfidmadd 19785 | The sum of two group sums ... |
| gsummptfidmadd2 19786 | The sum of two group sums ... |
| gsumzsplit 19787 | Split a group sum into two... |
| gsumsplit 19788 | Split a group sum into two... |
| gsumsplit2 19789 | Split a group sum into two... |
| gsummptfidmsplit 19790 | Split a group sum expresse... |
| gsummptfidmsplitres 19791 | Split a group sum expresse... |
| gsummptfzsplit 19792 | Split a group sum expresse... |
| gsummptfzsplitl 19793 | Split a group sum expresse... |
| gsumconst 19794 | Sum of a constant series. ... |
| gsumconstf 19795 | Sum of a constant series. ... |
| gsummptshft 19796 | Index shift of a finite gr... |
| gsumzmhm 19797 | Apply a group homomorphism... |
| gsummhm 19798 | Apply a group homomorphism... |
| gsummhm2 19799 | Apply a group homomorphism... |
| gsummptmhm 19800 | Apply a group homomorphism... |
| gsummulglem 19801 | Lemma for ~ gsummulg and ~... |
| gsummulg 19802 | Nonnegative multiple of a ... |
| gsummulgz 19803 | Integer multiple of a grou... |
| gsumzoppg 19804 | The opposite of a group su... |
| gsumzinv 19805 | Inverse of a group sum. (... |
| gsuminv 19806 | Inverse of a group sum. (... |
| gsummptfidminv 19807 | Inverse of a group sum exp... |
| gsumsub 19808 | The difference of two grou... |
| gsummptfssub 19809 | The difference of two grou... |
| gsummptfidmsub 19810 | The difference of two grou... |
| gsumsnfd 19811 | Group sum of a singleton, ... |
| gsumsnd 19812 | Group sum of a singleton, ... |
| gsumsnf 19813 | Group sum of a singleton, ... |
| gsumsn 19814 | Group sum of a singleton. ... |
| gsumpr 19815 | Group sum of a pair. (Con... |
| gsumzunsnd 19816 | Append an element to a fin... |
| gsumunsnfd 19817 | Append an element to a fin... |
| gsumunsnd 19818 | Append an element to a fin... |
| gsumunsnf 19819 | Append an element to a fin... |
| gsumunsn 19820 | Append an element to a fin... |
| gsumdifsnd 19821 | Extract a summand from a f... |
| gsumpt 19822 | Sum of a family that is no... |
| gsummptf1o 19823 | Re-index a finite group su... |
| gsummptun 19824 | Group sum of a disjoint un... |
| gsummpt1n0 19825 | If only one summand in a f... |
| gsummptif1n0 19826 | If only one summand in a f... |
| gsummptcl 19827 | Closure of a finite group ... |
| gsummptfif1o 19828 | Re-index a finite group su... |
| gsummptfzcl 19829 | Closure of a finite group ... |
| gsum2dlem1 19830 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19831 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19832 | Write a sum over a two-dim... |
| gsum2d2lem 19833 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19834 | Write a group sum over a t... |
| gsumcom2 19835 | Two-dimensional commutatio... |
| gsumxp 19836 | Write a group sum over a c... |
| gsumcom 19837 | Commute the arguments of a... |
| gsumcom3 19838 | A commutative law for fini... |
| gsumcom3fi 19839 | A commutative law for fini... |
| gsumxp2 19840 | Write a group sum over a c... |
| prdsgsum 19841 | Finite commutative sums in... |
| pwsgsum 19842 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19843 | Replacing a finitely suppo... |
| nn0gsumfz 19844 | Replacing a finitely suppo... |
| nn0gsumfz0 19845 | Replacing a finitely suppo... |
| gsummptnn0fz 19846 | A final group sum over a f... |
| gsummptnn0fzfv 19847 | A final group sum over a f... |
| telgsumfzslem 19848 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19849 | Telescoping group sum rang... |
| telgsumfz 19850 | Telescoping group sum rang... |
| telgsumfz0s 19851 | Telescoping finite group s... |
| telgsumfz0 19852 | Telescoping finite group s... |
| telgsums 19853 | Telescoping finitely suppo... |
| telgsum 19854 | Telescoping finitely suppo... |
| reldmdprd 19859 | The domain of the internal... |
| dmdprd 19860 | The domain of definition o... |
| dmdprdd 19861 | Show that a given family i... |
| dprddomprc 19862 | A family of subgroups inde... |
| dprddomcld 19863 | If a family of subgroups i... |
| dprdval0prc 19864 | The internal direct produc... |
| dprdval 19865 | The value of the internal ... |
| eldprd 19866 | A class ` A ` is an intern... |
| dprdgrp 19867 | Reverse closure for the in... |
| dprdf 19868 | The function ` S ` is a fa... |
| dprdf2 19869 | The function ` S ` is a fa... |
| dprdcntz 19870 | The function ` S ` is a fa... |
| dprddisj 19871 | The function ` S ` is a fa... |
| dprdw 19872 | The property of being a fi... |
| dprdwd 19873 | A mapping being a finitely... |
| dprdff 19874 | A finitely supported funct... |
| dprdfcl 19875 | A finitely supported funct... |
| dprdffsupp 19876 | A finitely supported funct... |
| dprdfcntz 19877 | A function on the elements... |
| dprdssv 19878 | The internal direct produc... |
| dprdfid 19879 | A function mapping all but... |
| eldprdi 19880 | The domain of definition o... |
| dprdfinv 19881 | Take the inverse of a grou... |
| dprdfadd 19882 | Take the sum of group sums... |
| dprdfsub 19883 | Take the difference of gro... |
| dprdfeq0 19884 | The zero function is the o... |
| dprdf11 19885 | Two group sums over a dire... |
| dprdsubg 19886 | The internal direct produc... |
| dprdub 19887 | Each factor is a subset of... |
| dprdlub 19888 | The direct product is smal... |
| dprdspan 19889 | The direct product is the ... |
| dprdres 19890 | Restriction of a direct pr... |
| dprdss 19891 | Create a direct product by... |
| dprdz 19892 | A family consisting entire... |
| dprd0 19893 | The empty family is an int... |
| dprdf1o 19894 | Rearrange the index set of... |
| dprdf1 19895 | Rearrange the index set of... |
| subgdmdprd 19896 | A direct product in a subg... |
| subgdprd 19897 | A direct product in a subg... |
| dprdsn 19898 | A singleton family is an i... |
| dmdprdsplitlem 19899 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 19900 | The function ` S ` is a fa... |
| dprddisj2 19901 | The function ` S ` is a fa... |
| dprd2dlem2 19902 | The direct product of a co... |
| dprd2dlem1 19903 | The direct product of a co... |
| dprd2da 19904 | The direct product of a co... |
| dprd2db 19905 | The direct product of a co... |
| dprd2d2 19906 | The direct product of a co... |
| dmdprdsplit2lem 19907 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 19908 | The direct product splits ... |
| dmdprdsplit 19909 | The direct product splits ... |
| dprdsplit 19910 | The direct product is the ... |
| dmdprdpr 19911 | A singleton family is an i... |
| dprdpr 19912 | A singleton family is an i... |
| dpjlem 19913 | Lemma for theorems about d... |
| dpjcntz 19914 | The two subgroups that app... |
| dpjdisj 19915 | The two subgroups that app... |
| dpjlsm 19916 | The two subgroups that app... |
| dpjfval 19917 | Value of the direct produc... |
| dpjval 19918 | Value of the direct produc... |
| dpjf 19919 | The ` X ` -th index projec... |
| dpjidcl 19920 | The key property of projec... |
| dpjeq 19921 | Decompose a group sum into... |
| dpjid 19922 | The key property of projec... |
| dpjlid 19923 | The ` X ` -th index projec... |
| dpjrid 19924 | The ` Y ` -th index projec... |
| dpjghm 19925 | The direct product is the ... |
| dpjghm2 19926 | The direct product is the ... |
| ablfacrplem 19927 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 19928 | A finite abelian group who... |
| ablfacrp2 19929 | The factors ` K , L ` of ~... |
| ablfac1lem 19930 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 19931 | The factors of ~ ablfac1b ... |
| ablfac1b 19932 | Any abelian group is the d... |
| ablfac1c 19933 | The factors of ~ ablfac1b ... |
| ablfac1eulem 19934 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 19935 | The factorization of ~ abl... |
| pgpfac1lem1 19936 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 19937 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 19938 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 19939 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 19940 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 19941 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 19942 | Factorization of a finite ... |
| pgpfaclem1 19943 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 19944 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 19945 | Lemma for ~ pgpfac . (Con... |
| pgpfac 19946 | Full factorization of a fi... |
| ablfaclem1 19947 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 19948 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 19949 | Lemma for ~ ablfac . (Con... |
| ablfac 19950 | The Fundamental Theorem of... |
| ablfac2 19951 | Choose generators for each... |
| issimpg 19954 | The predicate "is a simple... |
| issimpgd 19955 | Deduce a simple group from... |
| simpggrp 19956 | A simple group is a group.... |
| simpggrpd 19957 | A simple group is a group.... |
| simpg2nsg 19958 | A simple group has two nor... |
| trivnsimpgd 19959 | Trivial groups are not sim... |
| simpgntrivd 19960 | Simple groups are nontrivi... |
| simpgnideld 19961 | A simple group contains a ... |
| simpgnsgd 19962 | The only normal subgroups ... |
| simpgnsgeqd 19963 | A normal subgroup of a sim... |
| 2nsgsimpgd 19964 | If any normal subgroup of ... |
| simpgnsgbid 19965 | A nontrivial group is simp... |
| ablsimpnosubgd 19966 | A subgroup of an abelian s... |
| ablsimpg1gend 19967 | An abelian simple group is... |
| ablsimpgcygd 19968 | An abelian simple group is... |
| ablsimpgfindlem1 19969 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 19970 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 19971 | Relationship between the o... |
| ablsimpgfind 19972 | An abelian simple group is... |
| fincygsubgd 19973 | The subgroup referenced in... |
| fincygsubgodd 19974 | Calculate the order of a s... |
| fincygsubgodexd 19975 | A finite cyclic group has ... |
| prmgrpsimpgd 19976 | A group of prime order is ... |
| ablsimpgprmd 19977 | An abelian simple group ha... |
| ablsimpgd 19978 | An abelian group is simple... |
| fnmgp 19981 | The multiplicative group o... |
| mgpval 19982 | Value of the multiplicatio... |
| mgpplusg 19983 | Value of the group operati... |
| mgplemOLD 19984 | Obsolete version of ~ sets... |
| mgpbas 19985 | Base set of the multiplica... |
| mgpbasOLD 19986 | Obsolete version of ~ mgpb... |
| mgpsca 19987 | The multiplication monoid ... |
| mgpscaOLD 19988 | Obsolete version of ~ mgps... |
| mgptset 19989 | Topology component of the ... |
| mgptsetOLD 19990 | Obsolete version of ~ mgpt... |
| mgptopn 19991 | Topology of the multiplica... |
| mgpds 19992 | Distance function of the m... |
| mgpdsOLD 19993 | Obsolete version of ~ mgpd... |
| mgpress 19994 | Subgroup commutes with the... |
| mgpressOLD 19995 | Obsolete version of ~ mgpr... |
| ringidval 19998 | The value of the unity ele... |
| dfur2 19999 | The multiplicative identit... |
| issrg 20002 | The predicate "is a semiri... |
| srgcmn 20003 | A semiring is a commutativ... |
| srgmnd 20004 | A semiring is a monoid. (... |
| srgmgp 20005 | A semiring is a monoid und... |
| srgdilem 20006 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20007 | Closure of the multiplicat... |
| srgass 20008 | Associative law for the mu... |
| srgideu 20009 | The unity element of a sem... |
| srgfcl 20010 | Functionality of the multi... |
| srgdi 20011 | Distributive law for the m... |
| srgdir 20012 | Distributive law for the m... |
| srgidcl 20013 | The unity element of a sem... |
| srg0cl 20014 | The zero element of a semi... |
| srgidmlem 20015 | Lemma for ~ srglidm and ~ ... |
| srglidm 20016 | The unity element of a sem... |
| srgridm 20017 | The unity element of a sem... |
| issrgid 20018 | Properties showing that an... |
| srgacl 20019 | Closure of the addition op... |
| srgcom 20020 | Commutativity of the addit... |
| srgrz 20021 | The zero of a semiring is ... |
| srglz 20022 | The zero of a semiring is ... |
| srgisid 20023 | In a semiring, the only le... |
| o2timesd 20024 | An element of a ring-like ... |
| rglcom4d 20025 | Restricted commutativity o... |
| srgo2times 20026 | A semiring element plus it... |
| srgcom4lem 20027 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20028 | Restricted commutativity o... |
| srg1zr 20029 | The only semiring with a b... |
| srgen1zr 20030 | The only semiring with one... |
| srgmulgass 20031 | An associative property be... |
| srgpcomp 20032 | If two elements of a semir... |
| srgpcompp 20033 | If two elements of a semir... |
| srgpcomppsc 20034 | If two elements of a semir... |
| srglmhm 20035 | Left-multiplication in a s... |
| srgrmhm 20036 | Right-multiplication in a ... |
| srgsummulcr 20037 | A finite semiring sum mult... |
| sgsummulcl 20038 | A finite semiring sum mult... |
| srg1expzeq1 20039 | The exponentiation (by a n... |
| srgbinomlem1 20040 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20041 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20042 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20043 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20044 | Lemma for ~ srgbinom . In... |
| srgbinom 20045 | The binomial theorem for c... |
| csrgbinom 20046 | The binomial theorem for c... |
| isring 20051 | The predicate "is a (unita... |
| ringgrp 20052 | A ring is a group. (Contr... |
| ringmgp 20053 | A ring is a monoid under m... |
| iscrng 20054 | A commutative ring is a ri... |
| crngmgp 20055 | A commutative ring's multi... |
| ringgrpd 20056 | A ring is a group. (Contr... |
| ringmnd 20057 | A ring is a monoid under a... |
| ringmgm 20058 | A ring is a magma. (Contr... |
| crngring 20059 | A commutative ring is a ri... |
| crngringd 20060 | A commutative ring is a ri... |
| crnggrpd 20061 | A commutative ring is a gr... |
| mgpf 20062 | Restricted functionality o... |
| ringdilem 20063 | Properties of a unital rin... |
| ringcl 20064 | Closure of the multiplicat... |
| crngcom 20065 | A commutative ring's multi... |
| iscrng2 20066 | A commutative ring is a ri... |
| ringass 20067 | Associative law for multip... |
| ringideu 20068 | The unity element of a rin... |
| ringassd 20069 | Associative law for multip... |
| ringcld 20070 | Closure of the multiplicat... |
| ringdi 20071 | Distributive law for the m... |
| ringdir 20072 | Distributive law for the m... |
| ringidcl 20073 | The unity element of a rin... |
| ring0cl 20074 | The zero element of a ring... |
| ringidmlem 20075 | Lemma for ~ ringlidm and ~... |
| ringlidm 20076 | The unity element of a rin... |
| ringridm 20077 | The unity element of a rin... |
| isringid 20078 | Properties showing that an... |
| ringlidmd 20079 | The unity element of a rin... |
| ringridmd 20080 | The unity element of a rin... |
| ringid 20081 | The multiplication operati... |
| ringo2times 20082 | A ring element plus itself... |
| ringadd2 20083 | A ring element plus itself... |
| ringidss 20084 | A subset of the multiplica... |
| ringacl 20085 | Closure of the addition op... |
| ringcomlem 20086 | Lemma for ~ ringcom . Thi... |
| ringcom 20087 | Commutativity of the addit... |
| ringabl 20088 | A ring is an Abelian group... |
| ringcmn 20089 | A ring is a commutative mo... |
| ringabld 20090 | A ring is an Abelian group... |
| ringcmnd 20091 | A ring is a commutative mo... |
| ringpropd 20092 | If two structures have the... |
| crngpropd 20093 | If two structures have the... |
| ringprop 20094 | If two structures have the... |
| isringd 20095 | Properties that determine ... |
| iscrngd 20096 | Properties that determine ... |
| ringlz 20097 | The zero of a unital ring ... |
| ringrz 20098 | The zero of a unital ring ... |
| ringsrg 20099 | Any ring is also a semirin... |
| ring1eq0 20100 | If one and zero are equal,... |
| ring1ne0 20101 | If a ring has at least two... |
| ringinvnz1ne0 20102 | In a unital ring, a left i... |
| ringinvnzdiv 20103 | In a unital ring, a left i... |
| ringnegl 20104 | Negation in a ring is the ... |
| ringnegr 20105 | Negation in a ring is the ... |
| ringmneg1 20106 | Negation of a product in a... |
| ringmneg2 20107 | Negation of a product in a... |
| ringm2neg 20108 | Double negation of a produ... |
| ringsubdi 20109 | Ring multiplication distri... |
| ringsubdir 20110 | Ring multiplication distri... |
| mulgass2 20111 | An associative property be... |
| ring1 20112 | The (smallest) structure r... |
| ringn0 20113 | Rings exist. (Contributed... |
| ringlghm 20114 | Left-multiplication in a r... |
| ringrghm 20115 | Right-multiplication in a ... |
| gsummulc1OLD 20116 | Obsolete version of ~ gsum... |
| gsummulc2OLD 20117 | Obsolete version of ~ gsum... |
| gsummulc1 20118 | A finite ring sum multipli... |
| gsummulc2 20119 | A finite ring sum multipli... |
| gsummgp0 20120 | If one factor in a finite ... |
| gsumdixp 20121 | Distribute a binary produc... |
| prdsmgp 20122 | The multiplicative monoid ... |
| prdsmulrcl 20123 | A structure product of rin... |
| prdsringd 20124 | A product of rings is a ri... |
| prdscrngd 20125 | A product of commutative r... |
| prds1 20126 | Value of the ring unity in... |
| pwsring 20127 | A structure power of a rin... |
| pws1 20128 | Value of the ring unity in... |
| pwscrng 20129 | A structure power of a com... |
| pwsmgp 20130 | The multiplicative group o... |
| pwspjmhmmgpd 20131 | The projection given by ~ ... |
| pwsexpg 20132 | Value of a group exponenti... |
| imasring 20133 | The image structure of a r... |
| imasringf1 20134 | The image of a ring under ... |
| xpsringd 20135 | A product of two rings is ... |
| qusring2 20136 | The quotient structure of ... |
| crngbinom 20137 | The binomial theorem for c... |
| opprval 20140 | Value of the opposite ring... |
| opprmulfval 20141 | Value of the multiplicatio... |
| opprmul 20142 | Value of the multiplicatio... |
| crngoppr 20143 | In a commutative ring, the... |
| opprlem 20144 | Lemma for ~ opprbas and ~ ... |
| opprlemOLD 20145 | Obsolete version of ~ oppr... |
| opprbas 20146 | Base set of an opposite ri... |
| opprbasOLD 20147 | Obsolete proof of ~ opprba... |
| oppradd 20148 | Addition operation of an o... |
| oppraddOLD 20149 | Obsolete proof of ~ opprba... |
| opprring 20150 | An opposite ring is a ring... |
| opprringb 20151 | Bidirectional form of ~ op... |
| oppr0 20152 | Additive identity of an op... |
| oppr1 20153 | Multiplicative identity of... |
| opprneg 20154 | The negative function in a... |
| opprsubg 20155 | Being a subgroup is a symm... |
| mulgass3 20156 | An associative property be... |
| reldvdsr 20163 | The divides relation is a ... |
| dvdsrval 20164 | Value of the divides relat... |
| dvdsr 20165 | Value of the divides relat... |
| dvdsr2 20166 | Value of the divides relat... |
| dvdsrmul 20167 | A left-multiple of ` X ` i... |
| dvdsrcl 20168 | Closure of a dividing elem... |
| dvdsrcl2 20169 | Closure of a dividing elem... |
| dvdsrid 20170 | An element in a (unital) r... |
| dvdsrtr 20171 | Divisibility is transitive... |
| dvdsrmul1 20172 | The divisibility relation ... |
| dvdsrneg 20173 | An element divides its neg... |
| dvdsr01 20174 | In a ring, zero is divisib... |
| dvdsr02 20175 | Only zero is divisible by ... |
| isunit 20176 | Property of being a unit o... |
| 1unit 20177 | The multiplicative identit... |
| unitcl 20178 | A unit is an element of th... |
| unitss 20179 | The set of units is contai... |
| opprunit 20180 | Being a unit is a symmetri... |
| crngunit 20181 | Property of being a unit i... |
| dvdsunit 20182 | A divisor of a unit is a u... |
| unitmulcl 20183 | The product of units is a ... |
| unitmulclb 20184 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20185 | The base set of the group ... |
| unitgrp 20186 | The group of units is a gr... |
| unitabl 20187 | The group of units of a co... |
| unitgrpid 20188 | The identity of the group ... |
| unitsubm 20189 | The group of units is a su... |
| invrfval 20192 | Multiplicative inverse fun... |
| unitinvcl 20193 | The inverse of a unit exis... |
| unitinvinv 20194 | The inverse of the inverse... |
| ringinvcl 20195 | The inverse of a unit is a... |
| unitlinv 20196 | A unit times its inverse i... |
| unitrinv 20197 | A unit times its inverse i... |
| 1rinv 20198 | The inverse of the ring un... |
| 0unit 20199 | The additive identity is a... |
| unitnegcl 20200 | The negative of a unit is ... |
| ringunitnzdiv 20201 | In a unitary ring, a unit ... |
| ring1nzdiv 20202 | In a unitary ring, the rin... |
| dvrfval 20205 | Division operation in a ri... |
| dvrval 20206 | Division operation in a ri... |
| dvrcl 20207 | Closure of division operat... |
| unitdvcl 20208 | The units are closed under... |
| dvrid 20209 | A ring element divided by ... |
| dvr1 20210 | A ring element divided by ... |
| dvrass 20211 | An associative law for div... |
| dvrcan1 20212 | A cancellation law for div... |
| dvrcan3 20213 | A cancellation law for div... |
| dvreq1 20214 | Equality in terms of ratio... |
| dvrdir 20215 | Distributive law for the d... |
| rdivmuldivd 20216 | Multiplication of two rati... |
| ringinvdv 20217 | Write the inverse function... |
| rngidpropd 20218 | The ring unity depends onl... |
| dvdsrpropd 20219 | The divisibility relation ... |
| unitpropd 20220 | The set of units depends o... |
| invrpropd 20221 | The ring inverse function ... |
| isirred 20222 | An irreducible element of ... |
| isnirred 20223 | The property of being a no... |
| isirred2 20224 | Expand out the class diffe... |
| opprirred 20225 | Irreducibility is symmetri... |
| irredn0 20226 | The additive identity is n... |
| irredcl 20227 | An irreducible element is ... |
| irrednu 20228 | An irreducible element is ... |
| irredn1 20229 | The multiplicative identit... |
| irredrmul 20230 | The product of an irreduci... |
| irredlmul 20231 | The product of a unit and ... |
| irredmul 20232 | If product of two elements... |
| irredneg 20233 | The negative of an irreduc... |
| irrednegb 20234 | An element is irreducible ... |
| dfrhm2 20242 | The property of a ring hom... |
| rhmrcl1 20244 | Reverse closure of a ring ... |
| rhmrcl2 20245 | Reverse closure of a ring ... |
| isrhm 20246 | A function is a ring homom... |
| rhmmhm 20247 | A ring homomorphism is a h... |
| isrim0OLD 20248 | Obsolete version of ~ isri... |
| rimrcl 20249 | Reverse closure for an iso... |
| isrim0 20250 | A ring isomorphism is a ho... |
| rhmghm 20251 | A ring homomorphism is an ... |
| rhmf 20252 | A ring homomorphism is a f... |
| rhmmul 20253 | A homomorphism of rings pr... |
| isrhm2d 20254 | Demonstration of ring homo... |
| isrhmd 20255 | Demonstration of ring homo... |
| rhm1 20256 | Ring homomorphisms are req... |
| idrhm 20257 | The identity homomorphism ... |
| rhmf1o 20258 | A ring homomorphism is bij... |
| isrim 20259 | An isomorphism of rings is... |
| isrimOLD 20260 | Obsolete version of ~ isri... |
| rimf1o 20261 | An isomorphism of rings is... |
| rimrhmOLD 20262 | Obsolete version of ~ rimr... |
| rimrhm 20263 | A ring isomorphism is a ho... |
| rimgim 20264 | An isomorphism of rings is... |
| rhmco 20265 | The composition of ring ho... |
| pwsco1rhm 20266 | Right composition with a f... |
| pwsco2rhm 20267 | Left composition with a ri... |
| f1ghm0to0 20268 | If a group homomorphism ` ... |
| f1rhm0to0ALT 20269 | Alternate proof for ~ f1gh... |
| gim0to0 20270 | A group isomorphism maps t... |
| kerf1ghm 20271 | A group homomorphism ` F `... |
| brric 20272 | The relation "is isomorphi... |
| brrici 20273 | Prove isomorphic by an exp... |
| brric2 20274 | The relation "is isomorphi... |
| ricgic 20275 | If two rings are (ring) is... |
| rhmdvdsr 20276 | A ring homomorphism preser... |
| rhmopp 20277 | A ring homomorphism is als... |
| elrhmunit 20278 | Ring homomorphisms preserv... |
| rhmunitinv 20279 | Ring homomorphisms preserv... |
| isnzr 20282 | Property of a nonzero ring... |
| nzrnz 20283 | One and zero are different... |
| nzrring 20284 | A nonzero ring is a ring. ... |
| nzrringOLD 20285 | Obsolete version of ~ nzrr... |
| isnzr2 20286 | Equivalent characterizatio... |
| isnzr2hash 20287 | Equivalent characterizatio... |
| opprnzr 20288 | The opposite of a nonzero ... |
| ringelnzr 20289 | A ring is nonzero if it ha... |
| nzrunit 20290 | A unit is nonzero in any n... |
| 0ringnnzr 20291 | A ring is a zero ring iff ... |
| 0ring 20292 | If a ring has only one ele... |
| 0ring01eq 20293 | In a ring with only one el... |
| 01eq0ring 20294 | If the zero and the identi... |
| 01eq0ringOLD 20295 | Obsolete version of ~ 01eq... |
| 0ring01eqbi 20296 | In a unital ring the zero ... |
| islring 20299 | The predicate "is a local ... |
| lringnzr 20300 | A local ring is a nonzero ... |
| lringring 20301 | A local ring is a ring. (... |
| lringnz 20302 | A local ring is a nonzero ... |
| lringuplu 20303 | If the sum of two elements... |
| isdrng 20308 | The predicate "is a divisi... |
| drngunit 20309 | Elementhood in the set of ... |
| drngui 20310 | The set of units of a divi... |
| drngring 20311 | A division ring is a ring.... |
| drngringd 20312 | A division ring is a ring.... |
| drnggrpd 20313 | A division ring is a group... |
| drnggrp 20314 | A division ring is a group... |
| isfld 20315 | A field is a commutative d... |
| fldcrngd 20316 | A field is a commutative r... |
| isdrng2 20317 | A division ring can equiva... |
| drngprop 20318 | If two structures have the... |
| drngmgp 20319 | A division ring contains a... |
| drngmcl 20320 | The product of two nonzero... |
| drngid 20321 | A division ring's unity is... |
| drngunz 20322 | A division ring's unity is... |
| drngnzr 20323 | All division rings are non... |
| drngid2 20324 | Properties showing that an... |
| drnginvrcl 20325 | Closure of the multiplicat... |
| drnginvrn0 20326 | The multiplicative inverse... |
| drnginvrcld 20327 | Closure of the multiplicat... |
| drnginvrl 20328 | Property of the multiplica... |
| drnginvrr 20329 | Property of the multiplica... |
| drnginvrld 20330 | Property of the multiplica... |
| drnginvrrd 20331 | Property of the multiplica... |
| drngmul0or 20332 | A product is zero iff one ... |
| drngmulne0 20333 | A product is nonzero iff b... |
| drngmuleq0 20334 | An element is zero iff its... |
| opprdrng 20335 | The opposite of a division... |
| isdrngd 20336 | Properties that characteri... |
| isdrngrd 20337 | Properties that characteri... |
| isdrngdOLD 20338 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20339 | Obsolete version of ~ isdr... |
| drngpropd 20340 | If two structures have the... |
| fldpropd 20341 | If two structures have the... |
| rng1nnzr 20342 | The (smallest) structure r... |
| ring1zr 20343 | The only (unital) ring wit... |
| rngen1zr 20344 | The only (unital) ring wit... |
| ringen1zr 20345 | The only unital ring with ... |
| rng1nfld 20346 | The zero ring is not a fie... |
| issubrg 20351 | The subring predicate. (C... |
| subrgss 20352 | A subring is a subset. (C... |
| subrgid 20353 | Every ring is a subring of... |
| subrgring 20354 | A subring is a ring. (Con... |
| subrgcrng 20355 | A subring of a commutative... |
| subrgrcl 20356 | Reverse closure for a subr... |
| subrgsubg 20357 | A subring is a subgroup. ... |
| subrg0 20358 | A subring always has the s... |
| subrg1cl 20359 | A subring contains the mul... |
| subrgbas 20360 | Base set of a subring stru... |
| subrg1 20361 | A subring always has the s... |
| subrgacl 20362 | A subring is closed under ... |
| subrgmcl 20363 | A subgroup is closed under... |
| subrgsubm 20364 | A subring is a submonoid o... |
| subrgdvds 20365 | If an element divides anot... |
| subrguss 20366 | A unit of a subring is a u... |
| subrginv 20367 | A subring always has the s... |
| subrgdv 20368 | A subring always has the s... |
| subrgunit 20369 | An element of a ring is a ... |
| subrgugrp 20370 | The units of a subring for... |
| issubrg2 20371 | Characterize the subrings ... |
| opprsubrg 20372 | Being a subring is a symme... |
| subrgnzr 20373 | A subring of a nonzero rin... |
| subrgint 20374 | The intersection of a none... |
| subrgin 20375 | The intersection of two su... |
| subrgmre 20376 | The subrings of a ring are... |
| issubdrg 20377 | Characterize the subfields... |
| subsubrg 20378 | A subring of a subring is ... |
| subsubrg2 20379 | The set of subrings of a s... |
| issubrg3 20380 | A subring is an additive s... |
| resrhm 20381 | Restriction of a ring homo... |
| resrhm2b 20382 | Restriction of the codomai... |
| rhmeql 20383 | The equalizer of two ring ... |
| rhmima 20384 | The homomorphic image of a... |
| rnrhmsubrg 20385 | The range of a ring homomo... |
| cntzsubr 20386 | Centralizers in a ring are... |
| pwsdiagrhm 20387 | Diagonal homomorphism into... |
| subrgpropd 20388 | If two structures have the... |
| rhmpropd 20389 | Ring homomorphism depends ... |
| issdrg 20392 | Property of a division sub... |
| sdrgrcl 20393 | Reverse closure for a sub-... |
| sdrgdrng 20394 | A sub-division-ring is a d... |
| sdrgsubrg 20395 | A sub-division-ring is a s... |
| sdrgid 20396 | Every division ring is a d... |
| sdrgss 20397 | A division subring is a su... |
| sdrgbas 20398 | Base set of a sub-division... |
| issdrg2 20399 | Property of a division sub... |
| sdrgunit 20400 | A unit of a sub-division-r... |
| imadrhmcl 20401 | The image of a (nontrivial... |
| fldsdrgfld 20402 | A sub-division-ring of a f... |
| acsfn1p 20403 | Construction of a closure ... |
| subrgacs 20404 | Closure property of subrin... |
| sdrgacs 20405 | Closure property of divisi... |
| cntzsdrg 20406 | Centralizers in division r... |
| subdrgint 20407 | The intersection of a none... |
| sdrgint 20408 | The intersection of a none... |
| primefld 20409 | The smallest sub division ... |
| primefld0cl 20410 | The prime field contains t... |
| primefld1cl 20411 | The prime field contains t... |
| abvfval 20414 | Value of the set of absolu... |
| isabv 20415 | Elementhood in the set of ... |
| isabvd 20416 | Properties that determine ... |
| abvrcl 20417 | Reverse closure for the ab... |
| abvfge0 20418 | An absolute value is a fun... |
| abvf 20419 | An absolute value is a fun... |
| abvcl 20420 | An absolute value is a fun... |
| abvge0 20421 | The absolute value of a nu... |
| abveq0 20422 | The value of an absolute v... |
| abvne0 20423 | The absolute value of a no... |
| abvgt0 20424 | The absolute value of a no... |
| abvmul 20425 | An absolute value distribu... |
| abvtri 20426 | An absolute value satisfie... |
| abv0 20427 | The absolute value of zero... |
| abv1z 20428 | The absolute value of one ... |
| abv1 20429 | The absolute value of one ... |
| abvneg 20430 | The absolute value of a ne... |
| abvsubtri 20431 | An absolute value satisfie... |
| abvrec 20432 | The absolute value distrib... |
| abvdiv 20433 | The absolute value distrib... |
| abvdom 20434 | Any ring with an absolute ... |
| abvres 20435 | The restriction of an abso... |
| abvtrivd 20436 | The trivial absolute value... |
| abvtriv 20437 | The trivial absolute value... |
| abvpropd 20438 | If two structures have the... |
| staffval 20443 | The functionalization of t... |
| stafval 20444 | The functionalization of t... |
| staffn 20445 | The functionalization is e... |
| issrng 20446 | The predicate "is a star r... |
| srngrhm 20447 | The involution function in... |
| srngring 20448 | A star ring is a ring. (C... |
| srngcnv 20449 | The involution function in... |
| srngf1o 20450 | The involution function in... |
| srngcl 20451 | The involution function in... |
| srngnvl 20452 | The involution function in... |
| srngadd 20453 | The involution function in... |
| srngmul 20454 | The involution function in... |
| srng1 20455 | The conjugate of the ring ... |
| srng0 20456 | The conjugate of the ring ... |
| issrngd 20457 | Properties that determine ... |
| idsrngd 20458 | A commutative ring is a st... |
| islmod 20463 | The predicate "is a left m... |
| lmodlema 20464 | Lemma for properties of a ... |
| islmodd 20465 | Properties that determine ... |
| lmodgrp 20466 | A left module is a group. ... |
| lmodring 20467 | The scalar component of a ... |
| lmodfgrp 20468 | The scalar component of a ... |
| lmodgrpd 20469 | A left module is a group. ... |
| lmodbn0 20470 | The base set of a left mod... |
| lmodacl 20471 | Closure of ring addition f... |
| lmodmcl 20472 | Closure of ring multiplica... |
| lmodsn0 20473 | The set of scalars in a le... |
| lmodvacl 20474 | Closure of vector addition... |
| lmodass 20475 | Left module vector sum is ... |
| lmodlcan 20476 | Left cancellation law for ... |
| lmodvscl 20477 | Closure of scalar product ... |
| scaffval 20478 | The scalar multiplication ... |
| scafval 20479 | The scalar multiplication ... |
| scafeq 20480 | If the scalar multiplicati... |
| scaffn 20481 | The scalar multiplication ... |
| lmodscaf 20482 | The scalar multiplication ... |
| lmodvsdi 20483 | Distributive law for scala... |
| lmodvsdir 20484 | Distributive law for scala... |
| lmodvsass 20485 | Associative law for scalar... |
| lmod0cl 20486 | The ring zero in a left mo... |
| lmod1cl 20487 | The ring unity in a left m... |
| lmodvs1 20488 | Scalar product with the ri... |
| lmod0vcl 20489 | The zero vector is a vecto... |
| lmod0vlid 20490 | Left identity law for the ... |
| lmod0vrid 20491 | Right identity law for the... |
| lmod0vid 20492 | Identity equivalent to the... |
| lmod0vs 20493 | Zero times a vector is the... |
| lmodvs0 20494 | Anything times the zero ve... |
| lmodvsmmulgdi 20495 | Distributive law for a gro... |
| lmodfopnelem1 20496 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20497 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20498 | The (functionalized) opera... |
| lcomf 20499 | A linear-combination sum i... |
| lcomfsupp 20500 | A linear-combination sum i... |
| lmodvnegcl 20501 | Closure of vector negative... |
| lmodvnegid 20502 | Addition of a vector with ... |
| lmodvneg1 20503 | Minus 1 times a vector is ... |
| lmodvsneg 20504 | Multiplication of a vector... |
| lmodvsubcl 20505 | Closure of vector subtract... |
| lmodcom 20506 | Left module vector sum is ... |
| lmodabl 20507 | A left module is an abelia... |
| lmodcmn 20508 | A left module is a commuta... |
| lmodnegadd 20509 | Distribute negation throug... |
| lmod4 20510 | Commutative/associative la... |
| lmodvsubadd 20511 | Relationship between vecto... |
| lmodvaddsub4 20512 | Vector addition/subtractio... |
| lmodvpncan 20513 | Addition/subtraction cance... |
| lmodvnpcan 20514 | Cancellation law for vecto... |
| lmodvsubval2 20515 | Value of vector subtractio... |
| lmodsubvs 20516 | Subtraction of a scalar pr... |
| lmodsubdi 20517 | Scalar multiplication dist... |
| lmodsubdir 20518 | Scalar multiplication dist... |
| lmodsubeq0 20519 | If the difference between ... |
| lmodsubid 20520 | Subtraction of a vector fr... |
| lmodvsghm 20521 | Scalar multiplication of t... |
| lmodprop2d 20522 | If two structures have the... |
| lmodpropd 20523 | If two structures have the... |
| gsumvsmul 20524 | Pull a scalar multiplicati... |
| mptscmfsupp0 20525 | A mapping to a scalar prod... |
| mptscmfsuppd 20526 | A function mapping to a sc... |
| rmodislmodlem 20527 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20528 | The right module ` R ` ind... |
| rmodislmodOLD 20529 | Obsolete version of ~ rmod... |
| lssset 20532 | The set of all (not necess... |
| islss 20533 | The predicate "is a subspa... |
| islssd 20534 | Properties that determine ... |
| lssss 20535 | A subspace is a set of vec... |
| lssel 20536 | A subspace member is a vec... |
| lss1 20537 | The set of vectors in a le... |
| lssuni 20538 | The union of all subspaces... |
| lssn0 20539 | A subspace is not empty. ... |
| 00lss 20540 | The empty structure has no... |
| lsscl 20541 | Closure property of a subs... |
| lssvsubcl 20542 | Closure of vector subtract... |
| lssvancl1 20543 | Non-closure: if one vector... |
| lssvancl2 20544 | Non-closure: if one vector... |
| lss0cl 20545 | The zero vector belongs to... |
| lsssn0 20546 | The singleton of the zero ... |
| lss0ss 20547 | The zero subspace is inclu... |
| lssle0 20548 | No subspace is smaller tha... |
| lssne0 20549 | A nonzero subspace has a n... |
| lssvneln0 20550 | A vector ` X ` which doesn... |
| lssneln0 20551 | A vector ` X ` which doesn... |
| lssssr 20552 | Conclude subspace ordering... |
| lssvacl 20553 | Closure of vector addition... |
| lssvscl 20554 | Closure of scalar product ... |
| lssvnegcl 20555 | Closure of negative vector... |
| lsssubg 20556 | All subspaces are subgroup... |
| lsssssubg 20557 | All subspaces are subgroup... |
| islss3 20558 | A linear subspace of a mod... |
| lsslmod 20559 | A submodule is a module. ... |
| lsslss 20560 | The subspaces of a subspac... |
| islss4 20561 | A linear subspace is a sub... |
| lss1d 20562 | One-dimensional subspace (... |
| lssintcl 20563 | The intersection of a none... |
| lssincl 20564 | The intersection of two su... |
| lssmre 20565 | The subspaces of a module ... |
| lssacs 20566 | Submodules are an algebrai... |
| prdsvscacl 20567 | Pointwise scalar multiplic... |
| prdslmodd 20568 | The product of a family of... |
| pwslmod 20569 | A structure power of a lef... |
| lspfval 20572 | The span function for a le... |
| lspf 20573 | The span operator on a lef... |
| lspval 20574 | The span of a set of vecto... |
| lspcl 20575 | The span of a set of vecto... |
| lspsncl 20576 | The span of a singleton is... |
| lspprcl 20577 | The span of a pair is a su... |
| lsptpcl 20578 | The span of an unordered t... |
| lspsnsubg 20579 | The span of a singleton is... |
| 00lsp 20580 | ~ fvco4i lemma for linear ... |
| lspid 20581 | The span of a subspace is ... |
| lspssv 20582 | A span is a set of vectors... |
| lspss 20583 | Span preserves subset orde... |
| lspssid 20584 | A set of vectors is a subs... |
| lspidm 20585 | The span of a set of vecto... |
| lspun 20586 | The span of union is the s... |
| lspssp 20587 | If a set of vectors is a s... |
| mrclsp 20588 | Moore closure generalizes ... |
| lspsnss 20589 | The span of the singleton ... |
| lspsnel3 20590 | A member of the span of th... |
| lspprss 20591 | The span of a pair of vect... |
| lspsnid 20592 | A vector belongs to the sp... |
| lspsnel6 20593 | Relationship between a vec... |
| lspsnel5 20594 | Relationship between a vec... |
| lspsnel5a 20595 | Relationship between a vec... |
| lspprid1 20596 | A member of a pair of vect... |
| lspprid2 20597 | A member of a pair of vect... |
| lspprvacl 20598 | The sum of two vectors bel... |
| lssats2 20599 | A way to express atomistic... |
| lspsneli 20600 | A scalar product with a ve... |
| lspsn 20601 | Span of the singleton of a... |
| lspsnel 20602 | Member of span of the sing... |
| lspsnvsi 20603 | Span of a scalar product o... |
| lspsnss2 20604 | Comparable spans of single... |
| lspsnneg 20605 | Negation does not change t... |
| lspsnsub 20606 | Swapping subtraction order... |
| lspsn0 20607 | Span of the singleton of t... |
| lsp0 20608 | Span of the empty set. (C... |
| lspuni0 20609 | Union of the span of the e... |
| lspun0 20610 | The span of a union with t... |
| lspsneq0 20611 | Span of the singleton is t... |
| lspsneq0b 20612 | Equal singleton spans impl... |
| lmodindp1 20613 | Two independent (non-colin... |
| lsslsp 20614 | Spans in submodules corres... |
| lss0v 20615 | The zero vector in a submo... |
| lsspropd 20616 | If two structures have the... |
| lsppropd 20617 | If two structures have the... |
| reldmlmhm 20624 | Lemma for module homomorph... |
| lmimfn 20625 | Lemma for module isomorphi... |
| islmhm 20626 | Property of being a homomo... |
| islmhm3 20627 | Property of a module homom... |
| lmhmlem 20628 | Non-quantified consequence... |
| lmhmsca 20629 | A homomorphism of left mod... |
| lmghm 20630 | A homomorphism of left mod... |
| lmhmlmod2 20631 | A homomorphism of left mod... |
| lmhmlmod1 20632 | A homomorphism of left mod... |
| lmhmf 20633 | A homomorphism of left mod... |
| lmhmlin 20634 | A homomorphism of left mod... |
| lmodvsinv 20635 | Multiplication of a vector... |
| lmodvsinv2 20636 | Multiplying a negated vect... |
| islmhm2 20637 | A one-equation proof of li... |
| islmhmd 20638 | Deduction for a module hom... |
| 0lmhm 20639 | The constant zero linear f... |
| idlmhm 20640 | The identity function on a... |
| invlmhm 20641 | The negative function on a... |
| lmhmco 20642 | The composition of two mod... |
| lmhmplusg 20643 | The pointwise sum of two l... |
| lmhmvsca 20644 | The pointwise scalar produ... |
| lmhmf1o 20645 | A bijective module homomor... |
| lmhmima 20646 | The image of a subspace un... |
| lmhmpreima 20647 | The inverse image of a sub... |
| lmhmlsp 20648 | Homomorphisms preserve spa... |
| lmhmrnlss 20649 | The range of a homomorphis... |
| lmhmkerlss 20650 | The kernel of a homomorphi... |
| reslmhm 20651 | Restriction of a homomorph... |
| reslmhm2 20652 | Expansion of the codomain ... |
| reslmhm2b 20653 | Expansion of the codomain ... |
| lmhmeql 20654 | The equalizer of two modul... |
| lspextmo 20655 | A linear function is compl... |
| pwsdiaglmhm 20656 | Diagonal homomorphism into... |
| pwssplit0 20657 | Splitting for structure po... |
| pwssplit1 20658 | Splitting for structure po... |
| pwssplit2 20659 | Splitting for structure po... |
| pwssplit3 20660 | Splitting for structure po... |
| islmim 20661 | An isomorphism of left mod... |
| lmimf1o 20662 | An isomorphism of left mod... |
| lmimlmhm 20663 | An isomorphism of modules ... |
| lmimgim 20664 | An isomorphism of modules ... |
| islmim2 20665 | An isomorphism of left mod... |
| lmimcnv 20666 | The converse of a bijectiv... |
| brlmic 20667 | The relation "is isomorphi... |
| brlmici 20668 | Prove isomorphic by an exp... |
| lmiclcl 20669 | Isomorphism implies the le... |
| lmicrcl 20670 | Isomorphism implies the ri... |
| lmicsym 20671 | Module isomorphism is symm... |
| lmhmpropd 20672 | Module homomorphism depend... |
| islbs 20675 | The predicate " ` B ` is a... |
| lbsss 20676 | A basis is a set of vector... |
| lbsel 20677 | An element of a basis is a... |
| lbssp 20678 | The span of a basis is the... |
| lbsind 20679 | A basis is linearly indepe... |
| lbsind2 20680 | A basis is linearly indepe... |
| lbspss 20681 | No proper subset of a basi... |
| lsmcl 20682 | The sum of two subspaces i... |
| lsmspsn 20683 | Member of subspace sum of ... |
| lsmelval2 20684 | Subspace sum membership in... |
| lsmsp 20685 | Subspace sum in terms of s... |
| lsmsp2 20686 | Subspace sum of spans of s... |
| lsmssspx 20687 | Subspace sum (in its exten... |
| lsmpr 20688 | The span of a pair of vect... |
| lsppreli 20689 | A vector expressed as a su... |
| lsmelpr 20690 | Two ways to say that a vec... |
| lsppr0 20691 | The span of a vector paire... |
| lsppr 20692 | Span of a pair of vectors.... |
| lspprel 20693 | Member of the span of a pa... |
| lspprabs 20694 | Absorption of vector sum i... |
| lspvadd 20695 | The span of a vector sum i... |
| lspsntri 20696 | Triangle-type inequality f... |
| lspsntrim 20697 | Triangle-type inequality f... |
| lbspropd 20698 | If two structures have the... |
| pj1lmhm 20699 | The left projection functi... |
| pj1lmhm2 20700 | The left projection functi... |
| islvec 20703 | The predicate "is a left v... |
| lvecdrng 20704 | The set of scalars of a le... |
| lveclmod 20705 | A left vector space is a l... |
| lveclmodd 20706 | A vector space is a left m... |
| lsslvec 20707 | A vector subspace is a vec... |
| lmhmlvec 20708 | The property for modules t... |
| lvecvs0or 20709 | If a scalar product is zer... |
| lvecvsn0 20710 | A scalar product is nonzer... |
| lssvs0or 20711 | If a scalar product belong... |
| lvecvscan 20712 | Cancellation law for scala... |
| lvecvscan2 20713 | Cancellation law for scala... |
| lvecinv 20714 | Invert coefficient of scal... |
| lspsnvs 20715 | A nonzero scalar product d... |
| lspsneleq 20716 | Membership relation that i... |
| lspsncmp 20717 | Comparable spans of nonzer... |
| lspsnne1 20718 | Two ways to express that v... |
| lspsnne2 20719 | Two ways to express that v... |
| lspsnnecom 20720 | Swap two vectors with diff... |
| lspabs2 20721 | Absorption law for span of... |
| lspabs3 20722 | Absorption law for span of... |
| lspsneq 20723 | Equal spans of singletons ... |
| lspsneu 20724 | Nonzero vectors with equal... |
| lspsnel4 20725 | A member of the span of th... |
| lspdisj 20726 | The span of a vector not i... |
| lspdisjb 20727 | A nonzero vector is not in... |
| lspdisj2 20728 | Unequal spans are disjoint... |
| lspfixed 20729 | Show membership in the spa... |
| lspexch 20730 | Exchange property for span... |
| lspexchn1 20731 | Exchange property for span... |
| lspexchn2 20732 | Exchange property for span... |
| lspindpi 20733 | Partial independence prope... |
| lspindp1 20734 | Alternate way to say 3 vec... |
| lspindp2l 20735 | Alternate way to say 3 vec... |
| lspindp2 20736 | Alternate way to say 3 vec... |
| lspindp3 20737 | Independence of 2 vectors ... |
| lspindp4 20738 | (Partial) independence of ... |
| lvecindp 20739 | Compute the ` X ` coeffici... |
| lvecindp2 20740 | Sums of independent vector... |
| lspsnsubn0 20741 | Unequal singleton spans im... |
| lsmcv 20742 | Subspace sum has the cover... |
| lspsolvlem 20743 | Lemma for ~ lspsolv . (Co... |
| lspsolv 20744 | If ` X ` is in the span of... |
| lssacsex 20745 | In a vector space, subspac... |
| lspsnat 20746 | There is no subspace stric... |
| lspsncv0 20747 | The span of a singleton co... |
| lsppratlem1 20748 | Lemma for ~ lspprat . Let... |
| lsppratlem2 20749 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 20750 | Lemma for ~ lspprat . In ... |
| lsppratlem4 20751 | Lemma for ~ lspprat . In ... |
| lsppratlem5 20752 | Lemma for ~ lspprat . Com... |
| lsppratlem6 20753 | Lemma for ~ lspprat . Neg... |
| lspprat 20754 | A proper subspace of the s... |
| islbs2 20755 | An equivalent formulation ... |
| islbs3 20756 | An equivalent formulation ... |
| lbsacsbs 20757 | Being a basis in a vector ... |
| lvecdim 20758 | The dimension theorem for ... |
| lbsextlem1 20759 | Lemma for ~ lbsext . The ... |
| lbsextlem2 20760 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 20761 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 20762 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 20763 | For any linearly independe... |
| lbsext 20764 | For any linearly independe... |
| lbsexg 20765 | Every vector space has a b... |
| lbsex 20766 | Every vector space has a b... |
| lvecprop2d 20767 | If two structures have the... |
| lvecpropd 20768 | If two structures have the... |
| sraval 20777 | Lemma for ~ srabase throug... |
| sralem 20778 | Lemma for ~ srabase and si... |
| sralemOLD 20779 | Obsolete version of ~ sral... |
| srabase 20780 | Base set of a subring alge... |
| srabaseOLD 20781 | Obsolete proof of ~ srabas... |
| sraaddg 20782 | Additive operation of a su... |
| sraaddgOLD 20783 | Obsolete proof of ~ sraadd... |
| sramulr 20784 | Multiplicative operation o... |
| sramulrOLD 20785 | Obsolete proof of ~ sramul... |
| srasca 20786 | The set of scalars of a su... |
| srascaOLD 20787 | Obsolete proof of ~ srasca... |
| sravsca 20788 | The scalar product operati... |
| sravscaOLD 20789 | Obsolete proof of ~ sravsc... |
| sraip 20790 | The inner product operatio... |
| sratset 20791 | Topology component of a su... |
| sratsetOLD 20792 | Obsolete proof of ~ sratse... |
| sratopn 20793 | Topology component of a su... |
| srads 20794 | Distance function of a sub... |
| sradsOLD 20795 | Obsolete proof of ~ srads ... |
| sralmod 20796 | The subring algebra is a l... |
| sralmod0 20797 | The subring module inherit... |
| issubrgd 20798 | Prove a subring by closure... |
| rlmfn 20799 | ` ringLMod ` is a function... |
| rlmval 20800 | Value of the ring module. ... |
| lidlval 20801 | Value of the set of ring i... |
| rspval 20802 | Value of the ring span fun... |
| rlmval2 20803 | Value of the ring module e... |
| rlmbas 20804 | Base set of the ring modul... |
| rlmplusg 20805 | Vector addition in the rin... |
| rlm0 20806 | Zero vector in the ring mo... |
| rlmsub 20807 | Subtraction in the ring mo... |
| rlmmulr 20808 | Ring multiplication in the... |
| rlmsca 20809 | Scalars in the ring module... |
| rlmsca2 20810 | Scalars in the ring module... |
| rlmvsca 20811 | Scalar multiplication in t... |
| rlmtopn 20812 | Topology component of the ... |
| rlmds 20813 | Metric component of the ri... |
| rlmlmod 20814 | The ring module is a modul... |
| rlmlvec 20815 | The ring module over a div... |
| rlmlsm 20816 | Subgroup sum of the ring m... |
| rlmvneg 20817 | Vector negation in the rin... |
| rlmscaf 20818 | Functionalized scalar mult... |
| ixpsnbasval 20819 | The value of an infinite C... |
| lidlss 20820 | An ideal is a subset of th... |
| islidl 20821 | Predicate of being a (left... |
| lidl0cl 20822 | An ideal contains 0. (Con... |
| lidlacl 20823 | An ideal is closed under a... |
| lidlnegcl 20824 | An ideal contains negative... |
| lidlsubg 20825 | An ideal is a subgroup of ... |
| lidlsubcl 20826 | An ideal is closed under s... |
| lidlmcl 20827 | An ideal is closed under l... |
| lidl1el 20828 | An ideal contains 1 iff it... |
| dflidl2lem 20829 | Lemma for ~ dflidl2 : a su... |
| dflidl2 20830 | Alternate (the usual textb... |
| lidl0 20831 | Every ring contains a zero... |
| lidl1 20832 | Every ring contains a unit... |
| lidlacs 20833 | The ideal system is an alg... |
| rspcl 20834 | The span of a set of ring ... |
| rspssid 20835 | The span of a set of ring ... |
| rsp1 20836 | The span of the identity e... |
| rsp0 20837 | The span of the zero eleme... |
| rspssp 20838 | The ideal span of a set of... |
| mrcrsp 20839 | Moore closure generalizes ... |
| lidlnz 20840 | A nonzero ideal contains a... |
| drngnidl 20841 | A division ring has only t... |
| lidlrsppropd 20842 | The left ideals and ring s... |
| 2idlval 20845 | Definition of a two-sided ... |
| isridl 20846 | A right ideal is a left id... |
| df2idl2 20847 | Alternate (the usual textb... |
| ridl0 20848 | Every ring contains a zero... |
| ridl1 20849 | Every ring contains a unit... |
| 2idl0 20850 | Every ring contains a zero... |
| 2idl1 20851 | Every ring contains a unit... |
| 2idlelb 20852 | Membership in a two-sided ... |
| 2idllidld 20853 | A two-sided ideal is a lef... |
| 2idlridld 20854 | A two-sided ideal is a rig... |
| 2idlss 20855 | A two-sided ideal is a sub... |
| 2idlbas 20856 | The base set of a two-side... |
| 2idlelbas 20857 | The base set of a two-side... |
| 2idlcpbl 20858 | The coset equivalence rela... |
| qus1 20859 | The multiplicative identit... |
| qusring 20860 | If ` S ` is a two-sided id... |
| qusrhm 20861 | If ` S ` is a two-sided id... |
| qusmul2 20862 | Value of the ring operatio... |
| crngridl 20863 | In a commutative ring, the... |
| crng2idl 20864 | In a commutative ring, a t... |
| quscrng 20865 | The quotient of a commutat... |
| lpival 20870 | Value of the set of princi... |
| islpidl 20871 | Property of being a princi... |
| lpi0 20872 | The zero ideal is always p... |
| lpi1 20873 | The unit ideal is always p... |
| islpir 20874 | Principal ideal rings are ... |
| lpiss 20875 | Principal ideals are a sub... |
| islpir2 20876 | Principal ideal rings are ... |
| lpirring 20877 | Principal ideal rings are ... |
| drnglpir 20878 | Division rings are princip... |
| rspsn 20879 | Membership in principal id... |
| lidldvgen 20880 | An element generates an id... |
| lpigen 20881 | An ideal is principal iff ... |
| rrgval 20890 | Value of the set or left-r... |
| isrrg 20891 | Membership in the set of l... |
| rrgeq0i 20892 | Property of a left-regular... |
| rrgeq0 20893 | Left-multiplication by a l... |
| rrgsupp 20894 | Left multiplication by a l... |
| rrgss 20895 | Left-regular elements are ... |
| unitrrg 20896 | Units are regular elements... |
| isdomn 20897 | Expand definition of a dom... |
| domnnzr 20898 | A domain is a nonzero ring... |
| domnring 20899 | A domain is a ring. (Cont... |
| domneq0 20900 | In a domain, a product is ... |
| domnmuln0 20901 | In a domain, a product of ... |
| isdomn2 20902 | A ring is a domain iff all... |
| domnrrg 20903 | In a domain, any nonzero e... |
| opprdomn 20904 | The opposite of a domain i... |
| abvn0b 20905 | Another characterization o... |
| drngdomn 20906 | A division ring is a domai... |
| isidom 20907 | An integral domain is a co... |
| fldidom 20908 | A field is an integral dom... |
| fldidomOLD 20909 | Obsolete version of ~ fldi... |
| fidomndrnglem 20910 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20911 | A finite domain is a divis... |
| fiidomfld 20912 | A finite integral domain i... |
| cnfldstr 20931 | The field of complex numbe... |
| cnfldex 20932 | The field of complex numbe... |
| cnfldbas 20933 | The base set of the field ... |
| cnfldadd 20934 | The addition operation of ... |
| cnfldmul 20935 | The multiplication operati... |
| cnfldcj 20936 | The conjugation operation ... |
| cnfldtset 20937 | The topology component of ... |
| cnfldle 20938 | The ordering of the field ... |
| cnfldds 20939 | The metric of the field of... |
| cnfldunif 20940 | The uniform structure comp... |
| cnfldfun 20941 | The field of complex numbe... |
| cnfldfunALT 20942 | The field of complex numbe... |
| cnfldfunALTOLD 20943 | Obsolete proof of ~ cnfldf... |
| xrsstr 20944 | The extended real structur... |
| xrsex 20945 | The extended real structur... |
| xrsbas 20946 | The base set of the extend... |
| xrsadd 20947 | The addition operation of ... |
| xrsmul 20948 | The multiplication operati... |
| xrstset 20949 | The topology component of ... |
| xrsle 20950 | The ordering of the extend... |
| cncrng 20951 | The complex numbers form a... |
| cnring 20952 | The complex numbers form a... |
| xrsmcmn 20953 | The "multiplicative group"... |
| cnfld0 20954 | Zero is the zero element o... |
| cnfld1 20955 | One is the unity element o... |
| cnfldneg 20956 | The additive inverse in th... |
| cnfldplusf 20957 | The functionalized additio... |
| cnfldsub 20958 | The subtraction operator i... |
| cndrng 20959 | The complex numbers form a... |
| cnflddiv 20960 | The division operation in ... |
| cnfldinv 20961 | The multiplicative inverse... |
| cnfldmulg 20962 | The group multiple functio... |
| cnfldexp 20963 | The exponentiation operato... |
| cnsrng 20964 | The complex numbers form a... |
| xrsmgm 20965 | The "additive group" of th... |
| xrsnsgrp 20966 | The "additive group" of th... |
| xrsmgmdifsgrp 20967 | The "additive group" of th... |
| xrs1mnd 20968 | The extended real numbers,... |
| xrs10 20969 | The zero of the extended r... |
| xrs1cmn 20970 | The extended real numbers ... |
| xrge0subm 20971 | The nonnegative extended r... |
| xrge0cmn 20972 | The nonnegative extended r... |
| xrsds 20973 | The metric of the extended... |
| xrsdsval 20974 | The metric of the extended... |
| xrsdsreval 20975 | The metric of the extended... |
| xrsdsreclblem 20976 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 20977 | The metric of the extended... |
| cnsubmlem 20978 | Lemma for ~ nn0subm and fr... |
| cnsubglem 20979 | Lemma for ~ resubdrg and f... |
| cnsubrglem 20980 | Lemma for ~ resubdrg and f... |
| cnsubdrglem 20981 | Lemma for ~ resubdrg and f... |
| qsubdrg 20982 | The rational numbers form ... |
| zsubrg 20983 | The integers form a subrin... |
| gzsubrg 20984 | The gaussian integers form... |
| nn0subm 20985 | The nonnegative integers f... |
| rege0subm 20986 | The nonnegative reals form... |
| absabv 20987 | The regular absolute value... |
| zsssubrg 20988 | The integers are a subset ... |
| qsssubdrg 20989 | The rational numbers are a... |
| cnsubrg 20990 | There are no subrings of t... |
| cnmgpabl 20991 | The unit group of the comp... |
| cnmgpid 20992 | The group identity element... |
| cnmsubglem 20993 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 20994 | The positive reals form a ... |
| gzrngunitlem 20995 | Lemma for ~ gzrngunit . (... |
| gzrngunit 20996 | The units on ` ZZ [ _i ] `... |
| gsumfsum 20997 | Relate a group sum on ` CC... |
| regsumfsum 20998 | Relate a group sum on ` ( ... |
| expmhm 20999 | Exponentiation is a monoid... |
| nn0srg 21000 | The nonnegative integers f... |
| rge0srg 21001 | The nonnegative real numbe... |
| zringcrng 21004 | The ring of integers is a ... |
| zringring 21005 | The ring of integers is a ... |
| zringabl 21006 | The ring of integers is an... |
| zringgrp 21007 | The ring of integers is an... |
| zringbas 21008 | The integers are the base ... |
| zringplusg 21009 | The addition operation of ... |
| zringmulg 21010 | The multiplication (group ... |
| zringmulr 21011 | The multiplication operati... |
| zring0 21012 | The zero element of the ri... |
| zring1 21013 | The unity element of the r... |
| zringnzr 21014 | The ring of integers is a ... |
| dvdsrzring 21015 | Ring divisibility in the r... |
| zringlpirlem1 21016 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21017 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21018 | Lemma for ~ zringlpir . A... |
| zringinvg 21019 | The additive inverse of an... |
| zringunit 21020 | The units of ` ZZ ` are th... |
| zringlpir 21021 | The integers are a princip... |
| zringndrg 21022 | The integers are not a div... |
| zringcyg 21023 | The integers are a cyclic ... |
| zringsubgval 21024 | Subtraction in the ring of... |
| zringmpg 21025 | The multiplicative group o... |
| prmirredlem 21026 | A positive integer is irre... |
| dfprm2 21027 | The positive irreducible e... |
| prmirred 21028 | The irreducible elements o... |
| expghm 21029 | Exponentiation is a group ... |
| mulgghm2 21030 | The powers of a group elem... |
| mulgrhm 21031 | The powers of the element ... |
| mulgrhm2 21032 | The powers of the element ... |
| zrhval 21041 | Define the unique homomorp... |
| zrhval2 21042 | Alternate value of the ` Z... |
| zrhmulg 21043 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21044 | The ` ZRHom ` homomorphism... |
| zrhrhm 21045 | The ` ZRHom ` homomorphism... |
| zrh1 21046 | Interpretation of 1 in a r... |
| zrh0 21047 | Interpretation of 0 in a r... |
| zrhpropd 21048 | The ` ZZ ` ring homomorphi... |
| zlmval 21049 | Augment an abelian group w... |
| zlmlem 21050 | Lemma for ~ zlmbas and ~ z... |
| zlmlemOLD 21051 | Obsolete version of ~ zlml... |
| zlmbas 21052 | Base set of a ` ZZ ` -modu... |
| zlmbasOLD 21053 | Obsolete version of ~ zlmb... |
| zlmplusg 21054 | Group operation of a ` ZZ ... |
| zlmplusgOLD 21055 | Obsolete version of ~ zlmb... |
| zlmmulr 21056 | Ring operation of a ` ZZ `... |
| zlmmulrOLD 21057 | Obsolete version of ~ zlmb... |
| zlmsca 21058 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21059 | Scalar multiplication oper... |
| zlmlmod 21060 | The ` ZZ ` -module operati... |
| chrval 21061 | Definition substitution of... |
| chrcl 21062 | Closure of the characteris... |
| chrid 21063 | The canonical ` ZZ ` ring ... |
| chrdvds 21064 | The ` ZZ ` ring homomorphi... |
| chrcong 21065 | If two integers are congru... |
| chrnzr 21066 | Nonzero rings are precisel... |
| chrrhm 21067 | The characteristic restric... |
| domnchr 21068 | The characteristic of a do... |
| znlidl 21069 | The set ` n ZZ ` is an ide... |
| zncrng2 21070 | The value of the ` Z/nZ ` ... |
| znval 21071 | The value of the ` Z/nZ ` ... |
| znle 21072 | The value of the ` Z/nZ ` ... |
| znval2 21073 | Self-referential expressio... |
| znbaslem 21074 | Lemma for ~ znbas . (Cont... |
| znbaslemOLD 21075 | Obsolete version of ~ znba... |
| znbas2 21076 | The base set of ` Z/nZ ` i... |
| znbas2OLD 21077 | Obsolete version of ~ znba... |
| znadd 21078 | The additive structure of ... |
| znaddOLD 21079 | Obsolete version of ~ znad... |
| znmul 21080 | The multiplicative structu... |
| znmulOLD 21081 | Obsolete version of ~ znad... |
| znzrh 21082 | The ` ZZ ` ring homomorphi... |
| znbas 21083 | The base set of ` Z/nZ ` s... |
| zncrng 21084 | ` Z/nZ ` is a commutative ... |
| znzrh2 21085 | The ` ZZ ` ring homomorphi... |
| znzrhval 21086 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21087 | The ` ZZ ` ring homomorphi... |
| zncyg 21088 | The group ` ZZ / n ZZ ` is... |
| zndvds 21089 | Express equality of equiva... |
| zndvds0 21090 | Special case of ~ zndvds w... |
| znf1o 21091 | The function ` F ` enumera... |
| zzngim 21092 | The ` ZZ ` ring homomorphi... |
| znle2 21093 | The ordering of the ` Z/nZ... |
| znleval 21094 | The ordering of the ` Z/nZ... |
| znleval2 21095 | The ordering of the ` Z/nZ... |
| zntoslem 21096 | Lemma for ~ zntos . (Cont... |
| zntos 21097 | The ` Z/nZ ` structure is ... |
| znhash 21098 | The ` Z/nZ ` structure has... |
| znfi 21099 | The ` Z/nZ ` structure is ... |
| znfld 21100 | The ` Z/nZ ` structure is ... |
| znidomb 21101 | The ` Z/nZ ` structure is ... |
| znchr 21102 | Cyclic rings are defined b... |
| znunit 21103 | The units of ` Z/nZ ` are ... |
| znunithash 21104 | The size of the unit group... |
| znrrg 21105 | The regular elements of ` ... |
| cygznlem1 21106 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21107 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21108 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21109 | A cyclic group with ` n ` ... |
| cygzn 21110 | A cyclic group with ` n ` ... |
| cygth 21111 | The "fundamental theorem o... |
| cyggic 21112 | Cyclic groups are isomorph... |
| frgpcyg 21113 | A free group is cyclic iff... |
| cnmsgnsubg 21114 | The signs form a multiplic... |
| cnmsgnbas 21115 | The base set of the sign s... |
| cnmsgngrp 21116 | The group of signs under m... |
| psgnghm 21117 | The sign is a homomorphism... |
| psgnghm2 21118 | The sign is a homomorphism... |
| psgninv 21119 | The sign of a permutation ... |
| psgnco 21120 | Multiplicativity of the pe... |
| zrhpsgnmhm 21121 | Embedding of permutation s... |
| zrhpsgninv 21122 | The embedded sign of a per... |
| evpmss 21123 | Even permutations are perm... |
| psgnevpmb 21124 | A class is an even permuta... |
| psgnodpm 21125 | A permutation which is odd... |
| psgnevpm 21126 | A permutation which is eve... |
| psgnodpmr 21127 | If a permutation has sign ... |
| zrhpsgnevpm 21128 | The sign of an even permut... |
| zrhpsgnodpm 21129 | The sign of an odd permuta... |
| cofipsgn 21130 | Composition of any class `... |
| zrhpsgnelbas 21131 | Embedding of permutation s... |
| zrhcopsgnelbas 21132 | Embedding of permutation s... |
| evpmodpmf1o 21133 | The function for performin... |
| pmtrodpm 21134 | A transposition is an odd ... |
| psgnfix1 21135 | A permutation of a finite ... |
| psgnfix2 21136 | A permutation of a finite ... |
| psgndiflemB 21137 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21138 | Lemma 2 for ~ psgndif . (... |
| psgndif 21139 | Embedding of permutation s... |
| copsgndif 21140 | Embedding of permutation s... |
| rebase 21143 | The base of the field of r... |
| remulg 21144 | The multiplication (group ... |
| resubdrg 21145 | The real numbers form a di... |
| resubgval 21146 | Subtraction in the field o... |
| replusg 21147 | The addition operation of ... |
| remulr 21148 | The multiplication operati... |
| re0g 21149 | The zero element of the fi... |
| re1r 21150 | The unity element of the f... |
| rele2 21151 | The ordering relation of t... |
| relt 21152 | The ordering relation of t... |
| reds 21153 | The distance of the field ... |
| redvr 21154 | The division operation of ... |
| retos 21155 | The real numbers are a tot... |
| refld 21156 | The real numbers form a fi... |
| refldcj 21157 | The conjugation operation ... |
| resrng 21158 | The real numbers form a st... |
| regsumsupp 21159 | The group sum over the rea... |
| rzgrp 21160 | The quotient group ` RR / ... |
| isphl 21165 | The predicate "is a genera... |
| phllvec 21166 | A pre-Hilbert space is a l... |
| phllmod 21167 | A pre-Hilbert space is a l... |
| phlsrng 21168 | The scalar ring of a pre-H... |
| phllmhm 21169 | The inner product of a pre... |
| ipcl 21170 | Closure of the inner produ... |
| ipcj 21171 | Conjugate of an inner prod... |
| iporthcom 21172 | Orthogonality (meaning inn... |
| ip0l 21173 | Inner product with a zero ... |
| ip0r 21174 | Inner product with a zero ... |
| ipeq0 21175 | The inner product of a vec... |
| ipdir 21176 | Distributive law for inner... |
| ipdi 21177 | Distributive law for inner... |
| ip2di 21178 | Distributive law for inner... |
| ipsubdir 21179 | Distributive law for inner... |
| ipsubdi 21180 | Distributive law for inner... |
| ip2subdi 21181 | Distributive law for inner... |
| ipass 21182 | Associative law for inner ... |
| ipassr 21183 | "Associative" law for seco... |
| ipassr2 21184 | "Associative" law for inne... |
| ipffval 21185 | The inner product operatio... |
| ipfval 21186 | The inner product operatio... |
| ipfeq 21187 | If the inner product opera... |
| ipffn 21188 | The inner product operatio... |
| phlipf 21189 | The inner product operatio... |
| ip2eq 21190 | Two vectors are equal iff ... |
| isphld 21191 | Properties that determine ... |
| phlpropd 21192 | If two structures have the... |
| ssipeq 21193 | The inner product on a sub... |
| phssipval 21194 | The inner product on a sub... |
| phssip 21195 | The inner product (as a fu... |
| phlssphl 21196 | A subspace of an inner pro... |
| ocvfval 21203 | The orthocomplement operat... |
| ocvval 21204 | Value of the orthocompleme... |
| elocv 21205 | Elementhood in the orthoco... |
| ocvi 21206 | Property of a member of th... |
| ocvss 21207 | The orthocomplement of a s... |
| ocvocv 21208 | A set is contained in its ... |
| ocvlss 21209 | The orthocomplement of a s... |
| ocv2ss 21210 | Orthocomplements reverse s... |
| ocvin 21211 | An orthocomplement has tri... |
| ocvsscon 21212 | Two ways to say that ` S `... |
| ocvlsp 21213 | The orthocomplement of a l... |
| ocv0 21214 | The orthocomplement of the... |
| ocvz 21215 | The orthocomplement of the... |
| ocv1 21216 | The orthocomplement of the... |
| unocv 21217 | The orthocomplement of a u... |
| iunocv 21218 | The orthocomplement of an ... |
| cssval 21219 | The set of closed subspace... |
| iscss 21220 | The predicate "is a closed... |
| cssi 21221 | Property of a closed subsp... |
| cssss 21222 | A closed subspace is a sub... |
| iscss2 21223 | It is sufficient to prove ... |
| ocvcss 21224 | The orthocomplement of any... |
| cssincl 21225 | The zero subspace is a clo... |
| css0 21226 | The zero subspace is a clo... |
| css1 21227 | The whole space is a close... |
| csslss 21228 | A closed subspace of a pre... |
| lsmcss 21229 | A subset of a pre-Hilbert ... |
| cssmre 21230 | The closed subspaces of a ... |
| mrccss 21231 | The Moore closure correspo... |
| thlval 21232 | Value of the Hilbert latti... |
| thlbas 21233 | Base set of the Hilbert la... |
| thlbasOLD 21234 | Obsolete proof of ~ thlbas... |
| thlle 21235 | Ordering on the Hilbert la... |
| thlleOLD 21236 | Obsolete proof of ~ thlle ... |
| thlleval 21237 | Ordering on the Hilbert la... |
| thloc 21238 | Orthocomplement on the Hil... |
| pjfval 21245 | The value of the projectio... |
| pjdm 21246 | A subspace is in the domai... |
| pjpm 21247 | The projection map is a pa... |
| pjfval2 21248 | Value of the projection ma... |
| pjval 21249 | Value of the projection ma... |
| pjdm2 21250 | A subspace is in the domai... |
| pjff 21251 | A projection is a linear o... |
| pjf 21252 | A projection is a function... |
| pjf2 21253 | A projection is a function... |
| pjfo 21254 | A projection is a surjecti... |
| pjcss 21255 | A projection subspace is a... |
| ocvpj 21256 | The orthocomplement of a p... |
| ishil 21257 | The predicate "is a Hilber... |
| ishil2 21258 | The predicate "is a Hilber... |
| isobs 21259 | The predicate "is an ortho... |
| obsip 21260 | The inner product of two e... |
| obsipid 21261 | A basis element has length... |
| obsrcl 21262 | Reverse closure for an ort... |
| obsss 21263 | An orthonormal basis is a ... |
| obsne0 21264 | A basis element is nonzero... |
| obsocv 21265 | An orthonormal basis has t... |
| obs2ocv 21266 | The double orthocomplement... |
| obselocv 21267 | A basis element is in the ... |
| obs2ss 21268 | A basis has no proper subs... |
| obslbs 21269 | An orthogonal basis is a l... |
| reldmdsmm 21272 | The direct sum is a well-b... |
| dsmmval 21273 | Value of the module direct... |
| dsmmbase 21274 | Base set of the module dir... |
| dsmmval2 21275 | Self-referential definitio... |
| dsmmbas2 21276 | Base set of the direct sum... |
| dsmmfi 21277 | For finite products, the d... |
| dsmmelbas 21278 | Membership in the finitely... |
| dsmm0cl 21279 | The all-zero vector is con... |
| dsmmacl 21280 | The finite hull is closed ... |
| prdsinvgd2 21281 | Negation of a single coord... |
| dsmmsubg 21282 | The finite hull of a produ... |
| dsmmlss 21283 | The finite hull of a produ... |
| dsmmlmod 21284 | The direct sum of a family... |
| frlmval 21287 | Value of the "free module"... |
| frlmlmod 21288 | The free module is a modul... |
| frlmpws 21289 | The free module as a restr... |
| frlmlss 21290 | The base set of the free m... |
| frlmpwsfi 21291 | The finite free module is ... |
| frlmsca 21292 | The ring of scalars of a f... |
| frlm0 21293 | Zero in a free module (rin... |
| frlmbas 21294 | Base set of the free modul... |
| frlmelbas 21295 | Membership in the base set... |
| frlmrcl 21296 | If a free module is inhabi... |
| frlmbasfsupp 21297 | Elements of the free modul... |
| frlmbasmap 21298 | Elements of the free modul... |
| frlmbasf 21299 | Elements of the free modul... |
| frlmlvec 21300 | The free module over a div... |
| frlmfibas 21301 | The base set of the finite... |
| elfrlmbasn0 21302 | If the dimension of a free... |
| frlmplusgval 21303 | Addition in a free module.... |
| frlmsubgval 21304 | Subtraction in a free modu... |
| frlmvscafval 21305 | Scalar multiplication in a... |
| frlmvplusgvalc 21306 | Coordinates of a sum with ... |
| frlmvscaval 21307 | Coordinates of a scalar mu... |
| frlmplusgvalb 21308 | Addition in a free module ... |
| frlmvscavalb 21309 | Scalar multiplication in a... |
| frlmvplusgscavalb 21310 | Addition combined with sca... |
| frlmgsum 21311 | Finite commutative sums in... |
| frlmsplit2 21312 | Restriction is homomorphic... |
| frlmsslss 21313 | A subset of a free module ... |
| frlmsslss2 21314 | A subset of a free module ... |
| frlmbas3 21315 | An element of the base set... |
| mpofrlmd 21316 | Elements of the free modul... |
| frlmip 21317 | The inner product of a fre... |
| frlmipval 21318 | The inner product of a fre... |
| frlmphllem 21319 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21320 | Conditions for a free modu... |
| uvcfval 21323 | Value of the unit-vector g... |
| uvcval 21324 | Value of a single unit vec... |
| uvcvval 21325 | Value of a unit vector coo... |
| uvcvvcl 21326 | A coordinate of a unit vec... |
| uvcvvcl2 21327 | A unit vector coordinate i... |
| uvcvv1 21328 | The unit vector is one at ... |
| uvcvv0 21329 | The unit vector is zero at... |
| uvcff 21330 | Domain and codomain of the... |
| uvcf1 21331 | In a nonzero ring, each un... |
| uvcresum 21332 | Any element of a free modu... |
| frlmssuvc1 21333 | A scalar multiple of a uni... |
| frlmssuvc2 21334 | A nonzero scalar multiple ... |
| frlmsslsp 21335 | A subset of a free module ... |
| frlmlbs 21336 | The unit vectors comprise ... |
| frlmup1 21337 | Any assignment of unit vec... |
| frlmup2 21338 | The evaluation map has the... |
| frlmup3 21339 | The range of such an evalu... |
| frlmup4 21340 | Universal property of the ... |
| ellspd 21341 | The elements of the span o... |
| elfilspd 21342 | Simplified version of ~ el... |
| rellindf 21347 | The independent-family pre... |
| islinds 21348 | Property of an independent... |
| linds1 21349 | An independent set of vect... |
| linds2 21350 | An independent set of vect... |
| islindf 21351 | Property of an independent... |
| islinds2 21352 | Expanded property of an in... |
| islindf2 21353 | Property of an independent... |
| lindff 21354 | Functional property of a l... |
| lindfind 21355 | A linearly independent fam... |
| lindsind 21356 | A linearly independent set... |
| lindfind2 21357 | In a linearly independent ... |
| lindsind2 21358 | In a linearly independent ... |
| lindff1 21359 | A linearly independent fam... |
| lindfrn 21360 | The range of an independen... |
| f1lindf 21361 | Rearranging and deleting e... |
| lindfres 21362 | Any restriction of an inde... |
| lindsss 21363 | Any subset of an independe... |
| f1linds 21364 | A family constructed from ... |
| islindf3 21365 | In a nonzero ring, indepen... |
| lindfmm 21366 | Linear independence of a f... |
| lindsmm 21367 | Linear independence of a s... |
| lindsmm2 21368 | The monomorphic image of a... |
| lsslindf 21369 | Linear independence is unc... |
| lsslinds 21370 | Linear independence is unc... |
| islbs4 21371 | A basis is an independent ... |
| lbslinds 21372 | A basis is independent. (... |
| islinds3 21373 | A subset is linearly indep... |
| islinds4 21374 | A set is independent in a ... |
| lmimlbs 21375 | The isomorphic image of a ... |
| lmiclbs 21376 | Having a basis is an isomo... |
| islindf4 21377 | A family is independent if... |
| islindf5 21378 | A family is independent if... |
| indlcim 21379 | An independent, spanning f... |
| lbslcic 21380 | A module with a basis is i... |
| lmisfree 21381 | A module has a basis iff i... |
| lvecisfrlm 21382 | Every vector space is isom... |
| lmimco 21383 | The composition of two iso... |
| lmictra 21384 | Module isomorphism is tran... |
| uvcf1o 21385 | In a nonzero ring, the map... |
| uvcendim 21386 | In a nonzero ring, the num... |
| frlmisfrlm 21387 | A free module is isomorphi... |
| frlmiscvec 21388 | Every free module is isomo... |
| isassa 21395 | The properties of an assoc... |
| assalem 21396 | The properties of an assoc... |
| assaass 21397 | Left-associative property ... |
| assaassr 21398 | Right-associative property... |
| assalmod 21399 | An associative algebra is ... |
| assaring 21400 | An associative algebra is ... |
| assasca 21401 | The scalars of an associat... |
| assa2ass 21402 | Left- and right-associativ... |
| isassad 21403 | Sufficient condition for b... |
| issubassa3 21404 | A subring that is also a s... |
| issubassa 21405 | The subalgebras of an asso... |
| sraassa 21406 | The subring algebra over a... |
| rlmassa 21407 | The ring module over a com... |
| assapropd 21408 | If two structures have the... |
| aspval 21409 | Value of the algebraic clo... |
| asplss 21410 | The algebraic span of a se... |
| aspid 21411 | The algebraic span of a su... |
| aspsubrg 21412 | The algebraic span of a se... |
| aspss 21413 | Span preserves subset orde... |
| aspssid 21414 | A set of vectors is a subs... |
| asclfval 21415 | Function value of the alge... |
| asclval 21416 | Value of a mapped algebra ... |
| asclfn 21417 | Unconditional functionalit... |
| asclf 21418 | The algebra scalars functi... |
| asclghm 21419 | The algebra scalars functi... |
| ascl0 21420 | The scalar 0 embedded into... |
| ascl1 21421 | The scalar 1 embedded into... |
| asclmul1 21422 | Left multiplication by a l... |
| asclmul2 21423 | Right multiplication by a ... |
| ascldimul 21424 | The algebra scalars functi... |
| asclinvg 21425 | The group inverse (negatio... |
| asclrhm 21426 | The scalar injection is a ... |
| rnascl 21427 | The set of injected scalar... |
| issubassa2 21428 | A subring of a unital alge... |
| rnasclsubrg 21429 | The scalar multiples of th... |
| rnasclmulcl 21430 | (Vector) multiplication is... |
| rnasclassa 21431 | The scalar multiples of th... |
| ressascl 21432 | The injection of scalars i... |
| asclpropd 21433 | If two structures have the... |
| aspval2 21434 | The algebraic closure is t... |
| assamulgscmlem1 21435 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21436 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21437 | Exponentiation of a scalar... |
| zlmassa 21438 | The ` ZZ ` -module operati... |
| reldmpsr 21449 | The multivariate power ser... |
| psrval 21450 | Value of the multivariate ... |
| psrvalstr 21451 | The multivariate power ser... |
| psrbag 21452 | Elementhood in the set of ... |
| psrbagf 21453 | A finite bag is a function... |
| psrbagfOLD 21454 | Obsolete version of ~ psrb... |
| psrbagfsupp 21455 | Finite bags have finite su... |
| psrbagfsuppOLD 21456 | Obsolete version of ~ psrb... |
| snifpsrbag 21457 | A bag containing one eleme... |
| fczpsrbag 21458 | The constant function equa... |
| psrbaglesupp 21459 | The support of a dominated... |
| psrbaglesuppOLD 21460 | Obsolete version of ~ psrb... |
| psrbaglecl 21461 | The set of finite bags is ... |
| psrbagleclOLD 21462 | Obsolete version of ~ psrb... |
| psrbagaddcl 21463 | The sum of two finite bags... |
| psrbagaddclOLD 21464 | Obsolete version of ~ psrb... |
| psrbagcon 21465 | The analogue of the statem... |
| psrbagconOLD 21466 | Obsolete version of ~ psrb... |
| psrbaglefi 21467 | There are finitely many ba... |
| psrbaglefiOLD 21468 | Obsolete version of ~ psrb... |
| psrbagconcl 21469 | The complement of a bag is... |
| psrbagconclOLD 21470 | Obsolete version of ~ psrb... |
| psrbagconf1o 21471 | Bag complementation is a b... |
| psrbagconf1oOLD 21472 | Obsolete version of ~ psrb... |
| gsumbagdiaglemOLD 21473 | Obsolete version of ~ gsum... |
| gsumbagdiagOLD 21474 | Obsolete version of ~ gsum... |
| psrass1lemOLD 21475 | Obsolete version of ~ psra... |
| gsumbagdiaglem 21476 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21477 | Two-dimensional commutatio... |
| psrass1lem 21478 | A group sum commutation us... |
| psrbas 21479 | The base set of the multiv... |
| psrelbas 21480 | An element of the set of p... |
| psrelbasfun 21481 | An element of the set of p... |
| psrplusg 21482 | The addition operation of ... |
| psradd 21483 | The addition operation of ... |
| psraddcl 21484 | Closure of the power serie... |
| psrmulr 21485 | The multiplication operati... |
| psrmulfval 21486 | The multiplication operati... |
| psrmulval 21487 | The multiplication operati... |
| psrmulcllem 21488 | Closure of the power serie... |
| psrmulcl 21489 | Closure of the power serie... |
| psrsca 21490 | The scalar field of the mu... |
| psrvscafval 21491 | The scalar multiplication ... |
| psrvsca 21492 | The scalar multiplication ... |
| psrvscaval 21493 | The scalar multiplication ... |
| psrvscacl 21494 | Closure of the power serie... |
| psr0cl 21495 | The zero element of the ri... |
| psr0lid 21496 | The zero element of the ri... |
| psrnegcl 21497 | The negative function in t... |
| psrlinv 21498 | The negative function in t... |
| psrgrp 21499 | The ring of power series i... |
| psrgrpOLD 21500 | Obsolete proof of ~ psrgrp... |
| psr0 21501 | The zero element of the ri... |
| psrneg 21502 | The negative function of t... |
| psrlmod 21503 | The ring of power series i... |
| psr1cl 21504 | The identity element of th... |
| psrlidm 21505 | The identity element of th... |
| psrridm 21506 | The identity element of th... |
| psrass1 21507 | Associative identity for t... |
| psrdi 21508 | Distributive law for the r... |
| psrdir 21509 | Distributive law for the r... |
| psrass23l 21510 | Associative identity for t... |
| psrcom 21511 | Commutative law for the ri... |
| psrass23 21512 | Associative identities for... |
| psrring 21513 | The ring of power series i... |
| psr1 21514 | The identity element of th... |
| psrcrng 21515 | The ring of power series i... |
| psrassa 21516 | The ring of power series i... |
| resspsrbas 21517 | A restricted power series ... |
| resspsradd 21518 | A restricted power series ... |
| resspsrmul 21519 | A restricted power series ... |
| resspsrvsca 21520 | A restricted power series ... |
| subrgpsr 21521 | A subring of the base ring... |
| mvrfval 21522 | Value of the generating el... |
| mvrval 21523 | Value of the generating el... |
| mvrval2 21524 | Value of the generating el... |
| mvrid 21525 | The ` X i ` -th coefficien... |
| mvrf 21526 | The power series variable ... |
| mvrf1 21527 | The power series variable ... |
| mvrcl2 21528 | A power series variable is... |
| reldmmpl 21529 | The multivariate polynomia... |
| mplval 21530 | Value of the set of multiv... |
| mplbas 21531 | Base set of the set of mul... |
| mplelbas 21532 | Property of being a polyno... |
| mvrcl 21533 | A power series variable is... |
| mvrf2 21534 | The power series/polynomia... |
| mplrcl 21535 | Reverse closure for the po... |
| mplelsfi 21536 | A polynomial treated as a ... |
| mplval2 21537 | Self-referential expressio... |
| mplbasss 21538 | The set of polynomials is ... |
| mplelf 21539 | A polynomial is defined as... |
| mplsubglem 21540 | If ` A ` is an ideal of se... |
| mpllsslem 21541 | If ` A ` is an ideal of su... |
| mplsubglem2 21542 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 21543 | The set of polynomials is ... |
| mpllss 21544 | The set of polynomials is ... |
| mplsubrglem 21545 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 21546 | The set of polynomials is ... |
| mpl0 21547 | The zero polynomial. (Con... |
| mplplusg 21548 | Value of addition in a pol... |
| mplmulr 21549 | Value of multiplication in... |
| mpladd 21550 | The addition operation on ... |
| mplneg 21551 | The negative function on m... |
| mplmul 21552 | The multiplication operati... |
| mpl1 21553 | The identity element of th... |
| mplsca 21554 | The scalar field of a mult... |
| mplvsca2 21555 | The scalar multiplication ... |
| mplvsca 21556 | The scalar multiplication ... |
| mplvscaval 21557 | The scalar multiplication ... |
| mplgrp 21558 | The polynomial ring is a g... |
| mpllmod 21559 | The polynomial ring is a l... |
| mplring 21560 | The polynomial ring is a r... |
| mpllvec 21561 | The polynomial ring is a v... |
| mplcrng 21562 | The polynomial ring is a c... |
| mplassa 21563 | The polynomial ring is an ... |
| ressmplbas2 21564 | The base set of a restrict... |
| ressmplbas 21565 | A restricted polynomial al... |
| ressmpladd 21566 | A restricted polynomial al... |
| ressmplmul 21567 | A restricted polynomial al... |
| ressmplvsca 21568 | A restricted power series ... |
| subrgmpl 21569 | A subring of the base ring... |
| subrgmvr 21570 | The variables in a subring... |
| subrgmvrf 21571 | The variables in a polynom... |
| mplmon 21572 | A monomial is a polynomial... |
| mplmonmul 21573 | The product of two monomia... |
| mplcoe1 21574 | Decompose a polynomial int... |
| mplcoe3 21575 | Decompose a monomial in on... |
| mplcoe5lem 21576 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 21577 | Decompose a monomial into ... |
| mplcoe2 21578 | Decompose a monomial into ... |
| mplbas2 21579 | An alternative expression ... |
| ltbval 21580 | Value of the well-order on... |
| ltbwe 21581 | The finite bag order is a ... |
| reldmopsr 21582 | Lemma for ordered power se... |
| opsrval 21583 | The value of the "ordered ... |
| opsrle 21584 | An alternative expression ... |
| opsrval2 21585 | Self-referential expressio... |
| opsrbaslem 21586 | Get a component of the ord... |
| opsrbaslemOLD 21587 | Obsolete version of ~ opsr... |
| opsrbas 21588 | The base set of the ordere... |
| opsrbasOLD 21589 | Obsolete version of ~ opsr... |
| opsrplusg 21590 | The addition operation of ... |
| opsrplusgOLD 21591 | Obsolete version of ~ opsr... |
| opsrmulr 21592 | The multiplication operati... |
| opsrmulrOLD 21593 | Obsolete version of ~ opsr... |
| opsrvsca 21594 | The scalar product operati... |
| opsrvscaOLD 21595 | Obsolete version of ~ opsr... |
| opsrsca 21596 | The scalar ring of the ord... |
| opsrscaOLD 21597 | Obsolete version of ~ opsr... |
| opsrtoslem1 21598 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 21599 | Lemma for ~ opsrtos . (Co... |
| opsrtos 21600 | The ordered power series s... |
| opsrso 21601 | The ordered power series s... |
| opsrcrng 21602 | The ring of ordered power ... |
| opsrassa 21603 | The ring of ordered power ... |
| mplmon2 21604 | Express a scaled monomial.... |
| psrbag0 21605 | The empty bag is a bag. (... |
| psrbagsn 21606 | A singleton bag is a bag. ... |
| mplascl 21607 | Value of the scalar inject... |
| mplasclf 21608 | The scalar injection is a ... |
| subrgascl 21609 | The scalar injection funct... |
| subrgasclcl 21610 | The scalars in a polynomia... |
| mplmon2cl 21611 | A scaled monomial is a pol... |
| mplmon2mul 21612 | Product of scaled monomial... |
| mplind 21613 | Prove a property of polyno... |
| mplcoe4 21614 | Decompose a polynomial int... |
| evlslem4 21619 | The support of a tensor pr... |
| psrbagev1 21620 | A bag of multipliers provi... |
| psrbagev1OLD 21621 | Obsolete version of ~ psrb... |
| psrbagev2 21622 | Closure of a sum using a b... |
| psrbagev2OLD 21623 | Obsolete version of ~ psrb... |
| evlslem2 21624 | A linear function on the p... |
| evlslem3 21625 | Lemma for ~ evlseu . Poly... |
| evlslem6 21626 | Lemma for ~ evlseu . Fini... |
| evlslem1 21627 | Lemma for ~ evlseu , give ... |
| evlseu 21628 | For a given interpretation... |
| reldmevls 21629 | Well-behaved binary operat... |
| mpfrcl 21630 | Reverse closure for the se... |
| evlsval 21631 | Value of the polynomial ev... |
| evlsval2 21632 | Characterizing properties ... |
| evlsrhm 21633 | Polynomial evaluation is a... |
| evlssca 21634 | Polynomial evaluation maps... |
| evlsvar 21635 | Polynomial evaluation maps... |
| evlsgsumadd 21636 | Polynomial evaluation maps... |
| evlsgsummul 21637 | Polynomial evaluation maps... |
| evlspw 21638 | Polynomial evaluation for ... |
| evlsvarpw 21639 | Polynomial evaluation for ... |
| evlval 21640 | Value of the simple/same r... |
| evlrhm 21641 | The simple evaluation map ... |
| evlsscasrng 21642 | The evaluation of a scalar... |
| evlsca 21643 | Simple polynomial evaluati... |
| evlsvarsrng 21644 | The evaluation of the vari... |
| evlvar 21645 | Simple polynomial evaluati... |
| mpfconst 21646 | Constants are multivariate... |
| mpfproj 21647 | Projections are multivaria... |
| mpfsubrg 21648 | Polynomial functions are a... |
| mpff 21649 | Polynomial functions are f... |
| mpfaddcl 21650 | The sum of multivariate po... |
| mpfmulcl 21651 | The product of multivariat... |
| mpfind 21652 | Prove a property of polyno... |
| selvffval 21661 | Value of the "variable sel... |
| selvfval 21662 | Value of the "variable sel... |
| selvval 21663 | Value of the "variable sel... |
| mhpfval 21664 | Value of the "homogeneous ... |
| mhpval 21665 | Value of the "homogeneous ... |
| ismhp 21666 | Property of being a homoge... |
| ismhp2 21667 | Deduce a homogeneous polyn... |
| ismhp3 21668 | A polynomial is homogeneou... |
| mhpmpl 21669 | A homogeneous polynomial i... |
| mhpdeg 21670 | All nonzero terms of a hom... |
| mhp0cl 21671 | The zero polynomial is hom... |
| mhpsclcl 21672 | A scalar (or constant) pol... |
| mhpvarcl 21673 | A power series variable is... |
| mhpmulcl 21674 | A product of homogeneous p... |
| mhppwdeg 21675 | Degree of a homogeneous po... |
| mhpaddcl 21676 | Homogeneous polynomials ar... |
| mhpinvcl 21677 | Homogeneous polynomials ar... |
| mhpsubg 21678 | Homogeneous polynomials fo... |
| mhpvscacl 21679 | Homogeneous polynomials ar... |
| mhplss 21680 | Homogeneous polynomials fo... |
| psr1baslem 21691 | The set of finite bags on ... |
| psr1val 21692 | Value of the ring of univa... |
| psr1crng 21693 | The ring of univariate pow... |
| psr1assa 21694 | The ring of univariate pow... |
| psr1tos 21695 | The ordered power series s... |
| psr1bas2 21696 | The base set of the ring o... |
| psr1bas 21697 | The base set of the ring o... |
| vr1val 21698 | The value of the generator... |
| vr1cl2 21699 | The variable ` X ` is a me... |
| ply1val 21700 | The value of the set of un... |
| ply1bas 21701 | The value of the base set ... |
| ply1lss 21702 | Univariate polynomials for... |
| ply1subrg 21703 | Univariate polynomials for... |
| ply1crng 21704 | The ring of univariate pol... |
| ply1assa 21705 | The ring of univariate pol... |
| psr1bascl 21706 | A univariate power series ... |
| psr1basf 21707 | Univariate power series ba... |
| ply1basf 21708 | Univariate polynomial base... |
| ply1bascl 21709 | A univariate polynomial is... |
| ply1bascl2 21710 | A univariate polynomial is... |
| coe1fval 21711 | Value of the univariate po... |
| coe1fv 21712 | Value of an evaluated coef... |
| fvcoe1 21713 | Value of a multivariate co... |
| coe1fval3 21714 | Univariate power series co... |
| coe1f2 21715 | Functionality of univariat... |
| coe1fval2 21716 | Univariate polynomial coef... |
| coe1f 21717 | Functionality of univariat... |
| coe1fvalcl 21718 | A coefficient of a univari... |
| coe1sfi 21719 | Finite support of univaria... |
| coe1fsupp 21720 | The coefficient vector of ... |
| mptcoe1fsupp 21721 | A mapping involving coeffi... |
| coe1ae0 21722 | The coefficient vector of ... |
| vr1cl 21723 | The generator of a univari... |
| opsr0 21724 | Zero in the ordered power ... |
| opsr1 21725 | One in the ordered power s... |
| psr1plusg 21726 | Value of addition in a uni... |
| psr1vsca 21727 | Value of scalar multiplica... |
| psr1mulr 21728 | Value of multiplication in... |
| ply1plusg 21729 | Value of addition in a uni... |
| ply1vsca 21730 | Value of scalar multiplica... |
| ply1mulr 21731 | Value of multiplication in... |
| ressply1bas2 21732 | The base set of a restrict... |
| ressply1bas 21733 | A restricted polynomial al... |
| ressply1add 21734 | A restricted polynomial al... |
| ressply1mul 21735 | A restricted polynomial al... |
| ressply1vsca 21736 | A restricted power series ... |
| subrgply1 21737 | A subring of the base ring... |
| gsumply1subr 21738 | Evaluate a group sum in a ... |
| psrbaspropd 21739 | Property deduction for pow... |
| psrplusgpropd 21740 | Property deduction for pow... |
| mplbaspropd 21741 | Property deduction for pol... |
| psropprmul 21742 | Reversing multiplication i... |
| ply1opprmul 21743 | Reversing multiplication i... |
| 00ply1bas 21744 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 21745 | Protection compatibility o... |
| ply1plusgfvi 21746 | Protection compatibility o... |
| ply1baspropd 21747 | Property deduction for uni... |
| ply1plusgpropd 21748 | Property deduction for uni... |
| opsrring 21749 | Ordered power series form ... |
| opsrlmod 21750 | Ordered power series form ... |
| psr1ring 21751 | Univariate power series fo... |
| ply1ring 21752 | Univariate polynomials for... |
| psr1lmod 21753 | Univariate power series fo... |
| psr1sca 21754 | Scalars of a univariate po... |
| psr1sca2 21755 | Scalars of a univariate po... |
| ply1lmod 21756 | Univariate polynomials for... |
| ply1sca 21757 | Scalars of a univariate po... |
| ply1sca2 21758 | Scalars of a univariate po... |
| ply1mpl0 21759 | The univariate polynomial ... |
| ply10s0 21760 | Zero times a univariate po... |
| ply1mpl1 21761 | The univariate polynomial ... |
| ply1ascl 21762 | The univariate polynomial ... |
| subrg1ascl 21763 | The scalar injection funct... |
| subrg1asclcl 21764 | The scalars in a polynomia... |
| subrgvr1 21765 | The variables in a subring... |
| subrgvr1cl 21766 | The variables in a polynom... |
| coe1z 21767 | The coefficient vector of ... |
| coe1add 21768 | The coefficient vector of ... |
| coe1addfv 21769 | A particular coefficient o... |
| coe1subfv 21770 | A particular coefficient o... |
| coe1mul2lem1 21771 | An equivalence for ~ coe1m... |
| coe1mul2lem2 21772 | An equivalence for ~ coe1m... |
| coe1mul2 21773 | The coefficient vector of ... |
| coe1mul 21774 | The coefficient vector of ... |
| ply1moncl 21775 | Closure of the expression ... |
| ply1tmcl 21776 | Closure of the expression ... |
| coe1tm 21777 | Coefficient vector of a po... |
| coe1tmfv1 21778 | Nonzero coefficient of a p... |
| coe1tmfv2 21779 | Zero coefficient of a poly... |
| coe1tmmul2 21780 | Coefficient vector of a po... |
| coe1tmmul 21781 | Coefficient vector of a po... |
| coe1tmmul2fv 21782 | Function value of a right-... |
| coe1pwmul 21783 | Coefficient vector of a po... |
| coe1pwmulfv 21784 | Function value of a right-... |
| ply1scltm 21785 | A scalar is a term with ze... |
| coe1sclmul 21786 | Coefficient vector of a po... |
| coe1sclmulfv 21787 | A single coefficient of a ... |
| coe1sclmul2 21788 | Coefficient vector of a po... |
| ply1sclf 21789 | A scalar polynomial is a p... |
| ply1sclcl 21790 | The value of the algebra s... |
| coe1scl 21791 | Coefficient vector of a sc... |
| ply1sclid 21792 | Recover the base scalar fr... |
| ply1sclf1 21793 | The polynomial scalar func... |
| ply1scl0 21794 | The zero scalar is zero. ... |
| ply1scl0OLD 21795 | Obsolete version of ~ ply1... |
| ply1scln0 21796 | Nonzero scalars create non... |
| ply1scl1 21797 | The one scalar is the unit... |
| ply1scl1OLD 21798 | Obsolete version of ~ ply1... |
| ply1idvr1 21799 | The identity of a polynomi... |
| cply1mul 21800 | The product of two constan... |
| ply1coefsupp 21801 | The decomposition of a uni... |
| ply1coe 21802 | Decompose a univariate pol... |
| eqcoe1ply1eq 21803 | Two polynomials over the s... |
| ply1coe1eq 21804 | Two polynomials over the s... |
| cply1coe0 21805 | All but the first coeffici... |
| cply1coe0bi 21806 | A polynomial is constant (... |
| coe1fzgsumdlem 21807 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 21808 | Value of an evaluated coef... |
| gsumsmonply1 21809 | A finite group sum of scal... |
| gsummoncoe1 21810 | A coefficient of the polyn... |
| gsumply1eq 21811 | Two univariate polynomials... |
| lply1binom 21812 | The binomial theorem for l... |
| lply1binomsc 21813 | The binomial theorem for l... |
| reldmevls1 21818 | Well-behaved binary operat... |
| ply1frcl 21819 | Reverse closure for the se... |
| evls1fval 21820 | Value of the univariate po... |
| evls1val 21821 | Value of the univariate po... |
| evls1rhmlem 21822 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 21823 | Polynomial evaluation is a... |
| evls1sca 21824 | Univariate polynomial eval... |
| evls1gsumadd 21825 | Univariate polynomial eval... |
| evls1gsummul 21826 | Univariate polynomial eval... |
| evls1pw 21827 | Univariate polynomial eval... |
| evls1varpw 21828 | Univariate polynomial eval... |
| evl1fval 21829 | Value of the simple/same r... |
| evl1val 21830 | Value of the simple/same r... |
| evl1fval1lem 21831 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 21832 | Value of the simple/same r... |
| evl1rhm 21833 | Polynomial evaluation is a... |
| fveval1fvcl 21834 | The function value of the ... |
| evl1sca 21835 | Polynomial evaluation maps... |
| evl1scad 21836 | Polynomial evaluation buil... |
| evl1var 21837 | Polynomial evaluation maps... |
| evl1vard 21838 | Polynomial evaluation buil... |
| evls1var 21839 | Univariate polynomial eval... |
| evls1scasrng 21840 | The evaluation of a scalar... |
| evls1varsrng 21841 | The evaluation of the vari... |
| evl1addd 21842 | Polynomial evaluation buil... |
| evl1subd 21843 | Polynomial evaluation buil... |
| evl1muld 21844 | Polynomial evaluation buil... |
| evl1vsd 21845 | Polynomial evaluation buil... |
| evl1expd 21846 | Polynomial evaluation buil... |
| pf1const 21847 | Constants are polynomial f... |
| pf1id 21848 | The identity is a polynomi... |
| pf1subrg 21849 | Polynomial functions are a... |
| pf1rcl 21850 | Reverse closure for the se... |
| pf1f 21851 | Polynomial functions are f... |
| mpfpf1 21852 | Convert a multivariate pol... |
| pf1mpf 21853 | Convert a univariate polyn... |
| pf1addcl 21854 | The sum of multivariate po... |
| pf1mulcl 21855 | The product of multivariat... |
| pf1ind 21856 | Prove a property of polyno... |
| evl1gsumdlem 21857 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 21858 | Polynomial evaluation buil... |
| evl1gsumadd 21859 | Univariate polynomial eval... |
| evl1gsumaddval 21860 | Value of a univariate poly... |
| evl1gsummul 21861 | Univariate polynomial eval... |
| evl1varpw 21862 | Univariate polynomial eval... |
| evl1varpwval 21863 | Value of a univariate poly... |
| evl1scvarpw 21864 | Univariate polynomial eval... |
| evl1scvarpwval 21865 | Value of a univariate poly... |
| evl1gsummon 21866 | Value of a univariate poly... |
| mamufval 21869 | Functional value of the ma... |
| mamuval 21870 | Multiplication of two matr... |
| mamufv 21871 | A cell in the multiplicati... |
| mamudm 21872 | The domain of the matrix m... |
| mamufacex 21873 | Every solution of the equa... |
| mamures 21874 | Rows in a matrix product a... |
| mndvcl 21875 | Tuple-wise additive closur... |
| mndvass 21876 | Tuple-wise associativity i... |
| mndvlid 21877 | Tuple-wise left identity i... |
| mndvrid 21878 | Tuple-wise right identity ... |
| grpvlinv 21879 | Tuple-wise left inverse in... |
| grpvrinv 21880 | Tuple-wise right inverse i... |
| mhmvlin 21881 | Tuple extension of monoid ... |
| ringvcl 21882 | Tuple-wise multiplication ... |
| mamucl 21883 | Operation closure of matri... |
| mamuass 21884 | Matrix multiplication is a... |
| mamudi 21885 | Matrix multiplication dist... |
| mamudir 21886 | Matrix multiplication dist... |
| mamuvs1 21887 | Matrix multiplication dist... |
| mamuvs2 21888 | Matrix multiplication dist... |
| matbas0pc 21891 | There is no matrix with a ... |
| matbas0 21892 | There is no matrix for a n... |
| matval 21893 | Value of the matrix algebr... |
| matrcl 21894 | Reverse closure for the ma... |
| matbas 21895 | The matrix ring has the sa... |
| matplusg 21896 | The matrix ring has the sa... |
| matsca 21897 | The matrix ring has the sa... |
| matscaOLD 21898 | Obsolete proof of ~ matsca... |
| matvsca 21899 | The matrix ring has the sa... |
| matvscaOLD 21900 | Obsolete proof of ~ matvsc... |
| mat0 21901 | The matrix ring has the sa... |
| matinvg 21902 | The matrix ring has the sa... |
| mat0op 21903 | Value of a zero matrix as ... |
| matsca2 21904 | The scalars of the matrix ... |
| matbas2 21905 | The base set of the matrix... |
| matbas2i 21906 | A matrix is a function. (... |
| matbas2d 21907 | The base set of the matrix... |
| eqmat 21908 | Two square matrices of the... |
| matecl 21909 | Each entry (according to W... |
| matecld 21910 | Each entry (according to W... |
| matplusg2 21911 | Addition in the matrix rin... |
| matvsca2 21912 | Scalar multiplication in t... |
| matlmod 21913 | The matrix ring is a linea... |
| matgrp 21914 | The matrix ring is a group... |
| matvscl 21915 | Closure of the scalar mult... |
| matsubg 21916 | The matrix ring has the sa... |
| matplusgcell 21917 | Addition in the matrix rin... |
| matsubgcell 21918 | Subtraction in the matrix ... |
| matinvgcell 21919 | Additive inversion in the ... |
| matvscacell 21920 | Scalar multiplication in t... |
| matgsum 21921 | Finite commutative sums in... |
| matmulr 21922 | Multiplication in the matr... |
| mamumat1cl 21923 | The identity matrix (as op... |
| mat1comp 21924 | The components of the iden... |
| mamulid 21925 | The identity matrix (as op... |
| mamurid 21926 | The identity matrix (as op... |
| matring 21927 | Existence of the matrix ri... |
| matassa 21928 | Existence of the matrix al... |
| matmulcell 21929 | Multiplication in the matr... |
| mpomatmul 21930 | Multiplication of two N x ... |
| mat1 21931 | Value of an identity matri... |
| mat1ov 21932 | Entries of an identity mat... |
| mat1bas 21933 | The identity matrix is a m... |
| matsc 21934 | The identity matrix multip... |
| ofco2 21935 | Distribution law for the f... |
| oftpos 21936 | The transposition of the v... |
| mattposcl 21937 | The transpose of a square ... |
| mattpostpos 21938 | The transpose of the trans... |
| mattposvs 21939 | The transposition of a mat... |
| mattpos1 21940 | The transposition of the i... |
| tposmap 21941 | The transposition of an I ... |
| mamutpos 21942 | Behavior of transposes in ... |
| mattposm 21943 | Multiplying two transposed... |
| matgsumcl 21944 | Closure of a group sum ove... |
| madetsumid 21945 | The identity summand in th... |
| matepmcl 21946 | Each entry of a matrix wit... |
| matepm2cl 21947 | Each entry of a matrix wit... |
| madetsmelbas 21948 | A summand of the determina... |
| madetsmelbas2 21949 | A summand of the determina... |
| mat0dimbas0 21950 | The empty set is the one a... |
| mat0dim0 21951 | The zero of the algebra of... |
| mat0dimid 21952 | The identity of the algebr... |
| mat0dimscm 21953 | The scalar multiplication ... |
| mat0dimcrng 21954 | The algebra of matrices wi... |
| mat1dimelbas 21955 | A matrix with dimension 1 ... |
| mat1dimbas 21956 | A matrix with dimension 1 ... |
| mat1dim0 21957 | The zero of the algebra of... |
| mat1dimid 21958 | The identity of the algebr... |
| mat1dimscm 21959 | The scalar multiplication ... |
| mat1dimmul 21960 | The ring multiplication in... |
| mat1dimcrng 21961 | The algebra of matrices wi... |
| mat1f1o 21962 | There is a 1-1 function fr... |
| mat1rhmval 21963 | The value of the ring homo... |
| mat1rhmelval 21964 | The value of the ring homo... |
| mat1rhmcl 21965 | The value of the ring homo... |
| mat1f 21966 | There is a function from a... |
| mat1ghm 21967 | There is a group homomorph... |
| mat1mhm 21968 | There is a monoid homomorp... |
| mat1rhm 21969 | There is a ring homomorphi... |
| mat1rngiso 21970 | There is a ring isomorphis... |
| mat1ric 21971 | A ring is isomorphic to th... |
| dmatval 21976 | The set of ` N ` x ` N ` d... |
| dmatel 21977 | A ` N ` x ` N ` diagonal m... |
| dmatmat 21978 | An ` N ` x ` N ` diagonal ... |
| dmatid 21979 | The identity matrix is a d... |
| dmatelnd 21980 | An extradiagonal entry of ... |
| dmatmul 21981 | The product of two diagona... |
| dmatsubcl 21982 | The difference of two diag... |
| dmatsgrp 21983 | The set of diagonal matric... |
| dmatmulcl 21984 | The product of two diagona... |
| dmatsrng 21985 | The set of diagonal matric... |
| dmatcrng 21986 | The subring of diagonal ma... |
| dmatscmcl 21987 | The multiplication of a di... |
| scmatval 21988 | The set of ` N ` x ` N ` s... |
| scmatel 21989 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 21990 | A scalar matrix can be exp... |
| scmatscmide 21991 | An entry of a scalar matri... |
| scmatscmiddistr 21992 | Distributive law for scala... |
| scmatmat 21993 | An ` N ` x ` N ` scalar ma... |
| scmate 21994 | An entry of an ` N ` x ` N... |
| scmatmats 21995 | The set of an ` N ` x ` N ... |
| scmateALT 21996 | Alternate proof of ~ scmat... |
| scmatscm 21997 | The multiplication of a ma... |
| scmatid 21998 | The identity matrix is a s... |
| scmatdmat 21999 | A scalar matrix is a diago... |
| scmataddcl 22000 | The sum of two scalar matr... |
| scmatsubcl 22001 | The difference of two scal... |
| scmatmulcl 22002 | The product of two scalar ... |
| scmatsgrp 22003 | The set of scalar matrices... |
| scmatsrng 22004 | The set of scalar matrices... |
| scmatcrng 22005 | The subring of scalar matr... |
| scmatsgrp1 22006 | The set of scalar matrices... |
| scmatsrng1 22007 | The set of scalar matrices... |
| smatvscl 22008 | Closure of the scalar mult... |
| scmatlss 22009 | The set of scalar matrices... |
| scmatstrbas 22010 | The set of scalar matrices... |
| scmatrhmval 22011 | The value of the ring homo... |
| scmatrhmcl 22012 | The value of the ring homo... |
| scmatf 22013 | There is a function from a... |
| scmatfo 22014 | There is a function from a... |
| scmatf1 22015 | There is a 1-1 function fr... |
| scmatf1o 22016 | There is a bijection betwe... |
| scmatghm 22017 | There is a group homomorph... |
| scmatmhm 22018 | There is a monoid homomorp... |
| scmatrhm 22019 | There is a ring homomorphi... |
| scmatrngiso 22020 | There is a ring isomorphis... |
| scmatric 22021 | A ring is isomorphic to ev... |
| mat0scmat 22022 | The empty matrix over a ri... |
| mat1scmat 22023 | A 1-dimensional matrix ove... |
| mvmulfval 22026 | Functional value of the ma... |
| mvmulval 22027 | Multiplication of a vector... |
| mvmulfv 22028 | A cell/element in the vect... |
| mavmulval 22029 | Multiplication of a vector... |
| mavmulfv 22030 | A cell/element in the vect... |
| mavmulcl 22031 | Multiplication of an NxN m... |
| 1mavmul 22032 | Multiplication of the iden... |
| mavmulass 22033 | Associativity of the multi... |
| mavmuldm 22034 | The domain of the matrix v... |
| mavmulsolcl 22035 | Every solution of the equa... |
| mavmul0 22036 | Multiplication of a 0-dime... |
| mavmul0g 22037 | The result of the 0-dimens... |
| mvmumamul1 22038 | The multiplication of an M... |
| mavmumamul1 22039 | The multiplication of an N... |
| marrepfval 22044 | First substitution for the... |
| marrepval0 22045 | Second substitution for th... |
| marrepval 22046 | Third substitution for the... |
| marrepeval 22047 | An entry of a matrix with ... |
| marrepcl 22048 | Closure of the row replace... |
| marepvfval 22049 | First substitution for the... |
| marepvval0 22050 | Second substitution for th... |
| marepvval 22051 | Third substitution for the... |
| marepveval 22052 | An entry of a matrix with ... |
| marepvcl 22053 | Closure of the column repl... |
| ma1repvcl 22054 | Closure of the column repl... |
| ma1repveval 22055 | An entry of an identity ma... |
| mulmarep1el 22056 | Element by element multipl... |
| mulmarep1gsum1 22057 | The sum of element by elem... |
| mulmarep1gsum2 22058 | The sum of element by elem... |
| 1marepvmarrepid 22059 | Replacing the ith row by 0... |
| submabas 22062 | Any subset of the index se... |
| submafval 22063 | First substitution for a s... |
| submaval0 22064 | Second substitution for a ... |
| submaval 22065 | Third substitution for a s... |
| submaeval 22066 | An entry of a submatrix of... |
| 1marepvsma1 22067 | The submatrix of the ident... |
| mdetfval 22070 | First substitution for the... |
| mdetleib 22071 | Full substitution of our d... |
| mdetleib2 22072 | Leibniz' formula can also ... |
| nfimdetndef 22073 | The determinant is not def... |
| mdetfval1 22074 | First substitution of an a... |
| mdetleib1 22075 | Full substitution of an al... |
| mdet0pr 22076 | The determinant function f... |
| mdet0f1o 22077 | The determinant function f... |
| mdet0fv0 22078 | The determinant of the emp... |
| mdetf 22079 | Functionality of the deter... |
| mdetcl 22080 | The determinant evaluates ... |
| m1detdiag 22081 | The determinant of a 1-dim... |
| mdetdiaglem 22082 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22083 | The determinant of a diago... |
| mdetdiagid 22084 | The determinant of a diago... |
| mdet1 22085 | The determinant of the ide... |
| mdetrlin 22086 | The determinant function i... |
| mdetrsca 22087 | The determinant function i... |
| mdetrsca2 22088 | The determinant function i... |
| mdetr0 22089 | The determinant of a matri... |
| mdet0 22090 | The determinant of the zer... |
| mdetrlin2 22091 | The determinant function i... |
| mdetralt 22092 | The determinant function i... |
| mdetralt2 22093 | The determinant function i... |
| mdetero 22094 | The determinant function i... |
| mdettpos 22095 | Determinant is invariant u... |
| mdetunilem1 22096 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22097 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22098 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22099 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22100 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22101 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22102 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22103 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22104 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22105 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22106 | According to the definitio... |
| mdetmul 22107 | Multiplicativity of the de... |
| m2detleiblem1 22108 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22109 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22110 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22111 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22112 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22113 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22114 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22115 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22120 | Lemma for ~ maducoeval2 . ... |
| madufval 22121 | First substitution for the... |
| maduval 22122 | Second substitution for th... |
| maducoeval 22123 | An entry of the adjunct (c... |
| maducoeval2 22124 | An entry of the adjunct (c... |
| maduf 22125 | Creating the adjunct of ma... |
| madutpos 22126 | The adjuct of a transposed... |
| madugsum 22127 | The determinant of a matri... |
| madurid 22128 | Multiplying a matrix with ... |
| madulid 22129 | Multiplying the adjunct of... |
| minmar1fval 22130 | First substitution for the... |
| minmar1val0 22131 | Second substitution for th... |
| minmar1val 22132 | Third substitution for the... |
| minmar1eval 22133 | An entry of a matrix for a... |
| minmar1marrep 22134 | The minor matrix is a spec... |
| minmar1cl 22135 | Closure of the row replace... |
| maducoevalmin1 22136 | The coefficients of an adj... |
| symgmatr01lem 22137 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22138 | Applying a permutation tha... |
| gsummatr01lem1 22139 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22140 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22141 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22142 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22143 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22144 | Replacing a row of a squar... |
| smadiadetlem0 22145 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22146 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22147 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22148 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22149 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22150 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22151 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22152 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22153 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22154 | The determinant of a subma... |
| smadiadetglem1 22155 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22156 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22157 | The determinant of a squar... |
| smadiadetg0 22158 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22159 | The determinant of a squar... |
| invrvald 22160 | If a matrix multiplied wit... |
| matinv 22161 | The inverse of a matrix is... |
| matunit 22162 | A matrix is a unit in the ... |
| slesolvec 22163 | Every solution of a system... |
| slesolinv 22164 | The solution of a system o... |
| slesolinvbi 22165 | The solution of a system o... |
| slesolex 22166 | Every system of linear equ... |
| cramerimplem1 22167 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22168 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22169 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22170 | One direction of Cramer's ... |
| cramerlem1 22171 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22172 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22173 | Lemma 3 for ~ cramer . (C... |
| cramer0 22174 | Special case of Cramer's r... |
| cramer 22175 | Cramer's rule. According ... |
| pmatring 22176 | The set of polynomial matr... |
| pmatlmod 22177 | The set of polynomial matr... |
| pmatassa 22178 | The set of polynomial matr... |
| pmat0op 22179 | The zero polynomial matrix... |
| pmat1op 22180 | The identity polynomial ma... |
| pmat1ovd 22181 | Entries of the identity po... |
| pmat0opsc 22182 | The zero polynomial matrix... |
| pmat1opsc 22183 | The identity polynomial ma... |
| pmat1ovscd 22184 | Entries of the identity po... |
| pmatcoe1fsupp 22185 | For a polynomial matrix th... |
| 1pmatscmul 22186 | The scalar product of the ... |
| cpmat 22193 | Value of the constructor o... |
| cpmatpmat 22194 | A constant polynomial matr... |
| cpmatel 22195 | Property of a constant pol... |
| cpmatelimp 22196 | Implication of a set being... |
| cpmatel2 22197 | Another property of a cons... |
| cpmatelimp2 22198 | Another implication of a s... |
| 1elcpmat 22199 | The identity of the ring o... |
| cpmatacl 22200 | The set of all constant po... |
| cpmatinvcl 22201 | The set of all constant po... |
| cpmatmcllem 22202 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22203 | The set of all constant po... |
| cpmatsubgpmat 22204 | The set of all constant po... |
| cpmatsrgpmat 22205 | The set of all constant po... |
| 0elcpmat 22206 | The zero of the ring of al... |
| mat2pmatfval 22207 | Value of the matrix transf... |
| mat2pmatval 22208 | The result of a matrix tra... |
| mat2pmatvalel 22209 | A (matrix) element of the ... |
| mat2pmatbas 22210 | The result of a matrix tra... |
| mat2pmatbas0 22211 | The result of a matrix tra... |
| mat2pmatf 22212 | The matrix transformation ... |
| mat2pmatf1 22213 | The matrix transformation ... |
| mat2pmatghm 22214 | The transformation of matr... |
| mat2pmatmul 22215 | The transformation of matr... |
| mat2pmat1 22216 | The transformation of the ... |
| mat2pmatmhm 22217 | The transformation of matr... |
| mat2pmatrhm 22218 | The transformation of matr... |
| mat2pmatlin 22219 | The transformation of matr... |
| 0mat2pmat 22220 | The transformed zero matri... |
| idmatidpmat 22221 | The transformed identity m... |
| d0mat2pmat 22222 | The transformed empty set ... |
| d1mat2pmat 22223 | The transformation of a ma... |
| mat2pmatscmxcl 22224 | A transformed matrix multi... |
| m2cpm 22225 | The result of a matrix tra... |
| m2cpmf 22226 | The matrix transformation ... |
| m2cpmf1 22227 | The matrix transformation ... |
| m2cpmghm 22228 | The transformation of matr... |
| m2cpmmhm 22229 | The transformation of matr... |
| m2cpmrhm 22230 | The transformation of matr... |
| m2pmfzmap 22231 | The transformed values of ... |
| m2pmfzgsumcl 22232 | Closure of the sum of scal... |
| cpm2mfval 22233 | Value of the inverse matri... |
| cpm2mval 22234 | The result of an inverse m... |
| cpm2mvalel 22235 | A (matrix) element of the ... |
| cpm2mf 22236 | The inverse matrix transfo... |
| m2cpminvid 22237 | The inverse transformation... |
| m2cpminvid2lem 22238 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22239 | The transformation applied... |
| m2cpmfo 22240 | The matrix transformation ... |
| m2cpmf1o 22241 | The matrix transformation ... |
| m2cpmrngiso 22242 | The transformation of matr... |
| matcpmric 22243 | The ring of matrices over ... |
| m2cpminv 22244 | The inverse matrix transfo... |
| m2cpminv0 22245 | The inverse matrix transfo... |
| decpmatval0 22248 | The matrix consisting of t... |
| decpmatval 22249 | The matrix consisting of t... |
| decpmate 22250 | An entry of the matrix con... |
| decpmatcl 22251 | Closure of the decompositi... |
| decpmataa0 22252 | The matrix consisting of t... |
| decpmatfsupp 22253 | The mapping to the matrice... |
| decpmatid 22254 | The matrix consisting of t... |
| decpmatmullem 22255 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22256 | The matrix consisting of t... |
| decpmatmulsumfsupp 22257 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22258 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22259 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22260 | Write a polynomial matrix ... |
| pmatcollpw2lem 22261 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22262 | Write a polynomial matrix ... |
| monmatcollpw 22263 | The matrix consisting of t... |
| pmatcollpwlem 22264 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22265 | Write a polynomial matrix ... |
| pmatcollpwfi 22266 | Write a polynomial matrix ... |
| pmatcollpw3lem 22267 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22268 | Write a polynomial matrix ... |
| pmatcollpw3fi 22269 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22270 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22271 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22272 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22273 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22274 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22275 | Write a scalar matrix over... |
| pm2mpf1lem 22278 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22279 | Value of the transformatio... |
| pm2mpfval 22280 | A polynomial matrix transf... |
| pm2mpcl 22281 | The transformation of poly... |
| pm2mpf 22282 | The transformation of poly... |
| pm2mpf1 22283 | The transformation of poly... |
| pm2mpcoe1 22284 | A coefficient of the polyn... |
| idpm2idmp 22285 | The transformation of the ... |
| mptcoe1matfsupp 22286 | The mapping extracting the... |
| mply1topmatcllem 22287 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22288 | A polynomial over matrices... |
| mply1topmatcl 22289 | A polynomial over matrices... |
| mp2pm2mplem1 22290 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22291 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22292 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22293 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22294 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22295 | A polynomial over matrices... |
| pm2mpghmlem2 22296 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22297 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22298 | The transformation of poly... |
| pm2mpf1o 22299 | The transformation of poly... |
| pm2mpghm 22300 | The transformation of poly... |
| pm2mpgrpiso 22301 | The transformation of poly... |
| pm2mpmhmlem1 22302 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22303 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22304 | The transformation of poly... |
| pm2mprhm 22305 | The transformation of poly... |
| pm2mprngiso 22306 | The transformation of poly... |
| pmmpric 22307 | The ring of polynomial mat... |
| monmat2matmon 22308 | The transformation of a po... |
| pm2mp 22309 | The transformation of a su... |
| chmatcl 22312 | Closure of the characteris... |
| chmatval 22313 | The entries of the charact... |
| chpmatfval 22314 | Value of the characteristi... |
| chpmatval 22315 | The characteristic polynom... |
| chpmatply1 22316 | The characteristic polynom... |
| chpmatval2 22317 | The characteristic polynom... |
| chpmat0d 22318 | The characteristic polynom... |
| chpmat1dlem 22319 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22320 | The characteristic polynom... |
| chpdmatlem0 22321 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22322 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22323 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22324 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22325 | The characteristic polynom... |
| chpscmat 22326 | The characteristic polynom... |
| chpscmat0 22327 | The characteristic polynom... |
| chpscmatgsumbin 22328 | The characteristic polynom... |
| chpscmatgsummon 22329 | The characteristic polynom... |
| chp0mat 22330 | The characteristic polynom... |
| chpidmat 22331 | The characteristic polynom... |
| chmaidscmat 22332 | The characteristic polynom... |
| fvmptnn04if 22333 | The function values of a m... |
| fvmptnn04ifa 22334 | The function value of a ma... |
| fvmptnn04ifb 22335 | The function value of a ma... |
| fvmptnn04ifc 22336 | The function value of a ma... |
| fvmptnn04ifd 22337 | The function value of a ma... |
| chfacfisf 22338 | The "characteristic factor... |
| chfacfisfcpmat 22339 | The "characteristic factor... |
| chfacffsupp 22340 | The "characteristic factor... |
| chfacfscmulcl 22341 | Closure of a scaled value ... |
| chfacfscmul0 22342 | A scaled value of the "cha... |
| chfacfscmulfsupp 22343 | A mapping of scaled values... |
| chfacfscmulgsum 22344 | Breaking up a sum of value... |
| chfacfpmmulcl 22345 | Closure of the value of th... |
| chfacfpmmul0 22346 | The value of the "characte... |
| chfacfpmmulfsupp 22347 | A mapping of values of the... |
| chfacfpmmulgsum 22348 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22349 | Breaking up a sum of value... |
| cayhamlem1 22350 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22351 | The right-hand fundamental... |
| cpmidgsum 22352 | Representation of the iden... |
| cpmidgsumm2pm 22353 | Representation of the iden... |
| cpmidpmatlem1 22354 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22355 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22356 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22357 | Representation of the iden... |
| cpmadugsumlemB 22358 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22359 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22360 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22361 | The product of the charact... |
| cpmadugsum 22362 | The product of the charact... |
| cpmidgsum2 22363 | Representation of the iden... |
| cpmidg2sum 22364 | Equality of two sums repre... |
| cpmadumatpolylem1 22365 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22366 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22367 | The product of the charact... |
| cayhamlem2 22368 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22369 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22370 | The coefficients of the ch... |
| cayhamlem3 22371 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22372 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22373 | The Cayley-Hamilton theore... |
| cayleyhamilton 22374 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22375 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22376 | The Cayley-Hamilton theore... |
| istopg 22379 | Express the predicate " ` ... |
| istop2g 22380 | Express the predicate " ` ... |
| uniopn 22381 | The union of a subset of a... |
| iunopn 22382 | The indexed union of a sub... |
| inopn 22383 | The intersection of two op... |
| fitop 22384 | A topology is closed under... |
| fiinopn 22385 | The intersection of a none... |
| iinopn 22386 | The intersection of a none... |
| unopn 22387 | The union of two open sets... |
| 0opn 22388 | The empty set is an open s... |
| 0ntop 22389 | The empty set is not a top... |
| topopn 22390 | The underlying set of a to... |
| eltopss 22391 | A member of a topology is ... |
| riinopn 22392 | A finite indexed relative ... |
| rintopn 22393 | A finite relative intersec... |
| istopon 22396 | Property of being a topolo... |
| topontop 22397 | A topology on a given base... |
| toponuni 22398 | The base set of a topology... |
| topontopi 22399 | A topology on a given base... |
| toponunii 22400 | The base set of a topology... |
| toptopon 22401 | Alternative definition of ... |
| toptopon2 22402 | A topology is the same thi... |
| topontopon 22403 | A topology on a set is a t... |
| funtopon 22404 | The class ` TopOn ` is a f... |
| toponrestid 22405 | Given a topology on a set,... |
| toponsspwpw 22406 | The set of topologies on a... |
| dmtopon 22407 | The domain of ` TopOn ` is... |
| fntopon 22408 | The class ` TopOn ` is a f... |
| toprntopon 22409 | A topology is the same thi... |
| toponmax 22410 | The base set of a topology... |
| toponss 22411 | A member of a topology is ... |
| toponcom 22412 | If ` K ` is a topology on ... |
| toponcomb 22413 | Biconditional form of ~ to... |
| topgele 22414 | The topologies over the sa... |
| topsn 22415 | The only topology on a sin... |
| istps 22418 | Express the predicate "is ... |
| istps2 22419 | Express the predicate "is ... |
| tpsuni 22420 | The base set of a topologi... |
| tpstop 22421 | The topology extractor on ... |
| tpspropd 22422 | A topological space depend... |
| tpsprop2d 22423 | A topological space depend... |
| topontopn 22424 | Express the predicate "is ... |
| tsettps 22425 | If the topology component ... |
| istpsi 22426 | Properties that determine ... |
| eltpsg 22427 | Properties that determine ... |
| eltpsgOLD 22428 | Obsolete version of ~ eltp... |
| eltpsi 22429 | Properties that determine ... |
| isbasisg 22432 | Express the predicate "the... |
| isbasis2g 22433 | Express the predicate "the... |
| isbasis3g 22434 | Express the predicate "the... |
| basis1 22435 | Property of a basis. (Con... |
| basis2 22436 | Property of a basis. (Con... |
| fiinbas 22437 | If a set is closed under f... |
| basdif0 22438 | A basis is not affected by... |
| baspartn 22439 | A disjoint system of sets ... |
| tgval 22440 | The topology generated by ... |
| tgval2 22441 | Definition of a topology g... |
| eltg 22442 | Membership in a topology g... |
| eltg2 22443 | Membership in a topology g... |
| eltg2b 22444 | Membership in a topology g... |
| eltg4i 22445 | An open set in a topology ... |
| eltg3i 22446 | The union of a set of basi... |
| eltg3 22447 | Membership in a topology g... |
| tgval3 22448 | Alternate expression for t... |
| tg1 22449 | Property of a member of a ... |
| tg2 22450 | Property of a member of a ... |
| bastg 22451 | A member of a basis is a s... |
| unitg 22452 | The topology generated by ... |
| tgss 22453 | Subset relation for genera... |
| tgcl 22454 | Show that a basis generate... |
| tgclb 22455 | The property ~ tgcl can be... |
| tgtopon 22456 | A basis generates a topolo... |
| topbas 22457 | A topology is its own basi... |
| tgtop 22458 | A topology is its own basi... |
| eltop 22459 | Membership in a topology, ... |
| eltop2 22460 | Membership in a topology. ... |
| eltop3 22461 | Membership in a topology. ... |
| fibas 22462 | A collection of finite int... |
| tgdom 22463 | A space has no more open s... |
| tgiun 22464 | The indexed union of a set... |
| tgidm 22465 | The topology generator fun... |
| bastop 22466 | Two ways to express that a... |
| tgtop11 22467 | The topology generation fu... |
| 0top 22468 | The singleton of the empty... |
| en1top 22469 | ` { (/) } ` is the only to... |
| en2top 22470 | If a topology has two elem... |
| tgss3 22471 | A criterion for determinin... |
| tgss2 22472 | A criterion for determinin... |
| basgen 22473 | Given a topology ` J ` , s... |
| basgen2 22474 | Given a topology ` J ` , s... |
| 2basgen 22475 | Conditions that determine ... |
| tgfiss 22476 | If a subbase is included i... |
| tgdif0 22477 | A generated topology is no... |
| bastop1 22478 | A subset of a topology is ... |
| bastop2 22479 | A version of ~ bastop1 tha... |
| distop 22480 | The discrete topology on a... |
| topnex 22481 | The class of all topologie... |
| distopon 22482 | The discrete topology on a... |
| sn0topon 22483 | The singleton of the empty... |
| sn0top 22484 | The singleton of the empty... |
| indislem 22485 | A lemma to eliminate some ... |
| indistopon 22486 | The indiscrete topology on... |
| indistop 22487 | The indiscrete topology on... |
| indisuni 22488 | The base set of the indisc... |
| fctop 22489 | The finite complement topo... |
| fctop2 22490 | The finite complement topo... |
| cctop 22491 | The countable complement t... |
| ppttop 22492 | The particular point topol... |
| pptbas 22493 | The particular point topol... |
| epttop 22494 | The excluded point topolog... |
| indistpsx 22495 | The indiscrete topology on... |
| indistps 22496 | The indiscrete topology on... |
| indistps2 22497 | The indiscrete topology on... |
| indistpsALT 22498 | The indiscrete topology on... |
| indistpsALTOLD 22499 | Obsolete proof of ~ indist... |
| indistps2ALT 22500 | The indiscrete topology on... |
| distps 22501 | The discrete topology on a... |
| fncld 22508 | The closed-set generator i... |
| cldval 22509 | The set of closed sets of ... |
| ntrfval 22510 | The interior function on t... |
| clsfval 22511 | The closure function on th... |
| cldrcl 22512 | Reverse closure of the clo... |
| iscld 22513 | The predicate "the class `... |
| iscld2 22514 | A subset of the underlying... |
| cldss 22515 | A closed set is a subset o... |
| cldss2 22516 | The set of closed sets is ... |
| cldopn 22517 | The complement of a closed... |
| isopn2 22518 | A subset of the underlying... |
| opncld 22519 | The complement of an open ... |
| difopn 22520 | The difference of a closed... |
| topcld 22521 | The underlying set of a to... |
| ntrval 22522 | The interior of a subset o... |
| clsval 22523 | The closure of a subset of... |
| 0cld 22524 | The empty set is closed. ... |
| iincld 22525 | The indexed intersection o... |
| intcld 22526 | The intersection of a set ... |
| uncld 22527 | The union of two closed se... |
| cldcls 22528 | A closed subset equals its... |
| incld 22529 | The intersection of two cl... |
| riincld 22530 | An indexed relative inters... |
| iuncld 22531 | A finite indexed union of ... |
| unicld 22532 | A finite union of closed s... |
| clscld 22533 | The closure of a subset of... |
| clsf 22534 | The closure function is a ... |
| ntropn 22535 | The interior of a subset o... |
| clsval2 22536 | Express closure in terms o... |
| ntrval2 22537 | Interior expressed in term... |
| ntrdif 22538 | An interior of a complemen... |
| clsdif 22539 | A closure of a complement ... |
| clsss 22540 | Subset relationship for cl... |
| ntrss 22541 | Subset relationship for in... |
| sscls 22542 | A subset of a topology's u... |
| ntrss2 22543 | A subset includes its inte... |
| ssntr 22544 | An open subset of a set is... |
| clsss3 22545 | The closure of a subset of... |
| ntrss3 22546 | The interior of a subset o... |
| ntrin 22547 | A pairwise intersection of... |
| cmclsopn 22548 | The complement of a closur... |
| cmntrcld 22549 | The complement of an inter... |
| iscld3 22550 | A subset is closed iff it ... |
| iscld4 22551 | A subset is closed iff it ... |
| isopn3 22552 | A subset is open iff it eq... |
| clsidm 22553 | The closure operation is i... |
| ntridm 22554 | The interior operation is ... |
| clstop 22555 | The closure of a topology'... |
| ntrtop 22556 | The interior of a topology... |
| 0ntr 22557 | A subset with an empty int... |
| clsss2 22558 | If a subset is included in... |
| elcls 22559 | Membership in a closure. ... |
| elcls2 22560 | Membership in a closure. ... |
| clsndisj 22561 | Any open set containing a ... |
| ntrcls0 22562 | A subset whose closure has... |
| ntreq0 22563 | Two ways to say that a sub... |
| cldmre 22564 | The closed sets of a topol... |
| mrccls 22565 | Moore closure generalizes ... |
| cls0 22566 | The closure of the empty s... |
| ntr0 22567 | The interior of the empty ... |
| isopn3i 22568 | An open subset equals its ... |
| elcls3 22569 | Membership in a closure in... |
| opncldf1 22570 | A bijection useful for con... |
| opncldf2 22571 | The values of the open-clo... |
| opncldf3 22572 | The values of the converse... |
| isclo 22573 | A set ` A ` is clopen iff ... |
| isclo2 22574 | A set ` A ` is clopen iff ... |
| discld 22575 | The open sets of a discret... |
| sn0cld 22576 | The closed sets of the top... |
| indiscld 22577 | The closed sets of an indi... |
| mretopd 22578 | A Moore collection which i... |
| toponmre 22579 | The topologies over a give... |
| cldmreon 22580 | The closed sets of a topol... |
| iscldtop 22581 | A family is the closed set... |
| mreclatdemoBAD 22582 | The closed subspaces of a ... |
| neifval 22585 | Value of the neighborhood ... |
| neif 22586 | The neighborhood function ... |
| neiss2 22587 | A set with a neighborhood ... |
| neival 22588 | Value of the set of neighb... |
| isnei 22589 | The predicate "the class `... |
| neiint 22590 | An intuitive definition of... |
| isneip 22591 | The predicate "the class `... |
| neii1 22592 | A neighborhood is included... |
| neisspw 22593 | The neighborhoods of any s... |
| neii2 22594 | Property of a neighborhood... |
| neiss 22595 | Any neighborhood of a set ... |
| ssnei 22596 | A set is included in any o... |
| elnei 22597 | A point belongs to any of ... |
| 0nnei 22598 | The empty set is not a nei... |
| neips 22599 | A neighborhood of a set is... |
| opnneissb 22600 | An open set is a neighborh... |
| opnssneib 22601 | Any superset of an open se... |
| ssnei2 22602 | Any subset ` M ` of ` X ` ... |
| neindisj 22603 | Any neighborhood of an ele... |
| opnneiss 22604 | An open set is a neighborh... |
| opnneip 22605 | An open set is a neighborh... |
| opnnei 22606 | A set is open iff it is a ... |
| tpnei 22607 | The underlying set of a to... |
| neiuni 22608 | The union of the neighborh... |
| neindisj2 22609 | A point ` P ` belongs to t... |
| topssnei 22610 | A finer topology has more ... |
| innei 22611 | The intersection of two ne... |
| opnneiid 22612 | Only an open set is a neig... |
| neissex 22613 | For any neighborhood ` N `... |
| 0nei 22614 | The empty set is a neighbo... |
| neipeltop 22615 | Lemma for ~ neiptopreu . ... |
| neiptopuni 22616 | Lemma for ~ neiptopreu . ... |
| neiptoptop 22617 | Lemma for ~ neiptopreu . ... |
| neiptopnei 22618 | Lemma for ~ neiptopreu . ... |
| neiptopreu 22619 | If, to each element ` P ` ... |
| lpfval 22624 | The limit point function o... |
| lpval 22625 | The set of limit points of... |
| islp 22626 | The predicate "the class `... |
| lpsscls 22627 | The limit points of a subs... |
| lpss 22628 | The limit points of a subs... |
| lpdifsn 22629 | ` P ` is a limit point of ... |
| lpss3 22630 | Subset relationship for li... |
| islp2 22631 | The predicate " ` P ` is a... |
| islp3 22632 | The predicate " ` P ` is a... |
| maxlp 22633 | A point is a limit point o... |
| clslp 22634 | The closure of a subset of... |
| islpi 22635 | A point belonging to a set... |
| cldlp 22636 | A subset of a topological ... |
| isperf 22637 | Definition of a perfect sp... |
| isperf2 22638 | Definition of a perfect sp... |
| isperf3 22639 | A perfect space is a topol... |
| perflp 22640 | The limit points of a perf... |
| perfi 22641 | Property of a perfect spac... |
| perftop 22642 | A perfect space is a topol... |
| restrcl 22643 | Reverse closure for the su... |
| restbas 22644 | A subspace topology basis ... |
| tgrest 22645 | A subspace can be generate... |
| resttop 22646 | A subspace topology is a t... |
| resttopon 22647 | A subspace topology is a t... |
| restuni 22648 | The underlying set of a su... |
| stoig 22649 | The topological space buil... |
| restco 22650 | Composition of subspaces. ... |
| restabs 22651 | Equivalence of being a sub... |
| restin 22652 | When the subspace region i... |
| restuni2 22653 | The underlying set of a su... |
| resttopon2 22654 | The underlying set of a su... |
| rest0 22655 | The subspace topology indu... |
| restsn 22656 | The only subspace topology... |
| restsn2 22657 | The subspace topology indu... |
| restcld 22658 | A closed set of a subspace... |
| restcldi 22659 | A closed set is closed in ... |
| restcldr 22660 | A set which is closed in t... |
| restopnb 22661 | If ` B ` is an open subset... |
| ssrest 22662 | If ` K ` is a finer topolo... |
| restopn2 22663 | If ` A ` is open, then ` B... |
| restdis 22664 | A subspace of a discrete t... |
| restfpw 22665 | The restriction of the set... |
| neitr 22666 | The neighborhood of a trac... |
| restcls 22667 | A closure in a subspace to... |
| restntr 22668 | An interior in a subspace ... |
| restlp 22669 | The limit points of a subs... |
| restperf 22670 | Perfection of a subspace. ... |
| perfopn 22671 | An open subset of a perfec... |
| resstopn 22672 | The topology of a restrict... |
| resstps 22673 | A restricted topological s... |
| ordtbaslem 22674 | Lemma for ~ ordtbas . In ... |
| ordtval 22675 | Value of the order topolog... |
| ordtuni 22676 | Value of the order topolog... |
| ordtbas2 22677 | Lemma for ~ ordtbas . (Co... |
| ordtbas 22678 | In a total order, the fini... |
| ordttopon 22679 | Value of the order topolog... |
| ordtopn1 22680 | An upward ray ` ( P , +oo ... |
| ordtopn2 22681 | A downward ray ` ( -oo , P... |
| ordtopn3 22682 | An open interval ` ( A , B... |
| ordtcld1 22683 | A downward ray ` ( -oo , P... |
| ordtcld2 22684 | An upward ray ` [ P , +oo ... |
| ordtcld3 22685 | A closed interval ` [ A , ... |
| ordttop 22686 | The order topology is a to... |
| ordtcnv 22687 | The order dual generates t... |
| ordtrest 22688 | The subspace topology of a... |
| ordtrest2lem 22689 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 22690 | An interval-closed set ` A... |
| letopon 22691 | The topology of the extend... |
| letop 22692 | The topology of the extend... |
| letopuni 22693 | The topology of the extend... |
| xrstopn 22694 | The topology component of ... |
| xrstps 22695 | The extended real number s... |
| leordtvallem1 22696 | Lemma for ~ leordtval . (... |
| leordtvallem2 22697 | Lemma for ~ leordtval . (... |
| leordtval2 22698 | The topology of the extend... |
| leordtval 22699 | The topology of the extend... |
| iccordt 22700 | A closed interval is close... |
| iocpnfordt 22701 | An unbounded above open in... |
| icomnfordt 22702 | An unbounded above open in... |
| iooordt 22703 | An open interval is open i... |
| reordt 22704 | The real numbers are an op... |
| lecldbas 22705 | The set of closed interval... |
| pnfnei 22706 | A neighborhood of ` +oo ` ... |
| mnfnei 22707 | A neighborhood of ` -oo ` ... |
| ordtrestixx 22708 | The restriction of the les... |
| ordtresticc 22709 | The restriction of the les... |
| lmrel 22716 | The topological space conv... |
| lmrcl 22717 | Reverse closure for the co... |
| lmfval 22718 | The relation "sequence ` f... |
| cnfval 22719 | The set of all continuous ... |
| cnpfval 22720 | The function mapping the p... |
| iscn 22721 | The predicate "the class `... |
| cnpval 22722 | The set of all functions f... |
| iscnp 22723 | The predicate "the class `... |
| iscn2 22724 | The predicate "the class `... |
| iscnp2 22725 | The predicate "the class `... |
| cntop1 22726 | Reverse closure for a cont... |
| cntop2 22727 | Reverse closure for a cont... |
| cnptop1 22728 | Reverse closure for a func... |
| cnptop2 22729 | Reverse closure for a func... |
| iscnp3 22730 | The predicate "the class `... |
| cnprcl 22731 | Reverse closure for a func... |
| cnf 22732 | A continuous function is a... |
| cnpf 22733 | A continuous function at p... |
| cnpcl 22734 | The value of a continuous ... |
| cnf2 22735 | A continuous function is a... |
| cnpf2 22736 | A continuous function at p... |
| cnprcl2 22737 | Reverse closure for a func... |
| tgcn 22738 | The continuity predicate w... |
| tgcnp 22739 | The "continuous at a point... |
| subbascn 22740 | The continuity predicate w... |
| ssidcn 22741 | The identity function is a... |
| cnpimaex 22742 | Property of a function con... |
| idcn 22743 | A restricted identity func... |
| lmbr 22744 | Express the binary relatio... |
| lmbr2 22745 | Express the binary relatio... |
| lmbrf 22746 | Express the binary relatio... |
| lmconst 22747 | A constant sequence conver... |
| lmcvg 22748 | Convergence property of a ... |
| iscnp4 22749 | The predicate "the class `... |
| cnpnei 22750 | A condition for continuity... |
| cnima 22751 | An open subset of the codo... |
| cnco 22752 | The composition of two con... |
| cnpco 22753 | The composition of a funct... |
| cnclima 22754 | A closed subset of the cod... |
| iscncl 22755 | A characterization of a co... |
| cncls2i 22756 | Property of the preimage o... |
| cnntri 22757 | Property of the preimage o... |
| cnclsi 22758 | Property of the image of a... |
| cncls2 22759 | Continuity in terms of clo... |
| cncls 22760 | Continuity in terms of clo... |
| cnntr 22761 | Continuity in terms of int... |
| cnss1 22762 | If the topology ` K ` is f... |
| cnss2 22763 | If the topology ` K ` is f... |
| cncnpi 22764 | A continuous function is c... |
| cnsscnp 22765 | The set of continuous func... |
| cncnp 22766 | A continuous function is c... |
| cncnp2 22767 | A continuous function is c... |
| cnnei 22768 | Continuity in terms of nei... |
| cnconst2 22769 | A constant function is con... |
| cnconst 22770 | A constant function is con... |
| cnrest 22771 | Continuity of a restrictio... |
| cnrest2 22772 | Equivalence of continuity ... |
| cnrest2r 22773 | Equivalence of continuity ... |
| cnpresti 22774 | One direction of ~ cnprest... |
| cnprest 22775 | Equivalence of continuity ... |
| cnprest2 22776 | Equivalence of point-conti... |
| cndis 22777 | Every function is continuo... |
| cnindis 22778 | Every function is continuo... |
| cnpdis 22779 | If ` A ` is an isolated po... |
| paste 22780 | Pasting lemma. If ` A ` a... |
| lmfpm 22781 | If ` F ` converges, then `... |
| lmfss 22782 | Inclusion of a function ha... |
| lmcl 22783 | Closure of a limit. (Cont... |
| lmss 22784 | Limit on a subspace. (Con... |
| sslm 22785 | A finer topology has fewer... |
| lmres 22786 | A function converges iff i... |
| lmff 22787 | If ` F ` converges, there ... |
| lmcls 22788 | Any convergent sequence of... |
| lmcld 22789 | Any convergent sequence of... |
| lmcnp 22790 | The image of a convergent ... |
| lmcn 22791 | The image of a convergent ... |
| ist0 22806 | The predicate "is a T_0 sp... |
| ist1 22807 | The predicate "is a T_1 sp... |
| ishaus 22808 | The predicate "is a Hausdo... |
| iscnrm 22809 | The property of being comp... |
| t0sep 22810 | Any two topologically indi... |
| t0dist 22811 | Any two distinct points in... |
| t1sncld 22812 | In a T_1 space, singletons... |
| t1ficld 22813 | In a T_1 space, finite set... |
| hausnei 22814 | Neighborhood property of a... |
| t0top 22815 | A T_0 space is a topologic... |
| t1top 22816 | A T_1 space is a topologic... |
| haustop 22817 | A Hausdorff space is a top... |
| isreg 22818 | The predicate "is a regula... |
| regtop 22819 | A regular space is a topol... |
| regsep 22820 | In a regular space, every ... |
| isnrm 22821 | The predicate "is a normal... |
| nrmtop 22822 | A normal space is a topolo... |
| cnrmtop 22823 | A completely normal space ... |
| iscnrm2 22824 | The property of being comp... |
| ispnrm 22825 | The property of being perf... |
| pnrmnrm 22826 | A perfectly normal space i... |
| pnrmtop 22827 | A perfectly normal space i... |
| pnrmcld 22828 | A closed set in a perfectl... |
| pnrmopn 22829 | An open set in a perfectly... |
| ist0-2 22830 | The predicate "is a T_0 sp... |
| ist0-3 22831 | The predicate "is a T_0 sp... |
| cnt0 22832 | The preimage of a T_0 topo... |
| ist1-2 22833 | An alternate characterizat... |
| t1t0 22834 | A T_1 space is a T_0 space... |
| ist1-3 22835 | A space is T_1 iff every p... |
| cnt1 22836 | The preimage of a T_1 topo... |
| ishaus2 22837 | Express the predicate " ` ... |
| haust1 22838 | A Hausdorff space is a T_1... |
| hausnei2 22839 | The Hausdorff condition st... |
| cnhaus 22840 | The preimage of a Hausdorf... |
| nrmsep3 22841 | In a normal space, given a... |
| nrmsep2 22842 | In a normal space, any two... |
| nrmsep 22843 | In a normal space, disjoin... |
| isnrm2 22844 | An alternate characterizat... |
| isnrm3 22845 | A topological space is nor... |
| cnrmi 22846 | A subspace of a completely... |
| cnrmnrm 22847 | A completely normal space ... |
| restcnrm 22848 | A subspace of a completely... |
| resthauslem 22849 | Lemma for ~ resthaus and s... |
| lpcls 22850 | The limit points of the cl... |
| perfcls 22851 | A subset of a perfect spac... |
| restt0 22852 | A subspace of a T_0 topolo... |
| restt1 22853 | A subspace of a T_1 topolo... |
| resthaus 22854 | A subspace of a Hausdorff ... |
| t1sep2 22855 | Any two points in a T_1 sp... |
| t1sep 22856 | Any two distinct points in... |
| sncld 22857 | A singleton is closed in a... |
| sshauslem 22858 | Lemma for ~ sshaus and sim... |
| sst0 22859 | A topology finer than a T_... |
| sst1 22860 | A topology finer than a T_... |
| sshaus 22861 | A topology finer than a Ha... |
| regsep2 22862 | In a regular space, a clos... |
| isreg2 22863 | A topological space is reg... |
| dnsconst 22864 | If a continuous mapping to... |
| ordtt1 22865 | The order topology is T_1 ... |
| lmmo 22866 | A sequence in a Hausdorff ... |
| lmfun 22867 | The convergence relation i... |
| dishaus 22868 | A discrete topology is Hau... |
| ordthauslem 22869 | Lemma for ~ ordthaus . (C... |
| ordthaus 22870 | The order topology of a to... |
| xrhaus 22871 | The topology of the extend... |
| iscmp 22874 | The predicate "is a compac... |
| cmpcov 22875 | An open cover of a compact... |
| cmpcov2 22876 | Rewrite ~ cmpcov for the c... |
| cmpcovf 22877 | Combine ~ cmpcov with ~ ac... |
| cncmp 22878 | Compactness is respected b... |
| fincmp 22879 | A finite topology is compa... |
| 0cmp 22880 | The singleton of the empty... |
| cmptop 22881 | A compact topology is a to... |
| rncmp 22882 | The image of a compact set... |
| imacmp 22883 | The image of a compact set... |
| discmp 22884 | A discrete topology is com... |
| cmpsublem 22885 | Lemma for ~ cmpsub . (Con... |
| cmpsub 22886 | Two equivalent ways of des... |
| tgcmp 22887 | A topology generated by a ... |
| cmpcld 22888 | A closed subset of a compa... |
| uncmp 22889 | The union of two compact s... |
| fiuncmp 22890 | A finite union of compact ... |
| sscmp 22891 | A subset of a compact topo... |
| hauscmplem 22892 | Lemma for ~ hauscmp . (Co... |
| hauscmp 22893 | A compact subspace of a T2... |
| cmpfi 22894 | If a topology is compact a... |
| cmpfii 22895 | In a compact topology, a s... |
| bwth 22896 | The glorious Bolzano-Weier... |
| isconn 22899 | The predicate ` J ` is a c... |
| isconn2 22900 | The predicate ` J ` is a c... |
| connclo 22901 | The only nonempty clopen s... |
| conndisj 22902 | If a topology is connected... |
| conntop 22903 | A connected topology is a ... |
| indisconn 22904 | The indiscrete topology (o... |
| dfconn2 22905 | An alternate definition of... |
| connsuba 22906 | Connectedness for a subspa... |
| connsub 22907 | Two equivalent ways of say... |
| cnconn 22908 | Connectedness is respected... |
| nconnsubb 22909 | Disconnectedness for a sub... |
| connsubclo 22910 | If a clopen set meets a co... |
| connima 22911 | The image of a connected s... |
| conncn 22912 | A continuous function from... |
| iunconnlem 22913 | Lemma for ~ iunconn . (Co... |
| iunconn 22914 | The indexed union of conne... |
| unconn 22915 | The union of two connected... |
| clsconn 22916 | The closure of a connected... |
| conncompid 22917 | The connected component co... |
| conncompconn 22918 | The connected component co... |
| conncompss 22919 | The connected component co... |
| conncompcld 22920 | The connected component co... |
| conncompclo 22921 | The connected component co... |
| t1connperf 22922 | A connected T_1 space is p... |
| is1stc 22927 | The predicate "is a first-... |
| is1stc2 22928 | An equivalent way of sayin... |
| 1stctop 22929 | A first-countable topology... |
| 1stcclb 22930 | A property of points in a ... |
| 1stcfb 22931 | For any point ` A ` in a f... |
| is2ndc 22932 | The property of being seco... |
| 2ndctop 22933 | A second-countable topolog... |
| 2ndci 22934 | A countable basis generate... |
| 2ndcsb 22935 | Having a countable subbase... |
| 2ndcredom 22936 | A second-countable space h... |
| 2ndc1stc 22937 | A second-countable space i... |
| 1stcrestlem 22938 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 22939 | A subspace of a first-coun... |
| 2ndcrest 22940 | A subspace of a second-cou... |
| 2ndcctbss 22941 | If a topology is second-co... |
| 2ndcdisj 22942 | Any disjoint family of ope... |
| 2ndcdisj2 22943 | Any disjoint collection of... |
| 2ndcomap 22944 | A surjective continuous op... |
| 2ndcsep 22945 | A second-countable topolog... |
| dis2ndc 22946 | A discrete space is second... |
| 1stcelcls 22947 | A point belongs to the clo... |
| 1stccnp 22948 | A mapping is continuous at... |
| 1stccn 22949 | A mapping ` X --> Y ` , wh... |
| islly 22954 | The property of being a lo... |
| isnlly 22955 | The property of being an n... |
| llyeq 22956 | Equality theorem for the `... |
| nllyeq 22957 | Equality theorem for the `... |
| llytop 22958 | A locally ` A ` space is a... |
| nllytop 22959 | A locally ` A ` space is a... |
| llyi 22960 | The property of a locally ... |
| nllyi 22961 | The property of an n-local... |
| nlly2i 22962 | Eliminate the neighborhood... |
| llynlly 22963 | A locally ` A ` space is n... |
| llyssnlly 22964 | A locally ` A ` space is n... |
| llyss 22965 | The "locally" predicate re... |
| nllyss 22966 | The "n-locally" predicate ... |
| subislly 22967 | The property of a subspace... |
| restnlly 22968 | If the property ` A ` pass... |
| restlly 22969 | If the property ` A ` pass... |
| islly2 22970 | An alternative expression ... |
| llyrest 22971 | An open subspace of a loca... |
| nllyrest 22972 | An open subspace of an n-l... |
| loclly 22973 | If ` A ` is a local proper... |
| llyidm 22974 | Idempotence of the "locall... |
| nllyidm 22975 | Idempotence of the "n-loca... |
| toplly 22976 | A topology is locally a to... |
| topnlly 22977 | A topology is n-locally a ... |
| hauslly 22978 | A Hausdorff space is local... |
| hausnlly 22979 | A Hausdorff space is n-loc... |
| hausllycmp 22980 | A compact Hausdorff space ... |
| cldllycmp 22981 | A closed subspace of a loc... |
| lly1stc 22982 | First-countability is a lo... |
| dislly 22983 | The discrete space ` ~P X ... |
| disllycmp 22984 | A discrete space is locall... |
| dis1stc 22985 | A discrete space is first-... |
| hausmapdom 22986 | If ` X ` is a first-counta... |
| hauspwdom 22987 | Simplify the cardinal ` A ... |
| refrel 22994 | Refinement is a relation. ... |
| isref 22995 | The property of being a re... |
| refbas 22996 | A refinement covers the sa... |
| refssex 22997 | Every set in a refinement ... |
| ssref 22998 | A subcover is a refinement... |
| refref 22999 | Reflexivity of refinement.... |
| reftr 23000 | Refinement is transitive. ... |
| refun0 23001 | Adding the empty set prese... |
| isptfin 23002 | The statement "is a point-... |
| islocfin 23003 | The statement "is a locall... |
| finptfin 23004 | A finite cover is a point-... |
| ptfinfin 23005 | A point covered by a point... |
| finlocfin 23006 | A finite cover of a topolo... |
| locfintop 23007 | A locally finite cover cov... |
| locfinbas 23008 | A locally finite cover mus... |
| locfinnei 23009 | A point covered by a local... |
| lfinpfin 23010 | A locally finite cover is ... |
| lfinun 23011 | Adding a finite set preser... |
| locfincmp 23012 | For a compact space, the l... |
| unisngl 23013 | Taking the union of the se... |
| dissnref 23014 | The set of singletons is a... |
| dissnlocfin 23015 | The set of singletons is l... |
| locfindis 23016 | The locally finite covers ... |
| locfincf 23017 | A locally finite cover in ... |
| comppfsc 23018 | A space where every open c... |
| kgenval 23021 | Value of the compact gener... |
| elkgen 23022 | Value of the compact gener... |
| kgeni 23023 | Property of the open sets ... |
| kgentopon 23024 | The compact generator gene... |
| kgenuni 23025 | The base set of the compac... |
| kgenftop 23026 | The compact generator gene... |
| kgenf 23027 | The compact generator is a... |
| kgentop 23028 | A compactly generated spac... |
| kgenss 23029 | The compact generator gene... |
| kgenhaus 23030 | The compact generator gene... |
| kgencmp 23031 | The compact generator topo... |
| kgencmp2 23032 | The compact generator topo... |
| kgenidm 23033 | The compact generator is i... |
| iskgen2 23034 | A space is compactly gener... |
| iskgen3 23035 | Derive the usual definitio... |
| llycmpkgen2 23036 | A locally compact space is... |
| cmpkgen 23037 | A compact space is compact... |
| llycmpkgen 23038 | A locally compact space is... |
| 1stckgenlem 23039 | The one-point compactifica... |
| 1stckgen 23040 | A first-countable space is... |
| kgen2ss 23041 | The compact generator pres... |
| kgencn 23042 | A function from a compactl... |
| kgencn2 23043 | A function ` F : J --> K `... |
| kgencn3 23044 | The set of continuous func... |
| kgen2cn 23045 | A continuous function is a... |
| txval 23050 | Value of the binary topolo... |
| txuni2 23051 | The underlying set of the ... |
| txbasex 23052 | The basis for the product ... |
| txbas 23053 | The set of Cartesian produ... |
| eltx 23054 | A set in a product is open... |
| txtop 23055 | The product of two topolog... |
| ptval 23056 | The value of the product t... |
| ptpjpre1 23057 | The preimage of a projecti... |
| elpt 23058 | Elementhood in the bases o... |
| elptr 23059 | A basic open set in the pr... |
| elptr2 23060 | A basic open set in the pr... |
| ptbasid 23061 | The base set of the produc... |
| ptuni2 23062 | The base set for the produ... |
| ptbasin 23063 | The basis for a product to... |
| ptbasin2 23064 | The basis for a product to... |
| ptbas 23065 | The basis for a product to... |
| ptpjpre2 23066 | The basis for a product to... |
| ptbasfi 23067 | The basis for the product ... |
| pttop 23068 | The product topology is a ... |
| ptopn 23069 | A basic open set in the pr... |
| ptopn2 23070 | A sub-basic open set in th... |
| xkotf 23071 | Functionality of function ... |
| xkobval 23072 | Alternative expression for... |
| xkoval 23073 | Value of the compact-open ... |
| xkotop 23074 | The compact-open topology ... |
| xkoopn 23075 | A basic open set of the co... |
| txtopi 23076 | The product of two topolog... |
| txtopon 23077 | The underlying set of the ... |
| txuni 23078 | The underlying set of the ... |
| txunii 23079 | The underlying set of the ... |
| ptuni 23080 | The base set for the produ... |
| ptunimpt 23081 | Base set of a product topo... |
| pttopon 23082 | The base set for the produ... |
| pttoponconst 23083 | The base set for a product... |
| ptuniconst 23084 | The base set for a product... |
| xkouni 23085 | The base set of the compac... |
| xkotopon 23086 | The base set of the compac... |
| ptval2 23087 | The value of the product t... |
| txopn 23088 | The product of two open se... |
| txcld 23089 | The product of two closed ... |
| txcls 23090 | Closure of a rectangle in ... |
| txss12 23091 | Subset property of the top... |
| txbasval 23092 | It is sufficient to consid... |
| neitx 23093 | The Cartesian product of t... |
| txcnpi 23094 | Continuity of a two-argume... |
| tx1cn 23095 | Continuity of the first pr... |
| tx2cn 23096 | Continuity of the second p... |
| ptpjcn 23097 | Continuity of a projection... |
| ptpjopn 23098 | The projection map is an o... |
| ptcld 23099 | A closed box in the produc... |
| ptcldmpt 23100 | A closed box in the produc... |
| ptclsg 23101 | The closure of a box in th... |
| ptcls 23102 | The closure of a box in th... |
| dfac14lem 23103 | Lemma for ~ dfac14 . By e... |
| dfac14 23104 | Theorem ~ ptcls is an equi... |
| xkoccn 23105 | The "constant function" fu... |
| txcnp 23106 | If two functions are conti... |
| ptcnplem 23107 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23108 | If every projection of a f... |
| upxp 23109 | Universal property of the ... |
| txcnmpt 23110 | A map into the product of ... |
| uptx 23111 | Universal property of the ... |
| txcn 23112 | A map into the product of ... |
| ptcn 23113 | If every projection of a f... |
| prdstopn 23114 | Topology of a structure pr... |
| prdstps 23115 | A structure product of top... |
| pwstps 23116 | A structure power of a top... |
| txrest 23117 | The subspace of a topologi... |
| txdis 23118 | The topological product of... |
| txindislem 23119 | Lemma for ~ txindis . (Co... |
| txindis 23120 | The topological product of... |
| txdis1cn 23121 | A function is jointly cont... |
| txlly 23122 | If the property ` A ` is p... |
| txnlly 23123 | If the property ` A ` is p... |
| pthaus 23124 | The product of a collectio... |
| ptrescn 23125 | Restriction is a continuou... |
| txtube 23126 | The "tube lemma". If ` X ... |
| txcmplem1 23127 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23128 | Lemma for ~ txcmp . (Cont... |
| txcmp 23129 | The topological product of... |
| txcmpb 23130 | The topological product of... |
| hausdiag 23131 | A topology is Hausdorff if... |
| hauseqlcld 23132 | In a Hausdorff topology, t... |
| txhaus 23133 | The topological product of... |
| txlm 23134 | Two sequences converge iff... |
| lmcn2 23135 | The image of a convergent ... |
| tx1stc 23136 | The topological product of... |
| tx2ndc 23137 | The topological product of... |
| txkgen 23138 | The topological product of... |
| xkohaus 23139 | If the codomain space is H... |
| xkoptsub 23140 | The compact-open topology ... |
| xkopt 23141 | The compact-open topology ... |
| xkopjcn 23142 | Continuity of a projection... |
| xkoco1cn 23143 | If ` F ` is a continuous f... |
| xkoco2cn 23144 | If ` F ` is a continuous f... |
| xkococnlem 23145 | Continuity of the composit... |
| xkococn 23146 | Continuity of the composit... |
| cnmptid 23147 | The identity function is c... |
| cnmptc 23148 | A constant function is con... |
| cnmpt11 23149 | The composition of continu... |
| cnmpt11f 23150 | The composition of continu... |
| cnmpt1t 23151 | The composition of continu... |
| cnmpt12f 23152 | The composition of continu... |
| cnmpt12 23153 | The composition of continu... |
| cnmpt1st 23154 | The projection onto the fi... |
| cnmpt2nd 23155 | The projection onto the se... |
| cnmpt2c 23156 | A constant function is con... |
| cnmpt21 23157 | The composition of continu... |
| cnmpt21f 23158 | The composition of continu... |
| cnmpt2t 23159 | The composition of continu... |
| cnmpt22 23160 | The composition of continu... |
| cnmpt22f 23161 | The composition of continu... |
| cnmpt1res 23162 | The restriction of a conti... |
| cnmpt2res 23163 | The restriction of a conti... |
| cnmptcom 23164 | The argument converse of a... |
| cnmptkc 23165 | The curried first projecti... |
| cnmptkp 23166 | The evaluation of the inne... |
| cnmptk1 23167 | The composition of a curri... |
| cnmpt1k 23168 | The composition of a one-a... |
| cnmptkk 23169 | The composition of two cur... |
| xkofvcn 23170 | Joint continuity of the fu... |
| cnmptk1p 23171 | The evaluation of a currie... |
| cnmptk2 23172 | The uncurrying of a currie... |
| xkoinjcn 23173 | Continuity of "injection",... |
| cnmpt2k 23174 | The currying of a two-argu... |
| txconn 23175 | The topological product of... |
| imasnopn 23176 | If a relation graph is ope... |
| imasncld 23177 | If a relation graph is clo... |
| imasncls 23178 | If a relation graph is clo... |
| qtopval 23181 | Value of the quotient topo... |
| qtopval2 23182 | Value of the quotient topo... |
| elqtop 23183 | Value of the quotient topo... |
| qtopres 23184 | The quotient topology is u... |
| qtoptop2 23185 | The quotient topology is a... |
| qtoptop 23186 | The quotient topology is a... |
| elqtop2 23187 | Value of the quotient topo... |
| qtopuni 23188 | The base set of the quotie... |
| elqtop3 23189 | Value of the quotient topo... |
| qtoptopon 23190 | The base set of the quotie... |
| qtopid 23191 | A quotient map is a contin... |
| idqtop 23192 | The quotient topology indu... |
| qtopcmplem 23193 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23194 | A quotient of a compact sp... |
| qtopconn 23195 | A quotient of a connected ... |
| qtopkgen 23196 | A quotient of a compactly ... |
| basqtop 23197 | An injection maps bases to... |
| tgqtop 23198 | An injection maps generate... |
| qtopcld 23199 | The property of being a cl... |
| qtopcn 23200 | Universal property of a qu... |
| qtopss 23201 | A surjective continuous fu... |
| qtopeu 23202 | Universal property of the ... |
| qtoprest 23203 | If ` A ` is a saturated op... |
| qtopomap 23204 | If ` F ` is a surjective c... |
| qtopcmap 23205 | If ` F ` is a surjective c... |
| imastopn 23206 | The topology of an image s... |
| imastps 23207 | The image of a topological... |
| qustps 23208 | A quotient structure is a ... |
| kqfval 23209 | Value of the function appe... |
| kqfeq 23210 | Two points in the Kolmogor... |
| kqffn 23211 | The topological indistingu... |
| kqval 23212 | Value of the quotient topo... |
| kqtopon 23213 | The Kolmogorov quotient is... |
| kqid 23214 | The topological indistingu... |
| ist0-4 23215 | The topological indistingu... |
| kqfvima 23216 | When the image set is open... |
| kqsat 23217 | Any open set is saturated ... |
| kqdisj 23218 | A version of ~ imain for t... |
| kqcldsat 23219 | Any closed set is saturate... |
| kqopn 23220 | The topological indistingu... |
| kqcld 23221 | The topological indistingu... |
| kqt0lem 23222 | Lemma for ~ kqt0 . (Contr... |
| isr0 23223 | The property " ` J ` is an... |
| r0cld 23224 | The analogue of the T_1 ax... |
| regr1lem 23225 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23226 | A Kolmogorov quotient of a... |
| kqreglem1 23227 | A Kolmogorov quotient of a... |
| kqreglem2 23228 | If the Kolmogorov quotient... |
| kqnrmlem1 23229 | A Kolmogorov quotient of a... |
| kqnrmlem2 23230 | If the Kolmogorov quotient... |
| kqtop 23231 | The Kolmogorov quotient is... |
| kqt0 23232 | The Kolmogorov quotient is... |
| kqf 23233 | The Kolmogorov quotient is... |
| r0sep 23234 | The separation property of... |
| nrmr0reg 23235 | A normal R_0 space is also... |
| regr1 23236 | A regular space is R_1, wh... |
| kqreg 23237 | The Kolmogorov quotient of... |
| kqnrm 23238 | The Kolmogorov quotient of... |
| hmeofn 23243 | The set of homeomorphisms ... |
| hmeofval 23244 | The set of all the homeomo... |
| ishmeo 23245 | The predicate F is a homeo... |
| hmeocn 23246 | A homeomorphism is continu... |
| hmeocnvcn 23247 | The converse of a homeomor... |
| hmeocnv 23248 | The converse of a homeomor... |
| hmeof1o2 23249 | A homeomorphism is a 1-1-o... |
| hmeof1o 23250 | A homeomorphism is a 1-1-o... |
| hmeoima 23251 | The image of an open set b... |
| hmeoopn 23252 | Homeomorphisms preserve op... |
| hmeocld 23253 | Homeomorphisms preserve cl... |
| hmeocls 23254 | Homeomorphisms preserve cl... |
| hmeontr 23255 | Homeomorphisms preserve in... |
| hmeoimaf1o 23256 | The function mapping open ... |
| hmeores 23257 | The restriction of a homeo... |
| hmeoco 23258 | The composite of two homeo... |
| idhmeo 23259 | The identity function is a... |
| hmeocnvb 23260 | The converse of a homeomor... |
| hmeoqtop 23261 | A homeomorphism is a quoti... |
| hmph 23262 | Express the predicate ` J ... |
| hmphi 23263 | If there is a homeomorphis... |
| hmphtop 23264 | Reverse closure for the ho... |
| hmphtop1 23265 | The relation "being homeom... |
| hmphtop2 23266 | The relation "being homeom... |
| hmphref 23267 | "Is homeomorphic to" is re... |
| hmphsym 23268 | "Is homeomorphic to" is sy... |
| hmphtr 23269 | "Is homeomorphic to" is tr... |
| hmpher 23270 | "Is homeomorphic to" is an... |
| hmphen 23271 | Homeomorphisms preserve th... |
| hmphsymb 23272 | "Is homeomorphic to" is sy... |
| haushmphlem 23273 | Lemma for ~ haushmph and s... |
| cmphmph 23274 | Compactness is a topologic... |
| connhmph 23275 | Connectedness is a topolog... |
| t0hmph 23276 | T_0 is a topological prope... |
| t1hmph 23277 | T_1 is a topological prope... |
| haushmph 23278 | Hausdorff-ness is a topolo... |
| reghmph 23279 | Regularity is a topologica... |
| nrmhmph 23280 | Normality is a topological... |
| hmph0 23281 | A topology homeomorphic to... |
| hmphdis 23282 | Homeomorphisms preserve to... |
| hmphindis 23283 | Homeomorphisms preserve to... |
| indishmph 23284 | Equinumerous sets equipped... |
| hmphen2 23285 | Homeomorphisms preserve th... |
| cmphaushmeo 23286 | A continuous bijection fro... |
| ordthmeolem 23287 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23288 | An order isomorphism is a ... |
| txhmeo 23289 | Lift a pair of homeomorphi... |
| txswaphmeolem 23290 | Show inverse for the "swap... |
| txswaphmeo 23291 | There is a homeomorphism f... |
| pt1hmeo 23292 | The canonical homeomorphis... |
| ptuncnv 23293 | Exhibit the converse funct... |
| ptunhmeo 23294 | Define a homeomorphism fro... |
| xpstopnlem1 23295 | The function ` F ` used in... |
| xpstps 23296 | A binary product of topolo... |
| xpstopnlem2 23297 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23298 | The topology on a binary p... |
| ptcmpfi 23299 | A topological product of f... |
| xkocnv 23300 | The inverse of the "curryi... |
| xkohmeo 23301 | The Exponential Law for to... |
| qtopf1 23302 | If a quotient map is injec... |
| qtophmeo 23303 | If two functions on a base... |
| t0kq 23304 | A topological space is T_0... |
| kqhmph 23305 | A topological space is T_0... |
| ist1-5lem 23306 | Lemma for ~ ist1-5 and sim... |
| t1r0 23307 | A T_1 space is R_0. That ... |
| ist1-5 23308 | A topological space is T_1... |
| ishaus3 23309 | A topological space is Hau... |
| nrmreg 23310 | A normal T_1 space is regu... |
| reghaus 23311 | A regular T_0 space is Hau... |
| nrmhaus 23312 | A T_1 normal space is Haus... |
| elmptrab 23313 | Membership in a one-parame... |
| elmptrab2 23314 | Membership in a one-parame... |
| isfbas 23315 | The predicate " ` F ` is a... |
| fbasne0 23316 | There are no empty filter ... |
| 0nelfb 23317 | No filter base contains th... |
| fbsspw 23318 | A filter base on a set is ... |
| fbelss 23319 | An element of the filter b... |
| fbdmn0 23320 | The domain of a filter bas... |
| isfbas2 23321 | The predicate " ` F ` is a... |
| fbasssin 23322 | A filter base contains sub... |
| fbssfi 23323 | A filter base contains sub... |
| fbssint 23324 | A filter base contains sub... |
| fbncp 23325 | A filter base does not con... |
| fbun 23326 | A necessary and sufficient... |
| fbfinnfr 23327 | No filter base containing ... |
| opnfbas 23328 | The collection of open sup... |
| trfbas2 23329 | Conditions for the trace o... |
| trfbas 23330 | Conditions for the trace o... |
| isfil 23333 | The predicate "is a filter... |
| filfbas 23334 | A filter is a filter base.... |
| 0nelfil 23335 | The empty set doesn't belo... |
| fileln0 23336 | An element of a filter is ... |
| filsspw 23337 | A filter is a subset of th... |
| filelss 23338 | An element of a filter is ... |
| filss 23339 | A filter is closed under t... |
| filin 23340 | A filter is closed under t... |
| filtop 23341 | The underlying set belongs... |
| isfil2 23342 | Derive the standard axioms... |
| isfildlem 23343 | Lemma for ~ isfild . (Con... |
| isfild 23344 | Sufficient condition for a... |
| filfi 23345 | A filter is closed under t... |
| filinn0 23346 | The intersection of two el... |
| filintn0 23347 | A filter has the finite in... |
| filn0 23348 | The empty set is not a fil... |
| infil 23349 | The intersection of two fi... |
| snfil 23350 | A singleton is a filter. ... |
| fbasweak 23351 | A filter base on any set i... |
| snfbas 23352 | Condition for a singleton ... |
| fsubbas 23353 | A condition for a set to g... |
| fbasfip 23354 | A filter base has the fini... |
| fbunfip 23355 | A helpful lemma for showin... |
| fgval 23356 | The filter generating clas... |
| elfg 23357 | A condition for elements o... |
| ssfg 23358 | A filter base is a subset ... |
| fgss 23359 | A bigger base generates a ... |
| fgss2 23360 | A condition for a filter t... |
| fgfil 23361 | A filter generates itself.... |
| elfilss 23362 | An element belongs to a fi... |
| filfinnfr 23363 | No filter containing a fin... |
| fgcl 23364 | A generated filter is a fi... |
| fgabs 23365 | Absorption law for filter ... |
| neifil 23366 | The neighborhoods of a non... |
| filunibas 23367 | Recover the base set from ... |
| filunirn 23368 | Two ways to express a filt... |
| filconn 23369 | A filter gives rise to a c... |
| fbasrn 23370 | Given a filter on a domain... |
| filuni 23371 | The union of a nonempty se... |
| trfil1 23372 | Conditions for the trace o... |
| trfil2 23373 | Conditions for the trace o... |
| trfil3 23374 | Conditions for the trace o... |
| trfilss 23375 | If ` A ` is a member of th... |
| fgtr 23376 | If ` A ` is a member of th... |
| trfg 23377 | The trace operation and th... |
| trnei 23378 | The trace, over a set ` A ... |
| cfinfil 23379 | Relative complements of th... |
| csdfil 23380 | The set of all elements wh... |
| supfil 23381 | The supersets of a nonempt... |
| zfbas 23382 | The set of upper sets of i... |
| uzrest 23383 | The restriction of the set... |
| uzfbas 23384 | The set of upper sets of i... |
| isufil 23389 | The property of being an u... |
| ufilfil 23390 | An ultrafilter is a filter... |
| ufilss 23391 | For any subset of the base... |
| ufilb 23392 | The complement is in an ul... |
| ufilmax 23393 | Any filter finer than an u... |
| isufil2 23394 | The maximal property of an... |
| ufprim 23395 | An ultrafilter is a prime ... |
| trufil 23396 | Conditions for the trace o... |
| filssufilg 23397 | A filter is contained in s... |
| filssufil 23398 | A filter is contained in s... |
| isufl 23399 | Define the (strong) ultraf... |
| ufli 23400 | Property of a set that sat... |
| numufl 23401 | Consequence of ~ filssufil... |
| fiufl 23402 | A finite set satisfies the... |
| acufl 23403 | The axiom of choice implie... |
| ssufl 23404 | If ` Y ` is a subset of ` ... |
| ufileu 23405 | If the ultrafilter contain... |
| filufint 23406 | A filter is equal to the i... |
| uffix 23407 | Lemma for ~ fixufil and ~ ... |
| fixufil 23408 | The condition describing a... |
| uffixfr 23409 | An ultrafilter is either f... |
| uffix2 23410 | A classification of fixed ... |
| uffixsn 23411 | The singleton of the gener... |
| ufildom1 23412 | An ultrafilter is generate... |
| uffinfix 23413 | An ultrafilter containing ... |
| cfinufil 23414 | An ultrafilter is free iff... |
| ufinffr 23415 | An infinite subset is cont... |
| ufilen 23416 | Any infinite set has an ul... |
| ufildr 23417 | An ultrafilter gives rise ... |
| fin1aufil 23418 | There are no definable fre... |
| fmval 23429 | Introduce a function that ... |
| fmfil 23430 | A mapping filter is a filt... |
| fmf 23431 | Pushing-forward via a func... |
| fmss 23432 | A finer filter produces a ... |
| elfm 23433 | An element of a mapping fi... |
| elfm2 23434 | An element of a mapping fi... |
| fmfg 23435 | The image filter of a filt... |
| elfm3 23436 | An alternate formulation o... |
| imaelfm 23437 | An image of a filter eleme... |
| rnelfmlem 23438 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23439 | A condition for a filter t... |
| fmfnfmlem1 23440 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23441 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23442 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23443 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23444 | A filter finer than an ima... |
| fmufil 23445 | An image filter of an ultr... |
| fmid 23446 | The filter map applied to ... |
| fmco 23447 | Composition of image filte... |
| ufldom 23448 | The ultrafilter lemma prop... |
| flimval 23449 | The set of limit points of... |
| elflim2 23450 | The predicate "is a limit ... |
| flimtop 23451 | Reverse closure for the li... |
| flimneiss 23452 | A filter contains the neig... |
| flimnei 23453 | A filter contains all of t... |
| flimelbas 23454 | A limit point of a filter ... |
| flimfil 23455 | Reverse closure for the li... |
| flimtopon 23456 | Reverse closure for the li... |
| elflim 23457 | The predicate "is a limit ... |
| flimss2 23458 | A limit point of a filter ... |
| flimss1 23459 | A limit point of a filter ... |
| neiflim 23460 | A point is a limit point o... |
| flimopn 23461 | The condition for being a ... |
| fbflim 23462 | A condition for a filter t... |
| fbflim2 23463 | A condition for a filter b... |
| flimclsi 23464 | The convergent points of a... |
| hausflimlem 23465 | If ` A ` and ` B ` are bot... |
| hausflimi 23466 | One direction of ~ hausfli... |
| hausflim 23467 | A condition for a topology... |
| flimcf 23468 | Fineness is properly chara... |
| flimrest 23469 | The set of limit points in... |
| flimclslem 23470 | Lemma for ~ flimcls . (Co... |
| flimcls 23471 | Closure in terms of filter... |
| flimsncls 23472 | If ` A ` is a limit point ... |
| hauspwpwf1 23473 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 23474 | If ` X ` is a Hausdorff sp... |
| flffval 23475 | Given a topology and a fil... |
| flfval 23476 | Given a function from a fi... |
| flfnei 23477 | The property of being a li... |
| flfneii 23478 | A neighborhood of a limit ... |
| isflf 23479 | The property of being a li... |
| flfelbas 23480 | A limit point of a functio... |
| flffbas 23481 | Limit points of a function... |
| flftg 23482 | Limit points of a function... |
| hausflf 23483 | If a function has its valu... |
| hausflf2 23484 | If a convergent function h... |
| cnpflfi 23485 | Forward direction of ~ cnp... |
| cnpflf2 23486 | ` F ` is continuous at poi... |
| cnpflf 23487 | Continuity of a function a... |
| cnflf 23488 | A function is continuous i... |
| cnflf2 23489 | A function is continuous i... |
| flfcnp 23490 | A continuous function pres... |
| lmflf 23491 | The topological limit rela... |
| txflf 23492 | Two sequences converge in ... |
| flfcnp2 23493 | The image of a convergent ... |
| fclsval 23494 | The set of all cluster poi... |
| isfcls 23495 | A cluster point of a filte... |
| fclsfil 23496 | Reverse closure for the cl... |
| fclstop 23497 | Reverse closure for the cl... |
| fclstopon 23498 | Reverse closure for the cl... |
| isfcls2 23499 | A cluster point of a filte... |
| fclsopn 23500 | Write the cluster point co... |
| fclsopni 23501 | An open neighborhood of a ... |
| fclselbas 23502 | A cluster point is in the ... |
| fclsneii 23503 | A neighborhood of a cluste... |
| fclssscls 23504 | The set of cluster points ... |
| fclsnei 23505 | Cluster points in terms of... |
| supnfcls 23506 | The filter of supersets of... |
| fclsbas 23507 | Cluster points in terms of... |
| fclsss1 23508 | A finer topology has fewer... |
| fclsss2 23509 | A finer filter has fewer c... |
| fclsrest 23510 | The set of cluster points ... |
| fclscf 23511 | Characterization of finene... |
| flimfcls 23512 | A limit point is a cluster... |
| fclsfnflim 23513 | A filter clusters at a poi... |
| flimfnfcls 23514 | A filter converges to a po... |
| fclscmpi 23515 | Forward direction of ~ fcl... |
| fclscmp 23516 | A space is compact iff eve... |
| uffclsflim 23517 | The cluster points of an u... |
| ufilcmp 23518 | A space is compact iff eve... |
| fcfval 23519 | The set of cluster points ... |
| isfcf 23520 | The property of being a cl... |
| fcfnei 23521 | The property of being a cl... |
| fcfelbas 23522 | A cluster point of a funct... |
| fcfneii 23523 | A neighborhood of a cluste... |
| flfssfcf 23524 | A limit point of a functio... |
| uffcfflf 23525 | If the domain filter is an... |
| cnpfcfi 23526 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 23527 | A function ` F ` is contin... |
| cnfcf 23528 | Continuity of a function i... |
| flfcntr 23529 | A continuous function's va... |
| alexsublem 23530 | Lemma for ~ alexsub . (Co... |
| alexsub 23531 | The Alexander Subbase Theo... |
| alexsubb 23532 | Biconditional form of the ... |
| alexsubALTlem1 23533 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 23534 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 23535 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 23536 | Lemma for ~ alexsubALT . ... |
| alexsubALT 23537 | The Alexander Subbase Theo... |
| ptcmplem1 23538 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 23539 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 23540 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 23541 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 23542 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 23543 | Tychonoff's theorem: The ... |
| ptcmp 23544 | Tychonoff's theorem: The ... |
| cnextval 23547 | The function applying cont... |
| cnextfval 23548 | The continuous extension o... |
| cnextrel 23549 | In the general case, a con... |
| cnextfun 23550 | If the target space is Hau... |
| cnextfvval 23551 | The value of the continuou... |
| cnextf 23552 | Extension by continuity. ... |
| cnextcn 23553 | Extension by continuity. ... |
| cnextfres1 23554 | ` F ` and its extension by... |
| cnextfres 23555 | ` F ` and its extension by... |
| istmd 23560 | The predicate "is a topolo... |
| tmdmnd 23561 | A topological monoid is a ... |
| tmdtps 23562 | A topological monoid is a ... |
| istgp 23563 | The predicate "is a topolo... |
| tgpgrp 23564 | A topological group is a g... |
| tgptmd 23565 | A topological group is a t... |
| tgptps 23566 | A topological group is a t... |
| tmdtopon 23567 | The topology of a topologi... |
| tgptopon 23568 | The topology of a topologi... |
| tmdcn 23569 | In a topological monoid, t... |
| tgpcn 23570 | In a topological group, th... |
| tgpinv 23571 | In a topological group, th... |
| grpinvhmeo 23572 | The inverse function in a ... |
| cnmpt1plusg 23573 | Continuity of the group su... |
| cnmpt2plusg 23574 | Continuity of the group su... |
| tmdcn2 23575 | Write out the definition o... |
| tgpsubcn 23576 | In a topological group, th... |
| istgp2 23577 | A group with a topology is... |
| tmdmulg 23578 | In a topological monoid, t... |
| tgpmulg 23579 | In a topological group, th... |
| tgpmulg2 23580 | In a topological monoid, t... |
| tmdgsum 23581 | In a topological monoid, t... |
| tmdgsum2 23582 | For any neighborhood ` U `... |
| oppgtmd 23583 | The opposite of a topologi... |
| oppgtgp 23584 | The opposite of a topologi... |
| distgp 23585 | Any group equipped with th... |
| indistgp 23586 | Any group equipped with th... |
| efmndtmd 23587 | The monoid of endofunction... |
| tmdlactcn 23588 | The left group action of e... |
| tgplacthmeo 23589 | The left group action of e... |
| submtmd 23590 | A submonoid of a topologic... |
| subgtgp 23591 | A subgroup of a topologica... |
| symgtgp 23592 | The symmetric group is a t... |
| subgntr 23593 | A subgroup of a topologica... |
| opnsubg 23594 | An open subgroup of a topo... |
| clssubg 23595 | The closure of a subgroup ... |
| clsnsg 23596 | The closure of a normal su... |
| cldsubg 23597 | A subgroup of finite index... |
| tgpconncompeqg 23598 | The connected component co... |
| tgpconncomp 23599 | The identity component, th... |
| tgpconncompss 23600 | The identity component is ... |
| ghmcnp 23601 | A group homomorphism on to... |
| snclseqg 23602 | The coset of the closure o... |
| tgphaus 23603 | A topological group is Hau... |
| tgpt1 23604 | Hausdorff and T1 are equiv... |
| tgpt0 23605 | Hausdorff and T0 are equiv... |
| qustgpopn 23606 | A quotient map in a topolo... |
| qustgplem 23607 | Lemma for ~ qustgp . (Con... |
| qustgp 23608 | The quotient of a topologi... |
| qustgphaus 23609 | The quotient of a topologi... |
| prdstmdd 23610 | The product of a family of... |
| prdstgpd 23611 | The product of a family of... |
| tsmsfbas 23614 | The collection of all sets... |
| tsmslem1 23615 | The finite partial sums of... |
| tsmsval2 23616 | Definition of the topologi... |
| tsmsval 23617 | Definition of the topologi... |
| tsmspropd 23618 | The group sum depends only... |
| eltsms 23619 | The property of being a su... |
| tsmsi 23620 | The property of being a su... |
| tsmscl 23621 | A sum in a topological gro... |
| haustsms 23622 | In a Hausdorff topological... |
| haustsms2 23623 | In a Hausdorff topological... |
| tsmscls 23624 | One half of ~ tgptsmscls ,... |
| tsmsgsum 23625 | The convergent points of a... |
| tsmsid 23626 | If a sum is finite, the us... |
| haustsmsid 23627 | In a Hausdorff topological... |
| tsms0 23628 | The sum of zero is zero. ... |
| tsmssubm 23629 | Evaluate an infinite group... |
| tsmsres 23630 | Extend an infinite group s... |
| tsmsf1o 23631 | Re-index an infinite group... |
| tsmsmhm 23632 | Apply a continuous group h... |
| tsmsadd 23633 | The sum of two infinite gr... |
| tsmsinv 23634 | Inverse of an infinite gro... |
| tsmssub 23635 | The difference of two infi... |
| tgptsmscls 23636 | A sum in a topological gro... |
| tgptsmscld 23637 | The set of limit points to... |
| tsmssplit 23638 | Split a topological group ... |
| tsmsxplem1 23639 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 23640 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 23641 | Write a sum over a two-dim... |
| istrg 23650 | Express the predicate " ` ... |
| trgtmd 23651 | The multiplicative monoid ... |
| istdrg 23652 | Express the predicate " ` ... |
| tdrgunit 23653 | The unit group of a topolo... |
| trgtgp 23654 | A topological ring is a to... |
| trgtmd2 23655 | A topological ring is a to... |
| trgtps 23656 | A topological ring is a to... |
| trgring 23657 | A topological ring is a ri... |
| trggrp 23658 | A topological ring is a gr... |
| tdrgtrg 23659 | A topological division rin... |
| tdrgdrng 23660 | A topological division rin... |
| tdrgring 23661 | A topological division rin... |
| tdrgtmd 23662 | A topological division rin... |
| tdrgtps 23663 | A topological division rin... |
| istdrg2 23664 | A topological-ring divisio... |
| mulrcn 23665 | The functionalization of t... |
| invrcn2 23666 | The multiplicative inverse... |
| invrcn 23667 | The multiplicative inverse... |
| cnmpt1mulr 23668 | Continuity of ring multipl... |
| cnmpt2mulr 23669 | Continuity of ring multipl... |
| dvrcn 23670 | The division function is c... |
| istlm 23671 | The predicate " ` W ` is a... |
| vscacn 23672 | The scalar multiplication ... |
| tlmtmd 23673 | A topological module is a ... |
| tlmtps 23674 | A topological module is a ... |
| tlmlmod 23675 | A topological module is a ... |
| tlmtrg 23676 | The scalar ring of a topol... |
| tlmscatps 23677 | The scalar ring of a topol... |
| istvc 23678 | A topological vector space... |
| tvctdrg 23679 | The scalar field of a topo... |
| cnmpt1vsca 23680 | Continuity of scalar multi... |
| cnmpt2vsca 23681 | Continuity of scalar multi... |
| tlmtgp 23682 | A topological vector space... |
| tvctlm 23683 | A topological vector space... |
| tvclmod 23684 | A topological vector space... |
| tvclvec 23685 | A topological vector space... |
| ustfn 23688 | The defined uniform struct... |
| ustval 23689 | The class of all uniform s... |
| isust 23690 | The predicate " ` U ` is a... |
| ustssxp 23691 | Entourages are subsets of ... |
| ustssel 23692 | A uniform structure is upw... |
| ustbasel 23693 | The full set is always an ... |
| ustincl 23694 | A uniform structure is clo... |
| ustdiag 23695 | The diagonal set is includ... |
| ustinvel 23696 | If ` V ` is an entourage, ... |
| ustexhalf 23697 | For each entourage ` V ` t... |
| ustrel 23698 | The elements of uniform st... |
| ustfilxp 23699 | A uniform structure on a n... |
| ustne0 23700 | A uniform structure cannot... |
| ustssco 23701 | In an uniform structure, a... |
| ustexsym 23702 | In an uniform structure, f... |
| ustex2sym 23703 | In an uniform structure, f... |
| ustex3sym 23704 | In an uniform structure, f... |
| ustref 23705 | Any element of the base se... |
| ust0 23706 | The unique uniform structu... |
| ustn0 23707 | The empty set is not an un... |
| ustund 23708 | If two intersecting sets `... |
| ustelimasn 23709 | Any point ` A ` is near en... |
| ustneism 23710 | For a point ` A ` in ` X `... |
| elrnustOLD 23711 | Obsolete version of ~ elfv... |
| ustbas2 23712 | Second direction for ~ ust... |
| ustuni 23713 | The set union of a uniform... |
| ustbas 23714 | Recover the base of an uni... |
| ustimasn 23715 | Lemma for ~ ustuqtop . (C... |
| trust 23716 | The trace of a uniform str... |
| utopval 23719 | The topology induced by a ... |
| elutop 23720 | Open sets in the topology ... |
| utoptop 23721 | The topology induced by a ... |
| utopbas 23722 | The base of the topology i... |
| utoptopon 23723 | Topology induced by a unif... |
| restutop 23724 | Restriction of a topology ... |
| restutopopn 23725 | The restriction of the top... |
| ustuqtoplem 23726 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 23727 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 23728 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 23729 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 23730 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 23731 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 23732 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 23733 | For a given uniform struct... |
| utopsnneiplem 23734 | The neighborhoods of a poi... |
| utopsnneip 23735 | The neighborhoods of a poi... |
| utopsnnei 23736 | Images of singletons by en... |
| utop2nei 23737 | For any symmetrical entour... |
| utop3cls 23738 | Relation between a topolog... |
| utopreg 23739 | All Hausdorff uniform spac... |
| ussval 23746 | The uniform structure on u... |
| ussid 23747 | In case the base of the ` ... |
| isusp 23748 | The predicate ` W ` is a u... |
| ressuss 23749 | Value of the uniform struc... |
| ressust 23750 | The uniform structure of a... |
| ressusp 23751 | The restriction of a unifo... |
| tusval 23752 | The value of the uniform s... |
| tuslem 23753 | Lemma for ~ tusbas , ~ tus... |
| tuslemOLD 23754 | Obsolete proof of ~ tuslem... |
| tusbas 23755 | The base set of a construc... |
| tusunif 23756 | The uniform structure of a... |
| tususs 23757 | The uniform structure of a... |
| tustopn 23758 | The topology induced by a ... |
| tususp 23759 | A constructed uniform spac... |
| tustps 23760 | A constructed uniform spac... |
| uspreg 23761 | If a uniform space is Haus... |
| ucnval 23764 | The set of all uniformly c... |
| isucn 23765 | The predicate " ` F ` is a... |
| isucn2 23766 | The predicate " ` F ` is a... |
| ucnimalem 23767 | Reformulate the ` G ` func... |
| ucnima 23768 | An equivalent statement of... |
| ucnprima 23769 | The preimage by a uniforml... |
| iducn 23770 | The identity is uniformly ... |
| cstucnd 23771 | A constant function is uni... |
| ucncn 23772 | Uniform continuity implies... |
| iscfilu 23775 | The predicate " ` F ` is a... |
| cfilufbas 23776 | A Cauchy filter base is a ... |
| cfiluexsm 23777 | For a Cauchy filter base a... |
| fmucndlem 23778 | Lemma for ~ fmucnd . (Con... |
| fmucnd 23779 | The image of a Cauchy filt... |
| cfilufg 23780 | The filter generated by a ... |
| trcfilu 23781 | Condition for the trace of... |
| cfiluweak 23782 | A Cauchy filter base is al... |
| neipcfilu 23783 | In an uniform space, a nei... |
| iscusp 23786 | The predicate " ` W ` is a... |
| cuspusp 23787 | A complete uniform space i... |
| cuspcvg 23788 | In a complete uniform spac... |
| iscusp2 23789 | The predicate " ` W ` is a... |
| cnextucn 23790 | Extension by continuity. ... |
| ucnextcn 23791 | Extension by continuity. ... |
| ispsmet 23792 | Express the predicate " ` ... |
| psmetdmdm 23793 | Recover the base set from ... |
| psmetf 23794 | The distance function of a... |
| psmetcl 23795 | Closure of the distance fu... |
| psmet0 23796 | The distance function of a... |
| psmettri2 23797 | Triangle inequality for th... |
| psmetsym 23798 | The distance function of a... |
| psmettri 23799 | Triangle inequality for th... |
| psmetge0 23800 | The distance function of a... |
| psmetxrge0 23801 | The distance function of a... |
| psmetres2 23802 | Restriction of a pseudomet... |
| psmetlecl 23803 | Real closure of an extende... |
| distspace 23804 | A set ` X ` together with ... |
| ismet 23811 | Express the predicate " ` ... |
| isxmet 23812 | Express the predicate " ` ... |
| ismeti 23813 | Properties that determine ... |
| isxmetd 23814 | Properties that determine ... |
| isxmet2d 23815 | It is safe to only require... |
| metflem 23816 | Lemma for ~ metf and other... |
| xmetf 23817 | Mapping of the distance fu... |
| metf 23818 | Mapping of the distance fu... |
| xmetcl 23819 | Closure of the distance fu... |
| metcl 23820 | Closure of the distance fu... |
| ismet2 23821 | An extended metric is a me... |
| metxmet 23822 | A metric is an extended me... |
| xmetdmdm 23823 | Recover the base set from ... |
| metdmdm 23824 | Recover the base set from ... |
| xmetunirn 23825 | Two ways to express an ext... |
| xmeteq0 23826 | The value of an extended m... |
| meteq0 23827 | The value of a metric is z... |
| xmettri2 23828 | Triangle inequality for th... |
| mettri2 23829 | Triangle inequality for th... |
| xmet0 23830 | The distance function of a... |
| met0 23831 | The distance function of a... |
| xmetge0 23832 | The distance function of a... |
| metge0 23833 | The distance function of a... |
| xmetlecl 23834 | Real closure of an extende... |
| xmetsym 23835 | The distance function of a... |
| xmetpsmet 23836 | An extended metric is a ps... |
| xmettpos 23837 | The distance function of a... |
| metsym 23838 | The distance function of a... |
| xmettri 23839 | Triangle inequality for th... |
| mettri 23840 | Triangle inequality for th... |
| xmettri3 23841 | Triangle inequality for th... |
| mettri3 23842 | Triangle inequality for th... |
| xmetrtri 23843 | One half of the reverse tr... |
| xmetrtri2 23844 | The reverse triangle inequ... |
| metrtri 23845 | Reverse triangle inequalit... |
| xmetgt0 23846 | The distance function of a... |
| metgt0 23847 | The distance function of a... |
| metn0 23848 | A metric space is nonempty... |
| xmetres2 23849 | Restriction of an extended... |
| metreslem 23850 | Lemma for ~ metres . (Con... |
| metres2 23851 | Lemma for ~ metres . (Con... |
| xmetres 23852 | A restriction of an extend... |
| metres 23853 | A restriction of a metric ... |
| 0met 23854 | The empty metric. (Contri... |
| prdsdsf 23855 | The product metric is a fu... |
| prdsxmetlem 23856 | The product metric is an e... |
| prdsxmet 23857 | The product metric is an e... |
| prdsmet 23858 | The product metric is a me... |
| ressprdsds 23859 | Restriction of a product m... |
| resspwsds 23860 | Restriction of a power met... |
| imasdsf1olem 23861 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 23862 | The distance function is t... |
| imasf1oxmet 23863 | The image of an extended m... |
| imasf1omet 23864 | The image of a metric is a... |
| xpsdsfn 23865 | Closure of the metric in a... |
| xpsdsfn2 23866 | Closure of the metric in a... |
| xpsxmetlem 23867 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 23868 | A product metric of extend... |
| xpsdsval 23869 | Value of the metric in a b... |
| xpsmet 23870 | The direct product of two ... |
| blfvalps 23871 | The value of the ball func... |
| blfval 23872 | The value of the ball func... |
| blvalps 23873 | The ball around a point ` ... |
| blval 23874 | The ball around a point ` ... |
| elblps 23875 | Membership in a ball. (Co... |
| elbl 23876 | Membership in a ball. (Co... |
| elbl2ps 23877 | Membership in a ball. (Co... |
| elbl2 23878 | Membership in a ball. (Co... |
| elbl3ps 23879 | Membership in a ball, with... |
| elbl3 23880 | Membership in a ball, with... |
| blcomps 23881 | Commute the arguments to t... |
| blcom 23882 | Commute the arguments to t... |
| xblpnfps 23883 | The infinity ball in an ex... |
| xblpnf 23884 | The infinity ball in an ex... |
| blpnf 23885 | The infinity ball in a sta... |
| bldisj 23886 | Two balls are disjoint if ... |
| blgt0 23887 | A nonempty ball implies th... |
| bl2in 23888 | Two balls are disjoint if ... |
| xblss2ps 23889 | One ball is contained in a... |
| xblss2 23890 | One ball is contained in a... |
| blss2ps 23891 | One ball is contained in a... |
| blss2 23892 | One ball is contained in a... |
| blhalf 23893 | A ball of radius ` R / 2 `... |
| blfps 23894 | Mapping of a ball. (Contr... |
| blf 23895 | Mapping of a ball. (Contr... |
| blrnps 23896 | Membership in the range of... |
| blrn 23897 | Membership in the range of... |
| xblcntrps 23898 | A ball contains its center... |
| xblcntr 23899 | A ball contains its center... |
| blcntrps 23900 | A ball contains its center... |
| blcntr 23901 | A ball contains its center... |
| xbln0 23902 | A ball is nonempty iff the... |
| bln0 23903 | A ball is not empty. (Con... |
| blelrnps 23904 | A ball belongs to the set ... |
| blelrn 23905 | A ball belongs to the set ... |
| blssm 23906 | A ball is a subset of the ... |
| unirnblps 23907 | The union of the set of ba... |
| unirnbl 23908 | The union of the set of ba... |
| blin 23909 | The intersection of two ba... |
| ssblps 23910 | The size of a ball increas... |
| ssbl 23911 | The size of a ball increas... |
| blssps 23912 | Any point ` P ` in a ball ... |
| blss 23913 | Any point ` P ` in a ball ... |
| blssexps 23914 | Two ways to express the ex... |
| blssex 23915 | Two ways to express the ex... |
| ssblex 23916 | A nested ball exists whose... |
| blin2 23917 | Given any two balls and a ... |
| blbas 23918 | The balls of a metric spac... |
| blres 23919 | A ball in a restricted met... |
| xmeterval 23920 | Value of the "finitely sep... |
| xmeter 23921 | The "finitely separated" r... |
| xmetec 23922 | The equivalence classes un... |
| blssec 23923 | A ball centered at ` P ` i... |
| blpnfctr 23924 | The infinity ball in an ex... |
| xmetresbl 23925 | An extended metric restric... |
| mopnval 23926 | An open set is a subset of... |
| mopntopon 23927 | The set of open sets of a ... |
| mopntop 23928 | The set of open sets of a ... |
| mopnuni 23929 | The union of all open sets... |
| elmopn 23930 | The defining property of a... |
| mopnfss 23931 | The family of open sets of... |
| mopnm 23932 | The base set of a metric s... |
| elmopn2 23933 | A defining property of an ... |
| mopnss 23934 | An open set of a metric sp... |
| isxms 23935 | Express the predicate " ` ... |
| isxms2 23936 | Express the predicate " ` ... |
| isms 23937 | Express the predicate " ` ... |
| isms2 23938 | Express the predicate " ` ... |
| xmstopn 23939 | The topology component of ... |
| mstopn 23940 | The topology component of ... |
| xmstps 23941 | An extended metric space i... |
| msxms 23942 | A metric space is an exten... |
| mstps 23943 | A metric space is a topolo... |
| xmsxmet 23944 | The distance function, sui... |
| msmet 23945 | The distance function, sui... |
| msf 23946 | The distance function of a... |
| xmsxmet2 23947 | The distance function, sui... |
| msmet2 23948 | The distance function, sui... |
| mscl 23949 | Closure of the distance fu... |
| xmscl 23950 | Closure of the distance fu... |
| xmsge0 23951 | The distance function in a... |
| xmseq0 23952 | The distance between two p... |
| xmssym 23953 | The distance function in a... |
| xmstri2 23954 | Triangle inequality for th... |
| mstri2 23955 | Triangle inequality for th... |
| xmstri 23956 | Triangle inequality for th... |
| mstri 23957 | Triangle inequality for th... |
| xmstri3 23958 | Triangle inequality for th... |
| mstri3 23959 | Triangle inequality for th... |
| msrtri 23960 | Reverse triangle inequalit... |
| xmspropd 23961 | Property deduction for an ... |
| mspropd 23962 | Property deduction for a m... |
| setsmsbas 23963 | The base set of a construc... |
| setsmsbasOLD 23964 | Obsolete proof of ~ setsms... |
| setsmsds 23965 | The distance function of a... |
| setsmsdsOLD 23966 | Obsolete proof of ~ setsms... |
| setsmstset 23967 | The topology of a construc... |
| setsmstopn 23968 | The topology of a construc... |
| setsxms 23969 | The constructed metric spa... |
| setsms 23970 | The constructed metric spa... |
| tmsval 23971 | For any metric there is an... |
| tmslem 23972 | Lemma for ~ tmsbas , ~ tms... |
| tmslemOLD 23973 | Obsolete version of ~ tmsl... |
| tmsbas 23974 | The base set of a construc... |
| tmsds 23975 | The metric of a constructe... |
| tmstopn 23976 | The topology of a construc... |
| tmsxms 23977 | The constructed metric spa... |
| tmsms 23978 | The constructed metric spa... |
| imasf1obl 23979 | The image of a metric spac... |
| imasf1oxms 23980 | The image of a metric spac... |
| imasf1oms 23981 | The image of a metric spac... |
| prdsbl 23982 | A ball in the product metr... |
| mopni 23983 | An open set of a metric sp... |
| mopni2 23984 | An open set of a metric sp... |
| mopni3 23985 | An open set of a metric sp... |
| blssopn 23986 | The balls of a metric spac... |
| unimopn 23987 | The union of a collection ... |
| mopnin 23988 | The intersection of two op... |
| mopn0 23989 | The empty set is an open s... |
| rnblopn 23990 | A ball of a metric space i... |
| blopn 23991 | A ball of a metric space i... |
| neibl 23992 | The neighborhoods around a... |
| blnei 23993 | A ball around a point is a... |
| lpbl 23994 | Every ball around a limit ... |
| blsscls2 23995 | A smaller closed ball is c... |
| blcld 23996 | A "closed ball" in a metri... |
| blcls 23997 | The closure of an open bal... |
| blsscls 23998 | If two concentric balls ha... |
| metss 23999 | Two ways of saying that me... |
| metequiv 24000 | Two ways of saying that tw... |
| metequiv2 24001 | If there is a sequence of ... |
| metss2lem 24002 | Lemma for ~ metss2 . (Con... |
| metss2 24003 | If the metric ` D ` is "st... |
| comet 24004 | The composition of an exte... |
| stdbdmetval 24005 | Value of the standard boun... |
| stdbdxmet 24006 | The standard bounded metri... |
| stdbdmet 24007 | The standard bounded metri... |
| stdbdbl 24008 | The standard bounded metri... |
| stdbdmopn 24009 | The standard bounded metri... |
| mopnex 24010 | The topology generated by ... |
| methaus 24011 | The topology generated by ... |
| met1stc 24012 | The topology generated by ... |
| met2ndci 24013 | A separable metric space (... |
| met2ndc 24014 | A metric space is second-c... |
| metrest 24015 | Two alternate formulations... |
| ressxms 24016 | The restriction of a metri... |
| ressms 24017 | The restriction of a metri... |
| prdsmslem1 24018 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24019 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24020 | Lemma for ~ prdsxms . The... |
| prdsxms 24021 | The indexed product struct... |
| prdsms 24022 | The indexed product struct... |
| pwsxms 24023 | A power of an extended met... |
| pwsms 24024 | A power of a metric space ... |
| xpsxms 24025 | A binary product of metric... |
| xpsms 24026 | A binary product of metric... |
| tmsxps 24027 | Express the product of two... |
| tmsxpsmopn 24028 | Express the product of two... |
| tmsxpsval 24029 | Value of the product of tw... |
| tmsxpsval2 24030 | Value of the product of tw... |
| metcnp3 24031 | Two ways to express that `... |
| metcnp 24032 | Two ways to say a mapping ... |
| metcnp2 24033 | Two ways to say a mapping ... |
| metcn 24034 | Two ways to say a mapping ... |
| metcnpi 24035 | Epsilon-delta property of ... |
| metcnpi2 24036 | Epsilon-delta property of ... |
| metcnpi3 24037 | Epsilon-delta property of ... |
| txmetcnp 24038 | Continuity of a binary ope... |
| txmetcn 24039 | Continuity of a binary ope... |
| metuval 24040 | Value of the uniform struc... |
| metustel 24041 | Define a filter base ` F `... |
| metustss 24042 | Range of the elements of t... |
| metustrel 24043 | Elements of the filter bas... |
| metustto 24044 | Any two elements of the fi... |
| metustid 24045 | The identity diagonal is i... |
| metustsym 24046 | Elements of the filter bas... |
| metustexhalf 24047 | For any element ` A ` of t... |
| metustfbas 24048 | The filter base generated ... |
| metust 24049 | The uniform structure gene... |
| cfilucfil 24050 | Given a metric ` D ` and a... |
| metuust 24051 | The uniform structure gene... |
| cfilucfil2 24052 | Given a metric ` D ` and a... |
| blval2 24053 | The ball around a point ` ... |
| elbl4 24054 | Membership in a ball, alte... |
| metuel 24055 | Elementhood in the uniform... |
| metuel2 24056 | Elementhood in the uniform... |
| metustbl 24057 | The "section" image of an ... |
| psmetutop 24058 | The topology induced by a ... |
| xmetutop 24059 | The topology induced by a ... |
| xmsusp 24060 | If the uniform set of a me... |
| restmetu 24061 | The uniform structure gene... |
| metucn 24062 | Uniform continuity in metr... |
| dscmet 24063 | The discrete metric on any... |
| dscopn 24064 | The discrete metric genera... |
| nrmmetd 24065 | Show that a group norm gen... |
| abvmet 24066 | An absolute value ` F ` ge... |
| nmfval 24079 | The value of the norm func... |
| nmval 24080 | The value of the norm as t... |
| nmfval0 24081 | The value of the norm func... |
| nmfval2 24082 | The value of the norm func... |
| nmval2 24083 | The value of the norm on a... |
| nmf2 24084 | The norm on a metric group... |
| nmpropd 24085 | Weak property deduction fo... |
| nmpropd2 24086 | Strong property deduction ... |
| isngp 24087 | The property of being a no... |
| isngp2 24088 | The property of being a no... |
| isngp3 24089 | The property of being a no... |
| ngpgrp 24090 | A normed group is a group.... |
| ngpms 24091 | A normed group is a metric... |
| ngpxms 24092 | A normed group is an exten... |
| ngptps 24093 | A normed group is a topolo... |
| ngpmet 24094 | The (induced) metric of a ... |
| ngpds 24095 | Value of the distance func... |
| ngpdsr 24096 | Value of the distance func... |
| ngpds2 24097 | Write the distance between... |
| ngpds2r 24098 | Write the distance between... |
| ngpds3 24099 | Write the distance between... |
| ngpds3r 24100 | Write the distance between... |
| ngprcan 24101 | Cancel right addition insi... |
| ngplcan 24102 | Cancel left addition insid... |
| isngp4 24103 | Express the property of be... |
| ngpinvds 24104 | Two elements are the same ... |
| ngpsubcan 24105 | Cancel right subtraction i... |
| nmf 24106 | The norm on a normed group... |
| nmcl 24107 | The norm of a normed group... |
| nmge0 24108 | The norm of a normed group... |
| nmeq0 24109 | The identity is the only e... |
| nmne0 24110 | The norm of a nonzero elem... |
| nmrpcl 24111 | The norm of a nonzero elem... |
| nminv 24112 | The norm of a negated elem... |
| nmmtri 24113 | The triangle inequality fo... |
| nmsub 24114 | The norm of the difference... |
| nmrtri 24115 | Reverse triangle inequalit... |
| nm2dif 24116 | Inequality for the differe... |
| nmtri 24117 | The triangle inequality fo... |
| nmtri2 24118 | Triangle inequality for th... |
| ngpi 24119 | The properties of a normed... |
| nm0 24120 | Norm of the identity eleme... |
| nmgt0 24121 | The norm of a nonzero elem... |
| sgrim 24122 | The induced metric on a su... |
| sgrimval 24123 | The induced metric on a su... |
| subgnm 24124 | The norm in a subgroup. (... |
| subgnm2 24125 | A substructure assigns the... |
| subgngp 24126 | A normed group restricted ... |
| ngptgp 24127 | A normed abelian group is ... |
| ngppropd 24128 | Property deduction for a n... |
| reldmtng 24129 | The function ` toNrmGrp ` ... |
| tngval 24130 | Value of the function whic... |
| tnglem 24131 | Lemma for ~ tngbas and sim... |
| tnglemOLD 24132 | Obsolete version of ~ tngl... |
| tngbas 24133 | The base set of a structur... |
| tngbasOLD 24134 | Obsolete proof of ~ tngbas... |
| tngplusg 24135 | The group addition of a st... |
| tngplusgOLD 24136 | Obsolete proof of ~ tngplu... |
| tng0 24137 | The group identity of a st... |
| tngmulr 24138 | The ring multiplication of... |
| tngmulrOLD 24139 | Obsolete proof of ~ tngmul... |
| tngsca 24140 | The scalar ring of a struc... |
| tngscaOLD 24141 | Obsolete proof of ~ tngsca... |
| tngvsca 24142 | The scalar multiplication ... |
| tngvscaOLD 24143 | Obsolete proof of ~ tngvsc... |
| tngip 24144 | The inner product operatio... |
| tngipOLD 24145 | Obsolete proof of ~ tngip ... |
| tngds 24146 | The metric function of a s... |
| tngdsOLD 24147 | Obsolete proof of ~ tngds ... |
| tngtset 24148 | The topology generated by ... |
| tngtopn 24149 | The topology generated by ... |
| tngnm 24150 | The topology generated by ... |
| tngngp2 24151 | A norm turns a group into ... |
| tngngpd 24152 | Derive the axioms for a no... |
| tngngp 24153 | Derive the axioms for a no... |
| tnggrpr 24154 | If a structure equipped wi... |
| tngngp3 24155 | Alternate definition of a ... |
| nrmtngdist 24156 | The augmentation of a norm... |
| nrmtngnrm 24157 | The augmentation of a norm... |
| tngngpim 24158 | The induced metric of a no... |
| isnrg 24159 | A normed ring is a ring wi... |
| nrgabv 24160 | The norm of a normed ring ... |
| nrgngp 24161 | A normed ring is a normed ... |
| nrgring 24162 | A normed ring is a ring. ... |
| nmmul 24163 | The norm of a product in a... |
| nrgdsdi 24164 | Distribute a distance calc... |
| nrgdsdir 24165 | Distribute a distance calc... |
| nm1 24166 | The norm of one in a nonze... |
| unitnmn0 24167 | The norm of a unit is nonz... |
| nminvr 24168 | The norm of an inverse in ... |
| nmdvr 24169 | The norm of a division in ... |
| nrgdomn 24170 | A nonzero normed ring is a... |
| nrgtgp 24171 | A normed ring is a topolog... |
| subrgnrg 24172 | A normed ring restricted t... |
| tngnrg 24173 | Given any absolute value o... |
| isnlm 24174 | A normed (left) module is ... |
| nmvs 24175 | Defining property of a nor... |
| nlmngp 24176 | A normed module is a norme... |
| nlmlmod 24177 | A normed module is a left ... |
| nlmnrg 24178 | The scalar component of a ... |
| nlmngp2 24179 | The scalar component of a ... |
| nlmdsdi 24180 | Distribute a distance calc... |
| nlmdsdir 24181 | Distribute a distance calc... |
| nlmmul0or 24182 | If a scalar product is zer... |
| sranlm 24183 | The subring algebra over a... |
| nlmvscnlem2 24184 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24185 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24186 | The scalar multiplication ... |
| rlmnlm 24187 | The ring module over a nor... |
| rlmnm 24188 | The norm function in the r... |
| nrgtrg 24189 | A normed ring is a topolog... |
| nrginvrcnlem 24190 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24191 | The ring inverse function ... |
| nrgtdrg 24192 | A normed division ring is ... |
| nlmtlm 24193 | A normed module is a topol... |
| isnvc 24194 | A normed vector space is j... |
| nvcnlm 24195 | A normed vector space is a... |
| nvclvec 24196 | A normed vector space is a... |
| nvclmod 24197 | A normed vector space is a... |
| isnvc2 24198 | A normed vector space is j... |
| nvctvc 24199 | A normed vector space is a... |
| lssnlm 24200 | A subspace of a normed mod... |
| lssnvc 24201 | A subspace of a normed vec... |
| rlmnvc 24202 | The ring module over a nor... |
| ngpocelbl 24203 | Membership of an off-cente... |
| nmoffn 24210 | The function producing ope... |
| reldmnghm 24211 | Lemma for normed group hom... |
| reldmnmhm 24212 | Lemma for module homomorph... |
| nmofval 24213 | Value of the operator norm... |
| nmoval 24214 | Value of the operator norm... |
| nmogelb 24215 | Property of the operator n... |
| nmolb 24216 | Any upper bound on the val... |
| nmolb2d 24217 | Any upper bound on the val... |
| nmof 24218 | The operator norm is a fun... |
| nmocl 24219 | The operator norm of an op... |
| nmoge0 24220 | The operator norm of an op... |
| nghmfval 24221 | A normed group homomorphis... |
| isnghm 24222 | A normed group homomorphis... |
| isnghm2 24223 | A normed group homomorphis... |
| isnghm3 24224 | A normed group homomorphis... |
| bddnghm 24225 | A bounded group homomorphi... |
| nghmcl 24226 | A normed group homomorphis... |
| nmoi 24227 | The operator norm achieves... |
| nmoix 24228 | The operator norm is a bou... |
| nmoi2 24229 | The operator norm is a bou... |
| nmoleub 24230 | The operator norm, defined... |
| nghmrcl1 24231 | Reverse closure for a norm... |
| nghmrcl2 24232 | Reverse closure for a norm... |
| nghmghm 24233 | A normed group homomorphis... |
| nmo0 24234 | The operator norm of the z... |
| nmoeq0 24235 | The operator norm is zero ... |
| nmoco 24236 | An upper bound on the oper... |
| nghmco 24237 | The composition of normed ... |
| nmotri 24238 | Triangle inequality for th... |
| nghmplusg 24239 | The sum of two bounded lin... |
| 0nghm 24240 | The zero operator is a nor... |
| nmoid 24241 | The operator norm of the i... |
| idnghm 24242 | The identity operator is a... |
| nmods 24243 | Upper bound for the distan... |
| nghmcn 24244 | A normed group homomorphis... |
| isnmhm 24245 | A normed module homomorphi... |
| nmhmrcl1 24246 | Reverse closure for a norm... |
| nmhmrcl2 24247 | Reverse closure for a norm... |
| nmhmlmhm 24248 | A normed module homomorphi... |
| nmhmnghm 24249 | A normed module homomorphi... |
| nmhmghm 24250 | A normed module homomorphi... |
| isnmhm2 24251 | A normed module homomorphi... |
| nmhmcl 24252 | A normed module homomorphi... |
| idnmhm 24253 | The identity operator is a... |
| 0nmhm 24254 | The zero operator is a bou... |
| nmhmco 24255 | The composition of bounded... |
| nmhmplusg 24256 | The sum of two bounded lin... |
| qtopbaslem 24257 | The set of open intervals ... |
| qtopbas 24258 | The set of open intervals ... |
| retopbas 24259 | A basis for the standard t... |
| retop 24260 | The standard topology on t... |
| uniretop 24261 | The underlying set of the ... |
| retopon 24262 | The standard topology on t... |
| retps 24263 | The standard topological s... |
| iooretop 24264 | Open intervals are open se... |
| icccld 24265 | Closed intervals are close... |
| icopnfcld 24266 | Right-unbounded closed int... |
| iocmnfcld 24267 | Left-unbounded closed inte... |
| qdensere 24268 | ` QQ ` is dense in the sta... |
| cnmetdval 24269 | Value of the distance func... |
| cnmet 24270 | The absolute value metric ... |
| cnxmet 24271 | The absolute value metric ... |
| cnbl0 24272 | Two ways to write the open... |
| cnblcld 24273 | Two ways to write the clos... |
| cnfldms 24274 | The complex number field i... |
| cnfldxms 24275 | The complex number field i... |
| cnfldtps 24276 | The complex number field i... |
| cnfldnm 24277 | The norm of the field of c... |
| cnngp 24278 | The complex numbers form a... |
| cnnrg 24279 | The complex numbers form a... |
| cnfldtopn 24280 | The topology of the comple... |
| cnfldtopon 24281 | The topology of the comple... |
| cnfldtop 24282 | The topology of the comple... |
| cnfldhaus 24283 | The topology of the comple... |
| unicntop 24284 | The underlying set of the ... |
| cnopn 24285 | The set of complex numbers... |
| zringnrg 24286 | The ring of integers is a ... |
| remetdval 24287 | Value of the distance func... |
| remet 24288 | The absolute value metric ... |
| rexmet 24289 | The absolute value metric ... |
| bl2ioo 24290 | A ball in terms of an open... |
| ioo2bl 24291 | An open interval of reals ... |
| ioo2blex 24292 | An open interval of reals ... |
| blssioo 24293 | The balls of the standard ... |
| tgioo 24294 | The topology generated by ... |
| qdensere2 24295 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24296 | An open ball in the comple... |
| rehaus 24297 | The standard topology on t... |
| tgqioo 24298 | The topology generated by ... |
| re2ndc 24299 | The standard topology on t... |
| resubmet 24300 | The subspace topology indu... |
| tgioo2 24301 | The standard topology on t... |
| rerest 24302 | The subspace topology indu... |
| tgioo3 24303 | The standard topology on t... |
| xrtgioo 24304 | The topology on the extend... |
| xrrest 24305 | The subspace topology indu... |
| xrrest2 24306 | The subspace topology indu... |
| xrsxmet 24307 | The metric on the extended... |
| xrsdsre 24308 | The metric on the extended... |
| xrsblre 24309 | Any ball of the metric of ... |
| xrsmopn 24310 | The metric on the extended... |
| zcld 24311 | The integers are a closed ... |
| recld2 24312 | The real numbers are a clo... |
| zcld2 24313 | The integers are a closed ... |
| zdis 24314 | The integers are a discret... |
| sszcld 24315 | Every subset of the intege... |
| reperflem 24316 | A subset of the real numbe... |
| reperf 24317 | The real numbers are a per... |
| cnperf 24318 | The complex numbers are a ... |
| iccntr 24319 | The interior of a closed i... |
| icccmplem1 24320 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24321 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24322 | Lemma for ~ icccmp . (Con... |
| icccmp 24323 | A closed interval in ` RR ... |
| reconnlem1 24324 | Lemma for ~ reconn . Conn... |
| reconnlem2 24325 | Lemma for ~ reconn . (Con... |
| reconn 24326 | A subset of the reals is c... |
| retopconn 24327 | Corollary of ~ reconn . T... |
| iccconn 24328 | A closed interval is conne... |
| opnreen 24329 | Every nonempty open set is... |
| rectbntr0 24330 | A countable subset of the ... |
| xrge0gsumle 24331 | A finite sum in the nonneg... |
| xrge0tsms 24332 | Any finite or infinite sum... |
| xrge0tsms2 24333 | Any finite or infinite sum... |
| metdcnlem 24334 | The metric function of a m... |
| xmetdcn2 24335 | The metric function of an ... |
| xmetdcn 24336 | The metric function of an ... |
| metdcn2 24337 | The metric function of a m... |
| metdcn 24338 | The metric function of a m... |
| msdcn 24339 | The metric function of a m... |
| cnmpt1ds 24340 | Continuity of the metric f... |
| cnmpt2ds 24341 | Continuity of the metric f... |
| nmcn 24342 | The norm of a normed group... |
| ngnmcncn 24343 | The norm of a normed group... |
| abscn 24344 | The absolute value functio... |
| metdsval 24345 | Value of the "distance to ... |
| metdsf 24346 | The distance from a point ... |
| metdsge 24347 | The distance from the poin... |
| metds0 24348 | If a point is in a set, it... |
| metdstri 24349 | A generalization of the tr... |
| metdsle 24350 | The distance from a point ... |
| metdsre 24351 | The distance from a point ... |
| metdseq0 24352 | The distance from a point ... |
| metdscnlem 24353 | Lemma for ~ metdscn . (Co... |
| metdscn 24354 | The function ` F ` which g... |
| metdscn2 24355 | The function ` F ` which g... |
| metnrmlem1a 24356 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24357 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24358 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24359 | Lemma for ~ metnrm . (Con... |
| metnrm 24360 | A metric space is normal. ... |
| metreg 24361 | A metric space is regular.... |
| addcnlem 24362 | Lemma for ~ addcn , ~ subc... |
| addcn 24363 | Complex number addition is... |
| subcn 24364 | Complex number subtraction... |
| mulcn 24365 | Complex number multiplicat... |
| divcn 24366 | Complex number division is... |
| cnfldtgp 24367 | The complex numbers form a... |
| fsumcn 24368 | A finite sum of functions ... |
| fsum2cn 24369 | Version of ~ fsumcn for tw... |
| expcn 24370 | The power function on comp... |
| divccn 24371 | Division by a nonzero cons... |
| sqcn 24372 | The square function on com... |
| iitopon 24377 | The unit interval is a top... |
| iitop 24378 | The unit interval is a top... |
| iiuni 24379 | The base set of the unit i... |
| dfii2 24380 | Alternate definition of th... |
| dfii3 24381 | Alternate definition of th... |
| dfii4 24382 | Alternate definition of th... |
| dfii5 24383 | The unit interval expresse... |
| iicmp 24384 | The unit interval is compa... |
| iiconn 24385 | The unit interval is conne... |
| cncfval 24386 | The value of the continuou... |
| elcncf 24387 | Membership in the set of c... |
| elcncf2 24388 | Version of ~ elcncf with a... |
| cncfrss 24389 | Reverse closure of the con... |
| cncfrss2 24390 | Reverse closure of the con... |
| cncff 24391 | A continuous complex funct... |
| cncfi 24392 | Defining property of a con... |
| elcncf1di 24393 | Membership in the set of c... |
| elcncf1ii 24394 | Membership in the set of c... |
| rescncf 24395 | A continuous complex funct... |
| cncfcdm 24396 | Change the codomain of a c... |
| cncfss 24397 | The set of continuous func... |
| climcncf 24398 | Image of a limit under a c... |
| abscncf 24399 | Absolute value is continuo... |
| recncf 24400 | Real part is continuous. ... |
| imcncf 24401 | Imaginary part is continuo... |
| cjcncf 24402 | Complex conjugate is conti... |
| mulc1cncf 24403 | Multiplication by a consta... |
| divccncf 24404 | Division by a constant is ... |
| cncfco 24405 | The composition of two con... |
| cncfcompt2 24406 | Composition of continuous ... |
| cncfmet 24407 | Relate complex function co... |
| cncfcn 24408 | Relate complex function co... |
| cncfcn1 24409 | Relate complex function co... |
| cncfmptc 24410 | A constant function is a c... |
| cncfmptid 24411 | The identity function is a... |
| cncfmpt1f 24412 | Composition of continuous ... |
| cncfmpt2f 24413 | Composition of continuous ... |
| cncfmpt2ss 24414 | Composition of continuous ... |
| addccncf 24415 | Adding a constant is a con... |
| idcncf 24416 | The identity function is a... |
| sub1cncf 24417 | Subtracting a constant is ... |
| sub2cncf 24418 | Subtraction from a constan... |
| cdivcncf 24419 | Division with a constant n... |
| negcncf 24420 | The negative function is c... |
| negfcncf 24421 | The negative of a continuo... |
| abscncfALT 24422 | Absolute value is continuo... |
| cncfcnvcn 24423 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24424 | The power function on comp... |
| cnmptre 24425 | Lemma for ~ iirevcn and re... |
| cnmpopc 24426 | Piecewise definition of a ... |
| iirev 24427 | Reverse the unit interval.... |
| iirevcn 24428 | The reversion function is ... |
| iihalf1 24429 | Map the first half of ` II... |
| iihalf1cn 24430 | The first half function is... |
| iihalf2 24431 | Map the second half of ` I... |
| iihalf2cn 24432 | The second half function i... |
| elii1 24433 | Divide the unit interval i... |
| elii2 24434 | Divide the unit interval i... |
| iimulcl 24435 | The unit interval is close... |
| iimulcn 24436 | Multiplication is a contin... |
| icoopnst 24437 | A half-open interval start... |
| iocopnst 24438 | A half-open interval endin... |
| icchmeo 24439 | The natural bijection from... |
| icopnfcnv 24440 | Define a bijection from ` ... |
| icopnfhmeo 24441 | The defined bijection from... |
| iccpnfcnv 24442 | Define a bijection from ` ... |
| iccpnfhmeo 24443 | The defined bijection from... |
| xrhmeo 24444 | The bijection from ` [ -u ... |
| xrhmph 24445 | The extended reals are hom... |
| xrcmp 24446 | The topology of the extend... |
| xrconn 24447 | The topology of the extend... |
| icccvx 24448 | A linear combination of tw... |
| oprpiece1res1 24449 | Restriction to the first p... |
| oprpiece1res2 24450 | Restriction to the second ... |
| cnrehmeo 24451 | The canonical bijection fr... |
| cnheiborlem 24452 | Lemma for ~ cnheibor . (C... |
| cnheibor 24453 | Heine-Borel theorem for co... |
| cnllycmp 24454 | The topology on the comple... |
| rellycmp 24455 | The topology on the reals ... |
| bndth 24456 | The Boundedness Theorem. ... |
| evth 24457 | The Extreme Value Theorem.... |
| evth2 24458 | The Extreme Value Theorem,... |
| lebnumlem1 24459 | Lemma for ~ lebnum . The ... |
| lebnumlem2 24460 | Lemma for ~ lebnum . As a... |
| lebnumlem3 24461 | Lemma for ~ lebnum . By t... |
| lebnum 24462 | The Lebesgue number lemma,... |
| xlebnum 24463 | Generalize ~ lebnum to ext... |
| lebnumii 24464 | Specialize the Lebesgue nu... |
| ishtpy 24470 | Membership in the class of... |
| htpycn 24471 | A homotopy is a continuous... |
| htpyi 24472 | A homotopy evaluated at it... |
| ishtpyd 24473 | Deduction for membership i... |
| htpycom 24474 | Given a homotopy from ` F ... |
| htpyid 24475 | A homotopy from a function... |
| htpyco1 24476 | Compose a homotopy with a ... |
| htpyco2 24477 | Compose a homotopy with a ... |
| htpycc 24478 | Concatenate two homotopies... |
| isphtpy 24479 | Membership in the class of... |
| phtpyhtpy 24480 | A path homotopy is a homot... |
| phtpycn 24481 | A path homotopy is a conti... |
| phtpyi 24482 | Membership in the class of... |
| phtpy01 24483 | Two path-homotopic paths h... |
| isphtpyd 24484 | Deduction for membership i... |
| isphtpy2d 24485 | Deduction for membership i... |
| phtpycom 24486 | Given a homotopy from ` F ... |
| phtpyid 24487 | A homotopy from a path to ... |
| phtpyco2 24488 | Compose a path homotopy wi... |
| phtpycc 24489 | Concatenate two path homot... |
| phtpcrel 24491 | The path homotopy relation... |
| isphtpc 24492 | The relation "is path homo... |
| phtpcer 24493 | Path homotopy is an equiva... |
| phtpc01 24494 | Path homotopic paths have ... |
| reparphti 24495 | Lemma for ~ reparpht . (C... |
| reparpht 24496 | Reparametrization lemma. ... |
| phtpcco2 24497 | Compose a path homotopy wi... |
| pcofval 24508 | The value of the path conc... |
| pcoval 24509 | The concatenation of two p... |
| pcovalg 24510 | Evaluate the concatenation... |
| pcoval1 24511 | Evaluate the concatenation... |
| pco0 24512 | The starting point of a pa... |
| pco1 24513 | The ending point of a path... |
| pcoval2 24514 | Evaluate the concatenation... |
| pcocn 24515 | The concatenation of two p... |
| copco 24516 | The composition of a conca... |
| pcohtpylem 24517 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 24518 | Homotopy invariance of pat... |
| pcoptcl 24519 | A constant function is a p... |
| pcopt 24520 | Concatenation with a point... |
| pcopt2 24521 | Concatenation with a point... |
| pcoass 24522 | Order of concatenation doe... |
| pcorevcl 24523 | Closure for a reversed pat... |
| pcorevlem 24524 | Lemma for ~ pcorev . Prov... |
| pcorev 24525 | Concatenation with the rev... |
| pcorev2 24526 | Concatenation with the rev... |
| pcophtb 24527 | The path homotopy equivale... |
| om1val 24528 | The definition of the loop... |
| om1bas 24529 | The base set of the loop s... |
| om1elbas 24530 | Elementhood in the base se... |
| om1addcl 24531 | Closure of the group opera... |
| om1plusg 24532 | The group operation (which... |
| om1tset 24533 | The topology of the loop s... |
| om1opn 24534 | The topology of the loop s... |
| pi1val 24535 | The definition of the fund... |
| pi1bas 24536 | The base set of the fundam... |
| pi1blem 24537 | Lemma for ~ pi1buni . (Co... |
| pi1buni 24538 | Another way to write the l... |
| pi1bas2 24539 | The base set of the fundam... |
| pi1eluni 24540 | Elementhood in the base se... |
| pi1bas3 24541 | The base set of the fundam... |
| pi1cpbl 24542 | The group operation, loop ... |
| elpi1 24543 | The elements of the fundam... |
| elpi1i 24544 | The elements of the fundam... |
| pi1addf 24545 | The group operation of ` p... |
| pi1addval 24546 | The concatenation of two p... |
| pi1grplem 24547 | Lemma for ~ pi1grp . (Con... |
| pi1grp 24548 | The fundamental group is a... |
| pi1id 24549 | The identity element of th... |
| pi1inv 24550 | An inverse in the fundamen... |
| pi1xfrf 24551 | Functionality of the loop ... |
| pi1xfrval 24552 | The value of the loop tran... |
| pi1xfr 24553 | Given a path ` F ` and its... |
| pi1xfrcnvlem 24554 | Given a path ` F ` between... |
| pi1xfrcnv 24555 | Given a path ` F ` between... |
| pi1xfrgim 24556 | The mapping ` G ` between ... |
| pi1cof 24557 | Functionality of the loop ... |
| pi1coval 24558 | The value of the loop tran... |
| pi1coghm 24559 | The mapping ` G ` between ... |
| isclm 24562 | A subcomplex module is a l... |
| clmsca 24563 | The ring of scalars ` F ` ... |
| clmsubrg 24564 | The base set of the ring o... |
| clmlmod 24565 | A subcomplex module is a l... |
| clmgrp 24566 | A subcomplex module is an ... |
| clmabl 24567 | A subcomplex module is an ... |
| clmring 24568 | The scalar ring of a subco... |
| clmfgrp 24569 | The scalar ring of a subco... |
| clm0 24570 | The zero of the scalar rin... |
| clm1 24571 | The identity of the scalar... |
| clmadd 24572 | The addition of the scalar... |
| clmmul 24573 | The multiplication of the ... |
| clmcj 24574 | The conjugation of the sca... |
| isclmi 24575 | Reverse direction of ~ isc... |
| clmzss 24576 | The scalar ring of a subco... |
| clmsscn 24577 | The scalar ring of a subco... |
| clmsub 24578 | Subtraction in the scalar ... |
| clmneg 24579 | Negation in the scalar rin... |
| clmneg1 24580 | Minus one is in the scalar... |
| clmabs 24581 | Norm in the scalar ring of... |
| clmacl 24582 | Closure of ring addition f... |
| clmmcl 24583 | Closure of ring multiplica... |
| clmsubcl 24584 | Closure of ring subtractio... |
| lmhmclm 24585 | The domain of a linear ope... |
| clmvscl 24586 | Closure of scalar product ... |
| clmvsass 24587 | Associative law for scalar... |
| clmvscom 24588 | Commutative law for the sc... |
| clmvsdir 24589 | Distributive law for scala... |
| clmvsdi 24590 | Distributive law for scala... |
| clmvs1 24591 | Scalar product with ring u... |
| clmvs2 24592 | A vector plus itself is tw... |
| clm0vs 24593 | Zero times a vector is the... |
| clmopfne 24594 | The (functionalized) opera... |
| isclmp 24595 | The predicate "is a subcom... |
| isclmi0 24596 | Properties that determine ... |
| clmvneg1 24597 | Minus 1 times a vector is ... |
| clmvsneg 24598 | Multiplication of a vector... |
| clmmulg 24599 | The group multiple functio... |
| clmsubdir 24600 | Scalar multiplication dist... |
| clmpm1dir 24601 | Subtractive distributive l... |
| clmnegneg 24602 | Double negative of a vecto... |
| clmnegsubdi2 24603 | Distribution of negative o... |
| clmsub4 24604 | Rearrangement of 4 terms i... |
| clmvsrinv 24605 | A vector minus itself. (C... |
| clmvslinv 24606 | Minus a vector plus itself... |
| clmvsubval 24607 | Value of vector subtractio... |
| clmvsubval2 24608 | Value of vector subtractio... |
| clmvz 24609 | Two ways to express the ne... |
| zlmclm 24610 | The ` ZZ ` -module operati... |
| clmzlmvsca 24611 | The scalar product of a su... |
| nmoleub2lem 24612 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 24613 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 24614 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 24615 | The operator norm is the s... |
| nmoleub2b 24616 | The operator norm is the s... |
| nmoleub3 24617 | The operator norm is the s... |
| nmhmcn 24618 | A linear operator over a n... |
| cmodscexp 24619 | The powers of ` _i ` belon... |
| cmodscmulexp 24620 | The scalar product of a ve... |
| cvslvec 24623 | A subcomplex vector space ... |
| cvsclm 24624 | A subcomplex vector space ... |
| iscvs 24625 | A subcomplex vector space ... |
| iscvsp 24626 | The predicate "is a subcom... |
| iscvsi 24627 | Properties that determine ... |
| cvsi 24628 | The properties of a subcom... |
| cvsunit 24629 | Unit group of the scalar r... |
| cvsdiv 24630 | Division of the scalar rin... |
| cvsdivcl 24631 | The scalar field of a subc... |
| cvsmuleqdivd 24632 | An equality involving rati... |
| cvsdiveqd 24633 | An equality involving rati... |
| cnlmodlem1 24634 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 24635 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 24636 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 24637 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 24638 | The set of complex numbers... |
| cnstrcvs 24639 | The set of complex numbers... |
| cnrbas 24640 | The set of complex numbers... |
| cnrlmod 24641 | The complex left module of... |
| cnrlvec 24642 | The complex left module of... |
| cncvs 24643 | The complex left module of... |
| recvs 24644 | The field of the real numb... |
| recvsOLD 24645 | Obsolete version of ~ recv... |
| qcvs 24646 | The field of rational numb... |
| zclmncvs 24647 | The ring of integers as le... |
| isncvsngp 24648 | A normed subcomplex vector... |
| isncvsngpd 24649 | Properties that determine ... |
| ncvsi 24650 | The properties of a normed... |
| ncvsprp 24651 | Proportionality property o... |
| ncvsge0 24652 | The norm of a scalar produ... |
| ncvsm1 24653 | The norm of the opposite o... |
| ncvsdif 24654 | The norm of the difference... |
| ncvspi 24655 | The norm of a vector plus ... |
| ncvs1 24656 | From any nonzero vector of... |
| cnrnvc 24657 | The module of complex numb... |
| cnncvs 24658 | The module of complex numb... |
| cnnm 24659 | The norm of the normed sub... |
| ncvspds 24660 | Value of the distance func... |
| cnindmet 24661 | The metric induced on the ... |
| cnncvsaddassdemo 24662 | Derive the associative law... |
| cnncvsmulassdemo 24663 | Derive the associative law... |
| cnncvsabsnegdemo 24664 | Derive the absolute value ... |
| iscph 24669 | A subcomplex pre-Hilbert s... |
| cphphl 24670 | A subcomplex pre-Hilbert s... |
| cphnlm 24671 | A subcomplex pre-Hilbert s... |
| cphngp 24672 | A subcomplex pre-Hilbert s... |
| cphlmod 24673 | A subcomplex pre-Hilbert s... |
| cphlvec 24674 | A subcomplex pre-Hilbert s... |
| cphnvc 24675 | A subcomplex pre-Hilbert s... |
| cphsubrglem 24676 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 24677 | Lemma for ~ cphreccl . (C... |
| cphsca 24678 | A subcomplex pre-Hilbert s... |
| cphsubrg 24679 | The scalar field of a subc... |
| cphreccl 24680 | The scalar field of a subc... |
| cphdivcl 24681 | The scalar field of a subc... |
| cphcjcl 24682 | The scalar field of a subc... |
| cphsqrtcl 24683 | The scalar field of a subc... |
| cphabscl 24684 | The scalar field of a subc... |
| cphsqrtcl2 24685 | The scalar field of a subc... |
| cphsqrtcl3 24686 | If the scalar field of a s... |
| cphqss 24687 | The scalar field of a subc... |
| cphclm 24688 | A subcomplex pre-Hilbert s... |
| cphnmvs 24689 | Norm of a scalar product. ... |
| cphipcl 24690 | An inner product is a memb... |
| cphnmfval 24691 | The value of the norm in a... |
| cphnm 24692 | The square of the norm is ... |
| nmsq 24693 | The square of the norm is ... |
| cphnmf 24694 | The norm of a vector is a ... |
| cphnmcl 24695 | The norm of a vector is a ... |
| reipcl 24696 | An inner product of an ele... |
| ipge0 24697 | The inner product in a sub... |
| cphipcj 24698 | Conjugate of an inner prod... |
| cphipipcj 24699 | An inner product times its... |
| cphorthcom 24700 | Orthogonality (meaning inn... |
| cphip0l 24701 | Inner product with a zero ... |
| cphip0r 24702 | Inner product with a zero ... |
| cphipeq0 24703 | The inner product of a vec... |
| cphdir 24704 | Distributive law for inner... |
| cphdi 24705 | Distributive law for inner... |
| cph2di 24706 | Distributive law for inner... |
| cphsubdir 24707 | Distributive law for inner... |
| cphsubdi 24708 | Distributive law for inner... |
| cph2subdi 24709 | Distributive law for inner... |
| cphass 24710 | Associative law for inner ... |
| cphassr 24711 | "Associative" law for seco... |
| cph2ass 24712 | Move scalar multiplication... |
| cphassi 24713 | Associative law for the fi... |
| cphassir 24714 | "Associative" law for the ... |
| cphpyth 24715 | The pythagorean theorem fo... |
| tcphex 24716 | Lemma for ~ tcphbas and si... |
| tcphval 24717 | Define a function to augme... |
| tcphbas 24718 | The base set of a subcompl... |
| tchplusg 24719 | The addition operation of ... |
| tcphsub 24720 | The subtraction operation ... |
| tcphmulr 24721 | The ring operation of a su... |
| tcphsca 24722 | The scalar field of a subc... |
| tcphvsca 24723 | The scalar multiplication ... |
| tcphip 24724 | The inner product of a sub... |
| tcphtopn 24725 | The topology of a subcompl... |
| tcphphl 24726 | Augmentation of a subcompl... |
| tchnmfval 24727 | The norm of a subcomplex p... |
| tcphnmval 24728 | The norm of a subcomplex p... |
| cphtcphnm 24729 | The norm of a norm-augment... |
| tcphds 24730 | The distance of a pre-Hilb... |
| phclm 24731 | A pre-Hilbert space whose ... |
| tcphcphlem3 24732 | Lemma for ~ tcphcph : real... |
| ipcau2 24733 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 24734 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 24735 | Lemma for ~ tcphcph : homo... |
| tcphcph 24736 | The standard definition of... |
| ipcau 24737 | The Cauchy-Schwarz inequal... |
| nmparlem 24738 | Lemma for ~ nmpar . (Cont... |
| nmpar 24739 | A subcomplex pre-Hilbert s... |
| cphipval2 24740 | Value of the inner product... |
| 4cphipval2 24741 | Four times the inner produ... |
| cphipval 24742 | Value of the inner product... |
| ipcnlem2 24743 | The inner product operatio... |
| ipcnlem1 24744 | The inner product operatio... |
| ipcn 24745 | The inner product operatio... |
| cnmpt1ip 24746 | Continuity of inner produc... |
| cnmpt2ip 24747 | Continuity of inner produc... |
| csscld 24748 | A "closed subspace" in a s... |
| clsocv 24749 | The orthogonal complement ... |
| cphsscph 24750 | A subspace of a subcomplex... |
| lmmbr 24757 | Express the binary relatio... |
| lmmbr2 24758 | Express the binary relatio... |
| lmmbr3 24759 | Express the binary relatio... |
| lmmcvg 24760 | Convergence property of a ... |
| lmmbrf 24761 | Express the binary relatio... |
| lmnn 24762 | A condition that implies c... |
| cfilfval 24763 | The set of Cauchy filters ... |
| iscfil 24764 | The property of being a Ca... |
| iscfil2 24765 | The property of being a Ca... |
| cfilfil 24766 | A Cauchy filter is a filte... |
| cfili 24767 | Property of a Cauchy filte... |
| cfil3i 24768 | A Cauchy filter contains b... |
| cfilss 24769 | A filter finer than a Cauc... |
| fgcfil 24770 | The Cauchy filter conditio... |
| fmcfil 24771 | The Cauchy filter conditio... |
| iscfil3 24772 | A filter is Cauchy iff it ... |
| cfilfcls 24773 | Similar to ultrafilters ( ... |
| caufval 24774 | The set of Cauchy sequence... |
| iscau 24775 | Express the property " ` F... |
| iscau2 24776 | Express the property " ` F... |
| iscau3 24777 | Express the Cauchy sequenc... |
| iscau4 24778 | Express the property " ` F... |
| iscauf 24779 | Express the property " ` F... |
| caun0 24780 | A metric with a Cauchy seq... |
| caufpm 24781 | Inclusion of a Cauchy sequ... |
| caucfil 24782 | A Cauchy sequence predicat... |
| iscmet 24783 | The property " ` D ` is a ... |
| cmetcvg 24784 | The convergence of a Cauch... |
| cmetmet 24785 | A complete metric space is... |
| cmetmeti 24786 | A complete metric space is... |
| cmetcaulem 24787 | Lemma for ~ cmetcau . (Co... |
| cmetcau 24788 | The convergence of a Cauch... |
| iscmet3lem3 24789 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 24790 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 24791 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 24792 | The property " ` D ` is a ... |
| iscmet2 24793 | A metric ` D ` is complete... |
| cfilresi 24794 | A Cauchy filter on a metri... |
| cfilres 24795 | Cauchy filter on a metric ... |
| caussi 24796 | Cauchy sequence on a metri... |
| causs 24797 | Cauchy sequence on a metri... |
| equivcfil 24798 | If the metric ` D ` is "st... |
| equivcau 24799 | If the metric ` D ` is "st... |
| lmle 24800 | If the distance from each ... |
| nglmle 24801 | If the norm of each member... |
| lmclim 24802 | Relate a limit on the metr... |
| lmclimf 24803 | Relate a limit on the metr... |
| metelcls 24804 | A point belongs to the clo... |
| metcld 24805 | A subset of a metric space... |
| metcld2 24806 | A subset of a metric space... |
| caubl 24807 | Sufficient condition to en... |
| caublcls 24808 | The convergent point of a ... |
| metcnp4 24809 | Two ways to say a mapping ... |
| metcn4 24810 | Two ways to say a mapping ... |
| iscmet3i 24811 | Properties that determine ... |
| lmcau 24812 | Every convergent sequence ... |
| flimcfil 24813 | Every convergent filter in... |
| metsscmetcld 24814 | A complete subspace of a m... |
| cmetss 24815 | A subspace of a complete m... |
| equivcmet 24816 | If two metrics are strongl... |
| relcmpcmet 24817 | If ` D ` is a metric space... |
| cmpcmet 24818 | A compact metric space is ... |
| cfilucfil3 24819 | Given a metric ` D ` and a... |
| cfilucfil4 24820 | Given a metric ` D ` and a... |
| cncmet 24821 | The set of complex numbers... |
| recmet 24822 | The real numbers are a com... |
| bcthlem1 24823 | Lemma for ~ bcth . Substi... |
| bcthlem2 24824 | Lemma for ~ bcth . The ba... |
| bcthlem3 24825 | Lemma for ~ bcth . The li... |
| bcthlem4 24826 | Lemma for ~ bcth . Given ... |
| bcthlem5 24827 | Lemma for ~ bcth . The pr... |
| bcth 24828 | Baire's Category Theorem. ... |
| bcth2 24829 | Baire's Category Theorem, ... |
| bcth3 24830 | Baire's Category Theorem, ... |
| isbn 24837 | A Banach space is a normed... |
| bnsca 24838 | The scalar field of a Bana... |
| bnnvc 24839 | A Banach space is a normed... |
| bnnlm 24840 | A Banach space is a normed... |
| bnngp 24841 | A Banach space is a normed... |
| bnlmod 24842 | A Banach space is a left m... |
| bncms 24843 | A Banach space is a comple... |
| iscms 24844 | A complete metric space is... |
| cmscmet 24845 | The induced metric on a co... |
| bncmet 24846 | The induced metric on Bana... |
| cmsms 24847 | A complete metric space is... |
| cmspropd 24848 | Property deduction for a c... |
| cmssmscld 24849 | The restriction of a metri... |
| cmsss 24850 | The restriction of a compl... |
| lssbn 24851 | A subspace of a Banach spa... |
| cmetcusp1 24852 | If the uniform set of a co... |
| cmetcusp 24853 | The uniform space generate... |
| cncms 24854 | The field of complex numbe... |
| cnflduss 24855 | The uniform structure of t... |
| cnfldcusp 24856 | The field of complex numbe... |
| resscdrg 24857 | The real numbers are a sub... |
| cncdrg 24858 | The only complete subfield... |
| srabn 24859 | The subring algebra over a... |
| rlmbn 24860 | The ring module over a com... |
| ishl 24861 | The predicate "is a subcom... |
| hlbn 24862 | Every subcomplex Hilbert s... |
| hlcph 24863 | Every subcomplex Hilbert s... |
| hlphl 24864 | Every subcomplex Hilbert s... |
| hlcms 24865 | Every subcomplex Hilbert s... |
| hlprlem 24866 | Lemma for ~ hlpr . (Contr... |
| hlress 24867 | The scalar field of a subc... |
| hlpr 24868 | The scalar field of a subc... |
| ishl2 24869 | A Hilbert space is a compl... |
| cphssphl 24870 | A Banach subspace of a sub... |
| cmslssbn 24871 | A complete linear subspace... |
| cmscsscms 24872 | A closed subspace of a com... |
| bncssbn 24873 | A closed subspace of a Ban... |
| cssbn 24874 | A complete subspace of a n... |
| csschl 24875 | A complete subspace of a c... |
| cmslsschl 24876 | A complete linear subspace... |
| chlcsschl 24877 | A closed subspace of a sub... |
| retopn 24878 | The topology of the real n... |
| recms 24879 | The real numbers form a co... |
| reust 24880 | The Uniform structure of t... |
| recusp 24881 | The real numbers form a co... |
| rrxval 24886 | Value of the generalized E... |
| rrxbase 24887 | The base of the generalize... |
| rrxprds 24888 | Expand the definition of t... |
| rrxip 24889 | The inner product of the g... |
| rrxnm 24890 | The norm of the generalize... |
| rrxcph 24891 | Generalized Euclidean real... |
| rrxds 24892 | The distance over generali... |
| rrxvsca 24893 | The scalar product over ge... |
| rrxplusgvscavalb 24894 | The result of the addition... |
| rrxsca 24895 | The field of real numbers ... |
| rrx0 24896 | The zero ("origin") in a g... |
| rrx0el 24897 | The zero ("origin") in a g... |
| csbren 24898 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 24899 | Triangle inequality in R^n... |
| rrxf 24900 | Euclidean vectors as funct... |
| rrxfsupp 24901 | Euclidean vectors are of f... |
| rrxsuppss 24902 | Support of Euclidean vecto... |
| rrxmvallem 24903 | Support of the function us... |
| rrxmval 24904 | The value of the Euclidean... |
| rrxmfval 24905 | The value of the Euclidean... |
| rrxmetlem 24906 | Lemma for ~ rrxmet . (Con... |
| rrxmet 24907 | Euclidean space is a metri... |
| rrxdstprj1 24908 | The distance between two p... |
| rrxbasefi 24909 | The base of the generalize... |
| rrxdsfi 24910 | The distance over generali... |
| rrxmetfi 24911 | Euclidean space is a metri... |
| rrxdsfival 24912 | The value of the Euclidean... |
| ehlval 24913 | Value of the Euclidean spa... |
| ehlbase 24914 | The base of the Euclidean ... |
| ehl0base 24915 | The base of the Euclidean ... |
| ehl0 24916 | The Euclidean space of dim... |
| ehleudis 24917 | The Euclidean distance fun... |
| ehleudisval 24918 | The value of the Euclidean... |
| ehl1eudis 24919 | The Euclidean distance fun... |
| ehl1eudisval 24920 | The value of the Euclidean... |
| ehl2eudis 24921 | The Euclidean distance fun... |
| ehl2eudisval 24922 | The value of the Euclidean... |
| minveclem1 24923 | Lemma for ~ minvec . The ... |
| minveclem4c 24924 | Lemma for ~ minvec . The ... |
| minveclem2 24925 | Lemma for ~ minvec . Any ... |
| minveclem3a 24926 | Lemma for ~ minvec . ` D `... |
| minveclem3b 24927 | Lemma for ~ minvec . The ... |
| minveclem3 24928 | Lemma for ~ minvec . The ... |
| minveclem4a 24929 | Lemma for ~ minvec . ` F `... |
| minveclem4b 24930 | Lemma for ~ minvec . The ... |
| minveclem4 24931 | Lemma for ~ minvec . The ... |
| minveclem5 24932 | Lemma for ~ minvec . Disc... |
| minveclem6 24933 | Lemma for ~ minvec . Any ... |
| minveclem7 24934 | Lemma for ~ minvec . Sinc... |
| minvec 24935 | Minimizing vector theorem,... |
| pjthlem1 24936 | Lemma for ~ pjth . (Contr... |
| pjthlem2 24937 | Lemma for ~ pjth . (Contr... |
| pjth 24938 | Projection Theorem: Any H... |
| pjth2 24939 | Projection Theorem with ab... |
| cldcss 24940 | Corollary of the Projectio... |
| cldcss2 24941 | Corollary of the Projectio... |
| hlhil 24942 | Corollary of the Projectio... |
| addcncf 24943 | The addition of two contin... |
| subcncf 24944 | The addition of two contin... |
| mulcncf 24945 | The multiplication of two ... |
| divcncf 24946 | The quotient of two contin... |
| pmltpclem1 24947 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 24948 | Lemma for ~ pmltpc . (Con... |
| pmltpc 24949 | Any function on the reals ... |
| ivthlem1 24950 | Lemma for ~ ivth . The se... |
| ivthlem2 24951 | Lemma for ~ ivth . Show t... |
| ivthlem3 24952 | Lemma for ~ ivth , the int... |
| ivth 24953 | The intermediate value the... |
| ivth2 24954 | The intermediate value the... |
| ivthle 24955 | The intermediate value the... |
| ivthle2 24956 | The intermediate value the... |
| ivthicc 24957 | The interval between any t... |
| evthicc 24958 | Specialization of the Extr... |
| evthicc2 24959 | Combine ~ ivthicc with ~ e... |
| cniccbdd 24960 | A continuous function on a... |
| ovolfcl 24965 | Closure for the interval e... |
| ovolfioo 24966 | Unpack the interval coveri... |
| ovolficc 24967 | Unpack the interval coveri... |
| ovolficcss 24968 | Any (closed) interval cove... |
| ovolfsval 24969 | The value of the interval ... |
| ovolfsf 24970 | Closure for the interval l... |
| ovolsf 24971 | Closure for the partial su... |
| ovolval 24972 | The value of the outer mea... |
| elovolmlem 24973 | Lemma for ~ elovolm and re... |
| elovolm 24974 | Elementhood in the set ` M... |
| elovolmr 24975 | Sufficient condition for e... |
| ovolmge0 24976 | The set ` M ` is composed ... |
| ovolcl 24977 | The volume of a set is an ... |
| ovollb 24978 | The outer volume is a lowe... |
| ovolgelb 24979 | The outer volume is the gr... |
| ovolge0 24980 | The volume of a set is alw... |
| ovolf 24981 | The domain and codomain of... |
| ovollecl 24982 | If an outer volume is boun... |
| ovolsslem 24983 | Lemma for ~ ovolss . (Con... |
| ovolss 24984 | The volume of a set is mon... |
| ovolsscl 24985 | If a set is contained in a... |
| ovolssnul 24986 | A subset of a nullset is n... |
| ovollb2lem 24987 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 24988 | It is often more convenien... |
| ovolctb 24989 | The volume of a denumerabl... |
| ovolq 24990 | The rational numbers have ... |
| ovolctb2 24991 | The volume of a countable ... |
| ovol0 24992 | The empty set has 0 outer ... |
| ovolfi 24993 | A finite set has 0 outer L... |
| ovolsn 24994 | A singleton has 0 outer Le... |
| ovolunlem1a 24995 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 24996 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 24997 | Lemma for ~ ovolun . (Con... |
| ovolun 24998 | The Lebesgue outer measure... |
| ovolunnul 24999 | Adding a nullset does not ... |
| ovolfiniun 25000 | The Lebesgue outer measure... |
| ovoliunlem1 25001 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25002 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25003 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25004 | The Lebesgue outer measure... |
| ovoliun2 25005 | The Lebesgue outer measure... |
| ovoliunnul 25006 | A countable union of nulls... |
| shft2rab 25007 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25008 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25009 | Lemma for ~ ovolshft . (C... |
| ovolshft 25010 | The Lebesgue outer measure... |
| sca2rab 25011 | If ` B ` is a scale of ` A... |
| ovolscalem1 25012 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25013 | Lemma for ~ ovolshft . (C... |
| ovolsca 25014 | The Lebesgue outer measure... |
| ovolicc1 25015 | The measure of a closed in... |
| ovolicc2lem1 25016 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25017 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25018 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25019 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25020 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25021 | The measure of a closed in... |
| ovolicc 25022 | The measure of a closed in... |
| ovolicopnf 25023 | The measure of a right-unb... |
| ovolre 25024 | The measure of the real nu... |
| ismbl 25025 | The predicate " ` A ` is L... |
| ismbl2 25026 | From ~ ovolun , it suffice... |
| volres 25027 | A self-referencing abbrevi... |
| volf 25028 | The domain and codomain of... |
| mblvol 25029 | The volume of a measurable... |
| mblss 25030 | A measurable set is a subs... |
| mblsplit 25031 | The defining property of m... |
| volss 25032 | The Lebesgue measure is mo... |
| cmmbl 25033 | The complement of a measur... |
| nulmbl 25034 | A nullset is measurable. ... |
| nulmbl2 25035 | A set of outer measure zer... |
| unmbl 25036 | A union of measurable sets... |
| shftmbl 25037 | A shift of a measurable se... |
| 0mbl 25038 | The empty set is measurabl... |
| rembl 25039 | The set of all real number... |
| unidmvol 25040 | The union of the Lebesgue ... |
| inmbl 25041 | An intersection of measura... |
| difmbl 25042 | A difference of measurable... |
| finiunmbl 25043 | A finite union of measurab... |
| volun 25044 | The Lebesgue measure funct... |
| volinun 25045 | Addition of non-disjoint s... |
| volfiniun 25046 | The volume of a disjoint f... |
| iundisj 25047 | Rewrite a countable union ... |
| iundisj2 25048 | A disjoint union is disjoi... |
| voliunlem1 25049 | Lemma for ~ voliun . (Con... |
| voliunlem2 25050 | Lemma for ~ voliun . (Con... |
| voliunlem3 25051 | Lemma for ~ voliun . (Con... |
| iunmbl 25052 | The measurable sets are cl... |
| voliun 25053 | The Lebesgue measure funct... |
| volsuplem 25054 | Lemma for ~ volsup . (Con... |
| volsup 25055 | The volume of the limit of... |
| iunmbl2 25056 | The measurable sets are cl... |
| ioombl1lem1 25057 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25058 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25059 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25060 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25061 | An open right-unbounded in... |
| icombl1 25062 | A closed unbounded-above i... |
| icombl 25063 | A closed-below, open-above... |
| ioombl 25064 | An open real interval is m... |
| iccmbl 25065 | A closed real interval is ... |
| iccvolcl 25066 | A closed real interval has... |
| ovolioo 25067 | The measure of an open int... |
| volioo 25068 | The measure of an open int... |
| ioovolcl 25069 | An open real interval has ... |
| ovolfs2 25070 | Alternative expression for... |
| ioorcl2 25071 | An open interval with fini... |
| ioorf 25072 | Define a function from ope... |
| ioorval 25073 | Define a function from ope... |
| ioorinv2 25074 | The function ` F ` is an "... |
| ioorinv 25075 | The function ` F ` is an "... |
| ioorcl 25076 | The function ` F ` does no... |
| uniiccdif 25077 | A union of closed interval... |
| uniioovol 25078 | A disjoint union of open i... |
| uniiccvol 25079 | An almost-disjoint union o... |
| uniioombllem1 25080 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25081 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25082 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25083 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25084 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25085 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25086 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25087 | Lemma for ~ uniioombl . (... |
| uniioombl 25088 | A disjoint union of open i... |
| uniiccmbl 25089 | An almost-disjoint union o... |
| dyadf 25090 | The function ` F ` returns... |
| dyadval 25091 | Value of the dyadic ration... |
| dyadovol 25092 | Volume of a dyadic rationa... |
| dyadss 25093 | Two closed dyadic rational... |
| dyaddisjlem 25094 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25095 | Two closed dyadic rational... |
| dyadmaxlem 25096 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25097 | Any nonempty set of dyadic... |
| dyadmbllem 25098 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25099 | Any union of dyadic ration... |
| opnmbllem 25100 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25101 | All open sets are measurab... |
| opnmblALT 25102 | All open sets are measurab... |
| subopnmbl 25103 | Sets which are open in a m... |
| volsup2 25104 | The volume of ` A ` is the... |
| volcn 25105 | The function formed by res... |
| volivth 25106 | The Intermediate Value The... |
| vitalilem1 25107 | Lemma for ~ vitali . (Con... |
| vitalilem2 25108 | Lemma for ~ vitali . (Con... |
| vitalilem3 25109 | Lemma for ~ vitali . (Con... |
| vitalilem4 25110 | Lemma for ~ vitali . (Con... |
| vitalilem5 25111 | Lemma for ~ vitali . (Con... |
| vitali 25112 | If the reals can be well-o... |
| ismbf1 25123 | The predicate " ` F ` is a... |
| mbff 25124 | A measurable function is a... |
| mbfdm 25125 | The domain of a measurable... |
| mbfconstlem 25126 | Lemma for ~ mbfconst and r... |
| ismbf 25127 | The predicate " ` F ` is a... |
| ismbfcn 25128 | A complex function is meas... |
| mbfima 25129 | Definitional property of a... |
| mbfimaicc 25130 | The preimage of any closed... |
| mbfimasn 25131 | The preimage of a point un... |
| mbfconst 25132 | A constant function is mea... |
| mbf0 25133 | The empty function is meas... |
| mbfid 25134 | The identity function is m... |
| mbfmptcl 25135 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25136 | The domain of a measurable... |
| ismbfcn2 25137 | A complex function is meas... |
| ismbfd 25138 | Deduction to prove measura... |
| ismbf2d 25139 | Deduction to prove measura... |
| mbfeqalem1 25140 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25141 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25142 | If two functions are equal... |
| mbfres 25143 | The restriction of a measu... |
| mbfres2 25144 | Measurability of a piecewi... |
| mbfss 25145 | Change the domain of a mea... |
| mbfmulc2lem 25146 | Multiplication by a consta... |
| mbfmulc2re 25147 | Multiplication by a consta... |
| mbfmax 25148 | The maximum of two functio... |
| mbfneg 25149 | The negative of a measurab... |
| mbfpos 25150 | The positive part of a mea... |
| mbfposr 25151 | Converse to ~ mbfpos . (C... |
| mbfposb 25152 | A function is measurable i... |
| ismbf3d 25153 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25154 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25155 | The preimage of any open s... |
| mbfimaopn2 25156 | The preimage of any set op... |
| cncombf 25157 | The composition of a conti... |
| cnmbf 25158 | A continuous function is m... |
| mbfaddlem 25159 | The sum of two measurable ... |
| mbfadd 25160 | The sum of two measurable ... |
| mbfsub 25161 | The difference of two meas... |
| mbfmulc2 25162 | A complex constant times a... |
| mbfsup 25163 | The supremum of a sequence... |
| mbfinf 25164 | The infimum of a sequence ... |
| mbflimsup 25165 | The limit supremum of a se... |
| mbflimlem 25166 | The pointwise limit of a s... |
| mbflim 25167 | The pointwise limit of a s... |
| 0pval 25170 | The zero function evaluate... |
| 0plef 25171 | Two ways to say that the f... |
| 0pledm 25172 | Adjust the domain of the l... |
| isi1f 25173 | The predicate " ` F ` is a... |
| i1fmbf 25174 | Simple functions are measu... |
| i1ff 25175 | A simple function is a fun... |
| i1frn 25176 | A simple function has fini... |
| i1fima 25177 | Any preimage of a simple f... |
| i1fima2 25178 | Any preimage of a simple f... |
| i1fima2sn 25179 | Preimage of a singleton. ... |
| i1fd 25180 | A simplified set of assump... |
| i1f0rn 25181 | Any simple function takes ... |
| itg1val 25182 | The value of the integral ... |
| itg1val2 25183 | The value of the integral ... |
| itg1cl 25184 | Closure of the integral on... |
| itg1ge0 25185 | Closure of the integral on... |
| i1f0 25186 | The zero function is simpl... |
| itg10 25187 | The zero function has zero... |
| i1f1lem 25188 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25189 | Base case simple functions... |
| itg11 25190 | The integral of an indicat... |
| itg1addlem1 25191 | Decompose a preimage, whic... |
| i1faddlem 25192 | Decompose the preimage of ... |
| i1fmullem 25193 | Decompose the preimage of ... |
| i1fadd 25194 | The sum of two simple func... |
| i1fmul 25195 | The pointwise product of t... |
| itg1addlem2 25196 | Lemma for ~ itg1add . The... |
| itg1addlem3 25197 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25198 | Lemma for ~ itg1add . (Co... |
| itg1addlem4OLD 25199 | Obsolete version of ~ itg1... |
| itg1addlem5 25200 | Lemma for ~ itg1add . (Co... |
| itg1add 25201 | The integral of a sum of s... |
| i1fmulclem 25202 | Decompose the preimage of ... |
| i1fmulc 25203 | A nonnegative constant tim... |
| itg1mulc 25204 | The integral of a constant... |
| i1fres 25205 | The "restriction" of a sim... |
| i1fpos 25206 | The positive part of a sim... |
| i1fposd 25207 | Deduction form of ~ i1fpos... |
| i1fsub 25208 | The difference of two simp... |
| itg1sub 25209 | The integral of a differen... |
| itg10a 25210 | The integral of a simple f... |
| itg1ge0a 25211 | The integral of an almost ... |
| itg1lea 25212 | Approximate version of ~ i... |
| itg1le 25213 | If one simple function dom... |
| itg1climres 25214 | Restricting the simple fun... |
| mbfi1fseqlem1 25215 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25216 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25217 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25218 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25219 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25220 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25221 | A characterization of meas... |
| mbfi1flimlem 25222 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25223 | Any real measurable functi... |
| mbfmullem2 25224 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25225 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25226 | The product of two measura... |
| itg2lcl 25227 | The set of lower sums is a... |
| itg2val 25228 | Value of the integral on n... |
| itg2l 25229 | Elementhood in the set ` L... |
| itg2lr 25230 | Sufficient condition for e... |
| xrge0f 25231 | A real function is a nonne... |
| itg2cl 25232 | The integral of a nonnegat... |
| itg2ub 25233 | The integral of a nonnegat... |
| itg2leub 25234 | Any upper bound on the int... |
| itg2ge0 25235 | The integral of a nonnegat... |
| itg2itg1 25236 | The integral of a nonnegat... |
| itg20 25237 | The integral of the zero f... |
| itg2lecl 25238 | If an ` S.2 ` integral is ... |
| itg2le 25239 | If one function dominates ... |
| itg2const 25240 | Integral of a constant fun... |
| itg2const2 25241 | When the base set of a con... |
| itg2seq 25242 | Definitional property of t... |
| itg2uba 25243 | Approximate version of ~ i... |
| itg2lea 25244 | Approximate version of ~ i... |
| itg2eqa 25245 | Approximate equality of in... |
| itg2mulclem 25246 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25247 | The integral of a nonnegat... |
| itg2splitlem 25248 | Lemma for ~ itg2split . (... |
| itg2split 25249 | The ` S.2 ` integral split... |
| itg2monolem1 25250 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25251 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25252 | Lemma for ~ itg2mono . (C... |
| itg2mono 25253 | The Monotone Convergence T... |
| itg2i1fseqle 25254 | Subject to the conditions ... |
| itg2i1fseq 25255 | Subject to the conditions ... |
| itg2i1fseq2 25256 | In an extension to the res... |
| itg2i1fseq3 25257 | Special case of ~ itg2i1fs... |
| itg2addlem 25258 | Lemma for ~ itg2add . (Co... |
| itg2add 25259 | The ` S.2 ` integral is li... |
| itg2gt0 25260 | If the function ` F ` is s... |
| itg2cnlem1 25261 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25262 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25263 | A sort of absolute continu... |
| ibllem 25264 | Conditioned equality theor... |
| isibl 25265 | The predicate " ` F ` is i... |
| isibl2 25266 | The predicate " ` F ` is i... |
| iblmbf 25267 | An integrable function is ... |
| iblitg 25268 | If a function is integrabl... |
| dfitg 25269 | Evaluate the class substit... |
| itgex 25270 | An integral is a set. (Co... |
| itgeq1f 25271 | Equality theorem for an in... |
| itgeq1 25272 | Equality theorem for an in... |
| nfitg1 25273 | Bound-variable hypothesis ... |
| nfitg 25274 | Bound-variable hypothesis ... |
| cbvitg 25275 | Change bound variable in a... |
| cbvitgv 25276 | Change bound variable in a... |
| itgeq2 25277 | Equality theorem for an in... |
| itgresr 25278 | The domain of an integral ... |
| itg0 25279 | The integral of anything o... |
| itgz 25280 | The integral of zero on an... |
| itgeq2dv 25281 | Equality theorem for an in... |
| itgmpt 25282 | Change bound variable in a... |
| itgcl 25283 | The integral of an integra... |
| itgvallem 25284 | Substitution lemma. (Cont... |
| itgvallem3 25285 | Lemma for ~ itgposval and ... |
| ibl0 25286 | The zero function is integ... |
| iblcnlem1 25287 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25288 | Expand out the universal q... |
| itgcnlem 25289 | Expand out the sum in ~ df... |
| iblrelem 25290 | Integrability of a real fu... |
| iblposlem 25291 | Lemma for ~ iblpos . (Con... |
| iblpos 25292 | Integrability of a nonnega... |
| iblre 25293 | Integrability of a real fu... |
| itgrevallem1 25294 | Lemma for ~ itgposval and ... |
| itgposval 25295 | The integral of a nonnegat... |
| itgreval 25296 | Decompose the integral of ... |
| itgrecl 25297 | Real closure of an integra... |
| iblcn 25298 | Integrability of a complex... |
| itgcnval 25299 | Decompose the integral of ... |
| itgre 25300 | Real part of an integral. ... |
| itgim 25301 | Imaginary part of an integ... |
| iblneg 25302 | The negative of an integra... |
| itgneg 25303 | Negation of an integral. ... |
| iblss 25304 | A subset of an integrable ... |
| iblss2 25305 | Change the domain of an in... |
| itgitg2 25306 | Transfer an integral using... |
| i1fibl 25307 | A simple function is integ... |
| itgitg1 25308 | Transfer an integral using... |
| itgle 25309 | Monotonicity of an integra... |
| itgge0 25310 | The integral of a positive... |
| itgss 25311 | Expand the set of an integ... |
| itgss2 25312 | Expand the set of an integ... |
| itgeqa 25313 | Approximate equality of in... |
| itgss3 25314 | Expand the set of an integ... |
| itgioo 25315 | Equality of integrals on o... |
| itgless 25316 | Expand the integral of a n... |
| iblconst 25317 | A constant function is int... |
| itgconst 25318 | Integral of a constant fun... |
| ibladdlem 25319 | Lemma for ~ ibladd . (Con... |
| ibladd 25320 | Add two integrals over the... |
| iblsub 25321 | Subtract two integrals ove... |
| itgaddlem1 25322 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25323 | Lemma for ~ itgadd . (Con... |
| itgadd 25324 | Add two integrals over the... |
| itgsub 25325 | Subtract two integrals ove... |
| itgfsum 25326 | Take a finite sum of integ... |
| iblabslem 25327 | Lemma for ~ iblabs . (Con... |
| iblabs 25328 | The absolute value of an i... |
| iblabsr 25329 | A measurable function is i... |
| iblmulc2 25330 | Multiply an integral by a ... |
| itgmulc2lem1 25331 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25332 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25333 | Multiply an integral by a ... |
| itgabs 25334 | The triangle inequality fo... |
| itgsplit 25335 | The ` S. ` integral splits... |
| itgspliticc 25336 | The ` S. ` integral splits... |
| itgsplitioo 25337 | The ` S. ` integral splits... |
| bddmulibl 25338 | A bounded function times a... |
| bddibl 25339 | A bounded function is inte... |
| cniccibl 25340 | A continuous function on a... |
| bddiblnc 25341 | Choice-free proof of ~ bdd... |
| cnicciblnc 25342 | Choice-free proof of ~ cni... |
| itggt0 25343 | The integral of a strictly... |
| itgcn 25344 | Transfer ~ itg2cn to the f... |
| ditgeq1 25347 | Equality theorem for the d... |
| ditgeq2 25348 | Equality theorem for the d... |
| ditgeq3 25349 | Equality theorem for the d... |
| ditgeq3dv 25350 | Equality theorem for the d... |
| ditgex 25351 | A directed integral is a s... |
| ditg0 25352 | Value of the directed inte... |
| cbvditg 25353 | Change bound variable in a... |
| cbvditgv 25354 | Change bound variable in a... |
| ditgpos 25355 | Value of the directed inte... |
| ditgneg 25356 | Value of the directed inte... |
| ditgcl 25357 | Closure of a directed inte... |
| ditgswap 25358 | Reverse a directed integra... |
| ditgsplitlem 25359 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25360 | This theorem is the raison... |
| reldv 25369 | The derivative function is... |
| limcvallem 25370 | Lemma for ~ ellimc . (Con... |
| limcfval 25371 | Value and set bounds on th... |
| ellimc 25372 | Value of the limit predica... |
| limcrcl 25373 | Reverse closure for the li... |
| limccl 25374 | Closure of the limit opera... |
| limcdif 25375 | It suffices to consider fu... |
| ellimc2 25376 | Write the definition of a ... |
| limcnlp 25377 | If ` B ` is not a limit po... |
| ellimc3 25378 | Write the epsilon-delta de... |
| limcflflem 25379 | Lemma for ~ limcflf . (Co... |
| limcflf 25380 | The limit operator can be ... |
| limcmo 25381 | If ` B ` is a limit point ... |
| limcmpt 25382 | Express the limit operator... |
| limcmpt2 25383 | Express the limit operator... |
| limcresi 25384 | Any limit of ` F ` is also... |
| limcres 25385 | If ` B ` is an interior po... |
| cnplimc 25386 | A function is continuous a... |
| cnlimc 25387 | ` F ` is a continuous func... |
| cnlimci 25388 | If ` F ` is a continuous f... |
| cnmptlimc 25389 | If ` F ` is a continuous f... |
| limccnp 25390 | If the limit of ` F ` at `... |
| limccnp2 25391 | The image of a convergent ... |
| limcco 25392 | Composition of two limits.... |
| limciun 25393 | A point is a limit of ` F ... |
| limcun 25394 | A point is a limit of ` F ... |
| dvlem 25395 | Closure for a difference q... |
| dvfval 25396 | Value and set bounds on th... |
| eldv 25397 | The differentiable predica... |
| dvcl 25398 | The derivative function ta... |
| dvbssntr 25399 | The set of differentiable ... |
| dvbss 25400 | The set of differentiable ... |
| dvbsss 25401 | The set of differentiable ... |
| perfdvf 25402 | The derivative is a functi... |
| recnprss 25403 | Both ` RR ` and ` CC ` are... |
| recnperf 25404 | Both ` RR ` and ` CC ` are... |
| dvfg 25405 | Explicitly write out the f... |
| dvf 25406 | The derivative is a functi... |
| dvfcn 25407 | The derivative is a functi... |
| dvreslem 25408 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25409 | Lemma for ~ dvres2 . (Con... |
| dvres 25410 | Restriction of a derivativ... |
| dvres2 25411 | Restriction of the base se... |
| dvres3 25412 | Restriction of a complex d... |
| dvres3a 25413 | Restriction of a complex d... |
| dvidlem 25414 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25415 | Derivative of a function r... |
| dvconst 25416 | Derivative of a constant f... |
| dvid 25417 | Derivative of the identity... |
| dvcnp 25418 | The difference quotient is... |
| dvcnp2 25419 | A function is continuous a... |
| dvcn 25420 | A differentiable function ... |
| dvnfval 25421 | Value of the iterated deri... |
| dvnff 25422 | The iterated derivative is... |
| dvn0 25423 | Zero times iterated deriva... |
| dvnp1 25424 | Successor iterated derivat... |
| dvn1 25425 | One times iterated derivat... |
| dvnf 25426 | The N-times derivative is ... |
| dvnbss 25427 | The set of N-times differe... |
| dvnadd 25428 | The ` N ` -th derivative o... |
| dvn2bss 25429 | An N-times differentiable ... |
| dvnres 25430 | Multiple derivative versio... |
| cpnfval 25431 | Condition for n-times cont... |
| fncpn 25432 | The ` C^n ` object is a fu... |
| elcpn 25433 | Condition for n-times cont... |
| cpnord 25434 | ` C^n ` conditions are ord... |
| cpncn 25435 | A ` C^n ` function is cont... |
| cpnres 25436 | The restriction of a ` C^n... |
| dvaddbr 25437 | The sum rule for derivativ... |
| dvmulbr 25438 | The product rule for deriv... |
| dvadd 25439 | The sum rule for derivativ... |
| dvmul 25440 | The product rule for deriv... |
| dvaddf 25441 | The sum rule for everywher... |
| dvmulf 25442 | The product rule for every... |
| dvcmul 25443 | The product rule when one ... |
| dvcmulf 25444 | The product rule when one ... |
| dvcobr 25445 | The chain rule for derivat... |
| dvco 25446 | The chain rule for derivat... |
| dvcof 25447 | The chain rule for everywh... |
| dvcjbr 25448 | The derivative of the conj... |
| dvcj 25449 | The derivative of the conj... |
| dvfre 25450 | The derivative of a real f... |
| dvnfre 25451 | The ` N ` -th derivative o... |
| dvexp 25452 | Derivative of a power func... |
| dvexp2 25453 | Derivative of an exponenti... |
| dvrec 25454 | Derivative of the reciproc... |
| dvmptres3 25455 | Function-builder for deriv... |
| dvmptid 25456 | Function-builder for deriv... |
| dvmptc 25457 | Function-builder for deriv... |
| dvmptcl 25458 | Closure lemma for ~ dvmptc... |
| dvmptadd 25459 | Function-builder for deriv... |
| dvmptmul 25460 | Function-builder for deriv... |
| dvmptres2 25461 | Function-builder for deriv... |
| dvmptres 25462 | Function-builder for deriv... |
| dvmptcmul 25463 | Function-builder for deriv... |
| dvmptdivc 25464 | Function-builder for deriv... |
| dvmptneg 25465 | Function-builder for deriv... |
| dvmptsub 25466 | Function-builder for deriv... |
| dvmptcj 25467 | Function-builder for deriv... |
| dvmptre 25468 | Function-builder for deriv... |
| dvmptim 25469 | Function-builder for deriv... |
| dvmptntr 25470 | Function-builder for deriv... |
| dvmptco 25471 | Function-builder for deriv... |
| dvrecg 25472 | Derivative of the reciproc... |
| dvmptdiv 25473 | Function-builder for deriv... |
| dvmptfsum 25474 | Function-builder for deriv... |
| dvcnvlem 25475 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 25476 | A weak version of ~ dvcnvr... |
| dvexp3 25477 | Derivative of an exponenti... |
| dveflem 25478 | Derivative of the exponent... |
| dvef 25479 | Derivative of the exponent... |
| dvsincos 25480 | Derivative of the sine and... |
| dvsin 25481 | Derivative of the sine fun... |
| dvcos 25482 | Derivative of the cosine f... |
| dvferm1lem 25483 | Lemma for ~ dvferm . (Con... |
| dvferm1 25484 | One-sided version of ~ dvf... |
| dvferm2lem 25485 | Lemma for ~ dvferm . (Con... |
| dvferm2 25486 | One-sided version of ~ dvf... |
| dvferm 25487 | Fermat's theorem on statio... |
| rollelem 25488 | Lemma for ~ rolle . (Cont... |
| rolle 25489 | Rolle's theorem. If ` F `... |
| cmvth 25490 | Cauchy's Mean Value Theore... |
| mvth 25491 | The Mean Value Theorem. I... |
| dvlip 25492 | A function with derivative... |
| dvlipcn 25493 | A complex function with de... |
| dvlip2 25494 | Combine the results of ~ d... |
| c1liplem1 25495 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 25496 | C^1 functions are Lipschit... |
| c1lip2 25497 | C^1 functions are Lipschit... |
| c1lip3 25498 | C^1 functions are Lipschit... |
| dveq0 25499 | If a continuous function h... |
| dv11cn 25500 | Two functions defined on a... |
| dvgt0lem1 25501 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 25502 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 25503 | A function on a closed int... |
| dvlt0 25504 | A function on a closed int... |
| dvge0 25505 | A function on a closed int... |
| dvle 25506 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 25507 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 25508 | Lemma for ~ dvivth . (Con... |
| dvivth 25509 | Darboux' theorem, or the i... |
| dvne0 25510 | A function on a closed int... |
| dvne0f1 25511 | A function on a closed int... |
| lhop1lem 25512 | Lemma for ~ lhop1 . (Cont... |
| lhop1 25513 | L'Hôpital's Rule for... |
| lhop2 25514 | L'Hôpital's Rule for... |
| lhop 25515 | L'Hôpital's Rule. I... |
| dvcnvrelem1 25516 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 25517 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 25518 | The derivative rule for in... |
| dvcvx 25519 | A real function with stric... |
| dvfsumle 25520 | Compare a finite sum to an... |
| dvfsumge 25521 | Compare a finite sum to an... |
| dvfsumabs 25522 | Compare a finite sum to an... |
| dvmptrecl 25523 | Real closure of a derivati... |
| dvfsumrlimf 25524 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 25525 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 25526 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem3 25527 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 25528 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 25529 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 25530 | Compare a finite sum to an... |
| dvfsumrlim2 25531 | Compare a finite sum to an... |
| dvfsumrlim3 25532 | Conjoin the statements of ... |
| dvfsum2 25533 | The reverse of ~ dvfsumrli... |
| ftc1lem1 25534 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 25535 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 25536 | The Fundamental Theorem of... |
| ftc1lem3 25537 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 25538 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 25539 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 25540 | Lemma for ~ ftc1 . (Contr... |
| ftc1 25541 | The Fundamental Theorem of... |
| ftc1cn 25542 | Strengthen the assumptions... |
| ftc2 25543 | The Fundamental Theorem of... |
| ftc2ditglem 25544 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 25545 | Directed integral analogue... |
| itgparts 25546 | Integration by parts. If ... |
| itgsubstlem 25547 | Lemma for ~ itgsubst . (C... |
| itgsubst 25548 | Integration by ` u ` -subs... |
| itgpowd 25549 | The integral of a monomial... |
| reldmmdeg 25554 | Multivariate degree is a b... |
| tdeglem1 25555 | Functionality of the total... |
| tdeglem1OLD 25556 | Obsolete version of ~ tdeg... |
| tdeglem3 25557 | Additivity of the total de... |
| tdeglem3OLD 25558 | Obsolete version of ~ tdeg... |
| tdeglem4 25559 | There is only one multi-in... |
| tdeglem4OLD 25560 | Obsolete version of ~ tdeg... |
| tdeglem2 25561 | Simplification of total de... |
| mdegfval 25562 | Value of the multivariate ... |
| mdegval 25563 | Value of the multivariate ... |
| mdegleb 25564 | Property of being of limit... |
| mdeglt 25565 | If there is an upper limit... |
| mdegldg 25566 | A nonzero polynomial has s... |
| mdegxrcl 25567 | Closure of polynomial degr... |
| mdegxrf 25568 | Functionality of polynomia... |
| mdegcl 25569 | Sharp closure for multivar... |
| mdeg0 25570 | Degree of the zero polynom... |
| mdegnn0cl 25571 | Degree of a nonzero polyno... |
| degltlem1 25572 | Theorem on arithmetic of e... |
| degltp1le 25573 | Theorem on arithmetic of e... |
| mdegaddle 25574 | The degree of a sum is at ... |
| mdegvscale 25575 | The degree of a scalar mul... |
| mdegvsca 25576 | The degree of a scalar mul... |
| mdegle0 25577 | A polynomial has nonpositi... |
| mdegmullem 25578 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 25579 | The multivariate degree of... |
| deg1fval 25580 | Relate univariate polynomi... |
| deg1xrf 25581 | Functionality of univariat... |
| deg1xrcl 25582 | Closure of univariate poly... |
| deg1cl 25583 | Sharp closure of univariat... |
| mdegpropd 25584 | Property deduction for pol... |
| deg1fvi 25585 | Univariate polynomial degr... |
| deg1propd 25586 | Property deduction for pol... |
| deg1z 25587 | Degree of the zero univari... |
| deg1nn0cl 25588 | Degree of a nonzero univar... |
| deg1n0ima 25589 | Degree image of a set of p... |
| deg1nn0clb 25590 | A polynomial is nonzero if... |
| deg1lt0 25591 | A polynomial is zero iff i... |
| deg1ldg 25592 | A nonzero univariate polyn... |
| deg1ldgn 25593 | An index at which a polyno... |
| deg1ldgdomn 25594 | A nonzero univariate polyn... |
| deg1leb 25595 | Property of being of limit... |
| deg1val 25596 | Value of the univariate de... |
| deg1lt 25597 | If the degree of a univari... |
| deg1ge 25598 | Conversely, a nonzero coef... |
| coe1mul3 25599 | The coefficient vector of ... |
| coe1mul4 25600 | Value of the "leading" coe... |
| deg1addle 25601 | The degree of a sum is at ... |
| deg1addle2 25602 | If both factors have degre... |
| deg1add 25603 | Exact degree of a sum of t... |
| deg1vscale 25604 | The degree of a scalar tim... |
| deg1vsca 25605 | The degree of a scalar tim... |
| deg1invg 25606 | The degree of the negated ... |
| deg1suble 25607 | The degree of a difference... |
| deg1sub 25608 | Exact degree of a differen... |
| deg1mulle2 25609 | Produce a bound on the pro... |
| deg1sublt 25610 | Subtraction of two polynom... |
| deg1le0 25611 | A polynomial has nonpositi... |
| deg1sclle 25612 | A scalar polynomial has no... |
| deg1scl 25613 | A nonzero scalar polynomia... |
| deg1mul2 25614 | Degree of multiplication o... |
| deg1mul3 25615 | Degree of multiplication o... |
| deg1mul3le 25616 | Degree of multiplication o... |
| deg1tmle 25617 | Limiting degree of a polyn... |
| deg1tm 25618 | Exact degree of a polynomi... |
| deg1pwle 25619 | Limiting degree of a varia... |
| deg1pw 25620 | Exact degree of a variable... |
| ply1nz 25621 | Univariate polynomials ove... |
| ply1nzb 25622 | Univariate polynomials are... |
| ply1domn 25623 | Corollary of ~ deg1mul2 : ... |
| ply1idom 25624 | The ring of univariate pol... |
| ply1divmo 25635 | Uniqueness of a quotient i... |
| ply1divex 25636 | Lemma for ~ ply1divalg : e... |
| ply1divalg 25637 | The division algorithm for... |
| ply1divalg2 25638 | Reverse the order of multi... |
| uc1pval 25639 | Value of the set of unitic... |
| isuc1p 25640 | Being a unitic polynomial.... |
| mon1pval 25641 | Value of the set of monic ... |
| ismon1p 25642 | Being a monic polynomial. ... |
| uc1pcl 25643 | Unitic polynomials are pol... |
| mon1pcl 25644 | Monic polynomials are poly... |
| uc1pn0 25645 | Unitic polynomials are not... |
| mon1pn0 25646 | Monic polynomials are not ... |
| uc1pdeg 25647 | Unitic polynomials have no... |
| uc1pldg 25648 | Unitic polynomials have un... |
| mon1pldg 25649 | Unitic polynomials have on... |
| mon1puc1p 25650 | Monic polynomials are unit... |
| uc1pmon1p 25651 | Make a unitic polynomial m... |
| deg1submon1p 25652 | The difference of two moni... |
| q1pval 25653 | Value of the univariate po... |
| q1peqb 25654 | Characterizing property of... |
| q1pcl 25655 | Closure of the quotient by... |
| r1pval 25656 | Value of the polynomial re... |
| r1pcl 25657 | Closure of remainder follo... |
| r1pdeglt 25658 | The remainder has a degree... |
| r1pid 25659 | Express the original polyn... |
| dvdsq1p 25660 | Divisibility in a polynomi... |
| dvdsr1p 25661 | Divisibility in a polynomi... |
| ply1remlem 25662 | A term of the form ` x - N... |
| ply1rem 25663 | The polynomial remainder t... |
| facth1 25664 | The factor theorem and its... |
| fta1glem1 25665 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 25666 | Lemma for ~ fta1g . (Cont... |
| fta1g 25667 | The one-sided fundamental ... |
| fta1blem 25668 | Lemma for ~ fta1b . (Cont... |
| fta1b 25669 | The assumption that ` R ` ... |
| drnguc1p 25670 | Over a division ring, all ... |
| ig1peu 25671 | There is a unique monic po... |
| ig1pval 25672 | Substitutions for the poly... |
| ig1pval2 25673 | Generator of the zero idea... |
| ig1pval3 25674 | Characterizing properties ... |
| ig1pcl 25675 | The monic generator of an ... |
| ig1pdvds 25676 | The monic generator of an ... |
| ig1prsp 25677 | Any ideal of polynomials o... |
| ply1lpir 25678 | The ring of polynomials ov... |
| ply1pid 25679 | The polynomials over a fie... |
| plyco0 25688 | Two ways to say that a fun... |
| plyval 25689 | Value of the polynomial se... |
| plybss 25690 | Reverse closure of the par... |
| elply 25691 | Definition of a polynomial... |
| elply2 25692 | The coefficient function c... |
| plyun0 25693 | The set of polynomials is ... |
| plyf 25694 | The polynomial is a functi... |
| plyss 25695 | The polynomial set functio... |
| plyssc 25696 | Every polynomial ring is c... |
| elplyr 25697 | Sufficient condition for e... |
| elplyd 25698 | Sufficient condition for e... |
| ply1termlem 25699 | Lemma for ~ ply1term . (C... |
| ply1term 25700 | A one-term polynomial. (C... |
| plypow 25701 | A power is a polynomial. ... |
| plyconst 25702 | A constant function is a p... |
| ne0p 25703 | A test to show that a poly... |
| ply0 25704 | The zero function is a pol... |
| plyid 25705 | The identity function is a... |
| plyeq0lem 25706 | Lemma for ~ plyeq0 . If `... |
| plyeq0 25707 | If a polynomial is zero at... |
| plypf1 25708 | Write the set of complex p... |
| plyaddlem1 25709 | Derive the coefficient fun... |
| plymullem1 25710 | Derive the coefficient fun... |
| plyaddlem 25711 | Lemma for ~ plyadd . (Con... |
| plymullem 25712 | Lemma for ~ plymul . (Con... |
| plyadd 25713 | The sum of two polynomials... |
| plymul 25714 | The product of two polynom... |
| plysub 25715 | The difference of two poly... |
| plyaddcl 25716 | The sum of two polynomials... |
| plymulcl 25717 | The product of two polynom... |
| plysubcl 25718 | The difference of two poly... |
| coeval 25719 | Value of the coefficient f... |
| coeeulem 25720 | Lemma for ~ coeeu . (Cont... |
| coeeu 25721 | Uniqueness of the coeffici... |
| coelem 25722 | Lemma for properties of th... |
| coeeq 25723 | If ` A ` satisfies the pro... |
| dgrval 25724 | Value of the degree functi... |
| dgrlem 25725 | Lemma for ~ dgrcl and simi... |
| coef 25726 | The domain and codomain of... |
| coef2 25727 | The domain and codomain of... |
| coef3 25728 | The domain and codomain of... |
| dgrcl 25729 | The degree of any polynomi... |
| dgrub 25730 | If the ` M ` -th coefficie... |
| dgrub2 25731 | All the coefficients above... |
| dgrlb 25732 | If all the coefficients ab... |
| coeidlem 25733 | Lemma for ~ coeid . (Cont... |
| coeid 25734 | Reconstruct a polynomial a... |
| coeid2 25735 | Reconstruct a polynomial a... |
| coeid3 25736 | Reconstruct a polynomial a... |
| plyco 25737 | The composition of two pol... |
| coeeq2 25738 | Compute the coefficient fu... |
| dgrle 25739 | Given an explicit expressi... |
| dgreq 25740 | If the highest term in a p... |
| 0dgr 25741 | A constant function has de... |
| 0dgrb 25742 | A function has degree zero... |
| dgrnznn 25743 | A nonzero polynomial with ... |
| coefv0 25744 | The result of evaluating a... |
| coeaddlem 25745 | Lemma for ~ coeadd and ~ d... |
| coemullem 25746 | Lemma for ~ coemul and ~ d... |
| coeadd 25747 | The coefficient function o... |
| coemul 25748 | A coefficient of a product... |
| coe11 25749 | The coefficient function i... |
| coemulhi 25750 | The leading coefficient of... |
| coemulc 25751 | The coefficient function i... |
| coe0 25752 | The coefficients of the ze... |
| coesub 25753 | The coefficient function o... |
| coe1termlem 25754 | The coefficient function o... |
| coe1term 25755 | The coefficient function o... |
| dgr1term 25756 | The degree of a monomial. ... |
| plycn 25757 | A polynomial is a continuo... |
| dgr0 25758 | The degree of the zero pol... |
| coeidp 25759 | The coefficients of the id... |
| dgrid 25760 | The degree of the identity... |
| dgreq0 25761 | The leading coefficient of... |
| dgrlt 25762 | Two ways to say that the d... |
| dgradd 25763 | The degree of a sum of pol... |
| dgradd2 25764 | The degree of a sum of pol... |
| dgrmul2 25765 | The degree of a product of... |
| dgrmul 25766 | The degree of a product of... |
| dgrmulc 25767 | Scalar multiplication by a... |
| dgrsub 25768 | The degree of a difference... |
| dgrcolem1 25769 | The degree of a compositio... |
| dgrcolem2 25770 | Lemma for ~ dgrco . (Cont... |
| dgrco 25771 | The degree of a compositio... |
| plycjlem 25772 | Lemma for ~ plycj and ~ co... |
| plycj 25773 | The double conjugation of ... |
| coecj 25774 | Double conjugation of a po... |
| plyrecj 25775 | A polynomial with real coe... |
| plymul0or 25776 | Polynomial multiplication ... |
| ofmulrt 25777 | The set of roots of a prod... |
| plyreres 25778 | Real-coefficient polynomia... |
| dvply1 25779 | Derivative of a polynomial... |
| dvply2g 25780 | The derivative of a polyno... |
| dvply2 25781 | The derivative of a polyno... |
| dvnply2 25782 | Polynomials have polynomia... |
| dvnply 25783 | Polynomials have polynomia... |
| plycpn 25784 | Polynomials are smooth. (... |
| quotval 25787 | Value of the quotient func... |
| plydivlem1 25788 | Lemma for ~ plydivalg . (... |
| plydivlem2 25789 | Lemma for ~ plydivalg . (... |
| plydivlem3 25790 | Lemma for ~ plydivex . Ba... |
| plydivlem4 25791 | Lemma for ~ plydivex . In... |
| plydivex 25792 | Lemma for ~ plydivalg . (... |
| plydiveu 25793 | Lemma for ~ plydivalg . (... |
| plydivalg 25794 | The division algorithm on ... |
| quotlem 25795 | Lemma for properties of th... |
| quotcl 25796 | The quotient of two polyno... |
| quotcl2 25797 | Closure of the quotient fu... |
| quotdgr 25798 | Remainder property of the ... |
| plyremlem 25799 | Closure of a linear factor... |
| plyrem 25800 | The polynomial remainder t... |
| facth 25801 | The factor theorem. If a ... |
| fta1lem 25802 | Lemma for ~ fta1 . (Contr... |
| fta1 25803 | The easy direction of the ... |
| quotcan 25804 | Exact division with a mult... |
| vieta1lem1 25805 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 25806 | Lemma for ~ vieta1 : induc... |
| vieta1 25807 | The first-order Vieta's fo... |
| plyexmo 25808 | An infinite set of values ... |
| elaa 25811 | Elementhood in the set of ... |
| aacn 25812 | An algebraic number is a c... |
| aasscn 25813 | The algebraic numbers are ... |
| elqaalem1 25814 | Lemma for ~ elqaa . The f... |
| elqaalem2 25815 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 25816 | Lemma for ~ elqaa . (Cont... |
| elqaa 25817 | The set of numbers generat... |
| qaa 25818 | Every rational number is a... |
| qssaa 25819 | The rational numbers are c... |
| iaa 25820 | The imaginary unit is alge... |
| aareccl 25821 | The reciprocal of an algeb... |
| aacjcl 25822 | The conjugate of an algebr... |
| aannenlem1 25823 | Lemma for ~ aannen . (Con... |
| aannenlem2 25824 | Lemma for ~ aannen . (Con... |
| aannenlem3 25825 | The algebraic numbers are ... |
| aannen 25826 | The algebraic numbers are ... |
| aalioulem1 25827 | Lemma for ~ aaliou . An i... |
| aalioulem2 25828 | Lemma for ~ aaliou . (Con... |
| aalioulem3 25829 | Lemma for ~ aaliou . (Con... |
| aalioulem4 25830 | Lemma for ~ aaliou . (Con... |
| aalioulem5 25831 | Lemma for ~ aaliou . (Con... |
| aalioulem6 25832 | Lemma for ~ aaliou . (Con... |
| aaliou 25833 | Liouville's theorem on dio... |
| geolim3 25834 | Geometric series convergen... |
| aaliou2 25835 | Liouville's approximation ... |
| aaliou2b 25836 | Liouville's approximation ... |
| aaliou3lem1 25837 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 25838 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 25839 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 25840 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 25841 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 25842 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 25843 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 25844 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 25845 | Example of a "Liouville nu... |
| aaliou3 25846 | Example of a "Liouville nu... |
| taylfvallem1 25851 | Lemma for ~ taylfval . (C... |
| taylfvallem 25852 | Lemma for ~ taylfval . (C... |
| taylfval 25853 | Define the Taylor polynomi... |
| eltayl 25854 | Value of the Taylor series... |
| taylf 25855 | The Taylor series defines ... |
| tayl0 25856 | The Taylor series is alway... |
| taylplem1 25857 | Lemma for ~ taylpfval and ... |
| taylplem2 25858 | Lemma for ~ taylpfval and ... |
| taylpfval 25859 | Define the Taylor polynomi... |
| taylpf 25860 | The Taylor polynomial is a... |
| taylpval 25861 | Value of the Taylor polyno... |
| taylply2 25862 | The Taylor polynomial is a... |
| taylply 25863 | The Taylor polynomial is a... |
| dvtaylp 25864 | The derivative of the Tayl... |
| dvntaylp 25865 | The ` M ` -th derivative o... |
| dvntaylp0 25866 | The first ` N ` derivative... |
| taylthlem1 25867 | Lemma for ~ taylth . This... |
| taylthlem2 25868 | Lemma for ~ taylth . (Con... |
| taylth 25869 | Taylor's theorem. The Tay... |
| ulmrel 25872 | The uniform limit relation... |
| ulmscl 25873 | Closure of the base set in... |
| ulmval 25874 | Express the predicate: Th... |
| ulmcl 25875 | Closure of a uniform limit... |
| ulmf 25876 | Closure of a uniform limit... |
| ulmpm 25877 | Closure of a uniform limit... |
| ulmf2 25878 | Closure of a uniform limit... |
| ulm2 25879 | Simplify ~ ulmval when ` F... |
| ulmi 25880 | The uniform limit property... |
| ulmclm 25881 | A uniform limit of functio... |
| ulmres 25882 | A sequence of functions co... |
| ulmshftlem 25883 | Lemma for ~ ulmshft . (Co... |
| ulmshft 25884 | A sequence of functions co... |
| ulm0 25885 | Every function converges u... |
| ulmuni 25886 | A sequence of functions un... |
| ulmdm 25887 | Two ways to express that a... |
| ulmcaulem 25888 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 25889 | A sequence of functions co... |
| ulmcau2 25890 | A sequence of functions co... |
| ulmss 25891 | A uniform limit of functio... |
| ulmbdd 25892 | A uniform limit of bounded... |
| ulmcn 25893 | A uniform limit of continu... |
| ulmdvlem1 25894 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 25895 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 25896 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 25897 | If ` F ` is a sequence of ... |
| mtest 25898 | The Weierstrass M-test. I... |
| mtestbdd 25899 | Given the hypotheses of th... |
| mbfulm 25900 | A uniform limit of measura... |
| iblulm 25901 | A uniform limit of integra... |
| itgulm 25902 | A uniform limit of integra... |
| itgulm2 25903 | A uniform limit of integra... |
| pserval 25904 | Value of the function ` G ... |
| pserval2 25905 | Value of the function ` G ... |
| psergf 25906 | The sequence of terms in t... |
| radcnvlem1 25907 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 25908 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 25909 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 25910 | Zero is always a convergen... |
| radcnvcl 25911 | The radius of convergence ... |
| radcnvlt1 25912 | If ` X ` is within the ope... |
| radcnvlt2 25913 | If ` X ` is within the ope... |
| radcnvle 25914 | If ` X ` is a convergent p... |
| dvradcnv 25915 | The radius of convergence ... |
| pserulm 25916 | If ` S ` is a region conta... |
| psercn2 25917 | Since by ~ pserulm the ser... |
| psercnlem2 25918 | Lemma for ~ psercn . (Con... |
| psercnlem1 25919 | Lemma for ~ psercn . (Con... |
| psercn 25920 | An infinite series converg... |
| pserdvlem1 25921 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 25922 | Lemma for ~ pserdv . (Con... |
| pserdv 25923 | The derivative of a power ... |
| pserdv2 25924 | The derivative of a power ... |
| abelthlem1 25925 | Lemma for ~ abelth . (Con... |
| abelthlem2 25926 | Lemma for ~ abelth . The ... |
| abelthlem3 25927 | Lemma for ~ abelth . (Con... |
| abelthlem4 25928 | Lemma for ~ abelth . (Con... |
| abelthlem5 25929 | Lemma for ~ abelth . (Con... |
| abelthlem6 25930 | Lemma for ~ abelth . (Con... |
| abelthlem7a 25931 | Lemma for ~ abelth . (Con... |
| abelthlem7 25932 | Lemma for ~ abelth . (Con... |
| abelthlem8 25933 | Lemma for ~ abelth . (Con... |
| abelthlem9 25934 | Lemma for ~ abelth . By a... |
| abelth 25935 | Abel's theorem. If the po... |
| abelth2 25936 | Abel's theorem, restricted... |
| efcn 25937 | The exponential function i... |
| sincn 25938 | Sine is continuous. (Cont... |
| coscn 25939 | Cosine is continuous. (Co... |
| reeff1olem 25940 | Lemma for ~ reeff1o . (Co... |
| reeff1o 25941 | The real exponential funct... |
| reefiso 25942 | The exponential function o... |
| efcvx 25943 | The exponential function o... |
| reefgim 25944 | The exponential function i... |
| pilem1 25945 | Lemma for ~ pire , ~ pigt2... |
| pilem2 25946 | Lemma for ~ pire , ~ pigt2... |
| pilem3 25947 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 25948 | ` _pi ` is between 2 and 4... |
| sinpi 25949 | The sine of ` _pi ` is 0. ... |
| pire 25950 | ` _pi ` is a real number. ... |
| picn 25951 | ` _pi ` is a complex numbe... |
| pipos 25952 | ` _pi ` is positive. (Con... |
| pirp 25953 | ` _pi ` is a positive real... |
| negpicn 25954 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 25955 | Lemma for ~ sinhalfpi and ... |
| halfpire 25956 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 25957 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 25958 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 25959 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 25960 | The sine of ` _pi / 2 ` is... |
| coshalfpi 25961 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 25962 | The cosine of ` -u _pi / 2... |
| efhalfpi 25963 | The exponential of ` _i _p... |
| cospi 25964 | The cosine of ` _pi ` is `... |
| efipi 25965 | The exponential of ` _i x.... |
| eulerid 25966 | Euler's identity. (Contri... |
| sin2pi 25967 | The sine of ` 2 _pi ` is 0... |
| cos2pi 25968 | The cosine of ` 2 _pi ` is... |
| ef2pi 25969 | The exponential of ` 2 _pi... |
| ef2kpi 25970 | If ` K ` is an integer, th... |
| efper 25971 | The exponential function i... |
| sinperlem 25972 | Lemma for ~ sinper and ~ c... |
| sinper 25973 | The sine function is perio... |
| cosper 25974 | The cosine function is per... |
| sin2kpi 25975 | If ` K ` is an integer, th... |
| cos2kpi 25976 | If ` K ` is an integer, th... |
| sin2pim 25977 | Sine of a number subtracte... |
| cos2pim 25978 | Cosine of a number subtrac... |
| sinmpi 25979 | Sine of a number less ` _p... |
| cosmpi 25980 | Cosine of a number less ` ... |
| sinppi 25981 | Sine of a number plus ` _p... |
| cosppi 25982 | Cosine of a number plus ` ... |
| efimpi 25983 | The exponential function a... |
| sinhalfpip 25984 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 25985 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 25986 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 25987 | The cosine of ` _pi / 2 ` ... |
| ptolemy 25988 | Ptolemy's Theorem. This t... |
| sincosq1lem 25989 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 25990 | The signs of the sine and ... |
| sincosq2sgn 25991 | The signs of the sine and ... |
| sincosq3sgn 25992 | The signs of the sine and ... |
| sincosq4sgn 25993 | The signs of the sine and ... |
| coseq00topi 25994 | Location of the zeroes of ... |
| coseq0negpitopi 25995 | Location of the zeroes of ... |
| tanrpcl 25996 | Positive real closure of t... |
| tangtx 25997 | The tangent function is gr... |
| tanabsge 25998 | The tangent function is gr... |
| sinq12gt0 25999 | The sine of a number stric... |
| sinq12ge0 26000 | The sine of a number betwe... |
| sinq34lt0t 26001 | The sine of a number stric... |
| cosq14gt0 26002 | The cosine of a number str... |
| cosq14ge0 26003 | The cosine of a number bet... |
| sincosq1eq 26004 | Complementarity of the sin... |
| sincos4thpi 26005 | The sine and cosine of ` _... |
| tan4thpi 26006 | The tangent of ` _pi / 4 `... |
| sincos6thpi 26007 | The sine and cosine of ` _... |
| sincos3rdpi 26008 | The sine and cosine of ` _... |
| pigt3 26009 | ` _pi ` is greater than 3.... |
| pige3 26010 | ` _pi ` is greater than or... |
| pige3ALT 26011 | Alternate proof of ~ pige3... |
| abssinper 26012 | The absolute value of sine... |
| sinkpi 26013 | The sine of an integer mul... |
| coskpi 26014 | The absolute value of the ... |
| sineq0 26015 | A complex number whose sin... |
| coseq1 26016 | A complex number whose cos... |
| cos02pilt1 26017 | Cosine is less than one be... |
| cosq34lt1 26018 | Cosine is less than one in... |
| efeq1 26019 | A complex number whose exp... |
| cosne0 26020 | The cosine function has no... |
| cosordlem 26021 | Lemma for ~ cosord . (Con... |
| cosord 26022 | Cosine is decreasing over ... |
| cos0pilt1 26023 | Cosine is between minus on... |
| cos11 26024 | Cosine is one-to-one over ... |
| sinord 26025 | Sine is increasing over th... |
| recosf1o 26026 | The cosine function is a b... |
| resinf1o 26027 | The sine function is a bij... |
| tanord1 26028 | The tangent function is st... |
| tanord 26029 | The tangent function is st... |
| tanregt0 26030 | The real part of the tange... |
| negpitopissre 26031 | The interval ` ( -u _pi (,... |
| efgh 26032 | The exponential function o... |
| efif1olem1 26033 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26034 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26035 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26036 | The exponential function o... |
| efif1o 26037 | The exponential function o... |
| efifo 26038 | The exponential function o... |
| eff1olem 26039 | The exponential function m... |
| eff1o 26040 | The exponential function m... |
| efabl 26041 | The image of a subgroup of... |
| efsubm 26042 | The image of a subgroup of... |
| circgrp 26043 | The circle group ` T ` is ... |
| circsubm 26044 | The circle group ` T ` is ... |
| logrn 26049 | The range of the natural l... |
| ellogrn 26050 | Write out the property ` A... |
| dflog2 26051 | The natural logarithm func... |
| relogrn 26052 | The range of the natural l... |
| logrncn 26053 | The range of the natural l... |
| eff1o2 26054 | The exponential function r... |
| logf1o 26055 | The natural logarithm func... |
| dfrelog 26056 | The natural logarithm func... |
| relogf1o 26057 | The natural logarithm func... |
| logrncl 26058 | Closure of the natural log... |
| logcl 26059 | Closure of the natural log... |
| logimcl 26060 | Closure of the imaginary p... |
| logcld 26061 | The logarithm of a nonzero... |
| logimcld 26062 | The imaginary part of the ... |
| logimclad 26063 | The imaginary part of the ... |
| abslogimle 26064 | The imaginary part of the ... |
| logrnaddcl 26065 | The range of the natural l... |
| relogcl 26066 | Closure of the natural log... |
| eflog 26067 | Relationship between the n... |
| logeq0im1 26068 | If the logarithm of a numb... |
| logccne0 26069 | The logarithm isn't 0 if i... |
| logne0 26070 | Logarithm of a non-1 posit... |
| reeflog 26071 | Relationship between the n... |
| logef 26072 | Relationship between the n... |
| relogef 26073 | Relationship between the n... |
| logeftb 26074 | Relationship between the n... |
| relogeftb 26075 | Relationship between the n... |
| log1 26076 | The natural logarithm of `... |
| loge 26077 | The natural logarithm of `... |
| logneg 26078 | The natural logarithm of a... |
| logm1 26079 | The natural logarithm of n... |
| lognegb 26080 | If a number has imaginary ... |
| relogoprlem 26081 | Lemma for ~ relogmul and ~... |
| relogmul 26082 | The natural logarithm of t... |
| relogdiv 26083 | The natural logarithm of t... |
| explog 26084 | Exponentiation of a nonzer... |
| reexplog 26085 | Exponentiation of a positi... |
| relogexp 26086 | The natural logarithm of p... |
| relog 26087 | Real part of a logarithm. ... |
| relogiso 26088 | The natural logarithm func... |
| reloggim 26089 | The natural logarithm is a... |
| logltb 26090 | The natural logarithm func... |
| logfac 26091 | The logarithm of a factori... |
| eflogeq 26092 | Solve an equation involvin... |
| logleb 26093 | Natural logarithm preserve... |
| rplogcl 26094 | Closure of the logarithm f... |
| logge0 26095 | The logarithm of a number ... |
| logcj 26096 | The natural logarithm dist... |
| efiarg 26097 | The exponential of the "ar... |
| cosargd 26098 | The cosine of the argument... |
| cosarg0d 26099 | The cosine of the argument... |
| argregt0 26100 | Closure of the argument of... |
| argrege0 26101 | Closure of the argument of... |
| argimgt0 26102 | Closure of the argument of... |
| argimlt0 26103 | Closure of the argument of... |
| logimul 26104 | Multiplying a number by ` ... |
| logneg2 26105 | The logarithm of the negat... |
| logmul2 26106 | Generalization of ~ relogm... |
| logdiv2 26107 | Generalization of ~ relogd... |
| abslogle 26108 | Bound on the magnitude of ... |
| tanarg 26109 | The basic relation between... |
| logdivlti 26110 | The ` log x / x ` function... |
| logdivlt 26111 | The ` log x / x ` function... |
| logdivle 26112 | The ` log x / x ` function... |
| relogcld 26113 | Closure of the natural log... |
| reeflogd 26114 | Relationship between the n... |
| relogmuld 26115 | The natural logarithm of t... |
| relogdivd 26116 | The natural logarithm of t... |
| logled 26117 | Natural logarithm preserve... |
| relogefd 26118 | Relationship between the n... |
| rplogcld 26119 | Closure of the logarithm f... |
| logge0d 26120 | The logarithm of a number ... |
| logge0b 26121 | The logarithm of a number ... |
| loggt0b 26122 | The logarithm of a number ... |
| logle1b 26123 | The logarithm of a number ... |
| loglt1b 26124 | The logarithm of a number ... |
| divlogrlim 26125 | The inverse logarithm func... |
| logno1 26126 | The logarithm function is ... |
| dvrelog 26127 | The derivative of the real... |
| relogcn 26128 | The real logarithm functio... |
| ellogdm 26129 | Elementhood in the "contin... |
| logdmn0 26130 | A number in the continuous... |
| logdmnrp 26131 | A number in the continuous... |
| logdmss 26132 | The continuity domain of `... |
| logcnlem2 26133 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26134 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26135 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26136 | Lemma for ~ logcn . (Cont... |
| logcn 26137 | The logarithm function is ... |
| dvloglem 26138 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26139 | The "continuous domain" of... |
| logf1o2 26140 | The logarithm maps its con... |
| dvlog 26141 | The derivative of the comp... |
| dvlog2lem 26142 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26143 | The derivative of the comp... |
| advlog 26144 | The antiderivative of the ... |
| advlogexp 26145 | The antiderivative of a po... |
| efopnlem1 26146 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26147 | Lemma for ~ efopn . (Cont... |
| efopn 26148 | The exponential map is an ... |
| logtayllem 26149 | Lemma for ~ logtayl . (Co... |
| logtayl 26150 | The Taylor series for ` -u... |
| logtaylsum 26151 | The Taylor series for ` -u... |
| logtayl2 26152 | Power series expression fo... |
| logccv 26153 | The natural logarithm func... |
| cxpval 26154 | Value of the complex power... |
| cxpef 26155 | Value of the complex power... |
| 0cxp 26156 | Value of the complex power... |
| cxpexpz 26157 | Relate the complex power f... |
| cxpexp 26158 | Relate the complex power f... |
| logcxp 26159 | Logarithm of a complex pow... |
| cxp0 26160 | Value of the complex power... |
| cxp1 26161 | Value of the complex power... |
| 1cxp 26162 | Value of the complex power... |
| ecxp 26163 | Write the exponential func... |
| cxpcl 26164 | Closure of the complex pow... |
| recxpcl 26165 | Real closure of the comple... |
| rpcxpcl 26166 | Positive real closure of t... |
| cxpne0 26167 | Complex exponentiation is ... |
| cxpeq0 26168 | Complex exponentiation is ... |
| cxpadd 26169 | Sum of exponents law for c... |
| cxpp1 26170 | Value of a nonzero complex... |
| cxpneg 26171 | Value of a complex number ... |
| cxpsub 26172 | Exponent subtraction law f... |
| cxpge0 26173 | Nonnegative exponentiation... |
| mulcxplem 26174 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26175 | Complex exponentiation of ... |
| cxprec 26176 | Complex exponentiation of ... |
| divcxp 26177 | Complex exponentiation of ... |
| cxpmul 26178 | Product of exponents law f... |
| cxpmul2 26179 | Product of exponents law f... |
| cxproot 26180 | The complex power function... |
| cxpmul2z 26181 | Generalize ~ cxpmul2 to ne... |
| abscxp 26182 | Absolute value of a power,... |
| abscxp2 26183 | Absolute value of a power,... |
| cxplt 26184 | Ordering property for comp... |
| cxple 26185 | Ordering property for comp... |
| cxplea 26186 | Ordering property for comp... |
| cxple2 26187 | Ordering property for comp... |
| cxplt2 26188 | Ordering property for comp... |
| cxple2a 26189 | Ordering property for comp... |
| cxplt3 26190 | Ordering property for comp... |
| cxple3 26191 | Ordering property for comp... |
| cxpsqrtlem 26192 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26193 | The complex exponential fu... |
| logsqrt 26194 | Logarithm of a square root... |
| cxp0d 26195 | Value of the complex power... |
| cxp1d 26196 | Value of the complex power... |
| 1cxpd 26197 | Value of the complex power... |
| cxpcld 26198 | Closure of the complex pow... |
| cxpmul2d 26199 | Product of exponents law f... |
| 0cxpd 26200 | Value of the complex power... |
| cxpexpzd 26201 | Relate the complex power f... |
| cxpefd 26202 | Value of the complex power... |
| cxpne0d 26203 | Complex exponentiation is ... |
| cxpp1d 26204 | Value of a nonzero complex... |
| cxpnegd 26205 | Value of a complex number ... |
| cxpmul2zd 26206 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26207 | Sum of exponents law for c... |
| cxpsubd 26208 | Exponent subtraction law f... |
| cxpltd 26209 | Ordering property for comp... |
| cxpled 26210 | Ordering property for comp... |
| cxplead 26211 | Ordering property for comp... |
| divcxpd 26212 | Complex exponentiation of ... |
| recxpcld 26213 | Positive real closure of t... |
| cxpge0d 26214 | Nonnegative exponentiation... |
| cxple2ad 26215 | Ordering property for comp... |
| cxplt2d 26216 | Ordering property for comp... |
| cxple2d 26217 | Ordering property for comp... |
| mulcxpd 26218 | Complex exponentiation of ... |
| cxpsqrtth 26219 | Square root theorem over t... |
| 2irrexpq 26220 | There exist irrational num... |
| cxprecd 26221 | Complex exponentiation of ... |
| rpcxpcld 26222 | Positive real closure of t... |
| logcxpd 26223 | Logarithm of a complex pow... |
| cxplt3d 26224 | Ordering property for comp... |
| cxple3d 26225 | Ordering property for comp... |
| cxpmuld 26226 | Product of exponents law f... |
| cxpcom 26227 | Commutative law for real e... |
| dvcxp1 26228 | The derivative of a comple... |
| dvcxp2 26229 | The derivative of a comple... |
| dvsqrt 26230 | The derivative of the real... |
| dvcncxp1 26231 | Derivative of complex powe... |
| dvcnsqrt 26232 | Derivative of square root ... |
| cxpcn 26233 | Domain of continuity of th... |
| cxpcn2 26234 | Continuity of the complex ... |
| cxpcn3lem 26235 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26236 | Extend continuity of the c... |
| resqrtcn 26237 | Continuity of the real squ... |
| sqrtcn 26238 | Continuity of the square r... |
| cxpaddlelem 26239 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26240 | Ordering property for comp... |
| abscxpbnd 26241 | Bound on the absolute valu... |
| root1id 26242 | Property of an ` N ` -th r... |
| root1eq1 26243 | The only powers of an ` N ... |
| root1cj 26244 | Within the ` N ` -th roots... |
| cxpeq 26245 | Solve an equation involvin... |
| loglesqrt 26246 | An upper bound on the loga... |
| logreclem 26247 | Symmetry of the natural lo... |
| logrec 26248 | Logarithm of a reciprocal ... |
| logbval 26251 | Define the value of the ` ... |
| logbcl 26252 | General logarithm closure.... |
| logbid1 26253 | General logarithm is 1 whe... |
| logb1 26254 | The logarithm of ` 1 ` to ... |
| elogb 26255 | The general logarithm of a... |
| logbchbase 26256 | Change of base for logarit... |
| relogbval 26257 | Value of the general logar... |
| relogbcl 26258 | Closure of the general log... |
| relogbzcl 26259 | Closure of the general log... |
| relogbreexp 26260 | Power law for the general ... |
| relogbzexp 26261 | Power law for the general ... |
| relogbmul 26262 | The logarithm of the produ... |
| relogbmulexp 26263 | The logarithm of the produ... |
| relogbdiv 26264 | The logarithm of the quoti... |
| relogbexp 26265 | Identity law for general l... |
| nnlogbexp 26266 | Identity law for general l... |
| logbrec 26267 | Logarithm of a reciprocal ... |
| logbleb 26268 | The general logarithm func... |
| logblt 26269 | The general logarithm func... |
| relogbcxp 26270 | Identity law for the gener... |
| cxplogb 26271 | Identity law for the gener... |
| relogbcxpb 26272 | The logarithm is the inver... |
| logbmpt 26273 | The general logarithm to a... |
| logbf 26274 | The general logarithm to a... |
| logbfval 26275 | The general logarithm of a... |
| relogbf 26276 | The general logarithm to a... |
| logblog 26277 | The general logarithm to t... |
| logbgt0b 26278 | The logarithm of a positiv... |
| logbgcd1irr 26279 | The logarithm of an intege... |
| 2logb9irr 26280 | Example for ~ logbgcd1irr ... |
| logbprmirr 26281 | The logarithm of a prime t... |
| 2logb3irr 26282 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26283 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26284 | The square root of two to ... |
| 2irrexpqALT 26285 | Alternate proof of ~ 2irre... |
| angval 26286 | Define the angle function,... |
| angcan 26287 | Cancel a constant multipli... |
| angneg 26288 | Cancel a negative sign in ... |
| angvald 26289 | The (signed) angle between... |
| angcld 26290 | The (signed) angle between... |
| angrteqvd 26291 | Two vectors are at a right... |
| cosangneg2d 26292 | The cosine of the angle be... |
| angrtmuld 26293 | Perpendicularity of two ve... |
| ang180lem1 26294 | Lemma for ~ ang180 . Show... |
| ang180lem2 26295 | Lemma for ~ ang180 . Show... |
| ang180lem3 26296 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26297 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26298 | Lemma for ~ ang180 : Redu... |
| ang180 26299 | The sum of angles ` m A B ... |
| lawcoslem1 26300 | Lemma for ~ lawcos . Here... |
| lawcos 26301 | Law of cosines (also known... |
| pythag 26302 | Pythagorean theorem. Give... |
| isosctrlem1 26303 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26304 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26305 | Lemma for ~ isosctr . Cor... |
| isosctr 26306 | Isosceles triangle theorem... |
| ssscongptld 26307 | If two triangles have equa... |
| affineequiv 26308 | Equivalence between two wa... |
| affineequiv2 26309 | Equivalence between two wa... |
| affineequiv3 26310 | Equivalence between two wa... |
| affineequiv4 26311 | Equivalence between two wa... |
| affineequivne 26312 | Equivalence between two wa... |
| angpieqvdlem 26313 | Equivalence used in the pr... |
| angpieqvdlem2 26314 | Equivalence used in ~ angp... |
| angpined 26315 | If the angle at ABC is ` _... |
| angpieqvd 26316 | The angle ABC is ` _pi ` i... |
| chordthmlem 26317 | If ` M ` is the midpoint o... |
| chordthmlem2 26318 | If M is the midpoint of AB... |
| chordthmlem3 26319 | If M is the midpoint of AB... |
| chordthmlem4 26320 | If P is on the segment AB ... |
| chordthmlem5 26321 | If P is on the segment AB ... |
| chordthm 26322 | The intersecting chords th... |
| heron 26323 | Heron's formula gives the ... |
| quad2 26324 | The quadratic equation, wi... |
| quad 26325 | The quadratic equation. (... |
| 1cubrlem 26326 | The cube roots of unity. ... |
| 1cubr 26327 | The cube roots of unity. ... |
| dcubic1lem 26328 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26329 | Reverse direction of ~ dcu... |
| dcubic1 26330 | Forward direction of ~ dcu... |
| dcubic 26331 | Solutions to the depressed... |
| mcubic 26332 | Solutions to a monic cubic... |
| cubic2 26333 | The solution to the genera... |
| cubic 26334 | The cubic equation, which ... |
| binom4 26335 | Work out a quartic binomia... |
| dquartlem1 26336 | Lemma for ~ dquart . (Con... |
| dquartlem2 26337 | Lemma for ~ dquart . (Con... |
| dquart 26338 | Solve a depressed quartic ... |
| quart1cl 26339 | Closure lemmas for ~ quart... |
| quart1lem 26340 | Lemma for ~ quart1 . (Con... |
| quart1 26341 | Depress a quartic equation... |
| quartlem1 26342 | Lemma for ~ quart . (Cont... |
| quartlem2 26343 | Closure lemmas for ~ quart... |
| quartlem3 26344 | Closure lemmas for ~ quart... |
| quartlem4 26345 | Closure lemmas for ~ quart... |
| quart 26346 | The quartic equation, writ... |
| asinlem 26353 | The argument to the logari... |
| asinlem2 26354 | The argument to the logari... |
| asinlem3a 26355 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26356 | The argument to the logari... |
| asinf 26357 | Domain and codomain of the... |
| asincl 26358 | Closure for the arcsin fun... |
| acosf 26359 | Domain and codoamin of the... |
| acoscl 26360 | Closure for the arccos fun... |
| atandm 26361 | Since the property is a li... |
| atandm2 26362 | This form of ~ atandm is a... |
| atandm3 26363 | A compact form of ~ atandm... |
| atandm4 26364 | A compact form of ~ atandm... |
| atanf 26365 | Domain and codoamin of the... |
| atancl 26366 | Closure for the arctan fun... |
| asinval 26367 | Value of the arcsin functi... |
| acosval 26368 | Value of the arccos functi... |
| atanval 26369 | Value of the arctan functi... |
| atanre 26370 | A real number is in the do... |
| asinneg 26371 | The arcsine function is od... |
| acosneg 26372 | The negative symmetry rela... |
| efiasin 26373 | The exponential of the arc... |
| sinasin 26374 | The arcsine function is an... |
| cosacos 26375 | The arccosine function is ... |
| asinsinlem 26376 | Lemma for ~ asinsin . (Co... |
| asinsin 26377 | The arcsine function compo... |
| acoscos 26378 | The arccosine function is ... |
| asin1 26379 | The arcsine of ` 1 ` is ` ... |
| acos1 26380 | The arccosine of ` 1 ` is ... |
| reasinsin 26381 | The arcsine function compo... |
| asinsinb 26382 | Relationship between sine ... |
| acoscosb 26383 | Relationship between cosin... |
| asinbnd 26384 | The arcsine function has r... |
| acosbnd 26385 | The arccosine function has... |
| asinrebnd 26386 | Bounds on the arcsine func... |
| asinrecl 26387 | The arcsine function is re... |
| acosrecl 26388 | The arccosine function is ... |
| cosasin 26389 | The cosine of the arcsine ... |
| sinacos 26390 | The sine of the arccosine ... |
| atandmneg 26391 | The domain of the arctange... |
| atanneg 26392 | The arctangent function is... |
| atan0 26393 | The arctangent of zero is ... |
| atandmcj 26394 | The arctangent function di... |
| atancj 26395 | The arctangent function di... |
| atanrecl 26396 | The arctangent function is... |
| efiatan 26397 | Value of the exponential o... |
| atanlogaddlem 26398 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26399 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26400 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26401 | A variation on ~ atanlogad... |
| efiatan2 26402 | Value of the exponential o... |
| 2efiatan 26403 | Value of the exponential o... |
| tanatan 26404 | The arctangent function is... |
| atandmtan 26405 | The tangent function has r... |
| cosatan 26406 | The cosine of an arctangen... |
| cosatanne0 26407 | The arctangent function ha... |
| atantan 26408 | The arctangent function is... |
| atantanb 26409 | Relationship between tange... |
| atanbndlem 26410 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26411 | The arctangent function is... |
| atanord 26412 | The arctangent function is... |
| atan1 26413 | The arctangent of ` 1 ` is... |
| bndatandm 26414 | A point in the open unit d... |
| atans 26415 | The "domain of continuity"... |
| atans2 26416 | It suffices to show that `... |
| atansopn 26417 | The domain of continuity o... |
| atansssdm 26418 | The domain of continuity o... |
| ressatans 26419 | The real number line is a ... |
| dvatan 26420 | The derivative of the arct... |
| atancn 26421 | The arctangent is a contin... |
| atantayl 26422 | The Taylor series for ` ar... |
| atantayl2 26423 | The Taylor series for ` ar... |
| atantayl3 26424 | The Taylor series for ` ar... |
| leibpilem1 26425 | Lemma for ~ leibpi . (Con... |
| leibpilem2 26426 | The Leibniz formula for ` ... |
| leibpi 26427 | The Leibniz formula for ` ... |
| leibpisum 26428 | The Leibniz formula for ` ... |
| log2cnv 26429 | Using the Taylor series fo... |
| log2tlbnd 26430 | Bound the error term in th... |
| log2ublem1 26431 | Lemma for ~ log2ub . The ... |
| log2ublem2 26432 | Lemma for ~ log2ub . (Con... |
| log2ublem3 26433 | Lemma for ~ log2ub . In d... |
| log2ub 26434 | ` log 2 ` is less than ` 2... |
| log2le1 26435 | ` log 2 ` is less than ` 1... |
| birthdaylem1 26436 | Lemma for ~ birthday . (C... |
| birthdaylem2 26437 | For general ` N ` and ` K ... |
| birthdaylem3 26438 | For general ` N ` and ` K ... |
| birthday 26439 | The Birthday Problem. The... |
| dmarea 26442 | The domain of the area fun... |
| areambl 26443 | The fibers of a measurable... |
| areass 26444 | A measurable region is a s... |
| dfarea 26445 | Rewrite ~ df-area self-ref... |
| areaf 26446 | Area measurement is a func... |
| areacl 26447 | The area of a measurable r... |
| areage0 26448 | The area of a measurable r... |
| areaval 26449 | The area of a measurable r... |
| rlimcnp 26450 | Relate a limit of a real-v... |
| rlimcnp2 26451 | Relate a limit of a real-v... |
| rlimcnp3 26452 | Relate a limit of a real-v... |
| xrlimcnp 26453 | Relate a limit of a real-v... |
| efrlim 26454 | The limit of the sequence ... |
| dfef2 26455 | The limit of the sequence ... |
| cxplim 26456 | A power to a negative expo... |
| sqrtlim 26457 | The inverse square root fu... |
| rlimcxp 26458 | Any power to a positive ex... |
| o1cxp 26459 | An eventually bounded func... |
| cxp2limlem 26460 | A linear factor grows slow... |
| cxp2lim 26461 | Any power grows slower tha... |
| cxploglim 26462 | The logarithm grows slower... |
| cxploglim2 26463 | Every power of the logarit... |
| divsqrtsumlem 26464 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 26465 | The function ` F ` used in... |
| divsqrsum 26466 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 26467 | A bound on the distance of... |
| divsqrtsumo1 26468 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 26469 | Closure of a 0-1 linear co... |
| scvxcvx 26470 | A strictly convex function... |
| jensenlem1 26471 | Lemma for ~ jensen . (Con... |
| jensenlem2 26472 | Lemma for ~ jensen . (Con... |
| jensen 26473 | Jensen's inequality, a fin... |
| amgmlem 26474 | Lemma for ~ amgm . (Contr... |
| amgm 26475 | Inequality of arithmetic a... |
| logdifbnd 26478 | Bound on the difference of... |
| logdiflbnd 26479 | Lower bound on the differe... |
| emcllem1 26480 | Lemma for ~ emcl . The se... |
| emcllem2 26481 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 26482 | Lemma for ~ emcl . The fu... |
| emcllem4 26483 | Lemma for ~ emcl . The di... |
| emcllem5 26484 | Lemma for ~ emcl . The pa... |
| emcllem6 26485 | Lemma for ~ emcl . By the... |
| emcllem7 26486 | Lemma for ~ emcl and ~ har... |
| emcl 26487 | Closure and bounds for the... |
| harmonicbnd 26488 | A bound on the harmonic se... |
| harmonicbnd2 26489 | A bound on the harmonic se... |
| emre 26490 | The Euler-Mascheroni const... |
| emgt0 26491 | The Euler-Mascheroni const... |
| harmonicbnd3 26492 | A bound on the harmonic se... |
| harmoniclbnd 26493 | A bound on the harmonic se... |
| harmonicubnd 26494 | A bound on the harmonic se... |
| harmonicbnd4 26495 | The asymptotic behavior of... |
| fsumharmonic 26496 | Bound a finite sum based o... |
| zetacvg 26499 | The zeta series is converg... |
| eldmgm 26506 | Elementhood in the set of ... |
| dmgmaddn0 26507 | If ` A ` is not a nonposit... |
| dmlogdmgm 26508 | If ` A ` is in the continu... |
| rpdmgm 26509 | A positive real number is ... |
| dmgmn0 26510 | If ` A ` is not a nonposit... |
| dmgmaddnn0 26511 | If ` A ` is not a nonposit... |
| dmgmdivn0 26512 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 26513 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 26514 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 26515 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 26516 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 26517 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 26518 | The series ` G ` is unifor... |
| lgamgulm 26519 | The series ` G ` is unifor... |
| lgamgulm2 26520 | Rewrite the limit of the s... |
| lgambdd 26521 | The log-Gamma function is ... |
| lgamucov 26522 | The ` U ` regions used in ... |
| lgamucov2 26523 | The ` U ` regions used in ... |
| lgamcvglem 26524 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 26525 | The log-Gamma function is ... |
| lgamf 26526 | The log-Gamma function is ... |
| gamf 26527 | The Gamma function is a co... |
| gamcl 26528 | The exponential of the log... |
| eflgam 26529 | The exponential of the log... |
| gamne0 26530 | The Gamma function is neve... |
| igamval 26531 | Value of the inverse Gamma... |
| igamz 26532 | Value of the inverse Gamma... |
| igamgam 26533 | Value of the inverse Gamma... |
| igamlgam 26534 | Value of the inverse Gamma... |
| igamf 26535 | Closure of the inverse Gam... |
| igamcl 26536 | Closure of the inverse Gam... |
| gamigam 26537 | The Gamma function is the ... |
| lgamcvg 26538 | The series ` G ` converges... |
| lgamcvg2 26539 | The series ` G ` converges... |
| gamcvg 26540 | The pointwise exponential ... |
| lgamp1 26541 | The functional equation of... |
| gamp1 26542 | The functional equation of... |
| gamcvg2lem 26543 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 26544 | An infinite product expres... |
| regamcl 26545 | The Gamma function is real... |
| relgamcl 26546 | The log-Gamma function is ... |
| rpgamcl 26547 | The log-Gamma function is ... |
| lgam1 26548 | The log-Gamma function at ... |
| gam1 26549 | The log-Gamma function at ... |
| facgam 26550 | The Gamma function general... |
| gamfac 26551 | The Gamma function general... |
| wilthlem1 26552 | The only elements that are... |
| wilthlem2 26553 | Lemma for ~ wilth : induct... |
| wilthlem3 26554 | Lemma for ~ wilth . Here ... |
| wilth 26555 | Wilson's theorem. A numbe... |
| wilthimp 26556 | The forward implication of... |
| ftalem1 26557 | Lemma for ~ fta : "growth... |
| ftalem2 26558 | Lemma for ~ fta . There e... |
| ftalem3 26559 | Lemma for ~ fta . There e... |
| ftalem4 26560 | Lemma for ~ fta : Closure... |
| ftalem5 26561 | Lemma for ~ fta : Main pr... |
| ftalem6 26562 | Lemma for ~ fta : Dischar... |
| ftalem7 26563 | Lemma for ~ fta . Shift t... |
| fta 26564 | The Fundamental Theorem of... |
| basellem1 26565 | Lemma for ~ basel . Closu... |
| basellem2 26566 | Lemma for ~ basel . Show ... |
| basellem3 26567 | Lemma for ~ basel . Using... |
| basellem4 26568 | Lemma for ~ basel . By ~ ... |
| basellem5 26569 | Lemma for ~ basel . Using... |
| basellem6 26570 | Lemma for ~ basel . The f... |
| basellem7 26571 | Lemma for ~ basel . The f... |
| basellem8 26572 | Lemma for ~ basel . The f... |
| basellem9 26573 | Lemma for ~ basel . Since... |
| basel 26574 | The sum of the inverse squ... |
| efnnfsumcl 26587 | Finite sum closure in the ... |
| ppisval 26588 | The set of primes less tha... |
| ppisval2 26589 | The set of primes less tha... |
| ppifi 26590 | The set of primes less tha... |
| prmdvdsfi 26591 | The set of prime divisors ... |
| chtf 26592 | Domain and codoamin of the... |
| chtcl 26593 | Real closure of the Chebys... |
| chtval 26594 | Value of the Chebyshev fun... |
| efchtcl 26595 | The Chebyshev function is ... |
| chtge0 26596 | The Chebyshev function is ... |
| vmaval 26597 | Value of the von Mangoldt ... |
| isppw 26598 | Two ways to say that ` A `... |
| isppw2 26599 | Two ways to say that ` A `... |
| vmappw 26600 | Value of the von Mangoldt ... |
| vmaprm 26601 | Value of the von Mangoldt ... |
| vmacl 26602 | Closure for the von Mangol... |
| vmaf 26603 | Functionality of the von M... |
| efvmacl 26604 | The von Mangoldt is closed... |
| vmage0 26605 | The von Mangoldt function ... |
| chpval 26606 | Value of the second Chebys... |
| chpf 26607 | Functionality of the secon... |
| chpcl 26608 | Closure for the second Che... |
| efchpcl 26609 | The second Chebyshev funct... |
| chpge0 26610 | The second Chebyshev funct... |
| ppival 26611 | Value of the prime-countin... |
| ppival2 26612 | Value of the prime-countin... |
| ppival2g 26613 | Value of the prime-countin... |
| ppif 26614 | Domain and codomain of the... |
| ppicl 26615 | Real closure of the prime-... |
| muval 26616 | The value of the Möbi... |
| muval1 26617 | The value of the Möbi... |
| muval2 26618 | The value of the Möbi... |
| isnsqf 26619 | Two ways to say that a num... |
| issqf 26620 | Two ways to say that a num... |
| sqfpc 26621 | The prime count of a squar... |
| dvdssqf 26622 | A divisor of a squarefree ... |
| sqf11 26623 | A squarefree number is com... |
| muf 26624 | The Möbius function i... |
| mucl 26625 | Closure of the Möbius... |
| sgmval 26626 | The value of the divisor f... |
| sgmval2 26627 | The value of the divisor f... |
| 0sgm 26628 | The value of the sum-of-di... |
| sgmf 26629 | The divisor function is a ... |
| sgmcl 26630 | Closure of the divisor fun... |
| sgmnncl 26631 | Closure of the divisor fun... |
| mule1 26632 | The Möbius function t... |
| chtfl 26633 | The Chebyshev function doe... |
| chpfl 26634 | The second Chebyshev funct... |
| ppiprm 26635 | The prime-counting functio... |
| ppinprm 26636 | The prime-counting functio... |
| chtprm 26637 | The Chebyshev function at ... |
| chtnprm 26638 | The Chebyshev function at ... |
| chpp1 26639 | The second Chebyshev funct... |
| chtwordi 26640 | The Chebyshev function is ... |
| chpwordi 26641 | The second Chebyshev funct... |
| chtdif 26642 | The difference of the Cheb... |
| efchtdvds 26643 | The exponentiated Chebyshe... |
| ppifl 26644 | The prime-counting functio... |
| ppip1le 26645 | The prime-counting functio... |
| ppiwordi 26646 | The prime-counting functio... |
| ppidif 26647 | The difference of the prim... |
| ppi1 26648 | The prime-counting functio... |
| cht1 26649 | The Chebyshev function at ... |
| vma1 26650 | The von Mangoldt function ... |
| chp1 26651 | The second Chebyshev funct... |
| ppi1i 26652 | Inference form of ~ ppiprm... |
| ppi2i 26653 | Inference form of ~ ppinpr... |
| ppi2 26654 | The prime-counting functio... |
| ppi3 26655 | The prime-counting functio... |
| cht2 26656 | The Chebyshev function at ... |
| cht3 26657 | The Chebyshev function at ... |
| ppinncl 26658 | Closure of the prime-count... |
| chtrpcl 26659 | Closure of the Chebyshev f... |
| ppieq0 26660 | The prime-counting functio... |
| ppiltx 26661 | The prime-counting functio... |
| prmorcht 26662 | Relate the primorial (prod... |
| mumullem1 26663 | Lemma for ~ mumul . A mul... |
| mumullem2 26664 | Lemma for ~ mumul . The p... |
| mumul 26665 | The Möbius function i... |
| sqff1o 26666 | There is a bijection from ... |
| fsumdvdsdiaglem 26667 | A "diagonal commutation" o... |
| fsumdvdsdiag 26668 | A "diagonal commutation" o... |
| fsumdvdscom 26669 | A double commutation of di... |
| dvdsppwf1o 26670 | A bijection from the divis... |
| dvdsflf1o 26671 | A bijection from the numbe... |
| dvdsflsumcom 26672 | A sum commutation from ` s... |
| fsumfldivdiaglem 26673 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 26674 | The right-hand side of ~ d... |
| musum 26675 | The sum of the Möbius... |
| musumsum 26676 | Evaluate a collapsing sum ... |
| muinv 26677 | The Möbius inversion ... |
| dvdsmulf1o 26678 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 26679 | Product of two divisor sum... |
| sgmppw 26680 | The value of the divisor f... |
| 0sgmppw 26681 | A prime power ` P ^ K ` ha... |
| 1sgmprm 26682 | The sum of divisors for a ... |
| 1sgm2ppw 26683 | The sum of the divisors of... |
| sgmmul 26684 | The divisor function for f... |
| ppiublem1 26685 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 26686 | A prime greater than ` 3 `... |
| ppiub 26687 | An upper bound on the prim... |
| vmalelog 26688 | The von Mangoldt function ... |
| chtlepsi 26689 | The first Chebyshev functi... |
| chprpcl 26690 | Closure of the second Cheb... |
| chpeq0 26691 | The second Chebyshev funct... |
| chteq0 26692 | The first Chebyshev functi... |
| chtleppi 26693 | Upper bound on the ` theta... |
| chtublem 26694 | Lemma for ~ chtub . (Cont... |
| chtub 26695 | An upper bound on the Cheb... |
| fsumvma 26696 | Rewrite a sum over the von... |
| fsumvma2 26697 | Apply ~ fsumvma for the co... |
| pclogsum 26698 | The logarithmic analogue o... |
| vmasum 26699 | The sum of the von Mangold... |
| logfac2 26700 | Another expression for the... |
| chpval2 26701 | Express the second Chebysh... |
| chpchtsum 26702 | The second Chebyshev funct... |
| chpub 26703 | An upper bound on the seco... |
| logfacubnd 26704 | A simple upper bound on th... |
| logfaclbnd 26705 | A lower bound on the logar... |
| logfacbnd3 26706 | Show the stronger statemen... |
| logfacrlim 26707 | Combine the estimates ~ lo... |
| logexprlim 26708 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 26709 | Write out ~ logfacrlim as ... |
| mersenne 26710 | A Mersenne prime is a prim... |
| perfect1 26711 | Euclid's contribution to t... |
| perfectlem1 26712 | Lemma for ~ perfect . (Co... |
| perfectlem2 26713 | Lemma for ~ perfect . (Co... |
| perfect 26714 | The Euclid-Euler theorem, ... |
| dchrval 26717 | Value of the group of Diri... |
| dchrbas 26718 | Base set of the group of D... |
| dchrelbas 26719 | A Dirichlet character is a... |
| dchrelbas2 26720 | A Dirichlet character is a... |
| dchrelbas3 26721 | A Dirichlet character is a... |
| dchrelbasd 26722 | A Dirichlet character is a... |
| dchrrcl 26723 | Reverse closure for a Diri... |
| dchrmhm 26724 | A Dirichlet character is a... |
| dchrf 26725 | A Dirichlet character is a... |
| dchrelbas4 26726 | A Dirichlet character is a... |
| dchrzrh1 26727 | Value of a Dirichlet chara... |
| dchrzrhcl 26728 | A Dirichlet character take... |
| dchrzrhmul 26729 | A Dirichlet character is c... |
| dchrplusg 26730 | Group operation on the gro... |
| dchrmul 26731 | Group operation on the gro... |
| dchrmulcl 26732 | Closure of the group opera... |
| dchrn0 26733 | A Dirichlet character is n... |
| dchr1cl 26734 | Closure of the principal D... |
| dchrmullid 26735 | Left identity for the prin... |
| dchrinvcl 26736 | Closure of the group inver... |
| dchrabl 26737 | The set of Dirichlet chara... |
| dchrfi 26738 | The group of Dirichlet cha... |
| dchrghm 26739 | A Dirichlet character rest... |
| dchr1 26740 | Value of the principal Dir... |
| dchreq 26741 | A Dirichlet character is d... |
| dchrresb 26742 | A Dirichlet character is d... |
| dchrabs 26743 | A Dirichlet character take... |
| dchrinv 26744 | The inverse of a Dirichlet... |
| dchrabs2 26745 | A Dirichlet character take... |
| dchr1re 26746 | The principal Dirichlet ch... |
| dchrptlem1 26747 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 26748 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 26749 | Lemma for ~ dchrpt . (Con... |
| dchrpt 26750 | For any element other than... |
| dchrsum2 26751 | An orthogonality relation ... |
| dchrsum 26752 | An orthogonality relation ... |
| sumdchr2 26753 | Lemma for ~ sumdchr . (Co... |
| dchrhash 26754 | There are exactly ` phi ( ... |
| sumdchr 26755 | An orthogonality relation ... |
| dchr2sum 26756 | An orthogonality relation ... |
| sum2dchr 26757 | An orthogonality relation ... |
| bcctr 26758 | Value of the central binom... |
| pcbcctr 26759 | Prime count of a central b... |
| bcmono 26760 | The binomial coefficient i... |
| bcmax 26761 | The binomial coefficient t... |
| bcp1ctr 26762 | Ratio of two central binom... |
| bclbnd 26763 | A bound on the binomial co... |
| efexple 26764 | Convert a bound on a power... |
| bpos1lem 26765 | Lemma for ~ bpos1 . (Cont... |
| bpos1 26766 | Bertrand's postulate, chec... |
| bposlem1 26767 | An upper bound on the prim... |
| bposlem2 26768 | There are no odd primes in... |
| bposlem3 26769 | Lemma for ~ bpos . Since ... |
| bposlem4 26770 | Lemma for ~ bpos . (Contr... |
| bposlem5 26771 | Lemma for ~ bpos . Bound ... |
| bposlem6 26772 | Lemma for ~ bpos . By usi... |
| bposlem7 26773 | Lemma for ~ bpos . The fu... |
| bposlem8 26774 | Lemma for ~ bpos . Evalua... |
| bposlem9 26775 | Lemma for ~ bpos . Derive... |
| bpos 26776 | Bertrand's postulate: ther... |
| zabsle1 26779 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 26780 | When ` a ` is coprime to t... |
| lgslem2 26781 | The set ` Z ` of all integ... |
| lgslem3 26782 | The set ` Z ` of all integ... |
| lgslem4 26783 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 26784 | Value of the Legendre symb... |
| lgsfval 26785 | Value of the function ` F ... |
| lgsfcl2 26786 | The function ` F ` is clos... |
| lgscllem 26787 | The Legendre symbol is an ... |
| lgsfcl 26788 | Closure of the function ` ... |
| lgsfle1 26789 | The function ` F ` has mag... |
| lgsval2lem 26790 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 26791 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 26792 | The Legendre symbol is an ... |
| lgs0 26793 | The Legendre symbol when t... |
| lgscl 26794 | The Legendre symbol is an ... |
| lgsle1 26795 | The Legendre symbol has ab... |
| lgsval2 26796 | The Legendre symbol at a p... |
| lgs2 26797 | The Legendre symbol at ` 2... |
| lgsval3 26798 | The Legendre symbol at an ... |
| lgsvalmod 26799 | The Legendre symbol is equ... |
| lgsval4 26800 | Restate ~ lgsval for nonze... |
| lgsfcl3 26801 | Closure of the function ` ... |
| lgsval4a 26802 | Same as ~ lgsval4 for posi... |
| lgscl1 26803 | The value of the Legendre ... |
| lgsneg 26804 | The Legendre symbol is eit... |
| lgsneg1 26805 | The Legendre symbol for no... |
| lgsmod 26806 | The Legendre (Jacobi) symb... |
| lgsdilem 26807 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 26808 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 26809 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 26810 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 26811 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 26812 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 26813 | The Legendre symbol is com... |
| lgsdirprm 26814 | The Legendre symbol is com... |
| lgsdir 26815 | The Legendre symbol is com... |
| lgsdilem2 26816 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 26817 | The Legendre symbol is com... |
| lgsne0 26818 | The Legendre symbol is non... |
| lgsabs1 26819 | The Legendre symbol is non... |
| lgssq 26820 | The Legendre symbol at a s... |
| lgssq2 26821 | The Legendre symbol at a s... |
| lgsprme0 26822 | The Legendre symbol at any... |
| 1lgs 26823 | The Legendre symbol at ` 1... |
| lgs1 26824 | The Legendre symbol at ` 1... |
| lgsmodeq 26825 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 26826 | The Legendre (Jacobi) symb... |
| lgsdirnn0 26827 | Variation on ~ lgsdir vali... |
| lgsdinn0 26828 | Variation on ~ lgsdi valid... |
| lgsqrlem1 26829 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 26830 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 26831 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 26832 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 26833 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 26834 | The Legendre symbol for od... |
| lgsqrmod 26835 | If the Legendre symbol of ... |
| lgsqrmodndvds 26836 | If the Legendre symbol of ... |
| lgsdchrval 26837 | The Legendre symbol functi... |
| lgsdchr 26838 | The Legendre symbol functi... |
| gausslemma2dlem0a 26839 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 26840 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 26841 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 26842 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 26843 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 26844 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 26845 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 26846 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 26847 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 26848 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 26849 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 26850 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 26851 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 26852 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 26853 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 26854 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 26855 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 26856 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 26857 | Gauss' Lemma (see also the... |
| lgseisenlem1 26858 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 26859 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 26860 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 26861 | Lemma for ~ lgseisen . Th... |
| lgseisen 26862 | Eisenstein's lemma, an exp... |
| lgsquadlem1 26863 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 26864 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 26865 | Lemma for ~ lgsquad . (Co... |
| lgsquad 26866 | The Law of Quadratic Recip... |
| lgsquad2lem1 26867 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 26868 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 26869 | Extend ~ lgsquad to coprim... |
| lgsquad3 26870 | Extend ~ lgsquad2 to integ... |
| m1lgs 26871 | The first supplement to th... |
| 2lgslem1a1 26872 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 26873 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 26874 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 26875 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 26876 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 26877 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 26878 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 26879 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 26880 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 26881 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 26882 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 26883 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 26884 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 26885 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 26886 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 26887 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 26888 | The Legendre symbol for ` ... |
| 2lgslem4 26889 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 26890 | The second supplement to t... |
| 2lgsoddprmlem1 26891 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 26892 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 26893 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 26894 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 26895 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 26896 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 26897 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 26898 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 26899 | The second supplement to t... |
| 2sqlem1 26900 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 26901 | Lemma for ~ 2sq . (Contri... |
| mul2sq 26902 | Fibonacci's identity (actu... |
| 2sqlem3 26903 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 26904 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 26905 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 26906 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 26907 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 26908 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 26909 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 26910 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 26911 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 26912 | Lemma for ~ 2sq . (Contri... |
| 2sq 26913 | All primes of the form ` 4... |
| 2sqblem 26914 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 26915 | The converse to ~ 2sq . (... |
| 2sq2 26916 | ` 2 ` is the sum of square... |
| 2sqn0 26917 | If the sum of two squares ... |
| 2sqcoprm 26918 | If the sum of two squares ... |
| 2sqmod 26919 | Given two decompositions o... |
| 2sqmo 26920 | There exists at most one d... |
| 2sqnn0 26921 | All primes of the form ` 4... |
| 2sqnn 26922 | All primes of the form ` 4... |
| addsq2reu 26923 | For each complex number ` ... |
| addsqn2reu 26924 | For each complex number ` ... |
| addsqrexnreu 26925 | For each complex number, t... |
| addsqnreup 26926 | There is no unique decompo... |
| addsq2nreurex 26927 | For each complex number ` ... |
| addsqn2reurex2 26928 | For each complex number ` ... |
| 2sqreulem1 26929 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 26930 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 26931 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 26932 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 26933 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 26934 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 26935 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 26936 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 26937 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 26938 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 26939 | There exists a unique deco... |
| 2sqreunn 26940 | There exists a unique deco... |
| 2sqreult 26941 | There exists a unique deco... |
| 2sqreultb 26942 | There exists a unique deco... |
| 2sqreunnlt 26943 | There exists a unique deco... |
| 2sqreunnltb 26944 | There exists a unique deco... |
| 2sqreuop 26945 | There exists a unique deco... |
| 2sqreuopnn 26946 | There exists a unique deco... |
| 2sqreuoplt 26947 | There exists a unique deco... |
| 2sqreuopltb 26948 | There exists a unique deco... |
| 2sqreuopnnlt 26949 | There exists a unique deco... |
| 2sqreuopnnltb 26950 | There exists a unique deco... |
| 2sqreuopb 26951 | There exists a unique deco... |
| chebbnd1lem1 26952 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 26953 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 26954 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 26955 | The Chebyshev bound: The ... |
| chtppilimlem1 26956 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 26957 | Lemma for ~ chtppilim . (... |
| chtppilim 26958 | The ` theta ` function is ... |
| chto1ub 26959 | The ` theta ` function is ... |
| chebbnd2 26960 | The Chebyshev bound, part ... |
| chto1lb 26961 | The ` theta ` function is ... |
| chpchtlim 26962 | The ` psi ` and ` theta ` ... |
| chpo1ub 26963 | The ` psi ` function is up... |
| chpo1ubb 26964 | The ` psi ` function is up... |
| vmadivsum 26965 | The sum of the von Mangold... |
| vmadivsumb 26966 | Give a total bound on the ... |
| rplogsumlem1 26967 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 26968 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 26969 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 26970 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 26971 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 26972 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 26973 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 26974 | Lemma for ~ dchrisum . Le... |
| dchrisum 26975 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 26976 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 26977 | The sum of the Möbius... |
| dchrvmasumlem1 26978 | An alternative expression ... |
| dchrvmasum2lem 26979 | Give an expression for ` l... |
| dchrvmasum2if 26980 | Combine the results of ~ d... |
| dchrvmasumlem2 26981 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 26982 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 26983 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 26984 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 26985 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 26986 | An asymptotic approximatio... |
| dchrvmaeq0 26987 | The set ` W ` is the colle... |
| dchrisum0fval 26988 | Value of the function ` F ... |
| dchrisum0fmul 26989 | The function ` F ` , the d... |
| dchrisum0ff 26990 | The function ` F ` is a re... |
| dchrisum0flblem1 26991 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 26992 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 26993 | The divisor sum of a real ... |
| dchrisum0fno1 26994 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 26995 | A partial result along the... |
| dchrisum0re 26996 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 26997 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 26998 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 26999 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27000 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27001 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27002 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27003 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27004 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27005 | The sum of the Möbius... |
| dchrvmasumlem 27006 | The sum of the Möbius... |
| dchrmusum 27007 | The sum of the Möbius... |
| dchrvmasum 27008 | The sum of the von Mangold... |
| rpvmasum 27009 | The sum of the von Mangold... |
| rplogsum 27010 | The sum of ` log p / p ` o... |
| dirith2 27011 | Dirichlet's theorem: there... |
| dirith 27012 | Dirichlet's theorem: there... |
| mudivsum 27013 | Asymptotic formula for ` s... |
| mulogsumlem 27014 | Lemma for ~ mulogsum . (C... |
| mulogsum 27015 | Asymptotic formula for ... |
| logdivsum 27016 | Asymptotic analysis of ... |
| mulog2sumlem1 27017 | Asymptotic formula for ... |
| mulog2sumlem2 27018 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27019 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27020 | Asymptotic formula for ... |
| vmalogdivsum2 27021 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27022 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27023 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27024 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27025 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27026 | The Möbius inverse of... |
| log2sumbnd 27027 | Bound on the difference be... |
| selberglem1 27028 | Lemma for ~ selberg . Est... |
| selberglem2 27029 | Lemma for ~ selberg . (Co... |
| selberglem3 27030 | Lemma for ~ selberg . Est... |
| selberg 27031 | Selberg's symmetry formula... |
| selbergb 27032 | Convert eventual boundedne... |
| selberg2lem 27033 | Lemma for ~ selberg2 . Eq... |
| selberg2 27034 | Selberg's symmetry formula... |
| selberg2b 27035 | Convert eventual boundedne... |
| chpdifbndlem1 27036 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27037 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27038 | A bound on the difference ... |
| logdivbnd 27039 | A bound on a sum of logs, ... |
| selberg3lem1 27040 | Introduce a log weighting ... |
| selberg3lem2 27041 | Lemma for ~ selberg3 . Eq... |
| selberg3 27042 | Introduce a log weighting ... |
| selberg4lem1 27043 | Lemma for ~ selberg4 . Eq... |
| selberg4 27044 | The Selberg symmetry formu... |
| pntrval 27045 | Define the residual of the... |
| pntrf 27046 | Functionality of the resid... |
| pntrmax 27047 | There is a bound on the re... |
| pntrsumo1 27048 | A bound on a sum over ` R ... |
| pntrsumbnd 27049 | A bound on a sum over ` R ... |
| pntrsumbnd2 27050 | A bound on a sum over ` R ... |
| selbergr 27051 | Selberg's symmetry formula... |
| selberg3r 27052 | Selberg's symmetry formula... |
| selberg4r 27053 | Selberg's symmetry formula... |
| selberg34r 27054 | The sum of ~ selberg3r and... |
| pntsval 27055 | Define the "Selberg functi... |
| pntsf 27056 | Functionality of the Selbe... |
| selbergs 27057 | Selberg's symmetry formula... |
| selbergsb 27058 | Selberg's symmetry formula... |
| pntsval2 27059 | The Selberg function can b... |
| pntrlog2bndlem1 27060 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27061 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27062 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27063 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27064 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27065 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27066 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27067 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27068 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27069 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27070 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27071 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27072 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27073 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27074 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27075 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27076 | Lemma for ~ pnt . Establi... |
| pntlemd 27077 | Lemma for ~ pnt . Closure... |
| pntlemc 27078 | Lemma for ~ pnt . Closure... |
| pntlema 27079 | Lemma for ~ pnt . Closure... |
| pntlemb 27080 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27081 | Lemma for ~ pnt . Closure... |
| pntlemh 27082 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27083 | Lemma for ~ pnt . The "na... |
| pntlemq 27084 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27085 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27086 | Lemma for ~ pnt . The ind... |
| pntlemi 27087 | Lemma for ~ pnt . Elimina... |
| pntlemf 27088 | Lemma for ~ pnt . Add up ... |
| pntlemk 27089 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27090 | Lemma for ~ pnt . Combine... |
| pntleme 27091 | Lemma for ~ pnt . Package... |
| pntlem3 27092 | Lemma for ~ pnt . Equatio... |
| pntlemp 27093 | Lemma for ~ pnt . Wrappin... |
| pntleml 27094 | Lemma for ~ pnt . Equatio... |
| pnt3 27095 | The Prime Number Theorem, ... |
| pnt2 27096 | The Prime Number Theorem, ... |
| pnt 27097 | The Prime Number Theorem: ... |
| abvcxp 27098 | Raising an absolute value ... |
| padicfval 27099 | Value of the p-adic absolu... |
| padicval 27100 | Value of the p-adic absolu... |
| ostth2lem1 27101 | Lemma for ~ ostth2 , altho... |
| qrngbas 27102 | The base set of the field ... |
| qdrng 27103 | The rationals form a divis... |
| qrng0 27104 | The zero element of the fi... |
| qrng1 27105 | The unity element of the f... |
| qrngneg 27106 | The additive inverse in th... |
| qrngdiv 27107 | The division operation in ... |
| qabvle 27108 | By using induction on ` N ... |
| qabvexp 27109 | Induct the product rule ~ ... |
| ostthlem1 27110 | Lemma for ~ ostth . If tw... |
| ostthlem2 27111 | Lemma for ~ ostth . Refin... |
| qabsabv 27112 | The regular absolute value... |
| padicabv 27113 | The p-adic absolute value ... |
| padicabvf 27114 | The p-adic absolute value ... |
| padicabvcxp 27115 | All positive powers of the... |
| ostth1 27116 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27117 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27118 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27119 | Lemma for ~ ostth2 . (Con... |
| ostth2 27120 | - Lemma for ~ ostth : regu... |
| ostth3 27121 | - Lemma for ~ ostth : p-ad... |
| ostth 27122 | Ostrowski's theorem, which... |
| elno 27129 | Membership in the surreals... |
| sltval 27130 | The value of the surreal l... |
| bdayval 27131 | The value of the birthday ... |
| nofun 27132 | A surreal is a function. ... |
| nodmon 27133 | The domain of a surreal is... |
| norn 27134 | The range of a surreal is ... |
| nofnbday 27135 | A surreal is a function ov... |
| nodmord 27136 | The domain of a surreal ha... |
| elno2 27137 | An alternative condition f... |
| elno3 27138 | Another condition for memb... |
| sltval2 27139 | Alternate expression for s... |
| nofv 27140 | The function value of a su... |
| nosgnn0 27141 | ` (/) ` is not a surreal s... |
| nosgnn0i 27142 | If ` X ` is a surreal sign... |
| noreson 27143 | The restriction of a surre... |
| sltintdifex 27144 |
If ` A |
| sltres 27145 | If the restrictions of two... |
| noxp1o 27146 | The Cartesian product of a... |
| noseponlem 27147 | Lemma for ~ nosepon . Con... |
| nosepon 27148 | Given two unequal surreals... |
| noextend 27149 | Extending a surreal by one... |
| noextendseq 27150 | Extend a surreal by a sequ... |
| noextenddif 27151 | Calculate the place where ... |
| noextendlt 27152 | Extending a surreal with a... |
| noextendgt 27153 | Extending a surreal with a... |
| nolesgn2o 27154 | Given ` A ` less-than or e... |
| nolesgn2ores 27155 | Given ` A ` less-than or e... |
| nogesgn1o 27156 | Given ` A ` greater than o... |
| nogesgn1ores 27157 | Given ` A ` greater than o... |
| sltsolem1 27158 | Lemma for ~ sltso . The "... |
| sltso 27159 | Less-than totally orders t... |
| bdayfo 27160 | The birthday function maps... |
| fvnobday 27161 | The value of a surreal at ... |
| nosepnelem 27162 | Lemma for ~ nosepne . (Co... |
| nosepne 27163 | The value of two non-equal... |
| nosep1o 27164 | If the value of a surreal ... |
| nosep2o 27165 | If the value of a surreal ... |
| nosepdmlem 27166 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27167 | The first place two surrea... |
| nosepeq 27168 | The values of two surreals... |
| nosepssdm 27169 | Given two non-equal surrea... |
| nodenselem4 27170 | Lemma for ~ nodense . Sho... |
| nodenselem5 27171 | Lemma for ~ nodense . If ... |
| nodenselem6 27172 | The restriction of a surre... |
| nodenselem7 27173 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27174 | Lemma for ~ nodense . Giv... |
| nodense 27175 | Given two distinct surreal... |
| bdayimaon 27176 | Lemma for full-eta propert... |
| nolt02olem 27177 | Lemma for ~ nolt02o . If ... |
| nolt02o 27178 | Given ` A ` less-than ` B ... |
| nogt01o 27179 | Given ` A ` greater than `... |
| noresle 27180 | Restriction law for surrea... |
| nomaxmo 27181 | A class of surreals has at... |
| nominmo 27182 | A class of surreals has at... |
| nosupprefixmo 27183 | In any class of surreals, ... |
| noinfprefixmo 27184 | In any class of surreals, ... |
| nosupcbv 27185 | Lemma to change bound vari... |
| nosupno 27186 | The next several theorems ... |
| nosupdm 27187 | The domain of the surreal ... |
| nosupbday 27188 | Birthday bounding law for ... |
| nosupfv 27189 | The value of surreal supre... |
| nosupres 27190 | A restriction law for surr... |
| nosupbnd1lem1 27191 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27192 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27193 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27194 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27195 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27196 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27197 | Bounding law from below fo... |
| nosupbnd2lem1 27198 | Bounding law from above wh... |
| nosupbnd2 27199 | Bounding law from above fo... |
| noinfcbv 27200 | Change bound variables for... |
| noinfno 27201 | The next several theorems ... |
| noinfdm 27202 | Next, we calculate the dom... |
| noinfbday 27203 | Birthday bounding law for ... |
| noinffv 27204 | The value of surreal infim... |
| noinfres 27205 | The restriction of surreal... |
| noinfbnd1lem1 27206 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27207 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27208 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27209 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27210 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27211 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27212 | Bounding law from above fo... |
| noinfbnd2lem1 27213 | Bounding law from below wh... |
| noinfbnd2 27214 | Bounding law from below fo... |
| nosupinfsep 27215 | Given two sets of surreals... |
| noetasuplem1 27216 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27217 | Lemma for ~ noeta . The r... |
| noetasuplem3 27218 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27219 | Lemma for ~ noeta . When ... |
| noetainflem1 27220 | Lemma for ~ noeta . Estab... |
| noetainflem2 27221 | Lemma for ~ noeta . The r... |
| noetainflem3 27222 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27223 | Lemma for ~ noeta . If ` ... |
| noetalem1 27224 | Lemma for ~ noeta . Eithe... |
| noetalem2 27225 | Lemma for ~ noeta . The f... |
| noeta 27226 | The full-eta axiom for the... |
| sltirr 27229 | Surreal less-than is irref... |
| slttr 27230 | Surreal less-than is trans... |
| sltasym 27231 | Surreal less-than is asymm... |
| sltlin 27232 | Surreal less-than obeys tr... |
| slttrieq2 27233 | Trichotomy law for surreal... |
| slttrine 27234 | Trichotomy law for surreal... |
| slenlt 27235 | Surreal less-than or equal... |
| sltnle 27236 | Surreal less-than in terms... |
| sleloe 27237 | Surreal less-than or equal... |
| sletri3 27238 | Trichotomy law for surreal... |
| sltletr 27239 | Surreal transitive law. (... |
| slelttr 27240 | Surreal transitive law. (... |
| sletr 27241 | Surreal transitive law. (... |
| slttrd 27242 | Surreal less-than is trans... |
| sltletrd 27243 | Surreal less-than is trans... |
| slelttrd 27244 | Surreal less-than is trans... |
| sletrd 27245 | Surreal less-than or equal... |
| slerflex 27246 | Surreal less-than or equal... |
| sletric 27247 | Surreal trichotomy law. (... |
| maxs1 27248 | A surreal is less than or ... |
| maxs2 27249 | A surreal is less than or ... |
| mins1 27250 | The minimum of two surreal... |
| mins2 27251 | The minimum of two surreal... |
| sltled 27252 | Surreal less-than implies ... |
| sltne 27253 | Surreal less-than implies ... |
| bdayfun 27254 | The birthday function is a... |
| bdayfn 27255 | The birthday function is a... |
| bdaydm 27256 | The birthday function's do... |
| bdayrn 27257 | The birthday function's ra... |
| bdayelon 27258 | The value of the birthday ... |
| nocvxminlem 27259 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27260 | Given a nonempty convex cl... |
| noprc 27261 | The surreal numbers are a ... |
| noeta2 27266 | A version of ~ noeta with ... |
| brsslt 27267 | Binary relation form of th... |
| ssltex1 27268 | The first argument of surr... |
| ssltex2 27269 | The second argument of sur... |
| ssltss1 27270 | The first argument of surr... |
| ssltss2 27271 | The second argument of sur... |
| ssltsep 27272 | The separation property of... |
| ssltd 27273 | Deduce surreal set less-th... |
| ssltsepc 27274 | Two elements of separated ... |
| ssltsepcd 27275 | Two elements of separated ... |
| sssslt1 27276 | Relation between surreal s... |
| sssslt2 27277 | Relation between surreal s... |
| nulsslt 27278 | The empty set is less-than... |
| nulssgt 27279 | The empty set is greater t... |
| conway 27280 | Conway's Simplicity Theore... |
| scutval 27281 | The value of the surreal c... |
| scutcut 27282 | Cut properties of the surr... |
| scutcl 27283 | Closure law for surreal cu... |
| scutcld 27284 | Closure law for surreal cu... |
| scutbday 27285 | The birthday of the surrea... |
| eqscut 27286 | Condition for equality to ... |
| eqscut2 27287 | Condition for equality to ... |
| sslttr 27288 | Transitive law for surreal... |
| ssltun1 27289 | Union law for surreal set ... |
| ssltun2 27290 | Union law for surreal set ... |
| scutun12 27291 | Union law for surreal cuts... |
| dmscut 27292 | The domain of the surreal ... |
| scutf 27293 | Functionality statement fo... |
| etasslt 27294 | A restatement of ~ noeta u... |
| etasslt2 27295 | A version of ~ etasslt wit... |
| scutbdaybnd 27296 | An upper bound on the birt... |
| scutbdaybnd2 27297 | An upper bound on the birt... |
| scutbdaybnd2lim 27298 | An upper bound on the birt... |
| scutbdaylt 27299 | If a surreal lies in a gap... |
| slerec 27300 | A comparison law for surre... |
| sltrec 27301 | A comparison law for surre... |
| ssltdisj 27302 | If ` A ` preceeds ` B ` , ... |
| 0sno 27307 | Surreal zero is a surreal.... |
| 1sno 27308 | Surreal one is a surreal. ... |
| bday0s 27309 | Calculate the birthday of ... |
| 0slt1s 27310 | Surreal zero is less than ... |
| bday0b 27311 | The only surreal with birt... |
| bday1s 27312 | The birthday of surreal on... |
| cuteq0 27313 | Condition for a surreal cu... |
| cuteq1 27314 | Condition for a surreal cu... |
| sgt0ne0 27315 | A positive surreal is not ... |
| sgt0ne0d 27316 | A positive surreal is not ... |
| madeval 27327 | The value of the made by f... |
| madeval2 27328 | Alternative characterizati... |
| oldval 27329 | The value of the old optio... |
| newval 27330 | The value of the new optio... |
| madef 27331 | The made function is a fun... |
| oldf 27332 | The older function is a fu... |
| newf 27333 | The new function is a func... |
| old0 27334 | No surreal is older than `... |
| madessno 27335 | Made sets are surreals. (... |
| oldssno 27336 | Old sets are surreals. (C... |
| newssno 27337 | New sets are surreals. (C... |
| leftval 27338 | The value of the left opti... |
| rightval 27339 | The value of the right opt... |
| leftf 27340 | The functionality of the l... |
| rightf 27341 | The functionality of the r... |
| elmade 27342 | Membership in the made fun... |
| elmade2 27343 | Membership in the made fun... |
| elold 27344 | Membership in an old set. ... |
| ssltleft 27345 | A surreal is greater than ... |
| ssltright 27346 | A surreal is less than its... |
| lltropt 27347 | The left options of a surr... |
| made0 27348 | The only surreal made on d... |
| new0 27349 | The only surreal new on da... |
| old1 27350 | The only surreal older tha... |
| madess 27351 | If ` A ` is less than or e... |
| oldssmade 27352 | The older-than set is a su... |
| leftssold 27353 | The left options are a sub... |
| rightssold 27354 | The right options are a su... |
| leftssno 27355 | The left set of a surreal ... |
| rightssno 27356 | The right set of a surreal... |
| madecut 27357 | Given a section that is a ... |
| madeun 27358 | The made set is the union ... |
| madeoldsuc 27359 | The made set is the old se... |
| oldsuc 27360 | The value of the old set a... |
| oldlim 27361 | The value of the old set a... |
| madebdayim 27362 | If a surreal is a member o... |
| oldbdayim 27363 | If ` X ` is in the old set... |
| oldirr 27364 | No surreal is a member of ... |
| leftirr 27365 | No surreal is a member of ... |
| rightirr 27366 | No surreal is a member of ... |
| left0s 27367 | The left set of ` 0s ` is ... |
| right0s 27368 | The right set of ` 0s ` is... |
| left1s 27369 | The left set of ` 1s ` is ... |
| right1s 27370 | The right set of ` 1s ` is... |
| lrold 27371 | The union of the left and ... |
| madebdaylemold 27372 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27373 | Lemma for ~ madebday . If... |
| madebday 27374 | A surreal is part of the s... |
| oldbday 27375 | A surreal is part of the s... |
| newbday 27376 | A surreal is an element of... |
| lrcut 27377 | A surreal is equal to the ... |
| scutfo 27378 | The surreal cut function i... |
| sltn0 27379 | If ` X ` is less than ` Y ... |
| lruneq 27380 | If two surreals share a bi... |
| sltlpss 27381 | If two surreals share a bi... |
| 0elold 27382 | Zero is in the old set of ... |
| 0elleft 27383 | Zero is in the left set of... |
| 0elright 27384 | Zero is in the right set o... |
| cofsslt 27385 | If every element of ` A ` ... |
| coinitsslt 27386 | If ` B ` is coinitial with... |
| cofcut1 27387 | If ` C ` is cofinal with `... |
| cofcut1d 27388 | If ` C ` is cofinal with `... |
| cofcut2 27389 | If ` A ` and ` C ` are mut... |
| cofcut2d 27390 | If ` A ` and ` C ` are mut... |
| cofcutr 27391 | If ` X ` is the cut of ` A... |
| cofcutr1d 27392 | If ` X ` is the cut of ` A... |
| cofcutr2d 27393 | If ` X ` is the cut of ` A... |
| cofcutrtime 27394 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 27395 | If ` X ` is a timely cut o... |
| cofcutrtime2d 27396 | If ` X ` is a timely cut o... |
| cofss 27397 | Cofinality for a subset. ... |
| coiniss 27398 | Coinitiality for a subset.... |
| cutlt 27399 | Eliminating all elements b... |
| cutpos 27400 | Reduce the elements of a c... |
| lrrecval 27403 | The next step in the devel... |
| lrrecval2 27404 | Next, we establish an alte... |
| lrrecpo 27405 | Now, we establish that ` R... |
| lrrecse 27406 | Next, we show that ` R ` i... |
| lrrecfr 27407 | Now we show that ` R ` is ... |
| lrrecpred 27408 | Finally, we calculate the ... |
| noinds 27409 | Induction principle for a ... |
| norecfn 27410 | Surreal recursion over one... |
| norecov 27411 | Calculate the value of the... |
| noxpordpo 27414 | To get through most of the... |
| noxpordfr 27415 | Next we establish the foun... |
| noxpordse 27416 | Next we establish the set-... |
| noxpordpred 27417 | Next we calculate the pred... |
| no2indslem 27418 | Double induction on surrea... |
| no2inds 27419 | Double induction on surrea... |
| norec2fn 27420 | The double-recursion opera... |
| norec2ov 27421 | The value of the double-re... |
| no3inds 27422 | Triple induction over surr... |
| addsfn 27425 | Surreal addition is a func... |
| addsval 27426 | The value of surreal addit... |
| addsval2 27427 | The value of surreal addit... |
| addsrid 27428 | Surreal addition to zero i... |
| addsridd 27429 | Surreal addition to zero i... |
| addscom 27430 | Surreal addition commutes.... |
| addscomd 27431 | Surreal addition commutes.... |
| addslid 27432 | Surreal addition to zero i... |
| addsproplem1 27433 | Lemma for surreal addition... |
| addsproplem2 27434 | Lemma for surreal addition... |
| addsproplem3 27435 | Lemma for surreal addition... |
| addsproplem4 27436 | Lemma for surreal addition... |
| addsproplem5 27437 | Lemma for surreal addition... |
| addsproplem6 27438 | Lemma for surreal addition... |
| addsproplem7 27439 | Lemma for surreal addition... |
| addsprop 27440 | Inductively show that surr... |
| addscutlem 27441 | Lemma for ~ addscut . Sho... |
| addscut 27442 | Demonstrate the cut proper... |
| addscut2 27443 | Show that the cut involved... |
| addscld 27444 | Surreal numbers are closed... |
| addscl 27445 | Surreal numbers are closed... |
| addsf 27446 | Function statement for sur... |
| addsfo 27447 | Surreal addition is onto. ... |
| sltadd1im 27448 | Surreal less-than is prese... |
| sltadd2im 27449 | Surreal less-than is prese... |
| sleadd1im 27450 | Surreal less-than or equal... |
| sleadd2im 27451 | Surreal less-than or equal... |
| sleadd1 27452 | Addition to both sides of ... |
| sleadd2 27453 | Addition to both sides of ... |
| sltadd2 27454 | Addition to both sides of ... |
| sltadd1 27455 | Addition to both sides of ... |
| addscan2 27456 | Cancellation law for surre... |
| addscan1 27457 | Cancellation law for surre... |
| sleadd1d 27458 | Addition to both sides of ... |
| sleadd2d 27459 | Addition to both sides of ... |
| sltadd2d 27460 | Addition to both sides of ... |
| sltadd1d 27461 | Addition to both sides of ... |
| addscan2d 27462 | Cancellation law for surre... |
| addscan1d 27463 | Cancellation law for surre... |
| addsuniflem 27464 | Lemma for ~ addsunif . St... |
| addsunif 27465 | Uniformity theorem for sur... |
| addsasslem1 27466 | Lemma for addition associa... |
| addsasslem2 27467 | Lemma for addition associa... |
| addsass 27468 | Surreal addition is associ... |
| addsassd 27469 | Surreal addition is associ... |
| adds32d 27470 | Commutative/associative la... |
| adds12d 27471 | Commutative/associative la... |
| adds4d 27472 | Rearrangement of four term... |
| adds42d 27473 | Rearrangement of four term... |
| negsfn 27478 | Surreal negation is a func... |
| subsfn 27479 | Surreal subtraction is a f... |
| negsval 27480 | The value of the surreal n... |
| negs0s 27481 | Negative surreal zero is s... |
| negsproplem1 27482 | Lemma for surreal negation... |
| negsproplem2 27483 | Lemma for surreal negation... |
| negsproplem3 27484 | Lemma for surreal negation... |
| negsproplem4 27485 | Lemma for surreal negation... |
| negsproplem5 27486 | Lemma for surreal negation... |
| negsproplem6 27487 | Lemma for surreal negation... |
| negsproplem7 27488 | Lemma for surreal negation... |
| negsprop 27489 | Show closure and ordering ... |
| negscl 27490 | The surreals are closed un... |
| negscld 27491 | The surreals are closed un... |
| sltnegim 27492 | The forward direction of t... |
| negscut 27493 | The cut properties of surr... |
| negscut2 27494 | The cut that defines surre... |
| negsid 27495 | Surreal addition of a numb... |
| negsidd 27496 | Surreal addition of a numb... |
| negsex 27497 | Every surreal has a negati... |
| negnegs 27498 | A surreal is equal to the ... |
| sltneg 27499 | Negative of both sides of ... |
| sleneg 27500 | Negative of both sides of ... |
| sltnegd 27501 | Negative of both sides of ... |
| slenegd 27502 | Negative of both sides of ... |
| negs11 27503 | Surreal negation is one-to... |
| negsdi 27504 | Distribution of surreal ne... |
| slt0neg2d 27505 | Comparison of a surreal an... |
| negsf 27506 | Function statement for sur... |
| negsfo 27507 | Function statement for sur... |
| negsf1o 27508 | Surreal negation is a bije... |
| negsunif 27509 | Uniformity property for su... |
| negsbdaylem 27510 | Lemma for ~ negsbday . Bo... |
| negsbday 27511 | Negation of a surreal numb... |
| subsval 27512 | The value of surreal subtr... |
| subsvald 27513 | The value of surreal subtr... |
| subscl 27514 | Closure law for surreal su... |
| subscld 27515 | Closure law for surreal su... |
| subsid1 27516 | Identity law for subtracti... |
| subsid 27517 | Subtraction of a surreal f... |
| subadds 27518 | Relationship between addit... |
| subaddsd 27519 | Relationship between addit... |
| pncans 27520 | Cancellation law for surre... |
| pncan3s 27521 | Subtraction and addition o... |
| npcans 27522 | Cancellation law for surre... |
| sltsub1 27523 | Subtraction from both side... |
| sltsub2 27524 | Subtraction from both side... |
| sltsub1d 27525 | Subtraction from both side... |
| sltsub2d 27526 | Subtraction from both side... |
| negsubsdi2d 27527 | Distribution of negative o... |
| addsubsassd 27528 | Associative-type law for s... |
| addsubsd 27529 | Law for surreal addition a... |
| sltsubsubbd 27530 | Equivalence for the surrea... |
| sltsubsub2bd 27531 | Equivalence for the surrea... |
| sltsubsub3bd 27532 | Equivalence for the surrea... |
| slesubsubbd 27533 | Equivalence for the surrea... |
| slesubsub2bd 27534 | Equivalence for the surrea... |
| slesubsub3bd 27535 | Equivalence for the surrea... |
| sltsubaddd 27536 | Surreal less-than relation... |
| sltsubadd2d 27537 | Surreal less-than relation... |
| sltaddsubd 27538 | Surreal less-than relation... |
| sltaddsub2d 27539 | Surreal less-than relation... |
| subsubs4d 27540 | Law for double surreal sub... |
| posdifsd 27541 | Comparison of two surreals... |
| mulsfn 27544 | Surreal multiplication is ... |
| mulsval 27545 | The value of surreal multi... |
| mulsval2lem 27546 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 27547 | The value of surreal multi... |
| muls01 27548 | Surreal multiplication by ... |
| mulsrid 27549 | Surreal one is a right ide... |
| mulsridd 27550 | Surreal one is a right ide... |
| mulsproplemcbv 27551 | Lemma for surreal multipli... |
| mulsproplem1 27552 | Lemma for surreal multipli... |
| mulsproplem2 27553 | Lemma for surreal multipli... |
| mulsproplem3 27554 | Lemma for surreal multipli... |
| mulsproplem4 27555 | Lemma for surreal multipli... |
| mulsproplem5 27556 | Lemma for surreal multipli... |
| mulsproplem6 27557 | Lemma for surreal multipli... |
| mulsproplem7 27558 | Lemma for surreal multipli... |
| mulsproplem8 27559 | Lemma for surreal multipli... |
| mulsproplem9 27560 | Lemma for surreal multipli... |
| mulsproplem10 27561 | Lemma for surreal multipli... |
| mulsproplem11 27562 | Lemma for surreal multipli... |
| mulsproplem12 27563 | Lemma for surreal multipli... |
| mulsproplem13 27564 | Lemma for surreal multipli... |
| mulsproplem14 27565 | Lemma for surreal multipli... |
| mulsprop 27566 | Surreals are closed under ... |
| mulscutlem 27567 | Lemma for ~ mulscut . Sta... |
| mulscut 27568 | Show the cut properties of... |
| mulscut2 27569 | Show that the cut involved... |
| mulscl 27570 | The surreals are closed un... |
| mulscld 27571 | The surreals are closed un... |
| sltmul 27572 | An ordering relationship f... |
| sltmuld 27573 | An ordering relationship f... |
| slemuld 27574 | An ordering relationship f... |
| mulscom 27575 | Surreal multiplication com... |
| mulscomd 27576 | Surreal multiplication com... |
| muls02 27577 | Surreal multiplication by ... |
| mulslid 27578 | Surreal one is a left iden... |
| mulslidd 27579 | Surreal one is a left iden... |
| mulsgt0 27580 | The product of two positiv... |
| mulsgt0d 27581 | The product of two positiv... |
| ssltmul1 27582 | One surreal set less-than ... |
| ssltmul2 27583 | One surreal set less-than ... |
| mulsuniflem 27584 | Lemma for ~ mulsunif . St... |
| mulsunif 27585 | Surreal multiplication has... |
| addsdilem1 27586 | Lemma for surreal distribu... |
| addsdilem2 27587 | Lemma for surreal distribu... |
| addsdilem3 27588 | Lemma for ~ addsdi . Show... |
| addsdilem4 27589 | Lemma for ~ addsdi . Show... |
| addsdi 27590 | Distributive law for surre... |
| addsdid 27591 | Distributive law for surre... |
| addsdird 27592 | Distributive law for surre... |
| subsdid 27593 | Distribution of surreal mu... |
| subsdird 27594 | Distribution of surreal mu... |
| mulnegs1d 27595 | Product with negative is n... |
| mulnegs2d 27596 | Product with negative is n... |
| mul2negsd 27597 | Surreal product of two neg... |
| mulsasslem1 27598 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 27599 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 27600 | Lemma for ~ mulsass . Dem... |
| mulsass 27601 | Associative law for surrea... |
| mulsassd 27602 | Associative law for surrea... |
| sltmul2 27603 | Multiplication of both sid... |
| sltmul2d 27604 | Multiplication of both sid... |
| sltmul1d 27605 | Multiplication of both sid... |
| slemul2d 27606 | Multiplication of both sid... |
| slemul1d 27607 | Multiplication of both sid... |
| sltmulneg1d 27608 | Multiplication of both sid... |
| sltmulneg2d 27609 | Multiplication of both sid... |
| mulscan2dlem 27610 | Lemma for ~ mulscan2d . C... |
| mulscan2d 27611 | Cancellation of surreal mu... |
| mulscan1d 27612 | Cancellation of surreal mu... |
| muls12d 27613 | Commutative/associative la... |
| divsmo 27614 | Uniqueness of surreal inve... |
| divsval 27617 | The value of surreal divis... |
| norecdiv 27618 | If a surreal has a recipro... |
| noreceuw 27619 | If a surreal has a recipro... |
| divsmulw 27620 | Relationship between surre... |
| divsmulwd 27621 | Relationship between surre... |
| divsclw 27622 | Weak division closure law.... |
| divsclwd 27623 | Weak division closure law.... |
| divscan2wd 27624 | A weak cancellation law fo... |
| divscan1wd 27625 | A weak cancellation law fo... |
| sltdivmulwd 27626 | Surreal less-than relation... |
| sltdivmul2wd 27627 | Surreal less-than relation... |
| sltmuldivwd 27628 | Surreal less-than relation... |
| sltmuldiv2wd 27629 | Surreal less-than relation... |
| divsasswd 27630 | An associative law for sur... |
| divs1 27631 | A surreal divided by one i... |
| precsexlemcbv 27632 | Lemma for surreal reciproc... |
| precsexlem1 27633 | Lemma for surreal reciproc... |
| precsexlem2 27634 | Lemma for surreal reciproc... |
| precsexlem3 27635 | Lemma for surreal reciproc... |
| precsexlem4 27636 | Lemma for surreal reciproc... |
| precsexlem5 27637 | Lemma for surreal reciproc... |
| precsexlem6 27638 | Lemma for surreal reciproc... |
| precsexlem7 27639 | Lemma for surreal reciproc... |
| precsexlem8 27640 | Lemma for surreal reciproc... |
| precsexlem9 27641 | Lemma for surreal reciproc... |
| precsexlem10 27642 | Lemma for surreal reciproc... |
| precsexlem11 27643 | Lemma for surreal reciproc... |
| precsex 27644 | Every positive surreal has... |
| recsex 27645 | A non-zero surreal has a r... |
| recsexd 27646 | A non-zero surreal has a r... |
| divsmul 27647 | Relationship between surre... |
| divsmuld 27648 | Relationship between surre... |
| divscl 27649 | Surreal division closure l... |
| divscld 27650 | Surreal division closure l... |
| divscan2d 27651 | A cancellation law for sur... |
| divscan1d 27652 | A cancellation law for sur... |
| sltdivmuld 27653 | Surreal less-than relation... |
| sltdivmul2d 27654 | Surreal less-than relation... |
| sltmuldivd 27655 | Surreal less-than relation... |
| sltmuldiv2d 27656 | Surreal less-than relation... |
| divsassd 27657 | An associative law for sur... |
| itvndx 27668 | Index value of the Interva... |
| lngndx 27669 | Index value of the "line" ... |
| itvid 27670 | Utility theorem: index-ind... |
| lngid 27671 | Utility theorem: index-ind... |
| slotsinbpsd 27672 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 27673 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 27674 | The slot for the line is n... |
| trkgstr 27675 | Functionality of a Tarski ... |
| trkgbas 27676 | The base set of a Tarski g... |
| trkgdist 27677 | The measure of a distance ... |
| trkgitv 27678 | The congruence relation in... |
| istrkgc 27685 | Property of being a Tarski... |
| istrkgb 27686 | Property of being a Tarski... |
| istrkgcb 27687 | Property of being a Tarski... |
| istrkge 27688 | Property of fulfilling Euc... |
| istrkgl 27689 | Building lines from the se... |
| istrkgld 27690 | Property of fulfilling the... |
| istrkg2ld 27691 | Property of fulfilling the... |
| istrkg3ld 27692 | Property of fulfilling the... |
| axtgcgrrflx 27693 | Axiom of reflexivity of co... |
| axtgcgrid 27694 | Axiom of identity of congr... |
| axtgsegcon 27695 | Axiom of segment construct... |
| axtg5seg 27696 | Five segments axiom, Axiom... |
| axtgbtwnid 27697 | Identity of Betweenness. ... |
| axtgpasch 27698 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 27699 | Axiom of Continuity. Axio... |
| axtgcont 27700 | Axiom of Continuity. Axio... |
| axtglowdim2 27701 | Lower dimension axiom for ... |
| axtgupdim2 27702 | Upper dimension axiom for ... |
| axtgeucl 27703 | Euclid's Axiom. Axiom A10... |
| tgjustf 27704 | Given any function ` F ` ,... |
| tgjustr 27705 | Given any equivalence rela... |
| tgjustc1 27706 | A justification for using ... |
| tgjustc2 27707 | A justification for using ... |
| tgcgrcomimp 27708 | Congruence commutes on the... |
| tgcgrcomr 27709 | Congruence commutes on the... |
| tgcgrcoml 27710 | Congruence commutes on the... |
| tgcgrcomlr 27711 | Congruence commutes on bot... |
| tgcgreqb 27712 | Congruence and equality. ... |
| tgcgreq 27713 | Congruence and equality. ... |
| tgcgrneq 27714 | Congruence and equality. ... |
| tgcgrtriv 27715 | Degenerate segments are co... |
| tgcgrextend 27716 | Link congruence over a pai... |
| tgsegconeq 27717 | Two points that satisfy th... |
| tgbtwntriv2 27718 | Betweenness always holds f... |
| tgbtwncom 27719 | Betweenness commutes. The... |
| tgbtwncomb 27720 | Betweenness commutes, bico... |
| tgbtwnne 27721 | Betweenness and inequality... |
| tgbtwntriv1 27722 | Betweenness always holds f... |
| tgbtwnswapid 27723 | If you can swap the first ... |
| tgbtwnintr 27724 | Inner transitivity law for... |
| tgbtwnexch3 27725 | Exchange the first endpoin... |
| tgbtwnouttr2 27726 | Outer transitivity law for... |
| tgbtwnexch2 27727 | Exchange the outer point o... |
| tgbtwnouttr 27728 | Outer transitivity law for... |
| tgbtwnexch 27729 | Outer transitivity law for... |
| tgtrisegint 27730 | A line segment between two... |
| tglowdim1 27731 | Lower dimension axiom for ... |
| tglowdim1i 27732 | Lower dimension axiom for ... |
| tgldimor 27733 | Excluded-middle like state... |
| tgldim0eq 27734 | In dimension zero, any two... |
| tgldim0itv 27735 | In dimension zero, any two... |
| tgldim0cgr 27736 | In dimension zero, any two... |
| tgbtwndiff 27737 | There is always a ` c ` di... |
| tgdim01 27738 | In geometries of dimension... |
| tgifscgr 27739 | Inner five segment congrue... |
| tgcgrsub 27740 | Removing identical parts f... |
| iscgrg 27743 | The congruence property fo... |
| iscgrgd 27744 | The property for two seque... |
| iscgrglt 27745 | The property for two seque... |
| trgcgrg 27746 | The property for two trian... |
| trgcgr 27747 | Triangle congruence. (Con... |
| ercgrg 27748 | The shape congruence relat... |
| tgcgrxfr 27749 | A line segment can be divi... |
| cgr3id 27750 | Reflexivity law for three-... |
| cgr3simp1 27751 | Deduce segment congruence ... |
| cgr3simp2 27752 | Deduce segment congruence ... |
| cgr3simp3 27753 | Deduce segment congruence ... |
| cgr3swap12 27754 | Permutation law for three-... |
| cgr3swap23 27755 | Permutation law for three-... |
| cgr3swap13 27756 | Permutation law for three-... |
| cgr3rotr 27757 | Permutation law for three-... |
| cgr3rotl 27758 | Permutation law for three-... |
| trgcgrcom 27759 | Commutative law for three-... |
| cgr3tr 27760 | Transitivity law for three... |
| tgbtwnxfr 27761 | A condition for extending ... |
| tgcgr4 27762 | Two quadrilaterals to be c... |
| isismt 27765 | Property of being an isome... |
| ismot 27766 | Property of being an isome... |
| motcgr 27767 | Property of a motion: dist... |
| idmot 27768 | The identity is a motion. ... |
| motf1o 27769 | Motions are bijections. (... |
| motcl 27770 | Closure of motions. (Cont... |
| motco 27771 | The composition of two mot... |
| cnvmot 27772 | The converse of a motion i... |
| motplusg 27773 | The operation for motions ... |
| motgrp 27774 | The motions of a geometry ... |
| motcgrg 27775 | Property of a motion: dist... |
| motcgr3 27776 | Property of a motion: dist... |
| tglng 27777 | Lines of a Tarski Geometry... |
| tglnfn 27778 | Lines as functions. (Cont... |
| tglnunirn 27779 | Lines are sets of points. ... |
| tglnpt 27780 | Lines are sets of points. ... |
| tglngne 27781 | It takes two different poi... |
| tglngval 27782 | The line going through poi... |
| tglnssp 27783 | Lines are subset of the ge... |
| tgellng 27784 | Property of lying on the l... |
| tgcolg 27785 | We choose the notation ` (... |
| btwncolg1 27786 | Betweenness implies coline... |
| btwncolg2 27787 | Betweenness implies coline... |
| btwncolg3 27788 | Betweenness implies coline... |
| colcom 27789 | Swapping the points defini... |
| colrot1 27790 | Rotating the points defini... |
| colrot2 27791 | Rotating the points defini... |
| ncolcom 27792 | Swapping non-colinear poin... |
| ncolrot1 27793 | Rotating non-colinear poin... |
| ncolrot2 27794 | Rotating non-colinear poin... |
| tgdim01ln 27795 | In geometries of dimension... |
| ncoltgdim2 27796 | If there are three non-col... |
| lnxfr 27797 | Transfer law for colineari... |
| lnext 27798 | Extend a line with a missi... |
| tgfscgr 27799 | Congruence law for the gen... |
| lncgr 27800 | Congruence rule for lines.... |
| lnid 27801 | Identity law for points on... |
| tgidinside 27802 | Law for finding a point in... |
| tgbtwnconn1lem1 27803 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 27804 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 27805 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 27806 | Connectivity law for betwe... |
| tgbtwnconn2 27807 | Another connectivity law f... |
| tgbtwnconn3 27808 | Inner connectivity law for... |
| tgbtwnconnln3 27809 | Derive colinearity from be... |
| tgbtwnconn22 27810 | Double connectivity law fo... |
| tgbtwnconnln1 27811 | Derive colinearity from be... |
| tgbtwnconnln2 27812 | Derive colinearity from be... |
| legval 27815 | Value of the less-than rel... |
| legov 27816 | Value of the less-than rel... |
| legov2 27817 | An equivalent definition o... |
| legid 27818 | Reflexivity of the less-th... |
| btwnleg 27819 | Betweenness implies less-t... |
| legtrd 27820 | Transitivity of the less-t... |
| legtri3 27821 | Equality from the less-tha... |
| legtrid 27822 | Trichotomy law for the les... |
| leg0 27823 | Degenerated (zero-length) ... |
| legeq 27824 | Deduce equality from "less... |
| legbtwn 27825 | Deduce betweenness from "l... |
| tgcgrsub2 27826 | Removing identical parts f... |
| ltgseg 27827 | The set ` E ` denotes the ... |
| ltgov 27828 | Strict "shorter than" geom... |
| legov3 27829 | An equivalent definition o... |
| legso 27830 | The "shorter than" relatio... |
| ishlg 27833 | Rays : Definition 6.1 of ... |
| hlcomb 27834 | The half-line relation com... |
| hlcomd 27835 | The half-line relation com... |
| hlne1 27836 | The half-line relation imp... |
| hlne2 27837 | The half-line relation imp... |
| hlln 27838 | The half-line relation imp... |
| hleqnid 27839 | The endpoint does not belo... |
| hlid 27840 | The half-line relation is ... |
| hltr 27841 | The half-line relation is ... |
| hlbtwn 27842 | Betweenness is a sufficien... |
| btwnhl1 27843 | Deduce half-line from betw... |
| btwnhl2 27844 | Deduce half-line from betw... |
| btwnhl 27845 | Swap betweenness for a hal... |
| lnhl 27846 | Either a point ` C ` on th... |
| hlcgrex 27847 | Construct a point on a hal... |
| hlcgreulem 27848 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 27849 | The point constructed in ~... |
| btwnlng1 27850 | Betweenness implies coline... |
| btwnlng2 27851 | Betweenness implies coline... |
| btwnlng3 27852 | Betweenness implies coline... |
| lncom 27853 | Swapping the points defini... |
| lnrot1 27854 | Rotating the points defini... |
| lnrot2 27855 | Rotating the points defini... |
| ncolne1 27856 | Non-colinear points are di... |
| ncolne2 27857 | Non-colinear points are di... |
| tgisline 27858 | The property of being a pr... |
| tglnne 27859 | It takes two different poi... |
| tglndim0 27860 | There are no lines in dime... |
| tgelrnln 27861 | The property of being a pr... |
| tglineeltr 27862 | Transitivity law for lines... |
| tglineelsb2 27863 | If ` S ` lies on PQ , then... |
| tglinerflx1 27864 | Reflexivity law for line m... |
| tglinerflx2 27865 | Reflexivity law for line m... |
| tglinecom 27866 | Commutativity law for line... |
| tglinethru 27867 | If ` A ` is a line contain... |
| tghilberti1 27868 | There is a line through an... |
| tghilberti2 27869 | There is at most one line ... |
| tglinethrueu 27870 | There is a unique line goi... |
| tglnne0 27871 | A line ` A ` has at least ... |
| tglnpt2 27872 | Find a second point on a l... |
| tglineintmo 27873 | Two distinct lines interse... |
| tglineineq 27874 | Two distinct lines interse... |
| tglineneq 27875 | Given three non-colinear p... |
| tglineinteq 27876 | Two distinct lines interse... |
| ncolncol 27877 | Deduce non-colinearity fro... |
| coltr 27878 | A transitivity law for col... |
| coltr3 27879 | A transitivity law for col... |
| colline 27880 | Three points are colinear ... |
| tglowdim2l 27881 | Reformulation of the lower... |
| tglowdim2ln 27882 | There is always one point ... |
| mirreu3 27885 | Existential uniqueness of ... |
| mirval 27886 | Value of the point inversi... |
| mirfv 27887 | Value of the point inversi... |
| mircgr 27888 | Property of the image by t... |
| mirbtwn 27889 | Property of the image by t... |
| ismir 27890 | Property of the image by t... |
| mirf 27891 | Point inversion as functio... |
| mircl 27892 | Closure of the point inver... |
| mirmir 27893 | The point inversion functi... |
| mircom 27894 | Variation on ~ mirmir . (... |
| mirreu 27895 | Any point has a unique ant... |
| mireq 27896 | Equality deduction for poi... |
| mirinv 27897 | The only invariant point o... |
| mirne 27898 | Mirror of non-center point... |
| mircinv 27899 | The center point is invari... |
| mirf1o 27900 | The point inversion functi... |
| miriso 27901 | The point inversion functi... |
| mirbtwni 27902 | Point inversion preserves ... |
| mirbtwnb 27903 | Point inversion preserves ... |
| mircgrs 27904 | Point inversion preserves ... |
| mirmir2 27905 | Point inversion of a point... |
| mirmot 27906 | Point investion is a motio... |
| mirln 27907 | If two points are on the s... |
| mirln2 27908 | If a point and its mirror ... |
| mirconn 27909 | Point inversion of connect... |
| mirhl 27910 | If two points ` X ` and ` ... |
| mirbtwnhl 27911 | If the center of the point... |
| mirhl2 27912 | Deduce half-line relation ... |
| mircgrextend 27913 | Link congruence over a pai... |
| mirtrcgr 27914 | Point inversion of one poi... |
| mirauto 27915 | Point inversion preserves ... |
| miduniq 27916 | Uniqueness of the middle p... |
| miduniq1 27917 | Uniqueness of the middle p... |
| miduniq2 27918 | If two point inversions co... |
| colmid 27919 | Colinearity and equidistan... |
| symquadlem 27920 | Lemma of the symetrial qua... |
| krippenlem 27921 | Lemma for ~ krippen . We ... |
| krippen 27922 | Krippenlemma (German for c... |
| midexlem 27923 | Lemma for the existence of... |
| israg 27928 | Property for 3 points A, B... |
| ragcom 27929 | Commutative rule for right... |
| ragcol 27930 | The right angle property i... |
| ragmir 27931 | Right angle property is pr... |
| mirrag 27932 | Right angle is conserved b... |
| ragtrivb 27933 | Trivial right angle. Theo... |
| ragflat2 27934 | Deduce equality from two r... |
| ragflat 27935 | Deduce equality from two r... |
| ragtriva 27936 | Trivial right angle. Theo... |
| ragflat3 27937 | Right angle and colinearit... |
| ragcgr 27938 | Right angle and colinearit... |
| motrag 27939 | Right angles are preserved... |
| ragncol 27940 | Right angle implies non-co... |
| perpln1 27941 | Derive a line from perpend... |
| perpln2 27942 | Derive a line from perpend... |
| isperp 27943 | Property for 2 lines A, B ... |
| perpcom 27944 | The "perpendicular" relati... |
| perpneq 27945 | Two perpendicular lines ar... |
| isperp2 27946 | Property for 2 lines A, B,... |
| isperp2d 27947 | One direction of ~ isperp2... |
| ragperp 27948 | Deduce that two lines are ... |
| footexALT 27949 | Alternative version of ~ f... |
| footexlem1 27950 | Lemma for ~ footex . (Con... |
| footexlem2 27951 | Lemma for ~ footex . (Con... |
| footex 27952 | From a point ` C ` outside... |
| foot 27953 | From a point ` C ` outside... |
| footne 27954 | Uniqueness of the foot poi... |
| footeq 27955 | Uniqueness of the foot poi... |
| hlperpnel 27956 | A point on a half-line whi... |
| perprag 27957 | Deduce a right angle from ... |
| perpdragALT 27958 | Deduce a right angle from ... |
| perpdrag 27959 | Deduce a right angle from ... |
| colperp 27960 | Deduce a perpendicularity ... |
| colperpexlem1 27961 | Lemma for ~ colperp . Fir... |
| colperpexlem2 27962 | Lemma for ~ colperpex . S... |
| colperpexlem3 27963 | Lemma for ~ colperpex . C... |
| colperpex 27964 | In dimension 2 and above, ... |
| mideulem2 27965 | Lemma for ~ opphllem , whi... |
| opphllem 27966 | Lemma 8.24 of [Schwabhause... |
| mideulem 27967 | Lemma for ~ mideu . We ca... |
| midex 27968 | Existence of the midpoint,... |
| mideu 27969 | Existence and uniqueness o... |
| islnopp 27970 | The property for two point... |
| islnoppd 27971 | Deduce that ` A ` and ` B ... |
| oppne1 27972 | Points lying on opposite s... |
| oppne2 27973 | Points lying on opposite s... |
| oppne3 27974 | Points lying on opposite s... |
| oppcom 27975 | Commutativity rule for "op... |
| opptgdim2 27976 | If two points opposite to ... |
| oppnid 27977 | The "opposite to a line" r... |
| opphllem1 27978 | Lemma for ~ opphl . (Cont... |
| opphllem2 27979 | Lemma for ~ opphl . Lemma... |
| opphllem3 27980 | Lemma for ~ opphl : We as... |
| opphllem4 27981 | Lemma for ~ opphl . (Cont... |
| opphllem5 27982 | Second part of Lemma 9.4 o... |
| opphllem6 27983 | First part of Lemma 9.4 of... |
| oppperpex 27984 | Restating ~ colperpex usin... |
| opphl 27985 | If two points ` A ` and ` ... |
| outpasch 27986 | Axiom of Pasch, outer form... |
| hlpasch 27987 | An application of the axio... |
| ishpg 27990 | Value of the half-plane re... |
| hpgbr 27991 | Half-planes : property for... |
| hpgne1 27992 | Points on the open half pl... |
| hpgne2 27993 | Points on the open half pl... |
| lnopp2hpgb 27994 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 27995 | If two points lie on the o... |
| hpgerlem 27996 | Lemma for the proof that t... |
| hpgid 27997 | The half-plane relation is... |
| hpgcom 27998 | The half-plane relation co... |
| hpgtr 27999 | The half-plane relation is... |
| colopp 28000 | Opposite sides of a line f... |
| colhp 28001 | Half-plane relation for co... |
| hphl 28002 | If two points are on the s... |
| midf 28007 | Midpoint as a function. (... |
| midcl 28008 | Closure of the midpoint. ... |
| ismidb 28009 | Property of the midpoint. ... |
| midbtwn 28010 | Betweenness of midpoint. ... |
| midcgr 28011 | Congruence of midpoint. (... |
| midid 28012 | Midpoint of a null segment... |
| midcom 28013 | Commutativity rule for the... |
| mirmid 28014 | Point inversion preserves ... |
| lmieu 28015 | Uniqueness of the line mir... |
| lmif 28016 | Line mirror as a function.... |
| lmicl 28017 | Closure of the line mirror... |
| islmib 28018 | Property of the line mirro... |
| lmicom 28019 | The line mirroring functio... |
| lmilmi 28020 | Line mirroring is an invol... |
| lmireu 28021 | Any point has a unique ant... |
| lmieq 28022 | Equality deduction for lin... |
| lmiinv 28023 | The invariants of the line... |
| lmicinv 28024 | The mirroring line is an i... |
| lmimid 28025 | If we have a right angle, ... |
| lmif1o 28026 | The line mirroring functio... |
| lmiisolem 28027 | Lemma for ~ lmiiso . (Con... |
| lmiiso 28028 | The line mirroring functio... |
| lmimot 28029 | Line mirroring is a motion... |
| hypcgrlem1 28030 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 28031 | Lemma for ~ hypcgr , case ... |
| hypcgr 28032 | If the catheti of two righ... |
| lmiopp 28033 | Line mirroring produces po... |
| lnperpex 28034 | Existence of a perpendicul... |
| trgcopy 28035 | Triangle construction: a c... |
| trgcopyeulem 28036 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 28037 | Triangle construction: a c... |
| iscgra 28040 | Property for two angles AB... |
| iscgra1 28041 | A special version of ~ isc... |
| iscgrad 28042 | Sufficient conditions for ... |
| cgrane1 28043 | Angles imply inequality. ... |
| cgrane2 28044 | Angles imply inequality. ... |
| cgrane3 28045 | Angles imply inequality. ... |
| cgrane4 28046 | Angles imply inequality. ... |
| cgrahl1 28047 | Angle congruence is indepe... |
| cgrahl2 28048 | Angle congruence is indepe... |
| cgracgr 28049 | First direction of proposi... |
| cgraid 28050 | Angle congruence is reflex... |
| cgraswap 28051 | Swap rays in a congruence ... |
| cgrcgra 28052 | Triangle congruence implie... |
| cgracom 28053 | Angle congruence commutes.... |
| cgratr 28054 | Angle congruence is transi... |
| flatcgra 28055 | Flat angles are congruent.... |
| cgraswaplr 28056 | Swap both side of angle co... |
| cgrabtwn 28057 | Angle congruence preserves... |
| cgrahl 28058 | Angle congruence preserves... |
| cgracol 28059 | Angle congruence preserves... |
| cgrancol 28060 | Angle congruence preserves... |
| dfcgra2 28061 | This is the full statement... |
| sacgr 28062 | Supplementary angles of co... |
| oacgr 28063 | Vertical angle theorem. V... |
| acopy 28064 | Angle construction. Theor... |
| acopyeu 28065 | Angle construction. Theor... |
| isinag 28069 | Property for point ` X ` t... |
| isinagd 28070 | Sufficient conditions for ... |
| inagflat 28071 | Any point lies in a flat a... |
| inagswap 28072 | Swap the order of the half... |
| inagne1 28073 | Deduce inequality from the... |
| inagne2 28074 | Deduce inequality from the... |
| inagne3 28075 | Deduce inequality from the... |
| inaghl 28076 | The "point lie in angle" r... |
| isleag 28078 | Geometrical "less than" pr... |
| isleagd 28079 | Sufficient condition for "... |
| leagne1 28080 | Deduce inequality from the... |
| leagne2 28081 | Deduce inequality from the... |
| leagne3 28082 | Deduce inequality from the... |
| leagne4 28083 | Deduce inequality from the... |
| cgrg3col4 28084 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 28085 | First congruence theorem: ... |
| tgsas 28086 | First congruence theorem: ... |
| tgsas2 28087 | First congruence theorem: ... |
| tgsas3 28088 | First congruence theorem: ... |
| tgasa1 28089 | Second congruence theorem:... |
| tgasa 28090 | Second congruence theorem:... |
| tgsss1 28091 | Third congruence theorem: ... |
| tgsss2 28092 | Third congruence theorem: ... |
| tgsss3 28093 | Third congruence theorem: ... |
| dfcgrg2 28094 | Congruence for two triangl... |
| isoas 28095 | Congruence theorem for iso... |
| iseqlg 28098 | Property of a triangle bei... |
| iseqlgd 28099 | Condition for a triangle t... |
| f1otrgds 28100 | Convenient lemma for ~ f1o... |
| f1otrgitv 28101 | Convenient lemma for ~ f1o... |
| f1otrg 28102 | A bijection between bases ... |
| f1otrge 28103 | A bijection between bases ... |
| ttgval 28106 | Define a function to augme... |
| ttgvalOLD 28107 | Obsolete proof of ~ ttgval... |
| ttglem 28108 | Lemma for ~ ttgbas , ~ ttg... |
| ttglemOLD 28109 | Obsolete version of ~ ttgl... |
| ttgbas 28110 | The base set of a subcompl... |
| ttgbasOLD 28111 | Obsolete proof of ~ ttgbas... |
| ttgplusg 28112 | The addition operation of ... |
| ttgplusgOLD 28113 | Obsolete proof of ~ ttgplu... |
| ttgsub 28114 | The subtraction operation ... |
| ttgvsca 28115 | The scalar product of a su... |
| ttgvscaOLD 28116 | Obsolete proof of ~ ttgvsc... |
| ttgds 28117 | The metric of a subcomplex... |
| ttgdsOLD 28118 | Obsolete proof of ~ ttgds ... |
| ttgitvval 28119 | Betweenness for a subcompl... |
| ttgelitv 28120 | Betweenness for a subcompl... |
| ttgbtwnid 28121 | Any subcomplex module equi... |
| ttgcontlem1 28122 | Lemma for % ttgcont . (Co... |
| xmstrkgc 28123 | Any metric space fulfills ... |
| cchhllem 28124 | Lemma for chlbas and chlvs... |
| cchhllemOLD 28125 | Obsolete version of ~ cchh... |
| elee 28132 | Membership in a Euclidean ... |
| mptelee 28133 | A condition for a mapping ... |
| eleenn 28134 | If ` A ` is in ` ( EE `` N... |
| eleei 28135 | The forward direction of ~... |
| eedimeq 28136 | A point belongs to at most... |
| brbtwn 28137 | The binary relation form o... |
| brcgr 28138 | The binary relation form o... |
| fveere 28139 | The function value of a po... |
| fveecn 28140 | The function value of a po... |
| eqeefv 28141 | Two points are equal iff t... |
| eqeelen 28142 | Two points are equal iff t... |
| brbtwn2 28143 | Alternate characterization... |
| colinearalglem1 28144 | Lemma for ~ colinearalg . ... |
| colinearalglem2 28145 | Lemma for ~ colinearalg . ... |
| colinearalglem3 28146 | Lemma for ~ colinearalg . ... |
| colinearalglem4 28147 | Lemma for ~ colinearalg . ... |
| colinearalg 28148 | An algebraic characterizat... |
| eleesub 28149 | Membership of a subtractio... |
| eleesubd 28150 | Membership of a subtractio... |
| axdimuniq 28151 | The unique dimension axiom... |
| axcgrrflx 28152 | ` A ` is as far from ` B `... |
| axcgrtr 28153 | Congruence is transitive. ... |
| axcgrid 28154 | If there is no distance be... |
| axsegconlem1 28155 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 28156 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 28157 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 28158 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 28159 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 28160 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 28161 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 28162 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 28163 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 28164 | Lemma for ~ axsegcon . Sh... |
| axsegcon 28165 | Any segment ` A B ` can be... |
| ax5seglem1 28166 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 28167 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 28168 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 28169 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 28170 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 28171 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 28172 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 28173 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 28174 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 28175 | Lemma for ~ ax5seg . Take... |
| ax5seg 28176 | The five segment axiom. T... |
| axbtwnid 28177 | Points are indivisible. T... |
| axpaschlem 28178 | Lemma for ~ axpasch . Set... |
| axpasch 28179 | The inner Pasch axiom. Ta... |
| axlowdimlem1 28180 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 28181 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 28182 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 28183 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 28184 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 28185 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 28186 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 28187 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 28188 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 28189 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 28190 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 28191 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 28192 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 28193 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 28194 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 28195 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 28196 | Lemma for ~ axlowdim . Es... |
| axlowdim1 28197 | The lower dimension axiom ... |
| axlowdim2 28198 | The lower two-dimensional ... |
| axlowdim 28199 | The general lower dimensio... |
| axeuclidlem 28200 | Lemma for ~ axeuclid . Ha... |
| axeuclid 28201 | Euclid's axiom. Take an a... |
| axcontlem1 28202 | Lemma for ~ axcont . Chan... |
| axcontlem2 28203 | Lemma for ~ axcont . The ... |
| axcontlem3 28204 | Lemma for ~ axcont . Give... |
| axcontlem4 28205 | Lemma for ~ axcont . Give... |
| axcontlem5 28206 | Lemma for ~ axcont . Comp... |
| axcontlem6 28207 | Lemma for ~ axcont . Stat... |
| axcontlem7 28208 | Lemma for ~ axcont . Give... |
| axcontlem8 28209 | Lemma for ~ axcont . A po... |
| axcontlem9 28210 | Lemma for ~ axcont . Give... |
| axcontlem10 28211 | Lemma for ~ axcont . Give... |
| axcontlem11 28212 | Lemma for ~ axcont . Elim... |
| axcontlem12 28213 | Lemma for ~ axcont . Elim... |
| axcont 28214 | The axiom of continuity. ... |
| eengv 28217 | The value of the Euclidean... |
| eengstr 28218 | The Euclidean geometry as ... |
| eengbas 28219 | The Base of the Euclidean ... |
| ebtwntg 28220 | The betweenness relation u... |
| ecgrtg 28221 | The congruence relation us... |
| elntg 28222 | The line definition in the... |
| elntg2 28223 | The line definition in the... |
| eengtrkg 28224 | The geometry structure for... |
| eengtrkge 28225 | The geometry structure for... |
| edgfid 28228 | Utility theorem: index-ind... |
| edgfndx 28229 | Index value of the ~ df-ed... |
| edgfndxnn 28230 | The index value of the edg... |
| edgfndxid 28231 | The value of the edge func... |
| edgfndxidOLD 28232 | Obsolete version of ~ edgf... |
| basendxltedgfndx 28233 | The index value of the ` B... |
| baseltedgfOLD 28234 | Obsolete proof of ~ basend... |
| basendxnedgfndx 28235 | The slots ` Base ` and ` .... |
| vtxval 28240 | The set of vertices of a g... |
| iedgval 28241 | The set of indexed edges o... |
| 1vgrex 28242 | A graph with at least one ... |
| opvtxval 28243 | The set of vertices of a g... |
| opvtxfv 28244 | The set of vertices of a g... |
| opvtxov 28245 | The set of vertices of a g... |
| opiedgval 28246 | The set of indexed edges o... |
| opiedgfv 28247 | The set of indexed edges o... |
| opiedgov 28248 | The set of indexed edges o... |
| opvtxfvi 28249 | The set of vertices of a g... |
| opiedgfvi 28250 | The set of indexed edges o... |
| funvtxdmge2val 28251 | The set of vertices of an ... |
| funiedgdmge2val 28252 | The set of indexed edges o... |
| funvtxdm2val 28253 | The set of vertices of an ... |
| funiedgdm2val 28254 | The set of indexed edges o... |
| funvtxval0 28255 | The set of vertices of an ... |
| basvtxval 28256 | The set of vertices of a g... |
| edgfiedgval 28257 | The set of indexed edges o... |
| funvtxval 28258 | The set of vertices of a g... |
| funiedgval 28259 | The set of indexed edges o... |
| structvtxvallem 28260 | Lemma for ~ structvtxval a... |
| structvtxval 28261 | The set of vertices of an ... |
| structiedg0val 28262 | The set of indexed edges o... |
| structgrssvtxlem 28263 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 28264 | The set of vertices of a g... |
| structgrssiedg 28265 | The set of indexed edges o... |
| struct2grstr 28266 | A graph represented as an ... |
| struct2grvtx 28267 | The set of vertices of a g... |
| struct2griedg 28268 | The set of indexed edges o... |
| graop 28269 | Any representation of a gr... |
| grastruct 28270 | Any representation of a gr... |
| gropd 28271 | If any representation of a... |
| grstructd 28272 | If any representation of a... |
| gropeld 28273 | If any representation of a... |
| grstructeld 28274 | If any representation of a... |
| setsvtx 28275 | The vertices of a structur... |
| setsiedg 28276 | The (indexed) edges of a s... |
| snstrvtxval 28277 | The set of vertices of a g... |
| snstriedgval 28278 | The set of indexed edges o... |
| vtxval0 28279 | Degenerated case 1 for ver... |
| iedgval0 28280 | Degenerated case 1 for edg... |
| vtxvalsnop 28281 | Degenerated case 2 for ver... |
| iedgvalsnop 28282 | Degenerated case 2 for edg... |
| vtxval3sn 28283 | Degenerated case 3 for ver... |
| iedgval3sn 28284 | Degenerated case 3 for edg... |
| vtxvalprc 28285 | Degenerated case 4 for ver... |
| iedgvalprc 28286 | Degenerated case 4 for edg... |
| edgval 28289 | The edges of a graph. (Co... |
| iedgedg 28290 | An indexed edge is an edge... |
| edgopval 28291 | The edges of a graph repre... |
| edgov 28292 | The edges of a graph repre... |
| edgstruct 28293 | The edges of a graph repre... |
| edgiedgb 28294 | A set is an edge iff it is... |
| edg0iedg0 28295 | There is no edge in a grap... |
| isuhgr 28300 | The predicate "is an undir... |
| isushgr 28301 | The predicate "is an undir... |
| uhgrf 28302 | The edge function of an un... |
| ushgrf 28303 | The edge function of an un... |
| uhgrss 28304 | An edge is a subset of ver... |
| uhgreq12g 28305 | If two sets have the same ... |
| uhgrfun 28306 | The edge function of an un... |
| uhgrn0 28307 | An edge is a nonempty subs... |
| lpvtx 28308 | The endpoints of a loop (w... |
| ushgruhgr 28309 | An undirected simple hyper... |
| isuhgrop 28310 | The property of being an u... |
| uhgr0e 28311 | The empty graph, with vert... |
| uhgr0vb 28312 | The null graph, with no ve... |
| uhgr0 28313 | The null graph represented... |
| uhgrun 28314 | The union ` U ` of two (un... |
| uhgrunop 28315 | The union of two (undirect... |
| ushgrun 28316 | The union ` U ` of two (un... |
| ushgrunop 28317 | The union of two (undirect... |
| uhgrstrrepe 28318 | Replacing (or adding) the ... |
| incistruhgr 28319 | An _incidence structure_ `... |
| isupgr 28324 | The property of being an u... |
| wrdupgr 28325 | The property of being an u... |
| upgrf 28326 | The edge function of an un... |
| upgrfn 28327 | The edge function of an un... |
| upgrss 28328 | An edge is a subset of ver... |
| upgrn0 28329 | An edge is a nonempty subs... |
| upgrle 28330 | An edge of an undirected p... |
| upgrfi 28331 | An edge is a finite subset... |
| upgrex 28332 | An edge is an unordered pa... |
| upgrbi 28333 | Show that an unordered pai... |
| upgrop 28334 | A pseudograph represented ... |
| isumgr 28335 | The property of being an u... |
| isumgrs 28336 | The simplified property of... |
| wrdumgr 28337 | The property of being an u... |
| umgrf 28338 | The edge function of an un... |
| umgrfn 28339 | The edge function of an un... |
| umgredg2 28340 | An edge of a multigraph ha... |
| umgrbi 28341 | Show that an unordered pai... |
| upgruhgr 28342 | An undirected pseudograph ... |
| umgrupgr 28343 | An undirected multigraph i... |
| umgruhgr 28344 | An undirected multigraph i... |
| upgrle2 28345 | An edge of an undirected p... |
| umgrnloopv 28346 | In a multigraph, there is ... |
| umgredgprv 28347 | In a multigraph, an edge i... |
| umgrnloop 28348 | In a multigraph, there is ... |
| umgrnloop0 28349 | A multigraph has no loops.... |
| umgr0e 28350 | The empty graph, with vert... |
| upgr0e 28351 | The empty graph, with vert... |
| upgr1elem 28352 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 28353 | A pseudograph with one edg... |
| upgr0eop 28354 | The empty graph, with vert... |
| upgr1eop 28355 | A pseudograph with one edg... |
| upgr0eopALT 28356 | Alternate proof of ~ upgr0... |
| upgr1eopALT 28357 | Alternate proof of ~ upgr1... |
| upgrun 28358 | The union ` U ` of two pse... |
| upgrunop 28359 | The union of two pseudogra... |
| umgrun 28360 | The union ` U ` of two mul... |
| umgrunop 28361 | The union of two multigrap... |
| umgrislfupgrlem 28362 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 28363 | A multigraph is a loop-fre... |
| lfgredgge2 28364 | An edge of a loop-free gra... |
| lfgrnloop 28365 | A loop-free graph has no l... |
| uhgredgiedgb 28366 | In a hypergraph, a set is ... |
| uhgriedg0edg0 28367 | A hypergraph has no edges ... |
| uhgredgn0 28368 | An edge of a hypergraph is... |
| edguhgr 28369 | An edge of a hypergraph is... |
| uhgredgrnv 28370 | An edge of a hypergraph co... |
| uhgredgss 28371 | The set of edges of a hype... |
| upgredgss 28372 | The set of edges of a pseu... |
| umgredgss 28373 | The set of edges of a mult... |
| edgupgr 28374 | Properties of an edge of a... |
| edgumgr 28375 | Properties of an edge of a... |
| uhgrvtxedgiedgb 28376 | In a hypergraph, a vertex ... |
| upgredg 28377 | For each edge in a pseudog... |
| umgredg 28378 | For each edge in a multigr... |
| upgrpredgv 28379 | An edge of a pseudograph a... |
| umgrpredgv 28380 | An edge of a multigraph al... |
| upgredg2vtx 28381 | For a vertex incident to a... |
| upgredgpr 28382 | If a proper pair (of verti... |
| edglnl 28383 | The edges incident with a ... |
| numedglnl 28384 | The number of edges incide... |
| umgredgne 28385 | An edge of a multigraph al... |
| umgrnloop2 28386 | A multigraph has no loops.... |
| umgredgnlp 28387 | An edge of a multigraph is... |
| isuspgr 28392 | The property of being a si... |
| isusgr 28393 | The property of being a si... |
| uspgrf 28394 | The edge function of a sim... |
| usgrf 28395 | The edge function of a sim... |
| isusgrs 28396 | The property of being a si... |
| usgrfs 28397 | The edge function of a sim... |
| usgrfun 28398 | The edge function of a sim... |
| usgredgss 28399 | The set of edges of a simp... |
| edgusgr 28400 | An edge of a simple graph ... |
| isuspgrop 28401 | The property of being an u... |
| isusgrop 28402 | The property of being an u... |
| usgrop 28403 | A simple graph represented... |
| isausgr 28404 | The property of an unorder... |
| ausgrusgrb 28405 | The equivalence of the def... |
| usgrausgri 28406 | A simple graph represented... |
| ausgrumgri 28407 | If an alternatively define... |
| ausgrusgri 28408 | The equivalence of the def... |
| usgrausgrb 28409 | The equivalence of the def... |
| usgredgop 28410 | An edge of a simple graph ... |
| usgrf1o 28411 | The edge function of a sim... |
| usgrf1 28412 | The edge function of a sim... |
| uspgrf1oedg 28413 | The edge function of a sim... |
| usgrss 28414 | An edge is a subset of ver... |
| uspgrushgr 28415 | A simple pseudograph is an... |
| uspgrupgr 28416 | A simple pseudograph is an... |
| uspgrupgrushgr 28417 | A graph is a simple pseudo... |
| usgruspgr 28418 | A simple graph is a simple... |
| usgrumgr 28419 | A simple graph is an undir... |
| usgrumgruspgr 28420 | A graph is a simple graph ... |
| usgruspgrb 28421 | A class is a simple graph ... |
| usgrupgr 28422 | A simple graph is an undir... |
| usgruhgr 28423 | A simple graph is an undir... |
| usgrislfuspgr 28424 | A simple graph is a loop-f... |
| uspgrun 28425 | The union ` U ` of two sim... |
| uspgrunop 28426 | The union of two simple ps... |
| usgrun 28427 | The union ` U ` of two sim... |
| usgrunop 28428 | The union of two simple gr... |
| usgredg2 28429 | The value of the "edge fun... |
| usgredg2ALT 28430 | Alternate proof of ~ usgre... |
| usgredgprv 28431 | In a simple graph, an edge... |
| usgredgprvALT 28432 | Alternate proof of ~ usgre... |
| usgredgppr 28433 | An edge of a simple graph ... |
| usgrpredgv 28434 | An edge of a simple graph ... |
| edgssv2 28435 | An edge of a simple graph ... |
| usgredg 28436 | For each edge in a simple ... |
| usgrnloopv 28437 | In a simple graph, there i... |
| usgrnloopvALT 28438 | Alternate proof of ~ usgrn... |
| usgrnloop 28439 | In a simple graph, there i... |
| usgrnloopALT 28440 | Alternate proof of ~ usgrn... |
| usgrnloop0 28441 | A simple graph has no loop... |
| usgrnloop0ALT 28442 | Alternate proof of ~ usgrn... |
| usgredgne 28443 | An edge of a simple graph ... |
| usgrf1oedg 28444 | The edge function of a sim... |
| uhgr2edg 28445 | If a vertex is adjacent to... |
| umgr2edg 28446 | If a vertex is adjacent to... |
| usgr2edg 28447 | If a vertex is adjacent to... |
| umgr2edg1 28448 | If a vertex is adjacent to... |
| usgr2edg1 28449 | If a vertex is adjacent to... |
| umgrvad2edg 28450 | If a vertex is adjacent to... |
| umgr2edgneu 28451 | If a vertex is adjacent to... |
| usgrsizedg 28452 | In a simple graph, the siz... |
| usgredg3 28453 | The value of the "edge fun... |
| usgredg4 28454 | For a vertex incident to a... |
| usgredgreu 28455 | For a vertex incident to a... |
| usgredg2vtx 28456 | For a vertex incident to a... |
| uspgredg2vtxeu 28457 | For a vertex incident to a... |
| usgredg2vtxeu 28458 | For a vertex incident to a... |
| usgredg2vtxeuALT 28459 | Alternate proof of ~ usgre... |
| uspgredg2vlem 28460 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 28461 | In a simple pseudograph, t... |
| usgredg2vlem1 28462 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 28463 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 28464 | In a simple graph, the map... |
| usgriedgleord 28465 | Alternate version of ~ usg... |
| ushgredgedg 28466 | In a simple hypergraph the... |
| usgredgedg 28467 | In a simple graph there is... |
| ushgredgedgloop 28468 | In a simple hypergraph the... |
| uspgredgleord 28469 | In a simple pseudograph th... |
| usgredgleord 28470 | In a simple graph the numb... |
| usgredgleordALT 28471 | Alternate proof for ~ usgr... |
| usgrstrrepe 28472 | Replacing (or adding) the ... |
| usgr0e 28473 | The empty graph, with vert... |
| usgr0vb 28474 | The null graph, with no ve... |
| uhgr0v0e 28475 | The null graph, with no ve... |
| uhgr0vsize0 28476 | The size of a hypergraph w... |
| uhgr0edgfi 28477 | A graph of order 0 (i.e. w... |
| usgr0v 28478 | The null graph, with no ve... |
| uhgr0vusgr 28479 | The null graph, with no ve... |
| usgr0 28480 | The null graph represented... |
| uspgr1e 28481 | A simple pseudograph with ... |
| usgr1e 28482 | A simple graph with one ed... |
| usgr0eop 28483 | The empty graph, with vert... |
| uspgr1eop 28484 | A simple pseudograph with ... |
| uspgr1ewop 28485 | A simple pseudograph with ... |
| uspgr1v1eop 28486 | A simple pseudograph with ... |
| usgr1eop 28487 | A simple graph with (at le... |
| uspgr2v1e2w 28488 | A simple pseudograph with ... |
| usgr2v1e2w 28489 | A simple graph with two ve... |
| edg0usgr 28490 | A class without edges is a... |
| lfuhgr1v0e 28491 | A loop-free hypergraph wit... |
| usgr1vr 28492 | A simple graph with one ve... |
| usgr1v 28493 | A class with one (or no) v... |
| usgr1v0edg 28494 | A class with one (or no) v... |
| usgrexmpldifpr 28495 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 28496 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 28497 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 28498 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 28499 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 28500 | ` G ` is a simple graph of... |
| griedg0prc 28501 | The class of empty graphs ... |
| griedg0ssusgr 28502 | The class of all simple gr... |
| usgrprc 28503 | The class of simple graphs... |
| relsubgr 28506 | The class of the subgraph ... |
| subgrv 28507 | If a class is a subgraph o... |
| issubgr 28508 | The property of a set to b... |
| issubgr2 28509 | The property of a set to b... |
| subgrprop 28510 | The properties of a subgra... |
| subgrprop2 28511 | The properties of a subgra... |
| uhgrissubgr 28512 | The property of a hypergra... |
| subgrprop3 28513 | The properties of a subgra... |
| egrsubgr 28514 | An empty graph consisting ... |
| 0grsubgr 28515 | The null graph (represente... |
| 0uhgrsubgr 28516 | The null graph (as hypergr... |
| uhgrsubgrself 28517 | A hypergraph is a subgraph... |
| subgrfun 28518 | The edge function of a sub... |
| subgruhgrfun 28519 | The edge function of a sub... |
| subgreldmiedg 28520 | An element of the domain o... |
| subgruhgredgd 28521 | An edge of a subgraph of a... |
| subumgredg2 28522 | An edge of a subgraph of a... |
| subuhgr 28523 | A subgraph of a hypergraph... |
| subupgr 28524 | A subgraph of a pseudograp... |
| subumgr 28525 | A subgraph of a multigraph... |
| subusgr 28526 | A subgraph of a simple gra... |
| uhgrspansubgrlem 28527 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 28528 | A spanning subgraph ` S ` ... |
| uhgrspan 28529 | A spanning subgraph ` S ` ... |
| upgrspan 28530 | A spanning subgraph ` S ` ... |
| umgrspan 28531 | A spanning subgraph ` S ` ... |
| usgrspan 28532 | A spanning subgraph ` S ` ... |
| uhgrspanop 28533 | A spanning subgraph of a h... |
| upgrspanop 28534 | A spanning subgraph of a p... |
| umgrspanop 28535 | A spanning subgraph of a m... |
| usgrspanop 28536 | A spanning subgraph of a s... |
| uhgrspan1lem1 28537 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 28538 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 28539 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 28540 | The induced subgraph ` S `... |
| upgrreslem 28541 | Lemma for ~ upgrres . (Co... |
| umgrreslem 28542 | Lemma for ~ umgrres and ~ ... |
| upgrres 28543 | A subgraph obtained by rem... |
| umgrres 28544 | A subgraph obtained by rem... |
| usgrres 28545 | A subgraph obtained by rem... |
| upgrres1lem1 28546 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 28547 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 28548 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 28549 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 28550 | A pseudograph obtained by ... |
| umgrres1 28551 | A multigraph obtained by r... |
| usgrres1 28552 | Restricting a simple graph... |
| isfusgr 28555 | The property of being a fi... |
| fusgrvtxfi 28556 | A finite simple graph has ... |
| isfusgrf1 28557 | The property of being a fi... |
| isfusgrcl 28558 | The property of being a fi... |
| fusgrusgr 28559 | A finite simple graph is a... |
| opfusgr 28560 | A finite simple graph repr... |
| usgredgffibi 28561 | The number of edges in a s... |
| fusgredgfi 28562 | In a finite simple graph t... |
| usgr1v0e 28563 | The size of a (finite) sim... |
| usgrfilem 28564 | In a finite simple graph, ... |
| fusgrfisbase 28565 | Induction base for ~ fusgr... |
| fusgrfisstep 28566 | Induction step in ~ fusgrf... |
| fusgrfis 28567 | A finite simple graph is o... |
| fusgrfupgrfs 28568 | A finite simple graph is a... |
| nbgrprc0 28571 | The set of neighbors is em... |
| nbgrcl 28572 | If a class ` X ` has at le... |
| nbgrval 28573 | The set of neighbors of a ... |
| dfnbgr2 28574 | Alternate definition of th... |
| dfnbgr3 28575 | Alternate definition of th... |
| nbgrnvtx0 28576 | If a class ` X ` is not a ... |
| nbgrel 28577 | Characterization of a neig... |
| nbgrisvtx 28578 | Every neighbor ` N ` of a ... |
| nbgrssvtx 28579 | The neighbors of a vertex ... |
| nbuhgr 28580 | The set of neighbors of a ... |
| nbupgr 28581 | The set of neighbors of a ... |
| nbupgrel 28582 | A neighbor of a vertex in ... |
| nbumgrvtx 28583 | The set of neighbors of a ... |
| nbumgr 28584 | The set of neighbors of an... |
| nbusgrvtx 28585 | The set of neighbors of a ... |
| nbusgr 28586 | The set of neighbors of an... |
| nbgr2vtx1edg 28587 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 28588 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 28589 | If a hypergraph has two ve... |
| nbusgreledg 28590 | A class/vertex is a neighb... |
| uhgrnbgr0nb 28591 | A vertex which is not endp... |
| nbgr0vtxlem 28592 | Lemma for ~ nbgr0vtx and ~... |
| nbgr0vtx 28593 | In a null graph (with no v... |
| nbgr0edg 28594 | In an empty graph (with no... |
| nbgr1vtx 28595 | In a graph with one vertex... |
| nbgrnself 28596 | A vertex in a graph is not... |
| nbgrnself2 28597 | A class ` X ` is not a nei... |
| nbgrssovtx 28598 | The neighbors of a vertex ... |
| nbgrssvwo2 28599 | The neighbors of a vertex ... |
| nbgrsym 28600 | In a graph, the neighborho... |
| nbupgrres 28601 | The neighborhood of a vert... |
| usgrnbcnvfv 28602 | Applying the edge function... |
| nbusgredgeu 28603 | For each neighbor of a ver... |
| edgnbusgreu 28604 | For each edge incident to ... |
| nbusgredgeu0 28605 | For each neighbor of a ver... |
| nbusgrf1o0 28606 | The mapping of neighbors o... |
| nbusgrf1o1 28607 | The set of neighbors of a ... |
| nbusgrf1o 28608 | The set of neighbors of a ... |
| nbedgusgr 28609 | The number of neighbors of... |
| edgusgrnbfin 28610 | The number of neighbors of... |
| nbusgrfi 28611 | The class of neighbors of ... |
| nbfiusgrfi 28612 | The class of neighbors of ... |
| hashnbusgrnn0 28613 | The number of neighbors of... |
| nbfusgrlevtxm1 28614 | The number of neighbors of... |
| nbfusgrlevtxm2 28615 | If there is a vertex which... |
| nbusgrvtxm1 28616 | If the number of neighbors... |
| nb3grprlem1 28617 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 28618 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 28619 | The neighbors of a vertex ... |
| nb3grpr2 28620 | The neighbors of a vertex ... |
| nb3gr2nb 28621 | If the neighbors of two ve... |
| uvtxval 28624 | The set of all universal v... |
| uvtxel 28625 | A universal vertex, i.e. a... |
| uvtxisvtx 28626 | A universal vertex is a ve... |
| uvtxssvtx 28627 | The set of the universal v... |
| vtxnbuvtx 28628 | A universal vertex has all... |
| uvtxnbgrss 28629 | A universal vertex has all... |
| uvtxnbgrvtx 28630 | A universal vertex is neig... |
| uvtx0 28631 | There is no universal vert... |
| isuvtx 28632 | The set of all universal v... |
| uvtxel1 28633 | Characterization of a univ... |
| uvtx01vtx 28634 | If a graph/class has no ed... |
| uvtx2vtx1edg 28635 | If a graph has two vertice... |
| uvtx2vtx1edgb 28636 | If a hypergraph has two ve... |
| uvtxnbgr 28637 | A universal vertex has all... |
| uvtxnbgrb 28638 | A vertex is universal iff ... |
| uvtxusgr 28639 | The set of all universal v... |
| uvtxusgrel 28640 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 28641 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 28642 | If the number of neighbors... |
| uvtxnbvtxm1 28643 | A universal vertex has ` n... |
| nbupgruvtxres 28644 | The neighborhood of a univ... |
| uvtxupgrres 28645 | A universal vertex is univ... |
| cplgruvtxb 28650 | A graph ` G ` is complete ... |
| prcliscplgr 28651 | A proper class (representi... |
| iscplgr 28652 | The property of being a co... |
| iscplgrnb 28653 | A graph is complete iff al... |
| iscplgredg 28654 | A graph ` G ` is complete ... |
| iscusgr 28655 | The property of being a co... |
| cusgrusgr 28656 | A complete simple graph is... |
| cusgrcplgr 28657 | A complete simple graph is... |
| iscusgrvtx 28658 | A simple graph is complete... |
| cusgruvtxb 28659 | A simple graph is complete... |
| iscusgredg 28660 | A simple graph is complete... |
| cusgredg 28661 | In a complete simple graph... |
| cplgr0 28662 | The null graph (with no ve... |
| cusgr0 28663 | The null graph (with no ve... |
| cplgr0v 28664 | A null graph (with no vert... |
| cusgr0v 28665 | A graph with no vertices a... |
| cplgr1vlem 28666 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 28667 | A graph with one vertex is... |
| cusgr1v 28668 | A graph with one vertex an... |
| cplgr2v 28669 | An undirected hypergraph w... |
| cplgr2vpr 28670 | An undirected hypergraph w... |
| nbcplgr 28671 | In a complete graph, each ... |
| cplgr3v 28672 | A pseudograph with three (... |
| cusgr3vnbpr 28673 | The neighbors of a vertex ... |
| cplgrop 28674 | A complete graph represent... |
| cusgrop 28675 | A complete simple graph re... |
| cusgrexilem1 28676 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 28677 | Lemma for ~ usgrexi . (Co... |
| usgrexi 28678 | An arbitrary set regarded ... |
| cusgrexilem2 28679 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 28680 | An arbitrary set ` V ` reg... |
| cusgrexg 28681 | For each set there is a se... |
| structtousgr 28682 | Any (extensible) structure... |
| structtocusgr 28683 | Any (extensible) structure... |
| cffldtocusgr 28684 | The field of complex numbe... |
| cusgrres 28685 | Restricting a complete sim... |
| cusgrsizeindb0 28686 | Base case of the induction... |
| cusgrsizeindb1 28687 | Base case of the induction... |
| cusgrsizeindslem 28688 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 28689 | Part 1 of induction step i... |
| cusgrsize2inds 28690 | Induction step in ~ cusgrs... |
| cusgrsize 28691 | The size of a finite compl... |
| cusgrfilem1 28692 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 28693 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 28694 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 28695 | If the size of a complete ... |
| usgredgsscusgredg 28696 | A simple graph is a subgra... |
| usgrsscusgr 28697 | A simple graph is a subgra... |
| sizusglecusglem1 28698 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 28699 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 28700 | The size of a simple graph... |
| fusgrmaxsize 28701 | The maximum size of a fini... |
| vtxdgfval 28704 | The value of the vertex de... |
| vtxdgval 28705 | The degree of a vertex. (... |
| vtxdgfival 28706 | The degree of a vertex for... |
| vtxdgop 28707 | The vertex degree expresse... |
| vtxdgf 28708 | The vertex degree function... |
| vtxdgelxnn0 28709 | The degree of a vertex is ... |
| vtxdg0v 28710 | The degree of a vertex in ... |
| vtxdg0e 28711 | The degree of a vertex in ... |
| vtxdgfisnn0 28712 | The degree of a vertex in ... |
| vtxdgfisf 28713 | The vertex degree function... |
| vtxdeqd 28714 | Equality theorem for the v... |
| vtxduhgr0e 28715 | The degree of a vertex in ... |
| vtxdlfuhgr1v 28716 | The degree of the vertex i... |
| vdumgr0 28717 | A vertex in a multigraph h... |
| vtxdun 28718 | The degree of a vertex in ... |
| vtxdfiun 28719 | The degree of a vertex in ... |
| vtxduhgrun 28720 | The degree of a vertex in ... |
| vtxduhgrfiun 28721 | The degree of a vertex in ... |
| vtxdlfgrval 28722 | The value of the vertex de... |
| vtxdumgrval 28723 | The value of the vertex de... |
| vtxdusgrval 28724 | The value of the vertex de... |
| vtxd0nedgb 28725 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 28726 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 28727 | The value of the vertex de... |
| vtxdusgrfvedg 28728 | The value of the vertex de... |
| vtxduhgr0nedg 28729 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 28730 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 28731 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 28732 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 28733 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 28734 | The vertex degree function... |
| vtxdgfusgr 28735 | In a finite simple graph, ... |
| fusgrn0degnn0 28736 | In a nonempty, finite grap... |
| 1loopgruspgr 28737 | A graph with one edge whic... |
| 1loopgredg 28738 | The set of edges in a grap... |
| 1loopgrnb0 28739 | In a graph (simple pseudog... |
| 1loopgrvd2 28740 | The vertex degree of a one... |
| 1loopgrvd0 28741 | The vertex degree of a one... |
| 1hevtxdg0 28742 | The vertex degree of verte... |
| 1hevtxdg1 28743 | The vertex degree of verte... |
| 1hegrvtxdg1 28744 | The vertex degree of a gra... |
| 1hegrvtxdg1r 28745 | The vertex degree of a gra... |
| 1egrvtxdg1 28746 | The vertex degree of a one... |
| 1egrvtxdg1r 28747 | The vertex degree of a one... |
| 1egrvtxdg0 28748 | The vertex degree of a one... |
| p1evtxdeqlem 28749 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 28750 | If an edge ` E ` which doe... |
| p1evtxdp1 28751 | If an edge ` E ` (not bein... |
| uspgrloopvtx 28752 | The set of vertices in a g... |
| uspgrloopvtxel 28753 | A vertex in a graph (simpl... |
| uspgrloopiedg 28754 | The set of edges in a grap... |
| uspgrloopedg 28755 | The set of edges in a grap... |
| uspgrloopnb0 28756 | In a graph (simple pseudog... |
| uspgrloopvd2 28757 | The vertex degree of a one... |
| umgr2v2evtx 28758 | The set of vertices in a m... |
| umgr2v2evtxel 28759 | A vertex in a multigraph w... |
| umgr2v2eiedg 28760 | The edge function in a mul... |
| umgr2v2eedg 28761 | The set of edges in a mult... |
| umgr2v2e 28762 | A multigraph with two edge... |
| umgr2v2enb1 28763 | In a multigraph with two e... |
| umgr2v2evd2 28764 | In a multigraph with two e... |
| hashnbusgrvd 28765 | In a simple graph, the num... |
| usgruvtxvdb 28766 | In a finite simple graph w... |
| vdiscusgrb 28767 | A finite simple graph with... |
| vdiscusgr 28768 | In a finite complete simpl... |
| vtxdusgradjvtx 28769 | The degree of a vertex in ... |
| usgrvd0nedg 28770 | If a vertex in a simple gr... |
| uhgrvd00 28771 | If every vertex in a hyper... |
| usgrvd00 28772 | If every vertex in a simpl... |
| vdegp1ai 28773 | The induction step for a v... |
| vdegp1bi 28774 | The induction step for a v... |
| vdegp1ci 28775 | The induction step for a v... |
| vtxdginducedm1lem1 28776 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 28777 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 28778 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 28779 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 28780 | The degree of a vertex ` v... |
| vtxdginducedm1fi 28781 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 28782 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 28783 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 28784 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 28785 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 28786 | Induction step of ~ finsum... |
| finsumvtxdg2size 28787 | The sum of the degrees of ... |
| fusgr1th 28788 | The sum of the degrees of ... |
| finsumvtxdgeven 28789 | The sum of the degrees of ... |
| vtxdgoddnumeven 28790 | The number of vertices of ... |
| fusgrvtxdgonume 28791 | The number of vertices of ... |
| isrgr 28796 | The property of a class be... |
| rgrprop 28797 | The properties of a k-regu... |
| isrusgr 28798 | The property of being a k-... |
| rusgrprop 28799 | The properties of a k-regu... |
| rusgrrgr 28800 | A k-regular simple graph i... |
| rusgrusgr 28801 | A k-regular simple graph i... |
| finrusgrfusgr 28802 | A finite regular simple gr... |
| isrusgr0 28803 | The property of being a k-... |
| rusgrprop0 28804 | The properties of a k-regu... |
| usgreqdrusgr 28805 | If all vertices in a simpl... |
| fusgrregdegfi 28806 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 28807 | If all vertices in a nonem... |
| frusgrnn0 28808 | In a nonempty finite k-reg... |
| 0edg0rgr 28809 | A graph is 0-regular if it... |
| uhgr0edg0rgr 28810 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 28811 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 28812 | A simple graph is 0-regula... |
| 0vtxrgr 28813 | A null graph (with no vert... |
| 0vtxrusgr 28814 | A graph with no vertices a... |
| 0uhgrrusgr 28815 | The null graph as hypergra... |
| 0grrusgr 28816 | The null graph represented... |
| 0grrgr 28817 | The null graph represented... |
| cusgrrusgr 28818 | A complete simple graph wi... |
| cusgrm1rusgr 28819 | A finite simple graph with... |
| rusgrpropnb 28820 | The properties of a k-regu... |
| rusgrpropedg 28821 | The properties of a k-regu... |
| rusgrpropadjvtx 28822 | The properties of a k-regu... |
| rusgrnumwrdl2 28823 | In a k-regular simple grap... |
| rusgr1vtxlem 28824 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 28825 | If a k-regular simple grap... |
| rgrusgrprc 28826 | The class of 0-regular sim... |
| rusgrprc 28827 | The class of 0-regular sim... |
| rgrprc 28828 | The class of 0-regular gra... |
| rgrprcx 28829 | The class of 0-regular gra... |
| rgrx0ndm 28830 | 0 is not in the domain of ... |
| rgrx0nd 28831 | The potentially alternativ... |
| ewlksfval 28838 | The set of s-walks of edge... |
| isewlk 28839 | Conditions for a function ... |
| ewlkprop 28840 | Properties of an s-walk of... |
| ewlkinedg 28841 | The intersection (common v... |
| ewlkle 28842 | An s-walk of edges is also... |
| upgrewlkle2 28843 | In a pseudograph, there is... |
| wkslem1 28844 | Lemma 1 for walks to subst... |
| wkslem2 28845 | Lemma 2 for walks to subst... |
| wksfval 28846 | The set of walks (in an un... |
| iswlk 28847 | Properties of a pair of fu... |
| wlkprop 28848 | Properties of a walk. (Co... |
| wlkv 28849 | The classes involved in a ... |
| iswlkg 28850 | Generalization of ~ iswlk ... |
| wlkf 28851 | The mapping enumerating th... |
| wlkcl 28852 | A walk has length ` # ( F ... |
| wlkp 28853 | The mapping enumerating th... |
| wlkpwrd 28854 | The sequence of vertices o... |
| wlklenvp1 28855 | The number of vertices of ... |
| wksv 28856 | The class of walks is a se... |
| wksvOLD 28857 | Obsolete version of ~ wksv... |
| wlkn0 28858 | The sequence of vertices o... |
| wlklenvm1 28859 | The number of edges of a w... |
| ifpsnprss 28860 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 28861 | Each pair of adjacent vert... |
| wlkvtxiedg 28862 | The vertices of a walk are... |
| relwlk 28863 | The set ` ( Walks `` G ) `... |
| wlkvv 28864 | If there is at least one w... |
| wlkop 28865 | A walk is an ordered pair.... |
| wlkcpr 28866 | A walk as class with two c... |
| wlk2f 28867 | If there is a walk ` W ` t... |
| wlkcomp 28868 | A walk expressed by proper... |
| wlkcompim 28869 | Implications for the prope... |
| wlkelwrd 28870 | The components of a walk a... |
| wlkeq 28871 | Conditions for two walks (... |
| edginwlk 28872 | The value of the edge func... |
| upgredginwlk 28873 | The value of the edge func... |
| iedginwlk 28874 | The value of the edge func... |
| wlkl1loop 28875 | A walk of length 1 from a ... |
| wlk1walk 28876 | A walk is a 1-walk "on the... |
| wlk1ewlk 28877 | A walk is an s-walk "on th... |
| upgriswlk 28878 | Properties of a pair of fu... |
| upgrwlkedg 28879 | The edges of a walk in a p... |
| upgrwlkcompim 28880 | Implications for the prope... |
| wlkvtxedg 28881 | The vertices of a walk are... |
| upgrwlkvtxedg 28882 | The pairs of connected ver... |
| uspgr2wlkeq 28883 | Conditions for two walks w... |
| uspgr2wlkeq2 28884 | Conditions for two walks w... |
| uspgr2wlkeqi 28885 | Conditions for two walks w... |
| umgrwlknloop 28886 | In a multigraph, each walk... |
| wlkResOLD 28887 | Obsolete version of ~ opab... |
| wlkv0 28888 | If there is a walk in the ... |
| g0wlk0 28889 | There is no walk in a null... |
| 0wlk0 28890 | There is no walk for the e... |
| wlk0prc 28891 | There is no walk in a null... |
| wlklenvclwlk 28892 | The number of vertices in ... |
| wlkson 28893 | The set of walks between t... |
| iswlkon 28894 | Properties of a pair of fu... |
| wlkonprop 28895 | Properties of a walk betwe... |
| wlkpvtx 28896 | A walk connects vertices. ... |
| wlkepvtx 28897 | The endpoints of a walk ar... |
| wlkoniswlk 28898 | A walk between two vertice... |
| wlkonwlk 28899 | A walk is a walk between i... |
| wlkonwlk1l 28900 | A walk is a walk from its ... |
| wlksoneq1eq2 28901 | Two walks with identical s... |
| wlkonl1iedg 28902 | If there is a walk between... |
| wlkon2n0 28903 | The length of a walk betwe... |
| 2wlklem 28904 | Lemma for theorems for wal... |
| upgr2wlk 28905 | Properties of a pair of fu... |
| wlkreslem 28906 | Lemma for ~ wlkres . (Con... |
| wlkres 28907 | The restriction ` <. H , Q... |
| redwlklem 28908 | Lemma for ~ redwlk . (Con... |
| redwlk 28909 | A walk ending at the last ... |
| wlkp1lem1 28910 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 28911 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 28912 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 28913 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 28914 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 28915 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 28916 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 28917 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 28918 | Append one path segment (e... |
| wlkdlem1 28919 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 28920 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 28921 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 28922 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 28923 | Two words representing a w... |
| lfgrwlkprop 28924 | Two adjacent vertices in a... |
| lfgriswlk 28925 | Conditions for a pair of f... |
| lfgrwlknloop 28926 | In a loop-free graph, each... |
| reltrls 28931 | The set ` ( Trails `` G ) ... |
| trlsfval 28932 | The set of trails (in an u... |
| istrl 28933 | Conditions for a pair of c... |
| trliswlk 28934 | A trail is a walk. (Contr... |
| trlf1 28935 | The enumeration ` F ` of a... |
| trlreslem 28936 | Lemma for ~ trlres . Form... |
| trlres 28937 | The restriction ` <. H , Q... |
| upgrtrls 28938 | The set of trails in a pse... |
| upgristrl 28939 | Properties of a pair of fu... |
| upgrf1istrl 28940 | Properties of a pair of a ... |
| wksonproplem 28941 | Lemma for theorems for pro... |
| wksonproplemOLD 28942 | Obsolete version of ~ wkso... |
| trlsonfval 28943 | The set of trails between ... |
| istrlson 28944 | Properties of a pair of fu... |
| trlsonprop 28945 | Properties of a trail betw... |
| trlsonistrl 28946 | A trail between two vertic... |
| trlsonwlkon 28947 | A trail between two vertic... |
| trlontrl 28948 | A trail is a trail between... |
| relpths 28957 | The set ` ( Paths `` G ) `... |
| pthsfval 28958 | The set of paths (in an un... |
| spthsfval 28959 | The set of simple paths (i... |
| ispth 28960 | Conditions for a pair of c... |
| isspth 28961 | Conditions for a pair of c... |
| pthistrl 28962 | A path is a trail (in an u... |
| spthispth 28963 | A simple path is a path (i... |
| pthiswlk 28964 | A path is a walk (in an un... |
| spthiswlk 28965 | A simple path is a walk (i... |
| pthdivtx 28966 | The inner vertices of a pa... |
| pthdadjvtx 28967 | The adjacent vertices of a... |
| 2pthnloop 28968 | A path of length at least ... |
| upgr2pthnlp 28969 | A path of length at least ... |
| spthdifv 28970 | The vertices of a simple p... |
| spthdep 28971 | A simple path (at least of... |
| pthdepisspth 28972 | A path with different star... |
| upgrwlkdvdelem 28973 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 28974 | In a pseudograph, all edge... |
| upgrspthswlk 28975 | The set of simple paths in... |
| upgrwlkdvspth 28976 | A walk consisting of diffe... |
| pthsonfval 28977 | The set of paths between t... |
| spthson 28978 | The set of simple paths be... |
| ispthson 28979 | Properties of a pair of fu... |
| isspthson 28980 | Properties of a pair of fu... |
| pthsonprop 28981 | Properties of a path betwe... |
| spthonprop 28982 | Properties of a simple pat... |
| pthonispth 28983 | A path between two vertice... |
| pthontrlon 28984 | A path between two vertice... |
| pthonpth 28985 | A path is a path between i... |
| isspthonpth 28986 | A pair of functions is a s... |
| spthonisspth 28987 | A simple path between to v... |
| spthonpthon 28988 | A simple path between two ... |
| spthonepeq 28989 | The endpoints of a simple ... |
| uhgrwkspthlem1 28990 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 28991 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 28992 | Any walk of length 1 betwe... |
| usgr2wlkneq 28993 | The vertices and edges are... |
| usgr2wlkspthlem1 28994 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 28995 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 28996 | In a simple graph, any wal... |
| usgr2trlncl 28997 | In a simple graph, any tra... |
| usgr2trlspth 28998 | In a simple graph, any tra... |
| usgr2pthspth 28999 | In a simple graph, any pat... |
| usgr2pthlem 29000 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 29001 | In a simple graph, there i... |
| usgr2pth0 29002 | In a simply graph, there i... |
| pthdlem1 29003 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 29004 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 29005 | Lemma 2 for ~ pthd . (Con... |
| pthd 29006 | Two words representing a t... |
| clwlks 29009 | The set of closed walks (i... |
| isclwlk 29010 | A pair of functions repres... |
| clwlkiswlk 29011 | A closed walk is a walk (i... |
| clwlkwlk 29012 | Closed walks are walks (in... |
| clwlkswks 29013 | Closed walks are walks (in... |
| isclwlke 29014 | Properties of a pair of fu... |
| isclwlkupgr 29015 | Properties of a pair of fu... |
| clwlkcomp 29016 | A closed walk expressed by... |
| clwlkcompim 29017 | Implications for the prope... |
| upgrclwlkcompim 29018 | Implications for the prope... |
| clwlkcompbp 29019 | Basic properties of the co... |
| clwlkl1loop 29020 | A closed walk of length 1 ... |
| crcts 29025 | The set of circuits (in an... |
| cycls 29026 | The set of cycles (in an u... |
| iscrct 29027 | Sufficient and necessary c... |
| iscycl 29028 | Sufficient and necessary c... |
| crctprop 29029 | The properties of a circui... |
| cyclprop 29030 | The properties of a cycle:... |
| crctisclwlk 29031 | A circuit is a closed walk... |
| crctistrl 29032 | A circuit is a trail. (Co... |
| crctiswlk 29033 | A circuit is a walk. (Con... |
| cyclispth 29034 | A cycle is a path. (Contr... |
| cycliswlk 29035 | A cycle is a walk. (Contr... |
| cycliscrct 29036 | A cycle is a circuit. (Co... |
| cyclnspth 29037 | A (non-trivial) cycle is n... |
| cyclispthon 29038 | A cycle is a path starting... |
| lfgrn1cycl 29039 | In a loop-free graph there... |
| usgr2trlncrct 29040 | In a simple graph, any tra... |
| umgrn1cycl 29041 | In a multigraph graph (wit... |
| uspgrn2crct 29042 | In a simple pseudograph th... |
| usgrn2cycl 29043 | In a simple graph there ar... |
| crctcshwlkn0lem1 29044 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 29045 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 29046 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 29047 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 29048 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 29049 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 29050 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 29051 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 29052 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 29053 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 29054 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 29055 | Cyclically shifting the in... |
| crctcshwlk 29056 | Cyclically shifting the in... |
| crctcshtrl 29057 | Cyclically shifting the in... |
| crctcsh 29058 | Cyclically shifting the in... |
| wwlks 29069 | The set of walks (in an un... |
| iswwlks 29070 | A word over the set of ver... |
| wwlksn 29071 | The set of walks (in an un... |
| iswwlksn 29072 | A word over the set of ver... |
| wwlksnprcl 29073 | Derivation of the length o... |
| iswwlksnx 29074 | Properties of a word to re... |
| wwlkbp 29075 | Basic properties of a walk... |
| wwlknbp 29076 | Basic properties of a walk... |
| wwlknp 29077 | Properties of a set being ... |
| wwlknbp1 29078 | Other basic properties of ... |
| wwlknvtx 29079 | The symbols of a word ` W ... |
| wwlknllvtx 29080 | If a word ` W ` represents... |
| wwlknlsw 29081 | If a word represents a wal... |
| wspthsn 29082 | The set of simple paths of... |
| iswspthn 29083 | An element of the set of s... |
| wspthnp 29084 | Properties of a set being ... |
| wwlksnon 29085 | The set of walks of a fixe... |
| wspthsnon 29086 | The set of simple paths of... |
| iswwlksnon 29087 | The set of walks of a fixe... |
| wwlksnon0 29088 | Sufficient conditions for ... |
| wwlksonvtx 29089 | If a word ` W ` represents... |
| iswspthsnon 29090 | The set of simple paths of... |
| wwlknon 29091 | An element of the set of w... |
| wspthnon 29092 | An element of the set of s... |
| wspthnonp 29093 | Properties of a set being ... |
| wspthneq1eq2 29094 | Two simple paths with iden... |
| wwlksn0s 29095 | The set of all walks as wo... |
| wwlkssswrd 29096 | Walks (represented by word... |
| wwlksn0 29097 | A walk of length 0 is repr... |
| 0enwwlksnge1 29098 | In graphs without edges, t... |
| wwlkswwlksn 29099 | A walk of a fixed length a... |
| wwlkssswwlksn 29100 | The walks of a fixed lengt... |
| wlkiswwlks1 29101 | The sequence of vertices i... |
| wlklnwwlkln1 29102 | The sequence of vertices i... |
| wlkiswwlks2lem1 29103 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 29104 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 29105 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 29106 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 29107 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 29108 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 29109 | A walk as word corresponds... |
| wlkiswwlks 29110 | A walk as word corresponds... |
| wlkiswwlksupgr2 29111 | A walk as word corresponds... |
| wlkiswwlkupgr 29112 | A walk as word corresponds... |
| wlkswwlksf1o 29113 | The mapping of (ordinary) ... |
| wlkswwlksen 29114 | The set of walks as words ... |
| wwlksm1edg 29115 | Removing the trailing edge... |
| wlklnwwlkln2lem 29116 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 29117 | A walk of length ` N ` as ... |
| wlklnwwlkn 29118 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 29119 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 29120 | A walk of length ` N ` as ... |
| wlknewwlksn 29121 | If a walk in a pseudograph... |
| wlknwwlksnbij 29122 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 29123 | In a simple pseudograph, t... |
| wlknwwlksneqs 29124 | The set of walks of a fixe... |
| wwlkseq 29125 | Equality of two walks (as ... |
| wwlksnred 29126 | Reduction of a walk (as wo... |
| wwlksnext 29127 | Extension of a walk (as wo... |
| wwlksnextbi 29128 | Extension of a walk (as wo... |
| wwlksnredwwlkn 29129 | For each walk (as word) of... |
| wwlksnredwwlkn0 29130 | For each walk (as word) of... |
| wwlksnextwrd 29131 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 29132 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 29133 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 29134 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 29135 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 29136 | There is a bijection betwe... |
| wwlksnexthasheq 29137 | The number of the extensio... |
| disjxwwlksn 29138 | Sets of walks (as words) e... |
| wwlksnndef 29139 | Conditions for ` WWalksN `... |
| wwlksnfi 29140 | The number of walks repres... |
| wlksnfi 29141 | The number of walks of fix... |
| wlksnwwlknvbij 29142 | There is a bijection betwe... |
| wwlksnextproplem1 29143 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 29144 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 29145 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 29146 | Adding additional properti... |
| disjxwwlkn 29147 | Sets of walks (as words) e... |
| hashwwlksnext 29148 | Number of walks (as words)... |
| wwlksnwwlksnon 29149 | A walk of fixed length is ... |
| wspthsnwspthsnon 29150 | A simple path of fixed len... |
| wspthsnonn0vne 29151 | If the set of simple paths... |
| wspthsswwlkn 29152 | The set of simple paths of... |
| wspthnfi 29153 | In a finite graph, the set... |
| wwlksnonfi 29154 | In a finite graph, the set... |
| wspthsswwlknon 29155 | The set of simple paths of... |
| wspthnonfi 29156 | In a finite graph, the set... |
| wspniunwspnon 29157 | The set of nonempty simple... |
| wspn0 29158 | If there are no vertices, ... |
| 2wlkdlem1 29159 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 29160 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 29161 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 29162 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 29163 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 29164 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 29165 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 29166 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 29167 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 29168 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 29169 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 29170 | Construction of a walk fro... |
| 2wlkond 29171 | A walk of length 2 from on... |
| 2trld 29172 | Construction of a trail fr... |
| 2trlond 29173 | A trail of length 2 from o... |
| 2pthd 29174 | A path of length 2 from on... |
| 2spthd 29175 | A simple path of length 2 ... |
| 2pthond 29176 | A simple path of length 2 ... |
| 2pthon3v 29177 | For a vertex adjacent to t... |
| umgr2adedgwlklem 29178 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 29179 | In a multigraph, two adjac... |
| umgr2adedgwlkon 29180 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 29181 | Alternate proof for ~ umgr... |
| umgr2adedgspth 29182 | In a multigraph, two adjac... |
| umgr2wlk 29183 | In a multigraph, there is ... |
| umgr2wlkon 29184 | For each pair of adjacent ... |
| elwwlks2s3 29185 | A walk of length 2 as word... |
| midwwlks2s3 29186 | There is a vertex between ... |
| wwlks2onv 29187 | If a length 3 string repre... |
| elwwlks2ons3im 29188 | A walk as word of length 2... |
| elwwlks2ons3 29189 | For each walk of length 2 ... |
| s3wwlks2on 29190 | A length 3 string which re... |
| umgrwwlks2on 29191 | A walk of length 2 between... |
| wwlks2onsym 29192 | There is a walk of length ... |
| elwwlks2on 29193 | A walk of length 2 between... |
| elwspths2on 29194 | A simple path of length 2 ... |
| wpthswwlks2on 29195 | For two different vertices... |
| 2wspdisj 29196 | All simple paths of length... |
| 2wspiundisj 29197 | All simple paths of length... |
| usgr2wspthons3 29198 | A simple path of length 2 ... |
| usgr2wspthon 29199 | A simple path of length 2 ... |
| elwwlks2 29200 | A walk of length 2 between... |
| elwspths2spth 29201 | A simple path of length 2 ... |
| rusgrnumwwlkl1 29202 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 29203 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 29204 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 29205 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 29206 | Induction base 1 for ~ rus... |
| rusgr0edg 29207 | Special case for graphs wi... |
| rusgrnumwwlks 29208 | Induction step for ~ rusgr... |
| rusgrnumwwlk 29209 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 29210 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 29211 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 29212 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 29213 | In a ` K `-regular graph, ... |
| clwwlk 29216 | The set of closed walks (i... |
| isclwwlk 29217 | Properties of a word to re... |
| clwwlkbp 29218 | Basic properties of a clos... |
| clwwlkgt0 29219 | There is no empty closed w... |
| clwwlksswrd 29220 | Closed walks (represented ... |
| clwwlk1loop 29221 | A closed walk of length 1 ... |
| clwwlkccatlem 29222 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 29223 | The concatenation of two w... |
| umgrclwwlkge2 29224 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 29225 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 29226 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 29227 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 29228 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 29229 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 29230 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 29231 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 29232 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 29233 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 29234 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 29235 | A closed walk as word of l... |
| clwlkclwwlk2 29236 | A closed walk corresponds ... |
| clwlkclwwlkflem 29237 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 29238 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 29239 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 29240 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 29241 | ` F ` is a function from t... |
| clwlkclwwlkfo 29242 | ` F ` is a function from t... |
| clwlkclwwlkf1 29243 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 29244 | ` F ` is a bijection betwe... |
| clwlkclwwlken 29245 | The set of the nonempty cl... |
| clwwisshclwwslemlem 29246 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 29247 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 29248 | Cyclically shifting a clos... |
| clwwisshclwwsn 29249 | Cyclically shifting a clos... |
| erclwwlkrel 29250 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 29251 | Two classes are equivalent... |
| erclwwlkeqlen 29252 | If two classes are equival... |
| erclwwlkref 29253 | ` .~ ` is a reflexive rela... |
| erclwwlksym 29254 | ` .~ ` is a symmetric rela... |
| erclwwlktr 29255 | ` .~ ` is a transitive rel... |
| erclwwlk 29256 | ` .~ ` is an equivalence r... |
| clwwlkn 29259 | The set of closed walks of... |
| isclwwlkn 29260 | A word over the set of ver... |
| clwwlkn0 29261 | There is no closed walk of... |
| clwwlkneq0 29262 | Sufficient conditions for ... |
| clwwlkclwwlkn 29263 | A closed walk of a fixed l... |
| clwwlksclwwlkn 29264 | The closed walks of a fixe... |
| clwwlknlen 29265 | The length of a word repre... |
| clwwlknnn 29266 | The length of a closed wal... |
| clwwlknwrd 29267 | A closed walk of a fixed l... |
| clwwlknbp 29268 | Basic properties of a clos... |
| isclwwlknx 29269 | Characterization of a word... |
| clwwlknp 29270 | Properties of a set being ... |
| clwwlknwwlksn 29271 | A word representing a clos... |
| clwwlknlbonbgr1 29272 | The last but one vertex in... |
| clwwlkinwwlk 29273 | If the initial vertex of a... |
| clwwlkn1 29274 | A closed walk of length 1 ... |
| loopclwwlkn1b 29275 | The singleton word consist... |
| clwwlkn1loopb 29276 | A word represents a closed... |
| clwwlkn2 29277 | A closed walk of length 2 ... |
| clwwlknfi 29278 | If there is only a finite ... |
| clwwlkel 29279 | Obtaining a closed walk (a... |
| clwwlkf 29280 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 29281 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 29282 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 29283 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 29284 | F is a 1-1 onto function, ... |
| clwwlken 29285 | The set of closed walks of... |
| clwwlknwwlkncl 29286 | Obtaining a closed walk (a... |
| clwwlkwwlksb 29287 | A nonempty word over verti... |
| clwwlknwwlksnb 29288 | A word over vertices repre... |
| clwwlkext2edg 29289 | If a word concatenated wit... |
| wwlksext2clwwlk 29290 | If a word represents a wal... |
| wwlksubclwwlk 29291 | Any prefix of a word repre... |
| clwwnisshclwwsn 29292 | Cyclically shifting a clos... |
| eleclclwwlknlem1 29293 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 29294 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 29295 | The set of cyclical shifts... |
| clwwlknccat 29296 | The concatenation of two w... |
| umgr2cwwk2dif 29297 | If a word represents a clo... |
| umgr2cwwkdifex 29298 | If a word represents a clo... |
| erclwwlknrel 29299 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 29300 | Two classes are equivalent... |
| erclwwlkneqlen 29301 | If two classes are equival... |
| erclwwlknref 29302 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 29303 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 29304 | ` .~ ` is a transitive rel... |
| erclwwlkn 29305 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 29306 | The quotient set of the se... |
| hashclwwlkn0 29307 | The number of closed walks... |
| eclclwwlkn1 29308 | An equivalence class accor... |
| eleclclwwlkn 29309 | A member of an equivalence... |
| hashecclwwlkn1 29310 | The size of every equivale... |
| umgrhashecclwwlk 29311 | The size of every equivale... |
| fusgrhashclwwlkn 29312 | The size of the set of clo... |
| clwwlkndivn 29313 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 29314 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 29315 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 29316 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 29317 | There is a one-to-one onto... |
| clwlkssizeeq 29318 | The size of the set of clo... |
| clwlksndivn 29319 | The size of the set of clo... |
| clwwlknonmpo 29322 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 29323 | The set of closed walks on... |
| isclwwlknon 29324 | A word over the set of ver... |
| clwwlk0on0 29325 | There is no word over the ... |
| clwwlknon0 29326 | Sufficient conditions for ... |
| clwwlknonfin 29327 | In a finite graph ` G ` , ... |
| clwwlknonel 29328 | Characterization of a word... |
| clwwlknonccat 29329 | The concatenation of two w... |
| clwwlknon1 29330 | The set of closed walks on... |
| clwwlknon1loop 29331 | If there is a loop at vert... |
| clwwlknon1nloop 29332 | If there is no loop at ver... |
| clwwlknon1sn 29333 | The set of (closed) walks ... |
| clwwlknon1le1 29334 | There is at most one (clos... |
| clwwlknon2 29335 | The set of closed walks on... |
| clwwlknon2x 29336 | The set of closed walks on... |
| s2elclwwlknon2 29337 | Sufficient conditions of a... |
| clwwlknon2num 29338 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 29339 | A word over vertices repre... |
| clwwlknonex2lem1 29340 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 29341 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 29342 | Extending a closed walk ` ... |
| clwwlknonex2e 29343 | Extending a closed walk ` ... |
| clwwlknondisj 29344 | The sets of closed walks o... |
| clwwlknun 29345 | The set of closed walks of... |
| clwwlkvbij 29346 | There is a bijection betwe... |
| 0ewlk 29347 | The empty set (empty seque... |
| 1ewlk 29348 | A sequence of 1 edge is an... |
| 0wlk 29349 | A pair of an empty set (of... |
| is0wlk 29350 | A pair of an empty set (of... |
| 0wlkonlem1 29351 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 29352 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 29353 | A walk of length 0 from a ... |
| 0wlkons1 29354 | A walk of length 0 from a ... |
| 0trl 29355 | A pair of an empty set (of... |
| is0trl 29356 | A pair of an empty set (of... |
| 0trlon 29357 | A trail of length 0 from a... |
| 0pth 29358 | A pair of an empty set (of... |
| 0spth 29359 | A pair of an empty set (of... |
| 0pthon 29360 | A path of length 0 from a ... |
| 0pthon1 29361 | A path of length 0 from a ... |
| 0pthonv 29362 | For each vertex there is a... |
| 0clwlk 29363 | A pair of an empty set (of... |
| 0clwlkv 29364 | Any vertex (more precisely... |
| 0clwlk0 29365 | There is no closed walk in... |
| 0crct 29366 | A pair of an empty set (of... |
| 0cycl 29367 | A pair of an empty set (of... |
| 1pthdlem1 29368 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 29369 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 29370 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 29371 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 29372 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 29373 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 29374 | In a graph with two vertic... |
| 1trld 29375 | In a graph with two vertic... |
| 1pthd 29376 | In a graph with two vertic... |
| 1pthond 29377 | In a graph with two vertic... |
| upgr1wlkdlem1 29378 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 29379 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 29380 | In a pseudograph with two ... |
| upgr1trld 29381 | In a pseudograph with two ... |
| upgr1pthd 29382 | In a pseudograph with two ... |
| upgr1pthond 29383 | In a pseudograph with two ... |
| lppthon 29384 | A loop (which is an edge a... |
| lp1cycl 29385 | A loop (which is an edge a... |
| 1pthon2v 29386 | For each pair of adjacent ... |
| 1pthon2ve 29387 | For each pair of adjacent ... |
| wlk2v2elem1 29388 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 29389 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 29390 | In a graph with two vertic... |
| ntrl2v2e 29391 | A walk which is not a trai... |
| 3wlkdlem1 29392 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 29393 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 29394 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 29395 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 29396 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 29397 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 29398 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 29399 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 29400 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 29401 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 29402 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 29403 | Construction of a walk fro... |
| 3wlkond 29404 | A walk of length 3 from on... |
| 3trld 29405 | Construction of a trail fr... |
| 3trlond 29406 | A trail of length 3 from o... |
| 3pthd 29407 | A path of length 3 from on... |
| 3pthond 29408 | A path of length 3 from on... |
| 3spthd 29409 | A simple path of length 3 ... |
| 3spthond 29410 | A simple path of length 3 ... |
| 3cycld 29411 | Construction of a 3-cycle ... |
| 3cyclpd 29412 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 29413 | If there is a cycle of len... |
| uhgr3cyclexlem 29414 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 29415 | If there are three differe... |
| umgr3cyclex 29416 | If there are three (differ... |
| umgr3v3e3cycl 29417 | If and only if there is a ... |
| upgr4cycl4dv4e 29418 | If there is a cycle of len... |
| dfconngr1 29421 | Alternative definition of ... |
| isconngr 29422 | The property of being a co... |
| isconngr1 29423 | The property of being a co... |
| cusconngr 29424 | A complete hypergraph is c... |
| 0conngr 29425 | A graph without vertices i... |
| 0vconngr 29426 | A graph without vertices i... |
| 1conngr 29427 | A graph with (at most) one... |
| conngrv2edg 29428 | A vertex in a connected gr... |
| vdn0conngrumgrv2 29429 | A vertex in a connected mu... |
| releupth 29432 | The set ` ( EulerPaths `` ... |
| eupths 29433 | The Eulerian paths on the ... |
| iseupth 29434 | The property " ` <. F , P ... |
| iseupthf1o 29435 | The property " ` <. F , P ... |
| eupthi 29436 | Properties of an Eulerian ... |
| eupthf1o 29437 | The ` F ` function in an E... |
| eupthfi 29438 | Any graph with an Eulerian... |
| eupthseg 29439 | The ` N ` -th edge in an e... |
| upgriseupth 29440 | The property " ` <. F , P ... |
| upgreupthi 29441 | Properties of an Eulerian ... |
| upgreupthseg 29442 | The ` N ` -th edge in an e... |
| eupthcl 29443 | An Eulerian path has lengt... |
| eupthistrl 29444 | An Eulerian path is a trai... |
| eupthiswlk 29445 | An Eulerian path is a walk... |
| eupthpf 29446 | The ` P ` function in an E... |
| eupth0 29447 | There is an Eulerian path ... |
| eupthres 29448 | The restriction ` <. H , Q... |
| eupthp1 29449 | Append one path segment to... |
| eupth2eucrct 29450 | Append one path segment to... |
| eupth2lem1 29451 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 29452 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 29453 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 29454 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 29455 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 29456 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 29457 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 29458 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 29459 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 29460 | Formerly part of proof of ... |
| eupth2lem3lem1 29461 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 29462 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 29463 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 29464 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 29465 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 29466 | Formerly part of proof of ... |
| eupth2lem3lem7 29467 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 29468 | Formerly part of proof of ... |
| eupth2lem3 29469 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 29470 | Lemma for ~ eupth2 (induct... |
| eupth2lems 29471 | Lemma for ~ eupth2 (induct... |
| eupth2 29472 | The only vertices of odd d... |
| eulerpathpr 29473 | A graph with an Eulerian p... |
| eulerpath 29474 | A pseudograph with an Eule... |
| eulercrct 29475 | A pseudograph with an Eule... |
| eucrctshift 29476 | Cyclically shifting the in... |
| eucrct2eupth1 29477 | Removing one edge ` ( I ``... |
| eucrct2eupth 29478 | Removing one edge ` ( I ``... |
| konigsbergvtx 29479 | The set of vertices of the... |
| konigsbergiedg 29480 | The indexed edges of the K... |
| konigsbergiedgw 29481 | The indexed edges of the K... |
| konigsbergssiedgwpr 29482 | Each subset of the indexed... |
| konigsbergssiedgw 29483 | Each subset of the indexed... |
| konigsbergumgr 29484 | The Königsberg graph ... |
| konigsberglem1 29485 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 29486 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 29487 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 29488 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 29489 | Lemma 5 for ~ konigsberg :... |
| konigsberg 29490 | The Königsberg Bridge... |
| isfrgr 29493 | The property of being a fr... |
| frgrusgr 29494 | A friendship graph is a si... |
| frgr0v 29495 | Any null graph (set with n... |
| frgr0vb 29496 | Any null graph (without ve... |
| frgruhgr0v 29497 | Any null graph (without ve... |
| frgr0 29498 | The null graph (graph with... |
| frcond1 29499 | The friendship condition: ... |
| frcond2 29500 | The friendship condition: ... |
| frgreu 29501 | Variant of ~ frcond2 : An... |
| frcond3 29502 | The friendship condition, ... |
| frcond4 29503 | The friendship condition, ... |
| frgr1v 29504 | Any graph with (at most) o... |
| nfrgr2v 29505 | Any graph with two (differ... |
| frgr3vlem1 29506 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 29507 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 29508 | Any graph with three verti... |
| 1vwmgr 29509 | Every graph with one verte... |
| 3vfriswmgrlem 29510 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 29511 | Every friendship graph wit... |
| 1to2vfriswmgr 29512 | Every friendship graph wit... |
| 1to3vfriswmgr 29513 | Every friendship graph wit... |
| 1to3vfriendship 29514 | The friendship theorem for... |
| 2pthfrgrrn 29515 | Between any two (different... |
| 2pthfrgrrn2 29516 | Between any two (different... |
| 2pthfrgr 29517 | Between any two (different... |
| 3cyclfrgrrn1 29518 | Every vertex in a friendsh... |
| 3cyclfrgrrn 29519 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 29520 | Every vertex in a friendsh... |
| 3cyclfrgr 29521 | Every vertex in a friendsh... |
| 4cycl2v2nb 29522 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 29523 | In a 4-cycle, two distinct... |
| n4cyclfrgr 29524 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 29525 | A graph with a 4-cycle is ... |
| frgrnbnb 29526 | If two neighbors ` U ` and... |
| frgrconngr 29527 | A friendship graph is conn... |
| vdgn0frgrv2 29528 | A vertex in a friendship g... |
| vdgn1frgrv2 29529 | Any vertex in a friendship... |
| vdgn1frgrv3 29530 | Any vertex in a friendship... |
| vdgfrgrgt2 29531 | Any vertex in a friendship... |
| frgrncvvdeqlem1 29532 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 29533 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 29534 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 29535 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 29536 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 29537 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 29538 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 29539 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 29540 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 29541 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 29542 | In a friendship graph, two... |
| frgrwopreglem4a 29543 | In a friendship graph any ... |
| frgrwopreglem5a 29544 | If a friendship graph has ... |
| frgrwopreglem1 29545 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 29546 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 29547 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 29548 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 29549 | According to statement 5 i... |
| frgrwopregbsn 29550 | According to statement 5 i... |
| frgrwopreg1 29551 | According to statement 5 i... |
| frgrwopreg2 29552 | According to statement 5 i... |
| frgrwopreglem5lem 29553 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 29554 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 29555 | Alternate direct proof of ... |
| frgrwopreg 29556 | In a friendship graph ther... |
| frgrregorufr0 29557 | In a friendship graph ther... |
| frgrregorufr 29558 | If there is a vertex havin... |
| frgrregorufrg 29559 | If there is a vertex havin... |
| frgr2wwlkeu 29560 | For two different vertices... |
| frgr2wwlkn0 29561 | In a friendship graph, the... |
| frgr2wwlk1 29562 | In a friendship graph, the... |
| frgr2wsp1 29563 | In a friendship graph, the... |
| frgr2wwlkeqm 29564 | If there is a (simple) pat... |
| frgrhash2wsp 29565 | The number of simple paths... |
| fusgreg2wsplem 29566 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 29567 | The set of paths of length... |
| fusgreghash2wspv 29568 | According to statement 7 i... |
| fusgreg2wsp 29569 | In a finite simple graph, ... |
| 2wspmdisj 29570 | The sets of paths of lengt... |
| fusgreghash2wsp 29571 | In a finite k-regular grap... |
| frrusgrord0lem 29572 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 29573 | If a nonempty finite frien... |
| frrusgrord 29574 | If a nonempty finite frien... |
| numclwwlk2lem1lem 29575 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 29576 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 29577 | If the initial vertex of a... |
| clwwnonrepclwwnon 29578 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 29579 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 29580 | Value of operation ` C ` ,... |
| 2clwwlk2 29581 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 29582 | Characterization of an ele... |
| 2clwwlk2clwwlk 29583 | An element of the value of... |
| numclwwlk1lem2foalem 29584 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 29585 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 29586 | Characterization of an ele... |
| numclwwlk1lem2foa 29587 | Going forth and back from ... |
| numclwwlk1lem2f 29588 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 29589 | Value of the function ` T ... |
| numclwwlk1lem2f1 29590 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 29591 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 29592 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 29593 | The set of double loops of... |
| numclwwlk1 29594 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 29595 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 29596 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 29597 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 29598 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 29599 | The sets of the two repres... |
| wlkl0 29600 | There is exactly one walk ... |
| clwlknon2num 29601 | There are k walks of lengt... |
| numclwlk1lem1 29602 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 29603 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 29604 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 29605 | Value of operation ` H ` ,... |
| numclwwlkovh 29606 | Value of operation ` H ` ,... |
| numclwwlkovq 29607 | Value of operation ` Q ` ,... |
| numclwwlkqhash 29608 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 29609 | In a friendship graph, for... |
| numclwlk2lem2f 29610 | ` R ` is a function mappin... |
| numclwlk2lem2fv 29611 | Value of the function ` R ... |
| numclwlk2lem2f1o 29612 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 29613 | In a friendship graph, the... |
| numclwwlk2 29614 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 29615 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 29616 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 29617 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 29618 | Statement 12 in [Huneke] p... |
| numclwwlk4 29619 | The total number of closed... |
| numclwwlk5lem 29620 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 29621 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 29622 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 29623 | For a prime divisor ` P ` ... |
| numclwwlk7 29624 | Statement 14 in [Huneke] p... |
| numclwwlk8 29625 | The size of the set of clo... |
| frgrreggt1 29626 | If a finite nonempty frien... |
| frgrreg 29627 | If a finite nonempty frien... |
| frgrregord013 29628 | If a finite friendship gra... |
| frgrregord13 29629 | If a nonempty finite frien... |
| frgrogt3nreg 29630 | If a finite friendship gra... |
| friendshipgt3 29631 | The friendship theorem for... |
| friendship 29632 | The friendship theorem: I... |
| conventions 29633 |
H... |
| conventions-labels 29634 |
... |
| conventions-comments 29635 |
... |
| natded 29636 | Here are typical n... |
| ex-natded5.2 29637 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 29638 | A more efficient proof of ... |
| ex-natded5.2i 29639 | The same as ~ ex-natded5.2... |
| ex-natded5.3 29640 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 29641 | A more efficient proof of ... |
| ex-natded5.3i 29642 | The same as ~ ex-natded5.3... |
| ex-natded5.5 29643 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 29644 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 29645 | A more efficient proof of ... |
| ex-natded5.8 29646 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 29647 | A more efficient proof of ... |
| ex-natded5.13 29648 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 29649 | A more efficient proof of ... |
| ex-natded9.20 29650 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 29651 | A more efficient proof of ... |
| ex-natded9.26 29652 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 29653 | A more efficient proof of ... |
| ex-or 29654 | Example for ~ df-or . Exa... |
| ex-an 29655 | Example for ~ df-an . Exa... |
| ex-dif 29656 | Example for ~ df-dif . Ex... |
| ex-un 29657 | Example for ~ df-un . Exa... |
| ex-in 29658 | Example for ~ df-in . Exa... |
| ex-uni 29659 | Example for ~ df-uni . Ex... |
| ex-ss 29660 | Example for ~ df-ss . Exa... |
| ex-pss 29661 | Example for ~ df-pss . Ex... |
| ex-pw 29662 | Example for ~ df-pw . Exa... |
| ex-pr 29663 | Example for ~ df-pr . (Co... |
| ex-br 29664 | Example for ~ df-br . Exa... |
| ex-opab 29665 | Example for ~ df-opab . E... |
| ex-eprel 29666 | Example for ~ df-eprel . ... |
| ex-id 29667 | Example for ~ df-id . Exa... |
| ex-po 29668 | Example for ~ df-po . Exa... |
| ex-xp 29669 | Example for ~ df-xp . Exa... |
| ex-cnv 29670 | Example for ~ df-cnv . Ex... |
| ex-co 29671 | Example for ~ df-co . Exa... |
| ex-dm 29672 | Example for ~ df-dm . Exa... |
| ex-rn 29673 | Example for ~ df-rn . Exa... |
| ex-res 29674 | Example for ~ df-res . Ex... |
| ex-ima 29675 | Example for ~ df-ima . Ex... |
| ex-fv 29676 | Example for ~ df-fv . Exa... |
| ex-1st 29677 | Example for ~ df-1st . Ex... |
| ex-2nd 29678 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 29679 | Example for ~ df-dec , 100... |
| ex-fl 29680 | Example for ~ df-fl . Exa... |
| ex-ceil 29681 | Example for ~ df-ceil . (... |
| ex-mod 29682 | Example for ~ df-mod . (C... |
| ex-exp 29683 | Example for ~ df-exp . (C... |
| ex-fac 29684 | Example for ~ df-fac . (C... |
| ex-bc 29685 | Example for ~ df-bc . (Co... |
| ex-hash 29686 | Example for ~ df-hash . (... |
| ex-sqrt 29687 | Example for ~ df-sqrt . (... |
| ex-abs 29688 | Example for ~ df-abs . (C... |
| ex-dvds 29689 | Example for ~ df-dvds : 3 ... |
| ex-gcd 29690 | Example for ~ df-gcd . (C... |
| ex-lcm 29691 | Example for ~ df-lcm . (C... |
| ex-prmo 29692 | Example for ~ df-prmo : ` ... |
| aevdemo 29693 | Proof illustrating the com... |
| ex-ind-dvds 29694 | Example of a proof by indu... |
| ex-fpar 29695 | Formalized example provide... |
| avril1 29696 | Poisson d'Avril's Theorem.... |
| 2bornot2b 29697 | The law of excluded middle... |
| helloworld 29698 | The classic "Hello world" ... |
| 1p1e2apr1 29699 | One plus one equals two. ... |
| eqid1 29700 | Law of identity (reflexivi... |
| 1div0apr 29701 | Division by zero is forbid... |
| topnfbey 29702 | Nothing seems to be imposs... |
| 9p10ne21 29703 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 29704 | 9 + 10 equals 21. This as... |
| isplig 29707 | The predicate "is a planar... |
| ispligb 29708 | The predicate "is a planar... |
| tncp 29709 | In any planar incidence ge... |
| l2p 29710 | For any line in a planar i... |
| lpni 29711 | For any line in a planar i... |
| nsnlplig 29712 | There is no "one-point lin... |
| nsnlpligALT 29713 | Alternate version of ~ nsn... |
| n0lplig 29714 | There is no "empty line" i... |
| n0lpligALT 29715 | Alternate version of ~ n0l... |
| eulplig 29716 | Through two distinct point... |
| pliguhgr 29717 | Any planar incidence geome... |
| dummylink 29718 | Alias for ~ a1ii that may ... |
| id1 29719 | Alias for ~ idALT that may... |
| isgrpo 29728 | The predicate "is a group ... |
| isgrpoi 29729 | Properties that determine ... |
| grpofo 29730 | A group operation maps ont... |
| grpocl 29731 | Closure law for a group op... |
| grpolidinv 29732 | A group has a left identit... |
| grpon0 29733 | The base set of a group is... |
| grpoass 29734 | A group operation is assoc... |
| grpoidinvlem1 29735 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 29736 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 29737 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 29738 | Lemma for ~ grpoidinv . (... |
| grpoidinv 29739 | A group has a left and rig... |
| grpoideu 29740 | The left identity element ... |
| grporndm 29741 | A group's range in terms o... |
| 0ngrp 29742 | The empty set is not a gro... |
| gidval 29743 | The value of the identity ... |
| grpoidval 29744 | Lemma for ~ grpoidcl and o... |
| grpoidcl 29745 | The identity element of a ... |
| grpoidinv2 29746 | A group's properties using... |
| grpolid 29747 | The identity element of a ... |
| grporid 29748 | The identity element of a ... |
| grporcan 29749 | Right cancellation law for... |
| grpoinveu 29750 | The left inverse element o... |
| grpoid 29751 | Two ways of saying that an... |
| grporn 29752 | The range of a group opera... |
| grpoinvfval 29753 | The inverse function of a ... |
| grpoinvval 29754 | The inverse of a group ele... |
| grpoinvcl 29755 | A group element's inverse ... |
| grpoinv 29756 | The properties of a group ... |
| grpolinv 29757 | The left inverse of a grou... |
| grporinv 29758 | The right inverse of a gro... |
| grpoinvid1 29759 | The inverse of a group ele... |
| grpoinvid2 29760 | The inverse of a group ele... |
| grpolcan 29761 | Left cancellation law for ... |
| grpo2inv 29762 | Double inverse law for gro... |
| grpoinvf 29763 | Mapping of the inverse fun... |
| grpoinvop 29764 | The inverse of the group o... |
| grpodivfval 29765 | Group division (or subtrac... |
| grpodivval 29766 | Group division (or subtrac... |
| grpodivinv 29767 | Group division by an inver... |
| grpoinvdiv 29768 | Inverse of a group divisio... |
| grpodivf 29769 | Mapping for group division... |
| grpodivcl 29770 | Closure of group division ... |
| grpodivdiv 29771 | Double group division. (C... |
| grpomuldivass 29772 | Associative-type law for m... |
| grpodivid 29773 | Division of a group member... |
| grponpcan 29774 | Cancellation law for group... |
| isablo 29777 | The predicate "is an Abeli... |
| ablogrpo 29778 | An Abelian group operation... |
| ablocom 29779 | An Abelian group operation... |
| ablo32 29780 | Commutative/associative la... |
| ablo4 29781 | Commutative/associative la... |
| isabloi 29782 | Properties that determine ... |
| ablomuldiv 29783 | Law for group multiplicati... |
| ablodivdiv 29784 | Law for double group divis... |
| ablodivdiv4 29785 | Law for double group divis... |
| ablodiv32 29786 | Swap the second and third ... |
| ablonncan 29787 | Cancellation law for group... |
| ablonnncan1 29788 | Cancellation law for group... |
| vcrel 29791 | The class of all complex v... |
| vciOLD 29792 | Obsolete version of ~ cvsi... |
| vcsm 29793 | Functionality of th scalar... |
| vccl 29794 | Closure of the scalar prod... |
| vcidOLD 29795 | Identity element for the s... |
| vcdi 29796 | Distributive law for the s... |
| vcdir 29797 | Distributive law for the s... |
| vcass 29798 | Associative law for the sc... |
| vc2OLD 29799 | A vector plus itself is tw... |
| vcablo 29800 | Vector addition is an Abel... |
| vcgrp 29801 | Vector addition is a group... |
| vclcan 29802 | Left cancellation law for ... |
| vczcl 29803 | The zero vector is a vecto... |
| vc0rid 29804 | The zero vector is a right... |
| vc0 29805 | Zero times a vector is the... |
| vcz 29806 | Anything times the zero ve... |
| vcm 29807 | Minus 1 times a vector is ... |
| isvclem 29808 | Lemma for ~ isvcOLD . (Co... |
| vcex 29809 | The components of a comple... |
| isvcOLD 29810 | The predicate "is a comple... |
| isvciOLD 29811 | Properties that determine ... |
| cnaddabloOLD 29812 | Obsolete version of ~ cnad... |
| cnidOLD 29813 | Obsolete version of ~ cnad... |
| cncvcOLD 29814 | Obsolete version of ~ cncv... |
| nvss 29824 | Structure of the class of ... |
| nvvcop 29825 | A normed complex vector sp... |
| nvrel 29833 | The class of all normed co... |
| vafval 29834 | Value of the function for ... |
| bafval 29835 | Value of the function for ... |
| smfval 29836 | Value of the function for ... |
| 0vfval 29837 | Value of the function for ... |
| nmcvfval 29838 | Value of the norm function... |
| nvop2 29839 | A normed complex vector sp... |
| nvvop 29840 | The vector space component... |
| isnvlem 29841 | Lemma for ~ isnv . (Contr... |
| nvex 29842 | The components of a normed... |
| isnv 29843 | The predicate "is a normed... |
| isnvi 29844 | Properties that determine ... |
| nvi 29845 | The properties of a normed... |
| nvvc 29846 | The vector space component... |
| nvablo 29847 | The vector addition operat... |
| nvgrp 29848 | The vector addition operat... |
| nvgf 29849 | Mapping for the vector add... |
| nvsf 29850 | Mapping for the scalar mul... |
| nvgcl 29851 | Closure law for the vector... |
| nvcom 29852 | The vector addition (group... |
| nvass 29853 | The vector addition (group... |
| nvadd32 29854 | Commutative/associative la... |
| nvrcan 29855 | Right cancellation law for... |
| nvadd4 29856 | Rearrangement of 4 terms i... |
| nvscl 29857 | Closure law for the scalar... |
| nvsid 29858 | Identity element for the s... |
| nvsass 29859 | Associative law for the sc... |
| nvscom 29860 | Commutative law for the sc... |
| nvdi 29861 | Distributive law for the s... |
| nvdir 29862 | Distributive law for the s... |
| nv2 29863 | A vector plus itself is tw... |
| vsfval 29864 | Value of the function for ... |
| nvzcl 29865 | Closure law for the zero v... |
| nv0rid 29866 | The zero vector is a right... |
| nv0lid 29867 | The zero vector is a left ... |
| nv0 29868 | Zero times a vector is the... |
| nvsz 29869 | Anything times the zero ve... |
| nvinv 29870 | Minus 1 times a vector is ... |
| nvinvfval 29871 | Function for the negative ... |
| nvm 29872 | Vector subtraction in term... |
| nvmval 29873 | Value of vector subtractio... |
| nvmval2 29874 | Value of vector subtractio... |
| nvmfval 29875 | Value of the function for ... |
| nvmf 29876 | Mapping for the vector sub... |
| nvmcl 29877 | Closure law for the vector... |
| nvnnncan1 29878 | Cancellation law for vecto... |
| nvmdi 29879 | Distributive law for scala... |
| nvnegneg 29880 | Double negative of a vecto... |
| nvmul0or 29881 | If a scalar product is zer... |
| nvrinv 29882 | A vector minus itself. (C... |
| nvlinv 29883 | Minus a vector plus itself... |
| nvpncan2 29884 | Cancellation law for vecto... |
| nvpncan 29885 | Cancellation law for vecto... |
| nvaddsub 29886 | Commutative/associative la... |
| nvnpcan 29887 | Cancellation law for a nor... |
| nvaddsub4 29888 | Rearrangement of 4 terms i... |
| nvmeq0 29889 | The difference between two... |
| nvmid 29890 | A vector minus itself is t... |
| nvf 29891 | Mapping for the norm funct... |
| nvcl 29892 | The norm of a normed compl... |
| nvcli 29893 | The norm of a normed compl... |
| nvs 29894 | Proportionality property o... |
| nvsge0 29895 | The norm of a scalar produ... |
| nvm1 29896 | The norm of the negative o... |
| nvdif 29897 | The norm of the difference... |
| nvpi 29898 | The norm of a vector plus ... |
| nvz0 29899 | The norm of a zero vector ... |
| nvz 29900 | The norm of a vector is ze... |
| nvtri 29901 | Triangle inequality for th... |
| nvmtri 29902 | Triangle inequality for th... |
| nvabs 29903 | Norm difference property o... |
| nvge0 29904 | The norm of a normed compl... |
| nvgt0 29905 | A nonzero norm is positive... |
| nv1 29906 | From any nonzero vector, c... |
| nvop 29907 | A complex inner product sp... |
| cnnv 29908 | The set of complex numbers... |
| cnnvg 29909 | The vector addition (group... |
| cnnvba 29910 | The base set of the normed... |
| cnnvs 29911 | The scalar product operati... |
| cnnvnm 29912 | The norm operation of the ... |
| cnnvm 29913 | The vector subtraction ope... |
| elimnv 29914 | Hypothesis elimination lem... |
| elimnvu 29915 | Hypothesis elimination lem... |
| imsval 29916 | Value of the induced metri... |
| imsdval 29917 | Value of the induced metri... |
| imsdval2 29918 | Value of the distance func... |
| nvnd 29919 | The norm of a normed compl... |
| imsdf 29920 | Mapping for the induced me... |
| imsmetlem 29921 | Lemma for ~ imsmet . (Con... |
| imsmet 29922 | The induced metric of a no... |
| imsxmet 29923 | The induced metric of a no... |
| cnims 29924 | The metric induced on the ... |
| vacn 29925 | Vector addition is jointly... |
| nmcvcn 29926 | The norm of a normed compl... |
| nmcnc 29927 | The norm of a normed compl... |
| smcnlem 29928 | Lemma for ~ smcn . (Contr... |
| smcn 29929 | Scalar multiplication is j... |
| vmcn 29930 | Vector subtraction is join... |
| dipfval 29933 | The inner product function... |
| ipval 29934 | Value of the inner product... |
| ipval2lem2 29935 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 29936 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 29937 | Lemma for ~ ipval3 . (Con... |
| ipval2 29938 | Expansion of the inner pro... |
| 4ipval2 29939 | Four times the inner produ... |
| ipval3 29940 | Expansion of the inner pro... |
| ipidsq 29941 | The inner product of a vec... |
| ipnm 29942 | Norm expressed in terms of... |
| dipcl 29943 | An inner product is a comp... |
| ipf 29944 | Mapping for the inner prod... |
| dipcj 29945 | The complex conjugate of a... |
| ipipcj 29946 | An inner product times its... |
| diporthcom 29947 | Orthogonality (meaning inn... |
| dip0r 29948 | Inner product with a zero ... |
| dip0l 29949 | Inner product with a zero ... |
| ipz 29950 | The inner product of a vec... |
| dipcn 29951 | Inner product is jointly c... |
| sspval 29954 | The set of all subspaces o... |
| isssp 29955 | The predicate "is a subspa... |
| sspid 29956 | A normed complex vector sp... |
| sspnv 29957 | A subspace is a normed com... |
| sspba 29958 | The base set of a subspace... |
| sspg 29959 | Vector addition on a subsp... |
| sspgval 29960 | Vector addition on a subsp... |
| ssps 29961 | Scalar multiplication on a... |
| sspsval 29962 | Scalar multiplication on a... |
| sspmlem 29963 | Lemma for ~ sspm and other... |
| sspmval 29964 | Vector addition on a subsp... |
| sspm 29965 | Vector subtraction on a su... |
| sspz 29966 | The zero vector of a subsp... |
| sspn 29967 | The norm on a subspace is ... |
| sspnval 29968 | The norm on a subspace in ... |
| sspimsval 29969 | The induced metric on a su... |
| sspims 29970 | The induced metric on a su... |
| lnoval 29983 | The set of linear operator... |
| islno 29984 | The predicate "is a linear... |
| lnolin 29985 | Basic linearity property o... |
| lnof 29986 | A linear operator is a map... |
| lno0 29987 | The value of a linear oper... |
| lnocoi 29988 | The composition of two lin... |
| lnoadd 29989 | Addition property of a lin... |
| lnosub 29990 | Subtraction property of a ... |
| lnomul 29991 | Scalar multiplication prop... |
| nvo00 29992 | Two ways to express a zero... |
| nmoofval 29993 | The operator norm function... |
| nmooval 29994 | The operator norm function... |
| nmosetre 29995 | The set in the supremum of... |
| nmosetn0 29996 | The set in the supremum of... |
| nmoxr 29997 | The norm of an operator is... |
| nmooge0 29998 | The norm of an operator is... |
| nmorepnf 29999 | The norm of an operator is... |
| nmoreltpnf 30000 | The norm of any operator i... |
| nmogtmnf 30001 | The norm of an operator is... |
| nmoolb 30002 | A lower bound for an opera... |
| nmoubi 30003 | An upper bound for an oper... |
| nmoub3i 30004 | An upper bound for an oper... |
| nmoub2i 30005 | An upper bound for an oper... |
| nmobndi 30006 | Two ways to express that a... |
| nmounbi 30007 | Two ways two express that ... |
| nmounbseqi 30008 | An unbounded operator dete... |
| nmounbseqiALT 30009 | Alternate shorter proof of... |
| nmobndseqi 30010 | A bounded sequence determi... |
| nmobndseqiALT 30011 | Alternate shorter proof of... |
| bloval 30012 | The class of bounded linea... |
| isblo 30013 | The predicate "is a bounde... |
| isblo2 30014 | The predicate "is a bounde... |
| bloln 30015 | A bounded operator is a li... |
| blof 30016 | A bounded operator is an o... |
| nmblore 30017 | The norm of a bounded oper... |
| 0ofval 30018 | The zero operator between ... |
| 0oval 30019 | Value of the zero operator... |
| 0oo 30020 | The zero operator is an op... |
| 0lno 30021 | The zero operator is linea... |
| nmoo0 30022 | The operator norm of the z... |
| 0blo 30023 | The zero operator is a bou... |
| nmlno0lem 30024 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 30025 | The norm of a linear opera... |
| nmlno0 30026 | The norm of a linear opera... |
| nmlnoubi 30027 | An upper bound for the ope... |
| nmlnogt0 30028 | The norm of a nonzero line... |
| lnon0 30029 | The domain of a nonzero li... |
| nmblolbii 30030 | A lower bound for the norm... |
| nmblolbi 30031 | A lower bound for the norm... |
| isblo3i 30032 | The predicate "is a bounde... |
| blo3i 30033 | Properties that determine ... |
| blometi 30034 | Upper bound for the distan... |
| blocnilem 30035 | Lemma for ~ blocni and ~ l... |
| blocni 30036 | A linear operator is conti... |
| lnocni 30037 | If a linear operator is co... |
| blocn 30038 | A linear operator is conti... |
| blocn2 30039 | A bounded linear operator ... |
| ajfval 30040 | The adjoint function. (Co... |
| hmoval 30041 | The set of Hermitian (self... |
| ishmo 30042 | The predicate "is a hermit... |
| phnv 30045 | Every complex inner produc... |
| phrel 30046 | The class of all complex i... |
| phnvi 30047 | Every complex inner produc... |
| isphg 30048 | The predicate "is a comple... |
| phop 30049 | A complex inner product sp... |
| cncph 30050 | The set of complex numbers... |
| elimph 30051 | Hypothesis elimination lem... |
| elimphu 30052 | Hypothesis elimination lem... |
| isph 30053 | The predicate "is an inner... |
| phpar2 30054 | The parallelogram law for ... |
| phpar 30055 | The parallelogram law for ... |
| ip0i 30056 | A slight variant of Equati... |
| ip1ilem 30057 | Lemma for ~ ip1i . (Contr... |
| ip1i 30058 | Equation 6.47 of [Ponnusam... |
| ip2i 30059 | Equation 6.48 of [Ponnusam... |
| ipdirilem 30060 | Lemma for ~ ipdiri . (Con... |
| ipdiri 30061 | Distributive law for inner... |
| ipasslem1 30062 | Lemma for ~ ipassi . Show... |
| ipasslem2 30063 | Lemma for ~ ipassi . Show... |
| ipasslem3 30064 | Lemma for ~ ipassi . Show... |
| ipasslem4 30065 | Lemma for ~ ipassi . Show... |
| ipasslem5 30066 | Lemma for ~ ipassi . Show... |
| ipasslem7 30067 | Lemma for ~ ipassi . Show... |
| ipasslem8 30068 | Lemma for ~ ipassi . By ~... |
| ipasslem9 30069 | Lemma for ~ ipassi . Conc... |
| ipasslem10 30070 | Lemma for ~ ipassi . Show... |
| ipasslem11 30071 | Lemma for ~ ipassi . Show... |
| ipassi 30072 | Associative law for inner ... |
| dipdir 30073 | Distributive law for inner... |
| dipdi 30074 | Distributive law for inner... |
| ip2dii 30075 | Inner product of two sums.... |
| dipass 30076 | Associative law for inner ... |
| dipassr 30077 | "Associative" law for seco... |
| dipassr2 30078 | "Associative" law for inne... |
| dipsubdir 30079 | Distributive law for inner... |
| dipsubdi 30080 | Distributive law for inner... |
| pythi 30081 | The Pythagorean theorem fo... |
| siilem1 30082 | Lemma for ~ sii . (Contri... |
| siilem2 30083 | Lemma for ~ sii . (Contri... |
| siii 30084 | Inference from ~ sii . (C... |
| sii 30085 | Obsolete version of ~ ipca... |
| ipblnfi 30086 | A function ` F ` generated... |
| ip2eqi 30087 | Two vectors are equal iff ... |
| phoeqi 30088 | A condition implying that ... |
| ajmoi 30089 | Every operator has at most... |
| ajfuni 30090 | The adjoint function is a ... |
| ajfun 30091 | The adjoint function is a ... |
| ajval 30092 | Value of the adjoint funct... |
| iscbn 30095 | A complex Banach space is ... |
| cbncms 30096 | The induced metric on comp... |
| bnnv 30097 | Every complex Banach space... |
| bnrel 30098 | The class of all complex B... |
| bnsscmcl 30099 | A subspace of a Banach spa... |
| cnbn 30100 | The set of complex numbers... |
| ubthlem1 30101 | Lemma for ~ ubth . The fu... |
| ubthlem2 30102 | Lemma for ~ ubth . Given ... |
| ubthlem3 30103 | Lemma for ~ ubth . Prove ... |
| ubth 30104 | Uniform Boundedness Theore... |
| minvecolem1 30105 | Lemma for ~ minveco . The... |
| minvecolem2 30106 | Lemma for ~ minveco . Any... |
| minvecolem3 30107 | Lemma for ~ minveco . The... |
| minvecolem4a 30108 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 30109 | Lemma for ~ minveco . The... |
| minvecolem4c 30110 | Lemma for ~ minveco . The... |
| minvecolem4 30111 | Lemma for ~ minveco . The... |
| minvecolem5 30112 | Lemma for ~ minveco . Dis... |
| minvecolem6 30113 | Lemma for ~ minveco . Any... |
| minvecolem7 30114 | Lemma for ~ minveco . Sin... |
| minveco 30115 | Minimizing vector theorem,... |
| ishlo 30118 | The predicate "is a comple... |
| hlobn 30119 | Every complex Hilbert spac... |
| hlph 30120 | Every complex Hilbert spac... |
| hlrel 30121 | The class of all complex H... |
| hlnv 30122 | Every complex Hilbert spac... |
| hlnvi 30123 | Every complex Hilbert spac... |
| hlvc 30124 | Every complex Hilbert spac... |
| hlcmet 30125 | The induced metric on a co... |
| hlmet 30126 | The induced metric on a co... |
| hlpar2 30127 | The parallelogram law sati... |
| hlpar 30128 | The parallelogram law sati... |
| hlex 30129 | The base set of a Hilbert ... |
| hladdf 30130 | Mapping for Hilbert space ... |
| hlcom 30131 | Hilbert space vector addit... |
| hlass 30132 | Hilbert space vector addit... |
| hl0cl 30133 | The Hilbert space zero vec... |
| hladdid 30134 | Hilbert space addition wit... |
| hlmulf 30135 | Mapping for Hilbert space ... |
| hlmulid 30136 | Hilbert space scalar multi... |
| hlmulass 30137 | Hilbert space scalar multi... |
| hldi 30138 | Hilbert space scalar multi... |
| hldir 30139 | Hilbert space scalar multi... |
| hlmul0 30140 | Hilbert space scalar multi... |
| hlipf 30141 | Mapping for Hilbert space ... |
| hlipcj 30142 | Conjugate law for Hilbert ... |
| hlipdir 30143 | Distributive law for Hilbe... |
| hlipass 30144 | Associative law for Hilber... |
| hlipgt0 30145 | The inner product of a Hil... |
| hlcompl 30146 | Completeness of a Hilbert ... |
| cnchl 30147 | The set of complex numbers... |
| htthlem 30148 | Lemma for ~ htth . The co... |
| htth 30149 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 30205 | The group (addition) opera... |
| h2hsm 30206 | The scalar product operati... |
| h2hnm 30207 | The norm function of Hilbe... |
| h2hvs 30208 | The vector subtraction ope... |
| h2hmetdval 30209 | Value of the distance func... |
| h2hcau 30210 | The Cauchy sequences of Hi... |
| h2hlm 30211 | The limit sequences of Hil... |
| axhilex-zf 30212 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 30213 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 30214 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 30215 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 30216 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 30217 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 30218 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 30219 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 30220 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 30221 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 30222 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 30223 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 30224 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 30225 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 30226 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 30227 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 30228 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 30229 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 30242 | The Hilbert space scalar p... |
| hvaddcl 30243 | Closure of vector addition... |
| hvmulcl 30244 | Closure of scalar multipli... |
| hvmulcli 30245 | Closure inference for scal... |
| hvsubf 30246 | Mapping domain and codomai... |
| hvsubval 30247 | Value of vector subtractio... |
| hvsubcl 30248 | Closure of vector subtract... |
| hvaddcli 30249 | Closure of vector addition... |
| hvcomi 30250 | Commutation of vector addi... |
| hvsubvali 30251 | Value of vector subtractio... |
| hvsubcli 30252 | Closure of vector subtract... |
| ifhvhv0 30253 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 30254 | Addition with the zero vec... |
| hvmul0 30255 | Scalar multiplication with... |
| hvmul0or 30256 | If a scalar product is zer... |
| hvsubid 30257 | Subtraction of a vector fr... |
| hvnegid 30258 | Addition of negative of a ... |
| hv2neg 30259 | Two ways to express the ne... |
| hvaddlidi 30260 | Addition with the zero vec... |
| hvnegidi 30261 | Addition of negative of a ... |
| hv2negi 30262 | Two ways to express the ne... |
| hvm1neg 30263 | Convert minus one times a ... |
| hvaddsubval 30264 | Value of vector addition i... |
| hvadd32 30265 | Commutative/associative la... |
| hvadd12 30266 | Commutative/associative la... |
| hvadd4 30267 | Hilbert vector space addit... |
| hvsub4 30268 | Hilbert vector space addit... |
| hvaddsub12 30269 | Commutative/associative la... |
| hvpncan 30270 | Addition/subtraction cance... |
| hvpncan2 30271 | Addition/subtraction cance... |
| hvaddsubass 30272 | Associativity of sum and d... |
| hvpncan3 30273 | Subtraction and addition o... |
| hvmulcom 30274 | Scalar multiplication comm... |
| hvsubass 30275 | Hilbert vector space assoc... |
| hvsub32 30276 | Hilbert vector space commu... |
| hvmulassi 30277 | Scalar multiplication asso... |
| hvmulcomi 30278 | Scalar multiplication comm... |
| hvmul2negi 30279 | Double negative in scalar ... |
| hvsubdistr1 30280 | Scalar multiplication dist... |
| hvsubdistr2 30281 | Scalar multiplication dist... |
| hvdistr1i 30282 | Scalar multiplication dist... |
| hvsubdistr1i 30283 | Scalar multiplication dist... |
| hvassi 30284 | Hilbert vector space assoc... |
| hvadd32i 30285 | Hilbert vector space commu... |
| hvsubassi 30286 | Hilbert vector space assoc... |
| hvsub32i 30287 | Hilbert vector space commu... |
| hvadd12i 30288 | Hilbert vector space commu... |
| hvadd4i 30289 | Hilbert vector space addit... |
| hvsubsub4i 30290 | Hilbert vector space addit... |
| hvsubsub4 30291 | Hilbert vector space addit... |
| hv2times 30292 | Two times a vector. (Cont... |
| hvnegdii 30293 | Distribution of negative o... |
| hvsubeq0i 30294 | If the difference between ... |
| hvsubcan2i 30295 | Vector cancellation law. ... |
| hvaddcani 30296 | Cancellation law for vecto... |
| hvsubaddi 30297 | Relationship between vecto... |
| hvnegdi 30298 | Distribution of negative o... |
| hvsubeq0 30299 | If the difference between ... |
| hvaddeq0 30300 | If the sum of two vectors ... |
| hvaddcan 30301 | Cancellation law for vecto... |
| hvaddcan2 30302 | Cancellation law for vecto... |
| hvmulcan 30303 | Cancellation law for scala... |
| hvmulcan2 30304 | Cancellation law for scala... |
| hvsubcan 30305 | Cancellation law for vecto... |
| hvsubcan2 30306 | Cancellation law for vecto... |
| hvsub0 30307 | Subtraction of a zero vect... |
| hvsubadd 30308 | Relationship between vecto... |
| hvaddsub4 30309 | Hilbert vector space addit... |
| hicl 30311 | Closure of inner product. ... |
| hicli 30312 | Closure inference for inne... |
| his5 30317 | Associative law for inner ... |
| his52 30318 | Associative law for inner ... |
| his35 30319 | Move scalar multiplication... |
| his35i 30320 | Move scalar multiplication... |
| his7 30321 | Distributive law for inner... |
| hiassdi 30322 | Distributive/associative l... |
| his2sub 30323 | Distributive law for inner... |
| his2sub2 30324 | Distributive law for inner... |
| hire 30325 | A necessary and sufficient... |
| hiidrcl 30326 | Real closure of inner prod... |
| hi01 30327 | Inner product with the 0 v... |
| hi02 30328 | Inner product with the 0 v... |
| hiidge0 30329 | Inner product with self is... |
| his6 30330 | Zero inner product with se... |
| his1i 30331 | Conjugate law for inner pr... |
| abshicom 30332 | Commuted inner products ha... |
| hial0 30333 | A vector whose inner produ... |
| hial02 30334 | A vector whose inner produ... |
| hisubcomi 30335 | Two vector subtractions si... |
| hi2eq 30336 | Lemma used to prove equali... |
| hial2eq 30337 | Two vectors whose inner pr... |
| hial2eq2 30338 | Two vectors whose inner pr... |
| orthcom 30339 | Orthogonality commutes. (... |
| normlem0 30340 | Lemma used to derive prope... |
| normlem1 30341 | Lemma used to derive prope... |
| normlem2 30342 | Lemma used to derive prope... |
| normlem3 30343 | Lemma used to derive prope... |
| normlem4 30344 | Lemma used to derive prope... |
| normlem5 30345 | Lemma used to derive prope... |
| normlem6 30346 | Lemma used to derive prope... |
| normlem7 30347 | Lemma used to derive prope... |
| normlem8 30348 | Lemma used to derive prope... |
| normlem9 30349 | Lemma used to derive prope... |
| normlem7tALT 30350 | Lemma used to derive prope... |
| bcseqi 30351 | Equality case of Bunjakova... |
| normlem9at 30352 | Lemma used to derive prope... |
| dfhnorm2 30353 | Alternate definition of th... |
| normf 30354 | The norm function maps fro... |
| normval 30355 | The value of the norm of a... |
| normcl 30356 | Real closure of the norm o... |
| normge0 30357 | The norm of a vector is no... |
| normgt0 30358 | The norm of nonzero vector... |
| norm0 30359 | The norm of a zero vector.... |
| norm-i 30360 | Theorem 3.3(i) of [Beran] ... |
| normne0 30361 | A norm is nonzero iff its ... |
| normcli 30362 | Real closure of the norm o... |
| normsqi 30363 | The square of a norm. (Co... |
| norm-i-i 30364 | Theorem 3.3(i) of [Beran] ... |
| normsq 30365 | The square of a norm. (Co... |
| normsub0i 30366 | Two vectors are equal iff ... |
| normsub0 30367 | Two vectors are equal iff ... |
| norm-ii-i 30368 | Triangle inequality for no... |
| norm-ii 30369 | Triangle inequality for no... |
| norm-iii-i 30370 | Theorem 3.3(iii) of [Beran... |
| norm-iii 30371 | Theorem 3.3(iii) of [Beran... |
| normsubi 30372 | Negative doesn't change th... |
| normpythi 30373 | Analogy to Pythagorean the... |
| normsub 30374 | Swapping order of subtract... |
| normneg 30375 | The norm of a vector equal... |
| normpyth 30376 | Analogy to Pythagorean the... |
| normpyc 30377 | Corollary to Pythagorean t... |
| norm3difi 30378 | Norm of differences around... |
| norm3adifii 30379 | Norm of differences around... |
| norm3lem 30380 | Lemma involving norm of di... |
| norm3dif 30381 | Norm of differences around... |
| norm3dif2 30382 | Norm of differences around... |
| norm3lemt 30383 | Lemma involving norm of di... |
| norm3adifi 30384 | Norm of differences around... |
| normpari 30385 | Parallelogram law for norm... |
| normpar 30386 | Parallelogram law for norm... |
| normpar2i 30387 | Corollary of parallelogram... |
| polid2i 30388 | Generalized polarization i... |
| polidi 30389 | Polarization identity. Re... |
| polid 30390 | Polarization identity. Re... |
| hilablo 30391 | Hilbert space vector addit... |
| hilid 30392 | The group identity element... |
| hilvc 30393 | Hilbert space is a complex... |
| hilnormi 30394 | Hilbert space norm in term... |
| hilhhi 30395 | Deduce the structure of Hi... |
| hhnv 30396 | Hilbert space is a normed ... |
| hhva 30397 | The group (addition) opera... |
| hhba 30398 | The base set of Hilbert sp... |
| hh0v 30399 | The zero vector of Hilbert... |
| hhsm 30400 | The scalar product operati... |
| hhvs 30401 | The vector subtraction ope... |
| hhnm 30402 | The norm function of Hilbe... |
| hhims 30403 | The induced metric of Hilb... |
| hhims2 30404 | Hilbert space distance met... |
| hhmet 30405 | The induced metric of Hilb... |
| hhxmet 30406 | The induced metric of Hilb... |
| hhmetdval 30407 | Value of the distance func... |
| hhip 30408 | The inner product operatio... |
| hhph 30409 | The Hilbert space of the H... |
| bcsiALT 30410 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 30411 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 30412 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 30413 | Corollary of the Bunjakova... |
| bcs3 30414 | Corollary of the Bunjakova... |
| hcau 30415 | Member of the set of Cauch... |
| hcauseq 30416 | A Cauchy sequences on a Hi... |
| hcaucvg 30417 | A Cauchy sequence on a Hil... |
| seq1hcau 30418 | A sequence on a Hilbert sp... |
| hlimi 30419 | Express the predicate: Th... |
| hlimseqi 30420 | A sequence with a limit on... |
| hlimveci 30421 | Closure of the limit of a ... |
| hlimconvi 30422 | Convergence of a sequence ... |
| hlim2 30423 | The limit of a sequence on... |
| hlimadd 30424 | Limit of the sum of two se... |
| hilmet 30425 | The Hilbert space norm det... |
| hilxmet 30426 | The Hilbert space norm det... |
| hilmetdval 30427 | Value of the distance func... |
| hilims 30428 | Hilbert space distance met... |
| hhcau 30429 | The Cauchy sequences of Hi... |
| hhlm 30430 | The limit sequences of Hil... |
| hhcmpl 30431 | Lemma used for derivation ... |
| hilcompl 30432 | Lemma used for derivation ... |
| hhcms 30434 | The Hilbert space induced ... |
| hhhl 30435 | The Hilbert space structur... |
| hilcms 30436 | The Hilbert space norm det... |
| hilhl 30437 | The Hilbert space of the H... |
| issh 30439 | Subspace ` H ` of a Hilber... |
| issh2 30440 | Subspace ` H ` of a Hilber... |
| shss 30441 | A subspace is a subset of ... |
| shel 30442 | A member of a subspace of ... |
| shex 30443 | The set of subspaces of a ... |
| shssii 30444 | A closed subspace of a Hil... |
| sheli 30445 | A member of a subspace of ... |
| shelii 30446 | A member of a subspace of ... |
| sh0 30447 | The zero vector belongs to... |
| shaddcl 30448 | Closure of vector addition... |
| shmulcl 30449 | Closure of vector scalar m... |
| issh3 30450 | Subspace ` H ` of a Hilber... |
| shsubcl 30451 | Closure of vector subtract... |
| isch 30453 | Closed subspace ` H ` of a... |
| isch2 30454 | Closed subspace ` H ` of a... |
| chsh 30455 | A closed subspace is a sub... |
| chsssh 30456 | Closed subspaces are subsp... |
| chex 30457 | The set of closed subspace... |
| chshii 30458 | A closed subspace is a sub... |
| ch0 30459 | The zero vector belongs to... |
| chss 30460 | A closed subspace of a Hil... |
| chel 30461 | A member of a closed subsp... |
| chssii 30462 | A closed subspace of a Hil... |
| cheli 30463 | A member of a closed subsp... |
| chelii 30464 | A member of a closed subsp... |
| chlimi 30465 | The limit property of a cl... |
| hlim0 30466 | The zero sequence in Hilbe... |
| hlimcaui 30467 | If a sequence in Hilbert s... |
| hlimf 30468 | Function-like behavior of ... |
| hlimuni 30469 | A Hilbert space sequence c... |
| hlimreui 30470 | The limit of a Hilbert spa... |
| hlimeui 30471 | The limit of a Hilbert spa... |
| isch3 30472 | A Hilbert subspace is clos... |
| chcompl 30473 | Completeness of a closed s... |
| helch 30474 | The Hilbert lattice one (w... |
| ifchhv 30475 | Prove ` if ( A e. CH , A ,... |
| helsh 30476 | Hilbert space is a subspac... |
| shsspwh 30477 | Subspaces are subsets of H... |
| chsspwh 30478 | Closed subspaces are subse... |
| hsn0elch 30479 | The zero subspace belongs ... |
| norm1 30480 | From any nonzero Hilbert s... |
| norm1exi 30481 | A normalized vector exists... |
| norm1hex 30482 | A normalized vector can ex... |
| elch0 30485 | Membership in zero for clo... |
| h0elch 30486 | The zero subspace is a clo... |
| h0elsh 30487 | The zero subspace is a sub... |
| hhssva 30488 | The vector addition operat... |
| hhsssm 30489 | The scalar multiplication ... |
| hhssnm 30490 | The norm operation on a su... |
| issubgoilem 30491 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 30492 | Lemma for ~ hhssabloi . F... |
| hhssabloi 30493 | Abelian group property of ... |
| hhssablo 30494 | Abelian group property of ... |
| hhssnv 30495 | Normed complex vector spac... |
| hhssnvt 30496 | Normed complex vector spac... |
| hhsst 30497 | A member of ` SH ` is a su... |
| hhshsslem1 30498 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 30499 | Lemma for ~ hhsssh . (Con... |
| hhsssh 30500 | The predicate " ` H ` is a... |
| hhsssh2 30501 | The predicate " ` H ` is a... |
| hhssba 30502 | The base set of a subspace... |
| hhssvs 30503 | The vector subtraction ope... |
| hhssvsf 30504 | Mapping of the vector subt... |
| hhssims 30505 | Induced metric of a subspa... |
| hhssims2 30506 | Induced metric of a subspa... |
| hhssmet 30507 | Induced metric of a subspa... |
| hhssmetdval 30508 | Value of the distance func... |
| hhsscms 30509 | The induced metric of a cl... |
| hhssbnOLD 30510 | Obsolete version of ~ cssb... |
| ocval 30511 | Value of orthogonal comple... |
| ocel 30512 | Membership in orthogonal c... |
| shocel 30513 | Membership in orthogonal c... |
| ocsh 30514 | The orthogonal complement ... |
| shocsh 30515 | The orthogonal complement ... |
| ocss 30516 | An orthogonal complement i... |
| shocss 30517 | An orthogonal complement i... |
| occon 30518 | Contraposition law for ort... |
| occon2 30519 | Double contraposition for ... |
| occon2i 30520 | Double contraposition for ... |
| oc0 30521 | The zero vector belongs to... |
| ocorth 30522 | Members of a subset and it... |
| shocorth 30523 | Members of a subspace and ... |
| ococss 30524 | Inclusion in complement of... |
| shococss 30525 | Inclusion in complement of... |
| shorth 30526 | Members of orthogonal subs... |
| ocin 30527 | Intersection of a Hilbert ... |
| occon3 30528 | Hilbert lattice contraposi... |
| ocnel 30529 | A nonzero vector in the co... |
| chocvali 30530 | Value of the orthogonal co... |
| shuni 30531 | Two subspaces with trivial... |
| chocunii 30532 | Lemma for uniqueness part ... |
| pjhthmo 30533 | Projection Theorem, unique... |
| occllem 30534 | Lemma for ~ occl . (Contr... |
| occl 30535 | Closure of complement of H... |
| shoccl 30536 | Closure of complement of H... |
| choccl 30537 | Closure of complement of H... |
| choccli 30538 | Closure of ` CH ` orthocom... |
| shsval 30543 | Value of subspace sum of t... |
| shsss 30544 | The subspace sum is a subs... |
| shsel 30545 | Membership in the subspace... |
| shsel3 30546 | Membership in the subspace... |
| shseli 30547 | Membership in subspace sum... |
| shscli 30548 | Closure of subspace sum. ... |
| shscl 30549 | Closure of subspace sum. ... |
| shscom 30550 | Commutative law for subspa... |
| shsva 30551 | Vector sum belongs to subs... |
| shsel1 30552 | A subspace sum contains a ... |
| shsel2 30553 | A subspace sum contains a ... |
| shsvs 30554 | Vector subtraction belongs... |
| shsub1 30555 | Subspace sum is an upper b... |
| shsub2 30556 | Subspace sum is an upper b... |
| choc0 30557 | The orthocomplement of the... |
| choc1 30558 | The orthocomplement of the... |
| chocnul 30559 | Orthogonal complement of t... |
| shintcli 30560 | Closure of intersection of... |
| shintcl 30561 | The intersection of a none... |
| chintcli 30562 | The intersection of a none... |
| chintcl 30563 | The intersection (infimum)... |
| spanval 30564 | Value of the linear span o... |
| hsupval 30565 | Value of supremum of set o... |
| chsupval 30566 | The value of the supremum ... |
| spancl 30567 | The span of a subset of Hi... |
| elspancl 30568 | A member of a span is a ve... |
| shsupcl 30569 | Closure of the subspace su... |
| hsupcl 30570 | Closure of supremum of set... |
| chsupcl 30571 | Closure of supremum of sub... |
| hsupss 30572 | Subset relation for suprem... |
| chsupss 30573 | Subset relation for suprem... |
| hsupunss 30574 | The union of a set of Hilb... |
| chsupunss 30575 | The union of a set of clos... |
| spanss2 30576 | A subset of Hilbert space ... |
| shsupunss 30577 | The union of a set of subs... |
| spanid 30578 | A subspace of Hilbert spac... |
| spanss 30579 | Ordering relationship for ... |
| spanssoc 30580 | The span of a subset of Hi... |
| sshjval 30581 | Value of join for subsets ... |
| shjval 30582 | Value of join in ` SH ` . ... |
| chjval 30583 | Value of join in ` CH ` . ... |
| chjvali 30584 | Value of join in ` CH ` . ... |
| sshjval3 30585 | Value of join for subsets ... |
| sshjcl 30586 | Closure of join for subset... |
| shjcl 30587 | Closure of join in ` SH ` ... |
| chjcl 30588 | Closure of join in ` CH ` ... |
| shjcom 30589 | Commutative law for Hilber... |
| shless 30590 | Subset implies subset of s... |
| shlej1 30591 | Add disjunct to both sides... |
| shlej2 30592 | Add disjunct to both sides... |
| shincli 30593 | Closure of intersection of... |
| shscomi 30594 | Commutative law for subspa... |
| shsvai 30595 | Vector sum belongs to subs... |
| shsel1i 30596 | A subspace sum contains a ... |
| shsel2i 30597 | A subspace sum contains a ... |
| shsvsi 30598 | Vector subtraction belongs... |
| shunssi 30599 | Union is smaller than subs... |
| shunssji 30600 | Union is smaller than Hilb... |
| shsleji 30601 | Subspace sum is smaller th... |
| shjcomi 30602 | Commutative law for join i... |
| shsub1i 30603 | Subspace sum is an upper b... |
| shsub2i 30604 | Subspace sum is an upper b... |
| shub1i 30605 | Hilbert lattice join is an... |
| shjcli 30606 | Closure of ` CH ` join. (... |
| shjshcli 30607 | ` SH ` closure of join. (... |
| shlessi 30608 | Subset implies subset of s... |
| shlej1i 30609 | Add disjunct to both sides... |
| shlej2i 30610 | Add disjunct to both sides... |
| shslej 30611 | Subspace sum is smaller th... |
| shincl 30612 | Closure of intersection of... |
| shub1 30613 | Hilbert lattice join is an... |
| shub2 30614 | A subspace is a subset of ... |
| shsidmi 30615 | Idempotent law for Hilbert... |
| shslubi 30616 | The least upper bound law ... |
| shlesb1i 30617 | Hilbert lattice ordering i... |
| shsval2i 30618 | An alternate way to expres... |
| shsval3i 30619 | An alternate way to expres... |
| shmodsi 30620 | The modular law holds for ... |
| shmodi 30621 | The modular law is implied... |
| pjhthlem1 30622 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 30623 | Lemma for ~ pjhth . (Cont... |
| pjhth 30624 | Projection Theorem: Any H... |
| pjhtheu 30625 | Projection Theorem: Any H... |
| pjhfval 30627 | The value of the projectio... |
| pjhval 30628 | Value of a projection. (C... |
| pjpreeq 30629 | Equality with a projection... |
| pjeq 30630 | Equality with a projection... |
| axpjcl 30631 | Closure of a projection in... |
| pjhcl 30632 | Closure of a projection in... |
| omlsilem 30633 | Lemma for orthomodular law... |
| omlsii 30634 | Subspace inference form of... |
| omlsi 30635 | Subspace form of orthomodu... |
| ococi 30636 | Complement of complement o... |
| ococ 30637 | Complement of complement o... |
| dfch2 30638 | Alternate definition of th... |
| ococin 30639 | The double complement is t... |
| hsupval2 30640 | Alternate definition of su... |
| chsupval2 30641 | The value of the supremum ... |
| sshjval2 30642 | Value of join in the set o... |
| chsupid 30643 | A subspace is the supremum... |
| chsupsn 30644 | Value of supremum of subse... |
| shlub 30645 | Hilbert lattice join is th... |
| shlubi 30646 | Hilbert lattice join is th... |
| pjhtheu2 30647 | Uniqueness of ` y ` for th... |
| pjcli 30648 | Closure of a projection in... |
| pjhcli 30649 | Closure of a projection in... |
| pjpjpre 30650 | Decomposition of a vector ... |
| axpjpj 30651 | Decomposition of a vector ... |
| pjclii 30652 | Closure of a projection in... |
| pjhclii 30653 | Closure of a projection in... |
| pjpj0i 30654 | Decomposition of a vector ... |
| pjpji 30655 | Decomposition of a vector ... |
| pjpjhth 30656 | Projection Theorem: Any H... |
| pjpjhthi 30657 | Projection Theorem: Any H... |
| pjop 30658 | Orthocomplement projection... |
| pjpo 30659 | Projection in terms of ort... |
| pjopi 30660 | Orthocomplement projection... |
| pjpoi 30661 | Projection in terms of ort... |
| pjoc1i 30662 | Projection of a vector in ... |
| pjchi 30663 | Projection of a vector in ... |
| pjoccl 30664 | The part of a vector that ... |
| pjoc1 30665 | Projection of a vector in ... |
| pjomli 30666 | Subspace form of orthomodu... |
| pjoml 30667 | Subspace form of orthomodu... |
| pjococi 30668 | Proof of orthocomplement t... |
| pjoc2i 30669 | Projection of a vector in ... |
| pjoc2 30670 | Projection of a vector in ... |
| sh0le 30671 | The zero subspace is the s... |
| ch0le 30672 | The zero subspace is the s... |
| shle0 30673 | No subspace is smaller tha... |
| chle0 30674 | No Hilbert lattice element... |
| chnlen0 30675 | A Hilbert lattice element ... |
| ch0pss 30676 | The zero subspace is a pro... |
| orthin 30677 | The intersection of orthog... |
| ssjo 30678 | The lattice join of a subs... |
| shne0i 30679 | A nonzero subspace has a n... |
| shs0i 30680 | Hilbert subspace sum with ... |
| shs00i 30681 | Two subspaces are zero iff... |
| ch0lei 30682 | The closed subspace zero i... |
| chle0i 30683 | No Hilbert closed subspace... |
| chne0i 30684 | A nonzero closed subspace ... |
| chocini 30685 | Intersection of a closed s... |
| chj0i 30686 | Join with lattice zero in ... |
| chm1i 30687 | Meet with lattice one in `... |
| chjcli 30688 | Closure of ` CH ` join. (... |
| chsleji 30689 | Subspace sum is smaller th... |
| chseli 30690 | Membership in subspace sum... |
| chincli 30691 | Closure of Hilbert lattice... |
| chsscon3i 30692 | Hilbert lattice contraposi... |
| chsscon1i 30693 | Hilbert lattice contraposi... |
| chsscon2i 30694 | Hilbert lattice contraposi... |
| chcon2i 30695 | Hilbert lattice contraposi... |
| chcon1i 30696 | Hilbert lattice contraposi... |
| chcon3i 30697 | Hilbert lattice contraposi... |
| chunssji 30698 | Union is smaller than ` CH... |
| chjcomi 30699 | Commutative law for join i... |
| chub1i 30700 | ` CH ` join is an upper bo... |
| chub2i 30701 | ` CH ` join is an upper bo... |
| chlubi 30702 | Hilbert lattice join is th... |
| chlubii 30703 | Hilbert lattice join is th... |
| chlej1i 30704 | Add join to both sides of ... |
| chlej2i 30705 | Add join to both sides of ... |
| chlej12i 30706 | Add join to both sides of ... |
| chlejb1i 30707 | Hilbert lattice ordering i... |
| chdmm1i 30708 | De Morgan's law for meet i... |
| chdmm2i 30709 | De Morgan's law for meet i... |
| chdmm3i 30710 | De Morgan's law for meet i... |
| chdmm4i 30711 | De Morgan's law for meet i... |
| chdmj1i 30712 | De Morgan's law for join i... |
| chdmj2i 30713 | De Morgan's law for join i... |
| chdmj3i 30714 | De Morgan's law for join i... |
| chdmj4i 30715 | De Morgan's law for join i... |
| chnlei 30716 | Equivalent expressions for... |
| chjassi 30717 | Associative law for Hilber... |
| chj00i 30718 | Two Hilbert lattice elemen... |
| chjoi 30719 | The join of a closed subsp... |
| chj1i 30720 | Join with Hilbert lattice ... |
| chm0i 30721 | Meet with Hilbert lattice ... |
| chm0 30722 | Meet with Hilbert lattice ... |
| shjshsi 30723 | Hilbert lattice join equal... |
| shjshseli 30724 | A closed subspace sum equa... |
| chne0 30725 | A nonzero closed subspace ... |
| chocin 30726 | Intersection of a closed s... |
| chssoc 30727 | A closed subspace less tha... |
| chj0 30728 | Join with Hilbert lattice ... |
| chslej 30729 | Subspace sum is smaller th... |
| chincl 30730 | Closure of Hilbert lattice... |
| chsscon3 30731 | Hilbert lattice contraposi... |
| chsscon1 30732 | Hilbert lattice contraposi... |
| chsscon2 30733 | Hilbert lattice contraposi... |
| chpsscon3 30734 | Hilbert lattice contraposi... |
| chpsscon1 30735 | Hilbert lattice contraposi... |
| chpsscon2 30736 | Hilbert lattice contraposi... |
| chjcom 30737 | Commutative law for Hilber... |
| chub1 30738 | Hilbert lattice join is gr... |
| chub2 30739 | Hilbert lattice join is gr... |
| chlub 30740 | Hilbert lattice join is th... |
| chlej1 30741 | Add join to both sides of ... |
| chlej2 30742 | Add join to both sides of ... |
| chlejb1 30743 | Hilbert lattice ordering i... |
| chlejb2 30744 | Hilbert lattice ordering i... |
| chnle 30745 | Equivalent expressions for... |
| chjo 30746 | The join of a closed subsp... |
| chabs1 30747 | Hilbert lattice absorption... |
| chabs2 30748 | Hilbert lattice absorption... |
| chabs1i 30749 | Hilbert lattice absorption... |
| chabs2i 30750 | Hilbert lattice absorption... |
| chjidm 30751 | Idempotent law for Hilbert... |
| chjidmi 30752 | Idempotent law for Hilbert... |
| chj12i 30753 | A rearrangement of Hilbert... |
| chj4i 30754 | Rearrangement of the join ... |
| chjjdiri 30755 | Hilbert lattice join distr... |
| chdmm1 30756 | De Morgan's law for meet i... |
| chdmm2 30757 | De Morgan's law for meet i... |
| chdmm3 30758 | De Morgan's law for meet i... |
| chdmm4 30759 | De Morgan's law for meet i... |
| chdmj1 30760 | De Morgan's law for join i... |
| chdmj2 30761 | De Morgan's law for join i... |
| chdmj3 30762 | De Morgan's law for join i... |
| chdmj4 30763 | De Morgan's law for join i... |
| chjass 30764 | Associative law for Hilber... |
| chj12 30765 | A rearrangement of Hilbert... |
| chj4 30766 | Rearrangement of the join ... |
| ledii 30767 | An ortholattice is distrib... |
| lediri 30768 | An ortholattice is distrib... |
| lejdii 30769 | An ortholattice is distrib... |
| lejdiri 30770 | An ortholattice is distrib... |
| ledi 30771 | An ortholattice is distrib... |
| spansn0 30772 | The span of the singleton ... |
| span0 30773 | The span of the empty set ... |
| elspani 30774 | Membership in the span of ... |
| spanuni 30775 | The span of a union is the... |
| spanun 30776 | The span of a union is the... |
| sshhococi 30777 | The join of two Hilbert sp... |
| hne0 30778 | Hilbert space has a nonzer... |
| chsup0 30779 | The supremum of the empty ... |
| h1deoi 30780 | Membership in orthocomplem... |
| h1dei 30781 | Membership in 1-dimensiona... |
| h1did 30782 | A generating vector belong... |
| h1dn0 30783 | A nonzero vector generates... |
| h1de2i 30784 | Membership in 1-dimensiona... |
| h1de2bi 30785 | Membership in 1-dimensiona... |
| h1de2ctlem 30786 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 30787 | Membership in 1-dimensiona... |
| spansni 30788 | The span of a singleton in... |
| elspansni 30789 | Membership in the span of ... |
| spansn 30790 | The span of a singleton in... |
| spansnch 30791 | The span of a Hilbert spac... |
| spansnsh 30792 | The span of a Hilbert spac... |
| spansnchi 30793 | The span of a singleton in... |
| spansnid 30794 | A vector belongs to the sp... |
| spansnmul 30795 | A scalar product with a ve... |
| elspansncl 30796 | A member of a span of a si... |
| elspansn 30797 | Membership in the span of ... |
| elspansn2 30798 | Membership in the span of ... |
| spansncol 30799 | The singletons of collinea... |
| spansneleqi 30800 | Membership relation implie... |
| spansneleq 30801 | Membership relation that i... |
| spansnss 30802 | The span of the singleton ... |
| elspansn3 30803 | A member of the span of th... |
| elspansn4 30804 | A span membership conditio... |
| elspansn5 30805 | A vector belonging to both... |
| spansnss2 30806 | The span of the singleton ... |
| normcan 30807 | Cancellation-type law that... |
| pjspansn 30808 | A projection on the span o... |
| spansnpji 30809 | A subset of Hilbert space ... |
| spanunsni 30810 | The span of the union of a... |
| spanpr 30811 | The span of a pair of vect... |
| h1datomi 30812 | A 1-dimensional subspace i... |
| h1datom 30813 | A 1-dimensional subspace i... |
| cmbr 30815 | Binary relation expressing... |
| pjoml2i 30816 | Variation of orthomodular ... |
| pjoml3i 30817 | Variation of orthomodular ... |
| pjoml4i 30818 | Variation of orthomodular ... |
| pjoml5i 30819 | The orthomodular law. Rem... |
| pjoml6i 30820 | An equivalent of the ortho... |
| cmbri 30821 | Binary relation expressing... |
| cmcmlem 30822 | Commutation is symmetric. ... |
| cmcmi 30823 | Commutation is symmetric. ... |
| cmcm2i 30824 | Commutation with orthocomp... |
| cmcm3i 30825 | Commutation with orthocomp... |
| cmcm4i 30826 | Commutation with orthocomp... |
| cmbr2i 30827 | Alternate definition of th... |
| cmcmii 30828 | Commutation is symmetric. ... |
| cmcm2ii 30829 | Commutation with orthocomp... |
| cmcm3ii 30830 | Commutation with orthocomp... |
| cmbr3i 30831 | Alternate definition for t... |
| cmbr4i 30832 | Alternate definition for t... |
| lecmi 30833 | Comparable Hilbert lattice... |
| lecmii 30834 | Comparable Hilbert lattice... |
| cmj1i 30835 | A Hilbert lattice element ... |
| cmj2i 30836 | A Hilbert lattice element ... |
| cmm1i 30837 | A Hilbert lattice element ... |
| cmm2i 30838 | A Hilbert lattice element ... |
| cmbr3 30839 | Alternate definition for t... |
| cm0 30840 | The zero Hilbert lattice e... |
| cmidi 30841 | The commutes relation is r... |
| pjoml2 30842 | Variation of orthomodular ... |
| pjoml3 30843 | Variation of orthomodular ... |
| pjoml5 30844 | The orthomodular law. Rem... |
| cmcm 30845 | Commutation is symmetric. ... |
| cmcm3 30846 | Commutation with orthocomp... |
| cmcm2 30847 | Commutation with orthocomp... |
| lecm 30848 | Comparable Hilbert lattice... |
| fh1 30849 | Foulis-Holland Theorem. I... |
| fh2 30850 | Foulis-Holland Theorem. I... |
| cm2j 30851 | A lattice element that com... |
| fh1i 30852 | Foulis-Holland Theorem. I... |
| fh2i 30853 | Foulis-Holland Theorem. I... |
| fh3i 30854 | Variation of the Foulis-Ho... |
| fh4i 30855 | Variation of the Foulis-Ho... |
| cm2ji 30856 | A lattice element that com... |
| cm2mi 30857 | A lattice element that com... |
| qlax1i 30858 | One of the equations showi... |
| qlax2i 30859 | One of the equations showi... |
| qlax3i 30860 | One of the equations showi... |
| qlax4i 30861 | One of the equations showi... |
| qlax5i 30862 | One of the equations showi... |
| qlaxr1i 30863 | One of the conditions show... |
| qlaxr2i 30864 | One of the conditions show... |
| qlaxr4i 30865 | One of the conditions show... |
| qlaxr5i 30866 | One of the conditions show... |
| qlaxr3i 30867 | A variation of the orthomo... |
| chscllem1 30868 | Lemma for ~ chscl . (Cont... |
| chscllem2 30869 | Lemma for ~ chscl . (Cont... |
| chscllem3 30870 | Lemma for ~ chscl . (Cont... |
| chscllem4 30871 | Lemma for ~ chscl . (Cont... |
| chscl 30872 | The subspace sum of two cl... |
| osumi 30873 | If two closed subspaces of... |
| osumcori 30874 | Corollary of ~ osumi . (C... |
| osumcor2i 30875 | Corollary of ~ osumi , sho... |
| osum 30876 | If two closed subspaces of... |
| spansnji 30877 | The subspace sum of a clos... |
| spansnj 30878 | The subspace sum of a clos... |
| spansnscl 30879 | The subspace sum of a clos... |
| sumspansn 30880 | The sum of two vectors bel... |
| spansnm0i 30881 | The meet of different one-... |
| nonbooli 30882 | A Hilbert lattice with two... |
| spansncvi 30883 | Hilbert space has the cove... |
| spansncv 30884 | Hilbert space has the cove... |
| 5oalem1 30885 | Lemma for orthoarguesian l... |
| 5oalem2 30886 | Lemma for orthoarguesian l... |
| 5oalem3 30887 | Lemma for orthoarguesian l... |
| 5oalem4 30888 | Lemma for orthoarguesian l... |
| 5oalem5 30889 | Lemma for orthoarguesian l... |
| 5oalem6 30890 | Lemma for orthoarguesian l... |
| 5oalem7 30891 | Lemma for orthoarguesian l... |
| 5oai 30892 | Orthoarguesian law 5OA. Th... |
| 3oalem1 30893 | Lemma for 3OA (weak) ortho... |
| 3oalem2 30894 | Lemma for 3OA (weak) ortho... |
| 3oalem3 30895 | Lemma for 3OA (weak) ortho... |
| 3oalem4 30896 | Lemma for 3OA (weak) ortho... |
| 3oalem5 30897 | Lemma for 3OA (weak) ortho... |
| 3oalem6 30898 | Lemma for 3OA (weak) ortho... |
| 3oai 30899 | 3OA (weak) orthoarguesian ... |
| pjorthi 30900 | Projection components on o... |
| pjch1 30901 | Property of identity proje... |
| pjo 30902 | The orthogonal projection.... |
| pjcompi 30903 | Component of a projection.... |
| pjidmi 30904 | A projection is idempotent... |
| pjadjii 30905 | A projection is self-adjoi... |
| pjaddii 30906 | Projection of vector sum i... |
| pjinormii 30907 | The inner product of a pro... |
| pjmulii 30908 | Projection of (scalar) pro... |
| pjsubii 30909 | Projection of vector diffe... |
| pjsslem 30910 | Lemma for subset relations... |
| pjss2i 30911 | Subset relationship for pr... |
| pjssmii 30912 | Projection meet property. ... |
| pjssge0ii 30913 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 30914 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 30915 | The projection on a subspa... |
| pjadji 30916 | A projection is self-adjoi... |
| pjaddi 30917 | Projection of vector sum i... |
| pjinormi 30918 | The inner product of a pro... |
| pjsubi 30919 | Projection of vector diffe... |
| pjmuli 30920 | Projection of scalar produ... |
| pjige0i 30921 | The inner product of a pro... |
| pjige0 30922 | The inner product of a pro... |
| pjcjt2 30923 | The projection on a subspa... |
| pj0i 30924 | The projection of the zero... |
| pjch 30925 | Projection of a vector in ... |
| pjid 30926 | The projection of a vector... |
| pjvec 30927 | The set of vectors belongi... |
| pjocvec 30928 | The set of vectors belongi... |
| pjocini 30929 | Membership of projection i... |
| pjini 30930 | Membership of projection i... |
| pjjsi 30931 | A sufficient condition for... |
| pjfni 30932 | Functionality of a project... |
| pjrni 30933 | The range of a projection.... |
| pjfoi 30934 | A projection maps onto its... |
| pjfi 30935 | The mapping of a projectio... |
| pjvi 30936 | The value of a projection ... |
| pjhfo 30937 | A projection maps onto its... |
| pjrn 30938 | The range of a projection.... |
| pjhf 30939 | The mapping of a projectio... |
| pjfn 30940 | Functionality of a project... |
| pjsumi 30941 | The projection on a subspa... |
| pj11i 30942 | One-to-one correspondence ... |
| pjdsi 30943 | Vector decomposition into ... |
| pjds3i 30944 | Vector decomposition into ... |
| pj11 30945 | One-to-one correspondence ... |
| pjmfn 30946 | Functionality of the proje... |
| pjmf1 30947 | The projector function map... |
| pjoi0 30948 | The inner product of proje... |
| pjoi0i 30949 | The inner product of proje... |
| pjopythi 30950 | Pythagorean theorem for pr... |
| pjopyth 30951 | Pythagorean theorem for pr... |
| pjnormi 30952 | The norm of the projection... |
| pjpythi 30953 | Pythagorean theorem for pr... |
| pjneli 30954 | If a vector does not belon... |
| pjnorm 30955 | The norm of the projection... |
| pjpyth 30956 | Pythagorean theorem for pr... |
| pjnel 30957 | If a vector does not belon... |
| pjnorm2 30958 | A vector belongs to the su... |
| mayete3i 30959 | Mayet's equation E_3. Par... |
| mayetes3i 30960 | Mayet's equation E^*_3, de... |
| hosmval 30966 | Value of the sum of two Hi... |
| hommval 30967 | Value of the scalar produc... |
| hodmval 30968 | Value of the difference of... |
| hfsmval 30969 | Value of the sum of two Hi... |
| hfmmval 30970 | Value of the scalar produc... |
| hosval 30971 | Value of the sum of two Hi... |
| homval 30972 | Value of the scalar produc... |
| hodval 30973 | Value of the difference of... |
| hfsval 30974 | Value of the sum of two Hi... |
| hfmval 30975 | Value of the scalar produc... |
| hoscl 30976 | Closure of the sum of two ... |
| homcl 30977 | Closure of the scalar prod... |
| hodcl 30978 | Closure of the difference ... |
| ho0val 30981 | Value of the zero Hilbert ... |
| ho0f 30982 | Functionality of the zero ... |
| df0op2 30983 | Alternate definition of Hi... |
| dfiop2 30984 | Alternate definition of Hi... |
| hoif 30985 | Functionality of the Hilbe... |
| hoival 30986 | The value of the Hilbert s... |
| hoico1 30987 | Composition with the Hilbe... |
| hoico2 30988 | Composition with the Hilbe... |
| hoaddcl 30989 | The sum of Hilbert space o... |
| homulcl 30990 | The scalar product of a Hi... |
| hoeq 30991 | Equality of Hilbert space ... |
| hoeqi 30992 | Equality of Hilbert space ... |
| hoscli 30993 | Closure of Hilbert space o... |
| hodcli 30994 | Closure of Hilbert space o... |
| hocoi 30995 | Composition of Hilbert spa... |
| hococli 30996 | Closure of composition of ... |
| hocofi 30997 | Mapping of composition of ... |
| hocofni 30998 | Functionality of compositi... |
| hoaddcli 30999 | Mapping of sum of Hilbert ... |
| hosubcli 31000 | Mapping of difference of H... |
| hoaddfni 31001 | Functionality of sum of Hi... |
| hosubfni 31002 | Functionality of differenc... |
| hoaddcomi 31003 | Commutativity of sum of Hi... |
| hosubcl 31004 | Mapping of difference of H... |
| hoaddcom 31005 | Commutativity of sum of Hi... |
| hodsi 31006 | Relationship between Hilbe... |
| hoaddassi 31007 | Associativity of sum of Hi... |
| hoadd12i 31008 | Commutative/associative la... |
| hoadd32i 31009 | Commutative/associative la... |
| hocadddiri 31010 | Distributive law for Hilbe... |
| hocsubdiri 31011 | Distributive law for Hilbe... |
| ho2coi 31012 | Double composition of Hilb... |
| hoaddass 31013 | Associativity of sum of Hi... |
| hoadd32 31014 | Commutative/associative la... |
| hoadd4 31015 | Rearrangement of 4 terms i... |
| hocsubdir 31016 | Distributive law for Hilbe... |
| hoaddridi 31017 | Sum of a Hilbert space ope... |
| hodidi 31018 | Difference of a Hilbert sp... |
| ho0coi 31019 | Composition of the zero op... |
| hoid1i 31020 | Composition of Hilbert spa... |
| hoid1ri 31021 | Composition of Hilbert spa... |
| hoaddrid 31022 | Sum of a Hilbert space ope... |
| hodid 31023 | Difference of a Hilbert sp... |
| hon0 31024 | A Hilbert space operator i... |
| hodseqi 31025 | Subtraction and addition o... |
| ho0subi 31026 | Subtraction of Hilbert spa... |
| honegsubi 31027 | Relationship between Hilbe... |
| ho0sub 31028 | Subtraction of Hilbert spa... |
| hosubid1 31029 | The zero operator subtract... |
| honegsub 31030 | Relationship between Hilbe... |
| homullid 31031 | An operator equals its sca... |
| homco1 31032 | Associative law for scalar... |
| homulass 31033 | Scalar product associative... |
| hoadddi 31034 | Scalar product distributiv... |
| hoadddir 31035 | Scalar product reverse dis... |
| homul12 31036 | Swap first and second fact... |
| honegneg 31037 | Double negative of a Hilbe... |
| hosubneg 31038 | Relationship between opera... |
| hosubdi 31039 | Scalar product distributiv... |
| honegdi 31040 | Distribution of negative o... |
| honegsubdi 31041 | Distribution of negative o... |
| honegsubdi2 31042 | Distribution of negative o... |
| hosubsub2 31043 | Law for double subtraction... |
| hosub4 31044 | Rearrangement of 4 terms i... |
| hosubadd4 31045 | Rearrangement of 4 terms i... |
| hoaddsubass 31046 | Associative-type law for a... |
| hoaddsub 31047 | Law for operator addition ... |
| hosubsub 31048 | Law for double subtraction... |
| hosubsub4 31049 | Law for double subtraction... |
| ho2times 31050 | Two times a Hilbert space ... |
| hoaddsubassi 31051 | Associativity of sum and d... |
| hoaddsubi 31052 | Law for sum and difference... |
| hosd1i 31053 | Hilbert space operator sum... |
| hosd2i 31054 | Hilbert space operator sum... |
| hopncani 31055 | Hilbert space operator can... |
| honpcani 31056 | Hilbert space operator can... |
| hosubeq0i 31057 | If the difference between ... |
| honpncani 31058 | Hilbert space operator can... |
| ho01i 31059 | A condition implying that ... |
| ho02i 31060 | A condition implying that ... |
| hoeq1 31061 | A condition implying that ... |
| hoeq2 31062 | A condition implying that ... |
| adjmo 31063 | Every Hilbert space operat... |
| adjsym 31064 | Symmetry property of an ad... |
| eigrei 31065 | A necessary and sufficient... |
| eigre 31066 | A necessary and sufficient... |
| eigposi 31067 | A sufficient condition (fi... |
| eigorthi 31068 | A necessary and sufficient... |
| eigorth 31069 | A necessary and sufficient... |
| nmopval 31087 | Value of the norm of a Hil... |
| elcnop 31088 | Property defining a contin... |
| ellnop 31089 | Property defining a linear... |
| lnopf 31090 | A linear Hilbert space ope... |
| elbdop 31091 | Property defining a bounde... |
| bdopln 31092 | A bounded linear Hilbert s... |
| bdopf 31093 | A bounded linear Hilbert s... |
| nmopsetretALT 31094 | The set in the supremum of... |
| nmopsetretHIL 31095 | The set in the supremum of... |
| nmopsetn0 31096 | The set in the supremum of... |
| nmopxr 31097 | The norm of a Hilbert spac... |
| nmoprepnf 31098 | The norm of a Hilbert spac... |
| nmopgtmnf 31099 | The norm of a Hilbert spac... |
| nmopreltpnf 31100 | The norm of a Hilbert spac... |
| nmopre 31101 | The norm of a bounded oper... |
| elbdop2 31102 | Property defining a bounde... |
| elunop 31103 | Property defining a unitar... |
| elhmop 31104 | Property defining a Hermit... |
| hmopf 31105 | A Hermitian operator is a ... |
| hmopex 31106 | The class of Hermitian ope... |
| nmfnval 31107 | Value of the norm of a Hil... |
| nmfnsetre 31108 | The set in the supremum of... |
| nmfnsetn0 31109 | The set in the supremum of... |
| nmfnxr 31110 | The norm of any Hilbert sp... |
| nmfnrepnf 31111 | The norm of a Hilbert spac... |
| nlfnval 31112 | Value of the null space of... |
| elcnfn 31113 | Property defining a contin... |
| ellnfn 31114 | Property defining a linear... |
| lnfnf 31115 | A linear Hilbert space fun... |
| dfadj2 31116 | Alternate definition of th... |
| funadj 31117 | Functionality of the adjoi... |
| dmadjss 31118 | The domain of the adjoint ... |
| dmadjop 31119 | A member of the domain of ... |
| adjeu 31120 | Elementhood in the domain ... |
| adjval 31121 | Value of the adjoint funct... |
| adjval2 31122 | Value of the adjoint funct... |
| cnvadj 31123 | The adjoint function equal... |
| funcnvadj 31124 | The converse of the adjoin... |
| adj1o 31125 | The adjoint function maps ... |
| dmadjrn 31126 | The adjoint of an operator... |
| eigvecval 31127 | The set of eigenvectors of... |
| eigvalfval 31128 | The eigenvalues of eigenve... |
| specval 31129 | The value of the spectrum ... |
| speccl 31130 | The spectrum of an operato... |
| hhlnoi 31131 | The linear operators of Hi... |
| hhnmoi 31132 | The norm of an operator in... |
| hhbloi 31133 | A bounded linear operator ... |
| hh0oi 31134 | The zero operator in Hilbe... |
| hhcno 31135 | The continuous operators o... |
| hhcnf 31136 | The continuous functionals... |
| dmadjrnb 31137 | The adjoint of an operator... |
| nmoplb 31138 | A lower bound for an opera... |
| nmopub 31139 | An upper bound for an oper... |
| nmopub2tALT 31140 | An upper bound for an oper... |
| nmopub2tHIL 31141 | An upper bound for an oper... |
| nmopge0 31142 | The norm of any Hilbert sp... |
| nmopgt0 31143 | A linear Hilbert space ope... |
| cnopc 31144 | Basic continuity property ... |
| lnopl 31145 | Basic linearity property o... |
| unop 31146 | Basic inner product proper... |
| unopf1o 31147 | A unitary operator in Hilb... |
| unopnorm 31148 | A unitary operator is idem... |
| cnvunop 31149 | The inverse (converse) of ... |
| unopadj 31150 | The inverse (converse) of ... |
| unoplin 31151 | A unitary operator is line... |
| counop 31152 | The composition of two uni... |
| hmop 31153 | Basic inner product proper... |
| hmopre 31154 | The inner product of the v... |
| nmfnlb 31155 | A lower bound for a functi... |
| nmfnleub 31156 | An upper bound for the nor... |
| nmfnleub2 31157 | An upper bound for the nor... |
| nmfnge0 31158 | The norm of any Hilbert sp... |
| elnlfn 31159 | Membership in the null spa... |
| elnlfn2 31160 | Membership in the null spa... |
| cnfnc 31161 | Basic continuity property ... |
| lnfnl 31162 | Basic linearity property o... |
| adjcl 31163 | Closure of the adjoint of ... |
| adj1 31164 | Property of an adjoint Hil... |
| adj2 31165 | Property of an adjoint Hil... |
| adjeq 31166 | A property that determines... |
| adjadj 31167 | Double adjoint. Theorem 3... |
| adjvalval 31168 | Value of the value of the ... |
| unopadj2 31169 | The adjoint of a unitary o... |
| hmopadj 31170 | A Hermitian operator is se... |
| hmdmadj 31171 | Every Hermitian operator h... |
| hmopadj2 31172 | An operator is Hermitian i... |
| hmoplin 31173 | A Hermitian operator is li... |
| brafval 31174 | The bra of a vector, expre... |
| braval 31175 | A bra-ket juxtaposition, e... |
| braadd 31176 | Linearity property of bra ... |
| bramul 31177 | Linearity property of bra ... |
| brafn 31178 | The bra function is a func... |
| bralnfn 31179 | The Dirac bra function is ... |
| bracl 31180 | Closure of the bra functio... |
| bra0 31181 | The Dirac bra of the zero ... |
| brafnmul 31182 | Anti-linearity property of... |
| kbfval 31183 | The outer product of two v... |
| kbop 31184 | The outer product of two v... |
| kbval 31185 | The value of the operator ... |
| kbmul 31186 | Multiplication property of... |
| kbpj 31187 | If a vector ` A ` has norm... |
| eleigvec 31188 | Membership in the set of e... |
| eleigvec2 31189 | Membership in the set of e... |
| eleigveccl 31190 | Closure of an eigenvector ... |
| eigvalval 31191 | The eigenvalue of an eigen... |
| eigvalcl 31192 | An eigenvalue is a complex... |
| eigvec1 31193 | Property of an eigenvector... |
| eighmre 31194 | The eigenvalues of a Hermi... |
| eighmorth 31195 | Eigenvectors of a Hermitia... |
| nmopnegi 31196 | Value of the norm of the n... |
| lnop0 31197 | The value of a linear Hilb... |
| lnopmul 31198 | Multiplicative property of... |
| lnopli 31199 | Basic scalar product prope... |
| lnopfi 31200 | A linear Hilbert space ope... |
| lnop0i 31201 | The value of a linear Hilb... |
| lnopaddi 31202 | Additive property of a lin... |
| lnopmuli 31203 | Multiplicative property of... |
| lnopaddmuli 31204 | Sum/product property of a ... |
| lnopsubi 31205 | Subtraction property for a... |
| lnopsubmuli 31206 | Subtraction/product proper... |
| lnopmulsubi 31207 | Product/subtraction proper... |
| homco2 31208 | Move a scalar product out ... |
| idunop 31209 | The identity function (res... |
| 0cnop 31210 | The identically zero funct... |
| 0cnfn 31211 | The identically zero funct... |
| idcnop 31212 | The identity function (res... |
| idhmop 31213 | The Hilbert space identity... |
| 0hmop 31214 | The identically zero funct... |
| 0lnop 31215 | The identically zero funct... |
| 0lnfn 31216 | The identically zero funct... |
| nmop0 31217 | The norm of the zero opera... |
| nmfn0 31218 | The norm of the identicall... |
| hmopbdoptHIL 31219 | A Hermitian operator is a ... |
| hoddii 31220 | Distributive law for Hilbe... |
| hoddi 31221 | Distributive law for Hilbe... |
| nmop0h 31222 | The norm of any operator o... |
| idlnop 31223 | The identity function (res... |
| 0bdop 31224 | The identically zero opera... |
| adj0 31225 | Adjoint of the zero operat... |
| nmlnop0iALT 31226 | A linear operator with a z... |
| nmlnop0iHIL 31227 | A linear operator with a z... |
| nmlnopgt0i 31228 | A linear Hilbert space ope... |
| nmlnop0 31229 | A linear operator with a z... |
| nmlnopne0 31230 | A linear operator with a n... |
| lnopmi 31231 | The scalar product of a li... |
| lnophsi 31232 | The sum of two linear oper... |
| lnophdi 31233 | The difference of two line... |
| lnopcoi 31234 | The composition of two lin... |
| lnopco0i 31235 | The composition of a linea... |
| lnopeq0lem1 31236 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 31237 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 31238 | A condition implying that ... |
| lnopeqi 31239 | Two linear Hilbert space o... |
| lnopeq 31240 | Two linear Hilbert space o... |
| lnopunilem1 31241 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 31242 | Lemma for ~ lnopunii . (C... |
| lnopunii 31243 | If a linear operator (whos... |
| elunop2 31244 | An operator is unitary iff... |
| nmopun 31245 | Norm of a unitary Hilbert ... |
| unopbd 31246 | A unitary operator is a bo... |
| lnophmlem1 31247 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 31248 | Lemma for ~ lnophmi . (Co... |
| lnophmi 31249 | A linear operator is Hermi... |
| lnophm 31250 | A linear operator is Hermi... |
| hmops 31251 | The sum of two Hermitian o... |
| hmopm 31252 | The scalar product of a He... |
| hmopd 31253 | The difference of two Herm... |
| hmopco 31254 | The composition of two com... |
| nmbdoplbi 31255 | A lower bound for the norm... |
| nmbdoplb 31256 | A lower bound for the norm... |
| nmcexi 31257 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 31258 | The norm of a continuous l... |
| nmcoplbi 31259 | A lower bound for the norm... |
| nmcopex 31260 | The norm of a continuous l... |
| nmcoplb 31261 | A lower bound for the norm... |
| nmophmi 31262 | The norm of the scalar pro... |
| bdophmi 31263 | The scalar product of a bo... |
| lnconi 31264 | Lemma for ~ lnopconi and ~... |
| lnopconi 31265 | A condition equivalent to ... |
| lnopcon 31266 | A condition equivalent to ... |
| lnopcnbd 31267 | A linear operator is conti... |
| lncnopbd 31268 | A continuous linear operat... |
| lncnbd 31269 | A continuous linear operat... |
| lnopcnre 31270 | A linear operator is conti... |
| lnfnli 31271 | Basic property of a linear... |
| lnfnfi 31272 | A linear Hilbert space fun... |
| lnfn0i 31273 | The value of a linear Hilb... |
| lnfnaddi 31274 | Additive property of a lin... |
| lnfnmuli 31275 | Multiplicative property of... |
| lnfnaddmuli 31276 | Sum/product property of a ... |
| lnfnsubi 31277 | Subtraction property for a... |
| lnfn0 31278 | The value of a linear Hilb... |
| lnfnmul 31279 | Multiplicative property of... |
| nmbdfnlbi 31280 | A lower bound for the norm... |
| nmbdfnlb 31281 | A lower bound for the norm... |
| nmcfnexi 31282 | The norm of a continuous l... |
| nmcfnlbi 31283 | A lower bound for the norm... |
| nmcfnex 31284 | The norm of a continuous l... |
| nmcfnlb 31285 | A lower bound of the norm ... |
| lnfnconi 31286 | A condition equivalent to ... |
| lnfncon 31287 | A condition equivalent to ... |
| lnfncnbd 31288 | A linear functional is con... |
| imaelshi 31289 | The image of a subspace un... |
| rnelshi 31290 | The range of a linear oper... |
| nlelshi 31291 | The null space of a linear... |
| nlelchi 31292 | The null space of a contin... |
| riesz3i 31293 | A continuous linear functi... |
| riesz4i 31294 | A continuous linear functi... |
| riesz4 31295 | A continuous linear functi... |
| riesz1 31296 | Part 1 of the Riesz repres... |
| riesz2 31297 | Part 2 of the Riesz repres... |
| cnlnadjlem1 31298 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 31299 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 31300 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 31301 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 31302 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 31303 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 31304 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 31305 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 31306 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 31307 | Every continuous linear op... |
| cnlnadjeui 31308 | Every continuous linear op... |
| cnlnadjeu 31309 | Every continuous linear op... |
| cnlnadj 31310 | Every continuous linear op... |
| cnlnssadj 31311 | Every continuous linear Hi... |
| bdopssadj 31312 | Every bounded linear Hilbe... |
| bdopadj 31313 | Every bounded linear Hilbe... |
| adjbdln 31314 | The adjoint of a bounded l... |
| adjbdlnb 31315 | An operator is bounded and... |
| adjbd1o 31316 | The mapping of adjoints of... |
| adjlnop 31317 | The adjoint of an operator... |
| adjsslnop 31318 | Every operator with an adj... |
| nmopadjlei 31319 | Property of the norm of an... |
| nmopadjlem 31320 | Lemma for ~ nmopadji . (C... |
| nmopadji 31321 | Property of the norm of an... |
| adjeq0 31322 | An operator is zero iff it... |
| adjmul 31323 | The adjoint of the scalar ... |
| adjadd 31324 | The adjoint of the sum of ... |
| nmoptrii 31325 | Triangle inequality for th... |
| nmopcoi 31326 | Upper bound for the norm o... |
| bdophsi 31327 | The sum of two bounded lin... |
| bdophdi 31328 | The difference between two... |
| bdopcoi 31329 | The composition of two bou... |
| nmoptri2i 31330 | Triangle-type inequality f... |
| adjcoi 31331 | The adjoint of a compositi... |
| nmopcoadji 31332 | The norm of an operator co... |
| nmopcoadj2i 31333 | The norm of an operator co... |
| nmopcoadj0i 31334 | An operator composed with ... |
| unierri 31335 | If we approximate a chain ... |
| branmfn 31336 | The norm of the bra functi... |
| brabn 31337 | The bra of a vector is a b... |
| rnbra 31338 | The set of bras equals the... |
| bra11 31339 | The bra function maps vect... |
| bracnln 31340 | A bra is a continuous line... |
| cnvbraval 31341 | Value of the converse of t... |
| cnvbracl 31342 | Closure of the converse of... |
| cnvbrabra 31343 | The converse bra of the br... |
| bracnvbra 31344 | The bra of the converse br... |
| bracnlnval 31345 | The vector that a continuo... |
| cnvbramul 31346 | Multiplication property of... |
| kbass1 31347 | Dirac bra-ket associative ... |
| kbass2 31348 | Dirac bra-ket associative ... |
| kbass3 31349 | Dirac bra-ket associative ... |
| kbass4 31350 | Dirac bra-ket associative ... |
| kbass5 31351 | Dirac bra-ket associative ... |
| kbass6 31352 | Dirac bra-ket associative ... |
| leopg 31353 | Ordering relation for posi... |
| leop 31354 | Ordering relation for oper... |
| leop2 31355 | Ordering relation for oper... |
| leop3 31356 | Operator ordering in terms... |
| leoppos 31357 | Binary relation defining a... |
| leoprf2 31358 | The ordering relation for ... |
| leoprf 31359 | The ordering relation for ... |
| leopsq 31360 | The square of a Hermitian ... |
| 0leop 31361 | The zero operator is a pos... |
| idleop 31362 | The identity operator is a... |
| leopadd 31363 | The sum of two positive op... |
| leopmuli 31364 | The scalar product of a no... |
| leopmul 31365 | The scalar product of a po... |
| leopmul2i 31366 | Scalar product applied to ... |
| leoptri 31367 | The positive operator orde... |
| leoptr 31368 | The positive operator orde... |
| leopnmid 31369 | A bounded Hermitian operat... |
| nmopleid 31370 | A nonzero, bounded Hermiti... |
| opsqrlem1 31371 | Lemma for opsqri . (Contr... |
| opsqrlem2 31372 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 31373 | Lemma for opsqri . (Contr... |
| opsqrlem4 31374 | Lemma for opsqri . (Contr... |
| opsqrlem5 31375 | Lemma for opsqri . (Contr... |
| opsqrlem6 31376 | Lemma for opsqri . (Contr... |
| pjhmopi 31377 | A projector is a Hermitian... |
| pjlnopi 31378 | A projector is a linear op... |
| pjnmopi 31379 | The operator norm of a pro... |
| pjbdlni 31380 | A projector is a bounded l... |
| pjhmop 31381 | A projection is a Hermitia... |
| hmopidmchi 31382 | An idempotent Hermitian op... |
| hmopidmpji 31383 | An idempotent Hermitian op... |
| hmopidmch 31384 | An idempotent Hermitian op... |
| hmopidmpj 31385 | An idempotent Hermitian op... |
| pjsdii 31386 | Distributive law for Hilbe... |
| pjddii 31387 | Distributive law for Hilbe... |
| pjsdi2i 31388 | Chained distributive law f... |
| pjcoi 31389 | Composition of projections... |
| pjcocli 31390 | Closure of composition of ... |
| pjcohcli 31391 | Closure of composition of ... |
| pjadjcoi 31392 | Adjoint of composition of ... |
| pjcofni 31393 | Functionality of compositi... |
| pjss1coi 31394 | Subset relationship for pr... |
| pjss2coi 31395 | Subset relationship for pr... |
| pjssmi 31396 | Projection meet property. ... |
| pjssge0i 31397 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 31398 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 31399 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 31400 | Composition of projections... |
| pjscji 31401 | The projection of orthogon... |
| pjssumi 31402 | The projection on a subspa... |
| pjssposi 31403 | Projector ordering can be ... |
| pjordi 31404 | The definition of projecto... |
| pjssdif2i 31405 | The projection subspace of... |
| pjssdif1i 31406 | A necessary and sufficient... |
| pjimai 31407 | The image of a projection.... |
| pjidmcoi 31408 | A projection is idempotent... |
| pjoccoi 31409 | Composition of projections... |
| pjtoi 31410 | Subspace sum of projection... |
| pjoci 31411 | Projection of orthocomplem... |
| pjidmco 31412 | A projection operator is i... |
| dfpjop 31413 | Definition of projection o... |
| pjhmopidm 31414 | Two ways to express the se... |
| elpjidm 31415 | A projection operator is i... |
| elpjhmop 31416 | A projection operator is H... |
| 0leopj 31417 | A projector is a positive ... |
| pjadj2 31418 | A projector is self-adjoin... |
| pjadj3 31419 | A projector is self-adjoin... |
| elpjch 31420 | Reconstruction of the subs... |
| elpjrn 31421 | Reconstruction of the subs... |
| pjinvari 31422 | A closed subspace ` H ` wi... |
| pjin1i 31423 | Lemma for Theorem 1.22 of ... |
| pjin2i 31424 | Lemma for Theorem 1.22 of ... |
| pjin3i 31425 | Lemma for Theorem 1.22 of ... |
| pjclem1 31426 | Lemma for projection commu... |
| pjclem2 31427 | Lemma for projection commu... |
| pjclem3 31428 | Lemma for projection commu... |
| pjclem4a 31429 | Lemma for projection commu... |
| pjclem4 31430 | Lemma for projection commu... |
| pjci 31431 | Two subspaces commute iff ... |
| pjcmul1i 31432 | A necessary and sufficient... |
| pjcmul2i 31433 | The projection subspace of... |
| pjcohocli 31434 | Closure of composition of ... |
| pjadj2coi 31435 | Adjoint of double composit... |
| pj2cocli 31436 | Closure of double composit... |
| pj3lem1 31437 | Lemma for projection tripl... |
| pj3si 31438 | Stronger projection triple... |
| pj3i 31439 | Projection triplet theorem... |
| pj3cor1i 31440 | Projection triplet corolla... |
| pjs14i 31441 | Theorem S-14 of Watanabe, ... |
| isst 31444 | Property of a state. (Con... |
| ishst 31445 | Property of a complex Hilb... |
| sticl 31446 | ` [ 0 , 1 ] ` closure of t... |
| stcl 31447 | Real closure of the value ... |
| hstcl 31448 | Closure of the value of a ... |
| hst1a 31449 | Unit value of a Hilbert-sp... |
| hstel2 31450 | Properties of a Hilbert-sp... |
| hstorth 31451 | Orthogonality property of ... |
| hstosum 31452 | Orthogonal sum property of... |
| hstoc 31453 | Sum of a Hilbert-space-val... |
| hstnmoc 31454 | Sum of norms of a Hilbert-... |
| stge0 31455 | The value of a state is no... |
| stle1 31456 | The value of a state is le... |
| hstle1 31457 | The norm of the value of a... |
| hst1h 31458 | The norm of a Hilbert-spac... |
| hst0h 31459 | The norm of a Hilbert-spac... |
| hstpyth 31460 | Pythagorean property of a ... |
| hstle 31461 | Ordering property of a Hil... |
| hstles 31462 | Ordering property of a Hil... |
| hstoh 31463 | A Hilbert-space-valued sta... |
| hst0 31464 | A Hilbert-space-valued sta... |
| sthil 31465 | The value of a state at th... |
| stj 31466 | The value of a state on a ... |
| sto1i 31467 | The state of a subspace pl... |
| sto2i 31468 | The state of the orthocomp... |
| stge1i 31469 | If a state is greater than... |
| stle0i 31470 | If a state is less than or... |
| stlei 31471 | Ordering law for states. ... |
| stlesi 31472 | Ordering law for states. ... |
| stji1i 31473 | Join of components of Sasa... |
| stm1i 31474 | State of component of unit... |
| stm1ri 31475 | State of component of unit... |
| stm1addi 31476 | Sum of states whose meet i... |
| staddi 31477 | If the sum of 2 states is ... |
| stm1add3i 31478 | Sum of states whose meet i... |
| stadd3i 31479 | If the sum of 3 states is ... |
| st0 31480 | The state of the zero subs... |
| strlem1 31481 | Lemma for strong state the... |
| strlem2 31482 | Lemma for strong state the... |
| strlem3a 31483 | Lemma for strong state the... |
| strlem3 31484 | Lemma for strong state the... |
| strlem4 31485 | Lemma for strong state the... |
| strlem5 31486 | Lemma for strong state the... |
| strlem6 31487 | Lemma for strong state the... |
| stri 31488 | Strong state theorem. The... |
| strb 31489 | Strong state theorem (bidi... |
| hstrlem2 31490 | Lemma for strong set of CH... |
| hstrlem3a 31491 | Lemma for strong set of CH... |
| hstrlem3 31492 | Lemma for strong set of CH... |
| hstrlem4 31493 | Lemma for strong set of CH... |
| hstrlem5 31494 | Lemma for strong set of CH... |
| hstrlem6 31495 | Lemma for strong set of CH... |
| hstri 31496 | Hilbert space admits a str... |
| hstrbi 31497 | Strong CH-state theorem (b... |
| largei 31498 | A Hilbert lattice admits a... |
| jplem1 31499 | Lemma for Jauch-Piron theo... |
| jplem2 31500 | Lemma for Jauch-Piron theo... |
| jpi 31501 | The function ` S ` , that ... |
| golem1 31502 | Lemma for Godowski's equat... |
| golem2 31503 | Lemma for Godowski's equat... |
| goeqi 31504 | Godowski's equation, shown... |
| stcltr1i 31505 | Property of a strong class... |
| stcltr2i 31506 | Property of a strong class... |
| stcltrlem1 31507 | Lemma for strong classical... |
| stcltrlem2 31508 | Lemma for strong classical... |
| stcltrthi 31509 | Theorem for classically st... |
| cvbr 31513 | Binary relation expressing... |
| cvbr2 31514 | Binary relation expressing... |
| cvcon3 31515 | Contraposition law for the... |
| cvpss 31516 | The covers relation implie... |
| cvnbtwn 31517 | The covers relation implie... |
| cvnbtwn2 31518 | The covers relation implie... |
| cvnbtwn3 31519 | The covers relation implie... |
| cvnbtwn4 31520 | The covers relation implie... |
| cvnsym 31521 | The covers relation is not... |
| cvnref 31522 | The covers relation is not... |
| cvntr 31523 | The covers relation is not... |
| spansncv2 31524 | Hilbert space has the cove... |
| mdbr 31525 | Binary relation expressing... |
| mdi 31526 | Consequence of the modular... |
| mdbr2 31527 | Binary relation expressing... |
| mdbr3 31528 | Binary relation expressing... |
| mdbr4 31529 | Binary relation expressing... |
| dmdbr 31530 | Binary relation expressing... |
| dmdmd 31531 | The dual modular pair prop... |
| mddmd 31532 | The modular pair property ... |
| dmdi 31533 | Consequence of the dual mo... |
| dmdbr2 31534 | Binary relation expressing... |
| dmdi2 31535 | Consequence of the dual mo... |
| dmdbr3 31536 | Binary relation expressing... |
| dmdbr4 31537 | Binary relation expressing... |
| dmdi4 31538 | Consequence of the dual mo... |
| dmdbr5 31539 | Binary relation expressing... |
| mddmd2 31540 | Relationship between modul... |
| mdsl0 31541 | A sublattice condition tha... |
| ssmd1 31542 | Ordering implies the modul... |
| ssmd2 31543 | Ordering implies the modul... |
| ssdmd1 31544 | Ordering implies the dual ... |
| ssdmd2 31545 | Ordering implies the dual ... |
| dmdsl3 31546 | Sublattice mapping for a d... |
| mdsl3 31547 | Sublattice mapping for a m... |
| mdslle1i 31548 | Order preservation of the ... |
| mdslle2i 31549 | Order preservation of the ... |
| mdslj1i 31550 | Join preservation of the o... |
| mdslj2i 31551 | Meet preservation of the r... |
| mdsl1i 31552 | If the modular pair proper... |
| mdsl2i 31553 | If the modular pair proper... |
| mdsl2bi 31554 | If the modular pair proper... |
| cvmdi 31555 | The covering property impl... |
| mdslmd1lem1 31556 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 31557 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 31558 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 31559 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 31560 | Preservation of the modula... |
| mdslmd2i 31561 | Preservation of the modula... |
| mdsldmd1i 31562 | Preservation of the dual m... |
| mdslmd3i 31563 | Modular pair conditions th... |
| mdslmd4i 31564 | Modular pair condition tha... |
| csmdsymi 31565 | Cross-symmetry implies M-s... |
| mdexchi 31566 | An exchange lemma for modu... |
| cvmd 31567 | The covering property impl... |
| cvdmd 31568 | The covering property impl... |
| ela 31570 | Atoms in a Hilbert lattice... |
| elat2 31571 | Expanded membership relati... |
| elatcv0 31572 | A Hilbert lattice element ... |
| atcv0 31573 | An atom covers the zero su... |
| atssch 31574 | Atoms are a subset of the ... |
| atelch 31575 | An atom is a Hilbert latti... |
| atne0 31576 | An atom is not the Hilbert... |
| atss 31577 | A lattice element smaller ... |
| atsseq 31578 | Two atoms in a subset rela... |
| atcveq0 31579 | A Hilbert lattice element ... |
| h1da 31580 | A 1-dimensional subspace i... |
| spansna 31581 | The span of the singleton ... |
| sh1dle 31582 | A 1-dimensional subspace i... |
| ch1dle 31583 | A 1-dimensional subspace i... |
| atom1d 31584 | The 1-dimensional subspace... |
| superpos 31585 | Superposition Principle. ... |
| chcv1 31586 | The Hilbert lattice has th... |
| chcv2 31587 | The Hilbert lattice has th... |
| chjatom 31588 | The join of a closed subsp... |
| shatomici 31589 | The lattice of Hilbert sub... |
| hatomici 31590 | The Hilbert lattice is ato... |
| hatomic 31591 | A Hilbert lattice is atomi... |
| shatomistici 31592 | The lattice of Hilbert sub... |
| hatomistici 31593 | ` CH ` is atomistic, i.e. ... |
| chpssati 31594 | Two Hilbert lattice elemen... |
| chrelati 31595 | The Hilbert lattice is rel... |
| chrelat2i 31596 | A consequence of relative ... |
| cvati 31597 | If a Hilbert lattice eleme... |
| cvbr4i 31598 | An alternate way to expres... |
| cvexchlem 31599 | Lemma for ~ cvexchi . (Co... |
| cvexchi 31600 | The Hilbert lattice satisf... |
| chrelat2 31601 | A consequence of relative ... |
| chrelat3 31602 | A consequence of relative ... |
| chrelat3i 31603 | A consequence of the relat... |
| chrelat4i 31604 | A consequence of relative ... |
| cvexch 31605 | The Hilbert lattice satisf... |
| cvp 31606 | The Hilbert lattice satisf... |
| atnssm0 31607 | The meet of a Hilbert latt... |
| atnemeq0 31608 | The meet of distinct atoms... |
| atssma 31609 | The meet with an atom's su... |
| atcv0eq 31610 | Two atoms covering the zer... |
| atcv1 31611 | Two atoms covering the zer... |
| atexch 31612 | The Hilbert lattice satisf... |
| atomli 31613 | An assertion holding in at... |
| atoml2i 31614 | An assertion holding in at... |
| atordi 31615 | An ordering law for a Hilb... |
| atcvatlem 31616 | Lemma for ~ atcvati . (Co... |
| atcvati 31617 | A nonzero Hilbert lattice ... |
| atcvat2i 31618 | A Hilbert lattice element ... |
| atord 31619 | An ordering law for a Hilb... |
| atcvat2 31620 | A Hilbert lattice element ... |
| chirredlem1 31621 | Lemma for ~ chirredi . (C... |
| chirredlem2 31622 | Lemma for ~ chirredi . (C... |
| chirredlem3 31623 | Lemma for ~ chirredi . (C... |
| chirredlem4 31624 | Lemma for ~ chirredi . (C... |
| chirredi 31625 | The Hilbert lattice is irr... |
| chirred 31626 | The Hilbert lattice is irr... |
| atcvat3i 31627 | A condition implying that ... |
| atcvat4i 31628 | A condition implying exist... |
| atdmd 31629 | Two Hilbert lattice elemen... |
| atmd 31630 | Two Hilbert lattice elemen... |
| atmd2 31631 | Two Hilbert lattice elemen... |
| atabsi 31632 | Absorption of an incompara... |
| atabs2i 31633 | Absorption of an incompara... |
| mdsymlem1 31634 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 31635 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 31636 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 31637 | Lemma for ~ mdsymi . This... |
| mdsymlem5 31638 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 31639 | Lemma for ~ mdsymi . This... |
| mdsymlem7 31640 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 31641 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 31642 | M-symmetry of the Hilbert ... |
| mdsym 31643 | M-symmetry of the Hilbert ... |
| dmdsym 31644 | Dual M-symmetry of the Hil... |
| atdmd2 31645 | Two Hilbert lattice elemen... |
| sumdmdii 31646 | If the subspace sum of two... |
| cmmdi 31647 | Commuting subspaces form a... |
| cmdmdi 31648 | Commuting subspaces form a... |
| sumdmdlem 31649 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 31650 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 31651 | The subspace sum of two Hi... |
| dmdbr4ati 31652 | Dual modular pair property... |
| dmdbr5ati 31653 | Dual modular pair property... |
| dmdbr6ati 31654 | Dual modular pair property... |
| dmdbr7ati 31655 | Dual modular pair property... |
| mdoc1i 31656 | Orthocomplements form a mo... |
| mdoc2i 31657 | Orthocomplements form a mo... |
| dmdoc1i 31658 | Orthocomplements form a du... |
| dmdoc2i 31659 | Orthocomplements form a du... |
| mdcompli 31660 | A condition equivalent to ... |
| dmdcompli 31661 | A condition equivalent to ... |
| mddmdin0i 31662 | If dual modular implies mo... |
| cdjreui 31663 | A member of the sum of dis... |
| cdj1i 31664 | Two ways to express " ` A ... |
| cdj3lem1 31665 | A property of " ` A ` and ... |
| cdj3lem2 31666 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 31667 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 31668 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 31669 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 31670 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 31671 | Lemma for ~ cdj3i . The s... |
| cdj3i 31672 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 31673 | (_This theorem is a dummy ... |
| sa-abvi 31674 | A theorem about the univer... |
| xfree 31675 | A partial converse to ~ 19... |
| xfree2 31676 | A partial converse to ~ 19... |
| addltmulALT 31677 | A proof readability experi... |
| bian1d 31678 | Adding a superfluous conju... |
| or3di 31679 | Distributive law for disju... |
| or3dir 31680 | Distributive law for disju... |
| 3o1cs 31681 | Deduction eliminating disj... |
| 3o2cs 31682 | Deduction eliminating disj... |
| 3o3cs 31683 | Deduction eliminating disj... |
| 13an22anass 31684 | Associative law for four c... |
| sbc2iedf 31685 | Conversion of implicit sub... |
| rspc2daf 31686 | Double restricted speciali... |
| ralcom4f 31687 | Commutation of restricted ... |
| rexcom4f 31688 | Commutation of restricted ... |
| 19.9d2rf 31689 | A deduction version of one... |
| 19.9d2r 31690 | A deduction version of one... |
| r19.29ffa 31691 | A commonly used pattern ba... |
| eqtrb 31692 | A transposition of equalit... |
| eqelbid 31693 | A variable elimination law... |
| opsbc2ie 31694 | Conversion of implicit sub... |
| opreu2reuALT 31695 | Correspondence between uni... |
| 2reucom 31698 | Double restricted existent... |
| 2reu2rex1 31699 | Double restricted existent... |
| 2reureurex 31700 | Double restricted existent... |
| 2reu2reu2 31701 | Double restricted existent... |
| opreu2reu1 31702 | Equivalent definition of t... |
| sq2reunnltb 31703 | There exists a unique deco... |
| addsqnot2reu 31704 | For each complex number ` ... |
| sbceqbidf 31705 | Equality theorem for class... |
| sbcies 31706 | A special version of class... |
| mo5f 31707 | Alternate definition of "a... |
| nmo 31708 | Negation of "at most one".... |
| reuxfrdf 31709 | Transfer existential uniqu... |
| rexunirn 31710 | Restricted existential qua... |
| rmoxfrd 31711 | Transfer "at most one" res... |
| rmoun 31712 | "At most one" restricted e... |
| rmounid 31713 | A case where an "at most o... |
| riotaeqbidva 31714 | Equivalent wff's yield equ... |
| dmrab 31715 | Domain of a restricted cla... |
| difrab2 31716 | Difference of two restrict... |
| rabexgfGS 31717 | Separation Scheme in terms... |
| rabsnel 31718 | Truth implied by equality ... |
| eqrrabd 31719 | Deduce equality with a res... |
| foresf1o 31720 | From a surjective function... |
| rabfodom 31721 | Domination relation for re... |
| abrexdomjm 31722 | An indexed set is dominate... |
| abrexdom2jm 31723 | An indexed set is dominate... |
| abrexexd 31724 | Existence of a class abstr... |
| elabreximd 31725 | Class substitution in an i... |
| elabreximdv 31726 | Class substitution in an i... |
| abrexss 31727 | A necessary condition for ... |
| elunsn 31728 | Elementhood to a union wit... |
| nelun 31729 | Negated membership for a u... |
| snsssng 31730 | If a singleton is a subset... |
| inin 31731 | Intersection with an inter... |
| inindif 31732 | See ~ inundif . (Contribu... |
| difininv 31733 | Condition for the intersec... |
| difeq 31734 | Rewriting an equation with... |
| eqdif 31735 | If both set differences of... |
| indifbi 31736 | Two ways to express equali... |
| diffib 31737 | Case where ~ diffi is a bi... |
| difxp1ss 31738 | Difference law for Cartesi... |
| difxp2ss 31739 | Difference law for Cartesi... |
| indifundif 31740 | A remarkable equation with... |
| elpwincl1 31741 | Closure of intersection wi... |
| elpwdifcl 31742 | Closure of class differenc... |
| elpwiuncl 31743 | Closure of indexed union w... |
| eqsnd 31744 | Deduce that a set is a sin... |
| elpreq 31745 | Equality wihin a pair. (C... |
| nelpr 31746 | A set ` A ` not in a pair ... |
| inpr0 31747 | Rewrite an empty intersect... |
| neldifpr1 31748 | The first element of a pai... |
| neldifpr2 31749 | The second element of a pa... |
| unidifsnel 31750 | The other element of a pai... |
| unidifsnne 31751 | The other element of a pai... |
| ifeqeqx 31752 | An equality theorem tailor... |
| elimifd 31753 | Elimination of a condition... |
| elim2if 31754 | Elimination of two conditi... |
| elim2ifim 31755 | Elimination of two conditi... |
| ifeq3da 31756 | Given an expression ` C ` ... |
| ifnetrue 31757 | Deduce truth from a condit... |
| ifnefals 31758 | Deduce falsehood from a co... |
| ifnebib 31759 | The converse of ~ ifbi hol... |
| uniinn0 31760 | Sufficient and necessary c... |
| uniin1 31761 | Union of intersection. Ge... |
| uniin2 31762 | Union of intersection. Ge... |
| difuncomp 31763 | Express a class difference... |
| elpwunicl 31764 | Closure of a set union wit... |
| cbviunf 31765 | Rule used to change the bo... |
| iuneq12daf 31766 | Equality deduction for ind... |
| iunin1f 31767 | Indexed union of intersect... |
| ssiun3 31768 | Subset equivalence for an ... |
| ssiun2sf 31769 | Subset relationship for an... |
| iuninc 31770 | The union of an increasing... |
| iundifdifd 31771 | The intersection of a set ... |
| iundifdif 31772 | The intersection of a set ... |
| iunrdx 31773 | Re-index an indexed union.... |
| iunpreima 31774 | Preimage of an indexed uni... |
| iunrnmptss 31775 | A subset relation for an i... |
| iunxunsn 31776 | Appending a set to an inde... |
| iunxunpr 31777 | Appending two sets to an i... |
| iinabrex 31778 | Rewriting an indexed inter... |
| disjnf 31779 | In case ` x ` is not free ... |
| cbvdisjf 31780 | Change bound variables in ... |
| disjss1f 31781 | A subset of a disjoint col... |
| disjeq1f 31782 | Equality theorem for disjo... |
| disjxun0 31783 | Simplify a disjoint union.... |
| disjdifprg 31784 | A trivial partition into a... |
| disjdifprg2 31785 | A trivial partition of a s... |
| disji2f 31786 | Property of a disjoint col... |
| disjif 31787 | Property of a disjoint col... |
| disjorf 31788 | Two ways to say that a col... |
| disjorsf 31789 | Two ways to say that a col... |
| disjif2 31790 | Property of a disjoint col... |
| disjabrex 31791 | Rewriting a disjoint colle... |
| disjabrexf 31792 | Rewriting a disjoint colle... |
| disjpreima 31793 | A preimage of a disjoint s... |
| disjrnmpt 31794 | Rewriting a disjoint colle... |
| disjin 31795 | If a collection is disjoin... |
| disjin2 31796 | If a collection is disjoin... |
| disjxpin 31797 | Derive a disjunction over ... |
| iundisjf 31798 | Rewrite a countable union ... |
| iundisj2f 31799 | A disjoint union is disjoi... |
| disjrdx 31800 | Re-index a disjunct collec... |
| disjex 31801 | Two ways to say that two c... |
| disjexc 31802 | A variant of ~ disjex , ap... |
| disjunsn 31803 | Append an element to a dis... |
| disjun0 31804 | Adding the empty element p... |
| disjiunel 31805 | A set of elements B of a d... |
| disjuniel 31806 | A set of elements B of a d... |
| xpdisjres 31807 | Restriction of a constant ... |
| opeldifid 31808 | Ordered pair elementhood o... |
| difres 31809 | Case when class difference... |
| imadifxp 31810 | Image of the difference wi... |
| relfi 31811 | A relation (set) is finite... |
| 0res 31812 | Restriction of the empty f... |
| fcoinver 31813 | Build an equivalence relat... |
| fcoinvbr 31814 | Binary relation for the eq... |
| brabgaf 31815 | The law of concretion for ... |
| brelg 31816 | Two things in a binary rel... |
| br8d 31817 | Substitution for an eight-... |
| opabdm 31818 | Domain of an ordered-pair ... |
| opabrn 31819 | Range of an ordered-pair c... |
| opabssi 31820 | Sufficient condition for a... |
| opabid2ss 31821 | One direction of ~ opabid2... |
| ssrelf 31822 | A subclass relationship de... |
| eqrelrd2 31823 | A version of ~ eqrelrdv2 w... |
| erbr3b 31824 | Biconditional for equivale... |
| iunsnima 31825 | Image of a singleton by an... |
| iunsnima2 31826 | Version of ~ iunsnima with... |
| ac6sf2 31827 | Alternate version of ~ ac6... |
| fnresin 31828 | Restriction of a function ... |
| f1o3d 31829 | Describe an implicit one-t... |
| eldmne0 31830 | A function of nonempty dom... |
| f1rnen 31831 | Equinumerosity of the rang... |
| rinvf1o 31832 | Sufficient conditions for ... |
| fresf1o 31833 | Conditions for a restricti... |
| nfpconfp 31834 | The set of fixed points of... |
| fmptco1f1o 31835 | The action of composing (t... |
| cofmpt2 31836 | Express composition of a m... |
| f1mptrn 31837 | Express injection for a ma... |
| dfimafnf 31838 | Alternate definition of th... |
| funimass4f 31839 | Membership relation for th... |
| elimampt 31840 | Membership in the image of... |
| suppss2f 31841 | Show that the support of a... |
| ofrn 31842 | The range of the function ... |
| ofrn2 31843 | The range of the function ... |
| off2 31844 | The function operation pro... |
| ofresid 31845 | Applying an operation rest... |
| fimarab 31846 | Expressing the image of a ... |
| unipreima 31847 | Preimage of a class union.... |
| opfv 31848 | Value of a function produc... |
| xppreima 31849 | The preimage of a Cartesia... |
| 2ndimaxp 31850 | Image of a cartesian produ... |
| djussxp2 31851 | Stronger version of ~ djus... |
| 2ndresdju 31852 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 31853 | The ` 2nd ` function restr... |
| xppreima2 31854 | The preimage of a Cartesia... |
| abfmpunirn 31855 | Membership in a union of a... |
| rabfmpunirn 31856 | Membership in a union of a... |
| abfmpeld 31857 | Membership in an element o... |
| abfmpel 31858 | Membership in an element o... |
| fmptdF 31859 | Domain and codomain of the... |
| fmptcof2 31860 | Composition of two functio... |
| fcomptf 31861 | Express composition of two... |
| acunirnmpt 31862 | Axiom of choice for the un... |
| acunirnmpt2 31863 | Axiom of choice for the un... |
| acunirnmpt2f 31864 | Axiom of choice for the un... |
| aciunf1lem 31865 | Choice in an index union. ... |
| aciunf1 31866 | Choice in an index union. ... |
| ofoprabco 31867 | Function operation as a co... |
| ofpreima 31868 | Express the preimage of a ... |
| ofpreima2 31869 | Express the preimage of a ... |
| funcnvmpt 31870 | Condition for a function i... |
| funcnv5mpt 31871 | Two ways to say that a fun... |
| funcnv4mpt 31872 | Two ways to say that a fun... |
| preimane 31873 | Different elements have di... |
| fnpreimac 31874 | Choose a set ` x ` contain... |
| fgreu 31875 | Exactly one point of a fun... |
| fcnvgreu 31876 | If the converse of a relat... |
| rnmposs 31877 | The range of an operation ... |
| mptssALT 31878 | Deduce subset relation of ... |
| dfcnv2 31879 | Alternative definition of ... |
| fnimatp 31880 | The image of an unordered ... |
| rnexd 31881 | The range of a set is a se... |
| imaexd 31882 | The image of a set is a se... |
| mpomptxf 31883 | Express a two-argument fun... |
| suppovss 31884 | A bound for the support of... |
| fvdifsupp 31885 | Function value is zero out... |
| suppiniseg 31886 | Relation between the suppo... |
| fsuppinisegfi 31887 | The initial segment ` ( ``... |
| fressupp 31888 | The restriction of a funct... |
| fdifsuppconst 31889 | A function is a zero const... |
| ressupprn 31890 | The range of a function re... |
| supppreima 31891 | Express the support of a f... |
| fsupprnfi 31892 | Finite support implies fin... |
| mptiffisupp 31893 | Conditions for a mapping f... |
| cosnopne 31894 | Composition of two ordered... |
| cosnop 31895 | Composition of two ordered... |
| cnvprop 31896 | Converse of a pair of orde... |
| brprop 31897 | Binary relation for a pair... |
| mptprop 31898 | Rewrite pairs of ordered p... |
| coprprop 31899 | Composition of two pairs o... |
| gtiso 31900 | Two ways to write a strict... |
| isoun 31901 | Infer an isomorphism from ... |
| disjdsct 31902 | A disjoint collection is d... |
| df1stres 31903 | Definition for a restricti... |
| df2ndres 31904 | Definition for a restricti... |
| 1stpreimas 31905 | The preimage of a singleto... |
| 1stpreima 31906 | The preimage by ` 1st ` is... |
| 2ndpreima 31907 | The preimage by ` 2nd ` is... |
| curry2ima 31908 | The image of a curried fun... |
| preiman0 31909 | The preimage of a nonempty... |
| intimafv 31910 | The intersection of an ima... |
| ecref 31911 | All elements are in their ... |
| supssd 31912 | Inequality deduction for s... |
| infssd 31913 | Inequality deduction for i... |
| imafi2 31914 | The image by a finite set ... |
| unifi3 31915 | If a union is finite, then... |
| snct 31916 | A singleton is countable. ... |
| prct 31917 | An unordered pair is count... |
| mpocti 31918 | An operation is countable ... |
| abrexct 31919 | An image set of a countabl... |
| mptctf 31920 | A countable mapping set is... |
| abrexctf 31921 | An image set of a countabl... |
| padct 31922 | Index a countable set with... |
| cnvoprabOLD 31923 | The converse of a class ab... |
| f1od2 31924 | Sufficient condition for a... |
| fcobij 31925 | Composing functions with a... |
| fcobijfs 31926 | Composing finitely support... |
| suppss3 31927 | Deduce a function's suppor... |
| fsuppcurry1 31928 | Finite support of a currie... |
| fsuppcurry2 31929 | Finite support of a currie... |
| offinsupp1 31930 | Finite support for a funct... |
| ffs2 31931 | Rewrite a function's suppo... |
| ffsrn 31932 | The range of a finitely su... |
| resf1o 31933 | Restriction of functions t... |
| maprnin 31934 | Restricting the range of t... |
| fpwrelmapffslem 31935 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 31936 | Define a canonical mapping... |
| fpwrelmapffs 31937 | Define a canonical mapping... |
| creq0 31938 | The real representation of... |
| 1nei 31939 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 31940 | An integer unit times itse... |
| nnmulge 31941 | Multiplying by a positive ... |
| lt2addrd 31942 | If the right-hand side of ... |
| xrlelttric 31943 | Trichotomy law for extende... |
| xaddeq0 31944 | Two extended reals which a... |
| xrinfm 31945 | The extended real numbers ... |
| le2halvesd 31946 | A sum is less than the who... |
| xraddge02 31947 | A number is less than or e... |
| xrge0addge 31948 | A number is less than or e... |
| xlt2addrd 31949 | If the right-hand side of ... |
| xrsupssd 31950 | Inequality deduction for s... |
| xrge0infss 31951 | Any subset of nonnegative ... |
| xrge0infssd 31952 | Inequality deduction for i... |
| xrge0addcld 31953 | Nonnegative extended reals... |
| xrge0subcld 31954 | Condition for closure of n... |
| infxrge0lb 31955 | A member of a set of nonne... |
| infxrge0glb 31956 | The infimum of a set of no... |
| infxrge0gelb 31957 | The infimum of a set of no... |
| xrofsup 31958 | The supremum is preserved ... |
| supxrnemnf 31959 | The supremum of a nonempty... |
| xnn0gt0 31960 | Nonzero extended nonnegati... |
| xnn01gt 31961 | An extended nonnegative in... |
| nn0xmulclb 31962 | Finite multiplication in t... |
| joiniooico 31963 | Disjoint joining an open i... |
| ubico 31964 | A right-open interval does... |
| xeqlelt 31965 | Equality in terms of 'less... |
| eliccelico 31966 | Relate elementhood to a cl... |
| elicoelioo 31967 | Relate elementhood to a cl... |
| iocinioc2 31968 | Intersection between two o... |
| xrdifh 31969 | Class difference of a half... |
| iocinif 31970 | Relate intersection of two... |
| difioo 31971 | The difference between two... |
| difico 31972 | The difference between two... |
| uzssico 31973 | Upper integer sets are a s... |
| fz2ssnn0 31974 | A finite set of sequential... |
| nndiffz1 31975 | Upper set of the positive ... |
| ssnnssfz 31976 | For any finite subset of `... |
| fzne1 31977 | Elementhood in a finite se... |
| fzm1ne1 31978 | Elementhood of an integer ... |
| fzspl 31979 | Split the last element of ... |
| fzdif2 31980 | Split the last element of ... |
| fzodif2 31981 | Split the last element of ... |
| fzodif1 31982 | Set difference of two half... |
| fzsplit3 31983 | Split a finite interval of... |
| bcm1n 31984 | The proportion of one bino... |
| iundisjfi 31985 | Rewrite a countable union ... |
| iundisj2fi 31986 | A disjoint union is disjoi... |
| iundisjcnt 31987 | Rewrite a countable union ... |
| iundisj2cnt 31988 | A countable disjoint union... |
| fzone1 31989 | Elementhood in a half-open... |
| fzom1ne1 31990 | Elementhood in a half-open... |
| f1ocnt 31991 | Given a countable set ` A ... |
| fz1nnct 31992 | NN and integer ranges star... |
| fz1nntr 31993 | NN and integer ranges star... |
| nn0difffzod 31994 | A nonnegative integer that... |
| suppssnn0 31995 | Show that the support of a... |
| hashunif 31996 | The cardinality of a disjo... |
| hashxpe 31997 | The size of the Cartesian ... |
| hashgt1 31998 | Restate "set contains at l... |
| dvdszzq 31999 | Divisibility for an intege... |
| prmdvdsbc 32000 | Condition for a prime numb... |
| numdenneg 32001 | Numerator and denominator ... |
| divnumden2 32002 | Calculate the reduced form... |
| nnindf 32003 | Principle of Mathematical ... |
| nn0min 32004 | Extracting the minimum pos... |
| subne0nn 32005 | A nonnegative difference i... |
| ltesubnnd 32006 | Subtracting an integer num... |
| fprodeq02 32007 | If one of the factors is z... |
| pr01ssre 32008 | The range of the indicator... |
| fprodex01 32009 | A product of factors equal... |
| prodpr 32010 | A product over a pair is t... |
| prodtp 32011 | A product over a triple is... |
| fsumub 32012 | An upper bound for a term ... |
| fsumiunle 32013 | Upper bound for a sum of n... |
| dfdec100 32014 | Split the hundreds from a ... |
| dp2eq1 32017 | Equality theorem for the d... |
| dp2eq2 32018 | Equality theorem for the d... |
| dp2eq1i 32019 | Equality theorem for the d... |
| dp2eq2i 32020 | Equality theorem for the d... |
| dp2eq12i 32021 | Equality theorem for the d... |
| dp20u 32022 | Add a zero in the tenths (... |
| dp20h 32023 | Add a zero in the unit pla... |
| dp2cl 32024 | Closure for the decimal fr... |
| dp2clq 32025 | Closure for a decimal frac... |
| rpdp2cl 32026 | Closure for a decimal frac... |
| rpdp2cl2 32027 | Closure for a decimal frac... |
| dp2lt10 32028 | Decimal fraction builds re... |
| dp2lt 32029 | Comparing two decimal frac... |
| dp2ltsuc 32030 | Comparing a decimal fracti... |
| dp2ltc 32031 | Comparing two decimal expa... |
| dpval 32034 | Define the value of the de... |
| dpcl 32035 | Prove that the closure of ... |
| dpfrac1 32036 | Prove a simple equivalence... |
| dpval2 32037 | Value of the decimal point... |
| dpval3 32038 | Value of the decimal point... |
| dpmul10 32039 | Multiply by 10 a decimal e... |
| decdiv10 32040 | Divide a decimal number by... |
| dpmul100 32041 | Multiply by 100 a decimal ... |
| dp3mul10 32042 | Multiply by 10 a decimal e... |
| dpmul1000 32043 | Multiply by 1000 a decimal... |
| dpval3rp 32044 | Value of the decimal point... |
| dp0u 32045 | Add a zero in the tenths p... |
| dp0h 32046 | Remove a zero in the units... |
| rpdpcl 32047 | Closure of the decimal poi... |
| dplt 32048 | Comparing two decimal expa... |
| dplti 32049 | Comparing a decimal expans... |
| dpgti 32050 | Comparing a decimal expans... |
| dpltc 32051 | Comparing two decimal inte... |
| dpexpp1 32052 | Add one zero to the mantis... |
| 0dp2dp 32053 | Multiply by 10 a decimal e... |
| dpadd2 32054 | Addition with one decimal,... |
| dpadd 32055 | Addition with one decimal.... |
| dpadd3 32056 | Addition with two decimals... |
| dpmul 32057 | Multiplication with one de... |
| dpmul4 32058 | An upper bound to multipli... |
| threehalves 32059 | Example theorem demonstrat... |
| 1mhdrd 32060 | Example theorem demonstrat... |
| xdivval 32063 | Value of division: the (un... |
| xrecex 32064 | Existence of reciprocal of... |
| xmulcand 32065 | Cancellation law for exten... |
| xreceu 32066 | Existential uniqueness of ... |
| xdivcld 32067 | Closure law for the extend... |
| xdivcl 32068 | Closure law for the extend... |
| xdivmul 32069 | Relationship between divis... |
| rexdiv 32070 | The extended real division... |
| xdivrec 32071 | Relationship between divis... |
| xdivid 32072 | A number divided by itself... |
| xdiv0 32073 | Division into zero is zero... |
| xdiv0rp 32074 | Division into zero is zero... |
| eliccioo 32075 | Membership in a closed int... |
| elxrge02 32076 | Elementhood in the set of ... |
| xdivpnfrp 32077 | Plus infinity divided by a... |
| rpxdivcld 32078 | Closure law for extended d... |
| xrpxdivcld 32079 | Closure law for extended d... |
| wrdfd 32080 | A word is a zero-based seq... |
| wrdres 32081 | Condition for the restrict... |
| wrdsplex 32082 | Existence of a split of a ... |
| pfx1s2 32083 | The prefix of length 1 of ... |
| pfxrn2 32084 | The range of a prefix of a... |
| pfxrn3 32085 | Express the range of a pre... |
| pfxf1 32086 | Condition for a prefix to ... |
| s1f1 32087 | Conditions for a length 1 ... |
| s2rn 32088 | Range of a length 2 string... |
| s2f1 32089 | Conditions for a length 2 ... |
| s3rn 32090 | Range of a length 3 string... |
| s3f1 32091 | Conditions for a length 3 ... |
| s3clhash 32092 | Closure of the words of le... |
| ccatf1 32093 | Conditions for a concatena... |
| pfxlsw2ccat 32094 | Reconstruct a word from it... |
| wrdt2ind 32095 | Perform an induction over ... |
| swrdrn2 32096 | The range of a subword is ... |
| swrdrn3 32097 | Express the range of a sub... |
| swrdf1 32098 | Condition for a subword to... |
| swrdrndisj 32099 | Condition for the range of... |
| splfv3 32100 | Symbols to the right of a ... |
| 1cshid 32101 | Cyclically shifting a sing... |
| cshw1s2 32102 | Cyclically shifting a leng... |
| cshwrnid 32103 | Cyclically shifting a word... |
| cshf1o 32104 | Condition for the cyclic s... |
| ressplusf 32105 | The group operation functi... |
| ressnm 32106 | The norm in a restricted s... |
| abvpropd2 32107 | Weaker version of ~ abvpro... |
| oppgle 32108 | less-than relation of an o... |
| oppgleOLD 32109 | Obsolete version of ~ oppg... |
| oppglt 32110 | less-than relation of an o... |
| ressprs 32111 | The restriction of a prose... |
| oduprs 32112 | Being a proset is a self-d... |
| posrasymb 32113 | A poset ordering is asymet... |
| resspos 32114 | The restriction of a Poset... |
| resstos 32115 | The restriction of a Toset... |
| odutos 32116 | Being a toset is a self-du... |
| tlt2 32117 | In a Toset, two elements m... |
| tlt3 32118 | In a Toset, two elements m... |
| trleile 32119 | In a Toset, two elements m... |
| toslublem 32120 | Lemma for ~ toslub and ~ x... |
| toslub 32121 | In a toset, the lowest upp... |
| tosglblem 32122 | Lemma for ~ tosglb and ~ x... |
| tosglb 32123 | Same theorem as ~ toslub ,... |
| clatp0cl 32124 | The poset zero of a comple... |
| clatp1cl 32125 | The poset one of a complet... |
| mntoval 32130 | Operation value of the mon... |
| ismnt 32131 | Express the statement " ` ... |
| ismntd 32132 | Property of being a monoto... |
| mntf 32133 | A monotone function is a f... |
| mgcoval 32134 | Operation value of the mon... |
| mgcval 32135 | Monotone Galois connection... |
| mgcf1 32136 | The lower adjoint ` F ` of... |
| mgcf2 32137 | The upper adjoint ` G ` of... |
| mgccole1 32138 | An inequality for the kern... |
| mgccole2 32139 | Inequality for the closure... |
| mgcmnt1 32140 | The lower adjoint ` F ` of... |
| mgcmnt2 32141 | The upper adjoint ` G ` of... |
| mgcmntco 32142 | A Galois connection like s... |
| dfmgc2lem 32143 | Lemma for dfmgc2, backward... |
| dfmgc2 32144 | Alternate definition of th... |
| mgcmnt1d 32145 | Galois connection implies ... |
| mgcmnt2d 32146 | Galois connection implies ... |
| mgccnv 32147 | The inverse Galois connect... |
| pwrssmgc 32148 | Given a function ` F ` , e... |
| mgcf1olem1 32149 | Property of a Galois conne... |
| mgcf1olem2 32150 | Property of a Galois conne... |
| mgcf1o 32151 | Given a Galois connection,... |
| xrs0 32154 | The zero of the extended r... |
| xrslt 32155 | The "strictly less than" r... |
| xrsinvgval 32156 | The inversion operation in... |
| xrsmulgzz 32157 | The "multiple" function in... |
| xrstos 32158 | The extended real numbers ... |
| xrsclat 32159 | The extended real numbers ... |
| xrsp0 32160 | The poset 0 of the extende... |
| xrsp1 32161 | The poset 1 of the extende... |
| ressmulgnn 32162 | Values for the group multi... |
| ressmulgnn0 32163 | Values for the group multi... |
| xrge0base 32164 | The base of the extended n... |
| xrge00 32165 | The zero of the extended n... |
| xrge0plusg 32166 | The additive law of the ex... |
| xrge0le 32167 | The "less than or equal to... |
| xrge0mulgnn0 32168 | The group multiple functio... |
| xrge0addass 32169 | Associativity of extended ... |
| xrge0addgt0 32170 | The sum of nonnegative and... |
| xrge0adddir 32171 | Right-distributivity of ex... |
| xrge0adddi 32172 | Left-distributivity of ext... |
| xrge0npcan 32173 | Extended nonnegative real ... |
| fsumrp0cl 32174 | Closure of a finite sum of... |
| abliso 32175 | The image of an Abelian gr... |
| gsumsubg 32176 | The group sum in a subgrou... |
| gsumsra 32177 | The group sum in a subring... |
| gsummpt2co 32178 | Split a finite sum into a ... |
| gsummpt2d 32179 | Express a finite sum over ... |
| lmodvslmhm 32180 | Scalar multiplication in a... |
| gsumvsmul1 32181 | Pull a scalar multiplicati... |
| gsummptres 32182 | Extend a finite group sum ... |
| gsummptres2 32183 | Extend a finite group sum ... |
| gsumzresunsn 32184 | Append an element to a fin... |
| gsumpart 32185 | Express a group sum as a d... |
| gsumhashmul 32186 | Express a group sum by gro... |
| xrge0tsmsd 32187 | Any finite or infinite sum... |
| xrge0tsmsbi 32188 | Any limit of a finite or i... |
| xrge0tsmseq 32189 | Any limit of a finite or i... |
| cntzun 32190 | The centralizer of a union... |
| cntzsnid 32191 | The centralizer of the ide... |
| cntrcrng 32192 | The center of a ring is a ... |
| isomnd 32197 | A (left) ordered monoid is... |
| isogrp 32198 | A (left-)ordered group is ... |
| ogrpgrp 32199 | A left-ordered group is a ... |
| omndmnd 32200 | A left-ordered monoid is a... |
| omndtos 32201 | A left-ordered monoid is a... |
| omndadd 32202 | In an ordered monoid, the ... |
| omndaddr 32203 | In a right ordered monoid,... |
| omndadd2d 32204 | In a commutative left orde... |
| omndadd2rd 32205 | In a left- and right- orde... |
| submomnd 32206 | A submonoid of an ordered ... |
| xrge0omnd 32207 | The nonnegative extended r... |
| omndmul2 32208 | In an ordered monoid, the ... |
| omndmul3 32209 | In an ordered monoid, the ... |
| omndmul 32210 | In a commutative ordered m... |
| ogrpinv0le 32211 | In an ordered group, the o... |
| ogrpsub 32212 | In an ordered group, the o... |
| ogrpaddlt 32213 | In an ordered group, stric... |
| ogrpaddltbi 32214 | In a right ordered group, ... |
| ogrpaddltrd 32215 | In a right ordered group, ... |
| ogrpaddltrbid 32216 | In a right ordered group, ... |
| ogrpsublt 32217 | In an ordered group, stric... |
| ogrpinv0lt 32218 | In an ordered group, the o... |
| ogrpinvlt 32219 | In an ordered group, the o... |
| gsumle 32220 | A finite sum in an ordered... |
| symgfcoeu 32221 | Uniqueness property of per... |
| symgcom 32222 | Two permutations ` X ` and... |
| symgcom2 32223 | Two permutations ` X ` and... |
| symgcntz 32224 | All elements of a (finite)... |
| odpmco 32225 | The composition of two odd... |
| symgsubg 32226 | The value of the group sub... |
| pmtrprfv2 32227 | In a transposition of two ... |
| pmtrcnel 32228 | Composing a permutation ` ... |
| pmtrcnel2 32229 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 32230 | Composing a permutation ` ... |
| pmtridf1o 32231 | Transpositions of ` X ` an... |
| pmtridfv1 32232 | Value at X of the transpos... |
| pmtridfv2 32233 | Value at Y of the transpos... |
| psgnid 32234 | Permutation sign of the id... |
| psgndmfi 32235 | For a finite base set, the... |
| pmtrto1cl 32236 | Useful lemma for the follo... |
| psgnfzto1stlem 32237 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 32238 | Value of our permutation `... |
| fzto1st1 32239 | Special case where the per... |
| fzto1st 32240 | The function moving one el... |
| fzto1stinvn 32241 | Value of the inverse of ou... |
| psgnfzto1st 32242 | The permutation sign for m... |
| tocycval 32245 | Value of the cycle builder... |
| tocycfv 32246 | Function value of a permut... |
| tocycfvres1 32247 | A cyclic permutation is a ... |
| tocycfvres2 32248 | A cyclic permutation is th... |
| cycpmfvlem 32249 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 32250 | Value of a cycle function ... |
| cycpmfv2 32251 | Value of a cycle function ... |
| cycpmfv3 32252 | Values outside of the orbi... |
| cycpmcl 32253 | Cyclic permutations are pe... |
| tocycf 32254 | The permutation cycle buil... |
| tocyc01 32255 | Permutation cycles built f... |
| cycpm2tr 32256 | A cyclic permutation of 2 ... |
| cycpm2cl 32257 | Closure for the 2-cycles. ... |
| cyc2fv1 32258 | Function value of a 2-cycl... |
| cyc2fv2 32259 | Function value of a 2-cycl... |
| trsp2cyc 32260 | Exhibit the word a transpo... |
| cycpmco2f1 32261 | The word U used in ~ cycpm... |
| cycpmco2rn 32262 | The orbit of the compositi... |
| cycpmco2lem1 32263 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 32264 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 32265 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 32266 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 32267 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 32268 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 32269 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 32270 | The composition of a cycli... |
| cyc2fvx 32271 | Function value of a 2-cycl... |
| cycpm3cl 32272 | Closure of the 3-cycles in... |
| cycpm3cl2 32273 | Closure of the 3-cycles in... |
| cyc3fv1 32274 | Function value of a 3-cycl... |
| cyc3fv2 32275 | Function value of a 3-cycl... |
| cyc3fv3 32276 | Function value of a 3-cycl... |
| cyc3co2 32277 | Represent a 3-cycle as a c... |
| cycpmconjvlem 32278 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 32279 | A formula for computing co... |
| cycpmrn 32280 | The range of the word used... |
| tocyccntz 32281 | All elements of a (finite)... |
| evpmval 32282 | Value of the set of even p... |
| cnmsgn0g 32283 | The neutral element of the... |
| evpmsubg 32284 | The alternating group is a... |
| evpmid 32285 | The identity is an even pe... |
| altgnsg 32286 | The alternating group ` ( ... |
| cyc3evpm 32287 | 3-Cycles are even permutat... |
| cyc3genpmlem 32288 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 32289 | The alternating group ` A ... |
| cycpmgcl 32290 | Cyclic permutations are pe... |
| cycpmconjslem1 32291 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 32292 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 32293 | All cycles of the same len... |
| cyc3conja 32294 | All 3-cycles are conjugate... |
| sgnsv 32297 | The sign mapping. (Contri... |
| sgnsval 32298 | The sign value. (Contribu... |
| sgnsf 32299 | The sign function. (Contr... |
| inftmrel 32304 | The infinitesimal relation... |
| isinftm 32305 | Express ` x ` is infinites... |
| isarchi 32306 | Express the predicate " ` ... |
| pnfinf 32307 | Plus infinity is an infini... |
| xrnarchi 32308 | The completed real line is... |
| isarchi2 32309 | Alternative way to express... |
| submarchi 32310 | A submonoid is archimedean... |
| isarchi3 32311 | This is the usual definiti... |
| archirng 32312 | Property of Archimedean or... |
| archirngz 32313 | Property of Archimedean le... |
| archiexdiv 32314 | In an Archimedean group, g... |
| archiabllem1a 32315 | Lemma for ~ archiabl : In... |
| archiabllem1b 32316 | Lemma for ~ archiabl . (C... |
| archiabllem1 32317 | Archimedean ordered groups... |
| archiabllem2a 32318 | Lemma for ~ archiabl , whi... |
| archiabllem2c 32319 | Lemma for ~ archiabl . (C... |
| archiabllem2b 32320 | Lemma for ~ archiabl . (C... |
| archiabllem2 32321 | Archimedean ordered groups... |
| archiabl 32322 | Archimedean left- and righ... |
| isslmd 32325 | The predicate "is a semimo... |
| slmdlema 32326 | Lemma for properties of a ... |
| lmodslmd 32327 | Left semimodules generaliz... |
| slmdcmn 32328 | A semimodule is a commutat... |
| slmdmnd 32329 | A semimodule is a monoid. ... |
| slmdsrg 32330 | The scalar component of a ... |
| slmdbn0 32331 | The base set of a semimodu... |
| slmdacl 32332 | Closure of ring addition f... |
| slmdmcl 32333 | Closure of ring multiplica... |
| slmdsn0 32334 | The set of scalars in a se... |
| slmdvacl 32335 | Closure of vector addition... |
| slmdass 32336 | Semiring left module vecto... |
| slmdvscl 32337 | Closure of scalar product ... |
| slmdvsdi 32338 | Distributive law for scala... |
| slmdvsdir 32339 | Distributive law for scala... |
| slmdvsass 32340 | Associative law for scalar... |
| slmd0cl 32341 | The ring zero in a semimod... |
| slmd1cl 32342 | The ring unity in a semiri... |
| slmdvs1 32343 | Scalar product with ring u... |
| slmd0vcl 32344 | The zero vector is a vecto... |
| slmd0vlid 32345 | Left identity law for the ... |
| slmd0vrid 32346 | Right identity law for the... |
| slmd0vs 32347 | Zero times a vector is the... |
| slmdvs0 32348 | Anything times the zero ve... |
| gsumvsca1 32349 | Scalar product of a finite... |
| gsumvsca2 32350 | Scalar product of a finite... |
| prmsimpcyc 32351 | A group of prime order is ... |
| urpropd 32352 | Sufficient condition for r... |
| 0ringsubrg 32353 | A subring of a zero ring i... |
| rngurd 32354 | Deduce the unity element o... |
| dvdschrmulg 32355 | In a ring, any multiple of... |
| freshmansdream 32356 | For a prime number ` P ` ,... |
| frobrhm 32357 | In a commutative ring with... |
| ress1r 32358 | ` 1r ` is unaffected by re... |
| ringinvval 32359 | The ring inverse expressed... |
| dvrcan5 32360 | Cancellation law for commo... |
| subrgchr 32361 | If ` A ` is a subring of `... |
| rmfsupp2 32362 | A mapping of a multiplicat... |
| ringinveu 32363 | If a ring unit element ` X... |
| isdrng4 32364 | A division ring is a ring ... |
| rndrhmcl 32365 | The range of a ring homomo... |
| sdrgdvcl 32366 | A sub-division-ring is clo... |
| sdrginvcl 32367 | A sub-division-ring is clo... |
| primefldchr 32368 | The characteristic of a pr... |
| fldgenval 32371 | Value of the field generat... |
| fldgenssid 32372 | The field generated by a s... |
| fldgensdrg 32373 | A generated subfield is a ... |
| fldgenssv 32374 | A generated subfield is a ... |
| fldgenss 32375 | Generated subfields preser... |
| fldgenidfld 32376 | The subfield generated by ... |
| fldgenssp 32377 | The field generated by a s... |
| fldgenid 32378 | The subfield of a field ` ... |
| fldgenfld 32379 | A generated subfield is a ... |
| primefldgen1 32380 | The prime field of a divis... |
| 1fldgenq 32381 | The field of rational numb... |
| isorng 32386 | An ordered ring is a ring ... |
| orngring 32387 | An ordered ring is a ring.... |
| orngogrp 32388 | An ordered ring is an orde... |
| isofld 32389 | An ordered field is a fiel... |
| orngmul 32390 | In an ordered ring, the or... |
| orngsqr 32391 | In an ordered ring, all sq... |
| ornglmulle 32392 | In an ordered ring, multip... |
| orngrmulle 32393 | In an ordered ring, multip... |
| ornglmullt 32394 | In an ordered ring, multip... |
| orngrmullt 32395 | In an ordered ring, multip... |
| orngmullt 32396 | In an ordered ring, the st... |
| ofldfld 32397 | An ordered field is a fiel... |
| ofldtos 32398 | An ordered field is a tota... |
| orng0le1 32399 | In an ordered ring, the ri... |
| ofldlt1 32400 | In an ordered field, the r... |
| ofldchr 32401 | The characteristic of an o... |
| suborng 32402 | Every subring of an ordere... |
| subofld 32403 | Every subfield of an order... |
| isarchiofld 32404 | Axiom of Archimedes : a ch... |
| rhmdvd 32405 | A ring homomorphism preser... |
| kerunit 32406 | If a unit element lies in ... |
| reldmresv 32409 | The scalar restriction is ... |
| resvval 32410 | Value of structure restric... |
| resvid2 32411 | General behavior of trivia... |
| resvval2 32412 | Value of nontrivial struct... |
| resvsca 32413 | Base set of a structure re... |
| resvlem 32414 | Other elements of a scalar... |
| resvlemOLD 32415 | Obsolete version of ~ resv... |
| resvbas 32416 | ` Base ` is unaffected by ... |
| resvbasOLD 32417 | Obsolete proof of ~ resvba... |
| resvplusg 32418 | ` +g ` is unaffected by sc... |
| resvplusgOLD 32419 | Obsolete proof of ~ resvpl... |
| resvvsca 32420 | ` .s ` is unaffected by sc... |
| resvvscaOLD 32421 | Obsolete proof of ~ resvvs... |
| resvmulr 32422 | ` .r ` is unaffected by sc... |
| resvmulrOLD 32423 | Obsolete proof of ~ resvmu... |
| resv0g 32424 | ` 0g ` is unaffected by sc... |
| resv1r 32425 | ` 1r ` is unaffected by sc... |
| resvcmn 32426 | Scalar restriction preserv... |
| gzcrng 32427 | The gaussian integers form... |
| reofld 32428 | The real numbers form an o... |
| nn0omnd 32429 | The nonnegative integers f... |
| rearchi 32430 | The field of the real numb... |
| nn0archi 32431 | The monoid of the nonnegat... |
| xrge0slmod 32432 | The extended nonnegative r... |
| qusker 32433 | The kernel of a quotient m... |
| eqgvscpbl 32434 | The left coset equivalence... |
| qusvscpbl 32435 | The quotient map distribut... |
| qusvsval 32436 | Value of the scalar multip... |
| imaslmod 32437 | The image structure of a l... |
| quslmod 32438 | If ` G ` is a submodule in... |
| quslmhm 32439 | If ` G ` is a submodule of... |
| quslvec 32440 | If ` S ` is a vector subsp... |
| ecxpid 32441 | The equivalence class of a... |
| eqg0el 32442 | Equivalence class of a quo... |
| qsxpid 32443 | The quotient set of a cart... |
| qusxpid 32444 | The Group quotient equival... |
| qustriv 32445 | The quotient of a group ` ... |
| qustrivr 32446 | Converse of ~ qustriv . (... |
| fermltlchr 32447 | A generalization of Fermat... |
| znfermltl 32448 | Fermat's little theorem in... |
| islinds5 32449 | A set is linearly independ... |
| ellspds 32450 | Variation on ~ ellspd . (... |
| 0ellsp 32451 | Zero is in all spans. (Co... |
| 0nellinds 32452 | The group identity cannot ... |
| rspsnel 32453 | Membership in a principal ... |
| rspsnid 32454 | A principal ideal contains... |
| elrsp 32455 | Write the elements of a ri... |
| rspidlid 32456 | The ideal span of an ideal... |
| pidlnz 32457 | A principal ideal generate... |
| lbslsp 32458 | Any element of a left modu... |
| lindssn 32459 | Any singleton of a nonzero... |
| lindflbs 32460 | Conditions for an independ... |
| islbs5 32461 | An equivalent formulation ... |
| linds2eq 32462 | Deduce equality of element... |
| lindfpropd 32463 | Property deduction for lin... |
| lindspropd 32464 | Property deduction for lin... |
| elgrplsmsn 32465 | Membership in a sumset wit... |
| lsmsnorb 32466 | The sumset of a group with... |
| lsmsnorb2 32467 | The sumset of a single ele... |
| elringlsm 32468 | Membership in a product of... |
| elringlsmd 32469 | Membership in a product of... |
| ringlsmss 32470 | Closure of the product of ... |
| ringlsmss1 32471 | The product of an ideal ` ... |
| ringlsmss2 32472 | The product with an ideal ... |
| lsmsnpridl 32473 | The product of the ring wi... |
| lsmsnidl 32474 | The product of the ring wi... |
| lsmidllsp 32475 | The sum of two ideals is t... |
| lsmidl 32476 | The sum of two ideals is a... |
| lsmssass 32477 | Group sum is associative, ... |
| grplsm0l 32478 | Sumset with the identity s... |
| grplsmid 32479 | The direct sum of an eleme... |
| qusmul 32480 | Value of the ring operatio... |
| quslsm 32481 | Express the image by the q... |
| qus0g 32482 | The identity element of a ... |
| qusima 32483 | The image of a subgroup by... |
| nsgqus0 32484 | A normal subgroup ` N ` is... |
| nsgmgclem 32485 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 32486 | There is a monotone Galois... |
| nsgqusf1olem1 32487 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 32488 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 32489 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 32490 | The canonical projection h... |
| ghmquskerlem1 32491 | Lemma for ~ ghmqusker (Con... |
| ghmquskerco 32492 | In the case of theorem ~ g... |
| ghmquskerlem2 32493 | Lemma for ~ ghmqusker . (... |
| ghmqusker 32494 | A surjective group homomor... |
| gicqusker 32495 | The image ` H ` of a group... |
| lmhmqusker 32496 | A surjective module homomo... |
| lmicqusker 32497 | The image ` H ` of a modul... |
| intlidl 32498 | The intersection of a none... |
| rhmpreimaidl 32499 | The preimage of an ideal b... |
| kerlidl 32500 | The kernel of a ring homom... |
| 0ringidl 32501 | The zero ideal is the only... |
| rhmqusker 32502 | A surjective ring homomorp... |
| ricqusker 32503 | The image ` H ` of a ring ... |
| elrspunidl 32504 | Elementhood in the span of... |
| elrspunsn 32505 | Membership to the span of ... |
| lidlincl 32506 | Ideals are closed under in... |
| idlinsubrg 32507 | The intersection between a... |
| rhmimaidl 32508 | The image of an ideal ` I ... |
| drngidl 32509 | A ring is a division ring ... |
| drngidlhash 32510 | A ring is a division ring ... |
| prmidlval 32513 | The class of prime ideals ... |
| isprmidl 32514 | The predicate "is a prime ... |
| prmidlnr 32515 | A prime ideal is a proper ... |
| prmidl 32516 | The main property of a pri... |
| prmidl2 32517 | A condition that shows an ... |
| idlmulssprm 32518 | Let ` P ` be a prime ideal... |
| pridln1 32519 | A proper ideal cannot cont... |
| prmidlidl 32520 | A prime ideal is an ideal.... |
| prmidlssidl 32521 | Prime ideals as a subset o... |
| lidlnsg 32522 | An ideal is a normal subgr... |
| cringm4 32523 | Commutative/associative la... |
| isprmidlc 32524 | The predicate "is prime id... |
| prmidlc 32525 | Property of a prime ideal ... |
| 0ringprmidl 32526 | The trivial ring does not ... |
| prmidl0 32527 | The zero ideal of a commut... |
| rhmpreimaprmidl 32528 | The preimage of a prime id... |
| qsidomlem1 32529 | If the quotient ring of a ... |
| qsidomlem2 32530 | A quotient by a prime idea... |
| qsidom 32531 | An ideal ` I ` in the comm... |
| qsnzr 32532 | A quotient of a non-zero r... |
| mxidlval 32535 | The set of maximal ideals ... |
| ismxidl 32536 | The predicate "is a maxima... |
| mxidlidl 32537 | A maximal ideal is an idea... |
| mxidlnr 32538 | A maximal ideal is proper.... |
| mxidlmax 32539 | A maximal ideal is a maxim... |
| mxidln1 32540 | One is not contained in an... |
| mxidlnzr 32541 | A ring with a maximal idea... |
| mxidlmaxv 32542 | An ideal ` I ` strictly co... |
| crngmxidl 32543 | In a commutative ring, max... |
| mxidlprm 32544 | Every maximal ideal is pri... |
| ssmxidllem 32545 | The set ` P ` used in the ... |
| ssmxidl 32546 | Let ` R ` be a ring, and l... |
| krull 32547 | Krull's theorem: Any nonz... |
| mxidlnzrb 32548 | A ring is nonzero if and o... |
| opprabs 32549 | The opposite ring of the o... |
| oppreqg 32550 | Group coset equivalence re... |
| opprnsg 32551 | Normal subgroups of the op... |
| opprlidlabs 32552 | The ideals of the opposite... |
| oppr2idl 32553 | Two sided ideal of the opp... |
| opprmxidlabs 32554 | The maximal ideal of the o... |
| opprqusbas 32555 | The base of the quotient o... |
| opprqusplusg 32556 | The group operation of the... |
| opprqus0g 32557 | The group identity element... |
| opprqusmulr 32558 | The multiplication operati... |
| opprqus1r 32559 | The ring unity of the quot... |
| opprqusdrng 32560 | The quotient of the opposi... |
| qsdrngilem 32561 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 32562 | A quotient by a maximal le... |
| qsdrnglem2 32563 | Lemma for ~ qsdrng . (Con... |
| qsdrng 32564 | An ideal ` M ` is both lef... |
| qsfld 32565 | An ideal ` M ` in the comm... |
| idlsrgstr 32568 | A constructed semiring of ... |
| idlsrgval 32569 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 32570 | Base of the ideals of a ri... |
| idlsrgplusg 32571 | Additive operation of the ... |
| idlsrg0g 32572 | The zero ideal is the addi... |
| idlsrgmulr 32573 | Multiplicative operation o... |
| idlsrgtset 32574 | Topology component of the ... |
| idlsrgmulrval 32575 | Value of the ring multipli... |
| idlsrgmulrcl 32576 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 32577 | In a commutative ring, the... |
| idlsrgmulrss2 32578 | The product of two ideals ... |
| idlsrgmulrssin 32579 | In a commutative ring, the... |
| idlsrgmnd 32580 | The ideals of a ring form ... |
| idlsrgcmnd 32581 | The ideals of a ring form ... |
| isufd 32584 | The property of being a Un... |
| rprmval 32585 | The prime elements of a ri... |
| isrprm 32586 | Property for ` P ` to be a... |
| asclmulg 32587 | Apply group multiplication... |
| 0ringmon1p 32588 | There are no monic polynom... |
| fply1 32589 | Conditions for a function ... |
| ply1lvec 32590 | In a division ring, the un... |
| ply1scleq 32591 | Equality of a constant pol... |
| evls1fn 32592 | Functionality of the subri... |
| evls1scafv 32593 | Value of the univariate po... |
| evls1expd 32594 | Univariate polynomial eval... |
| evls1varpwval 32595 | Univariate polynomial eval... |
| evls1fpws 32596 | Evaluation of a univariate... |
| ressply1evl 32597 | Evaluation of a univariate... |
| evls1addd 32598 | Univariate polynomial eval... |
| evls1muld 32599 | Univariate polynomial eval... |
| evls1vsca 32600 | Univariate polynomial eval... |
| ressdeg1 32601 | The degree of a univariate... |
| ply1ascl0 32602 | The zero scalar as a polyn... |
| ressply10g 32603 | A restricted polynomial al... |
| ressply1mon1p 32604 | The monic polynomials of a... |
| ressply1invg 32605 | An element of a restricted... |
| ressply1sub 32606 | A restricted polynomial al... |
| asclply1subcl 32607 | Closure of the algebra sca... |
| ply1chr 32608 | The characteristic of a po... |
| ply1fermltlchr 32609 | Fermat's little theorem fo... |
| ply1fermltl 32610 | Fermat's little theorem fo... |
| coe1mon 32611 | Coefficient vector of a mo... |
| ply1moneq 32612 | Two monomials are equal if... |
| ply1degltel 32613 | Characterize elementhood t... |
| ply1degltlss 32614 | The space ` S ` of the uni... |
| gsummoncoe1fzo 32615 | A coefficient of the polyn... |
| ply1gsumz 32616 | If a polynomial given as a... |
| sra1r 32617 | The unity element of a sub... |
| sraring 32618 | Condition for a subring al... |
| sradrng 32619 | Condition for a subring al... |
| srasubrg 32620 | A subring of the original ... |
| sralvec 32621 | Given a sub division ring ... |
| srafldlvec 32622 | Given a subfield ` F ` of ... |
| drgext0g 32623 | The additive neutral eleme... |
| drgextvsca 32624 | The scalar multiplication ... |
| drgext0gsca 32625 | The additive neutral eleme... |
| drgextsubrg 32626 | The scalar field is a subr... |
| drgextlsp 32627 | The scalar field is a subs... |
| drgextgsum 32628 | Group sum in a division ri... |
| lvecdimfi 32629 | Finite version of ~ lvecdi... |
| dimval 32632 | The dimension of a vector ... |
| dimvalfi 32633 | The dimension of a vector ... |
| dimcl 32634 | Closure of the vector spac... |
| lmimdim 32635 | Module isomorphisms preser... |
| lvecdim0i 32636 | A vector space of dimensio... |
| lvecdim0 32637 | A vector space of dimensio... |
| lssdimle 32638 | The dimension of a linear ... |
| dimpropd 32639 | If two structures have the... |
| rgmoddim 32640 | The left vector space indu... |
| frlmdim 32641 | Dimension of a free left m... |
| tnglvec 32642 | Augmenting a structure wit... |
| tngdim 32643 | Dimension of a left vector... |
| rrxdim 32644 | Dimension of the generaliz... |
| matdim 32645 | Dimension of the space of ... |
| lbslsat 32646 | A nonzero vector ` X ` is ... |
| lsatdim 32647 | A line, spanned by a nonze... |
| drngdimgt0 32648 | The dimension of a vector ... |
| lmhmlvec2 32649 | A homomorphism of left vec... |
| kerlmhm 32650 | The kernel of a vector spa... |
| imlmhm 32651 | The image of a vector spac... |
| ply1degltdimlem 32652 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 32653 | The space ` S ` of the uni... |
| lindsunlem 32654 | Lemma for ~ lindsun . (Co... |
| lindsun 32655 | Condition for the union of... |
| lbsdiflsp0 32656 | The linear spans of two di... |
| dimkerim 32657 | Given a linear map ` F ` b... |
| qusdimsum 32658 | Let ` W ` be a vector spac... |
| fedgmullem1 32659 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 32660 | Lemma for ~ fedgmul . (Co... |
| fedgmul 32661 | The multiplicativity formu... |
| relfldext 32670 | The field extension is a r... |
| brfldext 32671 | The field extension relati... |
| ccfldextrr 32672 | The field of the complex n... |
| fldextfld1 32673 | A field extension is only ... |
| fldextfld2 32674 | A field extension is only ... |
| fldextsubrg 32675 | Field extension implies a ... |
| fldextress 32676 | Field extension implies a ... |
| brfinext 32677 | The finite field extension... |
| extdgval 32678 | Value of the field extensi... |
| fldextsralvec 32679 | The subring algebra associ... |
| extdgcl 32680 | Closure of the field exten... |
| extdggt0 32681 | Degrees of field extension... |
| fldexttr 32682 | Field extension is a trans... |
| fldextid 32683 | The field extension relati... |
| extdgid 32684 | A trivial field extension ... |
| extdgmul 32685 | The multiplicativity formu... |
| finexttrb 32686 | The extension ` E ` of ` K... |
| extdg1id 32687 | If the degree of the exten... |
| extdg1b 32688 | The degree of the extensio... |
| fldextchr 32689 | The characteristic of a su... |
| ccfldsrarelvec 32690 | The subring algebra of the... |
| ccfldextdgrr 32691 | The degree of the field ex... |
| irngval 32694 | The elements of a field ` ... |
| elirng 32695 | Property for an element ` ... |
| irngss 32696 | All elements of a subring ... |
| irngssv 32697 | An integral element is an ... |
| 0ringirng 32698 | A zero ring ` R ` has no i... |
| irngnzply1lem 32699 | In the case of a field ` E... |
| irngnzply1 32700 | In the case of a field ` E... |
| evls1maprhm 32703 | The function ` F ` mapping... |
| evls1maplmhm 32704 | The function ` F ` mapping... |
| evls1maprnss 32705 | The function ` F ` mapping... |
| ply1annidllem 32706 | Write the set ` Q ` of pol... |
| ply1annidl 32707 | The set ` Q ` of polynomia... |
| ply1annig1p 32708 | The ideal ` Q ` of polynom... |
| minplyval 32709 | Expand the value of the mi... |
| ply1annprmidl 32710 | The set ` Q ` of polynomia... |
| smatfval 32713 | Value of the submatrix. (... |
| smatrcl 32714 | Closure of the rectangular... |
| smatlem 32715 | Lemma for the next theorem... |
| smattl 32716 | Entries of a submatrix, to... |
| smattr 32717 | Entries of a submatrix, to... |
| smatbl 32718 | Entries of a submatrix, bo... |
| smatbr 32719 | Entries of a submatrix, bo... |
| smatcl 32720 | Closure of the square subm... |
| matmpo 32721 | Write a square matrix as a... |
| 1smat1 32722 | The submatrix of the ident... |
| submat1n 32723 | One case where the submatr... |
| submatres 32724 | Special case where the sub... |
| submateqlem1 32725 | Lemma for ~ submateq . (C... |
| submateqlem2 32726 | Lemma for ~ submateq . (C... |
| submateq 32727 | Sufficient condition for t... |
| submatminr1 32728 | If we take a submatrix by ... |
| lmatval 32731 | Value of the literal matri... |
| lmatfval 32732 | Entries of a literal matri... |
| lmatfvlem 32733 | Useful lemma to extract li... |
| lmatcl 32734 | Closure of the literal mat... |
| lmat22lem 32735 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 32736 | Entry of a 2x2 literal mat... |
| lmat22e12 32737 | Entry of a 2x2 literal mat... |
| lmat22e21 32738 | Entry of a 2x2 literal mat... |
| lmat22e22 32739 | Entry of a 2x2 literal mat... |
| lmat22det 32740 | The determinant of a liter... |
| mdetpmtr1 32741 | The determinant of a matri... |
| mdetpmtr2 32742 | The determinant of a matri... |
| mdetpmtr12 32743 | The determinant of a matri... |
| mdetlap1 32744 | A Laplace expansion of the... |
| madjusmdetlem1 32745 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 32746 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 32747 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 32748 | Lemma for ~ madjusmdet . ... |
| madjusmdet 32749 | Express the cofactor of th... |
| mdetlap 32750 | Laplace expansion of the d... |
| ist0cld 32751 | The predicate "is a T_0 sp... |
| txomap 32752 | Given two open maps ` F ` ... |
| qtopt1 32753 | If every equivalence class... |
| qtophaus 32754 | If an open map's graph in ... |
| circtopn 32755 | The topology of the unit c... |
| circcn 32756 | The function gluing the re... |
| reff 32757 | For any cover refinement, ... |
| locfinreflem 32758 | A locally finite refinemen... |
| locfinref 32759 | A locally finite refinemen... |
| iscref 32762 | The property that every op... |
| crefeq 32763 | Equality theorem for the "... |
| creftop 32764 | A space where every open c... |
| crefi 32765 | The property that every op... |
| crefdf 32766 | A formulation of ~ crefi e... |
| crefss 32767 | The "every open cover has ... |
| cmpcref 32768 | Equivalent definition of c... |
| cmpfiref 32769 | Every open cover of a Comp... |
| ldlfcntref 32772 | Every open cover of a Lind... |
| ispcmp 32775 | The predicate "is a paraco... |
| cmppcmp 32776 | Every compact space is par... |
| dispcmp 32777 | Every discrete space is pa... |
| pcmplfin 32778 | Given a paracompact topolo... |
| pcmplfinf 32779 | Given a paracompact topolo... |
| rspecval 32782 | Value of the spectrum of t... |
| rspecbas 32783 | The prime ideals form the ... |
| rspectset 32784 | Topology component of the ... |
| rspectopn 32785 | The topology component of ... |
| zarcls0 32786 | The closure of the identit... |
| zarcls1 32787 | The unit ideal ` B ` is th... |
| zarclsun 32788 | The union of two closed se... |
| zarclsiin 32789 | In a Zariski topology, the... |
| zarclsint 32790 | The intersection of a fami... |
| zarclssn 32791 | The closed points of Zaris... |
| zarcls 32792 | The open sets of the Zaris... |
| zartopn 32793 | The Zariski topology is a ... |
| zartop 32794 | The Zariski topology is a ... |
| zartopon 32795 | The points of the Zariski ... |
| zar0ring 32796 | The Zariski Topology of th... |
| zart0 32797 | The Zariski topology is T_... |
| zarmxt1 32798 | The Zariski topology restr... |
| zarcmplem 32799 | Lemma for ~ zarcmp . (Con... |
| zarcmp 32800 | The Zariski topology is co... |
| rspectps 32801 | The spectrum of a ring ` R... |
| rhmpreimacnlem 32802 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 32803 | The function mapping a pri... |
| metidval 32808 | Value of the metric identi... |
| metidss 32809 | As a relation, the metric ... |
| metidv 32810 | ` A ` and ` B ` identify b... |
| metideq 32811 | Basic property of the metr... |
| metider 32812 | The metric identification ... |
| pstmval 32813 | Value of the metric induce... |
| pstmfval 32814 | Function value of the metr... |
| pstmxmet 32815 | The metric induced by a ps... |
| hauseqcn 32816 | In a Hausdorff topology, t... |
| elunitge0 32817 | An element of the closed u... |
| unitssxrge0 32818 | The closed unit interval i... |
| unitdivcld 32819 | Necessary conditions for a... |
| iistmd 32820 | The closed unit interval f... |
| unicls 32821 | The union of the closed se... |
| tpr2tp 32822 | The usual topology on ` ( ... |
| tpr2uni 32823 | The usual topology on ` ( ... |
| xpinpreima 32824 | Rewrite the cartesian prod... |
| xpinpreima2 32825 | Rewrite the cartesian prod... |
| sqsscirc1 32826 | The complex square of side... |
| sqsscirc2 32827 | The complex square of side... |
| cnre2csqlem 32828 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 32829 | Image of a centered square... |
| tpr2rico 32830 | For any point of an open s... |
| cnvordtrestixx 32831 | The restriction of the 'gr... |
| prsdm 32832 | Domain of the relation of ... |
| prsrn 32833 | Range of the relation of a... |
| prsss 32834 | Relation of a subproset. ... |
| prsssdm 32835 | Domain of a subproset rela... |
| ordtprsval 32836 | Value of the order topolog... |
| ordtprsuni 32837 | Value of the order topolog... |
| ordtcnvNEW 32838 | The order dual generates t... |
| ordtrestNEW 32839 | The subspace topology of a... |
| ordtrest2NEWlem 32840 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 32841 | An interval-closed set ` A... |
| ordtconnlem1 32842 | Connectedness in the order... |
| ordtconn 32843 | Connectedness in the order... |
| mndpluscn 32844 | A mapping that is both a h... |
| mhmhmeotmd 32845 | Deduce a Topological Monoi... |
| rmulccn 32846 | Multiplication by a real c... |
| raddcn 32847 | Addition in the real numbe... |
| xrmulc1cn 32848 | The operation multiplying ... |
| fmcncfil 32849 | The image of a Cauchy filt... |
| xrge0hmph 32850 | The extended nonnegative r... |
| xrge0iifcnv 32851 | Define a bijection from ` ... |
| xrge0iifcv 32852 | The defined function's val... |
| xrge0iifiso 32853 | The defined bijection from... |
| xrge0iifhmeo 32854 | Expose a homeomorphism fro... |
| xrge0iifhom 32855 | The defined function from ... |
| xrge0iif1 32856 | Condition for the defined ... |
| xrge0iifmhm 32857 | The defined function from ... |
| xrge0pluscn 32858 | The addition operation of ... |
| xrge0mulc1cn 32859 | The operation multiplying ... |
| xrge0tps 32860 | The extended nonnegative r... |
| xrge0topn 32861 | The topology of the extend... |
| xrge0haus 32862 | The topology of the extend... |
| xrge0tmd 32863 | The extended nonnegative r... |
| xrge0tmdALT 32864 | Alternate proof of ~ xrge0... |
| lmlim 32865 | Relate a limit in a given ... |
| lmlimxrge0 32866 | Relate a limit in the nonn... |
| rge0scvg 32867 | Implication of convergence... |
| fsumcvg4 32868 | A serie with finite suppor... |
| pnfneige0 32869 | A neighborhood of ` +oo ` ... |
| lmxrge0 32870 | Express "sequence ` F ` co... |
| lmdvg 32871 | If a monotonic sequence of... |
| lmdvglim 32872 | If a monotonic real number... |
| pl1cn 32873 | A univariate polynomial is... |
| zringnm 32876 | The norm (function) for a ... |
| zzsnm 32877 | The norm of the ring of th... |
| zlm0 32878 | Zero of a ` ZZ ` -module. ... |
| zlm1 32879 | Unity element of a ` ZZ ` ... |
| zlmds 32880 | Distance in a ` ZZ ` -modu... |
| zlmdsOLD 32881 | Obsolete proof of ~ zlmds ... |
| zlmtset 32882 | Topology in a ` ZZ ` -modu... |
| zlmtsetOLD 32883 | Obsolete proof of ~ zlmtse... |
| zlmnm 32884 | Norm of a ` ZZ ` -module (... |
| zhmnrg 32885 | The ` ZZ ` -module built f... |
| nmmulg 32886 | The norm of a group produc... |
| zrhnm 32887 | The norm of the image by `... |
| cnzh 32888 | The ` ZZ ` -module of ` CC... |
| rezh 32889 | The ` ZZ ` -module of ` RR... |
| qqhval 32892 | Value of the canonical hom... |
| zrhf1ker 32893 | The kernel of the homomorp... |
| zrhchr 32894 | The kernel of the homomorp... |
| zrhker 32895 | The kernel of the homomorp... |
| zrhunitpreima 32896 | The preimage by ` ZRHom ` ... |
| elzrhunit 32897 | Condition for the image by... |
| elzdif0 32898 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 32899 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 32900 | Value of the canonical hom... |
| qqhvval 32901 | Value of the canonical hom... |
| qqh0 32902 | The image of ` 0 ` by the ... |
| qqh1 32903 | The image of ` 1 ` by the ... |
| qqhf 32904 | ` QQHom ` as a function. ... |
| qqhvq 32905 | The image of a quotient by... |
| qqhghm 32906 | The ` QQHom ` homomorphism... |
| qqhrhm 32907 | The ` QQHom ` homomorphism... |
| qqhnm 32908 | The norm of the image by `... |
| qqhcn 32909 | The ` QQHom ` homomorphism... |
| qqhucn 32910 | The ` QQHom ` homomorphism... |
| rrhval 32914 | Value of the canonical hom... |
| rrhcn 32915 | If the topology of ` R ` i... |
| rrhf 32916 | If the topology of ` R ` i... |
| isrrext 32918 | Express the property " ` R... |
| rrextnrg 32919 | An extension of ` RR ` is ... |
| rrextdrg 32920 | An extension of ` RR ` is ... |
| rrextnlm 32921 | The norm of an extension o... |
| rrextchr 32922 | The ring characteristic of... |
| rrextcusp 32923 | An extension of ` RR ` is ... |
| rrexttps 32924 | An extension of ` RR ` is ... |
| rrexthaus 32925 | The topology of an extensi... |
| rrextust 32926 | The uniformity of an exten... |
| rerrext 32927 | The field of the real numb... |
| cnrrext 32928 | The field of the complex n... |
| qqtopn 32929 | The topology of the field ... |
| rrhfe 32930 | If ` R ` is an extension o... |
| rrhcne 32931 | If ` R ` is an extension o... |
| rrhqima 32932 | The ` RRHom ` homomorphism... |
| rrh0 32933 | The image of ` 0 ` by the ... |
| xrhval 32936 | The value of the embedding... |
| zrhre 32937 | The ` ZRHom ` homomorphism... |
| qqhre 32938 | The ` QQHom ` homomorphism... |
| rrhre 32939 | The ` RRHom ` homomorphism... |
| relmntop 32942 | Manifold is a relation. (... |
| ismntoplly 32943 | Property of being a manifo... |
| ismntop 32944 | Property of being a manifo... |
| nexple 32945 | A lower bound for an expon... |
| indv 32948 | Value of the indicator fun... |
| indval 32949 | Value of the indicator fun... |
| indval2 32950 | Alternate value of the ind... |
| indf 32951 | An indicator function as a... |
| indfval 32952 | Value of the indicator fun... |
| ind1 32953 | Value of the indicator fun... |
| ind0 32954 | Value of the indicator fun... |
| ind1a 32955 | Value of the indicator fun... |
| indpi1 32956 | Preimage of the singleton ... |
| indsum 32957 | Finite sum of a product wi... |
| indsumin 32958 | Finite sum of a product wi... |
| prodindf 32959 | The product of indicators ... |
| indf1o 32960 | The bijection between a po... |
| indpreima 32961 | A function with range ` { ... |
| indf1ofs 32962 | The bijection between fini... |
| esumex 32965 | An extended sum is a set b... |
| esumcl 32966 | Closure for extended sum i... |
| esumeq12dvaf 32967 | Equality deduction for ext... |
| esumeq12dva 32968 | Equality deduction for ext... |
| esumeq12d 32969 | Equality deduction for ext... |
| esumeq1 32970 | Equality theorem for an ex... |
| esumeq1d 32971 | Equality theorem for an ex... |
| esumeq2 32972 | Equality theorem for exten... |
| esumeq2d 32973 | Equality deduction for ext... |
| esumeq2dv 32974 | Equality deduction for ext... |
| esumeq2sdv 32975 | Equality deduction for ext... |
| nfesum1 32976 | Bound-variable hypothesis ... |
| nfesum2 32977 | Bound-variable hypothesis ... |
| cbvesum 32978 | Change bound variable in a... |
| cbvesumv 32979 | Change bound variable in a... |
| esumid 32980 | Identify the extended sum ... |
| esumgsum 32981 | A finite extended sum is t... |
| esumval 32982 | Develop the value of the e... |
| esumel 32983 | The extended sum is a limi... |
| esumnul 32984 | Extended sum over the empt... |
| esum0 32985 | Extended sum of zero. (Co... |
| esumf1o 32986 | Re-index an extended sum u... |
| esumc 32987 | Convert from the collectio... |
| esumrnmpt 32988 | Rewrite an extended sum in... |
| esumsplit 32989 | Split an extended sum into... |
| esummono 32990 | Extended sum is monotonic.... |
| esumpad 32991 | Extend an extended sum by ... |
| esumpad2 32992 | Remove zeroes from an exte... |
| esumadd 32993 | Addition of infinite sums.... |
| esumle 32994 | If all of the terms of an ... |
| gsumesum 32995 | Relate a group sum on ` ( ... |
| esumlub 32996 | The extended sum is the lo... |
| esumaddf 32997 | Addition of infinite sums.... |
| esumlef 32998 | If all of the terms of an ... |
| esumcst 32999 | The extended sum of a cons... |
| esumsnf 33000 | The extended sum of a sing... |
| esumsn 33001 | The extended sum of a sing... |
| esumpr 33002 | Extended sum over a pair. ... |
| esumpr2 33003 | Extended sum over a pair, ... |
| esumrnmpt2 33004 | Rewrite an extended sum in... |
| esumfzf 33005 | Formulating a partial exte... |
| esumfsup 33006 | Formulating an extended su... |
| esumfsupre 33007 | Formulating an extended su... |
| esumss 33008 | Change the index set to a ... |
| esumpinfval 33009 | The value of the extended ... |
| esumpfinvallem 33010 | Lemma for ~ esumpfinval . ... |
| esumpfinval 33011 | The value of the extended ... |
| esumpfinvalf 33012 | Same as ~ esumpfinval , mi... |
| esumpinfsum 33013 | The value of the extended ... |
| esumpcvgval 33014 | The value of the extended ... |
| esumpmono 33015 | The partial sums in an ext... |
| esumcocn 33016 | Lemma for ~ esummulc2 and ... |
| esummulc1 33017 | An extended sum multiplied... |
| esummulc2 33018 | An extended sum multiplied... |
| esumdivc 33019 | An extended sum divided by... |
| hashf2 33020 | Lemma for ~ hasheuni . (C... |
| hasheuni 33021 | The cardinality of a disjo... |
| esumcvg 33022 | The sequence of partial su... |
| esumcvg2 33023 | Simpler version of ~ esumc... |
| esumcvgsum 33024 | The value of the extended ... |
| esumsup 33025 | Express an extended sum as... |
| esumgect 33026 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 33027 | All terms of a converging ... |
| esum2dlem 33028 | Lemma for ~ esum2d (finite... |
| esum2d 33029 | Write a double extended su... |
| esumiun 33030 | Sum over a nonnecessarily ... |
| ofceq 33033 | Equality theorem for funct... |
| ofcfval 33034 | Value of an operation appl... |
| ofcval 33035 | Evaluate a function/consta... |
| ofcfn 33036 | The function operation pro... |
| ofcfeqd2 33037 | Equality theorem for funct... |
| ofcfval3 33038 | General value of ` ( F oFC... |
| ofcf 33039 | The function/constant oper... |
| ofcfval2 33040 | The function operation exp... |
| ofcfval4 33041 | The function/constant oper... |
| ofcc 33042 | Left operation by a consta... |
| ofcof 33043 | Relate function operation ... |
| sigaex 33046 | Lemma for ~ issiga and ~ i... |
| sigaval 33047 | The set of sigma-algebra w... |
| issiga 33048 | An alternative definition ... |
| isrnsiga 33049 | The property of being a si... |
| 0elsiga 33050 | A sigma-algebra contains t... |
| baselsiga 33051 | A sigma-algebra contains i... |
| sigasspw 33052 | A sigma-algebra is a set o... |
| sigaclcu 33053 | A sigma-algebra is closed ... |
| sigaclcuni 33054 | A sigma-algebra is closed ... |
| sigaclfu 33055 | A sigma-algebra is closed ... |
| sigaclcu2 33056 | A sigma-algebra is closed ... |
| sigaclfu2 33057 | A sigma-algebra is closed ... |
| sigaclcu3 33058 | A sigma-algebra is closed ... |
| issgon 33059 | Property of being a sigma-... |
| sgon 33060 | A sigma-algebra is a sigma... |
| elsigass 33061 | An element of a sigma-alge... |
| elrnsiga 33062 | Dropping the base informat... |
| isrnsigau 33063 | The property of being a si... |
| unielsiga 33064 | A sigma-algebra contains i... |
| dmvlsiga 33065 | Lebesgue-measurable subset... |
| pwsiga 33066 | Any power set forms a sigm... |
| prsiga 33067 | The smallest possible sigm... |
| sigaclci 33068 | A sigma-algebra is closed ... |
| difelsiga 33069 | A sigma-algebra is closed ... |
| unelsiga 33070 | A sigma-algebra is closed ... |
| inelsiga 33071 | A sigma-algebra is closed ... |
| sigainb 33072 | Building a sigma-algebra f... |
| insiga 33073 | The intersection of a coll... |
| sigagenval 33076 | Value of the generated sig... |
| sigagensiga 33077 | A generated sigma-algebra ... |
| sgsiga 33078 | A generated sigma-algebra ... |
| unisg 33079 | The sigma-algebra generate... |
| dmsigagen 33080 | A sigma-algebra can be gen... |
| sssigagen 33081 | A set is a subset of the s... |
| sssigagen2 33082 | A subset of the generating... |
| elsigagen 33083 | Any element of a set is al... |
| elsigagen2 33084 | Any countable union of ele... |
| sigagenss 33085 | The generated sigma-algebr... |
| sigagenss2 33086 | Sufficient condition for i... |
| sigagenid 33087 | The sigma-algebra generate... |
| ispisys 33088 | The property of being a pi... |
| ispisys2 33089 | The property of being a pi... |
| inelpisys 33090 | Pi-systems are closed unde... |
| sigapisys 33091 | All sigma-algebras are pi-... |
| isldsys 33092 | The property of being a la... |
| pwldsys 33093 | The power set of the unive... |
| unelldsys 33094 | Lambda-systems are closed ... |
| sigaldsys 33095 | All sigma-algebras are lam... |
| ldsysgenld 33096 | The intersection of all la... |
| sigapildsyslem 33097 | Lemma for ~ sigapildsys . ... |
| sigapildsys 33098 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 33099 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 33100 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 33101 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 33102 | The lambda system ` E ` ge... |
| dynkin 33103 | Dynkin's lambda-pi theorem... |
| isros 33104 | The property of being a ri... |
| rossspw 33105 | A ring of sets is a collec... |
| 0elros 33106 | A ring of sets contains th... |
| unelros 33107 | A ring of sets is closed u... |
| difelros 33108 | A ring of sets is closed u... |
| inelros 33109 | A ring of sets is closed u... |
| fiunelros 33110 | A ring of sets is closed u... |
| issros 33111 | The property of being a se... |
| srossspw 33112 | A semiring of sets is a co... |
| 0elsros 33113 | A semiring of sets contain... |
| inelsros 33114 | A semiring of sets is clos... |
| diffiunisros 33115 | In semiring of sets, compl... |
| rossros 33116 | Rings of sets are semiring... |
| brsiga 33119 | The Borel Algebra on real ... |
| brsigarn 33120 | The Borel Algebra is a sig... |
| brsigasspwrn 33121 | The Borel Algebra is a set... |
| unibrsiga 33122 | The union of the Borel Alg... |
| cldssbrsiga 33123 | A Borel Algebra contains a... |
| sxval 33126 | Value of the product sigma... |
| sxsiga 33127 | A product sigma-algebra is... |
| sxsigon 33128 | A product sigma-algebra is... |
| sxuni 33129 | The base set of a product ... |
| elsx 33130 | The cartesian product of t... |
| measbase 33133 | The base set of a measure ... |
| measval 33134 | The value of the ` measure... |
| ismeas 33135 | The property of being a me... |
| isrnmeas 33136 | The property of being a me... |
| dmmeas 33137 | The domain of a measure is... |
| measbasedom 33138 | The base set of a measure ... |
| measfrge0 33139 | A measure is a function ov... |
| measfn 33140 | A measure is a function on... |
| measvxrge0 33141 | The values of a measure ar... |
| measvnul 33142 | The measure of the empty s... |
| measge0 33143 | A measure is nonnegative. ... |
| measle0 33144 | If the measure of a given ... |
| measvun 33145 | The measure of a countable... |
| measxun2 33146 | The measure the union of t... |
| measun 33147 | The measure the union of t... |
| measvunilem 33148 | Lemma for ~ measvuni . (C... |
| measvunilem0 33149 | Lemma for ~ measvuni . (C... |
| measvuni 33150 | The measure of a countable... |
| measssd 33151 | A measure is monotone with... |
| measunl 33152 | A measure is sub-additive ... |
| measiuns 33153 | The measure of the union o... |
| measiun 33154 | A measure is sub-additive.... |
| meascnbl 33155 | A measure is continuous fr... |
| measinblem 33156 | Lemma for ~ measinb . (Co... |
| measinb 33157 | Building a measure restric... |
| measres 33158 | Building a measure restric... |
| measinb2 33159 | Building a measure restric... |
| measdivcst 33160 | Division of a measure by a... |
| measdivcstALTV 33161 | Alternate version of ~ mea... |
| cntmeas 33162 | The Counting measure is a ... |
| pwcntmeas 33163 | The counting measure is a ... |
| cntnevol 33164 | Counting and Lebesgue meas... |
| voliune 33165 | The Lebesgue measure funct... |
| volfiniune 33166 | The Lebesgue measure funct... |
| volmeas 33167 | The Lebesgue measure is a ... |
| ddeval1 33170 | Value of the delta measure... |
| ddeval0 33171 | Value of the delta measure... |
| ddemeas 33172 | The Dirac delta measure is... |
| relae 33176 | 'almost everywhere' is a r... |
| brae 33177 | 'almost everywhere' relati... |
| braew 33178 | 'almost everywhere' relati... |
| truae 33179 | A truth holds almost every... |
| aean 33180 | A conjunction holds almost... |
| faeval 33182 | Value of the 'almost every... |
| relfae 33183 | The 'almost everywhere' bu... |
| brfae 33184 | 'almost everywhere' relati... |
| ismbfm 33187 | The predicate " ` F ` is a... |
| elunirnmbfm 33188 | The property of being a me... |
| mbfmfun 33189 | A measurable function is a... |
| mbfmf 33190 | A measurable function as a... |
| isanmbfmOLD 33191 | Obsolete version of ~ isan... |
| mbfmcnvima 33192 | The preimage by a measurab... |
| isanmbfm 33193 | The predicate to be a meas... |
| mbfmbfmOLD 33194 | A measurable function to a... |
| mbfmbfm 33195 | A measurable function to a... |
| mbfmcst 33196 | A constant function is mea... |
| 1stmbfm 33197 | The first projection map i... |
| 2ndmbfm 33198 | The second projection map ... |
| imambfm 33199 | If the sigma-algebra in th... |
| cnmbfm 33200 | A continuous function is m... |
| mbfmco 33201 | The composition of two mea... |
| mbfmco2 33202 | The pair building of two m... |
| mbfmvolf 33203 | Measurable functions with ... |
| elmbfmvol2 33204 | Measurable functions with ... |
| mbfmcnt 33205 | All functions are measurab... |
| br2base 33206 | The base set for the gener... |
| dya2ub 33207 | An upper bound for a dyadi... |
| sxbrsigalem0 33208 | The closed half-spaces of ... |
| sxbrsigalem3 33209 | The sigma-algebra generate... |
| dya2iocival 33210 | The function ` I ` returns... |
| dya2iocress 33211 | Dyadic intervals are subse... |
| dya2iocbrsiga 33212 | Dyadic intervals are Borel... |
| dya2icobrsiga 33213 | Dyadic intervals are Borel... |
| dya2icoseg 33214 | For any point and any clos... |
| dya2icoseg2 33215 | For any point and any open... |
| dya2iocrfn 33216 | The function returning dya... |
| dya2iocct 33217 | The dyadic rectangle set i... |
| dya2iocnrect 33218 | For any point of an open r... |
| dya2iocnei 33219 | For any point of an open s... |
| dya2iocuni 33220 | Every open set of ` ( RR X... |
| dya2iocucvr 33221 | The dyadic rectangular set... |
| sxbrsigalem1 33222 | The Borel algebra on ` ( R... |
| sxbrsigalem2 33223 | The sigma-algebra generate... |
| sxbrsigalem4 33224 | The Borel algebra on ` ( R... |
| sxbrsigalem5 33225 | First direction for ~ sxbr... |
| sxbrsigalem6 33226 | First direction for ~ sxbr... |
| sxbrsiga 33227 | The product sigma-algebra ... |
| omsval 33230 | Value of the function mapp... |
| omsfval 33231 | Value of the outer measure... |
| omscl 33232 | A closure lemma for the co... |
| omsf 33233 | A constructed outer measur... |
| oms0 33234 | A constructed outer measur... |
| omsmon 33235 | A constructed outer measur... |
| omssubaddlem 33236 | For any small margin ` E `... |
| omssubadd 33237 | A constructed outer measur... |
| carsgval 33240 | Value of the Caratheodory ... |
| carsgcl 33241 | Closure of the Caratheodor... |
| elcarsg 33242 | Property of being a Carath... |
| baselcarsg 33243 | The universe set, ` O ` , ... |
| 0elcarsg 33244 | The empty set is Caratheod... |
| carsguni 33245 | The union of all Caratheod... |
| elcarsgss 33246 | Caratheodory measurable se... |
| difelcarsg 33247 | The Caratheodory measurabl... |
| inelcarsg 33248 | The Caratheodory measurabl... |
| unelcarsg 33249 | The Caratheodory-measurabl... |
| difelcarsg2 33250 | The Caratheodory-measurabl... |
| carsgmon 33251 | Utility lemma: Apply mono... |
| carsgsigalem 33252 | Lemma for the following th... |
| fiunelcarsg 33253 | The Caratheodory measurabl... |
| carsgclctunlem1 33254 | Lemma for ~ carsgclctun . ... |
| carsggect 33255 | The outer measure is count... |
| carsgclctunlem2 33256 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 33257 | Lemma for ~ carsgclctun . ... |
| carsgclctun 33258 | The Caratheodory measurabl... |
| carsgsiga 33259 | The Caratheodory measurabl... |
| omsmeas 33260 | The restriction of a const... |
| pmeasmono 33261 | This theorem's hypotheses ... |
| pmeasadd 33262 | A premeasure on a ring of ... |
| itgeq12dv 33263 | Equality theorem for an in... |
| sitgval 33269 | Value of the simple functi... |
| issibf 33270 | The predicate " ` F ` is a... |
| sibf0 33271 | The constant zero function... |
| sibfmbl 33272 | A simple function is measu... |
| sibff 33273 | A simple function is a fun... |
| sibfrn 33274 | A simple function has fini... |
| sibfima 33275 | Any preimage of a singleto... |
| sibfinima 33276 | The measure of the interse... |
| sibfof 33277 | Applying function operatio... |
| sitgfval 33278 | Value of the Bochner integ... |
| sitgclg 33279 | Closure of the Bochner int... |
| sitgclbn 33280 | Closure of the Bochner int... |
| sitgclcn 33281 | Closure of the Bochner int... |
| sitgclre 33282 | Closure of the Bochner int... |
| sitg0 33283 | The integral of the consta... |
| sitgf 33284 | The integral for simple fu... |
| sitgaddlemb 33285 | Lemma for * sitgadd . (Co... |
| sitmval 33286 | Value of the simple functi... |
| sitmfval 33287 | Value of the integral dist... |
| sitmcl 33288 | Closure of the integral di... |
| sitmf 33289 | The integral metric as a f... |
| oddpwdc 33291 | Lemma for ~ eulerpart . T... |
| oddpwdcv 33292 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 33293 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 33294 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 33295 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 33296 | Lemma for ~ eulerpart . (... |
| eulerpartlems 33297 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 33298 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 33299 | Lemma for ~ eulerpart . (... |
| eulerpartleme 33300 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 33301 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 33302 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 33303 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 33304 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 33305 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 33306 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 33307 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 33308 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 33309 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 33310 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 33311 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 33312 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 33313 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 33314 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 33315 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 33316 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 33317 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 33318 | Lemma for ~ eulerpart . (... |
| eulerpart 33319 | Euler's theorem on partiti... |
| subiwrd 33322 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 33323 | Length of a subword of an ... |
| iwrdsplit 33324 | Lemma for ~ sseqp1 . (Con... |
| sseqval 33325 | Value of the strong sequen... |
| sseqfv1 33326 | Value of the strong sequen... |
| sseqfn 33327 | A strong recursive sequenc... |
| sseqmw 33328 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 33329 | A strong recursive sequenc... |
| sseqfres 33330 | The first elements in the ... |
| sseqfv2 33331 | Value of the strong sequen... |
| sseqp1 33332 | Value of the strong sequen... |
| fiblem 33335 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 33336 | Value of the Fibonacci seq... |
| fib1 33337 | Value of the Fibonacci seq... |
| fibp1 33338 | Value of the Fibonacci seq... |
| fib2 33339 | Value of the Fibonacci seq... |
| fib3 33340 | Value of the Fibonacci seq... |
| fib4 33341 | Value of the Fibonacci seq... |
| fib5 33342 | Value of the Fibonacci seq... |
| fib6 33343 | Value of the Fibonacci seq... |
| elprob 33346 | The property of being a pr... |
| domprobmeas 33347 | A probability measure is a... |
| domprobsiga 33348 | The domain of a probabilit... |
| probtot 33349 | The probability of the uni... |
| prob01 33350 | A probability is an elemen... |
| probnul 33351 | The probability of the emp... |
| unveldomd 33352 | The universe is an element... |
| unveldom 33353 | The universe is an element... |
| nuleldmp 33354 | The empty set is an elemen... |
| probcun 33355 | The probability of the uni... |
| probun 33356 | The probability of the uni... |
| probdif 33357 | The probability of the dif... |
| probinc 33358 | A probability law is incre... |
| probdsb 33359 | The probability of the com... |
| probmeasd 33360 | A probability measure is a... |
| probvalrnd 33361 | The value of a probability... |
| probtotrnd 33362 | The probability of the uni... |
| totprobd 33363 | Law of total probability, ... |
| totprob 33364 | Law of total probability. ... |
| probfinmeasb 33365 | Build a probability measur... |
| probfinmeasbALTV 33366 | Alternate version of ~ pro... |
| probmeasb 33367 | Build a probability from a... |
| cndprobval 33370 | The value of the condition... |
| cndprobin 33371 | An identity linking condit... |
| cndprob01 33372 | The conditional probabilit... |
| cndprobtot 33373 | The conditional probabilit... |
| cndprobnul 33374 | The conditional probabilit... |
| cndprobprob 33375 | The conditional probabilit... |
| bayesth 33376 | Bayes Theorem. (Contribut... |
| rrvmbfm 33379 | A real-valued random varia... |
| isrrvv 33380 | Elementhood to the set of ... |
| rrvvf 33381 | A real-valued random varia... |
| rrvfn 33382 | A real-valued random varia... |
| rrvdm 33383 | The domain of a random var... |
| rrvrnss 33384 | The range of a random vari... |
| rrvf2 33385 | A real-valued random varia... |
| rrvdmss 33386 | The domain of a random var... |
| rrvfinvima 33387 | For a real-value random va... |
| 0rrv 33388 | The constant function equa... |
| rrvadd 33389 | The sum of two random vari... |
| rrvmulc 33390 | A random variable multipli... |
| rrvsum 33391 | An indexed sum of random v... |
| orvcval 33394 | Value of the preimage mapp... |
| orvcval2 33395 | Another way to express the... |
| elorvc 33396 | Elementhood of a preimage.... |
| orvcval4 33397 | The value of the preimage ... |
| orvcoel 33398 | If the relation produces o... |
| orvccel 33399 | If the relation produces c... |
| elorrvc 33400 | Elementhood of a preimage ... |
| orrvcval4 33401 | The value of the preimage ... |
| orrvcoel 33402 | If the relation produces o... |
| orrvccel 33403 | If the relation produces c... |
| orvcgteel 33404 | Preimage maps produced by ... |
| orvcelval 33405 | Preimage maps produced by ... |
| orvcelel 33406 | Preimage maps produced by ... |
| dstrvval 33407 | The value of the distribut... |
| dstrvprob 33408 | The distribution of a rand... |
| orvclteel 33409 | Preimage maps produced by ... |
| dstfrvel 33410 | Elementhood of preimage ma... |
| dstfrvunirn 33411 | The limit of all preimage ... |
| orvclteinc 33412 | Preimage maps produced by ... |
| dstfrvinc 33413 | A cumulative distribution ... |
| dstfrvclim1 33414 | The limit of the cumulativ... |
| coinfliplem 33415 | Division in the extended r... |
| coinflipprob 33416 | The ` P ` we defined for c... |
| coinflipspace 33417 | The space of our coin-flip... |
| coinflipuniv 33418 | The universe of our coin-f... |
| coinfliprv 33419 | The ` X ` we defined for c... |
| coinflippv 33420 | The probability of heads i... |
| coinflippvt 33421 | The probability of tails i... |
| ballotlemoex 33422 | ` O ` is a set. (Contribu... |
| ballotlem1 33423 | The size of the universe i... |
| ballotlemelo 33424 | Elementhood in ` O ` . (C... |
| ballotlem2 33425 | The probability that the f... |
| ballotlemfval 33426 | The value of ` F ` . (Con... |
| ballotlemfelz 33427 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 33428 | If the ` J ` th ballot is ... |
| ballotlemfc0 33429 | ` F ` takes value 0 betwee... |
| ballotlemfcc 33430 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 33431 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 33432 | ` ( F `` C ) ` always star... |
| ballotleme 33433 | Elements of ` E ` . (Cont... |
| ballotlemodife 33434 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 33435 | If the first pick is a vot... |
| ballotlem5 33436 | If A is not ahead througho... |
| ballotlemi 33437 | Value of ` I ` for a given... |
| ballotlemiex 33438 | Properties of ` ( I `` C )... |
| ballotlemi1 33439 | The first tie cannot be re... |
| ballotlemii 33440 | The first tie cannot be re... |
| ballotlemsup 33441 | The set of zeroes of ` F `... |
| ballotlemimin 33442 | ` ( I `` C ) ` is the firs... |
| ballotlemic 33443 | If the first vote is for B... |
| ballotlem1c 33444 | If the first vote is for A... |
| ballotlemsval 33445 | Value of ` S ` . (Contrib... |
| ballotlemsv 33446 | Value of ` S ` evaluated a... |
| ballotlemsgt1 33447 | ` S ` maps values less tha... |
| ballotlemsdom 33448 | Domain of ` S ` for a give... |
| ballotlemsel1i 33449 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 33450 | The defined ` S ` is a bij... |
| ballotlemsi 33451 | The image by ` S ` of the ... |
| ballotlemsima 33452 | The image by ` S ` of an i... |
| ballotlemieq 33453 | If two countings share the... |
| ballotlemrval 33454 | Value of ` R ` . (Contrib... |
| ballotlemscr 33455 | The image of ` ( R `` C ) ... |
| ballotlemrv 33456 | Value of ` R ` evaluated a... |
| ballotlemrv1 33457 | Value of ` R ` before the ... |
| ballotlemrv2 33458 | Value of ` R ` after the t... |
| ballotlemro 33459 | Range of ` R ` is included... |
| ballotlemgval 33460 | Expand the value of ` .^ `... |
| ballotlemgun 33461 | A property of the defined ... |
| ballotlemfg 33462 | Express the value of ` ( F... |
| ballotlemfrc 33463 | Express the value of ` ( F... |
| ballotlemfrci 33464 | Reverse counting preserves... |
| ballotlemfrceq 33465 | Value of ` F ` for a rever... |
| ballotlemfrcn0 33466 | Value of ` F ` for a rever... |
| ballotlemrc 33467 | Range of ` R ` . (Contrib... |
| ballotlemirc 33468 | Applying ` R ` does not ch... |
| ballotlemrinv0 33469 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 33470 | ` R ` is its own inverse :... |
| ballotlem1ri 33471 | When the vote on the first... |
| ballotlem7 33472 | ` R ` is a bijection betwe... |
| ballotlem8 33473 | There are as many counting... |
| ballotth 33474 | Bertrand's ballot problem ... |
| sgncl 33475 | Closure of the signum. (C... |
| sgnclre 33476 | Closure of the signum. (C... |
| sgnneg 33477 | Negation of the signum. (... |
| sgn3da 33478 | A conditional containing a... |
| sgnmul 33479 | Signum of a product. (Con... |
| sgnmulrp2 33480 | Multiplication by a positi... |
| sgnsub 33481 | Subtraction of a number of... |
| sgnnbi 33482 | Negative signum. (Contrib... |
| sgnpbi 33483 | Positive signum. (Contrib... |
| sgn0bi 33484 | Zero signum. (Contributed... |
| sgnsgn 33485 | Signum is idempotent. (Co... |
| sgnmulsgn 33486 | If two real numbers are of... |
| sgnmulsgp 33487 | If two real numbers are of... |
| fzssfzo 33488 | Condition for an integer i... |
| gsumncl 33489 | Closure of a group sum in ... |
| gsumnunsn 33490 | Closure of a group sum in ... |
| ccatmulgnn0dir 33491 | Concatenation of words fol... |
| ofcccat 33492 | Letterwise operations on w... |
| ofcs1 33493 | Letterwise operations on a... |
| ofcs2 33494 | Letterwise operations on a... |
| plymul02 33495 | Product of a polynomial wi... |
| plymulx0 33496 | Coefficients of a polynomi... |
| plymulx 33497 | Coefficients of a polynomi... |
| plyrecld 33498 | Closure of a polynomial wi... |
| signsplypnf 33499 | The quotient of a polynomi... |
| signsply0 33500 | Lemma for the rule of sign... |
| signspval 33501 | The value of the skipping ... |
| signsw0glem 33502 | Neutral element property o... |
| signswbase 33503 | The base of ` W ` is the u... |
| signswplusg 33504 | The operation of ` W ` . ... |
| signsw0g 33505 | The neutral element of ` W... |
| signswmnd 33506 | ` W ` is a monoid structur... |
| signswrid 33507 | The zero-skipping operatio... |
| signswlid 33508 | The zero-skipping operatio... |
| signswn0 33509 | The zero-skipping operatio... |
| signswch 33510 | The zero-skipping operatio... |
| signslema 33511 | Computational part of ~~? ... |
| signstfv 33512 | Value of the zero-skipping... |
| signstfval 33513 | Value of the zero-skipping... |
| signstcl 33514 | Closure of the zero skippi... |
| signstf 33515 | The zero skipping sign wor... |
| signstlen 33516 | Length of the zero skippin... |
| signstf0 33517 | Sign of a single letter wo... |
| signstfvn 33518 | Zero-skipping sign in a wo... |
| signsvtn0 33519 | If the last letter is nonz... |
| signstfvp 33520 | Zero-skipping sign in a wo... |
| signstfvneq0 33521 | In case the first letter i... |
| signstfvcl 33522 | Closure of the zero skippi... |
| signstfvc 33523 | Zero-skipping sign in a wo... |
| signstres 33524 | Restriction of a zero skip... |
| signstfveq0a 33525 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 33526 | In case the last letter is... |
| signsvvfval 33527 | The value of ` V ` , which... |
| signsvvf 33528 | ` V ` is a function. (Con... |
| signsvf0 33529 | There is no change of sign... |
| signsvf1 33530 | In a single-letter word, w... |
| signsvfn 33531 | Number of changes in a wor... |
| signsvtp 33532 | Adding a letter of the sam... |
| signsvtn 33533 | Adding a letter of a diffe... |
| signsvfpn 33534 | Adding a letter of the sam... |
| signsvfnn 33535 | Adding a letter of a diffe... |
| signlem0 33536 | Adding a zero as the highe... |
| signshf 33537 | ` H ` , corresponding to t... |
| signshwrd 33538 | ` H ` , corresponding to t... |
| signshlen 33539 | Length of ` H ` , correspo... |
| signshnz 33540 | ` H ` is not the empty wor... |
| efcld 33541 | Closure law for the expone... |
| iblidicc 33542 | The identity function is i... |
| rpsqrtcn 33543 | Continuity of the real pos... |
| divsqrtid 33544 | A real number divided by i... |
| cxpcncf1 33545 | The power function on comp... |
| efmul2picn 33546 | Multiplying by ` ( _i x. (... |
| fct2relem 33547 | Lemma for ~ ftc2re . (Con... |
| ftc2re 33548 | The Fundamental Theorem of... |
| fdvposlt 33549 | Functions with a positive ... |
| fdvneggt 33550 | Functions with a negative ... |
| fdvposle 33551 | Functions with a nonnegati... |
| fdvnegge 33552 | Functions with a nonpositi... |
| prodfzo03 33553 | A product of three factors... |
| actfunsnf1o 33554 | The action ` F ` of extend... |
| actfunsnrndisj 33555 | The action ` F ` of extend... |
| itgexpif 33556 | The basis for the circle m... |
| fsum2dsub 33557 | Lemma for ~ breprexp - Re-... |
| reprval 33560 | Value of the representatio... |
| repr0 33561 | There is exactly one repre... |
| reprf 33562 | Members of the representat... |
| reprsum 33563 | Sums of values of the memb... |
| reprle 33564 | Upper bound to the terms i... |
| reprsuc 33565 | Express the representation... |
| reprfi 33566 | Bounded representations ar... |
| reprss 33567 | Representations with terms... |
| reprinrn 33568 | Representations with term ... |
| reprlt 33569 | There are no representatio... |
| hashreprin 33570 | Express a sum of represent... |
| reprgt 33571 | There are no representatio... |
| reprinfz1 33572 | For the representation of ... |
| reprfi2 33573 | Corollary of ~ reprinfz1 .... |
| reprfz1 33574 | Corollary of ~ reprinfz1 .... |
| hashrepr 33575 | Develop the number of repr... |
| reprpmtf1o 33576 | Transposing ` 0 ` and ` X ... |
| reprdifc 33577 | Express the representation... |
| chpvalz 33578 | Value of the second Chebys... |
| chtvalz 33579 | Value of the Chebyshev fun... |
| breprexplema 33580 | Lemma for ~ breprexp (indu... |
| breprexplemb 33581 | Lemma for ~ breprexp (clos... |
| breprexplemc 33582 | Lemma for ~ breprexp (indu... |
| breprexp 33583 | Express the ` S ` th power... |
| breprexpnat 33584 | Express the ` S ` th power... |
| vtsval 33587 | Value of the Vinogradov tr... |
| vtscl 33588 | Closure of the Vinogradov ... |
| vtsprod 33589 | Express the Vinogradov tri... |
| circlemeth 33590 | The Hardy, Littlewood and ... |
| circlemethnat 33591 | The Hardy, Littlewood and ... |
| circlevma 33592 | The Circle Method, where t... |
| circlemethhgt 33593 | The circle method, where t... |
| hgt750lemc 33597 | An upper bound to the summ... |
| hgt750lemd 33598 | An upper bound to the summ... |
| hgt749d 33599 | A deduction version of ~ a... |
| logdivsqrle 33600 | Conditions for ` ( ( log `... |
| hgt750lem 33601 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 33602 | Decimal multiplication gal... |
| hgt750lemf 33603 | Lemma for the statement 7.... |
| hgt750lemg 33604 | Lemma for the statement 7.... |
| oddprm2 33605 | Two ways to write the set ... |
| hgt750lemb 33606 | An upper bound on the cont... |
| hgt750lema 33607 | An upper bound on the cont... |
| hgt750leme 33608 | An upper bound on the cont... |
| tgoldbachgnn 33609 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 33610 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 33611 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 33612 | Odd integers greater than ... |
| tgoldbachgt 33613 | Odd integers greater than ... |
| istrkg2d 33616 | Property of fulfilling dim... |
| axtglowdim2ALTV 33617 | Alternate version of ~ axt... |
| axtgupdim2ALTV 33618 | Alternate version of ~ axt... |
| afsval 33621 | Value of the AFS relation ... |
| brafs 33622 | Binary relation form of th... |
| tg5segofs 33623 | Rephrase ~ axtg5seg using ... |
| lpadval 33626 | Value of the ` leftpad ` f... |
| lpadlem1 33627 | Lemma for the ` leftpad ` ... |
| lpadlem3 33628 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 33629 | Length of a left-padded wo... |
| lpadlem2 33630 | Lemma for the ` leftpad ` ... |
| lpadlen2 33631 | Length of a left-padded wo... |
| lpadmax 33632 | Length of a left-padded wo... |
| lpadleft 33633 | The contents of prefix of ... |
| lpadright 33634 | The suffix of a left-padde... |
| bnj170 33647 | ` /\ ` -manipulation. (Co... |
| bnj240 33648 | ` /\ ` -manipulation. (Co... |
| bnj248 33649 | ` /\ ` -manipulation. (Co... |
| bnj250 33650 | ` /\ ` -manipulation. (Co... |
| bnj251 33651 | ` /\ ` -manipulation. (Co... |
| bnj252 33652 | ` /\ ` -manipulation. (Co... |
| bnj253 33653 | ` /\ ` -manipulation. (Co... |
| bnj255 33654 | ` /\ ` -manipulation. (Co... |
| bnj256 33655 | ` /\ ` -manipulation. (Co... |
| bnj257 33656 | ` /\ ` -manipulation. (Co... |
| bnj258 33657 | ` /\ ` -manipulation. (Co... |
| bnj268 33658 | ` /\ ` -manipulation. (Co... |
| bnj290 33659 | ` /\ ` -manipulation. (Co... |
| bnj291 33660 | ` /\ ` -manipulation. (Co... |
| bnj312 33661 | ` /\ ` -manipulation. (Co... |
| bnj334 33662 | ` /\ ` -manipulation. (Co... |
| bnj345 33663 | ` /\ ` -manipulation. (Co... |
| bnj422 33664 | ` /\ ` -manipulation. (Co... |
| bnj432 33665 | ` /\ ` -manipulation. (Co... |
| bnj446 33666 | ` /\ ` -manipulation. (Co... |
| bnj23 33667 | First-order logic and set ... |
| bnj31 33668 | First-order logic and set ... |
| bnj62 33669 | First-order logic and set ... |
| bnj89 33670 | First-order logic and set ... |
| bnj90 33671 | First-order logic and set ... |
| bnj101 33672 | First-order logic and set ... |
| bnj105 33673 | First-order logic and set ... |
| bnj115 33674 | First-order logic and set ... |
| bnj132 33675 | First-order logic and set ... |
| bnj133 33676 | First-order logic and set ... |
| bnj156 33677 | First-order logic and set ... |
| bnj158 33678 | First-order logic and set ... |
| bnj168 33679 | First-order logic and set ... |
| bnj206 33680 | First-order logic and set ... |
| bnj216 33681 | First-order logic and set ... |
| bnj219 33682 | First-order logic and set ... |
| bnj226 33683 | First-order logic and set ... |
| bnj228 33684 | First-order logic and set ... |
| bnj519 33685 | First-order logic and set ... |
| bnj524 33686 | First-order logic and set ... |
| bnj525 33687 | First-order logic and set ... |
| bnj534 33688 | First-order logic and set ... |
| bnj538 33689 | First-order logic and set ... |
| bnj529 33690 | First-order logic and set ... |
| bnj551 33691 | First-order logic and set ... |
| bnj563 33692 | First-order logic and set ... |
| bnj564 33693 | First-order logic and set ... |
| bnj593 33694 | First-order logic and set ... |
| bnj596 33695 | First-order logic and set ... |
| bnj610 33696 | Pass from equality ( ` x =... |
| bnj642 33697 | ` /\ ` -manipulation. (Co... |
| bnj643 33698 | ` /\ ` -manipulation. (Co... |
| bnj645 33699 | ` /\ ` -manipulation. (Co... |
| bnj658 33700 | ` /\ ` -manipulation. (Co... |
| bnj667 33701 | ` /\ ` -manipulation. (Co... |
| bnj705 33702 | ` /\ ` -manipulation. (Co... |
| bnj706 33703 | ` /\ ` -manipulation. (Co... |
| bnj707 33704 | ` /\ ` -manipulation. (Co... |
| bnj708 33705 | ` /\ ` -manipulation. (Co... |
| bnj721 33706 | ` /\ ` -manipulation. (Co... |
| bnj832 33707 | ` /\ ` -manipulation. (Co... |
| bnj835 33708 | ` /\ ` -manipulation. (Co... |
| bnj836 33709 | ` /\ ` -manipulation. (Co... |
| bnj837 33710 | ` /\ ` -manipulation. (Co... |
| bnj769 33711 | ` /\ ` -manipulation. (Co... |
| bnj770 33712 | ` /\ ` -manipulation. (Co... |
| bnj771 33713 | ` /\ ` -manipulation. (Co... |
| bnj887 33714 | ` /\ ` -manipulation. (Co... |
| bnj918 33715 | First-order logic and set ... |
| bnj919 33716 | First-order logic and set ... |
| bnj923 33717 | First-order logic and set ... |
| bnj927 33718 | First-order logic and set ... |
| bnj931 33719 | First-order logic and set ... |
| bnj937 33720 | First-order logic and set ... |
| bnj941 33721 | First-order logic and set ... |
| bnj945 33722 | Technical lemma for ~ bnj6... |
| bnj946 33723 | First-order logic and set ... |
| bnj951 33724 | ` /\ ` -manipulation. (Co... |
| bnj956 33725 | First-order logic and set ... |
| bnj976 33726 | First-order logic and set ... |
| bnj982 33727 | First-order logic and set ... |
| bnj1019 33728 | First-order logic and set ... |
| bnj1023 33729 | First-order logic and set ... |
| bnj1095 33730 | First-order logic and set ... |
| bnj1096 33731 | First-order logic and set ... |
| bnj1098 33732 | First-order logic and set ... |
| bnj1101 33733 | First-order logic and set ... |
| bnj1113 33734 | First-order logic and set ... |
| bnj1109 33735 | First-order logic and set ... |
| bnj1131 33736 | First-order logic and set ... |
| bnj1138 33737 | First-order logic and set ... |
| bnj1142 33738 | First-order logic and set ... |
| bnj1143 33739 | First-order logic and set ... |
| bnj1146 33740 | First-order logic and set ... |
| bnj1149 33741 | First-order logic and set ... |
| bnj1185 33742 | First-order logic and set ... |
| bnj1196 33743 | First-order logic and set ... |
| bnj1198 33744 | First-order logic and set ... |
| bnj1209 33745 | First-order logic and set ... |
| bnj1211 33746 | First-order logic and set ... |
| bnj1213 33747 | First-order logic and set ... |
| bnj1212 33748 | First-order logic and set ... |
| bnj1219 33749 | First-order logic and set ... |
| bnj1224 33750 | First-order logic and set ... |
| bnj1230 33751 | First-order logic and set ... |
| bnj1232 33752 | First-order logic and set ... |
| bnj1235 33753 | First-order logic and set ... |
| bnj1239 33754 | First-order logic and set ... |
| bnj1238 33755 | First-order logic and set ... |
| bnj1241 33756 | First-order logic and set ... |
| bnj1247 33757 | First-order logic and set ... |
| bnj1254 33758 | First-order logic and set ... |
| bnj1262 33759 | First-order logic and set ... |
| bnj1266 33760 | First-order logic and set ... |
| bnj1265 33761 | First-order logic and set ... |
| bnj1275 33762 | First-order logic and set ... |
| bnj1276 33763 | First-order logic and set ... |
| bnj1292 33764 | First-order logic and set ... |
| bnj1293 33765 | First-order logic and set ... |
| bnj1294 33766 | First-order logic and set ... |
| bnj1299 33767 | First-order logic and set ... |
| bnj1304 33768 | First-order logic and set ... |
| bnj1316 33769 | First-order logic and set ... |
| bnj1317 33770 | First-order logic and set ... |
| bnj1322 33771 | First-order logic and set ... |
| bnj1340 33772 | First-order logic and set ... |
| bnj1345 33773 | First-order logic and set ... |
| bnj1350 33774 | First-order logic and set ... |
| bnj1351 33775 | First-order logic and set ... |
| bnj1352 33776 | First-order logic and set ... |
| bnj1361 33777 | First-order logic and set ... |
| bnj1366 33778 | First-order logic and set ... |
| bnj1379 33779 | First-order logic and set ... |
| bnj1383 33780 | First-order logic and set ... |
| bnj1385 33781 | First-order logic and set ... |
| bnj1386 33782 | First-order logic and set ... |
| bnj1397 33783 | First-order logic and set ... |
| bnj1400 33784 | First-order logic and set ... |
| bnj1405 33785 | First-order logic and set ... |
| bnj1422 33786 | First-order logic and set ... |
| bnj1424 33787 | First-order logic and set ... |
| bnj1436 33788 | First-order logic and set ... |
| bnj1441 33789 | First-order logic and set ... |
| bnj1441g 33790 | First-order logic and set ... |
| bnj1454 33791 | First-order logic and set ... |
| bnj1459 33792 | First-order logic and set ... |
| bnj1464 33793 | Conversion of implicit sub... |
| bnj1465 33794 | First-order logic and set ... |
| bnj1468 33795 | Conversion of implicit sub... |
| bnj1476 33796 | First-order logic and set ... |
| bnj1502 33797 | First-order logic and set ... |
| bnj1503 33798 | First-order logic and set ... |
| bnj1517 33799 | First-order logic and set ... |
| bnj1521 33800 | First-order logic and set ... |
| bnj1533 33801 | First-order logic and set ... |
| bnj1534 33802 | First-order logic and set ... |
| bnj1536 33803 | First-order logic and set ... |
| bnj1538 33804 | First-order logic and set ... |
| bnj1541 33805 | First-order logic and set ... |
| bnj1542 33806 | First-order logic and set ... |
| bnj110 33807 | Well-founded induction res... |
| bnj157 33808 | Well-founded induction res... |
| bnj66 33809 | Technical lemma for ~ bnj6... |
| bnj91 33810 | First-order logic and set ... |
| bnj92 33811 | First-order logic and set ... |
| bnj93 33812 | Technical lemma for ~ bnj9... |
| bnj95 33813 | Technical lemma for ~ bnj1... |
| bnj96 33814 | Technical lemma for ~ bnj1... |
| bnj97 33815 | Technical lemma for ~ bnj1... |
| bnj98 33816 | Technical lemma for ~ bnj1... |
| bnj106 33817 | First-order logic and set ... |
| bnj118 33818 | First-order logic and set ... |
| bnj121 33819 | First-order logic and set ... |
| bnj124 33820 | Technical lemma for ~ bnj1... |
| bnj125 33821 | Technical lemma for ~ bnj1... |
| bnj126 33822 | Technical lemma for ~ bnj1... |
| bnj130 33823 | Technical lemma for ~ bnj1... |
| bnj149 33824 | Technical lemma for ~ bnj1... |
| bnj150 33825 | Technical lemma for ~ bnj1... |
| bnj151 33826 | Technical lemma for ~ bnj1... |
| bnj154 33827 | Technical lemma for ~ bnj1... |
| bnj155 33828 | Technical lemma for ~ bnj1... |
| bnj153 33829 | Technical lemma for ~ bnj8... |
| bnj207 33830 | Technical lemma for ~ bnj8... |
| bnj213 33831 | First-order logic and set ... |
| bnj222 33832 | Technical lemma for ~ bnj2... |
| bnj229 33833 | Technical lemma for ~ bnj5... |
| bnj517 33834 | Technical lemma for ~ bnj5... |
| bnj518 33835 | Technical lemma for ~ bnj8... |
| bnj523 33836 | Technical lemma for ~ bnj8... |
| bnj526 33837 | Technical lemma for ~ bnj8... |
| bnj528 33838 | Technical lemma for ~ bnj8... |
| bnj535 33839 | Technical lemma for ~ bnj8... |
| bnj539 33840 | Technical lemma for ~ bnj8... |
| bnj540 33841 | Technical lemma for ~ bnj8... |
| bnj543 33842 | Technical lemma for ~ bnj8... |
| bnj544 33843 | Technical lemma for ~ bnj8... |
| bnj545 33844 | Technical lemma for ~ bnj8... |
| bnj546 33845 | Technical lemma for ~ bnj8... |
| bnj548 33846 | Technical lemma for ~ bnj8... |
| bnj553 33847 | Technical lemma for ~ bnj8... |
| bnj554 33848 | Technical lemma for ~ bnj8... |
| bnj556 33849 | Technical lemma for ~ bnj8... |
| bnj557 33850 | Technical lemma for ~ bnj8... |
| bnj558 33851 | Technical lemma for ~ bnj8... |
| bnj561 33852 | Technical lemma for ~ bnj8... |
| bnj562 33853 | Technical lemma for ~ bnj8... |
| bnj570 33854 | Technical lemma for ~ bnj8... |
| bnj571 33855 | Technical lemma for ~ bnj8... |
| bnj605 33856 | Technical lemma. This lem... |
| bnj581 33857 | Technical lemma for ~ bnj5... |
| bnj589 33858 | Technical lemma for ~ bnj8... |
| bnj590 33859 | Technical lemma for ~ bnj8... |
| bnj591 33860 | Technical lemma for ~ bnj8... |
| bnj594 33861 | Technical lemma for ~ bnj8... |
| bnj580 33862 | Technical lemma for ~ bnj5... |
| bnj579 33863 | Technical lemma for ~ bnj8... |
| bnj602 33864 | Equality theorem for the `... |
| bnj607 33865 | Technical lemma for ~ bnj8... |
| bnj609 33866 | Technical lemma for ~ bnj8... |
| bnj611 33867 | Technical lemma for ~ bnj8... |
| bnj600 33868 | Technical lemma for ~ bnj8... |
| bnj601 33869 | Technical lemma for ~ bnj8... |
| bnj852 33870 | Technical lemma for ~ bnj6... |
| bnj864 33871 | Technical lemma for ~ bnj6... |
| bnj865 33872 | Technical lemma for ~ bnj6... |
| bnj873 33873 | Technical lemma for ~ bnj6... |
| bnj849 33874 | Technical lemma for ~ bnj6... |
| bnj882 33875 | Definition (using hypothes... |
| bnj18eq1 33876 | Equality theorem for trans... |
| bnj893 33877 | Property of ` _trCl ` . U... |
| bnj900 33878 | Technical lemma for ~ bnj6... |
| bnj906 33879 | Property of ` _trCl ` . (... |
| bnj908 33880 | Technical lemma for ~ bnj6... |
| bnj911 33881 | Technical lemma for ~ bnj6... |
| bnj916 33882 | Technical lemma for ~ bnj6... |
| bnj917 33883 | Technical lemma for ~ bnj6... |
| bnj934 33884 | Technical lemma for ~ bnj6... |
| bnj929 33885 | Technical lemma for ~ bnj6... |
| bnj938 33886 | Technical lemma for ~ bnj6... |
| bnj944 33887 | Technical lemma for ~ bnj6... |
| bnj953 33888 | Technical lemma for ~ bnj6... |
| bnj958 33889 | Technical lemma for ~ bnj6... |
| bnj1000 33890 | Technical lemma for ~ bnj8... |
| bnj965 33891 | Technical lemma for ~ bnj8... |
| bnj964 33892 | Technical lemma for ~ bnj6... |
| bnj966 33893 | Technical lemma for ~ bnj6... |
| bnj967 33894 | Technical lemma for ~ bnj6... |
| bnj969 33895 | Technical lemma for ~ bnj6... |
| bnj970 33896 | Technical lemma for ~ bnj6... |
| bnj910 33897 | Technical lemma for ~ bnj6... |
| bnj978 33898 | Technical lemma for ~ bnj6... |
| bnj981 33899 | Technical lemma for ~ bnj6... |
| bnj983 33900 | Technical lemma for ~ bnj6... |
| bnj984 33901 | Technical lemma for ~ bnj6... |
| bnj985v 33902 | Version of ~ bnj985 with a... |
| bnj985 33903 | Technical lemma for ~ bnj6... |
| bnj986 33904 | Technical lemma for ~ bnj6... |
| bnj996 33905 | Technical lemma for ~ bnj6... |
| bnj998 33906 | Technical lemma for ~ bnj6... |
| bnj999 33907 | Technical lemma for ~ bnj6... |
| bnj1001 33908 | Technical lemma for ~ bnj6... |
| bnj1006 33909 | Technical lemma for ~ bnj6... |
| bnj1014 33910 | Technical lemma for ~ bnj6... |
| bnj1015 33911 | Technical lemma for ~ bnj6... |
| bnj1018g 33912 | Version of ~ bnj1018 with ... |
| bnj1018 33913 | Technical lemma for ~ bnj6... |
| bnj1020 33914 | Technical lemma for ~ bnj6... |
| bnj1021 33915 | Technical lemma for ~ bnj6... |
| bnj907 33916 | Technical lemma for ~ bnj6... |
| bnj1029 33917 | Property of ` _trCl ` . (... |
| bnj1033 33918 | Technical lemma for ~ bnj6... |
| bnj1034 33919 | Technical lemma for ~ bnj6... |
| bnj1039 33920 | Technical lemma for ~ bnj6... |
| bnj1040 33921 | Technical lemma for ~ bnj6... |
| bnj1047 33922 | Technical lemma for ~ bnj6... |
| bnj1049 33923 | Technical lemma for ~ bnj6... |
| bnj1052 33924 | Technical lemma for ~ bnj6... |
| bnj1053 33925 | Technical lemma for ~ bnj6... |
| bnj1071 33926 | Technical lemma for ~ bnj6... |
| bnj1083 33927 | Technical lemma for ~ bnj6... |
| bnj1090 33928 | Technical lemma for ~ bnj6... |
| bnj1093 33929 | Technical lemma for ~ bnj6... |
| bnj1097 33930 | Technical lemma for ~ bnj6... |
| bnj1110 33931 | Technical lemma for ~ bnj6... |
| bnj1112 33932 | Technical lemma for ~ bnj6... |
| bnj1118 33933 | Technical lemma for ~ bnj6... |
| bnj1121 33934 | Technical lemma for ~ bnj6... |
| bnj1123 33935 | Technical lemma for ~ bnj6... |
| bnj1030 33936 | Technical lemma for ~ bnj6... |
| bnj1124 33937 | Property of ` _trCl ` . (... |
| bnj1133 33938 | Technical lemma for ~ bnj6... |
| bnj1128 33939 | Technical lemma for ~ bnj6... |
| bnj1127 33940 | Property of ` _trCl ` . (... |
| bnj1125 33941 | Property of ` _trCl ` . (... |
| bnj1145 33942 | Technical lemma for ~ bnj6... |
| bnj1147 33943 | Property of ` _trCl ` . (... |
| bnj1137 33944 | Property of ` _trCl ` . (... |
| bnj1148 33945 | Property of ` _pred ` . (... |
| bnj1136 33946 | Technical lemma for ~ bnj6... |
| bnj1152 33947 | Technical lemma for ~ bnj6... |
| bnj1154 33948 | Property of ` Fr ` . (Con... |
| bnj1171 33949 | Technical lemma for ~ bnj6... |
| bnj1172 33950 | Technical lemma for ~ bnj6... |
| bnj1173 33951 | Technical lemma for ~ bnj6... |
| bnj1174 33952 | Technical lemma for ~ bnj6... |
| bnj1175 33953 | Technical lemma for ~ bnj6... |
| bnj1176 33954 | Technical lemma for ~ bnj6... |
| bnj1177 33955 | Technical lemma for ~ bnj6... |
| bnj1186 33956 | Technical lemma for ~ bnj6... |
| bnj1190 33957 | Technical lemma for ~ bnj6... |
| bnj1189 33958 | Technical lemma for ~ bnj6... |
| bnj69 33959 | Existence of a minimal ele... |
| bnj1228 33960 | Existence of a minimal ele... |
| bnj1204 33961 | Well-founded induction. T... |
| bnj1234 33962 | Technical lemma for ~ bnj6... |
| bnj1245 33963 | Technical lemma for ~ bnj6... |
| bnj1256 33964 | Technical lemma for ~ bnj6... |
| bnj1259 33965 | Technical lemma for ~ bnj6... |
| bnj1253 33966 | Technical lemma for ~ bnj6... |
| bnj1279 33967 | Technical lemma for ~ bnj6... |
| bnj1286 33968 | Technical lemma for ~ bnj6... |
| bnj1280 33969 | Technical lemma for ~ bnj6... |
| bnj1296 33970 | Technical lemma for ~ bnj6... |
| bnj1309 33971 | Technical lemma for ~ bnj6... |
| bnj1307 33972 | Technical lemma for ~ bnj6... |
| bnj1311 33973 | Technical lemma for ~ bnj6... |
| bnj1318 33974 | Technical lemma for ~ bnj6... |
| bnj1326 33975 | Technical lemma for ~ bnj6... |
| bnj1321 33976 | Technical lemma for ~ bnj6... |
| bnj1364 33977 | Property of ` _FrSe ` . (... |
| bnj1371 33978 | Technical lemma for ~ bnj6... |
| bnj1373 33979 | Technical lemma for ~ bnj6... |
| bnj1374 33980 | Technical lemma for ~ bnj6... |
| bnj1384 33981 | Technical lemma for ~ bnj6... |
| bnj1388 33982 | Technical lemma for ~ bnj6... |
| bnj1398 33983 | Technical lemma for ~ bnj6... |
| bnj1413 33984 | Property of ` _trCl ` . (... |
| bnj1408 33985 | Technical lemma for ~ bnj1... |
| bnj1414 33986 | Property of ` _trCl ` . (... |
| bnj1415 33987 | Technical lemma for ~ bnj6... |
| bnj1416 33988 | Technical lemma for ~ bnj6... |
| bnj1418 33989 | Property of ` _pred ` . (... |
| bnj1417 33990 | Technical lemma for ~ bnj6... |
| bnj1421 33991 | Technical lemma for ~ bnj6... |
| bnj1444 33992 | Technical lemma for ~ bnj6... |
| bnj1445 33993 | Technical lemma for ~ bnj6... |
| bnj1446 33994 | Technical lemma for ~ bnj6... |
| bnj1447 33995 | Technical lemma for ~ bnj6... |
| bnj1448 33996 | Technical lemma for ~ bnj6... |
| bnj1449 33997 | Technical lemma for ~ bnj6... |
| bnj1442 33998 | Technical lemma for ~ bnj6... |
| bnj1450 33999 | Technical lemma for ~ bnj6... |
| bnj1423 34000 | Technical lemma for ~ bnj6... |
| bnj1452 34001 | Technical lemma for ~ bnj6... |
| bnj1466 34002 | Technical lemma for ~ bnj6... |
| bnj1467 34003 | Technical lemma for ~ bnj6... |
| bnj1463 34004 | Technical lemma for ~ bnj6... |
| bnj1489 34005 | Technical lemma for ~ bnj6... |
| bnj1491 34006 | Technical lemma for ~ bnj6... |
| bnj1312 34007 | Technical lemma for ~ bnj6... |
| bnj1493 34008 | Technical lemma for ~ bnj6... |
| bnj1497 34009 | Technical lemma for ~ bnj6... |
| bnj1498 34010 | Technical lemma for ~ bnj6... |
| bnj60 34011 | Well-founded recursion, pa... |
| bnj1514 34012 | Technical lemma for ~ bnj1... |
| bnj1518 34013 | Technical lemma for ~ bnj1... |
| bnj1519 34014 | Technical lemma for ~ bnj1... |
| bnj1520 34015 | Technical lemma for ~ bnj1... |
| bnj1501 34016 | Technical lemma for ~ bnj1... |
| bnj1500 34017 | Well-founded recursion, pa... |
| bnj1525 34018 | Technical lemma for ~ bnj1... |
| bnj1529 34019 | Technical lemma for ~ bnj1... |
| bnj1523 34020 | Technical lemma for ~ bnj1... |
| bnj1522 34021 | Well-founded recursion, pa... |
| exdifsn 34022 | There exists an element in... |
| srcmpltd 34023 | If a statement is true for... |
| prsrcmpltd 34024 | If a statement is true for... |
| dff15 34025 | A one-to-one function in t... |
| f1resveqaeq 34026 | If a function restricted t... |
| f1resrcmplf1dlem 34027 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 34028 | If a function's restrictio... |
| funen1cnv 34029 | If a function is equinumer... |
| fnrelpredd 34030 | A function that preserves ... |
| cardpred 34031 | The cardinality function p... |
| nummin 34032 | Every nonempty class of nu... |
| fineqvrep 34033 | If the Axiom of Infinity i... |
| fineqvpow 34034 | If the Axiom of Infinity i... |
| fineqvac 34035 | If the Axiom of Infinity i... |
| fineqvacALT 34036 | Shorter proof of ~ fineqva... |
| zltp1ne 34037 | Integer ordering relation.... |
| nnltp1ne 34038 | Positive integer ordering ... |
| nn0ltp1ne 34039 | Nonnegative integer orderi... |
| 0nn0m1nnn0 34040 | A number is zero if and on... |
| f1resfz0f1d 34041 | If a function with a seque... |
| fisshasheq 34042 | A finite set is equal to i... |
| hashf1dmcdm 34043 | The size of the domain of ... |
| revpfxsfxrev 34044 | The reverse of a prefix of... |
| swrdrevpfx 34045 | A subword expressed in ter... |
| lfuhgr 34046 | A hypergraph is loop-free ... |
| lfuhgr2 34047 | A hypergraph is loop-free ... |
| lfuhgr3 34048 | A hypergraph is loop-free ... |
| cplgredgex 34049 | Any two (distinct) vertice... |
| cusgredgex 34050 | Any two (distinct) vertice... |
| cusgredgex2 34051 | Any two distinct vertices ... |
| pfxwlk 34052 | A prefix of a walk is a wa... |
| revwlk 34053 | The reverse of a walk is a... |
| revwlkb 34054 | Two words represent a walk... |
| swrdwlk 34055 | Two matching subwords of a... |
| pthhashvtx 34056 | A graph containing a path ... |
| pthisspthorcycl 34057 | A path is either a simple ... |
| spthcycl 34058 | A walk is a trivial path i... |
| usgrgt2cycl 34059 | A non-trivial cycle in a s... |
| usgrcyclgt2v 34060 | A simple graph with a non-... |
| subgrwlk 34061 | If a walk exists in a subg... |
| subgrtrl 34062 | If a trail exists in a sub... |
| subgrpth 34063 | If a path exists in a subg... |
| subgrcycl 34064 | If a cycle exists in a sub... |
| cusgr3cyclex 34065 | Every complete simple grap... |
| loop1cycl 34066 | A hypergraph has a cycle o... |
| 2cycld 34067 | Construction of a 2-cycle ... |
| 2cycl2d 34068 | Construction of a 2-cycle ... |
| umgr2cycllem 34069 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 34070 | A multigraph with two dist... |
| dfacycgr1 34073 | An alternate definition of... |
| isacycgr 34074 | The property of being an a... |
| isacycgr1 34075 | The property of being an a... |
| acycgrcycl 34076 | Any cycle in an acyclic gr... |
| acycgr0v 34077 | A null graph (with no vert... |
| acycgr1v 34078 | A multigraph with one vert... |
| acycgr2v 34079 | A simple graph with two ve... |
| prclisacycgr 34080 | A proper class (representi... |
| acycgrislfgr 34081 | An acyclic hypergraph is a... |
| upgracycumgr 34082 | An acyclic pseudograph is ... |
| umgracycusgr 34083 | An acyclic multigraph is a... |
| upgracycusgr 34084 | An acyclic pseudograph is ... |
| cusgracyclt3v 34085 | A complete simple graph is... |
| pthacycspth 34086 | A path in an acyclic graph... |
| acycgrsubgr 34087 | The subgraph of an acyclic... |
| quartfull 34094 | The quartic equation, writ... |
| deranglem 34095 | Lemma for derangements. (... |
| derangval 34096 | Define the derangement fun... |
| derangf 34097 | The derangement number is ... |
| derang0 34098 | The derangement number of ... |
| derangsn 34099 | The derangement number of ... |
| derangenlem 34100 | One half of ~ derangen . ... |
| derangen 34101 | The derangement number is ... |
| subfacval 34102 | The subfactorial is define... |
| derangen2 34103 | Write the derangement numb... |
| subfacf 34104 | The subfactorial is a func... |
| subfaclefac 34105 | The subfactorial is less t... |
| subfac0 34106 | The subfactorial at zero. ... |
| subfac1 34107 | The subfactorial at one. ... |
| subfacp1lem1 34108 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 34109 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 34110 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 34111 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 34112 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 34113 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 34114 | Lemma for ~ subfacp1 . By... |
| subfacp1 34115 | A two-term recurrence for ... |
| subfacval2 34116 | A closed-form expression f... |
| subfaclim 34117 | The subfactorial converges... |
| subfacval3 34118 | Another closed form expres... |
| derangfmla 34119 | The derangements formula, ... |
| erdszelem1 34120 | Lemma for ~ erdsze . (Con... |
| erdszelem2 34121 | Lemma for ~ erdsze . (Con... |
| erdszelem3 34122 | Lemma for ~ erdsze . (Con... |
| erdszelem4 34123 | Lemma for ~ erdsze . (Con... |
| erdszelem5 34124 | Lemma for ~ erdsze . (Con... |
| erdszelem6 34125 | Lemma for ~ erdsze . (Con... |
| erdszelem7 34126 | Lemma for ~ erdsze . (Con... |
| erdszelem8 34127 | Lemma for ~ erdsze . (Con... |
| erdszelem9 34128 | Lemma for ~ erdsze . (Con... |
| erdszelem10 34129 | Lemma for ~ erdsze . (Con... |
| erdszelem11 34130 | Lemma for ~ erdsze . (Con... |
| erdsze 34131 | The Erdős-Szekeres th... |
| erdsze2lem1 34132 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 34133 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 34134 | Generalize the statement o... |
| kur14lem1 34135 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 34136 | Lemma for ~ kur14 . Write... |
| kur14lem3 34137 | Lemma for ~ kur14 . A clo... |
| kur14lem4 34138 | Lemma for ~ kur14 . Compl... |
| kur14lem5 34139 | Lemma for ~ kur14 . Closu... |
| kur14lem6 34140 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 34141 | Lemma for ~ kur14 : main p... |
| kur14lem8 34142 | Lemma for ~ kur14 . Show ... |
| kur14lem9 34143 | Lemma for ~ kur14 . Since... |
| kur14lem10 34144 | Lemma for ~ kur14 . Disch... |
| kur14 34145 | Kuratowski's closure-compl... |
| ispconn 34152 | The property of being a pa... |
| pconncn 34153 | The property of being a pa... |
| pconntop 34154 | A simply connected space i... |
| issconn 34155 | The property of being a si... |
| sconnpconn 34156 | A simply connected space i... |
| sconntop 34157 | A simply connected space i... |
| sconnpht 34158 | A closed path in a simply ... |
| cnpconn 34159 | An image of a path-connect... |
| pconnconn 34160 | A path-connected space is ... |
| txpconn 34161 | The topological product of... |
| ptpconn 34162 | The topological product of... |
| indispconn 34163 | The indiscrete topology (o... |
| connpconn 34164 | A connected and locally pa... |
| qtoppconn 34165 | A quotient of a path-conne... |
| pconnpi1 34166 | All fundamental groups in ... |
| sconnpht2 34167 | Any two paths in a simply ... |
| sconnpi1 34168 | A path-connected topologic... |
| txsconnlem 34169 | Lemma for ~ txsconn . (Co... |
| txsconn 34170 | The topological product of... |
| cvxpconn 34171 | A convex subset of the com... |
| cvxsconn 34172 | A convex subset of the com... |
| blsconn 34173 | An open ball in the comple... |
| cnllysconn 34174 | The topology of the comple... |
| resconn 34175 | A subset of ` RR ` is simp... |
| ioosconn 34176 | An open interval is simply... |
| iccsconn 34177 | A closed interval is simpl... |
| retopsconn 34178 | The real numbers are simpl... |
| iccllysconn 34179 | A closed interval is local... |
| rellysconn 34180 | The real numbers are local... |
| iisconn 34181 | The unit interval is simpl... |
| iillysconn 34182 | The unit interval is local... |
| iinllyconn 34183 | The unit interval is local... |
| fncvm 34186 | Lemma for covering maps. ... |
| cvmscbv 34187 | Change bound variables in ... |
| iscvm 34188 | The property of being a co... |
| cvmtop1 34189 | Reverse closure for a cove... |
| cvmtop2 34190 | Reverse closure for a cove... |
| cvmcn 34191 | A covering map is a contin... |
| cvmcov 34192 | Property of a covering map... |
| cvmsrcl 34193 | Reverse closure for an eve... |
| cvmsi 34194 | One direction of ~ cvmsval... |
| cvmsval 34195 | Elementhood in the set ` S... |
| cvmsss 34196 | An even covering is a subs... |
| cvmsn0 34197 | An even covering is nonemp... |
| cvmsuni 34198 | An even covering of ` U ` ... |
| cvmsdisj 34199 | An even covering of ` U ` ... |
| cvmshmeo 34200 | Every element of an even c... |
| cvmsf1o 34201 | ` F ` , localized to an el... |
| cvmscld 34202 | The sets of an even coveri... |
| cvmsss2 34203 | An open subset of an evenl... |
| cvmcov2 34204 | The covering map property ... |
| cvmseu 34205 | Every element in ` U. T ` ... |
| cvmsiota 34206 | Identify the unique elemen... |
| cvmopnlem 34207 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 34208 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 34209 | A covering map is an open ... |
| cvmliftmolem1 34210 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 34211 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 34212 | A lift of a continuous fun... |
| cvmliftmo 34213 | A lift of a continuous fun... |
| cvmliftlem1 34214 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 34215 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 34216 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 34217 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 34218 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 34219 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 34220 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 34221 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 34222 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 34223 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 34224 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 34225 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 34226 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 34227 | Lemma for ~ cvmlift . Dis... |
| cvmlift 34228 | One of the important prope... |
| cvmfo 34229 | A covering map is an onto ... |
| cvmliftiota 34230 | Write out a function ` H `... |
| cvmlift2lem1 34231 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 34232 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 34233 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 34234 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 34235 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 34236 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 34237 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 34238 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 34239 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 34240 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 34241 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 34242 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 34243 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 34244 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 34245 | A two-dimensional version ... |
| cvmliftphtlem 34246 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 34247 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 34248 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 34249 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 34250 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 34251 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 34252 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 34253 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 34254 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 34255 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 34256 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 34257 | A general version of ~ cvm... |
| snmlff 34258 | The function ` F ` from ~ ... |
| snmlfval 34259 | The function ` F ` from ~ ... |
| snmlval 34260 | The property " ` A ` is si... |
| snmlflim 34261 | If ` A ` is simply normal,... |
| goel 34276 | A "Godel-set of membership... |
| goelel3xp 34277 | A "Godel-set of membership... |
| goeleq12bg 34278 | Two "Godel-set of membersh... |
| gonafv 34279 | The "Godel-set for the She... |
| goaleq12d 34280 | Equality of the "Godel-set... |
| gonanegoal 34281 | The Godel-set for the Shef... |
| satf 34282 | The satisfaction predicate... |
| satfsucom 34283 | The satisfaction predicate... |
| satfn 34284 | The satisfaction predicate... |
| satom 34285 | The satisfaction predicate... |
| satfvsucom 34286 | The satisfaction predicate... |
| satfv0 34287 | The value of the satisfact... |
| satfvsuclem1 34288 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 34289 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 34290 | The value of the satisfact... |
| satfv1lem 34291 | Lemma for ~ satfv1 . (Con... |
| satfv1 34292 | The value of the satisfact... |
| satfsschain 34293 | The binary relation of a s... |
| satfvsucsuc 34294 | The satisfaction predicate... |
| satfbrsuc 34295 | The binary relation of a s... |
| satfrel 34296 | The value of the satisfact... |
| satfdmlem 34297 | Lemma for ~ satfdm . (Con... |
| satfdm 34298 | The domain of the satisfac... |
| satfrnmapom 34299 | The range of the satisfact... |
| satfv0fun 34300 | The value of the satisfact... |
| satf0 34301 | The satisfaction predicate... |
| satf0sucom 34302 | The satisfaction predicate... |
| satf00 34303 | The value of the satisfact... |
| satf0suclem 34304 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 34305 | The value of the satisfact... |
| satf0op 34306 | An element of a value of t... |
| satf0n0 34307 | The value of the satisfact... |
| sat1el2xp 34308 | The first component of an ... |
| fmlafv 34309 | The valid Godel formulas o... |
| fmla 34310 | The set of all valid Godel... |
| fmla0 34311 | The valid Godel formulas o... |
| fmla0xp 34312 | The valid Godel formulas o... |
| fmlasuc0 34313 | The valid Godel formulas o... |
| fmlafvel 34314 | A class is a valid Godel f... |
| fmlasuc 34315 | The valid Godel formulas o... |
| fmla1 34316 | The valid Godel formulas o... |
| isfmlasuc 34317 | The characterization of a ... |
| fmlasssuc 34318 | The Godel formulas of heig... |
| fmlaomn0 34319 | The empty set is not a God... |
| fmlan0 34320 | The empty set is not a God... |
| gonan0 34321 | The "Godel-set of NAND" is... |
| goaln0 34322 | The "Godel-set of universa... |
| gonarlem 34323 | Lemma for ~ gonar (inducti... |
| gonar 34324 | If the "Godel-set of NAND"... |
| goalrlem 34325 | Lemma for ~ goalr (inducti... |
| goalr 34326 | If the "Godel-set of unive... |
| fmla0disjsuc 34327 | The set of valid Godel for... |
| fmlasucdisj 34328 | The valid Godel formulas o... |
| satfdmfmla 34329 | The domain of the satisfac... |
| satffunlem 34330 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 34331 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 34332 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 34333 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 34334 | The domain of an ordered p... |
| satffunlem2lem2 34335 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 34336 | Lemma 1 for ~ satffun : in... |
| satffunlem2 34337 | Lemma 2 for ~ satffun : in... |
| satffun 34338 | The value of the satisfact... |
| satff 34339 | The satisfaction predicate... |
| satfun 34340 | The satisfaction predicate... |
| satfvel 34341 | An element of the value of... |
| satfv0fvfmla0 34342 | The value of the satisfact... |
| satefv 34343 | The simplified satisfactio... |
| sate0 34344 | The simplified satisfactio... |
| satef 34345 | The simplified satisfactio... |
| sate0fv0 34346 | A simplified satisfaction ... |
| satefvfmla0 34347 | The simplified satisfactio... |
| sategoelfvb 34348 | Characterization of a valu... |
| sategoelfv 34349 | Condition of a valuation `... |
| ex-sategoelel 34350 | Example of a valuation of ... |
| ex-sategoel 34351 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 34352 | The value of the satisfact... |
| 2goelgoanfmla1 34353 | Two Godel-sets of membersh... |
| satefvfmla1 34354 | The simplified satisfactio... |
| ex-sategoelelomsuc 34355 | Example of a valuation of ... |
| ex-sategoelel12 34356 | Example of a valuation of ... |
| prv 34357 | The "proves" relation on a... |
| elnanelprv 34358 | The wff ` ( A e. B -/\ B e... |
| prv0 34359 | Every wff encoded as ` U `... |
| prv1n 34360 | No wff encoded as a Godel-... |
| mvtval 34429 | The set of variable typeco... |
| mrexval 34430 | The set of "raw expression... |
| mexval 34431 | The set of expressions, wh... |
| mexval2 34432 | The set of expressions, wh... |
| mdvval 34433 | The set of disjoint variab... |
| mvrsval 34434 | The set of variables in an... |
| mvrsfpw 34435 | The set of variables in an... |
| mrsubffval 34436 | The substitution of some v... |
| mrsubfval 34437 | The substitution of some v... |
| mrsubval 34438 | The substitution of some v... |
| mrsubcv 34439 | The value of a substituted... |
| mrsubvr 34440 | The value of a substituted... |
| mrsubff 34441 | A substitution is a functi... |
| mrsubrn 34442 | Although it is defined for... |
| mrsubff1 34443 | When restricted to complet... |
| mrsubff1o 34444 | When restricted to complet... |
| mrsub0 34445 | The value of the substitut... |
| mrsubf 34446 | A substitution is a functi... |
| mrsubccat 34447 | Substitution distributes o... |
| mrsubcn 34448 | A substitution does not ch... |
| elmrsubrn 34449 | Characterization of the su... |
| mrsubco 34450 | The composition of two sub... |
| mrsubvrs 34451 | The set of variables in a ... |
| msubffval 34452 | A substitution applied to ... |
| msubfval 34453 | A substitution applied to ... |
| msubval 34454 | A substitution applied to ... |
| msubrsub 34455 | A substitution applied to ... |
| msubty 34456 | The type of a substituted ... |
| elmsubrn 34457 | Characterization of substi... |
| msubrn 34458 | Although it is defined for... |
| msubff 34459 | A substitution is a functi... |
| msubco 34460 | The composition of two sub... |
| msubf 34461 | A substitution is a functi... |
| mvhfval 34462 | Value of the function mapp... |
| mvhval 34463 | Value of the function mapp... |
| mpstval 34464 | A pre-statement is an orde... |
| elmpst 34465 | Property of being a pre-st... |
| msrfval 34466 | Value of the reduct of a p... |
| msrval 34467 | Value of the reduct of a p... |
| mpstssv 34468 | A pre-statement is an orde... |
| mpst123 34469 | Decompose a pre-statement ... |
| mpstrcl 34470 | The elements of a pre-stat... |
| msrf 34471 | The reduct of a pre-statem... |
| msrrcl 34472 | If ` X ` and ` Y ` have th... |
| mstaval 34473 | Value of the set of statem... |
| msrid 34474 | The reduct of a statement ... |
| msrfo 34475 | The reduct of a pre-statem... |
| mstapst 34476 | A statement is a pre-state... |
| elmsta 34477 | Property of being a statem... |
| ismfs 34478 | A formal system is a tuple... |
| mfsdisj 34479 | The constants and variable... |
| mtyf2 34480 | The type function maps var... |
| mtyf 34481 | The type function maps var... |
| mvtss 34482 | The set of variable typeco... |
| maxsta 34483 | An axiom is a statement. ... |
| mvtinf 34484 | Each variable typecode has... |
| msubff1 34485 | When restricted to complet... |
| msubff1o 34486 | When restricted to complet... |
| mvhf 34487 | The function mapping varia... |
| mvhf1 34488 | The function mapping varia... |
| msubvrs 34489 | The set of variables in a ... |
| mclsrcl 34490 | Reverse closure for the cl... |
| mclsssvlem 34491 | Lemma for ~ mclsssv . (Co... |
| mclsval 34492 | The function mapping varia... |
| mclsssv 34493 | The closure of a set of ex... |
| ssmclslem 34494 | Lemma for ~ ssmcls . (Con... |
| vhmcls 34495 | All variable hypotheses ar... |
| ssmcls 34496 | The original expressions a... |
| ss2mcls 34497 | The closure is monotonic u... |
| mclsax 34498 | The closure is closed unde... |
| mclsind 34499 | Induction theorem for clos... |
| mppspstlem 34500 | Lemma for ~ mppspst . (Co... |
| mppsval 34501 | Definition of a provable p... |
| elmpps 34502 | Definition of a provable p... |
| mppspst 34503 | A provable pre-statement i... |
| mthmval 34504 | A theorem is a pre-stateme... |
| elmthm 34505 | A theorem is a pre-stateme... |
| mthmi 34506 | A statement whose reduct i... |
| mthmsta 34507 | A theorem is a pre-stateme... |
| mppsthm 34508 | A provable pre-statement i... |
| mthmblem 34509 | Lemma for ~ mthmb . (Cont... |
| mthmb 34510 | If two statements have the... |
| mthmpps 34511 | Given a theorem, there is ... |
| mclsppslem 34512 | The closure is closed unde... |
| mclspps 34513 | The closure is closed unde... |
| problem1 34588 | Practice problem 1. Clues... |
| problem2 34589 | Practice problem 2. Clues... |
| problem3 34590 | Practice problem 3. Clues... |
| problem4 34591 | Practice problem 4. Clues... |
| problem5 34592 | Practice problem 5. Clues... |
| quad3 34593 | Variant of quadratic equat... |
| climuzcnv 34594 | Utility lemma to convert b... |
| sinccvglem 34595 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 34596 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 34597 | The circumference of a cir... |
| elfzm12 34598 | Membership in a curtailed ... |
| nn0seqcvg 34599 | A strictly-decreasing nonn... |
| lediv2aALT 34600 | Division of both sides of ... |
| abs2sqlei 34601 | The absolute values of two... |
| abs2sqlti 34602 | The absolute values of two... |
| abs2sqle 34603 | The absolute values of two... |
| abs2sqlt 34604 | The absolute values of two... |
| abs2difi 34605 | Difference of absolute val... |
| abs2difabsi 34606 | Absolute value of differen... |
| currybi 34607 | Biconditional version of C... |
| axextprim 34608 | ~ ax-ext without distinct ... |
| axrepprim 34609 | ~ ax-rep without distinct ... |
| axunprim 34610 | ~ ax-un without distinct v... |
| axpowprim 34611 | ~ ax-pow without distinct ... |
| axregprim 34612 | ~ ax-reg without distinct ... |
| axinfprim 34613 | ~ ax-inf without distinct ... |
| axacprim 34614 | ~ ax-ac without distinct v... |
| untelirr 34615 | We call a class "untanged"... |
| untuni 34616 | The union of a class is un... |
| untsucf 34617 | If a class is untangled, t... |
| unt0 34618 | The null set is untangled.... |
| untint 34619 | If there is an untangled e... |
| efrunt 34620 | If ` A ` is well-founded b... |
| untangtr 34621 | A transitive class is unta... |
| 3jaodd 34622 | Double deduction form of ~... |
| 3orit 34623 | Closed form of ~ 3ori . (... |
| biimpexp 34624 | A biconditional in the ant... |
| nepss 34625 | Two classes are unequal if... |
| 3ccased 34626 | Triple disjunction form of... |
| dfso3 34627 | Expansion of the definitio... |
| brtpid1 34628 | A binary relation involvin... |
| brtpid2 34629 | A binary relation involvin... |
| brtpid3 34630 | A binary relation involvin... |
| iota5f 34631 | A method for computing iot... |
| jath 34632 | Closed form of ~ ja . Pro... |
| xpab 34633 | Cartesian product of two c... |
| nnuni 34634 | The union of a finite ordi... |
| sqdivzi 34635 | Distribution of square ove... |
| supfz 34636 | The supremum of a finite s... |
| inffz 34637 | The infimum of a finite se... |
| fz0n 34638 | The sequence ` ( 0 ... ( N... |
| shftvalg 34639 | Value of a sequence shifte... |
| divcnvlin 34640 | Limit of the ratio of two ... |
| climlec3 34641 | Comparison of a constant t... |
| logi 34642 | Calculate the logarithm of... |
| iexpire 34643 | ` _i ` raised to itself is... |
| bcneg1 34644 | The binomial coefficent ov... |
| bcm1nt 34645 | The proportion of one bion... |
| bcprod 34646 | A product identity for bin... |
| bccolsum 34647 | A column-sum rule for bino... |
| iprodefisumlem 34648 | Lemma for ~ iprodefisum . ... |
| iprodefisum 34649 | Applying the exponential f... |
| iprodgam 34650 | An infinite product versio... |
| faclimlem1 34651 | Lemma for ~ faclim . Clos... |
| faclimlem2 34652 | Lemma for ~ faclim . Show... |
| faclimlem3 34653 | Lemma for ~ faclim . Alge... |
| faclim 34654 | An infinite product expres... |
| iprodfac 34655 | An infinite product expres... |
| faclim2 34656 | Another factorial limit du... |
| gcd32 34657 | Swap the second and third ... |
| gcdabsorb 34658 | Absorption law for gcd. (... |
| dftr6 34659 | A potential definition of ... |
| coep 34660 | Composition with the membe... |
| coepr 34661 | Composition with the conve... |
| dffr5 34662 | A quantifier-free definiti... |
| dfso2 34663 | Quantifier-free definition... |
| br8 34664 | Substitution for an eight-... |
| br6 34665 | Substitution for a six-pla... |
| br4 34666 | Substitution for a four-pl... |
| cnvco1 34667 | Another distributive law o... |
| cnvco2 34668 | Another distributive law o... |
| eldm3 34669 | Quantifier-free definition... |
| elrn3 34670 | Quantifier-free definition... |
| pocnv 34671 | The converse of a partial ... |
| socnv 34672 | The converse of a strict o... |
| sotrd 34673 | Transitivity law for stric... |
| elintfv 34674 | Membership in an intersect... |
| funpsstri 34675 | A condition for subset tri... |
| fundmpss 34676 | If a class ` F ` is a prop... |
| funsseq 34677 | Given two functions with e... |
| fununiq 34678 | The uniqueness condition o... |
| funbreq 34679 | An equality condition for ... |
| br1steq 34680 | Uniqueness condition for t... |
| br2ndeq 34681 | Uniqueness condition for t... |
| dfdm5 34682 | Definition of domain in te... |
| dfrn5 34683 | Definition of range in ter... |
| opelco3 34684 | Alternate way of saying th... |
| elima4 34685 | Quantifier-free expression... |
| fv1stcnv 34686 | The value of the converse ... |
| fv2ndcnv 34687 | The value of the converse ... |
| setinds 34688 | Principle of set induction... |
| setinds2f 34689 | ` _E ` induction schema, u... |
| setinds2 34690 | ` _E ` induction schema, u... |
| elpotr 34691 | A class of transitive sets... |
| dford5reg 34692 | Given ~ ax-reg , an ordina... |
| dfon2lem1 34693 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 34694 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 34695 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 34696 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 34697 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 34698 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 34699 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 34700 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 34701 | Lemma for ~ dfon2 . A cla... |
| dfon2 34702 | ` On ` consists of all set... |
| rdgprc0 34703 | The value of the recursive... |
| rdgprc 34704 | The value of the recursive... |
| dfrdg2 34705 | Alternate definition of th... |
| dfrdg3 34706 | Generalization of ~ dfrdg2... |
| axextdfeq 34707 | A version of ~ ax-ext for ... |
| ax8dfeq 34708 | A version of ~ ax-8 for us... |
| axextdist 34709 | ~ ax-ext with distinctors ... |
| axextbdist 34710 | ~ axextb with distinctors ... |
| 19.12b 34711 | Version of ~ 19.12vv with ... |
| exnel 34712 | There is always a set not ... |
| distel 34713 | Distinctors in terms of me... |
| axextndbi 34714 | ~ axextnd as a bicondition... |
| hbntg 34715 | A more general form of ~ h... |
| hbimtg 34716 | A more general and closed ... |
| hbaltg 34717 | A more general and closed ... |
| hbng 34718 | A more general form of ~ h... |
| hbimg 34719 | A more general form of ~ h... |
| wsuceq123 34724 | Equality theorem for well-... |
| wsuceq1 34725 | Equality theorem for well-... |
| wsuceq2 34726 | Equality theorem for well-... |
| wsuceq3 34727 | Equality theorem for well-... |
| nfwsuc 34728 | Bound-variable hypothesis ... |
| wlimeq12 34729 | Equality theorem for the l... |
| wlimeq1 34730 | Equality theorem for the l... |
| wlimeq2 34731 | Equality theorem for the l... |
| nfwlim 34732 | Bound-variable hypothesis ... |
| elwlim 34733 | Membership in the limit cl... |
| wzel 34734 | The zero of a well-founded... |
| wsuclem 34735 | Lemma for the supremum pro... |
| wsucex 34736 | Existence theorem for well... |
| wsuccl 34737 | If ` X ` is a set with an ... |
| wsuclb 34738 | A well-founded successor i... |
| wlimss 34739 | The class of limit points ... |
| txpss3v 34788 | A tail Cartesian product i... |
| txprel 34789 | A tail Cartesian product i... |
| brtxp 34790 | Characterize a ternary rel... |
| brtxp2 34791 | The binary relation over a... |
| dfpprod2 34792 | Expanded definition of par... |
| pprodcnveq 34793 | A converse law for paralle... |
| pprodss4v 34794 | The parallel product is a ... |
| brpprod 34795 | Characterize a quaternary ... |
| brpprod3a 34796 | Condition for parallel pro... |
| brpprod3b 34797 | Condition for parallel pro... |
| relsset 34798 | The subset class is a bina... |
| brsset 34799 | For sets, the ` SSet ` bin... |
| idsset 34800 | ` _I ` is equal to the int... |
| eltrans 34801 | Membership in the class of... |
| dfon3 34802 | A quantifier-free definiti... |
| dfon4 34803 | Another quantifier-free de... |
| brtxpsd 34804 | Expansion of a common form... |
| brtxpsd2 34805 | Another common abbreviatio... |
| brtxpsd3 34806 | A third common abbreviatio... |
| relbigcup 34807 | The ` Bigcup ` relationshi... |
| brbigcup 34808 | Binary relation over ` Big... |
| dfbigcup2 34809 | ` Bigcup ` using maps-to n... |
| fobigcup 34810 | ` Bigcup ` maps the univer... |
| fnbigcup 34811 | ` Bigcup ` is a function o... |
| fvbigcup 34812 | For sets, ` Bigcup ` yield... |
| elfix 34813 | Membership in the fixpoint... |
| elfix2 34814 | Alternative membership in ... |
| dffix2 34815 | The fixpoints of a class i... |
| fixssdm 34816 | The fixpoints of a class a... |
| fixssrn 34817 | The fixpoints of a class a... |
| fixcnv 34818 | The fixpoints of a class a... |
| fixun 34819 | The fixpoint operator dist... |
| ellimits 34820 | Membership in the class of... |
| limitssson 34821 | The class of all limit ord... |
| dfom5b 34822 | A quantifier-free definiti... |
| sscoid 34823 | A condition for subset and... |
| dffun10 34824 | Another potential definiti... |
| elfuns 34825 | Membership in the class of... |
| elfunsg 34826 | Closed form of ~ elfuns . ... |
| brsingle 34827 | The binary relation form o... |
| elsingles 34828 | Membership in the class of... |
| fnsingle 34829 | The singleton relationship... |
| fvsingle 34830 | The value of the singleton... |
| dfsingles2 34831 | Alternate definition of th... |
| snelsingles 34832 | A singleton is a member of... |
| dfiota3 34833 | A definition of iota using... |
| dffv5 34834 | Another quantifier-free de... |
| unisnif 34835 | Express union of singleton... |
| brimage 34836 | Binary relation form of th... |
| brimageg 34837 | Closed form of ~ brimage .... |
| funimage 34838 | ` Image A ` is a function.... |
| fnimage 34839 | ` Image R ` is a function ... |
| imageval 34840 | The image functor in maps-... |
| fvimage 34841 | Value of the image functor... |
| brcart 34842 | Binary relation form of th... |
| brdomain 34843 | Binary relation form of th... |
| brrange 34844 | Binary relation form of th... |
| brdomaing 34845 | Closed form of ~ brdomain ... |
| brrangeg 34846 | Closed form of ~ brrange .... |
| brimg 34847 | Binary relation form of th... |
| brapply 34848 | Binary relation form of th... |
| brcup 34849 | Binary relation form of th... |
| brcap 34850 | Binary relation form of th... |
| brsuccf 34851 | Binary relation form of th... |
| funpartlem 34852 | Lemma for ~ funpartfun . ... |
| funpartfun 34853 | The functional part of ` F... |
| funpartss 34854 | The functional part of ` F... |
| funpartfv 34855 | The function value of the ... |
| fullfunfnv 34856 | The full functional part o... |
| fullfunfv 34857 | The function value of the ... |
| brfullfun 34858 | A binary relation form con... |
| brrestrict 34859 | Binary relation form of th... |
| dfrecs2 34860 | A quantifier-free definiti... |
| dfrdg4 34861 | A quantifier-free definiti... |
| dfint3 34862 | Quantifier-free definition... |
| imagesset 34863 | The Image functor applied ... |
| brub 34864 | Binary relation form of th... |
| brlb 34865 | Binary relation form of th... |
| altopex 34870 | Alternative ordered pairs ... |
| altopthsn 34871 | Two alternate ordered pair... |
| altopeq12 34872 | Equality for alternate ord... |
| altopeq1 34873 | Equality for alternate ord... |
| altopeq2 34874 | Equality for alternate ord... |
| altopth1 34875 | Equality of the first memb... |
| altopth2 34876 | Equality of the second mem... |
| altopthg 34877 | Alternate ordered pair the... |
| altopthbg 34878 | Alternate ordered pair the... |
| altopth 34879 | The alternate ordered pair... |
| altopthb 34880 | Alternate ordered pair the... |
| altopthc 34881 | Alternate ordered pair the... |
| altopthd 34882 | Alternate ordered pair the... |
| altxpeq1 34883 | Equality for alternate Car... |
| altxpeq2 34884 | Equality for alternate Car... |
| elaltxp 34885 | Membership in alternate Ca... |
| altopelaltxp 34886 | Alternate ordered pair mem... |
| altxpsspw 34887 | An inclusion rule for alte... |
| altxpexg 34888 | The alternate Cartesian pr... |
| rankaltopb 34889 | Compute the rank of an alt... |
| nfaltop 34890 | Bound-variable hypothesis ... |
| sbcaltop 34891 | Distribution of class subs... |
| cgrrflx2d 34894 | Deduction form of ~ axcgrr... |
| cgrtr4d 34895 | Deduction form of ~ axcgrt... |
| cgrtr4and 34896 | Deduction form of ~ axcgrt... |
| cgrrflx 34897 | Reflexivity law for congru... |
| cgrrflxd 34898 | Deduction form of ~ cgrrfl... |
| cgrcomim 34899 | Congruence commutes on the... |
| cgrcom 34900 | Congruence commutes betwee... |
| cgrcomand 34901 | Deduction form of ~ cgrcom... |
| cgrtr 34902 | Transitivity law for congr... |
| cgrtrand 34903 | Deduction form of ~ cgrtr ... |
| cgrtr3 34904 | Transitivity law for congr... |
| cgrtr3and 34905 | Deduction form of ~ cgrtr3... |
| cgrcoml 34906 | Congruence commutes on the... |
| cgrcomr 34907 | Congruence commutes on the... |
| cgrcomlr 34908 | Congruence commutes on bot... |
| cgrcomland 34909 | Deduction form of ~ cgrcom... |
| cgrcomrand 34910 | Deduction form of ~ cgrcom... |
| cgrcomlrand 34911 | Deduction form of ~ cgrcom... |
| cgrtriv 34912 | Degenerate segments are co... |
| cgrid2 34913 | Identity law for congruenc... |
| cgrdegen 34914 | Two congruent segments are... |
| brofs 34915 | Binary relation form of th... |
| 5segofs 34916 | Rephrase ~ ax5seg using th... |
| ofscom 34917 | The outer five segment pre... |
| cgrextend 34918 | Link congruence over a pai... |
| cgrextendand 34919 | Deduction form of ~ cgrext... |
| segconeq 34920 | Two points that satisfy th... |
| segconeu 34921 | Existential uniqueness ver... |
| btwntriv2 34922 | Betweenness always holds f... |
| btwncomim 34923 | Betweenness commutes. Imp... |
| btwncom 34924 | Betweenness commutes. (Co... |
| btwncomand 34925 | Deduction form of ~ btwnco... |
| btwntriv1 34926 | Betweenness always holds f... |
| btwnswapid 34927 | If you can swap the first ... |
| btwnswapid2 34928 | If you can swap arguments ... |
| btwnintr 34929 | Inner transitivity law for... |
| btwnexch3 34930 | Exchange the first endpoin... |
| btwnexch3and 34931 | Deduction form of ~ btwnex... |
| btwnouttr2 34932 | Outer transitivity law for... |
| btwnexch2 34933 | Exchange the outer point o... |
| btwnouttr 34934 | Outer transitivity law for... |
| btwnexch 34935 | Outer transitivity law for... |
| btwnexchand 34936 | Deduction form of ~ btwnex... |
| btwndiff 34937 | There is always a ` c ` di... |
| trisegint 34938 | A line segment between two... |
| funtransport 34941 | The ` TransportTo ` relati... |
| fvtransport 34942 | Calculate the value of the... |
| transportcl 34943 | Closure law for segment tr... |
| transportprops 34944 | Calculate the defining pro... |
| brifs 34953 | Binary relation form of th... |
| ifscgr 34954 | Inner five segment congrue... |
| cgrsub 34955 | Removing identical parts f... |
| brcgr3 34956 | Binary relation form of th... |
| cgr3permute3 34957 | Permutation law for three-... |
| cgr3permute1 34958 | Permutation law for three-... |
| cgr3permute2 34959 | Permutation law for three-... |
| cgr3permute4 34960 | Permutation law for three-... |
| cgr3permute5 34961 | Permutation law for three-... |
| cgr3tr4 34962 | Transitivity law for three... |
| cgr3com 34963 | Commutativity law for thre... |
| cgr3rflx 34964 | Identity law for three-pla... |
| cgrxfr 34965 | A line segment can be divi... |
| btwnxfr 34966 | A condition for extending ... |
| colinrel 34967 | Colinearity is a relations... |
| brcolinear2 34968 | Alternate colinearity bina... |
| brcolinear 34969 | The binary relation form o... |
| colinearex 34970 | The colinear predicate exi... |
| colineardim1 34971 | If ` A ` is colinear with ... |
| colinearperm1 34972 | Permutation law for coline... |
| colinearperm3 34973 | Permutation law for coline... |
| colinearperm2 34974 | Permutation law for coline... |
| colinearperm4 34975 | Permutation law for coline... |
| colinearperm5 34976 | Permutation law for coline... |
| colineartriv1 34977 | Trivial case of colinearit... |
| colineartriv2 34978 | Trivial case of colinearit... |
| btwncolinear1 34979 | Betweenness implies coline... |
| btwncolinear2 34980 | Betweenness implies coline... |
| btwncolinear3 34981 | Betweenness implies coline... |
| btwncolinear4 34982 | Betweenness implies coline... |
| btwncolinear5 34983 | Betweenness implies coline... |
| btwncolinear6 34984 | Betweenness implies coline... |
| colinearxfr 34985 | Transfer law for colineari... |
| lineext 34986 | Extend a line with a missi... |
| brofs2 34987 | Change some conditions for... |
| brifs2 34988 | Change some conditions for... |
| brfs 34989 | Binary relation form of th... |
| fscgr 34990 | Congruence law for the gen... |
| linecgr 34991 | Congruence rule for lines.... |
| linecgrand 34992 | Deduction form of ~ linecg... |
| lineid 34993 | Identity law for points on... |
| idinside 34994 | Law for finding a point in... |
| endofsegid 34995 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 34996 | Deduction form of ~ endofs... |
| btwnconn1lem1 34997 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 34998 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 34999 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 35000 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 35001 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 35002 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 35003 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 35004 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 35005 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 35006 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 35007 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 35008 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 35009 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 35010 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 35011 | Connectitivy law for betwe... |
| btwnconn2 35012 | Another connectivity law f... |
| btwnconn3 35013 | Inner connectivity law for... |
| midofsegid 35014 | If two points fall in the ... |
| segcon2 35015 | Generalization of ~ axsegc... |
| brsegle 35018 | Binary relation form of th... |
| brsegle2 35019 | Alternate characterization... |
| seglecgr12im 35020 | Substitution law for segme... |
| seglecgr12 35021 | Substitution law for segme... |
| seglerflx 35022 | Segment comparison is refl... |
| seglemin 35023 | Any segment is at least as... |
| segletr 35024 | Segment less than is trans... |
| segleantisym 35025 | Antisymmetry law for segme... |
| seglelin 35026 | Linearity law for segment ... |
| btwnsegle 35027 | If ` B ` falls between ` A... |
| colinbtwnle 35028 | Given three colinear point... |
| broutsideof 35031 | Binary relation form of ` ... |
| broutsideof2 35032 | Alternate form of ` Outsid... |
| outsidene1 35033 | Outsideness implies inequa... |
| outsidene2 35034 | Outsideness implies inequa... |
| btwnoutside 35035 | A principle linking outsid... |
| broutsideof3 35036 | Characterization of outsid... |
| outsideofrflx 35037 | Reflexivity of outsideness... |
| outsideofcom 35038 | Commutativity law for outs... |
| outsideoftr 35039 | Transitivity law for outsi... |
| outsideofeq 35040 | Uniqueness law for ` Outsi... |
| outsideofeu 35041 | Given a nondegenerate ray,... |
| outsidele 35042 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 35043 | Outside of implies colinea... |
| funray 35050 | Show that the ` Ray ` rela... |
| fvray 35051 | Calculate the value of the... |
| funline 35052 | Show that the ` Line ` rel... |
| linedegen 35053 | When ` Line ` is applied w... |
| fvline 35054 | Calculate the value of the... |
| liness 35055 | A line is a subset of the ... |
| fvline2 35056 | Alternate definition of a ... |
| lineunray 35057 | A line is composed of a po... |
| lineelsb2 35058 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 35059 | Reflexivity law for line m... |
| linecom 35060 | Commutativity law for line... |
| linerflx2 35061 | Reflexivity law for line m... |
| ellines 35062 | Membership in the set of a... |
| linethru 35063 | If ` A ` is a line contain... |
| hilbert1.1 35064 | There is a line through an... |
| hilbert1.2 35065 | There is at most one line ... |
| linethrueu 35066 | There is a unique line goi... |
| lineintmo 35067 | Two distinct lines interse... |
| fwddifval 35072 | Calculate the value of the... |
| fwddifnval 35073 | The value of the forward d... |
| fwddifn0 35074 | The value of the n-iterate... |
| fwddifnp1 35075 | The value of the n-iterate... |
| rankung 35076 | The rank of the union of t... |
| ranksng 35077 | The rank of a singleton. ... |
| rankelg 35078 | The membership relation is... |
| rankpwg 35079 | The rank of a power set. ... |
| rank0 35080 | The rank of the empty set ... |
| rankeq1o 35081 | The only set with rank ` 1... |
| elhf 35084 | Membership in the heredita... |
| elhf2 35085 | Alternate form of membersh... |
| elhf2g 35086 | Hereditarily finiteness vi... |
| 0hf 35087 | The empty set is a heredit... |
| hfun 35088 | The union of two HF sets i... |
| hfsn 35089 | The singleton of an HF set... |
| hfadj 35090 | Adjoining one HF element t... |
| hfelhf 35091 | Any member of an HF set is... |
| hftr 35092 | The class of all hereditar... |
| hfext 35093 | Extensionality for HF sets... |
| hfuni 35094 | The union of an HF set is ... |
| hfpw 35095 | The power class of an HF s... |
| hfninf 35096 | ` _om ` is not hereditaril... |
| mpomulf 35097 | Multiplication is an opera... |
| ovmul 35098 | Multiplication of complex ... |
| mpomulex 35099 | The multiplication operati... |
| mpomulcn 35100 | Complex number multiplicat... |
| gg-divcn 35101 | Complex number division is... |
| gg-expcn 35102 | The power function on comp... |
| gg-divccn 35103 | Division by a nonzero cons... |
| gg-negcncf 35104 | The negative function is c... |
| gg-iihalf1cn 35105 | The first half function is... |
| gg-iihalf2cn 35106 | The second half function i... |
| gg-iimulcn 35107 | Multiplication is a contin... |
| gg-icchmeo 35108 | The natural bijection from... |
| gg-cnrehmeo 35109 | The canonical bijection fr... |
| gg-reparphti 35110 | Lemma for ~ reparpht . (C... |
| gg-mulcncf 35111 | The multiplication of two ... |
| gg-dvcnp2 35112 | A function is continuous a... |
| gg-dvmulbr 35113 | The product rule for deriv... |
| gg-dvcobr 35114 | The chain rule for derivat... |
| gg-plycn 35115 | A polynomial is a continuo... |
| gg-psercn2 35116 | Since by ~ pserulm the ser... |
| gg-rmulccn 35117 | Multiplication by a real c... |
| gg-cnfldex 35118 | The field of complex numbe... |
| gg-cmvth 35119 | Cauchy's Mean Value Theore... |
| gg-dvfsumle 35120 | Compare a finite sum to an... |
| gg-dvfsumlem2 35121 | Lemma for ~ dvfsumrlim . ... |
| gg-cxpcn 35122 | Domain of continuity of th... |
| a1i14 35123 | Add two antecedents to a w... |
| a1i24 35124 | Add two antecedents to a w... |
| exp5d 35125 | An exportation inference. ... |
| exp5g 35126 | An exportation inference. ... |
| exp5k 35127 | An exportation inference. ... |
| exp56 35128 | An exportation inference. ... |
| exp58 35129 | An exportation inference. ... |
| exp510 35130 | An exportation inference. ... |
| exp511 35131 | An exportation inference. ... |
| exp512 35132 | An exportation inference. ... |
| 3com12d 35133 | Commutation in consequent.... |
| imp5p 35134 | A triple importation infer... |
| imp5q 35135 | A triple importation infer... |
| ecase13d 35136 | Deduction for elimination ... |
| subtr 35137 | Transitivity of implicit s... |
| subtr2 35138 | Transitivity of implicit s... |
| trer 35139 | A relation intersected wit... |
| elicc3 35140 | An equivalent membership c... |
| finminlem 35141 | A useful lemma about finit... |
| gtinf 35142 | Any number greater than an... |
| opnrebl 35143 | A set is open in the stand... |
| opnrebl2 35144 | A set is open in the stand... |
| nn0prpwlem 35145 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 35146 | Two nonnegative integers a... |
| topbnd 35147 | Two equivalent expressions... |
| opnbnd 35148 | A set is open iff it is di... |
| cldbnd 35149 | A set is closed iff it con... |
| ntruni 35150 | A union of interiors is a ... |
| clsun 35151 | A pairwise union of closur... |
| clsint2 35152 | The closure of an intersec... |
| opnregcld 35153 | A set is regularly closed ... |
| cldregopn 35154 | A set if regularly open if... |
| neiin 35155 | Two neighborhoods intersec... |
| hmeoclda 35156 | Homeomorphisms preserve cl... |
| hmeocldb 35157 | Homeomorphisms preserve cl... |
| ivthALT 35158 | An alternate proof of the ... |
| fnerel 35161 | Fineness is a relation. (... |
| isfne 35162 | The predicate " ` B ` is f... |
| isfne4 35163 | The predicate " ` B ` is f... |
| isfne4b 35164 | A condition for a topology... |
| isfne2 35165 | The predicate " ` B ` is f... |
| isfne3 35166 | The predicate " ` B ` is f... |
| fnebas 35167 | A finer cover covers the s... |
| fnetg 35168 | A finer cover generates a ... |
| fnessex 35169 | If ` B ` is finer than ` A... |
| fneuni 35170 | If ` B ` is finer than ` A... |
| fneint 35171 | If a cover is finer than a... |
| fness 35172 | A cover is finer than its ... |
| fneref 35173 | Reflexivity of the finenes... |
| fnetr 35174 | Transitivity of the finene... |
| fneval 35175 | Two covers are finer than ... |
| fneer 35176 | Fineness intersected with ... |
| topfne 35177 | Fineness for covers corres... |
| topfneec 35178 | A cover is equivalent to a... |
| topfneec2 35179 | A topology is precisely id... |
| fnessref 35180 | A cover is finer iff it ha... |
| refssfne 35181 | A cover is a refinement if... |
| neibastop1 35182 | A collection of neighborho... |
| neibastop2lem 35183 | Lemma for ~ neibastop2 . ... |
| neibastop2 35184 | In the topology generated ... |
| neibastop3 35185 | The topology generated by ... |
| topmtcl 35186 | The meet of a collection o... |
| topmeet 35187 | Two equivalent formulation... |
| topjoin 35188 | Two equivalent formulation... |
| fnemeet1 35189 | The meet of a collection o... |
| fnemeet2 35190 | The meet of equivalence cl... |
| fnejoin1 35191 | Join of equivalence classe... |
| fnejoin2 35192 | Join of equivalence classe... |
| fgmin 35193 | Minimality property of a g... |
| neifg 35194 | The neighborhood filter of... |
| tailfval 35195 | The tail function for a di... |
| tailval 35196 | The tail of an element in ... |
| eltail 35197 | An element of a tail. (Co... |
| tailf 35198 | The tail function of a dir... |
| tailini 35199 | A tail contains its initia... |
| tailfb 35200 | The collection of tails of... |
| filnetlem1 35201 | Lemma for ~ filnet . Chan... |
| filnetlem2 35202 | Lemma for ~ filnet . The ... |
| filnetlem3 35203 | Lemma for ~ filnet . (Con... |
| filnetlem4 35204 | Lemma for ~ filnet . (Con... |
| filnet 35205 | A filter has the same conv... |
| tb-ax1 35206 | The first of three axioms ... |
| tb-ax2 35207 | The second of three axioms... |
| tb-ax3 35208 | The third of three axioms ... |
| tbsyl 35209 | The weak syllogism from Ta... |
| re1ax2lem 35210 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 35211 | ~ ax-2 rederived from the ... |
| naim1 35212 | Constructor theorem for ` ... |
| naim2 35213 | Constructor theorem for ` ... |
| naim1i 35214 | Constructor rule for ` -/\... |
| naim2i 35215 | Constructor rule for ` -/\... |
| naim12i 35216 | Constructor rule for ` -/\... |
| nabi1i 35217 | Constructor rule for ` -/\... |
| nabi2i 35218 | Constructor rule for ` -/\... |
| nabi12i 35219 | Constructor rule for ` -/\... |
| df3nandALT1 35222 | The double nand expressed ... |
| df3nandALT2 35223 | The double nand expressed ... |
| andnand1 35224 | Double and in terms of dou... |
| imnand2 35225 | An ` -> ` nand relation. ... |
| nalfal 35226 | Not all sets hold ` F. ` a... |
| nexntru 35227 | There does not exist a set... |
| nexfal 35228 | There does not exist a set... |
| neufal 35229 | There does not exist exact... |
| neutru 35230 | There does not exist exact... |
| nmotru 35231 | There does not exist at mo... |
| mofal 35232 | There exist at most one se... |
| nrmo 35233 | "At most one" restricted e... |
| meran1 35234 | A single axiom for proposi... |
| meran2 35235 | A single axiom for proposi... |
| meran3 35236 | A single axiom for proposi... |
| waj-ax 35237 | A single axiom for proposi... |
| lukshef-ax2 35238 | A single axiom for proposi... |
| arg-ax 35239 | A single axiom for proposi... |
| negsym1 35240 | In the paper "On Variable ... |
| imsym1 35241 | A symmetry with ` -> ` . ... |
| bisym1 35242 | A symmetry with ` <-> ` . ... |
| consym1 35243 | A symmetry with ` /\ ` . ... |
| dissym1 35244 | A symmetry with ` \/ ` . ... |
| nandsym1 35245 | A symmetry with ` -/\ ` . ... |
| unisym1 35246 | A symmetry with ` A. ` . ... |
| exisym1 35247 | A symmetry with ` E. ` . ... |
| unqsym1 35248 | A symmetry with ` E! ` . ... |
| amosym1 35249 | A symmetry with ` E* ` . ... |
| subsym1 35250 | A symmetry with ` [ x / y ... |
| ontopbas 35251 | An ordinal number is a top... |
| onsstopbas 35252 | The class of ordinal numbe... |
| onpsstopbas 35253 | The class of ordinal numbe... |
| ontgval 35254 | The topology generated fro... |
| ontgsucval 35255 | The topology generated fro... |
| onsuctop 35256 | A successor ordinal number... |
| onsuctopon 35257 | One of the topologies on a... |
| ordtoplem 35258 | Membership of the class of... |
| ordtop 35259 | An ordinal is a topology i... |
| onsucconni 35260 | A successor ordinal number... |
| onsucconn 35261 | A successor ordinal number... |
| ordtopconn 35262 | An ordinal topology is con... |
| onintopssconn 35263 | An ordinal topology is con... |
| onsuct0 35264 | A successor ordinal number... |
| ordtopt0 35265 | An ordinal topology is T_0... |
| onsucsuccmpi 35266 | The successor of a success... |
| onsucsuccmp 35267 | The successor of a success... |
| limsucncmpi 35268 | The successor of a limit o... |
| limsucncmp 35269 | The successor of a limit o... |
| ordcmp 35270 | An ordinal topology is com... |
| ssoninhaus 35271 | The ordinal topologies ` 1... |
| onint1 35272 | The ordinal T_1 spaces are... |
| oninhaus 35273 | The ordinal Hausdorff spac... |
| fveleq 35274 | Please add description her... |
| findfvcl 35275 | Please add description her... |
| findreccl 35276 | Please add description her... |
| findabrcl 35277 | Please add description her... |
| nnssi2 35278 | Convert a theorem for real... |
| nnssi3 35279 | Convert a theorem for real... |
| nndivsub 35280 | Please add description her... |
| nndivlub 35281 | A factor of a positive int... |
| ee7.2aOLD 35284 | Lemma for Euclid's Element... |
| dnival 35285 | Value of the "distance to ... |
| dnicld1 35286 | Closure theorem for the "d... |
| dnicld2 35287 | Closure theorem for the "d... |
| dnif 35288 | The "distance to nearest i... |
| dnizeq0 35289 | The distance to nearest in... |
| dnizphlfeqhlf 35290 | The distance to nearest in... |
| rddif2 35291 | Variant of ~ rddif . (Con... |
| dnibndlem1 35292 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 35293 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 35294 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 35295 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 35296 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 35297 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 35298 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 35299 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 35300 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 35301 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 35302 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 35303 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 35304 | Lemma for ~ dnibnd . (Con... |
| dnibnd 35305 | The "distance to nearest i... |
| dnicn 35306 | The "distance to nearest i... |
| knoppcnlem1 35307 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 35308 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 35309 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 35310 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 35311 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 35312 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 35313 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 35314 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 35315 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 35316 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 35317 | Lemma for ~ knoppcn . (Co... |
| knoppcn 35318 | The continuous nowhere dif... |
| knoppcld 35319 | Closure theorem for Knopp'... |
| unblimceq0lem 35320 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 35321 | If ` F ` is unbounded near... |
| unbdqndv1 35322 | If the difference quotient... |
| unbdqndv2lem1 35323 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 35324 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 35325 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 35326 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 35327 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 35328 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 35329 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 35330 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 35331 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 35332 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 35333 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 35334 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 35335 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 35336 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 35337 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 35338 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 35339 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 35340 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 35341 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 35342 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 35343 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 35344 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 35345 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 35346 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 35347 | Lemma for ~ knoppndv . (C... |
| knoppndv 35348 | The continuous nowhere dif... |
| knoppf 35349 | Knopp's function is a func... |
| knoppcn2 35350 | Variant of ~ knoppcn with ... |
| cnndvlem1 35351 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 35352 | Lemma for ~ cnndv . (Cont... |
| cnndv 35353 | There exists a continuous ... |
| bj-mp2c 35354 | A double modus ponens infe... |
| bj-mp2d 35355 | A double modus ponens infe... |
| bj-0 35356 | A syntactic theorem. See ... |
| bj-1 35357 | In this proof, the use of ... |
| bj-a1k 35358 | Weakening of ~ ax-1 . As ... |
| bj-poni 35359 | Inference associated with ... |
| bj-nnclav 35360 | When ` F. ` is substituted... |
| bj-nnclavi 35361 | Inference associated with ... |
| bj-nnclavc 35362 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 35363 | Inference associated with ... |
| bj-jarrii 35364 | Inference associated with ... |
| bj-imim21 35365 | The propositional function... |
| bj-imim21i 35366 | Inference associated with ... |
| bj-peircestab 35367 | Over minimal implicational... |
| bj-stabpeirce 35368 | This minimal implicational... |
| bj-syl66ib 35369 | A mixed syllogism inferenc... |
| bj-orim2 35370 | Proof of ~ orim2 from the ... |
| bj-currypeirce 35371 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 35372 | Peirce's axiom ~ peirce im... |
| bj-animbi 35373 | Conjunction in terms of im... |
| bj-currypara 35374 | Curry's paradox. Note tha... |
| bj-con2com 35375 | A commuted form of the con... |
| bj-con2comi 35376 | Inference associated with ... |
| bj-pm2.01i 35377 | Inference associated with ... |
| bj-nimn 35378 | If a formula is true, then... |
| bj-nimni 35379 | Inference associated with ... |
| bj-peircei 35380 | Inference associated with ... |
| bj-looinvi 35381 | Inference associated with ... |
| bj-looinvii 35382 | Inference associated with ... |
| bj-mt2bi 35383 | Version of ~ mt2 where the... |
| bj-ntrufal 35384 | The negation of a theorem ... |
| bj-fal 35385 | Shortening of ~ fal using ... |
| bj-jaoi1 35386 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 35387 | Shortens ~ consensus (110>... |
| bj-dfbi4 35388 | Alternate definition of th... |
| bj-dfbi5 35389 | Alternate definition of th... |
| bj-dfbi6 35390 | Alternate definition of th... |
| bj-bijust0ALT 35391 | Alternate proof of ~ bijus... |
| bj-bijust00 35392 | A self-implication does no... |
| bj-consensus 35393 | Version of ~ consensus exp... |
| bj-consensusALT 35394 | Alternate proof of ~ bj-co... |
| bj-df-ifc 35395 | Candidate definition for t... |
| bj-dfif 35396 | Alternate definition of th... |
| bj-ififc 35397 | A biconditional connecting... |
| bj-imbi12 35398 | Uncurried (imported) form ... |
| bj-biorfi 35399 | This should be labeled "bi... |
| bj-falor 35400 | Dual of ~ truan (which has... |
| bj-falor2 35401 | Dual of ~ truan . (Contri... |
| bj-bibibi 35402 | A property of the bicondit... |
| bj-imn3ani 35403 | Duplication of ~ bnj1224 .... |
| bj-andnotim 35404 | Two ways of expressing a c... |
| bj-bi3ant 35405 | This used to be in the mai... |
| bj-bisym 35406 | This used to be in the mai... |
| bj-bixor 35407 | Equivalence of two ternary... |
| bj-axdd2 35408 | This implication, proved u... |
| bj-axd2d 35409 | This implication, proved u... |
| bj-axtd 35410 | This implication, proved f... |
| bj-gl4 35411 | In a normal modal logic, t... |
| bj-axc4 35412 | Over minimal calculus, the... |
| prvlem1 35417 | An elementary property of ... |
| prvlem2 35418 | An elementary property of ... |
| bj-babygodel 35419 | See the section header com... |
| bj-babylob 35420 | See the section header com... |
| bj-godellob 35421 | Proof of Gödel's theo... |
| bj-genr 35422 | Generalization rule on the... |
| bj-genl 35423 | Generalization rule on the... |
| bj-genan 35424 | Generalization rule on a c... |
| bj-mpgs 35425 | From a closed form theorem... |
| bj-2alim 35426 | Closed form of ~ 2alimi . ... |
| bj-2exim 35427 | Closed form of ~ 2eximi . ... |
| bj-alanim 35428 | Closed form of ~ alanimi .... |
| bj-2albi 35429 | Closed form of ~ 2albii . ... |
| bj-notalbii 35430 | Equivalence of universal q... |
| bj-2exbi 35431 | Closed form of ~ 2exbii . ... |
| bj-3exbi 35432 | Closed form of ~ 3exbii . ... |
| bj-sylgt2 35433 | Uncurried (imported) form ... |
| bj-alrimg 35434 | The general form of the *a... |
| bj-alrimd 35435 | A slightly more general ~ ... |
| bj-sylget 35436 | Dual statement of ~ sylgt ... |
| bj-sylget2 35437 | Uncurried (imported) form ... |
| bj-exlimg 35438 | The general form of the *e... |
| bj-sylge 35439 | Dual statement of ~ sylg (... |
| bj-exlimd 35440 | A slightly more general ~ ... |
| bj-nfimexal 35441 | A weak from of nonfreeness... |
| bj-alexim 35442 | Closed form of ~ aleximi .... |
| bj-nexdh 35443 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 35444 | Uncurried (imported) form ... |
| bj-hbxfrbi 35445 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 35446 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 35447 | Distribute quantifiers ove... |
| bj-exalimi 35448 | An inference for distribut... |
| bj-exalims 35449 | Distributing quantifiers o... |
| bj-exalimsi 35450 | An inference for distribut... |
| bj-ax12ig 35451 | A lemma used to prove a we... |
| bj-ax12i 35452 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 35453 | Closed form of ~ nfim and ... |
| bj-cbvalimt 35454 | A lemma in closed form use... |
| bj-cbveximt 35455 | A lemma in closed form use... |
| bj-eximALT 35456 | Alternate proof of ~ exim ... |
| bj-aleximiALT 35457 | Alternate proof of ~ alexi... |
| bj-eximcom 35458 | A commuted form of ~ exim ... |
| bj-ax12wlem 35459 | A lemma used to prove a we... |
| bj-cbvalim 35460 | A lemma used to prove ~ bj... |
| bj-cbvexim 35461 | A lemma used to prove ~ bj... |
| bj-cbvalimi 35462 | An equality-free general i... |
| bj-cbveximi 35463 | An equality-free general i... |
| bj-cbval 35464 | Changing a bound variable ... |
| bj-cbvex 35465 | Changing a bound variable ... |
| bj-ssbeq 35468 | Substitution in an equalit... |
| bj-ssblem1 35469 | A lemma for the definiens ... |
| bj-ssblem2 35470 | An instance of ~ ax-11 pro... |
| bj-ax12v 35471 | A weaker form of ~ ax-12 a... |
| bj-ax12 35472 | Remove a DV condition from... |
| bj-ax12ssb 35473 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 35474 | Special case of ~ 19.41 pr... |
| bj-equsexval 35475 | Special case of ~ equsexv ... |
| bj-subst 35476 | Proof of ~ sbalex from cor... |
| bj-ssbid2 35477 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 35478 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 35479 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 35480 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 35481 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 35482 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 35483 | Proof of ~ ax6e (hence ~ a... |
| bj-spimvwt 35484 | Closed form of ~ spimvw . ... |
| bj-spnfw 35485 | Theorem close to a closed ... |
| bj-cbvexiw 35486 | Change bound variable. Th... |
| bj-cbvexivw 35487 | Change bound variable. Th... |
| bj-modald 35488 | A short form of the axiom ... |
| bj-denot 35489 | A weakening of ~ ax-6 and ... |
| bj-eqs 35490 | A lemma for substitutions,... |
| bj-cbvexw 35491 | Change bound variable. Th... |
| bj-ax12w 35492 | The general statement that... |
| bj-ax89 35493 | A theorem which could be u... |
| bj-elequ12 35494 | An identity law for the no... |
| bj-cleljusti 35495 | One direction of ~ cleljus... |
| bj-alcomexcom 35496 | Commutation of two existen... |
| bj-hbalt 35497 | Closed form of ~ hbal . W... |
| axc11n11 35498 | Proof of ~ axc11n from { ~... |
| axc11n11r 35499 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 35500 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 35501 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 35502 | Alternate proof of ~ bj-ax... |
| bj-sb 35503 | A weak variant of ~ sbid2 ... |
| bj-modalbe 35504 | The predicate-calculus ver... |
| bj-spst 35505 | Closed form of ~ sps . On... |
| bj-19.21bit 35506 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 35507 | Closed form of ~ 19.23bi .... |
| bj-nexrt 35508 | Closed form of ~ nexr . C... |
| bj-alrim 35509 | Closed form of ~ alrimi . ... |
| bj-alrim2 35510 | Uncurried (imported) form ... |
| bj-nfdt0 35511 | A theorem close to a close... |
| bj-nfdt 35512 | Closed form of ~ nf5d and ... |
| bj-nexdt 35513 | Closed form of ~ nexd . (... |
| bj-nexdvt 35514 | Closed form of ~ nexdv . ... |
| bj-alexbiex 35515 | Adding a second quantifier... |
| bj-exexbiex 35516 | Adding a second quantifier... |
| bj-alalbial 35517 | Adding a second quantifier... |
| bj-exalbial 35518 | Adding a second quantifier... |
| bj-19.9htbi 35519 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 35520 | Strengthening ~ hbnt by re... |
| bj-biexal1 35521 | A general FOL biconditiona... |
| bj-biexal2 35522 | When ` ph ` is substituted... |
| bj-biexal3 35523 | When ` ph ` is substituted... |
| bj-bialal 35524 | When ` ph ` is substituted... |
| bj-biexex 35525 | When ` ph ` is substituted... |
| bj-hbext 35526 | Closed form of ~ hbex . (... |
| bj-nfalt 35527 | Closed form of ~ nfal . (... |
| bj-nfext 35528 | Closed form of ~ nfex . (... |
| bj-eeanvw 35529 | Version of ~ exdistrv with... |
| bj-modal4 35530 | First-order logic form of ... |
| bj-modal4e 35531 | First-order logic form of ... |
| bj-modalb 35532 | A short form of the axiom ... |
| bj-wnf1 35533 | When ` ph ` is substituted... |
| bj-wnf2 35534 | When ` ph ` is substituted... |
| bj-wnfanf 35535 | When ` ph ` is substituted... |
| bj-wnfenf 35536 | When ` ph ` is substituted... |
| bj-substax12 35537 | Equivalent form of the axi... |
| bj-substw 35538 | Weak form of the LHS of ~ ... |
| bj-nnfbi 35541 | If two formulas are equiva... |
| bj-nnfbd 35542 | If two formulas are equiva... |
| bj-nnfbii 35543 | If two formulas are equiva... |
| bj-nnfa 35544 | Nonfreeness implies the eq... |
| bj-nnfad 35545 | Nonfreeness implies the eq... |
| bj-nnfai 35546 | Nonfreeness implies the eq... |
| bj-nnfe 35547 | Nonfreeness implies the eq... |
| bj-nnfed 35548 | Nonfreeness implies the eq... |
| bj-nnfei 35549 | Nonfreeness implies the eq... |
| bj-nnfea 35550 | Nonfreeness implies the eq... |
| bj-nnfead 35551 | Nonfreeness implies the eq... |
| bj-nnfeai 35552 | Nonfreeness implies the eq... |
| bj-dfnnf2 35553 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 35554 | New nonfreeness implies ol... |
| bj-wnfnf 35555 | When ` ph ` is substituted... |
| bj-nnfnt 35556 | A variable is nonfree in a... |
| bj-nnftht 35557 | A variable is nonfree in a... |
| bj-nnfth 35558 | A variable is nonfree in a... |
| bj-nnfnth 35559 | A variable is nonfree in t... |
| bj-nnfim1 35560 | A consequence of nonfreene... |
| bj-nnfim2 35561 | A consequence of nonfreene... |
| bj-nnfim 35562 | Nonfreeness in the anteced... |
| bj-nnfimd 35563 | Nonfreeness in the anteced... |
| bj-nnfan 35564 | Nonfreeness in both conjun... |
| bj-nnfand 35565 | Nonfreeness in both conjun... |
| bj-nnfor 35566 | Nonfreeness in both disjun... |
| bj-nnford 35567 | Nonfreeness in both disjun... |
| bj-nnfbit 35568 | Nonfreeness in both sides ... |
| bj-nnfbid 35569 | Nonfreeness in both sides ... |
| bj-nnfv 35570 | A non-occurring variable i... |
| bj-nnf-alrim 35571 | Proof of the closed form o... |
| bj-nnf-exlim 35572 | Proof of the closed form o... |
| bj-dfnnf3 35573 | Alternate definition of no... |
| bj-nfnnfTEMP 35574 | New nonfreeness is equival... |
| bj-nnfa1 35575 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 35576 | See ~ nfe1 . (Contributed... |
| bj-19.12 35577 | See ~ 19.12 . Could be la... |
| bj-nnflemaa 35578 | One of four lemmas for non... |
| bj-nnflemee 35579 | One of four lemmas for non... |
| bj-nnflemae 35580 | One of four lemmas for non... |
| bj-nnflemea 35581 | One of four lemmas for non... |
| bj-nnfalt 35582 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 35583 | See ~ nfex and ~ bj-nfext ... |
| bj-stdpc5t 35584 | Alias of ~ bj-nnf-alrim fo... |
| bj-19.21t 35585 | Statement ~ 19.21t proved ... |
| bj-19.23t 35586 | Statement ~ 19.23t proved ... |
| bj-19.36im 35587 | One direction of ~ 19.36 f... |
| bj-19.37im 35588 | One direction of ~ 19.37 f... |
| bj-19.42t 35589 | Closed form of ~ 19.42 fro... |
| bj-19.41t 35590 | Closed form of ~ 19.41 fro... |
| bj-sbft 35591 | Version of ~ sbft using ` ... |
| bj-pm11.53vw 35592 | Version of ~ pm11.53v with... |
| bj-pm11.53v 35593 | Version of ~ pm11.53v with... |
| bj-pm11.53a 35594 | A variant of ~ pm11.53v . ... |
| bj-equsvt 35595 | A variant of ~ equsv . (C... |
| bj-equsalvwd 35596 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 35597 | Variant of ~ equsexvw . (... |
| bj-sbievwd 35598 | Variant of ~ sbievw . (Co... |
| bj-axc10 35599 | Alternate proof of ~ axc10... |
| bj-alequex 35600 | A fol lemma. See ~ aleque... |
| bj-spimt2 35601 | A step in the proof of ~ s... |
| bj-cbv3ta 35602 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 35603 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 35604 | A theorem close to a close... |
| bj-hbsb3 35605 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 35606 | A theorem close to a close... |
| bj-nfs1t2 35607 | A theorem close to a close... |
| bj-nfs1 35608 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 35609 | Version of ~ axc10 with a ... |
| bj-spimtv 35610 | Version of ~ spimt with a ... |
| bj-cbv3hv2 35611 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 35612 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 35613 | Version of ~ cbv2h with a ... |
| bj-cbv2v 35614 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 35615 | Version of ~ cbvald with a... |
| bj-cbvexdv 35616 | Version of ~ cbvexd with a... |
| bj-cbval2vv 35617 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 35618 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 35619 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 35620 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 35621 | Version of ~ cbvex4v with ... |
| bj-equsalhv 35622 | Version of ~ equsalh with ... |
| bj-axc11nv 35623 | Version of ~ axc11n with a... |
| bj-aecomsv 35624 | Version of ~ aecoms with a... |
| bj-axc11v 35625 | Version of ~ axc11 with a ... |
| bj-drnf2v 35626 | Version of ~ drnf2 with a ... |
| bj-equs45fv 35627 | Version of ~ equs45f with ... |
| bj-hbs1 35628 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 35629 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 35630 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 35631 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 35632 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 35633 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 35634 | Biconditional version of a... |
| bj-hbaeb 35635 | Biconditional version of ~... |
| bj-hbnaeb 35636 | Biconditional version of ~... |
| bj-dvv 35637 | A special instance of ~ bj... |
| bj-equsal1t 35638 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 35639 | Inference associated with ... |
| bj-equsal1 35640 | One direction of ~ equsal ... |
| bj-equsal2 35641 | One direction of ~ equsal ... |
| bj-equsal 35642 | Shorter proof of ~ equsal ... |
| stdpc5t 35643 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 35644 | More direct proof of ~ std... |
| 2stdpc5 35645 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 35646 | Proof of ~ 19.21t from ~ s... |
| exlimii 35647 | Inference associated with ... |
| ax11-pm 35648 | Proof of ~ ax-11 similar t... |
| ax6er 35649 | Commuted form of ~ ax6e . ... |
| exlimiieq1 35650 | Inferring a theorem when i... |
| exlimiieq2 35651 | Inferring a theorem when i... |
| ax11-pm2 35652 | Proof of ~ ax-11 from the ... |
| bj-sbsb 35653 | Biconditional showing two ... |
| bj-dfsb2 35654 | Alternate (dual) definitio... |
| bj-sbf3 35655 | Substitution has no effect... |
| bj-sbf4 35656 | Substitution has no effect... |
| bj-sbnf 35657 | Move nonfree predicate in ... |
| bj-eu3f 35658 | Version of ~ eu3v where th... |
| bj-sblem1 35659 | Lemma for substitution. (... |
| bj-sblem2 35660 | Lemma for substitution. (... |
| bj-sblem 35661 | Lemma for substitution. (... |
| bj-sbievw1 35662 | Lemma for substitution. (... |
| bj-sbievw2 35663 | Lemma for substitution. (... |
| bj-sbievw 35664 | Lemma for substitution. C... |
| bj-sbievv 35665 | Version of ~ sbie with a s... |
| bj-moeub 35666 | Uniqueness is equivalent t... |
| bj-sbidmOLD 35667 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 35668 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 35669 | Curried (exported) form of... |
| bj-dvelimv 35670 | A version of ~ dvelim usin... |
| bj-nfeel2 35671 | Nonfreeness in a membershi... |
| bj-axc14nf 35672 | Proof of a version of ~ ax... |
| bj-axc14 35673 | Alternate proof of ~ axc14... |
| mobidvALT 35674 | Alternate proof of ~ mobid... |
| sbn1ALT 35675 | Alternate proof of ~ sbn1 ... |
| eliminable1 35676 | A theorem used to prove th... |
| eliminable2a 35677 | A theorem used to prove th... |
| eliminable2b 35678 | A theorem used to prove th... |
| eliminable2c 35679 | A theorem used to prove th... |
| eliminable3a 35680 | A theorem used to prove th... |
| eliminable3b 35681 | A theorem used to prove th... |
| eliminable-velab 35682 | A theorem used to prove th... |
| eliminable-veqab 35683 | A theorem used to prove th... |
| eliminable-abeqv 35684 | A theorem used to prove th... |
| eliminable-abeqab 35685 | A theorem used to prove th... |
| eliminable-abelv 35686 | A theorem used to prove th... |
| eliminable-abelab 35687 | A theorem used to prove th... |
| bj-denoteslem 35688 | Lemma for ~ bj-denotes . ... |
| bj-denotes 35689 | This would be the justific... |
| bj-issettru 35690 | Weak version of ~ isset wi... |
| bj-elabtru 35691 | This is as close as we can... |
| bj-issetwt 35692 | Closed form of ~ bj-issetw... |
| bj-issetw 35693 | The closest one can get to... |
| bj-elissetALT 35694 | Alternate proof of ~ eliss... |
| bj-issetiv 35695 | Version of ~ bj-isseti wit... |
| bj-isseti 35696 | Version of ~ isseti with a... |
| bj-ralvw 35697 | A weak version of ~ ralv n... |
| bj-rexvw 35698 | A weak version of ~ rexv n... |
| bj-rababw 35699 | A weak version of ~ rabab ... |
| bj-rexcom4bv 35700 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 35701 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 35702 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 35703 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 35704 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 35705 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 35706 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 35707 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 35708 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 35709 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 35710 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 35711 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 35712 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 35713 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 35714 | Remove from ~ spcimdv depe... |
| elelb 35715 | Equivalence between two co... |
| bj-pwvrelb 35716 | Characterization of the el... |
| bj-nfcsym 35717 | The nonfreeness quantifier... |
| bj-sbeqALT 35718 | Substitution in an equalit... |
| bj-sbeq 35719 | Distribute proper substitu... |
| bj-sbceqgALT 35720 | Distribute proper substitu... |
| bj-csbsnlem 35721 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 35722 | Substitution in a singleto... |
| bj-sbel1 35723 | Version of ~ sbcel1g when ... |
| bj-abv 35724 | The class of sets verifyin... |
| bj-abvALT 35725 | Alternate version of ~ bj-... |
| bj-ab0 35726 | The class of sets verifyin... |
| bj-abf 35727 | Shorter proof of ~ abf (wh... |
| bj-csbprc 35728 | More direct proof of ~ csb... |
| bj-exlimvmpi 35729 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 35730 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 35731 | Lemma for theorems of the ... |
| bj-exlimmpbir 35732 | Lemma for theorems of the ... |
| bj-vtoclf 35733 | Remove dependency on ~ ax-... |
| bj-vtocl 35734 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 35735 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 35736 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 35737 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 35738 | A version of ~ vtoclg with... |
| bj-rabeqbid 35739 | Version of ~ rabeqbidv wit... |
| bj-seex 35740 | Version of ~ seex with a d... |
| bj-nfcf 35741 | Version of ~ df-nfc with a... |
| bj-zfauscl 35742 | General version of ~ zfaus... |
| bj-elabd2ALT 35743 | Alternate proof of ~ elabd... |
| bj-unrab 35744 | Generalization of ~ unrab ... |
| bj-inrab 35745 | Generalization of ~ inrab ... |
| bj-inrab2 35746 | Shorter proof of ~ inrab .... |
| bj-inrab3 35747 | Generalization of ~ dfrab3... |
| bj-rabtr 35748 | Restricted class abstracti... |
| bj-rabtrALT 35749 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 35750 | Proof of ~ bj-rabtr found ... |
| bj-gabss 35753 | Inclusion of generalized c... |
| bj-gabssd 35754 | Inclusion of generalized c... |
| bj-gabeqd 35755 | Equality of generalized cl... |
| bj-gabeqis 35756 | Equality of generalized cl... |
| bj-elgab 35757 | Elements of a generalized ... |
| bj-gabima 35758 | Generalized class abstract... |
| bj-ru0 35761 | The FOL part of Russell's ... |
| bj-ru1 35762 | A version of Russell's par... |
| bj-ru 35763 | Remove dependency on ~ ax-... |
| currysetlem 35764 | Lemma for ~ currysetlem , ... |
| curryset 35765 | Curry's paradox in set the... |
| currysetlem1 35766 | Lemma for ~ currysetALT . ... |
| currysetlem2 35767 | Lemma for ~ currysetALT . ... |
| currysetlem3 35768 | Lemma for ~ currysetALT . ... |
| currysetALT 35769 | Alternate proof of ~ curry... |
| bj-n0i 35770 | Inference associated with ... |
| bj-disjsn01 35771 | Disjointness of the single... |
| bj-0nel1 35772 | The empty set does not bel... |
| bj-1nel0 35773 | ` 1o ` does not belong to ... |
| bj-xpimasn 35774 | The image of a singleton, ... |
| bj-xpima1sn 35775 | The image of a singleton b... |
| bj-xpima1snALT 35776 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 35777 | The image of a singleton b... |
| bj-xpnzex 35778 | If the first factor of a p... |
| bj-xpexg2 35779 | Curried (exported) form of... |
| bj-xpnzexb 35780 | If the first factor of a p... |
| bj-cleq 35781 | Substitution property for ... |
| bj-snsetex 35782 | The class of sets "whose s... |
| bj-clexab 35783 | Sethood of certain classes... |
| bj-sngleq 35786 | Substitution property for ... |
| bj-elsngl 35787 | Characterization of the el... |
| bj-snglc 35788 | Characterization of the el... |
| bj-snglss 35789 | The singletonization of a ... |
| bj-0nelsngl 35790 | The empty set is not a mem... |
| bj-snglinv 35791 | Inverse of singletonizatio... |
| bj-snglex 35792 | A class is a set if and on... |
| bj-tageq 35795 | Substitution property for ... |
| bj-eltag 35796 | Characterization of the el... |
| bj-0eltag 35797 | The empty set belongs to t... |
| bj-tagn0 35798 | The tagging of a class is ... |
| bj-tagss 35799 | The tagging of a class is ... |
| bj-snglsstag 35800 | The singletonization is in... |
| bj-sngltagi 35801 | The singletonization is in... |
| bj-sngltag 35802 | The singletonization and t... |
| bj-tagci 35803 | Characterization of the el... |
| bj-tagcg 35804 | Characterization of the el... |
| bj-taginv 35805 | Inverse of tagging. (Cont... |
| bj-tagex 35806 | A class is a set if and on... |
| bj-xtageq 35807 | The products of a given cl... |
| bj-xtagex 35808 | The product of a set and t... |
| bj-projeq 35811 | Substitution property for ... |
| bj-projeq2 35812 | Substitution property for ... |
| bj-projun 35813 | The class projection on a ... |
| bj-projex 35814 | Sethood of the class proje... |
| bj-projval 35815 | Value of the class project... |
| bj-1upleq 35818 | Substitution property for ... |
| bj-pr1eq 35821 | Substitution property for ... |
| bj-pr1un 35822 | The first projection prese... |
| bj-pr1val 35823 | Value of the first project... |
| bj-pr11val 35824 | Value of the first project... |
| bj-pr1ex 35825 | Sethood of the first proje... |
| bj-1uplth 35826 | The characteristic propert... |
| bj-1uplex 35827 | A monuple is a set if and ... |
| bj-1upln0 35828 | A monuple is nonempty. (C... |
| bj-2upleq 35831 | Substitution property for ... |
| bj-pr21val 35832 | Value of the first project... |
| bj-pr2eq 35835 | Substitution property for ... |
| bj-pr2un 35836 | The second projection pres... |
| bj-pr2val 35837 | Value of the second projec... |
| bj-pr22val 35838 | Value of the second projec... |
| bj-pr2ex 35839 | Sethood of the second proj... |
| bj-2uplth 35840 | The characteristic propert... |
| bj-2uplex 35841 | A couple is a set if and o... |
| bj-2upln0 35842 | A couple is nonempty. (Co... |
| bj-2upln1upl 35843 | A couple is never equal to... |
| bj-rcleqf 35844 | Relative version of ~ cleq... |
| bj-rcleq 35845 | Relative version of ~ dfcl... |
| bj-reabeq 35846 | Relative form of ~ eqabb .... |
| bj-disj2r 35847 | Relative version of ~ ssdi... |
| bj-sscon 35848 | Contraposition law for rel... |
| bj-abex 35849 | Two ways of stating that t... |
| bj-clex 35850 | Two ways of stating that a... |
| bj-axsn 35851 | Two ways of stating the ax... |
| bj-snexg 35853 | A singleton built on a set... |
| bj-snex 35854 | A singleton is a set. See... |
| bj-axbun 35855 | Two ways of stating the ax... |
| bj-unexg 35857 | Existence of binary unions... |
| bj-prexg 35858 | Existence of unordered pai... |
| bj-prex 35859 | Existence of unordered pai... |
| bj-axadj 35860 | Two ways of stating the ax... |
| bj-adjg1 35862 | Existence of the result of... |
| bj-snfromadj 35863 | Singleton from adjunction ... |
| bj-prfromadj 35864 | Unordered pair from adjunc... |
| bj-adjfrombun 35865 | Adjunction from singleton ... |
| eleq2w2ALT 35866 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 35867 | Alternate proof of ~ clel3... |
| bj-pw0ALT 35868 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 35869 | Quantitative version of ~ ... |
| bj-unirel 35870 | Quantitative version of ~ ... |
| bj-elpwg 35871 | If the intersection of two... |
| bj-velpwALT 35872 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 35873 | Alternate proof of ~ elpwg... |
| bj-vjust 35874 | Justification theorem for ... |
| bj-nul 35875 | Two formulations of the ax... |
| bj-nuliota 35876 | Definition of the empty se... |
| bj-nuliotaALT 35877 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 35878 | Alternate proof of ~ vtocl... |
| bj-elsn12g 35879 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 35880 | Biconditional version of ~... |
| bj-pwcfsdom 35881 | Remove hypothesis from ~ p... |
| bj-grur1 35882 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 35883 | The extension of a predica... |
| bj-dfid2ALT 35884 | Alternate version of ~ dfi... |
| bj-0nelopab 35885 | The empty set is never an ... |
| bj-brrelex12ALT 35886 | Two classes related by a b... |
| bj-epelg 35887 | The membership relation an... |
| bj-epelb 35888 | Two classes are related by... |
| bj-nsnid 35889 | A set does not contain the... |
| bj-rdg0gALT 35890 | Alternate proof of ~ rdg0g... |
| bj-evaleq 35891 | Equality theorem for the `... |
| bj-evalfun 35892 | The evaluation at a class ... |
| bj-evalfn 35893 | The evaluation at a class ... |
| bj-evalval 35894 | Value of the evaluation at... |
| bj-evalid 35895 | The evaluation at a set of... |
| bj-ndxarg 35896 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 35897 | Closed general form of ~ s... |
| bj-rest00 35900 | An elementwise intersectio... |
| bj-restsn 35901 | An elementwise intersectio... |
| bj-restsnss 35902 | Special case of ~ bj-rests... |
| bj-restsnss2 35903 | Special case of ~ bj-rests... |
| bj-restsn0 35904 | An elementwise intersectio... |
| bj-restsn10 35905 | Special case of ~ bj-rests... |
| bj-restsnid 35906 | The elementwise intersecti... |
| bj-rest10 35907 | An elementwise intersectio... |
| bj-rest10b 35908 | Alternate version of ~ bj-... |
| bj-restn0 35909 | An elementwise intersectio... |
| bj-restn0b 35910 | Alternate version of ~ bj-... |
| bj-restpw 35911 | The elementwise intersecti... |
| bj-rest0 35912 | An elementwise intersectio... |
| bj-restb 35913 | An elementwise intersectio... |
| bj-restv 35914 | An elementwise intersectio... |
| bj-resta 35915 | An elementwise intersectio... |
| bj-restuni 35916 | The union of an elementwis... |
| bj-restuni2 35917 | The union of an elementwis... |
| bj-restreg 35918 | A reformulation of the axi... |
| bj-raldifsn 35919 | All elements in a set sati... |
| bj-0int 35920 | If ` A ` is a collection o... |
| bj-mooreset 35921 | A Moore collection is a se... |
| bj-ismoore 35924 | Characterization of Moore ... |
| bj-ismoored0 35925 | Necessary condition to be ... |
| bj-ismoored 35926 | Necessary condition to be ... |
| bj-ismoored2 35927 | Necessary condition to be ... |
| bj-ismooredr 35928 | Sufficient condition to be... |
| bj-ismooredr2 35929 | Sufficient condition to be... |
| bj-discrmoore 35930 | The powerclass ` ~P A ` is... |
| bj-0nmoore 35931 | The empty set is not a Moo... |
| bj-snmoore 35932 | A singleton is a Moore col... |
| bj-snmooreb 35933 | A singleton is a Moore col... |
| bj-prmoore 35934 | A pair formed of two neste... |
| bj-0nelmpt 35935 | The empty set is not an el... |
| bj-mptval 35936 | Value of a function given ... |
| bj-dfmpoa 35937 | An equivalent definition o... |
| bj-mpomptALT 35938 | Alternate proof of ~ mpomp... |
| setsstrset 35955 | Relation between ~ df-sets... |
| bj-nfald 35956 | Variant of ~ nfald . (Con... |
| bj-nfexd 35957 | Variant of ~ nfexd . (Con... |
| copsex2d 35958 | Implicit substitution dedu... |
| copsex2b 35959 | Biconditional form of ~ co... |
| opelopabd 35960 | Membership of an ordere pa... |
| opelopabb 35961 | Membership of an ordered p... |
| opelopabbv 35962 | Membership of an ordered p... |
| bj-opelrelex 35963 | The coordinates of an orde... |
| bj-opelresdm 35964 | If an ordered pair is in a... |
| bj-brresdm 35965 | If two classes are related... |
| brabd0 35966 | Expressing that two sets a... |
| brabd 35967 | Expressing that two sets a... |
| bj-brab2a1 35968 | "Unbounded" version of ~ b... |
| bj-opabssvv 35969 | A variant of ~ relopabiv (... |
| bj-funidres 35970 | The restricted identity re... |
| bj-opelidb 35971 | Characterization of the or... |
| bj-opelidb1 35972 | Characterization of the or... |
| bj-inexeqex 35973 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 35974 | If the intersection of two... |
| bj-opelid 35975 | Characterization of the or... |
| bj-ideqg 35976 | Characterization of the cl... |
| bj-ideqgALT 35977 | Alternate proof of ~ bj-id... |
| bj-ideqb 35978 | Characterization of classe... |
| bj-idres 35979 | Alternate expression for t... |
| bj-opelidres 35980 | Characterization of the or... |
| bj-idreseq 35981 | Sufficient condition for t... |
| bj-idreseqb 35982 | Characterization for two c... |
| bj-ideqg1 35983 | For sets, the identity rel... |
| bj-ideqg1ALT 35984 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 35985 | Characterization of the co... |
| bj-elid3 35986 | Characterization of the co... |
| bj-elid4 35987 | Characterization of the el... |
| bj-elid5 35988 | Characterization of the el... |
| bj-elid6 35989 | Characterization of the el... |
| bj-elid7 35990 | Characterization of the el... |
| bj-diagval 35993 | Value of the functionalize... |
| bj-diagval2 35994 | Value of the functionalize... |
| bj-eldiag 35995 | Characterization of the el... |
| bj-eldiag2 35996 | Characterization of the el... |
| bj-imdirvallem 35999 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 36000 | Value of the functionalize... |
| bj-imdirval2lem 36001 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 36002 | Value of the functionalize... |
| bj-imdirval3 36003 | Value of the functionalize... |
| bj-imdiridlem 36004 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 36005 | Functorial property of the... |
| bj-opelopabid 36006 | Membership in an ordered-p... |
| bj-opabco 36007 | Composition of ordered-pai... |
| bj-xpcossxp 36008 | The composition of two Car... |
| bj-imdirco 36009 | Functorial property of the... |
| bj-iminvval 36012 | Value of the functionalize... |
| bj-iminvval2 36013 | Value of the functionalize... |
| bj-iminvid 36014 | Functorial property of the... |
| bj-inftyexpitaufo 36021 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 36024 | An element of the circle a... |
| bj-inftyexpiinv 36027 | Utility theorem for the in... |
| bj-inftyexpiinj 36028 | Injectivity of the paramet... |
| bj-inftyexpidisj 36029 | An element of the circle a... |
| bj-ccinftydisj 36032 | The circle at infinity is ... |
| bj-elccinfty 36033 | A lemma for infinite exten... |
| bj-ccssccbar 36036 | Complex numbers are extend... |
| bj-ccinftyssccbar 36037 | Infinite extended complex ... |
| bj-pinftyccb 36040 | The class ` pinfty ` is an... |
| bj-pinftynrr 36041 | The extended complex numbe... |
| bj-minftyccb 36044 | The class ` minfty ` is an... |
| bj-minftynrr 36045 | The extended complex numbe... |
| bj-pinftynminfty 36046 | The extended complex numbe... |
| bj-rrhatsscchat 36055 | The real projective line i... |
| bj-imafv 36070 | If the direct image of a s... |
| bj-funun 36071 | Value of a function expres... |
| bj-fununsn1 36072 | Value of a function expres... |
| bj-fununsn2 36073 | Value of a function expres... |
| bj-fvsnun1 36074 | The value of a function wi... |
| bj-fvsnun2 36075 | The value of a function wi... |
| bj-fvmptunsn1 36076 | Value of a function expres... |
| bj-fvmptunsn2 36077 | Value of a function expres... |
| bj-iomnnom 36078 | The canonical bijection fr... |
| bj-smgrpssmgm 36087 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 36088 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 36089 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 36090 | Monoids are semigroups (el... |
| bj-cmnssmnd 36091 | Commutative monoids are mo... |
| bj-cmnssmndel 36092 | Commutative monoids are mo... |
| bj-grpssmnd 36093 | Groups are monoids. (Cont... |
| bj-grpssmndel 36094 | Groups are monoids (elemen... |
| bj-ablssgrp 36095 | Abelian groups are groups.... |
| bj-ablssgrpel 36096 | Abelian groups are groups ... |
| bj-ablsscmn 36097 | Abelian groups are commuta... |
| bj-ablsscmnel 36098 | Abelian groups are commuta... |
| bj-modssabl 36099 | (The additive groups of) m... |
| bj-vecssmod 36100 | Vector spaces are modules.... |
| bj-vecssmodel 36101 | Vector spaces are modules ... |
| bj-finsumval0 36104 | Value of a finite sum. (C... |
| bj-fvimacnv0 36105 | Variant of ~ fvimacnv wher... |
| bj-isvec 36106 | The predicate "is a vector... |
| bj-fldssdrng 36107 | Fields are division rings.... |
| bj-flddrng 36108 | Fields are division rings ... |
| bj-rrdrg 36109 | The field of real numbers ... |
| bj-isclm 36110 | The predicate "is a subcom... |
| bj-isrvec 36113 | The predicate "is a real v... |
| bj-rvecmod 36114 | Real vector spaces are mod... |
| bj-rvecssmod 36115 | Real vector spaces are mod... |
| bj-rvecrr 36116 | The field of scalars of a ... |
| bj-isrvecd 36117 | The predicate "is a real v... |
| bj-rvecvec 36118 | Real vector spaces are vec... |
| bj-isrvec2 36119 | The predicate "is a real v... |
| bj-rvecssvec 36120 | Real vector spaces are vec... |
| bj-rveccmod 36121 | Real vector spaces are sub... |
| bj-rvecsscmod 36122 | Real vector spaces are sub... |
| bj-rvecsscvec 36123 | Real vector spaces are sub... |
| bj-rveccvec 36124 | Real vector spaces are sub... |
| bj-rvecssabl 36125 | (The additive groups of) r... |
| bj-rvecabl 36126 | (The additive groups of) r... |
| bj-subcom 36127 | A consequence of commutati... |
| bj-lineqi 36128 | Solution of a (scalar) lin... |
| bj-bary1lem 36129 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 36130 | Lemma for bj-bary1: comput... |
| bj-bary1 36131 | Barycentric coordinates in... |
| bj-endval 36134 | Value of the monoid of end... |
| bj-endbase 36135 | Base set of the monoid of ... |
| bj-endcomp 36136 | Composition law of the mon... |
| bj-endmnd 36137 | The monoid of endomorphism... |
| taupilem3 36138 | Lemma for tau-related theo... |
| taupilemrplb 36139 | A set of positive reals ha... |
| taupilem1 36140 | Lemma for ~ taupi . A pos... |
| taupilem2 36141 | Lemma for ~ taupi . The s... |
| taupi 36142 | Relationship between ` _ta... |
| dfgcd3 36143 | Alternate definition of th... |
| irrdifflemf 36144 | Lemma for ~ irrdiff . The... |
| irrdiff 36145 | The irrationals are exactl... |
| iccioo01 36146 | The closed unit interval i... |
| csbrecsg 36147 | Move class substitution in... |
| csbrdgg 36148 | Move class substitution in... |
| csboprabg 36149 | Move class substitution in... |
| csbmpo123 36150 | Move class substitution in... |
| con1bii2 36151 | A contraposition inference... |
| con2bii2 36152 | A contraposition inference... |
| vtoclefex 36153 | Implicit substitution of a... |
| rnmptsn 36154 | The range of a function ma... |
| f1omptsnlem 36155 | This is the core of the pr... |
| f1omptsn 36156 | A function mapping to sing... |
| mptsnunlem 36157 | This is the core of the pr... |
| mptsnun 36158 | A class ` B ` is equal to ... |
| dissneqlem 36159 | This is the core of the pr... |
| dissneq 36160 | Any topology that contains... |
| exlimim 36161 | Closed form of ~ exlimimd ... |
| exlimimd 36162 | Existential elimination ru... |
| exellim 36163 | Closed form of ~ exellimdd... |
| exellimddv 36164 | Eliminate an antecedent wh... |
| topdifinfindis 36165 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 36166 | This is the core of the pr... |
| topdifinffin 36167 | Part of Exercise 3 of [Mun... |
| topdifinf 36168 | Part of Exercise 3 of [Mun... |
| topdifinfeq 36169 | Two different ways of defi... |
| icorempo 36170 | Closed-below, open-above i... |
| icoreresf 36171 | Closed-below, open-above i... |
| icoreval 36172 | Value of the closed-below,... |
| icoreelrnab 36173 | Elementhood in the set of ... |
| isbasisrelowllem1 36174 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 36175 | Lemma for ~ isbasisrelowl ... |
| icoreclin 36176 | The set of closed-below, o... |
| isbasisrelowl 36177 | The set of all closed-belo... |
| icoreunrn 36178 | The union of all closed-be... |
| istoprelowl 36179 | The set of all closed-belo... |
| icoreelrn 36180 | A class abstraction which ... |
| iooelexlt 36181 | An element of an open inte... |
| relowlssretop 36182 | The lower limit topology o... |
| relowlpssretop 36183 | The lower limit topology o... |
| sucneqond 36184 | Inequality of an ordinal s... |
| sucneqoni 36185 | Inequality of an ordinal s... |
| onsucuni3 36186 | If an ordinal number has a... |
| 1oequni2o 36187 | The ordinal number ` 1o ` ... |
| rdgsucuni 36188 | If an ordinal number has a... |
| rdgeqoa 36189 | If a recursive function wi... |
| elxp8 36190 | Membership in a Cartesian ... |
| cbveud 36191 | Deduction used to change b... |
| cbvreud 36192 | Deduction used to change b... |
| difunieq 36193 | The difference of unions i... |
| inunissunidif 36194 | Theorem about subsets of t... |
| rdgellim 36195 | Elementhood in a recursive... |
| rdglimss 36196 | A recursive definition at ... |
| rdgssun 36197 | In a recursive definition ... |
| exrecfnlem 36198 | Lemma for ~ exrecfn . (Co... |
| exrecfn 36199 | Theorem about the existenc... |
| exrecfnpw 36200 | For any base set, a set wh... |
| finorwe 36201 | If the Axiom of Infinity i... |
| dffinxpf 36204 | This theorem is the same a... |
| finxpeq1 36205 | Equality theorem for Carte... |
| finxpeq2 36206 | Equality theorem for Carte... |
| csbfinxpg 36207 | Distribute proper substitu... |
| finxpreclem1 36208 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 36209 | Lemma for ` ^^ ` recursion... |
| finxp0 36210 | The value of Cartesian exp... |
| finxp1o 36211 | The value of Cartesian exp... |
| finxpreclem3 36212 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 36213 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 36214 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 36215 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 36216 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 36217 | The value of Cartesian exp... |
| finxp2o 36218 | The value of Cartesian exp... |
| finxp3o 36219 | The value of Cartesian exp... |
| finxpnom 36220 | Cartesian exponentiation w... |
| finxp00 36221 | Cartesian exponentiation o... |
| iunctb2 36222 | Using the axiom of countab... |
| domalom 36223 | A class which dominates ev... |
| isinf2 36224 | The converse of ~ isinf . ... |
| ctbssinf 36225 | Using the axiom of choice,... |
| ralssiun 36226 | The index set of an indexe... |
| nlpineqsn 36227 | For every point ` p ` of a... |
| nlpfvineqsn 36228 | Given a subset ` A ` of ` ... |
| fvineqsnf1 36229 | A theorem about functions ... |
| fvineqsneu 36230 | A theorem about functions ... |
| fvineqsneq 36231 | A theorem about functions ... |
| pibp16 36232 | Property P000016 of pi-bas... |
| pibp19 36233 | Property P000019 of pi-bas... |
| pibp21 36234 | Property P000021 of pi-bas... |
| pibt1 36235 | Theorem T000001 of pi-base... |
| pibt2 36236 | Theorem T000002 of pi-base... |
| wl-section-prop 36237 | Intuitionistic logic is no... |
| wl-section-boot 36241 | In this section, I provide... |
| wl-luk-imim1i 36242 | Inference adding common co... |
| wl-luk-syl 36243 | An inference version of th... |
| wl-luk-imtrid 36244 | A syllogism rule of infere... |
| wl-luk-pm2.18d 36245 | Deduction based on reducti... |
| wl-luk-con4i 36246 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 36247 | Inference rule. Copy of ~... |
| wl-luk-a1i 36248 | Inference rule. Copy of ~... |
| wl-luk-mpi 36249 | A nested modus ponens infe... |
| wl-luk-imim2i 36250 | Inference adding common an... |
| wl-luk-imtrdi 36251 | A syllogism rule of infere... |
| wl-luk-ax3 36252 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 36253 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 36254 | This theorem, called "Asse... |
| wl-luk-com12 36255 | Inference that swaps (comm... |
| wl-luk-pm2.21 36256 | From a wff and its negatio... |
| wl-luk-con1i 36257 | A contraposition inference... |
| wl-luk-ja 36258 | Inference joining the ante... |
| wl-luk-imim2 36259 | A closed form of syllogism... |
| wl-luk-a1d 36260 | Deduction introducing an e... |
| wl-luk-ax2 36261 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 36262 | Principle of identity. Th... |
| wl-luk-notnotr 36263 | Converse of double negatio... |
| wl-luk-pm2.04 36264 | Swap antecedents. Theorem... |
| wl-section-impchain 36265 | An implication like ` ( ps... |
| wl-impchain-mp-x 36266 | This series of theorems pr... |
| wl-impchain-mp-0 36267 | This theorem is the start ... |
| wl-impchain-mp-1 36268 | This theorem is in fact a ... |
| wl-impchain-mp-2 36269 | This theorem is in fact a ... |
| wl-impchain-com-1.x 36270 | It is often convenient to ... |
| wl-impchain-com-1.1 36271 | A degenerate form of antec... |
| wl-impchain-com-1.2 36272 | This theorem is in fact a ... |
| wl-impchain-com-1.3 36273 | This theorem is in fact a ... |
| wl-impchain-com-1.4 36274 | This theorem is in fact a ... |
| wl-impchain-com-n.m 36275 | This series of theorems al... |
| wl-impchain-com-2.3 36276 | This theorem is in fact a ... |
| wl-impchain-com-2.4 36277 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 36278 | This theorem is in fact a ... |
| wl-impchain-a1-x 36279 | If an implication chain is... |
| wl-impchain-a1-1 36280 | Inference rule, a copy of ... |
| wl-impchain-a1-2 36281 | Inference rule, a copy of ... |
| wl-impchain-a1-3 36282 | Inference rule, a copy of ... |
| wl-ifp-ncond1 36283 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 36284 | If one case of an ` if- ` ... |
| wl-ifpimpr 36285 | If one case of an ` if- ` ... |
| wl-ifp4impr 36286 | If one case of an ` if- ` ... |
| wl-df-3xor 36287 | Alternative definition of ... |
| wl-df3xor2 36288 | Alternative definition of ... |
| wl-df3xor3 36289 | Alternative form of ~ wl-d... |
| wl-3xortru 36290 | If the first input is true... |
| wl-3xorfal 36291 | If the first input is fals... |
| wl-3xorbi 36292 | Triple xor can be replaced... |
| wl-3xorbi2 36293 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 36294 | Equivalence theorem for tr... |
| wl-3xorbi123i 36295 | Equivalence theorem for tr... |
| wl-3xorrot 36296 | Rotation law for triple xo... |
| wl-3xorcoma 36297 | Commutative law for triple... |
| wl-3xorcomb 36298 | Commutative law for triple... |
| wl-3xornot1 36299 | Flipping the first input f... |
| wl-3xornot 36300 | Triple xor distributes ove... |
| wl-1xor 36301 | In the recursive scheme ... |
| wl-2xor 36302 | In the recursive scheme ... |
| wl-df-3mintru2 36303 | Alternative definition of ... |
| wl-df2-3mintru2 36304 | The adder carry in disjunc... |
| wl-df3-3mintru2 36305 | The adder carry in conjunc... |
| wl-df4-3mintru2 36306 | An alternative definition ... |
| wl-1mintru1 36307 | Using the recursion formul... |
| wl-1mintru2 36308 | Using the recursion formul... |
| wl-2mintru1 36309 | Using the recursion formul... |
| wl-2mintru2 36310 | Using the recursion formul... |
| wl-df3maxtru1 36311 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 36313 | A version of ~ ax-wl-13v w... |
| wl-mps 36314 | Replacing a nested consequ... |
| wl-syls1 36315 | Replacing a nested consequ... |
| wl-syls2 36316 | Replacing a nested anteced... |
| wl-embant 36317 | A true wff can always be a... |
| wl-orel12 36318 | In a conjunctive normal fo... |
| wl-cases2-dnf 36319 | A particular instance of ~... |
| wl-cbvmotv 36320 | Change bound variable. Us... |
| wl-moteq 36321 | Change bound variable. Us... |
| wl-motae 36322 | Change bound variable. Us... |
| wl-moae 36323 | Two ways to express "at mo... |
| wl-euae 36324 | Two ways to express "exact... |
| wl-nax6im 36325 | The following series of th... |
| wl-hbae1 36326 | This specialization of ~ h... |
| wl-naevhba1v 36327 | An instance of ~ hbn1w app... |
| wl-spae 36328 | Prove an instance of ~ sp ... |
| wl-speqv 36329 | Under the assumption ` -. ... |
| wl-19.8eqv 36330 | Under the assumption ` -. ... |
| wl-19.2reqv 36331 | Under the assumption ` -. ... |
| wl-nfalv 36332 | If ` x ` is not present in... |
| wl-nfimf1 36333 | An antecedent is irrelevan... |
| wl-nfae1 36334 | Unlike ~ nfae , this speci... |
| wl-nfnae1 36335 | Unlike ~ nfnae , this spec... |
| wl-aetr 36336 | A transitive law for varia... |
| wl-axc11r 36337 | Same as ~ axc11r , but usi... |
| wl-dral1d 36338 | A version of ~ dral1 with ... |
| wl-cbvalnaed 36339 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 36340 | A more general version of ... |
| wl-exeq 36341 | The semantics of ` E. x y ... |
| wl-aleq 36342 | The semantics of ` A. x y ... |
| wl-nfeqfb 36343 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 36344 | If ` y ` is not free in ` ... |
| wl-equsalvw 36345 | Version of ~ equsalv with ... |
| wl-equsald 36346 | Deduction version of ~ equ... |
| wl-equsal 36347 | A useful equivalence relat... |
| wl-equsal1t 36348 | The expression ` x = y ` i... |
| wl-equsalcom 36349 | This simple equivalence ea... |
| wl-equsal1i 36350 | The antecedent ` x = y ` i... |
| wl-sb6rft 36351 | A specialization of ~ wl-e... |
| wl-cbvalsbi 36352 | Change bounded variables i... |
| wl-sbrimt 36353 | Substitution with a variab... |
| wl-sblimt 36354 | Substitution with a variab... |
| wl-sb8t 36355 | Substitution of variable i... |
| wl-sb8et 36356 | Substitution of variable i... |
| wl-sbhbt 36357 | Closed form of ~ sbhb . C... |
| wl-sbnf1 36358 | Two ways expressing that `... |
| wl-equsb3 36359 | ~ equsb3 with a distinctor... |
| wl-equsb4 36360 | Substitution applied to an... |
| wl-2sb6d 36361 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 36362 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 36363 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 36364 | Version of ~ sbcom2 with a... |
| wl-sbalnae 36365 | A theorem used in eliminat... |
| wl-sbal1 36366 | A theorem used in eliminat... |
| wl-sbal2 36367 | Move quantifier in and out... |
| wl-2spsbbi 36368 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 36369 | This theorem provides a ba... |
| wl-lem-nexmo 36370 | This theorem provides a ba... |
| wl-lem-moexsb 36371 | The antecedent ` A. x ( ph... |
| wl-alanbii 36372 | This theorem extends ~ ala... |
| wl-mo2df 36373 | Version of ~ mof with a co... |
| wl-mo2tf 36374 | Closed form of ~ mof with ... |
| wl-eudf 36375 | Version of ~ eu6 with a co... |
| wl-eutf 36376 | Closed form of ~ eu6 with ... |
| wl-euequf 36377 | ~ euequ proved with a dist... |
| wl-mo2t 36378 | Closed form of ~ mof . (C... |
| wl-mo3t 36379 | Closed form of ~ mo3 . (C... |
| wl-sb8eut 36380 | Substitution of variable i... |
| wl-sb8mot 36381 | Substitution of variable i... |
| wl-issetft 36382 | A closed form of ~ issetf ... |
| wl-axc11rc11 36383 | Proving ~ axc11r from ~ ax... |
| wl-ax11-lem1 36385 | A transitive law for varia... |
| wl-ax11-lem2 36386 | Lemma. (Contributed by Wo... |
| wl-ax11-lem3 36387 | Lemma. (Contributed by Wo... |
| wl-ax11-lem4 36388 | Lemma. (Contributed by Wo... |
| wl-ax11-lem5 36389 | Lemma. (Contributed by Wo... |
| wl-ax11-lem6 36390 | Lemma. (Contributed by Wo... |
| wl-ax11-lem7 36391 | Lemma. (Contributed by Wo... |
| wl-ax11-lem8 36392 | Lemma. (Contributed by Wo... |
| wl-ax11-lem9 36393 | The easy part when ` x ` c... |
| wl-ax11-lem10 36394 | We now have prepared every... |
| wl-clabv 36395 | Variant of ~ df-clab , whe... |
| wl-dfclab 36396 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 36397 | Using class abstraction in... |
| wl-clabt 36398 | Using class abstraction in... |
| rabiun 36399 | Abstraction restricted to ... |
| iundif1 36400 | Indexed union of class dif... |
| imadifss 36401 | The difference of images i... |
| cureq 36402 | Equality theorem for curry... |
| unceq 36403 | Equality theorem for uncur... |
| curf 36404 | Functional property of cur... |
| uncf 36405 | Functional property of unc... |
| curfv 36406 | Value of currying. (Contr... |
| uncov 36407 | Value of uncurrying. (Con... |
| curunc 36408 | Currying of uncurrying. (... |
| unccur 36409 | Uncurrying of currying. (... |
| phpreu 36410 | Theorem related to pigeonh... |
| finixpnum 36411 | A finite Cartesian product... |
| fin2solem 36412 | Lemma for ~ fin2so . (Con... |
| fin2so 36413 | Any totally ordered Tarski... |
| ltflcei 36414 | Theorem to move the floor ... |
| leceifl 36415 | Theorem to move the floor ... |
| sin2h 36416 | Half-angle rule for sine. ... |
| cos2h 36417 | Half-angle rule for cosine... |
| tan2h 36418 | Half-angle rule for tangen... |
| lindsadd 36419 | In a vector space, the uni... |
| lindsdom 36420 | A linearly independent set... |
| lindsenlbs 36421 | A maximal linearly indepen... |
| matunitlindflem1 36422 | One direction of ~ matunit... |
| matunitlindflem2 36423 | One direction of ~ matunit... |
| matunitlindf 36424 | A matrix over a field is i... |
| ptrest 36425 | Expressing a restriction o... |
| ptrecube 36426 | Any point in an open set o... |
| poimirlem1 36427 | Lemma for ~ poimir - the v... |
| poimirlem2 36428 | Lemma for ~ poimir - conse... |
| poimirlem3 36429 | Lemma for ~ poimir to add ... |
| poimirlem4 36430 | Lemma for ~ poimir connect... |
| poimirlem5 36431 | Lemma for ~ poimir to esta... |
| poimirlem6 36432 | Lemma for ~ poimir establi... |
| poimirlem7 36433 | Lemma for ~ poimir , simil... |
| poimirlem8 36434 | Lemma for ~ poimir , estab... |
| poimirlem9 36435 | Lemma for ~ poimir , estab... |
| poimirlem10 36436 | Lemma for ~ poimir establi... |
| poimirlem11 36437 | Lemma for ~ poimir connect... |
| poimirlem12 36438 | Lemma for ~ poimir connect... |
| poimirlem13 36439 | Lemma for ~ poimir - for a... |
| poimirlem14 36440 | Lemma for ~ poimir - for a... |
| poimirlem15 36441 | Lemma for ~ poimir , that ... |
| poimirlem16 36442 | Lemma for ~ poimir establi... |
| poimirlem17 36443 | Lemma for ~ poimir establi... |
| poimirlem18 36444 | Lemma for ~ poimir stating... |
| poimirlem19 36445 | Lemma for ~ poimir establi... |
| poimirlem20 36446 | Lemma for ~ poimir establi... |
| poimirlem21 36447 | Lemma for ~ poimir stating... |
| poimirlem22 36448 | Lemma for ~ poimir , that ... |
| poimirlem23 36449 | Lemma for ~ poimir , two w... |
| poimirlem24 36450 | Lemma for ~ poimir , two w... |
| poimirlem25 36451 | Lemma for ~ poimir stating... |
| poimirlem26 36452 | Lemma for ~ poimir showing... |
| poimirlem27 36453 | Lemma for ~ poimir showing... |
| poimirlem28 36454 | Lemma for ~ poimir , a var... |
| poimirlem29 36455 | Lemma for ~ poimir connect... |
| poimirlem30 36456 | Lemma for ~ poimir combini... |
| poimirlem31 36457 | Lemma for ~ poimir , assig... |
| poimirlem32 36458 | Lemma for ~ poimir , combi... |
| poimir 36459 | Poincare-Miranda theorem. ... |
| broucube 36460 | Brouwer - or as Kulpa call... |
| heicant 36461 | Heine-Cantor theorem: a co... |
| opnmbllem0 36462 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 36463 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 36464 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 36465 | The difference between two... |
| mblfinlem4 36466 | Backward direction of ~ is... |
| ismblfin 36467 | Measurability in terms of ... |
| ovoliunnfl 36468 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 36469 | Demonstration of ~ ovoliun... |
| voliunnfl 36470 | ~ voliun is incompatible w... |
| volsupnfl 36471 | ~ volsup is incompatible w... |
| mbfresfi 36472 | Measurability of a piecewi... |
| mbfposadd 36473 | If the sum of two measurab... |
| cnambfre 36474 | A real-valued, a.e. contin... |
| dvtanlem 36475 | Lemma for ~ dvtan - the do... |
| dvtan 36476 | Derivative of tangent. (C... |
| itg2addnclem 36477 | An alternate expression fo... |
| itg2addnclem2 36478 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 36479 | Lemma incomprehensible in ... |
| itg2addnc 36480 | Alternate proof of ~ itg2a... |
| itg2gt0cn 36481 | ~ itg2gt0 holds on functio... |
| ibladdnclem 36482 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 36483 | Choice-free analogue of ~ ... |
| itgaddnclem1 36484 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 36485 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 36486 | Choice-free analogue of ~ ... |
| iblsubnc 36487 | Choice-free analogue of ~ ... |
| itgsubnc 36488 | Choice-free analogue of ~ ... |
| iblabsnclem 36489 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 36490 | Choice-free analogue of ~ ... |
| iblmulc2nc 36491 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 36492 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 36493 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 36494 | Choice-free analogue of ~ ... |
| itgabsnc 36495 | Choice-free analogue of ~ ... |
| itggt0cn 36496 | ~ itggt0 holds for continu... |
| ftc1cnnclem 36497 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 36498 | Choice-free proof of ~ ftc... |
| ftc1anclem1 36499 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 36500 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 36501 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 36502 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 36503 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 36504 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 36505 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 36506 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 36507 | ~ ftc1a holds for function... |
| ftc2nc 36508 | Choice-free proof of ~ ftc... |
| asindmre 36509 | Real part of domain of dif... |
| dvasin 36510 | Derivative of arcsine. (C... |
| dvacos 36511 | Derivative of arccosine. ... |
| dvreasin 36512 | Real derivative of arcsine... |
| dvreacos 36513 | Real derivative of arccosi... |
| areacirclem1 36514 | Antiderivative of cross-se... |
| areacirclem2 36515 | Endpoint-inclusive continu... |
| areacirclem3 36516 | Integrability of cross-sec... |
| areacirclem4 36517 | Endpoint-inclusive continu... |
| areacirclem5 36518 | Finding the cross-section ... |
| areacirc 36519 | The area of a circle of ra... |
| unirep 36520 | Define a quantity whose de... |
| cover2 36521 | Two ways of expressing the... |
| cover2g 36522 | Two ways of expressing the... |
| brabg2 36523 | Relation by a binary relat... |
| opelopab3 36524 | Ordered pair membership in... |
| cocanfo 36525 | Cancellation of a surjecti... |
| brresi2 36526 | Restriction of a binary re... |
| fnopabeqd 36527 | Equality deduction for fun... |
| fvopabf4g 36528 | Function value of an opera... |
| fnopabco 36529 | Composition of a function ... |
| opropabco 36530 | Composition of an operator... |
| cocnv 36531 | Composition with a functio... |
| f1ocan1fv 36532 | Cancel a composition by a ... |
| f1ocan2fv 36533 | Cancel a composition by th... |
| inixp 36534 | Intersection of Cartesian ... |
| upixp 36535 | Universal property of the ... |
| abrexdom 36536 | An indexed set is dominate... |
| abrexdom2 36537 | An indexed set is dominate... |
| ac6gf 36538 | Axiom of Choice. (Contrib... |
| indexa 36539 | If for every element of an... |
| indexdom 36540 | If for every element of an... |
| frinfm 36541 | A subset of a well-founded... |
| welb 36542 | A nonempty subset of a wel... |
| supex2g 36543 | Existence of supremum. (C... |
| supclt 36544 | Closure of supremum. (Con... |
| supubt 36545 | Upper bound property of su... |
| filbcmb 36546 | Combine a finite set of lo... |
| fzmul 36547 | Membership of a product in... |
| sdclem2 36548 | Lemma for ~ sdc . (Contri... |
| sdclem1 36549 | Lemma for ~ sdc . (Contri... |
| sdc 36550 | Strong dependent choice. ... |
| fdc 36551 | Finite version of dependen... |
| fdc1 36552 | Variant of ~ fdc with no s... |
| seqpo 36553 | Two ways to say that a seq... |
| incsequz 36554 | An increasing sequence of ... |
| incsequz2 36555 | An increasing sequence of ... |
| nnubfi 36556 | A bounded above set of pos... |
| nninfnub 36557 | An infinite set of positiv... |
| subspopn 36558 | An open set is open in the... |
| neificl 36559 | Neighborhoods are closed u... |
| lpss2 36560 | Limit points of a subset a... |
| metf1o 36561 | Use a bijection with a met... |
| blssp 36562 | A ball in the subspace met... |
| mettrifi 36563 | Generalized triangle inequ... |
| lmclim2 36564 | A sequence in a metric spa... |
| geomcau 36565 | If the distance between co... |
| caures 36566 | The restriction of a Cauch... |
| caushft 36567 | A shifted Cauchy sequence ... |
| constcncf 36568 | A constant function is a c... |
| cnres2 36569 | The restriction of a conti... |
| cnresima 36570 | A continuous function is c... |
| cncfres 36571 | A continuous function on c... |
| istotbnd 36575 | The predicate "is a totall... |
| istotbnd2 36576 | The predicate "is a totall... |
| istotbnd3 36577 | A metric space is totally ... |
| totbndmet 36578 | The predicate "totally bou... |
| 0totbnd 36579 | The metric (there is only ... |
| sstotbnd2 36580 | Condition for a subset of ... |
| sstotbnd 36581 | Condition for a subset of ... |
| sstotbnd3 36582 | Use a net that is not nece... |
| totbndss 36583 | A subset of a totally boun... |
| equivtotbnd 36584 | If the metric ` M ` is "st... |
| isbnd 36586 | The predicate "is a bounde... |
| bndmet 36587 | A bounded metric space is ... |
| isbndx 36588 | A "bounded extended metric... |
| isbnd2 36589 | The predicate "is a bounde... |
| isbnd3 36590 | A metric space is bounded ... |
| isbnd3b 36591 | A metric space is bounded ... |
| bndss 36592 | A subset of a bounded metr... |
| blbnd 36593 | A ball is bounded. (Contr... |
| ssbnd 36594 | A subset of a metric space... |
| totbndbnd 36595 | A totally bounded metric s... |
| equivbnd 36596 | If the metric ` M ` is "st... |
| bnd2lem 36597 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 36598 | If balls are totally bound... |
| prdsbnd 36599 | The product metric over fi... |
| prdstotbnd 36600 | The product metric over fi... |
| prdsbnd2 36601 | If balls are totally bound... |
| cntotbnd 36602 | A subset of the complex nu... |
| cnpwstotbnd 36603 | A subset of ` A ^ I ` , wh... |
| ismtyval 36606 | The set of isometries betw... |
| isismty 36607 | The condition "is an isome... |
| ismtycnv 36608 | The inverse of an isometry... |
| ismtyima 36609 | The image of a ball under ... |
| ismtyhmeolem 36610 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 36611 | An isometry is a homeomorp... |
| ismtybndlem 36612 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 36613 | Isometries preserve bounde... |
| ismtyres 36614 | A restriction of an isomet... |
| heibor1lem 36615 | Lemma for ~ heibor1 . A c... |
| heibor1 36616 | One half of ~ heibor , tha... |
| heiborlem1 36617 | Lemma for ~ heibor . We w... |
| heiborlem2 36618 | Lemma for ~ heibor . Subs... |
| heiborlem3 36619 | Lemma for ~ heibor . Usin... |
| heiborlem4 36620 | Lemma for ~ heibor . Usin... |
| heiborlem5 36621 | Lemma for ~ heibor . The ... |
| heiborlem6 36622 | Lemma for ~ heibor . Sinc... |
| heiborlem7 36623 | Lemma for ~ heibor . Sinc... |
| heiborlem8 36624 | Lemma for ~ heibor . The ... |
| heiborlem9 36625 | Lemma for ~ heibor . Disc... |
| heiborlem10 36626 | Lemma for ~ heibor . The ... |
| heibor 36627 | Generalized Heine-Borel Th... |
| bfplem1 36628 | Lemma for ~ bfp . The seq... |
| bfplem2 36629 | Lemma for ~ bfp . Using t... |
| bfp 36630 | Banach fixed point theorem... |
| rrnval 36633 | The n-dimensional Euclidea... |
| rrnmval 36634 | The value of the Euclidean... |
| rrnmet 36635 | Euclidean space is a metri... |
| rrndstprj1 36636 | The distance between two p... |
| rrndstprj2 36637 | Bound on the distance betw... |
| rrncmslem 36638 | Lemma for ~ rrncms . (Con... |
| rrncms 36639 | Euclidean space is complet... |
| repwsmet 36640 | The supremum metric on ` R... |
| rrnequiv 36641 | The supremum metric on ` R... |
| rrntotbnd 36642 | A set in Euclidean space i... |
| rrnheibor 36643 | Heine-Borel theorem for Eu... |
| ismrer1 36644 | An isometry between ` RR `... |
| reheibor 36645 | Heine-Borel theorem for re... |
| iccbnd 36646 | A closed interval in ` RR ... |
| icccmpALT 36647 | A closed interval in ` RR ... |
| isass 36652 | The predicate "is an assoc... |
| isexid 36653 | The predicate ` G ` has a ... |
| ismgmOLD 36656 | Obsolete version of ~ ismg... |
| clmgmOLD 36657 | Obsolete version of ~ mgmc... |
| opidonOLD 36658 | Obsolete version of ~ mndp... |
| rngopidOLD 36659 | Obsolete version of ~ mndp... |
| opidon2OLD 36660 | Obsolete version of ~ mndp... |
| isexid2 36661 | If ` G e. ( Magma i^i ExId... |
| exidu1 36662 | Uniqueness of the left and... |
| idrval 36663 | The value of the identity ... |
| iorlid 36664 | A magma right and left ide... |
| cmpidelt 36665 | A magma right and left ide... |
| smgrpismgmOLD 36668 | Obsolete version of ~ sgrp... |
| issmgrpOLD 36669 | Obsolete version of ~ issg... |
| smgrpmgm 36670 | A semigroup is a magma. (... |
| smgrpassOLD 36671 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 36674 | Obsolete version of ~ mnds... |
| mndoisexid 36675 | A monoid has an identity e... |
| mndoismgmOLD 36676 | Obsolete version of ~ mndm... |
| mndomgmid 36677 | A monoid is a magma with a... |
| ismndo 36678 | The predicate "is a monoid... |
| ismndo1 36679 | The predicate "is a monoid... |
| ismndo2 36680 | The predicate "is a monoid... |
| grpomndo 36681 | A group is a monoid. (Con... |
| exidcl 36682 | Closure of the binary oper... |
| exidreslem 36683 | Lemma for ~ exidres and ~ ... |
| exidres 36684 | The restriction of a binar... |
| exidresid 36685 | The restriction of a binar... |
| ablo4pnp 36686 | A commutative/associative ... |
| grpoeqdivid 36687 | Two group elements are equ... |
| grposnOLD 36688 | The group operation for th... |
| elghomlem1OLD 36691 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 36692 | Obsolete as of 15-Mar-2020... |
| elghomOLD 36693 | Obsolete version of ~ isgh... |
| ghomlinOLD 36694 | Obsolete version of ~ ghml... |
| ghomidOLD 36695 | Obsolete version of ~ ghmi... |
| ghomf 36696 | Mapping property of a grou... |
| ghomco 36697 | The composition of two gro... |
| ghomdiv 36698 | Group homomorphisms preser... |
| grpokerinj 36699 | A group homomorphism is in... |
| relrngo 36702 | The class of all unital ri... |
| isrngo 36703 | The predicate "is a (unita... |
| isrngod 36704 | Conditions that determine ... |
| rngoi 36705 | The properties of a unital... |
| rngosm 36706 | Functionality of the multi... |
| rngocl 36707 | Closure of the multiplicat... |
| rngoid 36708 | The multiplication operati... |
| rngoideu 36709 | The unity element of a rin... |
| rngodi 36710 | Distributive law for the m... |
| rngodir 36711 | Distributive law for the m... |
| rngoass 36712 | Associative law for the mu... |
| rngo2 36713 | A ring element plus itself... |
| rngoablo 36714 | A ring's addition operatio... |
| rngoablo2 36715 | In a unital ring the addit... |
| rngogrpo 36716 | A ring's addition operatio... |
| rngone0 36717 | The base set of a ring is ... |
| rngogcl 36718 | Closure law for the additi... |
| rngocom 36719 | The addition operation of ... |
| rngoaass 36720 | The addition operation of ... |
| rngoa32 36721 | The addition operation of ... |
| rngoa4 36722 | Rearrangement of 4 terms i... |
| rngorcan 36723 | Right cancellation law for... |
| rngolcan 36724 | Left cancellation law for ... |
| rngo0cl 36725 | A ring has an additive ide... |
| rngo0rid 36726 | The additive identity of a... |
| rngo0lid 36727 | The additive identity of a... |
| rngolz 36728 | The zero of a unital ring ... |
| rngorz 36729 | The zero of a unital ring ... |
| rngosn3 36730 | Obsolete as of 25-Jan-2020... |
| rngosn4 36731 | Obsolete as of 25-Jan-2020... |
| rngosn6 36732 | Obsolete as of 25-Jan-2020... |
| rngonegcl 36733 | A ring is closed under neg... |
| rngoaddneg1 36734 | Adding the negative in a r... |
| rngoaddneg2 36735 | Adding the negative in a r... |
| rngosub 36736 | Subtraction in a ring, in ... |
| rngmgmbs4 36737 | The range of an internal o... |
| rngodm1dm2 36738 | In a unital ring the domai... |
| rngorn1 36739 | In a unital ring the range... |
| rngorn1eq 36740 | In a unital ring the range... |
| rngomndo 36741 | In a unital ring the multi... |
| rngoidmlem 36742 | The unity element of a rin... |
| rngolidm 36743 | The unity element of a rin... |
| rngoridm 36744 | The unity element of a rin... |
| rngo1cl 36745 | The unity element of a rin... |
| rngoueqz 36746 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 36747 | Negation in a ring is the ... |
| rngonegmn1r 36748 | Negation in a ring is the ... |
| rngoneglmul 36749 | Negation of a product in a... |
| rngonegrmul 36750 | Negation of a product in a... |
| rngosubdi 36751 | Ring multiplication distri... |
| rngosubdir 36752 | Ring multiplication distri... |
| zerdivemp1x 36753 | In a unital ring a left in... |
| isdivrngo 36756 | The predicate "is a divisi... |
| drngoi 36757 | The properties of a divisi... |
| gidsn 36758 | Obsolete as of 23-Jan-2020... |
| zrdivrng 36759 | The zero ring is not a div... |
| dvrunz 36760 | In a division ring the rin... |
| isgrpda 36761 | Properties that determine ... |
| isdrngo1 36762 | The predicate "is a divisi... |
| divrngcl 36763 | The product of two nonzero... |
| isdrngo2 36764 | A division ring is a ring ... |
| isdrngo3 36765 | A division ring is a ring ... |
| rngohomval 36770 | The set of ring homomorphi... |
| isrngohom 36771 | The predicate "is a ring h... |
| rngohomf 36772 | A ring homomorphism is a f... |
| rngohomcl 36773 | Closure law for a ring hom... |
| rngohom1 36774 | A ring homomorphism preser... |
| rngohomadd 36775 | Ring homomorphisms preserv... |
| rngohommul 36776 | Ring homomorphisms preserv... |
| rngogrphom 36777 | A ring homomorphism is a g... |
| rngohom0 36778 | A ring homomorphism preser... |
| rngohomsub 36779 | Ring homomorphisms preserv... |
| rngohomco 36780 | The composition of two rin... |
| rngokerinj 36781 | A ring homomorphism is inj... |
| rngoisoval 36783 | The set of ring isomorphis... |
| isrngoiso 36784 | The predicate "is a ring i... |
| rngoiso1o 36785 | A ring isomorphism is a bi... |
| rngoisohom 36786 | A ring isomorphism is a ri... |
| rngoisocnv 36787 | The inverse of a ring isom... |
| rngoisoco 36788 | The composition of two rin... |
| isriscg 36790 | The ring isomorphism relat... |
| isrisc 36791 | The ring isomorphism relat... |
| risc 36792 | The ring isomorphism relat... |
| risci 36793 | Determine that two rings a... |
| riscer 36794 | Ring isomorphism is an equ... |
| iscom2 36801 | A device to add commutativ... |
| iscrngo 36802 | The predicate "is a commut... |
| iscrngo2 36803 | The predicate "is a commut... |
| iscringd 36804 | Conditions that determine ... |
| flddivrng 36805 | A field is a division ring... |
| crngorngo 36806 | A commutative ring is a ri... |
| crngocom 36807 | The multiplication operati... |
| crngm23 36808 | Commutative/associative la... |
| crngm4 36809 | Commutative/associative la... |
| fldcrngo 36810 | A field is a commutative r... |
| isfld2 36811 | The predicate "is a field"... |
| crngohomfo 36812 | The image of a homomorphis... |
| idlval 36819 | The class of ideals of a r... |
| isidl 36820 | The predicate "is an ideal... |
| isidlc 36821 | The predicate "is an ideal... |
| idlss 36822 | An ideal of ` R ` is a sub... |
| idlcl 36823 | An element of an ideal is ... |
| idl0cl 36824 | An ideal contains ` 0 ` . ... |
| idladdcl 36825 | An ideal is closed under a... |
| idllmulcl 36826 | An ideal is closed under m... |
| idlrmulcl 36827 | An ideal is closed under m... |
| idlnegcl 36828 | An ideal is closed under n... |
| idlsubcl 36829 | An ideal is closed under s... |
| rngoidl 36830 | A ring ` R ` is an ` R ` i... |
| 0idl 36831 | The set containing only ` ... |
| 1idl 36832 | Two ways of expressing the... |
| 0rngo 36833 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 36834 | The only ideals in a divis... |
| intidl 36835 | The intersection of a none... |
| inidl 36836 | The intersection of two id... |
| unichnidl 36837 | The union of a nonempty ch... |
| keridl 36838 | The kernel of a ring homom... |
| pridlval 36839 | The class of prime ideals ... |
| ispridl 36840 | The predicate "is a prime ... |
| pridlidl 36841 | A prime ideal is an ideal.... |
| pridlnr 36842 | A prime ideal is a proper ... |
| pridl 36843 | The main property of a pri... |
| ispridl2 36844 | A condition that shows an ... |
| maxidlval 36845 | The set of maximal ideals ... |
| ismaxidl 36846 | The predicate "is a maxima... |
| maxidlidl 36847 | A maximal ideal is an idea... |
| maxidlnr 36848 | A maximal ideal is proper.... |
| maxidlmax 36849 | A maximal ideal is a maxim... |
| maxidln1 36850 | One is not contained in an... |
| maxidln0 36851 | A ring with a maximal idea... |
| isprrngo 36856 | The predicate "is a prime ... |
| prrngorngo 36857 | A prime ring is a ring. (... |
| smprngopr 36858 | A simple ring (one whose o... |
| divrngpr 36859 | A division ring is a prime... |
| isdmn 36860 | The predicate "is a domain... |
| isdmn2 36861 | The predicate "is a domain... |
| dmncrng 36862 | A domain is a commutative ... |
| dmnrngo 36863 | A domain is a ring. (Cont... |
| flddmn 36864 | A field is a domain. (Con... |
| igenval 36867 | The ideal generated by a s... |
| igenss 36868 | A set is a subset of the i... |
| igenidl 36869 | The ideal generated by a s... |
| igenmin 36870 | The ideal generated by a s... |
| igenidl2 36871 | The ideal generated by an ... |
| igenval2 36872 | The ideal generated by a s... |
| prnc 36873 | A principal ideal (an idea... |
| isfldidl 36874 | Determine if a ring is a f... |
| isfldidl2 36875 | Determine if a ring is a f... |
| ispridlc 36876 | The predicate "is a prime ... |
| pridlc 36877 | Property of a prime ideal ... |
| pridlc2 36878 | Property of a prime ideal ... |
| pridlc3 36879 | Property of a prime ideal ... |
| isdmn3 36880 | The predicate "is a domain... |
| dmnnzd 36881 | A domain has no zero-divis... |
| dmncan1 36882 | Cancellation law for domai... |
| dmncan2 36883 | Cancellation law for domai... |
| efald2 36884 | A proof by contradiction. ... |
| notbinot1 36885 | Simplification rule of neg... |
| bicontr 36886 | Biconditional of its own n... |
| impor 36887 | An equivalent formula for ... |
| orfa 36888 | The falsum ` F. ` can be r... |
| notbinot2 36889 | Commutation rule between n... |
| biimpor 36890 | A rewriting rule for bicon... |
| orfa1 36891 | Add a contradicting disjun... |
| orfa2 36892 | Remove a contradicting dis... |
| bifald 36893 | Infer the equivalence to a... |
| orsild 36894 | A lemma for not-or-not eli... |
| orsird 36895 | A lemma for not-or-not eli... |
| cnf1dd 36896 | A lemma for Conjunctive No... |
| cnf2dd 36897 | A lemma for Conjunctive No... |
| cnfn1dd 36898 | A lemma for Conjunctive No... |
| cnfn2dd 36899 | A lemma for Conjunctive No... |
| or32dd 36900 | A rearrangement of disjunc... |
| notornotel1 36901 | A lemma for not-or-not eli... |
| notornotel2 36902 | A lemma for not-or-not eli... |
| contrd 36903 | A proof by contradiction, ... |
| an12i 36904 | An inference from commutin... |
| exmid2 36905 | An excluded middle law. (... |
| selconj 36906 | An inference for selecting... |
| truconj 36907 | Add true as a conjunct. (... |
| orel 36908 | An inference for disjuncti... |
| negel 36909 | An inference for negation ... |
| botel 36910 | An inference for bottom el... |
| tradd 36911 | Add top ad a conjunct. (C... |
| gm-sbtru 36912 | Substitution does not chan... |
| sbfal 36913 | Substitution does not chan... |
| sbcani 36914 | Distribution of class subs... |
| sbcori 36915 | Distribution of class subs... |
| sbcimi 36916 | Distribution of class subs... |
| sbcni 36917 | Move class substitution in... |
| sbali 36918 | Discard class substitution... |
| sbexi 36919 | Discard class substitution... |
| sbcalf 36920 | Move universal quantifier ... |
| sbcexf 36921 | Move existential quantifie... |
| sbcalfi 36922 | Move universal quantifier ... |
| sbcexfi 36923 | Move existential quantifie... |
| spsbcdi 36924 | A lemma for eliminating a ... |
| alrimii 36925 | A lemma for introducing a ... |
| spesbcdi 36926 | A lemma for introducing an... |
| exlimddvf 36927 | A lemma for eliminating an... |
| exlimddvfi 36928 | A lemma for eliminating an... |
| sbceq1ddi 36929 | A lemma for eliminating in... |
| sbccom2lem 36930 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 36931 | Commutative law for double... |
| sbccom2f 36932 | Commutative law for double... |
| sbccom2fi 36933 | Commutative law for double... |
| csbcom2fi 36934 | Commutative law for double... |
| fald 36935 | Refutation of falsity, in ... |
| tsim1 36936 | A Tseitin axiom for logica... |
| tsim2 36937 | A Tseitin axiom for logica... |
| tsim3 36938 | A Tseitin axiom for logica... |
| tsbi1 36939 | A Tseitin axiom for logica... |
| tsbi2 36940 | A Tseitin axiom for logica... |
| tsbi3 36941 | A Tseitin axiom for logica... |
| tsbi4 36942 | A Tseitin axiom for logica... |
| tsxo1 36943 | A Tseitin axiom for logica... |
| tsxo2 36944 | A Tseitin axiom for logica... |
| tsxo3 36945 | A Tseitin axiom for logica... |
| tsxo4 36946 | A Tseitin axiom for logica... |
| tsan1 36947 | A Tseitin axiom for logica... |
| tsan2 36948 | A Tseitin axiom for logica... |
| tsan3 36949 | A Tseitin axiom for logica... |
| tsna1 36950 | A Tseitin axiom for logica... |
| tsna2 36951 | A Tseitin axiom for logica... |
| tsna3 36952 | A Tseitin axiom for logica... |
| tsor1 36953 | A Tseitin axiom for logica... |
| tsor2 36954 | A Tseitin axiom for logica... |
| tsor3 36955 | A Tseitin axiom for logica... |
| ts3an1 36956 | A Tseitin axiom for triple... |
| ts3an2 36957 | A Tseitin axiom for triple... |
| ts3an3 36958 | A Tseitin axiom for triple... |
| ts3or1 36959 | A Tseitin axiom for triple... |
| ts3or2 36960 | A Tseitin axiom for triple... |
| ts3or3 36961 | A Tseitin axiom for triple... |
| iuneq2f 36962 | Equality deduction for ind... |
| rabeq12f 36963 | Equality deduction for res... |
| csbeq12 36964 | Equality deduction for sub... |
| sbeqi 36965 | Equality deduction for sub... |
| ralbi12f 36966 | Equality deduction for res... |
| oprabbi 36967 | Equality deduction for cla... |
| mpobi123f 36968 | Equality deduction for map... |
| iuneq12f 36969 | Equality deduction for ind... |
| iineq12f 36970 | Equality deduction for ind... |
| opabbi 36971 | Equality deduction for cla... |
| mptbi12f 36972 | Equality deduction for map... |
| orcomdd 36973 | Commutativity of logic dis... |
| scottexf 36974 | A version of ~ scottex wit... |
| scott0f 36975 | A version of ~ scott0 with... |
| scottn0f 36976 | A version of ~ scott0f wit... |
| ac6s3f 36977 | Generalization of the Axio... |
| ac6s6 36978 | Generalization of the Axio... |
| ac6s6f 36979 | Generalization of the Axio... |
| el2v1 37023 | New way ( ~ elv , and the ... |
| el3v 37024 | New way ( ~ elv , and the ... |
| el3v1 37025 | New way ( ~ elv , and the ... |
| el3v2 37026 | New way ( ~ elv , and the ... |
| el3v3 37027 | New way ( ~ elv , and the ... |
| el3v12 37028 | New way ( ~ elv , and the ... |
| el3v13 37029 | New way ( ~ elv , and the ... |
| el3v23 37030 | New way ( ~ elv , and the ... |
| anan 37031 | Multiple commutations in c... |
| triantru3 37032 | A wff is equivalent to its... |
| bianbi 37033 | Exchanging conjunction in ... |
| bianim 37034 | Exchanging conjunction in ... |
| biorfd 37035 | A wff is equivalent to its... |
| eqbrtr 37036 | Substitution of equal clas... |
| eqbrb 37037 | Substitution of equal clas... |
| eqeltr 37038 | Substitution of equal clas... |
| eqelb 37039 | Substitution of equal clas... |
| eqeqan2d 37040 | Implication of introducing... |
| suceqsneq 37041 | One-to-one relationship be... |
| sucdifsn2 37042 | Absorption of union with a... |
| sucdifsn 37043 | The difference between the... |
| disjresin 37044 | The restriction to a disjo... |
| disjresdisj 37045 | The intersection of restri... |
| disjresdif 37046 | The difference between res... |
| disjresundif 37047 | Lemma for ~ ressucdifsn2 .... |
| ressucdifsn2 37048 | The difference between res... |
| ressucdifsn 37049 | The difference between res... |
| inres2 37050 | Two ways of expressing the... |
| coideq 37051 | Equality theorem for compo... |
| nexmo1 37052 | If there is no case where ... |
| ralin 37053 | Restricted universal quant... |
| r2alan 37054 | Double restricted universa... |
| ssrabi 37055 | Inference of restricted ab... |
| rabbieq 37056 | Equivalent wff's correspon... |
| rabimbieq 37057 | Restricted equivalent wff'... |
| abeqin 37058 | Intersection with class ab... |
| abeqinbi 37059 | Intersection with class ab... |
| rabeqel 37060 | Class element of a restric... |
| eqrelf 37061 | The equality connective be... |
| br1cnvinxp 37062 | Binary relation on the con... |
| releleccnv 37063 | Elementhood in a converse ... |
| releccnveq 37064 | Equality of converse ` R `... |
| opelvvdif 37065 | Negated elementhood of ord... |
| vvdifopab 37066 | Ordered-pair class abstrac... |
| brvdif 37067 | Binary relation with unive... |
| brvdif2 37068 | Binary relation with unive... |
| brvvdif 37069 | Binary relation with the c... |
| brvbrvvdif 37070 | Binary relation with the c... |
| brcnvep 37071 | The converse of the binary... |
| elecALTV 37072 | Elementhood in the ` R ` -... |
| brcnvepres 37073 | Restricted converse epsilo... |
| brres2 37074 | Binary relation on a restr... |
| br1cnvres 37075 | Binary relation on the con... |
| eldmres 37076 | Elementhood in the domain ... |
| elrnres 37077 | Element of the range of a ... |
| eldmressnALTV 37078 | Element of the domain of a... |
| elrnressn 37079 | Element of the range of a ... |
| eldm4 37080 | Elementhood in a domain. ... |
| eldmres2 37081 | Elementhood in the domain ... |
| eceq1i 37082 | Equality theorem for ` C `... |
| elecres 37083 | Elementhood in the restric... |
| ecres 37084 | Restricted coset of ` B ` ... |
| ecres2 37085 | The restricted coset of ` ... |
| eccnvepres 37086 | Restricted converse epsilo... |
| eleccnvep 37087 | Elementhood in the convers... |
| eccnvep 37088 | The converse epsilon coset... |
| extep 37089 | Property of epsilon relati... |
| disjeccnvep 37090 | Property of the epsilon re... |
| eccnvepres2 37091 | The restricted converse ep... |
| eccnvepres3 37092 | Condition for a restricted... |
| eldmqsres 37093 | Elementhood in a restricte... |
| eldmqsres2 37094 | Elementhood in a restricte... |
| qsss1 37095 | Subclass theorem for quoti... |
| qseq1i 37096 | Equality theorem for quoti... |
| qseq1d 37097 | Equality theorem for quoti... |
| brinxprnres 37098 | Binary relation on a restr... |
| inxprnres 37099 | Restriction of a class as ... |
| dfres4 37100 | Alternate definition of th... |
| exan3 37101 | Equivalent expressions wit... |
| exanres 37102 | Equivalent expressions wit... |
| exanres3 37103 | Equivalent expressions wit... |
| exanres2 37104 | Equivalent expressions wit... |
| cnvepres 37105 | Restricted converse epsilo... |
| eqrel2 37106 | Equality of relations. (C... |
| rncnv 37107 | Range of converse is the d... |
| dfdm6 37108 | Alternate definition of do... |
| dfrn6 37109 | Alternate definition of ra... |
| rncnvepres 37110 | The range of the restricte... |
| dmecd 37111 | Equality of the coset of `... |
| dmec2d 37112 | Equality of the coset of `... |
| brid 37113 | Property of the identity b... |
| ideq2 37114 | For sets, the identity bin... |
| idresssidinxp 37115 | Condition for the identity... |
| idreseqidinxp 37116 | Condition for the identity... |
| extid 37117 | Property of identity relat... |
| inxpss 37118 | Two ways to say that an in... |
| idinxpss 37119 | Two ways to say that an in... |
| ref5 37120 | Two ways to say that an in... |
| inxpss3 37121 | Two ways to say that an in... |
| inxpss2 37122 | Two ways to say that inter... |
| inxpssidinxp 37123 | Two ways to say that inter... |
| idinxpssinxp 37124 | Two ways to say that inter... |
| idinxpssinxp2 37125 | Identity intersection with... |
| idinxpssinxp3 37126 | Identity intersection with... |
| idinxpssinxp4 37127 | Identity intersection with... |
| relcnveq3 37128 | Two ways of saying a relat... |
| relcnveq 37129 | Two ways of saying a relat... |
| relcnveq2 37130 | Two ways of saying a relat... |
| relcnveq4 37131 | Two ways of saying a relat... |
| qsresid 37132 | Simplification of a specia... |
| n0elqs 37133 | Two ways of expressing tha... |
| n0elqs2 37134 | Two ways of expressing tha... |
| ecex2 37135 | Condition for a coset to b... |
| uniqsALTV 37136 | The union of a quotient se... |
| imaexALTV 37137 | Existence of an image of a... |
| ecexALTV 37138 | Existence of a coset, like... |
| rnresequniqs 37139 | The range of a restriction... |
| n0el2 37140 | Two ways of expressing tha... |
| cnvepresex 37141 | Sethood condition for the ... |
| eccnvepex 37142 | The converse epsilon coset... |
| cnvepimaex 37143 | The image of converse epsi... |
| cnvepima 37144 | The image of converse epsi... |
| inex3 37145 | Sufficient condition for t... |
| inxpex 37146 | Sufficient condition for a... |
| eqres 37147 | Converting a class constan... |
| brrabga 37148 | The law of concretion for ... |
| brcnvrabga 37149 | The law of concretion for ... |
| opideq 37150 | Equality conditions for or... |
| iss2 37151 | A subclass of the identity... |
| eldmcnv 37152 | Elementhood in a domain of... |
| dfrel5 37153 | Alternate definition of th... |
| dfrel6 37154 | Alternate definition of th... |
| cnvresrn 37155 | Converse restricted to ran... |
| relssinxpdmrn 37156 | Subset of restriction, spe... |
| cnvref4 37157 | Two ways to say that a rel... |
| cnvref5 37158 | Two ways to say that a rel... |
| ecin0 37159 | Two ways of saying that th... |
| ecinn0 37160 | Two ways of saying that th... |
| ineleq 37161 | Equivalence of restricted ... |
| inecmo 37162 | Equivalence of a double re... |
| inecmo2 37163 | Equivalence of a double re... |
| ineccnvmo 37164 | Equivalence of a double re... |
| alrmomorn 37165 | Equivalence of an "at most... |
| alrmomodm 37166 | Equivalence of an "at most... |
| ineccnvmo2 37167 | Equivalence of a double un... |
| inecmo3 37168 | Equivalence of a double un... |
| moeu2 37169 | Uniqueness is equivalent t... |
| mopickr 37170 | "At most one" picks a vari... |
| moantr 37171 | Sufficient condition for t... |
| brabidgaw 37172 | The law of concretion for ... |
| brabidga 37173 | The law of concretion for ... |
| inxp2 37174 | Intersection with a Cartes... |
| opabf 37175 | A class abstraction of a c... |
| ec0 37176 | The empty-coset of a class... |
| 0qs 37177 | Quotient set with the empt... |
| brcnvin 37178 | Intersection with a conver... |
| xrnss3v 37180 | A range Cartesian product ... |
| xrnrel 37181 | A range Cartesian product ... |
| brxrn 37182 | Characterize a ternary rel... |
| brxrn2 37183 | A characterization of the ... |
| dfxrn2 37184 | Alternate definition of th... |
| xrneq1 37185 | Equality theorem for the r... |
| xrneq1i 37186 | Equality theorem for the r... |
| xrneq1d 37187 | Equality theorem for the r... |
| xrneq2 37188 | Equality theorem for the r... |
| xrneq2i 37189 | Equality theorem for the r... |
| xrneq2d 37190 | Equality theorem for the r... |
| xrneq12 37191 | Equality theorem for the r... |
| xrneq12i 37192 | Equality theorem for the r... |
| xrneq12d 37193 | Equality theorem for the r... |
| elecxrn 37194 | Elementhood in the ` ( R |... |
| ecxrn 37195 | The ` ( R |X. S ) ` -coset... |
| disjressuc2 37196 | Double restricted quantifi... |
| disjecxrn 37197 | Two ways of saying that ` ... |
| disjecxrncnvep 37198 | Two ways of saying that co... |
| disjsuc2 37199 | Double restricted quantifi... |
| xrninxp 37200 | Intersection of a range Ca... |
| xrninxp2 37201 | Intersection of a range Ca... |
| xrninxpex 37202 | Sufficient condition for t... |
| inxpxrn 37203 | Two ways to express the in... |
| br1cnvxrn2 37204 | The converse of a binary r... |
| elec1cnvxrn2 37205 | Elementhood in the convers... |
| rnxrn 37206 | Range of the range Cartesi... |
| rnxrnres 37207 | Range of a range Cartesian... |
| rnxrncnvepres 37208 | Range of a range Cartesian... |
| rnxrnidres 37209 | Range of a range Cartesian... |
| xrnres 37210 | Two ways to express restri... |
| xrnres2 37211 | Two ways to express restri... |
| xrnres3 37212 | Two ways to express restri... |
| xrnres4 37213 | Two ways to express restri... |
| xrnresex 37214 | Sufficient condition for a... |
| xrnidresex 37215 | Sufficient condition for a... |
| xrncnvepresex 37216 | Sufficient condition for a... |
| brin2 37217 | Binary relation on an inte... |
| brin3 37218 | Binary relation on an inte... |
| dfcoss2 37221 | Alternate definition of th... |
| dfcoss3 37222 | Alternate definition of th... |
| dfcoss4 37223 | Alternate definition of th... |
| cosscnv 37224 | Class of cosets by the con... |
| coss1cnvres 37225 | Class of cosets by the con... |
| coss2cnvepres 37226 | Special case of ~ coss1cnv... |
| cossex 37227 | If ` A ` is a set then the... |
| cosscnvex 37228 | If ` A ` is a set then the... |
| 1cosscnvepresex 37229 | Sufficient condition for a... |
| 1cossxrncnvepresex 37230 | Sufficient condition for a... |
| relcoss 37231 | Cosets by ` R ` is a relat... |
| relcoels 37232 | Coelements on ` A ` is a r... |
| cossss 37233 | Subclass theorem for the c... |
| cosseq 37234 | Equality theorem for the c... |
| cosseqi 37235 | Equality theorem for the c... |
| cosseqd 37236 | Equality theorem for the c... |
| 1cossres 37237 | The class of cosets by a r... |
| dfcoels 37238 | Alternate definition of th... |
| brcoss 37239 | ` A ` and ` B ` are cosets... |
| brcoss2 37240 | Alternate form of the ` A ... |
| brcoss3 37241 | Alternate form of the ` A ... |
| brcosscnvcoss 37242 | For sets, the ` A ` and ` ... |
| brcoels 37243 | ` B ` and ` C ` are coelem... |
| cocossss 37244 | Two ways of saying that co... |
| cnvcosseq 37245 | The converse of cosets by ... |
| br2coss 37246 | Cosets by ` ,~ R ` binary ... |
| br1cossres 37247 | ` B ` and ` C ` are cosets... |
| br1cossres2 37248 | ` B ` and ` C ` are cosets... |
| brressn 37249 | Binary relation on a restr... |
| ressn2 37250 | A class ' R ' restricted t... |
| refressn 37251 | Any class ' R ' restricted... |
| antisymressn 37252 | Every class ' R ' restrict... |
| trressn 37253 | Any class ' R ' restricted... |
| relbrcoss 37254 | ` A ` and ` B ` are cosets... |
| br1cossinres 37255 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 37256 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 37257 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 37258 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 37259 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 37260 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 37261 | The domain of cosets is th... |
| dmcoss2 37262 | The domain of cosets is th... |
| rncossdmcoss 37263 | The range of cosets is the... |
| dm1cosscnvepres 37264 | The domain of cosets of th... |
| dmcoels 37265 | The domain of coelements i... |
| eldmcoss 37266 | Elementhood in the domain ... |
| eldmcoss2 37267 | Elementhood in the domain ... |
| eldm1cossres 37268 | Elementhood in the domain ... |
| eldm1cossres2 37269 | Elementhood in the domain ... |
| refrelcosslem 37270 | Lemma for the left side of... |
| refrelcoss3 37271 | The class of cosets by ` R... |
| refrelcoss2 37272 | The class of cosets by ` R... |
| symrelcoss3 37273 | The class of cosets by ` R... |
| symrelcoss2 37274 | The class of cosets by ` R... |
| cossssid 37275 | Equivalent expressions for... |
| cossssid2 37276 | Equivalent expressions for... |
| cossssid3 37277 | Equivalent expressions for... |
| cossssid4 37278 | Equivalent expressions for... |
| cossssid5 37279 | Equivalent expressions for... |
| brcosscnv 37280 | ` A ` and ` B ` are cosets... |
| brcosscnv2 37281 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 37282 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 37283 | Cosets by the converse ran... |
| cosscnvssid3 37284 | Equivalent expressions for... |
| cosscnvssid4 37285 | Equivalent expressions for... |
| cosscnvssid5 37286 | Equivalent expressions for... |
| coss0 37287 | Cosets by the empty set ar... |
| cossid 37288 | Cosets by the identity rel... |
| cosscnvid 37289 | Cosets by the converse ide... |
| trcoss 37290 | Sufficient condition for t... |
| eleccossin 37291 | Two ways of saying that th... |
| trcoss2 37292 | Equivalent expressions for... |
| elrels2 37294 | The element of the relatio... |
| elrelsrel 37295 | The element of the relatio... |
| elrelsrelim 37296 | The element of the relatio... |
| elrels5 37297 | Equivalent expressions for... |
| elrels6 37298 | Equivalent expressions for... |
| elrelscnveq3 37299 | Two ways of saying a relat... |
| elrelscnveq 37300 | Two ways of saying a relat... |
| elrelscnveq2 37301 | Two ways of saying a relat... |
| elrelscnveq4 37302 | Two ways of saying a relat... |
| cnvelrels 37303 | The converse of a set is a... |
| cosselrels 37304 | Cosets of sets are element... |
| cosscnvelrels 37305 | Cosets of converse sets ar... |
| dfssr2 37307 | Alternate definition of th... |
| relssr 37308 | The subset relation is a r... |
| brssr 37309 | The subset relation and su... |
| brssrid 37310 | Any set is a subset of its... |
| issetssr 37311 | Two ways of expressing set... |
| brssrres 37312 | Restricted subset binary r... |
| br1cnvssrres 37313 | Restricted converse subset... |
| brcnvssr 37314 | The converse of a subset r... |
| brcnvssrid 37315 | Any set is a converse subs... |
| br1cossxrncnvssrres 37316 | ` <. B , C >. ` and ` <. D... |
| extssr 37317 | Property of subset relatio... |
| dfrefrels2 37321 | Alternate definition of th... |
| dfrefrels3 37322 | Alternate definition of th... |
| dfrefrel2 37323 | Alternate definition of th... |
| dfrefrel3 37324 | Alternate definition of th... |
| dfrefrel5 37325 | Alternate definition of th... |
| elrefrels2 37326 | Element of the class of re... |
| elrefrels3 37327 | Element of the class of re... |
| elrefrelsrel 37328 | For sets, being an element... |
| refreleq 37329 | Equality theorem for refle... |
| refrelid 37330 | Identity relation is refle... |
| refrelcoss 37331 | The class of cosets by ` R... |
| refrelressn 37332 | Any class ' R ' restricted... |
| dfcnvrefrels2 37336 | Alternate definition of th... |
| dfcnvrefrels3 37337 | Alternate definition of th... |
| dfcnvrefrel2 37338 | Alternate definition of th... |
| dfcnvrefrel3 37339 | Alternate definition of th... |
| dfcnvrefrel4 37340 | Alternate definition of th... |
| dfcnvrefrel5 37341 | Alternate definition of th... |
| elcnvrefrels2 37342 | Element of the class of co... |
| elcnvrefrels3 37343 | Element of the class of co... |
| elcnvrefrelsrel 37344 | For sets, being an element... |
| cnvrefrelcoss2 37345 | Necessary and sufficient c... |
| cosselcnvrefrels2 37346 | Necessary and sufficient c... |
| cosselcnvrefrels3 37347 | Necessary and sufficient c... |
| cosselcnvrefrels4 37348 | Necessary and sufficient c... |
| cosselcnvrefrels5 37349 | Necessary and sufficient c... |
| dfsymrels2 37353 | Alternate definition of th... |
| dfsymrels3 37354 | Alternate definition of th... |
| dfsymrels4 37355 | Alternate definition of th... |
| dfsymrels5 37356 | Alternate definition of th... |
| dfsymrel2 37357 | Alternate definition of th... |
| dfsymrel3 37358 | Alternate definition of th... |
| dfsymrel4 37359 | Alternate definition of th... |
| dfsymrel5 37360 | Alternate definition of th... |
| elsymrels2 37361 | Element of the class of sy... |
| elsymrels3 37362 | Element of the class of sy... |
| elsymrels4 37363 | Element of the class of sy... |
| elsymrels5 37364 | Element of the class of sy... |
| elsymrelsrel 37365 | For sets, being an element... |
| symreleq 37366 | Equality theorem for symme... |
| symrelim 37367 | Symmetric relation implies... |
| symrelcoss 37368 | The class of cosets by ` R... |
| idsymrel 37369 | The identity relation is s... |
| epnsymrel 37370 | The membership (epsilon) r... |
| symrefref2 37371 | Symmetry is a sufficient c... |
| symrefref3 37372 | Symmetry is a sufficient c... |
| refsymrels2 37373 | Elements of the class of r... |
| refsymrels3 37374 | Elements of the class of r... |
| refsymrel2 37375 | A relation which is reflex... |
| refsymrel3 37376 | A relation which is reflex... |
| elrefsymrels2 37377 | Elements of the class of r... |
| elrefsymrels3 37378 | Elements of the class of r... |
| elrefsymrelsrel 37379 | For sets, being an element... |
| dftrrels2 37383 | Alternate definition of th... |
| dftrrels3 37384 | Alternate definition of th... |
| dftrrel2 37385 | Alternate definition of th... |
| dftrrel3 37386 | Alternate definition of th... |
| eltrrels2 37387 | Element of the class of tr... |
| eltrrels3 37388 | Element of the class of tr... |
| eltrrelsrel 37389 | For sets, being an element... |
| trreleq 37390 | Equality theorem for the t... |
| trrelressn 37391 | Any class ' R ' restricted... |
| dfeqvrels2 37396 | Alternate definition of th... |
| dfeqvrels3 37397 | Alternate definition of th... |
| dfeqvrel2 37398 | Alternate definition of th... |
| dfeqvrel3 37399 | Alternate definition of th... |
| eleqvrels2 37400 | Element of the class of eq... |
| eleqvrels3 37401 | Element of the class of eq... |
| eleqvrelsrel 37402 | For sets, being an element... |
| elcoeleqvrels 37403 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 37404 | For sets, being an element... |
| eqvrelrel 37405 | An equivalence relation is... |
| eqvrelrefrel 37406 | An equivalence relation is... |
| eqvrelsymrel 37407 | An equivalence relation is... |
| eqvreltrrel 37408 | An equivalence relation is... |
| eqvrelim 37409 | Equivalence relation impli... |
| eqvreleq 37410 | Equality theorem for equiv... |
| eqvreleqi 37411 | Equality theorem for equiv... |
| eqvreleqd 37412 | Equality theorem for equiv... |
| eqvrelsym 37413 | An equivalence relation is... |
| eqvrelsymb 37414 | An equivalence relation is... |
| eqvreltr 37415 | An equivalence relation is... |
| eqvreltrd 37416 | A transitivity relation fo... |
| eqvreltr4d 37417 | A transitivity relation fo... |
| eqvrelref 37418 | An equivalence relation is... |
| eqvrelth 37419 | Basic property of equivale... |
| eqvrelcl 37420 | Elementhood in the field o... |
| eqvrelthi 37421 | Basic property of equivale... |
| eqvreldisj 37422 | Equivalence classes do not... |
| qsdisjALTV 37423 | Elements of a quotient set... |
| eqvrelqsel 37424 | If an element of a quotien... |
| eqvrelcoss 37425 | Two ways to express equiva... |
| eqvrelcoss3 37426 | Two ways to express equiva... |
| eqvrelcoss2 37427 | Two ways to express equiva... |
| eqvrelcoss4 37428 | Two ways to express equiva... |
| dfcoeleqvrels 37429 | Alternate definition of th... |
| dfcoeleqvrel 37430 | Alternate definition of th... |
| brredunds 37434 | Binary relation on the cla... |
| brredundsredund 37435 | For sets, binary relation ... |
| redundss3 37436 | Implication of redundancy ... |
| redundeq1 37437 | Equivalence of redundancy ... |
| redundpim3 37438 | Implication of redundancy ... |
| redundpbi1 37439 | Equivalence of redundancy ... |
| refrelsredund4 37440 | The naive version of the c... |
| refrelsredund2 37441 | The naive version of the c... |
| refrelsredund3 37442 | The naive version of the c... |
| refrelredund4 37443 | The naive version of the d... |
| refrelredund2 37444 | The naive version of the d... |
| refrelredund3 37445 | The naive version of the d... |
| dmqseq 37448 | Equality theorem for domai... |
| dmqseqi 37449 | Equality theorem for domai... |
| dmqseqd 37450 | Equality theorem for domai... |
| dmqseqeq1 37451 | Equality theorem for domai... |
| dmqseqeq1i 37452 | Equality theorem for domai... |
| dmqseqeq1d 37453 | Equality theorem for domai... |
| brdmqss 37454 | The domain quotient binary... |
| brdmqssqs 37455 | If ` A ` and ` R ` are set... |
| n0eldmqs 37456 | The empty set is not an el... |
| n0eldmqseq 37457 | The empty set is not an el... |
| n0elim 37458 | Implication of that the em... |
| n0el3 37459 | Two ways of expressing tha... |
| cnvepresdmqss 37460 | The domain quotient binary... |
| cnvepresdmqs 37461 | The domain quotient predic... |
| unidmqs 37462 | The range of a relation is... |
| unidmqseq 37463 | The union of the domain qu... |
| dmqseqim 37464 | If the domain quotient of ... |
| dmqseqim2 37465 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 37466 | Elementhood in the domain ... |
| eldmqs1cossres 37467 | Elementhood in the domain ... |
| releldmqscoss 37468 | Elementhood in the domain ... |
| dmqscoelseq 37469 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 37470 | Two ways to express the eq... |
| brers 37475 | Binary equivalence relatio... |
| dferALTV2 37476 | Equivalence relation with ... |
| erALTVeq1 37477 | Equality theorem for equiv... |
| erALTVeq1i 37478 | Equality theorem for equiv... |
| erALTVeq1d 37479 | Equality theorem for equiv... |
| dfcomember 37480 | Alternate definition of th... |
| dfcomember2 37481 | Alternate definition of th... |
| dfcomember3 37482 | Alternate definition of th... |
| eqvreldmqs 37483 | Two ways to express comemb... |
| eqvreldmqs2 37484 | Two ways to express comemb... |
| brerser 37485 | Binary equivalence relatio... |
| erimeq2 37486 | Equivalence relation on it... |
| erimeq 37487 | Equivalence relation on it... |
| dffunsALTV 37491 | Alternate definition of th... |
| dffunsALTV2 37492 | Alternate definition of th... |
| dffunsALTV3 37493 | Alternate definition of th... |
| dffunsALTV4 37494 | Alternate definition of th... |
| dffunsALTV5 37495 | Alternate definition of th... |
| dffunALTV2 37496 | Alternate definition of th... |
| dffunALTV3 37497 | Alternate definition of th... |
| dffunALTV4 37498 | Alternate definition of th... |
| dffunALTV5 37499 | Alternate definition of th... |
| elfunsALTV 37500 | Elementhood in the class o... |
| elfunsALTV2 37501 | Elementhood in the class o... |
| elfunsALTV3 37502 | Elementhood in the class o... |
| elfunsALTV4 37503 | Elementhood in the class o... |
| elfunsALTV5 37504 | Elementhood in the class o... |
| elfunsALTVfunALTV 37505 | The element of the class o... |
| funALTVfun 37506 | Our definition of the func... |
| funALTVss 37507 | Subclass theorem for funct... |
| funALTVeq 37508 | Equality theorem for funct... |
| funALTVeqi 37509 | Equality inference for the... |
| funALTVeqd 37510 | Equality deduction for the... |
| dfdisjs 37516 | Alternate definition of th... |
| dfdisjs2 37517 | Alternate definition of th... |
| dfdisjs3 37518 | Alternate definition of th... |
| dfdisjs4 37519 | Alternate definition of th... |
| dfdisjs5 37520 | Alternate definition of th... |
| dfdisjALTV 37521 | Alternate definition of th... |
| dfdisjALTV2 37522 | Alternate definition of th... |
| dfdisjALTV3 37523 | Alternate definition of th... |
| dfdisjALTV4 37524 | Alternate definition of th... |
| dfdisjALTV5 37525 | Alternate definition of th... |
| dfeldisj2 37526 | Alternate definition of th... |
| dfeldisj3 37527 | Alternate definition of th... |
| dfeldisj4 37528 | Alternate definition of th... |
| dfeldisj5 37529 | Alternate definition of th... |
| eldisjs 37530 | Elementhood in the class o... |
| eldisjs2 37531 | Elementhood in the class o... |
| eldisjs3 37532 | Elementhood in the class o... |
| eldisjs4 37533 | Elementhood in the class o... |
| eldisjs5 37534 | Elementhood in the class o... |
| eldisjsdisj 37535 | The element of the class o... |
| eleldisjs 37536 | Elementhood in the disjoin... |
| eleldisjseldisj 37537 | The element of the disjoin... |
| disjrel 37538 | Disjoint relation is a rel... |
| disjss 37539 | Subclass theorem for disjo... |
| disjssi 37540 | Subclass theorem for disjo... |
| disjssd 37541 | Subclass theorem for disjo... |
| disjeq 37542 | Equality theorem for disjo... |
| disjeqi 37543 | Equality theorem for disjo... |
| disjeqd 37544 | Equality theorem for disjo... |
| disjdmqseqeq1 37545 | Lemma for the equality the... |
| eldisjss 37546 | Subclass theorem for disjo... |
| eldisjssi 37547 | Subclass theorem for disjo... |
| eldisjssd 37548 | Subclass theorem for disjo... |
| eldisjeq 37549 | Equality theorem for disjo... |
| eldisjeqi 37550 | Equality theorem for disjo... |
| eldisjeqd 37551 | Equality theorem for disjo... |
| disjres 37552 | Disjoint restriction. (Co... |
| eldisjn0elb 37553 | Two forms of disjoint elem... |
| disjxrn 37554 | Two ways of saying that a ... |
| disjxrnres5 37555 | Disjoint range Cartesian p... |
| disjorimxrn 37556 | Disjointness condition for... |
| disjimxrn 37557 | Disjointness condition for... |
| disjimres 37558 | Disjointness condition for... |
| disjimin 37559 | Disjointness condition for... |
| disjiminres 37560 | Disjointness condition for... |
| disjimxrnres 37561 | Disjointness condition for... |
| disjALTV0 37562 | The null class is disjoint... |
| disjALTVid 37563 | The class of identity rela... |
| disjALTVidres 37564 | The class of identity rela... |
| disjALTVinidres 37565 | The intersection with rest... |
| disjALTVxrnidres 37566 | The class of range Cartesi... |
| disjsuc 37567 | Disjoint range Cartesian p... |
| dfantisymrel4 37569 | Alternate definition of th... |
| dfantisymrel5 37570 | Alternate definition of th... |
| antisymrelres 37571 | (Contributed by Peter Mazs... |
| antisymrelressn 37572 | (Contributed by Peter Mazs... |
| dfpart2 37577 | Alternate definition of th... |
| dfmembpart2 37578 | Alternate definition of th... |
| brparts 37579 | Binary partitions relation... |
| brparts2 37580 | Binary partitions relation... |
| brpartspart 37581 | Binary partition and the p... |
| parteq1 37582 | Equality theorem for parti... |
| parteq2 37583 | Equality theorem for parti... |
| parteq12 37584 | Equality theorem for parti... |
| parteq1i 37585 | Equality theorem for parti... |
| parteq1d 37586 | Equality theorem for parti... |
| partsuc2 37587 | Property of the partition.... |
| partsuc 37588 | Property of the partition.... |
| disjim 37589 | The "Divide et Aequivalere... |
| disjimi 37590 | Every disjoint relation ge... |
| detlem 37591 | If a relation is disjoint,... |
| eldisjim 37592 | If the elements of ` A ` a... |
| eldisjim2 37593 | Alternate form of ~ eldisj... |
| eqvrel0 37594 | The null class is an equiv... |
| det0 37595 | The cosets by the null cla... |
| eqvrelcoss0 37596 | The cosets by the null cla... |
| eqvrelid 37597 | The identity relation is a... |
| eqvrel1cossidres 37598 | The cosets by a restricted... |
| eqvrel1cossinidres 37599 | The cosets by an intersect... |
| eqvrel1cossxrnidres 37600 | The cosets by a range Cart... |
| detid 37601 | The cosets by the identity... |
| eqvrelcossid 37602 | The cosets by the identity... |
| detidres 37603 | The cosets by the restrict... |
| detinidres 37604 | The cosets by the intersec... |
| detxrnidres 37605 | The cosets by the range Ca... |
| disjlem14 37606 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 37607 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 37608 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 37609 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 37610 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 37611 | Lemma for ~ disjdmqseq via... |
| disjdmqs 37612 | If a relation is disjoint,... |
| disjdmqseq 37613 | If a relation is disjoint,... |
| eldisjn0el 37614 | Special case of ~ disjdmqs... |
| partim2 37615 | Disjoint relation on its n... |
| partim 37616 | Partition implies equivale... |
| partimeq 37617 | Partition implies that the... |
| eldisjlem19 37618 | Special case of ~ disjlem1... |
| membpartlem19 37619 | Together with ~ disjlem19 ... |
| petlem 37620 | If you can prove that the ... |
| petlemi 37621 | If you can prove disjointn... |
| pet02 37622 | Class ` A ` is a partition... |
| pet0 37623 | Class ` A ` is a partition... |
| petid2 37624 | Class ` A ` is a partition... |
| petid 37625 | A class is a partition by ... |
| petidres2 37626 | Class ` A ` is a partition... |
| petidres 37627 | A class is a partition by ... |
| petinidres2 37628 | Class ` A ` is a partition... |
| petinidres 37629 | A class is a partition by ... |
| petxrnidres2 37630 | Class ` A ` is a partition... |
| petxrnidres 37631 | A class is a partition by ... |
| eqvreldisj1 37632 | The elements of the quotie... |
| eqvreldisj2 37633 | The elements of the quotie... |
| eqvreldisj3 37634 | The elements of the quotie... |
| eqvreldisj4 37635 | Intersection with the conv... |
| eqvreldisj5 37636 | Range Cartesian product wi... |
| eqvrelqseqdisj2 37637 | Implication of ~ eqvreldis... |
| fences3 37638 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 37639 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 37640 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 37641 | Lemma for the Partition-Eq... |
| mainer 37642 | The Main Theorem of Equiva... |
| partimcomember 37643 | Partition with general ` R... |
| mpet3 37644 | Member Partition-Equivalen... |
| cpet2 37645 | The conventional form of t... |
| cpet 37646 | The conventional form of M... |
| mpet 37647 | Member Partition-Equivalen... |
| mpet2 37648 | Member Partition-Equivalen... |
| mpets2 37649 | Member Partition-Equivalen... |
| mpets 37650 | Member Partition-Equivalen... |
| mainpart 37651 | Partition with general ` R... |
| fences 37652 | The Theorem of Fences by E... |
| fences2 37653 | The Theorem of Fences by E... |
| mainer2 37654 | The Main Theorem of Equiva... |
| mainerim 37655 | Every equivalence relation... |
| petincnvepres2 37656 | A partition-equivalence th... |
| petincnvepres 37657 | The shortest form of a par... |
| pet2 37658 | Partition-Equivalence Theo... |
| pet 37659 | Partition-Equivalence Theo... |
| pets 37660 | Partition-Equivalence Theo... |
| prtlem60 37661 | Lemma for ~ prter3 . (Con... |
| bicomdd 37662 | Commute two sides of a bic... |
| jca2r 37663 | Inference conjoining the c... |
| jca3 37664 | Inference conjoining the c... |
| prtlem70 37665 | Lemma for ~ prter3 : a rea... |
| ibdr 37666 | Reverse of ~ ibd . (Contr... |
| prtlem100 37667 | Lemma for ~ prter3 . (Con... |
| prtlem5 37668 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 37669 | Lemma for ~ prter2 . (Con... |
| brabsb2 37670 | A closed form of ~ brabsb ... |
| eqbrrdv2 37671 | Other version of ~ eqbrrdi... |
| prtlem9 37672 | Lemma for ~ prter3 . (Con... |
| prtlem10 37673 | Lemma for ~ prter3 . (Con... |
| prtlem11 37674 | Lemma for ~ prter2 . (Con... |
| prtlem12 37675 | Lemma for ~ prtex and ~ pr... |
| prtlem13 37676 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 37677 | Lemma for ~ prtex , ~ prte... |
| prtlem400 37678 | Lemma for ~ prter2 and als... |
| erprt 37681 | The quotient set of an equ... |
| prtlem14 37682 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 37683 | Lemma for ~ prter1 and ~ p... |
| prtlem17 37684 | Lemma for ~ prter2 . (Con... |
| prtlem18 37685 | Lemma for ~ prter2 . (Con... |
| prtlem19 37686 | Lemma for ~ prter2 . (Con... |
| prter1 37687 | Every partition generates ... |
| prtex 37688 | The equivalence relation g... |
| prter2 37689 | The quotient set of the eq... |
| prter3 37690 | For every partition there ... |
| axc5 37701 | This theorem repeats ~ sp ... |
| ax4fromc4 37702 | Rederivation of Axiom ~ ax... |
| ax10fromc7 37703 | Rederivation of Axiom ~ ax... |
| ax6fromc10 37704 | Rederivation of Axiom ~ ax... |
| hba1-o 37705 | The setvar ` x ` is not fr... |
| axc4i-o 37706 | Inference version of ~ ax-... |
| equid1 37707 | Proof of ~ equid from our ... |
| equcomi1 37708 | Proof of ~ equcomi from ~ ... |
| aecom-o 37709 | Commutation law for identi... |
| aecoms-o 37710 | A commutation rule for ide... |
| hbae-o 37711 | All variables are effectiv... |
| dral1-o 37712 | Formula-building lemma for... |
| ax12fromc15 37713 | Rederivation of Axiom ~ ax... |
| ax13fromc9 37714 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 37715 | Axiom to quantify a variab... |
| sps-o 37716 | Generalization of antecede... |
| hbequid 37717 | Bound-variable hypothesis ... |
| nfequid-o 37718 | Bound-variable hypothesis ... |
| axc5c7 37719 | Proof of a single axiom th... |
| axc5c7toc5 37720 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 37721 | Rederivation of ~ ax-c7 fr... |
| axc711 37722 | Proof of a single axiom th... |
| nfa1-o 37723 | ` x ` is not free in ` A. ... |
| axc711toc7 37724 | Rederivation of ~ ax-c7 fr... |
| axc711to11 37725 | Rederivation of ~ ax-11 fr... |
| axc5c711 37726 | Proof of a single axiom th... |
| axc5c711toc5 37727 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 37728 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 37729 | Rederivation of ~ ax-11 fr... |
| equidqe 37730 | ~ equid with existential q... |
| axc5sp1 37731 | A special case of ~ ax-c5 ... |
| equidq 37732 | ~ equid with universal qua... |
| equid1ALT 37733 | Alternate proof of ~ equid... |
| axc11nfromc11 37734 | Rederivation of ~ ax-c11n ... |
| naecoms-o 37735 | A commutation rule for dis... |
| hbnae-o 37736 | All variables are effectiv... |
| dvelimf-o 37737 | Proof of ~ dvelimh that us... |
| dral2-o 37738 | Formula-building lemma for... |
| aev-o 37739 | A "distinctor elimination"... |
| ax5eq 37740 | Theorem to add distinct qu... |
| dveeq2-o 37741 | Quantifier introduction wh... |
| axc16g-o 37742 | A generalization of Axiom ... |
| dveeq1-o 37743 | Quantifier introduction wh... |
| dveeq1-o16 37744 | Version of ~ dveeq1 using ... |
| ax5el 37745 | Theorem to add distinct qu... |
| axc11n-16 37746 | This theorem shows that, g... |
| dveel2ALT 37747 | Alternate proof of ~ dveel... |
| ax12f 37748 | Basis step for constructin... |
| ax12eq 37749 | Basis step for constructin... |
| ax12el 37750 | Basis step for constructin... |
| ax12indn 37751 | Induction step for constru... |
| ax12indi 37752 | Induction step for constru... |
| ax12indalem 37753 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 37754 | Alternate proof of ~ ax12i... |
| ax12inda2 37755 | Induction step for constru... |
| ax12inda 37756 | Induction step for constru... |
| ax12v2-o 37757 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 37758 | Derive ~ ax-c15 from a hyp... |
| axc11-o 37759 | Show that ~ ax-c11 can be ... |
| fsumshftd 37760 | Index shift of a finite su... |
| riotaclbgBAD 37762 | Closure of restricted iota... |
| riotaclbBAD 37763 | Closure of restricted iota... |
| riotasvd 37764 | Deduction version of ~ rio... |
| riotasv2d 37765 | Value of description binde... |
| riotasv2s 37766 | The value of description b... |
| riotasv 37767 | Value of description binde... |
| riotasv3d 37768 | A property ` ch ` holding ... |
| elimhyps 37769 | A version of ~ elimhyp usi... |
| dedths 37770 | A version of weak deductio... |
| renegclALT 37771 | Closure law for negative o... |
| elimhyps2 37772 | Generalization of ~ elimhy... |
| dedths2 37773 | Generalization of ~ dedths... |
| nfcxfrdf 37774 | A utility lemma to transfe... |
| nfded 37775 | A deduction theorem that c... |
| nfded2 37776 | A deduction theorem that c... |
| nfunidALT2 37777 | Deduction version of ~ nfu... |
| nfunidALT 37778 | Deduction version of ~ nfu... |
| nfopdALT 37779 | Deduction version of bound... |
| cnaddcom 37780 | Recover the commutative la... |
| toycom 37781 | Show the commutative law f... |
| lshpset 37786 | The set of all hyperplanes... |
| islshp 37787 | The predicate "is a hyperp... |
| islshpsm 37788 | Hyperplane properties expr... |
| lshplss 37789 | A hyperplane is a subspace... |
| lshpne 37790 | A hyperplane is not equal ... |
| lshpnel 37791 | A hyperplane's generating ... |
| lshpnelb 37792 | The subspace sum of a hype... |
| lshpnel2N 37793 | Condition that determines ... |
| lshpne0 37794 | The member of the span in ... |
| lshpdisj 37795 | A hyperplane and the span ... |
| lshpcmp 37796 | If two hyperplanes are com... |
| lshpinN 37797 | The intersection of two di... |
| lsatset 37798 | The set of all 1-dim subsp... |
| islsat 37799 | The predicate "is a 1-dim ... |
| lsatlspsn2 37800 | The span of a nonzero sing... |
| lsatlspsn 37801 | The span of a nonzero sing... |
| islsati 37802 | A 1-dim subspace (atom) (o... |
| lsateln0 37803 | A 1-dim subspace (atom) (o... |
| lsatlss 37804 | The set of 1-dim subspaces... |
| lsatlssel 37805 | An atom is a subspace. (C... |
| lsatssv 37806 | An atom is a set of vector... |
| lsatn0 37807 | A 1-dim subspace (atom) of... |
| lsatspn0 37808 | The span of a vector is an... |
| lsator0sp 37809 | The span of a vector is ei... |
| lsatssn0 37810 | A subspace (or any class) ... |
| lsatcmp 37811 | If two atoms are comparabl... |
| lsatcmp2 37812 | If an atom is included in ... |
| lsatel 37813 | A nonzero vector in an ato... |
| lsatelbN 37814 | A nonzero vector in an ato... |
| lsat2el 37815 | Two atoms sharing a nonzer... |
| lsmsat 37816 | Convert comparison of atom... |
| lsatfixedN 37817 | Show equality with the spa... |
| lsmsatcv 37818 | Subspace sum has the cover... |
| lssatomic 37819 | The lattice of subspaces i... |
| lssats 37820 | The lattice of subspaces i... |
| lpssat 37821 | Two subspaces in a proper ... |
| lrelat 37822 | Subspaces are relatively a... |
| lssatle 37823 | The ordering of two subspa... |
| lssat 37824 | Two subspaces in a proper ... |
| islshpat 37825 | Hyperplane properties expr... |
| lcvfbr 37828 | The covers relation for a ... |
| lcvbr 37829 | The covers relation for a ... |
| lcvbr2 37830 | The covers relation for a ... |
| lcvbr3 37831 | The covers relation for a ... |
| lcvpss 37832 | The covers relation implie... |
| lcvnbtwn 37833 | The covers relation implie... |
| lcvntr 37834 | The covers relation is not... |
| lcvnbtwn2 37835 | The covers relation implie... |
| lcvnbtwn3 37836 | The covers relation implie... |
| lsmcv2 37837 | Subspace sum has the cover... |
| lcvat 37838 | If a subspace covers anoth... |
| lsatcv0 37839 | An atom covers the zero su... |
| lsatcveq0 37840 | A subspace covered by an a... |
| lsat0cv 37841 | A subspace is an atom iff ... |
| lcvexchlem1 37842 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 37843 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 37844 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 37845 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 37846 | Lemma for ~ lcvexch . (Co... |
| lcvexch 37847 | Subspaces satisfy the exch... |
| lcvp 37848 | Covering property of Defin... |
| lcv1 37849 | Covering property of a sub... |
| lcv2 37850 | Covering property of a sub... |
| lsatexch 37851 | The atom exchange property... |
| lsatnle 37852 | The meet of a subspace and... |
| lsatnem0 37853 | The meet of distinct atoms... |
| lsatexch1 37854 | The atom exch1ange propert... |
| lsatcv0eq 37855 | If the sum of two atoms co... |
| lsatcv1 37856 | Two atoms covering the zer... |
| lsatcvatlem 37857 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 37858 | A nonzero subspace less th... |
| lsatcvat2 37859 | A subspace covered by the ... |
| lsatcvat3 37860 | A condition implying that ... |
| islshpcv 37861 | Hyperplane properties expr... |
| l1cvpat 37862 | A subspace covered by the ... |
| l1cvat 37863 | Create an atom under an el... |
| lshpat 37864 | Create an atom under a hyp... |
| lflset 37867 | The set of linear function... |
| islfl 37868 | The predicate "is a linear... |
| lfli 37869 | Property of a linear funct... |
| islfld 37870 | Properties that determine ... |
| lflf 37871 | A linear functional is a f... |
| lflcl 37872 | A linear functional value ... |
| lfl0 37873 | A linear functional is zer... |
| lfladd 37874 | Property of a linear funct... |
| lflsub 37875 | Property of a linear funct... |
| lflmul 37876 | Property of a linear funct... |
| lfl0f 37877 | The zero function is a fun... |
| lfl1 37878 | A nonzero functional has a... |
| lfladdcl 37879 | Closure of addition of two... |
| lfladdcom 37880 | Commutativity of functiona... |
| lfladdass 37881 | Associativity of functiona... |
| lfladd0l 37882 | Functional addition with t... |
| lflnegcl 37883 | Closure of the negative of... |
| lflnegl 37884 | A functional plus its nega... |
| lflvscl 37885 | Closure of a scalar produc... |
| lflvsdi1 37886 | Distributive law for (righ... |
| lflvsdi2 37887 | Reverse distributive law f... |
| lflvsdi2a 37888 | Reverse distributive law f... |
| lflvsass 37889 | Associative law for (right... |
| lfl0sc 37890 | The (right vector space) s... |
| lflsc0N 37891 | The scalar product with th... |
| lfl1sc 37892 | The (right vector space) s... |
| lkrfval 37895 | The kernel of a functional... |
| lkrval 37896 | Value of the kernel of a f... |
| ellkr 37897 | Membership in the kernel o... |
| lkrval2 37898 | Value of the kernel of a f... |
| ellkr2 37899 | Membership in the kernel o... |
| lkrcl 37900 | A member of the kernel of ... |
| lkrf0 37901 | The value of a functional ... |
| lkr0f 37902 | The kernel of the zero fun... |
| lkrlss 37903 | The kernel of a linear fun... |
| lkrssv 37904 | The kernel of a linear fun... |
| lkrsc 37905 | The kernel of a nonzero sc... |
| lkrscss 37906 | The kernel of a scalar pro... |
| eqlkr 37907 | Two functionals with the s... |
| eqlkr2 37908 | Two functionals with the s... |
| eqlkr3 37909 | Two functionals with the s... |
| lkrlsp 37910 | The subspace sum of a kern... |
| lkrlsp2 37911 | The subspace sum of a kern... |
| lkrlsp3 37912 | The subspace sum of a kern... |
| lkrshp 37913 | The kernel of a nonzero fu... |
| lkrshp3 37914 | The kernels of nonzero fun... |
| lkrshpor 37915 | The kernel of a functional... |
| lkrshp4 37916 | A kernel is a hyperplane i... |
| lshpsmreu 37917 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 37918 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 37919 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 37920 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 37921 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 37922 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 37923 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 37924 | The set ` G ` defined by h... |
| lshpkr 37925 | The kernel of functional `... |
| lshpkrex 37926 | There exists a functional ... |
| lshpset2N 37927 | The set of all hyperplanes... |
| islshpkrN 37928 | The predicate "is a hyperp... |
| lfl1dim 37929 | Equivalent expressions for... |
| lfl1dim2N 37930 | Equivalent expressions for... |
| ldualset 37933 | Define the (left) dual of ... |
| ldualvbase 37934 | The vectors of a dual spac... |
| ldualelvbase 37935 | Utility theorem for conver... |
| ldualfvadd 37936 | Vector addition in the dua... |
| ldualvadd 37937 | Vector addition in the dua... |
| ldualvaddcl 37938 | The value of vector additi... |
| ldualvaddval 37939 | The value of the value of ... |
| ldualsca 37940 | The ring of scalars of the... |
| ldualsbase 37941 | Base set of scalar ring fo... |
| ldualsaddN 37942 | Scalar addition for the du... |
| ldualsmul 37943 | Scalar multiplication for ... |
| ldualfvs 37944 | Scalar product operation f... |
| ldualvs 37945 | Scalar product operation v... |
| ldualvsval 37946 | Value of scalar product op... |
| ldualvscl 37947 | The scalar product operati... |
| ldualvaddcom 37948 | Commutative law for vector... |
| ldualvsass 37949 | Associative law for scalar... |
| ldualvsass2 37950 | Associative law for scalar... |
| ldualvsdi1 37951 | Distributive law for scala... |
| ldualvsdi2 37952 | Reverse distributive law f... |
| ldualgrplem 37953 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 37954 | The dual of a vector space... |
| ldual0 37955 | The zero scalar of the dua... |
| ldual1 37956 | The unit scalar of the dua... |
| ldualneg 37957 | The negative of a scalar o... |
| ldual0v 37958 | The zero vector of the dua... |
| ldual0vcl 37959 | The dual zero vector is a ... |
| lduallmodlem 37960 | Lemma for ~ lduallmod . (... |
| lduallmod 37961 | The dual of a left module ... |
| lduallvec 37962 | The dual of a left vector ... |
| ldualvsub 37963 | The value of vector subtra... |
| ldualvsubcl 37964 | Closure of vector subtract... |
| ldualvsubval 37965 | The value of the value of ... |
| ldualssvscl 37966 | Closure of scalar product ... |
| ldualssvsubcl 37967 | Closure of vector subtract... |
| ldual0vs 37968 | Scalar zero times a functi... |
| lkr0f2 37969 | The kernel of the zero fun... |
| lduallkr3 37970 | The kernels of nonzero fun... |
| lkrpssN 37971 | Proper subset relation bet... |
| lkrin 37972 | Intersection of the kernel... |
| eqlkr4 37973 | Two functionals with the s... |
| ldual1dim 37974 | Equivalent expressions for... |
| ldualkrsc 37975 | The kernel of a nonzero sc... |
| lkrss 37976 | The kernel of a scalar pro... |
| lkrss2N 37977 | Two functionals with kerne... |
| lkreqN 37978 | Proportional functionals h... |
| lkrlspeqN 37979 | Condition for colinear fun... |
| isopos 37988 | The predicate "is an ortho... |
| opposet 37989 | Every orthoposet is a pose... |
| oposlem 37990 | Lemma for orthoposet prope... |
| op01dm 37991 | Conditions necessary for z... |
| op0cl 37992 | An orthoposet has a zero e... |
| op1cl 37993 | An orthoposet has a unity ... |
| op0le 37994 | Orthoposet zero is less th... |
| ople0 37995 | An element less than or eq... |
| opnlen0 37996 | An element not less than a... |
| lub0N 37997 | The least upper bound of t... |
| opltn0 37998 | A lattice element greater ... |
| ople1 37999 | Any element is less than t... |
| op1le 38000 | If the orthoposet unity is... |
| glb0N 38001 | The greatest lower bound o... |
| opoccl 38002 | Closure of orthocomplement... |
| opococ 38003 | Double negative law for or... |
| opcon3b 38004 | Contraposition law for ort... |
| opcon2b 38005 | Orthocomplement contraposi... |
| opcon1b 38006 | Orthocomplement contraposi... |
| oplecon3 38007 | Contraposition law for ort... |
| oplecon3b 38008 | Contraposition law for ort... |
| oplecon1b 38009 | Contraposition law for str... |
| opoc1 38010 | Orthocomplement of orthopo... |
| opoc0 38011 | Orthocomplement of orthopo... |
| opltcon3b 38012 | Contraposition law for str... |
| opltcon1b 38013 | Contraposition law for str... |
| opltcon2b 38014 | Contraposition law for str... |
| opexmid 38015 | Law of excluded middle for... |
| opnoncon 38016 | Law of contradiction for o... |
| riotaocN 38017 | The orthocomplement of the... |
| cmtfvalN 38018 | Value of commutes relation... |
| cmtvalN 38019 | Equivalence for commutes r... |
| isolat 38020 | The predicate "is an ortho... |
| ollat 38021 | An ortholattice is a latti... |
| olop 38022 | An ortholattice is an orth... |
| olposN 38023 | An ortholattice is a poset... |
| isolatiN 38024 | Properties that determine ... |
| oldmm1 38025 | De Morgan's law for meet i... |
| oldmm2 38026 | De Morgan's law for meet i... |
| oldmm3N 38027 | De Morgan's law for meet i... |
| oldmm4 38028 | De Morgan's law for meet i... |
| oldmj1 38029 | De Morgan's law for join i... |
| oldmj2 38030 | De Morgan's law for join i... |
| oldmj3 38031 | De Morgan's law for join i... |
| oldmj4 38032 | De Morgan's law for join i... |
| olj01 38033 | An ortholattice element jo... |
| olj02 38034 | An ortholattice element jo... |
| olm11 38035 | The meet of an ortholattic... |
| olm12 38036 | The meet of an ortholattic... |
| latmassOLD 38037 | Ortholattice meet is assoc... |
| latm12 38038 | A rearrangement of lattice... |
| latm32 38039 | A rearrangement of lattice... |
| latmrot 38040 | Rotate lattice meet of 3 c... |
| latm4 38041 | Rearrangement of lattice m... |
| latmmdiN 38042 | Lattice meet distributes o... |
| latmmdir 38043 | Lattice meet distributes o... |
| olm01 38044 | Meet with lattice zero is ... |
| olm02 38045 | Meet with lattice zero is ... |
| isoml 38046 | The predicate "is an ortho... |
| isomliN 38047 | Properties that determine ... |
| omlol 38048 | An orthomodular lattice is... |
| omlop 38049 | An orthomodular lattice is... |
| omllat 38050 | An orthomodular lattice is... |
| omllaw 38051 | The orthomodular law. (Co... |
| omllaw2N 38052 | Variation of orthomodular ... |
| omllaw3 38053 | Orthomodular law equivalen... |
| omllaw4 38054 | Orthomodular law equivalen... |
| omllaw5N 38055 | The orthomodular law. Rem... |
| cmtcomlemN 38056 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 38057 | Commutation is symmetric. ... |
| cmt2N 38058 | Commutation with orthocomp... |
| cmt3N 38059 | Commutation with orthocomp... |
| cmt4N 38060 | Commutation with orthocomp... |
| cmtbr2N 38061 | Alternate definition of th... |
| cmtbr3N 38062 | Alternate definition for t... |
| cmtbr4N 38063 | Alternate definition for t... |
| lecmtN 38064 | Ordered elements commute. ... |
| cmtidN 38065 | Any element commutes with ... |
| omlfh1N 38066 | Foulis-Holland Theorem, pa... |
| omlfh3N 38067 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 38068 | Analogue of modular law ~ ... |
| omlspjN 38069 | Contraction of a Sasaki pr... |
| cvrfval 38076 | Value of covers relation "... |
| cvrval 38077 | Binary relation expressing... |
| cvrlt 38078 | The covers relation implie... |
| cvrnbtwn 38079 | There is no element betwee... |
| ncvr1 38080 | No element covers the latt... |
| cvrletrN 38081 | Property of an element abo... |
| cvrval2 38082 | Binary relation expressing... |
| cvrnbtwn2 38083 | The covers relation implie... |
| cvrnbtwn3 38084 | The covers relation implie... |
| cvrcon3b 38085 | Contraposition law for the... |
| cvrle 38086 | The covers relation implie... |
| cvrnbtwn4 38087 | The covers relation implie... |
| cvrnle 38088 | The covers relation implie... |
| cvrne 38089 | The covers relation implie... |
| cvrnrefN 38090 | The covers relation is not... |
| cvrcmp 38091 | If two lattice elements th... |
| cvrcmp2 38092 | If two lattice elements co... |
| pats 38093 | The set of atoms in a pose... |
| isat 38094 | The predicate "is an atom"... |
| isat2 38095 | The predicate "is an atom"... |
| atcvr0 38096 | An atom covers zero. ( ~ ... |
| atbase 38097 | An atom is a member of the... |
| atssbase 38098 | The set of atoms is a subs... |
| 0ltat 38099 | An atom is greater than ze... |
| leatb 38100 | A poset element less than ... |
| leat 38101 | A poset element less than ... |
| leat2 38102 | A nonzero poset element le... |
| leat3 38103 | A poset element less than ... |
| meetat 38104 | The meet of any element wi... |
| meetat2 38105 | The meet of any element wi... |
| isatl 38107 | The predicate "is an atomi... |
| atllat 38108 | An atomic lattice is a lat... |
| atlpos 38109 | An atomic lattice is a pos... |
| atl0dm 38110 | Condition necessary for ze... |
| atl0cl 38111 | An atomic lattice has a ze... |
| atl0le 38112 | Orthoposet zero is less th... |
| atlle0 38113 | An element less than or eq... |
| atlltn0 38114 | A lattice element greater ... |
| isat3 38115 | The predicate "is an atom"... |
| atn0 38116 | An atom is not zero. ( ~ ... |
| atnle0 38117 | An atom is not less than o... |
| atlen0 38118 | A lattice element is nonze... |
| atcmp 38119 | If two atoms are comparabl... |
| atncmp 38120 | Frequently-used variation ... |
| atnlt 38121 | Two atoms cannot satisfy t... |
| atcvreq0 38122 | An element covered by an a... |
| atncvrN 38123 | Two atoms cannot satisfy t... |
| atlex 38124 | Every nonzero element of a... |
| atnle 38125 | Two ways of expressing "an... |
| atnem0 38126 | The meet of distinct atoms... |
| atlatmstc 38127 | An atomic, complete, ortho... |
| atlatle 38128 | The ordering of two Hilber... |
| atlrelat1 38129 | An atomistic lattice with ... |
| iscvlat 38131 | The predicate "is an atomi... |
| iscvlat2N 38132 | The predicate "is an atomi... |
| cvlatl 38133 | An atomic lattice with the... |
| cvllat 38134 | An atomic lattice with the... |
| cvlposN 38135 | An atomic lattice with the... |
| cvlexch1 38136 | An atomic covering lattice... |
| cvlexch2 38137 | An atomic covering lattice... |
| cvlexchb1 38138 | An atomic covering lattice... |
| cvlexchb2 38139 | An atomic covering lattice... |
| cvlexch3 38140 | An atomic covering lattice... |
| cvlexch4N 38141 | An atomic covering lattice... |
| cvlatexchb1 38142 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 38143 | A version of ~ cvlexchb2 f... |
| cvlatexch1 38144 | Atom exchange property. (... |
| cvlatexch2 38145 | Atom exchange property. (... |
| cvlatexch3 38146 | Atom exchange property. (... |
| cvlcvr1 38147 | The covering property. Pr... |
| cvlcvrp 38148 | A Hilbert lattice satisfie... |
| cvlatcvr1 38149 | An atom is covered by its ... |
| cvlatcvr2 38150 | An atom is covered by its ... |
| cvlsupr2 38151 | Two equivalent ways of exp... |
| cvlsupr3 38152 | Two equivalent ways of exp... |
| cvlsupr4 38153 | Consequence of superpositi... |
| cvlsupr5 38154 | Consequence of superpositi... |
| cvlsupr6 38155 | Consequence of superpositi... |
| cvlsupr7 38156 | Consequence of superpositi... |
| cvlsupr8 38157 | Consequence of superpositi... |
| ishlat1 38160 | The predicate "is a Hilber... |
| ishlat2 38161 | The predicate "is a Hilber... |
| ishlat3N 38162 | The predicate "is a Hilber... |
| ishlatiN 38163 | Properties that determine ... |
| hlomcmcv 38164 | A Hilbert lattice is ortho... |
| hloml 38165 | A Hilbert lattice is ortho... |
| hlclat 38166 | A Hilbert lattice is compl... |
| hlcvl 38167 | A Hilbert lattice is an at... |
| hlatl 38168 | A Hilbert lattice is atomi... |
| hlol 38169 | A Hilbert lattice is an or... |
| hlop 38170 | A Hilbert lattice is an or... |
| hllat 38171 | A Hilbert lattice is a lat... |
| hllatd 38172 | Deduction form of ~ hllat ... |
| hlomcmat 38173 | A Hilbert lattice is ortho... |
| hlpos 38174 | A Hilbert lattice is a pos... |
| hlatjcl 38175 | Closure of join operation.... |
| hlatjcom 38176 | Commutatitivity of join op... |
| hlatjidm 38177 | Idempotence of join operat... |
| hlatjass 38178 | Lattice join is associativ... |
| hlatj12 38179 | Swap 1st and 2nd members o... |
| hlatj32 38180 | Swap 2nd and 3rd members o... |
| hlatjrot 38181 | Rotate lattice join of 3 c... |
| hlatj4 38182 | Rearrangement of lattice j... |
| hlatlej1 38183 | A join's first argument is... |
| hlatlej2 38184 | A join's second argument i... |
| glbconN 38185 | De Morgan's law for GLB an... |
| glbconNOLD 38186 | Obsolete version of ~ glbc... |
| glbconxN 38187 | De Morgan's law for GLB an... |
| atnlej1 38188 | If an atom is not less tha... |
| atnlej2 38189 | If an atom is not less tha... |
| hlsuprexch 38190 | A Hilbert lattice has the ... |
| hlexch1 38191 | A Hilbert lattice has the ... |
| hlexch2 38192 | A Hilbert lattice has the ... |
| hlexchb1 38193 | A Hilbert lattice has the ... |
| hlexchb2 38194 | A Hilbert lattice has the ... |
| hlsupr 38195 | A Hilbert lattice has the ... |
| hlsupr2 38196 | A Hilbert lattice has the ... |
| hlhgt4 38197 | A Hilbert lattice has a he... |
| hlhgt2 38198 | A Hilbert lattice has a he... |
| hl0lt1N 38199 | Lattice 0 is less than lat... |
| hlexch3 38200 | A Hilbert lattice has the ... |
| hlexch4N 38201 | A Hilbert lattice has the ... |
| hlatexchb1 38202 | A version of ~ hlexchb1 fo... |
| hlatexchb2 38203 | A version of ~ hlexchb2 fo... |
| hlatexch1 38204 | Atom exchange property. (... |
| hlatexch2 38205 | Atom exchange property. (... |
| hlatmstcOLDN 38206 | An atomic, complete, ortho... |
| hlatle 38207 | The ordering of two Hilber... |
| hlateq 38208 | The equality of two Hilber... |
| hlrelat1 38209 | An atomistic lattice with ... |
| hlrelat5N 38210 | An atomistic lattice with ... |
| hlrelat 38211 | A Hilbert lattice is relat... |
| hlrelat2 38212 | A consequence of relative ... |
| exatleN 38213 | A condition for an atom to... |
| hl2at 38214 | A Hilbert lattice has at l... |
| atex 38215 | At least one atom exists. ... |
| intnatN 38216 | If the intersection with a... |
| 2llnne2N 38217 | Condition implying that tw... |
| 2llnneN 38218 | Condition implying that tw... |
| cvr1 38219 | A Hilbert lattice has the ... |
| cvr2N 38220 | Less-than and covers equiv... |
| hlrelat3 38221 | The Hilbert lattice is rel... |
| cvrval3 38222 | Binary relation expressing... |
| cvrval4N 38223 | Binary relation expressing... |
| cvrval5 38224 | Binary relation expressing... |
| cvrp 38225 | A Hilbert lattice satisfie... |
| atcvr1 38226 | An atom is covered by its ... |
| atcvr2 38227 | An atom is covered by its ... |
| cvrexchlem 38228 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 38229 | A Hilbert lattice satisfie... |
| cvratlem 38230 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 38231 | A nonzero Hilbert lattice ... |
| ltltncvr 38232 | A chained strong ordering ... |
| ltcvrntr 38233 | Non-transitive condition f... |
| cvrntr 38234 | The covers relation is not... |
| atcvr0eq 38235 | The covers relation is not... |
| lnnat 38236 | A line (the join of two di... |
| atcvrj0 38237 | Two atoms covering the zer... |
| cvrat2 38238 | A Hilbert lattice element ... |
| atcvrneN 38239 | Inequality derived from at... |
| atcvrj1 38240 | Condition for an atom to b... |
| atcvrj2b 38241 | Condition for an atom to b... |
| atcvrj2 38242 | Condition for an atom to b... |
| atleneN 38243 | Inequality derived from at... |
| atltcvr 38244 | An equivalence of less-tha... |
| atle 38245 | Any nonzero element has an... |
| atlt 38246 | Two atoms are unequal iff ... |
| atlelt 38247 | Transfer less-than relatio... |
| 2atlt 38248 | Given an atom less than an... |
| atexchcvrN 38249 | Atom exchange property. V... |
| atexchltN 38250 | Atom exchange property. V... |
| cvrat3 38251 | A condition implying that ... |
| cvrat4 38252 | A condition implying exist... |
| cvrat42 38253 | Commuted version of ~ cvra... |
| 2atjm 38254 | The meet of a line (expres... |
| atbtwn 38255 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 38256 | There exists a 3rd atom ` ... |
| atbtwnex 38257 | Given atoms ` P ` in ` X `... |
| 3noncolr2 38258 | Two ways to express 3 non-... |
| 3noncolr1N 38259 | Two ways to express 3 non-... |
| hlatcon3 38260 | Atom exchange combined wit... |
| hlatcon2 38261 | Atom exchange combined wit... |
| 4noncolr3 38262 | A way to express 4 non-col... |
| 4noncolr2 38263 | A way to express 4 non-col... |
| 4noncolr1 38264 | A way to express 4 non-col... |
| athgt 38265 | A Hilbert lattice, whose h... |
| 3dim0 38266 | There exists a 3-dimension... |
| 3dimlem1 38267 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 38268 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 38269 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 38270 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 38271 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 38272 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 38273 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 38274 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 38275 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 38276 | Construct a 3-dimensional ... |
| 3dim2 38277 | Construct 2 new layers on ... |
| 3dim3 38278 | Construct a new layer on t... |
| 2dim 38279 | Generate a height-3 elemen... |
| 1dimN 38280 | An atom is covered by a he... |
| 1cvrco 38281 | The orthocomplement of an ... |
| 1cvratex 38282 | There exists an atom less ... |
| 1cvratlt 38283 | An atom less than or equal... |
| 1cvrjat 38284 | An element covered by the ... |
| 1cvrat 38285 | Create an atom under an el... |
| ps-1 38286 | The join of two atoms ` R ... |
| ps-2 38287 | Lattice analogue for the p... |
| 2atjlej 38288 | Two atoms are different if... |
| hlatexch3N 38289 | Rearrange join of atoms in... |
| hlatexch4 38290 | Exchange 2 atoms. (Contri... |
| ps-2b 38291 | Variation of projective ge... |
| 3atlem1 38292 | Lemma for ~ 3at . (Contri... |
| 3atlem2 38293 | Lemma for ~ 3at . (Contri... |
| 3atlem3 38294 | Lemma for ~ 3at . (Contri... |
| 3atlem4 38295 | Lemma for ~ 3at . (Contri... |
| 3atlem5 38296 | Lemma for ~ 3at . (Contri... |
| 3atlem6 38297 | Lemma for ~ 3at . (Contri... |
| 3atlem7 38298 | Lemma for ~ 3at . (Contri... |
| 3at 38299 | Any three non-colinear ato... |
| llnset 38314 | The set of lattice lines i... |
| islln 38315 | The predicate "is a lattic... |
| islln4 38316 | The predicate "is a lattic... |
| llni 38317 | Condition implying a latti... |
| llnbase 38318 | A lattice line is a lattic... |
| islln3 38319 | The predicate "is a lattic... |
| islln2 38320 | The predicate "is a lattic... |
| llni2 38321 | The join of two different ... |
| llnnleat 38322 | An atom cannot majorize a ... |
| llnneat 38323 | A lattice line is not an a... |
| 2atneat 38324 | The join of two distinct a... |
| llnn0 38325 | A lattice line is nonzero.... |
| islln2a 38326 | The predicate "is a lattic... |
| llnle 38327 | Any element greater than 0... |
| atcvrlln2 38328 | An atom under a line is co... |
| atcvrlln 38329 | An element covering an ato... |
| llnexatN 38330 | Given an atom on a line, t... |
| llncmp 38331 | If two lattice lines are c... |
| llnnlt 38332 | Two lattice lines cannot s... |
| 2llnmat 38333 | Two intersecting lines int... |
| 2at0mat0 38334 | Special case of ~ 2atmat0 ... |
| 2atmat0 38335 | The meet of two unequal li... |
| 2atm 38336 | An atom majorized by two d... |
| ps-2c 38337 | Variation of projective ge... |
| lplnset 38338 | The set of lattice planes ... |
| islpln 38339 | The predicate "is a lattic... |
| islpln4 38340 | The predicate "is a lattic... |
| lplni 38341 | Condition implying a latti... |
| islpln3 38342 | The predicate "is a lattic... |
| lplnbase 38343 | A lattice plane is a latti... |
| islpln5 38344 | The predicate "is a lattic... |
| islpln2 38345 | The predicate "is a lattic... |
| lplni2 38346 | The join of 3 different at... |
| lvolex3N 38347 | There is an atom outside o... |
| llnmlplnN 38348 | The intersection of a line... |
| lplnle 38349 | Any element greater than 0... |
| lplnnle2at 38350 | A lattice line (or atom) c... |
| lplnnleat 38351 | A lattice plane cannot maj... |
| lplnnlelln 38352 | A lattice plane is not les... |
| 2atnelpln 38353 | The join of two atoms is n... |
| lplnneat 38354 | No lattice plane is an ato... |
| lplnnelln 38355 | No lattice plane is a latt... |
| lplnn0N 38356 | A lattice plane is nonzero... |
| islpln2a 38357 | The predicate "is a lattic... |
| islpln2ah 38358 | The predicate "is a lattic... |
| lplnriaN 38359 | Property of a lattice plan... |
| lplnribN 38360 | Property of a lattice plan... |
| lplnric 38361 | Property of a lattice plan... |
| lplnri1 38362 | Property of a lattice plan... |
| lplnri2N 38363 | Property of a lattice plan... |
| lplnri3N 38364 | Property of a lattice plan... |
| lplnllnneN 38365 | Two lattice lines defined ... |
| llncvrlpln2 38366 | A lattice line under a lat... |
| llncvrlpln 38367 | An element covering a latt... |
| 2lplnmN 38368 | If the join of two lattice... |
| 2llnmj 38369 | The meet of two lattice li... |
| 2atmat 38370 | The meet of two intersecti... |
| lplncmp 38371 | If two lattice planes are ... |
| lplnexatN 38372 | Given a lattice line on a ... |
| lplnexllnN 38373 | Given an atom on a lattice... |
| lplnnlt 38374 | Two lattice planes cannot ... |
| 2llnjaN 38375 | The join of two different ... |
| 2llnjN 38376 | The join of two different ... |
| 2llnm2N 38377 | The meet of two different ... |
| 2llnm3N 38378 | Two lattice lines in a lat... |
| 2llnm4 38379 | Two lattice lines that maj... |
| 2llnmeqat 38380 | An atom equals the interse... |
| lvolset 38381 | The set of 3-dim lattice v... |
| islvol 38382 | The predicate "is a 3-dim ... |
| islvol4 38383 | The predicate "is a 3-dim ... |
| lvoli 38384 | Condition implying a 3-dim... |
| islvol3 38385 | The predicate "is a 3-dim ... |
| lvoli3 38386 | Condition implying a 3-dim... |
| lvolbase 38387 | A 3-dim lattice volume is ... |
| islvol5 38388 | The predicate "is a 3-dim ... |
| islvol2 38389 | The predicate "is a 3-dim ... |
| lvoli2 38390 | The join of 4 different at... |
| lvolnle3at 38391 | A lattice plane (or lattic... |
| lvolnleat 38392 | An atom cannot majorize a ... |
| lvolnlelln 38393 | A lattice line cannot majo... |
| lvolnlelpln 38394 | A lattice plane cannot maj... |
| 3atnelvolN 38395 | The join of 3 atoms is not... |
| 2atnelvolN 38396 | The join of two atoms is n... |
| lvolneatN 38397 | No lattice volume is an at... |
| lvolnelln 38398 | No lattice volume is a lat... |
| lvolnelpln 38399 | No lattice volume is a lat... |
| lvoln0N 38400 | A lattice volume is nonzer... |
| islvol2aN 38401 | The predicate "is a lattic... |
| 4atlem0a 38402 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 38403 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 38404 | Lemma for ~ 4at . (Contri... |
| 4atlem3 38405 | Lemma for ~ 4at . Break i... |
| 4atlem3a 38406 | Lemma for ~ 4at . Break i... |
| 4atlem3b 38407 | Lemma for ~ 4at . Break i... |
| 4atlem4a 38408 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 38409 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 38410 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 38411 | Lemma for ~ 4at . Frequen... |
| 4atlem9 38412 | Lemma for ~ 4at . Substit... |
| 4atlem10a 38413 | Lemma for ~ 4at . Substit... |
| 4atlem10b 38414 | Lemma for ~ 4at . Substit... |
| 4atlem10 38415 | Lemma for ~ 4at . Combine... |
| 4atlem11a 38416 | Lemma for ~ 4at . Substit... |
| 4atlem11b 38417 | Lemma for ~ 4at . Substit... |
| 4atlem11 38418 | Lemma for ~ 4at . Combine... |
| 4atlem12a 38419 | Lemma for ~ 4at . Substit... |
| 4atlem12b 38420 | Lemma for ~ 4at . Substit... |
| 4atlem12 38421 | Lemma for ~ 4at . Combine... |
| 4at 38422 | Four atoms determine a lat... |
| 4at2 38423 | Four atoms determine a lat... |
| lplncvrlvol2 38424 | A lattice line under a lat... |
| lplncvrlvol 38425 | An element covering a latt... |
| lvolcmp 38426 | If two lattice planes are ... |
| lvolnltN 38427 | Two lattice volumes cannot... |
| 2lplnja 38428 | The join of two different ... |
| 2lplnj 38429 | The join of two different ... |
| 2lplnm2N 38430 | The meet of two different ... |
| 2lplnmj 38431 | The meet of two lattice pl... |
| dalemkehl 38432 | Lemma for ~ dath . Freque... |
| dalemkelat 38433 | Lemma for ~ dath . Freque... |
| dalemkeop 38434 | Lemma for ~ dath . Freque... |
| dalempea 38435 | Lemma for ~ dath . Freque... |
| dalemqea 38436 | Lemma for ~ dath . Freque... |
| dalemrea 38437 | Lemma for ~ dath . Freque... |
| dalemsea 38438 | Lemma for ~ dath . Freque... |
| dalemtea 38439 | Lemma for ~ dath . Freque... |
| dalemuea 38440 | Lemma for ~ dath . Freque... |
| dalemyeo 38441 | Lemma for ~ dath . Freque... |
| dalemzeo 38442 | Lemma for ~ dath . Freque... |
| dalemclpjs 38443 | Lemma for ~ dath . Freque... |
| dalemclqjt 38444 | Lemma for ~ dath . Freque... |
| dalemclrju 38445 | Lemma for ~ dath . Freque... |
| dalem-clpjq 38446 | Lemma for ~ dath . Freque... |
| dalemceb 38447 | Lemma for ~ dath . Freque... |
| dalempeb 38448 | Lemma for ~ dath . Freque... |
| dalemqeb 38449 | Lemma for ~ dath . Freque... |
| dalemreb 38450 | Lemma for ~ dath . Freque... |
| dalemseb 38451 | Lemma for ~ dath . Freque... |
| dalemteb 38452 | Lemma for ~ dath . Freque... |
| dalemueb 38453 | Lemma for ~ dath . Freque... |
| dalempjqeb 38454 | Lemma for ~ dath . Freque... |
| dalemsjteb 38455 | Lemma for ~ dath . Freque... |
| dalemtjueb 38456 | Lemma for ~ dath . Freque... |
| dalemqrprot 38457 | Lemma for ~ dath . Freque... |
| dalemyeb 38458 | Lemma for ~ dath . Freque... |
| dalemcnes 38459 | Lemma for ~ dath . Freque... |
| dalempnes 38460 | Lemma for ~ dath . Freque... |
| dalemqnet 38461 | Lemma for ~ dath . Freque... |
| dalempjsen 38462 | Lemma for ~ dath . Freque... |
| dalemply 38463 | Lemma for ~ dath . Freque... |
| dalemsly 38464 | Lemma for ~ dath . Freque... |
| dalemswapyz 38465 | Lemma for ~ dath . Swap t... |
| dalemrot 38466 | Lemma for ~ dath . Rotate... |
| dalemrotyz 38467 | Lemma for ~ dath . Rotate... |
| dalem1 38468 | Lemma for ~ dath . Show t... |
| dalemcea 38469 | Lemma for ~ dath . Freque... |
| dalem2 38470 | Lemma for ~ dath . Show t... |
| dalemdea 38471 | Lemma for ~ dath . Freque... |
| dalemeea 38472 | Lemma for ~ dath . Freque... |
| dalem3 38473 | Lemma for ~ dalemdnee . (... |
| dalem4 38474 | Lemma for ~ dalemdnee . (... |
| dalemdnee 38475 | Lemma for ~ dath . Axis o... |
| dalem5 38476 | Lemma for ~ dath . Atom `... |
| dalem6 38477 | Lemma for ~ dath . Analog... |
| dalem7 38478 | Lemma for ~ dath . Analog... |
| dalem8 38479 | Lemma for ~ dath . Plane ... |
| dalem-cly 38480 | Lemma for ~ dalem9 . Cent... |
| dalem9 38481 | Lemma for ~ dath . Since ... |
| dalem10 38482 | Lemma for ~ dath . Atom `... |
| dalem11 38483 | Lemma for ~ dath . Analog... |
| dalem12 38484 | Lemma for ~ dath . Analog... |
| dalem13 38485 | Lemma for ~ dalem14 . (Co... |
| dalem14 38486 | Lemma for ~ dath . Planes... |
| dalem15 38487 | Lemma for ~ dath . The ax... |
| dalem16 38488 | Lemma for ~ dath . The at... |
| dalem17 38489 | Lemma for ~ dath . When p... |
| dalem18 38490 | Lemma for ~ dath . Show t... |
| dalem19 38491 | Lemma for ~ dath . Show t... |
| dalemccea 38492 | Lemma for ~ dath . Freque... |
| dalemddea 38493 | Lemma for ~ dath . Freque... |
| dalem-ccly 38494 | Lemma for ~ dath . Freque... |
| dalem-ddly 38495 | Lemma for ~ dath . Freque... |
| dalemccnedd 38496 | Lemma for ~ dath . Freque... |
| dalemclccjdd 38497 | Lemma for ~ dath . Freque... |
| dalemcceb 38498 | Lemma for ~ dath . Freque... |
| dalemswapyzps 38499 | Lemma for ~ dath . Swap t... |
| dalemrotps 38500 | Lemma for ~ dath . Rotate... |
| dalemcjden 38501 | Lemma for ~ dath . Show t... |
| dalem20 38502 | Lemma for ~ dath . Show t... |
| dalem21 38503 | Lemma for ~ dath . Show t... |
| dalem22 38504 | Lemma for ~ dath . Show t... |
| dalem23 38505 | Lemma for ~ dath . Show t... |
| dalem24 38506 | Lemma for ~ dath . Show t... |
| dalem25 38507 | Lemma for ~ dath . Show t... |
| dalem27 38508 | Lemma for ~ dath . Show t... |
| dalem28 38509 | Lemma for ~ dath . Lemma ... |
| dalem29 38510 | Lemma for ~ dath . Analog... |
| dalem30 38511 | Lemma for ~ dath . Analog... |
| dalem31N 38512 | Lemma for ~ dath . Analog... |
| dalem32 38513 | Lemma for ~ dath . Analog... |
| dalem33 38514 | Lemma for ~ dath . Analog... |
| dalem34 38515 | Lemma for ~ dath . Analog... |
| dalem35 38516 | Lemma for ~ dath . Analog... |
| dalem36 38517 | Lemma for ~ dath . Analog... |
| dalem37 38518 | Lemma for ~ dath . Analog... |
| dalem38 38519 | Lemma for ~ dath . Plane ... |
| dalem39 38520 | Lemma for ~ dath . Auxili... |
| dalem40 38521 | Lemma for ~ dath . Analog... |
| dalem41 38522 | Lemma for ~ dath . (Contr... |
| dalem42 38523 | Lemma for ~ dath . Auxili... |
| dalem43 38524 | Lemma for ~ dath . Planes... |
| dalem44 38525 | Lemma for ~ dath . Dummy ... |
| dalem45 38526 | Lemma for ~ dath . Dummy ... |
| dalem46 38527 | Lemma for ~ dath . Analog... |
| dalem47 38528 | Lemma for ~ dath . Analog... |
| dalem48 38529 | Lemma for ~ dath . Analog... |
| dalem49 38530 | Lemma for ~ dath . Analog... |
| dalem50 38531 | Lemma for ~ dath . Analog... |
| dalem51 38532 | Lemma for ~ dath . Constr... |
| dalem52 38533 | Lemma for ~ dath . Lines ... |
| dalem53 38534 | Lemma for ~ dath . The au... |
| dalem54 38535 | Lemma for ~ dath . Line `... |
| dalem55 38536 | Lemma for ~ dath . Lines ... |
| dalem56 38537 | Lemma for ~ dath . Analog... |
| dalem57 38538 | Lemma for ~ dath . Axis o... |
| dalem58 38539 | Lemma for ~ dath . Analog... |
| dalem59 38540 | Lemma for ~ dath . Analog... |
| dalem60 38541 | Lemma for ~ dath . ` B ` i... |
| dalem61 38542 | Lemma for ~ dath . Show t... |
| dalem62 38543 | Lemma for ~ dath . Elimin... |
| dalem63 38544 | Lemma for ~ dath . Combin... |
| dath 38545 | Desargues's theorem of pro... |
| dath2 38546 | Version of Desargues's the... |
| lineset 38547 | The set of lines in a Hilb... |
| isline 38548 | The predicate "is a line".... |
| islinei 38549 | Condition implying "is a l... |
| pointsetN 38550 | The set of points in a Hil... |
| ispointN 38551 | The predicate "is a point"... |
| atpointN 38552 | The singleton of an atom i... |
| psubspset 38553 | The set of projective subs... |
| ispsubsp 38554 | The predicate "is a projec... |
| ispsubsp2 38555 | The predicate "is a projec... |
| psubspi 38556 | Property of a projective s... |
| psubspi2N 38557 | Property of a projective s... |
| 0psubN 38558 | The empty set is a project... |
| snatpsubN 38559 | The singleton of an atom i... |
| pointpsubN 38560 | A point (singleton of an a... |
| linepsubN 38561 | A line is a projective sub... |
| atpsubN 38562 | The set of all atoms is a ... |
| psubssat 38563 | A projective subspace cons... |
| psubatN 38564 | A member of a projective s... |
| pmapfval 38565 | The projective map of a Hi... |
| pmapval 38566 | Value of the projective ma... |
| elpmap 38567 | Member of a projective map... |
| pmapssat 38568 | The projective map of a Hi... |
| pmapssbaN 38569 | A weakening of ~ pmapssat ... |
| pmaple 38570 | The projective map of a Hi... |
| pmap11 38571 | The projective map of a Hi... |
| pmapat 38572 | The projective map of an a... |
| elpmapat 38573 | Member of the projective m... |
| pmap0 38574 | Value of the projective ma... |
| pmapeq0 38575 | A projective map value is ... |
| pmap1N 38576 | Value of the projective ma... |
| pmapsub 38577 | The projective map of a Hi... |
| pmapglbx 38578 | The projective map of the ... |
| pmapglb 38579 | The projective map of the ... |
| pmapglb2N 38580 | The projective map of the ... |
| pmapglb2xN 38581 | The projective map of the ... |
| pmapmeet 38582 | The projective map of a me... |
| isline2 38583 | Definition of line in term... |
| linepmap 38584 | A line described with a pr... |
| isline3 38585 | Definition of line in term... |
| isline4N 38586 | Definition of line in term... |
| lneq2at 38587 | A line equals the join of ... |
| lnatexN 38588 | There is an atom in a line... |
| lnjatN 38589 | Given an atom in a line, t... |
| lncvrelatN 38590 | A lattice element covered ... |
| lncvrat 38591 | A line covers the atoms it... |
| lncmp 38592 | If two lines are comparabl... |
| 2lnat 38593 | Two intersecting lines int... |
| 2atm2atN 38594 | Two joins with a common at... |
| 2llnma1b 38595 | Generalization of ~ 2llnma... |
| 2llnma1 38596 | Two different intersecting... |
| 2llnma3r 38597 | Two different intersecting... |
| 2llnma2 38598 | Two different intersecting... |
| 2llnma2rN 38599 | Two different intersecting... |
| cdlema1N 38600 | A condition for required f... |
| cdlema2N 38601 | A condition for required f... |
| cdlemblem 38602 | Lemma for ~ cdlemb . (Con... |
| cdlemb 38603 | Given two atoms not less t... |
| paddfval 38606 | Projective subspace sum op... |
| paddval 38607 | Projective subspace sum op... |
| elpadd 38608 | Member of a projective sub... |
| elpaddn0 38609 | Member of projective subsp... |
| paddvaln0N 38610 | Projective subspace sum op... |
| elpaddri 38611 | Condition implying members... |
| elpaddatriN 38612 | Condition implying members... |
| elpaddat 38613 | Membership in a projective... |
| elpaddatiN 38614 | Consequence of membership ... |
| elpadd2at 38615 | Membership in a projective... |
| elpadd2at2 38616 | Membership in a projective... |
| paddunssN 38617 | Projective subspace sum in... |
| elpadd0 38618 | Member of projective subsp... |
| paddval0 38619 | Projective subspace sum wi... |
| padd01 38620 | Projective subspace sum wi... |
| padd02 38621 | Projective subspace sum wi... |
| paddcom 38622 | Projective subspace sum co... |
| paddssat 38623 | A projective subspace sum ... |
| sspadd1 38624 | A projective subspace sum ... |
| sspadd2 38625 | A projective subspace sum ... |
| paddss1 38626 | Subset law for projective ... |
| paddss2 38627 | Subset law for projective ... |
| paddss12 38628 | Subset law for projective ... |
| paddasslem1 38629 | Lemma for ~ paddass . (Co... |
| paddasslem2 38630 | Lemma for ~ paddass . (Co... |
| paddasslem3 38631 | Lemma for ~ paddass . Res... |
| paddasslem4 38632 | Lemma for ~ paddass . Com... |
| paddasslem5 38633 | Lemma for ~ paddass . Sho... |
| paddasslem6 38634 | Lemma for ~ paddass . (Co... |
| paddasslem7 38635 | Lemma for ~ paddass . Com... |
| paddasslem8 38636 | Lemma for ~ paddass . (Co... |
| paddasslem9 38637 | Lemma for ~ paddass . Com... |
| paddasslem10 38638 | Lemma for ~ paddass . Use... |
| paddasslem11 38639 | Lemma for ~ paddass . The... |
| paddasslem12 38640 | Lemma for ~ paddass . The... |
| paddasslem13 38641 | Lemma for ~ paddass . The... |
| paddasslem14 38642 | Lemma for ~ paddass . Rem... |
| paddasslem15 38643 | Lemma for ~ paddass . Use... |
| paddasslem16 38644 | Lemma for ~ paddass . Use... |
| paddasslem17 38645 | Lemma for ~ paddass . The... |
| paddasslem18 38646 | Lemma for ~ paddass . Com... |
| paddass 38647 | Projective subspace sum is... |
| padd12N 38648 | Commutative/associative la... |
| padd4N 38649 | Rearrangement of 4 terms i... |
| paddidm 38650 | Projective subspace sum is... |
| paddclN 38651 | The projective sum of two ... |
| paddssw1 38652 | Subset law for projective ... |
| paddssw2 38653 | Subset law for projective ... |
| paddss 38654 | Subset law for projective ... |
| pmodlem1 38655 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 38656 | Lemma for ~ pmod1i . (Con... |
| pmod1i 38657 | The modular law holds in a... |
| pmod2iN 38658 | Dual of the modular law. ... |
| pmodN 38659 | The modular law for projec... |
| pmodl42N 38660 | Lemma derived from modular... |
| pmapjoin 38661 | The projective map of the ... |
| pmapjat1 38662 | The projective map of the ... |
| pmapjat2 38663 | The projective map of the ... |
| pmapjlln1 38664 | The projective map of the ... |
| hlmod1i 38665 | A version of the modular l... |
| atmod1i1 38666 | Version of modular law ~ p... |
| atmod1i1m 38667 | Version of modular law ~ p... |
| atmod1i2 38668 | Version of modular law ~ p... |
| llnmod1i2 38669 | Version of modular law ~ p... |
| atmod2i1 38670 | Version of modular law ~ p... |
| atmod2i2 38671 | Version of modular law ~ p... |
| llnmod2i2 38672 | Version of modular law ~ p... |
| atmod3i1 38673 | Version of modular law tha... |
| atmod3i2 38674 | Version of modular law tha... |
| atmod4i1 38675 | Version of modular law tha... |
| atmod4i2 38676 | Version of modular law tha... |
| llnexchb2lem 38677 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 38678 | Line exchange property (co... |
| llnexch2N 38679 | Line exchange property (co... |
| dalawlem1 38680 | Lemma for ~ dalaw . Speci... |
| dalawlem2 38681 | Lemma for ~ dalaw . Utili... |
| dalawlem3 38682 | Lemma for ~ dalaw . First... |
| dalawlem4 38683 | Lemma for ~ dalaw . Secon... |
| dalawlem5 38684 | Lemma for ~ dalaw . Speci... |
| dalawlem6 38685 | Lemma for ~ dalaw . First... |
| dalawlem7 38686 | Lemma for ~ dalaw . Secon... |
| dalawlem8 38687 | Lemma for ~ dalaw . Speci... |
| dalawlem9 38688 | Lemma for ~ dalaw . Speci... |
| dalawlem10 38689 | Lemma for ~ dalaw . Combi... |
| dalawlem11 38690 | Lemma for ~ dalaw . First... |
| dalawlem12 38691 | Lemma for ~ dalaw . Secon... |
| dalawlem13 38692 | Lemma for ~ dalaw . Speci... |
| dalawlem14 38693 | Lemma for ~ dalaw . Combi... |
| dalawlem15 38694 | Lemma for ~ dalaw . Swap ... |
| dalaw 38695 | Desargues's law, derived f... |
| pclfvalN 38698 | The projective subspace cl... |
| pclvalN 38699 | Value of the projective su... |
| pclclN 38700 | Closure of the projective ... |
| elpclN 38701 | Membership in the projecti... |
| elpcliN 38702 | Implication of membership ... |
| pclssN 38703 | Ordering is preserved by s... |
| pclssidN 38704 | A set of atoms is included... |
| pclidN 38705 | The projective subspace cl... |
| pclbtwnN 38706 | A projective subspace sand... |
| pclunN 38707 | The projective subspace cl... |
| pclun2N 38708 | The projective subspace cl... |
| pclfinN 38709 | The projective subspace cl... |
| pclcmpatN 38710 | The set of projective subs... |
| polfvalN 38713 | The projective subspace po... |
| polvalN 38714 | Value of the projective su... |
| polval2N 38715 | Alternate expression for v... |
| polsubN 38716 | The polarity of a set of a... |
| polssatN 38717 | The polarity of a set of a... |
| pol0N 38718 | The polarity of the empty ... |
| pol1N 38719 | The polarity of the whole ... |
| 2pol0N 38720 | The closed subspace closur... |
| polpmapN 38721 | The polarity of a projecti... |
| 2polpmapN 38722 | Double polarity of a proje... |
| 2polvalN 38723 | Value of double polarity. ... |
| 2polssN 38724 | A set of atoms is a subset... |
| 3polN 38725 | Triple polarity cancels to... |
| polcon3N 38726 | Contraposition law for pol... |
| 2polcon4bN 38727 | Contraposition law for pol... |
| polcon2N 38728 | Contraposition law for pol... |
| polcon2bN 38729 | Contraposition law for pol... |
| pclss2polN 38730 | The projective subspace cl... |
| pcl0N 38731 | The projective subspace cl... |
| pcl0bN 38732 | The projective subspace cl... |
| pmaplubN 38733 | The LUB of a projective ma... |
| sspmaplubN 38734 | A set of atoms is a subset... |
| 2pmaplubN 38735 | Double projective map of a... |
| paddunN 38736 | The closure of the project... |
| poldmj1N 38737 | De Morgan's law for polari... |
| pmapj2N 38738 | The projective map of the ... |
| pmapocjN 38739 | The projective map of the ... |
| polatN 38740 | The polarity of the single... |
| 2polatN 38741 | Double polarity of the sin... |
| pnonsingN 38742 | The intersection of a set ... |
| psubclsetN 38745 | The set of closed projecti... |
| ispsubclN 38746 | The predicate "is a closed... |
| psubcliN 38747 | Property of a closed proje... |
| psubcli2N 38748 | Property of a closed proje... |
| psubclsubN 38749 | A closed projective subspa... |
| psubclssatN 38750 | A closed projective subspa... |
| pmapidclN 38751 | Projective map of the LUB ... |
| 0psubclN 38752 | The empty set is a closed ... |
| 1psubclN 38753 | The set of all atoms is a ... |
| atpsubclN 38754 | A point (singleton of an a... |
| pmapsubclN 38755 | A projective map value is ... |
| ispsubcl2N 38756 | Alternate predicate for "i... |
| psubclinN 38757 | The intersection of two cl... |
| paddatclN 38758 | The projective sum of a cl... |
| pclfinclN 38759 | The projective subspace cl... |
| linepsubclN 38760 | A line is a closed project... |
| polsubclN 38761 | A polarity is a closed pro... |
| poml4N 38762 | Orthomodular law for proje... |
| poml5N 38763 | Orthomodular law for proje... |
| poml6N 38764 | Orthomodular law for proje... |
| osumcllem1N 38765 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 38766 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 38767 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 38768 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 38769 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 38770 | Lemma for ~ osumclN . Use... |
| osumcllem7N 38771 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 38772 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 38773 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 38774 | Lemma for ~ osumclN . Con... |
| osumcllem11N 38775 | Lemma for ~ osumclN . (Co... |
| osumclN 38776 | Closure of orthogonal sum.... |
| pmapojoinN 38777 | For orthogonal elements, p... |
| pexmidN 38778 | Excluded middle law for cl... |
| pexmidlem1N 38779 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 38780 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 38781 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 38782 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 38783 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 38784 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 38785 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 38786 | Lemma for ~ pexmidN . The... |
| pexmidALTN 38787 | Excluded middle law for cl... |
| pl42lem1N 38788 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 38789 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 38790 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 38791 | Lemma for ~ pl42N . (Cont... |
| pl42N 38792 | Law holding in a Hilbert l... |
| watfvalN 38801 | The W atoms function. (Co... |
| watvalN 38802 | Value of the W atoms funct... |
| iswatN 38803 | The predicate "is a W atom... |
| lhpset 38804 | The set of co-atoms (latti... |
| islhp 38805 | The predicate "is a co-ato... |
| islhp2 38806 | The predicate "is a co-ato... |
| lhpbase 38807 | A co-atom is a member of t... |
| lhp1cvr 38808 | The lattice unity covers a... |
| lhplt 38809 | An atom under a co-atom is... |
| lhp2lt 38810 | The join of two atoms unde... |
| lhpexlt 38811 | There exists an atom less ... |
| lhp0lt 38812 | A co-atom is greater than ... |
| lhpn0 38813 | A co-atom is nonzero. TOD... |
| lhpexle 38814 | There exists an atom under... |
| lhpexnle 38815 | There exists an atom not u... |
| lhpexle1lem 38816 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 38817 | There exists an atom under... |
| lhpexle2lem 38818 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 38819 | There exists atom under a ... |
| lhpexle3lem 38820 | There exists atom under a ... |
| lhpexle3 38821 | There exists atom under a ... |
| lhpex2leN 38822 | There exist at least two d... |
| lhpoc 38823 | The orthocomplement of a c... |
| lhpoc2N 38824 | The orthocomplement of an ... |
| lhpocnle 38825 | The orthocomplement of a c... |
| lhpocat 38826 | The orthocomplement of a c... |
| lhpocnel 38827 | The orthocomplement of a c... |
| lhpocnel2 38828 | The orthocomplement of a c... |
| lhpjat1 38829 | The join of a co-atom (hyp... |
| lhpjat2 38830 | The join of a co-atom (hyp... |
| lhpj1 38831 | The join of a co-atom (hyp... |
| lhpmcvr 38832 | The meet of a lattice hype... |
| lhpmcvr2 38833 | Alternate way to express t... |
| lhpmcvr3 38834 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 38835 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 38836 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 38837 | Specialization of ~ lhpmcv... |
| lhpm0atN 38838 | If the meet of a lattice h... |
| lhpmat 38839 | An element covered by the ... |
| lhpmatb 38840 | An element covered by the ... |
| lhp2at0 38841 | Join and meet with differe... |
| lhp2atnle 38842 | Inequality for 2 different... |
| lhp2atne 38843 | Inequality for joins with ... |
| lhp2at0nle 38844 | Inequality for 2 different... |
| lhp2at0ne 38845 | Inequality for joins with ... |
| lhpelim 38846 | Eliminate an atom not unde... |
| lhpmod2i2 38847 | Modular law for hyperplane... |
| lhpmod6i1 38848 | Modular law for hyperplane... |
| lhprelat3N 38849 | The Hilbert lattice is rel... |
| cdlemb2 38850 | Given two atoms not under ... |
| lhple 38851 | Property of a lattice elem... |
| lhpat 38852 | Create an atom under a co-... |
| lhpat4N 38853 | Property of an atom under ... |
| lhpat2 38854 | Create an atom under a co-... |
| lhpat3 38855 | There is only one atom und... |
| 4atexlemk 38856 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 38857 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 38858 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 38859 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 38860 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 38861 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 38862 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 38863 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 38864 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 38865 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 38866 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 38867 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 38868 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 38869 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 38870 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 38871 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 38872 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 38873 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 38874 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 38875 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 38876 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 38877 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 38878 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 38879 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 38880 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 38881 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 38882 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 38883 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 38884 | Whenever there are at leas... |
| 4atex 38885 | Whenever there are at leas... |
| 4atex2 38886 | More general version of ~ ... |
| 4atex2-0aOLDN 38887 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 38888 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 38889 | Same as ~ 4atex2 except th... |
| 4atex3 38890 | More general version of ~ ... |
| lautset 38891 | The set of lattice automor... |
| islaut 38892 | The predicate "is a lattic... |
| lautle 38893 | Less-than or equal propert... |
| laut1o 38894 | A lattice automorphism is ... |
| laut11 38895 | One-to-one property of a l... |
| lautcl 38896 | A lattice automorphism val... |
| lautcnvclN 38897 | Reverse closure of a latti... |
| lautcnvle 38898 | Less-than or equal propert... |
| lautcnv 38899 | The converse of a lattice ... |
| lautlt 38900 | Less-than property of a la... |
| lautcvr 38901 | Covering property of a lat... |
| lautj 38902 | Meet property of a lattice... |
| lautm 38903 | Meet property of a lattice... |
| lauteq 38904 | A lattice automorphism arg... |
| idlaut 38905 | The identity function is a... |
| lautco 38906 | The composition of two lat... |
| pautsetN 38907 | The set of projective auto... |
| ispautN 38908 | The predicate "is a projec... |
| ldilfset 38917 | The mapping from fiducial ... |
| ldilset 38918 | The set of lattice dilatio... |
| isldil 38919 | The predicate "is a lattic... |
| ldillaut 38920 | A lattice dilation is an a... |
| ldil1o 38921 | A lattice dilation is a on... |
| ldilval 38922 | Value of a lattice dilatio... |
| idldil 38923 | The identity function is a... |
| ldilcnv 38924 | The converse of a lattice ... |
| ldilco 38925 | The composition of two lat... |
| ltrnfset 38926 | The set of all lattice tra... |
| ltrnset 38927 | The set of lattice transla... |
| isltrn 38928 | The predicate "is a lattic... |
| isltrn2N 38929 | The predicate "is a lattic... |
| ltrnu 38930 | Uniqueness property of a l... |
| ltrnldil 38931 | A lattice translation is a... |
| ltrnlaut 38932 | A lattice translation is a... |
| ltrn1o 38933 | A lattice translation is a... |
| ltrncl 38934 | Closure of a lattice trans... |
| ltrn11 38935 | One-to-one property of a l... |
| ltrncnvnid 38936 | If a translation is differ... |
| ltrncoidN 38937 | Two translations are equal... |
| ltrnle 38938 | Less-than or equal propert... |
| ltrncnvleN 38939 | Less-than or equal propert... |
| ltrnm 38940 | Lattice translation of a m... |
| ltrnj 38941 | Lattice translation of a m... |
| ltrncvr 38942 | Covering property of a lat... |
| ltrnval1 38943 | Value of a lattice transla... |
| ltrnid 38944 | A lattice translation is t... |
| ltrnnid 38945 | If a lattice translation i... |
| ltrnatb 38946 | The lattice translation of... |
| ltrncnvatb 38947 | The converse of the lattic... |
| ltrnel 38948 | The lattice translation of... |
| ltrnat 38949 | The lattice translation of... |
| ltrncnvat 38950 | The converse of the lattic... |
| ltrncnvel 38951 | The converse of the lattic... |
| ltrncoelN 38952 | Composition of lattice tra... |
| ltrncoat 38953 | Composition of lattice tra... |
| ltrncoval 38954 | Two ways to express value ... |
| ltrncnv 38955 | The converse of a lattice ... |
| ltrn11at 38956 | Frequently used one-to-one... |
| ltrneq2 38957 | The equality of two transl... |
| ltrneq 38958 | The equality of two transl... |
| idltrn 38959 | The identity function is a... |
| ltrnmw 38960 | Property of lattice transl... |
| dilfsetN 38961 | The mapping from fiducial ... |
| dilsetN 38962 | The set of dilations for a... |
| isdilN 38963 | The predicate "is a dilati... |
| trnfsetN 38964 | The mapping from fiducial ... |
| trnsetN 38965 | The set of translations fo... |
| istrnN 38966 | The predicate "is a transl... |
| trlfset 38969 | The set of all traces of l... |
| trlset 38970 | The set of traces of latti... |
| trlval 38971 | The value of the trace of ... |
| trlval2 38972 | The value of the trace of ... |
| trlcl 38973 | Closure of the trace of a ... |
| trlcnv 38974 | The trace of the converse ... |
| trljat1 38975 | The value of a translation... |
| trljat2 38976 | The value of a translation... |
| trljat3 38977 | The value of a translation... |
| trlat 38978 | If an atom differs from it... |
| trl0 38979 | If an atom not under the f... |
| trlator0 38980 | The trace of a lattice tra... |
| trlatn0 38981 | The trace of a lattice tra... |
| trlnidat 38982 | The trace of a lattice tra... |
| ltrnnidn 38983 | If a lattice translation i... |
| ltrnideq 38984 | Property of the identity l... |
| trlid0 38985 | The trace of the identity ... |
| trlnidatb 38986 | A lattice translation is n... |
| trlid0b 38987 | A lattice translation is t... |
| trlnid 38988 | Different translations wit... |
| ltrn2ateq 38989 | Property of the equality o... |
| ltrnateq 38990 | If any atom (under ` W ` )... |
| ltrnatneq 38991 | If any atom (under ` W ` )... |
| ltrnatlw 38992 | If the value of an atom eq... |
| trlle 38993 | The trace of a lattice tra... |
| trlne 38994 | The trace of a lattice tra... |
| trlnle 38995 | The atom not under the fid... |
| trlval3 38996 | The value of the trace of ... |
| trlval4 38997 | The value of the trace of ... |
| trlval5 38998 | The value of the trace of ... |
| arglem1N 38999 | Lemma for Desargues's law.... |
| cdlemc1 39000 | Part of proof of Lemma C i... |
| cdlemc2 39001 | Part of proof of Lemma C i... |
| cdlemc3 39002 | Part of proof of Lemma C i... |
| cdlemc4 39003 | Part of proof of Lemma C i... |
| cdlemc5 39004 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 39005 | Lemma for ~ cdlemc . (Con... |
| cdlemc 39006 | Lemma C in [Crawley] p. 11... |
| cdlemd1 39007 | Part of proof of Lemma D i... |
| cdlemd2 39008 | Part of proof of Lemma D i... |
| cdlemd3 39009 | Part of proof of Lemma D i... |
| cdlemd4 39010 | Part of proof of Lemma D i... |
| cdlemd5 39011 | Part of proof of Lemma D i... |
| cdlemd6 39012 | Part of proof of Lemma D i... |
| cdlemd7 39013 | Part of proof of Lemma D i... |
| cdlemd8 39014 | Part of proof of Lemma D i... |
| cdlemd9 39015 | Part of proof of Lemma D i... |
| cdlemd 39016 | If two translations agree ... |
| ltrneq3 39017 | Two translations agree at ... |
| cdleme00a 39018 | Part of proof of Lemma E i... |
| cdleme0aa 39019 | Part of proof of Lemma E i... |
| cdleme0a 39020 | Part of proof of Lemma E i... |
| cdleme0b 39021 | Part of proof of Lemma E i... |
| cdleme0c 39022 | Part of proof of Lemma E i... |
| cdleme0cp 39023 | Part of proof of Lemma E i... |
| cdleme0cq 39024 | Part of proof of Lemma E i... |
| cdleme0dN 39025 | Part of proof of Lemma E i... |
| cdleme0e 39026 | Part of proof of Lemma E i... |
| cdleme0fN 39027 | Part of proof of Lemma E i... |
| cdleme0gN 39028 | Part of proof of Lemma E i... |
| cdlemeulpq 39029 | Part of proof of Lemma E i... |
| cdleme01N 39030 | Part of proof of Lemma E i... |
| cdleme02N 39031 | Part of proof of Lemma E i... |
| cdleme0ex1N 39032 | Part of proof of Lemma E i... |
| cdleme0ex2N 39033 | Part of proof of Lemma E i... |
| cdleme0moN 39034 | Part of proof of Lemma E i... |
| cdleme1b 39035 | Part of proof of Lemma E i... |
| cdleme1 39036 | Part of proof of Lemma E i... |
| cdleme2 39037 | Part of proof of Lemma E i... |
| cdleme3b 39038 | Part of proof of Lemma E i... |
| cdleme3c 39039 | Part of proof of Lemma E i... |
| cdleme3d 39040 | Part of proof of Lemma E i... |
| cdleme3e 39041 | Part of proof of Lemma E i... |
| cdleme3fN 39042 | Part of proof of Lemma E i... |
| cdleme3g 39043 | Part of proof of Lemma E i... |
| cdleme3h 39044 | Part of proof of Lemma E i... |
| cdleme3fa 39045 | Part of proof of Lemma E i... |
| cdleme3 39046 | Part of proof of Lemma E i... |
| cdleme4 39047 | Part of proof of Lemma E i... |
| cdleme4a 39048 | Part of proof of Lemma E i... |
| cdleme5 39049 | Part of proof of Lemma E i... |
| cdleme6 39050 | Part of proof of Lemma E i... |
| cdleme7aa 39051 | Part of proof of Lemma E i... |
| cdleme7a 39052 | Part of proof of Lemma E i... |
| cdleme7b 39053 | Part of proof of Lemma E i... |
| cdleme7c 39054 | Part of proof of Lemma E i... |
| cdleme7d 39055 | Part of proof of Lemma E i... |
| cdleme7e 39056 | Part of proof of Lemma E i... |
| cdleme7ga 39057 | Part of proof of Lemma E i... |
| cdleme7 39058 | Part of proof of Lemma E i... |
| cdleme8 39059 | Part of proof of Lemma E i... |
| cdleme9a 39060 | Part of proof of Lemma E i... |
| cdleme9b 39061 | Utility lemma for Lemma E ... |
| cdleme9 39062 | Part of proof of Lemma E i... |
| cdleme10 39063 | Part of proof of Lemma E i... |
| cdleme8tN 39064 | Part of proof of Lemma E i... |
| cdleme9taN 39065 | Part of proof of Lemma E i... |
| cdleme9tN 39066 | Part of proof of Lemma E i... |
| cdleme10tN 39067 | Part of proof of Lemma E i... |
| cdleme16aN 39068 | Part of proof of Lemma E i... |
| cdleme11a 39069 | Part of proof of Lemma E i... |
| cdleme11c 39070 | Part of proof of Lemma E i... |
| cdleme11dN 39071 | Part of proof of Lemma E i... |
| cdleme11e 39072 | Part of proof of Lemma E i... |
| cdleme11fN 39073 | Part of proof of Lemma E i... |
| cdleme11g 39074 | Part of proof of Lemma E i... |
| cdleme11h 39075 | Part of proof of Lemma E i... |
| cdleme11j 39076 | Part of proof of Lemma E i... |
| cdleme11k 39077 | Part of proof of Lemma E i... |
| cdleme11l 39078 | Part of proof of Lemma E i... |
| cdleme11 39079 | Part of proof of Lemma E i... |
| cdleme12 39080 | Part of proof of Lemma E i... |
| cdleme13 39081 | Part of proof of Lemma E i... |
| cdleme14 39082 | Part of proof of Lemma E i... |
| cdleme15a 39083 | Part of proof of Lemma E i... |
| cdleme15b 39084 | Part of proof of Lemma E i... |
| cdleme15c 39085 | Part of proof of Lemma E i... |
| cdleme15d 39086 | Part of proof of Lemma E i... |
| cdleme15 39087 | Part of proof of Lemma E i... |
| cdleme16b 39088 | Part of proof of Lemma E i... |
| cdleme16c 39089 | Part of proof of Lemma E i... |
| cdleme16d 39090 | Part of proof of Lemma E i... |
| cdleme16e 39091 | Part of proof of Lemma E i... |
| cdleme16f 39092 | Part of proof of Lemma E i... |
| cdleme16g 39093 | Part of proof of Lemma E i... |
| cdleme16 39094 | Part of proof of Lemma E i... |
| cdleme17a 39095 | Part of proof of Lemma E i... |
| cdleme17b 39096 | Lemma leading to ~ cdleme1... |
| cdleme17c 39097 | Part of proof of Lemma E i... |
| cdleme17d1 39098 | Part of proof of Lemma E i... |
| cdleme0nex 39099 | Part of proof of Lemma E i... |
| cdleme18a 39100 | Part of proof of Lemma E i... |
| cdleme18b 39101 | Part of proof of Lemma E i... |
| cdleme18c 39102 | Part of proof of Lemma E i... |
| cdleme22gb 39103 | Utility lemma for Lemma E ... |
| cdleme18d 39104 | Part of proof of Lemma E i... |
| cdlemesner 39105 | Part of proof of Lemma E i... |
| cdlemedb 39106 | Part of proof of Lemma E i... |
| cdlemeda 39107 | Part of proof of Lemma E i... |
| cdlemednpq 39108 | Part of proof of Lemma E i... |
| cdlemednuN 39109 | Part of proof of Lemma E i... |
| cdleme20zN 39110 | Part of proof of Lemma E i... |
| cdleme20y 39111 | Part of proof of Lemma E i... |
| cdleme19a 39112 | Part of proof of Lemma E i... |
| cdleme19b 39113 | Part of proof of Lemma E i... |
| cdleme19c 39114 | Part of proof of Lemma E i... |
| cdleme19d 39115 | Part of proof of Lemma E i... |
| cdleme19e 39116 | Part of proof of Lemma E i... |
| cdleme19f 39117 | Part of proof of Lemma E i... |
| cdleme20aN 39118 | Part of proof of Lemma E i... |
| cdleme20bN 39119 | Part of proof of Lemma E i... |
| cdleme20c 39120 | Part of proof of Lemma E i... |
| cdleme20d 39121 | Part of proof of Lemma E i... |
| cdleme20e 39122 | Part of proof of Lemma E i... |
| cdleme20f 39123 | Part of proof of Lemma E i... |
| cdleme20g 39124 | Part of proof of Lemma E i... |
| cdleme20h 39125 | Part of proof of Lemma E i... |
| cdleme20i 39126 | Part of proof of Lemma E i... |
| cdleme20j 39127 | Part of proof of Lemma E i... |
| cdleme20k 39128 | Part of proof of Lemma E i... |
| cdleme20l1 39129 | Part of proof of Lemma E i... |
| cdleme20l2 39130 | Part of proof of Lemma E i... |
| cdleme20l 39131 | Part of proof of Lemma E i... |
| cdleme20m 39132 | Part of proof of Lemma E i... |
| cdleme20 39133 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 39134 | Part of proof of Lemma E i... |
| cdleme21b 39135 | Part of proof of Lemma E i... |
| cdleme21c 39136 | Part of proof of Lemma E i... |
| cdleme21at 39137 | Part of proof of Lemma E i... |
| cdleme21ct 39138 | Part of proof of Lemma E i... |
| cdleme21d 39139 | Part of proof of Lemma E i... |
| cdleme21e 39140 | Part of proof of Lemma E i... |
| cdleme21f 39141 | Part of proof of Lemma E i... |
| cdleme21g 39142 | Part of proof of Lemma E i... |
| cdleme21h 39143 | Part of proof of Lemma E i... |
| cdleme21i 39144 | Part of proof of Lemma E i... |
| cdleme21j 39145 | Combine ~ cdleme20 and ~ c... |
| cdleme21 39146 | Part of proof of Lemma E i... |
| cdleme21k 39147 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 39148 | Part of proof of Lemma E i... |
| cdleme22a 39149 | Part of proof of Lemma E i... |
| cdleme22b 39150 | Part of proof of Lemma E i... |
| cdleme22cN 39151 | Part of proof of Lemma E i... |
| cdleme22d 39152 | Part of proof of Lemma E i... |
| cdleme22e 39153 | Part of proof of Lemma E i... |
| cdleme22eALTN 39154 | Part of proof of Lemma E i... |
| cdleme22f 39155 | Part of proof of Lemma E i... |
| cdleme22f2 39156 | Part of proof of Lemma E i... |
| cdleme22g 39157 | Part of proof of Lemma E i... |
| cdleme23a 39158 | Part of proof of Lemma E i... |
| cdleme23b 39159 | Part of proof of Lemma E i... |
| cdleme23c 39160 | Part of proof of Lemma E i... |
| cdleme24 39161 | Quantified version of ~ cd... |
| cdleme25a 39162 | Lemma for ~ cdleme25b . (... |
| cdleme25b 39163 | Transform ~ cdleme24 . TO... |
| cdleme25c 39164 | Transform ~ cdleme25b . (... |
| cdleme25dN 39165 | Transform ~ cdleme25c . (... |
| cdleme25cl 39166 | Show closure of the unique... |
| cdleme25cv 39167 | Change bound variables in ... |
| cdleme26e 39168 | Part of proof of Lemma E i... |
| cdleme26ee 39169 | Part of proof of Lemma E i... |
| cdleme26eALTN 39170 | Part of proof of Lemma E i... |
| cdleme26fALTN 39171 | Part of proof of Lemma E i... |
| cdleme26f 39172 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 39173 | Part of proof of Lemma E i... |
| cdleme26f2 39174 | Part of proof of Lemma E i... |
| cdleme27cl 39175 | Part of proof of Lemma E i... |
| cdleme27a 39176 | Part of proof of Lemma E i... |
| cdleme27b 39177 | Lemma for ~ cdleme27N . (... |
| cdleme27N 39178 | Part of proof of Lemma E i... |
| cdleme28a 39179 | Lemma for ~ cdleme25b . T... |
| cdleme28b 39180 | Lemma for ~ cdleme25b . T... |
| cdleme28c 39181 | Part of proof of Lemma E i... |
| cdleme28 39182 | Quantified version of ~ cd... |
| cdleme29ex 39183 | Lemma for ~ cdleme29b . (... |
| cdleme29b 39184 | Transform ~ cdleme28 . (C... |
| cdleme29c 39185 | Transform ~ cdleme28b . (... |
| cdleme29cl 39186 | Show closure of the unique... |
| cdleme30a 39187 | Part of proof of Lemma E i... |
| cdleme31so 39188 | Part of proof of Lemma E i... |
| cdleme31sn 39189 | Part of proof of Lemma E i... |
| cdleme31sn1 39190 | Part of proof of Lemma E i... |
| cdleme31se 39191 | Part of proof of Lemma D i... |
| cdleme31se2 39192 | Part of proof of Lemma D i... |
| cdleme31sc 39193 | Part of proof of Lemma E i... |
| cdleme31sde 39194 | Part of proof of Lemma D i... |
| cdleme31snd 39195 | Part of proof of Lemma D i... |
| cdleme31sdnN 39196 | Part of proof of Lemma E i... |
| cdleme31sn1c 39197 | Part of proof of Lemma E i... |
| cdleme31sn2 39198 | Part of proof of Lemma E i... |
| cdleme31fv 39199 | Part of proof of Lemma E i... |
| cdleme31fv1 39200 | Part of proof of Lemma E i... |
| cdleme31fv1s 39201 | Part of proof of Lemma E i... |
| cdleme31fv2 39202 | Part of proof of Lemma E i... |
| cdleme31id 39203 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 39204 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 39205 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 39206 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 39207 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 39208 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 39209 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 39210 | Part of proof of Lemma E i... |
| cdlemefr29exN 39211 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 39212 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 39213 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 39214 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 39215 | TODO fix comment. (Contri... |
| cdlemefr29clN 39216 | Show closure of the unique... |
| cdleme43frv1snN 39217 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 39218 | Part of proof of Lemma E i... |
| cdlemefr32fva1 39219 | Part of proof of Lemma E i... |
| cdlemefr31fv1 39220 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 39221 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 39222 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 39223 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 39224 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 39225 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 39226 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 39227 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 39228 | Show closure of the unique... |
| cdleme43fsv1snlem 39229 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 39230 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 39231 | Part of proof of Lemma E i... |
| cdlemefs32fva1 39232 | Part of proof of Lemma E i... |
| cdlemefs31fv1 39233 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 39234 | Value of f(r) when r is an... |
| cdlemefs44 39235 | Value of f_s(r) when r is ... |
| cdlemefr45 39236 | Value of f(r) when r is an... |
| cdlemefr45e 39237 | Explicit expansion of ~ cd... |
| cdlemefs45 39238 | Value of f_s(r) when r is ... |
| cdlemefs45ee 39239 | Explicit expansion of ~ cd... |
| cdlemefs45eN 39240 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 39241 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 39242 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 39243 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 39244 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 39245 | Show closure of ` [_ R / s... |
| cdleme32fva 39246 | Part of proof of Lemma D i... |
| cdleme32fva1 39247 | Part of proof of Lemma D i... |
| cdleme32fvaw 39248 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 39249 | Part of proof of Lemma D i... |
| cdleme32a 39250 | Part of proof of Lemma D i... |
| cdleme32b 39251 | Part of proof of Lemma D i... |
| cdleme32c 39252 | Part of proof of Lemma D i... |
| cdleme32d 39253 | Part of proof of Lemma D i... |
| cdleme32e 39254 | Part of proof of Lemma D i... |
| cdleme32f 39255 | Part of proof of Lemma D i... |
| cdleme32le 39256 | Part of proof of Lemma D i... |
| cdleme35a 39257 | Part of proof of Lemma E i... |
| cdleme35fnpq 39258 | Part of proof of Lemma E i... |
| cdleme35b 39259 | Part of proof of Lemma E i... |
| cdleme35c 39260 | Part of proof of Lemma E i... |
| cdleme35d 39261 | Part of proof of Lemma E i... |
| cdleme35e 39262 | Part of proof of Lemma E i... |
| cdleme35f 39263 | Part of proof of Lemma E i... |
| cdleme35g 39264 | Part of proof of Lemma E i... |
| cdleme35h 39265 | Part of proof of Lemma E i... |
| cdleme35h2 39266 | Part of proof of Lemma E i... |
| cdleme35sn2aw 39267 | Part of proof of Lemma E i... |
| cdleme35sn3a 39268 | Part of proof of Lemma E i... |
| cdleme36a 39269 | Part of proof of Lemma E i... |
| cdleme36m 39270 | Part of proof of Lemma E i... |
| cdleme37m 39271 | Part of proof of Lemma E i... |
| cdleme38m 39272 | Part of proof of Lemma E i... |
| cdleme38n 39273 | Part of proof of Lemma E i... |
| cdleme39a 39274 | Part of proof of Lemma E i... |
| cdleme39n 39275 | Part of proof of Lemma E i... |
| cdleme40m 39276 | Part of proof of Lemma E i... |
| cdleme40n 39277 | Part of proof of Lemma E i... |
| cdleme40v 39278 | Part of proof of Lemma E i... |
| cdleme40w 39279 | Part of proof of Lemma E i... |
| cdleme42a 39280 | Part of proof of Lemma E i... |
| cdleme42c 39281 | Part of proof of Lemma E i... |
| cdleme42d 39282 | Part of proof of Lemma E i... |
| cdleme41sn3aw 39283 | Part of proof of Lemma E i... |
| cdleme41sn4aw 39284 | Part of proof of Lemma E i... |
| cdleme41snaw 39285 | Part of proof of Lemma E i... |
| cdleme41fva11 39286 | Part of proof of Lemma E i... |
| cdleme42b 39287 | Part of proof of Lemma E i... |
| cdleme42e 39288 | Part of proof of Lemma E i... |
| cdleme42f 39289 | Part of proof of Lemma E i... |
| cdleme42g 39290 | Part of proof of Lemma E i... |
| cdleme42h 39291 | Part of proof of Lemma E i... |
| cdleme42i 39292 | Part of proof of Lemma E i... |
| cdleme42k 39293 | Part of proof of Lemma E i... |
| cdleme42ke 39294 | Part of proof of Lemma E i... |
| cdleme42keg 39295 | Part of proof of Lemma E i... |
| cdleme42mN 39296 | Part of proof of Lemma E i... |
| cdleme42mgN 39297 | Part of proof of Lemma E i... |
| cdleme43aN 39298 | Part of proof of Lemma E i... |
| cdleme43bN 39299 | Lemma for Lemma E in [Craw... |
| cdleme43cN 39300 | Part of proof of Lemma E i... |
| cdleme43dN 39301 | Part of proof of Lemma E i... |
| cdleme46f2g2 39302 | Conversion for ` G ` to re... |
| cdleme46f2g1 39303 | Conversion for ` G ` to re... |
| cdleme17d2 39304 | Part of proof of Lemma E i... |
| cdleme17d3 39305 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 39306 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 39307 | Part of proof of Lemma E i... |
| cdleme48fv 39308 | Part of proof of Lemma D i... |
| cdleme48fvg 39309 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 39310 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 39311 | TODO: fix comment. TODO: ... |
| cdleme48b 39312 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 39313 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 39314 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 39315 | TODO: fix comment. (Contr... |
| cdleme4gfv 39316 | Part of proof of Lemma D i... |
| cdlemeg47b 39317 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 39318 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 39319 | Value of g_s(r) when r is ... |
| cdlemeg49le 39320 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 39321 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 39322 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 39323 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 39324 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 39325 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 39326 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 39327 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 39328 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 39329 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 39330 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 39331 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 39332 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 39333 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 39334 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 39335 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 39336 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 39337 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 39338 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 39339 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 39340 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 39341 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 39342 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 39343 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 39344 | TODO: fix comment. (Contr... |
| cdleme48gfv1 39345 | TODO: fix comment. (Contr... |
| cdleme48gfv 39346 | TODO: fix comment. (Contr... |
| cdleme48fgv 39347 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 39348 | Part of proof of Lemma D i... |
| cdleme50lebi 39349 | Part of proof of Lemma D i... |
| cdleme50eq 39350 | Part of proof of Lemma D i... |
| cdleme50f 39351 | Part of proof of Lemma D i... |
| cdleme50f1 39352 | Part of proof of Lemma D i... |
| cdleme50rnlem 39353 | Part of proof of Lemma D i... |
| cdleme50rn 39354 | Part of proof of Lemma D i... |
| cdleme50f1o 39355 | Part of proof of Lemma D i... |
| cdleme50laut 39356 | Part of proof of Lemma D i... |
| cdleme50ldil 39357 | Part of proof of Lemma D i... |
| cdleme50trn1 39358 | Part of proof that ` F ` i... |
| cdleme50trn2a 39359 | Part of proof that ` F ` i... |
| cdleme50trn2 39360 | Part of proof that ` F ` i... |
| cdleme50trn12 39361 | Part of proof that ` F ` i... |
| cdleme50trn3 39362 | Part of proof that ` F ` i... |
| cdleme50trn123 39363 | Part of proof that ` F ` i... |
| cdleme51finvfvN 39364 | Part of proof of Lemma E i... |
| cdleme51finvN 39365 | Part of proof of Lemma E i... |
| cdleme50ltrn 39366 | Part of proof of Lemma E i... |
| cdleme51finvtrN 39367 | Part of proof of Lemma E i... |
| cdleme50ex 39368 | Part of Lemma E in [Crawle... |
| cdleme 39369 | Lemma E in [Crawley] p. 11... |
| cdlemf1 39370 | Part of Lemma F in [Crawle... |
| cdlemf2 39371 | Part of Lemma F in [Crawle... |
| cdlemf 39372 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 39373 | ~ cdlemf with additional c... |
| cdlemftr3 39374 | Special case of ~ cdlemf s... |
| cdlemftr2 39375 | Special case of ~ cdlemf s... |
| cdlemftr1 39376 | Part of proof of Lemma G o... |
| cdlemftr0 39377 | Special case of ~ cdlemf s... |
| trlord 39378 | The ordering of two Hilber... |
| cdlemg1a 39379 | Shorter expression for ` G... |
| cdlemg1b2 39380 | This theorem can be used t... |
| cdlemg1idlemN 39381 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 39382 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 39383 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 39384 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 39385 | This theorem can be used t... |
| cdlemg1idN 39386 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 39387 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 39388 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 39389 | Version of ~ cdleme51finvt... |
| ltrniotaval 39390 | Value of the unique transl... |
| ltrniotacnvval 39391 | Converse value of the uniq... |
| ltrniotaidvalN 39392 | Value of the unique transl... |
| ltrniotavalbN 39393 | Value of the unique transl... |
| cdlemeiota 39394 | A translation is uniquely ... |
| cdlemg1ci2 39395 | Any function of the form o... |
| cdlemg1cN 39396 | Any translation belongs to... |
| cdlemg1cex 39397 | Any translation is one of ... |
| cdlemg2cN 39398 | Any translation belongs to... |
| cdlemg2dN 39399 | This theorem can be used t... |
| cdlemg2cex 39400 | Any translation is one of ... |
| cdlemg2ce 39401 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 39402 | Part of proof of Lemma E i... |
| cdlemg2fvlem 39403 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 39404 | ~ cdleme42keg with simpler... |
| cdlemg2idN 39405 | Version of ~ cdleme31id wi... |
| cdlemg3a 39406 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 39407 | TODO: Replace this with ~... |
| cdlemg2fv 39408 | Value of a translation in ... |
| cdlemg2fv2 39409 | Value of a translation in ... |
| cdlemg2k 39410 | ~ cdleme42keg with simpler... |
| cdlemg2kq 39411 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 39412 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 39413 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 39414 | TODO: Is there a simpler ... |
| cdlemb3 39415 | Given two atoms not under ... |
| cdlemg7fvbwN 39416 | Properties of a translatio... |
| cdlemg4a 39417 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 39418 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 39419 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 39420 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 39421 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 39422 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 39423 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 39424 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 39425 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 39426 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 39427 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 39428 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 39429 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 39430 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 39431 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 39432 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 39433 | Value of a translation com... |
| cdlemg7aN 39434 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 39435 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 39436 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 39437 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 39438 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 39439 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 39440 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 39441 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 39442 | The triples ` <. P , ( F `... |
| cdlemg9 39443 | The triples ` <. P , ( F `... |
| cdlemg10b 39444 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 39445 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 39446 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 39447 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 39448 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 39449 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 39450 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 39451 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 39452 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 39453 | The triples ` <. P , ( F `... |
| cdlemg12c 39454 | The triples ` <. P , ( F `... |
| cdlemg12d 39455 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 39456 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 39457 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 39458 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 39459 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 39460 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 39461 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 39462 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 39463 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 39464 | Eliminate the ` ( F `` P )... |
| cdlemg15 39465 | Eliminate the ` ( (... |
| cdlemg16 39466 | Part of proof of Lemma G o... |
| cdlemg16ALTN 39467 | This version of ~ cdlemg16... |
| cdlemg16z 39468 | Eliminate ` ( ( F `... |
| cdlemg16zz 39469 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 39470 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 39471 | Part of proof of Lemma G i... |
| cdlemg17dN 39472 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 39473 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 39474 | TODO: fix comment. (Contr... |
| cdlemg17f 39475 | TODO: fix comment. (Contr... |
| cdlemg17g 39476 | TODO: fix comment. (Contr... |
| cdlemg17h 39477 | TODO: fix comment. (Contr... |
| cdlemg17i 39478 | TODO: fix comment. (Contr... |
| cdlemg17ir 39479 | TODO: fix comment. (Contr... |
| cdlemg17j 39480 | TODO: fix comment. (Contr... |
| cdlemg17pq 39481 | Utility theorem for swappi... |
| cdlemg17bq 39482 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 39483 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 39484 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 39485 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 39486 | Part of Lemma G of [Crawle... |
| cdlemg18a 39487 | Show two lines are differe... |
| cdlemg18b 39488 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 39489 | Show two lines intersect a... |
| cdlemg18d 39490 | Show two lines intersect a... |
| cdlemg18 39491 | Show two lines intersect a... |
| cdlemg19a 39492 | Show two lines intersect a... |
| cdlemg19 39493 | Show two lines intersect a... |
| cdlemg20 39494 | Show two lines intersect a... |
| cdlemg21 39495 | Version of cdlemg19 with `... |
| cdlemg22 39496 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 39497 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 39498 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 39499 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 39500 | ~ cdlemg16zz restated for ... |
| cdlemg27a 39501 | For use with case when ` (... |
| cdlemg28a 39502 | Part of proof of Lemma G o... |
| cdlemg31b0N 39503 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 39504 | TODO: Fix comment. (Cont... |
| cdlemg27b 39505 | TODO: Fix comment. (Cont... |
| cdlemg31a 39506 | TODO: fix comment. (Contr... |
| cdlemg31b 39507 | TODO: fix comment. (Contr... |
| cdlemg31c 39508 | Show that when ` N ` is an... |
| cdlemg31d 39509 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 39510 | TODO: Fix comment. (Cont... |
| cdlemg33c0 39511 | TODO: Fix comment. (Cont... |
| cdlemg28b 39512 | Part of proof of Lemma G o... |
| cdlemg28 39513 | Part of proof of Lemma G o... |
| cdlemg29 39514 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 39515 | TODO: Fix comment. (Cont... |
| cdlemg33b 39516 | TODO: Fix comment. (Cont... |
| cdlemg33c 39517 | TODO: Fix comment. (Cont... |
| cdlemg33d 39518 | TODO: Fix comment. (Cont... |
| cdlemg33e 39519 | TODO: Fix comment. (Cont... |
| cdlemg33 39520 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 39521 | Use cdlemg33 to eliminate ... |
| cdlemg35 39522 | TODO: Fix comment. TODO:... |
| cdlemg36 39523 | Use cdlemg35 to eliminate ... |
| cdlemg38 39524 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 39525 | Eliminate ` =/= ` conditio... |
| cdlemg40 39526 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 39527 | Convert ~ cdlemg40 to func... |
| ltrnco 39528 | The composition of two tra... |
| trlcocnv 39529 | Swap the arguments of the ... |
| trlcoabs 39530 | Absorption into a composit... |
| trlcoabs2N 39531 | Absorption of the trace of... |
| trlcoat 39532 | The trace of a composition... |
| trlcocnvat 39533 | Commonly used special case... |
| trlconid 39534 | The composition of two dif... |
| trlcolem 39535 | Lemma for ~ trlco . (Cont... |
| trlco 39536 | The trace of a composition... |
| trlcone 39537 | If two translations have d... |
| cdlemg42 39538 | Part of proof of Lemma G o... |
| cdlemg43 39539 | Part of proof of Lemma G o... |
| cdlemg44a 39540 | Part of proof of Lemma G o... |
| cdlemg44b 39541 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 39542 | Part of proof of Lemma G o... |
| cdlemg47a 39543 | TODO: fix comment. TODO: ... |
| cdlemg46 39544 | Part of proof of Lemma G o... |
| cdlemg47 39545 | Part of proof of Lemma G o... |
| cdlemg48 39546 | Eliminate ` h ` from ~ cdl... |
| ltrncom 39547 | Composition is commutative... |
| ltrnco4 39548 | Rearrange a composition of... |
| trljco 39549 | Trace joined with trace of... |
| trljco2 39550 | Trace joined with trace of... |
| tgrpfset 39553 | The translation group maps... |
| tgrpset 39554 | The translation group for ... |
| tgrpbase 39555 | The base set of the transl... |
| tgrpopr 39556 | The group operation of the... |
| tgrpov 39557 | The group operation value ... |
| tgrpgrplem 39558 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 39559 | The translation group is a... |
| tgrpabl 39560 | The translation group is a... |
| tendofset 39567 | The set of all trace-prese... |
| tendoset 39568 | The set of trace-preservin... |
| istendo 39569 | The predicate "is a trace-... |
| tendotp 39570 | Trace-preserving property ... |
| istendod 39571 | Deduce the predicate "is a... |
| tendof 39572 | Functionality of a trace-p... |
| tendoeq1 39573 | Condition determining equa... |
| tendovalco 39574 | Value of composition of tr... |
| tendocoval 39575 | Value of composition of en... |
| tendocl 39576 | Closure of a trace-preserv... |
| tendoco2 39577 | Distribution of compositio... |
| tendoidcl 39578 | The identity is a trace-pr... |
| tendo1mul 39579 | Multiplicative identity mu... |
| tendo1mulr 39580 | Multiplicative identity mu... |
| tendococl 39581 | The composition of two tra... |
| tendoid 39582 | The identity value of a tr... |
| tendoeq2 39583 | Condition determining equa... |
| tendoplcbv 39584 | Define sum operation for t... |
| tendopl 39585 | Value of endomorphism sum ... |
| tendopl2 39586 | Value of result of endomor... |
| tendoplcl2 39587 | Value of result of endomor... |
| tendoplco2 39588 | Value of result of endomor... |
| tendopltp 39589 | Trace-preserving property ... |
| tendoplcl 39590 | Endomorphism sum is a trac... |
| tendoplcom 39591 | The endomorphism sum opera... |
| tendoplass 39592 | The endomorphism sum opera... |
| tendodi1 39593 | Endomorphism composition d... |
| tendodi2 39594 | Endomorphism composition d... |
| tendo0cbv 39595 | Define additive identity f... |
| tendo02 39596 | Value of additive identity... |
| tendo0co2 39597 | The additive identity trac... |
| tendo0tp 39598 | Trace-preserving property ... |
| tendo0cl 39599 | The additive identity is a... |
| tendo0pl 39600 | Property of the additive i... |
| tendo0plr 39601 | Property of the additive i... |
| tendoicbv 39602 | Define inverse function fo... |
| tendoi 39603 | Value of inverse endomorph... |
| tendoi2 39604 | Value of additive inverse ... |
| tendoicl 39605 | Closure of the additive in... |
| tendoipl 39606 | Property of the additive i... |
| tendoipl2 39607 | Property of the additive i... |
| erngfset 39608 | The division rings on trac... |
| erngset 39609 | The division ring on trace... |
| erngbase 39610 | The base set of the divisi... |
| erngfplus 39611 | Ring addition operation. ... |
| erngplus 39612 | Ring addition operation. ... |
| erngplus2 39613 | Ring addition operation. ... |
| erngfmul 39614 | Ring multiplication operat... |
| erngmul 39615 | Ring addition operation. ... |
| erngfset-rN 39616 | The division rings on trac... |
| erngset-rN 39617 | The division ring on trace... |
| erngbase-rN 39618 | The base set of the divisi... |
| erngfplus-rN 39619 | Ring addition operation. ... |
| erngplus-rN 39620 | Ring addition operation. ... |
| erngplus2-rN 39621 | Ring addition operation. ... |
| erngfmul-rN 39622 | Ring multiplication operat... |
| erngmul-rN 39623 | Ring addition operation. ... |
| cdlemh1 39624 | Part of proof of Lemma H o... |
| cdlemh2 39625 | Part of proof of Lemma H o... |
| cdlemh 39626 | Lemma H of [Crawley] p. 11... |
| cdlemi1 39627 | Part of proof of Lemma I o... |
| cdlemi2 39628 | Part of proof of Lemma I o... |
| cdlemi 39629 | Lemma I of [Crawley] p. 11... |
| cdlemj1 39630 | Part of proof of Lemma J o... |
| cdlemj2 39631 | Part of proof of Lemma J o... |
| cdlemj3 39632 | Part of proof of Lemma J o... |
| tendocan 39633 | Cancellation law: if the v... |
| tendoid0 39634 | A trace-preserving endomor... |
| tendo0mul 39635 | Additive identity multipli... |
| tendo0mulr 39636 | Additive identity multipli... |
| tendo1ne0 39637 | The identity (unity) is no... |
| tendoconid 39638 | The composition (product) ... |
| tendotr 39639 | The trace of the value of ... |
| cdlemk1 39640 | Part of proof of Lemma K o... |
| cdlemk2 39641 | Part of proof of Lemma K o... |
| cdlemk3 39642 | Part of proof of Lemma K o... |
| cdlemk4 39643 | Part of proof of Lemma K o... |
| cdlemk5a 39644 | Part of proof of Lemma K o... |
| cdlemk5 39645 | Part of proof of Lemma K o... |
| cdlemk6 39646 | Part of proof of Lemma K o... |
| cdlemk8 39647 | Part of proof of Lemma K o... |
| cdlemk9 39648 | Part of proof of Lemma K o... |
| cdlemk9bN 39649 | Part of proof of Lemma K o... |
| cdlemki 39650 | Part of proof of Lemma K o... |
| cdlemkvcl 39651 | Part of proof of Lemma K o... |
| cdlemk10 39652 | Part of proof of Lemma K o... |
| cdlemksv 39653 | Part of proof of Lemma K o... |
| cdlemksel 39654 | Part of proof of Lemma K o... |
| cdlemksat 39655 | Part of proof of Lemma K o... |
| cdlemksv2 39656 | Part of proof of Lemma K o... |
| cdlemk7 39657 | Part of proof of Lemma K o... |
| cdlemk11 39658 | Part of proof of Lemma K o... |
| cdlemk12 39659 | Part of proof of Lemma K o... |
| cdlemkoatnle 39660 | Utility lemma. (Contribut... |
| cdlemk13 39661 | Part of proof of Lemma K o... |
| cdlemkole 39662 | Utility lemma. (Contribut... |
| cdlemk14 39663 | Part of proof of Lemma K o... |
| cdlemk15 39664 | Part of proof of Lemma K o... |
| cdlemk16a 39665 | Part of proof of Lemma K o... |
| cdlemk16 39666 | Part of proof of Lemma K o... |
| cdlemk17 39667 | Part of proof of Lemma K o... |
| cdlemk1u 39668 | Part of proof of Lemma K o... |
| cdlemk5auN 39669 | Part of proof of Lemma K o... |
| cdlemk5u 39670 | Part of proof of Lemma K o... |
| cdlemk6u 39671 | Part of proof of Lemma K o... |
| cdlemkj 39672 | Part of proof of Lemma K o... |
| cdlemkuvN 39673 | Part of proof of Lemma K o... |
| cdlemkuel 39674 | Part of proof of Lemma K o... |
| cdlemkuat 39675 | Part of proof of Lemma K o... |
| cdlemkuv2 39676 | Part of proof of Lemma K o... |
| cdlemk18 39677 | Part of proof of Lemma K o... |
| cdlemk19 39678 | Part of proof of Lemma K o... |
| cdlemk7u 39679 | Part of proof of Lemma K o... |
| cdlemk11u 39680 | Part of proof of Lemma K o... |
| cdlemk12u 39681 | Part of proof of Lemma K o... |
| cdlemk21N 39682 | Part of proof of Lemma K o... |
| cdlemk20 39683 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 39684 | Utility lemma. (Contribut... |
| cdlemk13-2N 39685 | Part of proof of Lemma K o... |
| cdlemkole-2N 39686 | Utility lemma. (Contribut... |
| cdlemk14-2N 39687 | Part of proof of Lemma K o... |
| cdlemk15-2N 39688 | Part of proof of Lemma K o... |
| cdlemk16-2N 39689 | Part of proof of Lemma K o... |
| cdlemk17-2N 39690 | Part of proof of Lemma K o... |
| cdlemkj-2N 39691 | Part of proof of Lemma K o... |
| cdlemkuv-2N 39692 | Part of proof of Lemma K o... |
| cdlemkuel-2N 39693 | Part of proof of Lemma K o... |
| cdlemkuv2-2 39694 | Part of proof of Lemma K o... |
| cdlemk18-2N 39695 | Part of proof of Lemma K o... |
| cdlemk19-2N 39696 | Part of proof of Lemma K o... |
| cdlemk7u-2N 39697 | Part of proof of Lemma K o... |
| cdlemk11u-2N 39698 | Part of proof of Lemma K o... |
| cdlemk12u-2N 39699 | Part of proof of Lemma K o... |
| cdlemk21-2N 39700 | Part of proof of Lemma K o... |
| cdlemk20-2N 39701 | Part of proof of Lemma K o... |
| cdlemk22 39702 | Part of proof of Lemma K o... |
| cdlemk30 39703 | Part of proof of Lemma K o... |
| cdlemkuu 39704 | Convert between function a... |
| cdlemk31 39705 | Part of proof of Lemma K o... |
| cdlemk32 39706 | Part of proof of Lemma K o... |
| cdlemkuel-3 39707 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 39708 | Part of proof of Lemma K o... |
| cdlemk18-3N 39709 | Part of proof of Lemma K o... |
| cdlemk22-3 39710 | Part of proof of Lemma K o... |
| cdlemk23-3 39711 | Part of proof of Lemma K o... |
| cdlemk24-3 39712 | Part of proof of Lemma K o... |
| cdlemk25-3 39713 | Part of proof of Lemma K o... |
| cdlemk26b-3 39714 | Part of proof of Lemma K o... |
| cdlemk26-3 39715 | Part of proof of Lemma K o... |
| cdlemk27-3 39716 | Part of proof of Lemma K o... |
| cdlemk28-3 39717 | Part of proof of Lemma K o... |
| cdlemk33N 39718 | Part of proof of Lemma K o... |
| cdlemk34 39719 | Part of proof of Lemma K o... |
| cdlemk29-3 39720 | Part of proof of Lemma K o... |
| cdlemk35 39721 | Part of proof of Lemma K o... |
| cdlemk36 39722 | Part of proof of Lemma K o... |
| cdlemk37 39723 | Part of proof of Lemma K o... |
| cdlemk38 39724 | Part of proof of Lemma K o... |
| cdlemk39 39725 | Part of proof of Lemma K o... |
| cdlemk40 39726 | TODO: fix comment. (Contr... |
| cdlemk40t 39727 | TODO: fix comment. (Contr... |
| cdlemk40f 39728 | TODO: fix comment. (Contr... |
| cdlemk41 39729 | Part of proof of Lemma K o... |
| cdlemkfid1N 39730 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 39731 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 39732 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 39733 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 39734 | TODO: is this useful or sh... |
| cdlemky 39735 | Part of proof of Lemma K o... |
| cdlemkyu 39736 | Convert between function a... |
| cdlemkyuu 39737 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 39738 | Part of proof of Lemma K o... |
| cdlemk19ylem 39739 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 39740 | Part of proof of Lemma K o... |
| cdlemk19y 39741 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 39742 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 39743 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 39744 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 39745 | The value of the tau funct... |
| cdlemk35s 39746 | Substitution version of ~ ... |
| cdlemk35s-id 39747 | Substitution version of ~ ... |
| cdlemk39s 39748 | Substitution version of ~ ... |
| cdlemk39s-id 39749 | Substitution version of ~ ... |
| cdlemk42 39750 | Part of proof of Lemma K o... |
| cdlemk19xlem 39751 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 39752 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 39753 | Part of proof of Lemma K o... |
| cdlemk11tc 39754 | Part of proof of Lemma K o... |
| cdlemk11t 39755 | Part of proof of Lemma K o... |
| cdlemk45 39756 | Part of proof of Lemma K o... |
| cdlemk46 39757 | Part of proof of Lemma K o... |
| cdlemk47 39758 | Part of proof of Lemma K o... |
| cdlemk48 39759 | Part of proof of Lemma K o... |
| cdlemk49 39760 | Part of proof of Lemma K o... |
| cdlemk50 39761 | Part of proof of Lemma K o... |
| cdlemk51 39762 | Part of proof of Lemma K o... |
| cdlemk52 39763 | Part of proof of Lemma K o... |
| cdlemk53a 39764 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 39765 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 39766 | Part of proof of Lemma K o... |
| cdlemk54 39767 | Part of proof of Lemma K o... |
| cdlemk55a 39768 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 39769 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 39770 | Part of proof of Lemma K o... |
| cdlemkyyN 39771 | Part of proof of Lemma K o... |
| cdlemk43N 39772 | Part of proof of Lemma K o... |
| cdlemk35u 39773 | Substitution version of ~ ... |
| cdlemk55u1 39774 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 39775 | Part of proof of Lemma K o... |
| cdlemk39u1 39776 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 39777 | Part of proof of Lemma K o... |
| cdlemk19u1 39778 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 39779 | Part of Lemma K of [Crawle... |
| cdlemk56 39780 | Part of Lemma K of [Crawle... |
| cdlemk19w 39781 | Use a fixed element to eli... |
| cdlemk56w 39782 | Use a fixed element to eli... |
| cdlemk 39783 | Lemma K of [Crawley] p. 11... |
| tendoex 39784 | Generalization of Lemma K ... |
| cdleml1N 39785 | Part of proof of Lemma L o... |
| cdleml2N 39786 | Part of proof of Lemma L o... |
| cdleml3N 39787 | Part of proof of Lemma L o... |
| cdleml4N 39788 | Part of proof of Lemma L o... |
| cdleml5N 39789 | Part of proof of Lemma L o... |
| cdleml6 39790 | Part of proof of Lemma L o... |
| cdleml7 39791 | Part of proof of Lemma L o... |
| cdleml8 39792 | Part of proof of Lemma L o... |
| cdleml9 39793 | Part of proof of Lemma L o... |
| dva1dim 39794 | Two expressions for the 1-... |
| dvhb1dimN 39795 | Two expressions for the 1-... |
| erng1lem 39796 | Value of the endomorphism ... |
| erngdvlem1 39797 | Lemma for ~ eringring . (... |
| erngdvlem2N 39798 | Lemma for ~ eringring . (... |
| erngdvlem3 39799 | Lemma for ~ eringring . (... |
| erngdvlem4 39800 | Lemma for ~ erngdv . (Con... |
| eringring 39801 | An endomorphism ring is a ... |
| erngdv 39802 | An endomorphism ring is a ... |
| erng0g 39803 | The division ring zero of ... |
| erng1r 39804 | The division ring unity of... |
| erngdvlem1-rN 39805 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 39806 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 39807 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 39808 | Lemma for ~ erngdv . (Con... |
| erngring-rN 39809 | An endomorphism ring is a ... |
| erngdv-rN 39810 | An endomorphism ring is a ... |
| dvafset 39813 | The constructed partial ve... |
| dvaset 39814 | The constructed partial ve... |
| dvasca 39815 | The ring base set of the c... |
| dvabase 39816 | The ring base set of the c... |
| dvafplusg 39817 | Ring addition operation fo... |
| dvaplusg 39818 | Ring addition operation fo... |
| dvaplusgv 39819 | Ring addition operation fo... |
| dvafmulr 39820 | Ring multiplication operat... |
| dvamulr 39821 | Ring multiplication operat... |
| dvavbase 39822 | The vectors (vector base s... |
| dvafvadd 39823 | The vector sum operation f... |
| dvavadd 39824 | Ring addition operation fo... |
| dvafvsca 39825 | Ring addition operation fo... |
| dvavsca 39826 | Ring addition operation fo... |
| tendospcl 39827 | Closure of endomorphism sc... |
| tendospass 39828 | Associative law for endomo... |
| tendospdi1 39829 | Forward distributive law f... |
| tendocnv 39830 | Converse of a trace-preser... |
| tendospdi2 39831 | Reverse distributive law f... |
| tendospcanN 39832 | Cancellation law for trace... |
| dvaabl 39833 | The constructed partial ve... |
| dvalveclem 39834 | Lemma for ~ dvalvec . (Co... |
| dvalvec 39835 | The constructed partial ve... |
| dva0g 39836 | The zero vector of partial... |
| diaffval 39839 | The partial isomorphism A ... |
| diafval 39840 | The partial isomorphism A ... |
| diaval 39841 | The partial isomorphism A ... |
| diaelval 39842 | Member of the partial isom... |
| diafn 39843 | Functionality and domain o... |
| diadm 39844 | Domain of the partial isom... |
| diaeldm 39845 | Member of domain of the pa... |
| diadmclN 39846 | A member of domain of the ... |
| diadmleN 39847 | A member of domain of the ... |
| dian0 39848 | The value of the partial i... |
| dia0eldmN 39849 | The lattice zero belongs t... |
| dia1eldmN 39850 | The fiducial hyperplane (t... |
| diass 39851 | The value of the partial i... |
| diael 39852 | A member of the value of t... |
| diatrl 39853 | Trace of a member of the p... |
| diaelrnN 39854 | Any value of the partial i... |
| dialss 39855 | The value of partial isomo... |
| diaord 39856 | The partial isomorphism A ... |
| dia11N 39857 | The partial isomorphism A ... |
| diaf11N 39858 | The partial isomorphism A ... |
| diaclN 39859 | Closure of partial isomorp... |
| diacnvclN 39860 | Closure of partial isomorp... |
| dia0 39861 | The value of the partial i... |
| dia1N 39862 | The value of the partial i... |
| dia1elN 39863 | The largest subspace in th... |
| diaglbN 39864 | Partial isomorphism A of a... |
| diameetN 39865 | Partial isomorphism A of a... |
| diainN 39866 | Inverse partial isomorphis... |
| diaintclN 39867 | The intersection of partia... |
| diasslssN 39868 | The partial isomorphism A ... |
| diassdvaN 39869 | The partial isomorphism A ... |
| dia1dim 39870 | Two expressions for the 1-... |
| dia1dim2 39871 | Two expressions for a 1-di... |
| dia1dimid 39872 | A vector (translation) bel... |
| dia2dimlem1 39873 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 39874 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 39875 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 39876 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 39877 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 39878 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 39879 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 39880 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 39881 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 39882 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 39883 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 39884 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 39885 | Lemma for ~ dia2dim . Eli... |
| dia2dim 39886 | A two-dimensional subspace... |
| dvhfset 39889 | The constructed full vecto... |
| dvhset 39890 | The constructed full vecto... |
| dvhsca 39891 | The ring of scalars of the... |
| dvhbase 39892 | The ring base set of the c... |
| dvhfplusr 39893 | Ring addition operation fo... |
| dvhfmulr 39894 | Ring multiplication operat... |
| dvhmulr 39895 | Ring multiplication operat... |
| dvhvbase 39896 | The vectors (vector base s... |
| dvhelvbasei 39897 | Vector membership in the c... |
| dvhvaddcbv 39898 | Change bound variables to ... |
| dvhvaddval 39899 | The vector sum operation f... |
| dvhfvadd 39900 | The vector sum operation f... |
| dvhvadd 39901 | The vector sum operation f... |
| dvhopvadd 39902 | The vector sum operation f... |
| dvhopvadd2 39903 | The vector sum operation f... |
| dvhvaddcl 39904 | Closure of the vector sum ... |
| dvhvaddcomN 39905 | Commutativity of vector su... |
| dvhvaddass 39906 | Associativity of vector su... |
| dvhvscacbv 39907 | Change bound variables to ... |
| dvhvscaval 39908 | The scalar product operati... |
| dvhfvsca 39909 | Scalar product operation f... |
| dvhvsca 39910 | Scalar product operation f... |
| dvhopvsca 39911 | Scalar product operation f... |
| dvhvscacl 39912 | Closure of the scalar prod... |
| tendoinvcl 39913 | Closure of multiplicative ... |
| tendolinv 39914 | Left multiplicative invers... |
| tendorinv 39915 | Right multiplicative inver... |
| dvhgrp 39916 | The full vector space ` U ... |
| dvhlveclem 39917 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 39918 | The full vector space ` U ... |
| dvhlmod 39919 | The full vector space ` U ... |
| dvh0g 39920 | The zero vector of vector ... |
| dvheveccl 39921 | Properties of a unit vecto... |
| dvhopclN 39922 | Closure of a ` DVecH ` vec... |
| dvhopaddN 39923 | Sum of ` DVecH ` vectors e... |
| dvhopspN 39924 | Scalar product of ` DVecH ... |
| dvhopN 39925 | Decompose a ` DVecH ` vect... |
| dvhopellsm 39926 | Ordered pair membership in... |
| cdlemm10N 39927 | The image of the map ` G `... |
| docaffvalN 39930 | Subspace orthocomplement f... |
| docafvalN 39931 | Subspace orthocomplement f... |
| docavalN 39932 | Subspace orthocomplement f... |
| docaclN 39933 | Closure of subspace orthoc... |
| diaocN 39934 | Value of partial isomorphi... |
| doca2N 39935 | Double orthocomplement of ... |
| doca3N 39936 | Double orthocomplement of ... |
| dvadiaN 39937 | Any closed subspace is a m... |
| diarnN 39938 | Partial isomorphism A maps... |
| diaf1oN 39939 | The partial isomorphism A ... |
| djaffvalN 39942 | Subspace join for ` DVecA ... |
| djafvalN 39943 | Subspace join for ` DVecA ... |
| djavalN 39944 | Subspace join for ` DVecA ... |
| djaclN 39945 | Closure of subspace join f... |
| djajN 39946 | Transfer lattice join to `... |
| dibffval 39949 | The partial isomorphism B ... |
| dibfval 39950 | The partial isomorphism B ... |
| dibval 39951 | The partial isomorphism B ... |
| dibopelvalN 39952 | Member of the partial isom... |
| dibval2 39953 | Value of the partial isomo... |
| dibopelval2 39954 | Member of the partial isom... |
| dibval3N 39955 | Value of the partial isomo... |
| dibelval3 39956 | Member of the partial isom... |
| dibopelval3 39957 | Member of the partial isom... |
| dibelval1st 39958 | Membership in value of the... |
| dibelval1st1 39959 | Membership in value of the... |
| dibelval1st2N 39960 | Membership in value of the... |
| dibelval2nd 39961 | Membership in value of the... |
| dibn0 39962 | The value of the partial i... |
| dibfna 39963 | Functionality and domain o... |
| dibdiadm 39964 | Domain of the partial isom... |
| dibfnN 39965 | Functionality and domain o... |
| dibdmN 39966 | Domain of the partial isom... |
| dibeldmN 39967 | Member of domain of the pa... |
| dibord 39968 | The isomorphism B for a la... |
| dib11N 39969 | The isomorphism B for a la... |
| dibf11N 39970 | The partial isomorphism A ... |
| dibclN 39971 | Closure of partial isomorp... |
| dibvalrel 39972 | The value of partial isomo... |
| dib0 39973 | The value of partial isomo... |
| dib1dim 39974 | Two expressions for the 1-... |
| dibglbN 39975 | Partial isomorphism B of a... |
| dibintclN 39976 | The intersection of partia... |
| dib1dim2 39977 | Two expressions for a 1-di... |
| dibss 39978 | The partial isomorphism B ... |
| diblss 39979 | The value of partial isomo... |
| diblsmopel 39980 | Membership in subspace sum... |
| dicffval 39983 | The partial isomorphism C ... |
| dicfval 39984 | The partial isomorphism C ... |
| dicval 39985 | The partial isomorphism C ... |
| dicopelval 39986 | Membership in value of the... |
| dicelvalN 39987 | Membership in value of the... |
| dicval2 39988 | The partial isomorphism C ... |
| dicelval3 39989 | Member of the partial isom... |
| dicopelval2 39990 | Membership in value of the... |
| dicelval2N 39991 | Membership in value of the... |
| dicfnN 39992 | Functionality and domain o... |
| dicdmN 39993 | Domain of the partial isom... |
| dicvalrelN 39994 | The value of partial isomo... |
| dicssdvh 39995 | The partial isomorphism C ... |
| dicelval1sta 39996 | Membership in value of the... |
| dicelval1stN 39997 | Membership in value of the... |
| dicelval2nd 39998 | Membership in value of the... |
| dicvaddcl 39999 | Membership in value of the... |
| dicvscacl 40000 | Membership in value of the... |
| dicn0 40001 | The value of the partial i... |
| diclss 40002 | The value of partial isomo... |
| diclspsn 40003 | The value of isomorphism C... |
| cdlemn2 40004 | Part of proof of Lemma N o... |
| cdlemn2a 40005 | Part of proof of Lemma N o... |
| cdlemn3 40006 | Part of proof of Lemma N o... |
| cdlemn4 40007 | Part of proof of Lemma N o... |
| cdlemn4a 40008 | Part of proof of Lemma N o... |
| cdlemn5pre 40009 | Part of proof of Lemma N o... |
| cdlemn5 40010 | Part of proof of Lemma N o... |
| cdlemn6 40011 | Part of proof of Lemma N o... |
| cdlemn7 40012 | Part of proof of Lemma N o... |
| cdlemn8 40013 | Part of proof of Lemma N o... |
| cdlemn9 40014 | Part of proof of Lemma N o... |
| cdlemn10 40015 | Part of proof of Lemma N o... |
| cdlemn11a 40016 | Part of proof of Lemma N o... |
| cdlemn11b 40017 | Part of proof of Lemma N o... |
| cdlemn11c 40018 | Part of proof of Lemma N o... |
| cdlemn11pre 40019 | Part of proof of Lemma N o... |
| cdlemn11 40020 | Part of proof of Lemma N o... |
| cdlemn 40021 | Lemma N of [Crawley] p. 12... |
| dihordlem6 40022 | Part of proof of Lemma N o... |
| dihordlem7 40023 | Part of proof of Lemma N o... |
| dihordlem7b 40024 | Part of proof of Lemma N o... |
| dihjustlem 40025 | Part of proof after Lemma ... |
| dihjust 40026 | Part of proof after Lemma ... |
| dihord1 40027 | Part of proof after Lemma ... |
| dihord2a 40028 | Part of proof after Lemma ... |
| dihord2b 40029 | Part of proof after Lemma ... |
| dihord2cN 40030 | Part of proof after Lemma ... |
| dihord11b 40031 | Part of proof after Lemma ... |
| dihord10 40032 | Part of proof after Lemma ... |
| dihord11c 40033 | Part of proof after Lemma ... |
| dihord2pre 40034 | Part of proof after Lemma ... |
| dihord2pre2 40035 | Part of proof after Lemma ... |
| dihord2 40036 | Part of proof after Lemma ... |
| dihffval 40039 | The isomorphism H for a la... |
| dihfval 40040 | Isomorphism H for a lattic... |
| dihval 40041 | Value of isomorphism H for... |
| dihvalc 40042 | Value of isomorphism H for... |
| dihlsscpre 40043 | Closure of isomorphism H f... |
| dihvalcqpre 40044 | Value of isomorphism H for... |
| dihvalcq 40045 | Value of isomorphism H for... |
| dihvalb 40046 | Value of isomorphism H for... |
| dihopelvalbN 40047 | Ordered pair member of the... |
| dihvalcqat 40048 | Value of isomorphism H for... |
| dih1dimb 40049 | Two expressions for a 1-di... |
| dih1dimb2 40050 | Isomorphism H at an atom u... |
| dih1dimc 40051 | Isomorphism H at an atom n... |
| dib2dim 40052 | Extend ~ dia2dim to partia... |
| dih2dimb 40053 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 40054 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 40055 | Ordered pair member of the... |
| dihvalcq2 40056 | Value of isomorphism H for... |
| dihopelvalcpre 40057 | Member of value of isomorp... |
| dihopelvalc 40058 | Member of value of isomorp... |
| dihlss 40059 | The value of isomorphism H... |
| dihss 40060 | The value of isomorphism H... |
| dihssxp 40061 | An isomorphism H value is ... |
| dihopcl 40062 | Closure of an ordered pair... |
| xihopellsmN 40063 | Ordered pair membership in... |
| dihopellsm 40064 | Ordered pair membership in... |
| dihord6apre 40065 | Part of proof that isomorp... |
| dihord3 40066 | The isomorphism H for a la... |
| dihord4 40067 | The isomorphism H for a la... |
| dihord5b 40068 | Part of proof that isomorp... |
| dihord6b 40069 | Part of proof that isomorp... |
| dihord6a 40070 | Part of proof that isomorp... |
| dihord5apre 40071 | Part of proof that isomorp... |
| dihord5a 40072 | Part of proof that isomorp... |
| dihord 40073 | The isomorphism H is order... |
| dih11 40074 | The isomorphism H is one-t... |
| dihf11lem 40075 | Functionality of the isomo... |
| dihf11 40076 | The isomorphism H for a la... |
| dihfn 40077 | Functionality and domain o... |
| dihdm 40078 | Domain of isomorphism H. (... |
| dihcl 40079 | Closure of isomorphism H. ... |
| dihcnvcl 40080 | Closure of isomorphism H c... |
| dihcnvid1 40081 | The converse isomorphism o... |
| dihcnvid2 40082 | The isomorphism of a conve... |
| dihcnvord 40083 | Ordering property for conv... |
| dihcnv11 40084 | The converse of isomorphis... |
| dihsslss 40085 | The isomorphism H maps to ... |
| dihrnlss 40086 | The isomorphism H maps to ... |
| dihrnss 40087 | The isomorphism H maps to ... |
| dihvalrel 40088 | The value of isomorphism H... |
| dih0 40089 | The value of isomorphism H... |
| dih0bN 40090 | A lattice element is zero ... |
| dih0vbN 40091 | A vector is zero iff its s... |
| dih0cnv 40092 | The isomorphism H converse... |
| dih0rn 40093 | The zero subspace belongs ... |
| dih0sb 40094 | A subspace is zero iff the... |
| dih1 40095 | The value of isomorphism H... |
| dih1rn 40096 | The full vector space belo... |
| dih1cnv 40097 | The isomorphism H converse... |
| dihwN 40098 | Value of isomorphism H at ... |
| dihmeetlem1N 40099 | Isomorphism H of a conjunc... |
| dihglblem5apreN 40100 | A conjunction property of ... |
| dihglblem5aN 40101 | A conjunction property of ... |
| dihglblem2aN 40102 | Lemma for isomorphism H of... |
| dihglblem2N 40103 | The GLB of a set of lattic... |
| dihglblem3N 40104 | Isomorphism H of a lattice... |
| dihglblem3aN 40105 | Isomorphism H of a lattice... |
| dihglblem4 40106 | Isomorphism H of a lattice... |
| dihglblem5 40107 | Isomorphism H of a lattice... |
| dihmeetlem2N 40108 | Isomorphism H of a conjunc... |
| dihglbcpreN 40109 | Isomorphism H of a lattice... |
| dihglbcN 40110 | Isomorphism H of a lattice... |
| dihmeetcN 40111 | Isomorphism H of a lattice... |
| dihmeetbN 40112 | Isomorphism H of a lattice... |
| dihmeetbclemN 40113 | Lemma for isomorphism H of... |
| dihmeetlem3N 40114 | Lemma for isomorphism H of... |
| dihmeetlem4preN 40115 | Lemma for isomorphism H of... |
| dihmeetlem4N 40116 | Lemma for isomorphism H of... |
| dihmeetlem5 40117 | Part of proof that isomorp... |
| dihmeetlem6 40118 | Lemma for isomorphism H of... |
| dihmeetlem7N 40119 | Lemma for isomorphism H of... |
| dihjatc1 40120 | Lemma for isomorphism H of... |
| dihjatc2N 40121 | Isomorphism H of join with... |
| dihjatc3 40122 | Isomorphism H of join with... |
| dihmeetlem8N 40123 | Lemma for isomorphism H of... |
| dihmeetlem9N 40124 | Lemma for isomorphism H of... |
| dihmeetlem10N 40125 | Lemma for isomorphism H of... |
| dihmeetlem11N 40126 | Lemma for isomorphism H of... |
| dihmeetlem12N 40127 | Lemma for isomorphism H of... |
| dihmeetlem13N 40128 | Lemma for isomorphism H of... |
| dihmeetlem14N 40129 | Lemma for isomorphism H of... |
| dihmeetlem15N 40130 | Lemma for isomorphism H of... |
| dihmeetlem16N 40131 | Lemma for isomorphism H of... |
| dihmeetlem17N 40132 | Lemma for isomorphism H of... |
| dihmeetlem18N 40133 | Lemma for isomorphism H of... |
| dihmeetlem19N 40134 | Lemma for isomorphism H of... |
| dihmeetlem20N 40135 | Lemma for isomorphism H of... |
| dihmeetALTN 40136 | Isomorphism H of a lattice... |
| dih1dimatlem0 40137 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 40138 | Lemma for ~ dih1dimat . (... |
| dih1dimat 40139 | Any 1-dimensional subspace... |
| dihlsprn 40140 | The span of a vector belon... |
| dihlspsnssN 40141 | A subspace included in a 1... |
| dihlspsnat 40142 | The inverse isomorphism H ... |
| dihatlat 40143 | The isomorphism H of an at... |
| dihat 40144 | There exists at least one ... |
| dihpN 40145 | The value of isomorphism H... |
| dihlatat 40146 | The reverse isomorphism H ... |
| dihatexv 40147 | There is a nonzero vector ... |
| dihatexv2 40148 | There is a nonzero vector ... |
| dihglblem6 40149 | Isomorphism H of a lattice... |
| dihglb 40150 | Isomorphism H of a lattice... |
| dihglb2 40151 | Isomorphism H of a lattice... |
| dihmeet 40152 | Isomorphism H of a lattice... |
| dihintcl 40153 | The intersection of closed... |
| dihmeetcl 40154 | Closure of closed subspace... |
| dihmeet2 40155 | Reverse isomorphism H of a... |
| dochffval 40158 | Subspace orthocomplement f... |
| dochfval 40159 | Subspace orthocomplement f... |
| dochval 40160 | Subspace orthocomplement f... |
| dochval2 40161 | Subspace orthocomplement f... |
| dochcl 40162 | Closure of subspace orthoc... |
| dochlss 40163 | A subspace orthocomplement... |
| dochssv 40164 | A subspace orthocomplement... |
| dochfN 40165 | Domain and codomain of the... |
| dochvalr 40166 | Orthocomplement of a close... |
| doch0 40167 | Orthocomplement of the zer... |
| doch1 40168 | Orthocomplement of the uni... |
| dochoc0 40169 | The zero subspace is close... |
| dochoc1 40170 | The unit subspace (all vec... |
| dochvalr2 40171 | Orthocomplement of a close... |
| dochvalr3 40172 | Orthocomplement of a close... |
| doch2val2 40173 | Double orthocomplement for... |
| dochss 40174 | Subset law for orthocomple... |
| dochocss 40175 | Double negative law for or... |
| dochoc 40176 | Double negative law for or... |
| dochsscl 40177 | If a set of vectors is inc... |
| dochoccl 40178 | A set of vectors is closed... |
| dochord 40179 | Ordering law for orthocomp... |
| dochord2N 40180 | Ordering law for orthocomp... |
| dochord3 40181 | Ordering law for orthocomp... |
| doch11 40182 | Orthocomplement is one-to-... |
| dochsordN 40183 | Strict ordering law for or... |
| dochn0nv 40184 | An orthocomplement is nonz... |
| dihoml4c 40185 | Version of ~ dihoml4 with ... |
| dihoml4 40186 | Orthomodular law for const... |
| dochspss 40187 | The span of a set of vecto... |
| dochocsp 40188 | The span of an orthocomple... |
| dochspocN 40189 | The span of an orthocomple... |
| dochocsn 40190 | The double orthocomplement... |
| dochsncom 40191 | Swap vectors in an orthoco... |
| dochsat 40192 | The double orthocomplement... |
| dochshpncl 40193 | If a hyperplane is not clo... |
| dochlkr 40194 | Equivalent conditions for ... |
| dochkrshp 40195 | The closure of a kernel is... |
| dochkrshp2 40196 | Properties of the closure ... |
| dochkrshp3 40197 | Properties of the closure ... |
| dochkrshp4 40198 | Properties of the closure ... |
| dochdmj1 40199 | De Morgan-like law for sub... |
| dochnoncon 40200 | Law of noncontradiction. ... |
| dochnel2 40201 | A nonzero member of a subs... |
| dochnel 40202 | A nonzero vector doesn't b... |
| djhffval 40205 | Subspace join for ` DVecH ... |
| djhfval 40206 | Subspace join for ` DVecH ... |
| djhval 40207 | Subspace join for ` DVecH ... |
| djhval2 40208 | Value of subspace join for... |
| djhcl 40209 | Closure of subspace join f... |
| djhlj 40210 | Transfer lattice join to `... |
| djhljjN 40211 | Lattice join in terms of `... |
| djhjlj 40212 | ` DVecH ` vector space clo... |
| djhj 40213 | ` DVecH ` vector space clo... |
| djhcom 40214 | Subspace join commutes. (... |
| djhspss 40215 | Subspace span of union is ... |
| djhsumss 40216 | Subspace sum is a subset o... |
| dihsumssj 40217 | The subspace sum of two is... |
| djhunssN 40218 | Subspace union is a subset... |
| dochdmm1 40219 | De Morgan-like law for clo... |
| djhexmid 40220 | Excluded middle property o... |
| djh01 40221 | Closed subspace join with ... |
| djh02 40222 | Closed subspace join with ... |
| djhlsmcl 40223 | A closed subspace sum equa... |
| djhcvat42 40224 | A covering property. ( ~ ... |
| dihjatb 40225 | Isomorphism H of lattice j... |
| dihjatc 40226 | Isomorphism H of lattice j... |
| dihjatcclem1 40227 | Lemma for isomorphism H of... |
| dihjatcclem2 40228 | Lemma for isomorphism H of... |
| dihjatcclem3 40229 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 40230 | Lemma for isomorphism H of... |
| dihjatcc 40231 | Isomorphism H of lattice j... |
| dihjat 40232 | Isomorphism H of lattice j... |
| dihprrnlem1N 40233 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 40234 | Lemma for ~ dihprrn . (Co... |
| dihprrn 40235 | The span of a vector pair ... |
| djhlsmat 40236 | The sum of two subspace at... |
| dihjat1lem 40237 | Subspace sum of a closed s... |
| dihjat1 40238 | Subspace sum of a closed s... |
| dihsmsprn 40239 | Subspace sum of a closed s... |
| dihjat2 40240 | The subspace sum of a clos... |
| dihjat3 40241 | Isomorphism H of lattice j... |
| dihjat4 40242 | Transfer the subspace sum ... |
| dihjat6 40243 | Transfer the subspace sum ... |
| dihsmsnrn 40244 | The subspace sum of two si... |
| dihsmatrn 40245 | The subspace sum of a clos... |
| dihjat5N 40246 | Transfer lattice join with... |
| dvh4dimat 40247 | There is an atom that is o... |
| dvh3dimatN 40248 | There is an atom that is o... |
| dvh2dimatN 40249 | Given an atom, there exist... |
| dvh1dimat 40250 | There exists an atom. (Co... |
| dvh1dim 40251 | There exists a nonzero vec... |
| dvh4dimlem 40252 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 40253 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 40254 | There is a vector that is ... |
| dvh3dim 40255 | There is a vector that is ... |
| dvh4dimN 40256 | There is a vector that is ... |
| dvh3dim2 40257 | There is a vector that is ... |
| dvh3dim3N 40258 | There is a vector that is ... |
| dochsnnz 40259 | The orthocomplement of a s... |
| dochsatshp 40260 | The orthocomplement of a s... |
| dochsatshpb 40261 | The orthocomplement of a s... |
| dochsnshp 40262 | The orthocomplement of a n... |
| dochshpsat 40263 | A hyperplane is closed iff... |
| dochkrsat 40264 | The orthocomplement of a k... |
| dochkrsat2 40265 | The orthocomplement of a k... |
| dochsat0 40266 | The orthocomplement of a k... |
| dochkrsm 40267 | The subspace sum of a clos... |
| dochexmidat 40268 | Special case of excluded m... |
| dochexmidlem1 40269 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 40270 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 40271 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 40272 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 40273 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 40274 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 40275 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 40276 | Lemma for ~ dochexmid . T... |
| dochexmid 40277 | Excluded middle law for cl... |
| dochsnkrlem1 40278 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 40279 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 40280 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 40281 | A (closed) kernel expresse... |
| dochsnkr2 40282 | Kernel of the explicit fun... |
| dochsnkr2cl 40283 | The ` X ` determining func... |
| dochflcl 40284 | Closure of the explicit fu... |
| dochfl1 40285 | The value of the explicit ... |
| dochfln0 40286 | The value of a functional ... |
| dochkr1 40287 | A nonzero functional has a... |
| dochkr1OLDN 40288 | A nonzero functional has a... |
| lpolsetN 40291 | The set of polarities of a... |
| islpolN 40292 | The predicate "is a polari... |
| islpoldN 40293 | Properties that determine ... |
| lpolfN 40294 | Functionality of a polarit... |
| lpolvN 40295 | The polarity of the whole ... |
| lpolconN 40296 | Contraposition property of... |
| lpolsatN 40297 | The polarity of an atomic ... |
| lpolpolsatN 40298 | Property of a polarity. (... |
| dochpolN 40299 | The subspace orthocompleme... |
| lcfl1lem 40300 | Property of a functional w... |
| lcfl1 40301 | Property of a functional w... |
| lcfl2 40302 | Property of a functional w... |
| lcfl3 40303 | Property of a functional w... |
| lcfl4N 40304 | Property of a functional w... |
| lcfl5 40305 | Property of a functional w... |
| lcfl5a 40306 | Property of a functional w... |
| lcfl6lem 40307 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 40308 | Lemma for ~ lcfl7N . If t... |
| lcfl6 40309 | Property of a functional w... |
| lcfl7N 40310 | Property of a functional w... |
| lcfl8 40311 | Property of a functional w... |
| lcfl8a 40312 | Property of a functional w... |
| lcfl8b 40313 | Property of a nonzero func... |
| lcfl9a 40314 | Property implying that a f... |
| lclkrlem1 40315 | The set of functionals hav... |
| lclkrlem2a 40316 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 40317 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 40318 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 40319 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 40320 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 40321 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 40322 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 40323 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 40324 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 40325 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 40326 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 40327 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 40328 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 40329 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 40330 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 40331 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 40332 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 40333 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 40334 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 40335 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 40336 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 40337 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 40338 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 40339 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 40340 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 40341 | The set of functionals hav... |
| lclkr 40342 | The set of functionals wit... |
| lcfls1lem 40343 | Property of a functional w... |
| lcfls1N 40344 | Property of a functional w... |
| lcfls1c 40345 | Property of a functional w... |
| lclkrslem1 40346 | The set of functionals hav... |
| lclkrslem2 40347 | The set of functionals hav... |
| lclkrs 40348 | The set of functionals hav... |
| lclkrs2 40349 | The set of functionals wit... |
| lcfrvalsnN 40350 | Reconstruction from the du... |
| lcfrlem1 40351 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 40352 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 40353 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 40354 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 40355 | Lemma for ~ lcfr . The se... |
| lcfrlem6 40356 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 40357 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 40358 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 40359 | Lemma for ~ lcf1o . (This... |
| lcf1o 40360 | Define a function ` J ` th... |
| lcfrlem10 40361 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 40362 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 40363 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 40364 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 40365 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 40366 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 40367 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 40368 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 40369 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 40370 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 40371 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 40372 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 40373 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 40374 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 40375 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 40376 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 40377 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 40378 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 40379 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 40380 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 40381 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 40382 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 40383 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 40384 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 40385 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 40386 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 40387 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 40388 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 40389 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 40390 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 40391 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 40392 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 40393 | Lemma for ~ lcfr . Elimin... |
| lcfr 40394 | Reconstruction of a subspa... |
| lcdfval 40397 | Dual vector space of funct... |
| lcdval 40398 | Dual vector space of funct... |
| lcdval2 40399 | Dual vector space of funct... |
| lcdlvec 40400 | The dual vector space of f... |
| lcdlmod 40401 | The dual vector space of f... |
| lcdvbase 40402 | Vector base set of a dual ... |
| lcdvbasess 40403 | The vector base set of the... |
| lcdvbaselfl 40404 | A vector in the base set o... |
| lcdvbasecl 40405 | Closure of the value of a ... |
| lcdvadd 40406 | Vector addition for the cl... |
| lcdvaddval 40407 | The value of the value of ... |
| lcdsca 40408 | The ring of scalars of the... |
| lcdsbase 40409 | Base set of scalar ring fo... |
| lcdsadd 40410 | Scalar addition for the cl... |
| lcdsmul 40411 | Scalar multiplication for ... |
| lcdvs 40412 | Scalar product for the clo... |
| lcdvsval 40413 | Value of scalar product op... |
| lcdvscl 40414 | The scalar product operati... |
| lcdlssvscl 40415 | Closure of scalar product ... |
| lcdvsass 40416 | Associative law for scalar... |
| lcd0 40417 | The zero scalar of the clo... |
| lcd1 40418 | The unit scalar of the clo... |
| lcdneg 40419 | The unit scalar of the clo... |
| lcd0v 40420 | The zero functional in the... |
| lcd0v2 40421 | The zero functional in the... |
| lcd0vvalN 40422 | Value of the zero function... |
| lcd0vcl 40423 | Closure of the zero functi... |
| lcd0vs 40424 | A scalar zero times a func... |
| lcdvs0N 40425 | A scalar times the zero fu... |
| lcdvsub 40426 | The value of vector subtra... |
| lcdvsubval 40427 | The value of the value of ... |
| lcdlss 40428 | Subspaces of a dual vector... |
| lcdlss2N 40429 | Subspaces of a dual vector... |
| lcdlsp 40430 | Span in the set of functio... |
| lcdlkreqN 40431 | Colinear functionals have ... |
| lcdlkreq2N 40432 | Colinear functionals have ... |
| mapdffval 40435 | Projectivity from vector s... |
| mapdfval 40436 | Projectivity from vector s... |
| mapdval 40437 | Value of projectivity from... |
| mapdvalc 40438 | Value of projectivity from... |
| mapdval2N 40439 | Value of projectivity from... |
| mapdval3N 40440 | Value of projectivity from... |
| mapdval4N 40441 | Value of projectivity from... |
| mapdval5N 40442 | Value of projectivity from... |
| mapdordlem1a 40443 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 40444 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 40445 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 40446 | Lemma for ~ mapdord . Ord... |
| mapdord 40447 | Ordering property of the m... |
| mapd11 40448 | The map defined by ~ df-ma... |
| mapddlssN 40449 | The mapping of a subspace ... |
| mapdsn 40450 | Value of the map defined b... |
| mapdsn2 40451 | Value of the map defined b... |
| mapdsn3 40452 | Value of the map defined b... |
| mapd1dim2lem1N 40453 | Value of the map defined b... |
| mapdrvallem2 40454 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 40455 | Lemma for ~ mapdrval . (C... |
| mapdrval 40456 | Given a dual subspace ` R ... |
| mapd1o 40457 | The map defined by ~ df-ma... |
| mapdrn 40458 | Range of the map defined b... |
| mapdunirnN 40459 | Union of the range of the ... |
| mapdrn2 40460 | Range of the map defined b... |
| mapdcnvcl 40461 | Closure of the converse of... |
| mapdcl 40462 | Closure the value of the m... |
| mapdcnvid1N 40463 | Converse of the value of t... |
| mapdsord 40464 | Strong ordering property o... |
| mapdcl2 40465 | The mapping of a subspace ... |
| mapdcnvid2 40466 | Value of the converse of t... |
| mapdcnvordN 40467 | Ordering property of the c... |
| mapdcnv11N 40468 | The converse of the map de... |
| mapdcv 40469 | Covering property of the c... |
| mapdincl 40470 | Closure of dual subspace i... |
| mapdin 40471 | Subspace intersection is p... |
| mapdlsmcl 40472 | Closure of dual subspace s... |
| mapdlsm 40473 | Subspace sum is preserved ... |
| mapd0 40474 | Projectivity map of the ze... |
| mapdcnvatN 40475 | Atoms are preserved by the... |
| mapdat 40476 | Atoms are preserved by the... |
| mapdspex 40477 | The map of a span equals t... |
| mapdn0 40478 | Transfer nonzero property ... |
| mapdncol 40479 | Transfer non-colinearity f... |
| mapdindp 40480 | Transfer (part of) vector ... |
| mapdpglem1 40481 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 40482 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 40483 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 40484 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 40485 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 40486 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 40487 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 40488 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 40489 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 40490 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 40491 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 40492 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 40493 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 40494 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 40495 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 40496 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 40497 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 40498 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 40499 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 40500 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 40501 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 40502 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 40503 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 40504 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 40505 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 40506 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 40507 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 40508 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 40509 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 40510 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 40511 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 40512 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 40513 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 40514 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 40515 | Part 1 of proof of the fir... |
| baerlem3lem1 40516 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 40517 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 40518 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 40519 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 40520 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 40521 | Lemma for ~ baerlem5b . (... |
| baerlem3 40522 | An equality that holds whe... |
| baerlem5a 40523 | An equality that holds whe... |
| baerlem5b 40524 | An equality that holds whe... |
| baerlem5amN 40525 | An equality that holds whe... |
| baerlem5bmN 40526 | An equality that holds whe... |
| baerlem5abmN 40527 | An equality that holds whe... |
| mapdindp0 40528 | Vector independence lemma.... |
| mapdindp1 40529 | Vector independence lemma.... |
| mapdindp2 40530 | Vector independence lemma.... |
| mapdindp3 40531 | Vector independence lemma.... |
| mapdindp4 40532 | Vector independence lemma.... |
| mapdhval 40533 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 40534 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 40535 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 40536 | Lemmma for ~~? mapdh . (C... |
| mapdheq 40537 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 40538 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 40539 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 40540 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 40541 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 40542 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 40543 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 40544 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 40545 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 40546 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 40547 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 40548 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 40549 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 40550 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 40551 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 40552 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 40553 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 40554 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 40555 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 40556 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 40557 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 40558 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 40559 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 40560 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 40561 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 40562 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 40563 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 40564 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 40567 | Map from nonzero vectors t... |
| hvmapfval 40568 | Map from nonzero vectors t... |
| hvmapval 40569 | Value of map from nonzero ... |
| hvmapvalvalN 40570 | Value of value of map (i.e... |
| hvmapidN 40571 | The value of the vector to... |
| hvmap1o 40572 | The vector to functional m... |
| hvmapclN 40573 | Closure of the vector to f... |
| hvmap1o2 40574 | The vector to functional m... |
| hvmapcl2 40575 | Closure of the vector to f... |
| hvmaplfl 40576 | The vector to functional m... |
| hvmaplkr 40577 | Kernel of the vector to fu... |
| mapdhvmap 40578 | Relationship between ` map... |
| lspindp5 40579 | Obtain an independent vect... |
| hdmaplem1 40580 | Lemma to convert a frequen... |
| hdmaplem2N 40581 | Lemma to convert a frequen... |
| hdmaplem3 40582 | Lemma to convert a frequen... |
| hdmaplem4 40583 | Lemma to convert a frequen... |
| mapdh8a 40584 | Part of Part (8) in [Baer]... |
| mapdh8aa 40585 | Part of Part (8) in [Baer]... |
| mapdh8ab 40586 | Part of Part (8) in [Baer]... |
| mapdh8ac 40587 | Part of Part (8) in [Baer]... |
| mapdh8ad 40588 | Part of Part (8) in [Baer]... |
| mapdh8b 40589 | Part of Part (8) in [Baer]... |
| mapdh8c 40590 | Part of Part (8) in [Baer]... |
| mapdh8d0N 40591 | Part of Part (8) in [Baer]... |
| mapdh8d 40592 | Part of Part (8) in [Baer]... |
| mapdh8e 40593 | Part of Part (8) in [Baer]... |
| mapdh8g 40594 | Part of Part (8) in [Baer]... |
| mapdh8i 40595 | Part of Part (8) in [Baer]... |
| mapdh8j 40596 | Part of Part (8) in [Baer]... |
| mapdh8 40597 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 40598 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 40599 | Lemma for part (9) in [Bae... |
| hdmap1ffval 40604 | Preliminary map from vecto... |
| hdmap1fval 40605 | Preliminary map from vecto... |
| hdmap1vallem 40606 | Value of preliminary map f... |
| hdmap1val 40607 | Value of preliminary map f... |
| hdmap1val0 40608 | Value of preliminary map f... |
| hdmap1val2 40609 | Value of preliminary map f... |
| hdmap1eq 40610 | The defining equation for ... |
| hdmap1cbv 40611 | Frequently used lemma to c... |
| hdmap1valc 40612 | Connect the value of the p... |
| hdmap1cl 40613 | Convert closure theorem ~ ... |
| hdmap1eq2 40614 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 40615 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 40616 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 40617 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 40618 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 40619 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 40620 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 40621 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 40622 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 40623 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 40624 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 40625 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 40626 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 40627 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 40628 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 40629 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 40630 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 40631 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 40632 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 40633 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 40634 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 40635 | Map from vectors to functi... |
| hdmapfval 40636 | Map from vectors to functi... |
| hdmapval 40637 | Value of map from vectors ... |
| hdmapfnN 40638 | Functionality of map from ... |
| hdmapcl 40639 | Closure of map from vector... |
| hdmapval2lem 40640 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 40641 | Value of map from vectors ... |
| hdmapval0 40642 | Value of map from vectors ... |
| hdmapeveclem 40643 | Lemma for ~ hdmapevec . T... |
| hdmapevec 40644 | Value of map from vectors ... |
| hdmapevec2 40645 | The inner product of the r... |
| hdmapval3lemN 40646 | Value of map from vectors ... |
| hdmapval3N 40647 | Value of map from vectors ... |
| hdmap10lem 40648 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 40649 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 40650 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 40651 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 40652 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 40653 | Part of proof of part 12 i... |
| hdmapnzcl 40654 | Nonzero vector closure of ... |
| hdmapneg 40655 | Part of proof of part 12 i... |
| hdmapsub 40656 | Part of proof of part 12 i... |
| hdmap11 40657 | Part of proof of part 12 i... |
| hdmaprnlem1N 40658 | Part of proof of part 12 i... |
| hdmaprnlem3N 40659 | Part of proof of part 12 i... |
| hdmaprnlem3uN 40660 | Part of proof of part 12 i... |
| hdmaprnlem4tN 40661 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 40662 | Part of proof of part 12 i... |
| hdmaprnlem6N 40663 | Part of proof of part 12 i... |
| hdmaprnlem7N 40664 | Part of proof of part 12 i... |
| hdmaprnlem8N 40665 | Part of proof of part 12 i... |
| hdmaprnlem9N 40666 | Part of proof of part 12 i... |
| hdmaprnlem3eN 40667 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 40668 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 40669 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 40670 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 40671 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 40672 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 40673 | Part of proof of part 12 i... |
| hdmapf1oN 40674 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 40675 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 40676 | Prior to part 14 in [Baer]... |
| hdmap14lem1 40677 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 40678 | Prior to part 14 in [Baer]... |
| hdmap14lem3 40679 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 40680 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 40681 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 40682 | Case where ` F ` is zero. ... |
| hdmap14lem7 40683 | Combine cases of ` F ` . ... |
| hdmap14lem8 40684 | Part of proof of part 14 i... |
| hdmap14lem9 40685 | Part of proof of part 14 i... |
| hdmap14lem10 40686 | Part of proof of part 14 i... |
| hdmap14lem11 40687 | Part of proof of part 14 i... |
| hdmap14lem12 40688 | Lemma for proof of part 14... |
| hdmap14lem13 40689 | Lemma for proof of part 14... |
| hdmap14lem14 40690 | Part of proof of part 14 i... |
| hdmap14lem15 40691 | Part of proof of part 14 i... |
| hgmapffval 40694 | Map from the scalar divisi... |
| hgmapfval 40695 | Map from the scalar divisi... |
| hgmapval 40696 | Value of map from the scal... |
| hgmapfnN 40697 | Functionality of scalar si... |
| hgmapcl 40698 | Closure of scalar sigma ma... |
| hgmapdcl 40699 | Closure of the vector spac... |
| hgmapvs 40700 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 40701 | Value of the scalar sigma ... |
| hgmapval1 40702 | Value of the scalar sigma ... |
| hgmapadd 40703 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 40704 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 40705 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 40706 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 40707 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 40708 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 40709 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 40710 | Part of proof of part 16 i... |
| hgmap11 40711 | The scalar sigma map is on... |
| hgmapf1oN 40712 | The scalar sigma map is a ... |
| hgmapeq0 40713 | The scalar sigma map is ze... |
| hdmapipcl 40714 | The inner product (Hermiti... |
| hdmapln1 40715 | Linearity property that wi... |
| hdmaplna1 40716 | Additive property of first... |
| hdmaplns1 40717 | Subtraction property of fi... |
| hdmaplnm1 40718 | Multiplicative property of... |
| hdmaplna2 40719 | Additive property of secon... |
| hdmapglnm2 40720 | g-linear property of secon... |
| hdmapgln2 40721 | g-linear property that wil... |
| hdmaplkr 40722 | Kernel of the vector to du... |
| hdmapellkr 40723 | Membership in the kernel (... |
| hdmapip0 40724 | Zero property that will be... |
| hdmapip1 40725 | Construct a proportional v... |
| hdmapip0com 40726 | Commutation property of Ba... |
| hdmapinvlem1 40727 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 40728 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 40729 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 40730 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 40731 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 40732 | Involution property of sca... |
| hgmapvvlem2 40733 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 40734 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 40735 | Value of a double involuti... |
| hdmapglem7a 40736 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 40737 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 40738 | Lemma for ~ hdmapg . Line... |
| hdmapg 40739 | Apply the scalar sigma fun... |
| hdmapoc 40740 | Express our constructed or... |
| hlhilset 40743 | The final Hilbert space co... |
| hlhilsca 40744 | The scalar of the final co... |
| hlhilbase 40745 | The base set of the final ... |
| hlhilplus 40746 | The vector addition for th... |
| hlhilslem 40747 | Lemma for ~ hlhilsbase etc... |
| hlhilslemOLD 40748 | Obsolete version of ~ hlhi... |
| hlhilsbase 40749 | The scalar base set of the... |
| hlhilsbaseOLD 40750 | Obsolete version of ~ hlhi... |
| hlhilsplus 40751 | Scalar addition for the fi... |
| hlhilsplusOLD 40752 | Obsolete version of ~ hlhi... |
| hlhilsmul 40753 | Scalar multiplication for ... |
| hlhilsmulOLD 40754 | Obsolete version of ~ hlhi... |
| hlhilsbase2 40755 | The scalar base set of the... |
| hlhilsplus2 40756 | Scalar addition for the fi... |
| hlhilsmul2 40757 | Scalar multiplication for ... |
| hlhils0 40758 | The scalar ring zero for t... |
| hlhils1N 40759 | The scalar ring unity for ... |
| hlhilvsca 40760 | The scalar product for the... |
| hlhilip 40761 | Inner product operation fo... |
| hlhilipval 40762 | Value of inner product ope... |
| hlhilnvl 40763 | The involution operation o... |
| hlhillvec 40764 | The final constructed Hilb... |
| hlhildrng 40765 | The star division ring for... |
| hlhilsrnglem 40766 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 40767 | The star division ring for... |
| hlhil0 40768 | The zero vector for the fi... |
| hlhillsm 40769 | The vector sum operation f... |
| hlhilocv 40770 | The orthocomplement for th... |
| hlhillcs 40771 | The closed subspaces of th... |
| hlhilphllem 40772 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 40773 | Lemma for ~ hlhil . (Cont... |
| hlathil 40774 | Construction of a Hilbert ... |
| leexp1ad 40775 | Weak base ordering relatio... |
| relogbcld 40776 | Closure of the general log... |
| relogbexpd 40777 | Identity law for general l... |
| relogbzexpd 40778 | Power law for the general ... |
| logblebd 40779 | The general logarithm is m... |
| uzindd 40780 | Induction on the upper int... |
| fzadd2d 40781 | Membership of a sum in a f... |
| zltlem1d 40782 | Integer ordering relation,... |
| zltp1led 40783 | Integer ordering relation,... |
| fzne2d 40784 | Elementhood in a finite se... |
| eqfnfv2d2 40785 | Equality of functions is d... |
| fzsplitnd 40786 | Split a finite interval of... |
| fzsplitnr 40787 | Split a finite interval of... |
| addassnni 40788 | Associative law for additi... |
| addcomnni 40789 | Commutative law for additi... |
| mulassnni 40790 | Associative law for multip... |
| mulcomnni 40791 | Commutative law for multip... |
| gcdcomnni 40792 | Commutative law for gcd. ... |
| gcdnegnni 40793 | Negation invariance for gc... |
| neggcdnni 40794 | Negation invariance for gc... |
| bccl2d 40795 | Closure of the binomial co... |
| recbothd 40796 | Take reciprocal on both si... |
| gcdmultiplei 40797 | The GCD of a multiple of a... |
| gcdaddmzz2nni 40798 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 40799 | Adding a multiple of one o... |
| gcdnncli 40800 | Closure of the gcd operato... |
| muldvds1d 40801 | If a product divides an in... |
| muldvds2d 40802 | If a product divides an in... |
| nndivdvdsd 40803 | A positive integer divides... |
| nnproddivdvdsd 40804 | A product of natural numbe... |
| coprmdvds2d 40805 | If an integer is divisible... |
| 12gcd5e1 40806 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 40807 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 40808 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 40809 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 40810 | Calculate the least common... |
| 12lcm5e60 40811 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 40812 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 40813 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 40814 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 40815 | Useful equation to calcula... |
| lcm1un 40816 | Least common multiple of n... |
| lcm2un 40817 | Least common multiple of n... |
| lcm3un 40818 | Least common multiple of n... |
| lcm4un 40819 | Least common multiple of n... |
| lcm5un 40820 | Least common multiple of n... |
| lcm6un 40821 | Least common multiple of n... |
| lcm7un 40822 | Least common multiple of n... |
| lcm8un 40823 | Least common multiple of n... |
| 3factsumint1 40824 | Move constants out of inte... |
| 3factsumint2 40825 | Move constants out of inte... |
| 3factsumint3 40826 | Move constants out of inte... |
| 3factsumint4 40827 | Move constants out of inte... |
| 3factsumint 40828 | Helpful equation for lcm i... |
| resopunitintvd 40829 | Restrict continuous functi... |
| resclunitintvd 40830 | Restrict continuous functi... |
| resdvopclptsd 40831 | Restrict derivative on uni... |
| lcmineqlem1 40832 | Part of lcm inequality lem... |
| lcmineqlem2 40833 | Part of lcm inequality lem... |
| lcmineqlem3 40834 | Part of lcm inequality lem... |
| lcmineqlem4 40835 | Part of lcm inequality lem... |
| lcmineqlem5 40836 | Technical lemma for recipr... |
| lcmineqlem6 40837 | Part of lcm inequality lem... |
| lcmineqlem7 40838 | Derivative of 1-x for chai... |
| lcmineqlem8 40839 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 40840 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 40841 | Induction step of ~ lcmine... |
| lcmineqlem11 40842 | Induction step, continuati... |
| lcmineqlem12 40843 | Base case for induction. ... |
| lcmineqlem13 40844 | Induction proof for lcm in... |
| lcmineqlem14 40845 | Technical lemma for inequa... |
| lcmineqlem15 40846 | F times the least common m... |
| lcmineqlem16 40847 | Technical divisibility lem... |
| lcmineqlem17 40848 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 40849 | Technical lemma to shift f... |
| lcmineqlem19 40850 | Dividing implies inequalit... |
| lcmineqlem20 40851 | Inequality for lcm lemma. ... |
| lcmineqlem21 40852 | The lcm inequality lemma w... |
| lcmineqlem22 40853 | The lcm inequality lemma w... |
| lcmineqlem23 40854 | Penultimate step to the lc... |
| lcmineqlem 40855 | The least common multiple ... |
| 3exp7 40856 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 40857 | First inequality in inequa... |
| 3lexlogpow5ineq2 40858 | Second inequality in inequ... |
| 3lexlogpow5ineq4 40859 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 40860 | Combined inequality chain ... |
| 3lexlogpow2ineq1 40861 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 40862 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 40863 | Result for bound in AKS in... |
| intlewftc 40864 | Inequality inference by in... |
| aks4d1lem1 40865 | Technical lemma to reduce ... |
| aks4d1p1p1 40866 | Exponential law for finite... |
| dvrelog2 40867 | The derivative of the loga... |
| dvrelog3 40868 | The derivative of the loga... |
| dvrelog2b 40869 | Derivative of the binary l... |
| 0nonelalab 40870 | Technical lemma for open i... |
| dvrelogpow2b 40871 | Derivative of the power of... |
| aks4d1p1p3 40872 | Bound of a ceiling of the ... |
| aks4d1p1p2 40873 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 40874 | Technical step for inequal... |
| dvle2 40875 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 40876 | Inequality lift to differe... |
| aks4d1p1p7 40877 | Bound of intermediary of i... |
| aks4d1p1p5 40878 | Show inequality for existe... |
| aks4d1p1 40879 | Show inequality for existe... |
| aks4d1p2 40880 | Technical lemma for existe... |
| aks4d1p3 40881 | There exists a small enoug... |
| aks4d1p4 40882 | There exists a small enoug... |
| aks4d1p5 40883 | Show that ` N ` and ` R ` ... |
| aks4d1p6 40884 | The maximal prime power ex... |
| aks4d1p7d1 40885 | Technical step in AKS lemm... |
| aks4d1p7 40886 | Technical step in AKS lemm... |
| aks4d1p8d1 40887 | If a prime divides one num... |
| aks4d1p8d2 40888 | Any prime power dividing a... |
| aks4d1p8d3 40889 | The remainder of a divisio... |
| aks4d1p8 40890 | Show that ` N ` and ` R ` ... |
| aks4d1p9 40891 | Show that the order is bou... |
| aks4d1 40892 | Lemma 4.1 from ~ https://w... |
| fldhmf1 40893 | A field homomorphism is in... |
| aks6d1c2p1 40894 | In the AKS-theorem the sub... |
| aks6d1c2p2 40895 | Injective condition for co... |
| 5bc2eq10 40896 | The value of 5 choose 2. ... |
| facp2 40897 | The factorial of a success... |
| 2np3bcnp1 40898 | Part of induction step for... |
| 2ap1caineq 40899 | Inequality for Theorem 6.6... |
| sticksstones1 40900 | Different strictly monoton... |
| sticksstones2 40901 | The range function on stri... |
| sticksstones3 40902 | The range function on stri... |
| sticksstones4 40903 | Equinumerosity lemma for s... |
| sticksstones5 40904 | Count the number of strict... |
| sticksstones6 40905 | Function induces an order ... |
| sticksstones7 40906 | Closure property of sticks... |
| sticksstones8 40907 | Establish mapping between ... |
| sticksstones9 40908 | Establish mapping between ... |
| sticksstones10 40909 | Establish mapping between ... |
| sticksstones11 40910 | Establish bijective mappin... |
| sticksstones12a 40911 | Establish bijective mappin... |
| sticksstones12 40912 | Establish bijective mappin... |
| sticksstones13 40913 | Establish bijective mappin... |
| sticksstones14 40914 | Sticks and stones with def... |
| sticksstones15 40915 | Sticks and stones with alm... |
| sticksstones16 40916 | Sticks and stones with col... |
| sticksstones17 40917 | Extend sticks and stones t... |
| sticksstones18 40918 | Extend sticks and stones t... |
| sticksstones19 40919 | Extend sticks and stones t... |
| sticksstones20 40920 | Lift sticks and stones to ... |
| sticksstones21 40921 | Lift sticks and stones to ... |
| sticksstones22 40922 | Non-exhaustive sticks and ... |
| metakunt1 40923 | A is an endomapping. (Con... |
| metakunt2 40924 | A is an endomapping. (Con... |
| metakunt3 40925 | Value of A. (Contributed b... |
| metakunt4 40926 | Value of A. (Contributed b... |
| metakunt5 40927 | C is the left inverse for ... |
| metakunt6 40928 | C is the left inverse for ... |
| metakunt7 40929 | C is the left inverse for ... |
| metakunt8 40930 | C is the left inverse for ... |
| metakunt9 40931 | C is the left inverse for ... |
| metakunt10 40932 | C is the right inverse for... |
| metakunt11 40933 | C is the right inverse for... |
| metakunt12 40934 | C is the right inverse for... |
| metakunt13 40935 | C is the right inverse for... |
| metakunt14 40936 | A is a primitive permutati... |
| metakunt15 40937 | Construction of another pe... |
| metakunt16 40938 | Construction of another pe... |
| metakunt17 40939 | The union of three disjoin... |
| metakunt18 40940 | Disjoint domains and codom... |
| metakunt19 40941 | Domains on restrictions of... |
| metakunt20 40942 | Show that B coincides on t... |
| metakunt21 40943 | Show that B coincides on t... |
| metakunt22 40944 | Show that B coincides on t... |
| metakunt23 40945 | B coincides on the union o... |
| metakunt24 40946 | Technical condition such t... |
| metakunt25 40947 | B is a permutation. (Cont... |
| metakunt26 40948 | Construction of one soluti... |
| metakunt27 40949 | Construction of one soluti... |
| metakunt28 40950 | Construction of one soluti... |
| metakunt29 40951 | Construction of one soluti... |
| metakunt30 40952 | Construction of one soluti... |
| metakunt31 40953 | Construction of one soluti... |
| metakunt32 40954 | Construction of one soluti... |
| metakunt33 40955 | Construction of one soluti... |
| metakunt34 40956 | ` D ` is a permutation. (... |
| andiff 40957 | Adding biconditional when ... |
| fac2xp3 40958 | Factorial of 2x+3, sublemm... |
| prodsplit 40959 | Product split into two fac... |
| 2xp3dxp2ge1d 40960 | 2x+3 is greater than or eq... |
| factwoffsmonot 40961 | A factorial with offset is... |
| bicomdALT 40962 | Alternate proof of ~ bicom... |
| elabgw 40963 | Membership in a class abst... |
| elab2gw 40964 | Membership in a class abst... |
| elrab2w 40965 | Membership in a restricted... |
| ruvALT 40966 | Alternate proof of ~ ruv w... |
| sn-wcdeq 40967 | Alternative to ~ wcdeq and... |
| acos1half 40968 | The arccosine of ` 1 / 2 `... |
| isdomn5 40969 | The right conjunct in the ... |
| isdomn4 40970 | A ring is a domain iff it ... |
| ioin9i8 40971 | Miscellaneous inference cr... |
| jaodd 40972 | Double deduction form of ~... |
| syl3an12 40973 | A double syllogism inferen... |
| sbtd 40974 | A true statement is true u... |
| sbor2 40975 | One direction of ~ sbor , ... |
| 19.9dev 40976 | ~ 19.9d in the case of an ... |
| rspcedvdw 40977 | Version of ~ rspcedvd wher... |
| 3rspcedvdw 40978 | Triple application of ~ rs... |
| 3rspcedvd 40979 | Triple application of ~ rs... |
| rabdif 40980 | Move difference in and out... |
| sn-axrep5v 40981 | A condensed form of ~ axre... |
| sn-axprlem3 40982 | ~ axprlem3 using only Tars... |
| sn-exelALT 40983 | Alternate proof of ~ exel ... |
| ss2ab1 40984 | Class abstractions in a su... |
| ssabdv 40985 | Deduction of abstraction s... |
| sn-iotalem 40986 | An unused lemma showing th... |
| sn-iotalemcor 40987 | Corollary of ~ sn-iotalem ... |
| abbi1sn 40988 | Originally part of ~ uniab... |
| brif1 40989 | Move a relation inside and... |
| brif2 40990 | Move a relation inside and... |
| brif12 40991 | Move a relation inside and... |
| pssexg 40992 | The proper subset of a set... |
| pssn0 40993 | A proper superset is nonem... |
| psspwb 40994 | Classes are proper subclas... |
| xppss12 40995 | Proper subset theorem for ... |
| coexd 40996 | The composition of two set... |
| elpwbi 40997 | Membership in a power set,... |
| opelxpii 40998 | Ordered pair membership in... |
| imaopab 40999 | The image of a class of or... |
| fnsnbt 41000 | A function's domain is a s... |
| fnimasnd 41001 | The image of a function by... |
| fvmptd4 41002 | Deduction version of ~ fvm... |
| eqresfnbd 41003 | Property of being the rest... |
| f1o2d2 41004 | Sufficient condition for a... |
| fmpocos 41005 | Composition of two functio... |
| ovmpogad 41006 | Value of an operation give... |
| ofun 41007 | A function operation of un... |
| dfqs2 41008 | Alternate definition of qu... |
| dfqs3 41009 | Alternate definition of qu... |
| qseq12d 41010 | Equality theorem for quoti... |
| qsalrel 41011 | The quotient set is equal ... |
| fsuppfund 41012 | A finitely supported funct... |
| fsuppsssuppgd 41013 | If the support of a functi... |
| fsuppss 41014 | A subset of a finitely sup... |
| elmapssresd 41015 | A restricted mapping is a ... |
| mapcod 41016 | Compose two mappings. (Co... |
| fzosumm1 41017 | Separate out the last term... |
| ccatcan2d 41018 | Cancellation law for conca... |
| nelsubginvcld 41019 | The inverse of a non-subgr... |
| nelsubgcld 41020 | A non-subgroup-member plus... |
| nelsubgsubcld 41021 | A non-subgroup-member minu... |
| rnasclg 41022 | The set of injected scalar... |
| frlmfielbas 41023 | The vectors of a finite fr... |
| frlmfzwrd 41024 | A vector of a module with ... |
| frlmfzowrd 41025 | A vector of a module with ... |
| frlmfzolen 41026 | The dimension of a vector ... |
| frlmfzowrdb 41027 | The vectors of a module wi... |
| frlmfzoccat 41028 | The concatenation of two v... |
| frlmvscadiccat 41029 | Scalar multiplication dist... |
| grpasscan2d 41030 | An associative cancellatio... |
| grpcominv1 41031 | If two elements commute, t... |
| grpcominv2 41032 | If two elements commute, t... |
| finsubmsubg 41033 | A submonoid of a finite gr... |
| ringlzd 41034 | The zero of a unital ring ... |
| ringrzd 41035 | The zero of a unital ring ... |
| crngcomd 41036 | Multiplication is commutat... |
| crng12d 41037 | Commutative/associative la... |
| imacrhmcl 41038 | The image of a commutative... |
| rimrcl1 41039 | Reverse closure of a ring ... |
| rimrcl2 41040 | Reverse closure of a ring ... |
| rimcnv 41041 | The converse of a ring iso... |
| rimco 41042 | The composition of ring is... |
| ricsym 41043 | Ring isomorphism is symmet... |
| rictr 41044 | Ring isomorphism is transi... |
| riccrng1 41045 | Ring isomorphism preserves... |
| riccrng 41046 | A ring is commutative if a... |
| drnginvrn0d 41047 | A multiplicative inverse i... |
| drngmulcanad 41048 | Cancellation of a nonzero ... |
| drngmulcan2ad 41049 | Cancellation of a nonzero ... |
| drnginvmuld 41050 | Inverse of a nonzero produ... |
| ricdrng1 41051 | A ring isomorphism maps a ... |
| ricdrng 41052 | A ring is a division ring ... |
| lmodvscld 41053 | Closure of scalar product ... |
| flddrngd 41054 | A field is a division ring... |
| ricfld 41055 | A ring is a field if and o... |
| lvecgrp 41056 | A vector space is a group.... |
| lvecgrpd 41057 | A vector space is a group.... |
| lvecring 41058 | The scalar component of a ... |
| frlm0vald 41059 | All coordinates of the zer... |
| frlmsnic 41060 | Given a free module with a... |
| uvccl 41061 | A unit vector is a vector.... |
| uvcn0 41062 | A unit vector is nonzero. ... |
| pwselbasr 41063 | The reverse direction of ~... |
| pwsgprod 41064 | Finite products in a power... |
| psrbagres 41065 | Restrict a bag of variable... |
| mpllmodd 41066 | The polynomial ring is a l... |
| mplringd 41067 | The polynomial ring is a r... |
| mplcrngd 41068 | The polynomial ring is a c... |
| mplsubrgcl 41069 | An element of a polynomial... |
| mhmcompl 41070 | The composition of a monoi... |
| rhmmpllem1 41071 | Lemma for ~ rhmmpl . A su... |
| rhmmpllem2 41072 | Lemma for ~ rhmmpl . A su... |
| mhmcoaddmpl 41073 | Show that the ring homomor... |
| rhmcomulmpl 41074 | Show that the ring homomor... |
| rhmmpl 41075 | Provide a ring homomorphis... |
| mplascl0 41076 | The zero scalar as a polyn... |
| mplascl1 41077 | The one scalar as a polyno... |
| mplmapghm 41078 | The function ` H ` mapping... |
| evl0 41079 | The zero polynomial evalua... |
| evlscl 41080 | A polynomial over the ring... |
| evlsval3 41081 | Give a formula for the pol... |
| evlsvval 41082 | Give a formula for the eva... |
| evlsvvvallem 41083 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 41084 | Lemma for theorems using ~... |
| evlsvvval 41085 | Give a formula for the eva... |
| evlsscaval 41086 | Polynomial evaluation buil... |
| evlsvarval 41087 | Polynomial evaluation buil... |
| evlsbagval 41088 | Polynomial evaluation buil... |
| evlsexpval 41089 | Polynomial evaluation buil... |
| evlsaddval 41090 | Polynomial evaluation buil... |
| evlsmulval 41091 | Polynomial evaluation buil... |
| evlsmaprhm 41092 | The function ` F ` mapping... |
| evlsevl 41093 | Evaluation in a subring is... |
| evlcl 41094 | A polynomial over the ring... |
| evlvvval 41095 | Give a formula for the eva... |
| evlvvvallem 41096 | Lemma for theorems using ~... |
| evladdval 41097 | Polynomial evaluation buil... |
| evlmulval 41098 | Polynomial evaluation buil... |
| selvcllem1 41099 | ` T ` is an associative al... |
| selvcllem2 41100 | ` D ` is a ring homomorphi... |
| selvcllem3 41101 | The third argument passed ... |
| selvcllemh 41102 | Apply the third argument (... |
| selvcllem4 41103 | The fourth argument passed... |
| selvcllem5 41104 | The fifth argument passed ... |
| selvcl 41105 | Closure of the "variable s... |
| selvval2 41106 | Value of the "variable sel... |
| selvvvval 41107 | Recover the original polyn... |
| evlselvlem 41108 | Lemma for ~ evlselv . Use... |
| evlselv 41109 | Evaluating a selection of ... |
| selvadd 41110 | The "variable selection" f... |
| selvmul 41111 | The "variable selection" f... |
| fsuppind 41112 | Induction on functions ` F... |
| fsuppssindlem1 41113 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 41114 | Lemma for ~ fsuppssind . ... |
| fsuppssind 41115 | Induction on functions ` F... |
| mhpind 41116 | The homogeneous polynomial... |
| evlsmhpvvval 41117 | Give a formula for the eva... |
| mhphflem 41118 | Lemma for ~ mhphf . Add s... |
| mhphf 41119 | A homogeneous polynomial d... |
| mhphf2 41120 | A homogeneous polynomial d... |
| mhphf3 41121 | A homogeneous polynomial d... |
| mhphf4 41122 | A homogeneous polynomial d... |
| c0exALT 41123 | Alternate proof of ~ c0ex ... |
| 0cnALT3 41124 | Alternate proof of ~ 0cn u... |
| elre0re 41125 | Specialized version of ~ 0... |
| 1t1e1ALT 41126 | Alternate proof of ~ 1t1e1... |
| remulcan2d 41127 | ~ mulcan2d for real number... |
| readdridaddlidd 41128 | Given some real number ` B... |
| sn-1ne2 41129 | A proof of ~ 1ne2 without ... |
| nnn1suc 41130 | A positive integer that is... |
| nnadd1com 41131 | Addition with 1 is commuta... |
| nnaddcom 41132 | Addition is commutative fo... |
| nnaddcomli 41133 | Version of ~ addcomli for ... |
| nnadddir 41134 | Right-distributivity for n... |
| nnmul1com 41135 | Multiplication with 1 is c... |
| nnmulcom 41136 | Multiplication is commutat... |
| mvrrsubd 41137 | Move a subtraction in the ... |
| laddrotrd 41138 | Rotate the variables right... |
| raddcom12d 41139 | Swap the first two variabl... |
| lsubrotld 41140 | Rotate the variables left ... |
| lsubcom23d 41141 | Swap the second and third ... |
| addsubeq4com 41142 | Relation between sums and ... |
| sqsumi 41143 | A sum squared. (Contribut... |
| negn0nposznnd 41144 | Lemma for ~ dffltz . (Con... |
| sqmid3api 41145 | Value of the square of the... |
| decaddcom 41146 | Commute ones place in addi... |
| sqn5i 41147 | The square of a number end... |
| sqn5ii 41148 | The square of a number end... |
| decpmulnc 41149 | Partial products algorithm... |
| decpmul 41150 | Partial products algorithm... |
| sqdeccom12 41151 | The square of a number in ... |
| sq3deccom12 41152 | Variant of ~ sqdeccom12 wi... |
| 235t711 41153 | Calculate a product by lon... |
| ex-decpmul 41154 | Example usage of ~ decpmul... |
| oexpreposd 41155 | Lemma for ~ dffltz . TODO... |
| ltexp1d 41156 | ~ ltmul1d for exponentiati... |
| ltexp1dd 41157 | Raising both sides of 'les... |
| exp11nnd 41158 | ~ sq11d for positive real ... |
| exp11d 41159 | ~ exp11nnd for nonzero int... |
| 0dvds0 41160 | 0 divides 0. (Contributed... |
| absdvdsabsb 41161 | Divisibility is invariant ... |
| dvdsexpim 41162 | ~ dvdssqim generalized to ... |
| gcdnn0id 41163 | The ` gcd ` of a nonnegati... |
| gcdle1d 41164 | The greatest common diviso... |
| gcdle2d 41165 | The greatest common diviso... |
| dvdsexpad 41166 | Deduction associated with ... |
| nn0rppwr 41167 | If ` A ` and ` B ` are rel... |
| expgcd 41168 | Exponentiation distributes... |
| nn0expgcd 41169 | Exponentiation distributes... |
| zexpgcd 41170 | Exponentiation distributes... |
| numdenexp 41171 | ~ numdensq extended to non... |
| numexp 41172 | ~ numsq extended to nonneg... |
| denexp 41173 | ~ densq extended to nonneg... |
| dvdsexpnn 41174 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 41175 | ~ dvdsexpnn generalized to... |
| dvdsexpb 41176 | ~ dvdssq generalized to po... |
| posqsqznn 41177 | When a positive rational s... |
| cxpgt0d 41178 | A positive real raised to ... |
| zrtelqelz 41179 | ~ zsqrtelqelz generalized ... |
| zrtdvds 41180 | A positive integer root di... |
| rtprmirr 41181 | The root of a prime number... |
| resubval 41184 | Value of real subtraction,... |
| renegeulemv 41185 | Lemma for ~ renegeu and si... |
| renegeulem 41186 | Lemma for ~ renegeu and si... |
| renegeu 41187 | Existential uniqueness of ... |
| rernegcl 41188 | Closure law for negative r... |
| renegadd 41189 | Relationship between real ... |
| renegid 41190 | Addition of a real number ... |
| reneg0addlid 41191 | Negative zero is a left ad... |
| resubeulem1 41192 | Lemma for ~ resubeu . A v... |
| resubeulem2 41193 | Lemma for ~ resubeu . A v... |
| resubeu 41194 | Existential uniqueness of ... |
| rersubcl 41195 | Closure for real subtracti... |
| resubadd 41196 | Relation between real subt... |
| resubaddd 41197 | Relationship between subtr... |
| resubf 41198 | Real subtraction is an ope... |
| repncan2 41199 | Addition and subtraction o... |
| repncan3 41200 | Addition and subtraction o... |
| readdsub 41201 | Law for addition and subtr... |
| reladdrsub 41202 | Move LHS of a sum into RHS... |
| reltsub1 41203 | Subtraction from both side... |
| reltsubadd2 41204 | 'Less than' relationship b... |
| resubcan2 41205 | Cancellation law for real ... |
| resubsub4 41206 | Law for double subtraction... |
| rennncan2 41207 | Cancellation law for real ... |
| renpncan3 41208 | Cancellation law for real ... |
| repnpcan 41209 | Cancellation law for addit... |
| reppncan 41210 | Cancellation law for mixed... |
| resubidaddlidlem 41211 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 41212 | Any real number subtracted... |
| resubdi 41213 | Distribution of multiplica... |
| re1m1e0m0 41214 | Equality of two left-addit... |
| sn-00idlem1 41215 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 41216 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 41217 | Lemma for ~ sn-00id . (Co... |
| sn-00id 41218 | ~ 00id proven without ~ ax... |
| re0m0e0 41219 | Real number version of ~ 0... |
| readdlid 41220 | Real number version of ~ a... |
| sn-addlid 41221 | ~ addlid without ~ ax-mulc... |
| remul02 41222 | Real number version of ~ m... |
| sn-0ne2 41223 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 41224 | Real number version of ~ m... |
| resubid 41225 | Subtraction of a real numb... |
| readdrid 41226 | Real number version of ~ a... |
| resubid1 41227 | Real number version of ~ s... |
| renegneg 41228 | A real number is equal to ... |
| readdcan2 41229 | Commuted version of ~ read... |
| renegid2 41230 | Commuted version of ~ rene... |
| remulneg2d 41231 | Product with negative is n... |
| sn-it0e0 41232 | Proof of ~ it0e0 without ~... |
| sn-negex12 41233 | A combination of ~ cnegex ... |
| sn-negex 41234 | Proof of ~ cnegex without ... |
| sn-negex2 41235 | Proof of ~ cnegex2 without... |
| sn-addcand 41236 | ~ addcand without ~ ax-mul... |
| sn-addrid 41237 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 41238 | ~ addcan2d without ~ ax-mu... |
| reixi 41239 | ~ ixi without ~ ax-mulcom ... |
| rei4 41240 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 41241 | A number that sums to itse... |
| sn-mul01 41242 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 41243 | ~ negeu without ~ ax-mulco... |
| sn-subcl 41244 | ~ subcl without ~ ax-mulco... |
| sn-subf 41245 | ~ subf without ~ ax-mulcom... |
| resubeqsub 41246 | Equivalence between real s... |
| subresre 41247 | Subtraction restricted to ... |
| addinvcom 41248 | A number commutes with its... |
| remulinvcom 41249 | A left multiplicative inve... |
| remullid 41250 | Commuted version of ~ ax-1... |
| sn-1ticom 41251 | Lemma for ~ sn-mullid and ... |
| sn-mullid 41252 | ~ mullid without ~ ax-mulc... |
| it1ei 41253 | ` 1 ` is a multiplicative ... |
| ipiiie0 41254 | The multiplicative inverse... |
| remulcand 41255 | Commuted version of ~ remu... |
| sn-0tie0 41256 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 41257 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 41258 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 41259 | ~ ltaddneg without ~ ax-mu... |
| reposdif 41260 | Comparison of two numbers ... |
| relt0neg1 41261 | Comparison of a real and i... |
| relt0neg2 41262 | Comparison of a real and i... |
| sn-addlt0d 41263 | The sum of negative number... |
| sn-addgt0d 41264 | The sum of positive number... |
| sn-nnne0 41265 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 41266 | ~ elznn0nn restated using ... |
| nn0addcom 41267 | Addition is commutative fo... |
| zaddcomlem 41268 | Lemma for ~ zaddcom . (Co... |
| zaddcom 41269 | Addition is commutative fo... |
| renegmulnnass 41270 | Move multiplication by a n... |
| nn0mulcom 41271 | Multiplication is commutat... |
| zmulcomlem 41272 | Lemma for ~ zmulcom . (Co... |
| zmulcom 41273 | Multiplication is commutat... |
| mulgt0con1dlem 41274 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 41275 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 41276 | Lemma for ~ mulgt0b2d and ... |
| mulgt0b2d 41277 | Biconditional, deductive f... |
| sn-ltmul2d 41278 | ~ ltmul2d without ~ ax-mul... |
| sn-0lt1 41279 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 41280 | ~ ltp1 without ~ ax-mulcom... |
| reneg1lt0 41281 | Lemma for ~ sn-inelr . (C... |
| sn-inelr 41282 | ~ inelr without ~ ax-mulco... |
| itrere 41283 | ` _i ` times a real is rea... |
| retire 41284 | Commuted version of ~ itre... |
| cnreeu 41285 | The reals in the expressio... |
| sn-sup2 41286 | ~ sup2 with exactly the sa... |
| prjspval 41289 | Value of the projective sp... |
| prjsprel 41290 | Utility theorem regarding ... |
| prjspertr 41291 | The relation in ` PrjSp ` ... |
| prjsperref 41292 | The relation in ` PrjSp ` ... |
| prjspersym 41293 | The relation in ` PrjSp ` ... |
| prjsper 41294 | The relation used to defin... |
| prjspreln0 41295 | Two nonzero vectors are eq... |
| prjspvs 41296 | A nonzero multiple of a ve... |
| prjsprellsp 41297 | Two vectors are equivalent... |
| prjspeclsp 41298 | The vectors equivalent to ... |
| prjspval2 41299 | Alternate definition of pr... |
| prjspnval 41302 | Value of the n-dimensional... |
| prjspnerlem 41303 | A lemma showing that the e... |
| prjspnval2 41304 | Value of the n-dimensional... |
| prjspner 41305 | The relation used to defin... |
| prjspnvs 41306 | A nonzero multiple of a ve... |
| prjspnssbas 41307 | A projective point spans a... |
| prjspnn0 41308 | A projective point is none... |
| 0prjspnlem 41309 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 41310 | Any vector is equivalent t... |
| prjspner01 41311 | Any vector is equivalent t... |
| prjspner1 41312 | Two vectors whose zeroth c... |
| 0prjspnrel 41313 | In the zero-dimensional pr... |
| 0prjspn 41314 | A zero-dimensional project... |
| prjcrvfval 41317 | Value of the projective cu... |
| prjcrvval 41318 | Value of the projective cu... |
| prjcrv0 41319 | The "curve" (zero set) cor... |
| dffltz 41320 | Fermat's Last Theorem (FLT... |
| fltmul 41321 | A counterexample to FLT st... |
| fltdiv 41322 | A counterexample to FLT st... |
| flt0 41323 | A counterexample for FLT d... |
| fltdvdsabdvdsc 41324 | Any factor of both ` A ` a... |
| fltabcoprmex 41325 | A counterexample to FLT im... |
| fltaccoprm 41326 | A counterexample to FLT wi... |
| fltbccoprm 41327 | A counterexample to FLT wi... |
| fltabcoprm 41328 | A counterexample to FLT wi... |
| infdesc 41329 | Infinite descent. The hyp... |
| fltne 41330 | If a counterexample to FLT... |
| flt4lem 41331 | Raising a number to the fo... |
| flt4lem1 41332 | Satisfy the antecedent use... |
| flt4lem2 41333 | If ` A ` is even, ` B ` is... |
| flt4lem3 41334 | Equivalent to ~ pythagtrip... |
| flt4lem4 41335 | If the product of two copr... |
| flt4lem5 41336 | In the context of the lemm... |
| flt4lem5elem 41337 | Version of ~ fltaccoprm an... |
| flt4lem5a 41338 | Part 1 of Equation 1 of ... |
| flt4lem5b 41339 | Part 2 of Equation 1 of ... |
| flt4lem5c 41340 | Part 2 of Equation 2 of ... |
| flt4lem5d 41341 | Part 3 of Equation 2 of ... |
| flt4lem5e 41342 | Satisfy the hypotheses of ... |
| flt4lem5f 41343 | Final equation of ~... |
| flt4lem6 41344 | Remove shared factors in a... |
| flt4lem7 41345 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 41346 | Strengthening of Fermat's ... |
| fltltc 41347 | ` ( C ^ N ) ` is the large... |
| fltnltalem 41348 | Lemma for ~ fltnlta . A l... |
| fltnlta 41349 | In a Fermat counterexample... |
| binom2d 41350 | Deduction form of binom2. ... |
| cu3addd 41351 | Cube of sum of three numbe... |
| sqnegd 41352 | The square of the negative... |
| negexpidd 41353 | The sum of a real number t... |
| rexlimdv3d 41354 | An extended version of ~ r... |
| 3cubeslem1 41355 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 41356 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 41357 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 41358 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 41359 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 41360 | Lemma for ~ 3cubes . This... |
| 3cubes 41361 | Every rational number is a... |
| rntrclfvOAI 41362 | The range of the transitiv... |
| moxfr 41363 | Transfer at-most-one betwe... |
| imaiinfv 41364 | Indexed intersection of an... |
| elrfi 41365 | Elementhood in a set of re... |
| elrfirn 41366 | Elementhood in a set of re... |
| elrfirn2 41367 | Elementhood in a set of re... |
| cmpfiiin 41368 | In a compact topology, a s... |
| ismrcd1 41369 | Any function from the subs... |
| ismrcd2 41370 | Second half of ~ ismrcd1 .... |
| istopclsd 41371 | A closure function which s... |
| ismrc 41372 | A function is a Moore clos... |
| isnacs 41375 | Expand definition of Noeth... |
| nacsfg 41376 | In a Noetherian-type closu... |
| isnacs2 41377 | Express Noetherian-type cl... |
| mrefg2 41378 | Slight variation on finite... |
| mrefg3 41379 | Slight variation on finite... |
| nacsacs 41380 | A closure system of Noethe... |
| isnacs3 41381 | A choice-free order equiva... |
| incssnn0 41382 | Transitivity induction of ... |
| nacsfix 41383 | An increasing sequence of ... |
| constmap 41384 | A constant (represented wi... |
| mapco2g 41385 | Renaming indices in a tupl... |
| mapco2 41386 | Post-composition (renaming... |
| mapfzcons 41387 | Extending a one-based mapp... |
| mapfzcons1 41388 | Recover prefix mapping fro... |
| mapfzcons1cl 41389 | A nonempty mapping has a p... |
| mapfzcons2 41390 | Recover added element from... |
| mptfcl 41391 | Interpret range of a maps-... |
| mzpclval 41396 | Substitution lemma for ` m... |
| elmzpcl 41397 | Double substitution lemma ... |
| mzpclall 41398 | The set of all functions w... |
| mzpcln0 41399 | Corollary of ~ mzpclall : ... |
| mzpcl1 41400 | Defining property 1 of a p... |
| mzpcl2 41401 | Defining property 2 of a p... |
| mzpcl34 41402 | Defining properties 3 and ... |
| mzpval 41403 | Value of the ` mzPoly ` fu... |
| dmmzp 41404 | ` mzPoly ` is defined for ... |
| mzpincl 41405 | Polynomial closedness is a... |
| mzpconst 41406 | Constant functions are pol... |
| mzpf 41407 | A polynomial function is a... |
| mzpproj 41408 | A projection function is p... |
| mzpadd 41409 | The pointwise sum of two p... |
| mzpmul 41410 | The pointwise product of t... |
| mzpconstmpt 41411 | A constant function expres... |
| mzpaddmpt 41412 | Sum of polynomial function... |
| mzpmulmpt 41413 | Product of polynomial func... |
| mzpsubmpt 41414 | The difference of two poly... |
| mzpnegmpt 41415 | Negation of a polynomial f... |
| mzpexpmpt 41416 | Raise a polynomial functio... |
| mzpindd 41417 | "Structural" induction to ... |
| mzpmfp 41418 | Relationship between multi... |
| mzpsubst 41419 | Substituting polynomials f... |
| mzprename 41420 | Simplified version of ~ mz... |
| mzpresrename 41421 | A polynomial is a polynomi... |
| mzpcompact2lem 41422 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 41423 | Polynomials are finitary o... |
| coeq0i 41424 | ~ coeq0 but without explic... |
| fzsplit1nn0 41425 | Split a finite 1-based set... |
| eldiophb 41428 | Initial expression of Diop... |
| eldioph 41429 | Condition for a set to be ... |
| diophrw 41430 | Renaming and adding unused... |
| eldioph2lem1 41431 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 41432 | Lemma for ~ eldioph2 . Co... |
| eldioph2 41433 | Construct a Diophantine se... |
| eldioph2b 41434 | While Diophantine sets wer... |
| eldiophelnn0 41435 | Remove antecedent on ` B `... |
| eldioph3b 41436 | Define Diophantine sets in... |
| eldioph3 41437 | Inference version of ~ eld... |
| ellz1 41438 | Membership in a lower set ... |
| lzunuz 41439 | The union of a lower set o... |
| fz1eqin 41440 | Express a one-based finite... |
| lzenom 41441 | Lower integers are countab... |
| elmapresaunres2 41442 | ~ fresaunres2 transposed t... |
| diophin 41443 | If two sets are Diophantin... |
| diophun 41444 | If two sets are Diophantin... |
| eldiophss 41445 | Diophantine sets are sets ... |
| diophrex 41446 | Projecting a Diophantine s... |
| eq0rabdioph 41447 | This is the first of a num... |
| eqrabdioph 41448 | Diophantine set builder fo... |
| 0dioph 41449 | The null set is Diophantin... |
| vdioph 41450 | The "universal" set (as la... |
| anrabdioph 41451 | Diophantine set builder fo... |
| orrabdioph 41452 | Diophantine set builder fo... |
| 3anrabdioph 41453 | Diophantine set builder fo... |
| 3orrabdioph 41454 | Diophantine set builder fo... |
| 2sbcrex 41455 | Exchange an existential qu... |
| sbcrexgOLD 41456 | Interchange class substitu... |
| 2sbcrexOLD 41457 | Exchange an existential qu... |
| sbc2rex 41458 | Exchange a substitution wi... |
| sbc2rexgOLD 41459 | Exchange a substitution wi... |
| sbc4rex 41460 | Exchange a substitution wi... |
| sbc4rexgOLD 41461 | Exchange a substitution wi... |
| sbcrot3 41462 | Rotate a sequence of three... |
| sbcrot5 41463 | Rotate a sequence of five ... |
| sbccomieg 41464 | Commute two explicit subst... |
| rexrabdioph 41465 | Diophantine set builder fo... |
| rexfrabdioph 41466 | Diophantine set builder fo... |
| 2rexfrabdioph 41467 | Diophantine set builder fo... |
| 3rexfrabdioph 41468 | Diophantine set builder fo... |
| 4rexfrabdioph 41469 | Diophantine set builder fo... |
| 6rexfrabdioph 41470 | Diophantine set builder fo... |
| 7rexfrabdioph 41471 | Diophantine set builder fo... |
| rabdiophlem1 41472 | Lemma for arithmetic dioph... |
| rabdiophlem2 41473 | Lemma for arithmetic dioph... |
| elnn0rabdioph 41474 | Diophantine set builder fo... |
| rexzrexnn0 41475 | Rewrite an existential qua... |
| lerabdioph 41476 | Diophantine set builder fo... |
| eluzrabdioph 41477 | Diophantine set builder fo... |
| elnnrabdioph 41478 | Diophantine set builder fo... |
| ltrabdioph 41479 | Diophantine set builder fo... |
| nerabdioph 41480 | Diophantine set builder fo... |
| dvdsrabdioph 41481 | Divisibility is a Diophant... |
| eldioph4b 41482 | Membership in ` Dioph ` ex... |
| eldioph4i 41483 | Forward-only version of ~ ... |
| diophren 41484 | Change variables in a Diop... |
| rabrenfdioph 41485 | Change variable numbers in... |
| rabren3dioph 41486 | Change variable numbers in... |
| fphpd 41487 | Pigeonhole principle expre... |
| fphpdo 41488 | Pigeonhole principle for s... |
| ctbnfien 41489 | An infinite subset of a co... |
| fiphp3d 41490 | Infinite pigeonhole princi... |
| rencldnfilem 41491 | Lemma for ~ rencldnfi . (... |
| rencldnfi 41492 | A set of real numbers whic... |
| irrapxlem1 41493 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 41494 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 41495 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 41496 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 41497 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 41498 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 41499 | Dirichlet's approximation ... |
| pellexlem1 41500 | Lemma for ~ pellex . Arit... |
| pellexlem2 41501 | Lemma for ~ pellex . Arit... |
| pellexlem3 41502 | Lemma for ~ pellex . To e... |
| pellexlem4 41503 | Lemma for ~ pellex . Invo... |
| pellexlem5 41504 | Lemma for ~ pellex . Invo... |
| pellexlem6 41505 | Lemma for ~ pellex . Doin... |
| pellex 41506 | Every Pell equation has a ... |
| pell1qrval 41517 | Value of the set of first-... |
| elpell1qr 41518 | Membership in a first-quad... |
| pell14qrval 41519 | Value of the set of positi... |
| elpell14qr 41520 | Membership in the set of p... |
| pell1234qrval 41521 | Value of the set of genera... |
| elpell1234qr 41522 | Membership in the set of g... |
| pell1234qrre 41523 | General Pell solutions are... |
| pell1234qrne0 41524 | No solution to a Pell equa... |
| pell1234qrreccl 41525 | General solutions of the P... |
| pell1234qrmulcl 41526 | General solutions of the P... |
| pell14qrss1234 41527 | A positive Pell solution i... |
| pell14qrre 41528 | A positive Pell solution i... |
| pell14qrne0 41529 | A positive Pell solution i... |
| pell14qrgt0 41530 | A positive Pell solution i... |
| pell14qrrp 41531 | A positive Pell solution i... |
| pell1234qrdich 41532 | A general Pell solution is... |
| elpell14qr2 41533 | A number is a positive Pel... |
| pell14qrmulcl 41534 | Positive Pell solutions ar... |
| pell14qrreccl 41535 | Positive Pell solutions ar... |
| pell14qrdivcl 41536 | Positive Pell solutions ar... |
| pell14qrexpclnn0 41537 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 41538 | Positive Pell solutions ar... |
| pell1qrss14 41539 | First-quadrant Pell soluti... |
| pell14qrdich 41540 | A positive Pell solution i... |
| pell1qrge1 41541 | A Pell solution in the fir... |
| pell1qr1 41542 | 1 is a Pell solution and i... |
| elpell1qr2 41543 | The first quadrant solutio... |
| pell1qrgaplem 41544 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 41545 | First-quadrant Pell soluti... |
| pell14qrgap 41546 | Positive Pell solutions ar... |
| pell14qrgapw 41547 | Positive Pell solutions ar... |
| pellqrexplicit 41548 | Condition for a calculated... |
| infmrgelbi 41549 | Any lower bound of a nonem... |
| pellqrex 41550 | There is a nontrivial solu... |
| pellfundval 41551 | Value of the fundamental s... |
| pellfundre 41552 | The fundamental solution o... |
| pellfundge 41553 | Lower bound on the fundame... |
| pellfundgt1 41554 | Weak lower bound on the Pe... |
| pellfundlb 41555 | A nontrivial first quadran... |
| pellfundglb 41556 | If a real is larger than t... |
| pellfundex 41557 | The fundamental solution a... |
| pellfund14gap 41558 | There are no solutions bet... |
| pellfundrp 41559 | The fundamental Pell solut... |
| pellfundne1 41560 | The fundamental Pell solut... |
| reglogcl 41561 | General logarithm is a rea... |
| reglogltb 41562 | General logarithm preserve... |
| reglogleb 41563 | General logarithm preserve... |
| reglogmul 41564 | Multiplication law for gen... |
| reglogexp 41565 | Power law for general log.... |
| reglogbas 41566 | General log of the base is... |
| reglog1 41567 | General log of 1 is 0. (C... |
| reglogexpbas 41568 | General log of a power of ... |
| pellfund14 41569 | Every positive Pell soluti... |
| pellfund14b 41570 | The positive Pell solution... |
| rmxfval 41575 | Value of the X sequence. ... |
| rmyfval 41576 | Value of the Y sequence. ... |
| rmspecsqrtnq 41577 | The discriminant used to d... |
| rmspecnonsq 41578 | The discriminant used to d... |
| qirropth 41579 | This lemma implements the ... |
| rmspecfund 41580 | The base of exponent used ... |
| rmxyelqirr 41581 | The solutions used to cons... |
| rmxyelqirrOLD 41582 | Obsolete version of ~ rmxy... |
| rmxypairf1o 41583 | The function used to extra... |
| rmxyelxp 41584 | Lemma for ~ frmx and ~ frm... |
| frmx 41585 | The X sequence is a nonneg... |
| frmy 41586 | The Y sequence is an integ... |
| rmxyval 41587 | Main definition of the X a... |
| rmspecpos 41588 | The discriminant used to d... |
| rmxycomplete 41589 | The X and Y sequences take... |
| rmxynorm 41590 | The X and Y sequences defi... |
| rmbaserp 41591 | The base of exponentiation... |
| rmxyneg 41592 | Negation law for X and Y s... |
| rmxyadd 41593 | Addition formula for X and... |
| rmxy1 41594 | Value of the X and Y seque... |
| rmxy0 41595 | Value of the X and Y seque... |
| rmxneg 41596 | Negation law (even functio... |
| rmx0 41597 | Value of X sequence at 0. ... |
| rmx1 41598 | Value of X sequence at 1. ... |
| rmxadd 41599 | Addition formula for X seq... |
| rmyneg 41600 | Negation formula for Y seq... |
| rmy0 41601 | Value of Y sequence at 0. ... |
| rmy1 41602 | Value of Y sequence at 1. ... |
| rmyadd 41603 | Addition formula for Y seq... |
| rmxp1 41604 | Special addition-of-1 form... |
| rmyp1 41605 | Special addition of 1 form... |
| rmxm1 41606 | Subtraction of 1 formula f... |
| rmym1 41607 | Subtraction of 1 formula f... |
| rmxluc 41608 | The X sequence is a Lucas ... |
| rmyluc 41609 | The Y sequence is a Lucas ... |
| rmyluc2 41610 | Lucas sequence property of... |
| rmxdbl 41611 | "Double-angle formula" for... |
| rmydbl 41612 | "Double-angle formula" for... |
| monotuz 41613 | A function defined on an u... |
| monotoddzzfi 41614 | A function which is odd an... |
| monotoddzz 41615 | A function (given implicit... |
| oddcomabszz 41616 | An odd function which take... |
| 2nn0ind 41617 | Induction on nonnegative i... |
| zindbi 41618 | Inductively transfer a pro... |
| rmxypos 41619 | For all nonnegative indice... |
| ltrmynn0 41620 | The Y-sequence is strictly... |
| ltrmxnn0 41621 | The X-sequence is strictly... |
| lermxnn0 41622 | The X-sequence is monotoni... |
| rmxnn 41623 | The X-sequence is defined ... |
| ltrmy 41624 | The Y-sequence is strictly... |
| rmyeq0 41625 | Y is zero only at zero. (... |
| rmyeq 41626 | Y is one-to-one. (Contrib... |
| lermy 41627 | Y is monotonic (non-strict... |
| rmynn 41628 | ` rmY ` is positive for po... |
| rmynn0 41629 | ` rmY ` is nonnegative for... |
| rmyabs 41630 | ` rmY ` commutes with ` ab... |
| jm2.24nn 41631 | X(n) is strictly greater t... |
| jm2.17a 41632 | First half of lemma 2.17 o... |
| jm2.17b 41633 | Weak form of the second ha... |
| jm2.17c 41634 | Second half of lemma 2.17 ... |
| jm2.24 41635 | Lemma 2.24 of [JonesMatija... |
| rmygeid 41636 | Y(n) increases faster than... |
| congtr 41637 | A wff of the form ` A || (... |
| congadd 41638 | If two pairs of numbers ar... |
| congmul 41639 | If two pairs of numbers ar... |
| congsym 41640 | Congruence mod ` A ` is a ... |
| congneg 41641 | If two integers are congru... |
| congsub 41642 | If two pairs of numbers ar... |
| congid 41643 | Every integer is congruent... |
| mzpcong 41644 | Polynomials commute with c... |
| congrep 41645 | Every integer is congruent... |
| congabseq 41646 | If two integers are congru... |
| acongid 41647 | A wff like that in this th... |
| acongsym 41648 | Symmetry of alternating co... |
| acongneg2 41649 | Negate right side of alter... |
| acongtr 41650 | Transitivity of alternatin... |
| acongeq12d 41651 | Substitution deduction for... |
| acongrep 41652 | Every integer is alternati... |
| fzmaxdif 41653 | Bound on the difference be... |
| fzneg 41654 | Reflection of a finite ran... |
| acongeq 41655 | Two numbers in the fundame... |
| dvdsacongtr 41656 | Alternating congruence pas... |
| coprmdvdsb 41657 | Multiplication by a coprim... |
| modabsdifz 41658 | Divisibility in terms of m... |
| dvdsabsmod0 41659 | Divisibility in terms of m... |
| jm2.18 41660 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 41661 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 41662 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 41663 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 41664 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 41665 | Lemma 2.19 of [JonesMatija... |
| jm2.21 41666 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 41667 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 41668 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 41669 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 41670 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 41671 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 41672 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 41673 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 41674 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 41675 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 41676 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 41677 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 41678 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 41679 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 41680 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 41681 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 41682 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 41683 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 41684 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 41685 | Lemma for ~ rmydioph . Us... |
| rmydioph 41686 | ~ jm2.27 restated in terms... |
| rmxdiophlem 41687 | X can be expressed in term... |
| rmxdioph 41688 | X is a Diophantine functio... |
| jm3.1lem1 41689 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 41690 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 41691 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 41692 | Diophantine expression for... |
| expdiophlem1 41693 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 41694 | Lemma for ~ expdioph . Ex... |
| expdioph 41695 | The exponential function i... |
| setindtr 41696 | Set induction for sets con... |
| setindtrs 41697 | Set induction scheme witho... |
| dford3lem1 41698 | Lemma for ~ dford3 . (Con... |
| dford3lem2 41699 | Lemma for ~ dford3 . (Con... |
| dford3 41700 | Ordinals are precisely the... |
| dford4 41701 | ~ dford3 expressed in prim... |
| wopprc 41702 | Unrelated: Wiener pairs t... |
| rpnnen3lem 41703 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 41704 | Dedekind cut injection of ... |
| axac10 41705 | Characterization of choice... |
| harinf 41706 | The Hartogs number of an i... |
| wdom2d2 41707 | Deduction for weak dominan... |
| ttac 41708 | Tarski's theorem about cho... |
| pw2f1ocnv 41709 | Define a bijection between... |
| pw2f1o2 41710 | Define a bijection between... |
| pw2f1o2val 41711 | Function value of the ~ pw... |
| pw2f1o2val2 41712 | Membership in a mapped set... |
| soeq12d 41713 | Equality deduction for tot... |
| freq12d 41714 | Equality deduction for fou... |
| weeq12d 41715 | Equality deduction for wel... |
| limsuc2 41716 | Limit ordinals in the sens... |
| wepwsolem 41717 | Transfer an ordering on ch... |
| wepwso 41718 | A well-ordering induces a ... |
| dnnumch1 41719 | Define an enumeration of a... |
| dnnumch2 41720 | Define an enumeration (wea... |
| dnnumch3lem 41721 | Value of the ordinal injec... |
| dnnumch3 41722 | Define an injection from a... |
| dnwech 41723 | Define a well-ordering fro... |
| fnwe2val 41724 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 41725 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 41726 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 41727 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 41728 | A well-ordering can be con... |
| aomclem1 41729 | Lemma for ~ dfac11 . This... |
| aomclem2 41730 | Lemma for ~ dfac11 . Succ... |
| aomclem3 41731 | Lemma for ~ dfac11 . Succ... |
| aomclem4 41732 | Lemma for ~ dfac11 . Limi... |
| aomclem5 41733 | Lemma for ~ dfac11 . Comb... |
| aomclem6 41734 | Lemma for ~ dfac11 . Tran... |
| aomclem7 41735 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 41736 | Lemma for ~ dfac11 . Perf... |
| dfac11 41737 | The right-hand side of thi... |
| kelac1 41738 | Kelley's choice, basic for... |
| kelac2lem 41739 | Lemma for ~ kelac2 and ~ d... |
| kelac2 41740 | Kelley's choice, most comm... |
| dfac21 41741 | Tychonoff's theorem is a c... |
| islmodfg 41744 | Property of a finitely gen... |
| islssfg 41745 | Property of a finitely gen... |
| islssfg2 41746 | Property of a finitely gen... |
| islssfgi 41747 | Finitely spanned subspaces... |
| fglmod 41748 | Finitely generated left mo... |
| lsmfgcl 41749 | The sum of two finitely ge... |
| islnm 41752 | Property of being a Noethe... |
| islnm2 41753 | Property of being a Noethe... |
| lnmlmod 41754 | A Noetherian left module i... |
| lnmlssfg 41755 | A submodule of Noetherian ... |
| lnmlsslnm 41756 | All submodules of a Noethe... |
| lnmfg 41757 | A Noetherian left module i... |
| kercvrlsm 41758 | The domain of a linear fun... |
| lmhmfgima 41759 | A homomorphism maps finite... |
| lnmepi 41760 | Epimorphic images of Noeth... |
| lmhmfgsplit 41761 | If the kernel and range of... |
| lmhmlnmsplit 41762 | If the kernel and range of... |
| lnmlmic 41763 | Noetherian is an invariant... |
| pwssplit4 41764 | Splitting for structure po... |
| filnm 41765 | Finite left modules are No... |
| pwslnmlem0 41766 | Zeroeth powers are Noether... |
| pwslnmlem1 41767 | First powers are Noetheria... |
| pwslnmlem2 41768 | A sum of powers is Noether... |
| pwslnm 41769 | Finite powers of Noetheria... |
| unxpwdom3 41770 | Weaker version of ~ unxpwd... |
| pwfi2f1o 41771 | The ~ pw2f1o bijection rel... |
| pwfi2en 41772 | Finitely supported indicat... |
| frlmpwfi 41773 | Formal linear combinations... |
| gicabl 41774 | Being Abelian is a group i... |
| imasgim 41775 | A relabeling of the elemen... |
| isnumbasgrplem1 41776 | A set which is equipollent... |
| harn0 41777 | The Hartogs number of a se... |
| numinfctb 41778 | A numerable infinite set c... |
| isnumbasgrplem2 41779 | If the (to be thought of a... |
| isnumbasgrplem3 41780 | Every nonempty numerable s... |
| isnumbasabl 41781 | A set is numerable iff it ... |
| isnumbasgrp 41782 | A set is numerable iff it ... |
| dfacbasgrp 41783 | A choice equivalent in abs... |
| islnr 41786 | Property of a left-Noether... |
| lnrring 41787 | Left-Noetherian rings are ... |
| lnrlnm 41788 | Left-Noetherian rings have... |
| islnr2 41789 | Property of being a left-N... |
| islnr3 41790 | Relate left-Noetherian rin... |
| lnr2i 41791 | Given an ideal in a left-N... |
| lpirlnr 41792 | Left principal ideal rings... |
| lnrfrlm 41793 | Finite-dimensional free mo... |
| lnrfg 41794 | Finitely-generated modules... |
| lnrfgtr 41795 | A submodule of a finitely ... |
| hbtlem1 41798 | Value of the leading coeff... |
| hbtlem2 41799 | Leading coefficient ideals... |
| hbtlem7 41800 | Functionality of leading c... |
| hbtlem4 41801 | The leading ideal function... |
| hbtlem3 41802 | The leading ideal function... |
| hbtlem5 41803 | The leading ideal function... |
| hbtlem6 41804 | There is a finite set of p... |
| hbt 41805 | The Hilbert Basis Theorem ... |
| dgrsub2 41810 | Subtracting two polynomial... |
| elmnc 41811 | Property of a monic polyno... |
| mncply 41812 | A monic polynomial is a po... |
| mnccoe 41813 | A monic polynomial has lea... |
| mncn0 41814 | A monic polynomial is not ... |
| dgraaval 41819 | Value of the degree functi... |
| dgraalem 41820 | Properties of the degree o... |
| dgraacl 41821 | Closure of the degree func... |
| dgraaf 41822 | Degree function on algebra... |
| dgraaub 41823 | Upper bound on degree of a... |
| dgraa0p 41824 | A rational polynomial of d... |
| mpaaeu 41825 | An algebraic number has ex... |
| mpaaval 41826 | Value of the minimal polyn... |
| mpaalem 41827 | Properties of the minimal ... |
| mpaacl 41828 | Minimal polynomial is a po... |
| mpaadgr 41829 | Minimal polynomial has deg... |
| mpaaroot 41830 | The minimal polynomial of ... |
| mpaamn 41831 | Minimal polynomial is moni... |
| itgoval 41836 | Value of the integral-over... |
| aaitgo 41837 | The standard algebraic num... |
| itgoss 41838 | An integral element is int... |
| itgocn 41839 | All integral elements are ... |
| cnsrexpcl 41840 | Exponentiation is closed i... |
| fsumcnsrcl 41841 | Finite sums are closed in ... |
| cnsrplycl 41842 | Polynomials are closed in ... |
| rgspnval 41843 | Value of the ring-span of ... |
| rgspncl 41844 | The ring-span of a set is ... |
| rgspnssid 41845 | The ring-span of a set con... |
| rgspnmin 41846 | The ring-span is contained... |
| rgspnid 41847 | The span of a subring is i... |
| rngunsnply 41848 | Adjoining one element to a... |
| flcidc 41849 | Finite linear combinations... |
| algstr 41852 | Lemma to shorten proofs of... |
| algbase 41853 | The base set of a construc... |
| algaddg 41854 | The additive operation of ... |
| algmulr 41855 | The multiplicative operati... |
| algsca 41856 | The set of scalars of a co... |
| algvsca 41857 | The scalar product operati... |
| mendval 41858 | Value of the module endomo... |
| mendbas 41859 | Base set of the module end... |
| mendplusgfval 41860 | Addition in the module end... |
| mendplusg 41861 | A specific addition in the... |
| mendmulrfval 41862 | Multiplication in the modu... |
| mendmulr 41863 | A specific multiplication ... |
| mendsca 41864 | The module endomorphism al... |
| mendvscafval 41865 | Scalar multiplication in t... |
| mendvsca 41866 | A specific scalar multipli... |
| mendring 41867 | The module endomorphism al... |
| mendlmod 41868 | The module endomorphism al... |
| mendassa 41869 | The module endomorphism al... |
| idomrootle 41870 | No element of an integral ... |
| idomodle 41871 | Limit on the number of ` N... |
| fiuneneq 41872 | Two finite sets of equal s... |
| idomsubgmo 41873 | The units of an integral d... |
| proot1mul 41874 | Any primitive ` N ` -th ro... |
| proot1hash 41875 | If an integral domain has ... |
| proot1ex 41876 | The complex field has prim... |
| isdomn3 41879 | Nonzero elements form a mu... |
| mon1pid 41880 | Monicity and degree of the... |
| mon1psubm 41881 | Monic polynomials are a mu... |
| deg1mhm 41882 | Homomorphic property of th... |
| cytpfn 41883 | Functionality of the cyclo... |
| cytpval 41884 | Substitutions for the Nth ... |
| fgraphopab 41885 | Express a function as a su... |
| fgraphxp 41886 | Express a function as a su... |
| hausgraph 41887 | The graph of a continuous ... |
| r1sssucd 41892 | Deductive form of ~ r1sssu... |
| iocunico 41893 | Split an open interval int... |
| iocinico 41894 | The intersection of two se... |
| iocmbl 41895 | An open-below, closed-abov... |
| cnioobibld 41896 | A bounded, continuous func... |
| arearect 41897 | The area of a rectangle wh... |
| areaquad 41898 | The area of a quadrilatera... |
| uniel 41899 | Two ways to say a union is... |
| unielss 41900 | Two ways to say the union ... |
| unielid 41901 | Two ways to say the union ... |
| ssunib 41902 | Two ways to say a class is... |
| rp-intrabeq 41903 | Equality theorem for supre... |
| rp-unirabeq 41904 | Equality theorem for infim... |
| onmaxnelsup 41905 | Two ways to say the maximu... |
| onsupneqmaxlim0 41906 | If the supremum of a class... |
| onsupcl2 41907 | The supremum of a set of o... |
| onuniintrab 41908 | The union of a set of ordi... |
| onintunirab 41909 | The intersection of a non-... |
| onsupnmax 41910 | If the union of a class of... |
| onsupuni 41911 | The supremum of a set of o... |
| onsupuni2 41912 | The supremum of a set of o... |
| onsupintrab 41913 | The supremum of a set of o... |
| onsupintrab2 41914 | The supremum of a set of o... |
| onsupcl3 41915 | The supremum of a set of o... |
| onsupex3 41916 | The supremum of a set of o... |
| onuniintrab2 41917 | The union of a set of ordi... |
| oninfint 41918 | The infimum of a non-empty... |
| oninfunirab 41919 | The infimum of a non-empty... |
| oninfcl2 41920 | The infimum of a non-empty... |
| onsupmaxb 41921 | The union of a class of or... |
| onexgt 41922 | For any ordinal, there is ... |
| onexomgt 41923 | For any ordinal, there is ... |
| omlimcl2 41924 | The product of a limit ord... |
| onexlimgt 41925 | For any ordinal, there is ... |
| onexoegt 41926 | For any ordinal, there is ... |
| oninfex2 41927 | The infimum of a non-empty... |
| onsupeqmax 41928 | Condition when the supremu... |
| onsupeqnmax 41929 | Condition when the supremu... |
| onsuplub 41930 | The supremum of a set of o... |
| onsupnub 41931 | An upper bound of a set of... |
| onfisupcl 41932 | Sufficient condition when ... |
| onelord 41933 | Every element of a ordinal... |
| onepsuc 41934 | Every ordinal is less than... |
| epsoon 41935 | The ordinals are strictly ... |
| epirron 41936 | The strict order on the or... |
| oneptr 41937 | The strict order on the or... |
| oneltr 41938 | The elementhood relation o... |
| oneptri 41939 | The strict, complete (line... |
| oneltri 41940 | The elementhood relation o... |
| ordeldif 41941 | Membership in the differen... |
| ordeldifsucon 41942 | Membership in the differen... |
| ordeldif1o 41943 | Membership in the differen... |
| ordne0gt0 41944 | Ordinal zero is less than ... |
| ondif1i 41945 | Ordinal zero is less than ... |
| onsucelab 41946 | The successor of every ord... |
| dflim6 41947 | A limit ordinal is a non-z... |
| limnsuc 41948 | A limit ordinal is not an ... |
| onsucss 41949 | If one ordinal is less tha... |
| ordnexbtwnsuc 41950 | For any distinct pair of o... |
| orddif0suc 41951 | For any distinct pair of o... |
| onsucf1lem 41952 | For ordinals, the successo... |
| onsucf1olem 41953 | The successor operation is... |
| onsucrn 41954 | The successor operation is... |
| onsucf1o 41955 | The successor operation is... |
| dflim7 41956 | A limit ordinal is a non-z... |
| onov0suclim 41957 | Compactly express rules fo... |
| oa0suclim 41958 | Closed form expression of ... |
| om0suclim 41959 | Closed form expression of ... |
| oe0suclim 41960 | Closed form expression of ... |
| oaomoecl 41961 | The operations of addition... |
| onsupsucismax 41962 | If the union of a set of o... |
| onsssupeqcond 41963 | If for every element of a ... |
| limexissup 41964 | An ordinal which is a limi... |
| limiun 41965 | A limit ordinal is the uni... |
| limexissupab 41966 | An ordinal which is a limi... |
| om1om1r 41967 | Ordinal one is both a left... |
| oe0rif 41968 | Ordinal zero raised to any... |
| oasubex 41969 | While subtraction can't be... |
| nnamecl 41970 | Natural numbers are closed... |
| onsucwordi 41971 | The successor operation pr... |
| oalim2cl 41972 | The ordinal sum of any ord... |
| oaltublim 41973 | Given ` C ` is a limit ord... |
| oaordi3 41974 | Ordinal addition of the sa... |
| oaord3 41975 | When the same ordinal is a... |
| 1oaomeqom 41976 | Ordinal one plus omega is ... |
| oaabsb 41977 | The right addend absorbs t... |
| oaordnrex 41978 | When omega is added on the... |
| oaordnr 41979 | When the same ordinal is a... |
| omge1 41980 | Any non-zero ordinal produ... |
| omge2 41981 | Any non-zero ordinal produ... |
| omlim2 41982 | The non-zero product with ... |
| omord2lim 41983 | Given a limit ordinal, the... |
| omord2i 41984 | Ordinal multiplication of ... |
| omord2com 41985 | When the same non-zero ord... |
| 2omomeqom 41986 | Ordinal two times omega is... |
| omnord1ex 41987 | When omega is multiplied o... |
| omnord1 41988 | When the same non-zero ord... |
| oege1 41989 | Any non-zero ordinal power... |
| oege2 41990 | Any power of an ordinal at... |
| rp-oelim2 41991 | The power of an ordinal at... |
| oeord2lim 41992 | Given a limit ordinal, the... |
| oeord2i 41993 | Ordinal exponentiation of ... |
| oeord2com 41994 | When the same base at leas... |
| nnoeomeqom 41995 | Any natural number at leas... |
| df3o2 41996 | Ordinal 3 is the unordered... |
| df3o3 41997 | Ordinal 3, fully expanded.... |
| oenord1ex 41998 | When ordinals two and thre... |
| oenord1 41999 | When two ordinals (both at... |
| oaomoencom 42000 | Ordinal addition, multipli... |
| oenassex 42001 | Ordinal two raised to two ... |
| oenass 42002 | Ordinal exponentiation is ... |
| cantnftermord 42003 | For terms of the form of a... |
| cantnfub 42004 | Given a finite number of t... |
| cantnfub2 42005 | Given a finite number of t... |
| bropabg 42006 | Equivalence for two classe... |
| cantnfresb 42007 | A Cantor normal form which... |
| cantnf2 42008 | For every ordinal, ` A ` ,... |
| oawordex2 42009 | If ` C ` is between ` A ` ... |
| nnawordexg 42010 | If an ordinal, ` B ` , is ... |
| succlg 42011 | Closure law for ordinal su... |
| dflim5 42012 | A limit ordinal is either ... |
| oacl2g 42013 | Closure law for ordinal ad... |
| onmcl 42014 | If an ordinal is less than... |
| omabs2 42015 | Ordinal multiplication by ... |
| omcl2 42016 | Closure law for ordinal mu... |
| omcl3g 42017 | Closure law for ordinal mu... |
| ordsssucb 42018 | An ordinal number is less ... |
| tfsconcatlem 42019 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 42020 | The concatenation of two t... |
| tfsconcatfn 42021 | The concatenation of two t... |
| tfsconcatfv1 42022 | An early value of the conc... |
| tfsconcatfv2 42023 | A latter value of the conc... |
| tfsconcatfv 42024 | The value of the concatena... |
| tfsconcatrn 42025 | The range of the concatena... |
| tfsconcatfo 42026 | The concatenation of two t... |
| tfsconcatb0 42027 | The concatentation with th... |
| tfsconcat0i 42028 | The concatentation with th... |
| tfsconcat0b 42029 | The concatentation with th... |
| tfsconcat00 42030 | The concatentation of two ... |
| tfsconcatrev 42031 | If the domain of a transfi... |
| tfsconcatrnss12 42032 | The range of the concatena... |
| tfsconcatrnss 42033 | The concatenation of trans... |
| tfsconcatrnsson 42034 | The concatenation of trans... |
| tfsnfin 42035 | A transfinite sequence is ... |
| rp-tfslim 42036 | The limit of a sequence of... |
| ofoafg 42037 | Addition operator for func... |
| ofoaf 42038 | Addition operator for func... |
| ofoafo 42039 | Addition operator for func... |
| ofoacl 42040 | Closure law for component ... |
| ofoaid1 42041 | Identity law for component... |
| ofoaid2 42042 | Identity law for component... |
| ofoaass 42043 | Component-wise addition of... |
| ofoacom 42044 | Component-wise addition of... |
| naddcnff 42045 | Addition operator for Cant... |
| naddcnffn 42046 | Addition operator for Cant... |
| naddcnffo 42047 | Addition of Cantor normal ... |
| naddcnfcl 42048 | Closure law for component-... |
| naddcnfcom 42049 | Component-wise ordinal add... |
| naddcnfid1 42050 | Identity law for component... |
| naddcnfid2 42051 | Identity law for component... |
| naddcnfass 42052 | Component-wise addition of... |
| onsucunifi 42053 | The successor to the union... |
| sucunisn 42054 | The successor to the union... |
| onsucunipr 42055 | The successor to the union... |
| onsucunitp 42056 | The successor to the union... |
| oaun3lem1 42057 | The class of all ordinal s... |
| oaun3lem2 42058 | The class of all ordinal s... |
| oaun3lem3 42059 | The class of all ordinal s... |
| oaun3lem4 42060 | The class of all ordinal s... |
| rp-abid 42061 | Two ways to express a clas... |
| oadif1lem 42062 | Express the set difference... |
| oadif1 42063 | Express the set difference... |
| oaun2 42064 | Ordinal addition as a unio... |
| oaun3 42065 | Ordinal addition as a unio... |
| naddov4 42066 | Alternate expression for n... |
| nadd2rabtr 42067 | The set of ordinals which ... |
| nadd2rabord 42068 | The set of ordinals which ... |
| nadd2rabex 42069 | The class of ordinals whic... |
| nadd2rabon 42070 | The set of ordinals which ... |
| nadd1rabtr 42071 | The set of ordinals which ... |
| nadd1rabord 42072 | The set of ordinals which ... |
| nadd1rabex 42073 | The class of ordinals whic... |
| nadd1rabon 42074 | The set of ordinals which ... |
| nadd1suc 42075 | Natural addition with 1 is... |
| naddsuc2 42076 | Natural addition with succ... |
| naddass1 42077 | Natural addition of ordina... |
| naddgeoa 42078 | Natural addition results i... |
| naddonnn 42079 | Natural addition with a na... |
| naddwordnexlem0 42080 | When ` A ` is the sum of a... |
| naddwordnexlem1 42081 | When ` A ` is the sum of a... |
| naddwordnexlem2 42082 | When ` A ` is the sum of a... |
| naddwordnexlem3 42083 | When ` A ` is the sum of a... |
| oawordex3 42084 | When ` A ` is the sum of a... |
| naddwordnexlem4 42085 | When ` A ` is the sum of a... |
| ordsssucim 42086 | If an ordinal is less than... |
| insucid 42087 | The intersection of a clas... |
| om2 42088 | Two ways to double an ordi... |
| oaltom 42089 | Multiplication eventually ... |
| oe2 42090 | Two ways to square an ordi... |
| omltoe 42091 | Exponentiation eventually ... |
| abeqabi 42092 | Generalized condition for ... |
| abpr 42093 | Condition for a class abst... |
| abtp 42094 | Condition for a class abst... |
| ralopabb 42095 | Restricted universal quant... |
| fpwfvss 42096 | Functions into a powerset ... |
| sdomne0 42097 | A class that strictly domi... |
| sdomne0d 42098 | A class that strictly domi... |
| safesnsupfiss 42099 | If ` B ` is a finite subse... |
| safesnsupfiub 42100 | If ` B ` is a finite subse... |
| safesnsupfidom1o 42101 | If ` B ` is a finite subse... |
| safesnsupfilb 42102 | If ` B ` is a finite subse... |
| isoeq145d 42103 | Equality deduction for iso... |
| resisoeq45d 42104 | Equality deduction for equ... |
| negslem1 42105 | An equivalence between ide... |
| nvocnvb 42106 | Equivalence to saying the ... |
| rp-brsslt 42107 | Binary relation form of a ... |
| nla0002 42108 | Extending a linear order t... |
| nla0003 42109 | Extending a linear order t... |
| nla0001 42110 | Extending a linear order t... |
| faosnf0.11b 42111 | ` B ` is called a non-limi... |
| dfno2 42112 | A surreal number, in the f... |
| onnog 42113 | Every ordinal maps to a su... |
| onnobdayg 42114 | Every ordinal maps to a su... |
| bdaybndex 42115 | Bounds formed from the bir... |
| bdaybndbday 42116 | Bounds formed from the bir... |
| onno 42117 | Every ordinal maps to a su... |
| onnoi 42118 | Every ordinal maps to a su... |
| 0no 42119 | Ordinal zero maps to a sur... |
| 1no 42120 | Ordinal one maps to a surr... |
| 2no 42121 | Ordinal two maps to a surr... |
| 3no 42122 | Ordinal three maps to a su... |
| 4no 42123 | Ordinal four maps to a sur... |
| fnimafnex 42124 | The functional image of a ... |
| nlimsuc 42125 | A successor is not a limit... |
| nlim1NEW 42126 | 1 is not a limit ordinal. ... |
| nlim2NEW 42127 | 2 is not a limit ordinal. ... |
| nlim3 42128 | 3 is not a limit ordinal. ... |
| nlim4 42129 | 4 is not a limit ordinal. ... |
| oa1un 42130 | Given ` A e. On ` , let ` ... |
| oa1cl 42131 | ` A +o 1o ` is in ` On ` .... |
| 0finon 42132 | 0 is a finite ordinal. Se... |
| 1finon 42133 | 1 is a finite ordinal. Se... |
| 2finon 42134 | 2 is a finite ordinal. Se... |
| 3finon 42135 | 3 is a finite ordinal. Se... |
| 4finon 42136 | 4 is a finite ordinal. Se... |
| finona1cl 42137 | The finite ordinals are cl... |
| finonex 42138 | The finite ordinals are a ... |
| fzunt 42139 | Union of two adjacent fini... |
| fzuntd 42140 | Union of two adjacent fini... |
| fzunt1d 42141 | Union of two overlapping f... |
| fzuntgd 42142 | Union of two adjacent or o... |
| ifpan123g 42143 | Conjunction of conditional... |
| ifpan23 42144 | Conjunction of conditional... |
| ifpdfor2 42145 | Define or in terms of cond... |
| ifporcor 42146 | Corollary of commutation o... |
| ifpdfan2 42147 | Define and with conditiona... |
| ifpancor 42148 | Corollary of commutation o... |
| ifpdfor 42149 | Define or in terms of cond... |
| ifpdfan 42150 | Define and with conditiona... |
| ifpbi2 42151 | Equivalence theorem for co... |
| ifpbi3 42152 | Equivalence theorem for co... |
| ifpim1 42153 | Restate implication as con... |
| ifpnot 42154 | Restate negated wff as con... |
| ifpid2 42155 | Restate wff as conditional... |
| ifpim2 42156 | Restate implication as con... |
| ifpbi23 42157 | Equivalence theorem for co... |
| ifpbiidcor 42158 | Restatement of ~ biid . (... |
| ifpbicor 42159 | Corollary of commutation o... |
| ifpxorcor 42160 | Corollary of commutation o... |
| ifpbi1 42161 | Equivalence theorem for co... |
| ifpnot23 42162 | Negation of conditional lo... |
| ifpnotnotb 42163 | Factor conditional logic o... |
| ifpnorcor 42164 | Corollary of commutation o... |
| ifpnancor 42165 | Corollary of commutation o... |
| ifpnot23b 42166 | Negation of conditional lo... |
| ifpbiidcor2 42167 | Restatement of ~ biid . (... |
| ifpnot23c 42168 | Negation of conditional lo... |
| ifpnot23d 42169 | Negation of conditional lo... |
| ifpdfnan 42170 | Define nand as conditional... |
| ifpdfxor 42171 | Define xor as conditional ... |
| ifpbi12 42172 | Equivalence theorem for co... |
| ifpbi13 42173 | Equivalence theorem for co... |
| ifpbi123 42174 | Equivalence theorem for co... |
| ifpidg 42175 | Restate wff as conditional... |
| ifpid3g 42176 | Restate wff as conditional... |
| ifpid2g 42177 | Restate wff as conditional... |
| ifpid1g 42178 | Restate wff as conditional... |
| ifpim23g 42179 | Restate implication as con... |
| ifpim3 42180 | Restate implication as con... |
| ifpnim1 42181 | Restate negated implicatio... |
| ifpim4 42182 | Restate implication as con... |
| ifpnim2 42183 | Restate negated implicatio... |
| ifpim123g 42184 | Implication of conditional... |
| ifpim1g 42185 | Implication of conditional... |
| ifp1bi 42186 | Substitute the first eleme... |
| ifpbi1b 42187 | When the first variable is... |
| ifpimimb 42188 | Factor conditional logic o... |
| ifpororb 42189 | Factor conditional logic o... |
| ifpananb 42190 | Factor conditional logic o... |
| ifpnannanb 42191 | Factor conditional logic o... |
| ifpor123g 42192 | Disjunction of conditional... |
| ifpimim 42193 | Consequnce of implication.... |
| ifpbibib 42194 | Factor conditional logic o... |
| ifpxorxorb 42195 | Factor conditional logic o... |
| rp-fakeimass 42196 | A special case where impli... |
| rp-fakeanorass 42197 | A special case where a mix... |
| rp-fakeoranass 42198 | A special case where a mix... |
| rp-fakeinunass 42199 | A special case where a mix... |
| rp-fakeuninass 42200 | A special case where a mix... |
| rp-isfinite5 42201 | A set is said to be finite... |
| rp-isfinite6 42202 | A set is said to be finite... |
| intabssd 42203 | When for each element ` y ... |
| eu0 42204 | There is only one empty se... |
| epelon2 42205 | Over the ordinal numbers, ... |
| ontric3g 42206 | For all ` x , y e. On ` , ... |
| dfsucon 42207 | ` A ` is called a successo... |
| snen1g 42208 | A singleton is equinumerou... |
| snen1el 42209 | A singleton is equinumerou... |
| sn1dom 42210 | A singleton is dominated b... |
| pr2dom 42211 | An unordered pair is domin... |
| tr3dom 42212 | An unordered triple is dom... |
| ensucne0 42213 | A class equinumerous to a ... |
| ensucne0OLD 42214 | A class equinumerous to a ... |
| dfom6 42215 | Let ` _om ` be defined to ... |
| infordmin 42216 | ` _om ` is the smallest in... |
| iscard4 42217 | Two ways to express the pr... |
| minregex 42218 | Given any cardinal number ... |
| minregex2 42219 | Given any cardinal number ... |
| iscard5 42220 | Two ways to express the pr... |
| elrncard 42221 | Let us define a cardinal n... |
| harval3 42222 | ` ( har `` A ) ` is the le... |
| harval3on 42223 | For any ordinal number ` A... |
| omssrncard 42224 | All natural numbers are ca... |
| 0iscard 42225 | 0 is a cardinal number. (... |
| 1iscard 42226 | 1 is a cardinal number. (... |
| omiscard 42227 | ` _om ` is a cardinal numb... |
| sucomisnotcard 42228 | ` _om +o 1o ` is not a car... |
| nna1iscard 42229 | For any natural number, th... |
| har2o 42230 | The least cardinal greater... |
| en2pr 42231 | A class is equinumerous to... |
| pr2cv 42232 | If an unordered pair is eq... |
| pr2el1 42233 | If an unordered pair is eq... |
| pr2cv1 42234 | If an unordered pair is eq... |
| pr2el2 42235 | If an unordered pair is eq... |
| pr2cv2 42236 | If an unordered pair is eq... |
| pren2 42237 | An unordered pair is equin... |
| pr2eldif1 42238 | If an unordered pair is eq... |
| pr2eldif2 42239 | If an unordered pair is eq... |
| pren2d 42240 | A pair of two distinct set... |
| aleph1min 42241 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 42242 | ` aleph ` is a strictly or... |
| alephiso3 42243 | ` aleph ` is a strictly or... |
| pwelg 42244 | The powerclass is an eleme... |
| pwinfig 42245 | The powerclass of an infin... |
| pwinfi2 42246 | The powerclass of an infin... |
| pwinfi3 42247 | The powerclass of an infin... |
| pwinfi 42248 | The powerclass of an infin... |
| fipjust 42249 | A definition of the finite... |
| cllem0 42250 | The class of all sets with... |
| superficl 42251 | The class of all supersets... |
| superuncl 42252 | The class of all supersets... |
| ssficl 42253 | The class of all subsets o... |
| ssuncl 42254 | The class of all subsets o... |
| ssdifcl 42255 | The class of all subsets o... |
| sssymdifcl 42256 | The class of all subsets o... |
| fiinfi 42257 | If two classes have the fi... |
| rababg 42258 | Condition when restricted ... |
| elinintab 42259 | Two ways of saying a set i... |
| elmapintrab 42260 | Two ways to say a set is a... |
| elinintrab 42261 | Two ways of saying a set i... |
| inintabss 42262 | Upper bound on intersectio... |
| inintabd 42263 | Value of the intersection ... |
| xpinintabd 42264 | Value of the intersection ... |
| relintabex 42265 | If the intersection of a c... |
| elcnvcnvintab 42266 | Two ways of saying a set i... |
| relintab 42267 | Value of the intersection ... |
| nonrel 42268 | A non-relation is equal to... |
| elnonrel 42269 | Only an ordered pair where... |
| cnvssb 42270 | Subclass theorem for conve... |
| relnonrel 42271 | The non-relation part of a... |
| cnvnonrel 42272 | The converse of the non-re... |
| brnonrel 42273 | A non-relation cannot rela... |
| dmnonrel 42274 | The domain of the non-rela... |
| rnnonrel 42275 | The range of the non-relat... |
| resnonrel 42276 | A restriction of the non-r... |
| imanonrel 42277 | An image under the non-rel... |
| cononrel1 42278 | Composition with the non-r... |
| cononrel2 42279 | Composition with the non-r... |
| elmapintab 42280 | Two ways to say a set is a... |
| fvnonrel 42281 | The function value of any ... |
| elinlem 42282 | Two ways to say a set is a... |
| elcnvcnvlem 42283 | Two ways to say a set is a... |
| cnvcnvintabd 42284 | Value of the relationship ... |
| elcnvlem 42285 | Two ways to say a set is a... |
| elcnvintab 42286 | Two ways of saying a set i... |
| cnvintabd 42287 | Value of the converse of t... |
| undmrnresiss 42288 | Two ways of saying the ide... |
| reflexg 42289 | Two ways of saying a relat... |
| cnvssco 42290 | A condition weaker than re... |
| refimssco 42291 | Reflexive relations are su... |
| cleq2lem 42292 | Equality implies bijection... |
| cbvcllem 42293 | Change of bound variable i... |
| clublem 42294 | If a superset ` Y ` of ` X... |
| clss2lem 42295 | The closure of a property ... |
| dfid7 42296 | Definition of identity rel... |
| mptrcllem 42297 | Show two versions of a clo... |
| cotrintab 42298 | The intersection of a clas... |
| rclexi 42299 | The reflexive closure of a... |
| rtrclexlem 42300 | Existence of relation impl... |
| rtrclex 42301 | The reflexive-transitive c... |
| trclubgNEW 42302 | If a relation exists then ... |
| trclubNEW 42303 | If a relation exists then ... |
| trclexi 42304 | The transitive closure of ... |
| rtrclexi 42305 | The reflexive-transitive c... |
| clrellem 42306 | When the property ` ps ` h... |
| clcnvlem 42307 | When ` A ` , an upper boun... |
| cnvtrucl0 42308 | The converse of the trivia... |
| cnvrcl0 42309 | The converse of the reflex... |
| cnvtrcl0 42310 | The converse of the transi... |
| dmtrcl 42311 | The domain of the transiti... |
| rntrcl 42312 | The range of the transitiv... |
| dfrtrcl5 42313 | Definition of reflexive-tr... |
| trcleq2lemRP 42314 | Equality implies bijection... |
| sqrtcvallem1 42315 | Two ways of saying a compl... |
| reabsifneg 42316 | Alternate expression for t... |
| reabsifnpos 42317 | Alternate expression for t... |
| reabsifpos 42318 | Alternate expression for t... |
| reabsifnneg 42319 | Alternate expression for t... |
| reabssgn 42320 | Alternate expression for t... |
| sqrtcvallem2 42321 | Equivalent to saying that ... |
| sqrtcvallem3 42322 | Equivalent to saying that ... |
| sqrtcvallem4 42323 | Equivalent to saying that ... |
| sqrtcvallem5 42324 | Equivalent to saying that ... |
| sqrtcval 42325 | Explicit formula for the c... |
| sqrtcval2 42326 | Explicit formula for the c... |
| resqrtval 42327 | Real part of the complex s... |
| imsqrtval 42328 | Imaginary part of the comp... |
| resqrtvalex 42329 | Example for ~ resqrtval . ... |
| imsqrtvalex 42330 | Example for ~ imsqrtval . ... |
| al3im 42331 | Version of ~ ax-4 for a ne... |
| intima0 42332 | Two ways of expressing the... |
| elimaint 42333 | Element of image of inters... |
| cnviun 42334 | Converse of indexed union.... |
| imaiun1 42335 | The image of an indexed un... |
| coiun1 42336 | Composition with an indexe... |
| elintima 42337 | Element of intersection of... |
| intimass 42338 | The image under the inters... |
| intimass2 42339 | The image under the inters... |
| intimag 42340 | Requirement for the image ... |
| intimasn 42341 | Two ways to express the im... |
| intimasn2 42342 | Two ways to express the im... |
| ss2iundf 42343 | Subclass theorem for index... |
| ss2iundv 42344 | Subclass theorem for index... |
| cbviuneq12df 42345 | Rule used to change the bo... |
| cbviuneq12dv 42346 | Rule used to change the bo... |
| conrel1d 42347 | Deduction about compositio... |
| conrel2d 42348 | Deduction about compositio... |
| trrelind 42349 | The intersection of transi... |
| xpintrreld 42350 | The intersection of a tran... |
| restrreld 42351 | The restriction of a trans... |
| trrelsuperreldg 42352 | Concrete construction of a... |
| trficl 42353 | The class of all transitiv... |
| cnvtrrel 42354 | The converse of a transiti... |
| trrelsuperrel2dg 42355 | Concrete construction of a... |
| dfrcl2 42358 | Reflexive closure of a rel... |
| dfrcl3 42359 | Reflexive closure of a rel... |
| dfrcl4 42360 | Reflexive closure of a rel... |
| relexp2 42361 | A set operated on by the r... |
| relexpnul 42362 | If the domain and range of... |
| eliunov2 42363 | Membership in the indexed ... |
| eltrclrec 42364 | Membership in the indexed ... |
| elrtrclrec 42365 | Membership in the indexed ... |
| briunov2 42366 | Two classes related by the... |
| brmptiunrelexpd 42367 | If two elements are connec... |
| fvmptiunrelexplb0d 42368 | If the indexed union range... |
| fvmptiunrelexplb0da 42369 | If the indexed union range... |
| fvmptiunrelexplb1d 42370 | If the indexed union range... |
| brfvid 42371 | If two elements are connec... |
| brfvidRP 42372 | If two elements are connec... |
| fvilbd 42373 | A set is a subset of its i... |
| fvilbdRP 42374 | A set is a subset of its i... |
| brfvrcld 42375 | If two elements are connec... |
| brfvrcld2 42376 | If two elements are connec... |
| fvrcllb0d 42377 | A restriction of the ident... |
| fvrcllb0da 42378 | A restriction of the ident... |
| fvrcllb1d 42379 | A set is a subset of its i... |
| brtrclrec 42380 | Two classes related by the... |
| brrtrclrec 42381 | Two classes related by the... |
| briunov2uz 42382 | Two classes related by the... |
| eliunov2uz 42383 | Membership in the indexed ... |
| ov2ssiunov2 42384 | Any particular operator va... |
| relexp0eq 42385 | The zeroth power of relati... |
| iunrelexp0 42386 | Simplification of zeroth p... |
| relexpxpnnidm 42387 | Any positive power of a Ca... |
| relexpiidm 42388 | Any power of any restricti... |
| relexpss1d 42389 | The relational power of a ... |
| comptiunov2i 42390 | The composition two indexe... |
| corclrcl 42391 | The reflexive closure is i... |
| iunrelexpmin1 42392 | The indexed union of relat... |
| relexpmulnn 42393 | With exponents limited to ... |
| relexpmulg 42394 | With ordered exponents, th... |
| trclrelexplem 42395 | The union of relational po... |
| iunrelexpmin2 42396 | The indexed union of relat... |
| relexp01min 42397 | With exponents limited to ... |
| relexp1idm 42398 | Repeated raising a relatio... |
| relexp0idm 42399 | Repeated raising a relatio... |
| relexp0a 42400 | Absorption law for zeroth ... |
| relexpxpmin 42401 | The composition of powers ... |
| relexpaddss 42402 | The composition of two pow... |
| iunrelexpuztr 42403 | The indexed union of relat... |
| dftrcl3 42404 | Transitive closure of a re... |
| brfvtrcld 42405 | If two elements are connec... |
| fvtrcllb1d 42406 | A set is a subset of its i... |
| trclfvcom 42407 | The transitive closure of ... |
| cnvtrclfv 42408 | The converse of the transi... |
| cotrcltrcl 42409 | The transitive closure is ... |
| trclimalb2 42410 | Lower bound for image unde... |
| brtrclfv2 42411 | Two ways to indicate two e... |
| trclfvdecomr 42412 | The transitive closure of ... |
| trclfvdecoml 42413 | The transitive closure of ... |
| dmtrclfvRP 42414 | The domain of the transiti... |
| rntrclfvRP 42415 | The range of the transitiv... |
| rntrclfv 42416 | The range of the transitiv... |
| dfrtrcl3 42417 | Reflexive-transitive closu... |
| brfvrtrcld 42418 | If two elements are connec... |
| fvrtrcllb0d 42419 | A restriction of the ident... |
| fvrtrcllb0da 42420 | A restriction of the ident... |
| fvrtrcllb1d 42421 | A set is a subset of its i... |
| dfrtrcl4 42422 | Reflexive-transitive closu... |
| corcltrcl 42423 | The composition of the ref... |
| cortrcltrcl 42424 | Composition with the refle... |
| corclrtrcl 42425 | Composition with the refle... |
| cotrclrcl 42426 | The composition of the ref... |
| cortrclrcl 42427 | Composition with the refle... |
| cotrclrtrcl 42428 | Composition with the refle... |
| cortrclrtrcl 42429 | The reflexive-transitive c... |
| frege77d 42430 | If the images of both ` { ... |
| frege81d 42431 | If the image of ` U ` is a... |
| frege83d 42432 | If the image of the union ... |
| frege96d 42433 | If ` C ` follows ` A ` in ... |
| frege87d 42434 | If the images of both ` { ... |
| frege91d 42435 | If ` B ` follows ` A ` in ... |
| frege97d 42436 | If ` A ` contains all elem... |
| frege98d 42437 | If ` C ` follows ` A ` and... |
| frege102d 42438 | If either ` A ` and ` C ` ... |
| frege106d 42439 | If ` B ` follows ` A ` in ... |
| frege108d 42440 | If either ` A ` and ` C ` ... |
| frege109d 42441 | If ` A ` contains all elem... |
| frege114d 42442 | If either ` R ` relates ` ... |
| frege111d 42443 | If either ` A ` and ` C ` ... |
| frege122d 42444 | If ` F ` is a function, ` ... |
| frege124d 42445 | If ` F ` is a function, ` ... |
| frege126d 42446 | If ` F ` is a function, ` ... |
| frege129d 42447 | If ` F ` is a function and... |
| frege131d 42448 | If ` F ` is a function and... |
| frege133d 42449 | If ` F ` is a function and... |
| dfxor4 42450 | Express exclusive-or in te... |
| dfxor5 42451 | Express exclusive-or in te... |
| df3or2 42452 | Express triple-or in terms... |
| df3an2 42453 | Express triple-and in term... |
| nev 42454 | Express that not every set... |
| 0pssin 42455 | Express that an intersecti... |
| dfhe2 42458 | The property of relation `... |
| dfhe3 42459 | The property of relation `... |
| heeq12 42460 | Equality law for relations... |
| heeq1 42461 | Equality law for relations... |
| heeq2 42462 | Equality law for relations... |
| sbcheg 42463 | Distribute proper substitu... |
| hess 42464 | Subclass law for relations... |
| xphe 42465 | Any Cartesian product is h... |
| 0he 42466 | The empty relation is here... |
| 0heALT 42467 | The empty relation is here... |
| he0 42468 | Any relation is hereditary... |
| unhe1 42469 | The union of two relations... |
| snhesn 42470 | Any singleton is hereditar... |
| idhe 42471 | The identity relation is h... |
| psshepw 42472 | The relation between sets ... |
| sshepw 42473 | The relation between sets ... |
| rp-simp2-frege 42476 | Simplification of triple c... |
| rp-simp2 42477 | Simplification of triple c... |
| rp-frege3g 42478 | Add antecedent to ~ ax-fre... |
| frege3 42479 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 42480 | Double-use of ~ ax-frege2 ... |
| rp-frege24 42481 | Introducing an embedded an... |
| rp-frege4g 42482 | Deduction related to distr... |
| frege4 42483 | Special case of closed for... |
| frege5 42484 | A closed form of ~ syl . ... |
| rp-7frege 42485 | Distribute antecedent and ... |
| rp-4frege 42486 | Elimination of a nested an... |
| rp-6frege 42487 | Elimination of a nested an... |
| rp-8frege 42488 | Eliminate antecedent when ... |
| rp-frege25 42489 | Closed form for ~ a1dd . ... |
| frege6 42490 | A closed form of ~ imim2d ... |
| axfrege8 42491 | Swap antecedents. Identic... |
| frege7 42492 | A closed form of ~ syl6 . ... |
| frege26 42494 | Identical to ~ idd . Prop... |
| frege27 42495 | We cannot (at the same tim... |
| frege9 42496 | Closed form of ~ syl with ... |
| frege12 42497 | A closed form of ~ com23 .... |
| frege11 42498 | Elimination of a nested an... |
| frege24 42499 | Closed form for ~ a1d . D... |
| frege16 42500 | A closed form of ~ com34 .... |
| frege25 42501 | Closed form for ~ a1dd . ... |
| frege18 42502 | Closed form of a syllogism... |
| frege22 42503 | A closed form of ~ com45 .... |
| frege10 42504 | Result commuting anteceden... |
| frege17 42505 | A closed form of ~ com3l .... |
| frege13 42506 | A closed form of ~ com3r .... |
| frege14 42507 | Closed form of a deduction... |
| frege19 42508 | A closed form of ~ syl6 . ... |
| frege23 42509 | Syllogism followed by rota... |
| frege15 42510 | A closed form of ~ com4r .... |
| frege21 42511 | Replace antecedent in ante... |
| frege20 42512 | A closed form of ~ syl8 . ... |
| axfrege28 42513 | Contraposition. Identical... |
| frege29 42515 | Closed form of ~ con3d . ... |
| frege30 42516 | Commuted, closed form of ~... |
| axfrege31 42517 | Identical to ~ notnotr . ... |
| frege32 42519 | Deduce ~ con1 from ~ con3 ... |
| frege33 42520 | If ` ph ` or ` ps ` takes ... |
| frege34 42521 | If as a consequence of the... |
| frege35 42522 | Commuted, closed form of ~... |
| frege36 42523 | The case in which ` ps ` i... |
| frege37 42524 | If ` ch ` is a necessary c... |
| frege38 42525 | Identical to ~ pm2.21 . P... |
| frege39 42526 | Syllogism between ~ pm2.18... |
| frege40 42527 | Anything implies ~ pm2.18 ... |
| axfrege41 42528 | Identical to ~ notnot . A... |
| frege42 42530 | Not not ~ id . Propositio... |
| frege43 42531 | If there is a choice only ... |
| frege44 42532 | Similar to a commuted ~ pm... |
| frege45 42533 | Deduce ~ pm2.6 from ~ con1... |
| frege46 42534 | If ` ps ` holds when ` ph ... |
| frege47 42535 | Deduce consequence follows... |
| frege48 42536 | Closed form of syllogism w... |
| frege49 42537 | Closed form of deduction w... |
| frege50 42538 | Closed form of ~ jaoi . P... |
| frege51 42539 | Compare with ~ jaod . Pro... |
| axfrege52a 42540 | Justification for ~ ax-fre... |
| frege52aid 42542 | The case when the content ... |
| frege53aid 42543 | Specialization of ~ frege5... |
| frege53a 42544 | Lemma for ~ frege55a . Pr... |
| axfrege54a 42545 | Justification for ~ ax-fre... |
| frege54cor0a 42547 | Synonym for logical equiva... |
| frege54cor1a 42548 | Reflexive equality. (Cont... |
| frege55aid 42549 | Lemma for ~ frege57aid . ... |
| frege55lem1a 42550 | Necessary deduction regard... |
| frege55lem2a 42551 | Core proof of Proposition ... |
| frege55a 42552 | Proposition 55 of [Frege18... |
| frege55cor1a 42553 | Proposition 55 of [Frege18... |
| frege56aid 42554 | Lemma for ~ frege57aid . ... |
| frege56a 42555 | Proposition 56 of [Frege18... |
| frege57aid 42556 | This is the all imporant f... |
| frege57a 42557 | Analogue of ~ frege57aid .... |
| axfrege58a 42558 | Identical to ~ anifp . Ju... |
| frege58acor 42560 | Lemma for ~ frege59a . (C... |
| frege59a 42561 | A kind of Aristotelian inf... |
| frege60a 42562 | Swap antecedents of ~ ax-f... |
| frege61a 42563 | Lemma for ~ frege65a . Pr... |
| frege62a 42564 | A kind of Aristotelian inf... |
| frege63a 42565 | Proposition 63 of [Frege18... |
| frege64a 42566 | Lemma for ~ frege65a . Pr... |
| frege65a 42567 | A kind of Aristotelian inf... |
| frege66a 42568 | Swap antecedents of ~ freg... |
| frege67a 42569 | Lemma for ~ frege68a . Pr... |
| frege68a 42570 | Combination of applying a ... |
| axfrege52c 42571 | Justification for ~ ax-fre... |
| frege52b 42573 | The case when the content ... |
| frege53b 42574 | Lemma for frege102 (via ~ ... |
| axfrege54c 42575 | Reflexive equality of clas... |
| frege54b 42577 | Reflexive equality of sets... |
| frege54cor1b 42578 | Reflexive equality. (Cont... |
| frege55lem1b 42579 | Necessary deduction regard... |
| frege55lem2b 42580 | Lemma for ~ frege55b . Co... |
| frege55b 42581 | Lemma for ~ frege57b . Pr... |
| frege56b 42582 | Lemma for ~ frege57b . Pr... |
| frege57b 42583 | Analogue of ~ frege57aid .... |
| axfrege58b 42584 | If ` A. x ph ` is affirmed... |
| frege58bid 42586 | If ` A. x ph ` is affirmed... |
| frege58bcor 42587 | Lemma for ~ frege59b . (C... |
| frege59b 42588 | A kind of Aristotelian inf... |
| frege60b 42589 | Swap antecedents of ~ ax-f... |
| frege61b 42590 | Lemma for ~ frege65b . Pr... |
| frege62b 42591 | A kind of Aristotelian inf... |
| frege63b 42592 | Lemma for ~ frege91 . Pro... |
| frege64b 42593 | Lemma for ~ frege65b . Pr... |
| frege65b 42594 | A kind of Aristotelian inf... |
| frege66b 42595 | Swap antecedents of ~ freg... |
| frege67b 42596 | Lemma for ~ frege68b . Pr... |
| frege68b 42597 | Combination of applying a ... |
| frege53c 42598 | Proposition 53 of [Frege18... |
| frege54cor1c 42599 | Reflexive equality. (Cont... |
| frege55lem1c 42600 | Necessary deduction regard... |
| frege55lem2c 42601 | Core proof of Proposition ... |
| frege55c 42602 | Proposition 55 of [Frege18... |
| frege56c 42603 | Lemma for ~ frege57c . Pr... |
| frege57c 42604 | Swap order of implication ... |
| frege58c 42605 | Principle related to ~ sp ... |
| frege59c 42606 | A kind of Aristotelian inf... |
| frege60c 42607 | Swap antecedents of ~ freg... |
| frege61c 42608 | Lemma for ~ frege65c . Pr... |
| frege62c 42609 | A kind of Aristotelian inf... |
| frege63c 42610 | Analogue of ~ frege63b . ... |
| frege64c 42611 | Lemma for ~ frege65c . Pr... |
| frege65c 42612 | A kind of Aristotelian inf... |
| frege66c 42613 | Swap antecedents of ~ freg... |
| frege67c 42614 | Lemma for ~ frege68c . Pr... |
| frege68c 42615 | Combination of applying a ... |
| dffrege69 42616 | If from the proposition th... |
| frege70 42617 | Lemma for ~ frege72 . Pro... |
| frege71 42618 | Lemma for ~ frege72 . Pro... |
| frege72 42619 | If property ` A ` is hered... |
| frege73 42620 | Lemma for ~ frege87 . Pro... |
| frege74 42621 | If ` X ` has a property ` ... |
| frege75 42622 | If from the proposition th... |
| dffrege76 42623 | If from the two propositio... |
| frege77 42624 | If ` Y ` follows ` X ` in ... |
| frege78 42625 | Commuted form of of ~ freg... |
| frege79 42626 | Distributed form of ~ freg... |
| frege80 42627 | Add additional condition t... |
| frege81 42628 | If ` X ` has a property ` ... |
| frege82 42629 | Closed-form deduction base... |
| frege83 42630 | Apply commuted form of ~ f... |
| frege84 42631 | Commuted form of ~ frege81... |
| frege85 42632 | Commuted form of ~ frege77... |
| frege86 42633 | Conclusion about element o... |
| frege87 42634 | If ` Z ` is a result of an... |
| frege88 42635 | Commuted form of ~ frege87... |
| frege89 42636 | One direction of ~ dffrege... |
| frege90 42637 | Add antecedent to ~ frege8... |
| frege91 42638 | Every result of an applica... |
| frege92 42639 | Inference from ~ frege91 .... |
| frege93 42640 | Necessary condition for tw... |
| frege94 42641 | Looking one past a pair re... |
| frege95 42642 | Looking one past a pair re... |
| frege96 42643 | Every result of an applica... |
| frege97 42644 | The property of following ... |
| frege98 42645 | If ` Y ` follows ` X ` and... |
| dffrege99 42646 | If ` Z ` is identical with... |
| frege100 42647 | One direction of ~ dffrege... |
| frege101 42648 | Lemma for ~ frege102 . Pr... |
| frege102 42649 | If ` Z ` belongs to the ` ... |
| frege103 42650 | Proposition 103 of [Frege1... |
| frege104 42651 | Proposition 104 of [Frege1... |
| frege105 42652 | Proposition 105 of [Frege1... |
| frege106 42653 | Whatever follows ` X ` in ... |
| frege107 42654 | Proposition 107 of [Frege1... |
| frege108 42655 | If ` Y ` belongs to the ` ... |
| frege109 42656 | The property of belonging ... |
| frege110 42657 | Proposition 110 of [Frege1... |
| frege111 42658 | If ` Y ` belongs to the ` ... |
| frege112 42659 | Identity implies belonging... |
| frege113 42660 | Proposition 113 of [Frege1... |
| frege114 42661 | If ` X ` belongs to the ` ... |
| dffrege115 42662 | If from the circumstance t... |
| frege116 42663 | One direction of ~ dffrege... |
| frege117 42664 | Lemma for ~ frege118 . Pr... |
| frege118 42665 | Simplified application of ... |
| frege119 42666 | Lemma for ~ frege120 . Pr... |
| frege120 42667 | Simplified application of ... |
| frege121 42668 | Lemma for ~ frege122 . Pr... |
| frege122 42669 | If ` X ` is a result of an... |
| frege123 42670 | Lemma for ~ frege124 . Pr... |
| frege124 42671 | If ` X ` is a result of an... |
| frege125 42672 | Lemma for ~ frege126 . Pr... |
| frege126 42673 | If ` M ` follows ` Y ` in ... |
| frege127 42674 | Communte antecedents of ~ ... |
| frege128 42675 | Lemma for ~ frege129 . Pr... |
| frege129 42676 | If the procedure ` R ` is ... |
| frege130 42677 | Lemma for ~ frege131 . Pr... |
| frege131 42678 | If the procedure ` R ` is ... |
| frege132 42679 | Lemma for ~ frege133 . Pr... |
| frege133 42680 | If the procedure ` R ` is ... |
| enrelmap 42681 | The set of all possible re... |
| enrelmapr 42682 | The set of all possible re... |
| enmappw 42683 | The set of all mappings fr... |
| enmappwid 42684 | The set of all mappings fr... |
| rfovd 42685 | Value of the operator, ` (... |
| rfovfvd 42686 | Value of the operator, ` (... |
| rfovfvfvd 42687 | Value of the operator, ` (... |
| rfovcnvf1od 42688 | Properties of the operator... |
| rfovcnvd 42689 | Value of the converse of t... |
| rfovf1od 42690 | The value of the operator,... |
| rfovcnvfvd 42691 | Value of the converse of t... |
| fsovd 42692 | Value of the operator, ` (... |
| fsovrfovd 42693 | The operator which gives a... |
| fsovfvd 42694 | Value of the operator, ` (... |
| fsovfvfvd 42695 | Value of the operator, ` (... |
| fsovfd 42696 | The operator, ` ( A O B ) ... |
| fsovcnvlem 42697 | The ` O ` operator, which ... |
| fsovcnvd 42698 | The value of the converse ... |
| fsovcnvfvd 42699 | The value of the converse ... |
| fsovf1od 42700 | The value of ` ( A O B ) `... |
| dssmapfvd 42701 | Value of the duality opera... |
| dssmapfv2d 42702 | Value of the duality opera... |
| dssmapfv3d 42703 | Value of the duality opera... |
| dssmapnvod 42704 | For any base set ` B ` the... |
| dssmapf1od 42705 | For any base set ` B ` the... |
| dssmap2d 42706 | For any base set ` B ` the... |
| or3or 42707 | Decompose disjunction into... |
| andi3or 42708 | Distribute over triple dis... |
| uneqsn 42709 | If a union of classes is e... |
| brfvimex 42710 | If a binary relation holds... |
| brovmptimex 42711 | If a binary relation holds... |
| brovmptimex1 42712 | If a binary relation holds... |
| brovmptimex2 42713 | If a binary relation holds... |
| brcoffn 42714 | Conditions allowing the de... |
| brcofffn 42715 | Conditions allowing the de... |
| brco2f1o 42716 | Conditions allowing the de... |
| brco3f1o 42717 | Conditions allowing the de... |
| ntrclsbex 42718 | If (pseudo-)interior and (... |
| ntrclsrcomplex 42719 | The relative complement of... |
| neik0imk0p 42720 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 42721 | If an interior function is... |
| ntrkbimka 42722 | If the interiors of disjoi... |
| ntrk0kbimka 42723 | If the interiors of disjoi... |
| clsk3nimkb 42724 | If the base set is not emp... |
| clsk1indlem0 42725 | The ansatz closure functio... |
| clsk1indlem2 42726 | The ansatz closure functio... |
| clsk1indlem3 42727 | The ansatz closure functio... |
| clsk1indlem4 42728 | The ansatz closure functio... |
| clsk1indlem1 42729 | The ansatz closure functio... |
| clsk1independent 42730 | For generalized closure fu... |
| neik0pk1imk0 42731 | Kuratowski's K0' and K1 ax... |
| isotone1 42732 | Two different ways to say ... |
| isotone2 42733 | Two different ways to say ... |
| ntrk1k3eqk13 42734 | An interior function is bo... |
| ntrclsf1o 42735 | If (pseudo-)interior and (... |
| ntrclsnvobr 42736 | If (pseudo-)interior and (... |
| ntrclsiex 42737 | If (pseudo-)interior and (... |
| ntrclskex 42738 | If (pseudo-)interior and (... |
| ntrclsfv1 42739 | If (pseudo-)interior and (... |
| ntrclsfv2 42740 | If (pseudo-)interior and (... |
| ntrclselnel1 42741 | If (pseudo-)interior and (... |
| ntrclselnel2 42742 | If (pseudo-)interior and (... |
| ntrclsfv 42743 | The value of the interior ... |
| ntrclsfveq1 42744 | If interior and closure fu... |
| ntrclsfveq2 42745 | If interior and closure fu... |
| ntrclsfveq 42746 | If interior and closure fu... |
| ntrclsss 42747 | If interior and closure fu... |
| ntrclsneine0lem 42748 | If (pseudo-)interior and (... |
| ntrclsneine0 42749 | If (pseudo-)interior and (... |
| ntrclscls00 42750 | If (pseudo-)interior and (... |
| ntrclsiso 42751 | If (pseudo-)interior and (... |
| ntrclsk2 42752 | An interior function is co... |
| ntrclskb 42753 | The interiors of disjoint ... |
| ntrclsk3 42754 | The intersection of interi... |
| ntrclsk13 42755 | The interior of the inters... |
| ntrclsk4 42756 | Idempotence of the interio... |
| ntrneibex 42757 | If (pseudo-)interior and (... |
| ntrneircomplex 42758 | The relative complement of... |
| ntrneif1o 42759 | If (pseudo-)interior and (... |
| ntrneiiex 42760 | If (pseudo-)interior and (... |
| ntrneinex 42761 | If (pseudo-)interior and (... |
| ntrneicnv 42762 | If (pseudo-)interior and (... |
| ntrneifv1 42763 | If (pseudo-)interior and (... |
| ntrneifv2 42764 | If (pseudo-)interior and (... |
| ntrneiel 42765 | If (pseudo-)interior and (... |
| ntrneifv3 42766 | The value of the neighbors... |
| ntrneineine0lem 42767 | If (pseudo-)interior and (... |
| ntrneineine1lem 42768 | If (pseudo-)interior and (... |
| ntrneifv4 42769 | The value of the interior ... |
| ntrneiel2 42770 | Membership in iterated int... |
| ntrneineine0 42771 | If (pseudo-)interior and (... |
| ntrneineine1 42772 | If (pseudo-)interior and (... |
| ntrneicls00 42773 | If (pseudo-)interior and (... |
| ntrneicls11 42774 | If (pseudo-)interior and (... |
| ntrneiiso 42775 | If (pseudo-)interior and (... |
| ntrneik2 42776 | An interior function is co... |
| ntrneix2 42777 | An interior (closure) func... |
| ntrneikb 42778 | The interiors of disjoint ... |
| ntrneixb 42779 | The interiors (closures) o... |
| ntrneik3 42780 | The intersection of interi... |
| ntrneix3 42781 | The closure of the union o... |
| ntrneik13 42782 | The interior of the inters... |
| ntrneix13 42783 | The closure of the union o... |
| ntrneik4w 42784 | Idempotence of the interio... |
| ntrneik4 42785 | Idempotence of the interio... |
| clsneibex 42786 | If (pseudo-)closure and (p... |
| clsneircomplex 42787 | The relative complement of... |
| clsneif1o 42788 | If a (pseudo-)closure func... |
| clsneicnv 42789 | If a (pseudo-)closure func... |
| clsneikex 42790 | If closure and neighborhoo... |
| clsneinex 42791 | If closure and neighborhoo... |
| clsneiel1 42792 | If a (pseudo-)closure func... |
| clsneiel2 42793 | If a (pseudo-)closure func... |
| clsneifv3 42794 | Value of the neighborhoods... |
| clsneifv4 42795 | Value of the closure (inte... |
| neicvgbex 42796 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 42797 | The relative complement of... |
| neicvgf1o 42798 | If neighborhood and conver... |
| neicvgnvo 42799 | If neighborhood and conver... |
| neicvgnvor 42800 | If neighborhood and conver... |
| neicvgmex 42801 | If the neighborhoods and c... |
| neicvgnex 42802 | If the neighborhoods and c... |
| neicvgel1 42803 | A subset being an element ... |
| neicvgel2 42804 | The complement of a subset... |
| neicvgfv 42805 | The value of the neighborh... |
| ntrrn 42806 | The range of the interior ... |
| ntrf 42807 | The interior function of a... |
| ntrf2 42808 | The interior function is a... |
| ntrelmap 42809 | The interior function is a... |
| clsf2 42810 | The closure function is a ... |
| clselmap 42811 | The closure function is a ... |
| dssmapntrcls 42812 | The interior and closure o... |
| dssmapclsntr 42813 | The closure and interior o... |
| gneispa 42814 | Each point ` p ` of the ne... |
| gneispb 42815 | Given a neighborhood ` N `... |
| gneispace2 42816 | The predicate that ` F ` i... |
| gneispace3 42817 | The predicate that ` F ` i... |
| gneispace 42818 | The predicate that ` F ` i... |
| gneispacef 42819 | A generic neighborhood spa... |
| gneispacef2 42820 | A generic neighborhood spa... |
| gneispacefun 42821 | A generic neighborhood spa... |
| gneispacern 42822 | A generic neighborhood spa... |
| gneispacern2 42823 | A generic neighborhood spa... |
| gneispace0nelrn 42824 | A generic neighborhood spa... |
| gneispace0nelrn2 42825 | A generic neighborhood spa... |
| gneispace0nelrn3 42826 | A generic neighborhood spa... |
| gneispaceel 42827 | Every neighborhood of a po... |
| gneispaceel2 42828 | Every neighborhood of a po... |
| gneispacess 42829 | All supersets of a neighbo... |
| gneispacess2 42830 | All supersets of a neighbo... |
| k0004lem1 42831 | Application of ~ ssin to r... |
| k0004lem2 42832 | A mapping with a particula... |
| k0004lem3 42833 | When the value of a mappin... |
| k0004val 42834 | The topological simplex of... |
| k0004ss1 42835 | The topological simplex of... |
| k0004ss2 42836 | The topological simplex of... |
| k0004ss3 42837 | The topological simplex of... |
| k0004val0 42838 | The topological simplex of... |
| inductionexd 42839 | Simple induction example. ... |
| wwlemuld 42840 | Natural deduction form of ... |
| leeq1d 42841 | Specialization of ~ breq1d... |
| leeq2d 42842 | Specialization of ~ breq2d... |
| absmulrposd 42843 | Specialization of absmuld ... |
| imadisjld 42844 | Natural dduction form of o... |
| imadisjlnd 42845 | Natural deduction form of ... |
| wnefimgd 42846 | The image of a mapping fro... |
| fco2d 42847 | Natural deduction form of ... |
| wfximgfd 42848 | The value of a function on... |
| extoimad 42849 | If |f(x)| <= C for all x t... |
| imo72b2lem0 42850 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 42851 | Natural deduction form of ... |
| imo72b2lem2 42852 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 42853 | Natural deduction form of ... |
| imo72b2lem1 42854 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 42855 | 'Less than or equal to' re... |
| lemuldiv4d 42856 | 'Less than or equal to' re... |
| imo72b2 42857 | IMO 1972 B2. (14th Intern... |
| int-addcomd 42858 | AdditionCommutativity gene... |
| int-addassocd 42859 | AdditionAssociativity gene... |
| int-addsimpd 42860 | AdditionSimplification gen... |
| int-mulcomd 42861 | MultiplicationCommutativit... |
| int-mulassocd 42862 | MultiplicationAssociativit... |
| int-mulsimpd 42863 | MultiplicationSimplificati... |
| int-leftdistd 42864 | AdditionMultiplicationLeft... |
| int-rightdistd 42865 | AdditionMultiplicationRigh... |
| int-sqdefd 42866 | SquareDefinition generator... |
| int-mul11d 42867 | First MultiplicationOne ge... |
| int-mul12d 42868 | Second MultiplicationOne g... |
| int-add01d 42869 | First AdditionZero generat... |
| int-add02d 42870 | Second AdditionZero genera... |
| int-sqgeq0d 42871 | SquareGEQZero generator ru... |
| int-eqprincd 42872 | PrincipleOfEquality genera... |
| int-eqtransd 42873 | EqualityTransitivity gener... |
| int-eqmvtd 42874 | EquMoveTerm generator rule... |
| int-eqineqd 42875 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 42876 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 42877 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 42878 | SecondPrincipleOfInequalit... |
| int-ineqtransd 42879 | InequalityTransitivity gen... |
| unitadd 42880 | Theorem used in conjunctio... |
| gsumws3 42881 | Valuation of a length 3 wo... |
| gsumws4 42882 | Valuation of a length 4 wo... |
| amgm2d 42883 | Arithmetic-geometric mean ... |
| amgm3d 42884 | Arithmetic-geometric mean ... |
| amgm4d 42885 | Arithmetic-geometric mean ... |
| spALT 42886 | ~ sp can be proven from th... |
| elnelneqd 42887 | Two classes are not equal ... |
| elnelneq2d 42888 | Two classes are not equal ... |
| rr-spce 42889 | Prove an existential. (Co... |
| rexlimdvaacbv 42890 | Unpack a restricted existe... |
| rexlimddvcbvw 42891 | Unpack a restricted existe... |
| rexlimddvcbv 42892 | Unpack a restricted existe... |
| rr-elrnmpt3d 42893 | Elementhood in an image se... |
| finnzfsuppd 42894 | If a function is zero outs... |
| rr-phpd 42895 | Equivalent of ~ php withou... |
| suceqd 42896 | Deduction associated with ... |
| tfindsd 42897 | Deduction associated with ... |
| mnringvald 42900 | Value of the monoid ring f... |
| mnringnmulrd 42901 | Components of a monoid rin... |
| mnringnmulrdOLD 42902 | Obsolete version of ~ mnri... |
| mnringbased 42903 | The base set of a monoid r... |
| mnringbasedOLD 42904 | Obsolete version of ~ mnri... |
| mnringbaserd 42905 | The base set of a monoid r... |
| mnringelbased 42906 | Membership in the base set... |
| mnringbasefd 42907 | Elements of a monoid ring ... |
| mnringbasefsuppd 42908 | Elements of a monoid ring ... |
| mnringaddgd 42909 | The additive operation of ... |
| mnringaddgdOLD 42910 | Obsolete version of ~ mnri... |
| mnring0gd 42911 | The additive identity of a... |
| mnring0g2d 42912 | The additive identity of a... |
| mnringmulrd 42913 | The ring product of a mono... |
| mnringscad 42914 | The scalar ring of a monoi... |
| mnringscadOLD 42915 | Obsolete version of ~ mnri... |
| mnringvscad 42916 | The scalar product of a mo... |
| mnringvscadOLD 42917 | Obsolete version of ~ mnri... |
| mnringlmodd 42918 | Monoid rings are left modu... |
| mnringmulrvald 42919 | Value of multiplication in... |
| mnringmulrcld 42920 | Monoid rings are closed un... |
| gru0eld 42921 | A nonempty Grothendieck un... |
| grusucd 42922 | Grothendieck universes are... |
| r1rankcld 42923 | Any rank of the cumulative... |
| grur1cld 42924 | Grothendieck universes are... |
| grurankcld 42925 | Grothendieck universes are... |
| grurankrcld 42926 | If a Grothendieck universe... |
| scotteqd 42929 | Equality theorem for the S... |
| scotteq 42930 | Closed form of ~ scotteqd ... |
| nfscott 42931 | Bound-variable hypothesis ... |
| scottabf 42932 | Value of the Scott operati... |
| scottab 42933 | Value of the Scott operati... |
| scottabes 42934 | Value of the Scott operati... |
| scottss 42935 | Scott's trick produces a s... |
| elscottab 42936 | An element of the output o... |
| scottex2 42937 | ~ scottex expressed using ... |
| scotteld 42938 | The Scott operation sends ... |
| scottelrankd 42939 | Property of a Scott's tric... |
| scottrankd 42940 | Rank of a nonempty Scott's... |
| gruscottcld 42941 | If a Grothendieck universe... |
| dfcoll2 42944 | Alternate definition of th... |
| colleq12d 42945 | Equality theorem for the c... |
| colleq1 42946 | Equality theorem for the c... |
| colleq2 42947 | Equality theorem for the c... |
| nfcoll 42948 | Bound-variable hypothesis ... |
| collexd 42949 | The output of the collecti... |
| cpcolld 42950 | Property of the collection... |
| cpcoll2d 42951 | ~ cpcolld with an extra ex... |
| grucollcld 42952 | A Grothendieck universe co... |
| ismnu 42953 | The hypothesis of this the... |
| mnuop123d 42954 | Operations of a minimal un... |
| mnussd 42955 | Minimal universes are clos... |
| mnuss2d 42956 | ~ mnussd with arguments pr... |
| mnu0eld 42957 | A nonempty minimal univers... |
| mnuop23d 42958 | Second and third operation... |
| mnupwd 42959 | Minimal universes are clos... |
| mnusnd 42960 | Minimal universes are clos... |
| mnuprssd 42961 | A minimal universe contain... |
| mnuprss2d 42962 | Special case of ~ mnuprssd... |
| mnuop3d 42963 | Third operation of a minim... |
| mnuprdlem1 42964 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 42965 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 42966 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 42967 | Lemma for ~ mnuprd . Gene... |
| mnuprd 42968 | Minimal universes are clos... |
| mnuunid 42969 | Minimal universes are clos... |
| mnuund 42970 | Minimal universes are clos... |
| mnutrcld 42971 | Minimal universes contain ... |
| mnutrd 42972 | Minimal universes are tran... |
| mnurndlem1 42973 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 42974 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 42975 | Minimal universes contain ... |
| mnugrud 42976 | Minimal universes are Grot... |
| grumnudlem 42977 | Lemma for ~ grumnud . (Co... |
| grumnud 42978 | Grothendieck universes are... |
| grumnueq 42979 | The class of Grothendieck ... |
| expandan 42980 | Expand conjunction to prim... |
| expandexn 42981 | Expand an existential quan... |
| expandral 42982 | Expand a restricted univer... |
| expandrexn 42983 | Expand a restricted existe... |
| expandrex 42984 | Expand a restricted existe... |
| expanduniss 42985 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 42986 | Express the predicate on `... |
| rr-grothprimbi 42987 | Express "every set is cont... |
| inagrud 42988 | Inaccessible levels of the... |
| inaex 42989 | Assuming the Tarski-Grothe... |
| gruex 42990 | Assuming the Tarski-Grothe... |
| rr-groth 42991 | An equivalent of ~ ax-grot... |
| rr-grothprim 42992 | An equivalent of ~ ax-grot... |
| ismnushort 42993 | Express the predicate on `... |
| dfuniv2 42994 | Alternative definition of ... |
| rr-grothshortbi 42995 | Express "every set is cont... |
| rr-grothshort 42996 | A shorter equivalent of ~ ... |
| nanorxor 42997 | 'nand' is equivalent to th... |
| undisjrab 42998 | Union of two disjoint rest... |
| iso0 42999 | The empty set is an ` R , ... |
| ssrecnpr 43000 | ` RR ` is a subset of both... |
| seff 43001 | Let set ` S ` be the real ... |
| sblpnf 43002 | The infinity ball in the a... |
| prmunb2 43003 | The primes are unbounded. ... |
| dvgrat 43004 | Ratio test for divergence ... |
| cvgdvgrat 43005 | Ratio test for convergence... |
| radcnvrat 43006 | Let ` L ` be the limit, if... |
| reldvds 43007 | The divides relation is in... |
| nznngen 43008 | All positive integers in t... |
| nzss 43009 | The set of multiples of _m... |
| nzin 43010 | The intersection of the se... |
| nzprmdif 43011 | Subtract one prime's multi... |
| hashnzfz 43012 | Special case of ~ hashdvds... |
| hashnzfz2 43013 | Special case of ~ hashnzfz... |
| hashnzfzclim 43014 | As the upper bound ` K ` o... |
| caofcan 43015 | Transfer a cancellation la... |
| ofsubid 43016 | Function analogue of ~ sub... |
| ofmul12 43017 | Function analogue of ~ mul... |
| ofdivrec 43018 | Function analogue of ~ div... |
| ofdivcan4 43019 | Function analogue of ~ div... |
| ofdivdiv2 43020 | Function analogue of ~ div... |
| lhe4.4ex1a 43021 | Example of the Fundamental... |
| dvsconst 43022 | Derivative of a constant f... |
| dvsid 43023 | Derivative of the identity... |
| dvsef 43024 | Derivative of the exponent... |
| expgrowthi 43025 | Exponential growth and dec... |
| dvconstbi 43026 | The derivative of a functi... |
| expgrowth 43027 | Exponential growth and dec... |
| bccval 43030 | Value of the generalized b... |
| bcccl 43031 | Closure of the generalized... |
| bcc0 43032 | The generalized binomial c... |
| bccp1k 43033 | Generalized binomial coeff... |
| bccm1k 43034 | Generalized binomial coeff... |
| bccn0 43035 | Generalized binomial coeff... |
| bccn1 43036 | Generalized binomial coeff... |
| bccbc 43037 | The binomial coefficient a... |
| uzmptshftfval 43038 | When ` F ` is a maps-to fu... |
| dvradcnv2 43039 | The radius of convergence ... |
| binomcxplemwb 43040 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 43041 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 43042 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 43043 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 43044 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 43045 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 43046 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 43047 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 43048 | Lemma for ~ binomcxp . Wh... |
| binomcxp 43049 | Generalize the binomial th... |
| pm10.12 43050 | Theorem *10.12 in [Whitehe... |
| pm10.14 43051 | Theorem *10.14 in [Whitehe... |
| pm10.251 43052 | Theorem *10.251 in [Whiteh... |
| pm10.252 43053 | Theorem *10.252 in [Whiteh... |
| pm10.253 43054 | Theorem *10.253 in [Whiteh... |
| albitr 43055 | Theorem *10.301 in [Whiteh... |
| pm10.42 43056 | Theorem *10.42 in [Whitehe... |
| pm10.52 43057 | Theorem *10.52 in [Whitehe... |
| pm10.53 43058 | Theorem *10.53 in [Whitehe... |
| pm10.541 43059 | Theorem *10.541 in [Whiteh... |
| pm10.542 43060 | Theorem *10.542 in [Whiteh... |
| pm10.55 43061 | Theorem *10.55 in [Whitehe... |
| pm10.56 43062 | Theorem *10.56 in [Whitehe... |
| pm10.57 43063 | Theorem *10.57 in [Whitehe... |
| 2alanimi 43064 | Removes two universal quan... |
| 2al2imi 43065 | Removes two universal quan... |
| pm11.11 43066 | Theorem *11.11 in [Whitehe... |
| pm11.12 43067 | Theorem *11.12 in [Whitehe... |
| 19.21vv 43068 | Compare Theorem *11.3 in [... |
| 2alim 43069 | Theorem *11.32 in [Whitehe... |
| 2albi 43070 | Theorem *11.33 in [Whitehe... |
| 2exim 43071 | Theorem *11.34 in [Whitehe... |
| 2exbi 43072 | Theorem *11.341 in [Whiteh... |
| spsbce-2 43073 | Theorem *11.36 in [Whitehe... |
| 19.33-2 43074 | Theorem *11.421 in [Whiteh... |
| 19.36vv 43075 | Theorem *11.43 in [Whitehe... |
| 19.31vv 43076 | Theorem *11.44 in [Whitehe... |
| 19.37vv 43077 | Theorem *11.46 in [Whitehe... |
| 19.28vv 43078 | Theorem *11.47 in [Whitehe... |
| pm11.52 43079 | Theorem *11.52 in [Whitehe... |
| aaanv 43080 | Theorem *11.56 in [Whitehe... |
| pm11.57 43081 | Theorem *11.57 in [Whitehe... |
| pm11.58 43082 | Theorem *11.58 in [Whitehe... |
| pm11.59 43083 | Theorem *11.59 in [Whitehe... |
| pm11.6 43084 | Theorem *11.6 in [Whitehea... |
| pm11.61 43085 | Theorem *11.61 in [Whitehe... |
| pm11.62 43086 | Theorem *11.62 in [Whitehe... |
| pm11.63 43087 | Theorem *11.63 in [Whitehe... |
| pm11.7 43088 | Theorem *11.7 in [Whitehea... |
| pm11.71 43089 | Theorem *11.71 in [Whitehe... |
| sbeqal1 43090 | If ` x = y ` always implie... |
| sbeqal1i 43091 | Suppose you know ` x = y `... |
| sbeqal2i 43092 | If ` x = y ` implies ` x =... |
| axc5c4c711 43093 | Proof of a theorem that ca... |
| axc5c4c711toc5 43094 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 43095 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 43096 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 43097 | Rederivation of ~ ax-11 fr... |
| axc11next 43098 | This theorem shows that, g... |
| pm13.13a 43099 | One result of theorem *13.... |
| pm13.13b 43100 | Theorem *13.13 in [Whitehe... |
| pm13.14 43101 | Theorem *13.14 in [Whitehe... |
| pm13.192 43102 | Theorem *13.192 in [Whiteh... |
| pm13.193 43103 | Theorem *13.193 in [Whiteh... |
| pm13.194 43104 | Theorem *13.194 in [Whiteh... |
| pm13.195 43105 | Theorem *13.195 in [Whiteh... |
| pm13.196a 43106 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 43107 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 43108 | Theorem *13.22 in [Whitehe... |
| iotain 43109 | Equivalence between two di... |
| iotaexeu 43110 | The iota class exists. Th... |
| iotasbc 43111 | Definition *14.01 in [Whit... |
| iotasbc2 43112 | Theorem *14.111 in [Whiteh... |
| pm14.12 43113 | Theorem *14.12 in [Whitehe... |
| pm14.122a 43114 | Theorem *14.122 in [Whiteh... |
| pm14.122b 43115 | Theorem *14.122 in [Whiteh... |
| pm14.122c 43116 | Theorem *14.122 in [Whiteh... |
| pm14.123a 43117 | Theorem *14.123 in [Whiteh... |
| pm14.123b 43118 | Theorem *14.123 in [Whiteh... |
| pm14.123c 43119 | Theorem *14.123 in [Whiteh... |
| pm14.18 43120 | Theorem *14.18 in [Whitehe... |
| iotaequ 43121 | Theorem *14.2 in [Whitehea... |
| iotavalb 43122 | Theorem *14.202 in [Whiteh... |
| iotasbc5 43123 | Theorem *14.205 in [Whiteh... |
| pm14.24 43124 | Theorem *14.24 in [Whitehe... |
| iotavalsb 43125 | Theorem *14.242 in [Whiteh... |
| sbiota1 43126 | Theorem *14.25 in [Whitehe... |
| sbaniota 43127 | Theorem *14.26 in [Whitehe... |
| eubiOLD 43128 | Obsolete proof of ~ eubi a... |
| iotasbcq 43129 | Theorem *14.272 in [Whiteh... |
| elnev 43130 | Any set that contains one ... |
| rusbcALT 43131 | A version of Russell's par... |
| compeq 43132 | Equality between two ways ... |
| compne 43133 | The complement of ` A ` is... |
| compab 43134 | Two ways of saying "the co... |
| conss2 43135 | Contrapositive law for sub... |
| conss1 43136 | Contrapositive law for sub... |
| ralbidar 43137 | More general form of ~ ral... |
| rexbidar 43138 | More general form of ~ rex... |
| dropab1 43139 | Theorem to aid use of the ... |
| dropab2 43140 | Theorem to aid use of the ... |
| ipo0 43141 | If the identity relation p... |
| ifr0 43142 | A class that is founded by... |
| ordpss 43143 | ~ ordelpss with an anteced... |
| fvsb 43144 | Explicit substitution of a... |
| fveqsb 43145 | Implicit substitution of a... |
| xpexb 43146 | A Cartesian product exists... |
| trelpss 43147 | An element of a transitive... |
| addcomgi 43148 | Generalization of commutat... |
| addrval 43158 | Value of the operation of ... |
| subrval 43159 | Value of the operation of ... |
| mulvval 43160 | Value of the operation of ... |
| addrfv 43161 | Vector addition at a value... |
| subrfv 43162 | Vector subtraction at a va... |
| mulvfv 43163 | Scalar multiplication at a... |
| addrfn 43164 | Vector addition produces a... |
| subrfn 43165 | Vector subtraction produce... |
| mulvfn 43166 | Scalar multiplication prod... |
| addrcom 43167 | Vector addition is commuta... |
| idiALT 43171 | Placeholder for ~ idi . T... |
| exbir 43172 | Exportation implication al... |
| 3impexpbicom 43173 | Version of ~ 3impexp where... |
| 3impexpbicomi 43174 | Inference associated with ... |
| bi1imp 43175 | Importation inference simi... |
| bi2imp 43176 | Importation inference simi... |
| bi3impb 43177 | Similar to ~ 3impb with im... |
| bi3impa 43178 | Similar to ~ 3impa with im... |
| bi23impib 43179 | ~ 3impib with the inner im... |
| bi13impib 43180 | ~ 3impib with the outer im... |
| bi123impib 43181 | ~ 3impib with the implicat... |
| bi13impia 43182 | ~ 3impia with the outer im... |
| bi123impia 43183 | ~ 3impia with the implicat... |
| bi33imp12 43184 | ~ 3imp with innermost impl... |
| bi23imp13 43185 | ~ 3imp with middle implica... |
| bi13imp23 43186 | ~ 3imp with outermost impl... |
| bi13imp2 43187 | Similar to ~ 3imp except t... |
| bi12imp3 43188 | Similar to ~ 3imp except a... |
| bi23imp1 43189 | Similar to ~ 3imp except a... |
| bi123imp0 43190 | Similar to ~ 3imp except a... |
| 4animp1 43191 | A single hypothesis unific... |
| 4an31 43192 | A rearrangement of conjunc... |
| 4an4132 43193 | A rearrangement of conjunc... |
| expcomdg 43194 | Biconditional form of ~ ex... |
| iidn3 43195 | ~ idn3 without virtual ded... |
| ee222 43196 | ~ e222 without virtual ded... |
| ee3bir 43197 | Right-biconditional form o... |
| ee13 43198 | ~ e13 without virtual dedu... |
| ee121 43199 | ~ e121 without virtual ded... |
| ee122 43200 | ~ e122 without virtual ded... |
| ee333 43201 | ~ e333 without virtual ded... |
| ee323 43202 | ~ e323 without virtual ded... |
| 3ornot23 43203 | If the second and third di... |
| orbi1r 43204 | ~ orbi1 with order of disj... |
| 3orbi123 43205 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 43206 | Closed form of ~ syl5 . D... |
| impexpd 43207 | The following User's Proof... |
| com3rgbi 43208 | The following User's Proof... |
| impexpdcom 43209 | The following User's Proof... |
| ee1111 43210 | Non-virtual deduction form... |
| pm2.43bgbi 43211 | Logical equivalence of a 2... |
| pm2.43cbi 43212 | Logical equivalence of a 3... |
| ee233 43213 | Non-virtual deduction form... |
| imbi13 43214 | Join three logical equival... |
| ee33 43215 | Non-virtual deduction form... |
| con5 43216 | Biconditional contrapositi... |
| con5i 43217 | Inference form of ~ con5 .... |
| exlimexi 43218 | Inference similar to Theor... |
| sb5ALT 43219 | Equivalence for substituti... |
| eexinst01 43220 | ~ exinst01 without virtual... |
| eexinst11 43221 | ~ exinst11 without virtual... |
| vk15.4j 43222 | Excercise 4j of Unit 15 of... |
| notnotrALT 43223 | Converse of double negatio... |
| con3ALT2 43224 | Contraposition. Alternate... |
| ssralv2 43225 | Quantification restricted ... |
| sbc3or 43226 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 43227 | Closed form of ~ alrimi wi... |
| rspsbc2 43228 | ~ rspsbc with two quantify... |
| sbcoreleleq 43229 | Substitution of a setvar v... |
| tratrb 43230 | If a class is transitive a... |
| ordelordALT 43231 | An element of an ordinal c... |
| sbcim2g 43232 | Distribution of class subs... |
| sbcbi 43233 | Implication form of ~ sbcb... |
| trsbc 43234 | Formula-building inference... |
| truniALT 43235 | The union of a class of tr... |
| onfrALTlem5 43236 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 43237 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 43238 | Lemma for ~ onfrALT . (Co... |
| ggen31 43239 | ~ gen31 without virtual de... |
| onfrALTlem2 43240 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 43241 | A theorem pertaining to th... |
| onfrALTlem1 43242 | Lemma for ~ onfrALT . (Co... |
| onfrALT 43243 | The membership relation is... |
| 19.41rg 43244 | Closed form of right-to-le... |
| opelopab4 43245 | Ordered pair membership in... |
| 2pm13.193 43246 | ~ pm13.193 for two variabl... |
| hbntal 43247 | A closed form of ~ hbn . ~... |
| hbimpg 43248 | A closed form of ~ hbim . ... |
| hbalg 43249 | Closed form of ~ hbal . D... |
| hbexg 43250 | Closed form of ~ nfex . D... |
| ax6e2eq 43251 | Alternate form of ~ ax6e f... |
| ax6e2nd 43252 | If at least two sets exist... |
| ax6e2ndeq 43253 | "At least two sets exist" ... |
| 2sb5nd 43254 | Equivalence for double sub... |
| 2uasbanh 43255 | Distribute the unabbreviat... |
| 2uasban 43256 | Distribute the unabbreviat... |
| e2ebind 43257 | Absorption of an existenti... |
| elpwgded 43258 | ~ elpwgdedVD in convention... |
| trelded 43259 | Deduction form of ~ trel .... |
| jaoded 43260 | Deduction form of ~ jao . ... |
| sbtT 43261 | A substitution into a theo... |
| not12an2impnot1 43262 | If a double conjunction is... |
| in1 43265 | Inference form of ~ df-vd1... |
| iin1 43266 | ~ in1 without virtual dedu... |
| dfvd1ir 43267 | Inference form of ~ df-vd1... |
| idn1 43268 | Virtual deduction identity... |
| dfvd1imp 43269 | Left-to-right part of defi... |
| dfvd1impr 43270 | Right-to-left part of defi... |
| dfvd2 43273 | Definition of a 2-hypothes... |
| dfvd2an 43276 | Definition of a 2-hypothes... |
| dfvd2ani 43277 | Inference form of ~ dfvd2a... |
| dfvd2anir 43278 | Right-to-left inference fo... |
| dfvd2i 43279 | Inference form of ~ dfvd2 ... |
| dfvd2ir 43280 | Right-to-left inference fo... |
| dfvd3 43285 | Definition of a 3-hypothes... |
| dfvd3i 43286 | Inference form of ~ dfvd3 ... |
| dfvd3ir 43287 | Right-to-left inference fo... |
| dfvd3an 43288 | Definition of a 3-hypothes... |
| dfvd3ani 43289 | Inference form of ~ dfvd3a... |
| dfvd3anir 43290 | Right-to-left inference fo... |
| vd01 43291 | A virtual hypothesis virtu... |
| vd02 43292 | Two virtual hypotheses vir... |
| vd03 43293 | A theorem is virtually inf... |
| vd12 43294 | A virtual deduction with 1... |
| vd13 43295 | A virtual deduction with 1... |
| vd23 43296 | A virtual deduction with 2... |
| dfvd2imp 43297 | The virtual deduction form... |
| dfvd2impr 43298 | A 2-antecedent nested impl... |
| in2 43299 | The virtual deduction intr... |
| int2 43300 | The virtual deduction intr... |
| iin2 43301 | ~ in2 without virtual dedu... |
| in2an 43302 | The virtual deduction intr... |
| in3 43303 | The virtual deduction intr... |
| iin3 43304 | ~ in3 without virtual dedu... |
| in3an 43305 | The virtual deduction intr... |
| int3 43306 | The virtual deduction intr... |
| idn2 43307 | Virtual deduction identity... |
| iden2 43308 | Virtual deduction identity... |
| idn3 43309 | Virtual deduction identity... |
| gen11 43310 | Virtual deduction generali... |
| gen11nv 43311 | Virtual deduction generali... |
| gen12 43312 | Virtual deduction generali... |
| gen21 43313 | Virtual deduction generali... |
| gen21nv 43314 | Virtual deduction form of ... |
| gen31 43315 | Virtual deduction generali... |
| gen22 43316 | Virtual deduction generali... |
| ggen22 43317 | ~ gen22 without virtual de... |
| exinst 43318 | Existential Instantiation.... |
| exinst01 43319 | Existential Instantiation.... |
| exinst11 43320 | Existential Instantiation.... |
| e1a 43321 | A Virtual deduction elimin... |
| el1 43322 | A Virtual deduction elimin... |
| e1bi 43323 | Biconditional form of ~ e1... |
| e1bir 43324 | Right biconditional form o... |
| e2 43325 | A virtual deduction elimin... |
| e2bi 43326 | Biconditional form of ~ e2... |
| e2bir 43327 | Right biconditional form o... |
| ee223 43328 | ~ e223 without virtual ded... |
| e223 43329 | A virtual deduction elimin... |
| e222 43330 | A virtual deduction elimin... |
| e220 43331 | A virtual deduction elimin... |
| ee220 43332 | ~ e220 without virtual ded... |
| e202 43333 | A virtual deduction elimin... |
| ee202 43334 | ~ e202 without virtual ded... |
| e022 43335 | A virtual deduction elimin... |
| ee022 43336 | ~ e022 without virtual ded... |
| e002 43337 | A virtual deduction elimin... |
| ee002 43338 | ~ e002 without virtual ded... |
| e020 43339 | A virtual deduction elimin... |
| ee020 43340 | ~ e020 without virtual ded... |
| e200 43341 | A virtual deduction elimin... |
| ee200 43342 | ~ e200 without virtual ded... |
| e221 43343 | A virtual deduction elimin... |
| ee221 43344 | ~ e221 without virtual ded... |
| e212 43345 | A virtual deduction elimin... |
| ee212 43346 | ~ e212 without virtual ded... |
| e122 43347 | A virtual deduction elimin... |
| e112 43348 | A virtual deduction elimin... |
| ee112 43349 | ~ e112 without virtual ded... |
| e121 43350 | A virtual deduction elimin... |
| e211 43351 | A virtual deduction elimin... |
| ee211 43352 | ~ e211 without virtual ded... |
| e210 43353 | A virtual deduction elimin... |
| ee210 43354 | ~ e210 without virtual ded... |
| e201 43355 | A virtual deduction elimin... |
| ee201 43356 | ~ e201 without virtual ded... |
| e120 43357 | A virtual deduction elimin... |
| ee120 43358 | Virtual deduction rule ~ e... |
| e021 43359 | A virtual deduction elimin... |
| ee021 43360 | ~ e021 without virtual ded... |
| e012 43361 | A virtual deduction elimin... |
| ee012 43362 | ~ e012 without virtual ded... |
| e102 43363 | A virtual deduction elimin... |
| ee102 43364 | ~ e102 without virtual ded... |
| e22 43365 | A virtual deduction elimin... |
| e22an 43366 | Conjunction form of ~ e22 ... |
| ee22an 43367 | ~ e22an without virtual de... |
| e111 43368 | A virtual deduction elimin... |
| e1111 43369 | A virtual deduction elimin... |
| e110 43370 | A virtual deduction elimin... |
| ee110 43371 | ~ e110 without virtual ded... |
| e101 43372 | A virtual deduction elimin... |
| ee101 43373 | ~ e101 without virtual ded... |
| e011 43374 | A virtual deduction elimin... |
| ee011 43375 | ~ e011 without virtual ded... |
| e100 43376 | A virtual deduction elimin... |
| ee100 43377 | ~ e100 without virtual ded... |
| e010 43378 | A virtual deduction elimin... |
| ee010 43379 | ~ e010 without virtual ded... |
| e001 43380 | A virtual deduction elimin... |
| ee001 43381 | ~ e001 without virtual ded... |
| e11 43382 | A virtual deduction elimin... |
| e11an 43383 | Conjunction form of ~ e11 ... |
| ee11an 43384 | ~ e11an without virtual de... |
| e01 43385 | A virtual deduction elimin... |
| e01an 43386 | Conjunction form of ~ e01 ... |
| ee01an 43387 | ~ e01an without virtual de... |
| e10 43388 | A virtual deduction elimin... |
| e10an 43389 | Conjunction form of ~ e10 ... |
| ee10an 43390 | ~ e10an without virtual de... |
| e02 43391 | A virtual deduction elimin... |
| e02an 43392 | Conjunction form of ~ e02 ... |
| ee02an 43393 | ~ e02an without virtual de... |
| eel021old 43394 | ~ el021old without virtual... |
| el021old 43395 | A virtual deduction elimin... |
| eel132 43396 | ~ syl2an with antecedents ... |
| eel000cT 43397 | An elimination deduction. ... |
| eel0TT 43398 | An elimination deduction. ... |
| eelT00 43399 | An elimination deduction. ... |
| eelTTT 43400 | An elimination deduction. ... |
| eelT11 43401 | An elimination deduction. ... |
| eelT1 43402 | Syllogism inference combin... |
| eelT12 43403 | An elimination deduction. ... |
| eelTT1 43404 | An elimination deduction. ... |
| eelT01 43405 | An elimination deduction. ... |
| eel0T1 43406 | An elimination deduction. ... |
| eel12131 43407 | An elimination deduction. ... |
| eel2131 43408 | ~ syl2an with antecedents ... |
| eel3132 43409 | ~ syl2an with antecedents ... |
| eel0321old 43410 | ~ el0321old without virtua... |
| el0321old 43411 | A virtual deduction elimin... |
| eel2122old 43412 | ~ el2122old without virtua... |
| el2122old 43413 | A virtual deduction elimin... |
| eel0000 43414 | Elimination rule similar t... |
| eel00001 43415 | An elimination deduction. ... |
| eel00000 43416 | Elimination rule similar ~... |
| eel11111 43417 | Five-hypothesis eliminatio... |
| e12 43418 | A virtual deduction elimin... |
| e12an 43419 | Conjunction form of ~ e12 ... |
| el12 43420 | Virtual deduction form of ... |
| e20 43421 | A virtual deduction elimin... |
| e20an 43422 | Conjunction form of ~ e20 ... |
| ee20an 43423 | ~ e20an without virtual de... |
| e21 43424 | A virtual deduction elimin... |
| e21an 43425 | Conjunction form of ~ e21 ... |
| ee21an 43426 | ~ e21an without virtual de... |
| e333 43427 | A virtual deduction elimin... |
| e33 43428 | A virtual deduction elimin... |
| e33an 43429 | Conjunction form of ~ e33 ... |
| ee33an 43430 | ~ e33an without virtual de... |
| e3 43431 | Meta-connective form of ~ ... |
| e3bi 43432 | Biconditional form of ~ e3... |
| e3bir 43433 | Right biconditional form o... |
| e03 43434 | A virtual deduction elimin... |
| ee03 43435 | ~ e03 without virtual dedu... |
| e03an 43436 | Conjunction form of ~ e03 ... |
| ee03an 43437 | Conjunction form of ~ ee03... |
| e30 43438 | A virtual deduction elimin... |
| ee30 43439 | ~ e30 without virtual dedu... |
| e30an 43440 | A virtual deduction elimin... |
| ee30an 43441 | Conjunction form of ~ ee30... |
| e13 43442 | A virtual deduction elimin... |
| e13an 43443 | A virtual deduction elimin... |
| ee13an 43444 | ~ e13an without virtual de... |
| e31 43445 | A virtual deduction elimin... |
| ee31 43446 | ~ e31 without virtual dedu... |
| e31an 43447 | A virtual deduction elimin... |
| ee31an 43448 | ~ e31an without virtual de... |
| e23 43449 | A virtual deduction elimin... |
| e23an 43450 | A virtual deduction elimin... |
| ee23an 43451 | ~ e23an without virtual de... |
| e32 43452 | A virtual deduction elimin... |
| ee32 43453 | ~ e32 without virtual dedu... |
| e32an 43454 | A virtual deduction elimin... |
| ee32an 43455 | ~ e33an without virtual de... |
| e123 43456 | A virtual deduction elimin... |
| ee123 43457 | ~ e123 without virtual ded... |
| el123 43458 | A virtual deduction elimin... |
| e233 43459 | A virtual deduction elimin... |
| e323 43460 | A virtual deduction elimin... |
| e000 43461 | A virtual deduction elimin... |
| e00 43462 | Elimination rule identical... |
| e00an 43463 | Elimination rule identical... |
| eel00cT 43464 | An elimination deduction. ... |
| eelTT 43465 | An elimination deduction. ... |
| e0a 43466 | Elimination rule identical... |
| eelT 43467 | An elimination deduction. ... |
| eel0cT 43468 | An elimination deduction. ... |
| eelT0 43469 | An elimination deduction. ... |
| e0bi 43470 | Elimination rule identical... |
| e0bir 43471 | Elimination rule identical... |
| uun0.1 43472 | Convention notation form o... |
| un0.1 43473 | ` T. ` is the constant tru... |
| uunT1 43474 | A deduction unionizing a n... |
| uunT1p1 43475 | A deduction unionizing a n... |
| uunT21 43476 | A deduction unionizing a n... |
| uun121 43477 | A deduction unionizing a n... |
| uun121p1 43478 | A deduction unionizing a n... |
| uun132 43479 | A deduction unionizing a n... |
| uun132p1 43480 | A deduction unionizing a n... |
| anabss7p1 43481 | A deduction unionizing a n... |
| un10 43482 | A unionizing deduction. (... |
| un01 43483 | A unionizing deduction. (... |
| un2122 43484 | A deduction unionizing a n... |
| uun2131 43485 | A deduction unionizing a n... |
| uun2131p1 43486 | A deduction unionizing a n... |
| uunTT1 43487 | A deduction unionizing a n... |
| uunTT1p1 43488 | A deduction unionizing a n... |
| uunTT1p2 43489 | A deduction unionizing a n... |
| uunT11 43490 | A deduction unionizing a n... |
| uunT11p1 43491 | A deduction unionizing a n... |
| uunT11p2 43492 | A deduction unionizing a n... |
| uunT12 43493 | A deduction unionizing a n... |
| uunT12p1 43494 | A deduction unionizing a n... |
| uunT12p2 43495 | A deduction unionizing a n... |
| uunT12p3 43496 | A deduction unionizing a n... |
| uunT12p4 43497 | A deduction unionizing a n... |
| uunT12p5 43498 | A deduction unionizing a n... |
| uun111 43499 | A deduction unionizing a n... |
| 3anidm12p1 43500 | A deduction unionizing a n... |
| 3anidm12p2 43501 | A deduction unionizing a n... |
| uun123 43502 | A deduction unionizing a n... |
| uun123p1 43503 | A deduction unionizing a n... |
| uun123p2 43504 | A deduction unionizing a n... |
| uun123p3 43505 | A deduction unionizing a n... |
| uun123p4 43506 | A deduction unionizing a n... |
| uun2221 43507 | A deduction unionizing a n... |
| uun2221p1 43508 | A deduction unionizing a n... |
| uun2221p2 43509 | A deduction unionizing a n... |
| 3impdirp1 43510 | A deduction unionizing a n... |
| 3impcombi 43511 | A 1-hypothesis proposition... |
| trsspwALT 43512 | Virtual deduction proof of... |
| trsspwALT2 43513 | Virtual deduction proof of... |
| trsspwALT3 43514 | Short predicate calculus p... |
| sspwtr 43515 | Virtual deduction proof of... |
| sspwtrALT 43516 | Virtual deduction proof of... |
| sspwtrALT2 43517 | Short predicate calculus p... |
| pwtrVD 43518 | Virtual deduction proof of... |
| pwtrrVD 43519 | Virtual deduction proof of... |
| suctrALT 43520 | The successor of a transit... |
| snssiALTVD 43521 | Virtual deduction proof of... |
| snssiALT 43522 | If a class is an element o... |
| snsslVD 43523 | Virtual deduction proof of... |
| snssl 43524 | If a singleton is a subcla... |
| snelpwrVD 43525 | Virtual deduction proof of... |
| unipwrVD 43526 | Virtual deduction proof of... |
| unipwr 43527 | A class is a subclass of t... |
| sstrALT2VD 43528 | Virtual deduction proof of... |
| sstrALT2 43529 | Virtual deduction proof of... |
| suctrALT2VD 43530 | Virtual deduction proof of... |
| suctrALT2 43531 | Virtual deduction proof of... |
| elex2VD 43532 | Virtual deduction proof of... |
| elex22VD 43533 | Virtual deduction proof of... |
| eqsbc2VD 43534 | Virtual deduction proof of... |
| zfregs2VD 43535 | Virtual deduction proof of... |
| tpid3gVD 43536 | Virtual deduction proof of... |
| en3lplem1VD 43537 | Virtual deduction proof of... |
| en3lplem2VD 43538 | Virtual deduction proof of... |
| en3lpVD 43539 | Virtual deduction proof of... |
| simplbi2VD 43540 | Virtual deduction proof of... |
| 3ornot23VD 43541 | Virtual deduction proof of... |
| orbi1rVD 43542 | Virtual deduction proof of... |
| bitr3VD 43543 | Virtual deduction proof of... |
| 3orbi123VD 43544 | Virtual deduction proof of... |
| sbc3orgVD 43545 | Virtual deduction proof of... |
| 19.21a3con13vVD 43546 | Virtual deduction proof of... |
| exbirVD 43547 | Virtual deduction proof of... |
| exbiriVD 43548 | Virtual deduction proof of... |
| rspsbc2VD 43549 | Virtual deduction proof of... |
| 3impexpVD 43550 | Virtual deduction proof of... |
| 3impexpbicomVD 43551 | Virtual deduction proof of... |
| 3impexpbicomiVD 43552 | Virtual deduction proof of... |
| sbcoreleleqVD 43553 | Virtual deduction proof of... |
| hbra2VD 43554 | Virtual deduction proof of... |
| tratrbVD 43555 | Virtual deduction proof of... |
| al2imVD 43556 | Virtual deduction proof of... |
| syl5impVD 43557 | Virtual deduction proof of... |
| idiVD 43558 | Virtual deduction proof of... |
| ancomstVD 43559 | Closed form of ~ ancoms . ... |
| ssralv2VD 43560 | Quantification restricted ... |
| ordelordALTVD 43561 | An element of an ordinal c... |
| equncomVD 43562 | If a class equals the unio... |
| equncomiVD 43563 | Inference form of ~ equnco... |
| sucidALTVD 43564 | A set belongs to its succe... |
| sucidALT 43565 | A set belongs to its succe... |
| sucidVD 43566 | A set belongs to its succe... |
| imbi12VD 43567 | Implication form of ~ imbi... |
| imbi13VD 43568 | Join three logical equival... |
| sbcim2gVD 43569 | Distribution of class subs... |
| sbcbiVD 43570 | Implication form of ~ sbcb... |
| trsbcVD 43571 | Formula-building inference... |
| truniALTVD 43572 | The union of a class of tr... |
| ee33VD 43573 | Non-virtual deduction form... |
| trintALTVD 43574 | The intersection of a clas... |
| trintALT 43575 | The intersection of a clas... |
| undif3VD 43576 | The first equality of Exer... |
| sbcssgVD 43577 | Virtual deduction proof of... |
| csbingVD 43578 | Virtual deduction proof of... |
| onfrALTlem5VD 43579 | Virtual deduction proof of... |
| onfrALTlem4VD 43580 | Virtual deduction proof of... |
| onfrALTlem3VD 43581 | Virtual deduction proof of... |
| simplbi2comtVD 43582 | Virtual deduction proof of... |
| onfrALTlem2VD 43583 | Virtual deduction proof of... |
| onfrALTlem1VD 43584 | Virtual deduction proof of... |
| onfrALTVD 43585 | Virtual deduction proof of... |
| csbeq2gVD 43586 | Virtual deduction proof of... |
| csbsngVD 43587 | Virtual deduction proof of... |
| csbxpgVD 43588 | Virtual deduction proof of... |
| csbresgVD 43589 | Virtual deduction proof of... |
| csbrngVD 43590 | Virtual deduction proof of... |
| csbima12gALTVD 43591 | Virtual deduction proof of... |
| csbunigVD 43592 | Virtual deduction proof of... |
| csbfv12gALTVD 43593 | Virtual deduction proof of... |
| con5VD 43594 | Virtual deduction proof of... |
| relopabVD 43595 | Virtual deduction proof of... |
| 19.41rgVD 43596 | Virtual deduction proof of... |
| 2pm13.193VD 43597 | Virtual deduction proof of... |
| hbimpgVD 43598 | Virtual deduction proof of... |
| hbalgVD 43599 | Virtual deduction proof of... |
| hbexgVD 43600 | Virtual deduction proof of... |
| ax6e2eqVD 43601 | The following User's Proof... |
| ax6e2ndVD 43602 | The following User's Proof... |
| ax6e2ndeqVD 43603 | The following User's Proof... |
| 2sb5ndVD 43604 | The following User's Proof... |
| 2uasbanhVD 43605 | The following User's Proof... |
| e2ebindVD 43606 | The following User's Proof... |
| sb5ALTVD 43607 | The following User's Proof... |
| vk15.4jVD 43608 | The following User's Proof... |
| notnotrALTVD 43609 | The following User's Proof... |
| con3ALTVD 43610 | The following User's Proof... |
| elpwgdedVD 43611 | Membership in a power clas... |
| sspwimp 43612 | If a class is a subclass o... |
| sspwimpVD 43613 | The following User's Proof... |
| sspwimpcf 43614 | If a class is a subclass o... |
| sspwimpcfVD 43615 | The following User's Proof... |
| suctrALTcf 43616 | The sucessor of a transiti... |
| suctrALTcfVD 43617 | The following User's Proof... |
| suctrALT3 43618 | The successor of a transit... |
| sspwimpALT 43619 | If a class is a subclass o... |
| unisnALT 43620 | A set equals the union of ... |
| notnotrALT2 43621 | Converse of double negatio... |
| sspwimpALT2 43622 | If a class is a subclass o... |
| e2ebindALT 43623 | Absorption of an existenti... |
| ax6e2ndALT 43624 | If at least two sets exist... |
| ax6e2ndeqALT 43625 | "At least two sets exist" ... |
| 2sb5ndALT 43626 | Equivalence for double sub... |
| chordthmALT 43627 | The intersecting chords th... |
| isosctrlem1ALT 43628 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 43629 | The indexed union of conne... |
| iunconnALT 43630 | The indexed union of conne... |
| sineq0ALT 43631 | A complex number whose sin... |
| evth2f 43632 | A version of ~ evth2 using... |
| elunif 43633 | A version of ~ eluni using... |
| rzalf 43634 | A version of ~ rzal using ... |
| fvelrnbf 43635 | A version of ~ fvelrnb usi... |
| rfcnpre1 43636 | If F is a continuous funct... |
| ubelsupr 43637 | If U belongs to A and U is... |
| fsumcnf 43638 | A finite sum of functions ... |
| mulltgt0 43639 | The product of a negative ... |
| rspcegf 43640 | A version of ~ rspcev usin... |
| rabexgf 43641 | A version of ~ rabexg usin... |
| fcnre 43642 | A function continuous with... |
| sumsnd 43643 | A sum of a singleton is th... |
| evthf 43644 | A version of ~ evth using ... |
| cnfex 43645 | The class of continuous fu... |
| fnchoice 43646 | For a finite set, a choice... |
| refsumcn 43647 | A finite sum of continuous... |
| rfcnpre2 43648 | If ` F ` is a continuous f... |
| cncmpmax 43649 | When the hypothesis for th... |
| rfcnpre3 43650 | If F is a continuous funct... |
| rfcnpre4 43651 | If F is a continuous funct... |
| sumpair 43652 | Sum of two distinct comple... |
| rfcnnnub 43653 | Given a real continuous fu... |
| refsum2cnlem1 43654 | This is the core Lemma for... |
| refsum2cn 43655 | The sum of two continuus r... |
| adantlllr 43656 | Deduction adding a conjunc... |
| 3adantlr3 43657 | Deduction adding a conjunc... |
| 3adantll2 43658 | Deduction adding a conjunc... |
| 3adantll3 43659 | Deduction adding a conjunc... |
| ssnel 43660 | If not element of a set, t... |
| elabrexg 43661 | Elementhood in an image se... |
| sncldre 43662 | A singleton is closed w.r.... |
| n0p 43663 | A polynomial with a nonzer... |
| pm2.65ni 43664 | Inference rule for proof b... |
| pwssfi 43665 | Every element of the power... |
| iuneq2df 43666 | Equality deduction for ind... |
| nnfoctb 43667 | There exists a mapping fro... |
| ssinss1d 43668 | Intersection preserves sub... |
| elpwinss 43669 | An element of the powerset... |
| unidmex 43670 | If ` F ` is a set, then ` ... |
| ndisj2 43671 | A non-disjointness conditi... |
| zenom 43672 | The set of integer numbers... |
| uzwo4 43673 | Well-ordering principle: a... |
| unisn0 43674 | The union of the singleton... |
| ssin0 43675 | If two classes are disjoin... |
| inabs3 43676 | Absorption law for interse... |
| pwpwuni 43677 | Relationship between power... |
| disjiun2 43678 | In a disjoint collection, ... |
| 0pwfi 43679 | The empty set is in any po... |
| ssinss2d 43680 | Intersection preserves sub... |
| zct 43681 | The set of integer numbers... |
| pwfin0 43682 | A finite set always belong... |
| uzct 43683 | An upper integer set is co... |
| iunxsnf 43684 | A singleton index picks ou... |
| fiiuncl 43685 | If a set is closed under t... |
| iunp1 43686 | The addition of the next s... |
| fiunicl 43687 | If a set is closed under t... |
| ixpeq2d 43688 | Equality theorem for infin... |
| disjxp1 43689 | The sets of a cartesian pr... |
| disjsnxp 43690 | The sets in the cartesian ... |
| eliind 43691 | Membership in indexed inte... |
| rspcef 43692 | Restricted existential spe... |
| inn0f 43693 | A nonempty intersection. ... |
| ixpssmapc 43694 | An infinite Cartesian prod... |
| inn0 43695 | A nonempty intersection. ... |
| elintd 43696 | Membership in class inters... |
| ssdf 43697 | A sufficient condition for... |
| brneqtrd 43698 | Substitution of equal clas... |
| ssnct 43699 | A set containing an uncoun... |
| ssuniint 43700 | Sufficient condition for b... |
| elintdv 43701 | Membership in class inters... |
| ssd 43702 | A sufficient condition for... |
| ralimralim 43703 | Introducing any antecedent... |
| snelmap 43704 | Membership of the element ... |
| xrnmnfpnf 43705 | An extended real that is n... |
| nelrnmpt 43706 | Non-membership in the rang... |
| iuneq1i 43707 | Equality theorem for index... |
| nssrex 43708 | Negation of subclass relat... |
| ssinc 43709 | Inclusion relation for a m... |
| ssdec 43710 | Inclusion relation for a m... |
| elixpconstg 43711 | Membership in an infinite ... |
| iineq1d 43712 | Equality theorem for index... |
| metpsmet 43713 | A metric is a pseudometric... |
| ixpssixp 43714 | Subclass theorem for infin... |
| ballss3 43715 | A sufficient condition for... |
| iunincfi 43716 | Given a sequence of increa... |
| nsstr 43717 | If it's not a subclass, it... |
| rexanuz3 43718 | Combine two different uppe... |
| cbvmpo2 43719 | Rule to change the second ... |
| cbvmpo1 43720 | Rule to change the first b... |
| eliuniin 43721 | Indexed union of indexed i... |
| ssabf 43722 | Subclass of a class abstra... |
| pssnssi 43723 | A proper subclass does not... |
| rabidim2 43724 | Membership in a restricted... |
| eluni2f 43725 | Membership in class union.... |
| eliin2f 43726 | Membership in indexed inte... |
| nssd 43727 | Negation of subclass relat... |
| iineq12dv 43728 | Equality deduction for ind... |
| supxrcld 43729 | The supremum of an arbitra... |
| elrestd 43730 | A sufficient condition for... |
| eliuniincex 43731 | Counterexample to show tha... |
| eliincex 43732 | Counterexample to show tha... |
| eliinid 43733 | Membership in an indexed i... |
| abssf 43734 | Class abstraction in a sub... |
| supxrubd 43735 | A member of a set of exten... |
| ssrabf 43736 | Subclass of a restricted c... |
| ssrabdf 43737 | Subclass of a restricted c... |
| eliin2 43738 | Membership in indexed inte... |
| ssrab2f 43739 | Subclass relation for a re... |
| restuni3 43740 | The underlying set of a su... |
| rabssf 43741 | Restricted class abstracti... |
| eliuniin2 43742 | Indexed union of indexed i... |
| restuni4 43743 | The underlying set of a su... |
| restuni6 43744 | The underlying set of a su... |
| restuni5 43745 | The underlying set of a su... |
| unirestss 43746 | The union of an elementwis... |
| iniin1 43747 | Indexed intersection of in... |
| iniin2 43748 | Indexed intersection of in... |
| cbvrabv2 43749 | A more general version of ... |
| cbvrabv2w 43750 | A more general version of ... |
| iinssiin 43751 | Subset implication for an ... |
| eliind2 43752 | Membership in indexed inte... |
| iinssd 43753 | Subset implication for an ... |
| rabbida2 43754 | Equivalent wff's yield equ... |
| iinexd 43755 | The existence of an indexe... |
| rabexf 43756 | Separation Scheme in terms... |
| rabbida3 43757 | Equivalent wff's yield equ... |
| r19.36vf 43758 | Restricted quantifier vers... |
| raleqd 43759 | Equality deduction for res... |
| iinssf 43760 | Subset implication for an ... |
| iinssdf 43761 | Subset implication for an ... |
| resabs2i 43762 | Absorption law for restric... |
| ssdf2 43763 | A sufficient condition for... |
| rabssd 43764 | Restricted class abstracti... |
| rexnegd 43765 | Minus a real number. (Con... |
| rexlimd3 43766 | * Inference from Theorem 1... |
| resabs1i 43767 | Absorption law for restric... |
| nel1nelin 43768 | Membership in an intersect... |
| nel2nelin 43769 | Membership in an intersect... |
| nel1nelini 43770 | Membership in an intersect... |
| nel2nelini 43771 | Membership in an intersect... |
| eliunid 43772 | Membership in indexed unio... |
| reximddv3 43773 | Deduction from Theorem 19.... |
| reximdd 43774 | Deduction from Theorem 19.... |
| unfid 43775 | The union of two finite se... |
| inopnd 43776 | The intersection of two op... |
| ss2rabdf 43777 | Deduction of restricted ab... |
| restopn3 43778 | If ` A ` is open, then ` A... |
| restopnssd 43779 | A topology restricted to a... |
| restsubel 43780 | A subset belongs in the sp... |
| toprestsubel 43781 | A subset is open in the to... |
| rabidd 43782 | An "identity" law of concr... |
| iunssdf 43783 | Subset theorem for an inde... |
| iinss2d 43784 | Subset implication for an ... |
| r19.3rzf 43785 | Restricted quantification ... |
| r19.28zf 43786 | Restricted quantifier vers... |
| iindif2f 43787 | Indexed intersection of cl... |
| ralfal 43788 | Two ways of expressing emp... |
| archd 43789 | Archimedean property of re... |
| eliund 43790 | Membership in indexed unio... |
| nimnbi 43791 | If an implication is false... |
| nimnbi2 43792 | If an implication is false... |
| notbicom 43793 | Commutative law for the ne... |
| rexeqif 43794 | Equality inference for res... |
| rspced 43795 | Restricted existential spe... |
| feq1dd 43796 | Equality deduction for fun... |
| fnresdmss 43797 | A function does not change... |
| fmptsnxp 43798 | Maps-to notation and Carte... |
| fvmpt2bd 43799 | Value of a function given ... |
| rnmptfi 43800 | The range of a function wi... |
| fresin2 43801 | Restriction of a function ... |
| ffi 43802 | A function with finite dom... |
| suprnmpt 43803 | An explicit bound for the ... |
| rnffi 43804 | The range of a function wi... |
| mptelpm 43805 | A function in maps-to nota... |
| rnmptpr 43806 | Range of a function define... |
| resmpti 43807 | Restriction of the mapping... |
| founiiun 43808 | Union expressed as an inde... |
| rnresun 43809 | Distribution law for range... |
| dffo3f 43810 | An onto mapping expressed ... |
| elrnmptf 43811 | The range of a function in... |
| rnmptssrn 43812 | Inclusion relation for two... |
| disjf1 43813 | A 1 to 1 mapping built fro... |
| rnsnf 43814 | The range of a function wh... |
| wessf1ornlem 43815 | Given a function ` F ` on ... |
| wessf1orn 43816 | Given a function ` F ` on ... |
| foelrnf 43817 | Property of a surjective f... |
| nelrnres 43818 | If ` A ` is not in the ran... |
| disjrnmpt2 43819 | Disjointness of the range ... |
| elrnmpt1sf 43820 | Elementhood in an image se... |
| founiiun0 43821 | Union expressed as an inde... |
| disjf1o 43822 | A bijection built from dis... |
| fompt 43823 | Express being onto for a m... |
| disjinfi 43824 | Only a finite number of di... |
| fvovco 43825 | Value of the composition o... |
| ssnnf1octb 43826 | There exists a bijection b... |
| nnf1oxpnn 43827 | There is a bijection betwe... |
| rnmptssd 43828 | The range of a function gi... |
| projf1o 43829 | A biijection from a set to... |
| fvmap 43830 | Function value for a membe... |
| fvixp2 43831 | Projection of a factor of ... |
| choicefi 43832 | For a finite set, a choice... |
| mpct 43833 | The exponentiation of a co... |
| cnmetcoval 43834 | Value of the distance func... |
| fcomptss 43835 | Express composition of two... |
| elmapsnd 43836 | Membership in a set expone... |
| mapss2 43837 | Subset inheritance for set... |
| fsneq 43838 | Equality condition for two... |
| difmap 43839 | Difference of two sets exp... |
| unirnmap 43840 | Given a subset of a set ex... |
| inmap 43841 | Intersection of two sets e... |
| fcoss 43842 | Composition of two mapping... |
| fsneqrn 43843 | Equality condition for two... |
| difmapsn 43844 | Difference of two sets exp... |
| mapssbi 43845 | Subset inheritance for set... |
| unirnmapsn 43846 | Equality theorem for a sub... |
| iunmapss 43847 | The indexed union of set e... |
| ssmapsn 43848 | A subset ` C ` of a set ex... |
| iunmapsn 43849 | The indexed union of set e... |
| absfico 43850 | Mapping domain and codomai... |
| icof 43851 | The set of left-closed rig... |
| elpmrn 43852 | The range of a partial fun... |
| imaexi 43853 | The image of a set is a se... |
| axccdom 43854 | Relax the constraint on ax... |
| dmmptdff 43855 | The domain of the mapping ... |
| dmmptdf 43856 | The domain of the mapping ... |
| elpmi2 43857 | The domain of a partial fu... |
| dmrelrnrel 43858 | A relation preserving func... |
| fvcod 43859 | Value of a function compos... |
| elrnmpoid 43860 | Membership in the range of... |
| axccd 43861 | An alternative version of ... |
| axccd2 43862 | An alternative version of ... |
| funimassd 43863 | Sufficient condition for t... |
| fimassd 43864 | The image of a class is a ... |
| feqresmptf 43865 | Express a restricted funct... |
| elrnmpt1d 43866 | Elementhood in an image se... |
| dmresss 43867 | The domain of a restrictio... |
| dmmptssf 43868 | The domain of a mapping is... |
| dmmptdf2 43869 | The domain of the mapping ... |
| dmuz 43870 | Domain of the upper intege... |
| fmptd2f 43871 | Domain and codomain of the... |
| mpteq1df 43872 | An equality theorem for th... |
| mpteq1dfOLD 43873 | Obsolete version of ~ mpte... |
| mptexf 43874 | If the domain of a functio... |
| fvmpt4 43875 | Value of a function given ... |
| fmptf 43876 | Functionality of the mappi... |
| resimass 43877 | The image of a restriction... |
| mptssid 43878 | The mapping operation expr... |
| mptfnd 43879 | The maps-to notation defin... |
| mpteq12daOLD 43880 | Obsolete version of ~ mpte... |
| rnmptlb 43881 | Boundness below of the ran... |
| rnmptbddlem 43882 | Boundness of the range of ... |
| rnmptbdd 43883 | Boundness of the range of ... |
| mptima2 43884 | Image of a function in map... |
| funimaeq 43885 | Membership relation for th... |
| rnmptssf 43886 | The range of a function gi... |
| rnmptbd2lem 43887 | Boundness below of the ran... |
| rnmptbd2 43888 | Boundness below of the ran... |
| infnsuprnmpt 43889 | The indexed infimum of rea... |
| suprclrnmpt 43890 | Closure of the indexed sup... |
| suprubrnmpt2 43891 | A member of a nonempty ind... |
| suprubrnmpt 43892 | A member of a nonempty ind... |
| rnmptssdf 43893 | The range of a function gi... |
| rnmptbdlem 43894 | Boundness above of the ran... |
| rnmptbd 43895 | Boundness above of the ran... |
| rnmptss2 43896 | The range of a function gi... |
| elmptima 43897 | The image of a function in... |
| ralrnmpt3 43898 | A restricted quantifier ov... |
| fvelima2 43899 | Function value in an image... |
| rnmptssbi 43900 | The range of a function gi... |
| imass2d 43901 | Subset theorem for image. ... |
| imassmpt 43902 | Membership relation for th... |
| fpmd 43903 | A total function is a part... |
| fconst7 43904 | An alternative way to expr... |
| fnmptif 43905 | Functionality and domain o... |
| dmmptif 43906 | Domain of the mapping oper... |
| mpteq2dfa 43907 | Slightly more general equa... |
| dmmpt1 43908 | The domain of the mapping ... |
| fmptff 43909 | Functionality of the mappi... |
| fvmptelcdmf 43910 | The value of a function at... |
| fmptdff 43911 | A version of ~ fmptd using... |
| fvmpt2df 43912 | Deduction version of ~ fvm... |
| rn1st 43913 | The range of a function wi... |
| rnmptssff 43914 | The range of a function gi... |
| rnmptssdff 43915 | The range of a function gi... |
| fvmpt4d 43916 | Value of a function given ... |
| sub2times 43917 | Subtracting from a number,... |
| nnxrd 43918 | A natural number is an ext... |
| nnxr 43919 | A natural number is an ext... |
| abssubrp 43920 | The distance of two distin... |
| elfzfzo 43921 | Relationship between membe... |
| oddfl 43922 | Odd number representation ... |
| abscosbd 43923 | Bound for the absolute val... |
| mul13d 43924 | Commutative/associative la... |
| negpilt0 43925 | Negative ` _pi ` is negati... |
| dstregt0 43926 | A complex number ` A ` tha... |
| subadd4b 43927 | Rearrangement of 4 terms i... |
| xrlttri5d 43928 | Not equal and not larger i... |
| neglt 43929 | The negative of a positive... |
| zltlesub 43930 | If an integer ` N ` is les... |
| divlt0gt0d 43931 | The ratio of a negative nu... |
| subsub23d 43932 | Swap subtrahend and result... |
| 2timesgt 43933 | Double of a positive real ... |
| reopn 43934 | The reals are open with re... |
| sub31 43935 | Swap the first and third t... |
| nnne1ge2 43936 | A positive integer which i... |
| lefldiveq 43937 | A closed enough, smaller r... |
| negsubdi3d 43938 | Distribution of negative o... |
| ltdiv2dd 43939 | Division of a positive num... |
| abssinbd 43940 | Bound for the absolute val... |
| halffl 43941 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 43942 | Ordering relation for a st... |
| hashssle 43943 | The size of a subset of a ... |
| lttri5d 43944 | Not equal and not larger i... |
| fzisoeu 43945 | A finite ordered set has a... |
| lt3addmuld 43946 | If three real numbers are ... |
| absnpncan2d 43947 | Triangular inequality, com... |
| fperiodmullem 43948 | A function with period ` T... |
| fperiodmul 43949 | A function with period T i... |
| upbdrech 43950 | Choice of an upper bound f... |
| lt4addmuld 43951 | If four real numbers are l... |
| absnpncan3d 43952 | Triangular inequality, com... |
| upbdrech2 43953 | Choice of an upper bound f... |
| ssfiunibd 43954 | A finite union of bounded ... |
| fzdifsuc2 43955 | Remove a successor from th... |
| fzsscn 43956 | A finite sequence of integ... |
| divcan8d 43957 | A cancellation law for div... |
| dmmcand 43958 | Cancellation law for divis... |
| fzssre 43959 | A finite sequence of integ... |
| bccld 43960 | A binomial coefficient, in... |
| leadd12dd 43961 | Addition to both sides of ... |
| fzssnn0 43962 | A finite set of sequential... |
| xreqle 43963 | Equality implies 'less tha... |
| xaddlidd 43964 | ` 0 ` is a left identity f... |
| xadd0ge 43965 | A number is less than or e... |
| elfzolem1 43966 | A member in a half-open in... |
| xrgtned 43967 | 'Greater than' implies not... |
| xrleneltd 43968 | 'Less than or equal to' an... |
| xaddcomd 43969 | The extended real addition... |
| supxrre3 43970 | The supremum of a nonempty... |
| uzfissfz 43971 | For any finite subset of t... |
| xleadd2d 43972 | Addition of extended reals... |
| suprltrp 43973 | The supremum of a nonempty... |
| xleadd1d 43974 | Addition of extended reals... |
| xreqled 43975 | Equality implies 'less tha... |
| xrgepnfd 43976 | An extended real greater t... |
| xrge0nemnfd 43977 | A nonnegative extended rea... |
| supxrgere 43978 | If a real number can be ap... |
| iuneqfzuzlem 43979 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 43980 | If two unions indexed by u... |
| xle2addd 43981 | Adding both side of two in... |
| supxrgelem 43982 | If an extended real number... |
| supxrge 43983 | If an extended real number... |
| suplesup 43984 | If any element of ` A ` ca... |
| infxrglb 43985 | The infimum of a set of ex... |
| xadd0ge2 43986 | A number is less than or e... |
| nepnfltpnf 43987 | An extended real that is n... |
| ltadd12dd 43988 | Addition to both sides of ... |
| nemnftgtmnft 43989 | An extended real that is n... |
| xrgtso 43990 | 'Greater than' is a strict... |
| rpex 43991 | The positive reals form a ... |
| xrge0ge0 43992 | A nonnegative extended rea... |
| xrssre 43993 | A subset of extended reals... |
| ssuzfz 43994 | A finite subset of the upp... |
| absfun 43995 | The absolute value is a fu... |
| infrpge 43996 | The infimum of a nonempty,... |
| xrlexaddrp 43997 | If an extended real number... |
| supsubc 43998 | The supremum function dist... |
| xralrple2 43999 | Show that ` A ` is less th... |
| nnuzdisj 44000 | The first ` N ` elements o... |
| ltdivgt1 44001 | Divsion by a number greate... |
| xrltned 44002 | 'Less than' implies not eq... |
| nnsplit 44003 | Express the set of positiv... |
| divdiv3d 44004 | Division into a fraction. ... |
| abslt2sqd 44005 | Comparison of the square o... |
| qenom 44006 | The set of rational number... |
| qct 44007 | The set of rational number... |
| xrltnled 44008 | 'Less than' in terms of 'l... |
| lenlteq 44009 | 'less than or equal to' bu... |
| xrred 44010 | An extended real that is n... |
| rr2sscn2 44011 | The cartesian square of ` ... |
| infxr 44012 | The infimum of a set of ex... |
| infxrunb2 44013 | The infimum of an unbounde... |
| infxrbnd2 44014 | The infimum of a bounded-b... |
| infleinflem1 44015 | Lemma for ~ infleinf , cas... |
| infleinflem2 44016 | Lemma for ~ infleinf , whe... |
| infleinf 44017 | If any element of ` B ` ca... |
| xralrple4 44018 | Show that ` A ` is less th... |
| xralrple3 44019 | Show that ` A ` is less th... |
| eluzelzd 44020 | A member of an upper set o... |
| suplesup2 44021 | If any element of ` A ` is... |
| recnnltrp 44022 | ` N ` is a natural number ... |
| nnn0 44023 | The set of positive intege... |
| fzct 44024 | A finite set of sequential... |
| rpgtrecnn 44025 | Any positive real number i... |
| fzossuz 44026 | A half-open integer interv... |
| infxrrefi 44027 | The real and extended real... |
| xrralrecnnle 44028 | Show that ` A ` is less th... |
| fzoct 44029 | A finite set of sequential... |
| frexr 44030 | A function taking real val... |
| nnrecrp 44031 | The reciprocal of a positi... |
| reclt0d 44032 | The reciprocal of a negati... |
| lt0neg1dd 44033 | If a number is negative, i... |
| infxrcld 44034 | The infimum of an arbitrar... |
| xrralrecnnge 44035 | Show that ` A ` is less th... |
| reclt0 44036 | The reciprocal of a negati... |
| ltmulneg 44037 | Multiplying by a negative ... |
| allbutfi 44038 | For all but finitely many.... |
| ltdiv23neg 44039 | Swap denominator with othe... |
| xreqnltd 44040 | A consequence of trichotom... |
| mnfnre2 44041 | Minus infinity is not a re... |
| zssxr 44042 | The integers are a subset ... |
| fisupclrnmpt 44043 | A nonempty finite indexed ... |
| supxrunb3 44044 | The supremum of an unbound... |
| elfzod 44045 | Membership in a half-open ... |
| fimaxre4 44046 | A nonempty finite set of r... |
| ren0 44047 | The set of reals is nonemp... |
| eluzelz2 44048 | A member of an upper set o... |
| resabs2d 44049 | Absorption law for restric... |
| uzid2 44050 | Membership of the least me... |
| supxrleubrnmpt 44051 | The supremum of a nonempty... |
| uzssre2 44052 | An upper set of integers i... |
| uzssd 44053 | Subset relationship for tw... |
| eluzd 44054 | Membership in an upper set... |
| infxrlbrnmpt2 44055 | A member of a nonempty ind... |
| xrre4 44056 | An extended real is real i... |
| uz0 44057 | The upper integers functio... |
| eluzelz2d 44058 | A member of an upper set o... |
| infleinf2 44059 | If any element in ` B ` is... |
| unb2ltle 44060 | "Unbounded below" expresse... |
| uzidd2 44061 | Membership of the least me... |
| uzssd2 44062 | Subset relationship for tw... |
| rexabslelem 44063 | An indexed set of absolute... |
| rexabsle 44064 | An indexed set of absolute... |
| allbutfiinf 44065 | Given a "for all but finit... |
| supxrrernmpt 44066 | The real and extended real... |
| suprleubrnmpt 44067 | The supremum of a nonempty... |
| infrnmptle 44068 | An indexed infimum of exte... |
| infxrunb3 44069 | The infimum of an unbounde... |
| uzn0d 44070 | The upper integers are all... |
| uzssd3 44071 | Subset relationship for tw... |
| rexabsle2 44072 | An indexed set of absolute... |
| infxrunb3rnmpt 44073 | The infimum of an unbounde... |
| supxrre3rnmpt 44074 | The indexed supremum of a ... |
| uzublem 44075 | A set of reals, indexed by... |
| uzub 44076 | A set of reals, indexed by... |
| ssrexr 44077 | A subset of the reals is a... |
| supxrmnf2 44078 | Removing minus infinity fr... |
| supxrcli 44079 | The supremum of an arbitra... |
| uzid3 44080 | Membership of the least me... |
| infxrlesupxr 44081 | The supremum of a nonempty... |
| xnegeqd 44082 | Equality of two extended n... |
| xnegrecl 44083 | The extended real negative... |
| xnegnegi 44084 | Extended real version of ~... |
| xnegeqi 44085 | Equality of two extended n... |
| nfxnegd 44086 | Deduction version of ~ nfx... |
| xnegnegd 44087 | Extended real version of ~... |
| uzred 44088 | An upper integer is a real... |
| xnegcli 44089 | Closure of extended real n... |
| supminfrnmpt 44090 | The indexed supremum of a ... |
| infxrpnf 44091 | Adding plus infinity to a ... |
| infxrrnmptcl 44092 | The infimum of an arbitrar... |
| leneg2d 44093 | Negative of one side of 'l... |
| supxrltinfxr 44094 | The supremum of the empty ... |
| max1d 44095 | A number is less than or e... |
| supxrleubrnmptf 44096 | The supremum of a nonempty... |
| nleltd 44097 | 'Not less than or equal to... |
| zxrd 44098 | An integer is an extended ... |
| infxrgelbrnmpt 44099 | The infimum of an indexed ... |
| rphalfltd 44100 | Half of a positive real is... |
| uzssz2 44101 | An upper set of integers i... |
| leneg3d 44102 | Negative of one side of 'l... |
| max2d 44103 | A number is less than or e... |
| uzn0bi 44104 | The upper integers functio... |
| xnegrecl2 44105 | If the extended real negat... |
| nfxneg 44106 | Bound-variable hypothesis ... |
| uzxrd 44107 | An upper integer is an ext... |
| infxrpnf2 44108 | Removing plus infinity fro... |
| supminfxr 44109 | The extended real suprema ... |
| infrpgernmpt 44110 | The infimum of a nonempty,... |
| xnegre 44111 | An extended real is real i... |
| xnegrecl2d 44112 | If the extended real negat... |
| uzxr 44113 | An upper integer is an ext... |
| supminfxr2 44114 | The extended real suprema ... |
| xnegred 44115 | An extended real is real i... |
| supminfxrrnmpt 44116 | The indexed supremum of a ... |
| min1d 44117 | The minimum of two numbers... |
| min2d 44118 | The minimum of two numbers... |
| pnfged 44119 | Plus infinity is an upper ... |
| xrnpnfmnf 44120 | An extended real that is n... |
| uzsscn 44121 | An upper set of integers i... |
| absimnre 44122 | The absolute value of the ... |
| uzsscn2 44123 | An upper set of integers i... |
| xrtgcntopre 44124 | The standard topologies on... |
| absimlere 44125 | The absolute value of the ... |
| rpssxr 44126 | The positive reals are a s... |
| monoordxrv 44127 | Ordering relation for a mo... |
| monoordxr 44128 | Ordering relation for a mo... |
| monoord2xrv 44129 | Ordering relation for a mo... |
| monoord2xr 44130 | Ordering relation for a mo... |
| xrpnf 44131 | An extended real is plus i... |
| xlenegcon1 44132 | Extended real version of ~... |
| xlenegcon2 44133 | Extended real version of ~... |
| pimxrneun 44134 | The preimage of a set of e... |
| caucvgbf 44135 | A function is convergent i... |
| cvgcau 44136 | A convergent function is C... |
| cvgcaule 44137 | A convergent function is C... |
| rexanuz2nf 44138 | A simple counterexample re... |
| gtnelioc 44139 | A real number larger than ... |
| ioossioc 44140 | An open interval is a subs... |
| ioondisj2 44141 | A condition for two open i... |
| ioondisj1 44142 | A condition for two open i... |
| ioogtlb 44143 | An element of a closed int... |
| evthiccabs 44144 | Extreme Value Theorem on y... |
| ltnelicc 44145 | A real number smaller than... |
| eliood 44146 | Membership in an open real... |
| iooabslt 44147 | An upper bound for the dis... |
| gtnelicc 44148 | A real number greater than... |
| iooinlbub 44149 | An open interval has empty... |
| iocgtlb 44150 | An element of a left-open ... |
| iocleub 44151 | An element of a left-open ... |
| eliccd 44152 | Membership in a closed rea... |
| eliccre 44153 | A member of a closed inter... |
| eliooshift 44154 | Element of an open interva... |
| eliocd 44155 | Membership in a left-open ... |
| icoltub 44156 | An element of a left-close... |
| eliocre 44157 | A member of a left-open ri... |
| iooltub 44158 | An element of an open inte... |
| ioontr 44159 | The interior of an interva... |
| snunioo1 44160 | The closure of one end of ... |
| lbioc 44161 | A left-open right-closed i... |
| ioomidp 44162 | The midpoint is an element... |
| iccdifioo 44163 | If the open inverval is re... |
| iccdifprioo 44164 | An open interval is the cl... |
| ioossioobi 44165 | Biconditional form of ~ io... |
| iccshift 44166 | A closed interval shifted ... |
| iccsuble 44167 | An upper bound to the dist... |
| iocopn 44168 | A left-open right-closed i... |
| eliccelioc 44169 | Membership in a closed int... |
| iooshift 44170 | An open interval shifted b... |
| iccintsng 44171 | Intersection of two adiace... |
| icoiccdif 44172 | Left-closed right-open int... |
| icoopn 44173 | A left-closed right-open i... |
| icoub 44174 | A left-closed, right-open ... |
| eliccxrd 44175 | Membership in a closed rea... |
| pnfel0pnf 44176 | ` +oo ` is a nonnegative e... |
| eliccnelico 44177 | An element of a closed int... |
| eliccelicod 44178 | A member of a closed inter... |
| ge0xrre 44179 | A nonnegative extended rea... |
| ge0lere 44180 | A nonnegative extended Rea... |
| elicores 44181 | Membership in a left-close... |
| inficc 44182 | The infimum of a nonempty ... |
| qinioo 44183 | The rational numbers are d... |
| lenelioc 44184 | A real number smaller than... |
| ioonct 44185 | A nonempty open interval i... |
| xrgtnelicc 44186 | A real number greater than... |
| iccdificc 44187 | The difference of two clos... |
| iocnct 44188 | A nonempty left-open, righ... |
| iccnct 44189 | A closed interval, with mo... |
| iooiinicc 44190 | A closed interval expresse... |
| iccgelbd 44191 | An element of a closed int... |
| iooltubd 44192 | An element of an open inte... |
| icoltubd 44193 | An element of a left-close... |
| qelioo 44194 | The rational numbers are d... |
| tgqioo2 44195 | Every open set of reals is... |
| iccleubd 44196 | An element of a closed int... |
| elioored 44197 | A member of an open interv... |
| ioogtlbd 44198 | An element of a closed int... |
| ioofun 44199 | ` (,) ` is a function. (C... |
| icomnfinre 44200 | A left-closed, right-open,... |
| sqrlearg 44201 | The square compared with i... |
| ressiocsup 44202 | If the supremum belongs to... |
| ressioosup 44203 | If the supremum does not b... |
| iooiinioc 44204 | A left-open, right-closed ... |
| ressiooinf 44205 | If the infimum does not be... |
| icogelbd 44206 | An element of a left-close... |
| iocleubd 44207 | An element of a left-open ... |
| uzinico 44208 | An upper interval of integ... |
| preimaiocmnf 44209 | Preimage of a right-closed... |
| uzinico2 44210 | An upper interval of integ... |
| uzinico3 44211 | An upper interval of integ... |
| icossico2 44212 | Condition for a closed-bel... |
| dmico 44213 | The domain of the closed-b... |
| ndmico 44214 | The closed-below, open-abo... |
| uzubioo 44215 | The upper integers are unb... |
| uzubico 44216 | The upper integers are unb... |
| uzubioo2 44217 | The upper integers are unb... |
| uzubico2 44218 | The upper integers are unb... |
| iocgtlbd 44219 | An element of a left-open ... |
| xrtgioo2 44220 | The topology on the extend... |
| tgioo4 44221 | The standard topology on t... |
| fsummulc1f 44222 | Closure of a finite sum of... |
| fsumnncl 44223 | Closure of a nonempty, fin... |
| fsumge0cl 44224 | The finite sum of nonnegat... |
| fsumf1of 44225 | Re-index a finite sum usin... |
| fsumiunss 44226 | Sum over a disjoint indexe... |
| fsumreclf 44227 | Closure of a finite sum of... |
| fsumlessf 44228 | A shorter sum of nonnegati... |
| fsumsupp0 44229 | Finite sum of function val... |
| fsumsermpt 44230 | A finite sum expressed in ... |
| fmul01 44231 | Multiplying a finite numbe... |
| fmulcl 44232 | If ' Y ' is closed under t... |
| fmuldfeqlem1 44233 | induction step for the pro... |
| fmuldfeq 44234 | X and Z are two equivalent... |
| fmul01lt1lem1 44235 | Given a finite multiplicat... |
| fmul01lt1lem2 44236 | Given a finite multiplicat... |
| fmul01lt1 44237 | Given a finite multiplicat... |
| cncfmptss 44238 | A continuous complex funct... |
| rrpsscn 44239 | The positive reals are a s... |
| mulc1cncfg 44240 | A version of ~ mulc1cncf u... |
| infrglb 44241 | The infimum of a nonempty ... |
| expcnfg 44242 | If ` F ` is a complex cont... |
| prodeq2ad 44243 | Equality deduction for pro... |
| fprodsplit1 44244 | Separate out a term in a f... |
| fprodexp 44245 | Positive integer exponenti... |
| fprodabs2 44246 | The absolute value of a fi... |
| fprod0 44247 | A finite product with a ze... |
| mccllem 44248 | * Induction step for ~ mcc... |
| mccl 44249 | A multinomial coefficient,... |
| fprodcnlem 44250 | A finite product of functi... |
| fprodcn 44251 | A finite product of functi... |
| clim1fr1 44252 | A class of sequences of fr... |
| isumneg 44253 | Negation of a converging s... |
| climrec 44254 | Limit of the reciprocal of... |
| climmulf 44255 | A version of ~ climmul usi... |
| climexp 44256 | The limit of natural power... |
| climinf 44257 | A bounded monotonic noninc... |
| climsuselem1 44258 | The subsequence index ` I ... |
| climsuse 44259 | A subsequence ` G ` of a c... |
| climrecf 44260 | A version of ~ climrec usi... |
| climneg 44261 | Complex limit of the negat... |
| climinff 44262 | A version of ~ climinf usi... |
| climdivf 44263 | Limit of the ratio of two ... |
| climreeq 44264 | If ` F ` is a real functio... |
| ellimciota 44265 | An explicit value for the ... |
| climaddf 44266 | A version of ~ climadd usi... |
| mullimc 44267 | Limit of the product of tw... |
| ellimcabssub0 44268 | An equivalent condition fo... |
| limcdm0 44269 | If a function has empty do... |
| islptre 44270 | An equivalence condition f... |
| limccog 44271 | Limit of the composition o... |
| limciccioolb 44272 | The limit of a function at... |
| climf 44273 | Express the predicate: Th... |
| mullimcf 44274 | Limit of the multiplicatio... |
| constlimc 44275 | Limit of constant function... |
| rexlim2d 44276 | Inference removing two res... |
| idlimc 44277 | Limit of the identity func... |
| divcnvg 44278 | The sequence of reciprocal... |
| limcperiod 44279 | If ` F ` is a periodic fun... |
| limcrecl 44280 | If ` F ` is a real-valued ... |
| sumnnodd 44281 | A series indexed by ` NN `... |
| lptioo2 44282 | The upper bound of an open... |
| lptioo1 44283 | The lower bound of an open... |
| elprn1 44284 | A member of an unordered p... |
| elprn2 44285 | A member of an unordered p... |
| limcmptdm 44286 | The domain of a maps-to fu... |
| clim2f 44287 | Express the predicate: Th... |
| limcicciooub 44288 | The limit of a function at... |
| ltmod 44289 | A sufficient condition for... |
| islpcn 44290 | A characterization for a l... |
| lptre2pt 44291 | If a set in the real line ... |
| limsupre 44292 | If a sequence is bounded, ... |
| limcresiooub 44293 | The left limit doesn't cha... |
| limcresioolb 44294 | The right limit doesn't ch... |
| limcleqr 44295 | If the left and the right ... |
| lptioo2cn 44296 | The upper bound of an open... |
| lptioo1cn 44297 | The lower bound of an open... |
| neglimc 44298 | Limit of the negative func... |
| addlimc 44299 | Sum of two limits. (Contr... |
| 0ellimcdiv 44300 | If the numerator converges... |
| clim2cf 44301 | Express the predicate ` F ... |
| limclner 44302 | For a limit point, both fr... |
| sublimc 44303 | Subtraction of two limits.... |
| reclimc 44304 | Limit of the reciprocal of... |
| clim0cf 44305 | Express the predicate ` F ... |
| limclr 44306 | For a limit point, both fr... |
| divlimc 44307 | Limit of the quotient of t... |
| expfac 44308 | Factorial grows faster tha... |
| climconstmpt 44309 | A constant sequence conver... |
| climresmpt 44310 | A function restricted to u... |
| climsubmpt 44311 | Limit of the difference of... |
| climsubc2mpt 44312 | Limit of the difference of... |
| climsubc1mpt 44313 | Limit of the difference of... |
| fnlimfv 44314 | The value of the limit fun... |
| climreclf 44315 | The limit of a convergent ... |
| climeldmeq 44316 | Two functions that are eve... |
| climf2 44317 | Express the predicate: Th... |
| fnlimcnv 44318 | The sequence of function v... |
| climeldmeqmpt 44319 | Two functions that are eve... |
| climfveq 44320 | Two functions that are eve... |
| clim2f2 44321 | Express the predicate: Th... |
| climfveqmpt 44322 | Two functions that are eve... |
| climd 44323 | Express the predicate: Th... |
| clim2d 44324 | The limit of complex numbe... |
| fnlimfvre 44325 | The limit function of real... |
| allbutfifvre 44326 | Given a sequence of real-v... |
| climleltrp 44327 | The limit of complex numbe... |
| fnlimfvre2 44328 | The limit function of real... |
| fnlimf 44329 | The limit function of real... |
| fnlimabslt 44330 | A sequence of function val... |
| climfveqf 44331 | Two functions that are eve... |
| climmptf 44332 | Exhibit a function ` G ` w... |
| climfveqmpt3 44333 | Two functions that are eve... |
| climeldmeqf 44334 | Two functions that are eve... |
| climreclmpt 44335 | The limit of B convergent ... |
| limsupref 44336 | If a sequence is bounded, ... |
| limsupbnd1f 44337 | If a sequence is eventuall... |
| climbddf 44338 | A converging sequence of c... |
| climeqf 44339 | Two functions that are eve... |
| climeldmeqmpt3 44340 | Two functions that are eve... |
| limsupcld 44341 | Closure of the superior li... |
| climfv 44342 | The limit of a convergent ... |
| limsupval3 44343 | The superior limit of an i... |
| climfveqmpt2 44344 | Two functions that are eve... |
| limsup0 44345 | The superior limit of the ... |
| climeldmeqmpt2 44346 | Two functions that are eve... |
| limsupresre 44347 | The supremum limit of a fu... |
| climeqmpt 44348 | Two functions that are eve... |
| climfvd 44349 | The limit of a convergent ... |
| limsuplesup 44350 | An upper bound for the sup... |
| limsupresico 44351 | The superior limit doesn't... |
| limsuppnfdlem 44352 | If the restriction of a fu... |
| limsuppnfd 44353 | If the restriction of a fu... |
| limsupresuz 44354 | If the real part of the do... |
| limsupub 44355 | If the limsup is not ` +oo... |
| limsupres 44356 | The superior limit of a re... |
| climinf2lem 44357 | A convergent, nonincreasin... |
| climinf2 44358 | A convergent, nonincreasin... |
| limsupvaluz 44359 | The superior limit, when t... |
| limsupresuz2 44360 | If the domain of a functio... |
| limsuppnflem 44361 | If the restriction of a fu... |
| limsuppnf 44362 | If the restriction of a fu... |
| limsupubuzlem 44363 | If the limsup is not ` +oo... |
| limsupubuz 44364 | For a real-valued function... |
| climinf2mpt 44365 | A bounded below, monotonic... |
| climinfmpt 44366 | A bounded below, monotonic... |
| climinf3 44367 | A convergent, nonincreasin... |
| limsupvaluzmpt 44368 | The superior limit, when t... |
| limsupequzmpt2 44369 | Two functions that are eve... |
| limsupubuzmpt 44370 | If the limsup is not ` +oo... |
| limsupmnflem 44371 | The superior limit of a fu... |
| limsupmnf 44372 | The superior limit of a fu... |
| limsupequzlem 44373 | Two functions that are eve... |
| limsupequz 44374 | Two functions that are eve... |
| limsupre2lem 44375 | Given a function on the ex... |
| limsupre2 44376 | Given a function on the ex... |
| limsupmnfuzlem 44377 | The superior limit of a fu... |
| limsupmnfuz 44378 | The superior limit of a fu... |
| limsupequzmptlem 44379 | Two functions that are eve... |
| limsupequzmpt 44380 | Two functions that are eve... |
| limsupre2mpt 44381 | Given a function on the ex... |
| limsupequzmptf 44382 | Two functions that are eve... |
| limsupre3lem 44383 | Given a function on the ex... |
| limsupre3 44384 | Given a function on the ex... |
| limsupre3mpt 44385 | Given a function on the ex... |
| limsupre3uzlem 44386 | Given a function on the ex... |
| limsupre3uz 44387 | Given a function on the ex... |
| limsupreuz 44388 | Given a function on the re... |
| limsupvaluz2 44389 | The superior limit, when t... |
| limsupreuzmpt 44390 | Given a function on the re... |
| supcnvlimsup 44391 | If a function on a set of ... |
| supcnvlimsupmpt 44392 | If a function on a set of ... |
| 0cnv 44393 | If ` (/) ` is a complex nu... |
| climuzlem 44394 | Express the predicate: Th... |
| climuz 44395 | Express the predicate: Th... |
| lmbr3v 44396 | Express the binary relatio... |
| climisp 44397 | If a sequence converges to... |
| lmbr3 44398 | Express the binary relatio... |
| climrescn 44399 | A sequence converging w.r.... |
| climxrrelem 44400 | If a sequence ranging over... |
| climxrre 44401 | If a sequence ranging over... |
| limsuplt2 44404 | The defining property of t... |
| liminfgord 44405 | Ordering property of the i... |
| limsupvald 44406 | The superior limit of a se... |
| limsupresicompt 44407 | The superior limit doesn't... |
| limsupcli 44408 | Closure of the superior li... |
| liminfgf 44409 | Closure of the inferior li... |
| liminfval 44410 | The inferior limit of a se... |
| climlimsup 44411 | A sequence of real numbers... |
| limsupge 44412 | The defining property of t... |
| liminfgval 44413 | Value of the inferior limi... |
| liminfcl 44414 | Closure of the inferior li... |
| liminfvald 44415 | The inferior limit of a se... |
| liminfval5 44416 | The inferior limit of an i... |
| limsupresxr 44417 | The superior limit of a fu... |
| liminfresxr 44418 | The inferior limit of a fu... |
| liminfval2 44419 | The superior limit, relati... |
| climlimsupcex 44420 | Counterexample for ~ climl... |
| liminfcld 44421 | Closure of the inferior li... |
| liminfresico 44422 | The inferior limit doesn't... |
| limsup10exlem 44423 | The range of the given fun... |
| limsup10ex 44424 | The superior limit of a fu... |
| liminf10ex 44425 | The inferior limit of a fu... |
| liminflelimsuplem 44426 | The superior limit is grea... |
| liminflelimsup 44427 | The superior limit is grea... |
| limsupgtlem 44428 | For any positive real, the... |
| limsupgt 44429 | Given a sequence of real n... |
| liminfresre 44430 | The inferior limit of a fu... |
| liminfresicompt 44431 | The inferior limit doesn't... |
| liminfltlimsupex 44432 | An example where the ` lim... |
| liminfgelimsup 44433 | The inferior limit is grea... |
| liminfvalxr 44434 | Alternate definition of ` ... |
| liminfresuz 44435 | If the real part of the do... |
| liminflelimsupuz 44436 | The superior limit is grea... |
| liminfvalxrmpt 44437 | Alternate definition of ` ... |
| liminfresuz2 44438 | If the domain of a functio... |
| liminfgelimsupuz 44439 | The inferior limit is grea... |
| liminfval4 44440 | Alternate definition of ` ... |
| liminfval3 44441 | Alternate definition of ` ... |
| liminfequzmpt2 44442 | Two functions that are eve... |
| liminfvaluz 44443 | Alternate definition of ` ... |
| liminf0 44444 | The inferior limit of the ... |
| limsupval4 44445 | Alternate definition of ` ... |
| liminfvaluz2 44446 | Alternate definition of ` ... |
| liminfvaluz3 44447 | Alternate definition of ` ... |
| liminflelimsupcex 44448 | A counterexample for ~ lim... |
| limsupvaluz3 44449 | Alternate definition of ` ... |
| liminfvaluz4 44450 | Alternate definition of ` ... |
| limsupvaluz4 44451 | Alternate definition of ` ... |
| climliminflimsupd 44452 | If a sequence of real numb... |
| liminfreuzlem 44453 | Given a function on the re... |
| liminfreuz 44454 | Given a function on the re... |
| liminfltlem 44455 | Given a sequence of real n... |
| liminflt 44456 | Given a sequence of real n... |
| climliminf 44457 | A sequence of real numbers... |
| liminflimsupclim 44458 | A sequence of real numbers... |
| climliminflimsup 44459 | A sequence of real numbers... |
| climliminflimsup2 44460 | A sequence of real numbers... |
| climliminflimsup3 44461 | A sequence of real numbers... |
| climliminflimsup4 44462 | A sequence of real numbers... |
| limsupub2 44463 | A extended real valued fun... |
| limsupubuz2 44464 | A sequence with values in ... |
| xlimpnfxnegmnf 44465 | A sequence converges to ` ... |
| liminflbuz2 44466 | A sequence with values in ... |
| liminfpnfuz 44467 | The inferior limit of a fu... |
| liminflimsupxrre 44468 | A sequence with values in ... |
| xlimrel 44471 | The limit on extended real... |
| xlimres 44472 | A function converges iff i... |
| xlimcl 44473 | The limit of a sequence of... |
| rexlimddv2 44474 | Restricted existential eli... |
| xlimclim 44475 | Given a sequence of reals,... |
| xlimconst 44476 | A constant sequence conver... |
| climxlim 44477 | A converging sequence in t... |
| xlimbr 44478 | Express the binary relatio... |
| fuzxrpmcn 44479 | A function mapping from an... |
| cnrefiisplem 44480 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 44481 | A non-real, complex number... |
| xlimxrre 44482 | If a sequence ranging over... |
| xlimmnfvlem1 44483 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 44484 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 44485 | A function converges to mi... |
| xlimconst2 44486 | A sequence that eventually... |
| xlimpnfvlem1 44487 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 44488 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 44489 | A function converges to pl... |
| xlimclim2lem 44490 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 44491 | Given a sequence of extend... |
| xlimmnf 44492 | A function converges to mi... |
| xlimpnf 44493 | A function converges to pl... |
| xlimmnfmpt 44494 | A function converges to pl... |
| xlimpnfmpt 44495 | A function converges to pl... |
| climxlim2lem 44496 | In this lemma for ~ climxl... |
| climxlim2 44497 | A sequence of extended rea... |
| dfxlim2v 44498 | An alternative definition ... |
| dfxlim2 44499 | An alternative definition ... |
| climresd 44500 | A function restricted to u... |
| climresdm 44501 | A real function converges ... |
| dmclimxlim 44502 | A real valued sequence tha... |
| xlimmnflimsup2 44503 | A sequence of extended rea... |
| xlimuni 44504 | An infinite sequence conve... |
| xlimclimdm 44505 | A sequence of extended rea... |
| xlimfun 44506 | The convergence relation o... |
| xlimmnflimsup 44507 | If a sequence of extended ... |
| xlimdm 44508 | Two ways to express that a... |
| xlimpnfxnegmnf2 44509 | A sequence converges to ` ... |
| xlimresdm 44510 | A function converges in th... |
| xlimpnfliminf 44511 | If a sequence of extended ... |
| xlimpnfliminf2 44512 | A sequence of extended rea... |
| xlimliminflimsup 44513 | A sequence of extended rea... |
| xlimlimsupleliminf 44514 | A sequence of extended rea... |
| coseq0 44515 | A complex number whose cos... |
| sinmulcos 44516 | Multiplication formula for... |
| coskpi2 44517 | The cosine of an integer m... |
| cosnegpi 44518 | The cosine of negative ` _... |
| sinaover2ne0 44519 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 44520 | The cosine of an integer m... |
| mulcncff 44521 | The multiplication of two ... |
| cncfmptssg 44522 | A continuous complex funct... |
| constcncfg 44523 | A constant function is a c... |
| idcncfg 44524 | The identity function is a... |
| cncfshift 44525 | A periodic continuous func... |
| resincncf 44526 | ` sin ` restricted to real... |
| addccncf2 44527 | Adding a constant is a con... |
| 0cnf 44528 | The empty set is a continu... |
| fsumcncf 44529 | The finite sum of continuo... |
| cncfperiod 44530 | A periodic continuous func... |
| subcncff 44531 | The subtraction of two con... |
| negcncfg 44532 | The opposite of a continuo... |
| cnfdmsn 44533 | A function with a singleto... |
| cncfcompt 44534 | Composition of continuous ... |
| addcncff 44535 | The sum of two continuous ... |
| ioccncflimc 44536 | Limit at the upper bound o... |
| cncfuni 44537 | A complex function on a su... |
| icccncfext 44538 | A continuous function on a... |
| cncficcgt0 44539 | A the absolute value of a ... |
| icocncflimc 44540 | Limit at the lower bound, ... |
| cncfdmsn 44541 | A complex function with a ... |
| divcncff 44542 | The quotient of two contin... |
| cncfshiftioo 44543 | A periodic continuous func... |
| cncfiooicclem1 44544 | A continuous function ` F ... |
| cncfiooicc 44545 | A continuous function ` F ... |
| cncfiooiccre 44546 | A continuous function ` F ... |
| cncfioobdlem 44547 | ` G ` actually extends ` F... |
| cncfioobd 44548 | A continuous function ` F ... |
| jumpncnp 44549 | Jump discontinuity or disc... |
| cxpcncf2 44550 | The complex power function... |
| fprodcncf 44551 | The finite product of cont... |
| add1cncf 44552 | Addition to a constant is ... |
| add2cncf 44553 | Addition to a constant is ... |
| sub1cncfd 44554 | Subtracting a constant is ... |
| sub2cncfd 44555 | Subtraction from a constan... |
| fprodsub2cncf 44556 | ` F ` is continuous. (Con... |
| fprodadd2cncf 44557 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 44558 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 44559 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 44560 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 44561 | The sequence ` S ` of fini... |
| dvsinexp 44562 | The derivative of sin^N . ... |
| dvcosre 44563 | The real derivative of the... |
| dvsinax 44564 | Derivative exercise: the d... |
| dvsubf 44565 | The subtraction rule for e... |
| dvmptconst 44566 | Function-builder for deriv... |
| dvcnre 44567 | From complex differentiati... |
| dvmptidg 44568 | Function-builder for deriv... |
| dvresntr 44569 | Function-builder for deriv... |
| fperdvper 44570 | The derivative of a period... |
| dvasinbx 44571 | Derivative exercise: the d... |
| dvresioo 44572 | Restriction of a derivativ... |
| dvdivf 44573 | The quotient rule for ever... |
| dvdivbd 44574 | A sufficient condition for... |
| dvsubcncf 44575 | A sufficient condition for... |
| dvmulcncf 44576 | A sufficient condition for... |
| dvcosax 44577 | Derivative exercise: the d... |
| dvdivcncf 44578 | A sufficient condition for... |
| dvbdfbdioolem1 44579 | Given a function with boun... |
| dvbdfbdioolem2 44580 | A function on an open inte... |
| dvbdfbdioo 44581 | A function on an open inte... |
| ioodvbdlimc1lem1 44582 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 44583 | Limit at the lower bound o... |
| ioodvbdlimc1 44584 | A real function with bound... |
| ioodvbdlimc2lem 44585 | Limit at the upper bound o... |
| ioodvbdlimc2 44586 | A real function with bound... |
| dvdmsscn 44587 | ` X ` is a subset of ` CC ... |
| dvmptmulf 44588 | Function-builder for deriv... |
| dvnmptdivc 44589 | Function-builder for itera... |
| dvdsn1add 44590 | If ` K ` divides ` N ` but... |
| dvxpaek 44591 | Derivative of the polynomi... |
| dvnmptconst 44592 | The ` N ` -th derivative o... |
| dvnxpaek 44593 | The ` n ` -th derivative o... |
| dvnmul 44594 | Function-builder for the `... |
| dvmptfprodlem 44595 | Induction step for ~ dvmpt... |
| dvmptfprod 44596 | Function-builder for deriv... |
| dvnprodlem1 44597 | ` D ` is bijective. (Cont... |
| dvnprodlem2 44598 | Induction step for ~ dvnpr... |
| dvnprodlem3 44599 | The multinomial formula fo... |
| dvnprod 44600 | The multinomial formula fo... |
| itgsin0pilem1 44601 | Calculation of the integra... |
| ibliccsinexp 44602 | sin^n on a closed interval... |
| itgsin0pi 44603 | Calculation of the integra... |
| iblioosinexp 44604 | sin^n on an open integral ... |
| itgsinexplem1 44605 | Integration by parts is ap... |
| itgsinexp 44606 | A recursive formula for th... |
| iblconstmpt 44607 | A constant function is int... |
| itgeq1d 44608 | Equality theorem for an in... |
| mbfres2cn 44609 | Measurability of a piecewi... |
| vol0 44610 | The measure of the empty s... |
| ditgeqiooicc 44611 | A function ` F ` on an ope... |
| volge0 44612 | The volume of a set is alw... |
| cnbdibl 44613 | A continuous bounded funct... |
| snmbl 44614 | A singleton is measurable.... |
| ditgeq3d 44615 | Equality theorem for the d... |
| iblempty 44616 | The empty function is inte... |
| iblsplit 44617 | The union of two integrabl... |
| volsn 44618 | A singleton has 0 Lebesgue... |
| itgvol0 44619 | If the domani is negligibl... |
| itgcoscmulx 44620 | Exercise: the integral of ... |
| iblsplitf 44621 | A version of ~ iblsplit us... |
| ibliooicc 44622 | If a function is integrabl... |
| volioc 44623 | The measure of a left-open... |
| iblspltprt 44624 | If a function is integrabl... |
| itgsincmulx 44625 | Exercise: the integral of ... |
| itgsubsticclem 44626 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 44627 | Integration by u-substitut... |
| itgioocnicc 44628 | The integral of a piecewis... |
| iblcncfioo 44629 | A continuous function ` F ... |
| itgspltprt 44630 | The ` S. ` integral splits... |
| itgiccshift 44631 | The integral of a function... |
| itgperiod 44632 | The integral of a periodic... |
| itgsbtaddcnst 44633 | Integral substitution, add... |
| volico 44634 | The measure of left-closed... |
| sublevolico 44635 | The Lebesgue measure of a ... |
| dmvolss 44636 | Lebesgue measurable sets a... |
| ismbl3 44637 | The predicate " ` A ` is L... |
| volioof 44638 | The function that assigns ... |
| ovolsplit 44639 | The Lebesgue outer measure... |
| fvvolioof 44640 | The function value of the ... |
| volioore 44641 | The measure of an open int... |
| fvvolicof 44642 | The function value of the ... |
| voliooico 44643 | An open interval and a lef... |
| ismbl4 44644 | The predicate " ` A ` is L... |
| volioofmpt 44645 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 44646 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 44647 | The Lebesgue measure of op... |
| volicofmpt 44648 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 44649 | The Lebesgue measure of a ... |
| voliccico 44650 | A closed interval and a le... |
| mbfdmssre 44651 | The domain of a measurable... |
| stoweidlem1 44652 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 44653 | lemma for ~ stoweid : here... |
| stoweidlem3 44654 | Lemma for ~ stoweid : if `... |
| stoweidlem4 44655 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 44656 | There exists a δ as ... |
| stoweidlem6 44657 | Lemma for ~ stoweid : two ... |
| stoweidlem7 44658 | This lemma is used to prov... |
| stoweidlem8 44659 | Lemma for ~ stoweid : two ... |
| stoweidlem9 44660 | Lemma for ~ stoweid : here... |
| stoweidlem10 44661 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 44662 | This lemma is used to prov... |
| stoweidlem12 44663 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 44664 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 44665 | There exists a ` k ` as in... |
| stoweidlem15 44666 | This lemma is used to prov... |
| stoweidlem16 44667 | Lemma for ~ stoweid . The... |
| stoweidlem17 44668 | This lemma proves that the... |
| stoweidlem18 44669 | This theorem proves Lemma ... |
| stoweidlem19 44670 | If a set of real functions... |
| stoweidlem20 44671 | If a set A of real functio... |
| stoweidlem21 44672 | Once the Stone Weierstrass... |
| stoweidlem22 44673 | If a set of real functions... |
| stoweidlem23 44674 | This lemma is used to prov... |
| stoweidlem24 44675 | This lemma proves that for... |
| stoweidlem25 44676 | This lemma proves that for... |
| stoweidlem26 44677 | This lemma is used to prov... |
| stoweidlem27 44678 | This lemma is used to prov... |
| stoweidlem28 44679 | There exists a δ as ... |
| stoweidlem29 44680 | When the hypothesis for th... |
| stoweidlem30 44681 | This lemma is used to prov... |
| stoweidlem31 44682 | This lemma is used to prov... |
| stoweidlem32 44683 | If a set A of real functio... |
| stoweidlem33 44684 | If a set of real functions... |
| stoweidlem34 44685 | This lemma proves that for... |
| stoweidlem35 44686 | This lemma is used to prov... |
| stoweidlem36 44687 | This lemma is used to prov... |
| stoweidlem37 44688 | This lemma is used to prov... |
| stoweidlem38 44689 | This lemma is used to prov... |
| stoweidlem39 44690 | This lemma is used to prov... |
| stoweidlem40 44691 | This lemma proves that q_n... |
| stoweidlem41 44692 | This lemma is used to prov... |
| stoweidlem42 44693 | This lemma is used to prov... |
| stoweidlem43 44694 | This lemma is used to prov... |
| stoweidlem44 44695 | This lemma is used to prov... |
| stoweidlem45 44696 | This lemma proves that, gi... |
| stoweidlem46 44697 | This lemma proves that set... |
| stoweidlem47 44698 | Subtracting a constant fro... |
| stoweidlem48 44699 | This lemma is used to prov... |
| stoweidlem49 44700 | There exists a function q_... |
| stoweidlem50 44701 | This lemma proves that set... |
| stoweidlem51 44702 | There exists a function x ... |
| stoweidlem52 44703 | There exists a neighborhoo... |
| stoweidlem53 44704 | This lemma is used to prov... |
| stoweidlem54 44705 | There exists a function ` ... |
| stoweidlem55 44706 | This lemma proves the exis... |
| stoweidlem56 44707 | This theorem proves Lemma ... |
| stoweidlem57 44708 | There exists a function x ... |
| stoweidlem58 44709 | This theorem proves Lemma ... |
| stoweidlem59 44710 | This lemma proves that the... |
| stoweidlem60 44711 | This lemma proves that the... |
| stoweidlem61 44712 | This lemma proves that the... |
| stoweidlem62 44713 | This theorem proves the St... |
| stoweid 44714 | This theorem proves the St... |
| stowei 44715 | This theorem proves the St... |
| wallispilem1 44716 | ` I ` is monotone: increas... |
| wallispilem2 44717 | A first set of properties ... |
| wallispilem3 44718 | I maps to real values. (C... |
| wallispilem4 44719 | ` F ` maps to explicit exp... |
| wallispilem5 44720 | The sequence ` H ` converg... |
| wallispi 44721 | Wallis' formula for π :... |
| wallispi2lem1 44722 | An intermediate step betwe... |
| wallispi2lem2 44723 | Two expressions are proven... |
| wallispi2 44724 | An alternative version of ... |
| stirlinglem1 44725 | A simple limit of fraction... |
| stirlinglem2 44726 | ` A ` maps to positive rea... |
| stirlinglem3 44727 | Long but simple algebraic ... |
| stirlinglem4 44728 | Algebraic manipulation of ... |
| stirlinglem5 44729 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 44730 | A series that converges to... |
| stirlinglem7 44731 | Algebraic manipulation of ... |
| stirlinglem8 44732 | If ` A ` converges to ` C ... |
| stirlinglem9 44733 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 44734 | A bound for any B(N)-B(N +... |
| stirlinglem11 44735 | ` B ` is decreasing. (Con... |
| stirlinglem12 44736 | The sequence ` B ` is boun... |
| stirlinglem13 44737 | ` B ` is decreasing and ha... |
| stirlinglem14 44738 | The sequence ` A ` converg... |
| stirlinglem15 44739 | The Stirling's formula is ... |
| stirling 44740 | Stirling's approximation f... |
| stirlingr 44741 | Stirling's approximation f... |
| dirkerval 44742 | The N_th Dirichlet Kernel.... |
| dirker2re 44743 | The Dirichlet Kernel value... |
| dirkerdenne0 44744 | The Dirichlet Kernel denom... |
| dirkerval2 44745 | The N_th Dirichlet Kernel ... |
| dirkerre 44746 | The Dirichlet Kernel at an... |
| dirkerper 44747 | the Dirichlet Kernel has p... |
| dirkerf 44748 | For any natural number ` N... |
| dirkertrigeqlem1 44749 | Sum of an even number of a... |
| dirkertrigeqlem2 44750 | Trigonomic equality lemma ... |
| dirkertrigeqlem3 44751 | Trigonometric equality lem... |
| dirkertrigeq 44752 | Trigonometric equality for... |
| dirkeritg 44753 | The definite integral of t... |
| dirkercncflem1 44754 | If ` Y ` is a multiple of ... |
| dirkercncflem2 44755 | Lemma used to prove that t... |
| dirkercncflem3 44756 | The Dirichlet Kernel is co... |
| dirkercncflem4 44757 | The Dirichlet Kernel is co... |
| dirkercncf 44758 | For any natural number ` N... |
| fourierdlem1 44759 | A partition interval is a ... |
| fourierdlem2 44760 | Membership in a partition.... |
| fourierdlem3 44761 | Membership in a partition.... |
| fourierdlem4 44762 | ` E ` is a function that m... |
| fourierdlem5 44763 | ` S ` is a function. (Con... |
| fourierdlem6 44764 | ` X ` is in the periodic p... |
| fourierdlem7 44765 | The difference between the... |
| fourierdlem8 44766 | A partition interval is a ... |
| fourierdlem9 44767 | ` H ` is a complex functio... |
| fourierdlem10 44768 | Condition on the bounds of... |
| fourierdlem11 44769 | If there is a partition, t... |
| fourierdlem12 44770 | A point of a partition is ... |
| fourierdlem13 44771 | Value of ` V ` in terms of... |
| fourierdlem14 44772 | Given the partition ` V ` ... |
| fourierdlem15 44773 | The range of the partition... |
| fourierdlem16 44774 | The coefficients of the fo... |
| fourierdlem17 44775 | The defined ` L ` is actua... |
| fourierdlem18 44776 | The function ` S ` is cont... |
| fourierdlem19 44777 | If two elements of ` D ` h... |
| fourierdlem20 44778 | Every interval in the part... |
| fourierdlem21 44779 | The coefficients of the fo... |
| fourierdlem22 44780 | The coefficients of the fo... |
| fourierdlem23 44781 | If ` F ` is continuous and... |
| fourierdlem24 44782 | A sufficient condition for... |
| fourierdlem25 44783 | If ` C ` is not in the ran... |
| fourierdlem26 44784 | Periodic image of a point ... |
| fourierdlem27 44785 | A partition open interval ... |
| fourierdlem28 44786 | Derivative of ` ( F `` ( X... |
| fourierdlem29 44787 | Explicit function value fo... |
| fourierdlem30 44788 | Sum of three small pieces ... |
| fourierdlem31 44789 | If ` A ` is finite and for... |
| fourierdlem32 44790 | Limit of a continuous func... |
| fourierdlem33 44791 | Limit of a continuous func... |
| fourierdlem34 44792 | A partition is one to one.... |
| fourierdlem35 44793 | There is a single point in... |
| fourierdlem36 44794 | ` F ` is an isomorphism. ... |
| fourierdlem37 44795 | ` I ` is a function that m... |
| fourierdlem38 44796 | The function ` F ` is cont... |
| fourierdlem39 44797 | Integration by parts of ... |
| fourierdlem40 44798 | ` H ` is a continuous func... |
| fourierdlem41 44799 | Lemma used to prove that e... |
| fourierdlem42 44800 | The set of points in a mov... |
| fourierdlem43 44801 | ` K ` is a real function. ... |
| fourierdlem44 44802 | A condition for having ` (... |
| fourierdlem46 44803 | The function ` F ` has a l... |
| fourierdlem47 44804 | For ` r ` large enough, th... |
| fourierdlem48 44805 | The given periodic functio... |
| fourierdlem49 44806 | The given periodic functio... |
| fourierdlem50 44807 | Continuity of ` O ` and it... |
| fourierdlem51 44808 | ` X ` is in the periodic p... |
| fourierdlem52 44809 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 44810 | The limit of ` F ( s ) ` a... |
| fourierdlem54 44811 | Given a partition ` Q ` an... |
| fourierdlem55 44812 | ` U ` is a real function. ... |
| fourierdlem56 44813 | Derivative of the ` K ` fu... |
| fourierdlem57 44814 | The derivative of ` O ` . ... |
| fourierdlem58 44815 | The derivative of ` K ` is... |
| fourierdlem59 44816 | The derivative of ` H ` is... |
| fourierdlem60 44817 | Given a differentiable fun... |
| fourierdlem61 44818 | Given a differentiable fun... |
| fourierdlem62 44819 | The function ` K ` is cont... |
| fourierdlem63 44820 | The upper bound of interva... |
| fourierdlem64 44821 | The partition ` V ` is fin... |
| fourierdlem65 44822 | The distance of two adjace... |
| fourierdlem66 44823 | Value of the ` G ` functio... |
| fourierdlem67 44824 | ` G ` is a function. (Con... |
| fourierdlem68 44825 | The derivative of ` O ` is... |
| fourierdlem69 44826 | A piecewise continuous fun... |
| fourierdlem70 44827 | A piecewise continuous fun... |
| fourierdlem71 44828 | A periodic piecewise conti... |
| fourierdlem72 44829 | The derivative of ` O ` is... |
| fourierdlem73 44830 | A version of the Riemann L... |
| fourierdlem74 44831 | Given a piecewise smooth f... |
| fourierdlem75 44832 | Given a piecewise smooth f... |
| fourierdlem76 44833 | Continuity of ` O ` and it... |
| fourierdlem77 44834 | If ` H ` is bounded, then ... |
| fourierdlem78 44835 | ` G ` is continuous when r... |
| fourierdlem79 44836 | ` E ` projects every inter... |
| fourierdlem80 44837 | The derivative of ` O ` is... |
| fourierdlem81 44838 | The integral of a piecewis... |
| fourierdlem82 44839 | Integral by substitution, ... |
| fourierdlem83 44840 | The fourier partial sum fo... |
| fourierdlem84 44841 | If ` F ` is piecewise coni... |
| fourierdlem85 44842 | Limit of the function ` G ... |
| fourierdlem86 44843 | Continuity of ` O ` and it... |
| fourierdlem87 44844 | The integral of ` G ` goes... |
| fourierdlem88 44845 | Given a piecewise continuo... |
| fourierdlem89 44846 | Given a piecewise continuo... |
| fourierdlem90 44847 | Given a piecewise continuo... |
| fourierdlem91 44848 | Given a piecewise continuo... |
| fourierdlem92 44849 | The integral of a piecewis... |
| fourierdlem93 44850 | Integral by substitution (... |
| fourierdlem94 44851 | For a piecewise smooth fun... |
| fourierdlem95 44852 | Algebraic manipulation of ... |
| fourierdlem96 44853 | limit for ` F ` at the low... |
| fourierdlem97 44854 | ` F ` is continuous on the... |
| fourierdlem98 44855 | ` F ` is continuous on the... |
| fourierdlem99 44856 | limit for ` F ` at the upp... |
| fourierdlem100 44857 | A piecewise continuous fun... |
| fourierdlem101 44858 | Integral by substitution f... |
| fourierdlem102 44859 | For a piecewise smooth fun... |
| fourierdlem103 44860 | The half lower part of the... |
| fourierdlem104 44861 | The half upper part of the... |
| fourierdlem105 44862 | A piecewise continuous fun... |
| fourierdlem106 44863 | For a piecewise smooth fun... |
| fourierdlem107 44864 | The integral of a piecewis... |
| fourierdlem108 44865 | The integral of a piecewis... |
| fourierdlem109 44866 | The integral of a piecewis... |
| fourierdlem110 44867 | The integral of a piecewis... |
| fourierdlem111 44868 | The fourier partial sum fo... |
| fourierdlem112 44869 | Here abbreviations (local ... |
| fourierdlem113 44870 | Fourier series convergence... |
| fourierdlem114 44871 | Fourier series convergence... |
| fourierdlem115 44872 | Fourier serier convergence... |
| fourierd 44873 | Fourier series convergence... |
| fourierclimd 44874 | Fourier series convergence... |
| fourierclim 44875 | Fourier series convergence... |
| fourier 44876 | Fourier series convergence... |
| fouriercnp 44877 | If ` F ` is continuous at ... |
| fourier2 44878 | Fourier series convergence... |
| sqwvfoura 44879 | Fourier coefficients for t... |
| sqwvfourb 44880 | Fourier series ` B ` coeff... |
| fourierswlem 44881 | The Fourier series for the... |
| fouriersw 44882 | Fourier series convergence... |
| fouriercn 44883 | If the derivative of ` F `... |
| elaa2lem 44884 | Elementhood in the set of ... |
| elaa2 44885 | Elementhood in the set of ... |
| etransclem1 44886 | ` H ` is a function. (Con... |
| etransclem2 44887 | Derivative of ` G ` . (Co... |
| etransclem3 44888 | The given ` if ` term is a... |
| etransclem4 44889 | ` F ` expressed as a finit... |
| etransclem5 44890 | A change of bound variable... |
| etransclem6 44891 | A change of bound variable... |
| etransclem7 44892 | The given product is an in... |
| etransclem8 44893 | ` F ` is a function. (Con... |
| etransclem9 44894 | If ` K ` divides ` N ` but... |
| etransclem10 44895 | The given ` if ` term is a... |
| etransclem11 44896 | A change of bound variable... |
| etransclem12 44897 | ` C ` applied to ` N ` . ... |
| etransclem13 44898 | ` F ` applied to ` Y ` . ... |
| etransclem14 44899 | Value of the term ` T ` , ... |
| etransclem15 44900 | Value of the term ` T ` , ... |
| etransclem16 44901 | Every element in the range... |
| etransclem17 44902 | The ` N ` -th derivative o... |
| etransclem18 44903 | The given function is inte... |
| etransclem19 44904 | The ` N ` -th derivative o... |
| etransclem20 44905 | ` H ` is smooth. (Contrib... |
| etransclem21 44906 | The ` N ` -th derivative o... |
| etransclem22 44907 | The ` N ` -th derivative o... |
| etransclem23 44908 | This is the claim proof in... |
| etransclem24 44909 | ` P ` divides the I -th de... |
| etransclem25 44910 | ` P ` factorial divides th... |
| etransclem26 44911 | Every term in the sum of t... |
| etransclem27 44912 | The ` N ` -th derivative o... |
| etransclem28 44913 | ` ( P - 1 ) ` factorial di... |
| etransclem29 44914 | The ` N ` -th derivative o... |
| etransclem30 44915 | The ` N ` -th derivative o... |
| etransclem31 44916 | The ` N ` -th derivative o... |
| etransclem32 44917 | This is the proof for the ... |
| etransclem33 44918 | ` F ` is smooth. (Contrib... |
| etransclem34 44919 | The ` N ` -th derivative o... |
| etransclem35 44920 | ` P ` does not divide the ... |
| etransclem36 44921 | The ` N ` -th derivative o... |
| etransclem37 44922 | ` ( P - 1 ) ` factorial di... |
| etransclem38 44923 | ` P ` divides the I -th de... |
| etransclem39 44924 | ` G ` is a function. (Con... |
| etransclem40 44925 | The ` N ` -th derivative o... |
| etransclem41 44926 | ` P ` does not divide the ... |
| etransclem42 44927 | The ` N ` -th derivative o... |
| etransclem43 44928 | ` G ` is a continuous func... |
| etransclem44 44929 | The given finite sum is no... |
| etransclem45 44930 | ` K ` is an integer. (Con... |
| etransclem46 44931 | This is the proof for equa... |
| etransclem47 44932 | ` _e ` is transcendental. ... |
| etransclem48 44933 | ` _e ` is transcendental. ... |
| etransc 44934 | ` _e ` is transcendental. ... |
| rrxtopn 44935 | The topology of the genera... |
| rrxngp 44936 | Generalized Euclidean real... |
| rrxtps 44937 | Generalized Euclidean real... |
| rrxtopnfi 44938 | The topology of the n-dime... |
| rrxtopon 44939 | The topology on generalize... |
| rrxtop 44940 | The topology on generalize... |
| rrndistlt 44941 | Given two points in the sp... |
| rrxtoponfi 44942 | The topology on n-dimensio... |
| rrxunitopnfi 44943 | The base set of the standa... |
| rrxtopn0 44944 | The topology of the zero-d... |
| qndenserrnbllem 44945 | n-dimensional rational num... |
| qndenserrnbl 44946 | n-dimensional rational num... |
| rrxtopn0b 44947 | The topology of the zero-d... |
| qndenserrnopnlem 44948 | n-dimensional rational num... |
| qndenserrnopn 44949 | n-dimensional rational num... |
| qndenserrn 44950 | n-dimensional rational num... |
| rrxsnicc 44951 | A multidimensional singlet... |
| rrnprjdstle 44952 | The distance between two p... |
| rrndsmet 44953 | ` D ` is a metric for the ... |
| rrndsxmet 44954 | ` D ` is an extended metri... |
| ioorrnopnlem 44955 | The a point in an indexed ... |
| ioorrnopn 44956 | The indexed product of ope... |
| ioorrnopnxrlem 44957 | Given a point ` F ` that b... |
| ioorrnopnxr 44958 | The indexed product of ope... |
| issal 44965 | Express the predicate " ` ... |
| pwsal 44966 | The power set of a given s... |
| salunicl 44967 | SAlg sigma-algebra is clos... |
| saluncl 44968 | The union of two sets in a... |
| prsal 44969 | The pair of the empty set ... |
| saldifcl 44970 | The complement of an eleme... |
| 0sal 44971 | The empty set belongs to e... |
| salgenval 44972 | The sigma-algebra generate... |
| saliunclf 44973 | SAlg sigma-algebra is clos... |
| saliuncl 44974 | SAlg sigma-algebra is clos... |
| salincl 44975 | The intersection of two se... |
| saluni 44976 | A set is an element of any... |
| saliinclf 44977 | SAlg sigma-algebra is clos... |
| saliincl 44978 | SAlg sigma-algebra is clos... |
| saldifcl2 44979 | The difference of two elem... |
| intsaluni 44980 | The union of an arbitrary ... |
| intsal 44981 | The arbitrary intersection... |
| salgenn0 44982 | The set used in the defini... |
| salgencl 44983 | ` SalGen ` actually genera... |
| issald 44984 | Sufficient condition to pr... |
| salexct 44985 | An example of nontrivial s... |
| sssalgen 44986 | A set is a subset of the s... |
| salgenss 44987 | The sigma-algebra generate... |
| salgenuni 44988 | The base set of the sigma-... |
| issalgend 44989 | One side of ~ dfsalgen2 . ... |
| salexct2 44990 | An example of a subset tha... |
| unisalgen 44991 | The union of a set belongs... |
| dfsalgen2 44992 | Alternate characterization... |
| salexct3 44993 | An example of a sigma-alge... |
| salgencntex 44994 | This counterexample shows ... |
| salgensscntex 44995 | This counterexample shows ... |
| issalnnd 44996 | Sufficient condition to pr... |
| dmvolsal 44997 | Lebesgue measurable sets f... |
| saldifcld 44998 | The complement of an eleme... |
| saluncld 44999 | The union of two sets in a... |
| salgencld 45000 | ` SalGen ` actually genera... |
| 0sald 45001 | The empty set belongs to e... |
| iooborel 45002 | An open interval is a Bore... |
| salincld 45003 | The intersection of two se... |
| salunid 45004 | A set is an element of any... |
| unisalgen2 45005 | The union of a set belongs... |
| bor1sal 45006 | The Borel sigma-algebra on... |
| iocborel 45007 | A left-open, right-closed ... |
| subsaliuncllem 45008 | A subspace sigma-algebra i... |
| subsaliuncl 45009 | A subspace sigma-algebra i... |
| subsalsal 45010 | A subspace sigma-algebra i... |
| subsaluni 45011 | A set belongs to the subsp... |
| salrestss 45012 | A sigma-algebra restricted... |
| sge0rnre 45015 | When ` sum^ ` is applied t... |
| fge0icoicc 45016 | If ` F ` maps to nonnegati... |
| sge0val 45017 | The value of the sum of no... |
| fge0npnf 45018 | If ` F ` maps to nonnegati... |
| sge0rnn0 45019 | The range used in the defi... |
| sge0vald 45020 | The value of the sum of no... |
| fge0iccico 45021 | A range of nonnegative ext... |
| gsumge0cl 45022 | Closure of group sum, for ... |
| sge0reval 45023 | Value of the sum of nonneg... |
| sge0pnfval 45024 | If a term in the sum of no... |
| fge0iccre 45025 | A range of nonnegative ext... |
| sge0z 45026 | Any nonnegative extended s... |
| sge00 45027 | The sum of nonnegative ext... |
| fsumlesge0 45028 | Every finite subsum of non... |
| sge0revalmpt 45029 | Value of the sum of nonneg... |
| sge0sn 45030 | A sum of a nonnegative ext... |
| sge0tsms 45031 | ` sum^ ` applied to a nonn... |
| sge0cl 45032 | The arbitrary sum of nonne... |
| sge0f1o 45033 | Re-index a nonnegative ext... |
| sge0snmpt 45034 | A sum of a nonnegative ext... |
| sge0ge0 45035 | The sum of nonnegative ext... |
| sge0xrcl 45036 | The arbitrary sum of nonne... |
| sge0repnf 45037 | The of nonnegative extende... |
| sge0fsum 45038 | The arbitrary sum of a fin... |
| sge0rern 45039 | If the sum of nonnegative ... |
| sge0supre 45040 | If the arbitrary sum of no... |
| sge0fsummpt 45041 | The arbitrary sum of a fin... |
| sge0sup 45042 | The arbitrary sum of nonne... |
| sge0less 45043 | A shorter sum of nonnegati... |
| sge0rnbnd 45044 | The range used in the defi... |
| sge0pr 45045 | Sum of a pair of nonnegati... |
| sge0gerp 45046 | The arbitrary sum of nonne... |
| sge0pnffigt 45047 | If the sum of nonnegative ... |
| sge0ssre 45048 | If a sum of nonnegative ex... |
| sge0lefi 45049 | A sum of nonnegative exten... |
| sge0lessmpt 45050 | A shorter sum of nonnegati... |
| sge0ltfirp 45051 | If the sum of nonnegative ... |
| sge0prle 45052 | The sum of a pair of nonne... |
| sge0gerpmpt 45053 | The arbitrary sum of nonne... |
| sge0resrnlem 45054 | The sum of nonnegative ext... |
| sge0resrn 45055 | The sum of nonnegative ext... |
| sge0ssrempt 45056 | If a sum of nonnegative ex... |
| sge0resplit 45057 | ` sum^ ` splits into two p... |
| sge0le 45058 | If all of the terms of sum... |
| sge0ltfirpmpt 45059 | If the extended sum of non... |
| sge0split 45060 | Split a sum of nonnegative... |
| sge0lempt 45061 | If all of the terms of sum... |
| sge0splitmpt 45062 | Split a sum of nonnegative... |
| sge0ss 45063 | Change the index set to a ... |
| sge0iunmptlemfi 45064 | Sum of nonnegative extende... |
| sge0p1 45065 | The addition of the next t... |
| sge0iunmptlemre 45066 | Sum of nonnegative extende... |
| sge0fodjrnlem 45067 | Re-index a nonnegative ext... |
| sge0fodjrn 45068 | Re-index a nonnegative ext... |
| sge0iunmpt 45069 | Sum of nonnegative extende... |
| sge0iun 45070 | Sum of nonnegative extende... |
| sge0nemnf 45071 | The generalized sum of non... |
| sge0rpcpnf 45072 | The sum of an infinite num... |
| sge0rernmpt 45073 | If the sum of nonnegative ... |
| sge0lefimpt 45074 | A sum of nonnegative exten... |
| nn0ssge0 45075 | Nonnegative integers are n... |
| sge0clmpt 45076 | The generalized sum of non... |
| sge0ltfirpmpt2 45077 | If the extended sum of non... |
| sge0isum 45078 | If a series of nonnegative... |
| sge0xrclmpt 45079 | The generalized sum of non... |
| sge0xp 45080 | Combine two generalized su... |
| sge0isummpt 45081 | If a series of nonnegative... |
| sge0ad2en 45082 | The value of the infinite ... |
| sge0isummpt2 45083 | If a series of nonnegative... |
| sge0xaddlem1 45084 | The extended addition of t... |
| sge0xaddlem2 45085 | The extended addition of t... |
| sge0xadd 45086 | The extended addition of t... |
| sge0fsummptf 45087 | The generalized sum of a f... |
| sge0snmptf 45088 | A sum of a nonnegative ext... |
| sge0ge0mpt 45089 | The sum of nonnegative ext... |
| sge0repnfmpt 45090 | The of nonnegative extende... |
| sge0pnffigtmpt 45091 | If the generalized sum of ... |
| sge0splitsn 45092 | Separate out a term in a g... |
| sge0pnffsumgt 45093 | If the sum of nonnegative ... |
| sge0gtfsumgt 45094 | If the generalized sum of ... |
| sge0uzfsumgt 45095 | If a real number is smalle... |
| sge0pnfmpt 45096 | If a term in the sum of no... |
| sge0seq 45097 | A series of nonnegative re... |
| sge0reuz 45098 | Value of the generalized s... |
| sge0reuzb 45099 | Value of the generalized s... |
| ismea 45102 | Express the predicate " ` ... |
| dmmeasal 45103 | The domain of a measure is... |
| meaf 45104 | A measure is a function th... |
| mea0 45105 | The measure of the empty s... |
| nnfoctbdjlem 45106 | There exists a mapping fro... |
| nnfoctbdj 45107 | There exists a mapping fro... |
| meadjuni 45108 | The measure of the disjoin... |
| meacl 45109 | The measure of a set is a ... |
| iundjiunlem 45110 | The sets in the sequence `... |
| iundjiun 45111 | Given a sequence ` E ` of ... |
| meaxrcl 45112 | The measure of a set is an... |
| meadjun 45113 | The measure of the union o... |
| meassle 45114 | The measure of a set is gr... |
| meaunle 45115 | The measure of the union o... |
| meadjiunlem 45116 | The sum of nonnegative ext... |
| meadjiun 45117 | The measure of the disjoin... |
| ismeannd 45118 | Sufficient condition to pr... |
| meaiunlelem 45119 | The measure of the union o... |
| meaiunle 45120 | The measure of the union o... |
| psmeasurelem 45121 | ` M ` applied to a disjoin... |
| psmeasure 45122 | Point supported measure, R... |
| voliunsge0lem 45123 | The Lebesgue measure funct... |
| voliunsge0 45124 | The Lebesgue measure funct... |
| volmea 45125 | The Lebesgue measure on th... |
| meage0 45126 | If the measure of a measur... |
| meadjunre 45127 | The measure of the union o... |
| meassre 45128 | If the measure of a measur... |
| meale0eq0 45129 | A measure that is less tha... |
| meadif 45130 | The measure of the differe... |
| meaiuninclem 45131 | Measures are continuous fr... |
| meaiuninc 45132 | Measures are continuous fr... |
| meaiuninc2 45133 | Measures are continuous fr... |
| meaiunincf 45134 | Measures are continuous fr... |
| meaiuninc3v 45135 | Measures are continuous fr... |
| meaiuninc3 45136 | Measures are continuous fr... |
| meaiininclem 45137 | Measures are continuous fr... |
| meaiininc 45138 | Measures are continuous fr... |
| meaiininc2 45139 | Measures are continuous fr... |
| caragenval 45144 | The sigma-algebra generate... |
| isome 45145 | Express the predicate " ` ... |
| caragenel 45146 | Membership in the Caratheo... |
| omef 45147 | An outer measure is a func... |
| ome0 45148 | The outer measure of the e... |
| omessle 45149 | The outer measure of a set... |
| omedm 45150 | The domain of an outer mea... |
| caragensplit 45151 | If ` E ` is in the set gen... |
| caragenelss 45152 | An element of the Caratheo... |
| carageneld 45153 | Membership in the Caratheo... |
| omecl 45154 | The outer measure of a set... |
| caragenss 45155 | The sigma-algebra generate... |
| omeunile 45156 | The outer measure of the u... |
| caragen0 45157 | The empty set belongs to a... |
| omexrcl 45158 | The outer measure of a set... |
| caragenunidm 45159 | The base set of an outer m... |
| caragensspw 45160 | The sigma-algebra generate... |
| omessre 45161 | If the outer measure of a ... |
| caragenuni 45162 | The base set of the sigma-... |
| caragenuncllem 45163 | The Caratheodory's constru... |
| caragenuncl 45164 | The Caratheodory's constru... |
| caragendifcl 45165 | The Caratheodory's constru... |
| caragenfiiuncl 45166 | The Caratheodory's constru... |
| omeunle 45167 | The outer measure of the u... |
| omeiunle 45168 | The outer measure of the i... |
| omelesplit 45169 | The outer measure of a set... |
| omeiunltfirp 45170 | If the outer measure of a ... |
| omeiunlempt 45171 | The outer measure of the i... |
| carageniuncllem1 45172 | The outer measure of ` A i... |
| carageniuncllem2 45173 | The Caratheodory's constru... |
| carageniuncl 45174 | The Caratheodory's constru... |
| caragenunicl 45175 | The Caratheodory's constru... |
| caragensal 45176 | Caratheodory's method gene... |
| caratheodorylem1 45177 | Lemma used to prove that C... |
| caratheodorylem2 45178 | Caratheodory's constructio... |
| caratheodory 45179 | Caratheodory's constructio... |
| 0ome 45180 | The map that assigns 0 to ... |
| isomenndlem 45181 | ` O ` is sub-additive w.r.... |
| isomennd 45182 | Sufficient condition to pr... |
| caragenel2d 45183 | Membership in the Caratheo... |
| omege0 45184 | If the outer measure of a ... |
| omess0 45185 | If the outer measure of a ... |
| caragencmpl 45186 | A measure built with the C... |
| vonval 45191 | Value of the Lebesgue meas... |
| ovnval 45192 | Value of the Lebesgue oute... |
| elhoi 45193 | Membership in a multidimen... |
| icoresmbl 45194 | A closed-below, open-above... |
| hoissre 45195 | The projection of a half-o... |
| ovnval2 45196 | Value of the Lebesgue oute... |
| volicorecl 45197 | The Lebesgue measure of a ... |
| hoiprodcl 45198 | The pre-measure of half-op... |
| hoicvr 45199 | ` I ` is a countable set o... |
| hoissrrn 45200 | A half-open interval is a ... |
| ovn0val 45201 | The Lebesgue outer measure... |
| ovnn0val 45202 | The value of a (multidimen... |
| ovnval2b 45203 | Value of the Lebesgue oute... |
| volicorescl 45204 | The Lebesgue measure of a ... |
| ovnprodcl 45205 | The product used in the de... |
| hoiprodcl2 45206 | The pre-measure of half-op... |
| hoicvrrex 45207 | Any subset of the multidim... |
| ovnsupge0 45208 | The set used in the defini... |
| ovnlecvr 45209 | Given a subset of multidim... |
| ovnpnfelsup 45210 | ` +oo ` is an element of t... |
| ovnsslelem 45211 | The (multidimensional, non... |
| ovnssle 45212 | The (multidimensional) Leb... |
| ovnlerp 45213 | The Lebesgue outer measure... |
| ovnf 45214 | The Lebesgue outer measure... |
| ovncvrrp 45215 | The Lebesgue outer measure... |
| ovn0lem 45216 | For any finite dimension, ... |
| ovn0 45217 | For any finite dimension, ... |
| ovncl 45218 | The Lebesgue outer measure... |
| ovn02 45219 | For the zero-dimensional s... |
| ovnxrcl 45220 | The Lebesgue outer measure... |
| ovnsubaddlem1 45221 | The Lebesgue outer measure... |
| ovnsubaddlem2 45222 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 45223 | ` ( voln* `` X ) ` is suba... |
| ovnome 45224 | ` ( voln* `` X ) ` is an o... |
| vonmea 45225 | ` ( voln `` X ) ` is a mea... |
| volicon0 45226 | The measure of a nonempty ... |
| hsphoif 45227 | ` H ` is a function (that ... |
| hoidmvval 45228 | The dimensional volume of ... |
| hoissrrn2 45229 | A half-open interval is a ... |
| hsphoival 45230 | ` H ` is a function (that ... |
| hoiprodcl3 45231 | The pre-measure of half-op... |
| volicore 45232 | The Lebesgue measure of a ... |
| hoidmvcl 45233 | The dimensional volume of ... |
| hoidmv0val 45234 | The dimensional volume of ... |
| hoidmvn0val 45235 | The dimensional volume of ... |
| hsphoidmvle2 45236 | The dimensional volume of ... |
| hsphoidmvle 45237 | The dimensional volume of ... |
| hoidmvval0 45238 | The dimensional volume of ... |
| hoiprodp1 45239 | The dimensional volume of ... |
| sge0hsphoire 45240 | If the generalized sum of ... |
| hoidmvval0b 45241 | The dimensional volume of ... |
| hoidmv1lelem1 45242 | The supremum of ` U ` belo... |
| hoidmv1lelem2 45243 | This is the contradiction ... |
| hoidmv1lelem3 45244 | The dimensional volume of ... |
| hoidmv1le 45245 | The dimensional volume of ... |
| hoidmvlelem1 45246 | The supremum of ` U ` belo... |
| hoidmvlelem2 45247 | This is the contradiction ... |
| hoidmvlelem3 45248 | This is the contradiction ... |
| hoidmvlelem4 45249 | The dimensional volume of ... |
| hoidmvlelem5 45250 | The dimensional volume of ... |
| hoidmvle 45251 | The dimensional volume of ... |
| ovnhoilem1 45252 | The Lebesgue outer measure... |
| ovnhoilem2 45253 | The Lebesgue outer measure... |
| ovnhoi 45254 | The Lebesgue outer measure... |
| dmovn 45255 | The domain of the Lebesgue... |
| hoicoto2 45256 | The half-open interval exp... |
| dmvon 45257 | Lebesgue measurable n-dime... |
| hoi2toco 45258 | The half-open interval exp... |
| hoidifhspval 45259 | ` D ` is a function that r... |
| hspval 45260 | The value of the half-spac... |
| ovnlecvr2 45261 | Given a subset of multidim... |
| ovncvr2 45262 | ` B ` and ` T ` are the le... |
| dmovnsal 45263 | The domain of the Lebesgue... |
| unidmovn 45264 | Base set of the n-dimensio... |
| rrnmbl 45265 | The set of n-dimensional R... |
| hoidifhspval2 45266 | ` D ` is a function that r... |
| hspdifhsp 45267 | A n-dimensional half-open ... |
| unidmvon 45268 | Base set of the n-dimensio... |
| hoidifhspf 45269 | ` D ` is a function that r... |
| hoidifhspval3 45270 | ` D ` is a function that r... |
| hoidifhspdmvle 45271 | The dimensional volume of ... |
| voncmpl 45272 | The Lebesgue measure is co... |
| hoiqssbllem1 45273 | The center of the n-dimens... |
| hoiqssbllem2 45274 | The center of the n-dimens... |
| hoiqssbllem3 45275 | A n-dimensional ball conta... |
| hoiqssbl 45276 | A n-dimensional ball conta... |
| hspmbllem1 45277 | Any half-space of the n-di... |
| hspmbllem2 45278 | Any half-space of the n-di... |
| hspmbllem3 45279 | Any half-space of the n-di... |
| hspmbl 45280 | Any half-space of the n-di... |
| hoimbllem 45281 | Any n-dimensional half-ope... |
| hoimbl 45282 | Any n-dimensional half-ope... |
| opnvonmbllem1 45283 | The half-open interval exp... |
| opnvonmbllem2 45284 | An open subset of the n-di... |
| opnvonmbl 45285 | An open subset of the n-di... |
| opnssborel 45286 | Open sets of a generalized... |
| borelmbl 45287 | All Borel subsets of the n... |
| volicorege0 45288 | The Lebesgue measure of a ... |
| isvonmbl 45289 | The predicate " ` A ` is m... |
| mblvon 45290 | The n-dimensional Lebesgue... |
| vonmblss 45291 | n-dimensional Lebesgue mea... |
| volico2 45292 | The measure of left-closed... |
| vonmblss2 45293 | n-dimensional Lebesgue mea... |
| ovolval2lem 45294 | The value of the Lebesgue ... |
| ovolval2 45295 | The value of the Lebesgue ... |
| ovnsubadd2lem 45296 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 45297 | ` ( voln* `` X ) ` is suba... |
| ovolval3 45298 | The value of the Lebesgue ... |
| ovnsplit 45299 | The n-dimensional Lebesgue... |
| ovolval4lem1 45300 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 45301 | The value of the Lebesgue ... |
| ovolval4 45302 | The value of the Lebesgue ... |
| ovolval5lem1 45303 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 45304 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 45305 | The value of the Lebesgue ... |
| ovolval5 45306 | The value of the Lebesgue ... |
| ovnovollem1 45307 | if ` F ` is a cover of ` B... |
| ovnovollem2 45308 | if ` I ` is a cover of ` (... |
| ovnovollem3 45309 | The 1-dimensional Lebesgue... |
| ovnovol 45310 | The 1-dimensional Lebesgue... |
| vonvolmbllem 45311 | If a subset ` B ` of real ... |
| vonvolmbl 45312 | A subset of Real numbers i... |
| vonvol 45313 | The 1-dimensional Lebesgue... |
| vonvolmbl2 45314 | A subset ` X ` of the spac... |
| vonvol2 45315 | The 1-dimensional Lebesgue... |
| hoimbl2 45316 | Any n-dimensional half-ope... |
| voncl 45317 | The Lebesgue measure of a ... |
| vonhoi 45318 | The Lebesgue outer measure... |
| vonxrcl 45319 | The Lebesgue measure of a ... |
| ioosshoi 45320 | A n-dimensional open inter... |
| vonn0hoi 45321 | The Lebesgue outer measure... |
| von0val 45322 | The Lebesgue measure (for ... |
| vonhoire 45323 | The Lebesgue measure of a ... |
| iinhoiicclem 45324 | A n-dimensional closed int... |
| iinhoiicc 45325 | A n-dimensional closed int... |
| iunhoiioolem 45326 | A n-dimensional open inter... |
| iunhoiioo 45327 | A n-dimensional open inter... |
| ioovonmbl 45328 | Any n-dimensional open int... |
| iccvonmbllem 45329 | Any n-dimensional closed i... |
| iccvonmbl 45330 | Any n-dimensional closed i... |
| vonioolem1 45331 | The sequence of the measur... |
| vonioolem2 45332 | The n-dimensional Lebesgue... |
| vonioo 45333 | The n-dimensional Lebesgue... |
| vonicclem1 45334 | The sequence of the measur... |
| vonicclem2 45335 | The n-dimensional Lebesgue... |
| vonicc 45336 | The n-dimensional Lebesgue... |
| snvonmbl 45337 | A n-dimensional singleton ... |
| vonn0ioo 45338 | The n-dimensional Lebesgue... |
| vonn0icc 45339 | The n-dimensional Lebesgue... |
| ctvonmbl 45340 | Any n-dimensional countabl... |
| vonn0ioo2 45341 | The n-dimensional Lebesgue... |
| vonsn 45342 | The n-dimensional Lebesgue... |
| vonn0icc2 45343 | The n-dimensional Lebesgue... |
| vonct 45344 | The n-dimensional Lebesgue... |
| vitali2 45345 | There are non-measurable s... |
| pimltmnf2f 45348 | Given a real-valued functi... |
| pimltmnf2 45349 | Given a real-valued functi... |
| preimagelt 45350 | The preimage of a right-op... |
| preimalegt 45351 | The preimage of a left-ope... |
| pimconstlt0 45352 | Given a constant function,... |
| pimconstlt1 45353 | Given a constant function,... |
| pimltpnff 45354 | Given a real-valued functi... |
| pimltpnf 45355 | Given a real-valued functi... |
| pimgtpnf2f 45356 | Given a real-valued functi... |
| pimgtpnf2 45357 | Given a real-valued functi... |
| salpreimagelt 45358 | If all the preimages of le... |
| pimrecltpos 45359 | The preimage of an unbound... |
| salpreimalegt 45360 | If all the preimages of ri... |
| pimiooltgt 45361 | The preimage of an open in... |
| preimaicomnf 45362 | Preimage of an open interv... |
| pimltpnf2f 45363 | Given a real-valued functi... |
| pimltpnf2 45364 | Given a real-valued functi... |
| pimgtmnf2 45365 | Given a real-valued functi... |
| pimdecfgtioc 45366 | Given a nonincreasing func... |
| pimincfltioc 45367 | Given a nondecreasing func... |
| pimdecfgtioo 45368 | Given a nondecreasing func... |
| pimincfltioo 45369 | Given a nondecreasing func... |
| preimaioomnf 45370 | Preimage of an open interv... |
| preimageiingt 45371 | A preimage of a left-close... |
| preimaleiinlt 45372 | A preimage of a left-open,... |
| pimgtmnff 45373 | Given a real-valued functi... |
| pimgtmnf 45374 | Given a real-valued functi... |
| pimrecltneg 45375 | The preimage of an unbound... |
| salpreimagtge 45376 | If all the preimages of le... |
| salpreimaltle 45377 | If all the preimages of ri... |
| issmflem 45378 | The predicate " ` F ` is a... |
| issmf 45379 | The predicate " ` F ` is a... |
| salpreimalelt 45380 | If all the preimages of ri... |
| salpreimagtlt 45381 | If all the preimages of le... |
| smfpreimalt 45382 | Given a function measurabl... |
| smff 45383 | A function measurable w.r.... |
| smfdmss 45384 | The domain of a function m... |
| issmff 45385 | The predicate " ` F ` is a... |
| issmfd 45386 | A sufficient condition for... |
| smfpreimaltf 45387 | Given a function measurabl... |
| issmfdf 45388 | A sufficient condition for... |
| sssmf 45389 | The restriction of a sigma... |
| mbfresmf 45390 | A real-valued measurable f... |
| cnfsmf 45391 | A continuous function is m... |
| incsmflem 45392 | A nondecreasing function i... |
| incsmf 45393 | A real-valued, nondecreasi... |
| smfsssmf 45394 | If a function is measurabl... |
| issmflelem 45395 | The predicate " ` F ` is a... |
| issmfle 45396 | The predicate " ` F ` is a... |
| smfpimltmpt 45397 | Given a function measurabl... |
| smfpimltxr 45398 | Given a function measurabl... |
| issmfdmpt 45399 | A sufficient condition for... |
| smfconst 45400 | Given a sigma-algebra over... |
| sssmfmpt 45401 | The restriction of a sigma... |
| cnfrrnsmf 45402 | A function, continuous fro... |
| smfid 45403 | The identity function is B... |
| bormflebmf 45404 | A Borel measurable functio... |
| smfpreimale 45405 | Given a function measurabl... |
| issmfgtlem 45406 | The predicate " ` F ` is a... |
| issmfgt 45407 | The predicate " ` F ` is a... |
| issmfled 45408 | A sufficient condition for... |
| smfpimltxrmptf 45409 | Given a function measurabl... |
| smfpimltxrmpt 45410 | Given a function measurabl... |
| smfmbfcex 45411 | A constant function, with ... |
| issmfgtd 45412 | A sufficient condition for... |
| smfpreimagt 45413 | Given a function measurabl... |
| smfaddlem1 45414 | Given the sum of two funct... |
| smfaddlem2 45415 | The sum of two sigma-measu... |
| smfadd 45416 | The sum of two sigma-measu... |
| decsmflem 45417 | A nonincreasing function i... |
| decsmf 45418 | A real-valued, nonincreasi... |
| smfpreimagtf 45419 | Given a function measurabl... |
| issmfgelem 45420 | The predicate " ` F ` is a... |
| issmfge 45421 | The predicate " ` F ` is a... |
| smflimlem1 45422 | Lemma for the proof that t... |
| smflimlem2 45423 | Lemma for the proof that t... |
| smflimlem3 45424 | The limit of sigma-measura... |
| smflimlem4 45425 | Lemma for the proof that t... |
| smflimlem5 45426 | Lemma for the proof that t... |
| smflimlem6 45427 | Lemma for the proof that t... |
| smflim 45428 | The limit of sigma-measura... |
| nsssmfmbflem 45429 | The sigma-measurable funct... |
| nsssmfmbf 45430 | The sigma-measurable funct... |
| smfpimgtxr 45431 | Given a function measurabl... |
| smfpimgtmpt 45432 | Given a function measurabl... |
| smfpreimage 45433 | Given a function measurabl... |
| mbfpsssmf 45434 | Real-valued measurable fun... |
| smfpimgtxrmptf 45435 | Given a function measurabl... |
| smfpimgtxrmpt 45436 | Given a function measurabl... |
| smfpimioompt 45437 | Given a function measurabl... |
| smfpimioo 45438 | Given a function measurabl... |
| smfresal 45439 | Given a sigma-measurable f... |
| smfrec 45440 | The reciprocal of a sigma-... |
| smfres 45441 | The restriction of sigma-m... |
| smfmullem1 45442 | The multiplication of two ... |
| smfmullem2 45443 | The multiplication of two ... |
| smfmullem3 45444 | The multiplication of two ... |
| smfmullem4 45445 | The multiplication of two ... |
| smfmul 45446 | The multiplication of two ... |
| smfmulc1 45447 | A sigma-measurable functio... |
| smfdiv 45448 | The fraction of two sigma-... |
| smfpimbor1lem1 45449 | Every open set belongs to ... |
| smfpimbor1lem2 45450 | Given a sigma-measurable f... |
| smfpimbor1 45451 | Given a sigma-measurable f... |
| smf2id 45452 | Twice the identity functio... |
| smfco 45453 | The composition of a Borel... |
| smfneg 45454 | The negative of a sigma-me... |
| smffmptf 45455 | A function measurable w.r.... |
| smffmpt 45456 | A function measurable w.r.... |
| smflim2 45457 | The limit of a sequence of... |
| smfpimcclem 45458 | Lemma for ~ smfpimcc given... |
| smfpimcc 45459 | Given a countable set of s... |
| issmfle2d 45460 | A sufficient condition for... |
| smflimmpt 45461 | The limit of a sequence of... |
| smfsuplem1 45462 | The supremum of a countabl... |
| smfsuplem2 45463 | The supremum of a countabl... |
| smfsuplem3 45464 | The supremum of a countabl... |
| smfsup 45465 | The supremum of a countabl... |
| smfsupmpt 45466 | The supremum of a countabl... |
| smfsupxr 45467 | The supremum of a countabl... |
| smfinflem 45468 | The infimum of a countable... |
| smfinf 45469 | The infimum of a countable... |
| smfinfmpt 45470 | The infimum of a countable... |
| smflimsuplem1 45471 | If ` H ` converges, the ` ... |
| smflimsuplem2 45472 | The superior limit of a se... |
| smflimsuplem3 45473 | The limit of the ` ( H `` ... |
| smflimsuplem4 45474 | If ` H ` converges, the ` ... |
| smflimsuplem5 45475 | ` H ` converges to the sup... |
| smflimsuplem6 45476 | The superior limit of a se... |
| smflimsuplem7 45477 | The superior limit of a se... |
| smflimsuplem8 45478 | The superior limit of a se... |
| smflimsup 45479 | The superior limit of a se... |
| smflimsupmpt 45480 | The superior limit of a se... |
| smfliminflem 45481 | The inferior limit of a co... |
| smfliminf 45482 | The inferior limit of a co... |
| smfliminfmpt 45483 | The inferior limit of a co... |
| adddmmbl 45484 | If two functions have doma... |
| adddmmbl2 45485 | If two functions have doma... |
| muldmmbl 45486 | If two functions have doma... |
| muldmmbl2 45487 | If two functions have doma... |
| smfdmmblpimne 45488 | If a measurable function w... |
| smfdivdmmbl 45489 | If a functions and a sigma... |
| smfpimne 45490 | Given a function measurabl... |
| smfpimne2 45491 | Given a function measurabl... |
| smfdivdmmbl2 45492 | If a functions and a sigma... |
| fsupdm 45493 | The domain of the sup func... |
| fsupdm2 45494 | The domain of the sup func... |
| smfsupdmmbllem 45495 | If a countable set of sigm... |
| smfsupdmmbl 45496 | If a countable set of sigm... |
| finfdm 45497 | The domain of the inf func... |
| finfdm2 45498 | The domain of the inf func... |
| smfinfdmmbllem 45499 | If a countable set of sigm... |
| smfinfdmmbl 45500 | If a countable set of sigm... |
| sigarval 45501 | Define the signed area by ... |
| sigarim 45502 | Signed area takes value in... |
| sigarac 45503 | Signed area is anticommuta... |
| sigaraf 45504 | Signed area is additive by... |
| sigarmf 45505 | Signed area is additive (w... |
| sigaras 45506 | Signed area is additive by... |
| sigarms 45507 | Signed area is additive (w... |
| sigarls 45508 | Signed area is linear by t... |
| sigarid 45509 | Signed area of a flat para... |
| sigarexp 45510 | Expand the signed area for... |
| sigarperm 45511 | Signed area ` ( A - C ) G ... |
| sigardiv 45512 | If signed area between vec... |
| sigarimcd 45513 | Signed area takes value in... |
| sigariz 45514 | If signed area is zero, th... |
| sigarcol 45515 | Given three points ` A ` ,... |
| sharhght 45516 | Let ` A B C ` be a triangl... |
| sigaradd 45517 | Subtracting (double) area ... |
| cevathlem1 45518 | Ceva's theorem first lemma... |
| cevathlem2 45519 | Ceva's theorem second lemm... |
| cevath 45520 | Ceva's theorem. Let ` A B... |
| simpcntrab 45521 | The center of a simple gro... |
| et-ltneverrefl 45522 | Less-than class is never r... |
| et-equeucl 45523 | Alternative proof that equ... |
| et-sqrtnegnre 45524 | The square root of a negat... |
| natlocalincr 45525 | Global monotonicity on hal... |
| natglobalincr 45526 | Local monotonicity on half... |
| upwordnul 45529 | Empty set is an increasing... |
| upwordisword 45530 | Any increasing sequence is... |
| singoutnword 45531 | Singleton with character o... |
| singoutnupword 45532 | Singleton with character o... |
| upwordsing 45533 | Singleton is an increasing... |
| upwordsseti 45534 | Strictly increasing sequen... |
| tworepnotupword 45535 | Concatenation of identical... |
| upwrdfi 45536 | There is a finite number o... |
| hirstL-ax3 45537 | The third axiom of a syste... |
| ax3h 45538 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 45539 | A closed form showing (a i... |
| aibandbiaiaiffb 45540 | A closed form showing (a i... |
| notatnand 45541 | Do not use. Use intnanr i... |
| aistia 45542 | Given a is equivalent to `... |
| aisfina 45543 | Given a is equivalent to `... |
| bothtbothsame 45544 | Given both a, b are equiva... |
| bothfbothsame 45545 | Given both a, b are equiva... |
| aiffbbtat 45546 | Given a is equivalent to b... |
| aisbbisfaisf 45547 | Given a is equivalent to b... |
| axorbtnotaiffb 45548 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 45549 | Given a is equivalent to (... |
| axorbciffatcxorb 45550 | Given a is equivalent to (... |
| aibnbna 45551 | Given a implies b, (not b)... |
| aibnbaif 45552 | Given a implies b, not b, ... |
| aiffbtbat 45553 | Given a is equivalent to b... |
| astbstanbst 45554 | Given a is equivalent to T... |
| aistbistaandb 45555 | Given a is equivalent to T... |
| aisbnaxb 45556 | Given a is equivalent to b... |
| atbiffatnnb 45557 | If a implies b, then a imp... |
| bisaiaisb 45558 | Application of bicom1 with... |
| atbiffatnnbalt 45559 | If a implies b, then a imp... |
| abnotbtaxb 45560 | Assuming a, not b, there e... |
| abnotataxb 45561 | Assuming not a, b, there e... |
| conimpf 45562 | Assuming a, not b, and a i... |
| conimpfalt 45563 | Assuming a, not b, and a i... |
| aistbisfiaxb 45564 | Given a is equivalent to T... |
| aisfbistiaxb 45565 | Given a is equivalent to F... |
| aifftbifffaibif 45566 | Given a is equivalent to T... |
| aifftbifffaibifff 45567 | Given a is equivalent to T... |
| atnaiana 45568 | Given a, it is not the cas... |
| ainaiaandna 45569 | Given a, a implies it is n... |
| abcdta 45570 | Given (((a and b) and c) a... |
| abcdtb 45571 | Given (((a and b) and c) a... |
| abcdtc 45572 | Given (((a and b) and c) a... |
| abcdtd 45573 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 45574 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 45575 | Operands in a biconditiona... |
| nabctnabc 45576 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 45577 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 45578 | From one negated implicati... |
| twonotinotbothi 45579 | From these two negated imp... |
| clifte 45580 | show d is the same as an i... |
| cliftet 45581 | show d is the same as an i... |
| clifteta 45582 | show d is the same as an i... |
| cliftetb 45583 | show d is the same as an i... |
| confun 45584 | Given the hypotheses there... |
| confun2 45585 | Confun simplified to two p... |
| confun3 45586 | Confun's more complex form... |
| confun4 45587 | An attempt at derivative. ... |
| confun5 45588 | An attempt at derivative. ... |
| plcofph 45589 | Given, a,b and a "definiti... |
| pldofph 45590 | Given, a,b c, d, "definiti... |
| plvcofph 45591 | Given, a,b,d, and "definit... |
| plvcofphax 45592 | Given, a,b,d, and "definit... |
| plvofpos 45593 | rh is derivable because ON... |
| mdandyv0 45594 | Given the equivalences set... |
| mdandyv1 45595 | Given the equivalences set... |
| mdandyv2 45596 | Given the equivalences set... |
| mdandyv3 45597 | Given the equivalences set... |
| mdandyv4 45598 | Given the equivalences set... |
| mdandyv5 45599 | Given the equivalences set... |
| mdandyv6 45600 | Given the equivalences set... |
| mdandyv7 45601 | Given the equivalences set... |
| mdandyv8 45602 | Given the equivalences set... |
| mdandyv9 45603 | Given the equivalences set... |
| mdandyv10 45604 | Given the equivalences set... |
| mdandyv11 45605 | Given the equivalences set... |
| mdandyv12 45606 | Given the equivalences set... |
| mdandyv13 45607 | Given the equivalences set... |
| mdandyv14 45608 | Given the equivalences set... |
| mdandyv15 45609 | Given the equivalences set... |
| mdandyvr0 45610 | Given the equivalences set... |
| mdandyvr1 45611 | Given the equivalences set... |
| mdandyvr2 45612 | Given the equivalences set... |
| mdandyvr3 45613 | Given the equivalences set... |
| mdandyvr4 45614 | Given the equivalences set... |
| mdandyvr5 45615 | Given the equivalences set... |
| mdandyvr6 45616 | Given the equivalences set... |
| mdandyvr7 45617 | Given the equivalences set... |
| mdandyvr8 45618 | Given the equivalences set... |
| mdandyvr9 45619 | Given the equivalences set... |
| mdandyvr10 45620 | Given the equivalences set... |
| mdandyvr11 45621 | Given the equivalences set... |
| mdandyvr12 45622 | Given the equivalences set... |
| mdandyvr13 45623 | Given the equivalences set... |
| mdandyvr14 45624 | Given the equivalences set... |
| mdandyvr15 45625 | Given the equivalences set... |
| mdandyvrx0 45626 | Given the exclusivities se... |
| mdandyvrx1 45627 | Given the exclusivities se... |
| mdandyvrx2 45628 | Given the exclusivities se... |
| mdandyvrx3 45629 | Given the exclusivities se... |
| mdandyvrx4 45630 | Given the exclusivities se... |
| mdandyvrx5 45631 | Given the exclusivities se... |
| mdandyvrx6 45632 | Given the exclusivities se... |
| mdandyvrx7 45633 | Given the exclusivities se... |
| mdandyvrx8 45634 | Given the exclusivities se... |
| mdandyvrx9 45635 | Given the exclusivities se... |
| mdandyvrx10 45636 | Given the exclusivities se... |
| mdandyvrx11 45637 | Given the exclusivities se... |
| mdandyvrx12 45638 | Given the exclusivities se... |
| mdandyvrx13 45639 | Given the exclusivities se... |
| mdandyvrx14 45640 | Given the exclusivities se... |
| mdandyvrx15 45641 | Given the exclusivities se... |
| H15NH16TH15IH16 45642 | Given 15 hypotheses and a ... |
| dandysum2p2e4 45643 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 45644 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 45645 | Replacement of a nested an... |
| adh-minim 45646 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 45647 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 45648 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 45649 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 45650 | Fourth lemma for the deriv... |
| adh-minim-ax1 45651 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 45652 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 45653 | Sixth lemma for the deriva... |
| adh-minim-ax2c 45654 | Derivation of a commuted f... |
| adh-minim-ax2 45655 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 45656 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 45657 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 45658 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 45659 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 45660 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 45661 | Third lemma for the deriva... |
| adh-minimp-sylsimp 45662 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 45663 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 45664 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 45665 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 45666 | Fourth lemma for the deriv... |
| adh-minimp-ax2 45667 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 45668 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 45669 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 45670 | If a class with one elemen... |
| eusnsn 45671 | There is a unique element ... |
| absnsb 45672 | If the class abstraction `... |
| euabsneu 45673 | Another way to express exi... |
| elprneb 45674 | An element of a proper uno... |
| oppr 45675 | Equality for ordered pairs... |
| opprb 45676 | Equality for unordered pai... |
| or2expropbilem1 45677 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 45678 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 45679 | If two classes are strictl... |
| eubrv 45680 | If there is a unique set w... |
| eubrdm 45681 | If there is a unique set w... |
| eldmressn 45682 | Element of the domain of a... |
| iota0def 45683 | Example for a defined iota... |
| iota0ndef 45684 | Example for an undefined i... |
| fveqvfvv 45685 | If a function's value at a... |
| fnresfnco 45686 | Composition of two functio... |
| funcoressn 45687 | A composition restricted t... |
| funressnfv 45688 | A restriction to a singlet... |
| funressndmfvrn 45689 | The value of a function ` ... |
| funressnvmo 45690 | A function restricted to a... |
| funressnmo 45691 | A function restricted to a... |
| funressneu 45692 | There is exactly one value... |
| fresfo 45693 | Conditions for a restricti... |
| fsetsniunop 45694 | The class of all functions... |
| fsetabsnop 45695 | The class of all functions... |
| fsetsnf 45696 | The mapping of an element ... |
| fsetsnf1 45697 | The mapping of an element ... |
| fsetsnfo 45698 | The mapping of an element ... |
| fsetsnf1o 45699 | The mapping of an element ... |
| fsetsnprcnex 45700 | The class of all functions... |
| cfsetssfset 45701 | The class of constant func... |
| cfsetsnfsetfv 45702 | The function value of the ... |
| cfsetsnfsetf 45703 | The mapping of the class o... |
| cfsetsnfsetf1 45704 | The mapping of the class o... |
| cfsetsnfsetfo 45705 | The mapping of the class o... |
| cfsetsnfsetf1o 45706 | The mapping of the class o... |
| fsetprcnexALT 45707 | First version of proof for... |
| fcoreslem1 45708 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 45709 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 45710 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 45711 | Lemma 4 for ~ fcores . (C... |
| fcores 45712 | Every composite function `... |
| fcoresf1lem 45713 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 45714 | If a composition is inject... |
| fcoresf1b 45715 | A composition is injective... |
| fcoresfo 45716 | If a composition is surjec... |
| fcoresfob 45717 | A composition is surjectiv... |
| fcoresf1ob 45718 | A composition is bijective... |
| f1cof1blem 45719 | Lemma for ~ f1cof1b and ~ ... |
| f1cof1b 45720 | If the range of ` F ` equa... |
| funfocofob 45721 | If the domain of a functio... |
| fnfocofob 45722 | If the domain of a functio... |
| focofob 45723 | If the domain of a functio... |
| f1ocof1ob 45724 | If the range of ` F ` equa... |
| f1ocof1ob2 45725 | If the range of ` F ` equa... |
| aiotajust 45727 | Soundness justification th... |
| dfaiota2 45729 | Alternate definition of th... |
| reuabaiotaiota 45730 | The iota and the alternate... |
| reuaiotaiota 45731 | The iota and the alternate... |
| aiotaexb 45732 | The alternate iota over a ... |
| aiotavb 45733 | The alternate iota over a ... |
| aiotaint 45734 | This is to ~ df-aiota what... |
| dfaiota3 45735 | Alternate definition of ` ... |
| iotan0aiotaex 45736 | If the iota over a wff ` p... |
| aiotaexaiotaiota 45737 | The alternate iota over a ... |
| aiotaval 45738 | Theorem 8.19 in [Quine] p.... |
| aiota0def 45739 | Example for a defined alte... |
| aiota0ndef 45740 | Example for an undefined a... |
| r19.32 45741 | Theorem 19.32 of [Margaris... |
| rexsb 45742 | An equivalent expression f... |
| rexrsb 45743 | An equivalent expression f... |
| 2rexsb 45744 | An equivalent expression f... |
| 2rexrsb 45745 | An equivalent expression f... |
| cbvral2 45746 | Change bound variables of ... |
| cbvrex2 45747 | Change bound variables of ... |
| ralndv1 45748 | Example for a theorem abou... |
| ralndv2 45749 | Second example for a theor... |
| reuf1odnf 45750 | There is exactly one eleme... |
| reuf1od 45751 | There is exactly one eleme... |
| euoreqb 45752 | There is a set which is eq... |
| 2reu3 45753 | Double restricted existent... |
| 2reu7 45754 | Two equivalent expressions... |
| 2reu8 45755 | Two equivalent expressions... |
| 2reu8i 45756 | Implication of a double re... |
| 2reuimp0 45757 | Implication of a double re... |
| 2reuimp 45758 | Implication of a double re... |
| ralbinrald 45765 | Elemination of a restricte... |
| nvelim 45766 | If a class is the universa... |
| alneu 45767 | If a statement holds for a... |
| eu2ndop1stv 45768 | If there is a unique secon... |
| dfateq12d 45769 | Equality deduction for "de... |
| nfdfat 45770 | Bound-variable hypothesis ... |
| dfdfat2 45771 | Alternate definition of th... |
| fundmdfat 45772 | A function is defined at a... |
| dfatprc 45773 | A function is not defined ... |
| dfatelrn 45774 | The value of a function ` ... |
| dfafv2 45775 | Alternative definition of ... |
| afveq12d 45776 | Equality deduction for fun... |
| afveq1 45777 | Equality theorem for funct... |
| afveq2 45778 | Equality theorem for funct... |
| nfafv 45779 | Bound-variable hypothesis ... |
| csbafv12g 45780 | Move class substitution in... |
| afvfundmfveq 45781 | If a class is a function r... |
| afvnfundmuv 45782 | If a set is not in the dom... |
| ndmafv 45783 | The value of a class outsi... |
| afvvdm 45784 | If the function value of a... |
| nfunsnafv 45785 | If the restriction of a cl... |
| afvvfunressn 45786 | If the function value of a... |
| afvprc 45787 | A function's value at a pr... |
| afvvv 45788 | If a function's value at a... |
| afvpcfv0 45789 | If the value of the altern... |
| afvnufveq 45790 | The value of the alternati... |
| afvvfveq 45791 | The value of the alternati... |
| afv0fv0 45792 | If the value of the altern... |
| afvfvn0fveq 45793 | If the function's value at... |
| afv0nbfvbi 45794 | The function's value at an... |
| afvfv0bi 45795 | The function's value at an... |
| afveu 45796 | The value of a function at... |
| fnbrafvb 45797 | Equivalence of function va... |
| fnopafvb 45798 | Equivalence of function va... |
| funbrafvb 45799 | Equivalence of function va... |
| funopafvb 45800 | Equivalence of function va... |
| funbrafv 45801 | The second argument of a b... |
| funbrafv2b 45802 | Function value in terms of... |
| dfafn5a 45803 | Representation of a functi... |
| dfafn5b 45804 | Representation of a functi... |
| fnrnafv 45805 | The range of a function ex... |
| afvelrnb 45806 | A member of a function's r... |
| afvelrnb0 45807 | A member of a function's r... |
| dfaimafn 45808 | Alternate definition of th... |
| dfaimafn2 45809 | Alternate definition of th... |
| afvelima 45810 | Function value in an image... |
| afvelrn 45811 | A function's value belongs... |
| fnafvelrn 45812 | A function's value belongs... |
| fafvelcdm 45813 | A function's value belongs... |
| ffnafv 45814 | A function maps to a class... |
| afvres 45815 | The value of a restricted ... |
| tz6.12-afv 45816 | Function value. Theorem 6... |
| tz6.12-1-afv 45817 | Function value (Theorem 6.... |
| dmfcoafv 45818 | Domains of a function comp... |
| afvco2 45819 | Value of a function compos... |
| rlimdmafv 45820 | Two ways to express that a... |
| aoveq123d 45821 | Equality deduction for ope... |
| nfaov 45822 | Bound-variable hypothesis ... |
| csbaovg 45823 | Move class substitution in... |
| aovfundmoveq 45824 | If a class is a function r... |
| aovnfundmuv 45825 | If an ordered pair is not ... |
| ndmaov 45826 | The value of an operation ... |
| ndmaovg 45827 | The value of an operation ... |
| aovvdm 45828 | If the operation value of ... |
| nfunsnaov 45829 | If the restriction of a cl... |
| aovvfunressn 45830 | If the operation value of ... |
| aovprc 45831 | The value of an operation ... |
| aovrcl 45832 | Reverse closure for an ope... |
| aovpcov0 45833 | If the alternative value o... |
| aovnuoveq 45834 | The alternative value of t... |
| aovvoveq 45835 | The alternative value of t... |
| aov0ov0 45836 | If the alternative value o... |
| aovovn0oveq 45837 | If the operation's value a... |
| aov0nbovbi 45838 | The operation's value on a... |
| aovov0bi 45839 | The operation's value on a... |
| rspceaov 45840 | A frequently used special ... |
| fnotaovb 45841 | Equivalence of operation v... |
| ffnaov 45842 | An operation maps to a cla... |
| faovcl 45843 | Closure law for an operati... |
| aovmpt4g 45844 | Value of a function given ... |
| aoprssdm 45845 | Domain of closure of an op... |
| ndmaovcl 45846 | The "closure" of an operat... |
| ndmaovrcl 45847 | Reverse closure law, in co... |
| ndmaovcom 45848 | Any operation is commutati... |
| ndmaovass 45849 | Any operation is associati... |
| ndmaovdistr 45850 | Any operation is distribut... |
| dfatafv2iota 45853 | If a function is defined a... |
| ndfatafv2 45854 | The alternate function val... |
| ndfatafv2undef 45855 | The alternate function val... |
| dfatafv2ex 45856 | The alternate function val... |
| afv2ex 45857 | The alternate function val... |
| afv2eq12d 45858 | Equality deduction for fun... |
| afv2eq1 45859 | Equality theorem for funct... |
| afv2eq2 45860 | Equality theorem for funct... |
| nfafv2 45861 | Bound-variable hypothesis ... |
| csbafv212g 45862 | Move class substitution in... |
| fexafv2ex 45863 | The alternate function val... |
| ndfatafv2nrn 45864 | The alternate function val... |
| ndmafv2nrn 45865 | The value of a class outsi... |
| funressndmafv2rn 45866 | The alternate function val... |
| afv2ndefb 45867 | Two ways to say that an al... |
| nfunsnafv2 45868 | If the restriction of a cl... |
| afv2prc 45869 | A function's value at a pr... |
| dfatafv2rnb 45870 | The alternate function val... |
| afv2orxorb 45871 | If a set is in the range o... |
| dmafv2rnb 45872 | The alternate function val... |
| fundmafv2rnb 45873 | The alternate function val... |
| afv2elrn 45874 | An alternate function valu... |
| afv20defat 45875 | If the alternate function ... |
| fnafv2elrn 45876 | An alternate function valu... |
| fafv2elcdm 45877 | An alternate function valu... |
| fafv2elrnb 45878 | An alternate function valu... |
| fcdmvafv2v 45879 | If the codomain of a funct... |
| tz6.12-2-afv2 45880 | Function value when ` F ` ... |
| afv2eu 45881 | The value of a function at... |
| afv2res 45882 | The value of a restricted ... |
| tz6.12-afv2 45883 | Function value (Theorem 6.... |
| tz6.12-1-afv2 45884 | Function value (Theorem 6.... |
| tz6.12c-afv2 45885 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 45886 | Corollary of Theorem 6.12(... |
| funressnbrafv2 45887 | The second argument of a b... |
| dfatbrafv2b 45888 | Equivalence of function va... |
| dfatopafv2b 45889 | Equivalence of function va... |
| funbrafv2 45890 | The second argument of a b... |
| fnbrafv2b 45891 | Equivalence of function va... |
| fnopafv2b 45892 | Equivalence of function va... |
| funbrafv22b 45893 | Equivalence of function va... |
| funopafv2b 45894 | Equivalence of function va... |
| dfatsnafv2 45895 | Singleton of function valu... |
| dfafv23 45896 | A definition of function v... |
| dfatdmfcoafv2 45897 | Domain of a function compo... |
| dfatcolem 45898 | Lemma for ~ dfatco . (Con... |
| dfatco 45899 | The predicate "defined at"... |
| afv2co2 45900 | Value of a function compos... |
| rlimdmafv2 45901 | Two ways to express that a... |
| dfafv22 45902 | Alternate definition of ` ... |
| afv2ndeffv0 45903 | If the alternate function ... |
| dfatafv2eqfv 45904 | If a function is defined a... |
| afv2rnfveq 45905 | If the alternate function ... |
| afv20fv0 45906 | If the alternate function ... |
| afv2fvn0fveq 45907 | If the function's value at... |
| afv2fv0 45908 | If the function's value at... |
| afv2fv0b 45909 | The function's value at an... |
| afv2fv0xorb 45910 | If a set is in the range o... |
| an4com24 45911 | Rearrangement of 4 conjunc... |
| 3an4ancom24 45912 | Commutative law for a conj... |
| 4an21 45913 | Rearrangement of 4 conjunc... |
| dfnelbr2 45916 | Alternate definition of th... |
| nelbr 45917 | The binary relation of a s... |
| nelbrim 45918 | If a set is related to ano... |
| nelbrnel 45919 | A set is related to anothe... |
| nelbrnelim 45920 | If a set is related to ano... |
| ralralimp 45921 | Selecting one of two alter... |
| otiunsndisjX 45922 | The union of singletons co... |
| fvifeq 45923 | Equality of function value... |
| rnfdmpr 45924 | The range of a one-to-one ... |
| imarnf1pr 45925 | The image of the range of ... |
| funop1 45926 | A function is an ordered p... |
| fun2dmnopgexmpl 45927 | A function with a domain c... |
| opabresex0d 45928 | A collection of ordered pa... |
| opabbrfex0d 45929 | A collection of ordered pa... |
| opabresexd 45930 | A collection of ordered pa... |
| opabbrfexd 45931 | A collection of ordered pa... |
| f1oresf1orab 45932 | Build a bijection by restr... |
| f1oresf1o 45933 | Build a bijection by restr... |
| f1oresf1o2 45934 | Build a bijection by restr... |
| fvmptrab 45935 | Value of a function mappin... |
| fvmptrabdm 45936 | Value of a function mappin... |
| cnambpcma 45937 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 45938 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 45939 | The sum of two complex num... |
| leaddsuble 45940 | Addition and subtraction o... |
| 2leaddle2 45941 | If two real numbers are le... |
| ltnltne 45942 | Variant of trichotomy law ... |
| p1lep2 45943 | A real number increasd by ... |
| ltsubsubaddltsub 45944 | If the result of subtracti... |
| zm1nn 45945 | An integer minus 1 is posi... |
| readdcnnred 45946 | The sum of a real number a... |
| resubcnnred 45947 | The difference of a real n... |
| recnmulnred 45948 | The product of a real numb... |
| cndivrenred 45949 | The quotient of an imagina... |
| sqrtnegnre 45950 | The square root of a negat... |
| nn0resubcl 45951 | Closure law for subtractio... |
| zgeltp1eq 45952 | If an integer is between a... |
| 1t10e1p1e11 45953 | 11 is 1 times 10 to the po... |
| deccarry 45954 | Add 1 to a 2 digit number ... |
| eluzge0nn0 45955 | If an integer is greater t... |
| nltle2tri 45956 | Negated extended trichotom... |
| ssfz12 45957 | Subset relationship for fi... |
| elfz2z 45958 | Membership of an integer i... |
| 2elfz3nn0 45959 | If there are two elements ... |
| fz0addcom 45960 | The addition of two member... |
| 2elfz2melfz 45961 | If the sum of two integers... |
| fz0addge0 45962 | The sum of two integers in... |
| elfzlble 45963 | Membership of an integer i... |
| elfzelfzlble 45964 | Membership of an element o... |
| fzopred 45965 | Join a predecessor to the ... |
| fzopredsuc 45966 | Join a predecessor and a s... |
| 1fzopredsuc 45967 | Join 0 and a successor to ... |
| el1fzopredsuc 45968 | An element of an open inte... |
| subsubelfzo0 45969 | Subtracting a difference f... |
| fzoopth 45970 | A half-open integer range ... |
| 2ffzoeq 45971 | Two functions over a half-... |
| m1mod0mod1 45972 | An integer decreased by 1 ... |
| elmod2 45973 | An integer modulo 2 is eit... |
| smonoord 45974 | Ordering relation for a st... |
| fsummsndifre 45975 | A finite sum with one of i... |
| fsumsplitsndif 45976 | Separate out a term in a f... |
| fsummmodsndifre 45977 | A finite sum of summands m... |
| fsummmodsnunz 45978 | A finite sum of summands m... |
| setsidel 45979 | The injected slot is an el... |
| setsnidel 45980 | The injected slot is an el... |
| setsv 45981 | The value of the structure... |
| preimafvsnel 45982 | The preimage of a function... |
| preimafvn0 45983 | The preimage of a function... |
| uniimafveqt 45984 | The union of the image of ... |
| uniimaprimaeqfv 45985 | The union of the image of ... |
| setpreimafvex 45986 | The class ` P ` of all pre... |
| elsetpreimafvb 45987 | The characterization of an... |
| elsetpreimafv 45988 | An element of the class ` ... |
| elsetpreimafvssdm 45989 | An element of the class ` ... |
| fvelsetpreimafv 45990 | There is an element in a p... |
| preimafvelsetpreimafv 45991 | The preimage of a function... |
| preimafvsspwdm 45992 | The class ` P ` of all pre... |
| 0nelsetpreimafv 45993 | The empty set is not an el... |
| elsetpreimafvbi 45994 | An element of the preimage... |
| elsetpreimafveqfv 45995 | The elements of the preima... |
| eqfvelsetpreimafv 45996 | If an element of the domai... |
| elsetpreimafvrab 45997 | An element of the preimage... |
| imaelsetpreimafv 45998 | The image of an element of... |
| uniimaelsetpreimafv 45999 | The union of the image of ... |
| elsetpreimafveq 46000 | If two preimages of functi... |
| fundcmpsurinjlem1 46001 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 46002 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 46003 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 46004 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 46005 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 46006 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 46007 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 46008 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 46009 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 46010 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 46011 | Every function ` F : A -->... |
| fundcmpsurinj 46012 | Every function ` F : A -->... |
| fundcmpsurbijinj 46013 | Every function ` F : A -->... |
| fundcmpsurinjimaid 46014 | Every function ` F : A -->... |
| fundcmpsurinjALT 46015 | Alternate proof of ~ fundc... |
| iccpval 46018 | Partition consisting of a ... |
| iccpart 46019 | A special partition. Corr... |
| iccpartimp 46020 | Implications for a class b... |
| iccpartres 46021 | The restriction of a parti... |
| iccpartxr 46022 | If there is a partition, t... |
| iccpartgtprec 46023 | If there is a partition, t... |
| iccpartipre 46024 | If there is a partition, t... |
| iccpartiltu 46025 | If there is a partition, t... |
| iccpartigtl 46026 | If there is a partition, t... |
| iccpartlt 46027 | If there is a partition, t... |
| iccpartltu 46028 | If there is a partition, t... |
| iccpartgtl 46029 | If there is a partition, t... |
| iccpartgt 46030 | If there is a partition, t... |
| iccpartleu 46031 | If there is a partition, t... |
| iccpartgel 46032 | If there is a partition, t... |
| iccpartrn 46033 | If there is a partition, t... |
| iccpartf 46034 | The range of the partition... |
| iccpartel 46035 | If there is a partition, t... |
| iccelpart 46036 | An element of any partitio... |
| iccpartiun 46037 | A half-open interval of ex... |
| icceuelpartlem 46038 | Lemma for ~ icceuelpart . ... |
| icceuelpart 46039 | An element of a partitione... |
| iccpartdisj 46040 | The segments of a partitio... |
| iccpartnel 46041 | A point of a partition is ... |
| fargshiftfv 46042 | If a class is a function, ... |
| fargshiftf 46043 | If a class is a function, ... |
| fargshiftf1 46044 | If a function is 1-1, then... |
| fargshiftfo 46045 | If a function is onto, the... |
| fargshiftfva 46046 | The values of a shifted fu... |
| lswn0 46047 | The last symbol of a not e... |
| nfich1 46050 | The first interchangeable ... |
| nfich2 46051 | The second interchangeable... |
| ichv 46052 | Setvar variables are inter... |
| ichf 46053 | Setvar variables are inter... |
| ichid 46054 | A setvar variable is alway... |
| icht 46055 | A theorem is interchangeab... |
| ichbidv 46056 | Formula building rule for ... |
| ichcircshi 46057 | The setvar variables are i... |
| ichan 46058 | If two setvar variables ar... |
| ichn 46059 | Negation does not affect i... |
| ichim 46060 | Formula building rule for ... |
| dfich2 46061 | Alternate definition of th... |
| ichcom 46062 | The interchangeability of ... |
| ichbi12i 46063 | Equivalence for interchang... |
| icheqid 46064 | In an equality for the sam... |
| icheq 46065 | In an equality of setvar v... |
| ichnfimlem 46066 | Lemma for ~ ichnfim : A s... |
| ichnfim 46067 | If in an interchangeabilit... |
| ichnfb 46068 | If ` x ` and ` y ` are int... |
| ichal 46069 | Move a universal quantifie... |
| ich2al 46070 | Two setvar variables are a... |
| ich2ex 46071 | Two setvar variables are a... |
| ichexmpl1 46072 | Example for interchangeabl... |
| ichexmpl2 46073 | Example for interchangeabl... |
| ich2exprop 46074 | If the setvar variables ar... |
| ichnreuop 46075 | If the setvar variables ar... |
| ichreuopeq 46076 | If the setvar variables ar... |
| sprid 46077 | Two identical representati... |
| elsprel 46078 | An unordered pair is an el... |
| spr0nelg 46079 | The empty set is not an el... |
| sprval 46082 | The set of all unordered p... |
| sprvalpw 46083 | The set of all unordered p... |
| sprssspr 46084 | The set of all unordered p... |
| spr0el 46085 | The empty set is not an un... |
| sprvalpwn0 46086 | The set of all unordered p... |
| sprel 46087 | An element of the set of a... |
| prssspr 46088 | An element of a subset of ... |
| prelspr 46089 | An unordered pair of eleme... |
| prsprel 46090 | The elements of a pair fro... |
| prsssprel 46091 | The elements of a pair fro... |
| sprvalpwle2 46092 | The set of all unordered p... |
| sprsymrelfvlem 46093 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 46094 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 46095 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 46096 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 46097 | The value of the function ... |
| sprsymrelf 46098 | The mapping ` F ` is a fun... |
| sprsymrelf1 46099 | The mapping ` F ` is a one... |
| sprsymrelfo 46100 | The mapping ` F ` is a fun... |
| sprsymrelf1o 46101 | The mapping ` F ` is a bij... |
| sprbisymrel 46102 | There is a bijection betwe... |
| sprsymrelen 46103 | The class ` P ` of subsets... |
| prpair 46104 | Characterization of a prop... |
| prproropf1olem0 46105 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 46106 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 46107 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 46108 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 46109 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 46110 | There is a bijection betwe... |
| prproropen 46111 | The set of proper pairs an... |
| prproropreud 46112 | There is exactly one order... |
| pairreueq 46113 | Two equivalent representat... |
| paireqne 46114 | Two sets are not equal iff... |
| prprval 46117 | The set of all proper unor... |
| prprvalpw 46118 | The set of all proper unor... |
| prprelb 46119 | An element of the set of a... |
| prprelprb 46120 | A set is an element of the... |
| prprspr2 46121 | The set of all proper unor... |
| prprsprreu 46122 | There is a unique proper u... |
| prprreueq 46123 | There is a unique proper u... |
| sbcpr 46124 | The proper substitution of... |
| reupr 46125 | There is a unique unordere... |
| reuprpr 46126 | There is a unique proper u... |
| poprelb 46127 | Equality for unordered pai... |
| 2exopprim 46128 | The existence of an ordere... |
| reuopreuprim 46129 | There is a unique unordere... |
| fmtno 46132 | The ` N ` th Fermat number... |
| fmtnoge3 46133 | Each Fermat number is grea... |
| fmtnonn 46134 | Each Fermat number is a po... |
| fmtnom1nn 46135 | A Fermat number minus one ... |
| fmtnoodd 46136 | Each Fermat number is odd.... |
| fmtnorn 46137 | A Fermat number is a funct... |
| fmtnof1 46138 | The enumeration of the Fer... |
| fmtnoinf 46139 | The set of Fermat numbers ... |
| fmtnorec1 46140 | The first recurrence relat... |
| sqrtpwpw2p 46141 | The floor of the square ro... |
| fmtnosqrt 46142 | The floor of the square ro... |
| fmtno0 46143 | The ` 0 ` th Fermat number... |
| fmtno1 46144 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 46145 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 46146 | The second recurrence rela... |
| fmtnodvds 46147 | Any Fermat number divides ... |
| goldbachthlem1 46148 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 46149 | Lemma 2 for ~ goldbachth .... |
| goldbachth 46150 | Goldbach's theorem: Two d... |
| fmtnorec3 46151 | The third recurrence relat... |
| fmtnorec4 46152 | The fourth recurrence rela... |
| fmtno2 46153 | The ` 2 ` nd Fermat number... |
| fmtno3 46154 | The ` 3 ` rd Fermat number... |
| fmtno4 46155 | The ` 4 ` th Fermat number... |
| fmtno5lem1 46156 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 46157 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 46158 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 46159 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 46160 | The ` 5 ` th Fermat number... |
| fmtno0prm 46161 | The ` 0 ` th Fermat number... |
| fmtno1prm 46162 | The ` 1 ` st Fermat number... |
| fmtno2prm 46163 | The ` 2 ` nd Fermat number... |
| 257prm 46164 | 257 is a prime number (the... |
| fmtno3prm 46165 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 46166 | Any power of two is coprim... |
| fmtnoprmfac1lem 46167 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 46168 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 46169 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 46170 | Divisor of Fermat number (... |
| fmtnofac2lem 46171 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 46172 | Divisor of Fermat number (... |
| fmtnofac1 46173 | Divisor of Fermat number (... |
| fmtno4sqrt 46174 | The floor of the square ro... |
| fmtno4prmfac 46175 | If P was a (prime) factor ... |
| fmtno4prmfac193 46176 | If P was a (prime) factor ... |
| fmtno4nprmfac193 46177 | 193 is not a (prime) facto... |
| fmtno4prm 46178 | The ` 4 `-th Fermat number... |
| 65537prm 46179 | 65537 is a prime number (t... |
| fmtnofz04prm 46180 | The first five Fermat numb... |
| fmtnole4prm 46181 | The first five Fermat numb... |
| fmtno5faclem1 46182 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 46183 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 46184 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 46185 | The factorisation of the `... |
| fmtno5nprm 46186 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 46187 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 46188 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 46189 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 46190 | The mapping of a Fermat nu... |
| prminf2 46191 | The set of prime numbers i... |
| 2pwp1prm 46192 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 46193 | Every prime number of the ... |
| m2prm 46194 | The second Mersenne number... |
| m3prm 46195 | The third Mersenne number ... |
| flsqrt 46196 | A condition equivalent to ... |
| flsqrt5 46197 | The floor of the square ro... |
| 3ndvds4 46198 | 3 does not divide 4. (Con... |
| 139prmALT 46199 | 139 is a prime number. In... |
| 31prm 46200 | 31 is a prime number. In ... |
| m5prm 46201 | The fifth Mersenne number ... |
| 127prm 46202 | 127 is a prime number. (C... |
| m7prm 46203 | The seventh Mersenne numbe... |
| m11nprm 46204 | The eleventh Mersenne numb... |
| mod42tp1mod8 46205 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 46206 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 46207 | If ` P ` is a Sophie Germa... |
| lighneallem1 46208 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 46209 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 46210 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 46211 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 46212 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 46213 | Lemma 3 for ~ lighneal . ... |
| lighneal 46214 | If a power of a prime ` P ... |
| modexp2m1d 46215 | The square of an integer w... |
| proththdlem 46216 | Lemma for ~ proththd . (C... |
| proththd 46217 | Proth's theorem (1878). I... |
| 5tcu2e40 46218 | 5 times the cube of 2 is 4... |
| 3exp4mod41 46219 | 3 to the fourth power is -... |
| 41prothprmlem1 46220 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 46221 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 46222 | 41 is a _Proth prime_. (C... |
| quad1 46223 | A condition for a quadrati... |
| requad01 46224 | A condition for a quadrati... |
| requad1 46225 | A condition for a quadrati... |
| requad2 46226 | A condition for a quadrati... |
| iseven 46231 | The predicate "is an even ... |
| isodd 46232 | The predicate "is an odd n... |
| evenz 46233 | An even number is an integ... |
| oddz 46234 | An odd number is an intege... |
| evendiv2z 46235 | The result of dividing an ... |
| oddp1div2z 46236 | The result of dividing an ... |
| oddm1div2z 46237 | The result of dividing an ... |
| isodd2 46238 | The predicate "is an odd n... |
| dfodd2 46239 | Alternate definition for o... |
| dfodd6 46240 | Alternate definition for o... |
| dfeven4 46241 | Alternate definition for e... |
| evenm1odd 46242 | The predecessor of an even... |
| evenp1odd 46243 | The successor of an even n... |
| oddp1eveni 46244 | The successor of an odd nu... |
| oddm1eveni 46245 | The predecessor of an odd ... |
| evennodd 46246 | An even number is not an o... |
| oddneven 46247 | An odd number is not an ev... |
| enege 46248 | The negative of an even nu... |
| onego 46249 | The negative of an odd num... |
| m1expevenALTV 46250 | Exponentiation of -1 by an... |
| m1expoddALTV 46251 | Exponentiation of -1 by an... |
| dfeven2 46252 | Alternate definition for e... |
| dfodd3 46253 | Alternate definition for o... |
| iseven2 46254 | The predicate "is an even ... |
| isodd3 46255 | The predicate "is an odd n... |
| 2dvdseven 46256 | 2 divides an even number. ... |
| m2even 46257 | A multiple of 2 is an even... |
| 2ndvdsodd 46258 | 2 does not divide an odd n... |
| 2dvdsoddp1 46259 | 2 divides an odd number in... |
| 2dvdsoddm1 46260 | 2 divides an odd number de... |
| dfeven3 46261 | Alternate definition for e... |
| dfodd4 46262 | Alternate definition for o... |
| dfodd5 46263 | Alternate definition for o... |
| zefldiv2ALTV 46264 | The floor of an even numbe... |
| zofldiv2ALTV 46265 | The floor of an odd numer ... |
| oddflALTV 46266 | Odd number representation ... |
| iseven5 46267 | The predicate "is an even ... |
| isodd7 46268 | The predicate "is an odd n... |
| dfeven5 46269 | Alternate definition for e... |
| dfodd7 46270 | Alternate definition for o... |
| gcd2odd1 46271 | The greatest common diviso... |
| zneoALTV 46272 | No even integer equals an ... |
| zeoALTV 46273 | An integer is even or odd.... |
| zeo2ALTV 46274 | An integer is even or odd ... |
| nneoALTV 46275 | A positive integer is even... |
| nneoiALTV 46276 | A positive integer is even... |
| odd2np1ALTV 46277 | An integer is odd iff it i... |
| oddm1evenALTV 46278 | An integer is odd iff its ... |
| oddp1evenALTV 46279 | An integer is odd iff its ... |
| oexpnegALTV 46280 | The exponential of the neg... |
| oexpnegnz 46281 | The exponential of the neg... |
| bits0ALTV 46282 | Value of the zeroth bit. ... |
| bits0eALTV 46283 | The zeroth bit of an even ... |
| bits0oALTV 46284 | The zeroth bit of an odd n... |
| divgcdoddALTV 46285 | Either ` A / ( A gcd B ) `... |
| opoeALTV 46286 | The sum of two odds is eve... |
| opeoALTV 46287 | The sum of an odd and an e... |
| omoeALTV 46288 | The difference of two odds... |
| omeoALTV 46289 | The difference of an odd a... |
| oddprmALTV 46290 | A prime not equal to ` 2 `... |
| 0evenALTV 46291 | 0 is an even number. (Con... |
| 0noddALTV 46292 | 0 is not an odd number. (... |
| 1oddALTV 46293 | 1 is an odd number. (Cont... |
| 1nevenALTV 46294 | 1 is not an even number. ... |
| 2evenALTV 46295 | 2 is an even number. (Con... |
| 2noddALTV 46296 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 46297 | An odd nonnegative integer... |
| nnoALTV 46298 | An alternate characterizat... |
| nn0oALTV 46299 | An alternate characterizat... |
| nn0e 46300 | An alternate characterizat... |
| nneven 46301 | An alternate characterizat... |
| nn0onn0exALTV 46302 | For each odd nonnegative i... |
| nn0enn0exALTV 46303 | For each even nonnegative ... |
| nnennexALTV 46304 | For each even positive int... |
| nnpw2evenALTV 46305 | 2 to the power of a positi... |
| epoo 46306 | The sum of an even and an ... |
| emoo 46307 | The difference of an even ... |
| epee 46308 | The sum of two even number... |
| emee 46309 | The difference of two even... |
| evensumeven 46310 | If a summand is even, the ... |
| 3odd 46311 | 3 is an odd number. (Cont... |
| 4even 46312 | 4 is an even number. (Con... |
| 5odd 46313 | 5 is an odd number. (Cont... |
| 6even 46314 | 6 is an even number. (Con... |
| 7odd 46315 | 7 is an odd number. (Cont... |
| 8even 46316 | 8 is an even number. (Con... |
| evenprm2 46317 | A prime number is even iff... |
| oddprmne2 46318 | Every prime number not bei... |
| oddprmuzge3 46319 | A prime number which is od... |
| evenltle 46320 | If an even number is great... |
| odd2prm2 46321 | If an odd number is the su... |
| even3prm2 46322 | If an even number is the s... |
| mogoldbblem 46323 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 46324 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 46325 | Lemma for ~ perfectALTV . ... |
| perfectALTV 46326 | The Euclid-Euler theorem, ... |
| fppr 46329 | The set of Fermat pseudopr... |
| fpprmod 46330 | The set of Fermat pseudopr... |
| fpprel 46331 | A Fermat pseudoprime to th... |
| fpprbasnn 46332 | The base of a Fermat pseud... |
| fpprnn 46333 | A Fermat pseudoprime to th... |
| fppr2odd 46334 | A Fermat pseudoprime to th... |
| 11t31e341 46335 | 341 is the product of 11 a... |
| 2exp340mod341 46336 | Eight to the eighth power ... |
| 341fppr2 46337 | 341 is the (smallest) _Pou... |
| 4fppr1 46338 | 4 is the (smallest) Fermat... |
| 8exp8mod9 46339 | Eight to the eighth power ... |
| 9fppr8 46340 | 9 is the (smallest) Fermat... |
| dfwppr 46341 | Alternate definition of a ... |
| fpprwppr 46342 | A Fermat pseudoprime to th... |
| fpprwpprb 46343 | An integer ` X ` which is ... |
| fpprel2 46344 | An alternate definition fo... |
| nfermltl8rev 46345 | Fermat's little theorem wi... |
| nfermltl2rev 46346 | Fermat's little theorem wi... |
| nfermltlrev 46347 | Fermat's little theorem re... |
| isgbe 46354 | The predicate "is an even ... |
| isgbow 46355 | The predicate "is a weak o... |
| isgbo 46356 | The predicate "is an odd G... |
| gbeeven 46357 | An even Goldbach number is... |
| gbowodd 46358 | A weak odd Goldbach number... |
| gbogbow 46359 | A (strong) odd Goldbach nu... |
| gboodd 46360 | An odd Goldbach number is ... |
| gbepos 46361 | Any even Goldbach number i... |
| gbowpos 46362 | Any weak odd Goldbach numb... |
| gbopos 46363 | Any odd Goldbach number is... |
| gbegt5 46364 | Any even Goldbach number i... |
| gbowgt5 46365 | Any weak odd Goldbach numb... |
| gbowge7 46366 | Any weak odd Goldbach numb... |
| gboge9 46367 | Any odd Goldbach number is... |
| gbege6 46368 | Any even Goldbach number i... |
| gbpart6 46369 | The Goldbach partition of ... |
| gbpart7 46370 | The (weak) Goldbach partit... |
| gbpart8 46371 | The Goldbach partition of ... |
| gbpart9 46372 | The (strong) Goldbach part... |
| gbpart11 46373 | The (strong) Goldbach part... |
| 6gbe 46374 | 6 is an even Goldbach numb... |
| 7gbow 46375 | 7 is a weak odd Goldbach n... |
| 8gbe 46376 | 8 is an even Goldbach numb... |
| 9gbo 46377 | 9 is an odd Goldbach numbe... |
| 11gbo 46378 | 11 is an odd Goldbach numb... |
| stgoldbwt 46379 | If the strong ternary Gold... |
| sbgoldbwt 46380 | If the strong binary Goldb... |
| sbgoldbst 46381 | If the strong binary Goldb... |
| sbgoldbaltlem1 46382 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 46383 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 46384 | An alternate (related to t... |
| sbgoldbb 46385 | If the strong binary Goldb... |
| sgoldbeven3prm 46386 | If the binary Goldbach con... |
| sbgoldbm 46387 | If the strong binary Goldb... |
| mogoldbb 46388 | If the modern version of t... |
| sbgoldbmb 46389 | The strong binary Goldbach... |
| sbgoldbo 46390 | If the strong binary Goldb... |
| nnsum3primes4 46391 | 4 is the sum of at most 3 ... |
| nnsum4primes4 46392 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 46393 | Every prime is "the sum of... |
| nnsum4primesprm 46394 | Every prime is "the sum of... |
| nnsum3primesgbe 46395 | Any even Goldbach number i... |
| nnsum4primesgbe 46396 | Any even Goldbach number i... |
| nnsum3primesle9 46397 | Every integer greater than... |
| nnsum4primesle9 46398 | Every integer greater than... |
| nnsum4primesodd 46399 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 46400 | If the (strong) ternary Go... |
| evengpop3 46401 | If the (weak) ternary Gold... |
| evengpoap3 46402 | If the (strong) ternary Go... |
| nnsum4primeseven 46403 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 46404 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 46405 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 46406 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 46407 | If the binary Goldbach con... |
| bgoldbtbndlem1 46408 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 46409 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 46410 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 46411 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 46412 | If the binary Goldbach con... |
| tgoldbachgtALTV 46415 | Variant of Thierry Arnoux'... |
| bgoldbachlt 46416 | The binary Goldbach conjec... |
| tgblthelfgott 46418 | The ternary Goldbach conje... |
| tgoldbachlt 46419 | The ternary Goldbach conje... |
| tgoldbach 46420 | The ternary Goldbach conje... |
| isomgrrel 46425 | The isomorphy relation for... |
| isomgr 46426 | The isomorphy relation for... |
| isisomgr 46427 | Implications of two graphs... |
| isomgreqve 46428 | A set is isomorphic to a h... |
| isomushgr 46429 | The isomorphy relation for... |
| isomuspgrlem1 46430 | Lemma 1 for ~ isomuspgr . ... |
| isomuspgrlem2a 46431 | Lemma 1 for ~ isomuspgrlem... |
| isomuspgrlem2b 46432 | Lemma 2 for ~ isomuspgrlem... |
| isomuspgrlem2c 46433 | Lemma 3 for ~ isomuspgrlem... |
| isomuspgrlem2d 46434 | Lemma 4 for ~ isomuspgrlem... |
| isomuspgrlem2e 46435 | Lemma 5 for ~ isomuspgrlem... |
| isomuspgrlem2 46436 | Lemma 2 for ~ isomuspgr . ... |
| isomuspgr 46437 | The isomorphy relation for... |
| isomgrref 46438 | The isomorphy relation is ... |
| isomgrsym 46439 | The isomorphy relation is ... |
| isomgrsymb 46440 | The isomorphy relation is ... |
| isomgrtrlem 46441 | Lemma for ~ isomgrtr . (C... |
| isomgrtr 46442 | The isomorphy relation is ... |
| strisomgrop 46443 | A graph represented as an ... |
| ushrisomgr 46444 | A simple hypergraph (with ... |
| 1hegrlfgr 46445 | A graph ` G ` with one hyp... |
| upwlksfval 46448 | The set of simple walks (i... |
| isupwlk 46449 | Properties of a pair of fu... |
| isupwlkg 46450 | Generalization of ~ isupwl... |
| upwlkbprop 46451 | Basic properties of a simp... |
| upwlkwlk 46452 | A simple walk is a walk. ... |
| upgrwlkupwlk 46453 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 46454 | In a pseudograph, the defi... |
| upgrisupwlkALT 46455 | Alternate proof of ~ upgri... |
| upgredgssspr 46456 | The set of edges of a pseu... |
| uspgropssxp 46457 | The set ` G ` of "simple p... |
| uspgrsprfv 46458 | The value of the function ... |
| uspgrsprf 46459 | The mapping ` F ` is a fun... |
| uspgrsprf1 46460 | The mapping ` F ` is a one... |
| uspgrsprfo 46461 | The mapping ` F ` is a fun... |
| uspgrsprf1o 46462 | The mapping ` F ` is a bij... |
| uspgrex 46463 | The class ` G ` of all "si... |
| uspgrbispr 46464 | There is a bijection betwe... |
| uspgrspren 46465 | The set ` G ` of the "simp... |
| uspgrymrelen 46466 | The set ` G ` of the "simp... |
| uspgrbisymrel 46467 | There is a bijection betwe... |
| uspgrbisymrelALT 46468 | Alternate proof of ~ uspgr... |
| ovn0dmfun 46469 | If a class operation value... |
| xpsnopab 46470 | A Cartesian product with a... |
| xpiun 46471 | A Cartesian product expres... |
| ovn0ssdmfun 46472 | If a class' operation valu... |
| fnxpdmdm 46473 | The domain of the domain o... |
| cnfldsrngbas 46474 | The base set of a subring ... |
| cnfldsrngadd 46475 | The group addition operati... |
| cnfldsrngmul 46476 | The ring multiplication op... |
| plusfreseq 46477 | If the empty set is not co... |
| mgmplusfreseq 46478 | If the empty set is not co... |
| 0mgm 46479 | A set with an empty base s... |
| mgmpropd 46480 | If two structures have the... |
| ismgmd 46481 | Deduce a magma from its pr... |
| mgmhmrcl 46486 | Reverse closure of a magma... |
| submgmrcl 46487 | Reverse closure for submag... |
| ismgmhm 46488 | Property of a magma homomo... |
| mgmhmf 46489 | A magma homomorphism is a ... |
| mgmhmpropd 46490 | Magma homomorphism depends... |
| mgmhmlin 46491 | A magma homomorphism prese... |
| mgmhmf1o 46492 | A magma homomorphism is bi... |
| idmgmhm 46493 | The identity homomorphism ... |
| issubmgm 46494 | Expand definition of a sub... |
| issubmgm2 46495 | Submagmas are subsets that... |
| rabsubmgmd 46496 | Deduction for proving that... |
| submgmss 46497 | Submagmas are subsets of t... |
| submgmid 46498 | Every magma is trivially a... |
| submgmcl 46499 | Submagmas are closed under... |
| submgmmgm 46500 | Submagmas are themselves m... |
| submgmbas 46501 | The base set of a submagma... |
| subsubmgm 46502 | A submagma of a submagma i... |
| resmgmhm 46503 | Restriction of a magma hom... |
| resmgmhm2 46504 | One direction of ~ resmgmh... |
| resmgmhm2b 46505 | Restriction of the codomai... |
| mgmhmco 46506 | The composition of magma h... |
| mgmhmima 46507 | The homomorphic image of a... |
| mgmhmeql 46508 | The equalizer of two magma... |
| submgmacs 46509 | Submagmas are an algebraic... |
| ismhm0 46510 | Property of a monoid homom... |
| mhmismgmhm 46511 | Each monoid homomorphism i... |
| opmpoismgm 46512 | A structure with a group a... |
| copissgrp 46513 | A structure with a constan... |
| copisnmnd 46514 | A structure with a constan... |
| 0nodd 46515 | 0 is not an odd integer. ... |
| 1odd 46516 | 1 is an odd integer. (Con... |
| 2nodd 46517 | 2 is not an odd integer. ... |
| oddibas 46518 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 46519 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 46520 | The structure of all odd i... |
| nnsgrpmgm 46521 | The structure of positive ... |
| nnsgrp 46522 | The structure of positive ... |
| nnsgrpnmnd 46523 | The structure of positive ... |
| nn0mnd 46524 | The set of nonnegative int... |
| gsumsplit2f 46525 | Split a group sum into two... |
| gsumdifsndf 46526 | Extract a summand from a f... |
| gsumfsupp 46527 | A group sum of a family ca... |
| iscllaw 46534 | The predicate "is a closed... |
| iscomlaw 46535 | The predicate "is a commut... |
| clcllaw 46536 | Closure of a closed operat... |
| isasslaw 46537 | The predicate "is an assoc... |
| asslawass 46538 | Associativity of an associ... |
| mgmplusgiopALT 46539 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 46540 | Slot 2 (group operation) o... |
| intopval 46547 | The internal (binary) oper... |
| intop 46548 | An internal (binary) opera... |
| clintopval 46549 | The closed (internal binar... |
| assintopval 46550 | The associative (closed in... |
| assintopmap 46551 | The associative (closed in... |
| isclintop 46552 | The predicate "is a closed... |
| clintop 46553 | A closed (internal binary)... |
| assintop 46554 | An associative (closed int... |
| isassintop 46555 | The predicate "is an assoc... |
| clintopcllaw 46556 | The closure law holds for ... |
| assintopcllaw 46557 | The closure low holds for ... |
| assintopasslaw 46558 | The associative low holds ... |
| assintopass 46559 | An associative (closed int... |
| ismgmALT 46568 | The predicate "is a magma"... |
| iscmgmALT 46569 | The predicate "is a commut... |
| issgrpALT 46570 | The predicate "is a semigr... |
| iscsgrpALT 46571 | The predicate "is a commut... |
| mgm2mgm 46572 | Equivalence of the two def... |
| sgrp2sgrp 46573 | Equivalence of the two def... |
| idfusubc0 46574 | The identity functor for a... |
| idfusubc 46575 | The identity functor for a... |
| inclfusubc 46576 | The "inclusion functor" fr... |
| lmod0rng 46577 | If the scalar ring of a mo... |
| nzrneg1ne0 46578 | The additive inverse of th... |
| 0ringdif 46579 | A zero ring is a ring whic... |
| 0ringbas 46580 | The base set of a zero rin... |
| 0ring1eq0 46581 | In a zero ring, a ring whi... |
| nrhmzr 46582 | There is no ring homomorph... |
| isrng 46585 | The predicate "is a non-un... |
| rngabl 46586 | A non-unital ring is an (a... |
| rngmgp 46587 | A non-unital ring is a sem... |
| rngmgpf 46588 | Restricted functionality o... |
| rnggrp 46589 | A non-unital ring is a (ad... |
| ringrng 46590 | A unital ring is a non-uni... |
| ringssrng 46591 | The unital rings are non-u... |
| isringrng 46592 | The predicate "is a unital... |
| rngass 46593 | Associative law for the mu... |
| rngdi 46594 | Distributive law for the m... |
| rngdir 46595 | Distributive law for the m... |
| rngacl 46596 | Closure of the addition op... |
| rng0cl 46597 | The zero element of a non-... |
| rngcl 46598 | Closure of the multiplicat... |
| rnglz 46599 | The zero of a non-unital r... |
| rngrz 46600 | The zero of a non-unital r... |
| rngmneg1 46601 | Negation of a product in a... |
| rngmneg2 46602 | Negation of a product in a... |
| rngm2neg 46603 | Double negation of a produ... |
| rngansg 46604 | Every additive subgroup of... |
| rngsubdi 46605 | Ring multiplication distri... |
| rngsubdir 46606 | Ring multiplication distri... |
| isrngd 46607 | Properties that determine ... |
| rngpropd 46608 | If two structures have the... |
| opprrng 46609 | An opposite non-unital rin... |
| opprrngb 46610 | A class is a non-unital ri... |
| prdsmulrngcl 46611 | Closure of the multiplicat... |
| prdsrngd 46612 | A product of non-unital ri... |
| imasrng 46613 | The image structure of a n... |
| imasrngf1 46614 | The image of a non-unital ... |
| xpsrngd 46615 | A product of two non-unita... |
| qusrng 46616 | The quotient structure of ... |
| rnghmrcl 46621 | Reverse closure of a non-u... |
| rnghmfn 46622 | The mapping of two non-uni... |
| rnghmval 46623 | The set of the non-unital ... |
| isrnghm 46624 | A function is a non-unital... |
| isrnghmmul 46625 | A function is a non-unital... |
| rnghmmgmhm 46626 | A non-unital ring homomorp... |
| rnghmval2 46627 | The non-unital ring homomo... |
| isrngisom 46628 | An isomorphism of non-unit... |
| rngimrcl 46629 | Reverse closure for an iso... |
| rnghmghm 46630 | A non-unital ring homomorp... |
| rnghmf 46631 | A ring homomorphism is a f... |
| rnghmmul 46632 | A homomorphism of non-unit... |
| isrnghm2d 46633 | Demonstration of non-unita... |
| isrnghmd 46634 | Demonstration of non-unita... |
| rnghmf1o 46635 | A non-unital ring homomorp... |
| isrngim 46636 | An isomorphism of non-unit... |
| rngimf1o 46637 | An isomorphism of non-unit... |
| rngimrnghm 46638 | An isomorphism of non-unit... |
| rngimcnv 46639 | The converse of an isomorp... |
| rnghmco 46640 | The composition of non-uni... |
| idrnghm 46641 | The identity homomorphism ... |
| c0mgm 46642 | The constant mapping to ze... |
| c0mhm 46643 | The constant mapping to ze... |
| c0ghm 46644 | The constant mapping to ze... |
| c0rhm 46645 | The constant mapping to ze... |
| c0rnghm 46646 | The constant mapping to ze... |
| c0snmgmhm 46647 | The constant mapping to ze... |
| c0snmhm 46648 | The constant mapping to ze... |
| c0snghm 46649 | The constant mapping to ze... |
| zrrnghm 46650 | The constant mapping to ze... |
| rngisomfv1 46651 | If there is a non-unital r... |
| rngisom1 46652 | If there is a non-unital r... |
| rngisomring 46653 | If there is a non-unital r... |
| rhmfn 46654 | The mapping of two rings t... |
| rhmval 46655 | The ring homomorphisms bet... |
| rhmisrnghm 46656 | Each unital ring homomorph... |
| issubrng 46659 | The subring of non-unital ... |
| subrngss 46660 | A subring is a subset. (C... |
| subrngid 46661 | Every non-unital ring is a... |
| subrngrng 46662 | A subring is a non-unital ... |
| subrngrcl 46663 | Reverse closure for a subr... |
| subrngsubg 46664 | A subring is a subgroup. ... |
| subrngringnsg 46665 | A subring is a normal subg... |
| subrngbas 46666 | Base set of a subring stru... |
| subrng0 46667 | A subring always has the s... |
| subrngacl 46668 | A subring is closed under ... |
| subrngmcl 46669 | A subgroup is closed under... |
| issubrng2 46670 | Characterize the subrings ... |
| opprsubrng 46671 | Being a subring is a symme... |
| subrngint 46672 | The intersection of a none... |
| subrngin 46673 | The intersection of two su... |
| subrngmre 46674 | The subrings of a non-unit... |
| subsubrng 46675 | A subring of a subring is ... |
| subsubrng2 46676 | The set of subrings of a s... |
| rhmimasubrnglem 46677 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 46678 | The homomorphic image of a... |
| cntzsubrng 46679 | Centralizers in a non-unit... |
| subrngpropd 46680 | If two structures have the... |
| rnglidlmcl 46681 | A (left) ideal containing ... |
| rngridlmcl 46682 | A right ideal (which is a ... |
| rnglidl0 46683 | Every non-unital ring cont... |
| rnglidl1 46684 | The base set of every non-... |
| lidlssbas 46685 | The base set of the restri... |
| lidlbas 46686 | A (left) ideal of a ring i... |
| rnglidlmmgm 46687 | The multiplicative group o... |
| rnglidlmsgrp 46688 | The multiplicative group o... |
| rnglidlrng 46689 | A (left) ideal of a non-un... |
| rng2idlsubrng 46690 | A two-sided ideal of a non... |
| rng2idlnsg 46691 | A two-sided ideal of a non... |
| rng2idl0 46692 | The zero (additive identit... |
| rng2idlsubgsubrng 46693 | A two-sided ideal of a non... |
| rng2idlsubgnsg 46694 | A two-sided ideal of a non... |
| rng2idlsubg0 46695 | The zero (additive identit... |
| 2idlcpblrng 46696 | The coset equivalence rela... |
| qus2idrng 46697 | The quotient of a non-unit... |
| ecqusadd 46698 | Addition of equivalence cl... |
| ecqusaddcl 46699 | Closure of the addition in... |
| qusmulrng 46700 | Value of the multiplicatio... |
| rngqiprngghmlem1 46701 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 46702 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 46703 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 46704 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 46705 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 46706 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 46707 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 46708 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 46709 | The base set of the produc... |
| rngqiprng 46710 | The product of the quotien... |
| rngqiprngimf 46711 | ` F ` is a function from (... |
| rngqiprngimfv 46712 | The value of the function ... |
| rngqiprngghm 46713 | ` F ` is a homomorphism of... |
| rngqiprngimf1 46714 | ` F ` is a one-to-one func... |
| rngqiprngimfo 46715 | ` F ` is a function from (... |
| rngqiprnglin 46716 | ` F ` is linear with respe... |
| rngqiprngho 46717 | ` F ` is a homomorphism of... |
| rngqiprngim 46718 | ` F ` is an isomorphism of... |
| rng2idl1cntr 46719 | The unity of a two-sided i... |
| rngringbdlem1 46720 | In a unital ring, the quot... |
| rngringbdlem2 46721 | A non-unital ring is unita... |
| rngringbd 46722 | A non-unital ring is unita... |
| ring2idlqus 46723 | For every unital ring ther... |
| ring2idlqusb 46724 | A non-unital ring is unita... |
| lidldomn1 46725 | If a (left) ideal (which i... |
| lidlabl 46726 | A (left) ideal of a ring i... |
| lidlrng 46727 | A (left) ideal of a ring i... |
| zlidlring 46728 | The zero (left) ideal of a... |
| uzlidlring 46729 | Only the zero (left) ideal... |
| lidldomnnring 46730 | A (left) ideal of a domain... |
| 0even 46731 | 0 is an even integer. (Co... |
| 1neven 46732 | 1 is not an even integer. ... |
| 2even 46733 | 2 is an even integer. (Co... |
| 2zlidl 46734 | The even integers are a (l... |
| 2zrng 46735 | The ring of integers restr... |
| 2zrngbas 46736 | The base set of R is the s... |
| 2zrngadd 46737 | The group addition operati... |
| 2zrng0 46738 | The additive identity of R... |
| 2zrngamgm 46739 | R is an (additive) magma. ... |
| 2zrngasgrp 46740 | R is an (additive) semigro... |
| 2zrngamnd 46741 | R is an (additive) monoid.... |
| 2zrngacmnd 46742 | R is a commutative (additi... |
| 2zrngagrp 46743 | R is an (additive) group. ... |
| 2zrngaabl 46744 | R is an (additive) abelian... |
| 2zrngmul 46745 | The ring multiplication op... |
| 2zrngmmgm 46746 | R is a (multiplicative) ma... |
| 2zrngmsgrp 46747 | R is a (multiplicative) se... |
| 2zrngALT 46748 | The ring of integers restr... |
| 2zrngnmlid 46749 | R has no multiplicative (l... |
| 2zrngnmrid 46750 | R has no multiplicative (r... |
| 2zrngnmlid2 46751 | R has no multiplicative (l... |
| 2zrngnring 46752 | R is not a unital ring. (... |
| cznrnglem 46753 | Lemma for ~ cznrng : The ... |
| cznabel 46754 | The ring constructed from ... |
| cznrng 46755 | The ring constructed from ... |
| cznnring 46756 | The ring constructed from ... |
| rngcvalALTV 46761 | Value of the category of n... |
| rngcval 46762 | Value of the category of n... |
| rnghmresfn 46763 | The class of non-unital ri... |
| rnghmresel 46764 | An element of the non-unit... |
| rngcbas 46765 | Set of objects of the cate... |
| rngchomfval 46766 | Set of arrows of the categ... |
| rngchom 46767 | Set of arrows of the categ... |
| elrngchom 46768 | A morphism of non-unital r... |
| rngchomfeqhom 46769 | The functionalized Hom-set... |
| rngccofval 46770 | Composition in the categor... |
| rngcco 46771 | Composition in the categor... |
| dfrngc2 46772 | Alternate definition of th... |
| rnghmsscmap2 46773 | The non-unital ring homomo... |
| rnghmsscmap 46774 | The non-unital ring homomo... |
| rnghmsubcsetclem1 46775 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 46776 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 46777 | The non-unital ring homomo... |
| rngccat 46778 | The category of non-unital... |
| rngcid 46779 | The identity arrow in the ... |
| rngcsect 46780 | A section in the category ... |
| rngcinv 46781 | An inverse in the category... |
| rngciso 46782 | An isomorphism in the cate... |
| rngcbasALTV 46783 | Set of objects of the cate... |
| rngchomfvalALTV 46784 | Set of arrows of the categ... |
| rngchomALTV 46785 | Set of arrows of the categ... |
| elrngchomALTV 46786 | A morphism of non-unital r... |
| rngccofvalALTV 46787 | Composition in the categor... |
| rngccoALTV 46788 | Composition in the categor... |
| rngccatidALTV 46789 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 46790 | The category of non-unital... |
| rngcidALTV 46791 | The identity arrow in the ... |
| rngcsectALTV 46792 | A section in the category ... |
| rngcinvALTV 46793 | An inverse in the category... |
| rngcisoALTV 46794 | An isomorphism in the cate... |
| rngchomffvalALTV 46795 | The value of the functiona... |
| rngchomrnghmresALTV 46796 | The value of the functiona... |
| rngcifuestrc 46797 | The "inclusion functor" fr... |
| funcrngcsetc 46798 | The "natural forgetful fun... |
| funcrngcsetcALT 46799 | Alternate proof of ~ funcr... |
| zrinitorngc 46800 | The zero ring is an initia... |
| zrtermorngc 46801 | The zero ring is a termina... |
| zrzeroorngc 46802 | The zero ring is a zero ob... |
| ringcvalALTV 46807 | Value of the category of r... |
| ringcval 46808 | Value of the category of u... |
| rhmresfn 46809 | The class of unital ring h... |
| rhmresel 46810 | An element of the unital r... |
| ringcbas 46811 | Set of objects of the cate... |
| ringchomfval 46812 | Set of arrows of the categ... |
| ringchom 46813 | Set of arrows of the categ... |
| elringchom 46814 | A morphism of unital rings... |
| ringchomfeqhom 46815 | The functionalized Hom-set... |
| ringccofval 46816 | Composition in the categor... |
| ringcco 46817 | Composition in the categor... |
| dfringc2 46818 | Alternate definition of th... |
| rhmsscmap2 46819 | The unital ring homomorphi... |
| rhmsscmap 46820 | The unital ring homomorphi... |
| rhmsubcsetclem1 46821 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 46822 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 46823 | The unital ring homomorphi... |
| ringccat 46824 | The category of unital rin... |
| ringcid 46825 | The identity arrow in the ... |
| rhmsscrnghm 46826 | The unital ring homomorphi... |
| rhmsubcrngclem1 46827 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 46828 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 46829 | The unital ring homomorphi... |
| rngcresringcat 46830 | The restriction of the cat... |
| ringcsect 46831 | A section in the category ... |
| ringcinv 46832 | An inverse in the category... |
| ringciso 46833 | An isomorphism in the cate... |
| ringcbasbas 46834 | An element of the base set... |
| funcringcsetc 46835 | The "natural forgetful fun... |
| funcringcsetcALTV2lem1 46836 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 46837 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 46838 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 46839 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 46840 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 46841 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 46842 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 46843 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 46844 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 46845 | The "natural forgetful fun... |
| ringcbasALTV 46846 | Set of objects of the cate... |
| ringchomfvalALTV 46847 | Set of arrows of the categ... |
| ringchomALTV 46848 | Set of arrows of the categ... |
| elringchomALTV 46849 | A morphism of rings is a f... |
| ringccofvalALTV 46850 | Composition in the categor... |
| ringccoALTV 46851 | Composition in the categor... |
| ringccatidALTV 46852 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 46853 | The category of rings is a... |
| ringcidALTV 46854 | The identity arrow in the ... |
| ringcsectALTV 46855 | A section in the category ... |
| ringcinvALTV 46856 | An inverse in the category... |
| ringcisoALTV 46857 | An isomorphism in the cate... |
| ringcbasbasALTV 46858 | An element of the base set... |
| funcringcsetclem1ALTV 46859 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 46860 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 46861 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 46862 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 46863 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 46864 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 46865 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 46866 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 46867 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 46868 | The "natural forgetful fun... |
| irinitoringc 46869 | The ring of integers is an... |
| zrtermoringc 46870 | The zero ring is a termina... |
| zrninitoringc 46871 | The zero ring is not an in... |
| nzerooringczr 46872 | There is no zero object in... |
| srhmsubclem1 46873 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 46874 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 46875 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 46876 | According to ~ df-subc , t... |
| sringcat 46877 | The restriction of the cat... |
| crhmsubc 46878 | According to ~ df-subc , t... |
| cringcat 46879 | The restriction of the cat... |
| drhmsubc 46880 | According to ~ df-subc , t... |
| drngcat 46881 | The restriction of the cat... |
| fldcat 46882 | The restriction of the cat... |
| fldc 46883 | The restriction of the cat... |
| fldhmsubc 46884 | According to ~ df-subc , t... |
| rngcrescrhm 46885 | The category of non-unital... |
| rhmsubclem1 46886 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 46887 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 46888 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 46889 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 46890 | According to ~ df-subc , t... |
| rhmsubccat 46891 | The restriction of the cat... |
| srhmsubcALTVlem1 46892 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 46893 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 46894 | According to ~ df-subc , t... |
| sringcatALTV 46895 | The restriction of the cat... |
| crhmsubcALTV 46896 | According to ~ df-subc , t... |
| cringcatALTV 46897 | The restriction of the cat... |
| drhmsubcALTV 46898 | According to ~ df-subc , t... |
| drngcatALTV 46899 | The restriction of the cat... |
| fldcatALTV 46900 | The restriction of the cat... |
| fldcALTV 46901 | The restriction of the cat... |
| fldhmsubcALTV 46902 | According to ~ df-subc , t... |
| rngcrescrhmALTV 46903 | The category of non-unital... |
| rhmsubcALTVlem1 46904 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 46905 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 46906 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 46907 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 46908 | According to ~ df-subc , t... |
| rhmsubcALTVcat 46909 | The restriction of the cat... |
| opeliun2xp 46910 | Membership of an ordered p... |
| eliunxp2 46911 | Membership in a union of C... |
| mpomptx2 46912 | Express a two-argument fun... |
| cbvmpox2 46913 | Rule to change the bound v... |
| dmmpossx2 46914 | The domain of a mapping is... |
| mpoexxg2 46915 | Existence of an operation ... |
| ovmpordxf 46916 | Value of an operation give... |
| ovmpordx 46917 | Value of an operation give... |
| ovmpox2 46918 | The value of an operation ... |
| fdmdifeqresdif 46919 | The restriction of a condi... |
| offvalfv 46920 | The function operation exp... |
| ofaddmndmap 46921 | The function operation app... |
| mapsnop 46922 | A singleton of an ordered ... |
| fprmappr 46923 | A function with a domain o... |
| mapprop 46924 | An unordered pair containi... |
| ztprmneprm 46925 | A prime is not an integer ... |
| 2t6m3t4e0 46926 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 46927 | For any finite subset of `... |
| nn0sumltlt 46928 | If the sum of two nonnegat... |
| bcpascm1 46929 | Pascal's rule for the bino... |
| altgsumbc 46930 | The sum of binomial coeffi... |
| altgsumbcALT 46931 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 46932 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 46933 | An element of the (base se... |
| zlmodzxz0 46934 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 46935 | The scalar multiplication ... |
| zlmodzxzadd 46936 | The addition of the ` ZZ `... |
| zlmodzxzsubm 46937 | The subtraction of the ` Z... |
| zlmodzxzsub 46938 | The subtraction of the ` Z... |
| mgpsumunsn 46939 | Extract a summand/factor f... |
| mgpsumz 46940 | If the group sum for the m... |
| mgpsumn 46941 | If the group sum for the m... |
| exple2lt6 46942 | A nonnegative integer to t... |
| pgrple2abl 46943 | Every symmetric group on a... |
| pgrpgt2nabl 46944 | Every symmetric group on a... |
| invginvrid 46945 | Identity for a multiplicat... |
| rmsupp0 46946 | The support of a mapping o... |
| domnmsuppn0 46947 | The support of a mapping o... |
| rmsuppss 46948 | The support of a mapping o... |
| mndpsuppss 46949 | The support of a mapping o... |
| scmsuppss 46950 | The support of a mapping o... |
| rmsuppfi 46951 | The support of a mapping o... |
| rmfsupp 46952 | A mapping of a multiplicat... |
| mndpsuppfi 46953 | The support of a mapping o... |
| mndpfsupp 46954 | A mapping of a scalar mult... |
| scmsuppfi 46955 | The support of a mapping o... |
| scmfsupp 46956 | A mapping of a scalar mult... |
| suppmptcfin 46957 | The support of a mapping w... |
| mptcfsupp 46958 | A mapping with value 0 exc... |
| fsuppmptdmf 46959 | A mapping with a finite do... |
| lmodvsmdi 46960 | Multiple distributive law ... |
| gsumlsscl 46961 | Closure of a group sum in ... |
| assaascl0 46962 | The scalar 0 embedded into... |
| assaascl1 46963 | The scalar 1 embedded into... |
| ply1vr1smo 46964 | The variable in a polynomi... |
| ply1ass23l 46965 | Associative identity with ... |
| ply1sclrmsm 46966 | The ring multiplication of... |
| coe1id 46967 | Coefficient vector of the ... |
| coe1sclmulval 46968 | The value of the coefficie... |
| ply1mulgsumlem1 46969 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 46970 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 46971 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 46972 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 46973 | The product of two polynom... |
| evl1at0 46974 | Polynomial evaluation for ... |
| evl1at1 46975 | Polynomial evaluation for ... |
| linply1 46976 | A term of the form ` x - C... |
| lineval 46977 | A term of the form ` x - C... |
| linevalexample 46978 | The polynomial ` x - 3 ` o... |
| dmatALTval 46983 | The algebra of ` N ` x ` N... |
| dmatALTbas 46984 | The base set of the algebr... |
| dmatALTbasel 46985 | An element of the base set... |
| dmatbas 46986 | The set of all ` N ` x ` N... |
| lincop 46991 | A linear combination as op... |
| lincval 46992 | The value of a linear comb... |
| dflinc2 46993 | Alternative definition of ... |
| lcoop 46994 | A linear combination as op... |
| lcoval 46995 | The value of a linear comb... |
| lincfsuppcl 46996 | A linear combination of ve... |
| linccl 46997 | A linear combination of ve... |
| lincval0 46998 | The value of an empty line... |
| lincvalsng 46999 | The linear combination ove... |
| lincvalsn 47000 | The linear combination ove... |
| lincvalpr 47001 | The linear combination ove... |
| lincval1 47002 | The linear combination ove... |
| lcosn0 47003 | Properties of a linear com... |
| lincvalsc0 47004 | The linear combination whe... |
| lcoc0 47005 | Properties of a linear com... |
| linc0scn0 47006 | If a set contains the zero... |
| lincdifsn 47007 | A vector is a linear combi... |
| linc1 47008 | A vector is a linear combi... |
| lincellss 47009 | A linear combination of a ... |
| lco0 47010 | The set of empty linear co... |
| lcoel0 47011 | The zero vector is always ... |
| lincsum 47012 | The sum of two linear comb... |
| lincscm 47013 | A linear combinations mult... |
| lincsumcl 47014 | The sum of two linear comb... |
| lincscmcl 47015 | The multiplication of a li... |
| lincsumscmcl 47016 | The sum of a linear combin... |
| lincolss 47017 | According to the statement... |
| ellcoellss 47018 | Every linear combination o... |
| lcoss 47019 | A set of vectors of a modu... |
| lspsslco 47020 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 47021 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 47022 | Equivalence of a _span_ of... |
| rellininds 47026 | The class defining the rel... |
| linindsv 47028 | The classes of the module ... |
| islininds 47029 | The property of being a li... |
| linindsi 47030 | The implications of being ... |
| linindslinci 47031 | The implications of being ... |
| islinindfis 47032 | The property of being a li... |
| islinindfiss 47033 | The property of being a li... |
| linindscl 47034 | A linearly independent set... |
| lindepsnlininds 47035 | A linearly dependent subse... |
| islindeps 47036 | The property of being a li... |
| lincext1 47037 | Property 1 of an extension... |
| lincext2 47038 | Property 2 of an extension... |
| lincext3 47039 | Property 3 of an extension... |
| lindslinindsimp1 47040 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 47041 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 47042 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 47043 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 47044 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 47045 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 47046 | Implication 2 for ~ lindsl... |
| lindslininds 47047 | Equivalence of definitions... |
| linds0 47048 | The empty set is always a ... |
| el0ldep 47049 | A set containing the zero ... |
| el0ldepsnzr 47050 | A set containing the zero ... |
| lindsrng01 47051 | Any subset of a module is ... |
| lindszr 47052 | Any subset of a module ove... |
| snlindsntorlem 47053 | Lemma for ~ snlindsntor . ... |
| snlindsntor 47054 | A singleton is linearly in... |
| ldepsprlem 47055 | Lemma for ~ ldepspr . (Co... |
| ldepspr 47056 | If a vector is a scalar mu... |
| lincresunit3lem3 47057 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 47058 | Lemma 1 for properties of ... |
| lincresunitlem2 47059 | Lemma for properties of a ... |
| lincresunit1 47060 | Property 1 of a specially ... |
| lincresunit2 47061 | Property 2 of a specially ... |
| lincresunit3lem1 47062 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 47063 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 47064 | Property 3 of a specially ... |
| lincreslvec3 47065 | Property 3 of a specially ... |
| islindeps2 47066 | Conditions for being a lin... |
| islininds2 47067 | Implication of being a lin... |
| isldepslvec2 47068 | Alternative definition of ... |
| lindssnlvec 47069 | A singleton not containing... |
| lmod1lem1 47070 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 47071 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 47072 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 47073 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 47074 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 47075 | The (smallest) structure r... |
| lmod1zr 47076 | The (smallest) structure r... |
| lmod1zrnlvec 47077 | There is a (left) module (... |
| lmodn0 47078 | Left modules exist. (Cont... |
| zlmodzxzequa 47079 | Example of an equation wit... |
| zlmodzxznm 47080 | Example of a linearly depe... |
| zlmodzxzldeplem 47081 | A and B are not equal. (C... |
| zlmodzxzequap 47082 | Example of an equation wit... |
| zlmodzxzldeplem1 47083 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 47084 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 47085 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 47086 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 47087 | { A , B } is a linearly de... |
| ldepsnlinclem1 47088 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 47089 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 47090 | The class of all (left) ve... |
| ldepsnlinc 47091 | The reverse implication of... |
| ldepslinc 47092 | For (left) vector spaces, ... |
| suppdm 47093 | If the range of a function... |
| eluz2cnn0n1 47094 | An integer greater than 1 ... |
| divge1b 47095 | The ratio of a real number... |
| divgt1b 47096 | The ratio of a real number... |
| ltsubaddb 47097 | Equivalence for the "less ... |
| ltsubsubb 47098 | Equivalence for the "less ... |
| ltsubadd2b 47099 | Equivalence for the "less ... |
| divsub1dir 47100 | Distribution of division o... |
| expnegico01 47101 | An integer greater than 1 ... |
| elfzolborelfzop1 47102 | An element of a half-open ... |
| pw2m1lepw2m1 47103 | 2 to the power of a positi... |
| zgtp1leeq 47104 | If an integer is between a... |
| flsubz 47105 | An integer can be moved in... |
| fldivmod 47106 | Expressing the floor of a ... |
| mod0mul 47107 | If an integer is 0 modulo ... |
| modn0mul 47108 | If an integer is not 0 mod... |
| m1modmmod 47109 | An integer decreased by 1 ... |
| difmodm1lt 47110 | The difference between an ... |
| nn0onn0ex 47111 | For each odd nonnegative i... |
| nn0enn0ex 47112 | For each even nonnegative ... |
| nnennex 47113 | For each even positive int... |
| nneop 47114 | A positive integer is even... |
| nneom 47115 | A positive integer is even... |
| nn0eo 47116 | A nonnegative integer is e... |
| nnpw2even 47117 | 2 to the power of a positi... |
| zefldiv2 47118 | The floor of an even integ... |
| zofldiv2 47119 | The floor of an odd intege... |
| nn0ofldiv2 47120 | The floor of an odd nonneg... |
| flnn0div2ge 47121 | The floor of a positive in... |
| flnn0ohalf 47122 | The floor of the half of a... |
| logcxp0 47123 | Logarithm of a complex pow... |
| regt1loggt0 47124 | The natural logarithm for ... |
| fdivval 47127 | The quotient of two functi... |
| fdivmpt 47128 | The quotient of two functi... |
| fdivmptf 47129 | The quotient of two functi... |
| refdivmptf 47130 | The quotient of two functi... |
| fdivpm 47131 | The quotient of two functi... |
| refdivpm 47132 | The quotient of two functi... |
| fdivmptfv 47133 | The function value of a qu... |
| refdivmptfv 47134 | The function value of a qu... |
| bigoval 47137 | Set of functions of order ... |
| elbigofrcl 47138 | Reverse closure of the "bi... |
| elbigo 47139 | Properties of a function o... |
| elbigo2 47140 | Properties of a function o... |
| elbigo2r 47141 | Sufficient condition for a... |
| elbigof 47142 | A function of order G(x) i... |
| elbigodm 47143 | The domain of a function o... |
| elbigoimp 47144 | The defining property of a... |
| elbigolo1 47145 | A function (into the posit... |
| rege1logbrege0 47146 | The general logarithm, wit... |
| rege1logbzge0 47147 | The general logarithm, wit... |
| fllogbd 47148 | A real number is between t... |
| relogbmulbexp 47149 | The logarithm of the produ... |
| relogbdivb 47150 | The logarithm of the quoti... |
| logbge0b 47151 | The logarithm of a number ... |
| logblt1b 47152 | The logarithm of a number ... |
| fldivexpfllog2 47153 | The floor of a positive re... |
| nnlog2ge0lt1 47154 | A positive integer is 1 if... |
| logbpw2m1 47155 | The floor of the binary lo... |
| fllog2 47156 | The floor of the binary lo... |
| blenval 47159 | The binary length of an in... |
| blen0 47160 | The binary length of 0. (... |
| blenn0 47161 | The binary length of a "nu... |
| blenre 47162 | The binary length of a pos... |
| blennn 47163 | The binary length of a pos... |
| blennnelnn 47164 | The binary length of a pos... |
| blennn0elnn 47165 | The binary length of a non... |
| blenpw2 47166 | The binary length of a pow... |
| blenpw2m1 47167 | The binary length of a pow... |
| nnpw2blen 47168 | A positive integer is betw... |
| nnpw2blenfzo 47169 | A positive integer is betw... |
| nnpw2blenfzo2 47170 | A positive integer is eith... |
| nnpw2pmod 47171 | Every positive integer can... |
| blen1 47172 | The binary length of 1. (... |
| blen2 47173 | The binary length of 2. (... |
| nnpw2p 47174 | Every positive integer can... |
| nnpw2pb 47175 | A number is a positive int... |
| blen1b 47176 | The binary length of a non... |
| blennnt2 47177 | The binary length of a pos... |
| nnolog2flm1 47178 | The floor of the binary lo... |
| blennn0em1 47179 | The binary length of the h... |
| blennngt2o2 47180 | The binary length of an od... |
| blengt1fldiv2p1 47181 | The binary length of an in... |
| blennn0e2 47182 | The binary length of an ev... |
| digfval 47185 | Operation to obtain the ` ... |
| digval 47186 | The ` K ` th digit of a no... |
| digvalnn0 47187 | The ` K ` th digit of a no... |
| nn0digval 47188 | The ` K ` th digit of a no... |
| dignn0fr 47189 | The digits of the fraction... |
| dignn0ldlem 47190 | Lemma for ~ dignnld . (Co... |
| dignnld 47191 | The leading digits of a po... |
| dig2nn0ld 47192 | The leading digits of a po... |
| dig2nn1st 47193 | The first (relevant) digit... |
| dig0 47194 | All digits of 0 are 0. (C... |
| digexp 47195 | The ` K ` th digit of a po... |
| dig1 47196 | All but one digits of 1 ar... |
| 0dig1 47197 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 47198 | The integers 0 and 1 corre... |
| dig2nn0 47199 | A digit of a nonnegative i... |
| 0dig2nn0e 47200 | The last bit of an even in... |
| 0dig2nn0o 47201 | The last bit of an odd int... |
| dig2bits 47202 | The ` K ` th digit of a no... |
| dignn0flhalflem1 47203 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 47204 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 47205 | The digits of the half of ... |
| dignn0flhalf 47206 | The digits of the rounded ... |
| nn0sumshdiglemA 47207 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 47208 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 47209 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 47210 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 47211 | A nonnegative integer can ... |
| nn0mulfsum 47212 | Trivial algorithm to calcu... |
| nn0mullong 47213 | Standard algorithm (also k... |
| naryfval 47216 | The set of the n-ary (endo... |
| naryfvalixp 47217 | The set of the n-ary (endo... |
| naryfvalel 47218 | An n-ary (endo)function on... |
| naryrcl 47219 | Reverse closure for n-ary ... |
| naryfvalelfv 47220 | The value of an n-ary (end... |
| naryfvalelwrdf 47221 | An n-ary (endo)function on... |
| 0aryfvalel 47222 | A nullary (endo)function o... |
| 0aryfvalelfv 47223 | The value of a nullary (en... |
| 1aryfvalel 47224 | A unary (endo)function on ... |
| fv1arycl 47225 | Closure of a unary (endo)f... |
| 1arympt1 47226 | A unary (endo)function in ... |
| 1arympt1fv 47227 | The value of a unary (endo... |
| 1arymaptfv 47228 | The value of the mapping o... |
| 1arymaptf 47229 | The mapping of unary (endo... |
| 1arymaptf1 47230 | The mapping of unary (endo... |
| 1arymaptfo 47231 | The mapping of unary (endo... |
| 1arymaptf1o 47232 | The mapping of unary (endo... |
| 1aryenef 47233 | The set of unary (endo)fun... |
| 1aryenefmnd 47234 | The set of unary (endo)fun... |
| 2aryfvalel 47235 | A binary (endo)function on... |
| fv2arycl 47236 | Closure of a binary (endo)... |
| 2arympt 47237 | A binary (endo)function in... |
| 2arymptfv 47238 | The value of a binary (end... |
| 2arymaptfv 47239 | The value of the mapping o... |
| 2arymaptf 47240 | The mapping of binary (end... |
| 2arymaptf1 47241 | The mapping of binary (end... |
| 2arymaptfo 47242 | The mapping of binary (end... |
| 2arymaptf1o 47243 | The mapping of binary (end... |
| 2aryenef 47244 | The set of binary (endo)fu... |
| itcoval 47249 | The value of the function ... |
| itcoval0 47250 | A function iterated zero t... |
| itcoval1 47251 | A function iterated once. ... |
| itcoval2 47252 | A function iterated twice.... |
| itcoval3 47253 | A function iterated three ... |
| itcoval0mpt 47254 | A mapping iterated zero ti... |
| itcovalsuc 47255 | The value of the function ... |
| itcovalsucov 47256 | The value of the function ... |
| itcovalendof 47257 | The n-th iterate of an end... |
| itcovalpclem1 47258 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 47259 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 47260 | The value of the function ... |
| itcovalt2lem2lem1 47261 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 47262 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 47263 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 47264 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 47265 | The value of the function ... |
| ackvalsuc1mpt 47266 | The Ackermann function at ... |
| ackvalsuc1 47267 | The Ackermann function at ... |
| ackval0 47268 | The Ackermann function at ... |
| ackval1 47269 | The Ackermann function at ... |
| ackval2 47270 | The Ackermann function at ... |
| ackval3 47271 | The Ackermann function at ... |
| ackendofnn0 47272 | The Ackermann function at ... |
| ackfnnn0 47273 | The Ackermann function at ... |
| ackval0val 47274 | The Ackermann function at ... |
| ackvalsuc0val 47275 | The Ackermann function at ... |
| ackvalsucsucval 47276 | The Ackermann function at ... |
| ackval0012 47277 | The Ackermann function at ... |
| ackval1012 47278 | The Ackermann function at ... |
| ackval2012 47279 | The Ackermann function at ... |
| ackval3012 47280 | The Ackermann function at ... |
| ackval40 47281 | The Ackermann function at ... |
| ackval41a 47282 | The Ackermann function at ... |
| ackval41 47283 | The Ackermann function at ... |
| ackval42 47284 | The Ackermann function at ... |
| ackval42a 47285 | The Ackermann function at ... |
| ackval50 47286 | The Ackermann function at ... |
| fv1prop 47287 | The function value of unor... |
| fv2prop 47288 | The function value of unor... |
| submuladdmuld 47289 | Transformation of a sum of... |
| affinecomb1 47290 | Combination of two real af... |
| affinecomb2 47291 | Combination of two real af... |
| affineid 47292 | Identity of an affine comb... |
| 1subrec1sub 47293 | Subtract the reciprocal of... |
| resum2sqcl 47294 | The sum of two squares of ... |
| resum2sqgt0 47295 | The sum of the square of a... |
| resum2sqrp 47296 | The sum of the square of a... |
| resum2sqorgt0 47297 | The sum of the square of t... |
| reorelicc 47298 | Membership in and outside ... |
| rrx2pxel 47299 | The x-coordinate of a poin... |
| rrx2pyel 47300 | The y-coordinate of a poin... |
| prelrrx2 47301 | An unordered pair of order... |
| prelrrx2b 47302 | An unordered pair of order... |
| rrx2pnecoorneor 47303 | If two different points ` ... |
| rrx2pnedifcoorneor 47304 | If two different points ` ... |
| rrx2pnedifcoorneorr 47305 | If two different points ` ... |
| rrx2xpref1o 47306 | There is a bijection betwe... |
| rrx2xpreen 47307 | The set of points in the t... |
| rrx2plord 47308 | The lexicographical orderi... |
| rrx2plord1 47309 | The lexicographical orderi... |
| rrx2plord2 47310 | The lexicographical orderi... |
| rrx2plordisom 47311 | The set of points in the t... |
| rrx2plordso 47312 | The lexicographical orderi... |
| ehl2eudisval0 47313 | The Euclidean distance of ... |
| ehl2eudis0lt 47314 | An upper bound of the Eucl... |
| lines 47319 | The lines passing through ... |
| line 47320 | The line passing through t... |
| rrxlines 47321 | Definition of lines passin... |
| rrxline 47322 | The line passing through t... |
| rrxlinesc 47323 | Definition of lines passin... |
| rrxlinec 47324 | The line passing through t... |
| eenglngeehlnmlem1 47325 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 47326 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 47327 | The line definition in the... |
| rrx2line 47328 | The line passing through t... |
| rrx2vlinest 47329 | The vertical line passing ... |
| rrx2linest 47330 | The line passing through t... |
| rrx2linesl 47331 | The line passing through t... |
| rrx2linest2 47332 | The line passing through t... |
| elrrx2linest2 47333 | The line passing through t... |
| spheres 47334 | The spheres for given cent... |
| sphere 47335 | A sphere with center ` X `... |
| rrxsphere 47336 | The sphere with center ` M... |
| 2sphere 47337 | The sphere with center ` M... |
| 2sphere0 47338 | The sphere around the orig... |
| line2ylem 47339 | Lemma for ~ line2y . This... |
| line2 47340 | Example for a line ` G ` p... |
| line2xlem 47341 | Lemma for ~ line2x . This... |
| line2x 47342 | Example for a horizontal l... |
| line2y 47343 | Example for a vertical lin... |
| itsclc0lem1 47344 | Lemma for theorems about i... |
| itsclc0lem2 47345 | Lemma for theorems about i... |
| itsclc0lem3 47346 | Lemma for theorems about i... |
| itscnhlc0yqe 47347 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 47348 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 47349 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 47350 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 47351 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 47352 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 47353 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 47354 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 47355 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 47356 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 47357 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 47358 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 47359 | The intersection points of... |
| itsclc0b 47360 | The intersection points of... |
| itsclinecirc0 47361 | The intersection points of... |
| itsclinecirc0b 47362 | The intersection points of... |
| itsclinecirc0in 47363 | The intersection points of... |
| itsclquadb 47364 | Quadratic equation for the... |
| itsclquadeu 47365 | Quadratic equation for the... |
| 2itscplem1 47366 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 47367 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 47368 | Lemma D for ~ 2itscp . (C... |
| 2itscp 47369 | A condition for a quadrati... |
| itscnhlinecirc02plem1 47370 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 47371 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 47372 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 47373 | Intersection of a nonhoriz... |
| inlinecirc02plem 47374 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 47375 | Intersection of a line wit... |
| inlinecirc02preu 47376 | Intersection of a line wit... |
| pm4.71da 47377 | Deduction converting a bic... |
| logic1 47378 | Distribution of implicatio... |
| logic1a 47379 | Variant of ~ logic1 . (Co... |
| logic2 47380 | Variant of ~ logic1 . (Co... |
| pm5.32dav 47381 | Distribution of implicatio... |
| pm5.32dra 47382 | Reverse distribution of im... |
| exp12bd 47383 | The import-export theorem ... |
| mpbiran3d 47384 | Equivalence with a conjunc... |
| mpbiran4d 47385 | Equivalence with a conjunc... |
| dtrucor3 47386 | An example of how ~ ax-5 w... |
| ralbidb 47387 | Formula-building rule for ... |
| ralbidc 47388 | Formula-building rule for ... |
| r19.41dv 47389 | A complex deduction form o... |
| rspceb2dv 47390 | Restricted existential spe... |
| rmotru 47391 | Two ways of expressing "at... |
| reutru 47392 | Two ways of expressing "ex... |
| reutruALT 47393 | Alternate proof for ~ reut... |
| ssdisjd 47394 | Subset preserves disjointn... |
| ssdisjdr 47395 | Subset preserves disjointn... |
| disjdifb 47396 | Relative complement is ant... |
| predisj 47397 | Preimages of disjoint sets... |
| vsn 47398 | The singleton of the unive... |
| mosn 47399 | "At most one" element in a... |
| mo0 47400 | "At most one" element in a... |
| mosssn 47401 | "At most one" element in a... |
| mo0sn 47402 | Two ways of expressing "at... |
| mosssn2 47403 | Two ways of expressing "at... |
| unilbss 47404 | Superclass of the greatest... |
| inpw 47405 | Two ways of expressing a c... |
| mof0 47406 | There is at most one funct... |
| mof02 47407 | A variant of ~ mof0 . (Co... |
| mof0ALT 47408 | Alternate proof for ~ mof0... |
| eufsnlem 47409 | There is exactly one funct... |
| eufsn 47410 | There is exactly one funct... |
| eufsn2 47411 | There is exactly one funct... |
| mofsn 47412 | There is at most one funct... |
| mofsn2 47413 | There is at most one funct... |
| mofsssn 47414 | There is at most one funct... |
| mofmo 47415 | There is at most one funct... |
| mofeu 47416 | The uniqueness of a functi... |
| elfvne0 47417 | If a function value has a ... |
| fdomne0 47418 | A function with non-empty ... |
| f1sn2g 47419 | A function that maps a sin... |
| f102g 47420 | A function that maps the e... |
| f1mo 47421 | A function that maps a set... |
| f002 47422 | A function with an empty c... |
| map0cor 47423 | A function exists iff an e... |
| fvconstr 47424 | Two ways of expressing ` A... |
| fvconstrn0 47425 | Two ways of expressing ` A... |
| fvconstr2 47426 | Two ways of expressing ` A... |
| fvconst0ci 47427 | A constant function's valu... |
| fvconstdomi 47428 | A constant function's valu... |
| f1omo 47429 | There is at most one eleme... |
| f1omoALT 47430 | There is at most one eleme... |
| iccin 47431 | Intersection of two closed... |
| iccdisj2 47432 | If the upper bound of one ... |
| iccdisj 47433 | If the upper bound of one ... |
| mreuniss 47434 | The union of a collection ... |
| clduni 47435 | The union of closed sets i... |
| opncldeqv 47436 | Conditions on open sets ar... |
| opndisj 47437 | Two ways of saying that tw... |
| clddisj 47438 | Two ways of saying that tw... |
| neircl 47439 | Reverse closure of the nei... |
| opnneilem 47440 | Lemma factoring out common... |
| opnneir 47441 | If something is true for a... |
| opnneirv 47442 | A variant of ~ opnneir wit... |
| opnneilv 47443 | The converse of ~ opnneir ... |
| opnneil 47444 | A variant of ~ opnneilv . ... |
| opnneieqv 47445 | The equivalence between ne... |
| opnneieqvv 47446 | The equivalence between ne... |
| restcls2lem 47447 | A closed set in a subspace... |
| restcls2 47448 | A closed set in a subspace... |
| restclsseplem 47449 | Lemma for ~ restclssep . ... |
| restclssep 47450 | Two disjoint closed sets i... |
| cnneiima 47451 | Given a continuous functio... |
| iooii 47452 | Open intervals are open se... |
| icccldii 47453 | Closed intervals are close... |
| i0oii 47454 | ` ( 0 [,) A ) ` is open in... |
| io1ii 47455 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 47456 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 47457 | Open neighborhood and neig... |
| sepnsepo 47458 | Open neighborhood and neig... |
| sepdisj 47459 | Separated sets are disjoin... |
| seposep 47460 | If two sets are separated ... |
| sepcsepo 47461 | If two sets are separated ... |
| sepfsepc 47462 | If two sets are separated ... |
| seppsepf 47463 | If two sets are precisely ... |
| seppcld 47464 | If two sets are precisely ... |
| isnrm4 47465 | A topological space is nor... |
| dfnrm2 47466 | A topological space is nor... |
| dfnrm3 47467 | A topological space is nor... |
| iscnrm3lem1 47468 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 47469 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem3 47470 | Lemma for ~ iscnrm3lem4 . ... |
| iscnrm3lem4 47471 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 47472 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 47473 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 47474 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 47475 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 47476 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 47477 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 47478 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 47479 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 47480 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 47481 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 47482 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 47483 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 47484 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 47485 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 47486 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 47487 | A completely normal topolo... |
| iscnrm3v 47488 | A topology is completely n... |
| iscnrm4 47489 | A completely normal topolo... |
| isprsd 47490 | Property of being a preord... |
| lubeldm2 47491 | Member of the domain of th... |
| glbeldm2 47492 | Member of the domain of th... |
| lubeldm2d 47493 | Member of the domain of th... |
| glbeldm2d 47494 | Member of the domain of th... |
| lubsscl 47495 | If a subset of ` S ` conta... |
| glbsscl 47496 | If a subset of ` S ` conta... |
| lubprlem 47497 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 47498 | The set of two comparable ... |
| lubpr 47499 | The LUB of the set of two ... |
| glbprlem 47500 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 47501 | The set of two comparable ... |
| glbpr 47502 | The GLB of the set of two ... |
| joindm2 47503 | The join of any two elemen... |
| joindm3 47504 | The join of any two elemen... |
| meetdm2 47505 | The meet of any two elemen... |
| meetdm3 47506 | The meet of any two elemen... |
| posjidm 47507 | Poset join is idempotent. ... |
| posmidm 47508 | Poset meet is idempotent. ... |
| toslat 47509 | A toset is a lattice. (Co... |
| isclatd 47510 | The predicate "is a comple... |
| intubeu 47511 | Existential uniqueness of ... |
| unilbeu 47512 | Existential uniqueness of ... |
| ipolublem 47513 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 47514 | The domain of the LUB of t... |
| ipolub 47515 | The LUB of the inclusion p... |
| ipoglblem 47516 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 47517 | The domain of the GLB of t... |
| ipoglb 47518 | The GLB of the inclusion p... |
| ipolub0 47519 | The LUB of the empty set i... |
| ipolub00 47520 | The LUB of the empty set i... |
| ipoglb0 47521 | The GLB of the empty set i... |
| mrelatlubALT 47522 | Least upper bounds in a Mo... |
| mrelatglbALT 47523 | Greatest lower bounds in a... |
| mreclat 47524 | A Moore space is a complet... |
| topclat 47525 | A topology is a complete l... |
| toplatglb0 47526 | The empty intersection in ... |
| toplatlub 47527 | Least upper bounds in a to... |
| toplatglb 47528 | Greatest lower bounds in a... |
| toplatjoin 47529 | Joins in a topology are re... |
| toplatmeet 47530 | Meets in a topology are re... |
| topdlat 47531 | A topology is a distributi... |
| catprslem 47532 | Lemma for ~ catprs . (Con... |
| catprs 47533 | A preorder can be extracte... |
| catprs2 47534 | A category equipped with t... |
| catprsc 47535 | A construction of the preo... |
| catprsc2 47536 | An alternate construction ... |
| endmndlem 47537 | A diagonal hom-set in a ca... |
| idmon 47538 | An identity arrow, or an i... |
| idepi 47539 | An identity arrow, or an i... |
| funcf2lem 47540 | A utility theorem for prov... |
| isthinc 47543 | The predicate "is a thin c... |
| isthinc2 47544 | A thin category is a categ... |
| isthinc3 47545 | A thin category is a categ... |
| thincc 47546 | A thin category is a categ... |
| thinccd 47547 | A thin category is a categ... |
| thincssc 47548 | A thin category is a categ... |
| isthincd2lem1 47549 | Lemma for ~ isthincd2 and ... |
| thincmo2 47550 | Morphisms in the same hom-... |
| thincmo 47551 | There is at most one morph... |
| thincmoALT 47552 | Alternate proof for ~ thin... |
| thincmod 47553 | At most one morphism in ea... |
| thincn0eu 47554 | In a thin category, a hom-... |
| thincid 47555 | In a thin category, a morp... |
| thincmon 47556 | In a thin category, all mo... |
| thincepi 47557 | In a thin category, all mo... |
| isthincd2lem2 47558 | Lemma for ~ isthincd2 . (... |
| isthincd 47559 | The predicate "is a thin c... |
| isthincd2 47560 | The predicate " ` C ` is a... |
| oppcthin 47561 | The opposite category of a... |
| subthinc 47562 | A subcategory of a thin ca... |
| functhinclem1 47563 | Lemma for ~ functhinc . G... |
| functhinclem2 47564 | Lemma for ~ functhinc . (... |
| functhinclem3 47565 | Lemma for ~ functhinc . T... |
| functhinclem4 47566 | Lemma for ~ functhinc . O... |
| functhinc 47567 | A functor to a thin catego... |
| fullthinc 47568 | A functor to a thin catego... |
| fullthinc2 47569 | A full functor to a thin c... |
| thincfth 47570 | A functor from a thin cate... |
| thincciso 47571 | Two thin categories are is... |
| 0thincg 47572 | Any structure with an empt... |
| 0thinc 47573 | The empty category (see ~ ... |
| indthinc 47574 | An indiscrete category in ... |
| indthincALT 47575 | An alternate proof for ~ i... |
| prsthinc 47576 | Preordered sets as categor... |
| setcthin 47577 | A category of sets all of ... |
| setc2othin 47578 | The category ` ( SetCat ``... |
| thincsect 47579 | In a thin category, one mo... |
| thincsect2 47580 | In a thin category, ` F ` ... |
| thincinv 47581 | In a thin category, ` F ` ... |
| thinciso 47582 | In a thin category, ` F : ... |
| thinccic 47583 | In a thin category, two ob... |
| prstcval 47586 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 47587 | Lemma for ~ prstcnid and ~... |
| prstcnid 47588 | Components other than ` Ho... |
| prstcbas 47589 | The base set is unchanged.... |
| prstcleval 47590 | Value of the less-than-or-... |
| prstclevalOLD 47591 | Obsolete proof of ~ prstcl... |
| prstcle 47592 | Value of the less-than-or-... |
| prstcocval 47593 | Orthocomplementation is un... |
| prstcocvalOLD 47594 | Obsolete proof of ~ prstco... |
| prstcoc 47595 | Orthocomplementation is un... |
| prstchomval 47596 | Hom-sets of the constructe... |
| prstcprs 47597 | The category is a preorder... |
| prstcthin 47598 | The preordered set is equi... |
| prstchom 47599 | Hom-sets of the constructe... |
| prstchom2 47600 | Hom-sets of the constructe... |
| prstchom2ALT 47601 | Hom-sets of the constructe... |
| postcpos 47602 | The converted category is ... |
| postcposALT 47603 | Alternate proof for ~ post... |
| postc 47604 | The converted category is ... |
| mndtcval 47607 | Value of the category buil... |
| mndtcbasval 47608 | The base set of the catego... |
| mndtcbas 47609 | The category built from a ... |
| mndtcob 47610 | Lemma for ~ mndtchom and ~... |
| mndtcbas2 47611 | Two objects in a category ... |
| mndtchom 47612 | The only hom-set of the ca... |
| mndtcco 47613 | The composition of the cat... |
| mndtcco2 47614 | The composition of the cat... |
| mndtccatid 47615 | Lemma for ~ mndtccat and ~... |
| mndtccat 47616 | The function value is a ca... |
| mndtcid 47617 | The identity morphism, or ... |
| grptcmon 47618 | All morphisms in a categor... |
| grptcepi 47619 | All morphisms in a categor... |
| nfintd 47620 | Bound-variable hypothesis ... |
| nfiund 47621 | Bound-variable hypothesis ... |
| nfiundg 47622 | Bound-variable hypothesis ... |
| iunord 47623 | The indexed union of a col... |
| iunordi 47624 | The indexed union of a col... |
| spd 47625 | Specialization deduction, ... |
| spcdvw 47626 | A version of ~ spcdv where... |
| tfis2d 47627 | Transfinite Induction Sche... |
| bnd2d 47628 | Deduction form of ~ bnd2 .... |
| dffun3f 47629 | Alternate definition of fu... |
| setrecseq 47632 | Equality theorem for set r... |
| nfsetrecs 47633 | Bound-variable hypothesis ... |
| setrec1lem1 47634 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 47635 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 47636 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 47637 | Lemma for ~ setrec1 . If ... |
| setrec1 47638 | This is the first of two f... |
| setrec2fun 47639 | This is the second of two ... |
| setrec2lem1 47640 | Lemma for ~ setrec2 . The... |
| setrec2lem2 47641 | Lemma for ~ setrec2 . The... |
| setrec2 47642 | This is the second of two ... |
| setrec2v 47643 | Version of ~ setrec2 with ... |
| setrec2mpt 47644 | Version of ~ setrec2 where... |
| setis 47645 | Version of ~ setrec2 expre... |
| elsetrecslem 47646 | Lemma for ~ elsetrecs . A... |
| elsetrecs 47647 | A set ` A ` is an element ... |
| setrecsss 47648 | The ` setrecs ` operator r... |
| setrecsres 47649 | A recursively generated cl... |
| vsetrec 47650 | Construct ` _V ` using set... |
| 0setrec 47651 | If a function sends the em... |
| onsetreclem1 47652 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 47653 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 47654 | Lemma for ~ onsetrec . (C... |
| onsetrec 47655 | Construct ` On ` using set... |
| elpglem1 47658 | Lemma for ~ elpg . (Contr... |
| elpglem2 47659 | Lemma for ~ elpg . (Contr... |
| elpglem3 47660 | Lemma for ~ elpg . (Contr... |
| elpg 47661 | Membership in the class of... |
| pgindlem 47662 | Lemma for ~ pgind . (Cont... |
| pgindnf 47663 | Version of ~ pgind with ex... |
| pgind 47664 | Induction on partizan game... |
| sbidd 47665 | An identity theorem for su... |
| sbidd-misc 47666 | An identity theorem for su... |
| gte-lte 47671 | Simple relationship betwee... |
| gt-lt 47672 | Simple relationship betwee... |
| gte-lteh 47673 | Relationship between ` <_ ... |
| gt-lth 47674 | Relationship between ` < `... |
| ex-gt 47675 | Simple example of ` > ` , ... |
| ex-gte 47676 | Simple example of ` >_ ` ,... |
| sinhval-named 47683 | Value of the named sinh fu... |
| coshval-named 47684 | Value of the named cosh fu... |
| tanhval-named 47685 | Value of the named tanh fu... |
| sinh-conventional 47686 | Conventional definition of... |
| sinhpcosh 47687 | Prove that ` ( sinh `` A )... |
| secval 47694 | Value of the secant functi... |
| cscval 47695 | Value of the cosecant func... |
| cotval 47696 | Value of the cotangent fun... |
| seccl 47697 | The closure of the secant ... |
| csccl 47698 | The closure of the cosecan... |
| cotcl 47699 | The closure of the cotange... |
| reseccl 47700 | The closure of the secant ... |
| recsccl 47701 | The closure of the cosecan... |
| recotcl 47702 | The closure of the cotange... |
| recsec 47703 | The reciprocal of secant i... |
| reccsc 47704 | The reciprocal of cosecant... |
| reccot 47705 | The reciprocal of cotangen... |
| rectan 47706 | The reciprocal of tangent ... |
| sec0 47707 | The value of the secant fu... |
| onetansqsecsq 47708 | Prove the tangent squared ... |
| cotsqcscsq 47709 | Prove the tangent squared ... |
| ifnmfalse 47710 | If A is not a member of B,... |
| logb2aval 47711 | Define the value of the ` ... |
| comraddi 47718 | Commute RHS addition. See... |
| mvlraddi 47719 | Move the right term in a s... |
| mvrladdi 47720 | Move the left term in a su... |
| assraddsubi 47721 | Associate RHS addition-sub... |
| joinlmuladdmuli 47722 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 47723 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 47724 | Join AB-CB into (A-C) on L... |
| mvlrmuld 47725 | Move the right term in a p... |
| mvlrmuli 47726 | Move the right term in a p... |
| i2linesi 47727 | Solve for the intersection... |
| i2linesd 47728 | Solve for the intersection... |
| alimp-surprise 47729 | Demonstrate that when usin... |
| alimp-no-surprise 47730 | There is no "surprise" in ... |
| empty-surprise 47731 | Demonstrate that when usin... |
| empty-surprise2 47732 | "Prove" that false is true... |
| eximp-surprise 47733 | Show what implication insi... |
| eximp-surprise2 47734 | Show that "there exists" w... |
| alsconv 47739 | There is an equivalence be... |
| alsi1d 47740 | Deduction rule: Given "al... |
| alsi2d 47741 | Deduction rule: Given "al... |
| alsc1d 47742 | Deduction rule: Given "al... |
| alsc2d 47743 | Deduction rule: Given "al... |
| alscn0d 47744 | Deduction rule: Given "al... |
| alsi-no-surprise 47745 | Demonstrate that there is ... |
| 5m4e1 47746 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 47747 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 47748 | Resolution rule. This is ... |
| testable 47749 | In classical logic all wff... |
| aacllem 47750 | Lemma for other theorems a... |
| amgmwlem 47751 | Weighted version of ~ amgm... |
| amgmlemALT 47752 | Alternate proof of ~ amgml... |
| amgmw2d 47753 | Weighted arithmetic-geomet... |
| young2d 47754 | Young's inequality for ` n... |
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