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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... |
| imbitrrdi 251 | A mixed syllogism inferenc... |
| biimtrdi 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... |
| bibi2i 337 | Inference adding a bicondi... |
| bibi1i 338 | Inference adding a bicondi... |
| bibi12i 339 | The equivalence of two equ... |
| imbi2d 340 | Deduction adding an antece... |
| imbi1d 341 | Deduction adding a consequ... |
| bibi2d 342 | Deduction adding a bicondi... |
| bibi1d 343 | Deduction adding a bicondi... |
| imbi12d 344 | Deduction joining two equi... |
| bibi12d 345 | Deduction joining two equi... |
| imbi12 346 | Closed form of ~ imbi12i .... |
| imbi1 347 | Theorem *4.84 of [Whitehea... |
| imbi2 348 | Theorem *4.85 of [Whitehea... |
| imbi1i 349 | Introduce a consequent to ... |
| imbi12i 350 | Join two logical equivalen... |
| bibi1 351 | Theorem *4.86 of [Whitehea... |
| bitr3 352 | Closed nested implication ... |
| con2bi 353 | Contraposition. Theorem *... |
| con2bid 354 | A contraposition deduction... |
| con1bid 355 | A contraposition deduction... |
| con1bii 356 | A contraposition inference... |
| con2bii 357 | A contraposition inference... |
| con1b 358 | Contraposition. Bidirecti... |
| con2b 359 | Contraposition. Bidirecti... |
| biimt 360 | A wff is equivalent to its... |
| pm5.5 361 | Theorem *5.5 of [Whitehead... |
| a1bi 362 | Inference introducing a th... |
| mt2bi 363 | A false consequent falsifi... |
| mtt 364 | Modus-tollens-like theorem... |
| imnot 365 | If a proposition is false,... |
| pm5.501 366 | Theorem *5.501 of [Whitehe... |
| ibib 367 | Implication in terms of im... |
| ibibr 368 | Implication in terms of im... |
| tbt 369 | A wff is equivalent to its... |
| nbn2 370 | The negation of a wff is e... |
| bibif 371 | Transfer negation via an e... |
| nbn 372 | The negation of a wff is e... |
| nbn3 373 | Transfer falsehood via equ... |
| pm5.21im 374 | Two propositions are equiv... |
| 2false 375 | Two falsehoods are equival... |
| 2falsed 376 | Two falsehoods are equival... |
| pm5.21ni 377 | Two propositions implying ... |
| pm5.21nii 378 | Eliminate an antecedent im... |
| pm5.21ndd 379 | Eliminate an antecedent im... |
| bija 380 | Combine antecedents into a... |
| pm5.18 381 | Theorem *5.18 of [Whitehea... |
| xor3 382 | Two ways to express "exclu... |
| nbbn 383 | Move negation outside of b... |
| biass 384 | Associative law for the bi... |
| biluk 385 | Lukasiewicz's shortest axi... |
| pm5.19 386 | Theorem *5.19 of [Whitehea... |
| bi2.04 387 | Logical equivalence of com... |
| pm5.4 388 | Antecedent absorption impl... |
| imdi 389 | Distributive law for impli... |
| pm5.41 390 | Theorem *5.41 of [Whitehea... |
| imbibi 391 | The antecedent of one side... |
| pm4.8 392 | Theorem *4.8 of [Whitehead... |
| pm4.81 393 | A formula is equivalent to... |
| imim21b 394 | Simplify an implication be... |
| pm4.63 397 | Theorem *4.63 of [Whitehea... |
| pm4.67 398 | Theorem *4.67 of [Whitehea... |
| imnan 399 | Express an implication in ... |
| imnani 400 | Infer an implication from ... |
| iman 401 | Implication in terms of co... |
| pm3.24 402 | Law of noncontradiction. ... |
| annim 403 | Express a conjunction in t... |
| pm4.61 404 | Theorem *4.61 of [Whitehea... |
| pm4.65 405 | Theorem *4.65 of [Whitehea... |
| imp 406 | Importation inference. (C... |
| impcom 407 | Importation inference with... |
| con3dimp 408 | Variant of ~ con3d with im... |
| mpnanrd 409 | Eliminate the right side o... |
| impd 410 | Importation deduction. (C... |
| impcomd 411 | Importation deduction with... |
| ex 412 | Exportation inference. (T... |
| expcom 413 | Exportation inference with... |
| expdcom 414 | Commuted form of ~ expd . ... |
| expd 415 | Exportation deduction. (C... |
| expcomd 416 | Deduction form of ~ expcom... |
| imp31 417 | An importation inference. ... |
| imp32 418 | An importation inference. ... |
| exp31 419 | An exportation inference. ... |
| exp32 420 | An exportation inference. ... |
| imp4b 421 | An importation inference. ... |
| imp4a 422 | An importation inference. ... |
| imp4c 423 | An importation inference. ... |
| imp4d 424 | An importation inference. ... |
| imp41 425 | An importation inference. ... |
| imp42 426 | An importation inference. ... |
| imp43 427 | An importation inference. ... |
| imp44 428 | An importation inference. ... |
| imp45 429 | An importation inference. ... |
| exp4b 430 | An exportation inference. ... |
| exp4a 431 | An exportation inference. ... |
| exp4c 432 | An exportation inference. ... |
| exp4d 433 | An exportation inference. ... |
| exp41 434 | An exportation inference. ... |
| exp42 435 | An exportation inference. ... |
| exp43 436 | An exportation inference. ... |
| exp44 437 | An exportation inference. ... |
| exp45 438 | An exportation inference. ... |
| imp5d 439 | An importation inference. ... |
| imp5a 440 | An importation inference. ... |
| imp5g 441 | An importation inference. ... |
| imp55 442 | An importation inference. ... |
| imp511 443 | An importation inference. ... |
| exp5c 444 | An exportation inference. ... |
| exp5j 445 | An exportation inference. ... |
| exp5l 446 | An exportation inference. ... |
| exp53 447 | An exportation inference. ... |
| pm3.3 448 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 449 | Theorem *3.31 (Imp) of [Wh... |
| impexp 450 | Import-export theorem. Pa... |
| impancom 451 | Mixed importation/commutat... |
| expdimp 452 | A deduction version of exp... |
| expimpd 453 | Exportation followed by a ... |
| impr 454 | Import a wff into a right ... |
| impl 455 | Export a wff from a left c... |
| expr 456 | Export a wff from a right ... |
| expl 457 | Export a wff from a left c... |
| ancoms 458 | Inference commuting conjun... |
| pm3.22 459 | Theorem *3.22 of [Whitehea... |
| ancom 460 | Commutative law for conjun... |
| ancomd 461 | Commutation of conjuncts i... |
| biancomi 462 | Commuting conjunction in a... |
| biancomd 463 | Commuting conjunction in a... |
| ancomst 464 | Closed form of ~ ancoms . ... |
| ancomsd 465 | Deduction commuting conjun... |
| anasss 466 | Associative law for conjun... |
| anassrs 467 | Associative law for conjun... |
| anass 468 | Associative law for conjun... |
| pm3.2 469 | Join antecedents with conj... |
| pm3.2i 470 | Infer conjunction of premi... |
| pm3.21 471 | Join antecedents with conj... |
| pm3.43i 472 | Nested conjunction of ante... |
| pm3.43 473 | Theorem *3.43 (Comp) of [W... |
| dfbi2 474 | A theorem similar to the s... |
| dfbi 475 | Definition ~ df-bi rewritt... |
| biimpa 476 | Importation inference from... |
| biimpar 477 | Importation inference from... |
| biimpac 478 | Importation inference from... |
| biimparc 479 | Importation inference from... |
| adantr 480 | Inference adding a conjunc... |
| adantl 481 | Inference adding a conjunc... |
| simpl 482 | Elimination of a conjunct.... |
| simpli 483 | Inference eliminating a co... |
| simpr 484 | Elimination of a conjunct.... |
| simpri 485 | Inference eliminating a co... |
| intnan 486 | Introduction of conjunct i... |
| intnanr 487 | Introduction of conjunct i... |
| intnand 488 | Introduction of conjunct i... |
| intnanrd 489 | Introduction of conjunct i... |
| adantld 490 | Deduction adding a conjunc... |
| adantrd 491 | Deduction adding a conjunc... |
| pm3.41 492 | Theorem *3.41 of [Whitehea... |
| pm3.42 493 | Theorem *3.42 of [Whitehea... |
| simpld 494 | Deduction eliminating a co... |
| simprd 495 | Deduction eliminating a co... |
| simprbi 496 | Deduction eliminating a co... |
| simplbi 497 | Deduction eliminating a co... |
| simprbda 498 | Deduction eliminating a co... |
| simplbda 499 | Deduction eliminating a co... |
| simplbi2 500 | Deduction eliminating a co... |
| simplbi2comt 501 | Closed form of ~ simplbi2c... |
| simplbi2com 502 | A deduction eliminating a ... |
| simpl2im 503 | Implication from an elimin... |
| simplbiim 504 | Implication from an elimin... |
| impel 505 | An inference for implicati... |
| mpan9 506 | Modus ponens conjoining di... |
| sylan9 507 | Nested syllogism inference... |
| sylan9r 508 | Nested syllogism inference... |
| sylan9bb 509 | Nested syllogism inference... |
| sylan9bbr 510 | Nested syllogism inference... |
| jca 511 | Deduce conjunction of the ... |
| jcad 512 | Deduction conjoining the c... |
| jca2 513 | Inference conjoining the c... |
| jca31 514 | Join three consequents. (... |
| jca32 515 | Join three consequents. (... |
| jcai 516 | Deduction replacing implic... |
| jcab 517 | Distributive law for impli... |
| pm4.76 518 | Theorem *4.76 of [Whitehea... |
| jctil 519 | Inference conjoining a the... |
| jctir 520 | Inference conjoining a the... |
| jccir 521 | Inference conjoining a con... |
| jccil 522 | Inference conjoining a con... |
| jctl 523 | Inference conjoining a the... |
| jctr 524 | Inference conjoining a the... |
| jctild 525 | Deduction conjoining a the... |
| jctird 526 | Deduction conjoining a the... |
| iba 527 | Introduction of antecedent... |
| ibar 528 | Introduction of antecedent... |
| biantru 529 | A wff is equivalent to its... |
| biantrur 530 | A wff is equivalent to its... |
| biantrud 531 | A wff is equivalent to its... |
| biantrurd 532 | A wff is equivalent to its... |
| bianfi 533 | A wff conjoined with false... |
| bianfd 534 | A wff conjoined with false... |
| baib 535 | Move conjunction outside o... |
| baibr 536 | Move conjunction outside o... |
| rbaibr 537 | Move conjunction outside o... |
| rbaib 538 | Move conjunction outside o... |
| baibd 539 | Move conjunction outside o... |
| rbaibd 540 | Move conjunction outside o... |
| bianabs 541 | Absorb a hypothesis into t... |
| pm5.44 542 | Theorem *5.44 of [Whitehea... |
| pm5.42 543 | Theorem *5.42 of [Whitehea... |
| ancl 544 | Conjoin antecedent to left... |
| anclb 545 | Conjoin antecedent to left... |
| ancr 546 | Conjoin antecedent to righ... |
| ancrb 547 | Conjoin antecedent to righ... |
| ancli 548 | Deduction conjoining antec... |
| ancri 549 | Deduction conjoining antec... |
| ancld 550 | Deduction conjoining antec... |
| ancrd 551 | Deduction conjoining antec... |
| impac 552 | Importation with conjuncti... |
| anc2l 553 | Conjoin antecedent to left... |
| anc2r 554 | Conjoin antecedent to righ... |
| anc2li 555 | Deduction conjoining antec... |
| anc2ri 556 | Deduction conjoining antec... |
| pm4.71 557 | Implication in terms of bi... |
| pm4.71r 558 | Implication in terms of bi... |
| pm4.71i 559 | Inference converting an im... |
| pm4.71ri 560 | Inference converting an im... |
| pm4.71d 561 | Deduction converting an im... |
| pm4.71rd 562 | Deduction converting an im... |
| pm4.24 563 | Theorem *4.24 of [Whitehea... |
| anidm 564 | Idempotent law for conjunc... |
| anidmdbi 565 | Conjunction idempotence wi... |
| anidms 566 | Inference from idempotent ... |
| imdistan 567 | Distribution of implicatio... |
| imdistani 568 | Distribution of implicatio... |
| imdistanri 569 | Distribution of implicatio... |
| imdistand 570 | Distribution of implicatio... |
| imdistanda 571 | Distribution of implicatio... |
| pm5.3 572 | Theorem *5.3 of [Whitehead... |
| pm5.32 573 | Distribution of implicatio... |
| pm5.32i 574 | Distribution of implicatio... |
| pm5.32ri 575 | Distribution of implicatio... |
| pm5.32d 576 | Distribution of implicatio... |
| pm5.32rd 577 | Distribution of implicatio... |
| pm5.32da 578 | Distribution of implicatio... |
| sylan 579 | A syllogism inference. (C... |
| sylanb 580 | A syllogism inference. (C... |
| sylanbr 581 | A syllogism inference. (C... |
| sylanbrc 582 | Syllogism inference. (Con... |
| syl2anc 583 | Syllogism inference combin... |
| syl2anc2 584 | Double syllogism inference... |
| sylancl 585 | Syllogism inference combin... |
| sylancr 586 | Syllogism inference combin... |
| sylancom 587 | Syllogism inference with c... |
| sylanblc 588 | Syllogism inference combin... |
| sylanblrc 589 | Syllogism inference combin... |
| syldan 590 | A syllogism deduction with... |
| sylbida 591 | A syllogism deduction. (C... |
| sylan2 592 | A syllogism inference. (C... |
| sylan2b 593 | A syllogism inference. (C... |
| sylan2br 594 | A syllogism inference. (C... |
| syl2an 595 | A double syllogism inferen... |
| syl2anr 596 | A double syllogism inferen... |
| syl2anb 597 | A double syllogism inferen... |
| syl2anbr 598 | A double syllogism inferen... |
| sylancb 599 | A syllogism inference comb... |
| sylancbr 600 | A syllogism inference comb... |
| syldanl 601 | A syllogism deduction with... |
| syland 602 | A syllogism deduction. (C... |
| sylani 603 | A syllogism inference. (C... |
| sylan2d 604 | A syllogism deduction. (C... |
| sylan2i 605 | A syllogism inference. (C... |
| syl2ani 606 | A syllogism inference. (C... |
| syl2and 607 | A syllogism deduction. (C... |
| anim12d 608 | Conjoin antecedents and co... |
| anim12d1 609 | Variant of ~ anim12d where... |
| anim1d 610 | Add a conjunct to right of... |
| anim2d 611 | Add a conjunct to left of ... |
| anim12i 612 | Conjoin antecedents and co... |
| anim12ci 613 | Variant of ~ anim12i with ... |
| anim1i 614 | Introduce conjunct to both... |
| anim1ci 615 | Introduce conjunct to both... |
| anim2i 616 | Introduce conjunct to both... |
| anim12ii 617 | Conjoin antecedents and co... |
| anim12dan 618 | Conjoin antecedents and co... |
| im2anan9 619 | Deduction joining nested i... |
| im2anan9r 620 | Deduction joining nested i... |
| pm3.45 621 | Theorem *3.45 (Fact) of [W... |
| anbi2i 622 | Introduce a left conjunct ... |
| anbi1i 623 | Introduce a right conjunct... |
| anbi2ci 624 | Variant of ~ anbi2i with c... |
| anbi1ci 625 | Variant of ~ anbi1i with c... |
| anbi12i 626 | Conjoin both sides of two ... |
| anbi12ci 627 | Variant of ~ anbi12i with ... |
| anbi2d 628 | Deduction adding a left co... |
| anbi1d 629 | Deduction adding a right c... |
| anbi12d 630 | Deduction joining two equi... |
| anbi1 631 | Introduce a right conjunct... |
| anbi2 632 | Introduce a left conjunct ... |
| anbi1cd 633 | Introduce a proposition as... |
| bianbi 634 | Exchanging conjunction in ... |
| an2anr 635 | Double commutation in conj... |
| pm4.38 636 | Theorem *4.38 of [Whitehea... |
| bi2anan9 637 | Deduction joining two equi... |
| bi2anan9r 638 | Deduction joining two equi... |
| bi2bian9 639 | Deduction joining two bico... |
| anbiim 640 | Adding biconditional when ... |
| 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... |
| orcnd 877 | A lemma for Conjunctive No... |
| mtord 878 | A modus tollens deduction ... |
| pm3.2ni 879 | Infer negated disjunction ... |
| pm2.45 880 | Theorem *2.45 of [Whitehea... |
| pm2.46 881 | Theorem *2.46 of [Whitehea... |
| pm2.47 882 | Theorem *2.47 of [Whitehea... |
| pm2.48 883 | Theorem *2.48 of [Whitehea... |
| pm2.49 884 | Theorem *2.49 of [Whitehea... |
| norbi 885 | If neither of two proposit... |
| nbior 886 | If two propositions are no... |
| orel1 887 | Elimination of disjunction... |
| pm2.25 888 | Theorem *2.25 of [Whitehea... |
| orel2 889 | Elimination of disjunction... |
| pm2.67-2 890 | Slight generalization of T... |
| pm2.67 891 | Theorem *2.67 of [Whitehea... |
| curryax 892 | A non-intuitionistic posit... |
| exmid 893 | Law of excluded middle, al... |
| exmidd 894 | Law of excluded middle in ... |
| pm2.1 895 | Theorem *2.1 of [Whitehead... |
| pm2.13 896 | Theorem *2.13 of [Whitehea... |
| pm2.621 897 | Theorem *2.621 of [Whitehe... |
| pm2.62 898 | Theorem *2.62 of [Whitehea... |
| pm2.68 899 | Theorem *2.68 of [Whitehea... |
| dfor2 900 | Logical 'or' expressed in ... |
| pm2.07 901 | Theorem *2.07 of [Whitehea... |
| pm1.2 902 | Axiom *1.2 of [WhiteheadRu... |
| oridm 903 | Idempotent law for disjunc... |
| pm4.25 904 | Theorem *4.25 of [Whitehea... |
| pm2.4 905 | Theorem *2.4 of [Whitehead... |
| pm2.41 906 | Theorem *2.41 of [Whitehea... |
| orim12i 907 | Disjoin antecedents and co... |
| orim1i 908 | Introduce disjunct to both... |
| orim2i 909 | Introduce disjunct to both... |
| orim12dALT 910 | Alternate proof of ~ orim1... |
| orbi2i 911 | Inference adding a left di... |
| orbi1i 912 | Inference adding a right d... |
| orbi12i 913 | Infer the disjunction of t... |
| orbi2d 914 | Deduction adding a left di... |
| orbi1d 915 | Deduction adding a right d... |
| orbi1 916 | Theorem *4.37 of [Whitehea... |
| orbi12d 917 | Deduction joining two equi... |
| pm1.5 918 | Axiom *1.5 (Assoc) of [Whi... |
| or12 919 | Swap two disjuncts. (Cont... |
| orass 920 | Associative law for disjun... |
| pm2.31 921 | Theorem *2.31 of [Whitehea... |
| pm2.32 922 | Theorem *2.32 of [Whitehea... |
| pm2.3 923 | Theorem *2.3 of [Whitehead... |
| or32 924 | A rearrangement of disjunc... |
| or4 925 | Rearrangement of 4 disjunc... |
| or42 926 | Rearrangement of 4 disjunc... |
| orordi 927 | Distribution of disjunctio... |
| orordir 928 | Distribution of disjunctio... |
| orimdi 929 | Disjunction distributes ov... |
| pm2.76 930 | Theorem *2.76 of [Whitehea... |
| pm2.85 931 | Theorem *2.85 of [Whitehea... |
| pm2.75 932 | Theorem *2.75 of [Whitehea... |
| pm4.78 933 | Implication distributes ov... |
| biort 934 | A disjunction with a true ... |
| biorf 935 | A wff is equivalent to its... |
| biortn 936 | A wff is equivalent to its... |
| biorfi 937 | A wff is equivalent to its... |
| pm2.26 938 | Theorem *2.26 of [Whitehea... |
| pm2.63 939 | Theorem *2.63 of [Whitehea... |
| pm2.64 940 | Theorem *2.64 of [Whitehea... |
| pm2.42 941 | Theorem *2.42 of [Whitehea... |
| pm5.11g 942 | A general instance of Theo... |
| pm5.11 943 | Theorem *5.11 of [Whitehea... |
| pm5.12 944 | Theorem *5.12 of [Whitehea... |
| pm5.14 945 | Theorem *5.14 of [Whitehea... |
| pm5.13 946 | Theorem *5.13 of [Whitehea... |
| pm5.55 947 | Theorem *5.55 of [Whitehea... |
| pm4.72 948 | Implication in terms of bi... |
| imimorb 949 | Simplify an implication be... |
| oibabs 950 | Absorption of disjunction ... |
| orbidi 951 | Disjunction distributes ov... |
| pm5.7 952 | Disjunction distributes ov... |
| jaao 953 | Inference conjoining and d... |
| jaoa 954 | Inference disjoining and c... |
| jaoian 955 | Inference disjoining the a... |
| jaodan 956 | Deduction disjoining the a... |
| mpjaodan 957 | Eliminate a disjunction in... |
| pm3.44 958 | Theorem *3.44 of [Whitehea... |
| jao 959 | Disjunction of antecedents... |
| jaob 960 | Disjunction of antecedents... |
| pm4.77 961 | Theorem *4.77 of [Whitehea... |
| pm3.48 962 | Theorem *3.48 of [Whitehea... |
| orim12d 963 | Disjoin antecedents and co... |
| orim1d 964 | Disjoin antecedents and co... |
| orim2d 965 | Disjoin antecedents and co... |
| orim2 966 | Axiom *1.6 (Sum) of [White... |
| pm2.38 967 | Theorem *2.38 of [Whitehea... |
| pm2.36 968 | Theorem *2.36 of [Whitehea... |
| pm2.37 969 | Theorem *2.37 of [Whitehea... |
| pm2.81 970 | Theorem *2.81 of [Whitehea... |
| pm2.8 971 | Theorem *2.8 of [Whitehead... |
| pm2.73 972 | Theorem *2.73 of [Whitehea... |
| pm2.74 973 | Theorem *2.74 of [Whitehea... |
| pm2.82 974 | Theorem *2.82 of [Whitehea... |
| pm4.39 975 | Theorem *4.39 of [Whitehea... |
| animorl 976 | Conjunction implies disjun... |
| animorr 977 | Conjunction implies disjun... |
| animorlr 978 | Conjunction implies disjun... |
| animorrl 979 | Conjunction implies disjun... |
| ianor 980 | Negated conjunction in ter... |
| anor 981 | Conjunction in terms of di... |
| ioran 982 | Negated disjunction in ter... |
| pm4.52 983 | Theorem *4.52 of [Whitehea... |
| pm4.53 984 | Theorem *4.53 of [Whitehea... |
| pm4.54 985 | Theorem *4.54 of [Whitehea... |
| pm4.55 986 | Theorem *4.55 of [Whitehea... |
| pm4.56 987 | Theorem *4.56 of [Whitehea... |
| oran 988 | Disjunction in terms of co... |
| pm4.57 989 | Theorem *4.57 of [Whitehea... |
| pm3.1 990 | Theorem *3.1 of [Whitehead... |
| pm3.11 991 | Theorem *3.11 of [Whitehea... |
| pm3.12 992 | Theorem *3.12 of [Whitehea... |
| pm3.13 993 | Theorem *3.13 of [Whitehea... |
| pm3.14 994 | Theorem *3.14 of [Whitehea... |
| pm4.44 995 | Theorem *4.44 of [Whitehea... |
| pm4.45 996 | Theorem *4.45 of [Whitehea... |
| orabs 997 | Absorption of redundant in... |
| oranabs 998 | Absorb a disjunct into a c... |
| pm5.61 999 | Theorem *5.61 of [Whitehea... |
| pm5.6 1000 | Conjunction in antecedent ... |
| orcanai 1001 | Change disjunction in cons... |
| pm4.79 1002 | Theorem *4.79 of [Whitehea... |
| pm5.53 1003 | Theorem *5.53 of [Whitehea... |
| ordi 1004 | Distributive law for disju... |
| ordir 1005 | Distributive law for disju... |
| andi 1006 | Distributive law for conju... |
| andir 1007 | Distributive law for conju... |
| orddi 1008 | Double distributive law fo... |
| anddi 1009 | Double distributive law fo... |
| pm5.17 1010 | Theorem *5.17 of [Whitehea... |
| pm5.15 1011 | Theorem *5.15 of [Whitehea... |
| pm5.16 1012 | Theorem *5.16 of [Whitehea... |
| xor 1013 | Two ways to express exclus... |
| nbi2 1014 | Two ways to express "exclu... |
| xordi 1015 | Conjunction distributes ov... |
| pm5.54 1016 | Theorem *5.54 of [Whitehea... |
| pm5.62 1017 | Theorem *5.62 of [Whitehea... |
| pm5.63 1018 | Theorem *5.63 of [Whitehea... |
| niabn 1019 | Miscellaneous inference re... |
| ninba 1020 | Miscellaneous inference re... |
| pm4.43 1021 | Theorem *4.43 of [Whitehea... |
| pm4.82 1022 | Theorem *4.82 of [Whitehea... |
| pm4.83 1023 | Theorem *4.83 of [Whitehea... |
| pclem6 1024 | Negation inferred from emb... |
| bigolden 1025 | Dijkstra-Scholten's Golden... |
| pm5.71 1026 | Theorem *5.71 of [Whitehea... |
| pm5.75 1027 | Theorem *5.75 of [Whitehea... |
| ecase2d 1028 | Deduction for elimination ... |
| ecase2dOLD 1029 | Obsolete version of ~ ecas... |
| ecase3 1030 | Inference for elimination ... |
| ecase 1031 | Inference for elimination ... |
| ecase3d 1032 | Deduction for elimination ... |
| ecased 1033 | Deduction for elimination ... |
| ecase3ad 1034 | Deduction for elimination ... |
| ecase3adOLD 1035 | Obsolete version of ~ ecas... |
| ccase 1036 | Inference for combining ca... |
| ccased 1037 | Deduction for combining ca... |
| ccase2 1038 | Inference for combining ca... |
| 4cases 1039 | Inference eliminating two ... |
| 4casesdan 1040 | Deduction eliminating two ... |
| cases 1041 | Case disjunction according... |
| dedlem0a 1042 | Lemma for an alternate ver... |
| dedlem0b 1043 | Lemma for an alternate ver... |
| dedlema 1044 | Lemma for weak deduction t... |
| dedlemb 1045 | Lemma for weak deduction t... |
| cases2 1046 | Case disjunction according... |
| cases2ALT 1047 | Alternate proof of ~ cases... |
| dfbi3 1048 | An alternate definition of... |
| pm5.24 1049 | Theorem *5.24 of [Whitehea... |
| 4exmid 1050 | The disjunction of the fou... |
| consensus 1051 | The consensus theorem. Th... |
| pm4.42 1052 | Theorem *4.42 of [Whitehea... |
| prlem1 1053 | A specialized lemma for se... |
| prlem2 1054 | A specialized lemma for se... |
| oplem1 1055 | A specialized lemma for se... |
| dn1 1056 | A single axiom for Boolean... |
| bianir 1057 | A closed form of ~ mpbir ,... |
| jaoi2 1058 | Inference removing a negat... |
| jaoi3 1059 | Inference separating a dis... |
| ornld 1060 | Selecting one statement fr... |
| dfifp2 1063 | Alternate definition of th... |
| dfifp3 1064 | Alternate definition of th... |
| dfifp4 1065 | Alternate definition of th... |
| dfifp5 1066 | Alternate definition of th... |
| dfifp6 1067 | Alternate definition of th... |
| dfifp7 1068 | Alternate definition of th... |
| ifpdfbi 1069 | Define the biconditional a... |
| anifp 1070 | The conditional operator i... |
| ifpor 1071 | The conditional operator i... |
| ifpn 1072 | Conditional operator for t... |
| ifptru 1073 | Value of the conditional o... |
| ifpfal 1074 | Value of the conditional o... |
| ifpid 1075 | Value of the conditional o... |
| casesifp 1076 | Version of ~ cases express... |
| ifpbi123d 1077 | Equivalence deduction for ... |
| ifpbi23d 1078 | Equivalence deduction for ... |
| ifpimpda 1079 | Separation of the values o... |
| 1fpid3 1080 | The value of the condition... |
| elimh 1081 | Hypothesis builder for the... |
| dedt 1082 | The weak deduction theorem... |
| con3ALT 1083 | Proof of ~ con3 from its a... |
| 3orass 1088 | Associative law for triple... |
| 3orel1 1089 | Partial elimination of a t... |
| 3orrot 1090 | Rotation law for triple di... |
| 3orcoma 1091 | Commutation law for triple... |
| 3orcomb 1092 | Commutation law for triple... |
| 3anass 1093 | Associative law for triple... |
| 3anan12 1094 | Convert triple conjunction... |
| 3anan32 1095 | Convert triple conjunction... |
| 3ancoma 1096 | Commutation law for triple... |
| 3ancomb 1097 | Commutation law for triple... |
| 3anrot 1098 | Rotation law for triple co... |
| 3anrev 1099 | Reversal law for triple co... |
| anandi3 1100 | Distribution of triple con... |
| anandi3r 1101 | Distribution of triple con... |
| 3anidm 1102 | Idempotent law for conjunc... |
| 3an4anass 1103 | Associative law for four c... |
| 3ioran 1104 | Negated triple disjunction... |
| 3ianor 1105 | Negated triple conjunction... |
| 3anor 1106 | Triple conjunction express... |
| 3oran 1107 | Triple disjunction in term... |
| 3impa 1108 | Importation from double to... |
| 3imp 1109 | Importation inference. (C... |
| 3imp31 1110 | The importation inference ... |
| 3imp231 1111 | Importation inference. (C... |
| 3imp21 1112 | The importation inference ... |
| 3impb 1113 | Importation from double to... |
| 3impib 1114 | Importation to triple conj... |
| 3impia 1115 | Importation to triple conj... |
| 3expa 1116 | Exportation from triple to... |
| 3exp 1117 | Exportation inference. (C... |
| 3expb 1118 | Exportation from triple to... |
| 3expia 1119 | Exportation from triple co... |
| 3expib 1120 | Exportation from triple co... |
| 3com12 1121 | Commutation in antecedent.... |
| 3com13 1122 | Commutation in antecedent.... |
| 3comr 1123 | Commutation in antecedent.... |
| 3com23 1124 | Commutation in antecedent.... |
| 3coml 1125 | Commutation in antecedent.... |
| 3jca 1126 | Join consequents with conj... |
| 3jcad 1127 | Deduction conjoining the c... |
| 3adant1 1128 | Deduction adding a conjunc... |
| 3adant2 1129 | Deduction adding a conjunc... |
| 3adant3 1130 | Deduction adding a conjunc... |
| 3ad2ant1 1131 | Deduction adding conjuncts... |
| 3ad2ant2 1132 | Deduction adding conjuncts... |
| 3ad2ant3 1133 | Deduction adding conjuncts... |
| simp1 1134 | Simplification of triple c... |
| simp2 1135 | Simplification of triple c... |
| simp3 1136 | Simplification of triple c... |
| simp1i 1137 | Infer a conjunct from a tr... |
| simp2i 1138 | Infer a conjunct from a tr... |
| simp3i 1139 | Infer a conjunct from a tr... |
| simp1d 1140 | Deduce a conjunct from a t... |
| simp2d 1141 | Deduce a conjunct from a t... |
| simp3d 1142 | Deduce a conjunct from a t... |
| simp1bi 1143 | Deduce a conjunct from a t... |
| simp2bi 1144 | Deduce a conjunct from a t... |
| simp3bi 1145 | Deduce a conjunct from a t... |
| 3simpa 1146 | Simplification of triple c... |
| 3simpb 1147 | Simplification of triple c... |
| 3simpc 1148 | Simplification of triple c... |
| 3anim123i 1149 | Join antecedents and conse... |
| 3anim1i 1150 | Add two conjuncts to antec... |
| 3anim2i 1151 | Add two conjuncts to antec... |
| 3anim3i 1152 | Add two conjuncts to antec... |
| 3anbi123i 1153 | Join 3 biconditionals with... |
| 3orbi123i 1154 | Join 3 biconditionals with... |
| 3anbi1i 1155 | Inference adding two conju... |
| 3anbi2i 1156 | Inference adding two conju... |
| 3anbi3i 1157 | Inference adding two conju... |
| syl3an 1158 | A triple syllogism inferen... |
| syl3anb 1159 | A triple syllogism inferen... |
| syl3anbr 1160 | A triple syllogism inferen... |
| syl3an1 1161 | A syllogism inference. (C... |
| syl3an2 1162 | A syllogism inference. (C... |
| syl3an3 1163 | A syllogism inference. (C... |
| 3adantl1 1164 | Deduction adding a conjunc... |
| 3adantl2 1165 | Deduction adding a conjunc... |
| 3adantl3 1166 | Deduction adding a conjunc... |
| 3adantr1 1167 | Deduction adding a conjunc... |
| 3adantr2 1168 | Deduction adding a conjunc... |
| 3adantr3 1169 | Deduction adding a conjunc... |
| ad4ant123 1170 | Deduction adding conjuncts... |
| ad4ant124 1171 | Deduction adding conjuncts... |
| ad4ant134 1172 | Deduction adding conjuncts... |
| ad4ant234 1173 | Deduction adding conjuncts... |
| 3adant1l 1174 | Deduction adding a conjunc... |
| 3adant1r 1175 | Deduction adding a conjunc... |
| 3adant2l 1176 | Deduction adding a conjunc... |
| 3adant2r 1177 | Deduction adding a conjunc... |
| 3adant3l 1178 | Deduction adding a conjunc... |
| 3adant3r 1179 | Deduction adding a conjunc... |
| 3adant3r1 1180 | Deduction adding a conjunc... |
| 3adant3r2 1181 | Deduction adding a conjunc... |
| 3adant3r3 1182 | Deduction adding a conjunc... |
| 3ad2antl1 1183 | Deduction adding conjuncts... |
| 3ad2antl2 1184 | Deduction adding conjuncts... |
| 3ad2antl3 1185 | Deduction adding conjuncts... |
| 3ad2antr1 1186 | Deduction adding conjuncts... |
| 3ad2antr2 1187 | Deduction adding conjuncts... |
| 3ad2antr3 1188 | Deduction adding conjuncts... |
| simpl1 1189 | Simplification of conjunct... |
| simpl2 1190 | Simplification of conjunct... |
| simpl3 1191 | Simplification of conjunct... |
| simpr1 1192 | Simplification of conjunct... |
| simpr2 1193 | Simplification of conjunct... |
| simpr3 1194 | Simplification of conjunct... |
| simp1l 1195 | Simplification of triple c... |
| simp1r 1196 | Simplification of triple c... |
| simp2l 1197 | Simplification of triple c... |
| simp2r 1198 | Simplification of triple c... |
| simp3l 1199 | Simplification of triple c... |
| simp3r 1200 | Simplification of triple c... |
| simp11 1201 | Simplification of doubly t... |
| simp12 1202 | Simplification of doubly t... |
| simp13 1203 | Simplification of doubly t... |
| simp21 1204 | Simplification of doubly t... |
| simp22 1205 | Simplification of doubly t... |
| simp23 1206 | Simplification of doubly t... |
| simp31 1207 | Simplification of doubly t... |
| simp32 1208 | Simplification of doubly t... |
| simp33 1209 | Simplification of doubly t... |
| simpll1 1210 | Simplification of conjunct... |
| simpll2 1211 | Simplification of conjunct... |
| simpll3 1212 | Simplification of conjunct... |
| simplr1 1213 | Simplification of conjunct... |
| simplr2 1214 | Simplification of conjunct... |
| simplr3 1215 | Simplification of conjunct... |
| simprl1 1216 | Simplification of conjunct... |
| simprl2 1217 | Simplification of conjunct... |
| simprl3 1218 | Simplification of conjunct... |
| simprr1 1219 | Simplification of conjunct... |
| simprr2 1220 | Simplification of conjunct... |
| simprr3 1221 | Simplification of conjunct... |
| simpl1l 1222 | Simplification of conjunct... |
| simpl1r 1223 | Simplification of conjunct... |
| simpl2l 1224 | Simplification of conjunct... |
| simpl2r 1225 | Simplification of conjunct... |
| simpl3l 1226 | Simplification of conjunct... |
| simpl3r 1227 | Simplification of conjunct... |
| simpr1l 1228 | Simplification of conjunct... |
| simpr1r 1229 | Simplification of conjunct... |
| simpr2l 1230 | Simplification of conjunct... |
| simpr2r 1231 | Simplification of conjunct... |
| simpr3l 1232 | Simplification of conjunct... |
| simpr3r 1233 | Simplification of conjunct... |
| simp1ll 1234 | Simplification of conjunct... |
| simp1lr 1235 | Simplification of conjunct... |
| simp1rl 1236 | Simplification of conjunct... |
| simp1rr 1237 | Simplification of conjunct... |
| simp2ll 1238 | Simplification of conjunct... |
| simp2lr 1239 | Simplification of conjunct... |
| simp2rl 1240 | Simplification of conjunct... |
| simp2rr 1241 | Simplification of conjunct... |
| simp3ll 1242 | Simplification of conjunct... |
| simp3lr 1243 | Simplification of conjunct... |
| simp3rl 1244 | Simplification of conjunct... |
| simp3rr 1245 | Simplification of conjunct... |
| simpl11 1246 | Simplification of conjunct... |
| simpl12 1247 | Simplification of conjunct... |
| simpl13 1248 | Simplification of conjunct... |
| simpl21 1249 | Simplification of conjunct... |
| simpl22 1250 | Simplification of conjunct... |
| simpl23 1251 | Simplification of conjunct... |
| simpl31 1252 | Simplification of conjunct... |
| simpl32 1253 | Simplification of conjunct... |
| simpl33 1254 | Simplification of conjunct... |
| simpr11 1255 | Simplification of conjunct... |
| simpr12 1256 | Simplification of conjunct... |
| simpr13 1257 | Simplification of conjunct... |
| simpr21 1258 | Simplification of conjunct... |
| simpr22 1259 | Simplification of conjunct... |
| simpr23 1260 | Simplification of conjunct... |
| simpr31 1261 | Simplification of conjunct... |
| simpr32 1262 | Simplification of conjunct... |
| simpr33 1263 | Simplification of conjunct... |
| simp1l1 1264 | Simplification of conjunct... |
| simp1l2 1265 | Simplification of conjunct... |
| simp1l3 1266 | Simplification of conjunct... |
| simp1r1 1267 | Simplification of conjunct... |
| simp1r2 1268 | Simplification of conjunct... |
| simp1r3 1269 | Simplification of conjunct... |
| simp2l1 1270 | Simplification of conjunct... |
| simp2l2 1271 | Simplification of conjunct... |
| simp2l3 1272 | Simplification of conjunct... |
| simp2r1 1273 | Simplification of conjunct... |
| simp2r2 1274 | Simplification of conjunct... |
| simp2r3 1275 | Simplification of conjunct... |
| simp3l1 1276 | Simplification of conjunct... |
| simp3l2 1277 | Simplification of conjunct... |
| simp3l3 1278 | Simplification of conjunct... |
| simp3r1 1279 | Simplification of conjunct... |
| simp3r2 1280 | Simplification of conjunct... |
| simp3r3 1281 | Simplification of conjunct... |
| simp11l 1282 | Simplification of conjunct... |
| simp11r 1283 | Simplification of conjunct... |
| simp12l 1284 | Simplification of conjunct... |
| simp12r 1285 | Simplification of conjunct... |
| simp13l 1286 | Simplification of conjunct... |
| simp13r 1287 | Simplification of conjunct... |
| simp21l 1288 | Simplification of conjunct... |
| simp21r 1289 | Simplification of conjunct... |
| simp22l 1290 | Simplification of conjunct... |
| simp22r 1291 | Simplification of conjunct... |
| simp23l 1292 | Simplification of conjunct... |
| simp23r 1293 | Simplification of conjunct... |
| simp31l 1294 | Simplification of conjunct... |
| simp31r 1295 | Simplification of conjunct... |
| simp32l 1296 | Simplification of conjunct... |
| simp32r 1297 | Simplification of conjunct... |
| simp33l 1298 | Simplification of conjunct... |
| simp33r 1299 | Simplification of conjunct... |
| simp111 1300 | Simplification of conjunct... |
| simp112 1301 | Simplification of conjunct... |
| simp113 1302 | Simplification of conjunct... |
| simp121 1303 | Simplification of conjunct... |
| simp122 1304 | Simplification of conjunct... |
| simp123 1305 | Simplification of conjunct... |
| simp131 1306 | Simplification of conjunct... |
| simp132 1307 | Simplification of conjunct... |
| simp133 1308 | Simplification of conjunct... |
| simp211 1309 | Simplification of conjunct... |
| simp212 1310 | Simplification of conjunct... |
| simp213 1311 | Simplification of conjunct... |
| simp221 1312 | Simplification of conjunct... |
| simp222 1313 | Simplification of conjunct... |
| simp223 1314 | Simplification of conjunct... |
| simp231 1315 | Simplification of conjunct... |
| simp232 1316 | Simplification of conjunct... |
| simp233 1317 | Simplification of conjunct... |
| simp311 1318 | Simplification of conjunct... |
| simp312 1319 | Simplification of conjunct... |
| simp313 1320 | Simplification of conjunct... |
| simp321 1321 | Simplification of conjunct... |
| simp322 1322 | Simplification of conjunct... |
| simp323 1323 | Simplification of conjunct... |
| simp331 1324 | Simplification of conjunct... |
| simp332 1325 | Simplification of conjunct... |
| simp333 1326 | Simplification of conjunct... |
| 3anibar 1327 | Remove a hypothesis from t... |
| 3mix1 1328 | Introduction in triple dis... |
| 3mix2 1329 | Introduction in triple dis... |
| 3mix3 1330 | Introduction in triple dis... |
| 3mix1i 1331 | Introduction in triple dis... |
| 3mix2i 1332 | Introduction in triple dis... |
| 3mix3i 1333 | Introduction in triple dis... |
| 3mix1d 1334 | Deduction introducing trip... |
| 3mix2d 1335 | Deduction introducing trip... |
| 3mix3d 1336 | Deduction introducing trip... |
| 3pm3.2i 1337 | Infer conjunction of premi... |
| pm3.2an3 1338 | Version of ~ pm3.2 for a t... |
| mpbir3an 1339 | Detach a conjunction of tr... |
| mpbir3and 1340 | Detach a conjunction of tr... |
| syl3anbrc 1341 | Syllogism inference. (Con... |
| syl21anbrc 1342 | Syllogism inference. (Con... |
| 3imp3i2an 1343 | An elimination deduction. ... |
| ex3 1344 | Apply ~ ex to a hypothesis... |
| 3imp1 1345 | Importation to left triple... |
| 3impd 1346 | Importation deduction for ... |
| 3imp2 1347 | Importation to right tripl... |
| 3impdi 1348 | Importation inference (und... |
| 3impdir 1349 | Importation inference (und... |
| 3exp1 1350 | Exportation from left trip... |
| 3expd 1351 | Exportation deduction for ... |
| 3exp2 1352 | Exportation from right tri... |
| exp5o 1353 | A triple exportation infer... |
| exp516 1354 | A triple exportation infer... |
| exp520 1355 | A triple exportation infer... |
| 3impexp 1356 | Version of ~ impexp for a ... |
| 3an1rs 1357 | Swap conjuncts. (Contribu... |
| 3anassrs 1358 | Associative law for conjun... |
| ad5ant245 1359 | Deduction adding conjuncts... |
| ad5ant234 1360 | Deduction adding conjuncts... |
| ad5ant235 1361 | Deduction adding conjuncts... |
| ad5ant123 1362 | Deduction adding conjuncts... |
| ad5ant124 1363 | Deduction adding conjuncts... |
| ad5ant125 1364 | Deduction adding conjuncts... |
| ad5ant134 1365 | Deduction adding conjuncts... |
| ad5ant135 1366 | Deduction adding conjuncts... |
| ad5ant145 1367 | Deduction adding conjuncts... |
| ad5ant2345 1368 | Deduction adding conjuncts... |
| syl3anc 1369 | Syllogism combined with co... |
| syl13anc 1370 | Syllogism combined with co... |
| syl31anc 1371 | Syllogism combined with co... |
| syl112anc 1372 | Syllogism combined with co... |
| syl121anc 1373 | Syllogism combined with co... |
| syl211anc 1374 | Syllogism combined with co... |
| syl23anc 1375 | Syllogism combined with co... |
| syl32anc 1376 | Syllogism combined with co... |
| syl122anc 1377 | Syllogism combined with co... |
| syl212anc 1378 | Syllogism combined with co... |
| syl221anc 1379 | Syllogism combined with co... |
| syl113anc 1380 | Syllogism combined with co... |
| syl131anc 1381 | Syllogism combined with co... |
| syl311anc 1382 | Syllogism combined with co... |
| syl33anc 1383 | Syllogism combined with co... |
| syl222anc 1384 | Syllogism combined with co... |
| syl123anc 1385 | Syllogism combined with co... |
| syl132anc 1386 | Syllogism combined with co... |
| syl213anc 1387 | Syllogism combined with co... |
| syl231anc 1388 | Syllogism combined with co... |
| syl312anc 1389 | Syllogism combined with co... |
| syl321anc 1390 | Syllogism combined with co... |
| syl133anc 1391 | Syllogism combined with co... |
| syl313anc 1392 | Syllogism combined with co... |
| syl331anc 1393 | Syllogism combined with co... |
| syl223anc 1394 | Syllogism combined with co... |
| syl232anc 1395 | Syllogism combined with co... |
| syl322anc 1396 | Syllogism combined with co... |
| syl233anc 1397 | Syllogism combined with co... |
| syl323anc 1398 | Syllogism combined with co... |
| syl332anc 1399 | Syllogism combined with co... |
| syl333anc 1400 | A syllogism inference comb... |
| syl3an1b 1401 | A syllogism inference. (C... |
| syl3an2b 1402 | A syllogism inference. (C... |
| syl3an3b 1403 | A syllogism inference. (C... |
| syl3an1br 1404 | A syllogism inference. (C... |
| syl3an2br 1405 | A syllogism inference. (C... |
| syl3an3br 1406 | A syllogism inference. (C... |
| syld3an3 1407 | A syllogism inference. (C... |
| syld3an1 1408 | A syllogism inference. (C... |
| syld3an2 1409 | A syllogism inference. (C... |
| syl3anl1 1410 | A syllogism inference. (C... |
| syl3anl2 1411 | A syllogism inference. (C... |
| syl3anl3 1412 | A syllogism inference. (C... |
| syl3anl 1413 | A triple syllogism inferen... |
| syl3anr1 1414 | A syllogism inference. (C... |
| syl3anr2 1415 | A syllogism inference. (C... |
| syl3anr3 1416 | A syllogism inference. (C... |
| 3anidm12 1417 | Inference from idempotent ... |
| 3anidm13 1418 | Inference from idempotent ... |
| 3anidm23 1419 | Inference from idempotent ... |
| syl2an3an 1420 | ~ syl3an with antecedents ... |
| syl2an23an 1421 | Deduction related to ~ syl... |
| 3ori 1422 | Infer implication from tri... |
| 3jao 1423 | Disjunction of three antec... |
| 3jaob 1424 | Disjunction of three antec... |
| 3jaoi 1425 | Disjunction of three antec... |
| 3jaod 1426 | Disjunction of three antec... |
| 3jaoian 1427 | Disjunction of three antec... |
| 3jaodan 1428 | Disjunction of three antec... |
| mpjao3dan 1429 | Eliminate a three-way disj... |
| 3jaao 1430 | Inference conjoining and d... |
| syl3an9b 1431 | Nested syllogism inference... |
| 3orbi123d 1432 | Deduction joining 3 equiva... |
| 3anbi123d 1433 | Deduction joining 3 equiva... |
| 3anbi12d 1434 | Deduction conjoining and a... |
| 3anbi13d 1435 | Deduction conjoining and a... |
| 3anbi23d 1436 | Deduction conjoining and a... |
| 3anbi1d 1437 | Deduction adding conjuncts... |
| 3anbi2d 1438 | Deduction adding conjuncts... |
| 3anbi3d 1439 | Deduction adding conjuncts... |
| 3anim123d 1440 | Deduction joining 3 implic... |
| 3orim123d 1441 | Deduction joining 3 implic... |
| an6 1442 | Rearrangement of 6 conjunc... |
| 3an6 1443 | Analogue of ~ an4 for trip... |
| 3or6 1444 | Analogue of ~ or4 for trip... |
| mp3an1 1445 | An inference based on modu... |
| mp3an2 1446 | An inference based on modu... |
| mp3an3 1447 | An inference based on modu... |
| mp3an12 1448 | An inference based on modu... |
| mp3an13 1449 | An inference based on modu... |
| mp3an23 1450 | An inference based on modu... |
| mp3an1i 1451 | An inference based on modu... |
| mp3anl1 1452 | An inference based on modu... |
| mp3anl2 1453 | An inference based on modu... |
| mp3anl3 1454 | An inference based on modu... |
| mp3anr1 1455 | An inference based on modu... |
| mp3anr2 1456 | An inference based on modu... |
| mp3anr3 1457 | An inference based on modu... |
| mp3an 1458 | An inference based on modu... |
| mpd3an3 1459 | An inference based on modu... |
| mpd3an23 1460 | An inference based on modu... |
| mp3and 1461 | A deduction based on modus... |
| mp3an12i 1462 | ~ mp3an with antecedents i... |
| mp3an2i 1463 | ~ mp3an with antecedents i... |
| mp3an3an 1464 | ~ mp3an with antecedents i... |
| mp3an2ani 1465 | An elimination deduction. ... |
| biimp3a 1466 | Infer implication from a l... |
| biimp3ar 1467 | Infer implication from a l... |
| 3anandis 1468 | Inference that undistribut... |
| 3anandirs 1469 | Inference that undistribut... |
| ecase23d 1470 | Deduction for elimination ... |
| 3ecase 1471 | Inference for elimination ... |
| 3bior1fd 1472 | A disjunction is equivalen... |
| 3bior1fand 1473 | A disjunction is equivalen... |
| 3bior2fd 1474 | A wff is equivalent to its... |
| 3biant1d 1475 | A conjunction is equivalen... |
| intn3an1d 1476 | Introduction of a triple c... |
| intn3an2d 1477 | Introduction of a triple c... |
| intn3an3d 1478 | Introduction of a triple c... |
| an3andi 1479 | Distribution of conjunctio... |
| an33rean 1480 | Rearrange a 9-fold conjunc... |
| 3orel2 1481 | Partial elimination of a t... |
| 3orel3 1482 | Partial elimination of a t... |
| 3orel13 1483 | Elimination of two disjunc... |
| 3pm3.2ni 1484 | Triple negated disjunction... |
| nanan 1487 | Conjunction in terms of al... |
| dfnan2 1488 | Alternative denial in term... |
| nanor 1489 | Alternative denial in term... |
| nancom 1490 | Alternative denial is comm... |
| nannan 1491 | Nested alternative denials... |
| nanim 1492 | Implication in terms of al... |
| nannot 1493 | Negation in terms of alter... |
| nanbi 1494 | Biconditional in terms of ... |
| nanbi1 1495 | Introduce a right anti-con... |
| nanbi2 1496 | Introduce a left anti-conj... |
| nanbi12 1497 | Join two logical equivalen... |
| nanbi1i 1498 | Introduce a right anti-con... |
| nanbi2i 1499 | Introduce a left anti-conj... |
| nanbi12i 1500 | Join two logical equivalen... |
| nanbi1d 1501 | Introduce a right anti-con... |
| nanbi2d 1502 | Introduce a left anti-conj... |
| nanbi12d 1503 | Join two logical equivalen... |
| nanass 1504 | A characterization of when... |
| xnor 1507 | Two ways to write XNOR (ex... |
| xorcom 1508 | The connector ` \/_ ` is c... |
| xorass 1509 | The connector ` \/_ ` is a... |
| excxor 1510 | This tautology shows that ... |
| xor2 1511 | Two ways to express "exclu... |
| xoror 1512 | Exclusive disjunction impl... |
| xornan 1513 | Exclusive disjunction impl... |
| xornan2 1514 | XOR implies NAND (written ... |
| xorneg2 1515 | The connector ` \/_ ` is n... |
| xorneg1 1516 | The connector ` \/_ ` is n... |
| xorneg 1517 | The connector ` \/_ ` is u... |
| xorbi12i 1518 | Equality property for excl... |
| xorbi12d 1519 | Equality property for excl... |
| anxordi 1520 | Conjunction distributes ov... |
| xorexmid 1521 | Exclusive-or variant of th... |
| norcom 1524 | The connector ` -\/ ` is c... |
| nornot 1525 | ` -. ` is expressible via ... |
| noran 1526 | ` /\ ` is expressible via ... |
| noror 1527 | ` \/ ` is expressible via ... |
| norasslem1 1528 | This lemma shows the equiv... |
| norasslem2 1529 | This lemma specializes ~ b... |
| norasslem3 1530 | This lemma specializes ~ b... |
| norass 1531 | A characterization of when... |
| trujust 1536 | Soundness justification th... |
| tru 1538 | The truth value ` T. ` is ... |
| dftru2 1539 | An alternate definition of... |
| trut 1540 | A proposition is equivalen... |
| mptru 1541 | Eliminate ` T. ` as an ant... |
| tbtru 1542 | A proposition is equivalen... |
| bitru 1543 | A theorem is equivalent to... |
| trud 1544 | Anything implies ` T. ` . ... |
| truan 1545 | True can be removed from a... |
| fal 1548 | The truth value ` F. ` is ... |
| nbfal 1549 | The negation of a proposit... |
| bifal 1550 | A contradiction is equival... |
| falim 1551 | The truth value ` F. ` imp... |
| falimd 1552 | The truth value ` F. ` imp... |
| dfnot 1553 | Given falsum ` F. ` , we c... |
| inegd 1554 | Negation introduction rule... |
| efald 1555 | Deduction based on reducti... |
| pm2.21fal 1556 | If a wff and its negation ... |
| truimtru 1557 | A ` -> ` identity. (Contr... |
| truimfal 1558 | A ` -> ` identity. (Contr... |
| falimtru 1559 | A ` -> ` identity. (Contr... |
| falimfal 1560 | A ` -> ` identity. (Contr... |
| nottru 1561 | A ` -. ` identity. (Contr... |
| notfal 1562 | A ` -. ` identity. (Contr... |
| trubitru 1563 | A ` <-> ` identity. (Cont... |
| falbitru 1564 | A ` <-> ` identity. (Cont... |
| trubifal 1565 | A ` <-> ` identity. (Cont... |
| falbifal 1566 | A ` <-> ` identity. (Cont... |
| truantru 1567 | A ` /\ ` identity. (Contr... |
| truanfal 1568 | A ` /\ ` identity. (Contr... |
| falantru 1569 | A ` /\ ` identity. (Contr... |
| falanfal 1570 | A ` /\ ` identity. (Contr... |
| truortru 1571 | A ` \/ ` identity. (Contr... |
| truorfal 1572 | A ` \/ ` identity. (Contr... |
| falortru 1573 | A ` \/ ` identity. (Contr... |
| falorfal 1574 | A ` \/ ` identity. (Contr... |
| trunantru 1575 | A ` -/\ ` identity. (Cont... |
| trunanfal 1576 | A ` -/\ ` identity. (Cont... |
| falnantru 1577 | A ` -/\ ` identity. (Cont... |
| falnanfal 1578 | A ` -/\ ` identity. (Cont... |
| truxortru 1579 | A ` \/_ ` identity. (Cont... |
| truxorfal 1580 | A ` \/_ ` identity. (Cont... |
| falxortru 1581 | A ` \/_ ` identity. (Cont... |
| falxorfal 1582 | A ` \/_ ` identity. (Cont... |
| trunortru 1583 | A ` -\/ ` identity. (Cont... |
| trunorfal 1584 | A ` -\/ ` identity. (Cont... |
| falnortru 1585 | A ` -\/ ` identity. (Cont... |
| falnorfal 1586 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1589 | Equality theorem for the a... |
| hadbi123i 1590 | Equality theorem for the a... |
| hadass 1591 | Associative law for the ad... |
| hadbi 1592 | The adder sum is the same ... |
| hadcoma 1593 | Commutative law for the ad... |
| hadcomb 1594 | Commutative law for the ad... |
| hadrot 1595 | Rotation law for the adder... |
| hadnot 1596 | The adder sum distributes ... |
| had1 1597 | If the first input is true... |
| had0 1598 | If the first input is fals... |
| hadifp 1599 | The value of the adder sum... |
| cador 1602 | The adder carry in disjunc... |
| cadan 1603 | The adder carry in conjunc... |
| cadbi123d 1604 | Equality theorem for the a... |
| cadbi123i 1605 | Equality theorem for the a... |
| cadcoma 1606 | Commutative law for the ad... |
| cadcomb 1607 | Commutative law for the ad... |
| cadrot 1608 | Rotation law for the adder... |
| cadnot 1609 | The adder carry distribute... |
| cad11 1610 | If (at least) two inputs a... |
| cad1 1611 | If one input is true, then... |
| cad0 1612 | If one input is false, the... |
| cad0OLD 1613 | Obsolete version of ~ cad0... |
| cadifp 1614 | The value of the carry is,... |
| cadtru 1615 | The adder carry is true as... |
| minimp 1616 | A single axiom for minimal... |
| minimp-syllsimp 1617 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1618 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1619 | Derivation of a commuted f... |
| minimp-ax2 1620 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1621 | Derivation of ~ pm2.43 (al... |
| impsingle 1622 | The shortest single axiom ... |
| impsingle-step4 1623 | Derivation of impsingle-st... |
| impsingle-step8 1624 | Derivation of impsingle-st... |
| impsingle-ax1 1625 | Derivation of impsingle-ax... |
| impsingle-step15 1626 | Derivation of impsingle-st... |
| impsingle-step18 1627 | Derivation of impsingle-st... |
| impsingle-step19 1628 | Derivation of impsingle-st... |
| impsingle-step20 1629 | Derivation of impsingle-st... |
| impsingle-step21 1630 | Derivation of impsingle-st... |
| impsingle-step22 1631 | Derivation of impsingle-st... |
| impsingle-step25 1632 | Derivation of impsingle-st... |
| impsingle-imim1 1633 | Derivation of impsingle-im... |
| impsingle-peirce 1634 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1635 | Derivation of ~ ax-2 from ... |
| meredith 1636 | Carew Meredith's sole axio... |
| merlem1 1637 | Step 3 of Meredith's proof... |
| merlem2 1638 | Step 4 of Meredith's proof... |
| merlem3 1639 | Step 7 of Meredith's proof... |
| merlem4 1640 | Step 8 of Meredith's proof... |
| merlem5 1641 | Step 11 of Meredith's proo... |
| merlem6 1642 | Step 12 of Meredith's proo... |
| merlem7 1643 | Between steps 14 and 15 of... |
| merlem8 1644 | Step 15 of Meredith's proo... |
| merlem9 1645 | Step 18 of Meredith's proo... |
| merlem10 1646 | Step 19 of Meredith's proo... |
| merlem11 1647 | Step 20 of Meredith's proo... |
| merlem12 1648 | Step 28 of Meredith's proo... |
| merlem13 1649 | Step 35 of Meredith's proo... |
| luk-1 1650 | 1 of 3 axioms for proposit... |
| luk-2 1651 | 2 of 3 axioms for proposit... |
| luk-3 1652 | 3 of 3 axioms for proposit... |
| luklem1 1653 | Used to rederive standard ... |
| luklem2 1654 | Used to rederive standard ... |
| luklem3 1655 | Used to rederive standard ... |
| luklem4 1656 | Used to rederive standard ... |
| luklem5 1657 | Used to rederive standard ... |
| luklem6 1658 | Used to rederive standard ... |
| luklem7 1659 | Used to rederive standard ... |
| luklem8 1660 | Used to rederive standard ... |
| ax1 1661 | Standard propositional axi... |
| ax2 1662 | Standard propositional axi... |
| ax3 1663 | Standard propositional axi... |
| nic-dfim 1664 | This theorem "defines" imp... |
| nic-dfneg 1665 | This theorem "defines" neg... |
| nic-mp 1666 | Derive Nicod's rule of mod... |
| nic-mpALT 1667 | A direct proof of ~ nic-mp... |
| nic-ax 1668 | Nicod's axiom derived from... |
| nic-axALT 1669 | A direct proof of ~ nic-ax... |
| nic-imp 1670 | Inference for ~ nic-mp usi... |
| nic-idlem1 1671 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1672 | Lemma for ~ nic-id . Infe... |
| nic-id 1673 | Theorem ~ id expressed wit... |
| nic-swap 1674 | The connector ` -/\ ` is s... |
| nic-isw1 1675 | Inference version of ~ nic... |
| nic-isw2 1676 | Inference for swapping nes... |
| nic-iimp1 1677 | Inference version of ~ nic... |
| nic-iimp2 1678 | Inference version of ~ nic... |
| nic-idel 1679 | Inference to remove the tr... |
| nic-ich 1680 | Chained inference. (Contr... |
| nic-idbl 1681 | Double the terms. Since d... |
| nic-bijust 1682 | Biconditional justificatio... |
| nic-bi1 1683 | Inference to extract one s... |
| nic-bi2 1684 | Inference to extract the o... |
| nic-stdmp 1685 | Derive the standard modus ... |
| nic-luk1 1686 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1687 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1688 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1689 | This alternative axiom for... |
| lukshefth1 1690 | Lemma for ~ renicax . (Co... |
| lukshefth2 1691 | Lemma for ~ renicax . (Co... |
| renicax 1692 | A rederivation of ~ nic-ax... |
| tbw-bijust 1693 | Justification for ~ tbw-ne... |
| tbw-negdf 1694 | The definition of negation... |
| tbw-ax1 1695 | The first of four axioms i... |
| tbw-ax2 1696 | The second of four axioms ... |
| tbw-ax3 1697 | The third of four axioms i... |
| tbw-ax4 1698 | The fourth of four axioms ... |
| tbwsyl 1699 | Used to rederive the Lukas... |
| tbwlem1 1700 | Used to rederive the Lukas... |
| tbwlem2 1701 | Used to rederive the Lukas... |
| tbwlem3 1702 | Used to rederive the Lukas... |
| tbwlem4 1703 | Used to rederive the Lukas... |
| tbwlem5 1704 | Used to rederive the Lukas... |
| re1luk1 1705 | ~ luk-1 derived from the T... |
| re1luk2 1706 | ~ luk-2 derived from the T... |
| re1luk3 1707 | ~ luk-3 derived from the T... |
| merco1 1708 | A single axiom for proposi... |
| merco1lem1 1709 | Used to rederive the Tarsk... |
| retbwax4 1710 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1711 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1712 | Used to rederive the Tarsk... |
| merco1lem3 1713 | Used to rederive the Tarsk... |
| merco1lem4 1714 | Used to rederive the Tarsk... |
| merco1lem5 1715 | Used to rederive the Tarsk... |
| merco1lem6 1716 | Used to rederive the Tarsk... |
| merco1lem7 1717 | Used to rederive the Tarsk... |
| retbwax3 1718 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1719 | Used to rederive the Tarsk... |
| merco1lem9 1720 | Used to rederive the Tarsk... |
| merco1lem10 1721 | Used to rederive the Tarsk... |
| merco1lem11 1722 | Used to rederive the Tarsk... |
| merco1lem12 1723 | Used to rederive the Tarsk... |
| merco1lem13 1724 | Used to rederive the Tarsk... |
| merco1lem14 1725 | Used to rederive the Tarsk... |
| merco1lem15 1726 | Used to rederive the Tarsk... |
| merco1lem16 1727 | Used to rederive the Tarsk... |
| merco1lem17 1728 | Used to rederive the Tarsk... |
| merco1lem18 1729 | Used to rederive the Tarsk... |
| retbwax1 1730 | ~ tbw-ax1 rederived from ~... |
| merco2 1731 | A single axiom for proposi... |
| mercolem1 1732 | Used to rederive the Tarsk... |
| mercolem2 1733 | Used to rederive the Tarsk... |
| mercolem3 1734 | Used to rederive the Tarsk... |
| mercolem4 1735 | Used to rederive the Tarsk... |
| mercolem5 1736 | Used to rederive the Tarsk... |
| mercolem6 1737 | Used to rederive the Tarsk... |
| mercolem7 1738 | Used to rederive the Tarsk... |
| mercolem8 1739 | Used to rederive the Tarsk... |
| re1tbw1 1740 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1741 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1742 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1743 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1744 | Justification for ~ rb-imd... |
| rb-imdf 1745 | The definition of implicat... |
| anmp 1746 | Modus ponens for ` { \/ , ... |
| rb-ax1 1747 | The first of four axioms i... |
| rb-ax2 1748 | The second of four axioms ... |
| rb-ax3 1749 | The third of four axioms i... |
| rb-ax4 1750 | The fourth of four axioms ... |
| rbsyl 1751 | Used to rederive the Lukas... |
| rblem1 1752 | Used to rederive the Lukas... |
| rblem2 1753 | Used to rederive the Lukas... |
| rblem3 1754 | Used to rederive the Lukas... |
| rblem4 1755 | Used to rederive the Lukas... |
| rblem5 1756 | Used to rederive the Lukas... |
| rblem6 1757 | Used to rederive the Lukas... |
| rblem7 1758 | Used to rederive the Lukas... |
| re1axmp 1759 | ~ ax-mp derived from Russe... |
| re2luk1 1760 | ~ luk-1 derived from Russe... |
| re2luk2 1761 | ~ luk-2 derived from Russe... |
| re2luk3 1762 | ~ luk-3 derived from Russe... |
| mptnan 1763 | Modus ponendo tollens 1, o... |
| mptxor 1764 | Modus ponendo tollens 2, o... |
| mtpor 1765 | Modus tollendo ponens (inc... |
| mtpxor 1766 | Modus tollendo ponens (ori... |
| stoic1a 1767 | Stoic logic Thema 1 (part ... |
| stoic1b 1768 | Stoic logic Thema 1 (part ... |
| stoic2a 1769 | Stoic logic Thema 2 versio... |
| stoic2b 1770 | Stoic logic Thema 2 versio... |
| stoic3 1771 | Stoic logic Thema 3. Stat... |
| stoic4a 1772 | Stoic logic Thema 4 versio... |
| stoic4b 1773 | Stoic logic Thema 4 versio... |
| alnex 1776 | Universal quantification o... |
| eximal 1777 | An equivalence between an ... |
| nf2 1780 | Alternate definition of no... |
| nf3 1781 | Alternate definition of no... |
| nf4 1782 | Alternate definition of no... |
| nfi 1783 | Deduce that ` x ` is not f... |
| nfri 1784 | Consequence of the definit... |
| nfd 1785 | Deduce that ` x ` is not f... |
| nfrd 1786 | Consequence of the definit... |
| nftht 1787 | Closed form of ~ nfth . (... |
| nfntht 1788 | Closed form of ~ nfnth . ... |
| nfntht2 1789 | Closed form of ~ nfnth . ... |
| gen2 1791 | Generalization applied twi... |
| mpg 1792 | Modus ponens combined with... |
| mpgbi 1793 | Modus ponens on biconditio... |
| mpgbir 1794 | Modus ponens on biconditio... |
| nex 1795 | Generalization rule for ne... |
| nfth 1796 | No variable is (effectivel... |
| nfnth 1797 | No variable is (effectivel... |
| hbth 1798 | No variable is (effectivel... |
| nftru 1799 | The true constant has no f... |
| nffal 1800 | The false constant has no ... |
| sptruw 1801 | Version of ~ sp when ` ph ... |
| altru 1802 | For all sets, ` T. ` is tr... |
| alfal 1803 | For all sets, ` -. F. ` is... |
| alim 1805 | Restatement of Axiom ~ ax-... |
| alimi 1806 | Inference quantifying both... |
| 2alimi 1807 | Inference doubly quantifyi... |
| ala1 1808 | Add an antecedent in a uni... |
| al2im 1809 | Closed form of ~ al2imi . ... |
| al2imi 1810 | Inference quantifying ante... |
| alanimi 1811 | Variant of ~ al2imi with c... |
| alimdh 1812 | Deduction form of Theorem ... |
| albi 1813 | Theorem 19.15 of [Margaris... |
| albii 1814 | Inference adding universal... |
| 2albii 1815 | Inference adding two unive... |
| 3albii 1816 | Inference adding three uni... |
| sylgt 1817 | Closed form of ~ sylg . (... |
| sylg 1818 | A syllogism combined with ... |
| alrimih 1819 | Inference form of Theorem ... |
| hbxfrbi 1820 | A utility lemma to transfe... |
| alex 1821 | Universal quantifier in te... |
| exnal 1822 | Existential quantification... |
| 2nalexn 1823 | Part of theorem *11.5 in [... |
| 2exnaln 1824 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1825 | Theorem *11.25 in [Whitehe... |
| alimex 1826 | An equivalence between an ... |
| aleximi 1827 | A variant of ~ al2imi : in... |
| alexbii 1828 | Biconditional form of ~ al... |
| exim 1829 | Theorem 19.22 of [Margaris... |
| eximi 1830 | Inference adding existenti... |
| 2eximi 1831 | Inference adding two exist... |
| eximii 1832 | Inference associated with ... |
| exa1 1833 | Add an antecedent in an ex... |
| 19.38 1834 | Theorem 19.38 of [Margaris... |
| 19.38a 1835 | Under a nonfreeness hypoth... |
| 19.38b 1836 | Under a nonfreeness hypoth... |
| imnang 1837 | Quantified implication in ... |
| alinexa 1838 | A transformation of quanti... |
| exnalimn 1839 | Existential quantification... |
| alexn 1840 | A relationship between two... |
| 2exnexn 1841 | Theorem *11.51 in [Whitehe... |
| exbi 1842 | Theorem 19.18 of [Margaris... |
| exbii 1843 | Inference adding existenti... |
| 2exbii 1844 | Inference adding two exist... |
| 3exbii 1845 | Inference adding three exi... |
| nfbiit 1846 | Equivalence theorem for th... |
| nfbii 1847 | Equality theorem for the n... |
| nfxfr 1848 | A utility lemma to transfe... |
| nfxfrd 1849 | A utility lemma to transfe... |
| nfnbi 1850 | A variable is nonfree in a... |
| nfnbiOLD 1851 | Obsolete version of ~ nfnb... |
| nfnt 1852 | If a variable is nonfree i... |
| nfn 1853 | Inference associated with ... |
| nfnd 1854 | Deduction associated with ... |
| exanali 1855 | A transformation of quanti... |
| 2exanali 1856 | Theorem *11.521 in [Whiteh... |
| exancom 1857 | Commutation of conjunction... |
| exan 1858 | Place a conjunct in the sc... |
| alrimdh 1859 | Deduction form of Theorem ... |
| eximdh 1860 | Deduction from Theorem 19.... |
| nexdh 1861 | Deduction for generalizati... |
| albidh 1862 | Formula-building rule for ... |
| exbidh 1863 | Formula-building rule for ... |
| exsimpl 1864 | Simplification of an exist... |
| exsimpr 1865 | Simplification of an exist... |
| 19.26 1866 | Theorem 19.26 of [Margaris... |
| 19.26-2 1867 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1868 | Theorem ~ 19.26 with tripl... |
| 19.29 1869 | Theorem 19.29 of [Margaris... |
| 19.29r 1870 | Variation of ~ 19.29 . (C... |
| 19.29r2 1871 | Variation of ~ 19.29r with... |
| 19.29x 1872 | Variation of ~ 19.29 with ... |
| 19.35 1873 | Theorem 19.35 of [Margaris... |
| 19.35i 1874 | Inference associated with ... |
| 19.35ri 1875 | Inference associated with ... |
| 19.25 1876 | Theorem 19.25 of [Margaris... |
| 19.30 1877 | Theorem 19.30 of [Margaris... |
| 19.43 1878 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1879 | Obsolete proof of ~ 19.43 ... |
| 19.33 1880 | Theorem 19.33 of [Margaris... |
| 19.33b 1881 | The antecedent provides a ... |
| 19.40 1882 | Theorem 19.40 of [Margaris... |
| 19.40-2 1883 | Theorem *11.42 in [Whitehe... |
| 19.40b 1884 | The antecedent provides a ... |
| albiim 1885 | Split a biconditional and ... |
| 2albiim 1886 | Split a biconditional and ... |
| exintrbi 1887 | Add/remove a conjunct in t... |
| exintr 1888 | Introduce a conjunct in th... |
| alsyl 1889 | Universally quantified and... |
| nfimd 1890 | If in a context ` x ` is n... |
| nfimt 1891 | Closed form of ~ nfim and ... |
| nfim 1892 | If ` x ` is not free in ` ... |
| nfand 1893 | If in a context ` x ` is n... |
| nf3and 1894 | Deduction form of bound-va... |
| nfan 1895 | If ` x ` is not free in ` ... |
| nfnan 1896 | If ` x ` is not free in ` ... |
| nf3an 1897 | If ` x ` is not free in ` ... |
| nfbid 1898 | If in a context ` x ` is n... |
| nfbi 1899 | If ` x ` is not free in ` ... |
| nfor 1900 | If ` x ` is not free in ` ... |
| nf3or 1901 | If ` x ` is not free in ` ... |
| empty 1902 | Two characterizations of t... |
| emptyex 1903 | On the empty domain, any e... |
| emptyal 1904 | On the empty domain, any u... |
| emptynf 1905 | On the empty domain, any v... |
| ax5d 1907 | Version of ~ ax-5 with ant... |
| ax5e 1908 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1909 | If a formula holds for som... |
| nfv 1910 | If ` x ` is not present in... |
| nfvd 1911 | ~ nfv with antecedent. Us... |
| alimdv 1912 | Deduction form of Theorem ... |
| eximdv 1913 | Deduction form of Theorem ... |
| 2alimdv 1914 | Deduction form of Theorem ... |
| 2eximdv 1915 | Deduction form of Theorem ... |
| albidv 1916 | Formula-building rule for ... |
| exbidv 1917 | Formula-building rule for ... |
| nfbidv 1918 | An equality theorem for no... |
| 2albidv 1919 | Formula-building rule for ... |
| 2exbidv 1920 | Formula-building rule for ... |
| 3exbidv 1921 | Formula-building rule for ... |
| 4exbidv 1922 | Formula-building rule for ... |
| alrimiv 1923 | Inference form of Theorem ... |
| alrimivv 1924 | Inference form of Theorem ... |
| alrimdv 1925 | Deduction form of Theorem ... |
| exlimiv 1926 | Inference form of Theorem ... |
| exlimiiv 1927 | Inference (Rule C) associa... |
| exlimivv 1928 | Inference form of Theorem ... |
| exlimdv 1929 | Deduction form of Theorem ... |
| exlimdvv 1930 | Deduction form of Theorem ... |
| exlimddv 1931 | Existential elimination ru... |
| nexdv 1932 | Deduction for generalizati... |
| 2ax5 1933 | Quantification of two vari... |
| stdpc5v 1934 | Version of ~ stdpc5 with a... |
| 19.21v 1935 | Version of ~ 19.21 with a ... |
| 19.32v 1936 | Version of ~ 19.32 with a ... |
| 19.31v 1937 | Version of ~ 19.31 with a ... |
| 19.23v 1938 | Version of ~ 19.23 with a ... |
| 19.23vv 1939 | Theorem ~ 19.23v extended ... |
| pm11.53v 1940 | Version of ~ pm11.53 with ... |
| 19.36imv 1941 | One direction of ~ 19.36v ... |
| 19.36imvOLD 1942 | Obsolete version of ~ 19.3... |
| 19.36iv 1943 | Inference associated with ... |
| 19.37imv 1944 | One direction of ~ 19.37v ... |
| 19.37iv 1945 | Inference associated with ... |
| 19.41v 1946 | Version of ~ 19.41 with a ... |
| 19.41vv 1947 | Version of ~ 19.41 with tw... |
| 19.41vvv 1948 | Version of ~ 19.41 with th... |
| 19.41vvvv 1949 | Version of ~ 19.41 with fo... |
| 19.42v 1950 | Version of ~ 19.42 with a ... |
| exdistr 1951 | Distribution of existentia... |
| exdistrv 1952 | Distribute a pair of exist... |
| 4exdistrv 1953 | Distribute two pairs of ex... |
| 19.42vv 1954 | Version of ~ 19.42 with tw... |
| exdistr2 1955 | Distribution of existentia... |
| 19.42vvv 1956 | Version of ~ 19.42 with th... |
| 3exdistr 1957 | Distribution of existentia... |
| 4exdistr 1958 | Distribution of existentia... |
| weq 1959 | Extend wff definition to i... |
| speimfw 1960 | Specialization, with addit... |
| speimfwALT 1961 | Alternate proof of ~ speim... |
| spimfw 1962 | Specialization, with addit... |
| ax12i 1963 | Inference that has ~ ax-12... |
| ax6v 1965 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1966 | At least one individual ex... |
| spimw 1967 | Specialization. Lemma 8 o... |
| spimew 1968 | Existential introduction, ... |
| speiv 1969 | Inference from existential... |
| speivw 1970 | Version of ~ spei with a d... |
| exgen 1971 | Rule of existential genera... |
| extru 1972 | There exists a variable su... |
| 19.2 1973 | Theorem 19.2 of [Margaris]... |
| 19.2d 1974 | Deduction associated with ... |
| 19.8w 1975 | Weak version of ~ 19.8a an... |
| spnfw 1976 | Weak version of ~ sp . Us... |
| spvw 1977 | Version of ~ sp when ` x `... |
| 19.3v 1978 | Version of ~ 19.3 with a d... |
| 19.8v 1979 | Version of ~ 19.8a with a ... |
| 19.9v 1980 | Version of ~ 19.9 with a d... |
| 19.39 1981 | Theorem 19.39 of [Margaris... |
| 19.24 1982 | Theorem 19.24 of [Margaris... |
| 19.34 1983 | Theorem 19.34 of [Margaris... |
| 19.36v 1984 | Version of ~ 19.36 with a ... |
| 19.12vvv 1985 | Version of ~ 19.12vv with ... |
| 19.27v 1986 | Version of ~ 19.27 with a ... |
| 19.28v 1987 | Version of ~ 19.28 with a ... |
| 19.37v 1988 | Version of ~ 19.37 with a ... |
| 19.44v 1989 | Version of ~ 19.44 with a ... |
| 19.45v 1990 | Version of ~ 19.45 with a ... |
| spimevw 1991 | Existential introduction, ... |
| spimvw 1992 | A weak form of specializat... |
| spvv 1993 | Specialization, using impl... |
| spfalw 1994 | Version of ~ sp when ` ph ... |
| chvarvv 1995 | Implicit substitution of `... |
| equs4v 1996 | Version of ~ equs4 with a ... |
| alequexv 1997 | Version of ~ equs4v with i... |
| exsbim 1998 | One direction of the equiv... |
| equsv 1999 | If a formula does not cont... |
| equsalvw 2000 | Version of ~ equsalv with ... |
| equsexvw 2001 | Version of ~ equsexv with ... |
| cbvaliw 2002 | Change bound variable. Us... |
| cbvalivw 2003 | Change bound variable. Us... |
| ax7v 2005 | Weakened version of ~ ax-7... |
| ax7v1 2006 | First of two weakened vers... |
| ax7v2 2007 | Second of two weakened ver... |
| equid 2008 | Identity law for equality.... |
| nfequid 2009 | Bound-variable hypothesis ... |
| equcomiv 2010 | Weaker form of ~ equcomi w... |
| ax6evr 2011 | A commuted form of ~ ax6ev... |
| ax7 2012 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2013 | Commutative law for equali... |
| equcom 2014 | Commutative law for equali... |
| equcomd 2015 | Deduction form of ~ equcom... |
| equcoms 2016 | An inference commuting equ... |
| equtr 2017 | A transitive law for equal... |
| equtrr 2018 | A transitive law for equal... |
| equeuclr 2019 | Commuted version of ~ eque... |
| equeucl 2020 | Equality is a left-Euclide... |
| equequ1 2021 | An equivalence law for equ... |
| equequ2 2022 | An equivalence law for equ... |
| equtr2 2023 | Equality is a left-Euclide... |
| stdpc6 2024 | One of the two equality ax... |
| equvinv 2025 | A variable introduction la... |
| equvinva 2026 | A modified version of the ... |
| equvelv 2027 | A biconditional form of ~ ... |
| ax13b 2028 | An equivalence between two... |
| spfw 2029 | Weak version of ~ sp . Us... |
| spw 2030 | Weak version of the specia... |
| cbvalw 2031 | Change bound variable. Us... |
| cbvalvw 2032 | Change bound variable. Us... |
| cbvexvw 2033 | Change bound variable. Us... |
| cbvaldvaw 2034 | Rule used to change the bo... |
| cbvexdvaw 2035 | Rule used to change the bo... |
| cbval2vw 2036 | Rule used to change bound ... |
| cbvex2vw 2037 | Rule used to change bound ... |
| cbvex4vw 2038 | Rule used to change bound ... |
| alcomiw 2039 | Weak version of ~ ax-11 . ... |
| alcomw 2040 | Weak version of ~ alcom an... |
| hbn1fw 2041 | Weak version of ~ ax-10 fr... |
| hbn1w 2042 | Weak version of ~ hbn1 . ... |
| hba1w 2043 | Weak version of ~ hba1 . ... |
| hbe1w 2044 | Weak version of ~ hbe1 . ... |
| hbalw 2045 | Weak version of ~ hbal . ... |
| 19.8aw 2046 | If a formula is true, then... |
| exexw 2047 | Existential quantification... |
| spaev 2048 | A special instance of ~ sp... |
| cbvaev 2049 | Change bound variable in a... |
| aevlem0 2050 | Lemma for ~ aevlem . Inst... |
| aevlem 2051 | Lemma for ~ aev and ~ axc1... |
| aeveq 2052 | The antecedent ` A. x x = ... |
| aev 2053 | A "distinctor elimination"... |
| aev2 2054 | A version of ~ aev with tw... |
| hbaev 2055 | All variables are effectiv... |
| naev 2056 | If some set variables can ... |
| naev2 2057 | Generalization of ~ hbnaev... |
| hbnaev 2058 | Any variable is free in ` ... |
| sbjust 2059 | Justification theorem for ... |
| sbt 2062 | A substitution into a theo... |
| sbtru 2063 | The result of substituting... |
| stdpc4 2064 | The specialization axiom o... |
| sbtALT 2065 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2066 | A double specialization us... |
| sbi1 2067 | Distribute substitution ov... |
| spsbim 2068 | Distribute substitution ov... |
| spsbbi 2069 | Biconditional property for... |
| sbimi 2070 | Distribute substitution ov... |
| sb2imi 2071 | Distribute substitution ov... |
| sbbii 2072 | Infer substitution into bo... |
| 2sbbii 2073 | Infer double substitution ... |
| sbimdv 2074 | Deduction substituting bot... |
| sbbidv 2075 | Deduction substituting bot... |
| sban 2076 | Conjunction inside and out... |
| sb3an 2077 | Threefold conjunction insi... |
| spsbe 2078 | Existential generalization... |
| sbequ 2079 | Equality property for subs... |
| sbequi 2080 | An equality theorem for su... |
| sb6 2081 | Alternate definition of su... |
| 2sb6 2082 | Equivalence for double sub... |
| sb1v 2083 | One direction of ~ sb5 , p... |
| sbv 2084 | Substitution for a variabl... |
| sbcom4 2085 | Commutativity law for subs... |
| pm11.07 2086 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2087 | Substitution in an implica... |
| sbievw 2088 | Conversion of implicit sub... |
| sbiedvw 2089 | Conversion of implicit sub... |
| 2sbievw 2090 | Conversion of double impli... |
| sbcom3vv 2091 | Substituting ` y ` for ` x... |
| sbievw2 2092 | ~ sbievw applied twice, av... |
| sbco2vv 2093 | A composition law for subs... |
| equsb3 2094 | Substitution in an equalit... |
| equsb3r 2095 | Substitution applied to th... |
| equsb1v 2096 | Substitution applied to an... |
| nsb 2097 | Any substitution in an alw... |
| sbn1 2098 | One direction of ~ sbn , u... |
| wel 2100 | Extend wff definition to i... |
| ax8v 2102 | Weakened version of ~ ax-8... |
| ax8v1 2103 | First of two weakened vers... |
| ax8v2 2104 | Second of two weakened ver... |
| ax8 2105 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2106 | An identity law for the no... |
| elsb1 2107 | Substitution for the first... |
| cleljust 2108 | When the class variables i... |
| ax9v 2110 | Weakened version of ~ ax-9... |
| ax9v1 2111 | First of two weakened vers... |
| ax9v2 2112 | Second of two weakened ver... |
| ax9 2113 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2114 | An identity law for the no... |
| elequ2g 2115 | A form of ~ elequ2 with a ... |
| elsb2 2116 | Substitution for the secon... |
| ax6dgen 2117 | Tarski's system uses the w... |
| ax10w 2118 | Weak version of ~ ax-10 fr... |
| ax11w 2119 | Weak version of ~ ax-11 fr... |
| ax11dgen 2120 | Degenerate instance of ~ a... |
| ax12wlem 2121 | Lemma for weak version of ... |
| ax12w 2122 | Weak version of ~ ax-12 fr... |
| ax12dgen 2123 | Degenerate instance of ~ a... |
| ax12wdemo 2124 | Example of an application ... |
| ax13w 2125 | Weak version (principal in... |
| ax13dgen1 2126 | Degenerate instance of ~ a... |
| ax13dgen2 2127 | Degenerate instance of ~ a... |
| ax13dgen3 2128 | Degenerate instance of ~ a... |
| ax13dgen4 2129 | Degenerate instance of ~ a... |
| hbn1 2131 | Alias for ~ ax-10 to be us... |
| hbe1 2132 | The setvar ` x ` is not fr... |
| hbe1a 2133 | Dual statement of ~ hbe1 .... |
| nf5-1 2134 | One direction of ~ nf5 can... |
| nf5i 2135 | Deduce that ` x ` is not f... |
| nf5dh 2136 | Deduce that ` x ` is not f... |
| nf5dv 2137 | Apply the definition of no... |
| nfnaew 2138 | All variables are effectiv... |
| nfnaewOLD 2139 | Obsolete version of ~ nfna... |
| nfe1 2140 | The setvar ` x ` is not fr... |
| nfa1 2141 | The setvar ` x ` is not fr... |
| nfna1 2142 | A convenience theorem part... |
| nfia1 2143 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2144 | The setvar ` x ` is not fr... |
| modal5 2145 | The analogue in our predic... |
| nfs1v 2146 | The setvar ` x ` is not fr... |
| alcoms 2148 | Swap quantifiers in an ant... |
| alcom 2149 | Theorem 19.5 of [Margaris]... |
| alrot3 2150 | Theorem *11.21 in [Whitehe... |
| alrot4 2151 | Rotate four universal quan... |
| sbal 2152 | Move universal quantifier ... |
| sbalv 2153 | Quantify with new variable... |
| sbcom2 2154 | Commutativity law for subs... |
| excom 2155 | Theorem 19.11 of [Margaris... |
| excomim 2156 | One direction of Theorem 1... |
| excom13 2157 | Swap 1st and 3rd existenti... |
| exrot3 2158 | Rotate existential quantif... |
| exrot4 2159 | Rotate existential quantif... |
| hbal 2160 | If ` x ` is not free in ` ... |
| hbald 2161 | Deduction form of bound-va... |
| hbsbw 2162 | If ` z ` is not free in ` ... |
| nfa2 2163 | Lemma 24 of [Monk2] p. 114... |
| ax12v 2165 | This is essentially Axiom ... |
| ax12v2 2166 | It is possible to remove a... |
| 19.8a 2167 | If a wff is true, it is tr... |
| 19.8ad 2168 | If a wff is true, it is tr... |
| sp 2169 | Specialization. A univers... |
| spi 2170 | Inference rule of universa... |
| sps 2171 | Generalization of antecede... |
| 2sp 2172 | A double specialization (s... |
| spsd 2173 | Deduction generalizing ant... |
| 19.2g 2174 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2175 | Inference form of ~ 19.21 ... |
| 19.21bbi 2176 | Inference removing two uni... |
| 19.23bi 2177 | Inference form of Theorem ... |
| nexr 2178 | Inference associated with ... |
| qexmid 2179 | Quantified excluded middle... |
| nf5r 2180 | Consequence of the definit... |
| nf5ri 2181 | Consequence of the definit... |
| nf5rd 2182 | Consequence of the definit... |
| spimedv 2183 | Deduction version of ~ spi... |
| spimefv 2184 | Version of ~ spime with a ... |
| nfim1 2185 | A closed form of ~ nfim . ... |
| nfan1 2186 | A closed form of ~ nfan . ... |
| 19.3t 2187 | Closed form of ~ 19.3 and ... |
| 19.3 2188 | A wff may be quantified wi... |
| 19.9d 2189 | A deduction version of one... |
| 19.9t 2190 | Closed form of ~ 19.9 and ... |
| 19.9 2191 | A wff may be existentially... |
| 19.21t 2192 | Closed form of Theorem 19.... |
| 19.21 2193 | Theorem 19.21 of [Margaris... |
| stdpc5 2194 | An axiom scheme of standar... |
| 19.21-2 2195 | Version of ~ 19.21 with tw... |
| 19.23t 2196 | Closed form of Theorem 19.... |
| 19.23 2197 | Theorem 19.23 of [Margaris... |
| alimd 2198 | Deduction form of Theorem ... |
| alrimi 2199 | Inference form of Theorem ... |
| alrimdd 2200 | Deduction form of Theorem ... |
| alrimd 2201 | Deduction form of Theorem ... |
| eximd 2202 | Deduction form of Theorem ... |
| exlimi 2203 | Inference associated with ... |
| exlimd 2204 | Deduction form of Theorem ... |
| exlimimdd 2205 | Existential elimination ru... |
| exlimdd 2206 | Existential elimination ru... |
| nexd 2207 | Deduction for generalizati... |
| albid 2208 | Formula-building rule for ... |
| exbid 2209 | Formula-building rule for ... |
| nfbidf 2210 | An equality theorem for ef... |
| 19.16 2211 | Theorem 19.16 of [Margaris... |
| 19.17 2212 | Theorem 19.17 of [Margaris... |
| 19.27 2213 | Theorem 19.27 of [Margaris... |
| 19.28 2214 | Theorem 19.28 of [Margaris... |
| 19.19 2215 | Theorem 19.19 of [Margaris... |
| 19.36 2216 | Theorem 19.36 of [Margaris... |
| 19.36i 2217 | Inference associated with ... |
| 19.37 2218 | Theorem 19.37 of [Margaris... |
| 19.32 2219 | Theorem 19.32 of [Margaris... |
| 19.31 2220 | Theorem 19.31 of [Margaris... |
| 19.41 2221 | Theorem 19.41 of [Margaris... |
| 19.42 2222 | Theorem 19.42 of [Margaris... |
| 19.44 2223 | Theorem 19.44 of [Margaris... |
| 19.45 2224 | Theorem 19.45 of [Margaris... |
| spimfv 2225 | Specialization, using impl... |
| chvarfv 2226 | Implicit substitution of `... |
| cbv3v2 2227 | Version of ~ cbv3 with two... |
| sbalex 2228 | Equivalence of two ways to... |
| sb4av 2229 | Version of ~ sb4a with a d... |
| sbimd 2230 | Deduction substituting bot... |
| sbbid 2231 | Deduction substituting bot... |
| 2sbbid 2232 | Deduction doubly substitut... |
| sbequ1 2233 | An equality theorem for su... |
| sbequ2 2234 | An equality theorem for su... |
| stdpc7 2235 | One of the two equality ax... |
| sbequ12 2236 | An equality theorem for su... |
| sbequ12r 2237 | An equality theorem for su... |
| sbelx 2238 | Elimination of substitutio... |
| sbequ12a 2239 | An equality theorem for su... |
| sbid 2240 | An identity theorem for su... |
| sbcov 2241 | A composition law for subs... |
| sb6a 2242 | Equivalence for substituti... |
| sbid2vw 2243 | Reverting substitution yie... |
| axc16g 2244 | Generalization of ~ axc16 ... |
| axc16 2245 | Proof of older axiom ~ ax-... |
| axc16gb 2246 | Biconditional strengthenin... |
| axc16nf 2247 | If ~ dtru is false, then t... |
| axc11v 2248 | Version of ~ axc11 with a ... |
| axc11rv 2249 | Version of ~ axc11r with a... |
| drsb2 2250 | Formula-building lemma for... |
| equsalv 2251 | An equivalence related to ... |
| equsexv 2252 | An equivalence related to ... |
| equsexvOLD 2253 | Obsolete version of ~ equs... |
| sbft 2254 | Substitution has no effect... |
| sbf 2255 | Substitution for a variabl... |
| sbf2 2256 | Substitution has no effect... |
| sbh 2257 | Substitution for a variabl... |
| hbs1 2258 | The setvar ` x ` is not fr... |
| nfs1f 2259 | If ` x ` is not free in ` ... |
| sb5 2260 | Alternate definition of su... |
| sb5OLD 2261 | Obsolete version of ~ sb5 ... |
| sb56OLD 2262 | Obsolete version of ~ sbal... |
| equs5av 2263 | A property related to subs... |
| 2sb5 2264 | Equivalence for double sub... |
| sbco4lem 2265 | Lemma for ~ sbco4 . It re... |
| sbco4lemOLD 2266 | Obsolete version of ~ sbco... |
| sbco4 2267 | Two ways of exchanging two... |
| dfsb7 2268 | An alternate definition of... |
| sbn 2269 | Negation inside and outsid... |
| sbex 2270 | Move existential quantifie... |
| nf5 2271 | Alternate definition of ~ ... |
| nf6 2272 | An alternate definition of... |
| nf5d 2273 | Deduce that ` x ` is not f... |
| nf5di 2274 | Since the converse holds b... |
| 19.9h 2275 | A wff may be existentially... |
| 19.21h 2276 | Theorem 19.21 of [Margaris... |
| 19.23h 2277 | Theorem 19.23 of [Margaris... |
| exlimih 2278 | Inference associated with ... |
| exlimdh 2279 | Deduction form of Theorem ... |
| equsalhw 2280 | Version of ~ equsalh with ... |
| equsexhv 2281 | An equivalence related to ... |
| hba1 2282 | The setvar ` x ` is not fr... |
| hbnt 2283 | Closed theorem version of ... |
| hbn 2284 | If ` x ` is not free in ` ... |
| hbnd 2285 | Deduction form of bound-va... |
| hbim1 2286 | A closed form of ~ hbim . ... |
| hbimd 2287 | Deduction form of bound-va... |
| hbim 2288 | If ` x ` is not free in ` ... |
| hban 2289 | If ` x ` is not free in ` ... |
| hb3an 2290 | If ` x ` is not free in ` ... |
| sbi2 2291 | Introduction of implicatio... |
| sbim 2292 | Implication inside and out... |
| sbrim 2293 | Substitution in an implica... |
| sbrimOLD 2294 | Obsolete version of ~ sbri... |
| sblim 2295 | Substitution in an implica... |
| sbor 2296 | Disjunction inside and out... |
| sbbi 2297 | Equivalence inside and out... |
| sblbis 2298 | Introduce left bicondition... |
| sbrbis 2299 | Introduce right biconditio... |
| sbrbif 2300 | Introduce right biconditio... |
| sbnf 2301 | Move nonfree predicate in ... |
| sbnfOLD 2302 | Obsolete version of ~ sbnf... |
| sbiev 2303 | Conversion of implicit sub... |
| sbiedw 2304 | Conversion of implicit sub... |
| axc7 2305 | Show that the original axi... |
| axc7e 2306 | Abbreviated version of ~ a... |
| modal-b 2307 | The analogue in our predic... |
| 19.9ht 2308 | A closed version of ~ 19.9... |
| axc4 2309 | Show that the original axi... |
| axc4i 2310 | Inference version of ~ axc... |
| nfal 2311 | If ` x ` is not free in ` ... |
| nfex 2312 | If ` x ` is not free in ` ... |
| hbex 2313 | If ` x ` is not free in ` ... |
| nfnf 2314 | If ` x ` is not free in ` ... |
| 19.12 2315 | Theorem 19.12 of [Margaris... |
| nfald 2316 | Deduction form of ~ nfal .... |
| nfexd 2317 | If ` x ` is not free in ` ... |
| nfsbv 2318 | If ` z ` is not free in ` ... |
| nfsbvOLD 2319 | Obsolete version of ~ nfsb... |
| hbsbwOLD 2320 | Obsolete version of ~ hbsb... |
| sbco2v 2321 | A composition law for subs... |
| aaan 2322 | Distribute universal quant... |
| aaanOLD 2323 | Obsolete version of ~ aaan... |
| eeor 2324 | Distribute existential qua... |
| eeorOLD 2325 | Obsolete version of ~ eeor... |
| cbv3v 2326 | Rule used to change bound ... |
| cbv1v 2327 | Rule used to change bound ... |
| cbv2w 2328 | Rule used to change bound ... |
| cbvaldw 2329 | Deduction used to change b... |
| cbvexdw 2330 | Deduction used to change b... |
| cbv3hv 2331 | Rule used to change bound ... |
| cbvalv1 2332 | Rule used to change bound ... |
| cbvexv1 2333 | Rule used to change bound ... |
| cbval2v 2334 | Rule used to change bound ... |
| cbvex2v 2335 | Rule used to change bound ... |
| dvelimhw 2336 | Proof of ~ dvelimh without... |
| pm11.53 2337 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2338 | Special case of ~ 19.12 wh... |
| eean 2339 | Distribute existential qua... |
| eeanv 2340 | Distribute a pair of exist... |
| eeeanv 2341 | Distribute three existenti... |
| ee4anv 2342 | Distribute two pairs of ex... |
| sb8v 2343 | Substitution of variable i... |
| sb8f 2344 | Substitution of variable i... |
| sb8fOLD 2345 | Obsolete version of ~ sb8f... |
| sb8ef 2346 | Substitution of variable i... |
| 2sb8ef 2347 | An equivalent expression f... |
| sb6rfv 2348 | Reversed substitution. Ve... |
| sbnf2 2349 | Two ways of expressing " `... |
| exsb 2350 | An equivalent expression f... |
| 2exsb 2351 | An equivalent expression f... |
| sbbib 2352 | Reversal of substitution. ... |
| sbbibvv 2353 | Reversal of substitution. ... |
| cbvsbv 2354 | Change the bound variable ... |
| cbvsbvf 2355 | Change the bound variable ... |
| cleljustALT 2356 | Alternate proof of ~ clelj... |
| cleljustALT2 2357 | Alternate proof of ~ clelj... |
| equs5aALT 2358 | Alternate proof of ~ equs5... |
| equs5eALT 2359 | Alternate proof of ~ equs5... |
| axc11r 2360 | Same as ~ axc11 but with r... |
| dral1v 2361 | Formula-building lemma for... |
| dral1vOLD 2362 | Obsolete version of ~ dral... |
| drex1v 2363 | Formula-building lemma for... |
| drnf1v 2364 | Formula-building lemma for... |
| drnf1vOLD 2365 | Obsolete version of ~ drnf... |
| ax13v 2367 | A weaker version of ~ ax-1... |
| ax13lem1 2368 | A version of ~ ax13v with ... |
| ax13 2369 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2370 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2371 | An equation between setvar... |
| dveeq2 2372 | Quantifier introduction wh... |
| nfeqf1 2373 | An equation between setvar... |
| dveeq1 2374 | Quantifier introduction wh... |
| nfeqf 2375 | A variable is effectively ... |
| axc9 2376 | Derive set.mm's original ~... |
| ax6e 2377 | At least one individual ex... |
| ax6 2378 | Theorem showing that ~ ax-... |
| axc10 2379 | Show that the original axi... |
| spimt 2380 | Closed theorem form of ~ s... |
| spim 2381 | Specialization, using impl... |
| spimed 2382 | Deduction version of ~ spi... |
| spime 2383 | Existential introduction, ... |
| spimv 2384 | A version of ~ spim with a... |
| spimvALT 2385 | Alternate proof of ~ spimv... |
| spimev 2386 | Distinct-variable version ... |
| spv 2387 | Specialization, using impl... |
| spei 2388 | Inference from existential... |
| chvar 2389 | Implicit substitution of `... |
| chvarv 2390 | Implicit substitution of `... |
| cbv3 2391 | Rule used to change bound ... |
| cbval 2392 | Rule used to change bound ... |
| cbvex 2393 | Rule used to change bound ... |
| cbvalv 2394 | Rule used to change bound ... |
| cbvexv 2395 | Rule used to change bound ... |
| cbv1 2396 | Rule used to change bound ... |
| cbv2 2397 | Rule used to change bound ... |
| cbv3h 2398 | Rule used to change bound ... |
| cbv1h 2399 | Rule used to change bound ... |
| cbv2h 2400 | Rule used to change bound ... |
| cbvald 2401 | Deduction used to change b... |
| cbvexd 2402 | Deduction used to change b... |
| cbvaldva 2403 | Rule used to change the bo... |
| cbvexdva 2404 | Rule used to change the bo... |
| cbval2 2405 | Rule used to change bound ... |
| cbvex2 2406 | Rule used to change bound ... |
| cbval2vv 2407 | Rule used to change bound ... |
| cbvex2vv 2408 | Rule used to change bound ... |
| cbvex4v 2409 | Rule used to change bound ... |
| equs4 2410 | Lemma used in proofs of im... |
| equsal 2411 | An equivalence related to ... |
| equsex 2412 | An equivalence related to ... |
| equsexALT 2413 | Alternate proof of ~ equse... |
| equsalh 2414 | An equivalence related to ... |
| equsexh 2415 | An equivalence related to ... |
| axc15 2416 | Derivation of set.mm's ori... |
| ax12 2417 | Rederivation of Axiom ~ ax... |
| ax12b 2418 | A bidirectional version of... |
| ax13ALT 2419 | Alternate proof of ~ ax13 ... |
| axc11n 2420 | Derive set.mm's original ~... |
| aecom 2421 | Commutation law for identi... |
| aecoms 2422 | A commutation rule for ide... |
| naecoms 2423 | A commutation rule for dis... |
| axc11 2424 | Show that ~ ax-c11 can be ... |
| hbae 2425 | All variables are effectiv... |
| hbnae 2426 | All variables are effectiv... |
| nfae 2427 | All variables are effectiv... |
| nfnae 2428 | All variables are effectiv... |
| hbnaes 2429 | Rule that applies ~ hbnae ... |
| axc16i 2430 | Inference with ~ axc16 as ... |
| axc16nfALT 2431 | Alternate proof of ~ axc16... |
| dral2 2432 | Formula-building lemma for... |
| dral1 2433 | Formula-building lemma for... |
| dral1ALT 2434 | Alternate proof of ~ dral1... |
| drex1 2435 | Formula-building lemma for... |
| drex2 2436 | Formula-building lemma for... |
| drnf1 2437 | Formula-building lemma for... |
| drnf2 2438 | Formula-building lemma for... |
| nfald2 2439 | Variation on ~ nfald which... |
| nfexd2 2440 | Variation on ~ nfexd which... |
| exdistrf 2441 | Distribution of existentia... |
| dvelimf 2442 | Version of ~ dvelimv witho... |
| dvelimdf 2443 | Deduction form of ~ dvelim... |
| dvelimh 2444 | Version of ~ dvelim withou... |
| dvelim 2445 | This theorem can be used t... |
| dvelimv 2446 | Similar to ~ dvelim with f... |
| dvelimnf 2447 | Version of ~ dvelim using ... |
| dveeq2ALT 2448 | Alternate proof of ~ dveeq... |
| equvini 2449 | A variable introduction la... |
| equvel 2450 | A variable elimination law... |
| equs5a 2451 | A property related to subs... |
| equs5e 2452 | A property related to subs... |
| equs45f 2453 | Two ways of expressing sub... |
| equs5 2454 | Lemma used in proofs of su... |
| dveel1 2455 | Quantifier introduction wh... |
| dveel2 2456 | Quantifier introduction wh... |
| axc14 2457 | Axiom ~ ax-c14 is redundan... |
| sb6x 2458 | Equivalence involving subs... |
| sbequ5 2459 | Substitution does not chan... |
| sbequ6 2460 | Substitution does not chan... |
| sb5rf 2461 | Reversed substitution. Us... |
| sb6rf 2462 | Reversed substitution. Fo... |
| ax12vALT 2463 | Alternate proof of ~ ax12v... |
| 2ax6elem 2464 | We can always find values ... |
| 2ax6e 2465 | We can always find values ... |
| 2sb5rf 2466 | Reversed double substituti... |
| 2sb6rf 2467 | Reversed double substituti... |
| sbel2x 2468 | Elimination of double subs... |
| sb4b 2469 | Simplified definition of s... |
| sb3b 2470 | Simplified definition of s... |
| sb3 2471 | One direction of a simplif... |
| sb1 2472 | One direction of a simplif... |
| sb2 2473 | One direction of a simplif... |
| sb4a 2474 | A version of one implicati... |
| dfsb1 2475 | Alternate definition of su... |
| hbsb2 2476 | Bound-variable hypothesis ... |
| nfsb2 2477 | Bound-variable hypothesis ... |
| hbsb2a 2478 | Special case of a bound-va... |
| sb4e 2479 | One direction of a simplif... |
| hbsb2e 2480 | Special case of a bound-va... |
| hbsb3 2481 | If ` y ` is not free in ` ... |
| nfs1 2482 | If ` y ` is not free in ` ... |
| axc16ALT 2483 | Alternate proof of ~ axc16... |
| axc16gALT 2484 | Alternate proof of ~ axc16... |
| equsb1 2485 | Substitution applied to an... |
| equsb2 2486 | Substitution applied to an... |
| dfsb2 2487 | An alternate definition of... |
| dfsb3 2488 | An alternate definition of... |
| drsb1 2489 | Formula-building lemma for... |
| sb2ae 2490 | In the case of two success... |
| sb6f 2491 | Equivalence for substituti... |
| sb5f 2492 | Equivalence for substituti... |
| nfsb4t 2493 | A variable not free in a p... |
| nfsb4 2494 | A variable not free in a p... |
| sbequ8 2495 | Elimination of equality fr... |
| sbie 2496 | Conversion of implicit sub... |
| sbied 2497 | Conversion of implicit sub... |
| sbiedv 2498 | Conversion of implicit sub... |
| 2sbiev 2499 | Conversion of double impli... |
| sbcom3 2500 | Substituting ` y ` for ` x... |
| sbco 2501 | A composition law for subs... |
| sbid2 2502 | An identity law for substi... |
| sbid2v 2503 | An identity law for substi... |
| sbidm 2504 | An idempotent law for subs... |
| sbco2 2505 | A composition law for subs... |
| sbco2d 2506 | A composition law for subs... |
| sbco3 2507 | A composition law for subs... |
| sbcom 2508 | A commutativity law for su... |
| sbtrt 2509 | Partially closed form of ~... |
| sbtr 2510 | A partial converse to ~ sb... |
| sb8 2511 | Substitution of variable i... |
| sb8e 2512 | Substitution of variable i... |
| sb9 2513 | Commutation of quantificat... |
| sb9i 2514 | Commutation of quantificat... |
| sbhb 2515 | Two ways of expressing " `... |
| nfsbd 2516 | Deduction version of ~ nfs... |
| nfsb 2517 | If ` z ` is not free in ` ... |
| hbsb 2518 | If ` z ` is not free in ` ... |
| sb7f 2519 | This version of ~ dfsb7 do... |
| sb7h 2520 | This version of ~ dfsb7 do... |
| sb10f 2521 | Hao Wang's identity axiom ... |
| sbal1 2522 | Check out ~ sbal for a ver... |
| sbal2 2523 | Move quantifier in and out... |
| 2sb8e 2524 | An equivalent expression f... |
| dfmoeu 2525 | An elementary proof of ~ m... |
| dfeumo 2526 | An elementary proof showin... |
| mojust 2528 | Soundness justification th... |
| nexmo 2530 | Nonexistence implies uniqu... |
| exmo 2531 | Any proposition holds for ... |
| moabs 2532 | Absorption of existence co... |
| moim 2533 | The at-most-one quantifier... |
| moimi 2534 | The at-most-one quantifier... |
| moimdv 2535 | The at-most-one quantifier... |
| mobi 2536 | Equivalence theorem for th... |
| mobii 2537 | Formula-building rule for ... |
| mobidv 2538 | Formula-building rule for ... |
| mobid 2539 | Formula-building rule for ... |
| moa1 2540 | If an implication holds fo... |
| moan 2541 | "At most one" is still the... |
| moani 2542 | "At most one" is still tru... |
| moor 2543 | "At most one" is still the... |
| mooran1 2544 | "At most one" imports disj... |
| mooran2 2545 | "At most one" exports disj... |
| nfmo1 2546 | Bound-variable hypothesis ... |
| nfmod2 2547 | Bound-variable hypothesis ... |
| nfmodv 2548 | Bound-variable hypothesis ... |
| nfmov 2549 | Bound-variable hypothesis ... |
| nfmod 2550 | Bound-variable hypothesis ... |
| nfmo 2551 | Bound-variable hypothesis ... |
| mof 2552 | Version of ~ df-mo with di... |
| mo3 2553 | Alternate definition of th... |
| mo 2554 | Equivalent definitions of ... |
| mo4 2555 | At-most-one quantifier exp... |
| mo4f 2556 | At-most-one quantifier exp... |
| eu3v 2559 | An alternate way to expres... |
| eujust 2560 | Soundness justification th... |
| eujustALT 2561 | Alternate proof of ~ eujus... |
| eu6lem 2562 | Lemma of ~ eu6im . A diss... |
| eu6 2563 | Alternate definition of th... |
| eu6im 2564 | One direction of ~ eu6 nee... |
| euf 2565 | Version of ~ eu6 with disj... |
| euex 2566 | Existential uniqueness imp... |
| eumo 2567 | Existential uniqueness imp... |
| eumoi 2568 | Uniqueness inferred from e... |
| exmoeub 2569 | Existence implies that uni... |
| exmoeu 2570 | Existence is equivalent to... |
| moeuex 2571 | Uniqueness implies that ex... |
| moeu 2572 | Uniqueness is equivalent t... |
| eubi 2573 | Equivalence theorem for th... |
| eubii 2574 | Introduce unique existenti... |
| eubidv 2575 | Formula-building rule for ... |
| eubid 2576 | Formula-building rule for ... |
| nfeu1 2577 | Bound-variable hypothesis ... |
| nfeu1ALT 2578 | Alternate proof of ~ nfeu1... |
| nfeud2 2579 | Bound-variable hypothesis ... |
| nfeudw 2580 | Bound-variable hypothesis ... |
| nfeud 2581 | Bound-variable hypothesis ... |
| nfeuw 2582 | Bound-variable hypothesis ... |
| nfeu 2583 | Bound-variable hypothesis ... |
| dfeu 2584 | Rederive ~ df-eu from the ... |
| dfmo 2585 | Rederive ~ df-mo from the ... |
| euequ 2586 | There exists a unique set ... |
| sb8eulem 2587 | Lemma. Factor out the com... |
| sb8euv 2588 | Variable substitution in u... |
| sb8eu 2589 | Variable substitution in u... |
| sb8mo 2590 | Variable substitution for ... |
| cbvmovw 2591 | Change bound variable. Us... |
| cbvmow 2592 | Rule used to change bound ... |
| cbvmowOLD 2593 | Obsolete version of ~ cbvm... |
| cbvmo 2594 | Rule used to change bound ... |
| cbveuvw 2595 | Change bound variable. Us... |
| cbveuw 2596 | Version of ~ cbveu with a ... |
| cbveuwOLD 2597 | Obsolete version of ~ cbve... |
| cbveu 2598 | Rule used to change bound ... |
| cbveuALT 2599 | Alternative proof of ~ cbv... |
| eu2 2600 | An alternate way of defini... |
| eu1 2601 | An alternate way to expres... |
| euor 2602 | Introduce a disjunct into ... |
| euorv 2603 | Introduce a disjunct into ... |
| euor2 2604 | Introduce or eliminate a d... |
| sbmo 2605 | Substitution into an at-mo... |
| eu4 2606 | Uniqueness using implicit ... |
| euimmo 2607 | Existential uniqueness imp... |
| euim 2608 | Add unique existential qua... |
| moanimlem 2609 | Factor out the common proo... |
| moanimv 2610 | Introduction of a conjunct... |
| moanim 2611 | Introduction of a conjunct... |
| euan 2612 | Introduction of a conjunct... |
| moanmo 2613 | Nested at-most-one quantif... |
| moaneu 2614 | Nested at-most-one and uni... |
| euanv 2615 | Introduction of a conjunct... |
| mopick 2616 | "At most one" picks a vari... |
| moexexlem 2617 | Factor out the proof skele... |
| 2moexv 2618 | Double quantification with... |
| moexexvw 2619 | "At most one" double quant... |
| 2moswapv 2620 | A condition allowing to sw... |
| 2euswapv 2621 | A condition allowing to sw... |
| 2euexv 2622 | Double quantification with... |
| 2exeuv 2623 | Double existential uniquen... |
| eupick 2624 | Existential uniqueness "pi... |
| eupicka 2625 | Version of ~ eupick with c... |
| eupickb 2626 | Existential uniqueness "pi... |
| eupickbi 2627 | Theorem *14.26 in [Whitehe... |
| mopick2 2628 | "At most one" can show the... |
| moexex 2629 | "At most one" double quant... |
| moexexv 2630 | "At most one" double quant... |
| 2moex 2631 | Double quantification with... |
| 2euex 2632 | Double quantification with... |
| 2eumo 2633 | Nested unique existential ... |
| 2eu2ex 2634 | Double existential uniquen... |
| 2moswap 2635 | A condition allowing to sw... |
| 2euswap 2636 | A condition allowing to sw... |
| 2exeu 2637 | Double existential uniquen... |
| 2mo2 2638 | Two ways of expressing "th... |
| 2mo 2639 | Two ways of expressing "th... |
| 2mos 2640 | Double "there exists at mo... |
| 2eu1 2641 | Double existential uniquen... |
| 2eu1v 2642 | Double existential uniquen... |
| 2eu2 2643 | Double existential uniquen... |
| 2eu3 2644 | Double existential uniquen... |
| 2eu4 2645 | This theorem provides us w... |
| 2eu5 2646 | An alternate definition of... |
| 2eu6 2647 | Two equivalent expressions... |
| 2eu7 2648 | Two equivalent expressions... |
| 2eu8 2649 | Two equivalent expressions... |
| euae 2650 | Two ways to express "exact... |
| exists1 2651 | Two ways to express "exact... |
| exists2 2652 | A condition implying that ... |
| barbara 2653 | "Barbara", one of the fund... |
| celarent 2654 | "Celarent", one of the syl... |
| darii 2655 | "Darii", one of the syllog... |
| dariiALT 2656 | Alternate proof of ~ darii... |
| ferio 2657 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2658 | Lemma for ~ barbari and th... |
| barbari 2659 | "Barbari", one of the syll... |
| barbariALT 2660 | Alternate proof of ~ barba... |
| celaront 2661 | "Celaront", one of the syl... |
| cesare 2662 | "Cesare", one of the syllo... |
| camestres 2663 | "Camestres", one of the sy... |
| festino 2664 | "Festino", one of the syll... |
| festinoALT 2665 | Alternate proof of ~ festi... |
| baroco 2666 | "Baroco", one of the syllo... |
| barocoALT 2667 | Alternate proof of ~ festi... |
| cesaro 2668 | "Cesaro", one of the syllo... |
| camestros 2669 | "Camestros", one of the sy... |
| datisi 2670 | "Datisi", one of the syllo... |
| disamis 2671 | "Disamis", one of the syll... |
| ferison 2672 | "Ferison", one of the syll... |
| bocardo 2673 | "Bocardo", one of the syll... |
| darapti 2674 | "Darapti", one of the syll... |
| daraptiALT 2675 | Alternate proof of ~ darap... |
| felapton 2676 | "Felapton", one of the syl... |
| calemes 2677 | "Calemes", one of the syll... |
| dimatis 2678 | "Dimatis", one of the syll... |
| fresison 2679 | "Fresison", one of the syl... |
| calemos 2680 | "Calemos", one of the syll... |
| fesapo 2681 | "Fesapo", one of the syllo... |
| bamalip 2682 | "Bamalip", one of the syll... |
| axia1 2683 | Left 'and' elimination (in... |
| axia2 2684 | Right 'and' elimination (i... |
| axia3 2685 | 'And' introduction (intuit... |
| axin1 2686 | 'Not' introduction (intuit... |
| axin2 2687 | 'Not' elimination (intuiti... |
| axio 2688 | Definition of 'or' (intuit... |
| axi4 2689 | Specialization (intuitioni... |
| axi5r 2690 | Converse of ~ axc4 (intuit... |
| axial 2691 | The setvar ` x ` is not fr... |
| axie1 2692 | The setvar ` x ` is not fr... |
| axie2 2693 | A key property of existent... |
| axi9 2694 | Axiom of existence (intuit... |
| axi10 2695 | Axiom of Quantifier Substi... |
| axi12 2696 | Axiom of Quantifier Introd... |
| axbnd 2697 | Axiom of Bundling (intuiti... |
| axexte 2699 | The axiom of extensionalit... |
| axextg 2700 | A generalization of the ax... |
| axextb 2701 | A bidirectional version of... |
| axextmo 2702 | There exists at most one s... |
| nulmo 2703 | There exists at most one e... |
| eleq1ab 2706 | Extension (in the sense of... |
| cleljustab 2707 | Extension of ~ cleljust fr... |
| abid 2708 | Simplification of class ab... |
| vexwt 2709 | A standard theorem of pred... |
| vexw 2710 | If ` ph ` is a theorem, th... |
| vextru 2711 | Every setvar is a member o... |
| nfsab1 2712 | Bound-variable hypothesis ... |
| hbab1 2713 | Bound-variable hypothesis ... |
| hbab1OLD 2714 | Obsolete version of ~ hbab... |
| hbab 2715 | Bound-variable hypothesis ... |
| hbabg 2716 | Bound-variable hypothesis ... |
| nfsab 2717 | Bound-variable hypothesis ... |
| nfsabg 2718 | Bound-variable hypothesis ... |
| dfcleq 2720 | The defining characterizat... |
| cvjust 2721 | Every set is a class. Pro... |
| ax9ALT 2722 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2723 | A weaker version of ~ eleq... |
| eqriv 2724 | Infer equality of classes ... |
| eqrdv 2725 | Deduce equality of classes... |
| eqrdav 2726 | Deduce equality of classes... |
| eqid 2727 | Law of identity (reflexivi... |
| eqidd 2728 | Class identity law with an... |
| eqeq1d 2729 | Deduction from equality to... |
| eqeq1dALT 2730 | Alternate proof of ~ eqeq1... |
| eqeq1 2731 | Equality implies equivalen... |
| eqeq1i 2732 | Inference from equality to... |
| eqcomd 2733 | Deduction from commutative... |
| eqcom 2734 | Commutative law for class ... |
| eqcoms 2735 | Inference applying commuta... |
| eqcomi 2736 | Inference from commutative... |
| neqcomd 2737 | Commute an inequality. (C... |
| eqeq2d 2738 | Deduction from equality to... |
| eqeq2 2739 | Equality implies equivalen... |
| eqeq2i 2740 | Inference from equality to... |
| eqeqan12d 2741 | A useful inference for sub... |
| eqeqan12rd 2742 | A useful inference for sub... |
| eqeq12d 2743 | A useful inference for sub... |
| eqeq12 2744 | Equality relationship amon... |
| eqeq12i 2745 | A useful inference for sub... |
| eqeq12OLD 2746 | Obsolete version of ~ eqeq... |
| eqeq12dOLD 2747 | Obsolete version of ~ eqeq... |
| eqeqan12dOLD 2748 | Obsolete version of ~ eqeq... |
| eqeqan12dALT 2749 | Alternate proof of ~ eqeqa... |
| eqtr 2750 | Transitive law for class e... |
| eqtr2 2751 | A transitive law for class... |
| eqtr2OLD 2752 | Obsolete version of eqtr2 ... |
| eqtr3 2753 | A transitive law for class... |
| eqtr3OLD 2754 | Obsolete version of ~ eqtr... |
| eqtri 2755 | An equality transitivity i... |
| eqtr2i 2756 | An equality transitivity i... |
| eqtr3i 2757 | An equality transitivity i... |
| eqtr4i 2758 | An equality transitivity i... |
| 3eqtri 2759 | An inference from three ch... |
| 3eqtrri 2760 | An inference from three ch... |
| 3eqtr2i 2761 | An inference from three ch... |
| 3eqtr2ri 2762 | An inference from three ch... |
| 3eqtr3i 2763 | An inference from three ch... |
| 3eqtr3ri 2764 | An inference from three ch... |
| 3eqtr4i 2765 | An inference from three ch... |
| 3eqtr4ri 2766 | An inference from three ch... |
| eqtrd 2767 | An equality transitivity d... |
| eqtr2d 2768 | An equality transitivity d... |
| eqtr3d 2769 | An equality transitivity e... |
| eqtr4d 2770 | An equality transitivity e... |
| 3eqtrd 2771 | A deduction from three cha... |
| 3eqtrrd 2772 | A deduction from three cha... |
| 3eqtr2d 2773 | A deduction from three cha... |
| 3eqtr2rd 2774 | A deduction from three cha... |
| 3eqtr3d 2775 | A deduction from three cha... |
| 3eqtr3rd 2776 | A deduction from three cha... |
| 3eqtr4d 2777 | A deduction from three cha... |
| 3eqtr4rd 2778 | A deduction from three cha... |
| eqtrid 2779 | An equality transitivity d... |
| eqtr2id 2780 | An equality transitivity d... |
| eqtr3id 2781 | An equality transitivity d... |
| eqtr3di 2782 | An equality transitivity d... |
| eqtrdi 2783 | An equality transitivity d... |
| eqtr2di 2784 | An equality transitivity d... |
| eqtr4di 2785 | An equality transitivity d... |
| eqtr4id 2786 | An equality transitivity d... |
| sylan9eq 2787 | An equality transitivity d... |
| sylan9req 2788 | An equality transitivity d... |
| sylan9eqr 2789 | An equality transitivity d... |
| 3eqtr3g 2790 | A chained equality inferen... |
| 3eqtr3a 2791 | A chained equality inferen... |
| 3eqtr4g 2792 | A chained equality inferen... |
| 3eqtr4a 2793 | A chained equality inferen... |
| eq2tri 2794 | A compound transitive infe... |
| abbi 2795 | Equivalent formulas yield ... |
| abbidv 2796 | Equivalent wff's yield equ... |
| abbii 2797 | Equivalent wff's yield equ... |
| abbid 2798 | Equivalent wff's yield equ... |
| abbib 2799 | Equal class abstractions r... |
| cbvabv 2800 | Rule used to change bound ... |
| cbvabw 2801 | Rule used to change bound ... |
| cbvabwOLD 2802 | Obsolete version of ~ cbva... |
| cbvab 2803 | Rule used to change bound ... |
| eqabbw 2804 | Version of ~ eqabb using i... |
| dfclel 2806 | Characterization of the el... |
| elex2 2807 | If a class contains anothe... |
| issetlem 2808 | Lemma for ~ elisset and ~ ... |
| elissetv 2809 | An element of a class exis... |
| elisset 2810 | An element of a class exis... |
| eleq1w 2811 | Weaker version of ~ eleq1 ... |
| eleq2w 2812 | Weaker version of ~ eleq2 ... |
| eleq1d 2813 | Deduction from equality to... |
| eleq2d 2814 | Deduction from equality to... |
| eleq2dALT 2815 | Alternate proof of ~ eleq2... |
| eleq1 2816 | Equality implies equivalen... |
| eleq2 2817 | Equality implies equivalen... |
| eleq12 2818 | Equality implies equivalen... |
| eleq1i 2819 | Inference from equality to... |
| eleq2i 2820 | Inference from equality to... |
| eleq12i 2821 | Inference from equality to... |
| eleq12d 2822 | Deduction from equality to... |
| eleq1a 2823 | A transitive-type law rela... |
| eqeltri 2824 | Substitution of equal clas... |
| eqeltrri 2825 | Substitution of equal clas... |
| eleqtri 2826 | Substitution of equal clas... |
| eleqtrri 2827 | Substitution of equal clas... |
| eqeltrd 2828 | Substitution of equal clas... |
| eqeltrrd 2829 | Deduction that substitutes... |
| eleqtrd 2830 | Deduction that substitutes... |
| eleqtrrd 2831 | Deduction that substitutes... |
| eqeltrid 2832 | A membership and equality ... |
| eqeltrrid 2833 | A membership and equality ... |
| eleqtrid 2834 | A membership and equality ... |
| eleqtrrid 2835 | A membership and equality ... |
| eqeltrdi 2836 | A membership and equality ... |
| eqeltrrdi 2837 | A membership and equality ... |
| eleqtrdi 2838 | A membership and equality ... |
| eleqtrrdi 2839 | A membership and equality ... |
| 3eltr3i 2840 | Substitution of equal clas... |
| 3eltr4i 2841 | Substitution of equal clas... |
| 3eltr3d 2842 | Substitution of equal clas... |
| 3eltr4d 2843 | Substitution of equal clas... |
| 3eltr3g 2844 | Substitution of equal clas... |
| 3eltr4g 2845 | Substitution of equal clas... |
| eleq2s 2846 | Substitution of equal clas... |
| eqneltri 2847 | If a class is not an eleme... |
| eqneltrd 2848 | If a class is not an eleme... |
| eqneltrrd 2849 | If a class is not an eleme... |
| neleqtrd 2850 | If a class is not an eleme... |
| neleqtrrd 2851 | If a class is not an eleme... |
| nelneq 2852 | A way of showing two class... |
| nelneq2 2853 | A way of showing two class... |
| eqsb1 2854 | Substitution for the left-... |
| clelsb1 2855 | Substitution for the first... |
| clelsb2 2856 | Substitution for the secon... |
| clelsb2OLD 2857 | Obsolete version of ~ clel... |
| cleqh 2858 | Establish equality between... |
| hbxfreq 2859 | A utility lemma to transfe... |
| hblem 2860 | Change the free variable o... |
| hblemg 2861 | Change the free variable o... |
| eqabdv 2862 | Deduction from a wff to a ... |
| eqabcdv 2863 | Deduction from a wff to a ... |
| eqabi 2864 | Equality of a class variab... |
| abid1 2865 | Every class is equal to a ... |
| abid2 2866 | A simplification of class ... |
| eqab 2867 | One direction of ~ eqabb i... |
| eqabb 2868 | Equality of a class variab... |
| eqabbOLD 2869 | Obsolete version of ~ eqab... |
| eqabcb 2870 | Equality of a class variab... |
| eqabrd 2871 | Equality of a class variab... |
| eqabri 2872 | Equality of a class variab... |
| eqabcri 2873 | Equality of a class variab... |
| clelab 2874 | Membership of a class vari... |
| clelabOLD 2875 | Obsolete version of ~ clel... |
| clabel 2876 | Membership of a class abst... |
| sbab 2877 | The right-hand side of the... |
| nfcjust 2879 | Justification theorem for ... |
| nfci 2881 | Deduce that a class ` A ` ... |
| nfcii 2882 | Deduce that a class ` A ` ... |
| nfcr 2883 | Consequence of the not-fre... |
| nfcrALT 2884 | Alternate version of ~ nfc... |
| nfcri 2885 | Consequence of the not-fre... |
| nfcd 2886 | Deduce that a class ` A ` ... |
| nfcrd 2887 | Consequence of the not-fre... |
| nfcriOLD 2888 | Obsolete version of ~ nfcr... |
| nfcriOLDOLD 2889 | Obsolete version of ~ nfcr... |
| nfcrii 2890 | Consequence of the not-fre... |
| nfcriiOLD 2891 | Obsolete version of ~ nfcr... |
| nfcriOLDOLDOLD 2892 | Obsolete version of ~ nfcr... |
| nfceqdf 2893 | An equality theorem for ef... |
| nfceqdfOLD 2894 | Obsolete version of ~ nfce... |
| nfceqi 2895 | Equality theorem for class... |
| nfcxfr 2896 | A utility lemma to transfe... |
| nfcxfrd 2897 | A utility lemma to transfe... |
| nfcv 2898 | If ` x ` is disjoint from ... |
| nfcvd 2899 | If ` x ` is disjoint from ... |
| nfab1 2900 | Bound-variable hypothesis ... |
| nfnfc1 2901 | The setvar ` x ` is bound ... |
| clelsb1fw 2902 | Substitution for the first... |
| clelsb1f 2903 | Substitution for the first... |
| nfab 2904 | Bound-variable hypothesis ... |
| nfabg 2905 | Bound-variable hypothesis ... |
| nfaba1 2906 | Bound-variable hypothesis ... |
| nfaba1g 2907 | Bound-variable hypothesis ... |
| nfeqd 2908 | Hypothesis builder for equ... |
| nfeld 2909 | Hypothesis builder for ele... |
| nfnfc 2910 | Hypothesis builder for ` F... |
| nfeq 2911 | Hypothesis builder for equ... |
| nfel 2912 | Hypothesis builder for ele... |
| nfeq1 2913 | Hypothesis builder for equ... |
| nfel1 2914 | Hypothesis builder for ele... |
| nfeq2 2915 | Hypothesis builder for equ... |
| nfel2 2916 | Hypothesis builder for ele... |
| drnfc1 2917 | Formula-building lemma for... |
| drnfc1OLD 2918 | Obsolete version of ~ drnf... |
| drnfc2 2919 | Formula-building lemma for... |
| drnfc2OLD 2920 | Obsolete version of ~ drnf... |
| nfabdw 2921 | Bound-variable hypothesis ... |
| nfabdwOLD 2922 | Obsolete version of ~ nfab... |
| nfabd 2923 | Bound-variable hypothesis ... |
| nfabd2 2924 | Bound-variable hypothesis ... |
| dvelimdc 2925 | Deduction form of ~ dvelim... |
| dvelimc 2926 | Version of ~ dvelim for cl... |
| nfcvf 2927 | If ` x ` and ` y ` are dis... |
| nfcvf2 2928 | If ` x ` and ` y ` are dis... |
| cleqf 2929 | Establish equality between... |
| eqabf 2930 | Equality of a class variab... |
| abid2f 2931 | A simplification of class ... |
| abid2fOLD 2932 | Obsolete version of ~ abid... |
| sbabel 2933 | Theorem to move a substitu... |
| sbabelOLD 2934 | Obsolete version of ~ sbab... |
| neii 2937 | Inference associated with ... |
| neir 2938 | Inference associated with ... |
| nne 2939 | Negation of inequality. (... |
| neneqd 2940 | Deduction eliminating ineq... |
| neneq 2941 | From inequality to non-equ... |
| neqned 2942 | If it is not the case that... |
| neqne 2943 | From non-equality to inequ... |
| neirr 2944 | No class is unequal to its... |
| exmidne 2945 | Excluded middle with equal... |
| eqneqall 2946 | A contradiction concerning... |
| nonconne 2947 | Law of noncontradiction wi... |
| necon3ad 2948 | Contrapositive law deducti... |
| necon3bd 2949 | Contrapositive law deducti... |
| necon2ad 2950 | Contrapositive inference f... |
| necon2bd 2951 | Contrapositive inference f... |
| necon1ad 2952 | Contrapositive deduction f... |
| necon1bd 2953 | Contrapositive deduction f... |
| necon4ad 2954 | Contrapositive inference f... |
| necon4bd 2955 | Contrapositive inference f... |
| necon3d 2956 | Contrapositive law deducti... |
| necon1d 2957 | Contrapositive law deducti... |
| necon2d 2958 | Contrapositive inference f... |
| necon4d 2959 | Contrapositive inference f... |
| necon3ai 2960 | Contrapositive inference f... |
| necon3aiOLD 2961 | Obsolete version of ~ neco... |
| necon3bi 2962 | Contrapositive inference f... |
| necon1ai 2963 | Contrapositive inference f... |
| necon1bi 2964 | Contrapositive inference f... |
| necon2ai 2965 | Contrapositive inference f... |
| necon2bi 2966 | Contrapositive inference f... |
| necon4ai 2967 | Contrapositive inference f... |
| necon3i 2968 | Contrapositive inference f... |
| necon1i 2969 | Contrapositive inference f... |
| necon2i 2970 | Contrapositive inference f... |
| necon4i 2971 | Contrapositive inference f... |
| necon3abid 2972 | Deduction from equality to... |
| necon3bbid 2973 | Deduction from equality to... |
| necon1abid 2974 | Contrapositive deduction f... |
| necon1bbid 2975 | Contrapositive inference f... |
| necon4abid 2976 | Contrapositive law deducti... |
| necon4bbid 2977 | Contrapositive law deducti... |
| necon2abid 2978 | Contrapositive deduction f... |
| necon2bbid 2979 | Contrapositive deduction f... |
| necon3bid 2980 | Deduction from equality to... |
| necon4bid 2981 | Contrapositive law deducti... |
| necon3abii 2982 | Deduction from equality to... |
| necon3bbii 2983 | Deduction from equality to... |
| necon1abii 2984 | Contrapositive inference f... |
| necon1bbii 2985 | Contrapositive inference f... |
| necon2abii 2986 | Contrapositive inference f... |
| necon2bbii 2987 | Contrapositive inference f... |
| necon3bii 2988 | Inference from equality to... |
| necom 2989 | Commutation of inequality.... |
| necomi 2990 | Inference from commutative... |
| necomd 2991 | Deduction from commutative... |
| nesym 2992 | Characterization of inequa... |
| nesymi 2993 | Inference associated with ... |
| nesymir 2994 | Inference associated with ... |
| neeq1d 2995 | Deduction for inequality. ... |
| neeq2d 2996 | Deduction for inequality. ... |
| neeq12d 2997 | Deduction for inequality. ... |
| neeq1 2998 | Equality theorem for inequ... |
| neeq2 2999 | Equality theorem for inequ... |
| neeq1i 3000 | Inference for inequality. ... |
| neeq2i 3001 | Inference for inequality. ... |
| neeq12i 3002 | Inference for inequality. ... |
| eqnetrd 3003 | Substitution of equal clas... |
| eqnetrrd 3004 | Substitution of equal clas... |
| neeqtrd 3005 | Substitution of equal clas... |
| eqnetri 3006 | Substitution of equal clas... |
| eqnetrri 3007 | Substitution of equal clas... |
| neeqtri 3008 | Substitution of equal clas... |
| neeqtrri 3009 | Substitution of equal clas... |
| neeqtrrd 3010 | Substitution of equal clas... |
| eqnetrrid 3011 | A chained equality inferen... |
| 3netr3d 3012 | Substitution of equality i... |
| 3netr4d 3013 | Substitution of equality i... |
| 3netr3g 3014 | Substitution of equality i... |
| 3netr4g 3015 | Substitution of equality i... |
| nebi 3016 | Contraposition law for ine... |
| pm13.18 3017 | Theorem *13.18 in [Whitehe... |
| pm13.181 3018 | Theorem *13.181 in [Whiteh... |
| pm13.181OLD 3019 | Obsolete version of ~ pm13... |
| pm2.61ine 3020 | Inference eliminating an i... |
| pm2.21ddne 3021 | A contradiction implies an... |
| pm2.61ne 3022 | Deduction eliminating an i... |
| pm2.61dne 3023 | Deduction eliminating an i... |
| pm2.61dane 3024 | Deduction eliminating an i... |
| pm2.61da2ne 3025 | Deduction eliminating two ... |
| pm2.61da3ne 3026 | Deduction eliminating thre... |
| pm2.61iine 3027 | Equality version of ~ pm2.... |
| mteqand 3028 | A modus tollens deduction ... |
| neor 3029 | Logical OR with an equalit... |
| neanior 3030 | A De Morgan's law for ineq... |
| ne3anior 3031 | A De Morgan's law for ineq... |
| neorian 3032 | A De Morgan's law for ineq... |
| nemtbir 3033 | An inference from an inequ... |
| nelne1 3034 | Two classes are different ... |
| nelne2 3035 | Two classes are different ... |
| nelelne 3036 | Two classes are different ... |
| neneor 3037 | If two classes are differe... |
| nfne 3038 | Bound-variable hypothesis ... |
| nfned 3039 | Bound-variable hypothesis ... |
| nabbib 3040 | Not equivalent wff's corre... |
| neli 3043 | Inference associated with ... |
| nelir 3044 | Inference associated with ... |
| nelcon3d 3045 | Contrapositive law deducti... |
| neleq12d 3046 | Equality theorem for negat... |
| neleq1 3047 | Equality theorem for negat... |
| neleq2 3048 | Equality theorem for negat... |
| nfnel 3049 | Bound-variable hypothesis ... |
| nfneld 3050 | Bound-variable hypothesis ... |
| nnel 3051 | Negation of negated member... |
| elnelne1 3052 | Two classes are different ... |
| elnelne2 3053 | Two classes are different ... |
| pm2.24nel 3054 | A contradiction concerning... |
| pm2.61danel 3055 | Deduction eliminating an e... |
| rgen 3058 | Generalization rule for re... |
| ralel 3059 | All elements of a class ar... |
| rgenw 3060 | Generalization rule for re... |
| rgen2w 3061 | Generalization rule for re... |
| mprg 3062 | Modus ponens combined with... |
| mprgbir 3063 | Modus ponens on biconditio... |
| raln 3064 | Restricted universally qua... |
| ralnex 3067 | Relationship between restr... |
| dfrex2 3068 | Relationship between restr... |
| nrex 3069 | Inference adding restricte... |
| alral 3070 | Universal quantification i... |
| rexex 3071 | Restricted existence impli... |
| rextru 3072 | Two ways of expressing tha... |
| ralimi2 3073 | Inference quantifying both... |
| reximi2 3074 | Inference quantifying both... |
| ralimia 3075 | Inference quantifying both... |
| reximia 3076 | Inference quantifying both... |
| ralimiaa 3077 | Inference quantifying both... |
| ralimi 3078 | Inference quantifying both... |
| reximi 3079 | Inference quantifying both... |
| ral2imi 3080 | Inference quantifying ante... |
| ralim 3081 | Distribution of restricted... |
| rexim 3082 | Theorem 19.22 of [Margaris... |
| reximiaOLD 3083 | Obsolete version of ~ rexi... |
| ralbii2 3084 | Inference adding different... |
| rexbii2 3085 | Inference adding different... |
| ralbiia 3086 | Inference adding restricte... |
| rexbiia 3087 | Inference adding restricte... |
| ralbii 3088 | Inference adding restricte... |
| rexbii 3089 | Inference adding restricte... |
| ralanid 3090 | Cancellation law for restr... |
| rexanid 3091 | Cancellation law for restr... |
| ralcom3 3092 | A commutation law for rest... |
| ralcom3OLD 3093 | Obsolete version of ~ ralc... |
| dfral2 3094 | Relationship between restr... |
| rexnal 3095 | Relationship between restr... |
| ralinexa 3096 | A transformation of restri... |
| rexanali 3097 | A transformation of restri... |
| ralbi 3098 | Distribute a restricted un... |
| rexbi 3099 | Distribute restricted quan... |
| rexbiOLD 3100 | Obsolete version of ~ rexb... |
| ralrexbid 3101 | Formula-building rule for ... |
| ralrexbidOLD 3102 | Obsolete version of ~ ralr... |
| r19.35 3103 | Restricted quantifier vers... |
| r19.35OLD 3104 | Obsolete version of ~ 19.3... |
| r19.26m 3105 | Version of ~ 19.26 and ~ r... |
| r19.26 3106 | Restricted quantifier vers... |
| r19.26-3 3107 | Version of ~ r19.26 with t... |
| ralbiim 3108 | Split a biconditional and ... |
| r19.29 3109 | Restricted quantifier vers... |
| r19.29OLD 3110 | Obsolete version of ~ r19.... |
| r19.29r 3111 | Restricted quantifier vers... |
| r19.29rOLD 3112 | Obsolete version of ~ r19.... |
| r19.29imd 3113 | Theorem 19.29 of [Margaris... |
| r19.40 3114 | Restricted quantifier vers... |
| r19.30 3115 | Restricted quantifier vers... |
| r19.30OLD 3116 | Obsolete version of ~ 19.3... |
| r19.43 3117 | Restricted quantifier vers... |
| 2ralimi 3118 | Inference quantifying both... |
| 3ralimi 3119 | Inference quantifying both... |
| 4ralimi 3120 | Inference quantifying both... |
| 5ralimi 3121 | Inference quantifying both... |
| 6ralimi 3122 | Inference quantifying both... |
| 2ralbii 3123 | Inference adding two restr... |
| 2rexbii 3124 | Inference adding two restr... |
| 3ralbii 3125 | Inference adding three res... |
| 4ralbii 3126 | Inference adding four rest... |
| 2ralbiim 3127 | Split a biconditional and ... |
| ralnex2 3128 | Relationship between two r... |
| ralnex3 3129 | Relationship between three... |
| rexnal2 3130 | Relationship between two r... |
| rexnal3 3131 | Relationship between three... |
| nrexralim 3132 | Negation of a complex pred... |
| r19.26-2 3133 | Restricted quantifier vers... |
| 2r19.29 3134 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3135 | Theorem 19.29 of [Margaris... |
| r19.29d2rOLD 3136 | Obsolete version of ~ r19.... |
| r2allem 3137 | Lemma factoring out common... |
| r2exlem 3138 | Lemma factoring out common... |
| hbralrimi 3139 | Inference from Theorem 19.... |
| ralrimiv 3140 | Inference from Theorem 19.... |
| ralrimiva 3141 | Inference from Theorem 19.... |
| rexlimiva 3142 | Inference from Theorem 19.... |
| rexlimiv 3143 | Inference from Theorem 19.... |
| nrexdv 3144 | Deduction adding restricte... |
| ralrimivw 3145 | Inference from Theorem 19.... |
| rexlimivw 3146 | Weaker version of ~ rexlim... |
| ralrimdv 3147 | Inference from Theorem 19.... |
| rexlimdv 3148 | Inference from Theorem 19.... |
| ralrimdva 3149 | Inference from Theorem 19.... |
| rexlimdva 3150 | Inference from Theorem 19.... |
| rexlimdvaa 3151 | Inference from Theorem 19.... |
| rexlimdva2 3152 | Inference from Theorem 19.... |
| r19.29an 3153 | A commonly used pattern in... |
| rexlimdv3a 3154 | Inference from Theorem 19.... |
| rexlimdvw 3155 | Inference from Theorem 19.... |
| rexlimddv 3156 | Restricted existential eli... |
| r19.29a 3157 | A commonly used pattern in... |
| ralimdv2 3158 | Inference quantifying both... |
| reximdv2 3159 | Deduction quantifying both... |
| reximdvai 3160 | Deduction quantifying both... |
| reximdvaiOLD 3161 | Obsolete version of ~ rexi... |
| ralimdva 3162 | Deduction quantifying both... |
| reximdva 3163 | Deduction quantifying both... |
| ralimdv 3164 | Deduction quantifying both... |
| reximdv 3165 | Deduction from Theorem 19.... |
| reximddv 3166 | Deduction from Theorem 19.... |
| reximssdv 3167 | Derivation of a restricted... |
| ralbidv2 3168 | Formula-building rule for ... |
| rexbidv2 3169 | Formula-building rule for ... |
| ralbidva 3170 | Formula-building rule for ... |
| rexbidva 3171 | Formula-building rule for ... |
| ralbidv 3172 | Formula-building rule for ... |
| rexbidv 3173 | Formula-building rule for ... |
| r19.21v 3174 | Restricted quantifier vers... |
| r19.21vOLD 3175 | Obsolete version of ~ r19.... |
| r19.37v 3176 | Restricted quantifier vers... |
| r19.23v 3177 | Restricted quantifier vers... |
| r19.36v 3178 | Restricted quantifier vers... |
| rexlimivOLD 3179 | Obsolete version of ~ rexl... |
| rexlimivaOLD 3180 | Obsolete version of ~ rexl... |
| rexlimivwOLD 3181 | Obsolete version of ~ rexl... |
| r19.27v 3182 | Restricted quantitifer ver... |
| r19.41v 3183 | Restricted quantifier vers... |
| r19.28v 3184 | Restricted quantifier vers... |
| r19.42v 3185 | Restricted quantifier vers... |
| r19.32v 3186 | Restricted quantifier vers... |
| r19.45v 3187 | Restricted quantifier vers... |
| r19.44v 3188 | One direction of a restric... |
| r2al 3189 | Double restricted universa... |
| r2ex 3190 | Double restricted existent... |
| r3al 3191 | Triple restricted universa... |
| rgen2 3192 | Generalization rule for re... |
| ralrimivv 3193 | Inference from Theorem 19.... |
| rexlimivv 3194 | Inference from Theorem 19.... |
| ralrimivva 3195 | Inference from Theorem 19.... |
| ralrimdvv 3196 | Inference from Theorem 19.... |
| rgen3 3197 | Generalization rule for re... |
| ralrimivvva 3198 | Inference from Theorem 19.... |
| ralimdvva 3199 | Deduction doubly quantifyi... |
| reximdvva 3200 | Deduction doubly quantifyi... |
| ralimdvv 3201 | Deduction doubly quantifyi... |
| ralimd4v 3202 | Deduction quadrupally quan... |
| ralimd6v 3203 | Deduction sextupally quant... |
| ralrimdvva 3204 | Inference from Theorem 19.... |
| rexlimdvv 3205 | Inference from Theorem 19.... |
| rexlimdvva 3206 | Inference from Theorem 19.... |
| reximddv2 3207 | Double deduction from Theo... |
| r19.29vva 3208 | A commonly used pattern ba... |
| r19.29vvaOLD 3209 | Obsolete version of ~ r19.... |
| 2rexbiia 3210 | Inference adding two restr... |
| 2ralbidva 3211 | Formula-building rule for ... |
| 2rexbidva 3212 | Formula-building rule for ... |
| 2ralbidv 3213 | Formula-building rule for ... |
| 2rexbidv 3214 | Formula-building rule for ... |
| rexralbidv 3215 | Formula-building rule for ... |
| 3ralbidv 3216 | Formula-building rule for ... |
| 4ralbidv 3217 | Formula-building rule for ... |
| 6ralbidv 3218 | Formula-building rule for ... |
| r19.41vv 3219 | Version of ~ r19.41v with ... |
| reeanlem 3220 | Lemma factoring out common... |
| reeanv 3221 | Rearrange restricted exist... |
| 3reeanv 3222 | Rearrange three restricted... |
| 2ralor 3223 | Distribute restricted univ... |
| 2ralorOLD 3224 | Obsolete version of ~ 2ral... |
| risset 3225 | Two ways to say " ` A ` be... |
| nelb 3226 | A definition of ` -. A e. ... |
| nelbOLD 3227 | Obsolete version of ~ nelb... |
| rspw 3228 | Restricted specialization.... |
| cbvralvw 3229 | Change the bound variable ... |
| cbvrexvw 3230 | Change the bound variable ... |
| cbvraldva 3231 | Rule used to change the bo... |
| cbvrexdva 3232 | Rule used to change the bo... |
| cbvral2vw 3233 | Change bound variables of ... |
| cbvrex2vw 3234 | Change bound variables of ... |
| cbvral3vw 3235 | Change bound variables of ... |
| cbvral4vw 3236 | Change bound variables of ... |
| cbvral6vw 3237 | Change bound variables of ... |
| cbvral8vw 3238 | Change bound variables of ... |
| rsp 3239 | Restricted specialization.... |
| rspa 3240 | Restricted specialization.... |
| rspe 3241 | Restricted specialization.... |
| rspec 3242 | Specialization rule for re... |
| r19.21bi 3243 | Inference from Theorem 19.... |
| r19.21be 3244 | Inference from Theorem 19.... |
| r19.21t 3245 | Restricted quantifier vers... |
| r19.21 3246 | Restricted quantifier vers... |
| r19.23t 3247 | Closed theorem form of ~ r... |
| r19.23 3248 | Restricted quantifier vers... |
| ralrimi 3249 | Inference from Theorem 19.... |
| ralrimia 3250 | Inference from Theorem 19.... |
| rexlimi 3251 | Restricted quantifier vers... |
| ralimdaa 3252 | Deduction quantifying both... |
| reximdai 3253 | Deduction from Theorem 19.... |
| r19.37 3254 | Restricted quantifier vers... |
| r19.41 3255 | Restricted quantifier vers... |
| ralrimd 3256 | Inference from Theorem 19.... |
| rexlimd2 3257 | Version of ~ rexlimd with ... |
| rexlimd 3258 | Deduction form of ~ rexlim... |
| r19.29af2 3259 | A commonly used pattern ba... |
| r19.29af 3260 | A commonly used pattern ba... |
| reximd2a 3261 | Deduction quantifying both... |
| ralbida 3262 | Formula-building rule for ... |
| ralbidaOLD 3263 | Obsolete version of ~ ralb... |
| rexbida 3264 | Formula-building rule for ... |
| ralbid 3265 | Formula-building rule for ... |
| rexbid 3266 | Formula-building rule for ... |
| rexbidvALT 3267 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3268 | Alternate proof of ~ rexbi... |
| rsp2 3269 | Restricted specialization,... |
| rsp2e 3270 | Restricted specialization.... |
| rspec2 3271 | Specialization rule for re... |
| rspec3 3272 | Specialization rule for re... |
| r2alf 3273 | Double restricted universa... |
| r2exf 3274 | Double restricted existent... |
| 2ralbida 3275 | Formula-building rule for ... |
| nfra1 3276 | The setvar ` x ` is not fr... |
| nfre1 3277 | The setvar ` x ` is not fr... |
| ralcom4 3278 | Commutation of restricted ... |
| ralcom4OLD 3279 | Obsolete version of ~ ralc... |
| rexcom4 3280 | Commutation of restricted ... |
| ralcom 3281 | Commutation of restricted ... |
| rexcom 3282 | Commutation of restricted ... |
| rexcomOLD 3283 | Obsolete version of ~ rexc... |
| rexcom4a 3284 | Specialized existential co... |
| ralrot3 3285 | Rotate three restricted un... |
| ralcom13 3286 | Swap first and third restr... |
| ralcom13OLD 3287 | Obsolete version of ~ ralc... |
| rexcom13 3288 | Swap first and third restr... |
| rexrot4 3289 | Rotate four restricted exi... |
| 2ex2rexrot 3290 | Rotate two existential qua... |
| nfra2w 3291 | Similar to Lemma 24 of [Mo... |
| nfra2wOLD 3292 | Obsolete version of ~ nfra... |
| hbra1 3293 | The setvar ` x ` is not fr... |
| ralcomf 3294 | Commutation of restricted ... |
| rexcomf 3295 | Commutation of restricted ... |
| cbvralfw 3296 | Rule used to change bound ... |
| cbvrexfw 3297 | Rule used to change bound ... |
| cbvralw 3298 | Rule used to change bound ... |
| cbvrexw 3299 | Rule used to change bound ... |
| hbral 3300 | Bound-variable hypothesis ... |
| nfraldw 3301 | Deduction version of ~ nfr... |
| nfrexdw 3302 | Deduction version of ~ nfr... |
| nfralw 3303 | Bound-variable hypothesis ... |
| nfralwOLD 3304 | Obsolete version of ~ nfra... |
| nfrexw 3305 | Bound-variable hypothesis ... |
| r19.12 3306 | Restricted quantifier vers... |
| r19.12OLD 3307 | Obsolete version of ~ 19.1... |
| reean 3308 | Rearrange restricted exist... |
| cbvralsvw 3309 | Change bound variable by u... |
| cbvrexsvw 3310 | Change bound variable by u... |
| cbvralsvwOLD 3311 | Obsolete version of ~ cbvr... |
| cbvrexsvwOLD 3312 | Obsolete version of ~ cbvr... |
| nfraldwOLD 3313 | Obsolete version of ~ nfra... |
| nfra2wOLDOLD 3314 | Obsolete version of ~ nfra... |
| cbvralfwOLD 3315 | Obsolete version of ~ cbvr... |
| rexeq 3316 | Equality theorem for restr... |
| raleq 3317 | Equality theorem for restr... |
| raleqi 3318 | Equality inference for res... |
| rexeqi 3319 | Equality inference for res... |
| raleqdv 3320 | Equality deduction for res... |
| rexeqdv 3321 | Equality deduction for res... |
| raleqbidva 3322 | Equality deduction for res... |
| rexeqbidva 3323 | Equality deduction for res... |
| raleqbidvv 3324 | Version of ~ raleqbidv wit... |
| raleqbidvvOLD 3325 | Obsolete version of ~ rale... |
| rexeqbidvv 3326 | Version of ~ rexeqbidv wit... |
| rexeqbidvvOLD 3327 | Obsolete version of ~ rexe... |
| raleqbi1dv 3328 | Equality deduction for res... |
| rexeqbi1dv 3329 | Equality deduction for res... |
| raleqOLD 3330 | Obsolete version of ~ rale... |
| rexeqOLD 3331 | Obsolete version of ~ rale... |
| raleleq 3332 | All elements of a class ar... |
| raleqbii 3333 | Equality deduction for res... |
| rexeqbii 3334 | Equality deduction for res... |
| raleleqOLD 3335 | Obsolete version of ~ rale... |
| raleleqALT 3336 | Alternate proof of ~ ralel... |
| raleqbidv 3337 | Equality deduction for res... |
| rexeqbidv 3338 | Equality deduction for res... |
| cbvraldva2 3339 | Rule used to change the bo... |
| cbvrexdva2 3340 | Rule used to change the bo... |
| cbvrexdva2OLD 3341 | Obsolete version of ~ cbvr... |
| cbvraldvaOLD 3342 | Obsolete version of ~ cbvr... |
| cbvrexdvaOLD 3343 | Obsolete version of ~ cbvr... |
| raleqf 3344 | Equality theorem for restr... |
| rexeqf 3345 | Equality theorem for restr... |
| rexeqfOLD 3346 | Obsolete version of ~ rexe... |
| raleqbid 3347 | Equality deduction for res... |
| rexeqbid 3348 | Equality deduction for res... |
| sbralie 3349 | Implicit to explicit subst... |
| sbralieALT 3350 | Alternative shorter proof ... |
| cbvralf 3351 | Rule used to change bound ... |
| cbvrexf 3352 | Rule used to change bound ... |
| cbvral 3353 | Rule used to change bound ... |
| cbvrex 3354 | Rule used to change bound ... |
| cbvralv 3355 | Change the bound variable ... |
| cbvrexv 3356 | Change the bound variable ... |
| cbvralsv 3357 | Change bound variable by u... |
| cbvrexsv 3358 | Change bound variable by u... |
| cbvral2v 3359 | Change bound variables of ... |
| cbvrex2v 3360 | Change bound variables of ... |
| cbvral3v 3361 | Change bound variables of ... |
| rgen2a 3362 | Generalization rule for re... |
| nfrald 3363 | Deduction version of ~ nfr... |
| nfrexd 3364 | Deduction version of ~ nfr... |
| nfral 3365 | Bound-variable hypothesis ... |
| nfrex 3366 | Bound-variable hypothesis ... |
| nfra2 3367 | Similar to Lemma 24 of [Mo... |
| ralcom2 3368 | Commutation of restricted ... |
| reu5 3373 | Restricted uniqueness in t... |
| reurmo 3374 | Restricted existential uni... |
| reurex 3375 | Restricted unique existenc... |
| mormo 3376 | Unrestricted "at most one"... |
| rmobiia 3377 | Formula-building rule for ... |
| reubiia 3378 | Formula-building rule for ... |
| rmobii 3379 | Formula-building rule for ... |
| reubii 3380 | Formula-building rule for ... |
| rmoanid 3381 | Cancellation law for restr... |
| reuanid 3382 | Cancellation law for restr... |
| rmoanidOLD 3383 | Obsolete version of ~ rmoa... |
| reuanidOLD 3384 | Obsolete version of ~ reua... |
| 2reu2rex 3385 | Double restricted existent... |
| rmobidva 3386 | Formula-building rule for ... |
| reubidva 3387 | Formula-building rule for ... |
| rmobidv 3388 | Formula-building rule for ... |
| reubidv 3389 | Formula-building rule for ... |
| reueubd 3390 | Restricted existential uni... |
| rmo5 3391 | Restricted "at most one" i... |
| nrexrmo 3392 | Nonexistence implies restr... |
| moel 3393 | "At most one" element in a... |
| cbvrmovw 3394 | Change the bound variable ... |
| cbvreuvw 3395 | Change the bound variable ... |
| moelOLD 3396 | Obsolete version of ~ moel... |
| rmobida 3397 | Formula-building rule for ... |
| reubida 3398 | Formula-building rule for ... |
| rmobidvaOLD 3399 | Obsolete version of ~ rmob... |
| cbvrmow 3400 | Change the bound variable ... |
| cbvreuw 3401 | Change the bound variable ... |
| nfrmo1 3402 | The setvar ` x ` is not fr... |
| nfreu1 3403 | The setvar ` x ` is not fr... |
| nfrmow 3404 | Bound-variable hypothesis ... |
| nfreuw 3405 | Bound-variable hypothesis ... |
| cbvrmowOLD 3406 | Obsolete version of ~ cbvr... |
| cbvreuwOLD 3407 | Obsolete version of ~ cbvr... |
| cbvreuvwOLD 3408 | Obsolete version of ~ cbvr... |
| rmoeq1 3409 | Equality theorem for restr... |
| reueq1 3410 | Equality theorem for restr... |
| rmoeq1OLD 3411 | Obsolete version of ~ rmoe... |
| reueq1OLD 3412 | Obsolete version of ~ reue... |
| rmoeqd 3413 | Equality deduction for res... |
| reueqd 3414 | Equality deduction for res... |
| rmoeq1f 3415 | Equality theorem for restr... |
| reueq1f 3416 | Equality theorem for restr... |
| nfreuwOLD 3417 | Obsolete version of ~ nfre... |
| nfrmowOLD 3418 | Obsolete version of ~ nfrm... |
| cbvreu 3419 | Change the bound variable ... |
| cbvrmo 3420 | Change the bound variable ... |
| cbvrmov 3421 | Change the bound variable ... |
| cbvreuv 3422 | Change the bound variable ... |
| nfrmod 3423 | Deduction version of ~ nfr... |
| nfreud 3424 | Deduction version of ~ nfr... |
| nfrmo 3425 | Bound-variable hypothesis ... |
| nfreu 3426 | Bound-variable hypothesis ... |
| rabbidva2 3429 | Equivalent wff's yield equ... |
| rabbia2 3430 | Equivalent wff's yield equ... |
| rabbiia 3431 | Equivalent formulas yield ... |
| rabbiiaOLD 3432 | Obsolete version of ~ rabb... |
| rabbii 3433 | Equivalent wff's correspon... |
| rabbidva 3434 | Equivalent wff's yield equ... |
| rabbidv 3435 | Equivalent wff's yield equ... |
| rabswap 3436 | Swap with a membership rel... |
| cbvrabv 3437 | Rule to change the bound v... |
| rabeqcda 3438 | When ` ps ` is always true... |
| rabeqc 3439 | A restricted class abstrac... |
| rabeqi 3440 | Equality theorem for restr... |
| rabeq 3441 | Equality theorem for restr... |
| rabeqdv 3442 | Equality of restricted cla... |
| rabeqbidva 3443 | Equality of restricted cla... |
| rabeqbidv 3444 | Equality of restricted cla... |
| rabrabi 3445 | Abstract builder restricte... |
| nfrab1 3446 | The abstraction variable i... |
| rabid 3447 | An "identity" law of concr... |
| rabidim1 3448 | Membership in a restricted... |
| reqabi 3449 | Inference from equality of... |
| rabrab 3450 | Abstract builder restricte... |
| rabrabiOLD 3451 | Obsolete version of ~ rabr... |
| rabbida4 3452 | Version of ~ rabbidva2 wit... |
| rabbida 3453 | Equivalent wff's yield equ... |
| rabbid 3454 | Version of ~ rabbidv with ... |
| rabeqd 3455 | Deduction form of ~ rabeq ... |
| rabeqbida 3456 | Version of ~ rabeqbidva wi... |
| rabbi 3457 | Equivalent wff's correspon... |
| rabid2f 3458 | An "identity" law for rest... |
| rabid2 3459 | An "identity" law for rest... |
| rabid2OLD 3460 | Obsolete version of ~ rabi... |
| rabeqf 3461 | Equality theorem for restr... |
| cbvrabw 3462 | Rule to change the bound v... |
| nfrabw 3463 | A variable not free in a w... |
| nfrabwOLD 3464 | Obsolete version of ~ nfra... |
| rabbidaOLD 3465 | Obsolete version of ~ rabb... |
| rabeqiOLD 3466 | Obsolete version of ~ rabe... |
| nfrab 3467 | A variable not free in a w... |
| cbvrab 3468 | Rule to change the bound v... |
| vjust 3470 | Justification theorem for ... |
| dfv2 3472 | Alternate definition of th... |
| vex 3473 | All setvar variables are s... |
| vexOLD 3474 | Obsolete version of ~ vex ... |
| elv 3475 | If a proposition is implie... |
| elvd 3476 | If a proposition is implie... |
| el2v 3477 | If a proposition is implie... |
| eqv 3478 | The universe contains ever... |
| eqvf 3479 | The universe contains ever... |
| abv 3480 | The class of sets verifyin... |
| abvALT 3481 | Alternate proof of ~ abv ,... |
| isset 3482 | Two ways to express that "... |
| issetft 3483 | Closed theorem form of ~ i... |
| issetf 3484 | A version of ~ isset that ... |
| isseti 3485 | A way to say " ` A ` is a ... |
| issetri 3486 | A way to say " ` A ` is a ... |
| eqvisset 3487 | A class equal to a variabl... |
| elex 3488 | If a class is a member of ... |
| elexi 3489 | If a class is a member of ... |
| elexd 3490 | If a class is a member of ... |
| elex2OLD 3491 | Obsolete version of ~ elex... |
| elex22 3492 | If two classes each contai... |
| prcnel 3493 | A proper class doesn't bel... |
| ralv 3494 | A universal quantifier res... |
| rexv 3495 | An existential quantifier ... |
| reuv 3496 | A unique existential quant... |
| rmov 3497 | An at-most-one quantifier ... |
| rabab 3498 | A class abstraction restri... |
| rexcom4b 3499 | Specialized existential co... |
| ceqsal1t 3500 | One direction of ~ ceqsalt... |
| ceqsalt 3501 | Closed theorem version of ... |
| ceqsralt 3502 | Restricted quantifier vers... |
| ceqsalg 3503 | A representation of explic... |
| ceqsalgALT 3504 | Alternate proof of ~ ceqsa... |
| ceqsal 3505 | A representation of explic... |
| ceqsalALT 3506 | A representation of explic... |
| ceqsalv 3507 | A representation of explic... |
| ceqsalvOLD 3508 | Obsolete version of ~ ceqs... |
| ceqsralv 3509 | Restricted quantifier vers... |
| ceqsralvOLD 3510 | Obsolete version of ~ ceqs... |
| gencl 3511 | Implicit substitution for ... |
| 2gencl 3512 | Implicit substitution for ... |
| 3gencl 3513 | Implicit substitution for ... |
| cgsexg 3514 | Implicit substitution infe... |
| cgsex2g 3515 | Implicit substitution infe... |
| cgsex4g 3516 | An implicit substitution i... |
| cgsex4gOLD 3517 | Obsolete version of ~ cgse... |
| cgsex4gOLDOLD 3518 | Obsolete version of ~ cgse... |
| ceqsex 3519 | Elimination of an existent... |
| ceqsexOLD 3520 | Obsolete version of ~ ceqs... |
| ceqsexv 3521 | Elimination of an existent... |
| ceqsexvOLD 3522 | Obsolete version of ~ ceqs... |
| ceqsexvOLDOLD 3523 | Obsolete version of ~ ceqs... |
| ceqsexv2d 3524 | Elimination of an existent... |
| ceqsex2 3525 | Elimination of two existen... |
| ceqsex2v 3526 | Elimination of two existen... |
| ceqsex3v 3527 | Elimination of three exist... |
| ceqsex4v 3528 | Elimination of four existe... |
| ceqsex6v 3529 | Elimination of six existen... |
| ceqsex8v 3530 | Elimination of eight exist... |
| gencbvex 3531 | Change of bound variable u... |
| gencbvex2 3532 | Restatement of ~ gencbvex ... |
| gencbval 3533 | Change of bound variable u... |
| sbhypf 3534 | Introduce an explicit subs... |
| sbhypfOLD 3535 | Obsolete version of ~ sbhy... |
| vtoclgft 3536 | Closed theorem form of ~ v... |
| vtocleg 3537 | Implicit substitution of a... |
| vtoclg 3538 | Implicit substitution of a... |
| vtocle 3539 | Implicit substitution of a... |
| vtoclbg 3540 | Implicit substitution of a... |
| vtocl 3541 | Implicit substitution of a... |
| vtocldf 3542 | Implicit substitution of a... |
| vtocld 3543 | Implicit substitution of a... |
| vtocldOLD 3544 | Obsolete version of ~ vtoc... |
| vtocl2d 3545 | Implicit substitution of t... |
| vtoclef 3546 | Implicit substitution of a... |
| vtoclf 3547 | Implicit substitution of a... |
| vtoclfOLD 3548 | Obsolete version of ~ vtoc... |
| vtoclALT 3549 | Alternate proof of ~ vtocl... |
| vtocl2 3550 | Implicit substitution of c... |
| vtocl3 3551 | Implicit substitution of c... |
| vtoclb 3552 | Implicit substitution of a... |
| vtoclgf 3553 | Implicit substitution of a... |
| vtoclg1f 3554 | Version of ~ vtoclgf with ... |
| vtoclgOLD 3555 | Obsolete version of ~ vtoc... |
| vtocl2gf 3556 | Implicit substitution of a... |
| vtocl3gf 3557 | Implicit substitution of a... |
| vtocl2g 3558 | Implicit substitution of 2... |
| vtocl3g 3559 | Implicit substitution of a... |
| vtoclgaf 3560 | Implicit substitution of a... |
| vtoclga 3561 | Implicit substitution of a... |
| vtocl2ga 3562 | Implicit substitution of 2... |
| vtocl2gaf 3563 | Implicit substitution of 2... |
| vtocl3gaf 3564 | Implicit substitution of 3... |
| vtocl3ga 3565 | Implicit substitution of 3... |
| vtocl3gaOLD 3566 | Obsolete version of ~ vtoc... |
| vtocl4g 3567 | Implicit substitution of 4... |
| vtocl4ga 3568 | Implicit substitution of 4... |
| vtoclegft 3569 | Implicit substitution of a... |
| vtoclegftOLD 3570 | Obsolete version of ~ vtoc... |
| vtoclri 3571 | Implicit substitution of a... |
| spcimgft 3572 | A closed version of ~ spci... |
| spcgft 3573 | A closed version of ~ spcg... |
| spcimgf 3574 | Rule of specialization, us... |
| spcimegf 3575 | Existential specialization... |
| spcgf 3576 | Rule of specialization, us... |
| spcegf 3577 | Existential specialization... |
| spcimdv 3578 | Restricted specialization,... |
| spcdv 3579 | Rule of specialization, us... |
| spcimedv 3580 | Restricted existential spe... |
| spcgv 3581 | Rule of specialization, us... |
| spcegv 3582 | Existential specialization... |
| spcedv 3583 | Existential specialization... |
| spc2egv 3584 | Existential specialization... |
| spc2gv 3585 | Specialization with two qu... |
| spc2ed 3586 | Existential specialization... |
| spc2d 3587 | Specialization with 2 quan... |
| spc3egv 3588 | Existential specialization... |
| spc3gv 3589 | Specialization with three ... |
| spcv 3590 | Rule of specialization, us... |
| spcev 3591 | Existential specialization... |
| spc2ev 3592 | Existential specialization... |
| rspct 3593 | A closed version of ~ rspc... |
| rspcdf 3594 | Restricted specialization,... |
| rspc 3595 | Restricted specialization,... |
| rspce 3596 | Restricted existential spe... |
| rspcimdv 3597 | Restricted specialization,... |
| rspcimedv 3598 | Restricted existential spe... |
| rspcdv 3599 | Restricted specialization,... |
| rspcedv 3600 | Restricted existential spe... |
| rspcebdv 3601 | Restricted existential spe... |
| rspcdv2 3602 | Restricted specialization,... |
| rspcv 3603 | Restricted specialization,... |
| rspccv 3604 | Restricted specialization,... |
| rspcva 3605 | Restricted specialization,... |
| rspccva 3606 | Restricted specialization,... |
| rspcev 3607 | Restricted existential spe... |
| rspcdva 3608 | Restricted specialization,... |
| rspcedvd 3609 | Restricted existential spe... |
| rspcedvdw 3610 | Version of ~ rspcedvd wher... |
| rspceb2dv 3611 | Restricted existential spe... |
| rspcime 3612 | Prove a restricted existen... |
| rspceaimv 3613 | Restricted existential spe... |
| rspcedeq1vd 3614 | Restricted existential spe... |
| rspcedeq2vd 3615 | Restricted existential spe... |
| rspc2 3616 | Restricted specialization ... |
| rspc2gv 3617 | Restricted specialization ... |
| rspc2v 3618 | 2-variable restricted spec... |
| rspc2va 3619 | 2-variable restricted spec... |
| rspc2ev 3620 | 2-variable restricted exis... |
| 2rspcedvdw 3621 | Double application of ~ rs... |
| rspc2dv 3622 | 2-variable restricted spec... |
| rspc3v 3623 | 3-variable restricted spec... |
| rspc3ev 3624 | 3-variable restricted exis... |
| rspc3dv 3625 | 3-variable restricted spec... |
| rspc4v 3626 | 4-variable restricted spec... |
| rspc6v 3627 | 6-variable restricted spec... |
| rspc8v 3628 | 8-variable restricted spec... |
| rspceeqv 3629 | Restricted existential spe... |
| ralxpxfr2d 3630 | Transfer a universal quant... |
| rexraleqim 3631 | Statement following from e... |
| eqvincg 3632 | A variable introduction la... |
| eqvinc 3633 | A variable introduction la... |
| eqvincf 3634 | A variable introduction la... |
| alexeqg 3635 | Two ways to express substi... |
| ceqex 3636 | Equality implies equivalen... |
| ceqsexg 3637 | A representation of explic... |
| ceqsexgv 3638 | Elimination of an existent... |
| ceqsrexv 3639 | Elimination of a restricte... |
| ceqsrexbv 3640 | Elimination of a restricte... |
| ceqsralbv 3641 | Elimination of a restricte... |
| ceqsrex2v 3642 | Elimination of a restricte... |
| clel2g 3643 | Alternate definition of me... |
| clel2gOLD 3644 | Obsolete version of ~ clel... |
| clel2 3645 | Alternate definition of me... |
| clel3g 3646 | Alternate definition of me... |
| clel3 3647 | Alternate definition of me... |
| clel4g 3648 | Alternate definition of me... |
| clel4 3649 | Alternate definition of me... |
| clel4OLD 3650 | Obsolete version of ~ clel... |
| clel5 3651 | Alternate definition of cl... |
| pm13.183 3652 | Compare theorem *13.183 in... |
| rr19.3v 3653 | Restricted quantifier vers... |
| rr19.28v 3654 | Restricted quantifier vers... |
| elab6g 3655 | Membership in a class abst... |
| elabd2 3656 | Membership in a class abst... |
| elabd3 3657 | Membership in a class abst... |
| elabgt 3658 | Membership in a class abst... |
| elabgtOLD 3659 | Obsolete version of ~ elab... |
| elabgtOLDOLD 3660 | Obsolete version of ~ elab... |
| elabgf 3661 | Membership in a class abst... |
| elabf 3662 | Membership in a class abst... |
| elabg 3663 | Membership in a class abst... |
| elabgOLD 3664 | Obsolete version of ~ elab... |
| elab 3665 | Membership in a class abst... |
| elabOLD 3666 | Obsolete version of ~ elab... |
| elab2g 3667 | Membership in a class abst... |
| elabd 3668 | Explicit demonstration the... |
| elab2 3669 | Membership in a class abst... |
| elab4g 3670 | Membership in a class abst... |
| elab3gf 3671 | Membership in a class abst... |
| elab3g 3672 | Membership in a class abst... |
| elab3 3673 | Membership in a class abst... |
| elrabi 3674 | Implication for the member... |
| elrabiOLD 3675 | Obsolete version of ~ elra... |
| elrabf 3676 | Membership in a restricted... |
| rabtru 3677 | Abstract builder using the... |
| rabeqcOLD 3678 | Obsolete version of ~ rabe... |
| elrab3t 3679 | Membership in a restricted... |
| elrab 3680 | Membership in a restricted... |
| elrab3 3681 | Membership in a restricted... |
| elrabd 3682 | Membership in a restricted... |
| elrab2 3683 | Membership in a restricted... |
| ralab 3684 | Universal quantification o... |
| ralabOLD 3685 | Obsolete version of ~ rala... |
| ralrab 3686 | Universal quantification o... |
| rexab 3687 | Existential quantification... |
| rexabOLD 3688 | Obsolete version of ~ rexa... |
| rexrab 3689 | Existential quantification... |
| ralab2 3690 | Universal quantification o... |
| ralrab2 3691 | Universal quantification o... |
| rexab2 3692 | Existential quantification... |
| rexrab2 3693 | Existential quantification... |
| reurab 3694 | Restricted existential uni... |
| abidnf 3695 | Identity used to create cl... |
| dedhb 3696 | A deduction theorem for co... |
| class2seteq 3697 | Writing a set as a class a... |
| nelrdva 3698 | Deduce negative membership... |
| eqeu 3699 | A condition which implies ... |
| moeq 3700 | There exists at most one s... |
| eueq 3701 | A class is a set if and on... |
| eueqi 3702 | There exists a unique set ... |
| eueq2 3703 | Equality has existential u... |
| eueq3 3704 | Equality has existential u... |
| moeq3 3705 | "At most one" property of ... |
| mosub 3706 | "At most one" remains true... |
| mo2icl 3707 | Theorem for inferring "at ... |
| mob2 3708 | Consequence of "at most on... |
| moi2 3709 | Consequence of "at most on... |
| mob 3710 | Equality implied by "at mo... |
| moi 3711 | Equality implied by "at mo... |
| morex 3712 | Derive membership from uni... |
| euxfr2w 3713 | Transfer existential uniqu... |
| euxfrw 3714 | Transfer existential uniqu... |
| euxfr2 3715 | Transfer existential uniqu... |
| euxfr 3716 | Transfer existential uniqu... |
| euind 3717 | Existential uniqueness via... |
| reu2 3718 | A way to express restricte... |
| reu6 3719 | A way to express restricte... |
| reu3 3720 | A way to express restricte... |
| reu6i 3721 | A condition which implies ... |
| eqreu 3722 | A condition which implies ... |
| rmo4 3723 | Restricted "at most one" u... |
| reu4 3724 | Restricted uniqueness usin... |
| reu7 3725 | Restricted uniqueness usin... |
| reu8 3726 | Restricted uniqueness usin... |
| rmo3f 3727 | Restricted "at most one" u... |
| rmo4f 3728 | Restricted "at most one" u... |
| reu2eqd 3729 | Deduce equality from restr... |
| reueq 3730 | Equality has existential u... |
| rmoeq 3731 | Equality's restricted exis... |
| rmoan 3732 | Restricted "at most one" s... |
| rmoim 3733 | Restricted "at most one" i... |
| rmoimia 3734 | Restricted "at most one" i... |
| rmoimi 3735 | Restricted "at most one" i... |
| rmoimi2 3736 | Restricted "at most one" i... |
| 2reu5a 3737 | Double restricted existent... |
| reuimrmo 3738 | Restricted uniqueness impl... |
| 2reuswap 3739 | A condition allowing swap ... |
| 2reuswap2 3740 | A condition allowing swap ... |
| reuxfrd 3741 | Transfer existential uniqu... |
| reuxfr 3742 | Transfer existential uniqu... |
| reuxfr1d 3743 | Transfer existential uniqu... |
| reuxfr1ds 3744 | Transfer existential uniqu... |
| reuxfr1 3745 | Transfer existential uniqu... |
| reuind 3746 | Existential uniqueness via... |
| 2rmorex 3747 | Double restricted quantifi... |
| 2reu5lem1 3748 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3749 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3750 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3751 | Double restricted existent... |
| 2reurmo 3752 | Double restricted quantifi... |
| 2reurex 3753 | Double restricted quantifi... |
| 2rmoswap 3754 | A condition allowing to sw... |
| 2rexreu 3755 | Double restricted existent... |
| cdeqi 3758 | Deduce conditional equalit... |
| cdeqri 3759 | Property of conditional eq... |
| cdeqth 3760 | Deduce conditional equalit... |
| cdeqnot 3761 | Distribute conditional equ... |
| cdeqal 3762 | Distribute conditional equ... |
| cdeqab 3763 | Distribute conditional equ... |
| cdeqal1 3764 | Distribute conditional equ... |
| cdeqab1 3765 | Distribute conditional equ... |
| cdeqim 3766 | Distribute conditional equ... |
| cdeqcv 3767 | Conditional equality for s... |
| cdeqeq 3768 | Distribute conditional equ... |
| cdeqel 3769 | Distribute conditional equ... |
| nfcdeq 3770 | If we have a conditional e... |
| nfccdeq 3771 | Variation of ~ nfcdeq for ... |
| rru 3772 | Relative version of Russel... |
| ru 3773 | Russell's Paradox. Propos... |
| dfsbcq 3776 | Proper substitution of a c... |
| dfsbcq2 3777 | This theorem, which is sim... |
| sbsbc 3778 | Show that ~ df-sb and ~ df... |
| sbceq1d 3779 | Equality theorem for class... |
| sbceq1dd 3780 | Equality theorem for class... |
| sbceqbid 3781 | Equality theorem for class... |
| sbc8g 3782 | This is the closest we can... |
| sbc2or 3783 | The disjunction of two equ... |
| sbcex 3784 | By our definition of prope... |
| sbceq1a 3785 | Equality theorem for class... |
| sbceq2a 3786 | Equality theorem for class... |
| spsbc 3787 | Specialization: if a formu... |
| spsbcd 3788 | Specialization: if a formu... |
| sbcth 3789 | A substitution into a theo... |
| sbcthdv 3790 | Deduction version of ~ sbc... |
| sbcid 3791 | An identity theorem for su... |
| nfsbc1d 3792 | Deduction version of ~ nfs... |
| nfsbc1 3793 | Bound-variable hypothesis ... |
| nfsbc1v 3794 | Bound-variable hypothesis ... |
| nfsbcdw 3795 | Deduction version of ~ nfs... |
| nfsbcw 3796 | Bound-variable hypothesis ... |
| sbccow 3797 | A composition law for clas... |
| nfsbcd 3798 | Deduction version of ~ nfs... |
| nfsbc 3799 | Bound-variable hypothesis ... |
| sbcco 3800 | A composition law for clas... |
| sbcco2 3801 | A composition law for clas... |
| sbc5 3802 | An equivalence for class s... |
| sbc5ALT 3803 | Alternate proof of ~ sbc5 ... |
| sbc6g 3804 | An equivalence for class s... |
| sbc6gOLD 3805 | Obsolete version of ~ sbc6... |
| sbc6 3806 | An equivalence for class s... |
| sbc7 3807 | An equivalence for class s... |
| cbvsbcw 3808 | Change bound variables in ... |
| cbvsbcvw 3809 | Change the bound variable ... |
| cbvsbc 3810 | Change bound variables in ... |
| cbvsbcv 3811 | Change the bound variable ... |
| sbciegft 3812 | Conversion of implicit sub... |
| sbciegf 3813 | Conversion of implicit sub... |
| sbcieg 3814 | Conversion of implicit sub... |
| sbciegOLD 3815 | Obsolete version of ~ sbci... |
| sbcie2g 3816 | Conversion of implicit sub... |
| sbcie 3817 | Conversion of implicit sub... |
| sbciedf 3818 | Conversion of implicit sub... |
| sbcied 3819 | Conversion of implicit sub... |
| sbciedOLD 3820 | Obsolete version of ~ sbci... |
| sbcied2 3821 | Conversion of implicit sub... |
| elrabsf 3822 | Membership in a restricted... |
| eqsbc1 3823 | Substitution for the left-... |
| sbcng 3824 | Move negation in and out o... |
| sbcimg 3825 | Distribution of class subs... |
| sbcan 3826 | Distribution of class subs... |
| sbcor 3827 | Distribution of class subs... |
| sbcbig 3828 | Distribution of class subs... |
| sbcn1 3829 | Move negation in and out o... |
| sbcim1 3830 | Distribution of class subs... |
| sbcim1OLD 3831 | Obsolete version of ~ sbci... |
| sbcbid 3832 | Formula-building deduction... |
| sbcbidv 3833 | Formula-building deduction... |
| sbcbii 3834 | Formula-building inference... |
| sbcbi1 3835 | Distribution of class subs... |
| sbcbi2 3836 | Substituting into equivale... |
| sbcal 3837 | Move universal quantifier ... |
| sbcex2 3838 | Move existential quantifie... |
| sbceqal 3839 | Class version of one impli... |
| sbceqalOLD 3840 | Obsolete version of ~ sbce... |
| sbeqalb 3841 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3842 | Substitution for the right... |
| sbc3an 3843 | Distribution of class subs... |
| sbcel1v 3844 | Class substitution into a ... |
| sbcel2gv 3845 | Class substitution into a ... |
| sbcel21v 3846 | Class substitution into a ... |
| sbcimdv 3847 | Substitution analogue of T... |
| sbcimdvOLD 3848 | Obsolete version of ~ sbci... |
| sbctt 3849 | Substitution for a variabl... |
| sbcgf 3850 | Substitution for a variabl... |
| sbc19.21g 3851 | Substitution for a variabl... |
| sbcg 3852 | Substitution for a variabl... |
| sbcgOLD 3853 | Obsolete version of ~ sbcg... |
| sbcgfi 3854 | Substitution for a variabl... |
| sbc2iegf 3855 | Conversion of implicit sub... |
| sbc2ie 3856 | Conversion of implicit sub... |
| sbc2ieOLD 3857 | Obsolete version of ~ sbc2... |
| sbc2iedv 3858 | Conversion of implicit sub... |
| sbc3ie 3859 | Conversion of implicit sub... |
| sbccomlem 3860 | Lemma for ~ sbccom . (Con... |
| sbccom 3861 | Commutative law for double... |
| sbcralt 3862 | Interchange class substitu... |
| sbcrext 3863 | Interchange class substitu... |
| sbcralg 3864 | Interchange class substitu... |
| sbcrex 3865 | Interchange class substitu... |
| sbcreu 3866 | Interchange class substitu... |
| reu8nf 3867 | Restricted uniqueness usin... |
| sbcabel 3868 | Interchange class substitu... |
| rspsbc 3869 | Restricted quantifier vers... |
| rspsbca 3870 | Restricted quantifier vers... |
| rspesbca 3871 | Existence form of ~ rspsbc... |
| spesbc 3872 | Existence form of ~ spsbc ... |
| spesbcd 3873 | form of ~ spsbc . (Contri... |
| sbcth2 3874 | A substitution into a theo... |
| ra4v 3875 | Version of ~ ra4 with a di... |
| ra4 3876 | Restricted quantifier vers... |
| rmo2 3877 | Alternate definition of re... |
| rmo2i 3878 | Condition implying restric... |
| rmo3 3879 | Restricted "at most one" u... |
| rmob 3880 | Consequence of "at most on... |
| rmoi 3881 | Consequence of "at most on... |
| rmob2 3882 | Consequence of "restricted... |
| rmoi2 3883 | Consequence of "restricted... |
| rmoanim 3884 | Introduction of a conjunct... |
| rmoanimALT 3885 | Alternate proof of ~ rmoan... |
| reuan 3886 | Introduction of a conjunct... |
| 2reu1 3887 | Double restricted existent... |
| 2reu2 3888 | Double restricted existent... |
| csb2 3891 | Alternate expression for t... |
| csbeq1 3892 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3893 | Equality deduction for pro... |
| csbeq2 3894 | Substituting into equivale... |
| csbeq2d 3895 | Formula-building deduction... |
| csbeq2dv 3896 | Formula-building deduction... |
| csbeq2i 3897 | Formula-building inference... |
| csbeq12dv 3898 | Formula-building inference... |
| cbvcsbw 3899 | Change bound variables in ... |
| cbvcsb 3900 | Change bound variables in ... |
| cbvcsbv 3901 | Change the bound variable ... |
| csbid 3902 | Analogue of ~ sbid for pro... |
| csbeq1a 3903 | Equality theorem for prope... |
| csbcow 3904 | Composition law for chaine... |
| csbco 3905 | Composition law for chaine... |
| csbtt 3906 | Substitution doesn't affec... |
| csbconstgf 3907 | Substitution doesn't affec... |
| csbconstg 3908 | Substitution doesn't affec... |
| csbconstgOLD 3909 | Obsolete version of ~ csbc... |
| csbgfi 3910 | Substitution for a variabl... |
| csbconstgi 3911 | The proper substitution of... |
| nfcsb1d 3912 | Bound-variable hypothesis ... |
| nfcsb1 3913 | Bound-variable hypothesis ... |
| nfcsb1v 3914 | Bound-variable hypothesis ... |
| nfcsbd 3915 | Deduction version of ~ nfc... |
| nfcsbw 3916 | Bound-variable hypothesis ... |
| nfcsb 3917 | Bound-variable hypothesis ... |
| csbhypf 3918 | Introduce an explicit subs... |
| csbiebt 3919 | Conversion of implicit sub... |
| csbiedf 3920 | Conversion of implicit sub... |
| csbieb 3921 | Bidirectional conversion b... |
| csbiebg 3922 | Bidirectional conversion b... |
| csbiegf 3923 | Conversion of implicit sub... |
| csbief 3924 | Conversion of implicit sub... |
| csbie 3925 | Conversion of implicit sub... |
| csbieOLD 3926 | Obsolete version of ~ csbi... |
| csbied 3927 | Conversion of implicit sub... |
| csbiedOLD 3928 | Obsolete version of ~ csbi... |
| csbied2 3929 | Conversion of implicit sub... |
| csbie2t 3930 | Conversion of implicit sub... |
| csbie2 3931 | Conversion of implicit sub... |
| csbie2g 3932 | Conversion of implicit sub... |
| cbvrabcsfw 3933 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3934 | A more general version of ... |
| cbvrexcsf 3935 | A more general version of ... |
| cbvreucsf 3936 | A more general version of ... |
| cbvrabcsf 3937 | A more general version of ... |
| cbvralv2 3938 | Rule used to change the bo... |
| cbvrexv2 3939 | Rule used to change the bo... |
| rspc2vd 3940 | Deduction version of 2-var... |
| difjust 3946 | Soundness justification th... |
| unjust 3948 | Soundness justification th... |
| injust 3950 | Soundness justification th... |
| dfin5 3952 | Alternate definition for t... |
| dfdif2 3953 | Alternate definition of cl... |
| eldif 3954 | Expansion of membership in... |
| eldifd 3955 | If a class is in one class... |
| eldifad 3956 | If a class is in the diffe... |
| eldifbd 3957 | If a class is in the diffe... |
| elneeldif 3958 | The elements of a set diff... |
| velcomp 3959 | Characterization of setvar... |
| elin 3960 | Expansion of membership in... |
| dfss 3962 | Variant of subclass defini... |
| dfss2 3964 | Alternate definition of th... |
| dfss2OLD 3965 | Obsolete version of ~ dfss... |
| dfss3 3966 | Alternate definition of su... |
| dfss6 3967 | Alternate definition of su... |
| dfss2f 3968 | Equivalence for subclass r... |
| dfss3f 3969 | Equivalence for subclass r... |
| nfss 3970 | If ` x ` is not free in ` ... |
| ssel 3971 | Membership relationships f... |
| sselOLD 3972 | Obsolete version of ~ ssel... |
| ssel2 3973 | Membership relationships f... |
| sseli 3974 | Membership implication fro... |
| sselii 3975 | Membership inference from ... |
| sselid 3976 | Membership inference from ... |
| sseld 3977 | Membership deduction from ... |
| sselda 3978 | Membership deduction from ... |
| sseldd 3979 | Membership inference from ... |
| ssneld 3980 | If a class is not in anoth... |
| ssneldd 3981 | If an element is not in a ... |
| ssriv 3982 | Inference based on subclas... |
| ssrd 3983 | Deduction based on subclas... |
| ssrdv 3984 | Deduction based on subclas... |
| sstr2 3985 | Transitivity of subclass r... |
| sstr 3986 | Transitivity of subclass r... |
| sstri 3987 | Subclass transitivity infe... |
| sstrd 3988 | Subclass transitivity dedu... |
| sstrid 3989 | Subclass transitivity dedu... |
| sstrdi 3990 | Subclass transitivity dedu... |
| sylan9ss 3991 | A subclass transitivity de... |
| sylan9ssr 3992 | A subclass transitivity de... |
| eqss 3993 | The subclass relationship ... |
| eqssi 3994 | Infer equality from two su... |
| eqssd 3995 | Equality deduction from tw... |
| sssseq 3996 | If a class is a subclass o... |
| eqrd 3997 | Deduce equality of classes... |
| eqri 3998 | Infer equality of classes ... |
| eqelssd 3999 | Equality deduction from su... |
| ssid 4000 | Any class is a subclass of... |
| ssidd 4001 | Weakening of ~ ssid . (Co... |
| ssv 4002 | Any class is a subclass of... |
| sseq1 4003 | Equality theorem for subcl... |
| sseq2 4004 | Equality theorem for the s... |
| sseq12 4005 | Equality theorem for the s... |
| sseq1i 4006 | An equality inference for ... |
| sseq2i 4007 | An equality inference for ... |
| sseq12i 4008 | An equality inference for ... |
| sseq1d 4009 | An equality deduction for ... |
| sseq2d 4010 | An equality deduction for ... |
| sseq12d 4011 | An equality deduction for ... |
| eqsstri 4012 | Substitution of equality i... |
| eqsstrri 4013 | Substitution of equality i... |
| sseqtri 4014 | Substitution of equality i... |
| sseqtrri 4015 | Substitution of equality i... |
| eqsstrd 4016 | Substitution of equality i... |
| eqsstrrd 4017 | Substitution of equality i... |
| sseqtrd 4018 | Substitution of equality i... |
| sseqtrrd 4019 | Substitution of equality i... |
| 3sstr3i 4020 | Substitution of equality i... |
| 3sstr4i 4021 | Substitution of equality i... |
| 3sstr3g 4022 | Substitution of equality i... |
| 3sstr4g 4023 | Substitution of equality i... |
| 3sstr3d 4024 | Substitution of equality i... |
| 3sstr4d 4025 | Substitution of equality i... |
| eqsstrid 4026 | A chained subclass and equ... |
| eqsstrrid 4027 | A chained subclass and equ... |
| sseqtrdi 4028 | A chained subclass and equ... |
| sseqtrrdi 4029 | A chained subclass and equ... |
| sseqtrid 4030 | Subclass transitivity dedu... |
| sseqtrrid 4031 | Subclass transitivity dedu... |
| eqsstrdi 4032 | A chained subclass and equ... |
| eqsstrrdi 4033 | A chained subclass and equ... |
| eqimssd 4034 | Equality implies inclusion... |
| eqimsscd 4035 | Equality implies inclusion... |
| eqimss 4036 | Equality implies inclusion... |
| eqimss2 4037 | Equality implies inclusion... |
| eqimssi 4038 | Infer subclass relationshi... |
| eqimss2i 4039 | Infer subclass relationshi... |
| nssne1 4040 | Two classes are different ... |
| nssne2 4041 | Two classes are different ... |
| nss 4042 | Negation of subclass relat... |
| nelss 4043 | Demonstrate by witnesses t... |
| ssrexf 4044 | Restricted existential qua... |
| ssrmof 4045 | "At most one" existential ... |
| ssralv 4046 | Quantification restricted ... |
| ssrexv 4047 | Existential quantification... |
| ss2ralv 4048 | Two quantifications restri... |
| ss2rexv 4049 | Two existential quantifica... |
| ralss 4050 | Restricted universal quant... |
| rexss 4051 | Restricted existential qua... |
| ss2ab 4052 | Class abstractions in a su... |
| abss 4053 | Class abstraction in a sub... |
| ssab 4054 | Subclass of a class abstra... |
| ssabral 4055 | The relation for a subclas... |
| ss2abdv 4056 | Deduction of abstraction s... |
| ss2abdvALT 4057 | Alternate proof of ~ ss2ab... |
| ss2abdvOLD 4058 | Obsolete version of ~ ss2a... |
| ss2abi 4059 | Inference of abstraction s... |
| ss2abiOLD 4060 | Obsolete version of ~ ss2a... |
| abssdv 4061 | Deduction of abstraction s... |
| abssdvOLD 4062 | Obsolete version of ~ abss... |
| abssi 4063 | Inference of abstraction s... |
| ss2rab 4064 | Restricted abstraction cla... |
| rabss 4065 | Restricted class abstracti... |
| ssrab 4066 | Subclass of a restricted c... |
| ssrabdv 4067 | Subclass of a restricted c... |
| rabssdv 4068 | Subclass of a restricted c... |
| ss2rabdv 4069 | Deduction of restricted ab... |
| ss2rabi 4070 | Inference of restricted ab... |
| rabss2 4071 | Subclass law for restricte... |
| ssab2 4072 | Subclass relation for the ... |
| ssrab2 4073 | Subclass relation for a re... |
| ssrab2OLD 4074 | Obsolete version of ~ ssra... |
| rabss3d 4075 | Subclass law for restricte... |
| ssrab3 4076 | Subclass relation for a re... |
| rabssrabd 4077 | Subclass of a restricted c... |
| ssrabeq 4078 | If the restricting class o... |
| rabssab 4079 | A restricted class is a su... |
| uniiunlem 4080 | A subset relationship usef... |
| dfpss2 4081 | Alternate definition of pr... |
| dfpss3 4082 | Alternate definition of pr... |
| psseq1 4083 | Equality theorem for prope... |
| psseq2 4084 | Equality theorem for prope... |
| psseq1i 4085 | An equality inference for ... |
| psseq2i 4086 | An equality inference for ... |
| psseq12i 4087 | An equality inference for ... |
| psseq1d 4088 | An equality deduction for ... |
| psseq2d 4089 | An equality deduction for ... |
| psseq12d 4090 | An equality deduction for ... |
| pssss 4091 | A proper subclass is a sub... |
| pssne 4092 | Two classes in a proper su... |
| pssssd 4093 | Deduce subclass from prope... |
| pssned 4094 | Proper subclasses are uneq... |
| sspss 4095 | Subclass in terms of prope... |
| pssirr 4096 | Proper subclass is irrefle... |
| pssn2lp 4097 | Proper subclass has no 2-c... |
| sspsstri 4098 | Two ways of stating tricho... |
| ssnpss 4099 | Partial trichotomy law for... |
| psstr 4100 | Transitive law for proper ... |
| sspsstr 4101 | Transitive law for subclas... |
| psssstr 4102 | Transitive law for subclas... |
| psstrd 4103 | Proper subclass inclusion ... |
| sspsstrd 4104 | Transitivity involving sub... |
| psssstrd 4105 | Transitivity involving sub... |
| npss 4106 | A class is not a proper su... |
| ssnelpss 4107 | A subclass missing a membe... |
| ssnelpssd 4108 | Subclass inclusion with on... |
| ssexnelpss 4109 | If there is an element of ... |
| dfdif3 4110 | Alternate definition of cl... |
| difeq1 4111 | Equality theorem for class... |
| difeq2 4112 | Equality theorem for class... |
| difeq12 4113 | Equality theorem for class... |
| difeq1i 4114 | Inference adding differenc... |
| difeq2i 4115 | Inference adding differenc... |
| difeq12i 4116 | Equality inference for cla... |
| difeq1d 4117 | Deduction adding differenc... |
| difeq2d 4118 | Deduction adding differenc... |
| difeq12d 4119 | Equality deduction for cla... |
| difeqri 4120 | Inference from membership ... |
| nfdif 4121 | Bound-variable hypothesis ... |
| eldifi 4122 | Implication of membership ... |
| eldifn 4123 | Implication of membership ... |
| elndif 4124 | A set does not belong to a... |
| neldif 4125 | Implication of membership ... |
| difdif 4126 | Double class difference. ... |
| difss 4127 | Subclass relationship for ... |
| difssd 4128 | A difference of two classe... |
| difss2 4129 | If a class is contained in... |
| difss2d 4130 | If a class is contained in... |
| ssdifss 4131 | Preservation of a subclass... |
| ddif 4132 | Double complement under un... |
| ssconb 4133 | Contraposition law for sub... |
| sscon 4134 | Contraposition law for sub... |
| ssdif 4135 | Difference law for subsets... |
| ssdifd 4136 | If ` A ` is contained in `... |
| sscond 4137 | If ` A ` is contained in `... |
| ssdifssd 4138 | If ` A ` is contained in `... |
| ssdif2d 4139 | If ` A ` is contained in `... |
| raldifb 4140 | Restricted universal quant... |
| rexdifi 4141 | Restricted existential qua... |
| complss 4142 | Complementation reverses i... |
| compleq 4143 | Two classes are equal if a... |
| elun 4144 | Expansion of membership in... |
| elunnel1 4145 | A member of a union that i... |
| elunnel2 4146 | A member of a union that i... |
| uneqri 4147 | Inference from membership ... |
| unidm 4148 | Idempotent law for union o... |
| uncom 4149 | Commutative law for union ... |
| equncom 4150 | If a class equals the unio... |
| equncomi 4151 | Inference form of ~ equnco... |
| uneq1 4152 | Equality theorem for the u... |
| uneq2 4153 | Equality theorem for the u... |
| uneq12 4154 | Equality theorem for the u... |
| uneq1i 4155 | Inference adding union to ... |
| uneq2i 4156 | Inference adding union to ... |
| uneq12i 4157 | Equality inference for the... |
| uneq1d 4158 | Deduction adding union to ... |
| uneq2d 4159 | Deduction adding union to ... |
| uneq12d 4160 | Equality deduction for the... |
| nfun 4161 | Bound-variable hypothesis ... |
| unass 4162 | Associative law for union ... |
| un12 4163 | A rearrangement of union. ... |
| un23 4164 | A rearrangement of union. ... |
| un4 4165 | A rearrangement of the uni... |
| unundi 4166 | Union distributes over its... |
| unundir 4167 | Union distributes over its... |
| ssun1 4168 | Subclass relationship for ... |
| ssun2 4169 | Subclass relationship for ... |
| ssun3 4170 | Subclass law for union of ... |
| ssun4 4171 | Subclass law for union of ... |
| elun1 4172 | Membership law for union o... |
| elun2 4173 | Membership law for union o... |
| elunant 4174 | A statement is true for ev... |
| unss1 4175 | Subclass law for union of ... |
| ssequn1 4176 | A relationship between sub... |
| unss2 4177 | Subclass law for union of ... |
| unss12 4178 | Subclass law for union of ... |
| ssequn2 4179 | A relationship between sub... |
| unss 4180 | The union of two subclasse... |
| unssi 4181 | An inference showing the u... |
| unssd 4182 | A deduction showing the un... |
| unssad 4183 | If ` ( A u. B ) ` is conta... |
| unssbd 4184 | If ` ( A u. B ) ` is conta... |
| ssun 4185 | A condition that implies i... |
| rexun 4186 | Restricted existential qua... |
| ralunb 4187 | Restricted quantification ... |
| ralun 4188 | Restricted quantification ... |
| elini 4189 | Membership in an intersect... |
| elind 4190 | Deduce membership in an in... |
| elinel1 4191 | Membership in an intersect... |
| elinel2 4192 | Membership in an intersect... |
| elin2 4193 | Membership in a class defi... |
| elin1d 4194 | Elementhood in the first s... |
| elin2d 4195 | Elementhood in the first s... |
| elin3 4196 | Membership in a class defi... |
| incom 4197 | Commutative law for inters... |
| ineqcom 4198 | Two ways of expressing tha... |
| ineqcomi 4199 | Two ways of expressing tha... |
| ineqri 4200 | Inference from membership ... |
| ineq1 4201 | Equality theorem for inter... |
| ineq2 4202 | Equality theorem for inter... |
| ineq12 4203 | Equality theorem for inter... |
| ineq1i 4204 | Equality inference for int... |
| ineq2i 4205 | Equality inference for int... |
| ineq12i 4206 | Equality inference for int... |
| ineq1d 4207 | Equality deduction for int... |
| ineq2d 4208 | Equality deduction for int... |
| ineq12d 4209 | Equality deduction for int... |
| ineqan12d 4210 | Equality deduction for int... |
| sseqin2 4211 | A relationship between sub... |
| nfin 4212 | Bound-variable hypothesis ... |
| rabbi2dva 4213 | Deduction from a wff to a ... |
| inidm 4214 | Idempotent law for interse... |
| inass 4215 | Associative law for inters... |
| in12 4216 | A rearrangement of interse... |
| in32 4217 | A rearrangement of interse... |
| in13 4218 | A rearrangement of interse... |
| in31 4219 | A rearrangement of interse... |
| inrot 4220 | Rotate the intersection of... |
| in4 4221 | Rearrangement of intersect... |
| inindi 4222 | Intersection distributes o... |
| inindir 4223 | Intersection distributes o... |
| inss1 4224 | The intersection of two cl... |
| inss2 4225 | The intersection of two cl... |
| ssin 4226 | Subclass of intersection. ... |
| ssini 4227 | An inference showing that ... |
| ssind 4228 | A deduction showing that a... |
| ssrin 4229 | Add right intersection to ... |
| sslin 4230 | Add left intersection to s... |
| ssrind 4231 | Add right intersection to ... |
| ss2in 4232 | Intersection of subclasses... |
| ssinss1 4233 | Intersection preserves sub... |
| inss 4234 | Inclusion of an intersecti... |
| rexin 4235 | Restricted existential qua... |
| dfss7 4236 | Alternate definition of su... |
| symdifcom 4239 | Symmetric difference commu... |
| symdifeq1 4240 | Equality theorem for symme... |
| symdifeq2 4241 | Equality theorem for symme... |
| nfsymdif 4242 | Hypothesis builder for sym... |
| elsymdif 4243 | Membership in a symmetric ... |
| dfsymdif4 4244 | Alternate definition of th... |
| elsymdifxor 4245 | Membership in a symmetric ... |
| dfsymdif2 4246 | Alternate definition of th... |
| symdifass 4247 | Symmetric difference is as... |
| difsssymdif 4248 | The symmetric difference c... |
| difsymssdifssd 4249 | If the symmetric differenc... |
| unabs 4250 | Absorption law for union. ... |
| inabs 4251 | Absorption law for interse... |
| nssinpss 4252 | Negation of subclass expre... |
| nsspssun 4253 | Negation of subclass expre... |
| dfss4 4254 | Subclass defined in terms ... |
| dfun2 4255 | An alternate definition of... |
| dfin2 4256 | An alternate definition of... |
| difin 4257 | Difference with intersecti... |
| ssdifim 4258 | Implication of a class dif... |
| ssdifsym 4259 | Symmetric class difference... |
| dfss5 4260 | Alternate definition of su... |
| dfun3 4261 | Union defined in terms of ... |
| dfin3 4262 | Intersection defined in te... |
| dfin4 4263 | Alternate definition of th... |
| invdif 4264 | Intersection with universa... |
| indif 4265 | Intersection with class di... |
| indif2 4266 | Bring an intersection in a... |
| indif1 4267 | Bring an intersection in a... |
| indifcom 4268 | Commutation law for inters... |
| indi 4269 | Distributive law for inter... |
| undi 4270 | Distributive law for union... |
| indir 4271 | Distributive law for inter... |
| undir 4272 | Distributive law for union... |
| unineq 4273 | Infer equality from equali... |
| uneqin 4274 | Equality of union and inte... |
| difundi 4275 | Distributive law for class... |
| difundir 4276 | Distributive law for class... |
| difindi 4277 | Distributive law for class... |
| difindir 4278 | Distributive law for class... |
| indifdi 4279 | Distribute intersection ov... |
| indifdir 4280 | Distribute intersection ov... |
| indifdirOLD 4281 | Obsolete version of ~ indi... |
| difdif2 4282 | Class difference by a clas... |
| undm 4283 | De Morgan's law for union.... |
| indm 4284 | De Morgan's law for inters... |
| difun1 4285 | A relationship involving d... |
| undif3 4286 | An equality involving clas... |
| difin2 4287 | Represent a class differen... |
| dif32 4288 | Swap second and third argu... |
| difabs 4289 | Absorption-like law for cl... |
| sscon34b 4290 | Relative complementation r... |
| rcompleq 4291 | Two subclasses are equal i... |
| dfsymdif3 4292 | Alternate definition of th... |
| unabw 4293 | Union of two class abstrac... |
| unab 4294 | Union of two class abstrac... |
| inab 4295 | Intersection of two class ... |
| difab 4296 | Difference of two class ab... |
| abanssl 4297 | A class abstraction with a... |
| abanssr 4298 | A class abstraction with a... |
| notabw 4299 | A class abstraction define... |
| notab 4300 | A class abstraction define... |
| unrab 4301 | Union of two restricted cl... |
| inrab 4302 | Intersection of two restri... |
| inrab2 4303 | Intersection with a restri... |
| difrab 4304 | Difference of two restrict... |
| dfrab3 4305 | Alternate definition of re... |
| dfrab2 4306 | Alternate definition of re... |
| notrab 4307 | Complementation of restric... |
| dfrab3ss 4308 | Restricted class abstracti... |
| rabun2 4309 | Abstraction restricted to ... |
| reuun2 4310 | Transfer uniqueness to a s... |
| reuss2 4311 | Transfer uniqueness to a s... |
| reuss 4312 | Transfer uniqueness to a s... |
| reuun1 4313 | Transfer uniqueness to a s... |
| reupick 4314 | Restricted uniqueness "pic... |
| reupick3 4315 | Restricted uniqueness "pic... |
| reupick2 4316 | Restricted uniqueness "pic... |
| euelss 4317 | Transfer uniqueness of an ... |
| dfnul4 4320 | Alternate definition of th... |
| dfnul2 4321 | Alternate definition of th... |
| dfnul3 4322 | Alternate definition of th... |
| dfnul2OLD 4323 | Obsolete version of ~ dfnu... |
| dfnul3OLD 4324 | Obsolete version of ~ dfnu... |
| dfnul4OLD 4325 | Obsolete version of ~ dfnu... |
| noel 4326 | The empty set has no eleme... |
| noelOLD 4327 | Obsolete version of ~ noel... |
| nel02 4328 | The empty set has no eleme... |
| n0i 4329 | If a class has elements, t... |
| ne0i 4330 | If a class has elements, t... |
| ne0d 4331 | Deduction form of ~ ne0i .... |
| n0ii 4332 | If a class has elements, t... |
| ne0ii 4333 | If a class has elements, t... |
| vn0 4334 | The universal class is not... |
| vn0ALT 4335 | Alternate proof of ~ vn0 .... |
| eq0f 4336 | A class is equal to the em... |
| neq0f 4337 | A class is not empty if an... |
| n0f 4338 | A class is nonempty if and... |
| eq0 4339 | A class is equal to the em... |
| eq0ALT 4340 | Alternate proof of ~ eq0 .... |
| neq0 4341 | A class is not empty if an... |
| n0 4342 | A class is nonempty if and... |
| eq0OLDOLD 4343 | Obsolete version of ~ eq0 ... |
| neq0OLD 4344 | Obsolete version of ~ neq0... |
| n0OLD 4345 | Obsolete version of ~ n0 a... |
| nel0 4346 | From the general negation ... |
| reximdva0 4347 | Restricted existence deduc... |
| rspn0 4348 | Specialization for restric... |
| rspn0OLD 4349 | Obsolete version of ~ rspn... |
| n0rex 4350 | There is an element in a n... |
| ssn0rex 4351 | There is an element in a c... |
| n0moeu 4352 | A case of equivalence of "... |
| rex0 4353 | Vacuous restricted existen... |
| reu0 4354 | Vacuous restricted uniquen... |
| rmo0 4355 | Vacuous restricted at-most... |
| 0el 4356 | Membership of the empty se... |
| n0el 4357 | Negated membership of the ... |
| eqeuel 4358 | A condition which implies ... |
| ssdif0 4359 | Subclass expressed in term... |
| difn0 4360 | If the difference of two s... |
| pssdifn0 4361 | A proper subclass has a no... |
| pssdif 4362 | A proper subclass has a no... |
| ndisj 4363 | Express that an intersecti... |
| difin0ss 4364 | Difference, intersection, ... |
| inssdif0 4365 | Intersection, subclass, an... |
| difid 4366 | The difference between a c... |
| difidALT 4367 | Alternate proof of ~ difid... |
| dif0 4368 | The difference between a c... |
| ab0w 4369 | The class of sets verifyin... |
| ab0 4370 | The class of sets verifyin... |
| ab0OLD 4371 | Obsolete version of ~ ab0 ... |
| ab0ALT 4372 | Alternate proof of ~ ab0 ,... |
| dfnf5 4373 | Characterization of nonfre... |
| ab0orv 4374 | The class abstraction defi... |
| ab0orvALT 4375 | Alternate proof of ~ ab0or... |
| abn0 4376 | Nonempty class abstraction... |
| abn0OLD 4377 | Obsolete version of ~ abn0... |
| rab0 4378 | Any restricted class abstr... |
| rabeq0w 4379 | Condition for a restricted... |
| rabeq0 4380 | Condition for a restricted... |
| rabn0 4381 | Nonempty restricted class ... |
| rabxm 4382 | Law of excluded middle, in... |
| rabnc 4383 | Law of noncontradiction, i... |
| elneldisj 4384 | The set of elements ` s ` ... |
| elnelun 4385 | The union of the set of el... |
| un0 4386 | The union of a class with ... |
| in0 4387 | The intersection of a clas... |
| 0un 4388 | The union of the empty set... |
| 0in 4389 | The intersection of the em... |
| inv1 4390 | The intersection of a clas... |
| unv 4391 | The union of a class with ... |
| 0ss 4392 | The null set is a subset o... |
| ss0b 4393 | Any subset of the empty se... |
| ss0 4394 | Any subset of the empty se... |
| sseq0 4395 | A subclass of an empty cla... |
| ssn0 4396 | A class with a nonempty su... |
| 0dif 4397 | The difference between the... |
| abf 4398 | A class abstraction determ... |
| abfOLD 4399 | Obsolete version of ~ abf ... |
| eq0rdv 4400 | Deduction for equality to ... |
| eq0rdvALT 4401 | Alternate proof of ~ eq0rd... |
| csbprc 4402 | The proper substitution of... |
| csb0 4403 | The proper substitution of... |
| sbcel12 4404 | Distribute proper substitu... |
| sbceqg 4405 | Distribute proper substitu... |
| sbceqi 4406 | Distribution of class subs... |
| sbcnel12g 4407 | Distribute proper substitu... |
| sbcne12 4408 | Distribute proper substitu... |
| sbcel1g 4409 | Move proper substitution i... |
| sbceq1g 4410 | Move proper substitution t... |
| sbcel2 4411 | Move proper substitution i... |
| sbceq2g 4412 | Move proper substitution t... |
| csbcom 4413 | Commutative law for double... |
| sbcnestgfw 4414 | Nest the composition of tw... |
| csbnestgfw 4415 | Nest the composition of tw... |
| sbcnestgw 4416 | Nest the composition of tw... |
| csbnestgw 4417 | Nest the composition of tw... |
| sbcco3gw 4418 | Composition of two substit... |
| sbcnestgf 4419 | Nest the composition of tw... |
| csbnestgf 4420 | Nest the composition of tw... |
| sbcnestg 4421 | Nest the composition of tw... |
| csbnestg 4422 | Nest the composition of tw... |
| sbcco3g 4423 | Composition of two substit... |
| csbco3g 4424 | Composition of two class s... |
| csbnest1g 4425 | Nest the composition of tw... |
| csbidm 4426 | Idempotent law for class s... |
| csbvarg 4427 | The proper substitution of... |
| csbvargi 4428 | The proper substitution of... |
| sbccsb 4429 | Substitution into a wff ex... |
| sbccsb2 4430 | Substitution into a wff ex... |
| rspcsbela 4431 | Special case related to ~ ... |
| sbnfc2 4432 | Two ways of expressing " `... |
| csbab 4433 | Move substitution into a c... |
| csbun 4434 | Distribution of class subs... |
| csbin 4435 | Distribute proper substitu... |
| csbie2df 4436 | Conversion of implicit sub... |
| 2nreu 4437 | If there are two different... |
| un00 4438 | Two classes are empty iff ... |
| vss 4439 | Only the universal class h... |
| 0pss 4440 | The null set is a proper s... |
| npss0 4441 | No set is a proper subset ... |
| pssv 4442 | Any non-universal class is... |
| disj 4443 | Two ways of saying that tw... |
| disjOLD 4444 | Obsolete version of ~ disj... |
| disjr 4445 | Two ways of saying that tw... |
| disj1 4446 | Two ways of saying that tw... |
| reldisj 4447 | Two ways of saying that tw... |
| reldisjOLD 4448 | Obsolete version of ~ reld... |
| disj3 4449 | Two ways of saying that tw... |
| disjne 4450 | Members of disjoint sets a... |
| disjeq0 4451 | Two disjoint sets are equa... |
| disjel 4452 | A set can't belong to both... |
| disj2 4453 | Two ways of saying that tw... |
| disj4 4454 | Two ways of saying that tw... |
| ssdisj 4455 | Intersection with a subcla... |
| disjpss 4456 | A class is a proper subset... |
| undisj1 4457 | The union of disjoint clas... |
| undisj2 4458 | The union of disjoint clas... |
| ssindif0 4459 | Subclass expressed in term... |
| inelcm 4460 | The intersection of classe... |
| minel 4461 | A minimum element of a cla... |
| undif4 4462 | Distribute union over diff... |
| disjssun 4463 | Subset relation for disjoi... |
| vdif0 4464 | Universal class equality i... |
| difrab0eq 4465 | If the difference between ... |
| pssnel 4466 | A proper subclass has a me... |
| disjdif 4467 | A class and its relative c... |
| disjdifr 4468 | A class and its relative c... |
| difin0 4469 | The difference of a class ... |
| unvdif 4470 | The union of a class and i... |
| undif1 4471 | Absorption of difference b... |
| undif2 4472 | Absorption of difference b... |
| undifabs 4473 | Absorption of difference b... |
| inundif 4474 | The intersection and class... |
| disjdif2 4475 | The difference of a class ... |
| difun2 4476 | Absorption of union by dif... |
| undif 4477 | Union of complementary par... |
| undifr 4478 | Union of complementary par... |
| undifrOLD 4479 | Obsolete version of ~ undi... |
| undif5 4480 | An equality involving clas... |
| ssdifin0 4481 | A subset of a difference d... |
| ssdifeq0 4482 | A class is a subclass of i... |
| ssundif 4483 | A condition equivalent to ... |
| difcom 4484 | Swap the arguments of a cl... |
| pssdifcom1 4485 | Two ways to express overla... |
| pssdifcom2 4486 | Two ways to express non-co... |
| difdifdir 4487 | Distributive law for class... |
| uneqdifeq 4488 | Two ways to say that ` A `... |
| raldifeq 4489 | Equality theorem for restr... |
| r19.2z 4490 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4491 | A response to the notion t... |
| r19.3rz 4492 | Restricted quantification ... |
| r19.28z 4493 | Restricted quantifier vers... |
| r19.3rzv 4494 | Restricted quantification ... |
| r19.9rzv 4495 | Restricted quantification ... |
| r19.28zv 4496 | Restricted quantifier vers... |
| r19.37zv 4497 | Restricted quantifier vers... |
| r19.45zv 4498 | Restricted version of Theo... |
| r19.44zv 4499 | Restricted version of Theo... |
| r19.27z 4500 | Restricted quantifier vers... |
| r19.27zv 4501 | Restricted quantifier vers... |
| r19.36zv 4502 | Restricted quantifier vers... |
| ralidmw 4503 | Idempotent law for restric... |
| rzal 4504 | Vacuous quantification is ... |
| rzalALT 4505 | Alternate proof of ~ rzal ... |
| rexn0 4506 | Restricted existential qua... |
| ralidm 4507 | Idempotent law for restric... |
| ral0 4508 | Vacuous universal quantifi... |
| ralf0 4509 | The quantification of a fa... |
| rexn0OLD 4510 | Obsolete version of ~ rexn... |
| ralidmOLD 4511 | Obsolete version of ~ rali... |
| ral0OLD 4512 | Obsolete version of ~ ral0... |
| ralf0OLD 4513 | Obsolete version of ~ ralf... |
| ralnralall 4514 | A contradiction concerning... |
| falseral0 4515 | A false statement can only... |
| raaan 4516 | Rearrange restricted quant... |
| raaanv 4517 | Rearrange restricted quant... |
| sbss 4518 | Set substitution into the ... |
| sbcssg 4519 | Distribute proper substitu... |
| raaan2 4520 | Rearrange restricted quant... |
| 2reu4lem 4521 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4522 | Definition of double restr... |
| csbdif 4523 | Distribution of class subs... |
| dfif2 4526 | An alternate definition of... |
| dfif6 4527 | An alternate definition of... |
| ifeq1 4528 | Equality theorem for condi... |
| ifeq2 4529 | Equality theorem for condi... |
| iftrue 4530 | Value of the conditional o... |
| iftruei 4531 | Inference associated with ... |
| iftrued 4532 | Value of the conditional o... |
| iffalse 4533 | Value of the conditional o... |
| iffalsei 4534 | Inference associated with ... |
| iffalsed 4535 | Value of the conditional o... |
| ifnefalse 4536 | When values are unequal, b... |
| ifsb 4537 | Distribute a function over... |
| dfif3 4538 | Alternate definition of th... |
| dfif4 4539 | Alternate definition of th... |
| dfif5 4540 | Alternate definition of th... |
| ifssun 4541 | A conditional class is inc... |
| ifeq12 4542 | Equality theorem for condi... |
| ifeq1d 4543 | Equality deduction for con... |
| ifeq2d 4544 | Equality deduction for con... |
| ifeq12d 4545 | Equality deduction for con... |
| ifbi 4546 | Equivalence theorem for co... |
| ifbid 4547 | Equivalence deduction for ... |
| ifbieq1d 4548 | Equivalence/equality deduc... |
| ifbieq2i 4549 | Equivalence/equality infer... |
| ifbieq2d 4550 | Equivalence/equality deduc... |
| ifbieq12i 4551 | Equivalence deduction for ... |
| ifbieq12d 4552 | Equivalence deduction for ... |
| nfifd 4553 | Deduction form of ~ nfif .... |
| nfif 4554 | Bound-variable hypothesis ... |
| ifeq1da 4555 | Conditional equality. (Co... |
| ifeq2da 4556 | Conditional equality. (Co... |
| ifeq12da 4557 | Equivalence deduction for ... |
| ifbieq12d2 4558 | Equivalence deduction for ... |
| ifclda 4559 | Conditional closure. (Con... |
| ifeqda 4560 | Separation of the values o... |
| elimif 4561 | Elimination of a condition... |
| ifbothda 4562 | A wff ` th ` containing a ... |
| ifboth 4563 | A wff ` th ` containing a ... |
| ifid 4564 | Identical true and false a... |
| eqif 4565 | Expansion of an equality w... |
| ifval 4566 | Another expression of the ... |
| elif 4567 | Membership in a conditiona... |
| ifel 4568 | Membership of a conditiona... |
| ifcl 4569 | Membership (closure) of a ... |
| ifcld 4570 | Membership (closure) of a ... |
| ifcli 4571 | Inference associated with ... |
| ifexd 4572 | Existence of the condition... |
| ifexg 4573 | Existence of the condition... |
| ifex 4574 | Existence of the condition... |
| ifeqor 4575 | The possible values of a c... |
| ifnot 4576 | Negating the first argumen... |
| ifan 4577 | Rewrite a conjunction in a... |
| ifor 4578 | Rewrite a disjunction in a... |
| 2if2 4579 | Resolve two nested conditi... |
| ifcomnan 4580 | Commute the conditions in ... |
| csbif 4581 | Distribute proper substitu... |
| dedth 4582 | Weak deduction theorem tha... |
| dedth2h 4583 | Weak deduction theorem eli... |
| dedth3h 4584 | Weak deduction theorem eli... |
| dedth4h 4585 | Weak deduction theorem eli... |
| dedth2v 4586 | Weak deduction theorem for... |
| dedth3v 4587 | Weak deduction theorem for... |
| dedth4v 4588 | Weak deduction theorem for... |
| elimhyp 4589 | Eliminate a hypothesis con... |
| elimhyp2v 4590 | Eliminate a hypothesis con... |
| elimhyp3v 4591 | Eliminate a hypothesis con... |
| elimhyp4v 4592 | Eliminate a hypothesis con... |
| elimel 4593 | Eliminate a membership hyp... |
| elimdhyp 4594 | Version of ~ elimhyp where... |
| keephyp 4595 | Transform a hypothesis ` p... |
| keephyp2v 4596 | Keep a hypothesis containi... |
| keephyp3v 4597 | Keep a hypothesis containi... |
| pwjust 4599 | Soundness justification th... |
| elpwg 4601 | Membership in a power clas... |
| elpw 4602 | Membership in a power clas... |
| velpw 4603 | Setvar variable membership... |
| elpwd 4604 | Membership in a power clas... |
| elpwi 4605 | Subset relation implied by... |
| elpwb 4606 | Characterization of the el... |
| elpwid 4607 | An element of a power clas... |
| elelpwi 4608 | If ` A ` belongs to a part... |
| sspw 4609 | The powerclass preserves i... |
| sspwi 4610 | The powerclass preserves i... |
| sspwd 4611 | The powerclass preserves i... |
| pweq 4612 | Equality theorem for power... |
| pweqALT 4613 | Alternate proof of ~ pweq ... |
| pweqi 4614 | Equality inference for pow... |
| pweqd 4615 | Equality deduction for pow... |
| pwunss 4616 | The power class of the uni... |
| nfpw 4617 | Bound-variable hypothesis ... |
| pwidg 4618 | A set is an element of its... |
| pwidb 4619 | A class is an element of i... |
| pwid 4620 | A set is a member of its p... |
| pwss 4621 | Subclass relationship for ... |
| pwundif 4622 | Break up the power class o... |
| snjust 4623 | Soundness justification th... |
| sneq 4634 | Equality theorem for singl... |
| sneqi 4635 | Equality inference for sin... |
| sneqd 4636 | Equality deduction for sin... |
| dfsn2 4637 | Alternate definition of si... |
| elsng 4638 | There is exactly one eleme... |
| elsn 4639 | There is exactly one eleme... |
| velsn 4640 | There is only one element ... |
| elsni 4641 | There is at most one eleme... |
| absn 4642 | Condition for a class abst... |
| dfpr2 4643 | Alternate definition of a ... |
| dfsn2ALT 4644 | Alternate definition of si... |
| elprg 4645 | A member of a pair of clas... |
| elpri 4646 | If a class is an element o... |
| elpr 4647 | A member of a pair of clas... |
| elpr2g 4648 | A member of a pair of sets... |
| elpr2 4649 | A member of a pair of sets... |
| elpr2OLD 4650 | Obsolete version of ~ elpr... |
| nelpr2 4651 | If a class is not an eleme... |
| nelpr1 4652 | If a class is not an eleme... |
| nelpri 4653 | If an element doesn't matc... |
| prneli 4654 | If an element doesn't matc... |
| nelprd 4655 | If an element doesn't matc... |
| eldifpr 4656 | Membership in a set with t... |
| rexdifpr 4657 | Restricted existential qua... |
| snidg 4658 | A set is a member of its s... |
| snidb 4659 | A class is a set iff it is... |
| snid 4660 | A set is a member of its s... |
| vsnid 4661 | A setvar variable is a mem... |
| elsn2g 4662 | There is exactly one eleme... |
| elsn2 4663 | There is exactly one eleme... |
| nelsn 4664 | If a class is not equal to... |
| rabeqsn 4665 | Conditions for a restricte... |
| rabsssn 4666 | Conditions for a restricte... |
| rabeqsnd 4667 | Conditions for a restricte... |
| ralsnsg 4668 | Substitution expressed in ... |
| rexsns 4669 | Restricted existential qua... |
| rexsngf 4670 | Restricted existential qua... |
| ralsngf 4671 | Restricted universal quant... |
| reusngf 4672 | Restricted existential uni... |
| ralsng 4673 | Substitution expressed in ... |
| rexsng 4674 | Restricted existential qua... |
| reusng 4675 | Restricted existential uni... |
| 2ralsng 4676 | Substitution expressed in ... |
| ralsngOLD 4677 | Obsolete version of ~ rals... |
| rexsngOLD 4678 | Obsolete version of ~ rexs... |
| rexreusng 4679 | Restricted existential uni... |
| exsnrex 4680 | There is a set being the e... |
| ralsn 4681 | Convert a universal quanti... |
| rexsn 4682 | Convert an existential qua... |
| elpwunsn 4683 | Membership in an extension... |
| eqoreldif 4684 | An element of a set is eit... |
| eltpg 4685 | Members of an unordered tr... |
| eldiftp 4686 | Membership in a set with t... |
| eltpi 4687 | A member of an unordered t... |
| eltp 4688 | A member of an unordered t... |
| dftp2 4689 | Alternate definition of un... |
| nfpr 4690 | Bound-variable hypothesis ... |
| ifpr 4691 | Membership of a conditiona... |
| ralprgf 4692 | Convert a restricted unive... |
| rexprgf 4693 | Convert a restricted exist... |
| ralprg 4694 | Convert a restricted unive... |
| ralprgOLD 4695 | Obsolete version of ~ ralp... |
| rexprg 4696 | Convert a restricted exist... |
| rexprgOLD 4697 | Obsolete version of ~ rexp... |
| raltpg 4698 | Convert a restricted unive... |
| rextpg 4699 | Convert a restricted exist... |
| ralpr 4700 | Convert a restricted unive... |
| rexpr 4701 | Convert a restricted exist... |
| reuprg0 4702 | Convert a restricted exist... |
| reuprg 4703 | Convert a restricted exist... |
| reurexprg 4704 | Convert a restricted exist... |
| raltp 4705 | Convert a universal quanti... |
| rextp 4706 | Convert an existential qua... |
| nfsn 4707 | Bound-variable hypothesis ... |
| csbsng 4708 | Distribute proper substitu... |
| csbprg 4709 | Distribute proper substitu... |
| elinsn 4710 | If the intersection of two... |
| disjsn 4711 | Intersection with the sing... |
| disjsn2 4712 | Two distinct singletons ar... |
| disjpr2 4713 | Two completely distinct un... |
| disjprsn 4714 | The disjoint intersection ... |
| disjtpsn 4715 | The disjoint intersection ... |
| disjtp2 4716 | Two completely distinct un... |
| snprc 4717 | The singleton of a proper ... |
| snnzb 4718 | A singleton is nonempty if... |
| rmosn 4719 | A restricted at-most-one q... |
| r19.12sn 4720 | Special case of ~ r19.12 w... |
| rabsn 4721 | Condition where a restrict... |
| rabsnifsb 4722 | A restricted class abstrac... |
| rabsnif 4723 | A restricted class abstrac... |
| rabrsn 4724 | A restricted class abstrac... |
| euabsn2 4725 | Another way to express exi... |
| euabsn 4726 | Another way to express exi... |
| reusn 4727 | A way to express restricte... |
| absneu 4728 | Restricted existential uni... |
| rabsneu 4729 | Restricted existential uni... |
| eusn 4730 | Two ways to express " ` A ... |
| rabsnt 4731 | Truth implied by equality ... |
| prcom 4732 | Commutative law for unorde... |
| preq1 4733 | Equality theorem for unord... |
| preq2 4734 | Equality theorem for unord... |
| preq12 4735 | Equality theorem for unord... |
| preq1i 4736 | Equality inference for uno... |
| preq2i 4737 | Equality inference for uno... |
| preq12i 4738 | Equality inference for uno... |
| preq1d 4739 | Equality deduction for uno... |
| preq2d 4740 | Equality deduction for uno... |
| preq12d 4741 | Equality deduction for uno... |
| tpeq1 4742 | Equality theorem for unord... |
| tpeq2 4743 | Equality theorem for unord... |
| tpeq3 4744 | Equality theorem for unord... |
| tpeq1d 4745 | Equality theorem for unord... |
| tpeq2d 4746 | Equality theorem for unord... |
| tpeq3d 4747 | Equality theorem for unord... |
| tpeq123d 4748 | Equality theorem for unord... |
| tprot 4749 | Rotation of the elements o... |
| tpcoma 4750 | Swap 1st and 2nd members o... |
| tpcomb 4751 | Swap 2nd and 3rd members o... |
| tpass 4752 | Split off the first elemen... |
| qdass 4753 | Two ways to write an unord... |
| qdassr 4754 | Two ways to write an unord... |
| tpidm12 4755 | Unordered triple ` { A , A... |
| tpidm13 4756 | Unordered triple ` { A , B... |
| tpidm23 4757 | Unordered triple ` { A , B... |
| tpidm 4758 | Unordered triple ` { A , A... |
| tppreq3 4759 | An unordered triple is an ... |
| prid1g 4760 | An unordered pair contains... |
| prid2g 4761 | An unordered pair contains... |
| prid1 4762 | An unordered pair contains... |
| prid2 4763 | An unordered pair contains... |
| ifpprsnss 4764 | An unordered pair is a sin... |
| prprc1 4765 | A proper class vanishes in... |
| prprc2 4766 | A proper class vanishes in... |
| prprc 4767 | An unordered pair containi... |
| tpid1 4768 | One of the three elements ... |
| tpid1g 4769 | Closed theorem form of ~ t... |
| tpid2 4770 | One of the three elements ... |
| tpid2g 4771 | Closed theorem form of ~ t... |
| tpid3g 4772 | Closed theorem form of ~ t... |
| tpid3 4773 | One of the three elements ... |
| snnzg 4774 | The singleton of a set is ... |
| snn0d 4775 | The singleton of a set is ... |
| snnz 4776 | The singleton of a set is ... |
| prnz 4777 | A pair containing a set is... |
| prnzg 4778 | A pair containing a set is... |
| tpnz 4779 | An unordered triple contai... |
| tpnzd 4780 | An unordered triple contai... |
| raltpd 4781 | Convert a universal quanti... |
| snssb 4782 | Characterization of the in... |
| snssg 4783 | The singleton formed on a ... |
| snssgOLD 4784 | Obsolete version of ~ snss... |
| snss 4785 | The singleton of an elemen... |
| eldifsn 4786 | Membership in a set with a... |
| ssdifsn 4787 | Subset of a set with an el... |
| elpwdifsn 4788 | A subset of a set is an el... |
| eldifsni 4789 | Membership in a set with a... |
| eldifsnneq 4790 | An element of a difference... |
| neldifsn 4791 | The class ` A ` is not in ... |
| neldifsnd 4792 | The class ` A ` is not in ... |
| rexdifsn 4793 | Restricted existential qua... |
| raldifsni 4794 | Rearrangement of a propert... |
| raldifsnb 4795 | Restricted universal quant... |
| eldifvsn 4796 | A set is an element of the... |
| difsn 4797 | An element not in a set ca... |
| difprsnss 4798 | Removal of a singleton fro... |
| difprsn1 4799 | Removal of a singleton fro... |
| difprsn2 4800 | Removal of a singleton fro... |
| diftpsn3 4801 | Removal of a singleton fro... |
| difpr 4802 | Removing two elements as p... |
| tpprceq3 4803 | An unordered triple is an ... |
| tppreqb 4804 | An unordered triple is an ... |
| difsnb 4805 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4806 | ` ( B \ { A } ) ` is a pro... |
| snssi 4807 | The singleton of an elemen... |
| snssd 4808 | The singleton of an elemen... |
| difsnid 4809 | If we remove a single elem... |
| eldifeldifsn 4810 | An element of a difference... |
| pw0 4811 | Compute the power set of t... |
| pwpw0 4812 | Compute the power set of t... |
| snsspr1 4813 | A singleton is a subset of... |
| snsspr2 4814 | A singleton is a subset of... |
| snsstp1 4815 | A singleton is a subset of... |
| snsstp2 4816 | A singleton is a subset of... |
| snsstp3 4817 | A singleton is a subset of... |
| prssg 4818 | A pair of elements of a cl... |
| prss 4819 | A pair of elements of a cl... |
| prssi 4820 | A pair of elements of a cl... |
| prssd 4821 | Deduction version of ~ prs... |
| prsspwg 4822 | An unordered pair belongs ... |
| ssprss 4823 | A pair as subset of a pair... |
| ssprsseq 4824 | A proper pair is a subset ... |
| sssn 4825 | The subsets of a singleton... |
| ssunsn2 4826 | The property of being sand... |
| ssunsn 4827 | Possible values for a set ... |
| eqsn 4828 | Two ways to express that a... |
| issn 4829 | A sufficient condition for... |
| n0snor2el 4830 | A nonempty set is either a... |
| ssunpr 4831 | Possible values for a set ... |
| sspr 4832 | The subsets of a pair. (C... |
| sstp 4833 | The subsets of an unordere... |
| tpss 4834 | An unordered triple of ele... |
| tpssi 4835 | An unordered triple of ele... |
| sneqrg 4836 | Closed form of ~ sneqr . ... |
| sneqr 4837 | If the singletons of two s... |
| snsssn 4838 | If a singleton is a subset... |
| mosneq 4839 | There exists at most one s... |
| sneqbg 4840 | Two singletons of sets are... |
| snsspw 4841 | The singleton of a class i... |
| prsspw 4842 | An unordered pair belongs ... |
| preq1b 4843 | Biconditional equality lem... |
| preq2b 4844 | Biconditional equality lem... |
| preqr1 4845 | Reverse equality lemma for... |
| preqr2 4846 | Reverse equality lemma for... |
| preq12b 4847 | Equality relationship for ... |
| opthpr 4848 | An unordered pair has the ... |
| preqr1g 4849 | Reverse equality lemma for... |
| preq12bg 4850 | Closed form of ~ preq12b .... |
| prneimg 4851 | Two pairs are not equal if... |
| prnebg 4852 | A (proper) pair is not equ... |
| pr1eqbg 4853 | A (proper) pair is equal t... |
| pr1nebg 4854 | A (proper) pair is not equ... |
| preqsnd 4855 | Equivalence for a pair equ... |
| prnesn 4856 | A proper unordered pair is... |
| prneprprc 4857 | A proper unordered pair is... |
| preqsn 4858 | Equivalence for a pair equ... |
| preq12nebg 4859 | Equality relationship for ... |
| prel12g 4860 | Equality of two unordered ... |
| opthprneg 4861 | An unordered pair has the ... |
| elpreqprlem 4862 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4863 | Equality and membership ru... |
| elpreqprb 4864 | A set is an element of an ... |
| elpr2elpr 4865 | For an element ` A ` of an... |
| dfopif 4866 | Rewrite ~ df-op using ` if... |
| dfopg 4867 | Value of the ordered pair ... |
| dfop 4868 | Value of an ordered pair w... |
| opeq1 4869 | Equality theorem for order... |
| opeq2 4870 | Equality theorem for order... |
| opeq12 4871 | Equality theorem for order... |
| opeq1i 4872 | Equality inference for ord... |
| opeq2i 4873 | Equality inference for ord... |
| opeq12i 4874 | Equality inference for ord... |
| opeq1d 4875 | Equality deduction for ord... |
| opeq2d 4876 | Equality deduction for ord... |
| opeq12d 4877 | Equality deduction for ord... |
| oteq1 4878 | Equality theorem for order... |
| oteq2 4879 | Equality theorem for order... |
| oteq3 4880 | Equality theorem for order... |
| oteq1d 4881 | Equality deduction for ord... |
| oteq2d 4882 | Equality deduction for ord... |
| oteq3d 4883 | Equality deduction for ord... |
| oteq123d 4884 | Equality deduction for ord... |
| nfop 4885 | Bound-variable hypothesis ... |
| nfopd 4886 | Deduction version of bound... |
| csbopg 4887 | Distribution of class subs... |
| opidg 4888 | The ordered pair ` <. A , ... |
| opid 4889 | The ordered pair ` <. A , ... |
| ralunsn 4890 | Restricted quantification ... |
| 2ralunsn 4891 | Double restricted quantifi... |
| opprc 4892 | Expansion of an ordered pa... |
| opprc1 4893 | Expansion of an ordered pa... |
| opprc2 4894 | Expansion of an ordered pa... |
| oprcl 4895 | If an ordered pair has an ... |
| pwsn 4896 | The power set of a singlet... |
| pwpr 4897 | The power set of an unorde... |
| pwtp 4898 | The power set of an unorde... |
| pwpwpw0 4899 | Compute the power set of t... |
| pwv 4900 | The power class of the uni... |
| prproe 4901 | For an element of a proper... |
| 3elpr2eq 4902 | If there are three element... |
| dfuni2 4905 | Alternate definition of cl... |
| eluni 4906 | Membership in class union.... |
| eluni2 4907 | Membership in class union.... |
| elunii 4908 | Membership in class union.... |
| nfunid 4909 | Deduction version of ~ nfu... |
| nfuni 4910 | Bound-variable hypothesis ... |
| uniss 4911 | Subclass relationship for ... |
| unissi 4912 | Subclass relationship for ... |
| unissd 4913 | Subclass relationship for ... |
| unieq 4914 | Equality theorem for class... |
| unieqi 4915 | Inference of equality of t... |
| unieqd 4916 | Deduction of equality of t... |
| eluniab 4917 | Membership in union of a c... |
| elunirab 4918 | Membership in union of a c... |
| uniprg 4919 | The union of a pair is the... |
| unipr 4920 | The union of a pair is the... |
| uniprOLD 4921 | Obsolete version of ~ unip... |
| uniprgOLD 4922 | Obsolete version of ~ unip... |
| unisng 4923 | A set equals the union of ... |
| unisn 4924 | A set equals the union of ... |
| unisnv 4925 | A set equals the union of ... |
| unisn3 4926 | Union of a singleton in th... |
| dfnfc2 4927 | An alternative statement o... |
| uniun 4928 | The class union of the uni... |
| uniin 4929 | The class union of the int... |
| ssuni 4930 | Subclass relationship for ... |
| uni0b 4931 | The union of a set is empt... |
| uni0c 4932 | The union of a set is empt... |
| uni0 4933 | The union of the empty set... |
| csbuni 4934 | Distribute proper substitu... |
| elssuni 4935 | An element of a class is a... |
| unissel 4936 | Condition turning a subcla... |
| unissb 4937 | Relationship involving mem... |
| unissbOLD 4938 | Obsolete version of ~ unis... |
| uniss2 4939 | A subclass condition on th... |
| unidif 4940 | If the difference ` A \ B ... |
| ssunieq 4941 | Relationship implying unio... |
| unimax 4942 | Any member of a class is t... |
| pwuni 4943 | A class is a subclass of t... |
| dfint2 4946 | Alternate definition of cl... |
| inteq 4947 | Equality law for intersect... |
| inteqi 4948 | Equality inference for cla... |
| inteqd 4949 | Equality deduction for cla... |
| elint 4950 | Membership in class inters... |
| elint2 4951 | Membership in class inters... |
| elintg 4952 | Membership in class inters... |
| elinti 4953 | Membership in class inters... |
| nfint 4954 | Bound-variable hypothesis ... |
| elintabg 4955 | Two ways of saying a set i... |
| elintab 4956 | Membership in the intersec... |
| elintabOLD 4957 | Obsolete version of ~ elin... |
| elintrab 4958 | Membership in the intersec... |
| elintrabg 4959 | Membership in the intersec... |
| int0 4960 | The intersection of the em... |
| intss1 4961 | An element of a class incl... |
| ssint 4962 | Subclass of a class inters... |
| ssintab 4963 | Subclass of the intersecti... |
| ssintub 4964 | Subclass of the least uppe... |
| ssmin 4965 | Subclass of the minimum va... |
| intmin 4966 | Any member of a class is t... |
| intss 4967 | Intersection of subclasses... |
| intssuni 4968 | The intersection of a none... |
| ssintrab 4969 | Subclass of the intersecti... |
| unissint 4970 | If the union of a class is... |
| intssuni2 4971 | Subclass relationship for ... |
| intminss 4972 | Under subset ordering, the... |
| intmin2 4973 | Any set is the smallest of... |
| intmin3 4974 | Under subset ordering, the... |
| intmin4 4975 | Elimination of a conjunct ... |
| intab 4976 | The intersection of a spec... |
| int0el 4977 | The intersection of a clas... |
| intun 4978 | The class intersection of ... |
| intprg 4979 | The intersection of a pair... |
| intpr 4980 | The intersection of a pair... |
| intprOLD 4981 | Obsolete version of ~ intp... |
| intprgOLD 4982 | Obsolete version of ~ intp... |
| intsng 4983 | Intersection of a singleto... |
| intsn 4984 | The intersection of a sing... |
| uniintsn 4985 | Two ways to express " ` A ... |
| uniintab 4986 | The union and the intersec... |
| intunsn 4987 | Theorem joining a singleto... |
| rint0 4988 | Relative intersection of a... |
| elrint 4989 | Membership in a restricted... |
| elrint2 4990 | Membership in a restricted... |
| eliun 4995 | Membership in indexed unio... |
| eliin 4996 | Membership in indexed inte... |
| eliuni 4997 | Membership in an indexed u... |
| iuncom 4998 | Commutation of indexed uni... |
| iuncom4 4999 | Commutation of union with ... |
| iunconst 5000 | Indexed union of a constan... |
| iinconst 5001 | Indexed intersection of a ... |
| iuneqconst 5002 | Indexed union of identical... |
| iuniin 5003 | Law combining indexed unio... |
| iinssiun 5004 | An indexed intersection is... |
| iunss1 5005 | Subclass theorem for index... |
| iinss1 5006 | Subclass theorem for index... |
| iuneq1 5007 | Equality theorem for index... |
| iineq1 5008 | Equality theorem for index... |
| ss2iun 5009 | Subclass theorem for index... |
| iuneq2 5010 | Equality theorem for index... |
| iineq2 5011 | Equality theorem for index... |
| iuneq2i 5012 | Equality inference for ind... |
| iineq2i 5013 | Equality inference for ind... |
| iineq2d 5014 | Equality deduction for ind... |
| iuneq2dv 5015 | Equality deduction for ind... |
| iineq2dv 5016 | Equality deduction for ind... |
| iuneq12df 5017 | Equality deduction for ind... |
| iuneq1d 5018 | Equality theorem for index... |
| iuneq12d 5019 | Equality deduction for ind... |
| iuneq2d 5020 | Equality deduction for ind... |
| nfiun 5021 | Bound-variable hypothesis ... |
| nfiin 5022 | Bound-variable hypothesis ... |
| nfiung 5023 | Bound-variable hypothesis ... |
| nfiing 5024 | Bound-variable hypothesis ... |
| nfiu1 5025 | Bound-variable hypothesis ... |
| nfii1 5026 | Bound-variable hypothesis ... |
| dfiun2g 5027 | Alternate definition of in... |
| dfiun2gOLD 5028 | Obsolete version of ~ dfiu... |
| dfiin2g 5029 | Alternate definition of in... |
| dfiun2 5030 | Alternate definition of in... |
| dfiin2 5031 | Alternate definition of in... |
| dfiunv2 5032 | Define double indexed unio... |
| cbviun 5033 | Rule used to change the bo... |
| cbviin 5034 | Change bound variables in ... |
| cbviung 5035 | Rule used to change the bo... |
| cbviing 5036 | Change bound variables in ... |
| cbviunv 5037 | Rule used to change the bo... |
| cbviinv 5038 | Change bound variables in ... |
| cbviunvg 5039 | Rule used to change the bo... |
| cbviinvg 5040 | Change bound variables in ... |
| iunssf 5041 | Subset theorem for an inde... |
| iunss 5042 | Subset theorem for an inde... |
| ssiun 5043 | Subset implication for an ... |
| ssiun2 5044 | Identity law for subset of... |
| ssiun2s 5045 | Subset relationship for an... |
| iunss2 5046 | A subclass condition on th... |
| iunssd 5047 | Subset theorem for an inde... |
| iunab 5048 | The indexed union of a cla... |
| iunrab 5049 | The indexed union of a res... |
| iunxdif2 5050 | Indexed union with a class... |
| ssiinf 5051 | Subset theorem for an inde... |
| ssiin 5052 | Subset theorem for an inde... |
| iinss 5053 | Subset implication for an ... |
| iinss2 5054 | An indexed intersection is... |
| uniiun 5055 | Class union in terms of in... |
| intiin 5056 | Class intersection in term... |
| iunid 5057 | An indexed union of single... |
| iunidOLD 5058 | Obsolete version of ~ iuni... |
| iun0 5059 | An indexed union of the em... |
| 0iun 5060 | An empty indexed union is ... |
| 0iin 5061 | An empty indexed intersect... |
| viin 5062 | Indexed intersection with ... |
| iunsn 5063 | Indexed union of a singlet... |
| iunn0 5064 | There is a nonempty class ... |
| iinab 5065 | Indexed intersection of a ... |
| iinrab 5066 | Indexed intersection of a ... |
| iinrab2 5067 | Indexed intersection of a ... |
| iunin2 5068 | Indexed union of intersect... |
| iunin1 5069 | Indexed union of intersect... |
| iinun2 5070 | Indexed intersection of un... |
| iundif2 5071 | Indexed union of class dif... |
| iindif1 5072 | Indexed intersection of cl... |
| 2iunin 5073 | Rearrange indexed unions o... |
| iindif2 5074 | Indexed intersection of cl... |
| iinin2 5075 | Indexed intersection of in... |
| iinin1 5076 | Indexed intersection of in... |
| iinvdif 5077 | The indexed intersection o... |
| elriin 5078 | Elementhood in a relative ... |
| riin0 5079 | Relative intersection of a... |
| riinn0 5080 | Relative intersection of a... |
| riinrab 5081 | Relative intersection of a... |
| symdif0 5082 | Symmetric difference with ... |
| symdifv 5083 | The symmetric difference w... |
| symdifid 5084 | The symmetric difference o... |
| iinxsng 5085 | A singleton index picks ou... |
| iinxprg 5086 | Indexed intersection with ... |
| iunxsng 5087 | A singleton index picks ou... |
| iunxsn 5088 | A singleton index picks ou... |
| iunxsngf 5089 | A singleton index picks ou... |
| iunun 5090 | Separate a union in an ind... |
| iunxun 5091 | Separate a union in the in... |
| iunxdif3 5092 | An indexed union where som... |
| iunxprg 5093 | A pair index picks out two... |
| iunxiun 5094 | Separate an indexed union ... |
| iinuni 5095 | A relationship involving u... |
| iununi 5096 | A relationship involving u... |
| sspwuni 5097 | Subclass relationship for ... |
| pwssb 5098 | Two ways to express a coll... |
| elpwpw 5099 | Characterization of the el... |
| pwpwab 5100 | The double power class wri... |
| pwpwssunieq 5101 | The class of sets whose un... |
| elpwuni 5102 | Relationship for power cla... |
| iinpw 5103 | The power class of an inte... |
| iunpwss 5104 | Inclusion of an indexed un... |
| intss2 5105 | A nonempty intersection of... |
| rintn0 5106 | Relative intersection of a... |
| dfdisj2 5109 | Alternate definition for d... |
| disjss2 5110 | If each element of a colle... |
| disjeq2 5111 | Equality theorem for disjo... |
| disjeq2dv 5112 | Equality deduction for dis... |
| disjss1 5113 | A subset of a disjoint col... |
| disjeq1 5114 | Equality theorem for disjo... |
| disjeq1d 5115 | Equality theorem for disjo... |
| disjeq12d 5116 | Equality theorem for disjo... |
| cbvdisj 5117 | Change bound variables in ... |
| cbvdisjv 5118 | Change bound variables in ... |
| nfdisjw 5119 | Bound-variable hypothesis ... |
| nfdisj 5120 | Bound-variable hypothesis ... |
| nfdisj1 5121 | Bound-variable hypothesis ... |
| disjor 5122 | Two ways to say that a col... |
| disjors 5123 | Two ways to say that a col... |
| disji2 5124 | Property of a disjoint col... |
| disji 5125 | Property of a disjoint col... |
| invdisj 5126 | If there is a function ` C... |
| invdisjrabw 5127 | Version of ~ invdisjrab wi... |
| invdisjrab 5128 | The restricted class abstr... |
| disjiun 5129 | A disjoint collection yiel... |
| disjord 5130 | Conditions for a collectio... |
| disjiunb 5131 | Two ways to say that a col... |
| disjiund 5132 | Conditions for a collectio... |
| sndisj 5133 | Any collection of singleto... |
| 0disj 5134 | Any collection of empty se... |
| disjxsn 5135 | A singleton collection is ... |
| disjx0 5136 | An empty collection is dis... |
| disjprgw 5137 | Version of ~ disjprg with ... |
| disjprg 5138 | A pair collection is disjo... |
| disjxiun 5139 | An indexed union of a disj... |
| disjxun 5140 | The union of two disjoint ... |
| disjss3 5141 | Expand a disjoint collecti... |
| breq 5144 | Equality theorem for binar... |
| breq1 5145 | Equality theorem for a bin... |
| breq2 5146 | Equality theorem for a bin... |
| breq12 5147 | Equality theorem for a bin... |
| breqi 5148 | Equality inference for bin... |
| breq1i 5149 | Equality inference for a b... |
| breq2i 5150 | Equality inference for a b... |
| breq12i 5151 | Equality inference for a b... |
| breq1d 5152 | Equality deduction for a b... |
| breqd 5153 | Equality deduction for a b... |
| breq2d 5154 | Equality deduction for a b... |
| breq12d 5155 | Equality deduction for a b... |
| breq123d 5156 | Equality deduction for a b... |
| breqdi 5157 | Equality deduction for a b... |
| breqan12d 5158 | Equality deduction for a b... |
| breqan12rd 5159 | Equality deduction for a b... |
| eqnbrtrd 5160 | Substitution of equal clas... |
| nbrne1 5161 | Two classes are different ... |
| nbrne2 5162 | Two classes are different ... |
| eqbrtri 5163 | Substitution of equal clas... |
| eqbrtrd 5164 | Substitution of equal clas... |
| eqbrtrri 5165 | Substitution of equal clas... |
| eqbrtrrd 5166 | Substitution of equal clas... |
| breqtri 5167 | Substitution of equal clas... |
| breqtrd 5168 | Substitution of equal clas... |
| breqtrri 5169 | Substitution of equal clas... |
| breqtrrd 5170 | Substitution of equal clas... |
| 3brtr3i 5171 | Substitution of equality i... |
| 3brtr4i 5172 | Substitution of equality i... |
| 3brtr3d 5173 | Substitution of equality i... |
| 3brtr4d 5174 | Substitution of equality i... |
| 3brtr3g 5175 | Substitution of equality i... |
| 3brtr4g 5176 | Substitution of equality i... |
| eqbrtrid 5177 | A chained equality inferen... |
| eqbrtrrid 5178 | A chained equality inferen... |
| breqtrid 5179 | A chained equality inferen... |
| breqtrrid 5180 | A chained equality inferen... |
| eqbrtrdi 5181 | A chained equality inferen... |
| eqbrtrrdi 5182 | A chained equality inferen... |
| breqtrdi 5183 | A chained equality inferen... |
| breqtrrdi 5184 | A chained equality inferen... |
| ssbrd 5185 | Deduction from a subclass ... |
| ssbr 5186 | Implication from a subclas... |
| ssbri 5187 | Inference from a subclass ... |
| nfbrd 5188 | Deduction version of bound... |
| nfbr 5189 | Bound-variable hypothesis ... |
| brab1 5190 | Relationship between a bin... |
| br0 5191 | The empty binary relation ... |
| brne0 5192 | If two sets are in a binar... |
| brun 5193 | The union of two binary re... |
| brin 5194 | The intersection of two re... |
| brdif 5195 | The difference of two bina... |
| sbcbr123 5196 | Move substitution in and o... |
| sbcbr 5197 | Move substitution in and o... |
| sbcbr12g 5198 | Move substitution in and o... |
| sbcbr1g 5199 | Move substitution in and o... |
| sbcbr2g 5200 | Move substitution in and o... |
| brsymdif 5201 | Characterization of the sy... |
| brralrspcev 5202 | Restricted existential spe... |
| brimralrspcev 5203 | Restricted existential spe... |
| opabss 5206 | The collection of ordered ... |
| opabbid 5207 | Equivalent wff's yield equ... |
| opabbidv 5208 | Equivalent wff's yield equ... |
| opabbii 5209 | Equivalent wff's yield equ... |
| nfopabd 5210 | Bound-variable hypothesis ... |
| nfopab 5211 | Bound-variable hypothesis ... |
| nfopab1 5212 | The first abstraction vari... |
| nfopab2 5213 | The second abstraction var... |
| cbvopab 5214 | Rule used to change bound ... |
| cbvopabv 5215 | Rule used to change bound ... |
| cbvopabvOLD 5216 | Obsolete version of ~ cbvo... |
| cbvopab1 5217 | Change first bound variabl... |
| cbvopab1g 5218 | Change first bound variabl... |
| cbvopab2 5219 | Change second bound variab... |
| cbvopab1s 5220 | Change first bound variabl... |
| cbvopab1v 5221 | Rule used to change the fi... |
| cbvopab1vOLD 5222 | Obsolete version of ~ cbvo... |
| cbvopab2v 5223 | Rule used to change the se... |
| unopab 5224 | Union of two ordered pair ... |
| mpteq12da 5227 | An equality inference for ... |
| mpteq12df 5228 | An equality inference for ... |
| mpteq12dfOLD 5229 | Obsolete version of ~ mpte... |
| mpteq12f 5230 | An equality theorem for th... |
| mpteq12dva 5231 | An equality inference for ... |
| mpteq12dvaOLD 5232 | Obsolete version of ~ mpte... |
| mpteq12dv 5233 | An equality inference for ... |
| mpteq12 5234 | An equality theorem for th... |
| mpteq1 5235 | An equality theorem for th... |
| mpteq1OLD 5236 | Obsolete version of ~ mpte... |
| mpteq1d 5237 | An equality theorem for th... |
| mpteq1i 5238 | An equality theorem for th... |
| mpteq1iOLD 5239 | Obsolete version of ~ mpte... |
| mpteq2da 5240 | Slightly more general equa... |
| mpteq2daOLD 5241 | Obsolete version of ~ mpte... |
| mpteq2dva 5242 | Slightly more general equa... |
| mpteq2dvaOLD 5243 | Obsolete version of ~ mpte... |
| mpteq2dv 5244 | An equality inference for ... |
| mpteq2ia 5245 | An equality inference for ... |
| mpteq2iaOLD 5246 | Obsolete version of ~ mpte... |
| mpteq2i 5247 | An equality inference for ... |
| mpteq12i 5248 | An equality inference for ... |
| nfmpt 5249 | Bound-variable hypothesis ... |
| nfmpt1 5250 | Bound-variable hypothesis ... |
| cbvmptf 5251 | Rule to change the bound v... |
| cbvmptfg 5252 | Rule to change the bound v... |
| cbvmpt 5253 | Rule to change the bound v... |
| cbvmptg 5254 | Rule to change the bound v... |
| cbvmptv 5255 | Rule to change the bound v... |
| cbvmptvOLD 5256 | Obsolete version of ~ cbvm... |
| cbvmptvg 5257 | Rule to change the bound v... |
| mptv 5258 | Function with universal do... |
| dftr2 5261 | An alternate way of defini... |
| dftr2c 5262 | Variant of ~ dftr2 with co... |
| dftr5 5263 | An alternate way of defini... |
| dftr5OLD 5264 | Obsolete version of ~ dftr... |
| dftr3 5265 | An alternate way of defini... |
| dftr4 5266 | An alternate way of defini... |
| treq 5267 | Equality theorem for the t... |
| trel 5268 | In a transitive class, the... |
| trel3 5269 | In a transitive class, the... |
| trss 5270 | An element of a transitive... |
| trin 5271 | The intersection of transi... |
| tr0 5272 | The empty set is transitiv... |
| trv 5273 | The universe is transitive... |
| triun 5274 | An indexed union of a clas... |
| truni 5275 | The union of a class of tr... |
| triin 5276 | An indexed intersection of... |
| trint 5277 | The intersection of a clas... |
| trintss 5278 | Any nonempty transitive cl... |
| axrep1 5280 | The version of the Axiom o... |
| axreplem 5281 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5282 | Axiom of Replacement expre... |
| axrep3 5283 | Axiom of Replacement sligh... |
| axrep4 5284 | A more traditional version... |
| axrep5 5285 | Axiom of Replacement (simi... |
| axrep6 5286 | A condensed form of ~ ax-r... |
| axrep6g 5287 | ~ axrep6 in class notation... |
| zfrepclf 5288 | An inference based on the ... |
| zfrep3cl 5289 | An inference based on the ... |
| zfrep4 5290 | A version of Replacement u... |
| axsepgfromrep 5291 | A more general version ~ a... |
| axsep 5292 | Axiom scheme of separation... |
| axsepg 5294 | A more general version of ... |
| zfauscl 5295 | Separation Scheme (Aussond... |
| bm1.3ii 5296 | Convert implication to equ... |
| ax6vsep 5297 | Derive ~ ax6v (a weakened ... |
| axnulALT 5298 | Alternate proof of ~ axnul... |
| axnul 5299 | The Null Set Axiom of ZF s... |
| 0ex 5301 | The Null Set Axiom of ZF s... |
| al0ssb 5302 | The empty set is the uniqu... |
| sseliALT 5303 | Alternate proof of ~ sseli... |
| csbexg 5304 | The existence of proper su... |
| csbex 5305 | The existence of proper su... |
| unisn2 5306 | A version of ~ unisn witho... |
| nalset 5307 | No set contains all sets. ... |
| vnex 5308 | The universal class does n... |
| vprc 5309 | The universal class is not... |
| nvel 5310 | The universal class does n... |
| inex1 5311 | Separation Scheme (Aussond... |
| inex2 5312 | Separation Scheme (Aussond... |
| inex1g 5313 | Closed-form, generalized S... |
| inex2g 5314 | Sufficient condition for a... |
| ssex 5315 | The subset of a set is als... |
| ssexi 5316 | The subset of a set is als... |
| ssexg 5317 | The subset of a set is als... |
| ssexd 5318 | A subclass of a set is a s... |
| abexd 5319 | Conditions for a class abs... |
| abex 5320 | Conditions for a class abs... |
| prcssprc 5321 | The superclass of a proper... |
| sselpwd 5322 | Elementhood to a power set... |
| difexg 5323 | Existence of a difference.... |
| difexi 5324 | Existence of a difference,... |
| difexd 5325 | Existence of a difference.... |
| zfausab 5326 | Separation Scheme (Aussond... |
| rabexg 5327 | Separation Scheme in terms... |
| rabex 5328 | Separation Scheme in terms... |
| rabexd 5329 | Separation Scheme in terms... |
| rabex2 5330 | Separation Scheme in terms... |
| rab2ex 5331 | A class abstraction based ... |
| elssabg 5332 | Membership in a class abst... |
| intex 5333 | The intersection of a none... |
| intnex 5334 | If a class intersection is... |
| intexab 5335 | The intersection of a none... |
| intexrab 5336 | The intersection of a none... |
| iinexg 5337 | The existence of a class i... |
| intabs 5338 | Absorption of a redundant ... |
| inuni 5339 | The intersection of a unio... |
| elpw2g 5340 | Membership in a power clas... |
| elpw2 5341 | Membership in a power clas... |
| elpwi2 5342 | Membership in a power clas... |
| elpwi2OLD 5343 | Obsolete version of ~ elpw... |
| axpweq 5344 | Two equivalent ways to exp... |
| pwnss 5345 | The power set of a set is ... |
| pwne 5346 | No set equals its power se... |
| difelpw 5347 | A difference is an element... |
| rabelpw 5348 | A restricted class abstrac... |
| class2set 5349 | The class of elements of `... |
| 0elpw 5350 | Every power class contains... |
| pwne0 5351 | A power class is never emp... |
| 0nep0 5352 | The empty set and its powe... |
| 0inp0 5353 | Something cannot be equal ... |
| unidif0 5354 | The removal of the empty s... |
| eqsnuniex 5355 | If a class is equal to the... |
| iin0 5356 | An indexed intersection of... |
| notzfaus 5357 | In the Separation Scheme ~... |
| intv 5358 | The intersection of the un... |
| zfpow 5360 | Axiom of Power Sets expres... |
| axpow2 5361 | A variant of the Axiom of ... |
| axpow3 5362 | A variant of the Axiom of ... |
| elALT2 5363 | Alternate proof of ~ el us... |
| dtruALT2 5364 | Alternate proof of ~ dtru ... |
| dtrucor 5365 | Corollary of ~ dtru . Thi... |
| dtrucor2 5366 | The theorem form of the de... |
| dvdemo1 5367 | Demonstration of a theorem... |
| dvdemo2 5368 | Demonstration of a theorem... |
| nfnid 5369 | A setvar variable is not f... |
| nfcvb 5370 | The "distinctor" expressio... |
| vpwex 5371 | Power set axiom: the power... |
| pwexg 5372 | Power set axiom expressed ... |
| pwexd 5373 | Deduction version of the p... |
| pwex 5374 | Power set axiom expressed ... |
| pwel 5375 | Quantitative version of ~ ... |
| abssexg 5376 | Existence of a class of su... |
| snexALT 5377 | Alternate proof of ~ snex ... |
| p0ex 5378 | The power set of the empty... |
| p0exALT 5379 | Alternate proof of ~ p0ex ... |
| pp0ex 5380 | The power set of the power... |
| ord3ex 5381 | The ordinal number 3 is a ... |
| dtruALT 5382 | Alternate proof of ~ dtru ... |
| axc16b 5383 | This theorem shows that Ax... |
| eunex 5384 | Existential uniqueness imp... |
| eusv1 5385 | Two ways to express single... |
| eusvnf 5386 | Even if ` x ` is free in `... |
| eusvnfb 5387 | Two ways to say that ` A (... |
| eusv2i 5388 | Two ways to express single... |
| eusv2nf 5389 | Two ways to express single... |
| eusv2 5390 | Two ways to express single... |
| reusv1 5391 | Two ways to express single... |
| reusv2lem1 5392 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5393 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5394 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5395 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5396 | Lemma for ~ reusv2 . (Con... |
| reusv2 5397 | Two ways to express single... |
| reusv3i 5398 | Two ways of expressing exi... |
| reusv3 5399 | Two ways to express single... |
| eusv4 5400 | Two ways to express single... |
| alxfr 5401 | Transfer universal quantif... |
| ralxfrd 5402 | Transfer universal quantif... |
| rexxfrd 5403 | Transfer universal quantif... |
| ralxfr2d 5404 | Transfer universal quantif... |
| rexxfr2d 5405 | Transfer universal quantif... |
| ralxfrd2 5406 | Transfer universal quantif... |
| rexxfrd2 5407 | Transfer existence from a ... |
| ralxfr 5408 | Transfer universal quantif... |
| ralxfrALT 5409 | Alternate proof of ~ ralxf... |
| rexxfr 5410 | Transfer existence from a ... |
| rabxfrd 5411 | Membership in a restricted... |
| rabxfr 5412 | Membership in a restricted... |
| reuhypd 5413 | A theorem useful for elimi... |
| reuhyp 5414 | A theorem useful for elimi... |
| zfpair 5415 | The Axiom of Pairing of Ze... |
| axprALT 5416 | Alternate proof of ~ axpr ... |
| axprlem1 5417 | Lemma for ~ axpr . There ... |
| axprlem2 5418 | Lemma for ~ axpr . There ... |
| axprlem3 5419 | Lemma for ~ axpr . Elimin... |
| axprlem4 5420 | Lemma for ~ axpr . The fi... |
| axprlem5 5421 | Lemma for ~ axpr . The se... |
| axpr 5422 | Unabbreviated version of t... |
| zfpair2 5424 | Derive the abbreviated ver... |
| vsnex 5425 | A singleton built on a set... |
| snexg 5426 | A singleton built on a set... |
| snex 5427 | A singleton is a set. The... |
| prex 5428 | The Axiom of Pairing using... |
| exel 5429 | There exist two sets, one ... |
| exexneq 5430 | There exist two different ... |
| exneq 5431 | Given any set (the " ` y `... |
| dtru 5432 | Given any set (the " ` y `... |
| el 5433 | Any set is an element of s... |
| sels 5434 | If a class is a set, then ... |
| selsALT 5435 | Alternate proof of ~ sels ... |
| elALT 5436 | Alternate proof of ~ el , ... |
| dtruOLD 5437 | Obsolete proof of ~ dtru a... |
| snelpwg 5438 | A singleton of a set is a ... |
| snelpwi 5439 | If a set is a member of a ... |
| snelpwiOLD 5440 | Obsolete version of ~ snel... |
| snelpw 5441 | A singleton of a set is a ... |
| prelpw 5442 | An unordered pair of two s... |
| prelpwi 5443 | If two sets are members of... |
| rext 5444 | A theorem similar to exten... |
| sspwb 5445 | The powerclass constructio... |
| unipw 5446 | A class equals the union o... |
| univ 5447 | The union of the universe ... |
| pwtr 5448 | A class is transitive iff ... |
| ssextss 5449 | An extensionality-like pri... |
| ssext 5450 | An extensionality-like pri... |
| nssss 5451 | Negation of subclass relat... |
| pweqb 5452 | Classes are equal if and o... |
| intidg 5453 | The intersection of all se... |
| intidOLD 5454 | Obsolete version of ~ inti... |
| moabex 5455 | "At most one" existence im... |
| rmorabex 5456 | Restricted "at most one" e... |
| euabex 5457 | The abstraction of a wff w... |
| nnullss 5458 | A nonempty class (even if ... |
| exss 5459 | Restricted existence in a ... |
| opex 5460 | An ordered pair of classes... |
| otex 5461 | An ordered triple of class... |
| elopg 5462 | Characterization of the el... |
| elop 5463 | Characterization of the el... |
| opi1 5464 | One of the two elements in... |
| opi2 5465 | One of the two elements of... |
| opeluu 5466 | Each member of an ordered ... |
| op1stb 5467 | Extract the first member o... |
| brv 5468 | Two classes are always in ... |
| opnz 5469 | An ordered pair is nonempt... |
| opnzi 5470 | An ordered pair is nonempt... |
| opth1 5471 | Equality of the first memb... |
| opth 5472 | The ordered pair theorem. ... |
| opthg 5473 | Ordered pair theorem. ` C ... |
| opth1g 5474 | Equality of the first memb... |
| opthg2 5475 | Ordered pair theorem. (Co... |
| opth2 5476 | Ordered pair theorem. (Co... |
| opthneg 5477 | Two ordered pairs are not ... |
| opthne 5478 | Two ordered pairs are not ... |
| otth2 5479 | Ordered triple theorem, wi... |
| otth 5480 | Ordered triple theorem. (... |
| otthg 5481 | Ordered triple theorem, cl... |
| otthne 5482 | Contrapositive of the orde... |
| eqvinop 5483 | A variable introduction la... |
| sbcop1 5484 | The proper substitution of... |
| sbcop 5485 | The proper substitution of... |
| copsexgw 5486 | Version of ~ copsexg with ... |
| copsexg 5487 | Substitution of class ` A ... |
| copsex2t 5488 | Closed theorem form of ~ c... |
| copsex2g 5489 | Implicit substitution infe... |
| copsex2gOLD 5490 | Obsolete version of ~ cops... |
| copsex4g 5491 | An implicit substitution i... |
| 0nelop 5492 | A property of ordered pair... |
| opwo0id 5493 | An ordered pair is equal t... |
| opeqex 5494 | Equivalence of existence i... |
| oteqex2 5495 | Equivalence of existence i... |
| oteqex 5496 | Equivalence of existence i... |
| opcom 5497 | An ordered pair commutes i... |
| moop2 5498 | "At most one" property of ... |
| opeqsng 5499 | Equivalence for an ordered... |
| opeqsn 5500 | Equivalence for an ordered... |
| opeqpr 5501 | Equivalence for an ordered... |
| snopeqop 5502 | Equivalence for an ordered... |
| propeqop 5503 | Equivalence for an ordered... |
| propssopi 5504 | If a pair of ordered pairs... |
| snopeqopsnid 5505 | Equivalence for an ordered... |
| mosubopt 5506 | "At most one" remains true... |
| mosubop 5507 | "At most one" remains true... |
| euop2 5508 | Transfer existential uniqu... |
| euotd 5509 | Prove existential uniquene... |
| opthwiener 5510 | Justification theorem for ... |
| uniop 5511 | The union of an ordered pa... |
| uniopel 5512 | Ordered pair membership is... |
| opthhausdorff 5513 | Justification theorem for ... |
| opthhausdorff0 5514 | Justification theorem for ... |
| otsndisj 5515 | The singletons consisting ... |
| otiunsndisj 5516 | The union of singletons co... |
| iunopeqop 5517 | Implication of an ordered ... |
| brsnop 5518 | Binary relation for an ord... |
| brtp 5519 | A necessary and sufficient... |
| opabidw 5520 | The law of concretion. Sp... |
| opabid 5521 | The law of concretion. Sp... |
| elopabw 5522 | Membership in a class abst... |
| elopab 5523 | Membership in a class abst... |
| rexopabb 5524 | Restricted existential qua... |
| vopelopabsb 5525 | The law of concretion in t... |
| opelopabsb 5526 | The law of concretion in t... |
| brabsb 5527 | The law of concretion in t... |
| opelopabt 5528 | Closed theorem form of ~ o... |
| opelopabga 5529 | The law of concretion. Th... |
| brabga 5530 | The law of concretion for ... |
| opelopab2a 5531 | Ordered pair membership in... |
| opelopaba 5532 | The law of concretion. Th... |
| braba 5533 | The law of concretion for ... |
| opelopabg 5534 | The law of concretion. Th... |
| brabg 5535 | The law of concretion for ... |
| opelopabgf 5536 | The law of concretion. Th... |
| opelopab2 5537 | Ordered pair membership in... |
| opelopab 5538 | The law of concretion. Th... |
| brab 5539 | The law of concretion for ... |
| opelopabaf 5540 | The law of concretion. Th... |
| opelopabf 5541 | The law of concretion. Th... |
| ssopab2 5542 | Equivalence of ordered pai... |
| ssopab2bw 5543 | Equivalence of ordered pai... |
| eqopab2bw 5544 | Equivalence of ordered pai... |
| ssopab2b 5545 | Equivalence of ordered pai... |
| ssopab2i 5546 | Inference of ordered pair ... |
| ssopab2dv 5547 | Inference of ordered pair ... |
| eqopab2b 5548 | Equivalence of ordered pai... |
| opabn0 5549 | Nonempty ordered pair clas... |
| opab0 5550 | Empty ordered pair class a... |
| csbopab 5551 | Move substitution into a c... |
| csbopabgALT 5552 | Move substitution into a c... |
| csbmpt12 5553 | Move substitution into a m... |
| csbmpt2 5554 | Move substitution into the... |
| iunopab 5555 | Move indexed union inside ... |
| iunopabOLD 5556 | Obsolete version of ~ iuno... |
| elopabr 5557 | Membership in an ordered-p... |
| elopabran 5558 | Membership in an ordered-p... |
| elopabrOLD 5559 | Obsolete version of ~ elop... |
| rbropapd 5560 | Properties of a pair in an... |
| rbropap 5561 | Properties of a pair in a ... |
| 2rbropap 5562 | Properties of a pair in a ... |
| 0nelopab 5563 | The empty set is never an ... |
| 0nelopabOLD 5564 | Obsolete version of ~ 0nel... |
| brabv 5565 | If two classes are in a re... |
| pwin 5566 | The power class of the int... |
| pwssun 5567 | The power class of the uni... |
| pwun 5568 | The power class of the uni... |
| dfid4 5571 | The identity function expr... |
| dfid2 5572 | Alternate definition of th... |
| dfid3 5573 | A stronger version of ~ df... |
| dfid2OLD 5574 | Obsolete version of ~ dfid... |
| epelg 5577 | The membership relation an... |
| epeli 5578 | The membership relation an... |
| epel 5579 | The membership relation an... |
| 0sn0ep 5580 | An example for the members... |
| epn0 5581 | The membership relation is... |
| poss 5586 | Subset theorem for the par... |
| poeq1 5587 | Equality theorem for parti... |
| poeq2 5588 | Equality theorem for parti... |
| nfpo 5589 | Bound-variable hypothesis ... |
| nfso 5590 | Bound-variable hypothesis ... |
| pocl 5591 | Characteristic properties ... |
| poclOLD 5592 | Obsolete version of ~ pocl... |
| ispod 5593 | Sufficient conditions for ... |
| swopolem 5594 | Perform the substitutions ... |
| swopo 5595 | A strict weak order is a p... |
| poirr 5596 | A partial order is irrefle... |
| potr 5597 | A partial order is a trans... |
| po2nr 5598 | A partial order has no 2-c... |
| po3nr 5599 | A partial order has no 3-c... |
| po2ne 5600 | Two sets related by a part... |
| po0 5601 | Any relation is a partial ... |
| pofun 5602 | The inverse image of a par... |
| sopo 5603 | A strict linear order is a... |
| soss 5604 | Subset theorem for the str... |
| soeq1 5605 | Equality theorem for the s... |
| soeq2 5606 | Equality theorem for the s... |
| sonr 5607 | A strict order relation is... |
| sotr 5608 | A strict order relation is... |
| solin 5609 | A strict order relation is... |
| so2nr 5610 | A strict order relation ha... |
| so3nr 5611 | A strict order relation ha... |
| sotric 5612 | A strict order relation sa... |
| sotrieq 5613 | Trichotomy law for strict ... |
| sotrieq2 5614 | Trichotomy law for strict ... |
| soasym 5615 | Asymmetry law for strict o... |
| sotr2 5616 | A transitivity relation. ... |
| issod 5617 | An irreflexive, transitive... |
| issoi 5618 | An irreflexive, transitive... |
| isso2i 5619 | Deduce strict ordering fro... |
| so0 5620 | Any relation is a strict o... |
| somo 5621 | A totally ordered set has ... |
| sotrine 5622 | Trichotomy law for strict ... |
| sotr3 5623 | Transitivity law for stric... |
| dffr6 5630 | Alternate definition of ~ ... |
| frd 5631 | A nonempty subset of an ` ... |
| fri 5632 | A nonempty subset of an ` ... |
| friOLD 5633 | Obsolete version of ~ fri ... |
| seex 5634 | The ` R ` -preimage of an ... |
| exse 5635 | Any relation on a set is s... |
| dffr2 5636 | Alternate definition of we... |
| dffr2ALT 5637 | Alternate proof of ~ dffr2... |
| frc 5638 | Property of well-founded r... |
| frss 5639 | Subset theorem for the wel... |
| sess1 5640 | Subset theorem for the set... |
| sess2 5641 | Subset theorem for the set... |
| freq1 5642 | Equality theorem for the w... |
| freq2 5643 | Equality theorem for the w... |
| seeq1 5644 | Equality theorem for the s... |
| seeq2 5645 | Equality theorem for the s... |
| nffr 5646 | Bound-variable hypothesis ... |
| nfse 5647 | Bound-variable hypothesis ... |
| nfwe 5648 | Bound-variable hypothesis ... |
| frirr 5649 | A well-founded relation is... |
| fr2nr 5650 | A well-founded relation ha... |
| fr0 5651 | Any relation is well-found... |
| frminex 5652 | If an element of a well-fo... |
| efrirr 5653 | A well-founded class does ... |
| efrn2lp 5654 | A well-founded class conta... |
| epse 5655 | The membership relation is... |
| tz7.2 5656 | Similar to Theorem 7.2 of ... |
| dfepfr 5657 | An alternate way of saying... |
| epfrc 5658 | A subset of a well-founded... |
| wess 5659 | Subset theorem for the wel... |
| weeq1 5660 | Equality theorem for the w... |
| weeq2 5661 | Equality theorem for the w... |
| wefr 5662 | A well-ordering is well-fo... |
| weso 5663 | A well-ordering is a stric... |
| wecmpep 5664 | The elements of a class we... |
| wetrep 5665 | On a class well-ordered by... |
| wefrc 5666 | A nonempty subclass of a c... |
| we0 5667 | Any relation is a well-ord... |
| wereu 5668 | A nonempty subset of an ` ... |
| wereu2 5669 | A nonempty subclass of an ... |
| xpeq1 5686 | Equality theorem for Carte... |
| xpss12 5687 | Subset theorem for Cartesi... |
| xpss 5688 | A Cartesian product is inc... |
| inxpssres 5689 | Intersection with a Cartes... |
| relxp 5690 | A Cartesian product is a r... |
| xpss1 5691 | Subset relation for Cartes... |
| xpss2 5692 | Subset relation for Cartes... |
| xpeq2 5693 | Equality theorem for Carte... |
| elxpi 5694 | Membership in a Cartesian ... |
| elxp 5695 | Membership in a Cartesian ... |
| elxp2 5696 | Membership in a Cartesian ... |
| xpeq12 5697 | Equality theorem for Carte... |
| xpeq1i 5698 | Equality inference for Car... |
| xpeq2i 5699 | Equality inference for Car... |
| xpeq12i 5700 | Equality inference for Car... |
| xpeq1d 5701 | Equality deduction for Car... |
| xpeq2d 5702 | Equality deduction for Car... |
| xpeq12d 5703 | Equality deduction for Car... |
| sqxpeqd 5704 | Equality deduction for a C... |
| nfxp 5705 | Bound-variable hypothesis ... |
| 0nelxp 5706 | The empty set is not a mem... |
| 0nelelxp 5707 | A member of a Cartesian pr... |
| opelxp 5708 | Ordered pair membership in... |
| opelxpi 5709 | Ordered pair membership in... |
| opelxpii 5710 | Ordered pair membership in... |
| opelxpd 5711 | Ordered pair membership in... |
| opelvv 5712 | Ordered pair membership in... |
| opelvvg 5713 | Ordered pair membership in... |
| opelxp1 5714 | The first member of an ord... |
| opelxp2 5715 | The second member of an or... |
| otelxp 5716 | Ordered triple membership ... |
| otelxp1 5717 | The first member of an ord... |
| otel3xp 5718 | An ordered triple is an el... |
| opabssxpd 5719 | An ordered-pair class abst... |
| rabxp 5720 | Class abstraction restrict... |
| brxp 5721 | Binary relation on a Carte... |
| pwvrel 5722 | A set is a binary relation... |
| pwvabrel 5723 | The powerclass of the cart... |
| brrelex12 5724 | Two classes related by a b... |
| brrelex1 5725 | If two classes are related... |
| brrelex2 5726 | If two classes are related... |
| brrelex12i 5727 | Two classes that are relat... |
| brrelex1i 5728 | The first argument of a bi... |
| brrelex2i 5729 | The second argument of a b... |
| nprrel12 5730 | Proper classes are not rel... |
| nprrel 5731 | No proper class is related... |
| 0nelrel0 5732 | A binary relation does not... |
| 0nelrel 5733 | A binary relation does not... |
| fconstmpt 5734 | Representation of a consta... |
| vtoclr 5735 | Variable to class conversi... |
| opthprc 5736 | Justification theorem for ... |
| brel 5737 | Two things in a binary rel... |
| elxp3 5738 | Membership in a Cartesian ... |
| opeliunxp 5739 | Membership in a union of C... |
| xpundi 5740 | Distributive law for Carte... |
| xpundir 5741 | Distributive law for Carte... |
| xpiundi 5742 | Distributive law for Carte... |
| xpiundir 5743 | Distributive law for Carte... |
| iunxpconst 5744 | Membership in a union of C... |
| xpun 5745 | The Cartesian product of t... |
| elvv 5746 | Membership in universal cl... |
| elvvv 5747 | Membership in universal cl... |
| elvvuni 5748 | An ordered pair contains i... |
| brinxp2 5749 | Intersection of binary rel... |
| brinxp 5750 | Intersection of binary rel... |
| opelinxp 5751 | Ordered pair element in an... |
| poinxp 5752 | Intersection of partial or... |
| soinxp 5753 | Intersection of total orde... |
| frinxp 5754 | Intersection of well-found... |
| seinxp 5755 | Intersection of set-like r... |
| weinxp 5756 | Intersection of well-order... |
| posn 5757 | Partial ordering of a sing... |
| sosn 5758 | Strict ordering on a singl... |
| frsn 5759 | Founded relation on a sing... |
| wesn 5760 | Well-ordering of a singlet... |
| elopaelxp 5761 | Membership in an ordered-p... |
| elopaelxpOLD 5762 | Obsolete version of ~ elop... |
| bropaex12 5763 | Two classes related by an ... |
| opabssxp 5764 | An abstraction relation is... |
| brab2a 5765 | The law of concretion for ... |
| optocl 5766 | Implicit substitution of c... |
| 2optocl 5767 | Implicit substitution of c... |
| 3optocl 5768 | Implicit substitution of c... |
| opbrop 5769 | Ordered pair membership in... |
| 0xp 5770 | The Cartesian product with... |
| csbxp 5771 | Distribute proper substitu... |
| releq 5772 | Equality theorem for the r... |
| releqi 5773 | Equality inference for the... |
| releqd 5774 | Equality deduction for the... |
| nfrel 5775 | Bound-variable hypothesis ... |
| sbcrel 5776 | Distribute proper substitu... |
| relss 5777 | Subclass theorem for relat... |
| ssrel 5778 | A subclass relationship de... |
| ssrelOLD 5779 | Obsolete version of ~ ssre... |
| eqrel 5780 | Extensionality principle f... |
| ssrel2 5781 | A subclass relationship de... |
| ssrel3 5782 | Subclass relation in anoth... |
| relssi 5783 | Inference from subclass pr... |
| relssdv 5784 | Deduction from subclass pr... |
| eqrelriv 5785 | Inference from extensional... |
| eqrelriiv 5786 | Inference from extensional... |
| eqbrriv 5787 | Inference from extensional... |
| eqrelrdv 5788 | Deduce equality of relatio... |
| eqbrrdv 5789 | Deduction from extensional... |
| eqbrrdiv 5790 | Deduction from extensional... |
| eqrelrdv2 5791 | A version of ~ eqrelrdv . ... |
| ssrelrel 5792 | A subclass relationship de... |
| eqrelrel 5793 | Extensionality principle f... |
| elrel 5794 | A member of a relation is ... |
| rel0 5795 | The empty set is a relatio... |
| nrelv 5796 | The universal class is not... |
| relsng 5797 | A singleton is a relation ... |
| relsnb 5798 | An at-most-singleton is a ... |
| relsnopg 5799 | A singleton of an ordered ... |
| relsn 5800 | A singleton is a relation ... |
| relsnop 5801 | A singleton of an ordered ... |
| copsex2gb 5802 | Implicit substitution infe... |
| copsex2ga 5803 | Implicit substitution infe... |
| elopaba 5804 | Membership in an ordered-p... |
| xpsspw 5805 | A Cartesian product is inc... |
| unixpss 5806 | The double class union of ... |
| relun 5807 | The union of two relations... |
| relin1 5808 | The intersection with a re... |
| relin2 5809 | The intersection with a re... |
| relinxp 5810 | Intersection with a Cartes... |
| reldif 5811 | A difference cutting down ... |
| reliun 5812 | An indexed union is a rela... |
| reliin 5813 | An indexed intersection is... |
| reluni 5814 | The union of a class is a ... |
| relint 5815 | The intersection of a clas... |
| relopabiv 5816 | A class of ordered pairs i... |
| relopabv 5817 | A class of ordered pairs i... |
| relopabi 5818 | A class of ordered pairs i... |
| relopabiALT 5819 | Alternate proof of ~ relop... |
| relopab 5820 | A class of ordered pairs i... |
| mptrel 5821 | The maps-to notation alway... |
| reli 5822 | The identity relation is a... |
| rele 5823 | The membership relation is... |
| opabid2 5824 | A relation expressed as an... |
| inopab 5825 | Intersection of two ordere... |
| difopab 5826 | Difference of two ordered-... |
| difopabOLD 5827 | Obsolete version of ~ difo... |
| inxp 5828 | Intersection of two Cartes... |
| inxpOLD 5829 | Obsolete version of ~ inxp... |
| xpindi 5830 | Distributive law for Carte... |
| xpindir 5831 | Distributive law for Carte... |
| xpiindi 5832 | Distributive law for Carte... |
| xpriindi 5833 | Distributive law for Carte... |
| eliunxp 5834 | Membership in a union of C... |
| opeliunxp2 5835 | Membership in a union of C... |
| raliunxp 5836 | Write a double restricted ... |
| rexiunxp 5837 | Write a double restricted ... |
| ralxp 5838 | Universal quantification r... |
| rexxp 5839 | Existential quantification... |
| exopxfr 5840 | Transfer ordered-pair exis... |
| exopxfr2 5841 | Transfer ordered-pair exis... |
| djussxp 5842 | Disjoint union is a subset... |
| ralxpf 5843 | Version of ~ ralxp with bo... |
| rexxpf 5844 | Version of ~ rexxp with bo... |
| iunxpf 5845 | Indexed union on a Cartesi... |
| opabbi2dv 5846 | Deduce equality of a relat... |
| relop 5847 | A necessary and sufficient... |
| ideqg 5848 | For sets, the identity rel... |
| ideq 5849 | For sets, the identity rel... |
| ididg 5850 | A set is identical to itse... |
| issetid 5851 | Two ways of expressing set... |
| coss1 5852 | Subclass theorem for compo... |
| coss2 5853 | Subclass theorem for compo... |
| coeq1 5854 | Equality theorem for compo... |
| coeq2 5855 | Equality theorem for compo... |
| coeq1i 5856 | Equality inference for com... |
| coeq2i 5857 | Equality inference for com... |
| coeq1d 5858 | Equality deduction for com... |
| coeq2d 5859 | Equality deduction for com... |
| coeq12i 5860 | Equality inference for com... |
| coeq12d 5861 | Equality deduction for com... |
| nfco 5862 | Bound-variable hypothesis ... |
| brcog 5863 | Ordered pair membership in... |
| opelco2g 5864 | Ordered pair membership in... |
| brcogw 5865 | Ordered pair membership in... |
| eqbrrdva 5866 | Deduction from extensional... |
| brco 5867 | Binary relation on a compo... |
| opelco 5868 | Ordered pair membership in... |
| cnvss 5869 | Subset theorem for convers... |
| cnveq 5870 | Equality theorem for conve... |
| cnveqi 5871 | Equality inference for con... |
| cnveqd 5872 | Equality deduction for con... |
| elcnv 5873 | Membership in a converse r... |
| elcnv2 5874 | Membership in a converse r... |
| nfcnv 5875 | Bound-variable hypothesis ... |
| brcnvg 5876 | The converse of a binary r... |
| opelcnvg 5877 | Ordered-pair membership in... |
| opelcnv 5878 | Ordered-pair membership in... |
| brcnv 5879 | The converse of a binary r... |
| csbcnv 5880 | Move class substitution in... |
| csbcnvgALT 5881 | Move class substitution in... |
| cnvco 5882 | Distributive law of conver... |
| cnvuni 5883 | The converse of a class un... |
| dfdm3 5884 | Alternate definition of do... |
| dfrn2 5885 | Alternate definition of ra... |
| dfrn3 5886 | Alternate definition of ra... |
| elrn2g 5887 | Membership in a range. (C... |
| elrng 5888 | Membership in a range. (C... |
| elrn2 5889 | Membership in a range. (C... |
| elrn 5890 | Membership in a range. (C... |
| ssrelrn 5891 | If a relation is a subset ... |
| dfdm4 5892 | Alternate definition of do... |
| dfdmf 5893 | Definition of domain, usin... |
| csbdm 5894 | Distribute proper substitu... |
| eldmg 5895 | Domain membership. Theore... |
| eldm2g 5896 | Domain membership. Theore... |
| eldm 5897 | Membership in a domain. T... |
| eldm2 5898 | Membership in a domain. T... |
| dmss 5899 | Subset theorem for domain.... |
| dmeq 5900 | Equality theorem for domai... |
| dmeqi 5901 | Equality inference for dom... |
| dmeqd 5902 | Equality deduction for dom... |
| opeldmd 5903 | Membership of first of an ... |
| opeldm 5904 | Membership of first of an ... |
| breldm 5905 | Membership of first of a b... |
| breldmg 5906 | Membership of first of a b... |
| dmun 5907 | The domain of a union is t... |
| dmin 5908 | The domain of an intersect... |
| breldmd 5909 | Membership of first of a b... |
| dmiun 5910 | The domain of an indexed u... |
| dmuni 5911 | The domain of a union. Pa... |
| dmopab 5912 | The domain of a class of o... |
| dmopabelb 5913 | A set is an element of the... |
| dmopab2rex 5914 | The domain of an ordered p... |
| dmopabss 5915 | Upper bound for the domain... |
| dmopab3 5916 | The domain of a restricted... |
| dm0 5917 | The domain of the empty se... |
| dmi 5918 | The domain of the identity... |
| dmv 5919 | The domain of the universe... |
| dmep 5920 | The domain of the membersh... |
| dm0rn0 5921 | An empty domain is equival... |
| rn0 5922 | The range of the empty set... |
| rnep 5923 | The range of the membershi... |
| reldm0 5924 | A relation is empty iff it... |
| dmxp 5925 | The domain of a Cartesian ... |
| dmxpid 5926 | The domain of a Cartesian ... |
| dmxpin 5927 | The domain of the intersec... |
| xpid11 5928 | The Cartesian square is a ... |
| dmcnvcnv 5929 | The domain of the double c... |
| rncnvcnv 5930 | The range of the double co... |
| elreldm 5931 | The first member of an ord... |
| rneq 5932 | Equality theorem for range... |
| rneqi 5933 | Equality inference for ran... |
| rneqd 5934 | Equality deduction for ran... |
| rnss 5935 | Subset theorem for range. ... |
| rnssi 5936 | Subclass inference for ran... |
| brelrng 5937 | The second argument of a b... |
| brelrn 5938 | The second argument of a b... |
| opelrn 5939 | Membership of second membe... |
| releldm 5940 | The first argument of a bi... |
| relelrn 5941 | The second argument of a b... |
| releldmb 5942 | Membership in a domain. (... |
| relelrnb 5943 | Membership in a range. (C... |
| releldmi 5944 | The first argument of a bi... |
| relelrni 5945 | The second argument of a b... |
| dfrnf 5946 | Definition of range, using... |
| nfdm 5947 | Bound-variable hypothesis ... |
| nfrn 5948 | Bound-variable hypothesis ... |
| dmiin 5949 | Domain of an intersection.... |
| rnopab 5950 | The range of a class of or... |
| rnmpt 5951 | The range of a function in... |
| elrnmpt 5952 | The range of a function in... |
| elrnmpt1s 5953 | Elementhood in an image se... |
| elrnmpt1 5954 | Elementhood in an image se... |
| elrnmptg 5955 | Membership in the range of... |
| elrnmpti 5956 | Membership in the range of... |
| elrnmptd 5957 | The range of a function in... |
| elrnmpt1d 5958 | Elementhood in an image se... |
| 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... |
| mptimass 6070 | Image of a function in map... |
| imai 6071 | Image under the identity r... |
| rnresi 6072 | The range of the restricte... |
| resiima 6073 | The image of a restriction... |
| ima0 6074 | Image of the empty set. T... |
| 0ima 6075 | Image under the empty rela... |
| csbima12 6076 | Move class substitution in... |
| imadisj 6077 | A class whose image under ... |
| imadisjlnd 6078 | Deduction form of one nega... |
| cnvimass 6079 | A preimage under any class... |
| cnvimarndm 6080 | The preimage of the range ... |
| imasng 6081 | The image of a singleton. ... |
| relimasn 6082 | The image of a singleton. ... |
| elrelimasn 6083 | Elementhood in the image o... |
| elimasng1 6084 | Membership in an image of ... |
| elimasn1 6085 | Membership in an image of ... |
| elimasng 6086 | Membership in an image of ... |
| elimasn 6087 | Membership in an image of ... |
| elimasngOLD 6088 | Obsolete version of ~ elim... |
| elimasni 6089 | Membership in an image of ... |
| args 6090 | Two ways to express the cl... |
| elinisegg 6091 | Membership in the inverse ... |
| eliniseg 6092 | Membership in the inverse ... |
| epin 6093 | Any set is equal to its pr... |
| epini 6094 | Any set is equal to its pr... |
| iniseg 6095 | An idiom that signifies an... |
| inisegn0 6096 | Nonemptiness of an initial... |
| dffr3 6097 | Alternate definition of we... |
| dfse2 6098 | Alternate definition of se... |
| imass1 6099 | Subset theorem for image. ... |
| imass2 6100 | Subset theorem for image. ... |
| ndmima 6101 | The image of a singleton o... |
| relcnv 6102 | A converse is a relation. ... |
| relbrcnvg 6103 | When ` R ` is a relation, ... |
| eliniseg2 6104 | Eliminate the class existe... |
| relbrcnv 6105 | When ` R ` is a relation, ... |
| relco 6106 | A composition is a relatio... |
| cotrg 6107 | Two ways of saying that th... |
| cotrgOLD 6108 | Obsolete version of ~ cotr... |
| cotrgOLDOLD 6109 | Obsolete version of ~ cotr... |
| cotr 6110 | Two ways of saying a relat... |
| idrefALT 6111 | Alternate proof of ~ idref... |
| cnvsym 6112 | Two ways of saying a relat... |
| cnvsymOLD 6113 | Obsolete proof of ~ cnvsym... |
| cnvsymOLDOLD 6114 | Obsolete proof of ~ cnvsym... |
| intasym 6115 | Two ways of saying a relat... |
| asymref 6116 | Two ways of saying a relat... |
| asymref2 6117 | Two ways of saying a relat... |
| intirr 6118 | Two ways of saying a relat... |
| brcodir 6119 | Two ways of saying that tw... |
| codir 6120 | Two ways of saying a relat... |
| qfto 6121 | A quantifier-free way of e... |
| xpidtr 6122 | A Cartesian square is a tr... |
| trin2 6123 | The intersection of two tr... |
| poirr2 6124 | A partial order is irrefle... |
| trinxp 6125 | The relation induced by a ... |
| soirri 6126 | A strict order relation is... |
| sotri 6127 | A strict order relation is... |
| son2lpi 6128 | A strict order relation ha... |
| sotri2 6129 | A transitivity relation. ... |
| sotri3 6130 | A transitivity relation. ... |
| poleloe 6131 | Express "less than or equa... |
| poltletr 6132 | Transitive law for general... |
| somin1 6133 | Property of a minimum in a... |
| somincom 6134 | Commutativity of minimum i... |
| somin2 6135 | Property of a minimum in a... |
| soltmin 6136 | Being less than a minimum,... |
| cnvopab 6137 | The converse of a class ab... |
| mptcnv 6138 | The converse of a mapping ... |
| cnv0 6139 | The converse of the empty ... |
| cnvi 6140 | The converse of the identi... |
| cnvun 6141 | The converse of a union is... |
| cnvdif 6142 | Distributive law for conve... |
| cnvin 6143 | Distributive law for conve... |
| rnun 6144 | Distributive law for range... |
| rnin 6145 | The range of an intersecti... |
| rniun 6146 | The range of an indexed un... |
| rnuni 6147 | The range of a union. Par... |
| imaundi 6148 | Distributive law for image... |
| imaundir 6149 | The image of a union. (Co... |
| cnvimassrndm 6150 | The preimage of a superset... |
| dminss 6151 | An upper bound for interse... |
| imainss 6152 | An upper bound for interse... |
| inimass 6153 | The image of an intersecti... |
| inimasn 6154 | The intersection of the im... |
| cnvxp 6155 | The converse of a Cartesia... |
| xp0 6156 | The Cartesian product with... |
| xpnz 6157 | The Cartesian product of n... |
| xpeq0 6158 | At least one member of an ... |
| xpdisj1 6159 | Cartesian products with di... |
| xpdisj2 6160 | Cartesian products with di... |
| xpsndisj 6161 | Cartesian products with tw... |
| difxp 6162 | Difference of Cartesian pr... |
| difxp1 6163 | Difference law for Cartesi... |
| difxp2 6164 | Difference law for Cartesi... |
| djudisj 6165 | Disjoint unions with disjo... |
| xpdifid 6166 | The set of distinct couple... |
| resdisj 6167 | A double restriction to di... |
| rnxp 6168 | The range of a Cartesian p... |
| dmxpss 6169 | The domain of a Cartesian ... |
| rnxpss 6170 | The range of a Cartesian p... |
| rnxpid 6171 | The range of a Cartesian s... |
| ssxpb 6172 | A Cartesian product subcla... |
| xp11 6173 | The Cartesian product of n... |
| xpcan 6174 | Cancellation law for Carte... |
| xpcan2 6175 | Cancellation law for Carte... |
| ssrnres 6176 | Two ways to express surjec... |
| rninxp 6177 | Two ways to express surjec... |
| dminxp 6178 | Two ways to express totali... |
| imainrect 6179 | Image by a restricted and ... |
| xpima 6180 | Direct image by a Cartesia... |
| xpima1 6181 | Direct image by a Cartesia... |
| xpima2 6182 | Direct image by a Cartesia... |
| xpimasn 6183 | Direct image of a singleto... |
| sossfld 6184 | The base set of a strict o... |
| sofld 6185 | The base set of a nonempty... |
| cnvcnv3 6186 | The set of all ordered pai... |
| dfrel2 6187 | Alternate definition of re... |
| dfrel4v 6188 | A relation can be expresse... |
| dfrel4 6189 | A relation can be expresse... |
| cnvcnv 6190 | The double converse of a c... |
| cnvcnv2 6191 | The double converse of a c... |
| cnvcnvss 6192 | The double converse of a c... |
| cnvrescnv 6193 | Two ways to express the co... |
| cnveqb 6194 | Equality theorem for conve... |
| cnveq0 6195 | A relation empty iff its c... |
| dfrel3 6196 | Alternate definition of re... |
| elid 6197 | Characterization of the el... |
| dmresv 6198 | The domain of a universal ... |
| rnresv 6199 | The range of a universal r... |
| dfrn4 6200 | Range defined in terms of ... |
| csbrn 6201 | Distribute proper substitu... |
| rescnvcnv 6202 | The restriction of the dou... |
| cnvcnvres 6203 | The double converse of the... |
| imacnvcnv 6204 | The image of the double co... |
| dmsnn0 6205 | The domain of a singleton ... |
| rnsnn0 6206 | The range of a singleton i... |
| dmsn0 6207 | The domain of the singleto... |
| cnvsn0 6208 | The converse of the single... |
| dmsn0el 6209 | The domain of a singleton ... |
| relsn2 6210 | A singleton is a relation ... |
| dmsnopg 6211 | The domain of a singleton ... |
| dmsnopss 6212 | The domain of a singleton ... |
| dmpropg 6213 | The domain of an unordered... |
| dmsnop 6214 | The domain of a singleton ... |
| dmprop 6215 | The domain of an unordered... |
| dmtpop 6216 | The domain of an unordered... |
| cnvcnvsn 6217 | Double converse of a singl... |
| dmsnsnsn 6218 | The domain of the singleto... |
| rnsnopg 6219 | The range of a singleton o... |
| rnpropg 6220 | The range of a pair of ord... |
| cnvsng 6221 | Converse of a singleton of... |
| rnsnop 6222 | The range of a singleton o... |
| op1sta 6223 | Extract the first member o... |
| cnvsn 6224 | Converse of a singleton of... |
| op2ndb 6225 | Extract the second member ... |
| op2nda 6226 | Extract the second member ... |
| opswap 6227 | Swap the members of an ord... |
| cnvresima 6228 | An image under the convers... |
| resdm2 6229 | A class restricted to its ... |
| resdmres 6230 | Restriction to the domain ... |
| resresdm 6231 | A restriction by an arbitr... |
| imadmres 6232 | The image of the domain of... |
| resdmss 6233 | Subset relationship for th... |
| resdifdi 6234 | Distributive law for restr... |
| resdifdir 6235 | Distributive law for restr... |
| mptpreima 6236 | The preimage of a function... |
| mptiniseg 6237 | Converse singleton image o... |
| dmmpt 6238 | The domain of the mapping ... |
| dmmptss 6239 | The domain of a mapping is... |
| dmmptg 6240 | The domain of the mapping ... |
| rnmpt0f 6241 | The range of a function in... |
| rnmptn0 6242 | The range of a function in... |
| dfco2 6243 | Alternate definition of a ... |
| dfco2a 6244 | Generalization of ~ dfco2 ... |
| coundi 6245 | Class composition distribu... |
| coundir 6246 | Class composition distribu... |
| cores 6247 | Restricted first member of... |
| resco 6248 | Associative law for the re... |
| imaco 6249 | Image of the composition o... |
| rnco 6250 | The range of the compositi... |
| rnco2 6251 | The range of the compositi... |
| dmco 6252 | The domain of a compositio... |
| coeq0 6253 | A composition of two relat... |
| coiun 6254 | Composition with an indexe... |
| cocnvcnv1 6255 | A composition is not affec... |
| cocnvcnv2 6256 | A composition is not affec... |
| cores2 6257 | Absorption of a reverse (p... |
| co02 6258 | Composition with the empty... |
| co01 6259 | Composition with the empty... |
| coi1 6260 | Composition with the ident... |
| coi2 6261 | Composition with the ident... |
| coires1 6262 | Composition with a restric... |
| coass 6263 | Associative law for class ... |
| relcnvtrg 6264 | General form of ~ relcnvtr... |
| relcnvtr 6265 | A relation is transitive i... |
| relssdmrn 6266 | A relation is included in ... |
| relssdmrnOLD 6267 | Obsolete version of ~ rels... |
| resssxp 6268 | If the ` R ` -image of a c... |
| cnvssrndm 6269 | The converse is a subset o... |
| cossxp 6270 | Composition as a subset of... |
| relrelss 6271 | Two ways to describe the s... |
| unielrel 6272 | The membership relation fo... |
| relfld 6273 | The double union of a rela... |
| relresfld 6274 | Restriction of a relation ... |
| relcoi2 6275 | Composition with the ident... |
| relcoi1 6276 | Composition with the ident... |
| unidmrn 6277 | The double union of the co... |
| relcnvfld 6278 | if ` R ` is a relation, it... |
| dfdm2 6279 | Alternate definition of do... |
| unixp 6280 | The double class union of ... |
| unixp0 6281 | A Cartesian product is emp... |
| unixpid 6282 | Field of a Cartesian squar... |
| ressn 6283 | Restriction of a class to ... |
| cnviin 6284 | The converse of an interse... |
| cnvpo 6285 | The converse of a partial ... |
| cnvso 6286 | The converse of a strict o... |
| xpco 6287 | Composition of two Cartesi... |
| xpcoid 6288 | Composition of two Cartesi... |
| elsnxp 6289 | Membership in a Cartesian ... |
| reu3op 6290 | There is a unique ordered ... |
| reuop 6291 | There is a unique ordered ... |
| opreu2reurex 6292 | There is a unique ordered ... |
| opreu2reu 6293 | If there is a unique order... |
| dfpo2 6294 | Quantifier-free definition... |
| csbcog 6295 | Distribute proper substitu... |
| snres0 6296 | Condition for restriction ... |
| imaindm 6297 | The image is unaffected by... |
| predeq123 6300 | Equality theorem for the p... |
| predeq1 6301 | Equality theorem for the p... |
| predeq2 6302 | Equality theorem for the p... |
| predeq3 6303 | Equality theorem for the p... |
| nfpred 6304 | Bound-variable hypothesis ... |
| csbpredg 6305 | Move class substitution in... |
| predpredss 6306 | If ` A ` is a subset of ` ... |
| predss 6307 | The predecessor class of `... |
| sspred 6308 | Another subset/predecessor... |
| dfpred2 6309 | An alternate definition of... |
| dfpred3 6310 | An alternate definition of... |
| dfpred3g 6311 | An alternate definition of... |
| elpredgg 6312 | Membership in a predecesso... |
| elpredg 6313 | Membership in a predecesso... |
| elpredimg 6314 | Membership in a predecesso... |
| elpredim 6315 | Membership in a predecesso... |
| elpred 6316 | Membership in a predecesso... |
| predexg 6317 | The predecessor class exis... |
| predasetexOLD 6318 | Obsolete form of ~ predexg... |
| dffr4 6319 | Alternate definition of we... |
| predel 6320 | Membership in the predeces... |
| predbrg 6321 | Closed form of ~ elpredim ... |
| predtrss 6322 | If ` R ` is transitive ove... |
| predpo 6323 | Property of the predecesso... |
| predso 6324 | Property of the predecesso... |
| setlikespec 6325 | If ` R ` is set-like in ` ... |
| predidm 6326 | Idempotent law for the pre... |
| predin 6327 | Intersection law for prede... |
| predun 6328 | Union law for predecessor ... |
| preddif 6329 | Difference law for predece... |
| predep 6330 | The predecessor under the ... |
| trpred 6331 | The class of predecessors ... |
| preddowncl 6332 | A property of classes that... |
| predpoirr 6333 | Given a partial ordering, ... |
| predfrirr 6334 | Given a well-founded relat... |
| pred0 6335 | The predecessor class over... |
| dfse3 6336 | Alternate definition of se... |
| predrelss 6337 | Subset carries from relati... |
| predprc 6338 | The predecessor of a prope... |
| predres 6339 | Predecessor class is unaff... |
| frpomin 6340 | Every nonempty (possibly p... |
| frpomin2 6341 | Every nonempty (possibly p... |
| frpoind 6342 | The principle of well-foun... |
| frpoinsg 6343 | Well-Founded Induction Sch... |
| frpoins2fg 6344 | Well-Founded Induction sch... |
| frpoins2g 6345 | Well-Founded Induction sch... |
| frpoins3g 6346 | Well-Founded Induction sch... |
| tz6.26 6347 | All nonempty subclasses of... |
| tz6.26OLD 6348 | Obsolete proof of ~ tz6.26... |
| tz6.26i 6349 | All nonempty subclasses of... |
| wfi 6350 | The Principle of Well-Orde... |
| wfiOLD 6351 | Obsolete proof of ~ wfi as... |
| wfii 6352 | The Principle of Well-Orde... |
| wfisg 6353 | Well-Ordered Induction Sch... |
| wfisgOLD 6354 | Obsolete version of ~ wfis... |
| wfis 6355 | Well-Ordered Induction Sch... |
| wfis2fg 6356 | Well-Ordered Induction Sch... |
| wfis2fgOLD 6357 | Obsolete version of ~ wfis... |
| wfis2f 6358 | Well-Ordered Induction sch... |
| wfis2g 6359 | Well-Ordered Induction Sch... |
| wfis2 6360 | Well-Ordered Induction sch... |
| wfis3 6361 | Well-Ordered Induction sch... |
| ordeq 6370 | Equality theorem for the o... |
| elong 6371 | An ordinal number is an or... |
| elon 6372 | An ordinal number is an or... |
| eloni 6373 | An ordinal number has the ... |
| elon2 6374 | An ordinal number is an or... |
| limeq 6375 | Equality theorem for the l... |
| ordwe 6376 | Membership well-orders eve... |
| ordtr 6377 | An ordinal class is transi... |
| ordfr 6378 | Membership is well-founded... |
| ordelss 6379 | An element of an ordinal c... |
| trssord 6380 | A transitive subclass of a... |
| ordirr 6381 | No ordinal class is a memb... |
| nordeq 6382 | A member of an ordinal cla... |
| ordn2lp 6383 | An ordinal class cannot be... |
| tz7.5 6384 | A nonempty subclass of an ... |
| ordelord 6385 | An element of an ordinal c... |
| tron 6386 | The class of all ordinal n... |
| ordelon 6387 | An element of an ordinal c... |
| onelon 6388 | An element of an ordinal n... |
| tz7.7 6389 | A transitive class belongs... |
| ordelssne 6390 | For ordinal classes, membe... |
| ordelpss 6391 | For ordinal classes, membe... |
| ordsseleq 6392 | For ordinal classes, inclu... |
| ordin 6393 | The intersection of two or... |
| onin 6394 | The intersection of two or... |
| ordtri3or 6395 | A trichotomy law for ordin... |
| ordtri1 6396 | A trichotomy law for ordin... |
| ontri1 6397 | A trichotomy law for ordin... |
| ordtri2 6398 | A trichotomy law for ordin... |
| ordtri3 6399 | A trichotomy law for ordin... |
| ordtri4 6400 | A trichotomy law for ordin... |
| orddisj 6401 | An ordinal class and its s... |
| onfr 6402 | The ordinal class is well-... |
| onelpss 6403 | Relationship between membe... |
| onsseleq 6404 | Relationship between subse... |
| onelss 6405 | An element of an ordinal n... |
| ordtr1 6406 | Transitive law for ordinal... |
| ordtr2 6407 | Transitive law for ordinal... |
| ordtr3 6408 | Transitive law for ordinal... |
| ontr1 6409 | Transitive law for ordinal... |
| ontr2 6410 | Transitive law for ordinal... |
| onelssex 6411 | Ordinal less than is equiv... |
| ordunidif 6412 | The union of an ordinal st... |
| ordintdif 6413 | If ` B ` is smaller than `... |
| onintss 6414 | If a property is true for ... |
| oneqmini 6415 | A way to show that an ordi... |
| ord0 6416 | The empty set is an ordina... |
| 0elon 6417 | The empty set is an ordina... |
| ord0eln0 6418 | A nonempty ordinal contain... |
| on0eln0 6419 | An ordinal number contains... |
| dflim2 6420 | An alternate definition of... |
| inton 6421 | The intersection of the cl... |
| nlim0 6422 | The empty set is not a lim... |
| limord 6423 | A limit ordinal is ordinal... |
| limuni 6424 | A limit ordinal is its own... |
| limuni2 6425 | The union of a limit ordin... |
| 0ellim 6426 | A limit ordinal contains t... |
| limelon 6427 | A limit ordinal class that... |
| onn0 6428 | The class of all ordinal n... |
| suceq 6429 | Equality of successors. (... |
| elsuci 6430 | Membership in a successor.... |
| elsucg 6431 | Membership in a successor.... |
| elsuc2g 6432 | Variant of membership in a... |
| elsuc 6433 | Membership in a successor.... |
| elsuc2 6434 | Membership in a successor.... |
| nfsuc 6435 | Bound-variable hypothesis ... |
| elelsuc 6436 | Membership in a successor.... |
| sucel 6437 | Membership of a successor ... |
| suc0 6438 | The successor of the empty... |
| sucprc 6439 | A proper class is its own ... |
| unisucs 6440 | The union of the successor... |
| unisucg 6441 | A transitive class is equa... |
| unisuc 6442 | A transitive class is equa... |
| sssucid 6443 | A class is included in its... |
| sucidg 6444 | Part of Proposition 7.23 o... |
| sucid 6445 | A set belongs to its succe... |
| nsuceq0 6446 | No successor is empty. (C... |
| eqelsuc 6447 | A set belongs to the succe... |
| iunsuc 6448 | Inductive definition for t... |
| suctr 6449 | The successor of a transit... |
| trsuc 6450 | A set whose successor belo... |
| trsucss 6451 | A member of the successor ... |
| ordsssuc 6452 | An ordinal is a subset of ... |
| onsssuc 6453 | A subset of an ordinal num... |
| ordsssuc2 6454 | An ordinal subset of an or... |
| onmindif 6455 | When its successor is subt... |
| ordnbtwn 6456 | There is no set between an... |
| onnbtwn 6457 | There is no set between an... |
| sucssel 6458 | A set whose successor is a... |
| orddif 6459 | Ordinal derived from its s... |
| orduniss 6460 | An ordinal class includes ... |
| ordtri2or 6461 | A trichotomy law for ordin... |
| ordtri2or2 6462 | A trichotomy law for ordin... |
| ordtri2or3 6463 | A consequence of total ord... |
| ordelinel 6464 | The intersection of two or... |
| ordssun 6465 | Property of a subclass of ... |
| ordequn 6466 | The maximum (i.e. union) o... |
| ordun 6467 | The maximum (i.e., union) ... |
| onunel 6468 | The union of two ordinals ... |
| ordunisssuc 6469 | A subclass relationship fo... |
| suc11 6470 | The successor operation be... |
| onun2 6471 | The union of two ordinals ... |
| ontr 6472 | An ordinal number is a tra... |
| onunisuc 6473 | An ordinal number is equal... |
| onordi 6474 | An ordinal number is an or... |
| ontrciOLD 6475 | Obsolete version of ~ ontr... |
| onirri 6476 | An ordinal number is not a... |
| oneli 6477 | A member of an ordinal num... |
| onelssi 6478 | A member of an ordinal num... |
| onssneli 6479 | An ordering law for ordina... |
| onssnel2i 6480 | An ordering law for ordina... |
| onelini 6481 | An element of an ordinal n... |
| oneluni 6482 | An ordinal number equals i... |
| onunisuci 6483 | An ordinal number is equal... |
| onsseli 6484 | Subset is equivalent to me... |
| onun2i 6485 | The union of two ordinal n... |
| unizlim 6486 | An ordinal equal to its ow... |
| on0eqel 6487 | An ordinal number either e... |
| snsn0non 6488 | The singleton of the singl... |
| onxpdisj 6489 | Ordinal numbers and ordere... |
| onnev 6490 | The class of ordinal numbe... |
| onnevOLD 6491 | Obsolete version of ~ onne... |
| iotajust 6493 | Soundness justification th... |
| dfiota2 6495 | Alternate definition for d... |
| nfiota1 6496 | Bound-variable hypothesis ... |
| nfiotadw 6497 | Deduction version of ~ nfi... |
| nfiotaw 6498 | Bound-variable hypothesis ... |
| nfiotad 6499 | Deduction version of ~ nfi... |
| nfiota 6500 | Bound-variable hypothesis ... |
| cbviotaw 6501 | Change bound variables in ... |
| cbviotavw 6502 | Change bound variables in ... |
| cbviotavwOLD 6503 | Obsolete version of ~ cbvi... |
| cbviota 6504 | Change bound variables in ... |
| cbviotav 6505 | Change bound variables in ... |
| sb8iota 6506 | Variable substitution in d... |
| iotaeq 6507 | Equality theorem for descr... |
| iotabi 6508 | Equivalence theorem for de... |
| uniabio 6509 | Part of Theorem 8.17 in [Q... |
| iotaval2 6510 | Version of ~ iotaval using... |
| iotauni2 6511 | Version of ~ iotauni using... |
| iotanul2 6512 | Version of ~ iotanul using... |
| iotaval 6513 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6514 | The ` iota ` class is a su... |
| iotaex 6515 | Theorem 8.23 in [Quine] p.... |
| iotavalOLD 6516 | Obsolete version of ~ iota... |
| iotauni 6517 | Equivalence between two di... |
| iotaint 6518 | Equivalence between two di... |
| iota1 6519 | Property of iota. (Contri... |
| iotanul 6520 | Theorem 8.22 in [Quine] p.... |
| iotassuniOLD 6521 | Obsolete version of ~ iota... |
| iotaexOLD 6522 | Obsolete version of ~ iota... |
| iota4 6523 | Theorem *14.22 in [Whitehe... |
| iota4an 6524 | Theorem *14.23 in [Whitehe... |
| iota5 6525 | A method for computing iot... |
| iotabidv 6526 | Formula-building deduction... |
| iotabii 6527 | Formula-building deduction... |
| iotacl 6528 | Membership law for descrip... |
| iota2df 6529 | A condition that allows to... |
| iota2d 6530 | A condition that allows to... |
| iota2 6531 | The unique element such th... |
| iotan0 6532 | Representation of "the uni... |
| sniota 6533 | A class abstraction with a... |
| dfiota4 6534 | The ` iota ` operation usi... |
| csbiota 6535 | Class substitution within ... |
| dffun2 6552 | Alternate definition of a ... |
| dffun2OLD 6553 | Obsolete version of ~ dffu... |
| dffun2OLDOLD 6554 | Obsolete version of ~ dffu... |
| dffun6 6555 | Alternate definition of a ... |
| dffun3 6556 | Alternate definition of fu... |
| dffun3OLD 6557 | Obsolete version of ~ dffu... |
| dffun4 6558 | Alternate definition of a ... |
| dffun5 6559 | Alternate definition of fu... |
| dffun6f 6560 | Definition of function, us... |
| dffun6OLD 6561 | Obsolete version of ~ dffu... |
| funmo 6562 | A function has at most one... |
| funmoOLD 6563 | Obsolete version of ~ funm... |
| funrel 6564 | A function is a relation. ... |
| 0nelfun 6565 | A function does not contai... |
| funss 6566 | Subclass theorem for funct... |
| funeq 6567 | Equality theorem for funct... |
| funeqi 6568 | Equality inference for the... |
| funeqd 6569 | Equality deduction for the... |
| nffun 6570 | Bound-variable hypothesis ... |
| sbcfung 6571 | Distribute proper substitu... |
| funeu 6572 | There is exactly one value... |
| funeu2 6573 | There is exactly one value... |
| dffun7 6574 | Alternate definition of a ... |
| dffun8 6575 | Alternate definition of a ... |
| dffun9 6576 | Alternate definition of a ... |
| funfn 6577 | A class is a function if a... |
| funfnd 6578 | A function is a function o... |
| funi 6579 | The identity relation is a... |
| nfunv 6580 | The universal class is not... |
| funopg 6581 | A Kuratowski ordered pair ... |
| funopab 6582 | A class of ordered pairs i... |
| funopabeq 6583 | A class of ordered pairs o... |
| funopab4 6584 | A class of ordered pairs o... |
| funmpt 6585 | A function in maps-to nota... |
| funmpt2 6586 | Functionality of a class g... |
| funco 6587 | The composition of two fun... |
| funresfunco 6588 | Composition of two functio... |
| funres 6589 | A restriction of a functio... |
| funresd 6590 | A restriction of a functio... |
| funssres 6591 | The restriction of a funct... |
| fun2ssres 6592 | Equality of restrictions o... |
| funun 6593 | The union of functions wit... |
| fununmo 6594 | If the union of classes is... |
| fununfun 6595 | If the union of classes is... |
| fundif 6596 | A function with removed el... |
| funcnvsn 6597 | The converse singleton of ... |
| funsng 6598 | A singleton of an ordered ... |
| fnsng 6599 | Functionality and domain o... |
| funsn 6600 | A singleton of an ordered ... |
| funprg 6601 | A set of two pairs is a fu... |
| funtpg 6602 | A set of three pairs is a ... |
| funpr 6603 | A function with a domain o... |
| funtp 6604 | A function with a domain o... |
| fnsn 6605 | Functionality and domain o... |
| fnprg 6606 | Function with a domain of ... |
| fntpg 6607 | Function with a domain of ... |
| fntp 6608 | A function with a domain o... |
| funcnvpr 6609 | The converse pair of order... |
| funcnvtp 6610 | The converse triple of ord... |
| funcnvqp 6611 | The converse quadruple of ... |
| fun0 6612 | The empty set is a functio... |
| funcnv0 6613 | The converse of the empty ... |
| funcnvcnv 6614 | The double converse of a f... |
| funcnv2 6615 | A simpler equivalence for ... |
| funcnv 6616 | The converse of a class is... |
| funcnv3 6617 | A condition showing a clas... |
| fun2cnv 6618 | The double converse of a c... |
| svrelfun 6619 | A single-valued relation i... |
| fncnv 6620 | Single-rootedness (see ~ f... |
| fun11 6621 | Two ways of stating that `... |
| fununi 6622 | The union of a chain (with... |
| funin 6623 | The intersection with a fu... |
| funres11 6624 | The restriction of a one-t... |
| funcnvres 6625 | The converse of a restrict... |
| cnvresid 6626 | Converse of a restricted i... |
| funcnvres2 6627 | The converse of a restrict... |
| funimacnv 6628 | The image of the preimage ... |
| funimass1 6629 | A kind of contraposition l... |
| funimass2 6630 | A kind of contraposition l... |
| imadif 6631 | The image of a difference ... |
| imain 6632 | The image of an intersecti... |
| funimaexg 6633 | Axiom of Replacement using... |
| funimaexgOLD 6634 | Obsolete version of ~ funi... |
| funimaex 6635 | The image of a set under a... |
| isarep1 6636 | Part of a study of the Axi... |
| isarep1OLD 6637 | Obsolete version of ~ isar... |
| isarep2 6638 | Part of a study of the Axi... |
| fneq1 6639 | Equality theorem for funct... |
| fneq2 6640 | Equality theorem for funct... |
| fneq1d 6641 | Equality deduction for fun... |
| fneq2d 6642 | Equality deduction for fun... |
| fneq12d 6643 | Equality deduction for fun... |
| fneq12 6644 | Equality theorem for funct... |
| fneq1i 6645 | Equality inference for fun... |
| fneq2i 6646 | Equality inference for fun... |
| nffn 6647 | Bound-variable hypothesis ... |
| fnfun 6648 | A function with domain is ... |
| fnfund 6649 | A function with domain is ... |
| fnrel 6650 | A function with domain is ... |
| fndm 6651 | The domain of a function. ... |
| fndmi 6652 | The domain of a function. ... |
| fndmd 6653 | The domain of a function. ... |
| funfni 6654 | Inference to convert a fun... |
| fndmu 6655 | A function has a unique do... |
| fnbr 6656 | The first argument of bina... |
| fnop 6657 | The first argument of an o... |
| fneu 6658 | There is exactly one value... |
| fneu2 6659 | There is exactly one value... |
| fnunres1 6660 | Restriction of a disjoint ... |
| fnunres2 6661 | Restriction of a disjoint ... |
| fnun 6662 | The union of two functions... |
| fnund 6663 | The union of two functions... |
| fnunop 6664 | Extension of a function wi... |
| fncofn 6665 | Composition of a function ... |
| fnco 6666 | Composition of two functio... |
| fncoOLD 6667 | Obsolete version of ~ fnco... |
| fnresdm 6668 | A function does not change... |
| fnresdisj 6669 | A function restricted to a... |
| 2elresin 6670 | Membership in two function... |
| fnssresb 6671 | Restriction of a function ... |
| fnssres 6672 | Restriction of a function ... |
| fnssresd 6673 | Restriction of a function ... |
| fnresin1 6674 | Restriction of a function'... |
| fnresin2 6675 | Restriction of a function'... |
| fnres 6676 | An equivalence for functio... |
| idfn 6677 | The identity relation is a... |
| fnresi 6678 | The restricted identity re... |
| fnima 6679 | The image of a function's ... |
| fn0 6680 | A function with empty doma... |
| fnimadisj 6681 | A class that is disjoint w... |
| fnimaeq0 6682 | Images under a function ne... |
| dfmpt3 6683 | Alternate definition for t... |
| mptfnf 6684 | The maps-to notation defin... |
| fnmptf 6685 | The maps-to notation defin... |
| fnopabg 6686 | Functionality and domain o... |
| fnopab 6687 | Functionality and domain o... |
| mptfng 6688 | The maps-to notation defin... |
| fnmpt 6689 | The maps-to notation defin... |
| fnmptd 6690 | The maps-to notation defin... |
| mpt0 6691 | A mapping operation with e... |
| fnmpti 6692 | Functionality and domain o... |
| dmmpti 6693 | Domain of the mapping oper... |
| dmmptd 6694 | The domain of the mapping ... |
| mptun 6695 | Union of mappings which ar... |
| partfun 6696 | Rewrite a function defined... |
| feq1 6697 | Equality theorem for funct... |
| feq2 6698 | Equality theorem for funct... |
| feq3 6699 | Equality theorem for funct... |
| feq23 6700 | Equality theorem for funct... |
| feq1d 6701 | Equality deduction for fun... |
| feq2d 6702 | Equality deduction for fun... |
| feq3d 6703 | Equality deduction for fun... |
| feq12d 6704 | Equality deduction for fun... |
| feq123d 6705 | Equality deduction for fun... |
| feq123 6706 | Equality theorem for funct... |
| feq1i 6707 | Equality inference for fun... |
| feq2i 6708 | Equality inference for fun... |
| feq12i 6709 | Equality inference for fun... |
| feq23i 6710 | Equality inference for fun... |
| feq23d 6711 | Equality deduction for fun... |
| nff 6712 | Bound-variable hypothesis ... |
| sbcfng 6713 | Distribute proper substitu... |
| sbcfg 6714 | Distribute proper substitu... |
| elimf 6715 | Eliminate a mapping hypoth... |
| ffn 6716 | A mapping is a function wi... |
| ffnd 6717 | A mapping is a function wi... |
| dffn2 6718 | Any function is a mapping ... |
| ffun 6719 | A mapping is a function. ... |
| ffund 6720 | A mapping is a function, d... |
| frel 6721 | A mapping is a relation. ... |
| freld 6722 | A mapping is a relation. ... |
| frn 6723 | The range of a mapping. (... |
| frnd 6724 | Deduction form of ~ frn . ... |
| fdm 6725 | The domain of a mapping. ... |
| fdmOLD 6726 | Obsolete version of ~ fdm ... |
| fdmd 6727 | Deduction form of ~ fdm . ... |
| fdmi 6728 | Inference associated with ... |
| dffn3 6729 | A function maps to its ran... |
| ffrn 6730 | A function maps to its ran... |
| ffrnb 6731 | Characterization of a func... |
| ffrnbd 6732 | A function maps to its ran... |
| fss 6733 | Expanding the codomain of ... |
| fssd 6734 | Expanding the codomain of ... |
| fssdmd 6735 | Expressing that a class is... |
| fssdm 6736 | Expressing that a class is... |
| fimass 6737 | The image of a class under... |
| fimassd 6738 | The image of a class is a ... |
| fimacnv 6739 | The preimage of the codoma... |
| fcof 6740 | Composition of a function ... |
| fco 6741 | Composition of two functio... |
| fcoOLD 6742 | Obsolete version of ~ fco ... |
| fcod 6743 | Composition of two mapping... |
| fco2 6744 | Functionality of a composi... |
| fssxp 6745 | A mapping is a class of or... |
| funssxp 6746 | Two ways of specifying a p... |
| ffdm 6747 | A mapping is a partial fun... |
| ffdmd 6748 | The domain of a function. ... |
| fdmrn 6749 | A different way to write `... |
| funcofd 6750 | Composition of two functio... |
| fco3OLD 6751 | Obsolete version of ~ func... |
| opelf 6752 | The members of an ordered ... |
| fun 6753 | The union of two functions... |
| fun2 6754 | The union of two functions... |
| fun2d 6755 | The union of functions wit... |
| fnfco 6756 | Composition of two functio... |
| fssres 6757 | Restriction of a function ... |
| fssresd 6758 | Restriction of a function ... |
| fssres2 6759 | Restriction of a restricte... |
| fresin 6760 | An identity for the mappin... |
| resasplit 6761 | If two functions agree on ... |
| fresaun 6762 | The union of two functions... |
| fresaunres2 6763 | From the union of two func... |
| fresaunres1 6764 | From the union of two func... |
| fcoi1 6765 | Composition of a mapping a... |
| fcoi2 6766 | Composition of restricted ... |
| feu 6767 | There is exactly one value... |
| fcnvres 6768 | The converse of a restrict... |
| fimacnvdisj 6769 | The preimage of a class di... |
| fint 6770 | Function into an intersect... |
| fin 6771 | Mapping into an intersecti... |
| f0 6772 | The empty function. (Cont... |
| f00 6773 | A class is a function with... |
| f0bi 6774 | A function with empty doma... |
| f0dom0 6775 | A function is empty iff it... |
| f0rn0 6776 | If there is no element in ... |
| fconst 6777 | A Cartesian product with a... |
| fconstg 6778 | A Cartesian product with a... |
| fnconstg 6779 | A Cartesian product with a... |
| fconst6g 6780 | Constant function with loo... |
| fconst6 6781 | A constant function as a m... |
| f1eq1 6782 | Equality theorem for one-t... |
| f1eq2 6783 | Equality theorem for one-t... |
| f1eq3 6784 | Equality theorem for one-t... |
| nff1 6785 | Bound-variable hypothesis ... |
| dff12 6786 | Alternate definition of a ... |
| f1f 6787 | A one-to-one mapping is a ... |
| f1fn 6788 | A one-to-one mapping is a ... |
| f1fun 6789 | A one-to-one mapping is a ... |
| f1rel 6790 | A one-to-one onto mapping ... |
| f1dm 6791 | The domain of a one-to-one... |
| f1dmOLD 6792 | Obsolete version of ~ f1dm... |
| f1ss 6793 | A function that is one-to-... |
| f1ssr 6794 | A function that is one-to-... |
| f1ssres 6795 | A function that is one-to-... |
| f1resf1 6796 | The restriction of an inje... |
| f1cnvcnv 6797 | Two ways to express that a... |
| f1cof1 6798 | Composition of two one-to-... |
| f1co 6799 | Composition of one-to-one ... |
| f1coOLD 6800 | Obsolete version of ~ f1co... |
| foeq1 6801 | Equality theorem for onto ... |
| foeq2 6802 | Equality theorem for onto ... |
| foeq3 6803 | Equality theorem for onto ... |
| nffo 6804 | Bound-variable hypothesis ... |
| fof 6805 | An onto mapping is a mappi... |
| fofun 6806 | An onto mapping is a funct... |
| fofn 6807 | An onto mapping is a funct... |
| forn 6808 | The codomain of an onto fu... |
| dffo2 6809 | Alternate definition of an... |
| foima 6810 | The image of the domain of... |
| dffn4 6811 | A function maps onto its r... |
| funforn 6812 | A function maps its domain... |
| fodmrnu 6813 | An onto function has uniqu... |
| fimadmfo 6814 | A function is a function o... |
| fores 6815 | Restriction of an onto fun... |
| fimadmfoALT 6816 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6817 | The preimage of the codoma... |
| focofo 6818 | Composition of onto functi... |
| foco 6819 | Composition of onto functi... |
| foconst 6820 | A nonzero constant functio... |
| f1oeq1 6821 | Equality theorem for one-t... |
| f1oeq2 6822 | Equality theorem for one-t... |
| f1oeq3 6823 | Equality theorem for one-t... |
| f1oeq23 6824 | Equality theorem for one-t... |
| f1eq123d 6825 | Equality deduction for one... |
| foeq123d 6826 | Equality deduction for ont... |
| f1oeq123d 6827 | Equality deduction for one... |
| f1oeq1d 6828 | Equality deduction for one... |
| f1oeq2d 6829 | Equality deduction for one... |
| f1oeq3d 6830 | Equality deduction for one... |
| nff1o 6831 | Bound-variable hypothesis ... |
| f1of1 6832 | A one-to-one onto mapping ... |
| f1of 6833 | A one-to-one onto mapping ... |
| f1ofn 6834 | A one-to-one onto mapping ... |
| f1ofun 6835 | A one-to-one onto mapping ... |
| f1orel 6836 | A one-to-one onto mapping ... |
| f1odm 6837 | The domain of a one-to-one... |
| dff1o2 6838 | Alternate definition of on... |
| dff1o3 6839 | Alternate definition of on... |
| f1ofo 6840 | A one-to-one onto function... |
| dff1o4 6841 | Alternate definition of on... |
| dff1o5 6842 | Alternate definition of on... |
| f1orn 6843 | A one-to-one function maps... |
| f1f1orn 6844 | A one-to-one function maps... |
| f1ocnv 6845 | The converse of a one-to-o... |
| f1ocnvb 6846 | A relation is a one-to-one... |
| f1ores 6847 | The restriction of a one-t... |
| f1orescnv 6848 | The converse of a one-to-o... |
| f1imacnv 6849 | Preimage of an image. (Co... |
| foimacnv 6850 | A reverse version of ~ f1i... |
| foun 6851 | The union of two onto func... |
| f1oun 6852 | The union of two one-to-on... |
| f1un 6853 | The union of two one-to-on... |
| resdif 6854 | The restriction of a one-t... |
| resin 6855 | The restriction of a one-t... |
| f1oco 6856 | Composition of one-to-one ... |
| f1cnv 6857 | The converse of an injecti... |
| funcocnv2 6858 | Composition with the conve... |
| fococnv2 6859 | The composition of an onto... |
| f1ococnv2 6860 | The composition of a one-t... |
| f1cocnv2 6861 | Composition of an injectiv... |
| f1ococnv1 6862 | The composition of a one-t... |
| f1cocnv1 6863 | Composition of an injectiv... |
| funcoeqres 6864 | Express a constraint on a ... |
| f1ssf1 6865 | A subset of an injective f... |
| f10 6866 | The empty set maps one-to-... |
| f10d 6867 | The empty set maps one-to-... |
| f1o00 6868 | One-to-one onto mapping of... |
| fo00 6869 | Onto mapping of the empty ... |
| f1o0 6870 | One-to-one onto mapping of... |
| f1oi 6871 | A restriction of the ident... |
| f1ovi 6872 | The identity relation is a... |
| f1osn 6873 | A singleton of an ordered ... |
| f1osng 6874 | A singleton of an ordered ... |
| f1sng 6875 | A singleton of an ordered ... |
| fsnd 6876 | A singleton of an ordered ... |
| f1oprswap 6877 | A two-element swap is a bi... |
| f1oprg 6878 | An unordered pair of order... |
| tz6.12-2 6879 | Function value when ` F ` ... |
| fveu 6880 | The value of a function at... |
| brprcneu 6881 | If ` A ` is a proper class... |
| brprcneuALT 6882 | Alternate proof of ~ brprc... |
| fvprc 6883 | A function's value at a pr... |
| fvprcALT 6884 | Alternate proof of ~ fvprc... |
| rnfvprc 6885 | The range of a function va... |
| fv2 6886 | Alternate definition of fu... |
| dffv3 6887 | A definition of function v... |
| dffv4 6888 | The previous definition of... |
| elfv 6889 | Membership in a function v... |
| fveq1 6890 | Equality theorem for funct... |
| fveq2 6891 | Equality theorem for funct... |
| fveq1i 6892 | Equality inference for fun... |
| fveq1d 6893 | Equality deduction for fun... |
| fveq2i 6894 | Equality inference for fun... |
| fveq2d 6895 | Equality deduction for fun... |
| 2fveq3 6896 | Equality theorem for neste... |
| fveq12i 6897 | Equality deduction for fun... |
| fveq12d 6898 | Equality deduction for fun... |
| fveqeq2d 6899 | Equality deduction for fun... |
| fveqeq2 6900 | Equality deduction for fun... |
| nffv 6901 | Bound-variable hypothesis ... |
| nffvmpt1 6902 | Bound-variable hypothesis ... |
| nffvd 6903 | Deduction version of bound... |
| fvex 6904 | The value of a class exist... |
| fvexi 6905 | The value of a class exist... |
| fvexd 6906 | The value of a class exist... |
| fvif 6907 | Move a conditional outside... |
| iffv 6908 | Move a conditional outside... |
| fv3 6909 | Alternate definition of th... |
| fvres 6910 | The value of a restricted ... |
| fvresd 6911 | The value of a restricted ... |
| funssfv 6912 | The value of a member of t... |
| tz6.12c 6913 | Corollary of Theorem 6.12(... |
| tz6.12-1 6914 | Function value. Theorem 6... |
| tz6.12-1OLD 6915 | Obsolete version of ~ tz6.... |
| tz6.12 6916 | Function value. Theorem 6... |
| tz6.12f 6917 | Function value, using boun... |
| tz6.12cOLD 6918 | Obsolete version of ~ tz6.... |
| tz6.12i 6919 | Corollary of Theorem 6.12(... |
| fvbr0 6920 | Two possibilities for the ... |
| fvrn0 6921 | A function value is a memb... |
| fvn0fvelrn 6922 | If the value of a function... |
| elfvunirn 6923 | A function value is a subs... |
| fvssunirn 6924 | The result of a function v... |
| fvssunirnOLD 6925 | Obsolete version of ~ fvss... |
| ndmfv 6926 | The value of a class outsi... |
| ndmfvrcl 6927 | Reverse closure law for fu... |
| elfvdm 6928 | If a function value has a ... |
| elfvex 6929 | If a function value has a ... |
| elfvexd 6930 | If a function value has a ... |
| eliman0 6931 | A nonempty function value ... |
| nfvres 6932 | The value of a non-member ... |
| nfunsn 6933 | If the restriction of a cl... |
| fvfundmfvn0 6934 | If the "value of a class" ... |
| 0fv 6935 | Function value of the empt... |
| fv2prc 6936 | A function value of a func... |
| elfv2ex 6937 | If a function value of a f... |
| fveqres 6938 | Equal values imply equal v... |
| csbfv12 6939 | Move class substitution in... |
| csbfv2g 6940 | Move class substitution in... |
| csbfv 6941 | Substitution for a functio... |
| funbrfv 6942 | The second argument of a b... |
| funopfv 6943 | The second element in an o... |
| fnbrfvb 6944 | Equivalence of function va... |
| fnopfvb 6945 | Equivalence of function va... |
| funbrfvb 6946 | Equivalence of function va... |
| funopfvb 6947 | Equivalence of function va... |
| fnbrfvb2 6948 | Version of ~ fnbrfvb for f... |
| fdmeu 6949 | There is exactly one codom... |
| funbrfv2b 6950 | Function value in terms of... |
| dffn5 6951 | Representation of a functi... |
| fnrnfv 6952 | The range of a function ex... |
| fvelrnb 6953 | A member of a function's r... |
| foelcdmi 6954 | A member of a surjective f... |
| dfimafn 6955 | Alternate definition of th... |
| dfimafn2 6956 | Alternate definition of th... |
| funimass4 6957 | Membership relation for th... |
| fvelima 6958 | Function value in an image... |
| funimassd 6959 | Sufficient condition for t... |
| fvelimad 6960 | Function value in an image... |
| feqmptd 6961 | Deduction form of ~ dffn5 ... |
| feqresmpt 6962 | Express a restricted funct... |
| feqmptdf 6963 | Deduction form of ~ dffn5f... |
| dffn5f 6964 | Representation of a functi... |
| fvelimab 6965 | Function value in an image... |
| fvelimabd 6966 | Deduction form of ~ fvelim... |
| unima 6967 | Image of a union. (Contri... |
| fvi 6968 | The value of the identity ... |
| fviss 6969 | The value of the identity ... |
| fniinfv 6970 | The indexed intersection o... |
| fnsnfv 6971 | Singleton of function valu... |
| fnsnfvOLD 6972 | Obsolete version of ~ fnsn... |
| opabiotafun 6973 | Define a function whose va... |
| opabiotadm 6974 | Define a function whose va... |
| opabiota 6975 | Define a function whose va... |
| fnimapr 6976 | The image of a pair under ... |
| ssimaex 6977 | The existence of a subimag... |
| ssimaexg 6978 | The existence of a subimag... |
| funfv 6979 | A simplified expression fo... |
| funfv2 6980 | The value of a function. ... |
| funfv2f 6981 | The value of a function. ... |
| fvun 6982 | Value of the union of two ... |
| fvun1 6983 | The value of a union when ... |
| fvun2 6984 | The value of a union when ... |
| fvun1d 6985 | The value of a union when ... |
| fvun2d 6986 | The value of a union when ... |
| dffv2 6987 | Alternate definition of fu... |
| dmfco 6988 | Domains of a function comp... |
| fvco2 6989 | Value of a function compos... |
| fvco 6990 | Value of a function compos... |
| fvco3 6991 | Value of a function compos... |
| fvco3d 6992 | Value of a function compos... |
| fvco4i 6993 | Conditions for a compositi... |
| fvopab3g 6994 | Value of a function given ... |
| fvopab3ig 6995 | Value of a function given ... |
| brfvopabrbr 6996 | The binary relation of a f... |
| fvmptg 6997 | Value of a function given ... |
| fvmpti 6998 | Value of a function given ... |
| fvmpt 6999 | Value of a function given ... |
| fvmpt2f 7000 | Value of a function given ... |
| fvtresfn 7001 | Functionality of a tuple-r... |
| fvmpts 7002 | Value of a function given ... |
| fvmpt3 7003 | Value of a function given ... |
| fvmpt3i 7004 | Value of a function given ... |
| fvmptdf 7005 | Deduction version of ~ fvm... |
| fvmptd 7006 | Deduction version of ~ fvm... |
| fvmptd2 7007 | Deduction version of ~ fvm... |
| mptrcl 7008 | Reverse closure for a mapp... |
| fvmpt2i 7009 | Value of a function given ... |
| fvmpt2 7010 | Value of a function given ... |
| fvmptss 7011 | If all the values of the m... |
| fvmpt2d 7012 | Deduction version of ~ fvm... |
| fvmptex 7013 | Express a function ` F ` w... |
| fvmptd3f 7014 | Alternate deduction versio... |
| fvmptd2f 7015 | Alternate deduction versio... |
| fvmptdv 7016 | Alternate deduction versio... |
| fvmptdv2 7017 | Alternate deduction versio... |
| mpteqb 7018 | Bidirectional equality the... |
| fvmptt 7019 | Closed theorem form of ~ f... |
| fvmptf 7020 | Value of a function given ... |
| fvmptnf 7021 | The value of a function gi... |
| fvmptd3 7022 | Deduction version of ~ fvm... |
| fvmptd4 7023 | Deduction version of ~ fvm... |
| fvmptn 7024 | This somewhat non-intuitiv... |
| fvmptss2 7025 | A mapping always evaluates... |
| elfvmptrab1w 7026 | Implications for the value... |
| elfvmptrab1 7027 | Implications for the value... |
| elfvmptrab 7028 | Implications for the value... |
| fvopab4ndm 7029 | Value of a function given ... |
| fvmptndm 7030 | Value of a function given ... |
| fvmptrabfv 7031 | Value of a function mappin... |
| fvopab5 7032 | The value of a function th... |
| fvopab6 7033 | Value of a function given ... |
| eqfnfv 7034 | Equality of functions is d... |
| eqfnfv2 7035 | Equality of functions is d... |
| eqfnfv3 7036 | Derive equality of functio... |
| eqfnfvd 7037 | Deduction for equality of ... |
| eqfnfv2f 7038 | Equality of functions is d... |
| eqfunfv 7039 | Equality of functions is d... |
| eqfnun 7040 | Two functions on ` A u. B ... |
| fvreseq0 7041 | Equality of restricted fun... |
| fvreseq1 7042 | Equality of a function res... |
| fvreseq 7043 | Equality of restricted fun... |
| fnmptfvd 7044 | A function with a given do... |
| fndmdif 7045 | Two ways to express the lo... |
| fndmdifcom 7046 | The difference set between... |
| fndmdifeq0 7047 | The difference set of two ... |
| fndmin 7048 | Two ways to express the lo... |
| fneqeql 7049 | Two functions are equal if... |
| fneqeql2 7050 | Two functions are equal if... |
| fnreseql 7051 | Two functions are equal on... |
| chfnrn 7052 | The range of a choice func... |
| funfvop 7053 | Ordered pair with function... |
| funfvbrb 7054 | Two ways to say that ` A `... |
| fvimacnvi 7055 | A member of a preimage is ... |
| fvimacnv 7056 | The argument of a function... |
| funimass3 7057 | A kind of contraposition l... |
| funimass5 7058 | A subclass of a preimage i... |
| funconstss 7059 | Two ways of specifying tha... |
| fvimacnvALT 7060 | Alternate proof of ~ fvima... |
| elpreima 7061 | Membership in the preimage... |
| elpreimad 7062 | Membership in the preimage... |
| fniniseg 7063 | Membership in the preimage... |
| fncnvima2 7064 | Inverse images under funct... |
| fniniseg2 7065 | Inverse point images under... |
| unpreima 7066 | Preimage of a union. (Con... |
| inpreima 7067 | Preimage of an intersectio... |
| difpreima 7068 | Preimage of a difference. ... |
| respreima 7069 | The preimage of a restrict... |
| cnvimainrn 7070 | The preimage of the inters... |
| sspreima 7071 | The preimage of a subset i... |
| iinpreima 7072 | Preimage of an intersectio... |
| intpreima 7073 | Preimage of an intersectio... |
| fimacnvOLD 7074 | Obsolete version of ~ fima... |
| fimacnvinrn 7075 | Taking the converse image ... |
| fimacnvinrn2 7076 | Taking the converse image ... |
| rescnvimafod 7077 | The restriction of a funct... |
| fvn0ssdmfun 7078 | If a class' function value... |
| fnopfv 7079 | Ordered pair with function... |
| fvelrn 7080 | A function's value belongs... |
| nelrnfvne 7081 | A function value cannot be... |
| fveqdmss 7082 | If the empty set is not co... |
| fveqressseq 7083 | If the empty set is not co... |
| fnfvelrn 7084 | A function's value belongs... |
| ffvelcdm 7085 | A function's value belongs... |
| fnfvelrnd 7086 | A function's value belongs... |
| ffvelcdmi 7087 | A function's value belongs... |
| ffvelcdmda 7088 | A function's value belongs... |
| ffvelcdmd 7089 | A function's value belongs... |
| feldmfvelcdm 7090 | A class is an element of t... |
| rexrn 7091 | Restricted existential qua... |
| ralrn 7092 | Restricted universal quant... |
| elrnrexdm 7093 | For any element in the ran... |
| elrnrexdmb 7094 | For any element in the ran... |
| eldmrexrn 7095 | For any element in the dom... |
| eldmrexrnb 7096 | For any element in the dom... |
| fvcofneq 7097 | The values of two function... |
| ralrnmptw 7098 | A restricted quantifier ov... |
| rexrnmptw 7099 | A restricted quantifier ov... |
| ralrnmpt 7100 | A restricted quantifier ov... |
| rexrnmpt 7101 | A restricted quantifier ov... |
| f0cli 7102 | Unconditional closure of a... |
| dff2 7103 | Alternate definition of a ... |
| dff3 7104 | Alternate definition of a ... |
| dff4 7105 | Alternate definition of a ... |
| dffo3 7106 | An onto mapping expressed ... |
| dffo4 7107 | Alternate definition of an... |
| dffo5 7108 | Alternate definition of an... |
| exfo 7109 | A relation equivalent to t... |
| dffo3f 7110 | An onto mapping expressed ... |
| foelrn 7111 | Property of a surjective f... |
| foelrnf 7112 | Property of a surjective f... |
| foco2 7113 | If a composition of two fu... |
| fmpt 7114 | Functionality of the mappi... |
| f1ompt 7115 | Express bijection for a ma... |
| fmpti 7116 | Functionality of the mappi... |
| fvmptelcdm 7117 | The value of a function at... |
| fmptd 7118 | Domain and codomain of the... |
| fmpttd 7119 | Version of ~ fmptd with in... |
| fmpt3d 7120 | Domain and codomain of the... |
| fmptdf 7121 | A version of ~ fmptd using... |
| fompt 7122 | Express being onto for a m... |
| ffnfv 7123 | A function maps to a class... |
| ffnfvf 7124 | A function maps to a class... |
| fnfvrnss 7125 | An upper bound for range d... |
| fcdmssb 7126 | A function is a function i... |
| rnmptss 7127 | The range of an operation ... |
| fmpt2d 7128 | Domain and codomain of the... |
| ffvresb 7129 | A necessary and sufficient... |
| f1oresrab 7130 | Build a bijection between ... |
| f1ossf1o 7131 | Restricting a bijection, w... |
| fmptco 7132 | Composition of two functio... |
| fmptcof 7133 | Version of ~ fmptco where ... |
| fmptcos 7134 | Composition of two functio... |
| cofmpt 7135 | Express composition of a m... |
| fcompt 7136 | Express composition of two... |
| fcoconst 7137 | Composition with a constan... |
| fsn 7138 | A function maps a singleto... |
| fsn2 7139 | A function that maps a sin... |
| fsng 7140 | A function maps a singleto... |
| fsn2g 7141 | A function that maps a sin... |
| xpsng 7142 | The Cartesian product of t... |
| xpprsng 7143 | The Cartesian product of a... |
| xpsn 7144 | The Cartesian product of t... |
| f1o2sn 7145 | A singleton consisting in ... |
| residpr 7146 | Restriction of the identit... |
| dfmpt 7147 | Alternate definition for t... |
| fnasrn 7148 | A function expressed as th... |
| idref 7149 | Two ways to state that a r... |
| funiun 7150 | A function is a union of s... |
| funopsn 7151 | If a function is an ordere... |
| funop 7152 | An ordered pair is a funct... |
| funopdmsn 7153 | The domain of a function w... |
| funsndifnop 7154 | A singleton of an ordered ... |
| funsneqopb 7155 | A singleton of an ordered ... |
| ressnop0 7156 | If ` A ` is not in ` C ` ,... |
| fpr 7157 | A function with a domain o... |
| fprg 7158 | A function with a domain o... |
| ftpg 7159 | A function with a domain o... |
| ftp 7160 | A function with a domain o... |
| fnressn 7161 | A function restricted to a... |
| funressn 7162 | A function restricted to a... |
| fressnfv 7163 | The value of a function re... |
| fvrnressn 7164 | If the value of a function... |
| fvressn 7165 | The value of a function re... |
| fvn0fvelrnOLD 7166 | Obsolete version of ~ fvn0... |
| fvconst 7167 | The value of a constant fu... |
| fnsnr 7168 | If a class belongs to a fu... |
| fnsnb 7169 | A function whose domain is... |
| fmptsn 7170 | Express a singleton functi... |
| fmptsng 7171 | Express a singleton functi... |
| fmptsnd 7172 | Express a singleton functi... |
| fmptap 7173 | Append an additional value... |
| fmptapd 7174 | Append an additional value... |
| fmptpr 7175 | Express a pair function in... |
| fvresi 7176 | The value of a restricted ... |
| fninfp 7177 | Express the class of fixed... |
| fnelfp 7178 | Property of a fixed point ... |
| fndifnfp 7179 | Express the class of non-f... |
| fnelnfp 7180 | Property of a non-fixed po... |
| fnnfpeq0 7181 | A function is the identity... |
| fvunsn 7182 | Remove an ordered pair not... |
| fvsng 7183 | The value of a singleton o... |
| fvsn 7184 | The value of a singleton o... |
| fvsnun1 7185 | The value of a function wi... |
| fvsnun2 7186 | The value of a function wi... |
| fnsnsplit 7187 | Split a function into a si... |
| fsnunf 7188 | Adjoining a point to a fun... |
| fsnunf2 7189 | Adjoining a point to a pun... |
| fsnunfv 7190 | Recover the added point fr... |
| fsnunres 7191 | Recover the original funct... |
| funresdfunsn 7192 | Restricting a function to ... |
| fvpr1g 7193 | The value of a function wi... |
| fvpr2g 7194 | The value of a function wi... |
| fvpr2gOLD 7195 | Obsolete version of ~ fvpr... |
| fvpr1 7196 | The value of a function wi... |
| fvpr1OLD 7197 | Obsolete version of ~ fvpr... |
| fvpr2 7198 | The value of a function wi... |
| fvpr2OLD 7199 | Obsolete version of ~ fvpr... |
| fprb 7200 | A condition for functionho... |
| fvtp1 7201 | The first value of a funct... |
| fvtp2 7202 | The second value of a func... |
| fvtp3 7203 | The third value of a funct... |
| fvtp1g 7204 | The value of a function wi... |
| fvtp2g 7205 | The value of a function wi... |
| fvtp3g 7206 | The value of a function wi... |
| tpres 7207 | An unordered triple of ord... |
| fvconst2g 7208 | The value of a constant fu... |
| fconst2g 7209 | A constant function expres... |
| fvconst2 7210 | The value of a constant fu... |
| fconst2 7211 | A constant function expres... |
| fconst5 7212 | Two ways to express that a... |
| rnmptc 7213 | Range of a constant functi... |
| fnprb 7214 | A function whose domain ha... |
| fntpb 7215 | A function whose domain ha... |
| fnpr2g 7216 | A function whose domain ha... |
| fpr2g 7217 | A function that maps a pai... |
| fconstfv 7218 | A constant function expres... |
| fconst3 7219 | Two ways to express a cons... |
| fconst4 7220 | Two ways to express a cons... |
| resfunexg 7221 | The restriction of a funct... |
| resiexd 7222 | The restriction of the ide... |
| fnex 7223 | If the domain of a functio... |
| fnexd 7224 | If the domain of a functio... |
| funex 7225 | If the domain of a functio... |
| opabex 7226 | Existence of a function ex... |
| mptexg 7227 | If the domain of a functio... |
| mptexgf 7228 | If the domain of a functio... |
| mptex 7229 | If the domain of a functio... |
| mptexd 7230 | If the domain of a functio... |
| mptrabex 7231 | If the domain of a functio... |
| fex 7232 | If the domain of a mapping... |
| fexd 7233 | If the domain of a mapping... |
| mptfvmpt 7234 | A function in maps-to nota... |
| eufnfv 7235 | A function is uniquely det... |
| funfvima 7236 | A function's value in a pr... |
| funfvima2 7237 | A function's value in an i... |
| funfvima2d 7238 | A function's value in a pr... |
| fnfvima 7239 | The function value of an o... |
| fnfvimad 7240 | A function's value belongs... |
| resfvresima 7241 | The value of the function ... |
| funfvima3 7242 | A class including a functi... |
| rexima 7243 | Existential quantification... |
| ralima 7244 | Universal quantification u... |
| fvclss 7245 | Upper bound for the class ... |
| elabrex 7246 | Elementhood in an image se... |
| elabrexg 7247 | Elementhood in an image se... |
| abrexco 7248 | Composition of two image m... |
| imaiun 7249 | The image of an indexed un... |
| imauni 7250 | The image of a union is th... |
| fniunfv 7251 | The indexed union of a fun... |
| funiunfv 7252 | The indexed union of a fun... |
| funiunfvf 7253 | The indexed union of a fun... |
| eluniima 7254 | Membership in the union of... |
| elunirn 7255 | Membership in the union of... |
| elunirnALT 7256 | Alternate proof of ~ eluni... |
| elunirn2OLD 7257 | Obsolete version of ~ elfv... |
| fnunirn 7258 | Membership in a union of s... |
| dff13 7259 | A one-to-one function in t... |
| dff13f 7260 | A one-to-one function in t... |
| f1veqaeq 7261 | If the values of a one-to-... |
| f1cofveqaeq 7262 | If the values of a composi... |
| f1cofveqaeqALT 7263 | Alternate proof of ~ f1cof... |
| 2f1fvneq 7264 | If two one-to-one function... |
| f1mpt 7265 | Express injection for a ma... |
| f1fveq 7266 | Equality of function value... |
| f1elima 7267 | Membership in the image of... |
| f1imass 7268 | Taking images under a one-... |
| f1imaeq 7269 | Taking images under a one-... |
| f1imapss 7270 | Taking images under a one-... |
| fpropnf1 7271 | A function, given by an un... |
| f1dom3fv3dif 7272 | The function values for a ... |
| f1dom3el3dif 7273 | The codomain of a 1-1 func... |
| dff14a 7274 | A one-to-one function in t... |
| dff14b 7275 | A one-to-one function in t... |
| f12dfv 7276 | A one-to-one function with... |
| f13dfv 7277 | A one-to-one function with... |
| dff1o6 7278 | A one-to-one onto function... |
| f1ocnvfv1 7279 | The converse value of the ... |
| f1ocnvfv2 7280 | The value of the converse ... |
| f1ocnvfv 7281 | Relationship between the v... |
| f1ocnvfvb 7282 | Relationship between the v... |
| nvof1o 7283 | An involution is a bijecti... |
| nvocnv 7284 | The converse of an involut... |
| f1cdmsn 7285 | If a one-to-one function w... |
| fsnex 7286 | Relate a function with a s... |
| f1prex 7287 | Relate a one-to-one functi... |
| f1ocnvdm 7288 | The value of the converse ... |
| f1ocnvfvrneq 7289 | If the values of a one-to-... |
| fcof1 7290 | An application is injectiv... |
| fcofo 7291 | An application is surjecti... |
| cbvfo 7292 | Change bound variable betw... |
| cbvexfo 7293 | Change bound variable betw... |
| cocan1 7294 | An injection is left-cance... |
| cocan2 7295 | A surjection is right-canc... |
| fcof1oinvd 7296 | Show that a function is th... |
| fcof1od 7297 | A function is bijective if... |
| 2fcoidinvd 7298 | Show that a function is th... |
| fcof1o 7299 | Show that two functions ar... |
| 2fvcoidd 7300 | Show that the composition ... |
| 2fvidf1od 7301 | A function is bijective if... |
| 2fvidinvd 7302 | Show that two functions ar... |
| foeqcnvco 7303 | Condition for function equ... |
| f1eqcocnv 7304 | Condition for function equ... |
| f1eqcocnvOLD 7305 | Obsolete version of ~ f1eq... |
| fveqf1o 7306 | Given a bijection ` F ` , ... |
| nf1const 7307 | A constant function from a... |
| nf1oconst 7308 | A constant function from a... |
| f1ofvswap 7309 | Swapping two values in a b... |
| fliftrel 7310 | ` F ` , a function lift, i... |
| fliftel 7311 | Elementhood in the relatio... |
| fliftel1 7312 | Elementhood in the relatio... |
| fliftcnv 7313 | Converse of the relation `... |
| fliftfun 7314 | The function ` F ` is the ... |
| fliftfund 7315 | The function ` F ` is the ... |
| fliftfuns 7316 | The function ` F ` is the ... |
| fliftf 7317 | The domain and range of th... |
| fliftval 7318 | The value of the function ... |
| isoeq1 7319 | Equality theorem for isomo... |
| isoeq2 7320 | Equality theorem for isomo... |
| isoeq3 7321 | Equality theorem for isomo... |
| isoeq4 7322 | Equality theorem for isomo... |
| isoeq5 7323 | Equality theorem for isomo... |
| nfiso 7324 | Bound-variable hypothesis ... |
| isof1o 7325 | An isomorphism is a one-to... |
| isof1oidb 7326 | A function is a bijection ... |
| isof1oopb 7327 | A function is a bijection ... |
| isorel 7328 | An isomorphism connects bi... |
| soisores 7329 | Express the condition of i... |
| soisoi 7330 | Infer isomorphism from one... |
| isoid 7331 | Identity law for isomorphi... |
| isocnv 7332 | Converse law for isomorphi... |
| isocnv2 7333 | Converse law for isomorphi... |
| isocnv3 7334 | Complementation law for is... |
| isores2 7335 | An isomorphism from one we... |
| isores1 7336 | An isomorphism from one we... |
| isores3 7337 | Induced isomorphism on a s... |
| isotr 7338 | Composition (transitive) l... |
| isomin 7339 | Isomorphisms preserve mini... |
| isoini 7340 | Isomorphisms preserve init... |
| isoini2 7341 | Isomorphisms are isomorphi... |
| isofrlem 7342 | Lemma for ~ isofr . (Cont... |
| isoselem 7343 | Lemma for ~ isose . (Cont... |
| isofr 7344 | An isomorphism preserves w... |
| isose 7345 | An isomorphism preserves s... |
| isofr2 7346 | A weak form of ~ isofr tha... |
| isopolem 7347 | Lemma for ~ isopo . (Cont... |
| isopo 7348 | An isomorphism preserves t... |
| isosolem 7349 | Lemma for ~ isoso . (Cont... |
| isoso 7350 | An isomorphism preserves t... |
| isowe 7351 | An isomorphism preserves t... |
| isowe2 7352 | A weak form of ~ isowe tha... |
| f1oiso 7353 | Any one-to-one onto functi... |
| f1oiso2 7354 | Any one-to-one onto functi... |
| f1owe 7355 | Well-ordering of isomorphi... |
| weniso 7356 | A set-like well-ordering h... |
| weisoeq 7357 | Thus, there is at most one... |
| weisoeq2 7358 | Thus, there is at most one... |
| knatar 7359 | The Knaster-Tarski theorem... |
| fvresval 7360 | The value of a restricted ... |
| funeldmb 7361 | If ` (/) ` is not part of ... |
| eqfunresadj 7362 | Law for adjoining an eleme... |
| eqfunressuc 7363 | Law for equality of restri... |
| fnssintima 7364 | Condition for subset of an... |
| imaeqsexv 7365 | Substitute a function valu... |
| imaeqsalv 7366 | Substitute a function valu... |
| canth 7367 | No set ` A ` is equinumero... |
| ncanth 7368 | Cantor's theorem fails for... |
| riotaeqdv 7371 | Formula-building deduction... |
| riotabidv 7372 | Formula-building deduction... |
| riotaeqbidv 7373 | Equality deduction for res... |
| riotaex 7374 | Restricted iota is a set. ... |
| riotav 7375 | An iota restricted to the ... |
| riotauni 7376 | Restricted iota in terms o... |
| nfriota1 7377 | The abstraction variable i... |
| nfriotadw 7378 | Deduction version of ~ nfr... |
| cbvriotaw 7379 | Change bound variable in a... |
| cbvriotavw 7380 | Change bound variable in a... |
| cbvriotavwOLD 7381 | Obsolete version of ~ cbvr... |
| nfriotad 7382 | Deduction version of ~ nfr... |
| nfriota 7383 | A variable not free in a w... |
| cbvriota 7384 | Change bound variable in a... |
| cbvriotav 7385 | Change bound variable in a... |
| csbriota 7386 | Interchange class substitu... |
| riotacl2 7387 | Membership law for "the un... |
| riotacl 7388 | Closure of restricted iota... |
| riotasbc 7389 | Substitution law for descr... |
| riotabidva 7390 | Equivalent wff's yield equ... |
| riotabiia 7391 | Equivalent wff's yield equ... |
| riota1 7392 | Property of restricted iot... |
| riota1a 7393 | Property of iota. (Contri... |
| riota2df 7394 | A deduction version of ~ r... |
| riota2f 7395 | This theorem shows a condi... |
| riota2 7396 | This theorem shows a condi... |
| riotaeqimp 7397 | If two restricted iota des... |
| riotaprop 7398 | Properties of a restricted... |
| riota5f 7399 | A method for computing res... |
| riota5 7400 | A method for computing res... |
| riotass2 7401 | Restriction of a unique el... |
| riotass 7402 | Restriction of a unique el... |
| moriotass 7403 | Restriction of a unique el... |
| snriota 7404 | A restricted class abstrac... |
| riotaxfrd 7405 | Change the variable ` x ` ... |
| eusvobj2 7406 | Specify the same property ... |
| eusvobj1 7407 | Specify the same object in... |
| f1ofveu 7408 | There is one domain elemen... |
| f1ocnvfv3 7409 | Value of the converse of a... |
| riotaund 7410 | Restricted iota equals the... |
| riotassuni 7411 | The restricted iota class ... |
| riotaclb 7412 | Bidirectional closure of r... |
| riotarab 7413 | Restricted iota of a restr... |
| oveq 7420 | Equality theorem for opera... |
| oveq1 7421 | Equality theorem for opera... |
| oveq2 7422 | Equality theorem for opera... |
| oveq12 7423 | Equality theorem for opera... |
| oveq1i 7424 | Equality inference for ope... |
| oveq2i 7425 | Equality inference for ope... |
| oveq12i 7426 | Equality inference for ope... |
| oveqi 7427 | Equality inference for ope... |
| oveq123i 7428 | Equality inference for ope... |
| oveq1d 7429 | Equality deduction for ope... |
| oveq2d 7430 | Equality deduction for ope... |
| oveqd 7431 | Equality deduction for ope... |
| oveq12d 7432 | Equality deduction for ope... |
| oveqan12d 7433 | Equality deduction for ope... |
| oveqan12rd 7434 | Equality deduction for ope... |
| oveq123d 7435 | Equality deduction for ope... |
| fvoveq1d 7436 | Equality deduction for nes... |
| fvoveq1 7437 | Equality theorem for neste... |
| ovanraleqv 7438 | Equality theorem for a con... |
| imbrov2fvoveq 7439 | Equality theorem for neste... |
| ovrspc2v 7440 | If an operation value is e... |
| oveqrspc2v 7441 | Restricted specialization ... |
| oveqdr 7442 | Equality of two operations... |
| nfovd 7443 | Deduction version of bound... |
| nfov 7444 | Bound-variable hypothesis ... |
| oprabidw 7445 | The law of concretion. Sp... |
| oprabid 7446 | The law of concretion. Sp... |
| ovex 7447 | The result of an operation... |
| ovexi 7448 | The result of an operation... |
| ovexd 7449 | The result of an operation... |
| ovssunirn 7450 | The result of an operation... |
| 0ov 7451 | Operation value of the emp... |
| ovprc 7452 | The value of an operation ... |
| ovprc1 7453 | The value of an operation ... |
| ovprc2 7454 | The value of an operation ... |
| ovrcl 7455 | Reverse closure for an ope... |
| csbov123 7456 | Move class substitution in... |
| csbov 7457 | Move class substitution in... |
| csbov12g 7458 | Move class substitution in... |
| csbov1g 7459 | Move class substitution in... |
| csbov2g 7460 | Move class substitution in... |
| rspceov 7461 | A frequently used special ... |
| elovimad 7462 | Elementhood of the image s... |
| fnbrovb 7463 | Value of a binary operatio... |
| fnotovb 7464 | Equivalence of operation v... |
| opabbrex 7465 | A collection of ordered pa... |
| opabresex2 7466 | Restrictions of a collecti... |
| opabresex2d 7467 | Obsolete version of ~ opab... |
| fvmptopab 7468 | The function value of a ma... |
| fvmptopabOLD 7469 | Obsolete version of ~ fvmp... |
| f1opr 7470 | Condition for an operation... |
| brfvopab 7471 | The classes involved in a ... |
| dfoprab2 7472 | Class abstraction for oper... |
| reloprab 7473 | An operation class abstrac... |
| oprabv 7474 | If a pair and a class are ... |
| nfoprab1 7475 | The abstraction variables ... |
| nfoprab2 7476 | The abstraction variables ... |
| nfoprab3 7477 | The abstraction variables ... |
| nfoprab 7478 | Bound-variable hypothesis ... |
| oprabbid 7479 | Equivalent wff's yield equ... |
| oprabbidv 7480 | Equivalent wff's yield equ... |
| oprabbii 7481 | Equivalent wff's yield equ... |
| ssoprab2 7482 | Equivalence of ordered pai... |
| ssoprab2b 7483 | Equivalence of ordered pai... |
| eqoprab2bw 7484 | Equivalence of ordered pai... |
| eqoprab2b 7485 | Equivalence of ordered pai... |
| mpoeq123 7486 | An equality theorem for th... |
| mpoeq12 7487 | An equality theorem for th... |
| mpoeq123dva 7488 | An equality deduction for ... |
| mpoeq123dv 7489 | An equality deduction for ... |
| mpoeq123i 7490 | An equality inference for ... |
| mpoeq3dva 7491 | Slightly more general equa... |
| mpoeq3ia 7492 | An equality inference for ... |
| mpoeq3dv 7493 | An equality deduction for ... |
| nfmpo1 7494 | Bound-variable hypothesis ... |
| nfmpo2 7495 | Bound-variable hypothesis ... |
| nfmpo 7496 | Bound-variable hypothesis ... |
| 0mpo0 7497 | A mapping operation with e... |
| mpo0v 7498 | A mapping operation with e... |
| mpo0 7499 | A mapping operation with e... |
| oprab4 7500 | Two ways to state the doma... |
| cbvoprab1 7501 | Rule used to change first ... |
| cbvoprab2 7502 | Change the second bound va... |
| cbvoprab12 7503 | Rule used to change first ... |
| cbvoprab12v 7504 | Rule used to change first ... |
| cbvoprab3 7505 | Rule used to change the th... |
| cbvoprab3v 7506 | Rule used to change the th... |
| cbvmpox 7507 | Rule to change the bound v... |
| cbvmpo 7508 | Rule to change the bound v... |
| cbvmpov 7509 | Rule to change the bound v... |
| elimdelov 7510 | Eliminate a hypothesis whi... |
| brif1 7511 | Move a relation inside and... |
| ovif 7512 | Move a conditional outside... |
| ovif2 7513 | Move a conditional outside... |
| ovif12 7514 | Move a conditional outside... |
| ifov 7515 | Move a conditional outside... |
| dmoprab 7516 | The domain of an operation... |
| dmoprabss 7517 | The domain of an operation... |
| rnoprab 7518 | The range of an operation ... |
| rnoprab2 7519 | The range of a restricted ... |
| reldmoprab 7520 | The domain of an operation... |
| oprabss 7521 | Structure of an operation ... |
| eloprabga 7522 | The law of concretion for ... |
| eloprabgaOLD 7523 | Obsolete version of ~ elop... |
| eloprabg 7524 | The law of concretion for ... |
| ssoprab2i 7525 | Inference of operation cla... |
| mpov 7526 | Operation with universal d... |
| mpomptx 7527 | Express a two-argument fun... |
| mpompt 7528 | Express a two-argument fun... |
| mpodifsnif 7529 | A mapping with two argumen... |
| mposnif 7530 | A mapping with two argumen... |
| fconstmpo 7531 | Representation of a consta... |
| resoprab 7532 | Restriction of an operatio... |
| resoprab2 7533 | Restriction of an operator... |
| resmpo 7534 | Restriction of the mapping... |
| funoprabg 7535 | "At most one" is a suffici... |
| funoprab 7536 | "At most one" is a suffici... |
| fnoprabg 7537 | Functionality and domain o... |
| mpofun 7538 | The maps-to notation for a... |
| mpofunOLD 7539 | Obsolete version of ~ mpof... |
| fnoprab 7540 | Functionality and domain o... |
| ffnov 7541 | An operation maps to a cla... |
| fovcld 7542 | Closure law for an operati... |
| fovcl 7543 | Closure law for an operati... |
| eqfnov 7544 | Equality of two operations... |
| eqfnov2 7545 | Two operators with the sam... |
| fnov 7546 | Representation of a functi... |
| mpo2eqb 7547 | Bidirectional equality the... |
| rnmpo 7548 | The range of an operation ... |
| reldmmpo 7549 | The domain of an operation... |
| elrnmpog 7550 | Membership in the range of... |
| elrnmpo 7551 | Membership in the range of... |
| elimampo 7552 | Membership in the image of... |
| elrnmpores 7553 | Membership in the range of... |
| ralrnmpo 7554 | A restricted quantifier ov... |
| rexrnmpo 7555 | A restricted quantifier ov... |
| ovid 7556 | The value of an operation ... |
| ovidig 7557 | The value of an operation ... |
| ovidi 7558 | The value of an operation ... |
| ov 7559 | The value of an operation ... |
| ovigg 7560 | The value of an operation ... |
| ovig 7561 | The value of an operation ... |
| ovmpt4g 7562 | Value of a function given ... |
| ovmpos 7563 | Value of a function given ... |
| ov2gf 7564 | The value of an operation ... |
| ovmpodxf 7565 | Value of an operation give... |
| ovmpodx 7566 | Value of an operation give... |
| ovmpod 7567 | Value of an operation give... |
| ovmpox 7568 | The value of an operation ... |
| ovmpoga 7569 | Value of an operation give... |
| ovmpoa 7570 | Value of an operation give... |
| ovmpodf 7571 | Alternate deduction versio... |
| ovmpodv 7572 | Alternate deduction versio... |
| ovmpodv2 7573 | Alternate deduction versio... |
| ovmpog 7574 | Value of an operation give... |
| ovmpo 7575 | Value of an operation give... |
| ovmpot 7576 | The value of an operation ... |
| fvmpopr2d 7577 | Value of an operation give... |
| ov3 7578 | The value of an operation ... |
| ov6g 7579 | The value of an operation ... |
| ovg 7580 | The value of an operation ... |
| ovres 7581 | The value of a restricted ... |
| ovresd 7582 | Lemma for converting metri... |
| oprres 7583 | The restriction of an oper... |
| oprssov 7584 | The value of a member of t... |
| fovcdm 7585 | An operation's value belon... |
| fovcdmda 7586 | An operation's value belon... |
| fovcdmd 7587 | An operation's value belon... |
| fnrnov 7588 | The range of an operation ... |
| foov 7589 | An onto mapping of an oper... |
| fnovrn 7590 | An operation's value belon... |
| ovelrn 7591 | A member of an operation's... |
| funimassov 7592 | Membership relation for th... |
| ovelimab 7593 | Operation value in an imag... |
| ovima0 7594 | An operation value is a me... |
| ovconst2 7595 | The value of a constant op... |
| oprssdm 7596 | Domain of closure of an op... |
| nssdmovg 7597 | The value of an operation ... |
| ndmovg 7598 | The value of an operation ... |
| ndmov 7599 | The value of an operation ... |
| ndmovcl 7600 | The closure of an operatio... |
| ndmovrcl 7601 | Reverse closure law, when ... |
| ndmovcom 7602 | Any operation is commutati... |
| ndmovass 7603 | Any operation is associati... |
| ndmovdistr 7604 | Any operation is distribut... |
| ndmovord 7605 | Elimination of redundant a... |
| ndmovordi 7606 | Elimination of redundant a... |
| caovclg 7607 | Convert an operation closu... |
| caovcld 7608 | Convert an operation closu... |
| caovcl 7609 | Convert an operation closu... |
| caovcomg 7610 | Convert an operation commu... |
| caovcomd 7611 | Convert an operation commu... |
| caovcom 7612 | Convert an operation commu... |
| caovassg 7613 | Convert an operation assoc... |
| caovassd 7614 | Convert an operation assoc... |
| caovass 7615 | Convert an operation assoc... |
| caovcang 7616 | Convert an operation cance... |
| caovcand 7617 | Convert an operation cance... |
| caovcanrd 7618 | Commute the arguments of a... |
| caovcan 7619 | Convert an operation cance... |
| caovordig 7620 | Convert an operation order... |
| caovordid 7621 | Convert an operation order... |
| caovordg 7622 | Convert an operation order... |
| caovordd 7623 | Convert an operation order... |
| caovord2d 7624 | Operation ordering law wit... |
| caovord3d 7625 | Ordering law. (Contribute... |
| caovord 7626 | Convert an operation order... |
| caovord2 7627 | Operation ordering law wit... |
| caovord3 7628 | Ordering law. (Contribute... |
| caovdig 7629 | Convert an operation distr... |
| caovdid 7630 | Convert an operation distr... |
| caovdir2d 7631 | Convert an operation distr... |
| caovdirg 7632 | Convert an operation rever... |
| caovdird 7633 | Convert an operation distr... |
| caovdi 7634 | Convert an operation distr... |
| caov32d 7635 | Rearrange arguments in a c... |
| caov12d 7636 | Rearrange arguments in a c... |
| caov31d 7637 | Rearrange arguments in a c... |
| caov13d 7638 | Rearrange arguments in a c... |
| caov4d 7639 | Rearrange arguments in a c... |
| caov411d 7640 | Rearrange arguments in a c... |
| caov42d 7641 | Rearrange arguments in a c... |
| caov32 7642 | Rearrange arguments in a c... |
| caov12 7643 | Rearrange arguments in a c... |
| caov31 7644 | Rearrange arguments in a c... |
| caov13 7645 | Rearrange arguments in a c... |
| caov4 7646 | Rearrange arguments in a c... |
| caov411 7647 | Rearrange arguments in a c... |
| caov42 7648 | Rearrange arguments in a c... |
| caovdir 7649 | Reverse distributive law. ... |
| caovdilem 7650 | Lemma used by real number ... |
| caovlem2 7651 | Lemma used in real number ... |
| caovmo 7652 | Uniqueness of inverse elem... |
| imaeqexov 7653 | Substitute an operation va... |
| imaeqalov 7654 | Substitute an operation va... |
| mpondm0 7655 | The value of an operation ... |
| elmpocl 7656 | If a two-parameter class i... |
| elmpocl1 7657 | If a two-parameter class i... |
| elmpocl2 7658 | If a two-parameter class i... |
| elovmpod 7659 | Utility lemma for two-para... |
| elovmpo 7660 | Utility lemma for two-para... |
| elovmporab 7661 | Implications for the value... |
| elovmporab1w 7662 | Implications for the value... |
| elovmporab1 7663 | Implications for the value... |
| 2mpo0 7664 | If the operation value of ... |
| relmptopab 7665 | Any function to sets of or... |
| f1ocnvd 7666 | Describe an implicit one-t... |
| f1od 7667 | Describe an implicit one-t... |
| f1ocnv2d 7668 | Describe an implicit one-t... |
| f1o2d 7669 | Describe an implicit one-t... |
| f1opw2 7670 | A one-to-one mapping induc... |
| f1opw 7671 | A one-to-one mapping induc... |
| elovmpt3imp 7672 | If the value of a function... |
| ovmpt3rab1 7673 | The value of an operation ... |
| ovmpt3rabdm 7674 | If the value of a function... |
| elovmpt3rab1 7675 | Implications for the value... |
| elovmpt3rab 7676 | Implications for the value... |
| ofeqd 7681 | Equality theorem for funct... |
| ofeq 7682 | Equality theorem for funct... |
| ofreq 7683 | Equality theorem for funct... |
| ofexg 7684 | A function operation restr... |
| nfof 7685 | Hypothesis builder for fun... |
| nfofr 7686 | Hypothesis builder for fun... |
| ofrfvalg 7687 | Value of a relation applie... |
| offval 7688 | Value of an operation appl... |
| ofrfval 7689 | Value of a relation applie... |
| ofval 7690 | Evaluate a function operat... |
| ofrval 7691 | Exhibit a function relatio... |
| offn 7692 | The function operation pro... |
| offun 7693 | The function operation pro... |
| offval2f 7694 | The function operation exp... |
| ofmresval 7695 | Value of a restriction of ... |
| fnfvof 7696 | Function value of a pointw... |
| off 7697 | The function operation pro... |
| ofres 7698 | Restrict the operands of a... |
| offval2 7699 | The function operation exp... |
| ofrfval2 7700 | The function relation acti... |
| ofmpteq 7701 | Value of a pointwise opera... |
| ofco 7702 | The composition of a funct... |
| offveq 7703 | Convert an identity of the... |
| offveqb 7704 | Equivalent expressions for... |
| ofc1 7705 | Left operation by a consta... |
| ofc2 7706 | Right operation by a const... |
| ofc12 7707 | Function operation on two ... |
| caofref 7708 | Transfer a reflexive law t... |
| caofinvl 7709 | Transfer a left inverse la... |
| caofid0l 7710 | Transfer a left identity l... |
| caofid0r 7711 | Transfer a right identity ... |
| caofid1 7712 | Transfer a right absorptio... |
| caofid2 7713 | Transfer a right absorptio... |
| caofcom 7714 | Transfer a commutative law... |
| caofrss 7715 | Transfer a relation subset... |
| caofass 7716 | Transfer an associative la... |
| caoftrn 7717 | Transfer a transitivity la... |
| caofdi 7718 | Transfer a distributive la... |
| caofdir 7719 | Transfer a reverse distrib... |
| caonncan 7720 | Transfer ~ nncan -shaped l... |
| relrpss 7723 | The proper subset relation... |
| brrpssg 7724 | The proper subset relation... |
| brrpss 7725 | The proper subset relation... |
| porpss 7726 | Every class is partially o... |
| sorpss 7727 | Express strict ordering un... |
| sorpssi 7728 | Property of a chain of set... |
| sorpssun 7729 | A chain of sets is closed ... |
| sorpssin 7730 | A chain of sets is closed ... |
| sorpssuni 7731 | In a chain of sets, a maxi... |
| sorpssint 7732 | In a chain of sets, a mini... |
| sorpsscmpl 7733 | The componentwise compleme... |
| zfun 7735 | Axiom of Union expressed w... |
| axun2 7736 | A variant of the Axiom of ... |
| uniex2 7737 | The Axiom of Union using t... |
| vuniex 7738 | The union of a setvar is a... |
| uniexg 7739 | The ZF Axiom of Union in c... |
| uniex 7740 | The Axiom of Union in clas... |
| uniexd 7741 | Deduction version of the Z... |
| unex 7742 | The union of two sets is a... |
| tpex 7743 | An unordered triple of cla... |
| unexb 7744 | Existence of union is equi... |
| unexg 7745 | A union of two sets is a s... |
| xpexg 7746 | The Cartesian product of t... |
| xpexd 7747 | The Cartesian product of t... |
| 3xpexg 7748 | The Cartesian product of t... |
| xpex 7749 | The Cartesian product of t... |
| unexd 7750 | The union of two sets is a... |
| sqxpexg 7751 | The Cartesian square of a ... |
| abnexg 7752 | Sufficient condition for a... |
| abnex 7753 | Sufficient condition for a... |
| snnex 7754 | The class of all singleton... |
| pwnex 7755 | The class of all power set... |
| difex2 7756 | If the subtrahend of a cla... |
| difsnexi 7757 | If the difference of a cla... |
| uniuni 7758 | Expression for double unio... |
| uniexr 7759 | Converse of the Axiom of U... |
| uniexb 7760 | The Axiom of Union and its... |
| pwexr 7761 | Converse of the Axiom of P... |
| pwexb 7762 | The Axiom of Power Sets an... |
| elpwpwel 7763 | A class belongs to a doubl... |
| eldifpw 7764 | Membership in a power clas... |
| elpwun 7765 | Membership in the power cl... |
| pwuncl 7766 | Power classes are closed u... |
| iunpw 7767 | An indexed union of a powe... |
| fr3nr 7768 | A well-founded relation ha... |
| epne3 7769 | A well-founded class conta... |
| dfwe2 7770 | Alternate definition of we... |
| epweon 7771 | The membership relation we... |
| epweonALT 7772 | Alternate proof of ~ epweo... |
| ordon 7773 | The class of all ordinal n... |
| onprc 7774 | No set contains all ordina... |
| ssorduni 7775 | The union of a class of or... |
| ssonuni 7776 | The union of a set of ordi... |
| ssonunii 7777 | The union of a set of ordi... |
| ordeleqon 7778 | A way to express the ordin... |
| ordsson 7779 | Any ordinal class is a sub... |
| dford5 7780 | A class is ordinal iff it ... |
| onss 7781 | An ordinal number is a sub... |
| predon 7782 | The predecessor of an ordi... |
| predonOLD 7783 | Obsolete version of ~ pred... |
| ssonprc 7784 | Two ways of saying a class... |
| onuni 7785 | The union of an ordinal nu... |
| orduni 7786 | The union of an ordinal cl... |
| onint 7787 | The intersection (infimum)... |
| onint0 7788 | The intersection of a clas... |
| onssmin 7789 | A nonempty class of ordina... |
| onminesb 7790 | If a property is true for ... |
| onminsb 7791 | If a property is true for ... |
| oninton 7792 | The intersection of a none... |
| onintrab 7793 | The intersection of a clas... |
| onintrab2 7794 | An existence condition equ... |
| onnmin 7795 | No member of a set of ordi... |
| onnminsb 7796 | An ordinal number smaller ... |
| oneqmin 7797 | A way to show that an ordi... |
| uniordint 7798 | The union of a set of ordi... |
| onminex 7799 | If a wff is true for an or... |
| sucon 7800 | The class of all ordinal n... |
| sucexb 7801 | A successor exists iff its... |
| sucexg 7802 | The successor of a set is ... |
| sucex 7803 | The successor of a set is ... |
| onmindif2 7804 | The minimum of a class of ... |
| ordsuci 7805 | The successor of an ordina... |
| sucexeloni 7806 | If the successor of an ord... |
| sucexeloniOLD 7807 | Obsolete version of ~ suce... |
| onsuc 7808 | The successor of an ordina... |
| suceloniOLD 7809 | Obsolete version of ~ onsu... |
| ordsuc 7810 | A class is ordinal if and ... |
| ordsucOLD 7811 | Obsolete version of ~ ords... |
| ordpwsuc 7812 | The collection of ordinals... |
| onpwsuc 7813 | The collection of ordinal ... |
| onsucb 7814 | A class is an ordinal numb... |
| ordsucss 7815 | The successor of an elemen... |
| onpsssuc 7816 | An ordinal number is a pro... |
| ordelsuc 7817 | A set belongs to an ordina... |
| onsucmin 7818 | The successor of an ordina... |
| ordsucelsuc 7819 | Membership is inherited by... |
| ordsucsssuc 7820 | The subclass relationship ... |
| ordsucuniel 7821 | Given an element ` A ` of ... |
| ordsucun 7822 | The successor of the maxim... |
| ordunpr 7823 | The maximum of two ordinal... |
| ordunel 7824 | The maximum of two ordinal... |
| onsucuni 7825 | A class of ordinal numbers... |
| ordsucuni 7826 | An ordinal class is a subc... |
| orduniorsuc 7827 | An ordinal class is either... |
| unon 7828 | The class of all ordinal n... |
| ordunisuc 7829 | An ordinal class is equal ... |
| orduniss2 7830 | The union of the ordinal s... |
| onsucuni2 7831 | A successor ordinal is the... |
| 0elsuc 7832 | The successor of an ordina... |
| limon 7833 | The class of ordinal numbe... |
| onuniorsuc 7834 | An ordinal number is eithe... |
| onssi 7835 | An ordinal number is a sub... |
| onsuci 7836 | The successor of an ordina... |
| onuniorsuciOLD 7837 | Obsolete version of ~ onun... |
| onuninsuci 7838 | An ordinal is equal to its... |
| onsucssi 7839 | A set belongs to an ordina... |
| nlimsucg 7840 | A successor is not a limit... |
| orduninsuc 7841 | An ordinal class is equal ... |
| ordunisuc2 7842 | An ordinal equal to its un... |
| ordzsl 7843 | An ordinal is zero, a succ... |
| onzsl 7844 | An ordinal number is zero,... |
| dflim3 7845 | An alternate definition of... |
| dflim4 7846 | An alternate definition of... |
| limsuc 7847 | The successor of a member ... |
| limsssuc 7848 | A class includes a limit o... |
| nlimon 7849 | Two ways to express the cl... |
| limuni3 7850 | The union of a nonempty cl... |
| tfi 7851 | The Principle of Transfini... |
| tfisg 7852 | A closed form of ~ tfis . ... |
| tfis 7853 | Transfinite Induction Sche... |
| tfis2f 7854 | Transfinite Induction Sche... |
| tfis2 7855 | Transfinite Induction Sche... |
| tfis3 7856 | Transfinite Induction Sche... |
| tfisi 7857 | A transfinite induction sc... |
| tfinds 7858 | Principle of Transfinite I... |
| tfindsg 7859 | Transfinite Induction (inf... |
| tfindsg2 7860 | Transfinite Induction (inf... |
| tfindes 7861 | Transfinite Induction with... |
| tfinds2 7862 | Transfinite Induction (inf... |
| tfinds3 7863 | Principle of Transfinite I... |
| dfom2 7866 | An alternate definition of... |
| elom 7867 | Membership in omega. The ... |
| omsson 7868 | Omega is a subset of ` On ... |
| limomss 7869 | The class of natural numbe... |
| nnon 7870 | A natural number is an ord... |
| nnoni 7871 | A natural number is an ord... |
| nnord 7872 | A natural number is ordina... |
| trom 7873 | The class of finite ordina... |
| ordom 7874 | The class of finite ordina... |
| elnn 7875 | A member of a natural numb... |
| omon 7876 | The class of natural numbe... |
| omelon2 7877 | Omega is an ordinal number... |
| nnlim 7878 | A natural number is not a ... |
| omssnlim 7879 | The class of natural numbe... |
| limom 7880 | Omega is a limit ordinal. ... |
| peano2b 7881 | A class belongs to omega i... |
| nnsuc 7882 | A nonzero natural number i... |
| omsucne 7883 | A natural number is not th... |
| ssnlim 7884 | An ordinal subclass of non... |
| omsinds 7885 | Strong (or "total") induct... |
| omsindsOLD 7886 | Obsolete version of ~ omsi... |
| omun 7887 | The union of two finite or... |
| peano1 7888 | Zero is a natural number. ... |
| peano1OLD 7889 | Obsolete version of ~ pean... |
| peano2 7890 | The successor of any natur... |
| peano3 7891 | The successor of any natur... |
| peano4 7892 | Two natural numbers are eq... |
| peano5 7893 | The induction postulate: a... |
| peano5OLD 7894 | Obsolete version of ~ pean... |
| nn0suc 7895 | A natural number is either... |
| find 7896 | The Principle of Finite In... |
| findOLD 7897 | Obsolete version of ~ find... |
| finds 7898 | Principle of Finite Induct... |
| findsg 7899 | Principle of Finite Induct... |
| finds2 7900 | Principle of Finite Induct... |
| finds1 7901 | Principle of Finite Induct... |
| findes 7902 | Finite induction with expl... |
| dmexg 7903 | The domain of a set is a s... |
| rnexg 7904 | The range of a set is a se... |
| dmexd 7905 | The domain of a set is a s... |
| fndmexd 7906 | If a function is a set, it... |
| dmfex 7907 | If a mapping is a set, its... |
| fndmexb 7908 | The domain of a function i... |
| fdmexb 7909 | The domain of a function i... |
| dmfexALT 7910 | Alternate proof of ~ dmfex... |
| dmex 7911 | The domain of a set is a s... |
| rnex 7912 | The range of a set is a se... |
| iprc 7913 | The identity function is a... |
| resiexg 7914 | The existence of a restric... |
| imaexg 7915 | The image of a set is a se... |
| imaex 7916 | The image of a set is a se... |
| rnexd 7917 | The range of a set is a se... |
| imaexd 7918 | The image of a set is a se... |
| exse2 7919 | Any set relation is set-li... |
| xpexr 7920 | If a Cartesian product is ... |
| xpexr2 7921 | If a nonempty Cartesian pr... |
| xpexcnv 7922 | A condition where the conv... |
| soex 7923 | If the relation in a stric... |
| elxp4 7924 | Membership in a Cartesian ... |
| elxp5 7925 | Membership in a Cartesian ... |
| cnvexg 7926 | The converse of a set is a... |
| cnvex 7927 | The converse of a set is a... |
| relcnvexb 7928 | A relation is a set iff it... |
| f1oexrnex 7929 | If the range of a 1-1 onto... |
| f1oexbi 7930 | There is a one-to-one onto... |
| coexg 7931 | The composition of two set... |
| coex 7932 | The composition of two set... |
| funcnvuni 7933 | The union of a chain (with... |
| fun11uni 7934 | The union of a chain (with... |
| fex2 7935 | A function with bounded do... |
| fabexg 7936 | Existence of a set of func... |
| fabex 7937 | Existence of a set of func... |
| f1oabexg 7938 | The class of all 1-1-onto ... |
| fiunlem 7939 | Lemma for ~ fiun and ~ f1i... |
| fiun 7940 | The union of a chain (with... |
| f1iun 7941 | The union of a chain (with... |
| fviunfun 7942 | The function value of an i... |
| ffoss 7943 | Relationship between a map... |
| f11o 7944 | Relationship between one-t... |
| resfunexgALT 7945 | Alternate proof of ~ resfu... |
| cofunexg 7946 | Existence of a composition... |
| cofunex2g 7947 | Existence of a composition... |
| fnexALT 7948 | Alternate proof of ~ fnex ... |
| funexw 7949 | Weak version of ~ funex th... |
| mptexw 7950 | Weak version of ~ mptex th... |
| funrnex 7951 | If the domain of a functio... |
| zfrep6 7952 | A version of the Axiom of ... |
| focdmex 7953 | If the domain of an onto f... |
| f1dmex 7954 | If the codomain of a one-t... |
| f1ovv 7955 | The codomain/range of a 1-... |
| fvclex 7956 | Existence of the class of ... |
| fvresex 7957 | Existence of the class of ... |
| abrexexg 7958 | Existence of a class abstr... |
| abrexexgOLD 7959 | Obsolete version of ~ abre... |
| abrexex 7960 | Existence of a class abstr... |
| iunexg 7961 | The existence of an indexe... |
| abrexex2g 7962 | Existence of an existentia... |
| opabex3d 7963 | Existence of an ordered pa... |
| opabex3rd 7964 | Existence of an ordered pa... |
| opabex3 7965 | Existence of an ordered pa... |
| iunex 7966 | The existence of an indexe... |
| abrexex2 7967 | Existence of an existentia... |
| abexssex 7968 | Existence of a class abstr... |
| abexex 7969 | A condition where a class ... |
| f1oweALT 7970 | Alternate proof of ~ f1owe... |
| wemoiso 7971 | Thus, there is at most one... |
| wemoiso2 7972 | Thus, there is at most one... |
| oprabexd 7973 | Existence of an operator a... |
| oprabex 7974 | Existence of an operation ... |
| oprabex3 7975 | Existence of an operation ... |
| oprabrexex2 7976 | Existence of an existentia... |
| ab2rexex 7977 | Existence of a class abstr... |
| ab2rexex2 7978 | Existence of an existentia... |
| xpexgALT 7979 | Alternate proof of ~ xpexg... |
| offval3 7980 | General value of ` ( F oF ... |
| offres 7981 | Pointwise combination comm... |
| ofmres 7982 | Equivalent expressions for... |
| ofmresex 7983 | Existence of a restriction... |
| mptcnfimad 7984 | The converse of a mapping ... |
| 1stval 7989 | The value of the function ... |
| 2ndval 7990 | The value of the function ... |
| 1stnpr 7991 | Value of the first-member ... |
| 2ndnpr 7992 | Value of the second-member... |
| 1st0 7993 | The value of the first-mem... |
| 2nd0 7994 | The value of the second-me... |
| op1st 7995 | Extract the first member o... |
| op2nd 7996 | Extract the second member ... |
| op1std 7997 | Extract the first member o... |
| op2ndd 7998 | Extract the second member ... |
| op1stg 7999 | Extract the first member o... |
| op2ndg 8000 | Extract the second member ... |
| ot1stg 8001 | Extract the first member o... |
| ot2ndg 8002 | Extract the second member ... |
| ot3rdg 8003 | Extract the third member o... |
| 1stval2 8004 | Alternate value of the fun... |
| 2ndval2 8005 | Alternate value of the fun... |
| oteqimp 8006 | The components of an order... |
| fo1st 8007 | The ` 1st ` function maps ... |
| fo2nd 8008 | The ` 2nd ` function maps ... |
| br1steqg 8009 | Uniqueness condition for t... |
| br2ndeqg 8010 | Uniqueness condition for t... |
| f1stres 8011 | Mapping of a restriction o... |
| f2ndres 8012 | Mapping of a restriction o... |
| fo1stres 8013 | Onto mapping of a restrict... |
| fo2ndres 8014 | Onto mapping of a restrict... |
| 1st2val 8015 | Value of an alternate defi... |
| 2nd2val 8016 | Value of an alternate defi... |
| 1stcof 8017 | Composition of the first m... |
| 2ndcof 8018 | Composition of the second ... |
| xp1st 8019 | Location of the first elem... |
| xp2nd 8020 | Location of the second ele... |
| elxp6 8021 | Membership in a Cartesian ... |
| elxp7 8022 | Membership in a Cartesian ... |
| eqopi 8023 | Equality with an ordered p... |
| xp2 8024 | Representation of Cartesia... |
| unielxp 8025 | The membership relation fo... |
| 1st2nd2 8026 | Reconstruction of a member... |
| 1st2ndb 8027 | Reconstruction of an order... |
| xpopth 8028 | An ordered pair theorem fo... |
| eqop 8029 | Two ways to express equali... |
| eqop2 8030 | Two ways to express equali... |
| op1steq 8031 | Two ways of expressing tha... |
| opreuopreu 8032 | There is a unique ordered ... |
| el2xptp 8033 | A member of a nested Carte... |
| el2xptp0 8034 | A member of a nested Carte... |
| el2xpss 8035 | Version of ~ elrel for tri... |
| 2nd1st 8036 | Swap the members of an ord... |
| 1st2nd 8037 | Reconstruction of a member... |
| 1stdm 8038 | The first ordered pair com... |
| 2ndrn 8039 | The second ordered pair co... |
| 1st2ndbr 8040 | Express an element of a re... |
| releldm2 8041 | Two ways of expressing mem... |
| reldm 8042 | An expression for the doma... |
| releldmdifi 8043 | One way of expressing memb... |
| funfv1st2nd 8044 | The function value for the... |
| funelss 8045 | If the first component of ... |
| funeldmdif 8046 | Two ways of expressing mem... |
| sbcopeq1a 8047 | Equality theorem for subst... |
| csbopeq1a 8048 | Equality theorem for subst... |
| sbcoteq1a 8049 | Equality theorem for subst... |
| dfopab2 8050 | A way to define an ordered... |
| dfoprab3s 8051 | A way to define an operati... |
| dfoprab3 8052 | Operation class abstractio... |
| dfoprab4 8053 | Operation class abstractio... |
| dfoprab4f 8054 | Operation class abstractio... |
| opabex2 8055 | Condition for an operation... |
| opabn1stprc 8056 | An ordered-pair class abst... |
| opiota 8057 | The property of a uniquely... |
| cnvoprab 8058 | The converse of a class ab... |
| dfxp3 8059 | Define the Cartesian produ... |
| elopabi 8060 | A consequence of membershi... |
| eloprabi 8061 | A consequence of membershi... |
| mpomptsx 8062 | Express a two-argument fun... |
| mpompts 8063 | Express a two-argument fun... |
| dmmpossx 8064 | The domain of a mapping is... |
| fmpox 8065 | Functionality, domain and ... |
| fmpo 8066 | Functionality, domain and ... |
| fnmpo 8067 | Functionality and domain o... |
| fnmpoi 8068 | Functionality and domain o... |
| dmmpo 8069 | Domain of a class given by... |
| ovmpoelrn 8070 | An operation's value belon... |
| dmmpoga 8071 | Domain of an operation giv... |
| dmmpogaOLD 8072 | Obsolete version of ~ dmmp... |
| dmmpog 8073 | Domain of an operation giv... |
| mpoexxg 8074 | Existence of an operation ... |
| mpoexg 8075 | Existence of an operation ... |
| mpoexga 8076 | If the domain of an operat... |
| mpoexw 8077 | Weak version of ~ mpoex th... |
| mpoex 8078 | If the domain of an operat... |
| mptmpoopabbrd 8079 | The operation value of a f... |
| mptmpoopabbrdOLD 8080 | Obsolete version of ~ mptm... |
| mptmpoopabovd 8081 | The operation value of a f... |
| mptmpoopabbrdOLDOLD 8082 | Obsolete version of ~ mptm... |
| mptmpoopabovdOLD 8083 | Obsolete version of ~ mptm... |
| el2mpocsbcl 8084 | If the operation value of ... |
| el2mpocl 8085 | If the operation value of ... |
| fnmpoovd 8086 | A function with a Cartesia... |
| offval22 8087 | The function operation exp... |
| brovpreldm 8088 | If a binary relation holds... |
| bropopvvv 8089 | If a binary relation holds... |
| bropfvvvvlem 8090 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8091 | If a binary relation holds... |
| ovmptss 8092 | If all the values of the m... |
| relmpoopab 8093 | Any function to sets of or... |
| fmpoco 8094 | Composition of two functio... |
| oprabco 8095 | Composition of a function ... |
| oprab2co 8096 | Composition of operator ab... |
| df1st2 8097 | An alternate possible defi... |
| df2nd2 8098 | An alternate possible defi... |
| 1stconst 8099 | The mapping of a restricti... |
| 2ndconst 8100 | The mapping of a restricti... |
| dfmpo 8101 | Alternate definition for t... |
| mposn 8102 | An operation (in maps-to n... |
| curry1 8103 | Composition with ` ``' ( 2... |
| curry1val 8104 | The value of a curried fun... |
| curry1f 8105 | Functionality of a curried... |
| curry2 8106 | Composition with ` ``' ( 1... |
| curry2f 8107 | Functionality of a curried... |
| curry2val 8108 | The value of a curried fun... |
| cnvf1olem 8109 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8110 | Describe a function that m... |
| fparlem1 8111 | Lemma for ~ fpar . (Contr... |
| fparlem2 8112 | Lemma for ~ fpar . (Contr... |
| fparlem3 8113 | Lemma for ~ fpar . (Contr... |
| fparlem4 8114 | Lemma for ~ fpar . (Contr... |
| fpar 8115 | Merge two functions in par... |
| fsplit 8116 | A function that can be use... |
| fsplitfpar 8117 | Merge two functions with a... |
| offsplitfpar 8118 | Express the function opera... |
| f2ndf 8119 | The ` 2nd ` (second compon... |
| fo2ndf 8120 | The ` 2nd ` (second compon... |
| f1o2ndf1 8121 | The ` 2nd ` (second compon... |
| opco1 8122 | Value of an operation prec... |
| opco2 8123 | Value of an operation prec... |
| opco1i 8124 | Inference form of ~ opco1 ... |
| frxp 8125 | A lexicographical ordering... |
| xporderlem 8126 | Lemma for lexicographical ... |
| poxp 8127 | A lexicographical ordering... |
| soxp 8128 | A lexicographical ordering... |
| wexp 8129 | A lexicographical ordering... |
| fnwelem 8130 | Lemma for ~ fnwe . (Contr... |
| fnwe 8131 | A variant on lexicographic... |
| fnse 8132 | Condition for the well-ord... |
| fvproj 8133 | Value of a function on ord... |
| fimaproj 8134 | Image of a cartesian produ... |
| ralxpes 8135 | A version of ~ ralxp with ... |
| ralxp3f 8136 | Restricted for all over a ... |
| ralxp3 8137 | Restricted for all over a ... |
| ralxp3es 8138 | Restricted for-all over a ... |
| frpoins3xpg 8139 | Special case of founded pa... |
| frpoins3xp3g 8140 | Special case of founded pa... |
| xpord2lem 8141 | Lemma for Cartesian produc... |
| poxp2 8142 | Another way of partially o... |
| frxp2 8143 | Another way of giving a we... |
| xpord2pred 8144 | Calculate the predecessor ... |
| sexp2 8145 | Condition for the relation... |
| xpord2indlem 8146 | Induction over the Cartesi... |
| xpord2ind 8147 | Induction over the Cartesi... |
| xpord3lem 8148 | Lemma for triple ordering.... |
| poxp3 8149 | Triple Cartesian product p... |
| frxp3 8150 | Give well-foundedness over... |
| xpord3pred 8151 | Calculate the predecsessor... |
| sexp3 8152 | Show that the triple order... |
| xpord3inddlem 8153 | Induction over the triple ... |
| xpord3indd 8154 | Induction over the triple ... |
| xpord3ind 8155 | Induction over the triple ... |
| orderseqlem 8156 | Lemma for ~ poseq and ~ so... |
| poseq 8157 | A partial ordering of ordi... |
| soseq 8158 | A linear ordering of ordin... |
| suppval 8161 | The value of the operation... |
| supp0prc 8162 | The support of a class is ... |
| suppvalbr 8163 | The value of the operation... |
| supp0 8164 | The support of the empty s... |
| suppval1 8165 | The value of the operation... |
| suppvalfng 8166 | The value of the operation... |
| suppvalfn 8167 | The value of the operation... |
| elsuppfng 8168 | An element of the support ... |
| elsuppfn 8169 | An element of the support ... |
| cnvimadfsn 8170 | The support of functions "... |
| suppimacnvss 8171 | The support of functions "... |
| suppimacnv 8172 | Support sets of functions ... |
| fsuppeq 8173 | Two ways of writing the su... |
| fsuppeqg 8174 | Version of ~ fsuppeq avoid... |
| suppssdm 8175 | The support of a function ... |
| suppsnop 8176 | The support of a singleton... |
| snopsuppss 8177 | The support of a singleton... |
| fvn0elsupp 8178 | If the function value for ... |
| fvn0elsuppb 8179 | The function value for a g... |
| rexsupp 8180 | Existential quantification... |
| ressuppss 8181 | The support of the restric... |
| suppun 8182 | The support of a class/fun... |
| ressuppssdif 8183 | The support of the restric... |
| mptsuppdifd 8184 | The support of a function ... |
| mptsuppd 8185 | The support of a function ... |
| extmptsuppeq 8186 | The support of an extended... |
| suppfnss 8187 | The support of a function ... |
| funsssuppss 8188 | The support of a function ... |
| fnsuppres 8189 | Two ways to express restri... |
| fnsuppeq0 8190 | The support of a function ... |
| fczsupp0 8191 | The support of a constant ... |
| suppss 8192 | Show that the support of a... |
| suppssOLD 8193 | Obsolete version of ~ supp... |
| suppssr 8194 | A function is zero outside... |
| suppssrg 8195 | A function is zero outside... |
| suppssov1 8196 | Formula building theorem f... |
| suppssov2 8197 | Formula building theorem f... |
| suppssof1 8198 | Formula building theorem f... |
| suppss2 8199 | Show that the support of a... |
| suppsssn 8200 | Show that the support of a... |
| suppssfv 8201 | Formula building theorem f... |
| suppofssd 8202 | Condition for the support ... |
| suppofss1d 8203 | Condition for the support ... |
| suppofss2d 8204 | Condition for the support ... |
| suppco 8205 | The support of the composi... |
| suppcoss 8206 | The support of the composi... |
| supp0cosupp0 8207 | The support of the composi... |
| imacosupp 8208 | The image of the support o... |
| opeliunxp2f 8209 | Membership in a union of C... |
| mpoxeldm 8210 | If there is an element of ... |
| mpoxneldm 8211 | If the first argument of a... |
| mpoxopn0yelv 8212 | If there is an element of ... |
| mpoxopynvov0g 8213 | If the second argument of ... |
| mpoxopxnop0 8214 | If the first argument of a... |
| mpoxopx0ov0 8215 | If the first argument of a... |
| mpoxopxprcov0 8216 | If the components of the f... |
| mpoxopynvov0 8217 | If the second argument of ... |
| mpoxopoveq 8218 | Value of an operation give... |
| mpoxopovel 8219 | Element of the value of an... |
| mpoxopoveqd 8220 | Value of an operation give... |
| brovex 8221 | A binary relation of the v... |
| brovmpoex 8222 | A binary relation of the v... |
| sprmpod 8223 | The extension of a binary ... |
| tposss 8226 | Subset theorem for transpo... |
| tposeq 8227 | Equality theorem for trans... |
| tposeqd 8228 | Equality theorem for trans... |
| tposssxp 8229 | The transposition is a sub... |
| reltpos 8230 | The transposition is a rel... |
| brtpos2 8231 | Value of the transposition... |
| brtpos0 8232 | The behavior of ` tpos ` w... |
| reldmtpos 8233 | Necessary and sufficient c... |
| brtpos 8234 | The transposition swaps ar... |
| ottpos 8235 | The transposition swaps th... |
| relbrtpos 8236 | The transposition swaps ar... |
| dmtpos 8237 | The domain of ` tpos F ` w... |
| rntpos 8238 | The range of ` tpos F ` wh... |
| tposexg 8239 | The transposition of a set... |
| ovtpos 8240 | The transposition swaps th... |
| tposfun 8241 | The transposition of a fun... |
| dftpos2 8242 | Alternate definition of ` ... |
| dftpos3 8243 | Alternate definition of ` ... |
| dftpos4 8244 | Alternate definition of ` ... |
| tpostpos 8245 | Value of the double transp... |
| tpostpos2 8246 | Value of the double transp... |
| tposfn2 8247 | The domain of a transposit... |
| tposfo2 8248 | Condition for a surjective... |
| tposf2 8249 | The domain and codomain of... |
| tposf12 8250 | Condition for an injective... |
| tposf1o2 8251 | Condition of a bijective t... |
| tposfo 8252 | The domain and codomain/ra... |
| tposf 8253 | The domain and codomain of... |
| tposfn 8254 | Functionality of a transpo... |
| tpos0 8255 | Transposition of the empty... |
| tposco 8256 | Transposition of a composi... |
| tpossym 8257 | Two ways to say a function... |
| tposeqi 8258 | Equality theorem for trans... |
| tposex 8259 | A transposition is a set. ... |
| nftpos 8260 | Hypothesis builder for tra... |
| tposoprab 8261 | Transposition of a class o... |
| tposmpo 8262 | Transposition of a two-arg... |
| tposconst 8263 | The transposition of a con... |
| mpocurryd 8268 | The currying of an operati... |
| mpocurryvald 8269 | The value of a curried ope... |
| fvmpocurryd 8270 | The value of the value of ... |
| pwuninel2 8273 | Direct proof of ~ pwuninel... |
| pwuninel 8274 | The power set of the union... |
| undefval 8275 | Value of the undefined val... |
| undefnel2 8276 | The undefined value genera... |
| undefnel 8277 | The undefined value genera... |
| undefne0 8278 | The undefined value genera... |
| frecseq123 8281 | Equality theorem for the w... |
| nffrecs 8282 | Bound-variable hypothesis ... |
| csbfrecsg 8283 | Move class substitution in... |
| fpr3g 8284 | Functions defined by well-... |
| frrlem1 8285 | Lemma for well-founded rec... |
| frrlem2 8286 | Lemma for well-founded rec... |
| frrlem3 8287 | Lemma for well-founded rec... |
| frrlem4 8288 | Lemma for well-founded rec... |
| frrlem5 8289 | Lemma for well-founded rec... |
| frrlem6 8290 | Lemma for well-founded rec... |
| frrlem7 8291 | Lemma for well-founded rec... |
| frrlem8 8292 | Lemma for well-founded rec... |
| frrlem9 8293 | Lemma for well-founded rec... |
| frrlem10 8294 | Lemma for well-founded rec... |
| frrlem11 8295 | Lemma for well-founded rec... |
| frrlem12 8296 | Lemma for well-founded rec... |
| frrlem13 8297 | Lemma for well-founded rec... |
| frrlem14 8298 | Lemma for well-founded rec... |
| fprlem1 8299 | Lemma for well-founded rec... |
| fprlem2 8300 | Lemma for well-founded rec... |
| fpr2a 8301 | Weak version of ~ fpr2 whi... |
| fpr1 8302 | Law of well-founded recurs... |
| fpr2 8303 | Law of well-founded recurs... |
| fpr3 8304 | Law of well-founded recurs... |
| frrrel 8305 | Show without using the axi... |
| frrdmss 8306 | Show without using the axi... |
| frrdmcl 8307 | Show without using the axi... |
| fprfung 8308 | A "function" defined by we... |
| fprresex 8309 | The restriction of a funct... |
| dfwrecsOLD 8312 | Obsolete definition of the... |
| wrecseq123 8313 | General equality theorem f... |
| wrecseq123OLD 8314 | Obsolete version of ~ wrec... |
| nfwrecs 8315 | Bound-variable hypothesis ... |
| nfwrecsOLD 8316 | Obsolete proof of ~ nfwrec... |
| wrecseq1 8317 | Equality theorem for the w... |
| wrecseq2 8318 | Equality theorem for the w... |
| wrecseq3 8319 | Equality theorem for the w... |
| csbwrecsg 8320 | Move class substitution in... |
| wfr3g 8321 | Functions defined by well-... |
| wfrlem1OLD 8322 | Lemma for well-ordered rec... |
| wfrlem2OLD 8323 | Lemma for well-ordered rec... |
| wfrlem3OLD 8324 | Lemma for well-ordered rec... |
| wfrlem3OLDa 8325 | Lemma for well-ordered rec... |
| wfrlem4OLD 8326 | Lemma for well-ordered rec... |
| wfrlem5OLD 8327 | Lemma for well-ordered rec... |
| wfrrelOLD 8328 | Obsolete proof of ~ wfrrel... |
| wfrdmssOLD 8329 | Obsolete proof of ~ wfrdms... |
| wfrlem8OLD 8330 | Lemma for well-ordered rec... |
| wfrdmclOLD 8331 | Obsolete version of ~ wfrd... |
| wfrlem10OLD 8332 | Lemma for well-ordered rec... |
| wfrfunOLD 8333 | Obsolete proof of ~ wfrfun... |
| wfrlem12OLD 8334 | Lemma for well-ordered rec... |
| wfrlem13OLD 8335 | Lemma for well-ordered rec... |
| wfrlem14OLD 8336 | Lemma for well-ordered rec... |
| wfrlem15OLD 8337 | Lemma for well-ordered rec... |
| wfrlem16OLD 8338 | Lemma for well-ordered rec... |
| wfrlem17OLD 8339 | Without using ~ ax-rep , s... |
| wfr2aOLD 8340 | Obsolete version of ~ wfr2... |
| wfr1OLD 8341 | Obsolete version of ~ wfr1... |
| wfr2OLD 8342 | Obsolete version of ~ wfr2... |
| wfrrel 8343 | The well-ordered recursion... |
| wfrdmss 8344 | The domain of the well-ord... |
| wfrdmcl 8345 | The predecessor class of a... |
| wfrfun 8346 | The "function" generated b... |
| wfrresex 8347 | Show without using the axi... |
| wfr2a 8348 | A weak version of ~ wfr2 w... |
| wfr1 8349 | The Principle of Well-Orde... |
| wfr2 8350 | The Principle of Well-Orde... |
| wfr3 8351 | The principle of Well-Orde... |
| wfr3OLD 8352 | Obsolete form of ~ wfr3 as... |
| iunon 8353 | The indexed union of a set... |
| iinon 8354 | The nonempty indexed inter... |
| onfununi 8355 | A property of functions on... |
| onovuni 8356 | A variant of ~ onfununi fo... |
| onoviun 8357 | A variant of ~ onovuni wit... |
| onnseq 8358 | There are no length ` _om ... |
| dfsmo2 8361 | Alternate definition of a ... |
| issmo 8362 | Conditions for which ` A `... |
| issmo2 8363 | Alternate definition of a ... |
| smoeq 8364 | Equality theorem for stric... |
| smodm 8365 | The domain of a strictly m... |
| smores 8366 | A strictly monotone functi... |
| smores3 8367 | A strictly monotone functi... |
| smores2 8368 | A strictly monotone ordina... |
| smodm2 8369 | The domain of a strictly m... |
| smofvon2 8370 | The function values of a s... |
| iordsmo 8371 | The identity relation rest... |
| smo0 8372 | The null set is a strictly... |
| smofvon 8373 | If ` B ` is a strictly mon... |
| smoel 8374 | If ` x ` is less than ` y ... |
| smoiun 8375 | The value of a strictly mo... |
| smoiso 8376 | If ` F ` is an isomorphism... |
| smoel2 8377 | A strictly monotone ordina... |
| smo11 8378 | A strictly monotone ordina... |
| smoord 8379 | A strictly monotone ordina... |
| smoword 8380 | A strictly monotone ordina... |
| smogt 8381 | A strictly monotone ordina... |
| smocdmdom 8382 | The codomain of a strictly... |
| smoiso2 8383 | The strictly monotone ordi... |
| dfrecs3 8386 | The old definition of tran... |
| dfrecs3OLD 8387 | Obsolete version of ~ dfre... |
| recseq 8388 | Equality theorem for ` rec... |
| nfrecs 8389 | Bound-variable hypothesis ... |
| tfrlem1 8390 | A technical lemma for tran... |
| tfrlem3a 8391 | Lemma for transfinite recu... |
| tfrlem3 8392 | Lemma for transfinite recu... |
| tfrlem4 8393 | Lemma for transfinite recu... |
| tfrlem5 8394 | Lemma for transfinite recu... |
| recsfval 8395 | Lemma for transfinite recu... |
| tfrlem6 8396 | Lemma for transfinite recu... |
| tfrlem7 8397 | Lemma for transfinite recu... |
| tfrlem8 8398 | Lemma for transfinite recu... |
| tfrlem9 8399 | Lemma for transfinite recu... |
| tfrlem9a 8400 | Lemma for transfinite recu... |
| tfrlem10 8401 | Lemma for transfinite recu... |
| tfrlem11 8402 | Lemma for transfinite recu... |
| tfrlem12 8403 | Lemma for transfinite recu... |
| tfrlem13 8404 | Lemma for transfinite recu... |
| tfrlem14 8405 | Lemma for transfinite recu... |
| tfrlem15 8406 | Lemma for transfinite recu... |
| tfrlem16 8407 | Lemma for finite recursion... |
| tfr1a 8408 | A weak version of ~ tfr1 w... |
| tfr2a 8409 | A weak version of ~ tfr2 w... |
| tfr2b 8410 | Without assuming ~ ax-rep ... |
| tfr1 8411 | Principle of Transfinite R... |
| tfr2 8412 | Principle of Transfinite R... |
| tfr3 8413 | Principle of Transfinite R... |
| tfr1ALT 8414 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8415 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8416 | Alternate proof of ~ tfr3 ... |
| recsfnon 8417 | Strong transfinite recursi... |
| recsval 8418 | Strong transfinite recursi... |
| tz7.44lem1 8419 | The ordered pair abstracti... |
| tz7.44-1 8420 | The value of ` F ` at ` (/... |
| tz7.44-2 8421 | The value of ` F ` at a su... |
| tz7.44-3 8422 | The value of ` F ` at a li... |
| rdgeq1 8425 | Equality theorem for the r... |
| rdgeq2 8426 | Equality theorem for the r... |
| rdgeq12 8427 | Equality theorem for the r... |
| nfrdg 8428 | Bound-variable hypothesis ... |
| rdglem1 8429 | Lemma used with the recurs... |
| rdgfun 8430 | The recursive definition g... |
| rdgdmlim 8431 | The domain of the recursiv... |
| rdgfnon 8432 | The recursive definition g... |
| rdgvalg 8433 | Value of the recursive def... |
| rdgval 8434 | Value of the recursive def... |
| rdg0 8435 | The initial value of the r... |
| rdgseg 8436 | The initial segments of th... |
| rdgsucg 8437 | The value of the recursive... |
| rdgsuc 8438 | The value of the recursive... |
| rdglimg 8439 | The value of the recursive... |
| rdglim 8440 | The value of the recursive... |
| rdg0g 8441 | The initial value of the r... |
| rdgsucmptf 8442 | The value of the recursive... |
| rdgsucmptnf 8443 | The value of the recursive... |
| rdgsucmpt2 8444 | This version of ~ rdgsucmp... |
| rdgsucmpt 8445 | The value of the recursive... |
| rdglim2 8446 | The value of the recursive... |
| rdglim2a 8447 | The value of the recursive... |
| rdg0n 8448 | If ` A ` is a proper class... |
| frfnom 8449 | The function generated by ... |
| fr0g 8450 | The initial value resultin... |
| frsuc 8451 | The successor value result... |
| frsucmpt 8452 | The successor value result... |
| frsucmptn 8453 | The value of the finite re... |
| frsucmpt2 8454 | The successor value result... |
| tz7.48lem 8455 | A way of showing an ordina... |
| tz7.48-2 8456 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8457 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8458 | Proposition 7.48(3) of [Ta... |
| tz7.49 8459 | Proposition 7.49 of [Takeu... |
| tz7.49c 8460 | Corollary of Proposition 7... |
| seqomlem0 8463 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8464 | Lemma for ` seqom ` . The... |
| seqomlem2 8465 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8466 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8467 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8468 | Equality theorem for ` seq... |
| fnseqom 8469 | An index-aware recursive d... |
| seqom0g 8470 | Value of an index-aware re... |
| seqomsuc 8471 | Value of an index-aware re... |
| omsucelsucb 8472 | Membership is inherited by... |
| df1o2 8487 | Expanded value of the ordi... |
| df2o3 8488 | Expanded value of the ordi... |
| df2o2 8489 | Expanded value of the ordi... |
| 1oex 8490 | Ordinal 1 is a set. (Cont... |
| 2oex 8491 | ` 2o ` is a set. (Contrib... |
| 1on 8492 | Ordinal 1 is an ordinal nu... |
| 1onOLD 8493 | Obsolete version of ~ 1on ... |
| 2on 8494 | Ordinal 2 is an ordinal nu... |
| 2onOLD 8495 | Obsolete version of ~ 2on ... |
| 2on0 8496 | Ordinal two is not zero. ... |
| ord3 8497 | Ordinal 3 is an ordinal cl... |
| 3on 8498 | Ordinal 3 is an ordinal nu... |
| 4on 8499 | Ordinal 4 is an ordinal nu... |
| 1oexOLD 8500 | Obsolete version of ~ 1oex... |
| 2oexOLD 8501 | Obsolete version of ~ 2oex... |
| 1n0 8502 | Ordinal one is not equal t... |
| nlim1 8503 | 1 is not a limit ordinal. ... |
| nlim2 8504 | 2 is not a limit ordinal. ... |
| xp01disj 8505 | Cartesian products with th... |
| xp01disjl 8506 | Cartesian products with th... |
| ordgt0ge1 8507 | Two ways to express that a... |
| ordge1n0 8508 | An ordinal greater than or... |
| el1o 8509 | Membership in ordinal one.... |
| ord1eln01 8510 | An ordinal that is not 0 o... |
| ord2eln012 8511 | An ordinal that is not 0, ... |
| 1ellim 8512 | A limit ordinal contains 1... |
| 2ellim 8513 | A limit ordinal contains 2... |
| dif1o 8514 | Two ways to say that ` A `... |
| ondif1 8515 | Two ways to say that ` A `... |
| ondif2 8516 | Two ways to say that ` A `... |
| 2oconcl 8517 | Closure of the pair swappi... |
| 0lt1o 8518 | Ordinal zero is less than ... |
| dif20el 8519 | An ordinal greater than on... |
| 0we1 8520 | The empty set is a well-or... |
| brwitnlem 8521 | Lemma for relations which ... |
| fnoa 8522 | Functionality and domain o... |
| fnom 8523 | Functionality and domain o... |
| fnoe 8524 | Functionality and domain o... |
| oav 8525 | Value of ordinal addition.... |
| omv 8526 | Value of ordinal multiplic... |
| oe0lem 8527 | A helper lemma for ~ oe0 a... |
| oev 8528 | Value of ordinal exponenti... |
| oevn0 8529 | Value of ordinal exponenti... |
| oa0 8530 | Addition with zero. Propo... |
| om0 8531 | Ordinal multiplication wit... |
| oe0m 8532 | Value of zero raised to an... |
| om0x 8533 | Ordinal multiplication wit... |
| oe0m0 8534 | Ordinal exponentiation wit... |
| oe0m1 8535 | Ordinal exponentiation wit... |
| oe0 8536 | Ordinal exponentiation wit... |
| oev2 8537 | Alternate value of ordinal... |
| oasuc 8538 | Addition with successor. ... |
| oesuclem 8539 | Lemma for ~ oesuc . (Cont... |
| omsuc 8540 | Multiplication with succes... |
| oesuc 8541 | Ordinal exponentiation wit... |
| onasuc 8542 | Addition with successor. ... |
| onmsuc 8543 | Multiplication with succes... |
| onesuc 8544 | Exponentiation with a succ... |
| oa1suc 8545 | Addition with 1 is same as... |
| oalim 8546 | Ordinal addition with a li... |
| omlim 8547 | Ordinal multiplication wit... |
| oelim 8548 | Ordinal exponentiation wit... |
| oacl 8549 | Closure law for ordinal ad... |
| omcl 8550 | Closure law for ordinal mu... |
| oecl 8551 | Closure law for ordinal ex... |
| oa0r 8552 | Ordinal addition with zero... |
| om0r 8553 | Ordinal multiplication wit... |
| o1p1e2 8554 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8555 | 2 + 2 = 4 for ordinal numb... |
| om1 8556 | Ordinal multiplication wit... |
| om1r 8557 | Ordinal multiplication wit... |
| oe1 8558 | Ordinal exponentiation wit... |
| oe1m 8559 | Ordinal exponentiation wit... |
| oaordi 8560 | Ordering property of ordin... |
| oaord 8561 | Ordering property of ordin... |
| oacan 8562 | Left cancellation law for ... |
| oaword 8563 | Weak ordering property of ... |
| oawordri 8564 | Weak ordering property of ... |
| oaord1 8565 | An ordinal is less than it... |
| oaword1 8566 | An ordinal is less than or... |
| oaword2 8567 | An ordinal is less than or... |
| oawordeulem 8568 | Lemma for ~ oawordex . (C... |
| oawordeu 8569 | Existence theorem for weak... |
| oawordexr 8570 | Existence theorem for weak... |
| oawordex 8571 | Existence theorem for weak... |
| oaordex 8572 | Existence theorem for orde... |
| oa00 8573 | An ordinal sum is zero iff... |
| oalimcl 8574 | The ordinal sum with a lim... |
| oaass 8575 | Ordinal addition is associ... |
| oarec 8576 | Recursive definition of or... |
| oaf1o 8577 | Left addition by a constan... |
| oacomf1olem 8578 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8579 | Define a bijection from ` ... |
| omordi 8580 | Ordering property of ordin... |
| omord2 8581 | Ordering property of ordin... |
| omord 8582 | Ordering property of ordin... |
| omcan 8583 | Left cancellation law for ... |
| omword 8584 | Weak ordering property of ... |
| omwordi 8585 | Weak ordering property of ... |
| omwordri 8586 | Weak ordering property of ... |
| omword1 8587 | An ordinal is less than or... |
| omword2 8588 | An ordinal is less than or... |
| om00 8589 | The product of two ordinal... |
| om00el 8590 | The product of two nonzero... |
| omordlim 8591 | Ordering involving the pro... |
| omlimcl 8592 | The product of any nonzero... |
| odi 8593 | Distributive law for ordin... |
| omass 8594 | Multiplication of ordinal ... |
| oneo 8595 | If an ordinal number is ev... |
| omeulem1 8596 | Lemma for ~ omeu : existen... |
| omeulem2 8597 | Lemma for ~ omeu : uniquen... |
| omopth2 8598 | An ordered pair-like theor... |
| omeu 8599 | The division algorithm for... |
| oen0 8600 | Ordinal exponentiation wit... |
| oeordi 8601 | Ordering law for ordinal e... |
| oeord 8602 | Ordering property of ordin... |
| oecan 8603 | Left cancellation law for ... |
| oeword 8604 | Weak ordering property of ... |
| oewordi 8605 | Weak ordering property of ... |
| oewordri 8606 | Weak ordering property of ... |
| oeworde 8607 | Ordinal exponentiation com... |
| oeordsuc 8608 | Ordering property of ordin... |
| oelim2 8609 | Ordinal exponentiation wit... |
| oeoalem 8610 | Lemma for ~ oeoa . (Contr... |
| oeoa 8611 | Sum of exponents law for o... |
| oeoelem 8612 | Lemma for ~ oeoe . (Contr... |
| oeoe 8613 | Product of exponents law f... |
| oelimcl 8614 | The ordinal exponential wi... |
| oeeulem 8615 | Lemma for ~ oeeu . (Contr... |
| oeeui 8616 | The division algorithm for... |
| oeeu 8617 | The division algorithm for... |
| nna0 8618 | Addition with zero. Theor... |
| nnm0 8619 | Multiplication with zero. ... |
| nnasuc 8620 | Addition with successor. ... |
| nnmsuc 8621 | Multiplication with succes... |
| nnesuc 8622 | Exponentiation with a succ... |
| nna0r 8623 | Addition to zero. Remark ... |
| nnm0r 8624 | Multiplication with zero. ... |
| nnacl 8625 | Closure of addition of nat... |
| nnmcl 8626 | Closure of multiplication ... |
| nnecl 8627 | Closure of exponentiation ... |
| nnacli 8628 | ` _om ` is closed under ad... |
| nnmcli 8629 | ` _om ` is closed under mu... |
| nnarcl 8630 | Reverse closure law for ad... |
| nnacom 8631 | Addition of natural number... |
| nnaordi 8632 | Ordering property of addit... |
| nnaord 8633 | Ordering property of addit... |
| nnaordr 8634 | Ordering property of addit... |
| nnawordi 8635 | Adding to both sides of an... |
| nnaass 8636 | Addition of natural number... |
| nndi 8637 | Distributive law for natur... |
| nnmass 8638 | Multiplication of natural ... |
| nnmsucr 8639 | Multiplication with succes... |
| nnmcom 8640 | Multiplication of natural ... |
| nnaword 8641 | Weak ordering property of ... |
| nnacan 8642 | Cancellation law for addit... |
| nnaword1 8643 | Weak ordering property of ... |
| nnaword2 8644 | Weak ordering property of ... |
| nnmordi 8645 | Ordering property of multi... |
| nnmord 8646 | Ordering property of multi... |
| nnmword 8647 | Weak ordering property of ... |
| nnmcan 8648 | Cancellation law for multi... |
| nnmwordi 8649 | Weak ordering property of ... |
| nnmwordri 8650 | Weak ordering property of ... |
| nnawordex 8651 | Equivalence for weak order... |
| nnaordex 8652 | Equivalence for ordering. ... |
| nnaordex2 8653 | Equivalence for ordering. ... |
| 1onn 8654 | The ordinal 1 is a natural... |
| 1onnALT 8655 | Shorter proof of ~ 1onn us... |
| 2onn 8656 | The ordinal 2 is a natural... |
| 2onnALT 8657 | Shorter proof of ~ 2onn us... |
| 3onn 8658 | The ordinal 3 is a natural... |
| 4onn 8659 | The ordinal 4 is a natural... |
| 1one2o 8660 | Ordinal one is not ordinal... |
| oaabslem 8661 | Lemma for ~ oaabs . (Cont... |
| oaabs 8662 | Ordinal addition absorbs a... |
| oaabs2 8663 | The absorption law ~ oaabs... |
| omabslem 8664 | Lemma for ~ omabs . (Cont... |
| omabs 8665 | Ordinal multiplication is ... |
| nnm1 8666 | Multiply an element of ` _... |
| nnm2 8667 | Multiply an element of ` _... |
| nn2m 8668 | Multiply an element of ` _... |
| nnneo 8669 | If a natural number is eve... |
| nneob 8670 | A natural number is even i... |
| omsmolem 8671 | Lemma for ~ omsmo . (Cont... |
| omsmo 8672 | A strictly monotonic ordin... |
| omopthlem1 8673 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8674 | Lemma for ~ omopthi . (Co... |
| omopthi 8675 | An ordered pair theorem fo... |
| omopth 8676 | An ordered pair theorem fo... |
| nnasmo 8677 | There is at most one left ... |
| eldifsucnn 8678 | Condition for membership i... |
| on2recsfn 8681 | Show that double recursion... |
| on2recsov 8682 | Calculate the value of the... |
| on2ind 8683 | Double induction over ordi... |
| on3ind 8684 | Triple induction over ordi... |
| coflton 8685 | Cofinality theorem for ord... |
| cofon1 8686 | Cofinality theorem for ord... |
| cofon2 8687 | Cofinality theorem for ord... |
| cofonr 8688 | Inverse cofinality law for... |
| naddfn 8689 | Natural addition is a func... |
| naddcllem 8690 | Lemma for ordinal addition... |
| naddcl 8691 | Closure law for natural ad... |
| naddov 8692 | The value of natural addit... |
| naddov2 8693 | Alternate expression for n... |
| naddov3 8694 | Alternate expression for n... |
| naddf 8695 | Function statement for nat... |
| naddcom 8696 | Natural addition commutes.... |
| naddrid 8697 | Ordinal zero is the additi... |
| naddlid 8698 | Ordinal zero is the additi... |
| naddssim 8699 | Ordinal less-than-or-equal... |
| naddelim 8700 | Ordinal less-than is prese... |
| naddel1 8701 | Ordinal less-than is not a... |
| naddel2 8702 | Ordinal less-than is not a... |
| naddss1 8703 | Ordinal less-than-or-equal... |
| naddss2 8704 | Ordinal less-than-or-equal... |
| naddword1 8705 | Weak-ordering principle fo... |
| naddword2 8706 | Weak-ordering principle fo... |
| naddunif 8707 | Uniformity theorem for nat... |
| naddasslem1 8708 | Lemma for ~ naddass . Exp... |
| naddasslem2 8709 | Lemma for ~ naddass . Exp... |
| naddass 8710 | Natural ordinal addition i... |
| nadd32 8711 | Commutative/associative la... |
| nadd4 8712 | Rearragement of terms in a... |
| nadd42 8713 | Rearragement of terms in a... |
| naddel12 8714 | Natural addition to both s... |
| dfer2 8719 | Alternate definition of eq... |
| dfec2 8721 | Alternate definition of ` ... |
| ecexg 8722 | An equivalence class modul... |
| ecexr 8723 | A nonempty equivalence cla... |
| ereq1 8725 | Equality theorem for equiv... |
| ereq2 8726 | Equality theorem for equiv... |
| errel 8727 | An equivalence relation is... |
| erdm 8728 | The domain of an equivalen... |
| ercl 8729 | Elementhood in the field o... |
| ersym 8730 | An equivalence relation is... |
| ercl2 8731 | Elementhood in the field o... |
| ersymb 8732 | An equivalence relation is... |
| ertr 8733 | An equivalence relation is... |
| ertrd 8734 | A transitivity relation fo... |
| ertr2d 8735 | A transitivity relation fo... |
| ertr3d 8736 | A transitivity relation fo... |
| ertr4d 8737 | A transitivity relation fo... |
| erref 8738 | An equivalence relation is... |
| ercnv 8739 | The converse of an equival... |
| errn 8740 | The range and domain of an... |
| erssxp 8741 | An equivalence relation is... |
| erex 8742 | An equivalence relation is... |
| erexb 8743 | An equivalence relation is... |
| iserd 8744 | A reflexive, symmetric, tr... |
| iseri 8745 | A reflexive, symmetric, tr... |
| iseriALT 8746 | Alternate proof of ~ iseri... |
| brdifun 8747 | Evaluate the incomparabili... |
| swoer 8748 | Incomparability under a st... |
| swoord1 8749 | The incomparability equiva... |
| swoord2 8750 | The incomparability equiva... |
| swoso 8751 | If the incomparability rel... |
| eqerlem 8752 | Lemma for ~ eqer . (Contr... |
| eqer 8753 | Equivalence relation invol... |
| ider 8754 | The identity relation is a... |
| 0er 8755 | The empty set is an equiva... |
| eceq1 8756 | Equality theorem for equiv... |
| eceq1d 8757 | Equality theorem for equiv... |
| eceq2 8758 | Equality theorem for equiv... |
| eceq2i 8759 | Equality theorem for the `... |
| eceq2d 8760 | Equality theorem for the `... |
| elecg 8761 | Membership in an equivalen... |
| ecref 8762 | All elements are in their ... |
| elec 8763 | Membership in an equivalen... |
| relelec 8764 | Membership in an equivalen... |
| ecss 8765 | An equivalence class is a ... |
| ecdmn0 8766 | A representative of a none... |
| ereldm 8767 | Equality of equivalence cl... |
| erth 8768 | Basic property of equivale... |
| erth2 8769 | Basic property of equivale... |
| erthi 8770 | Basic property of equivale... |
| erdisj 8771 | Equivalence classes do not... |
| ecidsn 8772 | An equivalence class modul... |
| qseq1 8773 | Equality theorem for quoti... |
| qseq2 8774 | Equality theorem for quoti... |
| qseq2i 8775 | Equality theorem for quoti... |
| qseq2d 8776 | Equality theorem for quoti... |
| qseq12 8777 | Equality theorem for quoti... |
| elqsg 8778 | Closed form of ~ elqs . (... |
| elqs 8779 | Membership in a quotient s... |
| elqsi 8780 | Membership in a quotient s... |
| elqsecl 8781 | Membership in a quotient s... |
| ecelqsg 8782 | Membership of an equivalen... |
| ecelqsi 8783 | Membership of an equivalen... |
| ecopqsi 8784 | "Closure" law for equivale... |
| qsexg 8785 | A quotient set exists. (C... |
| qsex 8786 | A quotient set exists. (C... |
| uniqs 8787 | The union of a quotient se... |
| qsss 8788 | A quotient set is a set of... |
| uniqs2 8789 | The union of a quotient se... |
| snec 8790 | The singleton of an equiva... |
| ecqs 8791 | Equivalence class in terms... |
| ecid 8792 | A set is equal to its cose... |
| qsid 8793 | A set is equal to its quot... |
| ectocld 8794 | Implicit substitution of c... |
| ectocl 8795 | Implicit substitution of c... |
| elqsn0 8796 | A quotient set does not co... |
| ecelqsdm 8797 | Membership of an equivalen... |
| xpider 8798 | A Cartesian square is an e... |
| iiner 8799 | The intersection of a none... |
| riiner 8800 | The relative intersection ... |
| erinxp 8801 | A restricted equivalence r... |
| ecinxp 8802 | Restrict the relation in a... |
| qsinxp 8803 | Restrict the equivalence r... |
| qsdisj 8804 | Members of a quotient set ... |
| qsdisj2 8805 | A quotient set is a disjoi... |
| qsel 8806 | If an element of a quotien... |
| uniinqs 8807 | Class union distributes ov... |
| qliftlem 8808 | Lemma for theorems about a... |
| qliftrel 8809 | ` F ` , a function lift, i... |
| qliftel 8810 | Elementhood in the relatio... |
| qliftel1 8811 | Elementhood in the relatio... |
| qliftfun 8812 | The function ` F ` is the ... |
| qliftfund 8813 | The function ` F ` is the ... |
| qliftfuns 8814 | The function ` F ` is the ... |
| qliftf 8815 | The domain and codomain of... |
| qliftval 8816 | The value of the function ... |
| ecoptocl 8817 | Implicit substitution of c... |
| 2ecoptocl 8818 | Implicit substitution of c... |
| 3ecoptocl 8819 | Implicit substitution of c... |
| brecop 8820 | Binary relation on a quoti... |
| brecop2 8821 | Binary relation on a quoti... |
| eroveu 8822 | Lemma for ~ erov and ~ ero... |
| erovlem 8823 | Lemma for ~ erov and ~ ero... |
| erov 8824 | The value of an operation ... |
| eroprf 8825 | Functionality of an operat... |
| erov2 8826 | The value of an operation ... |
| eroprf2 8827 | Functionality of an operat... |
| ecopoveq 8828 | This is the first of sever... |
| ecopovsym 8829 | Assuming the operation ` F... |
| ecopovtrn 8830 | Assuming that operation ` ... |
| ecopover 8831 | Assuming that operation ` ... |
| eceqoveq 8832 | Equality of equivalence re... |
| ecovcom 8833 | Lemma used to transfer a c... |
| ecovass 8834 | Lemma used to transfer an ... |
| ecovdi 8835 | Lemma used to transfer a d... |
| mapprc 8840 | When ` A ` is a proper cla... |
| pmex 8841 | The class of all partial f... |
| mapex 8842 | The class of all functions... |
| fnmap 8843 | Set exponentiation has a u... |
| fnpm 8844 | Partial function exponenti... |
| reldmmap 8845 | Set exponentiation is a we... |
| mapvalg 8846 | The value of set exponenti... |
| pmvalg 8847 | The value of the partial m... |
| mapval 8848 | The value of set exponenti... |
| elmapg 8849 | Membership relation for se... |
| elmapd 8850 | Deduction form of ~ elmapg... |
| elmapdd 8851 | Deduction associated with ... |
| mapdm0 8852 | The empty set is the only ... |
| elpmg 8853 | The predicate "is a partia... |
| elpm2g 8854 | The predicate "is a partia... |
| elpm2r 8855 | Sufficient condition for b... |
| elpmi 8856 | A partial function is a fu... |
| pmfun 8857 | A partial function is a fu... |
| elmapex 8858 | Eliminate antecedent for m... |
| elmapi 8859 | A mapping is a function, f... |
| mapfset 8860 | If ` B ` is a set, the val... |
| mapssfset 8861 | The value of the set expon... |
| mapfoss 8862 | The value of the set expon... |
| fsetsspwxp 8863 | The class of all functions... |
| fset0 8864 | The set of functions from ... |
| fsetdmprc0 8865 | The set of functions with ... |
| fsetex 8866 | The set of functions betwe... |
| f1setex 8867 | The set of injections betw... |
| fosetex 8868 | The set of surjections bet... |
| f1osetex 8869 | The set of bijections betw... |
| fsetfcdm 8870 | The class of functions wit... |
| fsetfocdm 8871 | The class of functions wit... |
| fsetprcnex 8872 | The class of all functions... |
| fsetcdmex 8873 | The class of all functions... |
| fsetexb 8874 | The class of all functions... |
| elmapfn 8875 | A mapping is a function wi... |
| elmapfun 8876 | A mapping is always a func... |
| elmapssres 8877 | A restricted mapping is a ... |
| fpmg 8878 | A total function is a part... |
| pmss12g 8879 | Subset relation for the se... |
| pmresg 8880 | Elementhood of a restricte... |
| elmap 8881 | Membership relation for se... |
| mapval2 8882 | Alternate expression for t... |
| elpm 8883 | The predicate "is a partia... |
| elpm2 8884 | The predicate "is a partia... |
| fpm 8885 | A total function is a part... |
| mapsspm 8886 | Set exponentiation is a su... |
| pmsspw 8887 | Partial maps are a subset ... |
| mapsspw 8888 | Set exponentiation is a su... |
| mapfvd 8889 | The value of a function th... |
| elmapresaun 8890 | ~ fresaun transposed to ma... |
| fvmptmap 8891 | Special case of ~ fvmpt fo... |
| map0e 8892 | Set exponentiation with an... |
| map0b 8893 | Set exponentiation with an... |
| map0g 8894 | Set exponentiation is empt... |
| 0map0sn0 8895 | The set of mappings of the... |
| mapsnd 8896 | The value of set exponenti... |
| map0 8897 | Set exponentiation is empt... |
| mapsn 8898 | The value of set exponenti... |
| mapss 8899 | Subset inheritance for set... |
| fdiagfn 8900 | Functionality of the diago... |
| fvdiagfn 8901 | Functionality of the diago... |
| mapsnconst 8902 | Every singleton map is a c... |
| mapsncnv 8903 | Expression for the inverse... |
| mapsnf1o2 8904 | Explicit bijection between... |
| mapsnf1o3 8905 | Explicit bijection in the ... |
| ralxpmap 8906 | Quantification over functi... |
| dfixp 8909 | Eliminate the expression `... |
| ixpsnval 8910 | The value of an infinite C... |
| elixp2 8911 | Membership in an infinite ... |
| fvixp 8912 | Projection of a factor of ... |
| ixpfn 8913 | A nuple is a function. (C... |
| elixp 8914 | Membership in an infinite ... |
| elixpconst 8915 | Membership in an infinite ... |
| ixpconstg 8916 | Infinite Cartesian product... |
| ixpconst 8917 | Infinite Cartesian product... |
| ixpeq1 8918 | Equality theorem for infin... |
| ixpeq1d 8919 | Equality theorem for infin... |
| ss2ixp 8920 | Subclass theorem for infin... |
| ixpeq2 8921 | Equality theorem for infin... |
| ixpeq2dva 8922 | Equality theorem for infin... |
| ixpeq2dv 8923 | Equality theorem for infin... |
| cbvixp 8924 | Change bound variable in a... |
| cbvixpv 8925 | Change bound variable in a... |
| nfixpw 8926 | Bound-variable hypothesis ... |
| nfixp 8927 | Bound-variable hypothesis ... |
| nfixp1 8928 | The index variable in an i... |
| ixpprc 8929 | A cartesian product of pro... |
| ixpf 8930 | A member of an infinite Ca... |
| uniixp 8931 | The union of an infinite C... |
| ixpexg 8932 | The existence of an infini... |
| ixpin 8933 | The intersection of two in... |
| ixpiin 8934 | The indexed intersection o... |
| ixpint 8935 | The intersection of a coll... |
| ixp0x 8936 | An infinite Cartesian prod... |
| ixpssmap2g 8937 | An infinite Cartesian prod... |
| ixpssmapg 8938 | An infinite Cartesian prod... |
| 0elixp 8939 | Membership of the empty se... |
| ixpn0 8940 | The infinite Cartesian pro... |
| ixp0 8941 | The infinite Cartesian pro... |
| ixpssmap 8942 | An infinite Cartesian prod... |
| resixp 8943 | Restriction of an element ... |
| undifixp 8944 | Union of two projections o... |
| mptelixpg 8945 | Condition for an explicit ... |
| resixpfo 8946 | Restriction of elements of... |
| elixpsn 8947 | Membership in a class of s... |
| ixpsnf1o 8948 | A bijection between a clas... |
| mapsnf1o 8949 | A bijection between a set ... |
| boxriin 8950 | A rectangular subset of a ... |
| boxcutc 8951 | The relative complement of... |
| relen 8960 | Equinumerosity is a relati... |
| reldom 8961 | Dominance is a relation. ... |
| relsdom 8962 | Strict dominance is a rela... |
| encv 8963 | If two classes are equinum... |
| breng 8964 | Equinumerosity relation. ... |
| bren 8965 | Equinumerosity relation. ... |
| brenOLD 8966 | Obsolete version of ~ bren... |
| brdom2g 8967 | Dominance relation. This ... |
| brdomg 8968 | Dominance relation. (Cont... |
| brdomgOLD 8969 | Obsolete version of ~ brdo... |
| brdomi 8970 | Dominance relation. (Cont... |
| brdomiOLD 8971 | Obsolete version of ~ brdo... |
| brdom 8972 | Dominance relation. (Cont... |
| domen 8973 | Dominance in terms of equi... |
| domeng 8974 | Dominance in terms of equi... |
| ctex 8975 | A countable set is a set. ... |
| f1oen4g 8976 | The domain and range of a ... |
| f1dom4g 8977 | The domain of a one-to-one... |
| f1oen3g 8978 | The domain and range of a ... |
| f1dom3g 8979 | The domain of a one-to-one... |
| f1oen2g 8980 | The domain and range of a ... |
| f1dom2g 8981 | The domain of a one-to-one... |
| f1dom2gOLD 8982 | Obsolete version of ~ f1do... |
| f1oeng 8983 | The domain and range of a ... |
| f1domg 8984 | The domain of a one-to-one... |
| f1oen 8985 | The domain and range of a ... |
| f1dom 8986 | The domain of a one-to-one... |
| brsdom 8987 | Strict dominance relation,... |
| isfi 8988 | Express " ` A ` is finite"... |
| enssdom 8989 | Equinumerosity implies dom... |
| dfdom2 8990 | Alternate definition of do... |
| endom 8991 | Equinumerosity implies dom... |
| sdomdom 8992 | Strict dominance implies d... |
| sdomnen 8993 | Strict dominance implies n... |
| brdom2 8994 | Dominance in terms of stri... |
| bren2 8995 | Equinumerosity expressed i... |
| enrefg 8996 | Equinumerosity is reflexiv... |
| enref 8997 | Equinumerosity is reflexiv... |
| eqeng 8998 | Equality implies equinumer... |
| domrefg 8999 | Dominance is reflexive. (... |
| en2d 9000 | Equinumerosity inference f... |
| en3d 9001 | Equinumerosity inference f... |
| en2i 9002 | Equinumerosity inference f... |
| en3i 9003 | Equinumerosity inference f... |
| dom2lem 9004 | A mapping (first hypothesi... |
| dom2d 9005 | A mapping (first hypothesi... |
| dom3d 9006 | A mapping (first hypothesi... |
| dom2 9007 | A mapping (first hypothesi... |
| dom3 9008 | A mapping (first hypothesi... |
| idssen 9009 | Equality implies equinumer... |
| domssl 9010 | If ` A ` is a subset of ` ... |
| domssr 9011 | If ` C ` is a superset of ... |
| ssdomg 9012 | A set dominates its subset... |
| ener 9013 | Equinumerosity is an equiv... |
| ensymb 9014 | Symmetry of equinumerosity... |
| ensym 9015 | Symmetry of equinumerosity... |
| ensymi 9016 | Symmetry of equinumerosity... |
| ensymd 9017 | Symmetry of equinumerosity... |
| entr 9018 | Transitivity of equinumero... |
| domtr 9019 | Transitivity of dominance ... |
| entri 9020 | A chained equinumerosity i... |
| entr2i 9021 | A chained equinumerosity i... |
| entr3i 9022 | A chained equinumerosity i... |
| entr4i 9023 | A chained equinumerosity i... |
| endomtr 9024 | Transitivity of equinumero... |
| domentr 9025 | Transitivity of dominance ... |
| f1imaeng 9026 | If a function is one-to-on... |
| f1imaen2g 9027 | If a function is one-to-on... |
| f1imaen 9028 | If a function is one-to-on... |
| en0 9029 | The empty set is equinumer... |
| en0OLD 9030 | Obsolete version of ~ en0 ... |
| en0ALT 9031 | Shorter proof of ~ en0 , d... |
| en0r 9032 | The empty set is equinumer... |
| ensn1 9033 | A singleton is equinumerou... |
| ensn1OLD 9034 | Obsolete version of ~ ensn... |
| ensn1g 9035 | A singleton is equinumerou... |
| enpr1g 9036 | ` { A , A } ` has only one... |
| en1 9037 | A set is equinumerous to o... |
| en1OLD 9038 | Obsolete version of ~ en1 ... |
| en1b 9039 | A set is equinumerous to o... |
| en1bOLD 9040 | Obsolete version of ~ en1b... |
| reuen1 9041 | Two ways to express "exact... |
| euen1 9042 | Two ways to express "exact... |
| euen1b 9043 | Two ways to express " ` A ... |
| en1uniel 9044 | A singleton contains its s... |
| en1unielOLD 9045 | Obsolete version of ~ en1u... |
| 2dom 9046 | A set that dominates ordin... |
| fundmen 9047 | A function is equinumerous... |
| fundmeng 9048 | A function is equinumerous... |
| cnven 9049 | A relational set is equinu... |
| cnvct 9050 | If a set is countable, so ... |
| fndmeng 9051 | A function is equinumerate... |
| mapsnend 9052 | Set exponentiation to a si... |
| mapsnen 9053 | Set exponentiation to a si... |
| snmapen 9054 | Set exponentiation: a sing... |
| snmapen1 9055 | Set exponentiation: a sing... |
| map1 9056 | Set exponentiation: ordina... |
| en2sn 9057 | Two singletons are equinum... |
| en2snOLD 9058 | Obsolete version of ~ en2s... |
| en2snOLDOLD 9059 | Obsolete version of ~ en2s... |
| snfi 9060 | A singleton is finite. (C... |
| fiprc 9061 | The class of finite sets i... |
| unen 9062 | Equinumerosity of union of... |
| enrefnn 9063 | Equinumerosity is reflexiv... |
| en2prd 9064 | Two unordered pairs are eq... |
| enpr2d 9065 | A pair with distinct eleme... |
| enpr2dOLD 9066 | Obsolete version of ~ enpr... |
| ssct 9067 | Any subset of a countable ... |
| ssctOLD 9068 | Obsolete version of ~ ssct... |
| difsnen 9069 | All decrements of a set ar... |
| domdifsn 9070 | Dominance over a set with ... |
| xpsnen 9071 | A set is equinumerous to i... |
| xpsneng 9072 | A set is equinumerous to i... |
| xp1en 9073 | One times a cardinal numbe... |
| endisj 9074 | Any two sets are equinumer... |
| undom 9075 | Dominance law for union. ... |
| undomOLD 9076 | Obsolete version of ~ undo... |
| xpcomf1o 9077 | The canonical bijection fr... |
| xpcomco 9078 | Composition with the bijec... |
| xpcomen 9079 | Commutative law for equinu... |
| xpcomeng 9080 | Commutative law for equinu... |
| xpsnen2g 9081 | A set is equinumerous to i... |
| xpassen 9082 | Associative law for equinu... |
| xpdom2 9083 | Dominance law for Cartesia... |
| xpdom2g 9084 | Dominance law for Cartesia... |
| xpdom1g 9085 | Dominance law for Cartesia... |
| xpdom3 9086 | A set is dominated by its ... |
| xpdom1 9087 | Dominance law for Cartesia... |
| domunsncan 9088 | A singleton cancellation l... |
| omxpenlem 9089 | Lemma for ~ omxpen . (Con... |
| omxpen 9090 | The cardinal and ordinal p... |
| omf1o 9091 | Construct an explicit bije... |
| pw2f1olem 9092 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9093 | The power set of a set is ... |
| pw2eng 9094 | The power set of a set is ... |
| pw2en 9095 | The power set of a set is ... |
| fopwdom 9096 | Covering implies injection... |
| enfixsn 9097 | Given two equipollent sets... |
| sucdom2OLD 9098 | Obsolete version of ~ sucd... |
| sbthlem1 9099 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9100 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9101 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9102 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9103 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9104 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9105 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9106 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9107 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9108 | Lemma for ~ sbth . (Contr... |
| sbth 9109 | Schroeder-Bernstein Theore... |
| sbthb 9110 | Schroeder-Bernstein Theore... |
| sbthcl 9111 | Schroeder-Bernstein Theore... |
| dfsdom2 9112 | Alternate definition of st... |
| brsdom2 9113 | Alternate definition of st... |
| sdomnsym 9114 | Strict dominance is asymme... |
| domnsym 9115 | Theorem 22(i) of [Suppes] ... |
| 0domg 9116 | Any set dominates the empt... |
| 0domgOLD 9117 | Obsolete version of ~ 0dom... |
| dom0 9118 | A set dominated by the emp... |
| dom0OLD 9119 | Obsolete version of ~ dom0... |
| 0sdomg 9120 | A set strictly dominates t... |
| 0sdomgOLD 9121 | Obsolete version of ~ 0sdo... |
| 0dom 9122 | Any set dominates the empt... |
| 0sdom 9123 | A set strictly dominates t... |
| sdom0 9124 | The empty set does not str... |
| sdom0OLD 9125 | Obsolete version of ~ sdom... |
| sdomdomtr 9126 | Transitivity of strict dom... |
| sdomentr 9127 | Transitivity of strict dom... |
| domsdomtr 9128 | Transitivity of dominance ... |
| ensdomtr 9129 | Transitivity of equinumero... |
| sdomirr 9130 | Strict dominance is irrefl... |
| sdomtr 9131 | Strict dominance is transi... |
| sdomn2lp 9132 | Strict dominance has no 2-... |
| enen1 9133 | Equality-like theorem for ... |
| enen2 9134 | Equality-like theorem for ... |
| domen1 9135 | Equality-like theorem for ... |
| domen2 9136 | Equality-like theorem for ... |
| sdomen1 9137 | Equality-like theorem for ... |
| sdomen2 9138 | Equality-like theorem for ... |
| domtriord 9139 | Dominance is trichotomous ... |
| sdomel 9140 | For ordinals, strict domin... |
| sdomdif 9141 | The difference of a set fr... |
| onsdominel 9142 | An ordinal with more eleme... |
| domunsn 9143 | Dominance over a set with ... |
| fodomr 9144 | There exists a mapping fro... |
| pwdom 9145 | Injection of sets implies ... |
| canth2 9146 | Cantor's Theorem. No set ... |
| canth2g 9147 | Cantor's theorem with the ... |
| 2pwuninel 9148 | The power set of the power... |
| 2pwne 9149 | No set equals the power se... |
| disjen 9150 | A stronger form of ~ pwuni... |
| disjenex 9151 | Existence version of ~ dis... |
| domss2 9152 | A corollary of ~ disjenex ... |
| domssex2 9153 | A corollary of ~ disjenex ... |
| domssex 9154 | Weakening of ~ domssex2 to... |
| xpf1o 9155 | Construct a bijection on a... |
| xpen 9156 | Equinumerosity law for Car... |
| mapen 9157 | Two set exponentiations ar... |
| mapdom1 9158 | Order-preserving property ... |
| mapxpen 9159 | Equinumerosity law for dou... |
| xpmapenlem 9160 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9161 | Equinumerosity law for set... |
| mapunen 9162 | Equinumerosity law for set... |
| map2xp 9163 | A cardinal power with expo... |
| mapdom2 9164 | Order-preserving property ... |
| mapdom3 9165 | Set exponentiation dominat... |
| pwen 9166 | If two sets are equinumero... |
| ssenen 9167 | Equinumerosity of equinume... |
| limenpsi 9168 | A limit ordinal is equinum... |
| limensuci 9169 | A limit ordinal is equinum... |
| limensuc 9170 | A limit ordinal is equinum... |
| infensuc 9171 | Any infinite ordinal is eq... |
| dif1enlem 9172 | Lemma for ~ rexdif1en and ... |
| dif1enlemOLD 9173 | Obsolete version of ~ dif1... |
| rexdif1en 9174 | If a set is equinumerous t... |
| rexdif1enOLD 9175 | Obsolete version of ~ rexd... |
| dif1en 9176 | If a set ` A ` is equinume... |
| dif1ennn 9177 | If a set ` A ` is equinume... |
| dif1enOLD 9178 | Obsolete version of ~ dif1... |
| findcard 9179 | Schema for induction on th... |
| findcard2 9180 | Schema for induction on th... |
| findcard2s 9181 | Variation of ~ findcard2 r... |
| findcard2d 9182 | Deduction version of ~ fin... |
| nnfi 9183 | Natural numbers are finite... |
| pssnn 9184 | A proper subset of a natur... |
| ssnnfi 9185 | A subset of a natural numb... |
| ssnnfiOLD 9186 | Obsolete version of ~ ssnn... |
| 0fin 9187 | The empty set is finite. ... |
| unfi 9188 | The union of two finite se... |
| ssfi 9189 | A subset of a finite set i... |
| ssfiALT 9190 | Shorter proof of ~ ssfi us... |
| imafi 9191 | Images of finite sets are ... |
| pwfir 9192 | If the power set of a set ... |
| pwfilem 9193 | Lemma for ~ pwfi . (Contr... |
| pwfi 9194 | The power set of a finite ... |
| diffi 9195 | If ` A ` is finite, ` ( A ... |
| cnvfi 9196 | If a set is finite, its co... |
| fnfi 9197 | A version of ~ fnex for fi... |
| f1oenfi 9198 | If the domain of a one-to-... |
| f1oenfirn 9199 | If the range of a one-to-o... |
| f1domfi 9200 | If the codomain of a one-t... |
| f1domfi2 9201 | If the domain of a one-to-... |
| enreffi 9202 | Equinumerosity is reflexiv... |
| ensymfib 9203 | Symmetry of equinumerosity... |
| entrfil 9204 | Transitivity of equinumero... |
| enfii 9205 | A set equinumerous to a fi... |
| enfi 9206 | Equinumerous sets have the... |
| enfiALT 9207 | Shorter proof of ~ enfi us... |
| domfi 9208 | A set dominated by a finit... |
| entrfi 9209 | Transitivity of equinumero... |
| entrfir 9210 | Transitivity of equinumero... |
| domtrfil 9211 | Transitivity of dominance ... |
| domtrfi 9212 | Transitivity of dominance ... |
| domtrfir 9213 | Transitivity of dominance ... |
| f1imaenfi 9214 | If a function is one-to-on... |
| ssdomfi 9215 | A finite set dominates its... |
| ssdomfi2 9216 | A set dominates its finite... |
| sbthfilem 9217 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9218 | Schroeder-Bernstein Theore... |
| domnsymfi 9219 | If a set dominates a finit... |
| sdomdomtrfi 9220 | Transitivity of strict dom... |
| domsdomtrfi 9221 | Transitivity of dominance ... |
| sucdom2 9222 | Strict dominance of a set ... |
| phplem1 9223 | Lemma for Pigeonhole Princ... |
| phplem2 9224 | Lemma for Pigeonhole Princ... |
| nneneq 9225 | Two equinumerous natural n... |
| php 9226 | Pigeonhole Principle. A n... |
| php2 9227 | Corollary of Pigeonhole Pr... |
| php3 9228 | Corollary of Pigeonhole Pr... |
| php4 9229 | Corollary of the Pigeonhol... |
| php5 9230 | Corollary of the Pigeonhol... |
| phpeqd 9231 | Corollary of the Pigeonhol... |
| nndomog 9232 | Cardinal ordering agrees w... |
| phplem1OLD 9233 | Obsolete lemma for ~ php a... |
| phplem2OLD 9234 | Obsolete lemma for ~ php a... |
| phplem3OLD 9235 | Obsolete version of ~ phpl... |
| phplem4OLD 9236 | Obsolete version of ~ phpl... |
| nneneqOLD 9237 | Obsolete version of ~ nnen... |
| phpOLD 9238 | Obsolete version of ~ php ... |
| php2OLD 9239 | Obsolete version of ~ php2... |
| php3OLD 9240 | Obsolete version of ~ php3... |
| phpeqdOLD 9241 | Obsolete version of ~ phpe... |
| nndomogOLD 9242 | Obsolete version of ~ nndo... |
| snnen2oOLD 9243 | Obsolete version of ~ snne... |
| onomeneq 9244 | An ordinal number equinume... |
| onomeneqOLD 9245 | Obsolete version of ~ onom... |
| onfin 9246 | An ordinal number is finit... |
| onfin2 9247 | A set is a natural number ... |
| nnfiOLD 9248 | Obsolete version of ~ nnfi... |
| nndomo 9249 | Cardinal ordering agrees w... |
| nnsdomo 9250 | Cardinal ordering agrees w... |
| sucdom 9251 | Strict dominance of a set ... |
| sucdomOLD 9252 | Obsolete version of ~ sucd... |
| snnen2o 9253 | A singleton ` { A } ` is n... |
| 0sdom1dom 9254 | Strict dominance over 0 is... |
| 0sdom1domALT 9255 | Alternate proof of ~ 0sdom... |
| 1sdom2 9256 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9257 | Alternate proof of ~ 1sdom... |
| sdom1 9258 | A set has less than one me... |
| sdom1OLD 9259 | Obsolete version of ~ sdom... |
| modom 9260 | Two ways to express "at mo... |
| modom2 9261 | Two ways to express "at mo... |
| rex2dom 9262 | A set that has at least 2 ... |
| 1sdom2dom 9263 | Strict dominance over 1 is... |
| 1sdom 9264 | A set that strictly domina... |
| 1sdomOLD 9265 | Obsolete version of ~ 1sdo... |
| unxpdomlem1 9266 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9267 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9268 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9269 | Cartesian product dominate... |
| unxpdom2 9270 | Corollary of ~ unxpdom . ... |
| sucxpdom 9271 | Cartesian product dominate... |
| pssinf 9272 | A set equinumerous to a pr... |
| fisseneq 9273 | A finite set is equal to i... |
| ominf 9274 | The set of natural numbers... |
| ominfOLD 9275 | Obsolete version of ~ omin... |
| isinf 9276 | Any set that is not finite... |
| isinfOLD 9277 | Obsolete version of ~ isin... |
| fineqvlem 9278 | Lemma for ~ fineqv . (Con... |
| fineqv 9279 | If the Axiom of Infinity i... |
| enfiiOLD 9280 | Obsolete version of ~ enfi... |
| pssnnOLD 9281 | Obsolete version of ~ pssn... |
| xpfir 9282 | The components of a nonemp... |
| ssfid 9283 | A subset of a finite set i... |
| infi 9284 | The intersection of two se... |
| rabfi 9285 | A restricted class built f... |
| finresfin 9286 | The restriction of a finit... |
| f1finf1o 9287 | Any injection from one fin... |
| f1finf1oOLD 9288 | Obsolete version of ~ f1fi... |
| nfielex 9289 | If a class is not finite, ... |
| en1eqsn 9290 | A set with one element is ... |
| en1eqsnOLD 9291 | Obsolete version of ~ en1e... |
| en1eqsnbi 9292 | A set containing an elemen... |
| dif1ennnALT 9293 | Alternate proof of ~ dif1e... |
| enp1ilem 9294 | Lemma for uses of ~ enp1i ... |
| enp1i 9295 | Proof induction for ~ en2 ... |
| enp1iOLD 9296 | Obsolete version of ~ enp1... |
| en2 9297 | A set equinumerous to ordi... |
| en3 9298 | A set equinumerous to ordi... |
| en4 9299 | A set equinumerous to ordi... |
| findcard2OLD 9300 | Obsolete version of ~ find... |
| findcard3 9301 | Schema for strong inductio... |
| findcard3OLD 9302 | Obsolete version of ~ find... |
| ac6sfi 9303 | A version of ~ ac6s for fi... |
| frfi 9304 | A partial order is well-fo... |
| fimax2g 9305 | A finite set has a maximum... |
| fimaxg 9306 | A finite set has a maximum... |
| fisupg 9307 | Lemma showing existence an... |
| wofi 9308 | A total order on a finite ... |
| ordunifi 9309 | The maximum of a finite co... |
| nnunifi 9310 | The union (supremum) of a ... |
| unblem1 9311 | Lemma for ~ unbnn . After... |
| unblem2 9312 | Lemma for ~ unbnn . The v... |
| unblem3 9313 | Lemma for ~ unbnn . The v... |
| unblem4 9314 | Lemma for ~ unbnn . The f... |
| unbnn 9315 | Any unbounded subset of na... |
| unbnn2 9316 | Version of ~ unbnn that do... |
| isfinite2 9317 | Any set strictly dominated... |
| nnsdomg 9318 | Omega strictly dominates a... |
| nnsdomgOLD 9319 | Obsolete version of ~ nnsd... |
| isfiniteg 9320 | A set is finite iff it is ... |
| infsdomnn 9321 | An infinite set strictly d... |
| infsdomnnOLD 9322 | Obsolete version of ~ infs... |
| infn0 9323 | An infinite set is not emp... |
| infn0ALT 9324 | Shorter proof of ~ infn0 u... |
| fin2inf 9325 | This (useless) theorem, wh... |
| unfilem1 9326 | Lemma for proving that the... |
| unfilem2 9327 | Lemma for proving that the... |
| unfilem3 9328 | Lemma for proving that the... |
| unfiOLD 9329 | Obsolete version of ~ unfi... |
| unfir 9330 | If a union is finite, the ... |
| unfi2 9331 | The union of two finite se... |
| difinf 9332 | An infinite set ` A ` minu... |
| xpfi 9333 | The Cartesian product of t... |
| xpfiOLD 9334 | Obsolete version of ~ xpfi... |
| 3xpfi 9335 | The Cartesian product of t... |
| domunfican 9336 | A finite set union cancell... |
| infcntss 9337 | Every infinite set has a d... |
| prfi 9338 | An unordered pair is finit... |
| tpfi 9339 | An unordered triple is fin... |
| fiint 9340 | Equivalent ways of stating... |
| fodomfi 9341 | An onto function implies d... |
| fodomfib 9342 | Equivalence of an onto map... |
| fofinf1o 9343 | Any surjection from one fi... |
| rneqdmfinf1o 9344 | Any function from a finite... |
| fidomdm 9345 | Any finite set dominates i... |
| dmfi 9346 | The domain of a finite set... |
| fundmfibi 9347 | A function is finite if an... |
| resfnfinfin 9348 | The restriction of a funct... |
| residfi 9349 | A restricted identity func... |
| cnvfiALT 9350 | Shorter proof of ~ cnvfi u... |
| rnfi 9351 | The range of a finite set ... |
| f1dmvrnfibi 9352 | A one-to-one function whos... |
| f1vrnfibi 9353 | A one-to-one function whic... |
| fofi 9354 | If an onto function has a ... |
| f1fi 9355 | If a 1-to-1 function has a... |
| iunfi 9356 | The finite union of finite... |
| unifi 9357 | The finite union of finite... |
| unifi2 9358 | The finite union of finite... |
| infssuni 9359 | If an infinite set ` A ` i... |
| unirnffid 9360 | The union of the range of ... |
| imafiALT 9361 | Shorter proof of ~ imafi u... |
| pwfilemOLD 9362 | Obsolete version of ~ pwfi... |
| pwfiOLD 9363 | Obsolete version of ~ pwfi... |
| mapfi 9364 | Set exponentiation of fini... |
| ixpfi 9365 | A Cartesian product of fin... |
| ixpfi2 9366 | A Cartesian product of fin... |
| mptfi 9367 | A finite mapping set is fi... |
| abrexfi 9368 | An image set from a finite... |
| cnvimamptfin 9369 | A preimage of a mapping wi... |
| elfpw 9370 | Membership in a class of f... |
| unifpw 9371 | A set is the union of its ... |
| f1opwfi 9372 | A one-to-one mapping induc... |
| fissuni 9373 | A finite subset of a union... |
| fipreima 9374 | Given a finite subset ` A ... |
| finsschain 9375 | A finite subset of the uni... |
| indexfi 9376 | If for every element of a ... |
| relfsupp 9379 | The property of a function... |
| relprcnfsupp 9380 | A proper class is never fi... |
| isfsupp 9381 | The property of a class to... |
| isfsuppd 9382 | Deduction form of ~ isfsup... |
| funisfsupp 9383 | The property of a function... |
| fsuppimp 9384 | Implications of a class be... |
| fsuppimpd 9385 | A finitely supported funct... |
| fsuppfund 9386 | A finitely supported funct... |
| fisuppfi 9387 | A function on a finite set... |
| fidmfisupp 9388 | A function with a finite d... |
| fdmfisuppfi 9389 | The support of a function ... |
| fdmfifsupp 9390 | A function with a finite d... |
| fsuppmptdm 9391 | A mapping with a finite do... |
| fndmfisuppfi 9392 | The support of a function ... |
| fndmfifsupp 9393 | A function with a finite d... |
| suppeqfsuppbi 9394 | If two functions have the ... |
| suppssfifsupp 9395 | If the support of a functi... |
| fsuppsssupp 9396 | If the support of a functi... |
| fsuppsssuppgd 9397 | If the support of a functi... |
| fsuppss 9398 | A subset of a finitely sup... |
| fsuppssov1 9399 | Formula building theorem f... |
| fsuppxpfi 9400 | The cartesian product of t... |
| fczfsuppd 9401 | A constant function with v... |
| fsuppun 9402 | The union of two finitely ... |
| fsuppunfi 9403 | The union of the support o... |
| fsuppunbi 9404 | If the union of two classe... |
| 0fsupp 9405 | The empty set is a finitel... |
| snopfsupp 9406 | A singleton containing an ... |
| funsnfsupp 9407 | Finite support for a funct... |
| fsuppres 9408 | The restriction of a finit... |
| fmptssfisupp 9409 | The restriction of a mappi... |
| ressuppfi 9410 | If the support of the rest... |
| resfsupp 9411 | If the restriction of a fu... |
| resfifsupp 9412 | The restriction of a funct... |
| ffsuppbi 9413 | Two ways of saying that a ... |
| fsuppmptif 9414 | A function mapping an argu... |
| sniffsupp 9415 | A function mapping all but... |
| fsuppcolem 9416 | Lemma for ~ fsuppco . For... |
| fsuppco 9417 | The composition of a 1-1 f... |
| fsuppco2 9418 | The composition of a funct... |
| fsuppcor 9419 | The composition of a funct... |
| mapfienlem1 9420 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9421 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9422 | Lemma 3 for ~ mapfien . (... |
| mapfien 9423 | A bijection of the base se... |
| mapfien2 9424 | Equinumerousity relation f... |
| fival 9427 | The set of all the finite ... |
| elfi 9428 | Specific properties of an ... |
| elfi2 9429 | The empty intersection nee... |
| elfir 9430 | Sufficient condition for a... |
| intrnfi 9431 | Sufficient condition for t... |
| iinfi 9432 | An indexed intersection of... |
| inelfi 9433 | The intersection of two se... |
| ssfii 9434 | Any element of a set ` A `... |
| fi0 9435 | The set of finite intersec... |
| fieq0 9436 | A set is empty iff the cla... |
| fiin 9437 | The elements of ` ( fi `` ... |
| dffi2 9438 | The set of finite intersec... |
| fiss 9439 | Subset relationship for fu... |
| inficl 9440 | A set which is closed unde... |
| fipwuni 9441 | The set of finite intersec... |
| fisn 9442 | A singleton is closed unde... |
| fiuni 9443 | The union of the finite in... |
| fipwss 9444 | If a set is a family of su... |
| elfiun 9445 | A finite intersection of e... |
| dffi3 9446 | The set of finite intersec... |
| fifo 9447 | Describe a surjection from... |
| marypha1lem 9448 | Core induction for Philip ... |
| marypha1 9449 | (Philip) Hall's marriage t... |
| marypha2lem1 9450 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9451 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9452 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9453 | Lemma for ~ marypha2 . Pr... |
| marypha2 9454 | Version of ~ marypha1 usin... |
| dfsup2 9459 | Quantifier-free definition... |
| supeq1 9460 | Equality theorem for supre... |
| supeq1d 9461 | Equality deduction for sup... |
| supeq1i 9462 | Equality inference for sup... |
| supeq2 9463 | Equality theorem for supre... |
| supeq3 9464 | Equality theorem for supre... |
| supeq123d 9465 | Equality deduction for sup... |
| nfsup 9466 | Hypothesis builder for sup... |
| supmo 9467 | Any class ` B ` has at mos... |
| supexd 9468 | A supremum is a set. (Con... |
| supeu 9469 | A supremum is unique. Sim... |
| supval2 9470 | Alternate expression for t... |
| eqsup 9471 | Sufficient condition for a... |
| eqsupd 9472 | Sufficient condition for a... |
| supcl 9473 | A supremum belongs to its ... |
| supub 9474 | A supremum is an upper bou... |
| suplub 9475 | A supremum is the least up... |
| suplub2 9476 | Bidirectional form of ~ su... |
| supnub 9477 | An upper bound is not less... |
| supex 9478 | A supremum is a set. (Con... |
| sup00 9479 | The supremum under an empt... |
| sup0riota 9480 | The supremum of an empty s... |
| sup0 9481 | The supremum of an empty s... |
| supmax 9482 | The greatest element of a ... |
| fisup2g 9483 | A finite set satisfies the... |
| fisupcl 9484 | A nonempty finite set cont... |
| supgtoreq 9485 | The supremum of a finite s... |
| suppr 9486 | The supremum of a pair. (... |
| supsn 9487 | The supremum of a singleto... |
| supisolem 9488 | Lemma for ~ supiso . (Con... |
| supisoex 9489 | Lemma for ~ supiso . (Con... |
| supiso 9490 | Image of a supremum under ... |
| infeq1 9491 | Equality theorem for infim... |
| infeq1d 9492 | Equality deduction for inf... |
| infeq1i 9493 | Equality inference for inf... |
| infeq2 9494 | Equality theorem for infim... |
| infeq3 9495 | Equality theorem for infim... |
| infeq123d 9496 | Equality deduction for inf... |
| nfinf 9497 | Hypothesis builder for inf... |
| infexd 9498 | An infimum is a set. (Con... |
| eqinf 9499 | Sufficient condition for a... |
| eqinfd 9500 | Sufficient condition for a... |
| infval 9501 | Alternate expression for t... |
| infcllem 9502 | Lemma for ~ infcl , ~ infl... |
| infcl 9503 | An infimum belongs to its ... |
| inflb 9504 | An infimum is a lower boun... |
| infglb 9505 | An infimum is the greatest... |
| infglbb 9506 | Bidirectional form of ~ in... |
| infnlb 9507 | A lower bound is not great... |
| infex 9508 | An infimum is a set. (Con... |
| infmin 9509 | The smallest element of a ... |
| infmo 9510 | Any class ` B ` has at mos... |
| infeu 9511 | An infimum is unique. (Co... |
| fimin2g 9512 | A finite set has a minimum... |
| fiming 9513 | A finite set has a minimum... |
| fiinfg 9514 | Lemma showing existence an... |
| fiinf2g 9515 | A finite set satisfies the... |
| fiinfcl 9516 | A nonempty finite set cont... |
| infltoreq 9517 | The infimum of a finite se... |
| infpr 9518 | The infimum of a pair. (C... |
| infsupprpr 9519 | The infimum of a proper pa... |
| infsn 9520 | The infimum of a singleton... |
| inf00 9521 | The infimum regarding an e... |
| infempty 9522 | The infimum of an empty se... |
| infiso 9523 | Image of an infimum under ... |
| dfoi 9526 | Rewrite ~ df-oi with abbre... |
| oieq1 9527 | Equality theorem for ordin... |
| oieq2 9528 | Equality theorem for ordin... |
| nfoi 9529 | Hypothesis builder for ord... |
| ordiso2 9530 | Generalize ~ ordiso to pro... |
| ordiso 9531 | Order-isomorphic ordinal n... |
| ordtypecbv 9532 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9533 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9534 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9535 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9536 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9537 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9538 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9539 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9540 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9541 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9542 | Lemma for ~ ordtype . Usi... |
| oi0 9543 | Definition of the ordinal ... |
| oicl 9544 | The order type of the well... |
| oif 9545 | The order isomorphism of t... |
| oiiso2 9546 | The order isomorphism of t... |
| ordtype 9547 | For any set-like well-orde... |
| oiiniseg 9548 | ` ran F ` is an initial se... |
| ordtype2 9549 | For any set-like well-orde... |
| oiexg 9550 | The order isomorphism on a... |
| oion 9551 | The order type of the well... |
| oiiso 9552 | The order isomorphism of t... |
| oien 9553 | The order type of a well-o... |
| oieu 9554 | Uniqueness of the unique o... |
| oismo 9555 | When ` A ` is a subclass o... |
| oiid 9556 | The order type of an ordin... |
| hartogslem1 9557 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9558 | Lemma for ~ hartogs . (Co... |
| hartogs 9559 | The class of ordinals domi... |
| wofib 9560 | The only sets which are we... |
| wemaplem1 9561 | Value of the lexicographic... |
| wemaplem2 9562 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9563 | Lemma for ~ wemapso . Tra... |
| wemappo 9564 | Construct lexicographic or... |
| wemapsolem 9565 | Lemma for ~ wemapso . (Co... |
| wemapso 9566 | Construct lexicographic or... |
| wemapso2lem 9567 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9568 | An alternative to having a... |
| card2on 9569 | The alternate definition o... |
| card2inf 9570 | The alternate definition o... |
| harf 9573 | Functionality of the Harto... |
| harcl 9574 | Values of the Hartogs func... |
| harval 9575 | Function value of the Hart... |
| elharval 9576 | The Hartogs number of a se... |
| harndom 9577 | The Hartogs number of a se... |
| harword 9578 | Weak ordering property of ... |
| relwdom 9581 | Weak dominance is a relati... |
| brwdom 9582 | Property of weak dominance... |
| brwdomi 9583 | Property of weak dominance... |
| brwdomn0 9584 | Weak dominance over nonemp... |
| 0wdom 9585 | Any set weakly dominates t... |
| fowdom 9586 | An onto function implies w... |
| wdomref 9587 | Reflexivity of weak domina... |
| brwdom2 9588 | Alternate characterization... |
| domwdom 9589 | Weak dominance is implied ... |
| wdomtr 9590 | Transitivity of weak domin... |
| wdomen1 9591 | Equality-like theorem for ... |
| wdomen2 9592 | Equality-like theorem for ... |
| wdompwdom 9593 | Weak dominance strengthens... |
| canthwdom 9594 | Cantor's Theorem, stated u... |
| wdom2d 9595 | Deduce weak dominance from... |
| wdomd 9596 | Deduce weak dominance from... |
| brwdom3 9597 | Condition for weak dominan... |
| brwdom3i 9598 | Weak dominance implies exi... |
| unwdomg 9599 | Weak dominance of a (disjo... |
| xpwdomg 9600 | Weak dominance of a Cartes... |
| wdomima2g 9601 | A set is weakly dominant o... |
| wdomimag 9602 | A set is weakly dominant o... |
| unxpwdom2 9603 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9604 | If a Cartesian product is ... |
| ixpiunwdom 9605 | Describe an onto function ... |
| harwdom 9606 | The value of the Hartogs f... |
| axreg2 9608 | Axiom of Regularity expres... |
| zfregcl 9609 | The Axiom of Regularity wi... |
| zfreg 9610 | The Axiom of Regularity us... |
| elirrv 9611 | The membership relation is... |
| elirr 9612 | No class is a member of it... |
| elneq 9613 | A class is not equal to an... |
| nelaneq 9614 | A class is not an element ... |
| epinid0 9615 | The membership relation an... |
| sucprcreg 9616 | A class is equal to its su... |
| ruv 9617 | The Russell class is equal... |
| ruALT 9618 | Alternate proof of ~ ru , ... |
| disjcsn 9619 | A class is disjoint from i... |
| zfregfr 9620 | The membership relation is... |
| en2lp 9621 | No class has 2-cycle membe... |
| elnanel 9622 | Two classes are not elemen... |
| cnvepnep 9623 | The membership (epsilon) r... |
| epnsym 9624 | The membership (epsilon) r... |
| elnotel 9625 | A class cannot be an eleme... |
| elnel 9626 | A class cannot be an eleme... |
| en3lplem1 9627 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9628 | Lemma for ~ en3lp . (Cont... |
| en3lp 9629 | No class has 3-cycle membe... |
| preleqg 9630 | Equality of two unordered ... |
| preleq 9631 | Equality of two unordered ... |
| preleqALT 9632 | Alternate proof of ~ prele... |
| opthreg 9633 | Theorem for alternate repr... |
| suc11reg 9634 | The successor operation be... |
| dford2 9635 | Assuming ~ ax-reg , an ord... |
| inf0 9636 | Existence of ` _om ` impli... |
| inf1 9637 | Variation of Axiom of Infi... |
| inf2 9638 | Variation of Axiom of Infi... |
| inf3lema 9639 | Lemma for our Axiom of Inf... |
| inf3lemb 9640 | Lemma for our Axiom of Inf... |
| inf3lemc 9641 | Lemma for our Axiom of Inf... |
| inf3lemd 9642 | Lemma for our Axiom of Inf... |
| inf3lem1 9643 | Lemma for our Axiom of Inf... |
| inf3lem2 9644 | Lemma for our Axiom of Inf... |
| inf3lem3 9645 | Lemma for our Axiom of Inf... |
| inf3lem4 9646 | Lemma for our Axiom of Inf... |
| inf3lem5 9647 | Lemma for our Axiom of Inf... |
| inf3lem6 9648 | Lemma for our Axiom of Inf... |
| inf3lem7 9649 | Lemma for our Axiom of Inf... |
| inf3 9650 | Our Axiom of Infinity ~ ax... |
| infeq5i 9651 | Half of ~ infeq5 . (Contr... |
| infeq5 9652 | The statement "there exist... |
| zfinf 9654 | Axiom of Infinity expresse... |
| axinf2 9655 | A standard version of Axio... |
| zfinf2 9657 | A standard version of the ... |
| omex 9658 | The existence of omega (th... |
| axinf 9659 | The first version of the A... |
| inf5 9660 | The statement "there exist... |
| omelon 9661 | Omega is an ordinal number... |
| dfom3 9662 | The class of natural numbe... |
| elom3 9663 | A simplification of ~ elom... |
| dfom4 9664 | A simplification of ~ df-o... |
| dfom5 9665 | ` _om ` is the smallest li... |
| oancom 9666 | Ordinal addition is not co... |
| isfinite 9667 | A set is finite iff it is ... |
| fict 9668 | A finite set is countable ... |
| nnsdom 9669 | A natural number is strict... |
| omenps 9670 | Omega is equinumerous to a... |
| omensuc 9671 | The set of natural numbers... |
| infdifsn 9672 | Removing a singleton from ... |
| infdiffi 9673 | Removing a finite set from... |
| unbnn3 9674 | Any unbounded subset of na... |
| noinfep 9675 | Using the Axiom of Regular... |
| cantnffval 9678 | The value of the Cantor no... |
| cantnfdm 9679 | The domain of the Cantor n... |
| cantnfvalf 9680 | Lemma for ~ cantnf . The ... |
| cantnfs 9681 | Elementhood in the set of ... |
| cantnfcl 9682 | Basic properties of the or... |
| cantnfval 9683 | The value of the Cantor no... |
| cantnfval2 9684 | Alternate expression for t... |
| cantnfsuc 9685 | The value of the recursive... |
| cantnfle 9686 | A lower bound on the ` CNF... |
| cantnflt 9687 | An upper bound on the part... |
| cantnflt2 9688 | An upper bound on the ` CN... |
| cantnff 9689 | The ` CNF ` function is a ... |
| cantnf0 9690 | The value of the zero func... |
| cantnfrescl 9691 | A function is finitely sup... |
| cantnfres 9692 | The ` CNF ` function respe... |
| cantnfp1lem1 9693 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9694 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9695 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9696 | If ` F ` is created by add... |
| oemapso 9697 | The relation ` T ` is a st... |
| oemapval 9698 | Value of the relation ` T ... |
| oemapvali 9699 | If ` F < G ` , then there ... |
| cantnflem1a 9700 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9701 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9702 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9703 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9704 | Lemma for ~ cantnf . This... |
| cantnflem2 9705 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9706 | Lemma for ~ cantnf . Here... |
| cantnflem4 9707 | Lemma for ~ cantnf . Comp... |
| cantnf 9708 | The Cantor Normal Form the... |
| oemapwe 9709 | The lexicographic order on... |
| cantnffval2 9710 | An alternate definition of... |
| cantnff1o 9711 | Simplify the isomorphism o... |
| wemapwe 9712 | Construct lexicographic or... |
| oef1o 9713 | A bijection of the base se... |
| cnfcomlem 9714 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9715 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9716 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9717 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9718 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9719 | Any infinite ordinal ` B `... |
| cnfcom3clem 9720 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9721 | Wrap the construction of ~... |
| ttrcleq 9724 | Equality theorem for trans... |
| nfttrcld 9725 | Bound variable hypothesis ... |
| nfttrcl 9726 | Bound variable hypothesis ... |
| relttrcl 9727 | The transitive closure of ... |
| brttrcl 9728 | Characterization of elemen... |
| brttrcl2 9729 | Characterization of elemen... |
| ssttrcl 9730 | If ` R ` is a relation, th... |
| ttrcltr 9731 | The transitive closure of ... |
| ttrclresv 9732 | The transitive closure of ... |
| ttrclco 9733 | Composition law for the tr... |
| cottrcl 9734 | Composition law for the tr... |
| ttrclss 9735 | If ` R ` is a subclass of ... |
| dmttrcl 9736 | The domain of a transitive... |
| rnttrcl 9737 | The range of a transitive ... |
| ttrclexg 9738 | If ` R ` is a set, then so... |
| dfttrcl2 9739 | When ` R ` is a set and a ... |
| ttrclselem1 9740 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9741 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9742 | If ` R ` is set-like over ... |
| trcl 9743 | For any set ` A ` , show t... |
| tz9.1 9744 | Every set has a transitive... |
| tz9.1c 9745 | Alternate expression for t... |
| epfrs 9746 | The strong form of the Axi... |
| zfregs 9747 | The strong form of the Axi... |
| zfregs2 9748 | Alternate strong form of t... |
| setind 9749 | Set (epsilon) induction. ... |
| setind2 9750 | Set (epsilon) induction, s... |
| tcvalg 9753 | Value of the transitive cl... |
| tcid 9754 | Defining property of the t... |
| tctr 9755 | Defining property of the t... |
| tcmin 9756 | Defining property of the t... |
| tc2 9757 | A variant of the definitio... |
| tcsni 9758 | The transitive closure of ... |
| tcss 9759 | The transitive closure fun... |
| tcel 9760 | The transitive closure fun... |
| tcidm 9761 | The transitive closure fun... |
| tc0 9762 | The transitive closure of ... |
| tc00 9763 | The transitive closure is ... |
| frmin 9764 | Every (possibly proper) su... |
| frind 9765 | A subclass of a well-found... |
| frinsg 9766 | Well-Founded Induction Sch... |
| frins 9767 | Well-Founded Induction Sch... |
| frins2f 9768 | Well-Founded Induction sch... |
| frins2 9769 | Well-Founded Induction sch... |
| frins3 9770 | Well-Founded Induction sch... |
| frr3g 9771 | Functions defined by well-... |
| frrlem15 9772 | Lemma for general well-fou... |
| frrlem16 9773 | Lemma for general well-fou... |
| frr1 9774 | Law of general well-founde... |
| frr2 9775 | Law of general well-founde... |
| frr3 9776 | Law of general well-founde... |
| r1funlim 9781 | The cumulative hierarchy o... |
| r1fnon 9782 | The cumulative hierarchy o... |
| r10 9783 | Value of the cumulative hi... |
| r1sucg 9784 | Value of the cumulative hi... |
| r1suc 9785 | Value of the cumulative hi... |
| r1limg 9786 | Value of the cumulative hi... |
| r1lim 9787 | Value of the cumulative hi... |
| r1fin 9788 | The first ` _om ` levels o... |
| r1sdom 9789 | Each stage in the cumulati... |
| r111 9790 | The cumulative hierarchy i... |
| r1tr 9791 | The cumulative hierarchy o... |
| r1tr2 9792 | The union of a cumulative ... |
| r1ordg 9793 | Ordering relation for the ... |
| r1ord3g 9794 | Ordering relation for the ... |
| r1ord 9795 | Ordering relation for the ... |
| r1ord2 9796 | Ordering relation for the ... |
| r1ord3 9797 | Ordering relation for the ... |
| r1sssuc 9798 | The value of the cumulativ... |
| r1pwss 9799 | Each set of the cumulative... |
| r1sscl 9800 | Each set of the cumulative... |
| r1val1 9801 | The value of the cumulativ... |
| tz9.12lem1 9802 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9803 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9804 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9805 | A set is well-founded if a... |
| tz9.13 9806 | Every set is well-founded,... |
| tz9.13g 9807 | Every set is well-founded,... |
| rankwflemb 9808 | Two ways of saying a set i... |
| rankf 9809 | The domain and codomain of... |
| rankon 9810 | The rank of a set is an or... |
| r1elwf 9811 | Any member of the cumulati... |
| rankvalb 9812 | Value of the rank function... |
| rankr1ai 9813 | One direction of ~ rankr1a... |
| rankvaln 9814 | Value of the rank function... |
| rankidb 9815 | Identity law for the rank ... |
| rankdmr1 9816 | A rank is a member of the ... |
| rankr1ag 9817 | A version of ~ rankr1a tha... |
| rankr1bg 9818 | A relationship between ran... |
| r1rankidb 9819 | Any set is a subset of the... |
| r1elssi 9820 | The range of the ` R1 ` fu... |
| r1elss 9821 | The range of the ` R1 ` fu... |
| pwwf 9822 | A power set is well-founde... |
| sswf 9823 | A subset of a well-founded... |
| snwf 9824 | A singleton is well-founde... |
| unwf 9825 | A binary union is well-fou... |
| prwf 9826 | An unordered pair is well-... |
| opwf 9827 | An ordered pair is well-fo... |
| unir1 9828 | The cumulative hierarchy o... |
| jech9.3 9829 | Every set belongs to some ... |
| rankwflem 9830 | Every set is well-founded,... |
| rankval 9831 | Value of the rank function... |
| rankvalg 9832 | Value of the rank function... |
| rankval2 9833 | Value of an alternate defi... |
| uniwf 9834 | A union is well-founded if... |
| rankr1clem 9835 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9836 | A relationship between the... |
| rankidn 9837 | A relationship between the... |
| rankpwi 9838 | The rank of a power set. ... |
| rankelb 9839 | The membership relation is... |
| wfelirr 9840 | A well-founded set is not ... |
| rankval3b 9841 | The value of the rank func... |
| ranksnb 9842 | The rank of a singleton. ... |
| rankonidlem 9843 | Lemma for ~ rankonid . (C... |
| rankonid 9844 | The rank of an ordinal num... |
| onwf 9845 | The ordinals are all well-... |
| onssr1 9846 | Initial segments of the or... |
| rankr1g 9847 | A relationship between the... |
| rankid 9848 | Identity law for the rank ... |
| rankr1 9849 | A relationship between the... |
| ssrankr1 9850 | A relationship between an ... |
| rankr1a 9851 | A relationship between ran... |
| r1val2 9852 | The value of the cumulativ... |
| r1val3 9853 | The value of the cumulativ... |
| rankel 9854 | The membership relation is... |
| rankval3 9855 | The value of the rank func... |
| bndrank 9856 | Any class whose elements h... |
| unbndrank 9857 | The elements of a proper c... |
| rankpw 9858 | The rank of a power set. ... |
| ranklim 9859 | The rank of a set belongs ... |
| r1pw 9860 | A stronger property of ` R... |
| r1pwALT 9861 | Alternate shorter proof of... |
| r1pwcl 9862 | The cumulative hierarchy o... |
| rankssb 9863 | The subset relation is inh... |
| rankss 9864 | The subset relation is inh... |
| rankunb 9865 | The rank of the union of t... |
| rankprb 9866 | The rank of an unordered p... |
| rankopb 9867 | The rank of an ordered pai... |
| rankuni2b 9868 | The value of the rank func... |
| ranksn 9869 | The rank of a singleton. ... |
| rankuni2 9870 | The rank of a union. Part... |
| rankun 9871 | The rank of the union of t... |
| rankpr 9872 | The rank of an unordered p... |
| rankop 9873 | The rank of an ordered pai... |
| r1rankid 9874 | Any set is a subset of the... |
| rankeq0b 9875 | A set is empty iff its ran... |
| rankeq0 9876 | A set is empty iff its ran... |
| rankr1id 9877 | The rank of the hierarchy ... |
| rankuni 9878 | The rank of a union. Part... |
| rankr1b 9879 | A relationship between ran... |
| ranksuc 9880 | The rank of a successor. ... |
| rankuniss 9881 | Upper bound of the rank of... |
| rankval4 9882 | The rank of a set is the s... |
| rankbnd 9883 | The rank of a set is bound... |
| rankbnd2 9884 | The rank of a set is bound... |
| rankc1 9885 | A relationship that can be... |
| rankc2 9886 | A relationship that can be... |
| rankelun 9887 | Rank membership is inherit... |
| rankelpr 9888 | Rank membership is inherit... |
| rankelop 9889 | Rank membership is inherit... |
| rankxpl 9890 | A lower bound on the rank ... |
| rankxpu 9891 | An upper bound on the rank... |
| rankfu 9892 | An upper bound on the rank... |
| rankmapu 9893 | An upper bound on the rank... |
| rankxplim 9894 | The rank of a Cartesian pr... |
| rankxplim2 9895 | If the rank of a Cartesian... |
| rankxplim3 9896 | The rank of a Cartesian pr... |
| rankxpsuc 9897 | The rank of a Cartesian pr... |
| tcwf 9898 | The transitive closure fun... |
| tcrank 9899 | This theorem expresses two... |
| scottex 9900 | Scott's trick collects all... |
| scott0 9901 | Scott's trick collects all... |
| scottexs 9902 | Theorem scheme version of ... |
| scott0s 9903 | Theorem scheme version of ... |
| cplem1 9904 | Lemma for the Collection P... |
| cplem2 9905 | Lemma for the Collection P... |
| cp 9906 | Collection Principle. Thi... |
| bnd 9907 | A very strong generalizati... |
| bnd2 9908 | A variant of the Boundedne... |
| kardex 9909 | The collection of all sets... |
| karden 9910 | If we allow the Axiom of R... |
| htalem 9911 | Lemma for defining an emul... |
| hta 9912 | A ZFC emulation of Hilbert... |
| djueq12 9919 | Equality theorem for disjo... |
| djueq1 9920 | Equality theorem for disjo... |
| djueq2 9921 | Equality theorem for disjo... |
| nfdju 9922 | Bound-variable hypothesis ... |
| djuex 9923 | The disjoint union of sets... |
| djuexb 9924 | The disjoint union of two ... |
| djulcl 9925 | Left closure of disjoint u... |
| djurcl 9926 | Right closure of disjoint ... |
| djulf1o 9927 | The left injection functio... |
| djurf1o 9928 | The right injection functi... |
| inlresf 9929 | The left injection restric... |
| inlresf1 9930 | The left injection restric... |
| inrresf 9931 | The right injection restri... |
| inrresf1 9932 | The right injection restri... |
| djuin 9933 | The images of any classes ... |
| djur 9934 | A member of a disjoint uni... |
| djuss 9935 | A disjoint union is a subc... |
| djuunxp 9936 | The union of a disjoint un... |
| djuexALT 9937 | Alternate proof of ~ djuex... |
| eldju1st 9938 | The first component of an ... |
| eldju2ndl 9939 | The second component of an... |
| eldju2ndr 9940 | The second component of an... |
| djuun 9941 | The disjoint union of two ... |
| 1stinl 9942 | The first component of the... |
| 2ndinl 9943 | The second component of th... |
| 1stinr 9944 | The first component of the... |
| 2ndinr 9945 | The second component of th... |
| updjudhf 9946 | The mapping of an element ... |
| updjudhcoinlf 9947 | The composition of the map... |
| updjudhcoinrg 9948 | The composition of the map... |
| updjud 9949 | Universal property of the ... |
| cardf2 9958 | The cardinality function i... |
| cardon 9959 | The cardinal number of a s... |
| isnum2 9960 | A way to express well-orde... |
| isnumi 9961 | A set equinumerous to an o... |
| ennum 9962 | Equinumerous sets are equi... |
| finnum 9963 | Every finite set is numera... |
| onenon 9964 | Every ordinal number is nu... |
| tskwe 9965 | A Tarski set is well-order... |
| xpnum 9966 | The cartesian product of n... |
| cardval3 9967 | An alternate definition of... |
| cardid2 9968 | Any numerable set is equin... |
| isnum3 9969 | A set is numerable iff it ... |
| oncardval 9970 | The value of the cardinal ... |
| oncardid 9971 | Any ordinal number is equi... |
| cardonle 9972 | The cardinal of an ordinal... |
| card0 9973 | The cardinality of the emp... |
| cardidm 9974 | The cardinality function i... |
| oncard 9975 | A set is a cardinal number... |
| ficardom 9976 | The cardinal number of a f... |
| ficardid 9977 | A finite set is equinumero... |
| cardnn 9978 | The cardinality of a natur... |
| cardnueq0 9979 | The empty set is the only ... |
| cardne 9980 | No member of a cardinal nu... |
| carden2a 9981 | If two sets have equal non... |
| carden2b 9982 | If two sets are equinumero... |
| card1 9983 | A set has cardinality one ... |
| cardsn 9984 | A singleton has cardinalit... |
| carddomi2 9985 | Two sets have the dominanc... |
| sdomsdomcardi 9986 | A set strictly dominates i... |
| cardlim 9987 | An infinite cardinal is a ... |
| cardsdomelir 9988 | A cardinal strictly domina... |
| cardsdomel 9989 | A cardinal strictly domina... |
| iscard 9990 | Two ways to express the pr... |
| iscard2 9991 | Two ways to express the pr... |
| carddom2 9992 | Two numerable sets have th... |
| harcard 9993 | The class of ordinal numbe... |
| cardprclem 9994 | Lemma for ~ cardprc . (Co... |
| cardprc 9995 | The class of all cardinal ... |
| carduni 9996 | The union of a set of card... |
| cardiun 9997 | The indexed union of a set... |
| cardennn 9998 | If ` A ` is equinumerous t... |
| cardsucinf 9999 | The cardinality of the suc... |
| cardsucnn 10000 | The cardinality of the suc... |
| cardom 10001 | The set of natural numbers... |
| carden2 10002 | Two numerable sets are equ... |
| cardsdom2 10003 | A numerable set is strictl... |
| domtri2 10004 | Trichotomy of dominance fo... |
| nnsdomel 10005 | Strict dominance and eleme... |
| cardval2 10006 | An alternate version of th... |
| isinffi 10007 | An infinite set contains s... |
| fidomtri 10008 | Trichotomy of dominance wi... |
| fidomtri2 10009 | Trichotomy of dominance wi... |
| harsdom 10010 | The Hartogs number of a we... |
| onsdom 10011 | Any well-orderable set is ... |
| harval2 10012 | An alternate expression fo... |
| harsucnn 10013 | The next cardinal after a ... |
| cardmin2 10014 | The smallest ordinal that ... |
| pm54.43lem 10015 | In Theorem *54.43 of [Whit... |
| pm54.43 10016 | Theorem *54.43 of [Whitehe... |
| enpr2 10017 | An unordered pair with dis... |
| pr2nelemOLD 10018 | Obsolete version of ~ enpr... |
| pr2ne 10019 | If an unordered pair has t... |
| pr2neOLD 10020 | Obsolete version of ~ pr2n... |
| prdom2 10021 | An unordered pair has at m... |
| en2eqpr 10022 | Building a set with two el... |
| en2eleq 10023 | Express a set of pair card... |
| en2other2 10024 | Taking the other element t... |
| dif1card 10025 | The cardinality of a nonem... |
| leweon 10026 | Lexicographical order is a... |
| r0weon 10027 | A set-like well-ordering o... |
| infxpenlem 10028 | Lemma for ~ infxpen . (Co... |
| infxpen 10029 | Every infinite ordinal is ... |
| xpomen 10030 | The Cartesian product of o... |
| xpct 10031 | The cartesian product of t... |
| infxpidm2 10032 | Every infinite well-ordera... |
| infxpenc 10033 | A canonical version of ~ i... |
| infxpenc2lem1 10034 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 10035 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 10036 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 10037 | Existence form of ~ infxpe... |
| iunmapdisj 10038 | The union ` U_ n e. C ( A ... |
| fseqenlem1 10039 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 10040 | Lemma for ~ fseqen . (Con... |
| fseqdom 10041 | One half of ~ fseqen . (C... |
| fseqen 10042 | A set that is equinumerous... |
| infpwfidom 10043 | The collection of finite s... |
| dfac8alem 10044 | Lemma for ~ dfac8a . If t... |
| dfac8a 10045 | Numeration theorem: every ... |
| dfac8b 10046 | The well-ordering theorem:... |
| dfac8clem 10047 | Lemma for ~ dfac8c . (Con... |
| dfac8c 10048 | If the union of a set is w... |
| ac10ct 10049 | A proof of the well-orderi... |
| ween 10050 | A set is numerable iff it ... |
| ac5num 10051 | A version of ~ ac5b with t... |
| ondomen 10052 | If a set is dominated by a... |
| numdom 10053 | A set dominated by a numer... |
| ssnum 10054 | A subset of a numerable se... |
| onssnum 10055 | All subsets of the ordinal... |
| indcardi 10056 | Indirect strong induction ... |
| acnrcl 10057 | Reverse closure for the ch... |
| acneq 10058 | Equality theorem for the c... |
| isacn 10059 | The property of being a ch... |
| acni 10060 | The property of being a ch... |
| acni2 10061 | The property of being a ch... |
| acni3 10062 | The property of being a ch... |
| acnlem 10063 | Construct a mapping satisf... |
| numacn 10064 | A well-orderable set has c... |
| finacn 10065 | Every set has finite choic... |
| acndom 10066 | A set with long choice seq... |
| acnnum 10067 | A set ` X ` which has choi... |
| acnen 10068 | The class of choice sets o... |
| acndom2 10069 | A set smaller than one wit... |
| acnen2 10070 | The class of sets with cho... |
| fodomacn 10071 | A version of ~ fodom that ... |
| fodomnum 10072 | A version of ~ fodom that ... |
| fonum 10073 | A surjection maps numerabl... |
| numwdom 10074 | A surjection maps numerabl... |
| fodomfi2 10075 | Onto functions define domi... |
| wdomfil 10076 | Weak dominance agrees with... |
| infpwfien 10077 | Any infinite well-orderabl... |
| inffien 10078 | The set of finite intersec... |
| wdomnumr 10079 | Weak dominance agrees with... |
| alephfnon 10080 | The aleph function is a fu... |
| aleph0 10081 | The first infinite cardina... |
| alephlim 10082 | Value of the aleph functio... |
| alephsuc 10083 | Value of the aleph functio... |
| alephon 10084 | An aleph is an ordinal num... |
| alephcard 10085 | Every aleph is a cardinal ... |
| alephnbtwn 10086 | No cardinal can be sandwic... |
| alephnbtwn2 10087 | No set has equinumerosity ... |
| alephordilem1 10088 | Lemma for ~ alephordi . (... |
| alephordi 10089 | Strict ordering property o... |
| alephord 10090 | Ordering property of the a... |
| alephord2 10091 | Ordering property of the a... |
| alephord2i 10092 | Ordering property of the a... |
| alephord3 10093 | Ordering property of the a... |
| alephsucdom 10094 | A set dominated by an alep... |
| alephsuc2 10095 | An alternate representatio... |
| alephdom 10096 | Relationship between inclu... |
| alephgeom 10097 | Every aleph is greater tha... |
| alephislim 10098 | Every aleph is a limit ord... |
| aleph11 10099 | The aleph function is one-... |
| alephf1 10100 | The aleph function is a on... |
| alephsdom 10101 | If an ordinal is smaller t... |
| alephdom2 10102 | A dominated initial ordina... |
| alephle 10103 | The argument of the aleph ... |
| cardaleph 10104 | Given any transfinite card... |
| cardalephex 10105 | Every transfinite cardinal... |
| infenaleph 10106 | An infinite numerable set ... |
| isinfcard 10107 | Two ways to express the pr... |
| iscard3 10108 | Two ways to express the pr... |
| cardnum 10109 | Two ways to express the cl... |
| alephinit 10110 | An infinite initial ordina... |
| carduniima 10111 | The union of the image of ... |
| cardinfima 10112 | If a mapping to cardinals ... |
| alephiso 10113 | Aleph is an order isomorph... |
| alephprc 10114 | The class of all transfini... |
| alephsson 10115 | The class of transfinite c... |
| unialeph 10116 | The union of the class of ... |
| alephsmo 10117 | The aleph function is stri... |
| alephf1ALT 10118 | Alternate proof of ~ aleph... |
| alephfplem1 10119 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10120 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10121 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10122 | Lemma for ~ alephfp . (Co... |
| alephfp 10123 | The aleph function has a f... |
| alephfp2 10124 | The aleph function has at ... |
| alephval3 10125 | An alternate way to expres... |
| alephsucpw2 10126 | The power set of an aleph ... |
| mappwen 10127 | Power rule for cardinal ar... |
| finnisoeu 10128 | A finite totally ordered s... |
| iunfictbso 10129 | Countability of a countabl... |
| aceq1 10132 | Equivalence of two version... |
| aceq0 10133 | Equivalence of two version... |
| aceq2 10134 | Equivalence of two version... |
| aceq3lem 10135 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10136 | Equivalence of two version... |
| dfac4 10137 | Equivalence of two version... |
| dfac5lem1 10138 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10139 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10140 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10141 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10142 | Lemma for ~ dfac5 . (Cont... |
| dfac5 10143 | Equivalence of two version... |
| dfac2a 10144 | Our Axiom of Choice (in th... |
| dfac2b 10145 | Axiom of Choice (first for... |
| dfac2 10146 | Axiom of Choice (first for... |
| dfac7 10147 | Equivalence of the Axiom o... |
| dfac0 10148 | Equivalence of two version... |
| dfac1 10149 | Equivalence of two version... |
| dfac8 10150 | A proof of the equivalency... |
| dfac9 10151 | Equivalence of the axiom o... |
| dfac10 10152 | Axiom of Choice equivalent... |
| dfac10c 10153 | Axiom of Choice equivalent... |
| dfac10b 10154 | Axiom of Choice equivalent... |
| acacni 10155 | A choice equivalent: every... |
| dfacacn 10156 | A choice equivalent: every... |
| dfac13 10157 | The axiom of choice holds ... |
| dfac12lem1 10158 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10159 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10160 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10161 | The axiom of choice holds ... |
| dfac12k 10162 | Equivalence of ~ dfac12 an... |
| dfac12a 10163 | The axiom of choice holds ... |
| dfac12 10164 | The axiom of choice holds ... |
| kmlem1 10165 | Lemma for 5-quantifier AC ... |
| kmlem2 10166 | Lemma for 5-quantifier AC ... |
| kmlem3 10167 | Lemma for 5-quantifier AC ... |
| kmlem4 10168 | Lemma for 5-quantifier AC ... |
| kmlem5 10169 | Lemma for 5-quantifier AC ... |
| kmlem6 10170 | Lemma for 5-quantifier AC ... |
| kmlem7 10171 | Lemma for 5-quantifier AC ... |
| kmlem8 10172 | Lemma for 5-quantifier AC ... |
| kmlem9 10173 | Lemma for 5-quantifier AC ... |
| kmlem10 10174 | Lemma for 5-quantifier AC ... |
| kmlem11 10175 | Lemma for 5-quantifier AC ... |
| kmlem12 10176 | Lemma for 5-quantifier AC ... |
| kmlem13 10177 | Lemma for 5-quantifier AC ... |
| kmlem14 10178 | Lemma for 5-quantifier AC ... |
| kmlem15 10179 | Lemma for 5-quantifier AC ... |
| kmlem16 10180 | Lemma for 5-quantifier AC ... |
| dfackm 10181 | Equivalence of the Axiom o... |
| undjudom 10182 | Cardinal addition dominate... |
| endjudisj 10183 | Equinumerosity of a disjoi... |
| djuen 10184 | Disjoint unions of equinum... |
| djuenun 10185 | Disjoint union is equinume... |
| dju1en 10186 | Cardinal addition with car... |
| dju1dif 10187 | Adding and subtracting one... |
| dju1p1e2 10188 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10189 | Alternate proof of ~ dju1p... |
| dju0en 10190 | Cardinal addition with car... |
| xp2dju 10191 | Two times a cardinal numbe... |
| djucomen 10192 | Commutative law for cardin... |
| djuassen 10193 | Associative law for cardin... |
| xpdjuen 10194 | Cardinal multiplication di... |
| mapdjuen 10195 | Sum of exponents law for c... |
| pwdjuen 10196 | Sum of exponents law for c... |
| djudom1 10197 | Ordering law for cardinal ... |
| djudom2 10198 | Ordering law for cardinal ... |
| djudoml 10199 | A set is dominated by its ... |
| djuxpdom 10200 | Cartesian product dominate... |
| djufi 10201 | The disjoint union of two ... |
| cdainflem 10202 | Any partition of omega int... |
| djuinf 10203 | A set is infinite iff the ... |
| infdju1 10204 | An infinite set is equinum... |
| pwdju1 10205 | The sum of a powerset with... |
| pwdjuidm 10206 | If the natural numbers inj... |
| djulepw 10207 | If ` A ` is idempotent und... |
| onadju 10208 | The cardinal and ordinal s... |
| cardadju 10209 | The cardinal sum is equinu... |
| djunum 10210 | The disjoint union of two ... |
| unnum 10211 | The union of two numerable... |
| nnadju 10212 | The cardinal and ordinal s... |
| nnadjuALT 10213 | Shorter proof of ~ nnadju ... |
| ficardadju 10214 | The disjoint union of fini... |
| ficardun 10215 | The cardinality of the uni... |
| ficardunOLD 10216 | Obsolete version of ~ fica... |
| ficardun2 10217 | The cardinality of the uni... |
| ficardun2OLD 10218 | Obsolete version of ~ fica... |
| pwsdompw 10219 | Lemma for ~ domtriom . Th... |
| unctb 10220 | The union of two countable... |
| infdjuabs 10221 | Absorption law for additio... |
| infunabs 10222 | An infinite set is equinum... |
| infdju 10223 | The sum of two cardinal nu... |
| infdif 10224 | The cardinality of an infi... |
| infdif2 10225 | Cardinality ordering for a... |
| infxpdom 10226 | Dominance law for multipli... |
| infxpabs 10227 | Absorption law for multipl... |
| infunsdom1 10228 | The union of two sets that... |
| infunsdom 10229 | The union of two sets that... |
| infxp 10230 | Absorption law for multipl... |
| pwdjudom 10231 | A property of dominance ov... |
| infpss 10232 | Every infinite set has an ... |
| infmap2 10233 | An exponentiation law for ... |
| ackbij2lem1 10234 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10235 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10236 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10237 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10238 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10239 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10240 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10241 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10242 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10243 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10244 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10245 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10246 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10247 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10248 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10249 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10250 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10251 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10252 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10253 | The Ackermann bijection, p... |
| ackbij1b 10254 | The Ackermann bijection, p... |
| ackbij2lem2 10255 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10256 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10257 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10258 | The Ackermann bijection, p... |
| r1om 10259 | The set of hereditarily fi... |
| fictb 10260 | A set is countable iff its... |
| cflem 10261 | A lemma used to simplify c... |
| cfval 10262 | Value of the cofinality fu... |
| cff 10263 | Cofinality is a function o... |
| cfub 10264 | An upper bound on cofinali... |
| cflm 10265 | Value of the cofinality fu... |
| cf0 10266 | Value of the cofinality fu... |
| cardcf 10267 | Cofinality is a cardinal n... |
| cflecard 10268 | Cofinality is bounded by t... |
| cfle 10269 | Cofinality is bounded by i... |
| cfon 10270 | The cofinality of any set ... |
| cfeq0 10271 | Only the ordinal zero has ... |
| cfsuc 10272 | Value of the cofinality fu... |
| cff1 10273 | There is always a map from... |
| cfflb 10274 | If there is a cofinal map ... |
| cfval2 10275 | Another expression for the... |
| coflim 10276 | A simpler expression for t... |
| cflim3 10277 | Another expression for the... |
| cflim2 10278 | The cofinality function is... |
| cfom 10279 | Value of the cofinality fu... |
| cfss 10280 | There is a cofinal subset ... |
| cfslb 10281 | Any cofinal subset of ` A ... |
| cfslbn 10282 | Any subset of ` A ` smalle... |
| cfslb2n 10283 | Any small collection of sm... |
| cofsmo 10284 | Any cofinal map implies th... |
| cfsmolem 10285 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10286 | The map in ~ cff1 can be a... |
| cfcoflem 10287 | Lemma for ~ cfcof , showin... |
| coftr 10288 | If there is a cofinal map ... |
| cfcof 10289 | If there is a cofinal map ... |
| cfidm 10290 | The cofinality function is... |
| alephsing 10291 | The cofinality of a limit ... |
| sornom 10292 | The range of a single-step... |
| isfin1a 10307 | Definition of a Ia-finite ... |
| fin1ai 10308 | Property of a Ia-finite se... |
| isfin2 10309 | Definition of a II-finite ... |
| fin2i 10310 | Property of a II-finite se... |
| isfin3 10311 | Definition of a III-finite... |
| isfin4 10312 | Definition of a IV-finite ... |
| fin4i 10313 | Infer that a set is IV-inf... |
| isfin5 10314 | Definition of a V-finite s... |
| isfin6 10315 | Definition of a VI-finite ... |
| isfin7 10316 | Definition of a VII-finite... |
| sdom2en01 10317 | A set with less than two e... |
| infpssrlem1 10318 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10319 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10320 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10321 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10322 | Lemma for ~ infpssr . (Co... |
| infpssr 10323 | Dedekind infinity implies ... |
| fin4en1 10324 | Dedekind finite is a cardi... |
| ssfin4 10325 | Dedekind finite sets have ... |
| domfin4 10326 | A set dominated by a Dedek... |
| ominf4 10327 | ` _om ` is Dedekind infini... |
| infpssALT 10328 | Alternate proof of ~ infps... |
| isfin4-2 10329 | Alternate definition of IV... |
| isfin4p1 10330 | Alternate definition of IV... |
| fin23lem7 10331 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10332 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10333 | A II-finite set contains m... |
| isfin2-2 10334 | ` Fin2 ` expressed in term... |
| ssfin2 10335 | A subset of a II-finite se... |
| enfin2i 10336 | II-finiteness is a cardina... |
| fin23lem24 10337 | Lemma for ~ fin23 . In a ... |
| fincssdom 10338 | In a chain of finite sets,... |
| fin23lem25 10339 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10340 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10341 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10342 | Lemma for ~ fin23 but coul... |
| fin23lem27 10343 | The mapping constructed in... |
| isfin3ds 10344 | Property of a III-finite s... |
| ssfin3ds 10345 | A subset of a III-finite s... |
| fin23lem12 10346 | The beginning of the proof... |
| fin23lem13 10347 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10348 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10349 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10350 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10351 | Lemma for ~ fin23 . The f... |
| fin23lem20 10352 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10353 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10354 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10355 | Lemma for ~ fin23 . The r... |
| fin23lem29 10356 | Lemma for ~ fin23 . The r... |
| fin23lem30 10357 | Lemma for ~ fin23 . The r... |
| fin23lem31 10358 | Lemma for ~ fin23 . The r... |
| fin23lem32 10359 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10360 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10361 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10362 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10363 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10364 | Lemma for ~ fin23 . The c... |
| fin23lem39 10365 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10366 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10367 | Lemma for ~ fin23 . A set... |
| isf32lem1 10368 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10369 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10370 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10371 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10372 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10373 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10374 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10375 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10376 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10377 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10378 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10379 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10380 | One half of ~ isfin3-2 . ... |
| isf33lem 10381 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10382 | Weakly Dedekind-infinite s... |
| isfin3-3 10383 | Weakly Dedekind-infinite s... |
| fin33i 10384 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10385 | Lemma for ~ compsscnv . (... |
| compsscnv 10386 | Complementation on a power... |
| isf34lem1 10387 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10388 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10389 | Complementation is an anti... |
| isf34lem3 10390 | Lemma for ~ isfin3-4 . (C... |
| compss 10391 | Express image under of the... |
| isf34lem4 10392 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10393 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10394 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10395 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10396 | Inference from ~ isfin3-4 ... |
| isfin3-4 10397 | Weakly Dedekind-infinite s... |
| fin11a 10398 | Every I-finite set is Ia-f... |
| enfin1ai 10399 | Ia-finiteness is a cardina... |
| isfin1-2 10400 | A set is finite in the usu... |
| isfin1-3 10401 | A set is I-finite iff ever... |
| isfin1-4 10402 | A set is I-finite iff ever... |
| dffin1-5 10403 | Compact quantifier-free ve... |
| fin23 10404 | Every II-finite set (every... |
| fin34 10405 | Every III-finite set is IV... |
| isfin5-2 10406 | Alternate definition of V-... |
| fin45 10407 | Every IV-finite set is V-f... |
| fin56 10408 | Every V-finite set is VI-f... |
| fin17 10409 | Every I-finite set is VII-... |
| fin67 10410 | Every VI-finite set is VII... |
| isfin7-2 10411 | A set is VII-finite iff it... |
| fin71num 10412 | A well-orderable set is VI... |
| dffin7-2 10413 | Class form of ~ isfin7-2 .... |
| dfacfin7 10414 | Axiom of Choice equivalent... |
| fin1a2lem1 10415 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10416 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10417 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10418 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10419 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10420 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10421 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10422 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10423 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10424 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10425 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10426 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10427 | Lemma for ~ fin1a2 . (Con... |
| fin12 10428 | Weak theorem which skips I... |
| fin1a2s 10429 | An II-infinite set can hav... |
| fin1a2 10430 | Every Ia-finite set is II-... |
| itunifval 10431 | Function value of iterated... |
| itunifn 10432 | Functionality of the itera... |
| ituni0 10433 | A zero-fold iterated union... |
| itunisuc 10434 | Successor iterated union. ... |
| itunitc1 10435 | Each union iterate is a me... |
| itunitc 10436 | The union of all union ite... |
| ituniiun 10437 | Unwrap an iterated union f... |
| hsmexlem7 10438 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10439 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10440 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10441 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10442 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10443 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10444 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10445 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10446 | Lemma for ~ hsmex . (Cont... |
| hsmex 10447 | The collection of heredita... |
| hsmex2 10448 | The set of hereditary size... |
| hsmex3 10449 | The set of hereditary size... |
| axcc2lem 10451 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10452 | A possibly more useful ver... |
| axcc3 10453 | A possibly more useful ver... |
| axcc4 10454 | A version of ~ axcc3 that ... |
| acncc 10455 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10456 | Relax the constraint on ~ ... |
| domtriomlem 10457 | Lemma for ~ domtriom . (C... |
| domtriom 10458 | Trichotomy of equinumerosi... |
| fin41 10459 | Under countable choice, th... |
| dominf 10460 | A nonempty set that is a s... |
| dcomex 10462 | The Axiom of Dependent Cho... |
| axdc2lem 10463 | Lemma for ~ axdc2 . We co... |
| axdc2 10464 | An apparent strengthening ... |
| axdc3lem 10465 | The class ` S ` of finite ... |
| axdc3lem2 10466 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10467 | Simple substitution lemma ... |
| axdc3lem4 10468 | Lemma for ~ axdc3 . We ha... |
| axdc3 10469 | Dependent Choice. Axiom D... |
| axdc4lem 10470 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10471 | A more general version of ... |
| axcclem 10472 | Lemma for ~ axcc . (Contr... |
| axcc 10473 | Although CC can be proven ... |
| zfac 10475 | Axiom of Choice expressed ... |
| ac2 10476 | Axiom of Choice equivalent... |
| ac3 10477 | Axiom of Choice using abbr... |
| axac3 10479 | This theorem asserts that ... |
| ackm 10480 | A remarkable equivalent to... |
| axac2 10481 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10482 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10483 | Apply a choice equivalent.... |
| cardeqv 10484 | All sets are well-orderabl... |
| numth3 10485 | All sets are well-orderabl... |
| numth2 10486 | Numeration theorem: any se... |
| numth 10487 | Numeration theorem: every ... |
| ac7 10488 | An Axiom of Choice equival... |
| ac7g 10489 | An Axiom of Choice equival... |
| ac4 10490 | Equivalent of Axiom of Cho... |
| ac4c 10491 | Equivalent of Axiom of Cho... |
| ac5 10492 | An Axiom of Choice equival... |
| ac5b 10493 | Equivalent of Axiom of Cho... |
| ac6num 10494 | A version of ~ ac6 which t... |
| ac6 10495 | Equivalent of Axiom of Cho... |
| ac6c4 10496 | Equivalent of Axiom of Cho... |
| ac6c5 10497 | Equivalent of Axiom of Cho... |
| ac9 10498 | An Axiom of Choice equival... |
| ac6s 10499 | Equivalent of Axiom of Cho... |
| ac6n 10500 | Equivalent of Axiom of Cho... |
| ac6s2 10501 | Generalization of the Axio... |
| ac6s3 10502 | Generalization of the Axio... |
| ac6sg 10503 | ~ ac6s with sethood as ant... |
| ac6sf 10504 | Version of ~ ac6 with boun... |
| ac6s4 10505 | Generalization of the Axio... |
| ac6s5 10506 | Generalization of the Axio... |
| ac8 10507 | An Axiom of Choice equival... |
| ac9s 10508 | An Axiom of Choice equival... |
| numthcor 10509 | Any set is strictly domina... |
| weth 10510 | Well-ordering theorem: any... |
| zorn2lem1 10511 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10512 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10513 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10514 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10515 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10516 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10517 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10518 | Zorn's Lemma of [Monk1] p.... |
| zorng 10519 | Zorn's Lemma. If the unio... |
| zornn0g 10520 | Variant of Zorn's lemma ~ ... |
| zorn2 10521 | Zorn's Lemma of [Monk1] p.... |
| zorn 10522 | Zorn's Lemma. If the unio... |
| zornn0 10523 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10524 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10525 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10526 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10527 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10528 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10529 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10530 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10531 | The Teichmüller-Tukey... |
| ttukeyg 10532 | The Teichmüller-Tukey... |
| ttukey 10533 | The Teichmüller-Tukey... |
| axdclem 10534 | Lemma for ~ axdc . (Contr... |
| axdclem2 10535 | Lemma for ~ axdc . Using ... |
| axdc 10536 | This theorem derives ~ ax-... |
| fodomg 10537 | An onto function implies d... |
| fodom 10538 | An onto function implies d... |
| dmct 10539 | The domain of a countable ... |
| rnct 10540 | The range of a countable s... |
| fodomb 10541 | Equivalence of an onto map... |
| wdomac 10542 | When assuming AC, weak and... |
| brdom3 10543 | Equivalence to a dominance... |
| brdom5 10544 | An equivalence to a domina... |
| brdom4 10545 | An equivalence to a domina... |
| brdom7disj 10546 | An equivalence to a domina... |
| brdom6disj 10547 | An equivalence to a domina... |
| fin71ac 10548 | Once we allow AC, the "str... |
| imadomg 10549 | An image of a function und... |
| fimact 10550 | The image by a function of... |
| fnrndomg 10551 | The range of a function is... |
| fnct 10552 | If the domain of a functio... |
| mptct 10553 | A countable mapping set is... |
| iunfo 10554 | Existence of an onto funct... |
| iundom2g 10555 | An upper bound for the car... |
| iundomg 10556 | An upper bound for the car... |
| iundom 10557 | An upper bound for the car... |
| unidom 10558 | An upper bound for the car... |
| uniimadom 10559 | An upper bound for the car... |
| uniimadomf 10560 | An upper bound for the car... |
| cardval 10561 | The value of the cardinal ... |
| cardid 10562 | Any set is equinumerous to... |
| cardidg 10563 | Any set is equinumerous to... |
| cardidd 10564 | Any set is equinumerous to... |
| cardf 10565 | The cardinality function i... |
| carden 10566 | Two sets are equinumerous ... |
| cardeq0 10567 | Only the empty set has car... |
| unsnen 10568 | Equinumerosity of a set wi... |
| carddom 10569 | Two sets have the dominanc... |
| cardsdom 10570 | Two sets have the strict d... |
| domtri 10571 | Trichotomy law for dominan... |
| entric 10572 | Trichotomy of equinumerosi... |
| entri2 10573 | Trichotomy of dominance an... |
| entri3 10574 | Trichotomy of dominance. ... |
| sdomsdomcard 10575 | A set strictly dominates i... |
| canth3 10576 | Cantor's theorem in terms ... |
| infxpidm 10577 | Every infinite class is eq... |
| ondomon 10578 | The class of ordinals domi... |
| cardmin 10579 | The smallest ordinal that ... |
| ficard 10580 | A set is finite iff its ca... |
| infinf 10581 | Equivalence between two in... |
| unirnfdomd 10582 | The union of the range of ... |
| konigthlem 10583 | Lemma for ~ konigth . (Co... |
| konigth 10584 | Konig's Theorem. If ` m (... |
| alephsucpw 10585 | The power set of an aleph ... |
| aleph1 10586 | The set exponentiation of ... |
| alephval2 10587 | An alternate way to expres... |
| dominfac 10588 | A nonempty set that is a s... |
| iunctb 10589 | The countable union of cou... |
| unictb 10590 | The countable union of cou... |
| infmap 10591 | An exponentiation law for ... |
| alephadd 10592 | The sum of two alephs is t... |
| alephmul 10593 | The product of two alephs ... |
| alephexp1 10594 | An exponentiation law for ... |
| alephsuc3 10595 | An alternate representatio... |
| alephexp2 10596 | An expression equinumerous... |
| alephreg 10597 | A successor aleph is regul... |
| pwcfsdom 10598 | A corollary of Konig's The... |
| cfpwsdom 10599 | A corollary of Konig's The... |
| alephom 10600 | From ~ canth2 , we know th... |
| smobeth 10601 | The beth function is stric... |
| nd1 10602 | A lemma for proving condit... |
| nd2 10603 | A lemma for proving condit... |
| nd3 10604 | A lemma for proving condit... |
| nd4 10605 | A lemma for proving condit... |
| axextnd 10606 | A version of the Axiom of ... |
| axrepndlem1 10607 | Lemma for the Axiom of Rep... |
| axrepndlem2 10608 | Lemma for the Axiom of Rep... |
| axrepnd 10609 | A version of the Axiom of ... |
| axunndlem1 10610 | Lemma for the Axiom of Uni... |
| axunnd 10611 | A version of the Axiom of ... |
| axpowndlem1 10612 | Lemma for the Axiom of Pow... |
| axpowndlem2 10613 | Lemma for the Axiom of Pow... |
| axpowndlem3 10614 | Lemma for the Axiom of Pow... |
| axpowndlem4 10615 | Lemma for the Axiom of Pow... |
| axpownd 10616 | A version of the Axiom of ... |
| axregndlem1 10617 | Lemma for the Axiom of Reg... |
| axregndlem2 10618 | Lemma for the Axiom of Reg... |
| axregnd 10619 | A version of the Axiom of ... |
| axinfndlem1 10620 | Lemma for the Axiom of Inf... |
| axinfnd 10621 | A version of the Axiom of ... |
| axacndlem1 10622 | Lemma for the Axiom of Cho... |
| axacndlem2 10623 | Lemma for the Axiom of Cho... |
| axacndlem3 10624 | Lemma for the Axiom of Cho... |
| axacndlem4 10625 | Lemma for the Axiom of Cho... |
| axacndlem5 10626 | Lemma for the Axiom of Cho... |
| axacnd 10627 | A version of the Axiom of ... |
| zfcndext 10628 | Axiom of Extensionality ~ ... |
| zfcndrep 10629 | Axiom of Replacement ~ ax-... |
| zfcndun 10630 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10631 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10632 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10633 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10634 | Axiom of Choice ~ ax-ac , ... |
| elgch 10637 | Elementhood in the collect... |
| fingch 10638 | A finite set is a GCH-set.... |
| gchi 10639 | The only GCH-sets which ha... |
| gchen1 10640 | If ` A <_ B < ~P A ` , and... |
| gchen2 10641 | If ` A < B <_ ~P A ` , and... |
| gchor 10642 | If ` A <_ B <_ ~P A ` , an... |
| engch 10643 | The property of being a GC... |
| gchdomtri 10644 | Under certain conditions, ... |
| fpwwe2cbv 10645 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10646 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10647 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10648 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10649 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10650 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10651 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10652 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10653 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10654 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10655 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10656 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10657 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10658 | Given any function ` F ` f... |
| fpwwecbv 10659 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10660 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10661 | Given any function ` F ` f... |
| canth4 10662 | An "effective" form of Can... |
| canthnumlem 10663 | Lemma for ~ canthnum . (C... |
| canthnum 10664 | The set of well-orderable ... |
| canthwelem 10665 | Lemma for ~ canthwe . (Co... |
| canthwe 10666 | The set of well-orders of ... |
| canthp1lem1 10667 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10668 | Lemma for ~ canthp1 . (Co... |
| canthp1 10669 | A slightly stronger form o... |
| finngch 10670 | The exclusion of finite se... |
| gchdju1 10671 | An infinite GCH-set is ide... |
| gchinf 10672 | An infinite GCH-set is Ded... |
| pwfseqlem1 10673 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10674 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10675 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10676 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10677 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10678 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10679 | The powerset of a Dedekind... |
| pwxpndom2 10680 | The powerset of a Dedekind... |
| pwxpndom 10681 | The powerset of a Dedekind... |
| pwdjundom 10682 | The powerset of a Dedekind... |
| gchdjuidm 10683 | An infinite GCH-set is ide... |
| gchxpidm 10684 | An infinite GCH-set is ide... |
| gchpwdom 10685 | A relationship between dom... |
| gchaleph 10686 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10687 | If ` ( aleph `` A ) ` and ... |
| hargch 10688 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10689 | If ` ( aleph `` suc A ) ` ... |
| gch2 10690 | It is sufficient to requir... |
| gch3 10691 | An equivalent formulation ... |
| gch-kn 10692 | The equivalence of two ver... |
| gchaclem 10693 | Lemma for ~ gchac (obsolet... |
| gchhar 10694 | A "local" form of ~ gchac ... |
| gchacg 10695 | A "local" form of ~ gchac ... |
| gchac 10696 | The Generalized Continuum ... |
| elwina 10701 | Conditions of weak inacces... |
| elina 10702 | Conditions of strong inacc... |
| winaon 10703 | A weakly inaccessible card... |
| inawinalem 10704 | Lemma for ~ inawina . (Co... |
| inawina 10705 | Every strongly inaccessibl... |
| omina 10706 | ` _om ` is a strongly inac... |
| winacard 10707 | A weakly inaccessible card... |
| winainflem 10708 | A weakly inaccessible card... |
| winainf 10709 | A weakly inaccessible card... |
| winalim 10710 | A weakly inaccessible card... |
| winalim2 10711 | A nontrivial weakly inacce... |
| winafp 10712 | A nontrivial weakly inacce... |
| winafpi 10713 | This theorem, which states... |
| gchina 10714 | Assuming the GCH, weakly a... |
| iswun 10719 | Properties of a weak unive... |
| wuntr 10720 | A weak universe is transit... |
| wununi 10721 | A weak universe is closed ... |
| wunpw 10722 | A weak universe is closed ... |
| wunelss 10723 | The elements of a weak uni... |
| wunpr 10724 | A weak universe is closed ... |
| wunun 10725 | A weak universe is closed ... |
| wuntp 10726 | A weak universe is closed ... |
| wunss 10727 | A weak universe is closed ... |
| wunin 10728 | A weak universe is closed ... |
| wundif 10729 | A weak universe is closed ... |
| wunint 10730 | A weak universe is closed ... |
| wunsn 10731 | A weak universe is closed ... |
| wunsuc 10732 | A weak universe is closed ... |
| wun0 10733 | A weak universe contains t... |
| wunr1om 10734 | A weak universe is infinit... |
| wunom 10735 | A weak universe contains a... |
| wunfi 10736 | A weak universe contains a... |
| wunop 10737 | A weak universe is closed ... |
| wunot 10738 | A weak universe is closed ... |
| wunxp 10739 | A weak universe is closed ... |
| wunpm 10740 | A weak universe is closed ... |
| wunmap 10741 | A weak universe is closed ... |
| wunf 10742 | A weak universe is closed ... |
| wundm 10743 | A weak universe is closed ... |
| wunrn 10744 | A weak universe is closed ... |
| wuncnv 10745 | A weak universe is closed ... |
| wunres 10746 | A weak universe is closed ... |
| wunfv 10747 | A weak universe is closed ... |
| wunco 10748 | A weak universe is closed ... |
| wuntpos 10749 | A weak universe is closed ... |
| intwun 10750 | The intersection of a coll... |
| r1limwun 10751 | Each limit stage in the cu... |
| r1wunlim 10752 | The weak universes in the ... |
| wunex2 10753 | Construct a weak universe ... |
| wunex 10754 | Construct a weak universe ... |
| uniwun 10755 | Every set is contained in ... |
| wunex3 10756 | Construct a weak universe ... |
| wuncval 10757 | Value of the weak universe... |
| wuncid 10758 | The weak universe closure ... |
| wunccl 10759 | The weak universe closure ... |
| wuncss 10760 | The weak universe closure ... |
| wuncidm 10761 | The weak universe closure ... |
| wuncval2 10762 | Our earlier expression for... |
| eltskg 10765 | Properties of a Tarski cla... |
| eltsk2g 10766 | Properties of a Tarski cla... |
| tskpwss 10767 | First axiom of a Tarski cl... |
| tskpw 10768 | Second axiom of a Tarski c... |
| tsken 10769 | Third axiom of a Tarski cl... |
| 0tsk 10770 | The empty set is a (transi... |
| tsksdom 10771 | An element of a Tarski cla... |
| tskssel 10772 | A part of a Tarski class s... |
| tskss 10773 | The subsets of an element ... |
| tskin 10774 | The intersection of two el... |
| tsksn 10775 | A singleton of an element ... |
| tsktrss 10776 | A transitive element of a ... |
| tsksuc 10777 | If an element of a Tarski ... |
| tsk0 10778 | A nonempty Tarski class co... |
| tsk1 10779 | One is an element of a non... |
| tsk2 10780 | Two is an element of a non... |
| 2domtsk 10781 | If a Tarski class is not e... |
| tskr1om 10782 | A nonempty Tarski class is... |
| tskr1om2 10783 | A nonempty Tarski class co... |
| tskinf 10784 | A nonempty Tarski class is... |
| tskpr 10785 | If ` A ` and ` B ` are mem... |
| tskop 10786 | If ` A ` and ` B ` are mem... |
| tskxpss 10787 | A Cartesian product of two... |
| tskwe2 10788 | A Tarski class is well-ord... |
| inttsk 10789 | The intersection of a coll... |
| inar1 10790 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10791 | Alternate proof of ~ r1om ... |
| rankcf 10792 | Any set must be at least a... |
| inatsk 10793 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10794 | The set of hereditarily fi... |
| tskord 10795 | A Tarski class contains al... |
| tskcard 10796 | An even more direct relati... |
| r1tskina 10797 | There is a direct relation... |
| tskuni 10798 | The union of an element of... |
| tskwun 10799 | A nonempty transitive Tars... |
| tskint 10800 | The intersection of an ele... |
| tskun 10801 | The union of two elements ... |
| tskxp 10802 | The Cartesian product of t... |
| tskmap 10803 | Set exponentiation is an e... |
| tskurn 10804 | A transitive Tarski class ... |
| elgrug 10807 | Properties of a Grothendie... |
| grutr 10808 | A Grothendieck universe is... |
| gruelss 10809 | A Grothendieck universe is... |
| grupw 10810 | A Grothendieck universe co... |
| gruss 10811 | Any subset of an element o... |
| grupr 10812 | A Grothendieck universe co... |
| gruurn 10813 | A Grothendieck universe co... |
| gruiun 10814 | If ` B ( x ) ` is a family... |
| gruuni 10815 | A Grothendieck universe co... |
| grurn 10816 | A Grothendieck universe co... |
| gruima 10817 | A Grothendieck universe co... |
| gruel 10818 | Any element of an element ... |
| grusn 10819 | A Grothendieck universe co... |
| gruop 10820 | A Grothendieck universe co... |
| gruun 10821 | A Grothendieck universe co... |
| gruxp 10822 | A Grothendieck universe co... |
| grumap 10823 | A Grothendieck universe co... |
| gruixp 10824 | A Grothendieck universe co... |
| gruiin 10825 | A Grothendieck universe co... |
| gruf 10826 | A Grothendieck universe co... |
| gruen 10827 | A Grothendieck universe co... |
| gruwun 10828 | A nonempty Grothendieck un... |
| intgru 10829 | The intersection of a fami... |
| ingru 10830 | The intersection of a univ... |
| wfgru 10831 | The wellfounded part of a ... |
| grudomon 10832 | Each ordinal that is compa... |
| gruina 10833 | If a Grothendieck universe... |
| grur1a 10834 | A characterization of Grot... |
| grur1 10835 | A characterization of Grot... |
| grutsk1 10836 | Grothendieck universes are... |
| grutsk 10837 | Grothendieck universes are... |
| axgroth5 10839 | The Tarski-Grothendieck ax... |
| axgroth2 10840 | Alternate version of the T... |
| grothpw 10841 | Derive the Axiom of Power ... |
| grothpwex 10842 | Derive the Axiom of Power ... |
| axgroth6 10843 | The Tarski-Grothendieck ax... |
| grothomex 10844 | The Tarski-Grothendieck Ax... |
| grothac 10845 | The Tarski-Grothendieck Ax... |
| axgroth3 10846 | Alternate version of the T... |
| axgroth4 10847 | Alternate version of the T... |
| grothprimlem 10848 | Lemma for ~ grothprim . E... |
| grothprim 10849 | The Tarski-Grothendieck Ax... |
| grothtsk 10850 | The Tarski-Grothendieck Ax... |
| inaprc 10851 | An equivalent to the Tarsk... |
| tskmval 10854 | Value of our tarski map. ... |
| tskmid 10855 | The set ` A ` is an elemen... |
| tskmcl 10856 | A Tarski class that contai... |
| sstskm 10857 | Being a part of ` ( tarski... |
| eltskm 10858 | Belonging to ` ( tarskiMap... |
| elni 10891 | Membership in the class of... |
| elni2 10892 | Membership in the class of... |
| pinn 10893 | A positive integer is a na... |
| pion 10894 | A positive integer is an o... |
| piord 10895 | A positive integer is ordi... |
| niex 10896 | The class of positive inte... |
| 0npi 10897 | The empty set is not a pos... |
| 1pi 10898 | Ordinal 'one' is a positiv... |
| addpiord 10899 | Positive integer addition ... |
| mulpiord 10900 | Positive integer multiplic... |
| mulidpi 10901 | 1 is an identity element f... |
| ltpiord 10902 | Positive integer 'less tha... |
| ltsopi 10903 | Positive integer 'less tha... |
| ltrelpi 10904 | Positive integer 'less tha... |
| dmaddpi 10905 | Domain of addition on posi... |
| dmmulpi 10906 | Domain of multiplication o... |
| addclpi 10907 | Closure of addition of pos... |
| mulclpi 10908 | Closure of multiplication ... |
| addcompi 10909 | Addition of positive integ... |
| addasspi 10910 | Addition of positive integ... |
| mulcompi 10911 | Multiplication of positive... |
| mulasspi 10912 | Multiplication of positive... |
| distrpi 10913 | Multiplication of positive... |
| addcanpi 10914 | Addition cancellation law ... |
| mulcanpi 10915 | Multiplication cancellatio... |
| addnidpi 10916 | There is no identity eleme... |
| ltexpi 10917 | Ordering on positive integ... |
| ltapi 10918 | Ordering property of addit... |
| ltmpi 10919 | Ordering property of multi... |
| 1lt2pi 10920 | One is less than two (one ... |
| nlt1pi 10921 | No positive integer is les... |
| indpi 10922 | Principle of Finite Induct... |
| enqbreq 10934 | Equivalence relation for p... |
| enqbreq2 10935 | Equivalence relation for p... |
| enqer 10936 | The equivalence relation f... |
| enqex 10937 | The equivalence relation f... |
| nqex 10938 | The class of positive frac... |
| 0nnq 10939 | The empty set is not a pos... |
| elpqn 10940 | Each positive fraction is ... |
| ltrelnq 10941 | Positive fraction 'less th... |
| pinq 10942 | The representatives of pos... |
| 1nq 10943 | The positive fraction 'one... |
| nqereu 10944 | There is a unique element ... |
| nqerf 10945 | Corollary of ~ nqereu : th... |
| nqercl 10946 | Corollary of ~ nqereu : cl... |
| nqerrel 10947 | Any member of ` ( N. X. N.... |
| nqerid 10948 | Corollary of ~ nqereu : th... |
| enqeq 10949 | Corollary of ~ nqereu : if... |
| nqereq 10950 | The function ` /Q ` acts a... |
| addpipq2 10951 | Addition of positive fract... |
| addpipq 10952 | Addition of positive fract... |
| addpqnq 10953 | Addition of positive fract... |
| mulpipq2 10954 | Multiplication of positive... |
| mulpipq 10955 | Multiplication of positive... |
| mulpqnq 10956 | Multiplication of positive... |
| ordpipq 10957 | Ordering of positive fract... |
| ordpinq 10958 | Ordering of positive fract... |
| addpqf 10959 | Closure of addition on pos... |
| addclnq 10960 | Closure of addition on pos... |
| mulpqf 10961 | Closure of multiplication ... |
| mulclnq 10962 | Closure of multiplication ... |
| addnqf 10963 | Domain of addition on posi... |
| mulnqf 10964 | Domain of multiplication o... |
| addcompq 10965 | Addition of positive fract... |
| addcomnq 10966 | Addition of positive fract... |
| mulcompq 10967 | Multiplication of positive... |
| mulcomnq 10968 | Multiplication of positive... |
| adderpqlem 10969 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10970 | Lemma for ~ mulerpq . (Co... |
| adderpq 10971 | Addition is compatible wit... |
| mulerpq 10972 | Multiplication is compatib... |
| addassnq 10973 | Addition of positive fract... |
| mulassnq 10974 | Multiplication of positive... |
| mulcanenq 10975 | Lemma for distributive law... |
| distrnq 10976 | Multiplication of positive... |
| 1nqenq 10977 | The equivalence class of r... |
| mulidnq 10978 | Multiplication identity el... |
| recmulnq 10979 | Relationship between recip... |
| recidnq 10980 | A positive fraction times ... |
| recclnq 10981 | Closure law for positive f... |
| recrecnq 10982 | Reciprocal of reciprocal o... |
| dmrecnq 10983 | Domain of reciprocal on po... |
| ltsonq 10984 | 'Less than' is a strict or... |
| lterpq 10985 | Compatibility of ordering ... |
| ltanq 10986 | Ordering property of addit... |
| ltmnq 10987 | Ordering property of multi... |
| 1lt2nq 10988 | One is less than two (one ... |
| ltaddnq 10989 | The sum of two fractions i... |
| ltexnq 10990 | Ordering on positive fract... |
| halfnq 10991 | One-half of any positive f... |
| nsmallnq 10992 | The is no smallest positiv... |
| ltbtwnnq 10993 | There exists a number betw... |
| ltrnq 10994 | Ordering property of recip... |
| archnq 10995 | For any fraction, there is... |
| npex 11001 | The class of positive real... |
| elnp 11002 | Membership in positive rea... |
| elnpi 11003 | Membership in positive rea... |
| prn0 11004 | A positive real is not emp... |
| prpssnq 11005 | A positive real is a subse... |
| elprnq 11006 | A positive real is a set o... |
| 0npr 11007 | The empty set is not a pos... |
| prcdnq 11008 | A positive real is closed ... |
| prub 11009 | A positive fraction not in... |
| prnmax 11010 | A positive real has no lar... |
| npomex 11011 | A simplifying observation,... |
| prnmadd 11012 | A positive real has no lar... |
| ltrelpr 11013 | Positive real 'less than' ... |
| genpv 11014 | Value of general operation... |
| genpelv 11015 | Membership in value of gen... |
| genpprecl 11016 | Pre-closure law for genera... |
| genpdm 11017 | Domain of general operatio... |
| genpn0 11018 | The result of an operation... |
| genpss 11019 | The result of an operation... |
| genpnnp 11020 | The result of an operation... |
| genpcd 11021 | Downward closure of an ope... |
| genpnmax 11022 | An operation on positive r... |
| genpcl 11023 | Closure of an operation on... |
| genpass 11024 | Associativity of an operat... |
| plpv 11025 | Value of addition on posit... |
| mpv 11026 | Value of multiplication on... |
| dmplp 11027 | Domain of addition on posi... |
| dmmp 11028 | Domain of multiplication o... |
| nqpr 11029 | The canonical embedding of... |
| 1pr 11030 | The positive real number '... |
| addclprlem1 11031 | Lemma to prove downward cl... |
| addclprlem2 11032 | Lemma to prove downward cl... |
| addclpr 11033 | Closure of addition on pos... |
| mulclprlem 11034 | Lemma to prove downward cl... |
| mulclpr 11035 | Closure of multiplication ... |
| addcompr 11036 | Addition of positive reals... |
| addasspr 11037 | Addition of positive reals... |
| mulcompr 11038 | Multiplication of positive... |
| mulasspr 11039 | Multiplication of positive... |
| distrlem1pr 11040 | Lemma for distributive law... |
| distrlem4pr 11041 | Lemma for distributive law... |
| distrlem5pr 11042 | Lemma for distributive law... |
| distrpr 11043 | Multiplication of positive... |
| 1idpr 11044 | 1 is an identity element f... |
| ltprord 11045 | Positive real 'less than' ... |
| psslinpr 11046 | Proper subset is a linear ... |
| ltsopr 11047 | Positive real 'less than' ... |
| prlem934 11048 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 11049 | The sum of two positive re... |
| ltaddpr2 11050 | The sum of two positive re... |
| ltexprlem1 11051 | Lemma for Proposition 9-3.... |
| ltexprlem2 11052 | Lemma for Proposition 9-3.... |
| ltexprlem3 11053 | Lemma for Proposition 9-3.... |
| ltexprlem4 11054 | Lemma for Proposition 9-3.... |
| ltexprlem5 11055 | Lemma for Proposition 9-3.... |
| ltexprlem6 11056 | Lemma for Proposition 9-3.... |
| ltexprlem7 11057 | Lemma for Proposition 9-3.... |
| ltexpri 11058 | Proposition 9-3.5(iv) of [... |
| ltaprlem 11059 | Lemma for Proposition 9-3.... |
| ltapr 11060 | Ordering property of addit... |
| addcanpr 11061 | Addition cancellation law ... |
| prlem936 11062 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 11063 | Lemma for Proposition 9-3.... |
| reclem3pr 11064 | Lemma for Proposition 9-3.... |
| reclem4pr 11065 | Lemma for Proposition 9-3.... |
| recexpr 11066 | The reciprocal of a positi... |
| suplem1pr 11067 | The union of a nonempty, b... |
| suplem2pr 11068 | The union of a set of posi... |
| supexpr 11069 | The union of a nonempty, b... |
| enrer 11078 | The equivalence relation f... |
| nrex1 11079 | The class of signed reals ... |
| enrbreq 11080 | Equivalence relation for s... |
| enreceq 11081 | Equivalence class equality... |
| enrex 11082 | The equivalence relation f... |
| ltrelsr 11083 | Signed real 'less than' is... |
| addcmpblnr 11084 | Lemma showing compatibilit... |
| mulcmpblnrlem 11085 | Lemma used in lemma showin... |
| mulcmpblnr 11086 | Lemma showing compatibilit... |
| prsrlem1 11087 | Decomposing signed reals i... |
| addsrmo 11088 | There is at most one resul... |
| mulsrmo 11089 | There is at most one resul... |
| addsrpr 11090 | Addition of signed reals i... |
| mulsrpr 11091 | Multiplication of signed r... |
| ltsrpr 11092 | Ordering of signed reals i... |
| gt0srpr 11093 | Greater than zero in terms... |
| 0nsr 11094 | The empty set is not a sig... |
| 0r 11095 | The constant ` 0R ` is a s... |
| 1sr 11096 | The constant ` 1R ` is a s... |
| m1r 11097 | The constant ` -1R ` is a ... |
| addclsr 11098 | Closure of addition on sig... |
| mulclsr 11099 | Closure of multiplication ... |
| dmaddsr 11100 | Domain of addition on sign... |
| dmmulsr 11101 | Domain of multiplication o... |
| addcomsr 11102 | Addition of signed reals i... |
| addasssr 11103 | Addition of signed reals i... |
| mulcomsr 11104 | Multiplication of signed r... |
| mulasssr 11105 | Multiplication of signed r... |
| distrsr 11106 | Multiplication of signed r... |
| m1p1sr 11107 | Minus one plus one is zero... |
| m1m1sr 11108 | Minus one times minus one ... |
| ltsosr 11109 | Signed real 'less than' is... |
| 0lt1sr 11110 | 0 is less than 1 for signe... |
| 1ne0sr 11111 | 1 and 0 are distinct for s... |
| 0idsr 11112 | The signed real number 0 i... |
| 1idsr 11113 | 1 is an identity element f... |
| 00sr 11114 | A signed real times 0 is 0... |
| ltasr 11115 | Ordering property of addit... |
| pn0sr 11116 | A signed real plus its neg... |
| negexsr 11117 | Existence of negative sign... |
| recexsrlem 11118 | The reciprocal of a positi... |
| addgt0sr 11119 | The sum of two positive si... |
| mulgt0sr 11120 | The product of two positiv... |
| sqgt0sr 11121 | The square of a nonzero si... |
| recexsr 11122 | The reciprocal of a nonzer... |
| mappsrpr 11123 | Mapping from positive sign... |
| ltpsrpr 11124 | Mapping of order from posi... |
| map2psrpr 11125 | Equivalence for positive s... |
| supsrlem 11126 | Lemma for supremum theorem... |
| supsr 11127 | A nonempty, bounded set of... |
| opelcn 11144 | Ordered pair membership in... |
| opelreal 11145 | Ordered pair membership in... |
| elreal 11146 | Membership in class of rea... |
| elreal2 11147 | Ordered pair membership in... |
| 0ncn 11148 | The empty set is not a com... |
| ltrelre 11149 | 'Less than' is a relation ... |
| addcnsr 11150 | Addition of complex number... |
| mulcnsr 11151 | Multiplication of complex ... |
| eqresr 11152 | Equality of real numbers i... |
| addresr 11153 | Addition of real numbers i... |
| mulresr 11154 | Multiplication of real num... |
| ltresr 11155 | Ordering of real subset of... |
| ltresr2 11156 | Ordering of real subset of... |
| dfcnqs 11157 | Technical trick to permit ... |
| addcnsrec 11158 | Technical trick to permit ... |
| mulcnsrec 11159 | Technical trick to permit ... |
| axaddf 11160 | Addition is an operation o... |
| axmulf 11161 | Multiplication is an opera... |
| axcnex 11162 | The complex numbers form a... |
| axresscn 11163 | The real numbers are a sub... |
| ax1cn 11164 | 1 is a complex number. Ax... |
| axicn 11165 | ` _i ` is a complex number... |
| axaddcl 11166 | Closure law for addition o... |
| axaddrcl 11167 | Closure law for addition i... |
| axmulcl 11168 | Closure law for multiplica... |
| axmulrcl 11169 | Closure law for multiplica... |
| axmulcom 11170 | Multiplication of complex ... |
| axaddass 11171 | Addition of complex number... |
| axmulass 11172 | Multiplication of complex ... |
| axdistr 11173 | Distributive law for compl... |
| axi2m1 11174 | i-squared equals -1 (expre... |
| ax1ne0 11175 | 1 and 0 are distinct. Axi... |
| ax1rid 11176 | ` 1 ` is an identity eleme... |
| axrnegex 11177 | Existence of negative of r... |
| axrrecex 11178 | Existence of reciprocal of... |
| axcnre 11179 | A complex number can be ex... |
| axpre-lttri 11180 | Ordering on reals satisfie... |
| axpre-lttrn 11181 | Ordering on reals is trans... |
| axpre-ltadd 11182 | Ordering property of addit... |
| axpre-mulgt0 11183 | The product of two positiv... |
| axpre-sup 11184 | A nonempty, bounded-above ... |
| wuncn 11185 | A weak universe containing... |
| cnex 11211 | Alias for ~ ax-cnex . See... |
| addcl 11212 | Alias for ~ ax-addcl , for... |
| readdcl 11213 | Alias for ~ ax-addrcl , fo... |
| mulcl 11214 | Alias for ~ ax-mulcl , for... |
| remulcl 11215 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11216 | Alias for ~ ax-mulcom , fo... |
| addass 11217 | Alias for ~ ax-addass , fo... |
| mulass 11218 | Alias for ~ ax-mulass , fo... |
| adddi 11219 | Alias for ~ ax-distr , for... |
| recn 11220 | A real number is a complex... |
| reex 11221 | The real numbers form a se... |
| reelprrecn 11222 | Reals are a subset of the ... |
| cnelprrecn 11223 | Complex numbers are a subs... |
| mpoaddf 11224 | Addition is an operation o... |
| mpomulf 11225 | Multiplication is an opera... |
| elimne0 11226 | Hypothesis for weak deduct... |
| adddir 11227 | Distributive law for compl... |
| 0cn 11228 | Zero is a complex number. ... |
| 0cnd 11229 | Zero is a complex number, ... |
| c0ex 11230 | Zero is a set. (Contribut... |
| 1cnd 11231 | One is a complex number, d... |
| 1ex 11232 | One is a set. (Contribute... |
| cnre 11233 | Alias for ~ ax-cnre , for ... |
| mulrid 11234 | The number 1 is an identit... |
| mullid 11235 | Identity law for multiplic... |
| 1re 11236 | The number 1 is real. Thi... |
| 1red 11237 | The number 1 is real, dedu... |
| 0re 11238 | The number 0 is real. Rem... |
| 0red 11239 | The number 0 is real, dedu... |
| mulridi 11240 | Identity law for multiplic... |
| mullidi 11241 | Identity law for multiplic... |
| addcli 11242 | Closure law for addition. ... |
| mulcli 11243 | Closure law for multiplica... |
| mulcomi 11244 | Commutative law for multip... |
| mulcomli 11245 | Commutative law for multip... |
| addassi 11246 | Associative law for additi... |
| mulassi 11247 | Associative law for multip... |
| adddii 11248 | Distributive law (left-dis... |
| adddiri 11249 | Distributive law (right-di... |
| recni 11250 | A real number is a complex... |
| readdcli 11251 | Closure law for addition o... |
| remulcli 11252 | Closure law for multiplica... |
| mulridd 11253 | Identity law for multiplic... |
| mullidd 11254 | Identity law for multiplic... |
| addcld 11255 | Closure law for addition. ... |
| mulcld 11256 | Closure law for multiplica... |
| mulcomd 11257 | Commutative law for multip... |
| addassd 11258 | Associative law for additi... |
| mulassd 11259 | Associative law for multip... |
| adddid 11260 | Distributive law (left-dis... |
| adddird 11261 | Distributive law (right-di... |
| adddirp1d 11262 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11263 | Join AB+CB into (A+C) on L... |
| recnd 11264 | Deduction from real number... |
| readdcld 11265 | Closure law for addition o... |
| remulcld 11266 | Closure law for multiplica... |
| pnfnre 11277 | Plus infinity is not a rea... |
| pnfnre2 11278 | Plus infinity is not a rea... |
| mnfnre 11279 | Minus infinity is not a re... |
| ressxr 11280 | The standard reals are a s... |
| rexpssxrxp 11281 | The Cartesian product of s... |
| rexr 11282 | A standard real is an exte... |
| 0xr 11283 | Zero is an extended real. ... |
| renepnf 11284 | No (finite) real equals pl... |
| renemnf 11285 | No real equals minus infin... |
| rexrd 11286 | A standard real is an exte... |
| renepnfd 11287 | No (finite) real equals pl... |
| renemnfd 11288 | No real equals minus infin... |
| pnfex 11289 | Plus infinity exists. (Co... |
| pnfxr 11290 | Plus infinity belongs to t... |
| pnfnemnf 11291 | Plus and minus infinity ar... |
| mnfnepnf 11292 | Minus and plus infinity ar... |
| mnfxr 11293 | Minus infinity belongs to ... |
| rexri 11294 | A standard real is an exte... |
| 1xr 11295 | ` 1 ` is an extended real ... |
| renfdisj 11296 | The reals and the infiniti... |
| ltrelxr 11297 | "Less than" is a relation ... |
| ltrel 11298 | "Less than" is a relation.... |
| lerelxr 11299 | "Less than or equal to" is... |
| lerel 11300 | "Less than or equal to" is... |
| xrlenlt 11301 | "Less than or equal to" ex... |
| xrlenltd 11302 | "Less than or equal to" ex... |
| xrltnle 11303 | "Less than" expressed in t... |
| xrnltled 11304 | "Not less than" implies "l... |
| ssxr 11305 | The three (non-exclusive) ... |
| ltxrlt 11306 | The standard less-than ` <... |
| axlttri 11307 | Ordering on reals satisfie... |
| axlttrn 11308 | Ordering on reals is trans... |
| axltadd 11309 | Ordering property of addit... |
| axmulgt0 11310 | The product of two positiv... |
| axsup 11311 | A nonempty, bounded-above ... |
| lttr 11312 | Alias for ~ axlttrn , for ... |
| mulgt0 11313 | The product of two positiv... |
| lenlt 11314 | 'Less than or equal to' ex... |
| ltnle 11315 | 'Less than' expressed in t... |
| ltso 11316 | 'Less than' is a strict or... |
| gtso 11317 | 'Greater than' is a strict... |
| lttri2 11318 | Consequence of trichotomy.... |
| lttri3 11319 | Trichotomy law for 'less t... |
| lttri4 11320 | Trichotomy law for 'less t... |
| letri3 11321 | Trichotomy law. (Contribu... |
| leloe 11322 | 'Less than or equal to' ex... |
| eqlelt 11323 | Equality in terms of 'less... |
| ltle 11324 | 'Less than' implies 'less ... |
| leltne 11325 | 'Less than or equal to' im... |
| lelttr 11326 | Transitive law. (Contribu... |
| leltletr 11327 | Transitive law, weaker for... |
| ltletr 11328 | Transitive law. (Contribu... |
| ltleletr 11329 | Transitive law, weaker for... |
| letr 11330 | Transitive law. (Contribu... |
| ltnr 11331 | 'Less than' is irreflexive... |
| leid 11332 | 'Less than or equal to' is... |
| ltne 11333 | 'Less than' implies not eq... |
| ltnsym 11334 | 'Less than' is not symmetr... |
| ltnsym2 11335 | 'Less than' is antisymmetr... |
| letric 11336 | Trichotomy law. (Contribu... |
| ltlen 11337 | 'Less than' expressed in t... |
| eqle 11338 | Equality implies 'less tha... |
| eqled 11339 | Equality implies 'less tha... |
| ltadd2 11340 | Addition to both sides of ... |
| ne0gt0 11341 | A nonzero nonnegative numb... |
| lecasei 11342 | Ordering elimination by ca... |
| lelttric 11343 | Trichotomy law. (Contribu... |
| ltlecasei 11344 | Ordering elimination by ca... |
| ltnri 11345 | 'Less than' is irreflexive... |
| eqlei 11346 | Equality implies 'less tha... |
| eqlei2 11347 | Equality implies 'less tha... |
| gtneii 11348 | 'Less than' implies not eq... |
| ltneii 11349 | 'Greater than' implies not... |
| lttri2i 11350 | Consequence of trichotomy.... |
| lttri3i 11351 | Consequence of trichotomy.... |
| letri3i 11352 | Consequence of trichotomy.... |
| leloei 11353 | 'Less than or equal to' in... |
| ltleni 11354 | 'Less than' expressed in t... |
| ltnsymi 11355 | 'Less than' is not symmetr... |
| lenlti 11356 | 'Less than or equal to' in... |
| ltnlei 11357 | 'Less than' in terms of 'l... |
| ltlei 11358 | 'Less than' implies 'less ... |
| ltleii 11359 | 'Less than' implies 'less ... |
| ltnei 11360 | 'Less than' implies not eq... |
| letrii 11361 | Trichotomy law for 'less t... |
| lttri 11362 | 'Less than' is transitive.... |
| lelttri 11363 | 'Less than or equal to', '... |
| ltletri 11364 | 'Less than', 'less than or... |
| letri 11365 | 'Less than or equal to' is... |
| le2tri3i 11366 | Extended trichotomy law fo... |
| ltadd2i 11367 | Addition to both sides of ... |
| mulgt0i 11368 | The product of two positiv... |
| mulgt0ii 11369 | The product of two positiv... |
| ltnrd 11370 | 'Less than' is irreflexive... |
| gtned 11371 | 'Less than' implies not eq... |
| ltned 11372 | 'Greater than' implies not... |
| ne0gt0d 11373 | A nonzero nonnegative numb... |
| lttrid 11374 | Ordering on reals satisfie... |
| lttri2d 11375 | Consequence of trichotomy.... |
| lttri3d 11376 | Consequence of trichotomy.... |
| lttri4d 11377 | Trichotomy law for 'less t... |
| letri3d 11378 | Consequence of trichotomy.... |
| leloed 11379 | 'Less than or equal to' in... |
| eqleltd 11380 | Equality in terms of 'less... |
| ltlend 11381 | 'Less than' expressed in t... |
| lenltd 11382 | 'Less than or equal to' in... |
| ltnled 11383 | 'Less than' in terms of 'l... |
| ltled 11384 | 'Less than' implies 'less ... |
| ltnsymd 11385 | 'Less than' implies 'less ... |
| nltled 11386 | 'Not less than ' implies '... |
| lensymd 11387 | 'Less than or equal to' im... |
| letrid 11388 | Trichotomy law for 'less t... |
| leltned 11389 | 'Less than or equal to' im... |
| leneltd 11390 | 'Less than or equal to' an... |
| mulgt0d 11391 | The product of two positiv... |
| ltadd2d 11392 | Addition to both sides of ... |
| letrd 11393 | Transitive law deduction f... |
| lelttrd 11394 | Transitive law deduction f... |
| ltadd2dd 11395 | Addition to both sides of ... |
| ltletrd 11396 | Transitive law deduction f... |
| lttrd 11397 | Transitive law deduction f... |
| lelttrdi 11398 | If a number is less than a... |
| dedekind 11399 | The Dedekind cut theorem. ... |
| dedekindle 11400 | The Dedekind cut theorem, ... |
| mul12 11401 | Commutative/associative la... |
| mul32 11402 | Commutative/associative la... |
| mul31 11403 | Commutative/associative la... |
| mul4 11404 | Rearrangement of 4 factors... |
| mul4r 11405 | Rearrangement of 4 factors... |
| muladd11 11406 | A simple product of sums e... |
| 1p1times 11407 | Two times a number. (Cont... |
| peano2cn 11408 | A theorem for complex numb... |
| peano2re 11409 | A theorem for reals analog... |
| readdcan 11410 | Cancellation law for addit... |
| 00id 11411 | ` 0 ` is its own additive ... |
| mul02lem1 11412 | Lemma for ~ mul02 . If an... |
| mul02lem2 11413 | Lemma for ~ mul02 . Zero ... |
| mul02 11414 | Multiplication by ` 0 ` . ... |
| mul01 11415 | Multiplication by ` 0 ` . ... |
| addrid 11416 | ` 0 ` is an additive ident... |
| cnegex 11417 | Existence of the negative ... |
| cnegex2 11418 | Existence of a left invers... |
| addlid 11419 | ` 0 ` is a left identity f... |
| addcan 11420 | Cancellation law for addit... |
| addcan2 11421 | Cancellation law for addit... |
| addcom 11422 | Addition commutes. This u... |
| addridi 11423 | ` 0 ` is an additive ident... |
| addlidi 11424 | ` 0 ` is a left identity f... |
| mul02i 11425 | Multiplication by 0. Theo... |
| mul01i 11426 | Multiplication by ` 0 ` . ... |
| addcomi 11427 | Addition commutes. Based ... |
| addcomli 11428 | Addition commutes. (Contr... |
| addcani 11429 | Cancellation law for addit... |
| addcan2i 11430 | Cancellation law for addit... |
| mul12i 11431 | Commutative/associative la... |
| mul32i 11432 | Commutative/associative la... |
| mul4i 11433 | Rearrangement of 4 factors... |
| mul02d 11434 | Multiplication by 0. Theo... |
| mul01d 11435 | Multiplication by ` 0 ` . ... |
| addridd 11436 | ` 0 ` is an additive ident... |
| addlidd 11437 | ` 0 ` is a left identity f... |
| addcomd 11438 | Addition commutes. Based ... |
| addcand 11439 | Cancellation law for addit... |
| addcan2d 11440 | Cancellation law for addit... |
| addcanad 11441 | Cancelling a term on the l... |
| addcan2ad 11442 | Cancelling a term on the r... |
| addneintrd 11443 | Introducing a term on the ... |
| addneintr2d 11444 | Introducing a term on the ... |
| mul12d 11445 | Commutative/associative la... |
| mul32d 11446 | Commutative/associative la... |
| mul31d 11447 | Commutative/associative la... |
| mul4d 11448 | Rearrangement of 4 factors... |
| muladd11r 11449 | A simple product of sums e... |
| comraddd 11450 | Commute RHS addition, in d... |
| ltaddneg 11451 | Adding a negative number t... |
| ltaddnegr 11452 | Adding a negative number t... |
| add12 11453 | Commutative/associative la... |
| add32 11454 | Commutative/associative la... |
| add32r 11455 | Commutative/associative la... |
| add4 11456 | Rearrangement of 4 terms i... |
| add42 11457 | Rearrangement of 4 terms i... |
| add12i 11458 | Commutative/associative la... |
| add32i 11459 | Commutative/associative la... |
| add4i 11460 | Rearrangement of 4 terms i... |
| add42i 11461 | Rearrangement of 4 terms i... |
| add12d 11462 | Commutative/associative la... |
| add32d 11463 | Commutative/associative la... |
| add4d 11464 | Rearrangement of 4 terms i... |
| add42d 11465 | Rearrangement of 4 terms i... |
| 0cnALT 11470 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11471 | Alternate proof of ~ 0cnAL... |
| negeu 11472 | Existential uniqueness of ... |
| subval 11473 | Value of subtraction, whic... |
| negeq 11474 | Equality theorem for negat... |
| negeqi 11475 | Equality inference for neg... |
| negeqd 11476 | Equality deduction for neg... |
| nfnegd 11477 | Deduction version of ~ nfn... |
| nfneg 11478 | Bound-variable hypothesis ... |
| csbnegg 11479 | Move class substitution in... |
| negex 11480 | A negative is a set. (Con... |
| subcl 11481 | Closure law for subtractio... |
| negcl 11482 | Closure law for negative. ... |
| negicn 11483 | ` -u _i ` is a complex num... |
| subf 11484 | Subtraction is an operatio... |
| subadd 11485 | Relationship between subtr... |
| subadd2 11486 | Relationship between subtr... |
| subsub23 11487 | Swap subtrahend and result... |
| pncan 11488 | Cancellation law for subtr... |
| pncan2 11489 | Cancellation law for subtr... |
| pncan3 11490 | Subtraction and addition o... |
| npcan 11491 | Cancellation law for subtr... |
| addsubass 11492 | Associative-type law for a... |
| addsub 11493 | Law for addition and subtr... |
| subadd23 11494 | Commutative/associative la... |
| addsub12 11495 | Commutative/associative la... |
| 2addsub 11496 | Law for subtraction and ad... |
| addsubeq4 11497 | Relation between sums and ... |
| pncan3oi 11498 | Subtraction and addition o... |
| mvrraddi 11499 | Move the right term in a s... |
| mvlladdi 11500 | Move the left term in a su... |
| subid 11501 | Subtraction of a number fr... |
| subid1 11502 | Identity law for subtracti... |
| npncan 11503 | Cancellation law for subtr... |
| nppcan 11504 | Cancellation law for subtr... |
| nnpcan 11505 | Cancellation law for subtr... |
| nppcan3 11506 | Cancellation law for subtr... |
| subcan2 11507 | Cancellation law for subtr... |
| subeq0 11508 | If the difference between ... |
| npncan2 11509 | Cancellation law for subtr... |
| subsub2 11510 | Law for double subtraction... |
| nncan 11511 | Cancellation law for subtr... |
| subsub 11512 | Law for double subtraction... |
| nppcan2 11513 | Cancellation law for subtr... |
| subsub3 11514 | Law for double subtraction... |
| subsub4 11515 | Law for double subtraction... |
| sub32 11516 | Swap the second and third ... |
| nnncan 11517 | Cancellation law for subtr... |
| nnncan1 11518 | Cancellation law for subtr... |
| nnncan2 11519 | Cancellation law for subtr... |
| npncan3 11520 | Cancellation law for subtr... |
| pnpcan 11521 | Cancellation law for mixed... |
| pnpcan2 11522 | Cancellation law for mixed... |
| pnncan 11523 | Cancellation law for mixed... |
| ppncan 11524 | Cancellation law for mixed... |
| addsub4 11525 | Rearrangement of 4 terms i... |
| subadd4 11526 | Rearrangement of 4 terms i... |
| sub4 11527 | Rearrangement of 4 terms i... |
| neg0 11528 | Minus 0 equals 0. (Contri... |
| negid 11529 | Addition of a number and i... |
| negsub 11530 | Relationship between subtr... |
| subneg 11531 | Relationship between subtr... |
| negneg 11532 | A number is equal to the n... |
| neg11 11533 | Negative is one-to-one. (... |
| negcon1 11534 | Negative contraposition la... |
| negcon2 11535 | Negative contraposition la... |
| negeq0 11536 | A number is zero iff its n... |
| subcan 11537 | Cancellation law for subtr... |
| negsubdi 11538 | Distribution of negative o... |
| negdi 11539 | Distribution of negative o... |
| negdi2 11540 | Distribution of negative o... |
| negsubdi2 11541 | Distribution of negative o... |
| neg2sub 11542 | Relationship between subtr... |
| renegcli 11543 | Closure law for negative o... |
| resubcli 11544 | Closure law for subtractio... |
| renegcl 11545 | Closure law for negative o... |
| resubcl 11546 | Closure law for subtractio... |
| negreb 11547 | The negative of a real is ... |
| peano2cnm 11548 | "Reverse" second Peano pos... |
| peano2rem 11549 | "Reverse" second Peano pos... |
| negcli 11550 | Closure law for negative. ... |
| negidi 11551 | Addition of a number and i... |
| negnegi 11552 | A number is equal to the n... |
| subidi 11553 | Subtraction of a number fr... |
| subid1i 11554 | Identity law for subtracti... |
| negne0bi 11555 | A number is nonzero iff it... |
| negrebi 11556 | The negative of a real is ... |
| negne0i 11557 | The negative of a nonzero ... |
| subcli 11558 | Closure law for subtractio... |
| pncan3i 11559 | Subtraction and addition o... |
| negsubi 11560 | Relationship between subtr... |
| subnegi 11561 | Relationship between subtr... |
| subeq0i 11562 | If the difference between ... |
| neg11i 11563 | Negative is one-to-one. (... |
| negcon1i 11564 | Negative contraposition la... |
| negcon2i 11565 | Negative contraposition la... |
| negdii 11566 | Distribution of negative o... |
| negsubdii 11567 | Distribution of negative o... |
| negsubdi2i 11568 | Distribution of negative o... |
| subaddi 11569 | Relationship between subtr... |
| subadd2i 11570 | Relationship between subtr... |
| subaddrii 11571 | Relationship between subtr... |
| subsub23i 11572 | Swap subtrahend and result... |
| addsubassi 11573 | Associative-type law for s... |
| addsubi 11574 | Law for subtraction and ad... |
| subcani 11575 | Cancellation law for subtr... |
| subcan2i 11576 | Cancellation law for subtr... |
| pnncani 11577 | Cancellation law for mixed... |
| addsub4i 11578 | Rearrangement of 4 terms i... |
| 0reALT 11579 | Alternate proof of ~ 0re .... |
| negcld 11580 | Closure law for negative. ... |
| subidd 11581 | Subtraction of a number fr... |
| subid1d 11582 | Identity law for subtracti... |
| negidd 11583 | Addition of a number and i... |
| negnegd 11584 | A number is equal to the n... |
| negeq0d 11585 | A number is zero iff its n... |
| negne0bd 11586 | A number is nonzero iff it... |
| negcon1d 11587 | Contraposition law for una... |
| negcon1ad 11588 | Contraposition law for una... |
| neg11ad 11589 | The negatives of two compl... |
| negned 11590 | If two complex numbers are... |
| negne0d 11591 | The negative of a nonzero ... |
| negrebd 11592 | The negative of a real is ... |
| subcld 11593 | Closure law for subtractio... |
| pncand 11594 | Cancellation law for subtr... |
| pncan2d 11595 | Cancellation law for subtr... |
| pncan3d 11596 | Subtraction and addition o... |
| npcand 11597 | Cancellation law for subtr... |
| nncand 11598 | Cancellation law for subtr... |
| negsubd 11599 | Relationship between subtr... |
| subnegd 11600 | Relationship between subtr... |
| subeq0d 11601 | If the difference between ... |
| subne0d 11602 | Two unequal numbers have n... |
| subeq0ad 11603 | The difference of two comp... |
| subne0ad 11604 | If the difference of two c... |
| neg11d 11605 | If the difference between ... |
| negdid 11606 | Distribution of negative o... |
| negdi2d 11607 | Distribution of negative o... |
| negsubdid 11608 | Distribution of negative o... |
| negsubdi2d 11609 | Distribution of negative o... |
| neg2subd 11610 | Relationship between subtr... |
| subaddd 11611 | Relationship between subtr... |
| subadd2d 11612 | Relationship between subtr... |
| addsubassd 11613 | Associative-type law for s... |
| addsubd 11614 | Law for subtraction and ad... |
| subadd23d 11615 | Commutative/associative la... |
| addsub12d 11616 | Commutative/associative la... |
| npncand 11617 | Cancellation law for subtr... |
| nppcand 11618 | Cancellation law for subtr... |
| nppcan2d 11619 | Cancellation law for subtr... |
| nppcan3d 11620 | Cancellation law for subtr... |
| subsubd 11621 | Law for double subtraction... |
| subsub2d 11622 | Law for double subtraction... |
| subsub3d 11623 | Law for double subtraction... |
| subsub4d 11624 | Law for double subtraction... |
| sub32d 11625 | Swap the second and third ... |
| nnncand 11626 | Cancellation law for subtr... |
| nnncan1d 11627 | Cancellation law for subtr... |
| nnncan2d 11628 | Cancellation law for subtr... |
| npncan3d 11629 | Cancellation law for subtr... |
| pnpcand 11630 | Cancellation law for mixed... |
| pnpcan2d 11631 | Cancellation law for mixed... |
| pnncand 11632 | Cancellation law for mixed... |
| ppncand 11633 | Cancellation law for mixed... |
| subcand 11634 | Cancellation law for subtr... |
| subcan2d 11635 | Cancellation law for subtr... |
| subcanad 11636 | Cancellation law for subtr... |
| subneintrd 11637 | Introducing subtraction on... |
| subcan2ad 11638 | Cancellation law for subtr... |
| subneintr2d 11639 | Introducing subtraction on... |
| addsub4d 11640 | Rearrangement of 4 terms i... |
| subadd4d 11641 | Rearrangement of 4 terms i... |
| sub4d 11642 | Rearrangement of 4 terms i... |
| 2addsubd 11643 | Law for subtraction and ad... |
| addsubeq4d 11644 | Relation between sums and ... |
| subeqxfrd 11645 | Transfer two terms of a su... |
| mvlraddd 11646 | Move the right term in a s... |
| mvlladdd 11647 | Move the left term in a su... |
| mvrraddd 11648 | Move the right term in a s... |
| mvrladdd 11649 | Move the left term in a su... |
| assraddsubd 11650 | Associate RHS addition-sub... |
| subaddeqd 11651 | Transfer two terms of a su... |
| addlsub 11652 | Left-subtraction: Subtrac... |
| addrsub 11653 | Right-subtraction: Subtra... |
| subexsub 11654 | A subtraction law: Exchan... |
| addid0 11655 | If adding a number to a an... |
| addn0nid 11656 | Adding a nonzero number to... |
| pnpncand 11657 | Addition/subtraction cance... |
| subeqrev 11658 | Reverse the order of subtr... |
| addeq0 11659 | Two complex numbers add up... |
| pncan1 11660 | Cancellation law for addit... |
| npcan1 11661 | Cancellation law for subtr... |
| subeq0bd 11662 | If two complex numbers are... |
| renegcld 11663 | Closure law for negative o... |
| resubcld 11664 | Closure law for subtractio... |
| negn0 11665 | The image under negation o... |
| negf1o 11666 | Negation is an isomorphism... |
| kcnktkm1cn 11667 | k times k minus 1 is a com... |
| muladd 11668 | Product of two sums. (Con... |
| subdi 11669 | Distribution of multiplica... |
| subdir 11670 | Distribution of multiplica... |
| ine0 11671 | The imaginary unit ` _i ` ... |
| mulneg1 11672 | Product with negative is n... |
| mulneg2 11673 | The product with a negativ... |
| mulneg12 11674 | Swap the negative sign in ... |
| mul2neg 11675 | Product of two negatives. ... |
| submul2 11676 | Convert a subtraction to a... |
| mulm1 11677 | Product with minus one is ... |
| addneg1mul 11678 | Addition with product with... |
| mulsub 11679 | Product of two differences... |
| mulsub2 11680 | Swap the order of subtract... |
| mulm1i 11681 | Product with minus one is ... |
| mulneg1i 11682 | Product with negative is n... |
| mulneg2i 11683 | Product with negative is n... |
| mul2negi 11684 | Product of two negatives. ... |
| subdii 11685 | Distribution of multiplica... |
| subdiri 11686 | Distribution of multiplica... |
| muladdi 11687 | Product of two sums. (Con... |
| mulm1d 11688 | Product with minus one is ... |
| mulneg1d 11689 | Product with negative is n... |
| mulneg2d 11690 | Product with negative is n... |
| mul2negd 11691 | Product of two negatives. ... |
| subdid 11692 | Distribution of multiplica... |
| subdird 11693 | Distribution of multiplica... |
| muladdd 11694 | Product of two sums. (Con... |
| mulsubd 11695 | Product of two differences... |
| muls1d 11696 | Multiplication by one minu... |
| mulsubfacd 11697 | Multiplication followed by... |
| addmulsub 11698 | The product of a sum and a... |
| subaddmulsub 11699 | The difference with a prod... |
| mulsubaddmulsub 11700 | A special difference of a ... |
| gt0ne0 11701 | Positive implies nonzero. ... |
| lt0ne0 11702 | A number which is less tha... |
| ltadd1 11703 | Addition to both sides of ... |
| leadd1 11704 | Addition to both sides of ... |
| leadd2 11705 | Addition to both sides of ... |
| ltsubadd 11706 | 'Less than' relationship b... |
| ltsubadd2 11707 | 'Less than' relationship b... |
| lesubadd 11708 | 'Less than or equal to' re... |
| lesubadd2 11709 | 'Less than or equal to' re... |
| ltaddsub 11710 | 'Less than' relationship b... |
| ltaddsub2 11711 | 'Less than' relationship b... |
| leaddsub 11712 | 'Less than or equal to' re... |
| leaddsub2 11713 | 'Less than or equal to' re... |
| suble 11714 | Swap subtrahends in an ine... |
| lesub 11715 | Swap subtrahends in an ine... |
| ltsub23 11716 | 'Less than' relationship b... |
| ltsub13 11717 | 'Less than' relationship b... |
| le2add 11718 | Adding both sides of two '... |
| ltleadd 11719 | Adding both sides of two o... |
| leltadd 11720 | Adding both sides of two o... |
| lt2add 11721 | Adding both sides of two '... |
| addgt0 11722 | The sum of 2 positive numb... |
| addgegt0 11723 | The sum of nonnegative and... |
| addgtge0 11724 | The sum of nonnegative and... |
| addge0 11725 | The sum of 2 nonnegative n... |
| ltaddpos 11726 | Adding a positive number t... |
| ltaddpos2 11727 | Adding a positive number t... |
| ltsubpos 11728 | Subtracting a positive num... |
| posdif 11729 | Comparison of two numbers ... |
| lesub1 11730 | Subtraction from both side... |
| lesub2 11731 | Subtraction of both sides ... |
| ltsub1 11732 | Subtraction from both side... |
| ltsub2 11733 | Subtraction of both sides ... |
| lt2sub 11734 | Subtracting both sides of ... |
| le2sub 11735 | Subtracting both sides of ... |
| ltneg 11736 | Negative of both sides of ... |
| ltnegcon1 11737 | Contraposition of negative... |
| ltnegcon2 11738 | Contraposition of negative... |
| leneg 11739 | Negative of both sides of ... |
| lenegcon1 11740 | Contraposition of negative... |
| lenegcon2 11741 | Contraposition of negative... |
| lt0neg1 11742 | Comparison of a number and... |
| lt0neg2 11743 | Comparison of a number and... |
| le0neg1 11744 | Comparison of a number and... |
| le0neg2 11745 | Comparison of a number and... |
| addge01 11746 | A number is less than or e... |
| addge02 11747 | A number is less than or e... |
| add20 11748 | Two nonnegative numbers ar... |
| subge0 11749 | Nonnegative subtraction. ... |
| suble0 11750 | Nonpositive subtraction. ... |
| leaddle0 11751 | The sum of a real number a... |
| subge02 11752 | Nonnegative subtraction. ... |
| lesub0 11753 | Lemma to show a nonnegativ... |
| mulge0 11754 | The product of two nonnega... |
| mullt0 11755 | The product of two negativ... |
| msqgt0 11756 | A nonzero square is positi... |
| msqge0 11757 | A square is nonnegative. ... |
| 0lt1 11758 | 0 is less than 1. Theorem... |
| 0le1 11759 | 0 is less than or equal to... |
| relin01 11760 | An interval law for less t... |
| ltordlem 11761 | Lemma for ~ ltord1 . (Con... |
| ltord1 11762 | Infer an ordering relation... |
| leord1 11763 | Infer an ordering relation... |
| eqord1 11764 | A strictly increasing real... |
| ltord2 11765 | Infer an ordering relation... |
| leord2 11766 | Infer an ordering relation... |
| eqord2 11767 | A strictly decreasing real... |
| wloglei 11768 | Form of ~ wlogle where bot... |
| wlogle 11769 | If the predicate ` ch ( x ... |
| leidi 11770 | 'Less than or equal to' is... |
| gt0ne0i 11771 | Positive means nonzero (us... |
| gt0ne0ii 11772 | Positive implies nonzero. ... |
| msqgt0i 11773 | A nonzero square is positi... |
| msqge0i 11774 | A square is nonnegative. ... |
| addgt0i 11775 | Addition of 2 positive num... |
| addge0i 11776 | Addition of 2 nonnegative ... |
| addgegt0i 11777 | Addition of nonnegative an... |
| addgt0ii 11778 | Addition of 2 positive num... |
| add20i 11779 | Two nonnegative numbers ar... |
| ltnegi 11780 | Negative of both sides of ... |
| lenegi 11781 | Negative of both sides of ... |
| ltnegcon2i 11782 | Contraposition of negative... |
| mulge0i 11783 | The product of two nonnega... |
| lesub0i 11784 | Lemma to show a nonnegativ... |
| ltaddposi 11785 | Adding a positive number t... |
| posdifi 11786 | Comparison of two numbers ... |
| ltnegcon1i 11787 | Contraposition of negative... |
| lenegcon1i 11788 | Contraposition of negative... |
| subge0i 11789 | Nonnegative subtraction. ... |
| ltadd1i 11790 | Addition to both sides of ... |
| leadd1i 11791 | Addition to both sides of ... |
| leadd2i 11792 | Addition to both sides of ... |
| ltsubaddi 11793 | 'Less than' relationship b... |
| lesubaddi 11794 | 'Less than or equal to' re... |
| ltsubadd2i 11795 | 'Less than' relationship b... |
| lesubadd2i 11796 | 'Less than or equal to' re... |
| ltaddsubi 11797 | 'Less than' relationship b... |
| lt2addi 11798 | Adding both side of two in... |
| le2addi 11799 | Adding both side of two in... |
| gt0ne0d 11800 | Positive implies nonzero. ... |
| lt0ne0d 11801 | Something less than zero i... |
| leidd 11802 | 'Less than or equal to' is... |
| msqgt0d 11803 | A nonzero square is positi... |
| msqge0d 11804 | A square is nonnegative. ... |
| lt0neg1d 11805 | Comparison of a number and... |
| lt0neg2d 11806 | Comparison of a number and... |
| le0neg1d 11807 | Comparison of a number and... |
| le0neg2d 11808 | Comparison of a number and... |
| addgegt0d 11809 | Addition of nonnegative an... |
| addgtge0d 11810 | Addition of positive and n... |
| addgt0d 11811 | Addition of 2 positive num... |
| addge0d 11812 | Addition of 2 nonnegative ... |
| mulge0d 11813 | The product of two nonnega... |
| ltnegd 11814 | Negative of both sides of ... |
| lenegd 11815 | Negative of both sides of ... |
| ltnegcon1d 11816 | Contraposition of negative... |
| ltnegcon2d 11817 | Contraposition of negative... |
| lenegcon1d 11818 | Contraposition of negative... |
| lenegcon2d 11819 | Contraposition of negative... |
| ltaddposd 11820 | Adding a positive number t... |
| ltaddpos2d 11821 | Adding a positive number t... |
| ltsubposd 11822 | Subtracting a positive num... |
| posdifd 11823 | Comparison of two numbers ... |
| addge01d 11824 | A number is less than or e... |
| addge02d 11825 | A number is less than or e... |
| subge0d 11826 | Nonnegative subtraction. ... |
| suble0d 11827 | Nonpositive subtraction. ... |
| subge02d 11828 | Nonnegative subtraction. ... |
| ltadd1d 11829 | Addition to both sides of ... |
| leadd1d 11830 | Addition to both sides of ... |
| leadd2d 11831 | Addition to both sides of ... |
| ltsubaddd 11832 | 'Less than' relationship b... |
| lesubaddd 11833 | 'Less than or equal to' re... |
| ltsubadd2d 11834 | 'Less than' relationship b... |
| lesubadd2d 11835 | 'Less than or equal to' re... |
| ltaddsubd 11836 | 'Less than' relationship b... |
| ltaddsub2d 11837 | 'Less than' relationship b... |
| leaddsub2d 11838 | 'Less than or equal to' re... |
| subled 11839 | Swap subtrahends in an ine... |
| lesubd 11840 | Swap subtrahends in an ine... |
| ltsub23d 11841 | 'Less than' relationship b... |
| ltsub13d 11842 | 'Less than' relationship b... |
| lesub1d 11843 | Subtraction from both side... |
| lesub2d 11844 | Subtraction of both sides ... |
| ltsub1d 11845 | Subtraction from both side... |
| ltsub2d 11846 | Subtraction of both sides ... |
| ltadd1dd 11847 | Addition to both sides of ... |
| ltsub1dd 11848 | Subtraction from both side... |
| ltsub2dd 11849 | Subtraction of both sides ... |
| leadd1dd 11850 | Addition to both sides of ... |
| leadd2dd 11851 | Addition to both sides of ... |
| lesub1dd 11852 | Subtraction from both side... |
| lesub2dd 11853 | Subtraction of both sides ... |
| lesub3d 11854 | The result of subtracting ... |
| le2addd 11855 | Adding both side of two in... |
| le2subd 11856 | Subtracting both sides of ... |
| ltleaddd 11857 | Adding both sides of two o... |
| leltaddd 11858 | Adding both sides of two o... |
| lt2addd 11859 | Adding both side of two in... |
| lt2subd 11860 | Subtracting both sides of ... |
| possumd 11861 | Condition for a positive s... |
| sublt0d 11862 | When a subtraction gives a... |
| ltaddsublt 11863 | Addition and subtraction o... |
| 1le1 11864 | One is less than or equal ... |
| ixi 11865 | ` _i ` times itself is min... |
| recextlem1 11866 | Lemma for ~ recex . (Cont... |
| recextlem2 11867 | Lemma for ~ recex . (Cont... |
| recex 11868 | Existence of reciprocal of... |
| mulcand 11869 | Cancellation law for multi... |
| mulcan2d 11870 | Cancellation law for multi... |
| mulcanad 11871 | Cancellation of a nonzero ... |
| mulcan2ad 11872 | Cancellation of a nonzero ... |
| mulcan 11873 | Cancellation law for multi... |
| mulcan2 11874 | Cancellation law for multi... |
| mulcani 11875 | Cancellation law for multi... |
| mul0or 11876 | If a product is zero, one ... |
| mulne0b 11877 | The product of two nonzero... |
| mulne0 11878 | The product of two nonzero... |
| mulne0i 11879 | The product of two nonzero... |
| muleqadd 11880 | Property of numbers whose ... |
| receu 11881 | Existential uniqueness of ... |
| mulnzcnf 11882 | Multiplication maps nonzer... |
| msq0i 11883 | A number is zero iff its s... |
| mul0ori 11884 | If a product is zero, one ... |
| msq0d 11885 | A number is zero iff its s... |
| mul0ord 11886 | If a product is zero, one ... |
| mulne0bd 11887 | The product of two nonzero... |
| mulne0d 11888 | The product of two nonzero... |
| mulcan1g 11889 | A generalized form of the ... |
| mulcan2g 11890 | A generalized form of the ... |
| mulne0bad 11891 | A factor of a nonzero comp... |
| mulne0bbd 11892 | A factor of a nonzero comp... |
| 1div0 11895 | You can't divide by zero, ... |
| divval 11896 | Value of division: if ` A ... |
| divmul 11897 | Relationship between divis... |
| divmul2 11898 | Relationship between divis... |
| divmul3 11899 | Relationship between divis... |
| divcl 11900 | Closure law for division. ... |
| reccl 11901 | Closure law for reciprocal... |
| divcan2 11902 | A cancellation law for div... |
| divcan1 11903 | A cancellation law for div... |
| diveq0 11904 | A ratio is zero iff the nu... |
| divne0b 11905 | The ratio of nonzero numbe... |
| divne0 11906 | The ratio of nonzero numbe... |
| recne0 11907 | The reciprocal of a nonzer... |
| recid 11908 | Multiplication of a number... |
| recid2 11909 | Multiplication of a number... |
| divrec 11910 | Relationship between divis... |
| divrec2 11911 | Relationship between divis... |
| divass 11912 | An associative law for div... |
| div23 11913 | A commutative/associative ... |
| div32 11914 | A commutative/associative ... |
| div13 11915 | A commutative/associative ... |
| div12 11916 | A commutative/associative ... |
| divmulass 11917 | An associative law for div... |
| divmulasscom 11918 | An associative/commutative... |
| divdir 11919 | Distribution of division o... |
| divcan3 11920 | A cancellation law for div... |
| divcan4 11921 | A cancellation law for div... |
| div11 11922 | One-to-one relationship fo... |
| divid 11923 | A number divided by itself... |
| div0 11924 | Division into zero is zero... |
| div1 11925 | A number divided by 1 is i... |
| 1div1e1 11926 | 1 divided by 1 is 1. (Con... |
| diveq1 11927 | Equality in terms of unit ... |
| divneg 11928 | Move negative sign inside ... |
| muldivdir 11929 | Distribution of division o... |
| divsubdir 11930 | Distribution of division o... |
| subdivcomb1 11931 | Bring a term in a subtract... |
| subdivcomb2 11932 | Bring a term in a subtract... |
| recrec 11933 | A number is equal to the r... |
| rec11 11934 | Reciprocal is one-to-one. ... |
| rec11r 11935 | Mutual reciprocals. (Cont... |
| divmuldiv 11936 | Multiplication of two rati... |
| divdivdiv 11937 | Division of two ratios. T... |
| divcan5 11938 | Cancellation of common fac... |
| divmul13 11939 | Swap the denominators in t... |
| divmul24 11940 | Swap the numerators in the... |
| divmuleq 11941 | Cross-multiply in an equal... |
| recdiv 11942 | The reciprocal of a ratio.... |
| divcan6 11943 | Cancellation of inverted f... |
| divdiv32 11944 | Swap denominators in a div... |
| divcan7 11945 | Cancel equal divisors in a... |
| dmdcan 11946 | Cancellation law for divis... |
| divdiv1 11947 | Division into a fraction. ... |
| divdiv2 11948 | Division by a fraction. (... |
| recdiv2 11949 | Division into a reciprocal... |
| ddcan 11950 | Cancellation in a double d... |
| divadddiv 11951 | Addition of two ratios. T... |
| divsubdiv 11952 | Subtraction of two ratios.... |
| conjmul 11953 | Two numbers whose reciproc... |
| rereccl 11954 | Closure law for reciprocal... |
| redivcl 11955 | Closure law for division o... |
| eqneg 11956 | A number equal to its nega... |
| eqnegd 11957 | A complex number equals it... |
| eqnegad 11958 | If a complex number equals... |
| div2neg 11959 | Quotient of two negatives.... |
| divneg2 11960 | Move negative sign inside ... |
| recclzi 11961 | Closure law for reciprocal... |
| recne0zi 11962 | The reciprocal of a nonzer... |
| recidzi 11963 | Multiplication of a number... |
| div1i 11964 | A number divided by 1 is i... |
| eqnegi 11965 | A number equal to its nega... |
| reccli 11966 | Closure law for reciprocal... |
| recidi 11967 | Multiplication of a number... |
| recreci 11968 | A number is equal to the r... |
| dividi 11969 | A number divided by itself... |
| div0i 11970 | Division into zero is zero... |
| divclzi 11971 | Closure law for division. ... |
| divcan1zi 11972 | A cancellation law for div... |
| divcan2zi 11973 | A cancellation law for div... |
| divreczi 11974 | Relationship between divis... |
| divcan3zi 11975 | A cancellation law for div... |
| divcan4zi 11976 | A cancellation law for div... |
| rec11i 11977 | Reciprocal is one-to-one. ... |
| divcli 11978 | Closure law for division. ... |
| divcan2i 11979 | A cancellation law for div... |
| divcan1i 11980 | A cancellation law for div... |
| divreci 11981 | Relationship between divis... |
| divcan3i 11982 | A cancellation law for div... |
| divcan4i 11983 | A cancellation law for div... |
| divne0i 11984 | The ratio of nonzero numbe... |
| rec11ii 11985 | Reciprocal is one-to-one. ... |
| divasszi 11986 | An associative law for div... |
| divmulzi 11987 | Relationship between divis... |
| divdirzi 11988 | Distribution of division o... |
| divdiv23zi 11989 | Swap denominators in a div... |
| divmuli 11990 | Relationship between divis... |
| divdiv32i 11991 | Swap denominators in a div... |
| divassi 11992 | An associative law for div... |
| divdiri 11993 | Distribution of division o... |
| div23i 11994 | A commutative/associative ... |
| div11i 11995 | One-to-one relationship fo... |
| divmuldivi 11996 | Multiplication of two rati... |
| divmul13i 11997 | Swap denominators of two r... |
| divadddivi 11998 | Addition of two ratios. T... |
| divdivdivi 11999 | Division of two ratios. T... |
| rerecclzi 12000 | Closure law for reciprocal... |
| rereccli 12001 | Closure law for reciprocal... |
| redivclzi 12002 | Closure law for division o... |
| redivcli 12003 | Closure law for division o... |
| div1d 12004 | A number divided by 1 is i... |
| reccld 12005 | Closure law for reciprocal... |
| recne0d 12006 | The reciprocal of a nonzer... |
| recidd 12007 | Multiplication of a number... |
| recid2d 12008 | Multiplication of a number... |
| recrecd 12009 | A number is equal to the r... |
| dividd 12010 | A number divided by itself... |
| div0d 12011 | Division into zero is zero... |
| divcld 12012 | Closure law for division. ... |
| divcan1d 12013 | A cancellation law for div... |
| divcan2d 12014 | A cancellation law for div... |
| divrecd 12015 | Relationship between divis... |
| divrec2d 12016 | Relationship between divis... |
| divcan3d 12017 | A cancellation law for div... |
| divcan4d 12018 | A cancellation law for div... |
| diveq0d 12019 | A ratio is zero iff the nu... |
| diveq1d 12020 | Equality in terms of unit ... |
| diveq1ad 12021 | The quotient of two comple... |
| diveq0ad 12022 | A fraction of complex numb... |
| divne1d 12023 | If two complex numbers are... |
| divne0bd 12024 | A ratio is zero iff the nu... |
| divnegd 12025 | Move negative sign inside ... |
| divneg2d 12026 | Move negative sign inside ... |
| div2negd 12027 | Quotient of two negatives.... |
| divne0d 12028 | The ratio of nonzero numbe... |
| recdivd 12029 | The reciprocal of a ratio.... |
| recdiv2d 12030 | Division into a reciprocal... |
| divcan6d 12031 | Cancellation of inverted f... |
| ddcand 12032 | Cancellation in a double d... |
| rec11d 12033 | Reciprocal is one-to-one. ... |
| divmuld 12034 | Relationship between divis... |
| div32d 12035 | A commutative/associative ... |
| div13d 12036 | A commutative/associative ... |
| divdiv32d 12037 | Swap denominators in a div... |
| divcan5d 12038 | Cancellation of common fac... |
| divcan5rd 12039 | Cancellation of common fac... |
| divcan7d 12040 | Cancel equal divisors in a... |
| dmdcand 12041 | Cancellation law for divis... |
| dmdcan2d 12042 | Cancellation law for divis... |
| divdiv1d 12043 | Division into a fraction. ... |
| divdiv2d 12044 | Division by a fraction. (... |
| divmul2d 12045 | Relationship between divis... |
| divmul3d 12046 | Relationship between divis... |
| divassd 12047 | An associative law for div... |
| div12d 12048 | A commutative/associative ... |
| div23d 12049 | A commutative/associative ... |
| divdird 12050 | Distribution of division o... |
| divsubdird 12051 | Distribution of division o... |
| div11d 12052 | One-to-one relationship fo... |
| divmuldivd 12053 | Multiplication of two rati... |
| divmul13d 12054 | Swap denominators of two r... |
| divmul24d 12055 | Swap the numerators in the... |
| divadddivd 12056 | Addition of two ratios. T... |
| divsubdivd 12057 | Subtraction of two ratios.... |
| divmuleqd 12058 | Cross-multiply in an equal... |
| divdivdivd 12059 | Division of two ratios. T... |
| diveq1bd 12060 | If two complex numbers are... |
| div2sub 12061 | Swap the order of subtract... |
| div2subd 12062 | Swap subtrahend and minuen... |
| rereccld 12063 | Closure law for reciprocal... |
| redivcld 12064 | Closure law for division o... |
| subrec 12065 | Subtraction of reciprocals... |
| subreci 12066 | Subtraction of reciprocals... |
| subrecd 12067 | Subtraction of reciprocals... |
| mvllmuld 12068 | Move the left term in a pr... |
| mvllmuli 12069 | Move the left term in a pr... |
| ldiv 12070 | Left-division. (Contribut... |
| rdiv 12071 | Right-division. (Contribu... |
| mdiv 12072 | A division law. (Contribu... |
| lineq 12073 | Solution of a (scalar) lin... |
| elimgt0 12074 | Hypothesis for weak deduct... |
| elimge0 12075 | Hypothesis for weak deduct... |
| ltp1 12076 | A number is less than itse... |
| lep1 12077 | A number is less than or e... |
| ltm1 12078 | A number minus 1 is less t... |
| lem1 12079 | A number minus 1 is less t... |
| letrp1 12080 | A transitive property of '... |
| p1le 12081 | A transitive property of p... |
| recgt0 12082 | The reciprocal of a positi... |
| prodgt0 12083 | Infer that a multiplicand ... |
| prodgt02 12084 | Infer that a multiplier is... |
| ltmul1a 12085 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12086 | Multiplication of both sid... |
| ltmul2 12087 | Multiplication of both sid... |
| lemul1 12088 | Multiplication of both sid... |
| lemul2 12089 | Multiplication of both sid... |
| lemul1a 12090 | Multiplication of both sid... |
| lemul2a 12091 | Multiplication of both sid... |
| ltmul12a 12092 | Comparison of product of t... |
| lemul12b 12093 | Comparison of product of t... |
| lemul12a 12094 | Comparison of product of t... |
| mulgt1 12095 | The product of two numbers... |
| ltmulgt11 12096 | Multiplication by a number... |
| ltmulgt12 12097 | Multiplication by a number... |
| lemulge11 12098 | Multiplication by a number... |
| lemulge12 12099 | Multiplication by a number... |
| ltdiv1 12100 | Division of both sides of ... |
| lediv1 12101 | Division of both sides of ... |
| gt0div 12102 | Division of a positive num... |
| ge0div 12103 | Division of a nonnegative ... |
| divgt0 12104 | The ratio of two positive ... |
| divge0 12105 | The ratio of nonnegative a... |
| mulge0b 12106 | A condition for multiplica... |
| mulle0b 12107 | A condition for multiplica... |
| mulsuble0b 12108 | A condition for multiplica... |
| ltmuldiv 12109 | 'Less than' relationship b... |
| ltmuldiv2 12110 | 'Less than' relationship b... |
| ltdivmul 12111 | 'Less than' relationship b... |
| ledivmul 12112 | 'Less than or equal to' re... |
| ltdivmul2 12113 | 'Less than' relationship b... |
| lt2mul2div 12114 | 'Less than' relationship b... |
| ledivmul2 12115 | 'Less than or equal to' re... |
| lemuldiv 12116 | 'Less than or equal' relat... |
| lemuldiv2 12117 | 'Less than or equal' relat... |
| ltrec 12118 | The reciprocal of both sid... |
| lerec 12119 | The reciprocal of both sid... |
| lt2msq1 12120 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12121 | Two nonnegative numbers co... |
| ltdiv2 12122 | Division of a positive num... |
| ltrec1 12123 | Reciprocal swap in a 'less... |
| lerec2 12124 | Reciprocal swap in a 'less... |
| ledivdiv 12125 | Invert ratios of positive ... |
| lediv2 12126 | Division of a positive num... |
| ltdiv23 12127 | Swap denominator with othe... |
| lediv23 12128 | Swap denominator with othe... |
| lediv12a 12129 | Comparison of ratio of two... |
| lediv2a 12130 | Division of both sides of ... |
| reclt1 12131 | The reciprocal of a positi... |
| recgt1 12132 | The reciprocal of a positi... |
| recgt1i 12133 | The reciprocal of a number... |
| recp1lt1 12134 | Construct a number less th... |
| recreclt 12135 | Given a positive number ` ... |
| le2msq 12136 | The square function on non... |
| msq11 12137 | The square of a nonnegativ... |
| ledivp1 12138 | "Less than or equal to" an... |
| squeeze0 12139 | If a nonnegative number is... |
| ltp1i 12140 | A number is less than itse... |
| recgt0i 12141 | The reciprocal of a positi... |
| recgt0ii 12142 | The reciprocal of a positi... |
| prodgt0i 12143 | Infer that a multiplicand ... |
| divgt0i 12144 | The ratio of two positive ... |
| divge0i 12145 | The ratio of nonnegative a... |
| ltreci 12146 | The reciprocal of both sid... |
| lereci 12147 | The reciprocal of both sid... |
| lt2msqi 12148 | The square function on non... |
| le2msqi 12149 | The square function on non... |
| msq11i 12150 | The square of a nonnegativ... |
| divgt0i2i 12151 | The ratio of two positive ... |
| ltrecii 12152 | The reciprocal of both sid... |
| divgt0ii 12153 | The ratio of two positive ... |
| ltmul1i 12154 | Multiplication of both sid... |
| ltdiv1i 12155 | Division of both sides of ... |
| ltmuldivi 12156 | 'Less than' relationship b... |
| ltmul2i 12157 | Multiplication of both sid... |
| lemul1i 12158 | Multiplication of both sid... |
| lemul2i 12159 | Multiplication of both sid... |
| ltdiv23i 12160 | Swap denominator with othe... |
| ledivp1i 12161 | "Less than or equal to" an... |
| ltdivp1i 12162 | Less-than and division rel... |
| ltdiv23ii 12163 | Swap denominator with othe... |
| ltmul1ii 12164 | Multiplication of both sid... |
| ltdiv1ii 12165 | Division of both sides of ... |
| ltp1d 12166 | A number is less than itse... |
| lep1d 12167 | A number is less than or e... |
| ltm1d 12168 | A number minus 1 is less t... |
| lem1d 12169 | A number minus 1 is less t... |
| recgt0d 12170 | The reciprocal of a positi... |
| divgt0d 12171 | The ratio of two positive ... |
| mulgt1d 12172 | The product of two numbers... |
| lemulge11d 12173 | Multiplication by a number... |
| lemulge12d 12174 | Multiplication by a number... |
| lemul1ad 12175 | Multiplication of both sid... |
| lemul2ad 12176 | Multiplication of both sid... |
| ltmul12ad 12177 | Comparison of product of t... |
| lemul12ad 12178 | Comparison of product of t... |
| lemul12bd 12179 | Comparison of product of t... |
| fimaxre 12180 | A finite set of real numbe... |
| fimaxre2 12181 | A nonempty finite set of r... |
| fimaxre3 12182 | A nonempty finite set of r... |
| fiminre 12183 | A nonempty finite set of r... |
| fiminre2 12184 | A nonempty finite set of r... |
| negfi 12185 | The negation of a finite s... |
| lbreu 12186 | If a set of reals contains... |
| lbcl 12187 | If a set of reals contains... |
| lble 12188 | If a set of reals contains... |
| lbinf 12189 | If a set of reals contains... |
| lbinfcl 12190 | If a set of reals contains... |
| lbinfle 12191 | If a set of reals contains... |
| sup2 12192 | A nonempty, bounded-above ... |
| sup3 12193 | A version of the completen... |
| infm3lem 12194 | Lemma for ~ infm3 . (Cont... |
| infm3 12195 | The completeness axiom for... |
| suprcl 12196 | Closure of supremum of a n... |
| suprub 12197 | A member of a nonempty bou... |
| suprubd 12198 | Natural deduction form of ... |
| suprcld 12199 | Natural deduction form of ... |
| suprlub 12200 | The supremum of a nonempty... |
| suprnub 12201 | An upper bound is not less... |
| suprleub 12202 | The supremum of a nonempty... |
| supaddc 12203 | The supremum function dist... |
| supadd 12204 | The supremum function dist... |
| supmul1 12205 | The supremum function dist... |
| supmullem1 12206 | Lemma for ~ supmul . (Con... |
| supmullem2 12207 | Lemma for ~ supmul . (Con... |
| supmul 12208 | The supremum function dist... |
| sup3ii 12209 | A version of the completen... |
| suprclii 12210 | Closure of supremum of a n... |
| suprubii 12211 | A member of a nonempty bou... |
| suprlubii 12212 | The supremum of a nonempty... |
| suprnubii 12213 | An upper bound is not less... |
| suprleubii 12214 | The supremum of a nonempty... |
| riotaneg 12215 | The negative of the unique... |
| negiso 12216 | Negation is an order anti-... |
| dfinfre 12217 | The infimum of a set of re... |
| infrecl 12218 | Closure of infimum of a no... |
| infrenegsup 12219 | The infimum of a set of re... |
| infregelb 12220 | Any lower bound of a nonem... |
| infrelb 12221 | If a nonempty set of real ... |
| infrefilb 12222 | The infimum of a finite se... |
| supfirege 12223 | The supremum of a finite s... |
| inelr 12224 | The imaginary unit ` _i ` ... |
| rimul 12225 | A real number times the im... |
| cru 12226 | The representation of comp... |
| crne0 12227 | The real representation of... |
| creur 12228 | The real part of a complex... |
| creui 12229 | The imaginary part of a co... |
| cju 12230 | The complex conjugate of a... |
| ofsubeq0 12231 | Function analogue of ~ sub... |
| ofnegsub 12232 | Function analogue of ~ neg... |
| ofsubge0 12233 | Function analogue of ~ sub... |
| nnexALT 12236 | Alternate proof of ~ nnex ... |
| peano5nni 12237 | Peano's inductive postulat... |
| nnssre 12238 | The positive integers are ... |
| nnsscn 12239 | The positive integers are ... |
| nnex 12240 | The set of positive intege... |
| nnre 12241 | A positive integer is a re... |
| nncn 12242 | A positive integer is a co... |
| nnrei 12243 | A positive integer is a re... |
| nncni 12244 | A positive integer is a co... |
| 1nn 12245 | Peano postulate: 1 is a po... |
| peano2nn 12246 | Peano postulate: a success... |
| dfnn2 12247 | Alternate definition of th... |
| dfnn3 12248 | Alternate definition of th... |
| nnred 12249 | A positive integer is a re... |
| nncnd 12250 | A positive integer is a co... |
| peano2nnd 12251 | Peano postulate: a success... |
| nnind 12252 | Principle of Mathematical ... |
| nnindALT 12253 | Principle of Mathematical ... |
| nnindd 12254 | Principle of Mathematical ... |
| nn1m1nn 12255 | Every positive integer is ... |
| nn1suc 12256 | If a statement holds for 1... |
| nnaddcl 12257 | Closure of addition of pos... |
| nnmulcl 12258 | Closure of multiplication ... |
| nnmulcli 12259 | Closure of multiplication ... |
| nnmtmip 12260 | "Minus times minus is plus... |
| nn2ge 12261 | There exists a positive in... |
| nnge1 12262 | A positive integer is one ... |
| nngt1ne1 12263 | A positive integer is grea... |
| nnle1eq1 12264 | A positive integer is less... |
| nngt0 12265 | A positive integer is posi... |
| nnnlt1 12266 | A positive integer is not ... |
| nnnle0 12267 | A positive integer is not ... |
| nnne0 12268 | A positive integer is nonz... |
| nnneneg 12269 | No positive integer is equ... |
| 0nnn 12270 | Zero is not a positive int... |
| 0nnnALT 12271 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12272 | Alternate version of ~ nnn... |
| nngt0i 12273 | A positive integer is posi... |
| nnne0i 12274 | A positive integer is nonz... |
| nndivre 12275 | The quotient of a real and... |
| nnrecre 12276 | The reciprocal of a positi... |
| nnrecgt0 12277 | The reciprocal of a positi... |
| nnsub 12278 | Subtraction of positive in... |
| nnsubi 12279 | Subtraction of positive in... |
| nndiv 12280 | Two ways to express " ` A ... |
| nndivtr 12281 | Transitive property of div... |
| nnge1d 12282 | A positive integer is one ... |
| nngt0d 12283 | A positive integer is posi... |
| nnne0d 12284 | A positive integer is nonz... |
| nnrecred 12285 | The reciprocal of a positi... |
| nnaddcld 12286 | Closure of addition of pos... |
| nnmulcld 12287 | Closure of multiplication ... |
| nndivred 12288 | A positive integer is one ... |
| 0ne1 12305 | Zero is different from one... |
| 1m1e0 12306 | One minus one equals zero.... |
| 2nn 12307 | 2 is a positive integer. ... |
| 2re 12308 | The number 2 is real. (Co... |
| 2cn 12309 | The number 2 is a complex ... |
| 2cnALT 12310 | Alternate proof of ~ 2cn .... |
| 2ex 12311 | The number 2 is a set. (C... |
| 2cnd 12312 | The number 2 is a complex ... |
| 3nn 12313 | 3 is a positive integer. ... |
| 3re 12314 | The number 3 is real. (Co... |
| 3cn 12315 | The number 3 is a complex ... |
| 3ex 12316 | The number 3 is a set. (C... |
| 4nn 12317 | 4 is a positive integer. ... |
| 4re 12318 | The number 4 is real. (Co... |
| 4cn 12319 | The number 4 is a complex ... |
| 5nn 12320 | 5 is a positive integer. ... |
| 5re 12321 | The number 5 is real. (Co... |
| 5cn 12322 | The number 5 is a complex ... |
| 6nn 12323 | 6 is a positive integer. ... |
| 6re 12324 | The number 6 is real. (Co... |
| 6cn 12325 | The number 6 is a complex ... |
| 7nn 12326 | 7 is a positive integer. ... |
| 7re 12327 | The number 7 is real. (Co... |
| 7cn 12328 | The number 7 is a complex ... |
| 8nn 12329 | 8 is a positive integer. ... |
| 8re 12330 | The number 8 is real. (Co... |
| 8cn 12331 | The number 8 is a complex ... |
| 9nn 12332 | 9 is a positive integer. ... |
| 9re 12333 | The number 9 is real. (Co... |
| 9cn 12334 | The number 9 is a complex ... |
| 0le0 12335 | Zero is nonnegative. (Con... |
| 0le2 12336 | The number 0 is less than ... |
| 2pos 12337 | The number 2 is positive. ... |
| 2ne0 12338 | The number 2 is nonzero. ... |
| 3pos 12339 | The number 3 is positive. ... |
| 3ne0 12340 | The number 3 is nonzero. ... |
| 4pos 12341 | The number 4 is positive. ... |
| 4ne0 12342 | The number 4 is nonzero. ... |
| 5pos 12343 | The number 5 is positive. ... |
| 6pos 12344 | The number 6 is positive. ... |
| 7pos 12345 | The number 7 is positive. ... |
| 8pos 12346 | The number 8 is positive. ... |
| 9pos 12347 | The number 9 is positive. ... |
| neg1cn 12348 | -1 is a complex number. (... |
| neg1rr 12349 | -1 is a real number. (Con... |
| neg1ne0 12350 | -1 is nonzero. (Contribut... |
| neg1lt0 12351 | -1 is less than 0. (Contr... |
| negneg1e1 12352 | ` -u -u 1 ` is 1. (Contri... |
| 1pneg1e0 12353 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12354 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12355 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12356 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12357 | Function value at ` N + 1 ... |
| 1p0e1 12358 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12359 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12360 | 2 - 1 = 1. The result is ... |
| 1e2m1 12361 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12362 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12363 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12364 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12365 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12366 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12367 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12368 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12369 | Two plus two equals four. ... |
| 2times 12370 | Two times a number. (Cont... |
| times2 12371 | A number times 2. (Contri... |
| 2timesi 12372 | Two times a number. (Cont... |
| times2i 12373 | A number times 2. (Contri... |
| 2txmxeqx 12374 | Two times a complex number... |
| 2div2e1 12375 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12376 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12377 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12378 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12379 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12380 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12381 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12382 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12383 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12384 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12385 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12386 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12387 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12388 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12389 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12390 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12391 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12392 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12393 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12394 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12395 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12396 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12397 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12398 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12399 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12400 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12401 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12402 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12403 | 2 times 0 equals 0. (Cont... |
| 4d2e2 12404 | One half of four is two. ... |
| 1lt2 12405 | 1 is less than 2. (Contri... |
| 2lt3 12406 | 2 is less than 3. (Contri... |
| 1lt3 12407 | 1 is less than 3. (Contri... |
| 3lt4 12408 | 3 is less than 4. (Contri... |
| 2lt4 12409 | 2 is less than 4. (Contri... |
| 1lt4 12410 | 1 is less than 4. (Contri... |
| 4lt5 12411 | 4 is less than 5. (Contri... |
| 3lt5 12412 | 3 is less than 5. (Contri... |
| 2lt5 12413 | 2 is less than 5. (Contri... |
| 1lt5 12414 | 1 is less than 5. (Contri... |
| 5lt6 12415 | 5 is less than 6. (Contri... |
| 4lt6 12416 | 4 is less than 6. (Contri... |
| 3lt6 12417 | 3 is less than 6. (Contri... |
| 2lt6 12418 | 2 is less than 6. (Contri... |
| 1lt6 12419 | 1 is less than 6. (Contri... |
| 6lt7 12420 | 6 is less than 7. (Contri... |
| 5lt7 12421 | 5 is less than 7. (Contri... |
| 4lt7 12422 | 4 is less than 7. (Contri... |
| 3lt7 12423 | 3 is less than 7. (Contri... |
| 2lt7 12424 | 2 is less than 7. (Contri... |
| 1lt7 12425 | 1 is less than 7. (Contri... |
| 7lt8 12426 | 7 is less than 8. (Contri... |
| 6lt8 12427 | 6 is less than 8. (Contri... |
| 5lt8 12428 | 5 is less than 8. (Contri... |
| 4lt8 12429 | 4 is less than 8. (Contri... |
| 3lt8 12430 | 3 is less than 8. (Contri... |
| 2lt8 12431 | 2 is less than 8. (Contri... |
| 1lt8 12432 | 1 is less than 8. (Contri... |
| 8lt9 12433 | 8 is less than 9. (Contri... |
| 7lt9 12434 | 7 is less than 9. (Contri... |
| 6lt9 12435 | 6 is less than 9. (Contri... |
| 5lt9 12436 | 5 is less than 9. (Contri... |
| 4lt9 12437 | 4 is less than 9. (Contri... |
| 3lt9 12438 | 3 is less than 9. (Contri... |
| 2lt9 12439 | 2 is less than 9. (Contri... |
| 1lt9 12440 | 1 is less than 9. (Contri... |
| 0ne2 12441 | 0 is not equal to 2. (Con... |
| 1ne2 12442 | 1 is not equal to 2. (Con... |
| 1le2 12443 | 1 is less than or equal to... |
| 2cnne0 12444 | 2 is a nonzero complex num... |
| 2rene0 12445 | 2 is a nonzero real number... |
| 1le3 12446 | 1 is less than or equal to... |
| neg1mulneg1e1 12447 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12448 | One-half is real. (Contri... |
| halfcn 12449 | One-half is a complex numb... |
| halfgt0 12450 | One-half is greater than z... |
| halfge0 12451 | One-half is not negative. ... |
| halflt1 12452 | One-half is less than one.... |
| 1mhlfehlf 12453 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12454 | An eighth of four thirds i... |
| halfpm6th 12455 | One half plus or minus one... |
| it0e0 12456 | i times 0 equals 0. (Cont... |
| 2mulicn 12457 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12458 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12459 | Closure of half of a numbe... |
| rehalfcl 12460 | Real closure of half. (Co... |
| half0 12461 | Half of a number is zero i... |
| 2halves 12462 | Two halves make a whole. ... |
| halfpos2 12463 | A number is positive iff i... |
| halfpos 12464 | A positive number is great... |
| halfnneg2 12465 | A number is nonnegative if... |
| halfaddsubcl 12466 | Closure of half-sum and ha... |
| halfaddsub 12467 | Sum and difference of half... |
| subhalfhalf 12468 | Subtracting the half of a ... |
| lt2halves 12469 | A sum is less than the who... |
| addltmul 12470 | Sum is less than product f... |
| nominpos 12471 | There is no smallest posit... |
| avglt1 12472 | Ordering property for aver... |
| avglt2 12473 | Ordering property for aver... |
| avgle1 12474 | Ordering property for aver... |
| avgle2 12475 | Ordering property for aver... |
| avgle 12476 | The average of two numbers... |
| 2timesd 12477 | Two times a number. (Cont... |
| times2d 12478 | A number times 2. (Contri... |
| halfcld 12479 | Closure of half of a numbe... |
| 2halvesd 12480 | Two halves make a whole. ... |
| rehalfcld 12481 | Real closure of half. (Co... |
| lt2halvesd 12482 | A sum is less than the who... |
| rehalfcli 12483 | Half a real number is real... |
| lt2addmuld 12484 | If two real numbers are le... |
| add1p1 12485 | Adding two times 1 to a nu... |
| sub1m1 12486 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12487 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12488 | A complex number increased... |
| div4p1lem1div2 12489 | An integer greater than 5,... |
| nnunb 12490 | The set of positive intege... |
| arch 12491 | Archimedean property of re... |
| nnrecl 12492 | There exists a positive in... |
| bndndx 12493 | A bounded real sequence ` ... |
| elnn0 12496 | Nonnegative integers expre... |
| nnssnn0 12497 | Positive naturals are a su... |
| nn0ssre 12498 | Nonnegative integers are a... |
| nn0sscn 12499 | Nonnegative integers are a... |
| nn0ex 12500 | The set of nonnegative int... |
| nnnn0 12501 | A positive integer is a no... |
| nnnn0i 12502 | A positive integer is a no... |
| nn0re 12503 | A nonnegative integer is a... |
| nn0cn 12504 | A nonnegative integer is a... |
| nn0rei 12505 | A nonnegative integer is a... |
| nn0cni 12506 | A nonnegative integer is a... |
| dfn2 12507 | The set of positive intege... |
| elnnne0 12508 | The positive integer prope... |
| 0nn0 12509 | 0 is a nonnegative integer... |
| 1nn0 12510 | 1 is a nonnegative integer... |
| 2nn0 12511 | 2 is a nonnegative integer... |
| 3nn0 12512 | 3 is a nonnegative integer... |
| 4nn0 12513 | 4 is a nonnegative integer... |
| 5nn0 12514 | 5 is a nonnegative integer... |
| 6nn0 12515 | 6 is a nonnegative integer... |
| 7nn0 12516 | 7 is a nonnegative integer... |
| 8nn0 12517 | 8 is a nonnegative integer... |
| 9nn0 12518 | 9 is a nonnegative integer... |
| nn0ge0 12519 | A nonnegative integer is g... |
| nn0nlt0 12520 | A nonnegative integer is n... |
| nn0ge0i 12521 | Nonnegative integers are n... |
| nn0le0eq0 12522 | A nonnegative integer is l... |
| nn0p1gt0 12523 | A nonnegative integer incr... |
| nnnn0addcl 12524 | A positive integer plus a ... |
| nn0nnaddcl 12525 | A nonnegative integer plus... |
| 0mnnnnn0 12526 | The result of subtracting ... |
| un0addcl 12527 | If ` S ` is closed under a... |
| un0mulcl 12528 | If ` S ` is closed under m... |
| nn0addcl 12529 | Closure of addition of non... |
| nn0mulcl 12530 | Closure of multiplication ... |
| nn0addcli 12531 | Closure of addition of non... |
| nn0mulcli 12532 | Closure of multiplication ... |
| nn0p1nn 12533 | A nonnegative integer plus... |
| peano2nn0 12534 | Second Peano postulate for... |
| nnm1nn0 12535 | A positive integer minus 1... |
| elnn0nn 12536 | The nonnegative integer pr... |
| elnnnn0 12537 | The positive integer prope... |
| elnnnn0b 12538 | The positive integer prope... |
| elnnnn0c 12539 | The positive integer prope... |
| nn0addge1 12540 | A number is less than or e... |
| nn0addge2 12541 | A number is less than or e... |
| nn0addge1i 12542 | A number is less than or e... |
| nn0addge2i 12543 | A number is less than or e... |
| nn0sub 12544 | Subtraction of nonnegative... |
| ltsubnn0 12545 | Subtracting a nonnegative ... |
| nn0negleid 12546 | A nonnegative integer is g... |
| difgtsumgt 12547 | If the difference of a rea... |
| nn0le2xi 12548 | A nonnegative integer is l... |
| nn0lele2xi 12549 | 'Less than or equal to' im... |
| fcdmnn0supp 12550 | Two ways to write the supp... |
| fcdmnn0fsupp 12551 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12552 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12553 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12554 | A positive integer is a no... |
| nn0red 12555 | A nonnegative integer is a... |
| nn0cnd 12556 | A nonnegative integer is a... |
| nn0ge0d 12557 | A nonnegative integer is g... |
| nn0addcld 12558 | Closure of addition of non... |
| nn0mulcld 12559 | Closure of multiplication ... |
| nn0readdcl 12560 | Closure law for addition o... |
| nn0n0n1ge2 12561 | A nonnegative integer whic... |
| nn0n0n1ge2b 12562 | A nonnegative integer is n... |
| nn0ge2m1nn 12563 | If a nonnegative integer i... |
| nn0ge2m1nn0 12564 | If a nonnegative integer i... |
| nn0nndivcl 12565 | Closure law for dividing o... |
| elxnn0 12568 | An extended nonnegative in... |
| nn0ssxnn0 12569 | The standard nonnegative i... |
| nn0xnn0 12570 | A standard nonnegative int... |
| xnn0xr 12571 | An extended nonnegative in... |
| 0xnn0 12572 | Zero is an extended nonneg... |
| pnf0xnn0 12573 | Positive infinity is an ex... |
| nn0nepnf 12574 | No standard nonnegative in... |
| nn0xnn0d 12575 | A standard nonnegative int... |
| nn0nepnfd 12576 | No standard nonnegative in... |
| xnn0nemnf 12577 | No extended nonnegative in... |
| xnn0xrnemnf 12578 | The extended nonnegative i... |
| xnn0nnn0pnf 12579 | An extended nonnegative in... |
| elz 12582 | Membership in the set of i... |
| nnnegz 12583 | The negative of a positive... |
| zre 12584 | An integer is a real. (Co... |
| zcn 12585 | An integer is a complex nu... |
| zrei 12586 | An integer is a real numbe... |
| zssre 12587 | The integers are a subset ... |
| zsscn 12588 | The integers are a subset ... |
| zex 12589 | The set of integers exists... |
| elnnz 12590 | Positive integer property ... |
| 0z 12591 | Zero is an integer. (Cont... |
| 0zd 12592 | Zero is an integer, deduct... |
| elnn0z 12593 | Nonnegative integer proper... |
| elznn0nn 12594 | Integer property expressed... |
| elznn0 12595 | Integer property expressed... |
| elznn 12596 | Integer property expressed... |
| zle0orge1 12597 | There is no integer in the... |
| elz2 12598 | Membership in the set of i... |
| dfz2 12599 | Alternative definition of ... |
| zexALT 12600 | Alternate proof of ~ zex .... |
| nnz 12601 | A positive integer is an i... |
| nnssz 12602 | Positive integers are a su... |
| nn0ssz 12603 | Nonnegative integers are a... |
| nnzOLD 12604 | Obsolete version of ~ nnz ... |
| nn0z 12605 | A nonnegative integer is a... |
| nn0zd 12606 | A nonnegative integer is a... |
| nnzd 12607 | A positive integer is an i... |
| nnzi 12608 | A positive integer is an i... |
| nn0zi 12609 | A nonnegative integer is a... |
| elnnz1 12610 | Positive integer property ... |
| znnnlt1 12611 | An integer is not a positi... |
| nnzrab 12612 | Positive integers expresse... |
| nn0zrab 12613 | Nonnegative integers expre... |
| 1z 12614 | One is an integer. (Contr... |
| 1zzd 12615 | One is an integer, deducti... |
| 2z 12616 | 2 is an integer. (Contrib... |
| 3z 12617 | 3 is an integer. (Contrib... |
| 4z 12618 | 4 is an integer. (Contrib... |
| znegcl 12619 | Closure law for negative i... |
| neg1z 12620 | -1 is an integer. (Contri... |
| znegclb 12621 | A complex number is an int... |
| nn0negz 12622 | The negative of a nonnegat... |
| nn0negzi 12623 | The negative of a nonnegat... |
| zaddcl 12624 | Closure of addition of int... |
| peano2z 12625 | Second Peano postulate gen... |
| zsubcl 12626 | Closure of subtraction of ... |
| peano2zm 12627 | "Reverse" second Peano pos... |
| zletr 12628 | Transitive law of ordering... |
| zrevaddcl 12629 | Reverse closure law for ad... |
| znnsub 12630 | The positive difference of... |
| znn0sub 12631 | The nonnegative difference... |
| nzadd 12632 | The sum of a real number n... |
| zmulcl 12633 | Closure of multiplication ... |
| zltp1le 12634 | Integer ordering relation.... |
| zleltp1 12635 | Integer ordering relation.... |
| zlem1lt 12636 | Integer ordering relation.... |
| zltlem1 12637 | Integer ordering relation.... |
| zgt0ge1 12638 | An integer greater than ` ... |
| nnleltp1 12639 | Positive integer ordering ... |
| nnltp1le 12640 | Positive integer ordering ... |
| nnaddm1cl 12641 | Closure of addition of pos... |
| nn0ltp1le 12642 | Nonnegative integer orderi... |
| nn0leltp1 12643 | Nonnegative integer orderi... |
| nn0ltlem1 12644 | Nonnegative integer orderi... |
| nn0sub2 12645 | Subtraction of nonnegative... |
| nn0lt10b 12646 | A nonnegative integer less... |
| nn0lt2 12647 | A nonnegative integer less... |
| nn0le2is012 12648 | A nonnegative integer whic... |
| nn0lem1lt 12649 | Nonnegative integer orderi... |
| nnlem1lt 12650 | Positive integer ordering ... |
| nnltlem1 12651 | Positive integer ordering ... |
| nnm1ge0 12652 | A positive integer decreas... |
| nn0ge0div 12653 | Division of a nonnegative ... |
| zdiv 12654 | Two ways to express " ` M ... |
| zdivadd 12655 | Property of divisibility: ... |
| zdivmul 12656 | Property of divisibility: ... |
| zextle 12657 | An extensionality-like pro... |
| zextlt 12658 | An extensionality-like pro... |
| recnz 12659 | The reciprocal of a number... |
| btwnnz 12660 | A number between an intege... |
| gtndiv 12661 | A larger number does not d... |
| halfnz 12662 | One-half is not an integer... |
| 3halfnz 12663 | Three halves is not an int... |
| suprzcl 12664 | The supremum of a bounded-... |
| prime 12665 | Two ways to express " ` A ... |
| msqznn 12666 | The square of a nonzero in... |
| zneo 12667 | No even integer equals an ... |
| nneo 12668 | A positive integer is even... |
| nneoi 12669 | A positive integer is even... |
| zeo 12670 | An integer is even or odd.... |
| zeo2 12671 | An integer is even or odd ... |
| peano2uz2 12672 | Second Peano postulate for... |
| peano5uzi 12673 | Peano's inductive postulat... |
| peano5uzti 12674 | Peano's inductive postulat... |
| dfuzi 12675 | An expression for the uppe... |
| uzind 12676 | Induction on the upper int... |
| uzind2 12677 | Induction on the upper int... |
| uzind3 12678 | Induction on the upper int... |
| nn0ind 12679 | Principle of Mathematical ... |
| nn0indALT 12680 | Principle of Mathematical ... |
| nn0indd 12681 | Principle of Mathematical ... |
| fzind 12682 | Induction on the integers ... |
| fnn0ind 12683 | Induction on the integers ... |
| nn0ind-raph 12684 | Principle of Mathematical ... |
| zindd 12685 | Principle of Mathematical ... |
| fzindd 12686 | Induction on the integers ... |
| btwnz 12687 | Any real number can be san... |
| zred 12688 | An integer is a real numbe... |
| zcnd 12689 | An integer is a complex nu... |
| znegcld 12690 | Closure law for negative i... |
| peano2zd 12691 | Deduction from second Pean... |
| zaddcld 12692 | Closure of addition of int... |
| zsubcld 12693 | Closure of subtraction of ... |
| zmulcld 12694 | Closure of multiplication ... |
| znnn0nn 12695 | The negative of a negative... |
| zadd2cl 12696 | Increasing an integer by 2... |
| zriotaneg 12697 | The negative of the unique... |
| suprfinzcl 12698 | The supremum of a nonempty... |
| 9p1e10 12701 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12702 | Version of the definition ... |
| decex 12703 | A decimal number is a set.... |
| deceq1 12704 | Equality theorem for the d... |
| deceq2 12705 | Equality theorem for the d... |
| deceq1i 12706 | Equality theorem for the d... |
| deceq2i 12707 | Equality theorem for the d... |
| deceq12i 12708 | Equality theorem for the d... |
| numnncl 12709 | Closure for a numeral (wit... |
| num0u 12710 | Add a zero in the units pl... |
| num0h 12711 | Add a zero in the higher p... |
| numcl 12712 | Closure for a decimal inte... |
| numsuc 12713 | The successor of a decimal... |
| deccl 12714 | Closure for a numeral. (C... |
| 10nn 12715 | 10 is a positive integer. ... |
| 10pos 12716 | The number 10 is positive.... |
| 10nn0 12717 | 10 is a nonnegative intege... |
| 10re 12718 | The number 10 is real. (C... |
| decnncl 12719 | Closure for a numeral. (C... |
| dec0u 12720 | Add a zero in the units pl... |
| dec0h 12721 | Add a zero in the higher p... |
| numnncl2 12722 | Closure for a decimal inte... |
| decnncl2 12723 | Closure for a decimal inte... |
| numlt 12724 | Comparing two decimal inte... |
| numltc 12725 | Comparing two decimal inte... |
| le9lt10 12726 | A "decimal digit" (i.e. a ... |
| declt 12727 | Comparing two decimal inte... |
| decltc 12728 | Comparing two decimal inte... |
| declth 12729 | Comparing two decimal inte... |
| decsuc 12730 | The successor of a decimal... |
| 3declth 12731 | Comparing two decimal inte... |
| 3decltc 12732 | Comparing two decimal inte... |
| decle 12733 | Comparing two decimal inte... |
| decleh 12734 | Comparing two decimal inte... |
| declei 12735 | Comparing a digit to a dec... |
| numlti 12736 | Comparing a digit to a dec... |
| declti 12737 | Comparing a digit to a dec... |
| decltdi 12738 | Comparing a digit to a dec... |
| numsucc 12739 | The successor of a decimal... |
| decsucc 12740 | The successor of a decimal... |
| 1e0p1 12741 | The successor of zero. (C... |
| dec10p 12742 | Ten plus an integer. (Con... |
| numma 12743 | Perform a multiply-add of ... |
| nummac 12744 | Perform a multiply-add of ... |
| numma2c 12745 | Perform a multiply-add of ... |
| numadd 12746 | Add two decimal integers `... |
| numaddc 12747 | Add two decimal integers `... |
| nummul1c 12748 | The product of a decimal i... |
| nummul2c 12749 | The product of a decimal i... |
| decma 12750 | Perform a multiply-add of ... |
| decmac 12751 | Perform a multiply-add of ... |
| decma2c 12752 | Perform a multiply-add of ... |
| decadd 12753 | Add two numerals ` M ` and... |
| decaddc 12754 | Add two numerals ` M ` and... |
| decaddc2 12755 | Add two numerals ` M ` and... |
| decrmanc 12756 | Perform a multiply-add of ... |
| decrmac 12757 | Perform a multiply-add of ... |
| decaddm10 12758 | The sum of two multiples o... |
| decaddi 12759 | Add two numerals ` M ` and... |
| decaddci 12760 | Add two numerals ` M ` and... |
| decaddci2 12761 | Add two numerals ` M ` and... |
| decsubi 12762 | Difference between a numer... |
| decmul1 12763 | The product of a numeral w... |
| decmul1c 12764 | The product of a numeral w... |
| decmul2c 12765 | The product of a numeral w... |
| decmulnc 12766 | The product of a numeral w... |
| 11multnc 12767 | The product of 11 (as nume... |
| decmul10add 12768 | A multiplication of a numb... |
| 6p5lem 12769 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12770 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12771 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12772 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12773 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12774 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12775 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12776 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12777 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12778 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12779 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12780 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12781 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12782 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12783 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12784 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12785 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12786 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12787 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12788 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12789 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12790 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12791 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12792 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12793 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12794 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12795 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12796 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12797 | 4 times 3 equals 12. (Con... |
| 4t4e16 12798 | 4 times 4 equals 16. (Con... |
| 5t2e10 12799 | 5 times 2 equals 10. (Con... |
| 5t3e15 12800 | 5 times 3 equals 15. (Con... |
| 5t4e20 12801 | 5 times 4 equals 20. (Con... |
| 5t5e25 12802 | 5 times 5 equals 25. (Con... |
| 6t2e12 12803 | 6 times 2 equals 12. (Con... |
| 6t3e18 12804 | 6 times 3 equals 18. (Con... |
| 6t4e24 12805 | 6 times 4 equals 24. (Con... |
| 6t5e30 12806 | 6 times 5 equals 30. (Con... |
| 6t6e36 12807 | 6 times 6 equals 36. (Con... |
| 7t2e14 12808 | 7 times 2 equals 14. (Con... |
| 7t3e21 12809 | 7 times 3 equals 21. (Con... |
| 7t4e28 12810 | 7 times 4 equals 28. (Con... |
| 7t5e35 12811 | 7 times 5 equals 35. (Con... |
| 7t6e42 12812 | 7 times 6 equals 42. (Con... |
| 7t7e49 12813 | 7 times 7 equals 49. (Con... |
| 8t2e16 12814 | 8 times 2 equals 16. (Con... |
| 8t3e24 12815 | 8 times 3 equals 24. (Con... |
| 8t4e32 12816 | 8 times 4 equals 32. (Con... |
| 8t5e40 12817 | 8 times 5 equals 40. (Con... |
| 8t6e48 12818 | 8 times 6 equals 48. (Con... |
| 8t7e56 12819 | 8 times 7 equals 56. (Con... |
| 8t8e64 12820 | 8 times 8 equals 64. (Con... |
| 9t2e18 12821 | 9 times 2 equals 18. (Con... |
| 9t3e27 12822 | 9 times 3 equals 27. (Con... |
| 9t4e36 12823 | 9 times 4 equals 36. (Con... |
| 9t5e45 12824 | 9 times 5 equals 45. (Con... |
| 9t6e54 12825 | 9 times 6 equals 54. (Con... |
| 9t7e63 12826 | 9 times 7 equals 63. (Con... |
| 9t8e72 12827 | 9 times 8 equals 72. (Con... |
| 9t9e81 12828 | 9 times 9 equals 81. (Con... |
| 9t11e99 12829 | 9 times 11 equals 99. (Co... |
| 9lt10 12830 | 9 is less than 10. (Contr... |
| 8lt10 12831 | 8 is less than 10. (Contr... |
| 7lt10 12832 | 7 is less than 10. (Contr... |
| 6lt10 12833 | 6 is less than 10. (Contr... |
| 5lt10 12834 | 5 is less than 10. (Contr... |
| 4lt10 12835 | 4 is less than 10. (Contr... |
| 3lt10 12836 | 3 is less than 10. (Contr... |
| 2lt10 12837 | 2 is less than 10. (Contr... |
| 1lt10 12838 | 1 is less than 10. (Contr... |
| decbin0 12839 | Decompose base 4 into base... |
| decbin2 12840 | Decompose base 4 into base... |
| decbin3 12841 | Decompose base 4 into base... |
| halfthird 12842 | Half minus a third. (Cont... |
| 5recm6rec 12843 | One fifth minus one sixth.... |
| uzval 12846 | The value of the upper int... |
| uzf 12847 | The domain and codomain of... |
| eluz1 12848 | Membership in the upper se... |
| eluzel2 12849 | Implication of membership ... |
| eluz2 12850 | Membership in an upper set... |
| eluzmn 12851 | Membership in an earlier u... |
| eluz1i 12852 | Membership in an upper set... |
| eluzuzle 12853 | An integer in an upper set... |
| eluzelz 12854 | A member of an upper set o... |
| eluzelre 12855 | A member of an upper set o... |
| eluzelcn 12856 | A member of an upper set o... |
| eluzle 12857 | Implication of membership ... |
| eluz 12858 | Membership in an upper set... |
| uzid 12859 | Membership of the least me... |
| uzidd 12860 | Membership of the least me... |
| uzn0 12861 | The upper integers are all... |
| uztrn 12862 | Transitive law for sets of... |
| uztrn2 12863 | Transitive law for sets of... |
| uzneg 12864 | Contraposition law for upp... |
| uzssz 12865 | An upper set of integers i... |
| uzssre 12866 | An upper set of integers i... |
| uzss 12867 | Subset relationship for tw... |
| uztric 12868 | Totality of the ordering r... |
| uz11 12869 | The upper integers functio... |
| eluzp1m1 12870 | Membership in the next upp... |
| eluzp1l 12871 | Strict ordering implied by... |
| eluzp1p1 12872 | Membership in the next upp... |
| eluzadd 12873 | Membership in a later uppe... |
| eluzsub 12874 | Membership in an earlier u... |
| eluzaddi 12875 | Membership in a later uppe... |
| eluzaddiOLD 12876 | Obsolete version of ~ eluz... |
| eluzsubi 12877 | Membership in an earlier u... |
| eluzsubiOLD 12878 | Obsolete version of ~ eluz... |
| eluzaddOLD 12879 | Obsolete version of ~ eluz... |
| eluzsubOLD 12880 | Obsolete version of ~ eluz... |
| subeluzsub 12881 | Membership of a difference... |
| uzm1 12882 | Choices for an element of ... |
| uznn0sub 12883 | The nonnegative difference... |
| uzin 12884 | Intersection of two upper ... |
| uzp1 12885 | Choices for an element of ... |
| nn0uz 12886 | Nonnegative integers expre... |
| nnuz 12887 | Positive integers expresse... |
| elnnuz 12888 | A positive integer express... |
| elnn0uz 12889 | A nonnegative integer expr... |
| eluz2nn 12890 | An integer greater than or... |
| eluz4eluz2 12891 | An integer greater than or... |
| eluz4nn 12892 | An integer greater than or... |
| eluzge2nn0 12893 | If an integer is greater t... |
| eluz2n0 12894 | An integer greater than or... |
| uzuzle23 12895 | An integer in the upper se... |
| eluzge3nn 12896 | If an integer is greater t... |
| uz3m2nn 12897 | An integer greater than or... |
| 1eluzge0 12898 | 1 is an integer greater th... |
| 2eluzge0 12899 | 2 is an integer greater th... |
| 2eluzge1 12900 | 2 is an integer greater th... |
| uznnssnn 12901 | The upper integers startin... |
| raluz 12902 | Restricted universal quant... |
| raluz2 12903 | Restricted universal quant... |
| rexuz 12904 | Restricted existential qua... |
| rexuz2 12905 | Restricted existential qua... |
| 2rexuz 12906 | Double existential quantif... |
| peano2uz 12907 | Second Peano postulate for... |
| peano2uzs 12908 | Second Peano postulate for... |
| peano2uzr 12909 | Reversed second Peano axio... |
| uzaddcl 12910 | Addition closure law for a... |
| nn0pzuz 12911 | The sum of a nonnegative i... |
| uzind4 12912 | Induction on the upper set... |
| uzind4ALT 12913 | Induction on the upper set... |
| uzind4s 12914 | Induction on the upper set... |
| uzind4s2 12915 | Induction on the upper set... |
| uzind4i 12916 | Induction on the upper int... |
| uzwo 12917 | Well-ordering principle: a... |
| uzwo2 12918 | Well-ordering principle: a... |
| nnwo 12919 | Well-ordering principle: a... |
| nnwof 12920 | Well-ordering principle: a... |
| nnwos 12921 | Well-ordering principle: a... |
| indstr 12922 | Strong Mathematical Induct... |
| eluznn0 12923 | Membership in a nonnegativ... |
| eluznn 12924 | Membership in a positive u... |
| eluz2b1 12925 | Two ways to say "an intege... |
| eluz2gt1 12926 | An integer greater than or... |
| eluz2b2 12927 | Two ways to say "an intege... |
| eluz2b3 12928 | Two ways to say "an intege... |
| uz2m1nn 12929 | One less than an integer g... |
| 1nuz2 12930 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12931 | A positive integer is eith... |
| uz2mulcl 12932 | Closure of multiplication ... |
| indstr2 12933 | Strong Mathematical Induct... |
| uzinfi 12934 | Extract the lower bound of... |
| nninf 12935 | The infimum of the set of ... |
| nn0inf 12936 | The infimum of the set of ... |
| infssuzle 12937 | The infimum of a subset of... |
| infssuzcl 12938 | The infimum of a subset of... |
| ublbneg 12939 | The image under negation o... |
| eqreznegel 12940 | Two ways to express the im... |
| supminf 12941 | The supremum of a bounded-... |
| lbzbi 12942 | If a set of reals is bound... |
| zsupss 12943 | Any nonempty bounded subse... |
| suprzcl2 12944 | The supremum of a bounded-... |
| suprzub 12945 | The supremum of a bounded-... |
| uzsupss 12946 | Any bounded subset of an u... |
| nn01to3 12947 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12948 | Alternate proof of ~ nn0ge... |
| uzwo3 12949 | Well-ordering principle: a... |
| zmin 12950 | There is a unique smallest... |
| zmax 12951 | There is a unique largest ... |
| zbtwnre 12952 | There is a unique integer ... |
| rebtwnz 12953 | There is a unique greatest... |
| elq 12956 | Membership in the set of r... |
| qmulz 12957 | If ` A ` is rational, then... |
| znq 12958 | The ratio of an integer an... |
| qre 12959 | A rational number is a rea... |
| zq 12960 | An integer is a rational n... |
| qred 12961 | A rational number is a rea... |
| zssq 12962 | The integers are a subset ... |
| nn0ssq 12963 | The nonnegative integers a... |
| nnssq 12964 | The positive integers are ... |
| qssre 12965 | The rationals are a subset... |
| qsscn 12966 | The rationals are a subset... |
| qex 12967 | The set of rational number... |
| nnq 12968 | A positive integer is rati... |
| qcn 12969 | A rational number is a com... |
| qexALT 12970 | Alternate proof of ~ qex .... |
| qaddcl 12971 | Closure of addition of rat... |
| qnegcl 12972 | Closure law for the negati... |
| qmulcl 12973 | Closure of multiplication ... |
| qsubcl 12974 | Closure of subtraction of ... |
| qreccl 12975 | Closure of reciprocal of r... |
| qdivcl 12976 | Closure of division of rat... |
| qrevaddcl 12977 | Reverse closure law for ad... |
| nnrecq 12978 | The reciprocal of a positi... |
| irradd 12979 | The sum of an irrational n... |
| irrmul 12980 | The product of an irration... |
| elpq 12981 | A positive rational is the... |
| elpqb 12982 | A class is a positive rati... |
| rpnnen1lem2 12983 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12984 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12985 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12986 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12987 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12988 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12989 | One half of ~ rpnnen , whe... |
| reexALT 12990 | Alternate proof of ~ reex ... |
| cnref1o 12991 | There is a natural one-to-... |
| cnexALT 12992 | The set of complex numbers... |
| xrex 12993 | The set of extended reals ... |
| mpoaddex 12994 | The addition operation is ... |
| addex 12995 | The addition operation is ... |
| mpomulex 12996 | The multiplication operati... |
| mulex 12997 | The multiplication operati... |
| elrp 13000 | Membership in the set of p... |
| elrpii 13001 | Membership in the set of p... |
| 1rp 13002 | 1 is a positive real. (Co... |
| 2rp 13003 | 2 is a positive real. (Co... |
| 3rp 13004 | 3 is a positive real. (Co... |
| rpssre 13005 | The positive reals are a s... |
| rpre 13006 | A positive real is a real.... |
| rpxr 13007 | A positive real is an exte... |
| rpcn 13008 | A positive real is a compl... |
| nnrp 13009 | A positive integer is a po... |
| rpgt0 13010 | A positive real is greater... |
| rpge0 13011 | A positive real is greater... |
| rpregt0 13012 | A positive real is a posit... |
| rprege0 13013 | A positive real is a nonne... |
| rpne0 13014 | A positive real is nonzero... |
| rprene0 13015 | A positive real is a nonze... |
| rpcnne0 13016 | A positive real is a nonze... |
| rpcndif0 13017 | A positive real number is ... |
| ralrp 13018 | Quantification over positi... |
| rexrp 13019 | Quantification over positi... |
| rpaddcl 13020 | Closure law for addition o... |
| rpmulcl 13021 | Closure law for multiplica... |
| rpmtmip 13022 | "Minus times minus is plus... |
| rpdivcl 13023 | Closure law for division o... |
| rpreccl 13024 | Closure law for reciprocat... |
| rphalfcl 13025 | Closure law for half of a ... |
| rpgecl 13026 | A number greater than or e... |
| rphalflt 13027 | Half of a positive real is... |
| rerpdivcl 13028 | Closure law for division o... |
| ge0p1rp 13029 | A nonnegative number plus ... |
| rpneg 13030 | Either a nonzero real or i... |
| negelrp 13031 | Elementhood of a negation ... |
| negelrpd 13032 | The negation of a negative... |
| 0nrp 13033 | Zero is not a positive rea... |
| ltsubrp 13034 | Subtracting a positive rea... |
| ltaddrp 13035 | Adding a positive number t... |
| difrp 13036 | Two ways to say one number... |
| elrpd 13037 | Membership in the set of p... |
| nnrpd 13038 | A positive integer is a po... |
| zgt1rpn0n1 13039 | An integer greater than 1 ... |
| rpred 13040 | A positive real is a real.... |
| rpxrd 13041 | A positive real is an exte... |
| rpcnd 13042 | A positive real is a compl... |
| rpgt0d 13043 | A positive real is greater... |
| rpge0d 13044 | A positive real is greater... |
| rpne0d 13045 | A positive real is nonzero... |
| rpregt0d 13046 | A positive real is real an... |
| rprege0d 13047 | A positive real is real an... |
| rprene0d 13048 | A positive real is a nonze... |
| rpcnne0d 13049 | A positive real is a nonze... |
| rpreccld 13050 | Closure law for reciprocat... |
| rprecred 13051 | Closure law for reciprocat... |
| rphalfcld 13052 | Closure law for half of a ... |
| reclt1d 13053 | The reciprocal of a positi... |
| recgt1d 13054 | The reciprocal of a positi... |
| rpaddcld 13055 | Closure law for addition o... |
| rpmulcld 13056 | Closure law for multiplica... |
| rpdivcld 13057 | Closure law for division o... |
| ltrecd 13058 | The reciprocal of both sid... |
| lerecd 13059 | The reciprocal of both sid... |
| ltrec1d 13060 | Reciprocal swap in a 'less... |
| lerec2d 13061 | Reciprocal swap in a 'less... |
| lediv2ad 13062 | Division of both sides of ... |
| ltdiv2d 13063 | Division of a positive num... |
| lediv2d 13064 | Division of a positive num... |
| ledivdivd 13065 | Invert ratios of positive ... |
| divge1 13066 | The ratio of a number over... |
| divlt1lt 13067 | A real number divided by a... |
| divle1le 13068 | A real number divided by a... |
| ledivge1le 13069 | If a number is less than o... |
| ge0p1rpd 13070 | A nonnegative number plus ... |
| rerpdivcld 13071 | Closure law for division o... |
| ltsubrpd 13072 | Subtracting a positive rea... |
| ltaddrpd 13073 | Adding a positive number t... |
| ltaddrp2d 13074 | Adding a positive number t... |
| ltmulgt11d 13075 | Multiplication by a number... |
| ltmulgt12d 13076 | Multiplication by a number... |
| gt0divd 13077 | Division of a positive num... |
| ge0divd 13078 | Division of a nonnegative ... |
| rpgecld 13079 | A number greater than or e... |
| divge0d 13080 | The ratio of nonnegative a... |
| ltmul1d 13081 | The ratio of nonnegative a... |
| ltmul2d 13082 | Multiplication of both sid... |
| lemul1d 13083 | Multiplication of both sid... |
| lemul2d 13084 | Multiplication of both sid... |
| ltdiv1d 13085 | Division of both sides of ... |
| lediv1d 13086 | Division of both sides of ... |
| ltmuldivd 13087 | 'Less than' relationship b... |
| ltmuldiv2d 13088 | 'Less than' relationship b... |
| lemuldivd 13089 | 'Less than or equal to' re... |
| lemuldiv2d 13090 | 'Less than or equal to' re... |
| ltdivmuld 13091 | 'Less than' relationship b... |
| ltdivmul2d 13092 | 'Less than' relationship b... |
| ledivmuld 13093 | 'Less than or equal to' re... |
| ledivmul2d 13094 | 'Less than or equal to' re... |
| ltmul1dd 13095 | The ratio of nonnegative a... |
| ltmul2dd 13096 | Multiplication of both sid... |
| ltdiv1dd 13097 | Division of both sides of ... |
| lediv1dd 13098 | Division of both sides of ... |
| lediv12ad 13099 | Comparison of ratio of two... |
| mul2lt0rlt0 13100 | If the result of a multipl... |
| mul2lt0rgt0 13101 | If the result of a multipl... |
| mul2lt0llt0 13102 | If the result of a multipl... |
| mul2lt0lgt0 13103 | If the result of a multipl... |
| mul2lt0bi 13104 | If the result of a multipl... |
| prodge0rd 13105 | Infer that a multiplicand ... |
| prodge0ld 13106 | Infer that a multiplier is... |
| ltdiv23d 13107 | Swap denominator with othe... |
| lediv23d 13108 | Swap denominator with othe... |
| lt2mul2divd 13109 | The ratio of nonnegative a... |
| nnledivrp 13110 | Division of a positive int... |
| nn0ledivnn 13111 | Division of a nonnegative ... |
| addlelt 13112 | If the sum of a real numbe... |
| ltxr 13119 | The 'less than' binary rel... |
| elxr 13120 | Membership in the set of e... |
| xrnemnf 13121 | An extended real other tha... |
| xrnepnf 13122 | An extended real other tha... |
| xrltnr 13123 | The extended real 'less th... |
| ltpnf 13124 | Any (finite) real is less ... |
| ltpnfd 13125 | Any (finite) real is less ... |
| 0ltpnf 13126 | Zero is less than plus inf... |
| mnflt 13127 | Minus infinity is less tha... |
| mnfltd 13128 | Minus infinity is less tha... |
| mnflt0 13129 | Minus infinity is less tha... |
| mnfltpnf 13130 | Minus infinity is less tha... |
| mnfltxr 13131 | Minus infinity is less tha... |
| pnfnlt 13132 | No extended real is greate... |
| nltmnf 13133 | No extended real is less t... |
| pnfge 13134 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13135 | An extended nonnegative in... |
| 0lepnf 13136 | 0 less than or equal to po... |
| xnn0ge0 13137 | An extended nonnegative in... |
| mnfle 13138 | Minus infinity is less tha... |
| mnfled 13139 | Minus infinity is less tha... |
| xrltnsym 13140 | Ordering on the extended r... |
| xrltnsym2 13141 | 'Less than' is antisymmetr... |
| xrlttri 13142 | Ordering on the extended r... |
| xrlttr 13143 | Ordering on the extended r... |
| xrltso 13144 | 'Less than' is a strict or... |
| xrlttri2 13145 | Trichotomy law for 'less t... |
| xrlttri3 13146 | Trichotomy law for 'less t... |
| xrleloe 13147 | 'Less than or equal' expre... |
| xrleltne 13148 | 'Less than or equal to' im... |
| xrltlen 13149 | 'Less than' expressed in t... |
| dfle2 13150 | Alternative definition of ... |
| dflt2 13151 | Alternative definition of ... |
| xrltle 13152 | 'Less than' implies 'less ... |
| xrltled 13153 | 'Less than' implies 'less ... |
| xrleid 13154 | 'Less than or equal to' is... |
| xrleidd 13155 | 'Less than or equal to' is... |
| xrletri 13156 | Trichotomy law for extende... |
| xrletri3 13157 | Trichotomy law for extende... |
| xrletrid 13158 | Trichotomy law for extende... |
| xrlelttr 13159 | Transitive law for orderin... |
| xrltletr 13160 | Transitive law for orderin... |
| xrletr 13161 | Transitive law for orderin... |
| xrlttrd 13162 | Transitive law for orderin... |
| xrlelttrd 13163 | Transitive law for orderin... |
| xrltletrd 13164 | Transitive law for orderin... |
| xrletrd 13165 | Transitive law for orderin... |
| xrltne 13166 | 'Less than' implies not eq... |
| nltpnft 13167 | An extended real is not le... |
| xgepnf 13168 | An extended real which is ... |
| ngtmnft 13169 | An extended real is not gr... |
| xlemnf 13170 | An extended real which is ... |
| xrrebnd 13171 | An extended real is real i... |
| xrre 13172 | A way of proving that an e... |
| xrre2 13173 | An extended real between t... |
| xrre3 13174 | A way of proving that an e... |
| ge0gtmnf 13175 | A nonnegative extended rea... |
| ge0nemnf 13176 | A nonnegative extended rea... |
| xrrege0 13177 | A nonnegative extended rea... |
| xrmax1 13178 | An extended real is less t... |
| xrmax2 13179 | An extended real is less t... |
| xrmin1 13180 | The minimum of two extende... |
| xrmin2 13181 | The minimum of two extende... |
| xrmaxeq 13182 | The maximum of two extende... |
| xrmineq 13183 | The minimum of two extende... |
| xrmaxlt 13184 | Two ways of saying the max... |
| xrltmin 13185 | Two ways of saying an exte... |
| xrmaxle 13186 | Two ways of saying the max... |
| xrlemin 13187 | Two ways of saying a numbe... |
| max1 13188 | A number is less than or e... |
| max1ALT 13189 | A number is less than or e... |
| max2 13190 | A number is less than or e... |
| 2resupmax 13191 | The supremum of two real n... |
| min1 13192 | The minimum of two numbers... |
| min2 13193 | The minimum of two numbers... |
| maxle 13194 | Two ways of saying the max... |
| lemin 13195 | Two ways of saying a numbe... |
| maxlt 13196 | Two ways of saying the max... |
| ltmin 13197 | Two ways of saying a numbe... |
| lemaxle 13198 | A real number which is les... |
| max0sub 13199 | Decompose a real number in... |
| ifle 13200 | An if statement transforms... |
| z2ge 13201 | There exists an integer gr... |
| qbtwnre 13202 | The rational numbers are d... |
| qbtwnxr 13203 | The rational numbers are d... |
| qsqueeze 13204 | If a nonnegative real is l... |
| qextltlem 13205 | Lemma for ~ qextlt and qex... |
| qextlt 13206 | An extensionality-like pro... |
| qextle 13207 | An extensionality-like pro... |
| xralrple 13208 | Show that ` A ` is less th... |
| alrple 13209 | Show that ` A ` is less th... |
| xnegeq 13210 | Equality of two extended n... |
| xnegex 13211 | A negative extended real e... |
| xnegpnf 13212 | Minus ` +oo ` . Remark of... |
| xnegmnf 13213 | Minus ` -oo ` . Remark of... |
| rexneg 13214 | Minus a real number. Rema... |
| xneg0 13215 | The negative of zero. (Co... |
| xnegcl 13216 | Closure of extended real n... |
| xnegneg 13217 | Extended real version of ~... |
| xneg11 13218 | Extended real version of ~... |
| xltnegi 13219 | Forward direction of ~ xlt... |
| xltneg 13220 | Extended real version of ~... |
| xleneg 13221 | Extended real version of ~... |
| xlt0neg1 13222 | Extended real version of ~... |
| xlt0neg2 13223 | Extended real version of ~... |
| xle0neg1 13224 | Extended real version of ~... |
| xle0neg2 13225 | Extended real version of ~... |
| xaddval 13226 | Value of the extended real... |
| xaddf 13227 | The extended real addition... |
| xmulval 13228 | Value of the extended real... |
| xaddpnf1 13229 | Addition of positive infin... |
| xaddpnf2 13230 | Addition of positive infin... |
| xaddmnf1 13231 | Addition of negative infin... |
| xaddmnf2 13232 | Addition of negative infin... |
| pnfaddmnf 13233 | Addition of positive and n... |
| mnfaddpnf 13234 | Addition of negative and p... |
| rexadd 13235 | The extended real addition... |
| rexsub 13236 | Extended real subtraction ... |
| rexaddd 13237 | The extended real addition... |
| xnn0xaddcl 13238 | The extended nonnegative i... |
| xaddnemnf 13239 | Closure of extended real a... |
| xaddnepnf 13240 | Closure of extended real a... |
| xnegid 13241 | Extended real version of ~... |
| xaddcl 13242 | The extended real addition... |
| xaddcom 13243 | The extended real addition... |
| xaddrid 13244 | Extended real version of ~... |
| xaddlid 13245 | Extended real version of ~... |
| xaddridd 13246 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13247 | Extended nonnegative integ... |
| xnn0lenn0nn0 13248 | An extended nonnegative in... |
| xnn0le2is012 13249 | An extended nonnegative in... |
| xnn0xadd0 13250 | The sum of two extended no... |
| xnegdi 13251 | Extended real version of ~... |
| xaddass 13252 | Associativity of extended ... |
| xaddass2 13253 | Associativity of extended ... |
| xpncan 13254 | Extended real version of ~... |
| xnpcan 13255 | Extended real version of ~... |
| xleadd1a 13256 | Extended real version of ~... |
| xleadd2a 13257 | Commuted form of ~ xleadd1... |
| xleadd1 13258 | Weakened version of ~ xlea... |
| xltadd1 13259 | Extended real version of ~... |
| xltadd2 13260 | Extended real version of ~... |
| xaddge0 13261 | The sum of nonnegative ext... |
| xle2add 13262 | Extended real version of ~... |
| xlt2add 13263 | Extended real version of ~... |
| xsubge0 13264 | Extended real version of ~... |
| xposdif 13265 | Extended real version of ~... |
| xlesubadd 13266 | Under certain conditions, ... |
| xmullem 13267 | Lemma for ~ rexmul . (Con... |
| xmullem2 13268 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13269 | Extended real multiplicati... |
| xmul01 13270 | Extended real version of ~... |
| xmul02 13271 | Extended real version of ~... |
| xmulneg1 13272 | Extended real version of ~... |
| xmulneg2 13273 | Extended real version of ~... |
| rexmul 13274 | The extended real multipli... |
| xmulf 13275 | The extended real multipli... |
| xmulcl 13276 | Closure of extended real m... |
| xmulpnf1 13277 | Multiplication by plus inf... |
| xmulpnf2 13278 | Multiplication by plus inf... |
| xmulmnf1 13279 | Multiplication by minus in... |
| xmulmnf2 13280 | Multiplication by minus in... |
| xmulpnf1n 13281 | Multiplication by plus inf... |
| xmulrid 13282 | Extended real version of ~... |
| xmullid 13283 | Extended real version of ~... |
| xmulm1 13284 | Extended real version of ~... |
| xmulasslem2 13285 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13286 | Extended real version of ~... |
| xmulge0 13287 | Extended real version of ~... |
| xmulasslem 13288 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13289 | Lemma for ~ xmulass . (Co... |
| xmulass 13290 | Associativity of the exten... |
| xlemul1a 13291 | Extended real version of ~... |
| xlemul2a 13292 | Extended real version of ~... |
| xlemul1 13293 | Extended real version of ~... |
| xlemul2 13294 | Extended real version of ~... |
| xltmul1 13295 | Extended real version of ~... |
| xltmul2 13296 | Extended real version of ~... |
| xadddilem 13297 | Lemma for ~ xadddi . (Con... |
| xadddi 13298 | Distributive property for ... |
| xadddir 13299 | Commuted version of ~ xadd... |
| xadddi2 13300 | The assumption that the mu... |
| xadddi2r 13301 | Commuted version of ~ xadd... |
| x2times 13302 | Extended real version of ~... |
| xnegcld 13303 | Closure of extended real n... |
| xaddcld 13304 | The extended real addition... |
| xmulcld 13305 | Closure of extended real m... |
| xadd4d 13306 | Rearrangement of 4 terms i... |
| xnn0add4d 13307 | Rearrangement of 4 terms i... |
| xrsupexmnf 13308 | Adding minus infinity to a... |
| xrinfmexpnf 13309 | Adding plus infinity to a ... |
| xrsupsslem 13310 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13311 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13312 | Any subset of extended rea... |
| xrinfmss 13313 | Any subset of extended rea... |
| xrinfmss2 13314 | Any subset of extended rea... |
| xrub 13315 | By quantifying only over r... |
| supxr 13316 | The supremum of a set of e... |
| supxr2 13317 | The supremum of a set of e... |
| supxrcl 13318 | The supremum of an arbitra... |
| supxrun 13319 | The supremum of the union ... |
| supxrmnf 13320 | Adding minus infinity to a... |
| supxrpnf 13321 | The supremum of a set of e... |
| supxrunb1 13322 | The supremum of an unbound... |
| supxrunb2 13323 | The supremum of an unbound... |
| supxrbnd1 13324 | The supremum of a bounded-... |
| supxrbnd2 13325 | The supremum of a bounded-... |
| xrsup0 13326 | The supremum of an empty s... |
| supxrub 13327 | A member of a set of exten... |
| supxrlub 13328 | The supremum of a set of e... |
| supxrleub 13329 | The supremum of a set of e... |
| supxrre 13330 | The real and extended real... |
| supxrbnd 13331 | The supremum of a bounded-... |
| supxrgtmnf 13332 | The supremum of a nonempty... |
| supxrre1 13333 | The supremum of a nonempty... |
| supxrre2 13334 | The supremum of a nonempty... |
| supxrss 13335 | Smaller sets of extended r... |
| infxrcl 13336 | The infimum of an arbitrar... |
| infxrlb 13337 | A member of a set of exten... |
| infxrgelb 13338 | The infimum of a set of ex... |
| infxrre 13339 | The real and extended real... |
| infxrmnf 13340 | The infinimum of a set of ... |
| xrinf0 13341 | The infimum of the empty s... |
| infxrss 13342 | Larger sets of extended re... |
| reltre 13343 | For all real numbers there... |
| rpltrp 13344 | For all positive real numb... |
| reltxrnmnf 13345 | For all extended real numb... |
| infmremnf 13346 | The infimum of the reals i... |
| infmrp1 13347 | The infimum of the positiv... |
| ixxval 13356 | Value of the interval func... |
| elixx1 13357 | Membership in an interval ... |
| ixxf 13358 | The set of intervals of ex... |
| ixxex 13359 | The set of intervals of ex... |
| ixxssxr 13360 | The set of intervals of ex... |
| elixx3g 13361 | Membership in a set of ope... |
| ixxssixx 13362 | An interval is a subset of... |
| ixxdisj 13363 | Split an interval into dis... |
| ixxun 13364 | Split an interval into two... |
| ixxin 13365 | Intersection of two interv... |
| ixxss1 13366 | Subset relationship for in... |
| ixxss2 13367 | Subset relationship for in... |
| ixxss12 13368 | Subset relationship for in... |
| ixxub 13369 | Extract the upper bound of... |
| ixxlb 13370 | Extract the lower bound of... |
| iooex 13371 | The set of open intervals ... |
| iooval 13372 | Value of the open interval... |
| ioo0 13373 | An empty open interval of ... |
| ioon0 13374 | An open interval of extend... |
| ndmioo 13375 | The open interval function... |
| iooid 13376 | An open interval with iden... |
| elioo3g 13377 | Membership in a set of ope... |
| elioore 13378 | A member of an open interv... |
| lbioo 13379 | An open interval does not ... |
| ubioo 13380 | An open interval does not ... |
| iooval2 13381 | Value of the open interval... |
| iooin 13382 | Intersection of two open i... |
| iooss1 13383 | Subset relationship for op... |
| iooss2 13384 | Subset relationship for op... |
| iocval 13385 | Value of the open-below, c... |
| icoval 13386 | Value of the closed-below,... |
| iccval 13387 | Value of the closed interv... |
| elioo1 13388 | Membership in an open inte... |
| elioo2 13389 | Membership in an open inte... |
| elioc1 13390 | Membership in an open-belo... |
| elico1 13391 | Membership in a closed-bel... |
| elicc1 13392 | Membership in a closed int... |
| iccid 13393 | A closed interval with ide... |
| ico0 13394 | An empty open interval of ... |
| ioc0 13395 | An empty open interval of ... |
| icc0 13396 | An empty closed interval o... |
| dfrp2 13397 | Alternate definition of th... |
| elicod 13398 | Membership in a left-close... |
| icogelb 13399 | An element of a left-close... |
| elicore 13400 | A member of a left-closed ... |
| ubioc1 13401 | The upper bound belongs to... |
| lbico1 13402 | The lower bound belongs to... |
| iccleub 13403 | An element of a closed int... |
| iccgelb 13404 | An element of a closed int... |
| elioo5 13405 | Membership in an open inte... |
| eliooxr 13406 | A nonempty open interval s... |
| eliooord 13407 | Ordering implied by a memb... |
| elioo4g 13408 | Membership in an open inte... |
| ioossre 13409 | An open interval is a set ... |
| ioosscn 13410 | An open interval is a set ... |
| elioc2 13411 | Membership in an open-belo... |
| elico2 13412 | Membership in a closed-bel... |
| elicc2 13413 | Membership in a closed rea... |
| elicc2i 13414 | Inference for membership i... |
| elicc4 13415 | Membership in a closed rea... |
| iccss 13416 | Condition for a closed int... |
| iccssioo 13417 | Condition for a closed int... |
| icossico 13418 | Condition for a closed-bel... |
| iccss2 13419 | Condition for a closed int... |
| iccssico 13420 | Condition for a closed int... |
| iccssioo2 13421 | Condition for a closed int... |
| iccssico2 13422 | Condition for a closed int... |
| ioomax 13423 | The open interval from min... |
| iccmax 13424 | The closed interval from m... |
| ioopos 13425 | The set of positive reals ... |
| ioorp 13426 | The set of positive reals ... |
| iooshf 13427 | Shift the arguments of the... |
| iocssre 13428 | A closed-above interval wi... |
| icossre 13429 | A closed-below interval wi... |
| iccssre 13430 | A closed real interval is ... |
| iccssxr 13431 | A closed interval is a set... |
| iocssxr 13432 | An open-below, closed-abov... |
| icossxr 13433 | A closed-below, open-above... |
| ioossicc 13434 | An open interval is a subs... |
| iccssred 13435 | A closed real interval is ... |
| eliccxr 13436 | A member of a closed inter... |
| icossicc 13437 | A closed-below, open-above... |
| iocssicc 13438 | A closed-above, open-below... |
| ioossico 13439 | An open interval is a subs... |
| iocssioo 13440 | Condition for a closed int... |
| icossioo 13441 | Condition for a closed int... |
| ioossioo 13442 | Condition for an open inte... |
| iccsupr 13443 | A nonempty subset of a clo... |
| elioopnf 13444 | Membership in an unbounded... |
| elioomnf 13445 | Membership in an unbounded... |
| elicopnf 13446 | Membership in a closed unb... |
| repos 13447 | Two ways of saying that a ... |
| ioof 13448 | The set of open intervals ... |
| iccf 13449 | The set of closed interval... |
| unirnioo 13450 | The union of the range of ... |
| dfioo2 13451 | Alternate definition of th... |
| ioorebas 13452 | Open intervals are element... |
| xrge0neqmnf 13453 | A nonnegative extended rea... |
| xrge0nre 13454 | An extended real which is ... |
| elrege0 13455 | The predicate "is a nonneg... |
| nn0rp0 13456 | A nonnegative integer is a... |
| rge0ssre 13457 | Nonnegative real numbers a... |
| elxrge0 13458 | Elementhood in the set of ... |
| 0e0icopnf 13459 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13460 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13461 | The nonnegative reals are ... |
| ge0mulcl 13462 | The nonnegative reals are ... |
| ge0xaddcl 13463 | The nonnegative reals are ... |
| ge0xmulcl 13464 | The nonnegative extended r... |
| lbicc2 13465 | The lower bound of a close... |
| ubicc2 13466 | The upper bound of a close... |
| elicc01 13467 | Membership in the closed r... |
| elunitrn 13468 | The closed unit interval i... |
| elunitcn 13469 | The closed unit interval i... |
| 0elunit 13470 | Zero is an element of the ... |
| 1elunit 13471 | One is an element of the c... |
| iooneg 13472 | Membership in a negated op... |
| iccneg 13473 | Membership in a negated cl... |
| icoshft 13474 | A shifted real is a member... |
| icoshftf1o 13475 | Shifting a closed-below, o... |
| icoun 13476 | The union of two adjacent ... |
| icodisj 13477 | Adjacent left-closed right... |
| ioounsn 13478 | The union of an open inter... |
| snunioo 13479 | The closure of one end of ... |
| snunico 13480 | The closure of the open en... |
| snunioc 13481 | The closure of the open en... |
| prunioo 13482 | The closure of an open rea... |
| ioodisj 13483 | If the upper bound of one ... |
| ioojoin 13484 | Join two open intervals to... |
| difreicc 13485 | The class difference of ` ... |
| iccsplit 13486 | Split a closed interval in... |
| iccshftr 13487 | Membership in a shifted in... |
| iccshftri 13488 | Membership in a shifted in... |
| iccshftl 13489 | Membership in a shifted in... |
| iccshftli 13490 | Membership in a shifted in... |
| iccdil 13491 | Membership in a dilated in... |
| iccdili 13492 | Membership in a dilated in... |
| icccntr 13493 | Membership in a contracted... |
| icccntri 13494 | Membership in a contracted... |
| divelunit 13495 | A condition for a ratio to... |
| lincmb01cmp 13496 | A linear combination of tw... |
| iccf1o 13497 | Describe a bijection from ... |
| iccen 13498 | Any nontrivial closed inte... |
| xov1plusxeqvd 13499 | A complex number ` X ` is ... |
| unitssre 13500 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13501 | The closed unit interval i... |
| supicc 13502 | Supremum of a bounded set ... |
| supiccub 13503 | The supremum of a bounded ... |
| supicclub 13504 | The supremum of a bounded ... |
| supicclub2 13505 | The supremum of a bounded ... |
| zltaddlt1le 13506 | The sum of an integer and ... |
| xnn0xrge0 13507 | An extended nonnegative in... |
| fzval 13510 | The value of a finite set ... |
| fzval2 13511 | An alternative way of expr... |
| fzf 13512 | Establish the domain and c... |
| elfz1 13513 | Membership in a finite set... |
| elfz 13514 | Membership in a finite set... |
| elfz2 13515 | Membership in a finite set... |
| elfzd 13516 | Membership in a finite set... |
| elfz5 13517 | Membership in a finite set... |
| elfz4 13518 | Membership in a finite set... |
| elfzuzb 13519 | Membership in a finite set... |
| eluzfz 13520 | Membership in a finite set... |
| elfzuz 13521 | A member of a finite set o... |
| elfzuz3 13522 | Membership in a finite set... |
| elfzel2 13523 | Membership in a finite set... |
| elfzel1 13524 | Membership in a finite set... |
| elfzelz 13525 | A member of a finite set o... |
| elfzelzd 13526 | A member of a finite set o... |
| fzssz 13527 | A finite sequence of integ... |
| elfzle1 13528 | A member of a finite set o... |
| elfzle2 13529 | A member of a finite set o... |
| elfzuz2 13530 | Implication of membership ... |
| elfzle3 13531 | Membership in a finite set... |
| eluzfz1 13532 | Membership in a finite set... |
| eluzfz2 13533 | Membership in a finite set... |
| eluzfz2b 13534 | Membership in a finite set... |
| elfz3 13535 | Membership in a finite set... |
| elfz1eq 13536 | Membership in a finite set... |
| elfzubelfz 13537 | If there is a member in a ... |
| peano2fzr 13538 | A Peano-postulate-like the... |
| fzn0 13539 | Properties of a finite int... |
| fz0 13540 | A finite set of sequential... |
| fzn 13541 | A finite set of sequential... |
| fzen 13542 | A shifted finite set of se... |
| fz1n 13543 | A 1-based finite set of se... |
| 0nelfz1 13544 | 0 is not an element of a f... |
| 0fz1 13545 | Two ways to say a finite 1... |
| fz10 13546 | There are no integers betw... |
| uzsubsubfz 13547 | Membership of an integer g... |
| uzsubsubfz1 13548 | Membership of an integer g... |
| ige3m2fz 13549 | Membership of an integer g... |
| fzsplit2 13550 | Split a finite interval of... |
| fzsplit 13551 | Split a finite interval of... |
| fzdisj 13552 | Condition for two finite i... |
| fz01en 13553 | 0-based and 1-based finite... |
| elfznn 13554 | A member of a finite set o... |
| elfz1end 13555 | A nonempty finite range of... |
| fz1ssnn 13556 | A finite set of positive i... |
| fznn0sub 13557 | Subtraction closure for a ... |
| fzmmmeqm 13558 | Subtracting the difference... |
| fzaddel 13559 | Membership of a sum in a f... |
| fzadd2 13560 | Membership of a sum in a f... |
| fzsubel 13561 | Membership of a difference... |
| fzopth 13562 | A finite set of sequential... |
| fzass4 13563 | Two ways to express a nond... |
| fzss1 13564 | Subset relationship for fi... |
| fzss2 13565 | Subset relationship for fi... |
| fzssuz 13566 | A finite set of sequential... |
| fzsn 13567 | A finite interval of integ... |
| fzssp1 13568 | Subset relationship for fi... |
| fzssnn 13569 | Finite sets of sequential ... |
| ssfzunsnext 13570 | A subset of a finite seque... |
| ssfzunsn 13571 | A subset of a finite seque... |
| fzsuc 13572 | Join a successor to the en... |
| fzpred 13573 | Join a predecessor to the ... |
| fzpreddisj 13574 | A finite set of sequential... |
| elfzp1 13575 | Append an element to a fin... |
| fzp1ss 13576 | Subset relationship for fi... |
| fzelp1 13577 | Membership in a set of seq... |
| fzp1elp1 13578 | Add one to an element of a... |
| fznatpl1 13579 | Shift membership in a fini... |
| fzpr 13580 | A finite interval of integ... |
| fztp 13581 | A finite interval of integ... |
| fz12pr 13582 | An integer range between 1... |
| fzsuc2 13583 | Join a successor to the en... |
| fzp1disj 13584 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13585 | Remove a successor from th... |
| fzprval 13586 | Two ways of defining the f... |
| fztpval 13587 | Two ways of defining the f... |
| fzrev 13588 | Reversal of start and end ... |
| fzrev2 13589 | Reversal of start and end ... |
| fzrev2i 13590 | Reversal of start and end ... |
| fzrev3 13591 | The "complement" of a memb... |
| fzrev3i 13592 | The "complement" of a memb... |
| fznn 13593 | Finite set of sequential i... |
| elfz1b 13594 | Membership in a 1-based fi... |
| elfz1uz 13595 | Membership in a 1-based fi... |
| elfzm11 13596 | Membership in a finite set... |
| uzsplit 13597 | Express an upper integer s... |
| uzdisj 13598 | The first ` N ` elements o... |
| fseq1p1m1 13599 | Add/remove an item to/from... |
| fseq1m1p1 13600 | Add/remove an item to/from... |
| fz1sbc 13601 | Quantification over a one-... |
| elfzp1b 13602 | An integer is a member of ... |
| elfzm1b 13603 | An integer is a member of ... |
| elfzp12 13604 | Options for membership in ... |
| fzm1 13605 | Choices for an element of ... |
| fzneuz 13606 | No finite set of sequentia... |
| fznuz 13607 | Disjointness of the upper ... |
| uznfz 13608 | Disjointness of the upper ... |
| fzp1nel 13609 | One plus the upper bound o... |
| fzrevral 13610 | Reversal of scanning order... |
| fzrevral2 13611 | Reversal of scanning order... |
| fzrevral3 13612 | Reversal of scanning order... |
| fzshftral 13613 | Shift the scanning order i... |
| ige2m1fz1 13614 | Membership of an integer g... |
| ige2m1fz 13615 | Membership in a 0-based fi... |
| elfz2nn0 13616 | Membership in a finite set... |
| fznn0 13617 | Characterization of a fini... |
| elfznn0 13618 | A member of a finite set o... |
| elfz3nn0 13619 | The upper bound of a nonem... |
| fz0ssnn0 13620 | Finite sets of sequential ... |
| fz1ssfz0 13621 | Subset relationship for fi... |
| 0elfz 13622 | 0 is an element of a finit... |
| nn0fz0 13623 | A nonnegative integer is a... |
| elfz0add 13624 | An element of a finite set... |
| fz0sn 13625 | An integer range from 0 to... |
| fz0tp 13626 | An integer range from 0 to... |
| fz0to3un2pr 13627 | An integer range from 0 to... |
| fz0to4untppr 13628 | An integer range from 0 to... |
| elfz0ubfz0 13629 | An element of a finite set... |
| elfz0fzfz0 13630 | A member of a finite set o... |
| fz0fzelfz0 13631 | If a member of a finite se... |
| fznn0sub2 13632 | Subtraction closure for a ... |
| uzsubfz0 13633 | Membership of an integer g... |
| fz0fzdiffz0 13634 | The difference of an integ... |
| elfzmlbm 13635 | Subtracting the lower boun... |
| elfzmlbp 13636 | Subtracting the lower boun... |
| fzctr 13637 | Lemma for theorems about t... |
| difelfzle 13638 | The difference of two inte... |
| difelfznle 13639 | The difference of two inte... |
| nn0split 13640 | Express the set of nonnega... |
| nn0disj 13641 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13642 | A finite set of sequential... |
| fvffz0 13643 | The function value of a fu... |
| 1fv 13644 | A function on a singleton.... |
| 4fvwrd4 13645 | The first four function va... |
| 2ffzeq 13646 | Two functions over 0-based... |
| preduz 13647 | The value of the predecess... |
| prednn 13648 | The value of the predecess... |
| prednn0 13649 | The value of the predecess... |
| predfz 13650 | Calculate the predecessor ... |
| fzof 13653 | Functionality of the half-... |
| elfzoel1 13654 | Reverse closure for half-o... |
| elfzoel2 13655 | Reverse closure for half-o... |
| elfzoelz 13656 | Reverse closure for half-o... |
| fzoval 13657 | Value of the half-open int... |
| elfzo 13658 | Membership in a half-open ... |
| elfzo2 13659 | Membership in a half-open ... |
| elfzouz 13660 | Membership in a half-open ... |
| nelfzo 13661 | An integer not being a mem... |
| fzolb 13662 | The left endpoint of a hal... |
| fzolb2 13663 | The left endpoint of a hal... |
| elfzole1 13664 | A member in a half-open in... |
| elfzolt2 13665 | A member in a half-open in... |
| elfzolt3 13666 | Membership in a half-open ... |
| elfzolt2b 13667 | A member in a half-open in... |
| elfzolt3b 13668 | Membership in a half-open ... |
| elfzop1le2 13669 | A member in a half-open in... |
| fzonel 13670 | A half-open range does not... |
| elfzouz2 13671 | The upper bound of a half-... |
| elfzofz 13672 | A half-open range is conta... |
| elfzo3 13673 | Express membership in a ha... |
| fzon0 13674 | A half-open integer interv... |
| fzossfz 13675 | A half-open range is conta... |
| fzossz 13676 | A half-open integer interv... |
| fzon 13677 | A half-open set of sequent... |
| fzo0n 13678 | A half-open range of nonne... |
| fzonlt0 13679 | A half-open integer range ... |
| fzo0 13680 | Half-open sets with equal ... |
| fzonnsub 13681 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13682 | If ` M < N ` then ` N - M ... |
| fzoss1 13683 | Subset relationship for ha... |
| fzoss2 13684 | Subset relationship for ha... |
| fzossrbm1 13685 | Subset of a half-open rang... |
| fzo0ss1 13686 | Subset relationship for ha... |
| fzossnn0 13687 | A half-open integer range ... |
| fzospliti 13688 | One direction of splitting... |
| fzosplit 13689 | Split a half-open integer ... |
| fzodisj 13690 | Abutting half-open integer... |
| fzouzsplit 13691 | Split an upper integer set... |
| fzouzdisj 13692 | A half-open integer range ... |
| fzoun 13693 | A half-open integer range ... |
| fzodisjsn 13694 | A half-open integer range ... |
| prinfzo0 13695 | The intersection of a half... |
| lbfzo0 13696 | An integer is strictly gre... |
| elfzo0 13697 | Membership in a half-open ... |
| elfzo0z 13698 | Membership in a half-open ... |
| nn0p1elfzo 13699 | A nonnegative integer incr... |
| elfzo0le 13700 | A member in a half-open ra... |
| elfzonn0 13701 | A member of a half-open ra... |
| fzonmapblen 13702 | The result of subtracting ... |
| fzofzim 13703 | If a nonnegative integer i... |
| fz1fzo0m1 13704 | Translation of one between... |
| fzossnn 13705 | Half-open integer ranges s... |
| elfzo1 13706 | Membership in a half-open ... |
| fzo1fzo0n0 13707 | An integer between 1 and a... |
| fzo0n0 13708 | A half-open integer range ... |
| fzoaddel 13709 | Translate membership in a ... |
| fzo0addel 13710 | Translate membership in a ... |
| fzo0addelr 13711 | Translate membership in a ... |
| fzoaddel2 13712 | Translate membership in a ... |
| elfzoext 13713 | Membership of an integer i... |
| elincfzoext 13714 | Membership of an increased... |
| fzosubel 13715 | Translate membership in a ... |
| fzosubel2 13716 | Membership in a translated... |
| fzosubel3 13717 | Membership in a translated... |
| eluzgtdifelfzo 13718 | Membership of the differen... |
| ige2m2fzo 13719 | Membership of an integer g... |
| fzocatel 13720 | Translate membership in a ... |
| ubmelfzo 13721 | If an integer in a 1-based... |
| elfzodifsumelfzo 13722 | If an integer is in a half... |
| elfzom1elp1fzo 13723 | Membership of an integer i... |
| elfzom1elfzo 13724 | Membership in a half-open ... |
| fzval3 13725 | Expressing a closed intege... |
| fz0add1fz1 13726 | Translate membership in a ... |
| fzosn 13727 | Expressing a singleton as ... |
| elfzomin 13728 | Membership of an integer i... |
| zpnn0elfzo 13729 | Membership of an integer i... |
| zpnn0elfzo1 13730 | Membership of an integer i... |
| fzosplitsnm1 13731 | Removing a singleton from ... |
| elfzonlteqm1 13732 | If an element of a half-op... |
| fzonn0p1 13733 | A nonnegative integer is e... |
| fzossfzop1 13734 | A half-open range of nonne... |
| fzonn0p1p1 13735 | If a nonnegative integer i... |
| elfzom1p1elfzo 13736 | Increasing an element of a... |
| fzo0ssnn0 13737 | Half-open integer ranges s... |
| fzo01 13738 | Expressing the singleton o... |
| fzo12sn 13739 | A 1-based half-open intege... |
| fzo13pr 13740 | A 1-based half-open intege... |
| fzo0to2pr 13741 | A half-open integer range ... |
| fzo0to3tp 13742 | A half-open integer range ... |
| fzo0to42pr 13743 | A half-open integer range ... |
| fzo1to4tp 13744 | A half-open integer range ... |
| fzo0sn0fzo1 13745 | A half-open range of nonne... |
| elfzo0l 13746 | A member of a half-open ra... |
| fzoend 13747 | The endpoint of a half-ope... |
| fzo0end 13748 | The endpoint of a zero-bas... |
| ssfzo12 13749 | Subset relationship for ha... |
| ssfzoulel 13750 | If a half-open integer ran... |
| ssfzo12bi 13751 | Subset relationship for ha... |
| ubmelm1fzo 13752 | The result of subtracting ... |
| fzofzp1 13753 | If a point is in a half-op... |
| fzofzp1b 13754 | If a point is in a half-op... |
| elfzom1b 13755 | An integer is a member of ... |
| elfzom1elp1fzo1 13756 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13757 | Membership of a positive i... |
| elfzonelfzo 13758 | If an element of a half-op... |
| fzonfzoufzol 13759 | If an element of a half-op... |
| elfzomelpfzo 13760 | An integer increased by an... |
| elfznelfzo 13761 | A value in a finite set of... |
| elfznelfzob 13762 | A value in a finite set of... |
| peano2fzor 13763 | A Peano-postulate-like the... |
| fzosplitsn 13764 | Extending a half-open rang... |
| fzosplitpr 13765 | Extending a half-open inte... |
| fzosplitprm1 13766 | Extending a half-open inte... |
| fzosplitsni 13767 | Membership in a half-open ... |
| fzisfzounsn 13768 | A finite interval of integ... |
| elfzr 13769 | A member of a finite inter... |
| elfzlmr 13770 | A member of a finite inter... |
| elfz0lmr 13771 | A member of a finite inter... |
| fzostep1 13772 | Two possibilities for a nu... |
| fzoshftral 13773 | Shift the scanning order i... |
| fzind2 13774 | Induction on the integers ... |
| fvinim0ffz 13775 | The function values for th... |
| injresinjlem 13776 | Lemma for ~ injresinj . (... |
| injresinj 13777 | A function whose restricti... |
| subfzo0 13778 | The difference between two... |
| flval 13783 | Value of the floor (greate... |
| flcl 13784 | The floor (greatest intege... |
| reflcl 13785 | The floor (greatest intege... |
| fllelt 13786 | A basic property of the fl... |
| flcld 13787 | The floor (greatest intege... |
| flle 13788 | A basic property of the fl... |
| flltp1 13789 | A basic property of the fl... |
| fllep1 13790 | A basic property of the fl... |
| fraclt1 13791 | The fractional part of a r... |
| fracle1 13792 | The fractional part of a r... |
| fracge0 13793 | The fractional part of a r... |
| flge 13794 | The floor function value i... |
| fllt 13795 | The floor function value i... |
| flflp1 13796 | Move floor function betwee... |
| flid 13797 | An integer is its own floo... |
| flidm 13798 | The floor function is idem... |
| flidz 13799 | A real number equals its f... |
| flltnz 13800 | The floor of a non-integer... |
| flwordi 13801 | Ordering relation for the ... |
| flword2 13802 | Ordering relation for the ... |
| flval2 13803 | An alternate way to define... |
| flval3 13804 | An alternate way to define... |
| flbi 13805 | A condition equivalent to ... |
| flbi2 13806 | A condition equivalent to ... |
| adddivflid 13807 | The floor of a sum of an i... |
| ico01fl0 13808 | The floor of a real number... |
| flge0nn0 13809 | The floor of a number grea... |
| flge1nn 13810 | The floor of a number grea... |
| fldivnn0 13811 | The floor function of a di... |
| refldivcl 13812 | The floor function of a di... |
| divfl0 13813 | The floor of a fraction is... |
| fladdz 13814 | An integer can be moved in... |
| flzadd 13815 | An integer can be moved in... |
| flmulnn0 13816 | Move a nonnegative integer... |
| btwnzge0 13817 | A real bounded between an ... |
| 2tnp1ge0ge0 13818 | Two times an integer plus ... |
| flhalf 13819 | Ordering relation for the ... |
| fldivle 13820 | The floor function of a di... |
| fldivnn0le 13821 | The floor function of a di... |
| flltdivnn0lt 13822 | The floor function of a di... |
| ltdifltdiv 13823 | If the dividend of a divis... |
| fldiv4p1lem1div2 13824 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13825 | The floor of an integer gr... |
| fldiv4lem1div2 13826 | The floor of a positive in... |
| ceilval 13827 | The value of the ceiling f... |
| dfceil2 13828 | Alternative definition of ... |
| ceilval2 13829 | The value of the ceiling f... |
| ceicl 13830 | The ceiling function retur... |
| ceilcl 13831 | Closure of the ceiling fun... |
| ceilcld 13832 | Closure of the ceiling fun... |
| ceige 13833 | The ceiling of a real numb... |
| ceilge 13834 | The ceiling of a real numb... |
| ceilged 13835 | The ceiling of a real numb... |
| ceim1l 13836 | One less than the ceiling ... |
| ceilm1lt 13837 | One less than the ceiling ... |
| ceile 13838 | The ceiling of a real numb... |
| ceille 13839 | The ceiling of a real numb... |
| ceilid 13840 | An integer is its own ceil... |
| ceilidz 13841 | A real number equals its c... |
| flleceil 13842 | The floor of a real number... |
| fleqceilz 13843 | A real number is an intege... |
| quoremz 13844 | Quotient and remainder of ... |
| quoremnn0 13845 | Quotient and remainder of ... |
| quoremnn0ALT 13846 | Alternate proof of ~ quore... |
| intfrac2 13847 | Decompose a real into inte... |
| intfracq 13848 | Decompose a rational numbe... |
| fldiv 13849 | Cancellation of the embedd... |
| fldiv2 13850 | Cancellation of an embedde... |
| fznnfl 13851 | Finite set of sequential i... |
| uzsup 13852 | An upper set of integers i... |
| ioopnfsup 13853 | An upper set of reals is u... |
| icopnfsup 13854 | An upper set of reals is u... |
| rpsup 13855 | The positive reals are unb... |
| resup 13856 | The real numbers are unbou... |
| xrsup 13857 | The extended real numbers ... |
| modval 13860 | The value of the modulo op... |
| modvalr 13861 | The value of the modulo op... |
| modcl 13862 | Closure law for the modulo... |
| flpmodeq 13863 | Partition of a division in... |
| modcld 13864 | Closure law for the modulo... |
| mod0 13865 | ` A mod B ` is zero iff ` ... |
| mulmod0 13866 | The product of an integer ... |
| negmod0 13867 | ` A ` is divisible by ` B ... |
| modge0 13868 | The modulo operation is no... |
| modlt 13869 | The modulo operation is le... |
| modelico 13870 | Modular reduction produces... |
| moddiffl 13871 | Value of the modulo operat... |
| moddifz 13872 | The modulo operation diffe... |
| modfrac 13873 | The fractional part of a n... |
| flmod 13874 | The floor function express... |
| intfrac 13875 | Break a number into its in... |
| zmod10 13876 | An integer modulo 1 is 0. ... |
| zmod1congr 13877 | Two arbitrary integers are... |
| modmulnn 13878 | Move a positive integer in... |
| modvalp1 13879 | The value of the modulo op... |
| zmodcl 13880 | Closure law for the modulo... |
| zmodcld 13881 | Closure law for the modulo... |
| zmodfz 13882 | An integer mod ` B ` lies ... |
| zmodfzo 13883 | An integer mod ` B ` lies ... |
| zmodfzp1 13884 | An integer mod ` B ` lies ... |
| modid 13885 | Identity law for modulo. ... |
| modid0 13886 | A positive real number mod... |
| modid2 13887 | Identity law for modulo. ... |
| zmodid2 13888 | Identity law for modulo re... |
| zmodidfzo 13889 | Identity law for modulo re... |
| zmodidfzoimp 13890 | Identity law for modulo re... |
| 0mod 13891 | Special case: 0 modulo a p... |
| 1mod 13892 | Special case: 1 modulo a r... |
| modabs 13893 | Absorption law for modulo.... |
| modabs2 13894 | Absorption law for modulo.... |
| modcyc 13895 | The modulo operation is pe... |
| modcyc2 13896 | The modulo operation is pe... |
| modadd1 13897 | Addition property of the m... |
| modaddabs 13898 | Absorption law for modulo.... |
| modaddmod 13899 | The sum of a real number m... |
| muladdmodid 13900 | The sum of a positive real... |
| mulp1mod1 13901 | The product of an integer ... |
| modmuladd 13902 | Decomposition of an intege... |
| modmuladdim 13903 | Implication of a decomposi... |
| modmuladdnn0 13904 | Implication of a decomposi... |
| negmod 13905 | The negation of a number m... |
| m1modnnsub1 13906 | Minus one modulo a positiv... |
| m1modge3gt1 13907 | Minus one modulo an intege... |
| addmodid 13908 | The sum of a positive inte... |
| addmodidr 13909 | The sum of a positive inte... |
| modadd2mod 13910 | The sum of a real number m... |
| modm1p1mod0 13911 | If a real number modulo a ... |
| modltm1p1mod 13912 | If a real number modulo a ... |
| modmul1 13913 | Multiplication property of... |
| modmul12d 13914 | Multiplication property of... |
| modnegd 13915 | Negation property of the m... |
| modadd12d 13916 | Additive property of the m... |
| modsub12d 13917 | Subtraction property of th... |
| modsubmod 13918 | The difference of a real n... |
| modsubmodmod 13919 | The difference of a real n... |
| 2txmodxeq0 13920 | Two times a positive real ... |
| 2submod 13921 | If a real number is betwee... |
| modifeq2int 13922 | If a nonnegative integer i... |
| modaddmodup 13923 | The sum of an integer modu... |
| modaddmodlo 13924 | The sum of an integer modu... |
| modmulmod 13925 | The product of a real numb... |
| modmulmodr 13926 | The product of an integer ... |
| modaddmulmod 13927 | The sum of a real number a... |
| moddi 13928 | Distribute multiplication ... |
| modsubdir 13929 | Distribute the modulo oper... |
| modeqmodmin 13930 | A real number equals the d... |
| modirr 13931 | A number modulo an irratio... |
| modfzo0difsn 13932 | For a number within a half... |
| modsumfzodifsn 13933 | The sum of a number within... |
| modlteq 13934 | Two nonnegative integers l... |
| addmodlteq 13935 | Two nonnegative integers l... |
| om2uz0i 13936 | The mapping ` G ` is a one... |
| om2uzsuci 13937 | The value of ` G ` (see ~ ... |
| om2uzuzi 13938 | The value ` G ` (see ~ om2... |
| om2uzlti 13939 | Less-than relation for ` G... |
| om2uzlt2i 13940 | The mapping ` G ` (see ~ o... |
| om2uzrani 13941 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13942 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13943 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13944 | An alternative definition ... |
| om2uzrdg 13945 | A helper lemma for the val... |
| uzrdglem 13946 | A helper lemma for the val... |
| uzrdgfni 13947 | The recursive definition g... |
| uzrdg0i 13948 | Initial value of a recursi... |
| uzrdgsuci 13949 | Successor value of a recur... |
| ltweuz 13950 | ` < ` is a well-founded re... |
| ltwenn 13951 | Less than well-orders the ... |
| ltwefz 13952 | Less than well-orders a se... |
| uzenom 13953 | An upper integer set is de... |
| uzinf 13954 | An upper integer set is in... |
| nnnfi 13955 | The set of positive intege... |
| uzrdgxfr 13956 | Transfer the value of the ... |
| fzennn 13957 | The cardinality of a finit... |
| fzen2 13958 | The cardinality of a finit... |
| cardfz 13959 | The cardinality of a finit... |
| hashgf1o 13960 | ` G ` maps ` _om ` one-to-... |
| fzfi 13961 | A finite interval of integ... |
| fzfid 13962 | Commonly used special case... |
| fzofi 13963 | Half-open integer sets are... |
| fsequb 13964 | The values of a finite rea... |
| fsequb2 13965 | The values of a finite rea... |
| fseqsupcl 13966 | The values of a finite rea... |
| fseqsupubi 13967 | The values of a finite rea... |
| nn0ennn 13968 | The nonnegative integers a... |
| nnenom 13969 | The set of positive intege... |
| nnct 13970 | ` NN ` is countable. (Con... |
| uzindi 13971 | Indirect strong induction ... |
| axdc4uzlem 13972 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13973 | A version of ~ axdc4 that ... |
| ssnn0fi 13974 | A subset of the nonnegativ... |
| rabssnn0fi 13975 | A subset of the nonnegativ... |
| uzsinds 13976 | Strong (or "total") induct... |
| nnsinds 13977 | Strong (or "total") induct... |
| nn0sinds 13978 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 13979 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 13980 | If all functions of a fini... |
| fsuppmapnn0fiubex 13981 | If all functions of a fini... |
| fsuppmapnn0fiub0 13982 | If all functions of a fini... |
| suppssfz 13983 | Condition for a function o... |
| fsuppmapnn0ub 13984 | If a function over the non... |
| fsuppmapnn0fz 13985 | If a function over the non... |
| mptnn0fsupp 13986 | A mapping from the nonnega... |
| mptnn0fsuppd 13987 | A mapping from the nonnega... |
| mptnn0fsuppr 13988 | A finitely supported mappi... |
| f13idfv 13989 | A one-to-one function with... |
| seqex 13992 | Existence of the sequence ... |
| seqeq1 13993 | Equality theorem for the s... |
| seqeq2 13994 | Equality theorem for the s... |
| seqeq3 13995 | Equality theorem for the s... |
| seqeq1d 13996 | Equality deduction for the... |
| seqeq2d 13997 | Equality deduction for the... |
| seqeq3d 13998 | Equality deduction for the... |
| seqeq123d 13999 | Equality deduction for the... |
| nfseq 14000 | Hypothesis builder for the... |
| seqval 14001 | Value of the sequence buil... |
| seqfn 14002 | The sequence builder funct... |
| seq1 14003 | Value of the sequence buil... |
| seq1i 14004 | Value of the sequence buil... |
| seqp1 14005 | Value of the sequence buil... |
| seqexw 14006 | Weak version of ~ seqex th... |
| seqp1d 14007 | Value of the sequence buil... |
| seqm1 14008 | Value of the sequence buil... |
| seqcl2 14009 | Closure properties of the ... |
| seqf2 14010 | Range of the recursive seq... |
| seqcl 14011 | Closure properties of the ... |
| seqf 14012 | Range of the recursive seq... |
| seqfveq2 14013 | Equality of sequences. (C... |
| seqfeq2 14014 | Equality of sequences. (C... |
| seqfveq 14015 | Equality of sequences. (C... |
| seqfeq 14016 | Equality of sequences. (C... |
| seqshft2 14017 | Shifting the index set of ... |
| seqres 14018 | Restricting its characteri... |
| serf 14019 | An infinite series of comp... |
| serfre 14020 | An infinite series of real... |
| monoord 14021 | Ordering relation for a mo... |
| monoord2 14022 | Ordering relation for a mo... |
| sermono 14023 | The partial sums in an inf... |
| seqsplit 14024 | Split a sequence into two ... |
| seq1p 14025 | Removing the first term fr... |
| seqcaopr3 14026 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 14027 | The sum of two infinite se... |
| seqcaopr 14028 | The sum of two infinite se... |
| seqf1olem2a 14029 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 14030 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 14031 | Lemma for ~ seqf1o . (Con... |
| seqf1o 14032 | Rearrange a sum via an arb... |
| seradd 14033 | The sum of two infinite se... |
| sersub 14034 | The difference of two infi... |
| seqid3 14035 | A sequence that consists e... |
| seqid 14036 | Discarding the first few t... |
| seqid2 14037 | The last few partial sums ... |
| seqhomo 14038 | Apply a homomorphism to a ... |
| seqz 14039 | If the operation ` .+ ` ha... |
| seqfeq4 14040 | Equality of series under d... |
| seqfeq3 14041 | Equality of series under d... |
| seqdistr 14042 | The distributive property ... |
| ser0 14043 | The value of the partial s... |
| ser0f 14044 | A zero-valued infinite ser... |
| serge0 14045 | A finite sum of nonnegativ... |
| serle 14046 | Comparison of partial sums... |
| ser1const 14047 | Value of the partial serie... |
| seqof 14048 | Distribute function operat... |
| seqof2 14049 | Distribute function operat... |
| expval 14052 | Value of exponentiation to... |
| expnnval 14053 | Value of exponentiation to... |
| exp0 14054 | Value of a complex number ... |
| 0exp0e1 14055 | The zeroth power of zero e... |
| exp1 14056 | Value of a complex number ... |
| expp1 14057 | Value of a complex number ... |
| expneg 14058 | Value of a complex number ... |
| expneg2 14059 | Value of a complex number ... |
| expn1 14060 | A complex number raised to... |
| expcllem 14061 | Lemma for proving nonnegat... |
| expcl2lem 14062 | Lemma for proving integer ... |
| nnexpcl 14063 | Closure of exponentiation ... |
| nn0expcl 14064 | Closure of exponentiation ... |
| zexpcl 14065 | Closure of exponentiation ... |
| qexpcl 14066 | Closure of exponentiation ... |
| reexpcl 14067 | Closure of exponentiation ... |
| expcl 14068 | Closure law for nonnegativ... |
| rpexpcl 14069 | Closure law for integer ex... |
| qexpclz 14070 | Closure of integer exponen... |
| reexpclz 14071 | Closure of integer exponen... |
| expclzlem 14072 | Lemma for ~ expclz . (Con... |
| expclz 14073 | Closure law for integer ex... |
| m1expcl2 14074 | Closure of integer exponen... |
| m1expcl 14075 | Closure of exponentiation ... |
| zexpcld 14076 | Closure of exponentiation ... |
| nn0expcli 14077 | Closure of exponentiation ... |
| nn0sqcl 14078 | The square of a nonnegativ... |
| expm1t 14079 | Exponentiation in terms of... |
| 1exp 14080 | Value of 1 raised to an in... |
| expeq0 14081 | A positive integer power i... |
| expne0 14082 | A positive integer power i... |
| expne0i 14083 | An integer power is nonzer... |
| expgt0 14084 | A positive real raised to ... |
| expnegz 14085 | Value of a nonzero complex... |
| 0exp 14086 | Value of zero raised to a ... |
| expge0 14087 | A nonnegative real raised ... |
| expge1 14088 | A real greater than or equ... |
| expgt1 14089 | A real greater than 1 rais... |
| mulexp 14090 | Nonnegative integer expone... |
| mulexpz 14091 | Integer exponentiation of ... |
| exprec 14092 | Integer exponentiation of ... |
| expadd 14093 | Sum of exponents law for n... |
| expaddzlem 14094 | Lemma for ~ expaddz . (Co... |
| expaddz 14095 | Sum of exponents law for i... |
| expmul 14096 | Product of exponents law f... |
| expmulz 14097 | Product of exponents law f... |
| m1expeven 14098 | Exponentiation of negative... |
| expsub 14099 | Exponent subtraction law f... |
| expp1z 14100 | Value of a nonzero complex... |
| expm1 14101 | Value of a nonzero complex... |
| expdiv 14102 | Nonnegative integer expone... |
| sqval 14103 | Value of the square of a c... |
| sqneg 14104 | The square of the negative... |
| sqsubswap 14105 | Swap the order of subtract... |
| sqcl 14106 | Closure of square. (Contr... |
| sqmul 14107 | Distribution of squaring o... |
| sqeq0 14108 | A complex number is zero i... |
| sqdiv 14109 | Distribution of squaring o... |
| sqdivid 14110 | The square of a nonzero co... |
| sqne0 14111 | A complex number is nonzer... |
| resqcl 14112 | Closure of squaring in rea... |
| resqcld 14113 | Closure of squaring in rea... |
| sqgt0 14114 | The square of a nonzero re... |
| sqn0rp 14115 | The square of a nonzero re... |
| nnsqcl 14116 | The positive naturals are ... |
| zsqcl 14117 | Integers are closed under ... |
| qsqcl 14118 | The square of a rational i... |
| sq11 14119 | The square function is one... |
| nn0sq11 14120 | The square function is one... |
| lt2sq 14121 | The square function is inc... |
| le2sq 14122 | The square function is non... |
| le2sq2 14123 | The square function is non... |
| sqge0 14124 | The square of a real is no... |
| sqge0d 14125 | The square of a real is no... |
| zsqcl2 14126 | The square of an integer i... |
| 0expd 14127 | Value of zero raised to a ... |
| exp0d 14128 | Value of a complex number ... |
| exp1d 14129 | Value of a complex number ... |
| expeq0d 14130 | If a positive integer powe... |
| sqvald 14131 | Value of square. Inferenc... |
| sqcld 14132 | Closure of square. (Contr... |
| sqeq0d 14133 | A number is zero iff its s... |
| expcld 14134 | Closure law for nonnegativ... |
| expp1d 14135 | Value of a complex number ... |
| expaddd 14136 | Sum of exponents law for n... |
| expmuld 14137 | Product of exponents law f... |
| sqrecd 14138 | Square of reciprocal is re... |
| expclzd 14139 | Closure law for integer ex... |
| expne0d 14140 | A nonnegative integer powe... |
| expnegd 14141 | Value of a nonzero complex... |
| exprecd 14142 | An integer power of a reci... |
| expp1zd 14143 | Value of a nonzero complex... |
| expm1d 14144 | Value of a nonzero complex... |
| expsubd 14145 | Exponent subtraction law f... |
| sqmuld 14146 | Distribution of squaring o... |
| sqdivd 14147 | Distribution of squaring o... |
| expdivd 14148 | Nonnegative integer expone... |
| mulexpd 14149 | Nonnegative integer expone... |
| znsqcld 14150 | The square of a nonzero in... |
| reexpcld 14151 | Closure of exponentiation ... |
| expge0d 14152 | A nonnegative real raised ... |
| expge1d 14153 | A real greater than or equ... |
| ltexp2a 14154 | Exponent ordering relation... |
| expmordi 14155 | Base ordering relationship... |
| rpexpmord 14156 | Base ordering relationship... |
| expcan 14157 | Cancellation law for integ... |
| ltexp2 14158 | Strict ordering law for ex... |
| leexp2 14159 | Ordering law for exponenti... |
| leexp2a 14160 | Weak ordering relationship... |
| ltexp2r 14161 | The integer powers of a fi... |
| leexp2r 14162 | Weak ordering relationship... |
| leexp1a 14163 | Weak base ordering relatio... |
| exple1 14164 | A real between 0 and 1 inc... |
| expubnd 14165 | An upper bound on ` A ^ N ... |
| sumsqeq0 14166 | The sum of two squres of r... |
| sqvali 14167 | Value of square. Inferenc... |
| sqcli 14168 | Closure of square. (Contr... |
| sqeq0i 14169 | A complex number is zero i... |
| sqrecii 14170 | The square of a reciprocal... |
| sqmuli 14171 | Distribution of squaring o... |
| sqdivi 14172 | Distribution of squaring o... |
| resqcli 14173 | Closure of square in reals... |
| sqgt0i 14174 | The square of a nonzero re... |
| sqge0i 14175 | The square of a real is no... |
| lt2sqi 14176 | The square function on non... |
| le2sqi 14177 | The square function on non... |
| sq11i 14178 | The square function is one... |
| sq0 14179 | The square of 0 is 0. (Co... |
| sq0i 14180 | If a number is zero, then ... |
| sq0id 14181 | If a number is zero, then ... |
| sq1 14182 | The square of 1 is 1. (Co... |
| neg1sqe1 14183 | The square of ` -u 1 ` is ... |
| sq2 14184 | The square of 2 is 4. (Co... |
| sq3 14185 | The square of 3 is 9. (Co... |
| sq4e2t8 14186 | The square of 4 is 2 times... |
| cu2 14187 | The cube of 2 is 8. (Cont... |
| irec 14188 | The reciprocal of ` _i ` .... |
| i2 14189 | ` _i ` squared. (Contribu... |
| i3 14190 | ` _i ` cubed. (Contribute... |
| i4 14191 | ` _i ` to the fourth power... |
| nnlesq 14192 | A positive integer is less... |
| zzlesq 14193 | An integer is less than or... |
| iexpcyc 14194 | Taking ` _i ` to the ` K `... |
| expnass 14195 | A counterexample showing t... |
| sqlecan 14196 | Cancel one factor of a squ... |
| subsq 14197 | Factor the difference of t... |
| subsq2 14198 | Express the difference of ... |
| binom2i 14199 | The square of a binomial. ... |
| subsqi 14200 | Factor the difference of t... |
| sqeqori 14201 | The squares of two complex... |
| subsq0i 14202 | The two solutions to the d... |
| sqeqor 14203 | The squares of two complex... |
| binom2 14204 | The square of a binomial. ... |
| binom21 14205 | Special case of ~ binom2 w... |
| binom2sub 14206 | Expand the square of a sub... |
| binom2sub1 14207 | Special case of ~ binom2su... |
| binom2subi 14208 | Expand the square of a sub... |
| mulbinom2 14209 | The square of a binomial w... |
| binom3 14210 | The cube of a binomial. (... |
| sq01 14211 | If a complex number equals... |
| zesq 14212 | An integer is even iff its... |
| nnesq 14213 | A positive integer is even... |
| crreczi 14214 | Reciprocal of a complex nu... |
| bernneq 14215 | Bernoulli's inequality, du... |
| bernneq2 14216 | Variation of Bernoulli's i... |
| bernneq3 14217 | A corollary of ~ bernneq .... |
| expnbnd 14218 | Exponentiation with a base... |
| expnlbnd 14219 | The reciprocal of exponent... |
| expnlbnd2 14220 | The reciprocal of exponent... |
| expmulnbnd 14221 | Exponentiation with a base... |
| digit2 14222 | Two ways to express the ` ... |
| digit1 14223 | Two ways to express the ` ... |
| modexp 14224 | Exponentiation property of... |
| discr1 14225 | A nonnegative quadratic fo... |
| discr 14226 | If a quadratic polynomial ... |
| expnngt1 14227 | If an integer power with a... |
| expnngt1b 14228 | An integer power with an i... |
| sqoddm1div8 14229 | A squared odd number minus... |
| nnsqcld 14230 | The naturals are closed un... |
| nnexpcld 14231 | Closure of exponentiation ... |
| nn0expcld 14232 | Closure of exponentiation ... |
| rpexpcld 14233 | Closure law for exponentia... |
| ltexp2rd 14234 | The power of a positive nu... |
| reexpclzd 14235 | Closure of exponentiation ... |
| sqgt0d 14236 | The square of a nonzero re... |
| ltexp2d 14237 | Ordering relationship for ... |
| leexp2d 14238 | Ordering law for exponenti... |
| expcand 14239 | Ordering relationship for ... |
| leexp2ad 14240 | Ordering relationship for ... |
| leexp2rd 14241 | Ordering relationship for ... |
| lt2sqd 14242 | The square function on non... |
| le2sqd 14243 | The square function on non... |
| sq11d 14244 | The square function is one... |
| mulsubdivbinom2 14245 | The square of a binomial w... |
| muldivbinom2 14246 | The square of a binomial w... |
| sq10 14247 | The square of 10 is 100. ... |
| sq10e99m1 14248 | The square of 10 is 99 plu... |
| 3dec 14249 | A "decimal constructor" wh... |
| nn0le2msqi 14250 | The square function on non... |
| nn0opthlem1 14251 | A rather pretty lemma for ... |
| nn0opthlem2 14252 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14253 | An ordered pair theorem fo... |
| nn0opth2i 14254 | An ordered pair theorem fo... |
| nn0opth2 14255 | An ordered pair theorem fo... |
| facnn 14258 | Value of the factorial fun... |
| fac0 14259 | The factorial of 0. (Cont... |
| fac1 14260 | The factorial of 1. (Cont... |
| facp1 14261 | The factorial of a success... |
| fac2 14262 | The factorial of 2. (Cont... |
| fac3 14263 | The factorial of 3. (Cont... |
| fac4 14264 | The factorial of 4. (Cont... |
| facnn2 14265 | Value of the factorial fun... |
| faccl 14266 | Closure of the factorial f... |
| faccld 14267 | Closure of the factorial f... |
| facmapnn 14268 | The factorial function res... |
| facne0 14269 | The factorial function is ... |
| facdiv 14270 | A positive integer divides... |
| facndiv 14271 | No positive integer (great... |
| facwordi 14272 | Ordering property of facto... |
| faclbnd 14273 | A lower bound for the fact... |
| faclbnd2 14274 | A lower bound for the fact... |
| faclbnd3 14275 | A lower bound for the fact... |
| faclbnd4lem1 14276 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14277 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14278 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14279 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14280 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14281 | The factorial function gro... |
| faclbnd6 14282 | Geometric lower bound for ... |
| facubnd 14283 | An upper bound for the fac... |
| facavg 14284 | The product of two factori... |
| bcval 14287 | Value of the binomial coef... |
| bcval2 14288 | Value of the binomial coef... |
| bcval3 14289 | Value of the binomial coef... |
| bcval4 14290 | Value of the binomial coef... |
| bcrpcl 14291 | Closure of the binomial co... |
| bccmpl 14292 | "Complementing" its second... |
| bcn0 14293 | ` N ` choose 0 is 1. Rema... |
| bc0k 14294 | The binomial coefficient "... |
| bcnn 14295 | ` N ` choose ` N ` is 1. ... |
| bcn1 14296 | Binomial coefficient: ` N ... |
| bcnp1n 14297 | Binomial coefficient: ` N ... |
| bcm1k 14298 | The proportion of one bino... |
| bcp1n 14299 | The proportion of one bino... |
| bcp1nk 14300 | The proportion of one bino... |
| bcval5 14301 | Write out the top and bott... |
| bcn2 14302 | Binomial coefficient: ` N ... |
| bcp1m1 14303 | Compute the binomial coeff... |
| bcpasc 14304 | Pascal's rule for the bino... |
| bccl 14305 | A binomial coefficient, in... |
| bccl2 14306 | A binomial coefficient, in... |
| bcn2m1 14307 | Compute the binomial coeff... |
| bcn2p1 14308 | Compute the binomial coeff... |
| permnn 14309 | The number of permutations... |
| bcnm1 14310 | The binomial coefficent of... |
| 4bc3eq4 14311 | The value of four choose t... |
| 4bc2eq6 14312 | The value of four choose t... |
| hashkf 14315 | The finite part of the siz... |
| hashgval 14316 | The value of the ` # ` fun... |
| hashginv 14317 | The converse of ` G ` maps... |
| hashinf 14318 | The value of the ` # ` fun... |
| hashbnd 14319 | If ` A ` has size bounded ... |
| hashfxnn0 14320 | The size function is a fun... |
| hashf 14321 | The size function maps all... |
| hashxnn0 14322 | The value of the hash func... |
| hashresfn 14323 | Restriction of the domain ... |
| dmhashres 14324 | Restriction of the domain ... |
| hashnn0pnf 14325 | The value of the hash func... |
| hashnnn0genn0 14326 | If the size of a set is no... |
| hashnemnf 14327 | The size of a set is never... |
| hashv01gt1 14328 | The size of a set is eithe... |
| hashfz1 14329 | The set ` ( 1 ... N ) ` ha... |
| hashen 14330 | Two finite sets have the s... |
| hasheni 14331 | Equinumerous sets have the... |
| hasheqf1o 14332 | The size of two finite set... |
| fiinfnf1o 14333 | There is no bijection betw... |
| hasheqf1oi 14334 | The size of two sets is eq... |
| hashf1rn 14335 | The size of a finite set w... |
| hasheqf1od 14336 | The size of two sets is eq... |
| fz1eqb 14337 | Two possibly-empty 1-based... |
| hashcard 14338 | The size function of the c... |
| hashcl 14339 | Closure of the ` # ` funct... |
| hashxrcl 14340 | Extended real closure of t... |
| hashclb 14341 | Reverse closure of the ` #... |
| nfile 14342 | The size of any infinite s... |
| hashvnfin 14343 | A set of finite size is a ... |
| hashnfinnn0 14344 | The size of an infinite se... |
| isfinite4 14345 | A finite set is equinumero... |
| hasheq0 14346 | Two ways of saying a set i... |
| hashneq0 14347 | Two ways of saying a set i... |
| hashgt0n0 14348 | If the size of a set is gr... |
| hashnncl 14349 | Positive natural closure o... |
| hash0 14350 | The empty set has size zer... |
| hashelne0d 14351 | A set with an element has ... |
| hashsng 14352 | The size of a singleton. ... |
| hashen1 14353 | A set has size 1 if and on... |
| hash1elsn 14354 | A set of size 1 with a kno... |
| hashrabrsn 14355 | The size of a restricted c... |
| hashrabsn01 14356 | The size of a restricted c... |
| hashrabsn1 14357 | If the size of a restricte... |
| hashfn 14358 | A function is equinumerous... |
| fseq1hash 14359 | The value of the size func... |
| hashgadd 14360 | ` G ` maps ordinal additio... |
| hashgval2 14361 | A short expression for the... |
| hashdom 14362 | Dominance relation for the... |
| hashdomi 14363 | Non-strict order relation ... |
| hashsdom 14364 | Strict dominance relation ... |
| hashun 14365 | The size of the union of d... |
| hashun2 14366 | The size of the union of f... |
| hashun3 14367 | The size of the union of f... |
| hashinfxadd 14368 | The extended real addition... |
| hashunx 14369 | The size of the union of d... |
| hashge0 14370 | The cardinality of a set i... |
| hashgt0 14371 | The cardinality of a nonem... |
| hashge1 14372 | The cardinality of a nonem... |
| 1elfz0hash 14373 | 1 is an element of the fin... |
| hashnn0n0nn 14374 | If a nonnegative integer i... |
| hashunsng 14375 | The size of the union of a... |
| hashunsngx 14376 | The size of the union of a... |
| hashunsnggt 14377 | The size of a set is great... |
| hashprg 14378 | The size of an unordered p... |
| elprchashprn2 14379 | If one element of an unord... |
| hashprb 14380 | The size of an unordered p... |
| hashprdifel 14381 | The elements of an unorder... |
| prhash2ex 14382 | There is (at least) one se... |
| hashle00 14383 | If the size of a set is le... |
| hashgt0elex 14384 | If the size of a set is gr... |
| hashgt0elexb 14385 | The size of a set is great... |
| hashp1i 14386 | Size of a finite ordinal. ... |
| hash1 14387 | Size of a finite ordinal. ... |
| hash2 14388 | Size of a finite ordinal. ... |
| hash3 14389 | Size of a finite ordinal. ... |
| hash4 14390 | Size of a finite ordinal. ... |
| pr0hash2ex 14391 | There is (at least) one se... |
| hashss 14392 | The size of a subset is le... |
| prsshashgt1 14393 | The size of a superset of ... |
| hashin 14394 | The size of the intersecti... |
| hashssdif 14395 | The size of the difference... |
| hashdif 14396 | The size of the difference... |
| hashdifsn 14397 | The size of the difference... |
| hashdifpr 14398 | The size of the difference... |
| hashsn01 14399 | The size of a singleton is... |
| hashsnle1 14400 | The size of a singleton is... |
| hashsnlei 14401 | Get an upper bound on a co... |
| hash1snb 14402 | The size of a set is 1 if ... |
| euhash1 14403 | The size of a set is 1 in ... |
| hash1n0 14404 | If the size of a set is 1 ... |
| hashgt12el 14405 | In a set with more than on... |
| hashgt12el2 14406 | In a set with more than on... |
| hashgt23el 14407 | A set with more than two e... |
| hashunlei 14408 | Get an upper bound on a co... |
| hashsslei 14409 | Get an upper bound on a co... |
| hashfz 14410 | Value of the numeric cardi... |
| fzsdom2 14411 | Condition for finite range... |
| hashfzo 14412 | Cardinality of a half-open... |
| hashfzo0 14413 | Cardinality of a half-open... |
| hashfzp1 14414 | Value of the numeric cardi... |
| hashfz0 14415 | Value of the numeric cardi... |
| hashxplem 14416 | Lemma for ~ hashxp . (Con... |
| hashxp 14417 | The size of the Cartesian ... |
| hashmap 14418 | The size of the set expone... |
| hashpw 14419 | The size of the power set ... |
| hashfun 14420 | A finite set is a function... |
| hashres 14421 | The number of elements of ... |
| hashreshashfun 14422 | The number of elements of ... |
| hashimarn 14423 | The size of the image of a... |
| hashimarni 14424 | If the size of the image o... |
| hashfundm 14425 | The size of a set function... |
| hashf1dmrn 14426 | The size of the domain of ... |
| hashf1dmcdm 14427 | The size of the domain of ... |
| resunimafz0 14428 | TODO-AV: Revise using ` F... |
| fnfz0hash 14429 | The size of a function on ... |
| ffz0hash 14430 | The size of a function on ... |
| fnfz0hashnn0 14431 | The size of a function on ... |
| ffzo0hash 14432 | The size of a function on ... |
| fnfzo0hash 14433 | The size of a function on ... |
| fnfzo0hashnn0 14434 | The value of the size func... |
| hashbclem 14435 | Lemma for ~ hashbc : induc... |
| hashbc 14436 | The binomial coefficient c... |
| hashfacen 14437 | The number of bijections b... |
| hashfacenOLD 14438 | Obsolete version of ~ hash... |
| hashf1lem1 14439 | Lemma for ~ hashf1 . (Con... |
| hashf1lem1OLD 14440 | Obsolete version of ~ hash... |
| hashf1lem2 14441 | Lemma for ~ hashf1 . (Con... |
| hashf1 14442 | The permutation number ` |... |
| hashfac 14443 | A factorial counts the num... |
| leiso 14444 | Two ways to write a strict... |
| leisorel 14445 | Version of ~ isorel for st... |
| fz1isolem 14446 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14447 | Any finite ordered set has... |
| ishashinf 14448 | Any set that is not finite... |
| seqcoll 14449 | The function ` F ` contain... |
| seqcoll2 14450 | The function ` F ` contain... |
| phphashd 14451 | Corollary of the Pigeonhol... |
| phphashrd 14452 | Corollary of the Pigeonhol... |
| hashprlei 14453 | An unordered pair has at m... |
| hash2pr 14454 | A set of size two is an un... |
| hash2prde 14455 | A set of size two is an un... |
| hash2exprb 14456 | A set of size two is an un... |
| hash2prb 14457 | A set of size two is a pro... |
| prprrab 14458 | The set of proper pairs of... |
| nehash2 14459 | The cardinality of a set w... |
| hash2prd 14460 | A set of size two is an un... |
| hash2pwpr 14461 | If the size of a subset of... |
| hashle2pr 14462 | A nonempty set of size les... |
| hashle2prv 14463 | A nonempty subset of a pow... |
| pr2pwpr 14464 | The set of subsets of a pa... |
| hashge2el2dif 14465 | A set with size at least 2... |
| hashge2el2difr 14466 | A set with at least 2 diff... |
| hashge2el2difb 14467 | A set has size at least 2 ... |
| hashdmpropge2 14468 | The size of the domain of ... |
| hashtplei 14469 | An unordered triple has at... |
| hashtpg 14470 | The size of an unordered t... |
| hashge3el3dif 14471 | A set with size at least 3... |
| elss2prb 14472 | An element of the set of s... |
| hash2sspr 14473 | A subset of size two is an... |
| exprelprel 14474 | If there is an element of ... |
| hash3tr 14475 | A set of size three is an ... |
| hash1to3 14476 | If the size of a set is be... |
| fundmge2nop0 14477 | A function with a domain c... |
| fundmge2nop 14478 | A function with a domain c... |
| fun2dmnop0 14479 | A function with a domain c... |
| fun2dmnop 14480 | A function with a domain c... |
| hashdifsnp1 14481 | If the size of a set is a ... |
| fi1uzind 14482 | Properties of an ordered p... |
| brfi1uzind 14483 | Properties of a binary rel... |
| brfi1ind 14484 | Properties of a binary rel... |
| brfi1indALT 14485 | Alternate proof of ~ brfi1... |
| opfi1uzind 14486 | Properties of an ordered p... |
| opfi1ind 14487 | Properties of an ordered p... |
| iswrd 14490 | Property of being a word o... |
| wrdval 14491 | Value of the set of words ... |
| iswrdi 14492 | A zero-based sequence is a... |
| wrdf 14493 | A word is a zero-based seq... |
| iswrdb 14494 | A word over an alphabet is... |
| wrddm 14495 | The indices of a word (i.e... |
| sswrd 14496 | The set of words respects ... |
| snopiswrd 14497 | A singleton of an ordered ... |
| wrdexg 14498 | The set of words over a se... |
| wrdexb 14499 | The set of words over a se... |
| wrdexi 14500 | The set of words over a se... |
| wrdsymbcl 14501 | A symbol within a word ove... |
| wrdfn 14502 | A word is a function with ... |
| wrdv 14503 | A word over an alphabet is... |
| wrdlndm 14504 | The length of a word is no... |
| iswrdsymb 14505 | An arbitrary word is a wor... |
| wrdfin 14506 | A word is a finite set. (... |
| lencl 14507 | The length of a word is a ... |
| lennncl 14508 | The length of a nonempty w... |
| wrdffz 14509 | A word is a function from ... |
| wrdeq 14510 | Equality theorem for the s... |
| wrdeqi 14511 | Equality theorem for the s... |
| iswrddm0 14512 | A function with empty doma... |
| wrd0 14513 | The empty set is a word (t... |
| 0wrd0 14514 | The empty word is the only... |
| ffz0iswrd 14515 | A sequence with zero-based... |
| wrdsymb 14516 | A word is a word over the ... |
| nfwrd 14517 | Hypothesis builder for ` W... |
| csbwrdg 14518 | Class substitution for the... |
| wrdnval 14519 | Words of a fixed length ar... |
| wrdmap 14520 | Words as a mapping. (Cont... |
| hashwrdn 14521 | If there is only a finite ... |
| wrdnfi 14522 | If there is only a finite ... |
| wrdsymb0 14523 | A symbol at a position "ou... |
| wrdlenge1n0 14524 | A word with length at leas... |
| len0nnbi 14525 | The length of a word is a ... |
| wrdlenge2n0 14526 | A word with length at leas... |
| wrdsymb1 14527 | The first symbol of a none... |
| wrdlen1 14528 | A word of length 1 starts ... |
| fstwrdne 14529 | The first symbol of a none... |
| fstwrdne0 14530 | The first symbol of a none... |
| eqwrd 14531 | Two words are equal iff th... |
| elovmpowrd 14532 | Implications for the value... |
| elovmptnn0wrd 14533 | Implications for the value... |
| wrdred1 14534 | A word truncated by a symb... |
| wrdred1hash 14535 | The length of a word trunc... |
| lsw 14538 | Extract the last symbol of... |
| lsw0 14539 | The last symbol of an empt... |
| lsw0g 14540 | The last symbol of an empt... |
| lsw1 14541 | The last symbol of a word ... |
| lswcl 14542 | Closure of the last symbol... |
| lswlgt0cl 14543 | The last symbol of a nonem... |
| ccatfn 14546 | The concatenation operator... |
| ccatfval 14547 | Value of the concatenation... |
| ccatcl 14548 | The concatenation of two w... |
| ccatlen 14549 | The length of a concatenat... |
| ccat0 14550 | The concatenation of two w... |
| ccatval1 14551 | Value of a symbol in the l... |
| ccatval2 14552 | Value of a symbol in the r... |
| ccatval3 14553 | Value of a symbol in the r... |
| elfzelfzccat 14554 | An element of a finite set... |
| ccatvalfn 14555 | The concatenation of two w... |
| ccatsymb 14556 | The symbol at a given posi... |
| ccatfv0 14557 | The first symbol of a conc... |
| ccatval1lsw 14558 | The last symbol of the lef... |
| ccatval21sw 14559 | The first symbol of the ri... |
| ccatlid 14560 | Concatenation of a word by... |
| ccatrid 14561 | Concatenation of a word by... |
| ccatass 14562 | Associative law for concat... |
| ccatrn 14563 | The range of a concatenate... |
| ccatidid 14564 | Concatenation of the empty... |
| lswccatn0lsw 14565 | The last symbol of a word ... |
| lswccat0lsw 14566 | The last symbol of a word ... |
| ccatalpha 14567 | A concatenation of two arb... |
| ccatrcl1 14568 | Reverse closure of a conca... |
| ids1 14571 | Identity function protecti... |
| s1val 14572 | Value of a singleton word.... |
| s1rn 14573 | The range of a singleton w... |
| s1eq 14574 | Equality theorem for a sin... |
| s1eqd 14575 | Equality theorem for a sin... |
| s1cl 14576 | A singleton word is a word... |
| s1cld 14577 | A singleton word is a word... |
| s1prc 14578 | Value of a singleton word ... |
| s1cli 14579 | A singleton word is a word... |
| s1len 14580 | Length of a singleton word... |
| s1nz 14581 | A singleton word is not th... |
| s1dm 14582 | The domain of a singleton ... |
| s1dmALT 14583 | Alternate version of ~ s1d... |
| s1fv 14584 | Sole symbol of a singleton... |
| lsws1 14585 | The last symbol of a singl... |
| eqs1 14586 | A word of length 1 is a si... |
| wrdl1exs1 14587 | A word of length 1 is a si... |
| wrdl1s1 14588 | A word of length 1 is a si... |
| s111 14589 | The singleton word functio... |
| ccatws1cl 14590 | The concatenation of a wor... |
| ccatws1clv 14591 | The concatenation of a wor... |
| ccat2s1cl 14592 | The concatenation of two s... |
| ccats1alpha 14593 | A concatenation of a word ... |
| ccatws1len 14594 | The length of the concaten... |
| ccatws1lenp1b 14595 | The length of a word is ` ... |
| wrdlenccats1lenm1 14596 | The length of a word is th... |
| ccat2s1len 14597 | The length of the concaten... |
| ccatw2s1cl 14598 | The concatenation of a wor... |
| ccatw2s1len 14599 | The length of the concaten... |
| ccats1val1 14600 | Value of a symbol in the l... |
| ccats1val2 14601 | Value of the symbol concat... |
| ccat1st1st 14602 | The first symbol of a word... |
| ccat2s1p1 14603 | Extract the first of two c... |
| ccat2s1p2 14604 | Extract the second of two ... |
| ccatw2s1ass 14605 | Associative law for a conc... |
| ccatws1n0 14606 | The concatenation of a wor... |
| ccatws1ls 14607 | The last symbol of the con... |
| lswccats1 14608 | The last symbol of a word ... |
| lswccats1fst 14609 | The last symbol of a nonem... |
| ccatw2s1p1 14610 | Extract the symbol of the ... |
| ccatw2s1p2 14611 | Extract the second of two ... |
| ccat2s1fvw 14612 | Extract a symbol of a word... |
| ccat2s1fst 14613 | The first symbol of the co... |
| swrdnznd 14616 | The value of a subword ope... |
| swrdval 14617 | Value of a subword. (Cont... |
| swrd00 14618 | A zero length substring. ... |
| swrdcl 14619 | Closure of the subword ext... |
| swrdval2 14620 | Value of the subword extra... |
| swrdlen 14621 | Length of an extracted sub... |
| swrdfv 14622 | A symbol in an extracted s... |
| swrdfv0 14623 | The first symbol in an ext... |
| swrdf 14624 | A subword of a word is a f... |
| swrdvalfn 14625 | Value of the subword extra... |
| swrdrn 14626 | The range of a subword of ... |
| swrdlend 14627 | The value of the subword e... |
| swrdnd 14628 | The value of the subword e... |
| swrdnd2 14629 | Value of the subword extra... |
| swrdnnn0nd 14630 | The value of a subword ope... |
| swrdnd0 14631 | The value of a subword ope... |
| swrd0 14632 | A subword of an empty set ... |
| swrdrlen 14633 | Length of a right-anchored... |
| swrdlen2 14634 | Length of an extracted sub... |
| swrdfv2 14635 | A symbol in an extracted s... |
| swrdwrdsymb 14636 | A subword is a word over t... |
| swrdsb0eq 14637 | Two subwords with the same... |
| swrdsbslen 14638 | Two subwords with the same... |
| swrdspsleq 14639 | Two words have a common su... |
| swrds1 14640 | Extract a single symbol fr... |
| swrdlsw 14641 | Extract the last single sy... |
| ccatswrd 14642 | Joining two adjacent subwo... |
| swrdccat2 14643 | Recover the right half of ... |
| pfxnndmnd 14646 | The value of a prefix oper... |
| pfxval 14647 | Value of a prefix operatio... |
| pfx00 14648 | The zero length prefix is ... |
| pfx0 14649 | A prefix of an empty set i... |
| pfxval0 14650 | Value of a prefix operatio... |
| pfxcl 14651 | Closure of the prefix extr... |
| pfxmpt 14652 | Value of the prefix extrac... |
| pfxres 14653 | Value of the subword extra... |
| pfxf 14654 | A prefix of a word is a fu... |
| pfxfn 14655 | Value of the prefix extrac... |
| pfxfv 14656 | A symbol in a prefix of a ... |
| pfxlen 14657 | Length of a prefix. (Cont... |
| pfxid 14658 | A word is a prefix of itse... |
| pfxrn 14659 | The range of a prefix of a... |
| pfxn0 14660 | A prefix consisting of at ... |
| pfxnd 14661 | The value of a prefix oper... |
| pfxnd0 14662 | The value of a prefix oper... |
| pfxwrdsymb 14663 | A prefix of a word is a wo... |
| addlenrevpfx 14664 | The sum of the lengths of ... |
| addlenpfx 14665 | The sum of the lengths of ... |
| pfxfv0 14666 | The first symbol of a pref... |
| pfxtrcfv 14667 | A symbol in a word truncat... |
| pfxtrcfv0 14668 | The first symbol in a word... |
| pfxfvlsw 14669 | The last symbol in a nonem... |
| pfxeq 14670 | The prefixes of two words ... |
| pfxtrcfvl 14671 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14672 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14673 | Two (nonempty) words are e... |
| disjwrdpfx 14674 | Sets of words are disjoint... |
| ccatpfx 14675 | Concatenating a prefix wit... |
| pfxccat1 14676 | Recover the left half of a... |
| pfx1 14677 | The prefix of length one o... |
| swrdswrdlem 14678 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14679 | A subword of a subword is ... |
| pfxswrd 14680 | A prefix of a subword is a... |
| swrdpfx 14681 | A subword of a prefix is a... |
| pfxpfx 14682 | A prefix of a prefix is a ... |
| pfxpfxid 14683 | A prefix of a prefix with ... |
| pfxcctswrd 14684 | The concatenation of the p... |
| lenpfxcctswrd 14685 | The length of the concaten... |
| lenrevpfxcctswrd 14686 | The length of the concaten... |
| pfxlswccat 14687 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14688 | The last symbol of a word ... |
| ccats1pfxeqrex 14689 | There exists a symbol such... |
| ccatopth 14690 | An ~ opth -like theorem fo... |
| ccatopth2 14691 | An ~ opth -like theorem fo... |
| ccatlcan 14692 | Concatenation of words is ... |
| ccatrcan 14693 | Concatenation of words is ... |
| wrdeqs1cat 14694 | Decompose a nonempty word ... |
| cats1un 14695 | Express a word with an ext... |
| wrdind 14696 | Perform induction over the... |
| wrd2ind 14697 | Perform induction over the... |
| swrdccatfn 14698 | The subword of a concatena... |
| swrdccatin1 14699 | The subword of a concatena... |
| pfxccatin12lem4 14700 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14701 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14702 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14703 | The subword of a concatena... |
| pfxccatin12lem2c 14704 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14705 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14706 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14707 | The subword of a concatena... |
| pfxccat3 14708 | The subword of a concatena... |
| swrdccat 14709 | The subword of a concatena... |
| pfxccatpfx1 14710 | A prefix of a concatenatio... |
| pfxccatpfx2 14711 | A prefix of a concatenatio... |
| pfxccat3a 14712 | A prefix of a concatenatio... |
| swrdccat3blem 14713 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14714 | A suffix of a concatenatio... |
| pfxccatid 14715 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14716 | A word is a prefix of a wo... |
| swrdccatin1d 14717 | The subword of a concatena... |
| swrdccatin2d 14718 | The subword of a concatena... |
| pfxccatin12d 14719 | The subword of a concatena... |
| reuccatpfxs1lem 14720 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14721 | There is a unique word hav... |
| reuccatpfxs1v 14722 | There is a unique word hav... |
| splval 14725 | Value of the substring rep... |
| splcl 14726 | Closure of the substring r... |
| splid 14727 | Splicing a subword for the... |
| spllen 14728 | The length of a splice. (... |
| splfv1 14729 | Symbols to the left of a s... |
| splfv2a 14730 | Symbols within the replace... |
| splval2 14731 | Value of a splice, assumin... |
| revval 14734 | Value of the word reversin... |
| revcl 14735 | The reverse of a word is a... |
| revlen 14736 | The reverse of a word has ... |
| revfv 14737 | Reverse of a word at a poi... |
| rev0 14738 | The empty word is its own ... |
| revs1 14739 | Singleton words are their ... |
| revccat 14740 | Antiautomorphic property o... |
| revrev 14741 | Reversal is an involution ... |
| reps 14744 | Construct a function mappi... |
| repsundef 14745 | A function mapping a half-... |
| repsconst 14746 | Construct a function mappi... |
| repsf 14747 | The constructed function m... |
| repswsymb 14748 | The symbols of a "repeated... |
| repsw 14749 | A function mapping a half-... |
| repswlen 14750 | The length of a "repeated ... |
| repsw0 14751 | The "repeated symbol word"... |
| repsdf2 14752 | Alternative definition of ... |
| repswsymball 14753 | All the symbols of a "repe... |
| repswsymballbi 14754 | A word is a "repeated symb... |
| repswfsts 14755 | The first symbol of a none... |
| repswlsw 14756 | The last symbol of a nonem... |
| repsw1 14757 | The "repeated symbol word"... |
| repswswrd 14758 | A subword of a "repeated s... |
| repswpfx 14759 | A prefix of a repeated sym... |
| repswccat 14760 | The concatenation of two "... |
| repswrevw 14761 | The reverse of a "repeated... |
| cshfn 14764 | Perform a cyclical shift f... |
| cshword 14765 | Perform a cyclical shift f... |
| cshnz 14766 | A cyclical shift is the em... |
| 0csh0 14767 | Cyclically shifting an emp... |
| cshw0 14768 | A word cyclically shifted ... |
| cshwmodn 14769 | Cyclically shifting a word... |
| cshwsublen 14770 | Cyclically shifting a word... |
| cshwn 14771 | A word cyclically shifted ... |
| cshwcl 14772 | A cyclically shifted word ... |
| cshwlen 14773 | The length of a cyclically... |
| cshwf 14774 | A cyclically shifted word ... |
| cshwfn 14775 | A cyclically shifted word ... |
| cshwrn 14776 | The range of a cyclically ... |
| cshwidxmod 14777 | The symbol at a given inde... |
| cshwidxmodr 14778 | The symbol at a given inde... |
| cshwidx0mod 14779 | The symbol at index 0 of a... |
| cshwidx0 14780 | The symbol at index 0 of a... |
| cshwidxm1 14781 | The symbol at index ((n-N)... |
| cshwidxm 14782 | The symbol at index (n-N) ... |
| cshwidxn 14783 | The symbol at index (n-1) ... |
| cshf1 14784 | Cyclically shifting a word... |
| cshinj 14785 | If a word is injectiv (reg... |
| repswcshw 14786 | A cyclically shifted "repe... |
| 2cshw 14787 | Cyclically shifting a word... |
| 2cshwid 14788 | Cyclically shifting a word... |
| lswcshw 14789 | The last symbol of a word ... |
| 2cshwcom 14790 | Cyclically shifting a word... |
| cshwleneq 14791 | If the results of cyclical... |
| 3cshw 14792 | Cyclically shifting a word... |
| cshweqdif2 14793 | If cyclically shifting two... |
| cshweqdifid 14794 | If cyclically shifting a w... |
| cshweqrep 14795 | If cyclically shifting a w... |
| cshw1 14796 | If cyclically shifting a w... |
| cshw1repsw 14797 | If cyclically shifting a w... |
| cshwsexa 14798 | The class of (different!) ... |
| cshwsexaOLD 14799 | Obsolete version of ~ cshw... |
| 2cshwcshw 14800 | If a word is a cyclically ... |
| scshwfzeqfzo 14801 | For a nonempty word the se... |
| cshwcshid 14802 | A cyclically shifted word ... |
| cshwcsh2id 14803 | A cyclically shifted word ... |
| cshimadifsn 14804 | The image of a cyclically ... |
| cshimadifsn0 14805 | The image of a cyclically ... |
| wrdco 14806 | Mapping a word by a functi... |
| lenco 14807 | Length of a mapped word is... |
| s1co 14808 | Mapping of a singleton wor... |
| revco 14809 | Mapping of words (i.e., a ... |
| ccatco 14810 | Mapping of words commutes ... |
| cshco 14811 | Mapping of words commutes ... |
| swrdco 14812 | Mapping of words commutes ... |
| pfxco 14813 | Mapping of words commutes ... |
| lswco 14814 | Mapping of (nonempty) word... |
| repsco 14815 | Mapping of words commutes ... |
| cats1cld 14830 | Closure of concatenation w... |
| cats1co 14831 | Closure of concatenation w... |
| cats1cli 14832 | Closure of concatenation w... |
| cats1fvn 14833 | The last symbol of a conca... |
| cats1fv 14834 | A symbol other than the la... |
| cats1len 14835 | The length of concatenatio... |
| cats1cat 14836 | Closure of concatenation w... |
| cats2cat 14837 | Closure of concatenation o... |
| s2eqd 14838 | Equality theorem for a dou... |
| s3eqd 14839 | Equality theorem for a len... |
| s4eqd 14840 | Equality theorem for a len... |
| s5eqd 14841 | Equality theorem for a len... |
| s6eqd 14842 | Equality theorem for a len... |
| s7eqd 14843 | Equality theorem for a len... |
| s8eqd 14844 | Equality theorem for a len... |
| s3eq2 14845 | Equality theorem for a len... |
| s2cld 14846 | A doubleton word is a word... |
| s3cld 14847 | A length 3 string is a wor... |
| s4cld 14848 | A length 4 string is a wor... |
| s5cld 14849 | A length 5 string is a wor... |
| s6cld 14850 | A length 6 string is a wor... |
| s7cld 14851 | A length 7 string is a wor... |
| s8cld 14852 | A length 7 string is a wor... |
| s2cl 14853 | A doubleton word is a word... |
| s3cl 14854 | A length 3 string is a wor... |
| s2cli 14855 | A doubleton word is a word... |
| s3cli 14856 | A length 3 string is a wor... |
| s4cli 14857 | A length 4 string is a wor... |
| s5cli 14858 | A length 5 string is a wor... |
| s6cli 14859 | A length 6 string is a wor... |
| s7cli 14860 | A length 7 string is a wor... |
| s8cli 14861 | A length 8 string is a wor... |
| s2fv0 14862 | Extract the first symbol f... |
| s2fv1 14863 | Extract the second symbol ... |
| s2len 14864 | The length of a doubleton ... |
| s2dm 14865 | The domain of a doubleton ... |
| s3fv0 14866 | Extract the first symbol f... |
| s3fv1 14867 | Extract the second symbol ... |
| s3fv2 14868 | Extract the third symbol f... |
| s3len 14869 | The length of a length 3 s... |
| s4fv0 14870 | Extract the first symbol f... |
| s4fv1 14871 | Extract the second symbol ... |
| s4fv2 14872 | Extract the third symbol f... |
| s4fv3 14873 | Extract the fourth symbol ... |
| s4len 14874 | The length of a length 4 s... |
| s5len 14875 | The length of a length 5 s... |
| s6len 14876 | The length of a length 6 s... |
| s7len 14877 | The length of a length 7 s... |
| s8len 14878 | The length of a length 8 s... |
| lsws2 14879 | The last symbol of a doubl... |
| lsws3 14880 | The last symbol of a 3 let... |
| lsws4 14881 | The last symbol of a 4 let... |
| s2prop 14882 | A length 2 word is an unor... |
| s2dmALT 14883 | Alternate version of ~ s2d... |
| s3tpop 14884 | A length 3 word is an unor... |
| s4prop 14885 | A length 4 word is a union... |
| s3fn 14886 | A length 3 word is a funct... |
| funcnvs1 14887 | The converse of a singleto... |
| funcnvs2 14888 | The converse of a length 2... |
| funcnvs3 14889 | The converse of a length 3... |
| funcnvs4 14890 | The converse of a length 4... |
| s2f1o 14891 | A length 2 word with mutua... |
| f1oun2prg 14892 | A union of unordered pairs... |
| s4f1o 14893 | A length 4 word with mutua... |
| s4dom 14894 | The domain of a length 4 w... |
| s2co 14895 | Mapping a doubleton word b... |
| s3co 14896 | Mapping a length 3 string ... |
| s0s1 14897 | Concatenation of fixed len... |
| s1s2 14898 | Concatenation of fixed len... |
| s1s3 14899 | Concatenation of fixed len... |
| s1s4 14900 | Concatenation of fixed len... |
| s1s5 14901 | Concatenation of fixed len... |
| s1s6 14902 | Concatenation of fixed len... |
| s1s7 14903 | Concatenation of fixed len... |
| s2s2 14904 | Concatenation of fixed len... |
| s4s2 14905 | Concatenation of fixed len... |
| s4s3 14906 | Concatenation of fixed len... |
| s4s4 14907 | Concatenation of fixed len... |
| s3s4 14908 | Concatenation of fixed len... |
| s2s5 14909 | Concatenation of fixed len... |
| s5s2 14910 | Concatenation of fixed len... |
| s2eq2s1eq 14911 | Two length 2 words are equ... |
| s2eq2seq 14912 | Two length 2 words are equ... |
| s3eqs2s1eq 14913 | Two length 3 words are equ... |
| s3eq3seq 14914 | Two length 3 words are equ... |
| swrds2 14915 | Extract two adjacent symbo... |
| swrds2m 14916 | Extract two adjacent symbo... |
| wrdlen2i 14917 | Implications of a word of ... |
| wrd2pr2op 14918 | A word of length two repre... |
| wrdlen2 14919 | A word of length two. (Co... |
| wrdlen2s2 14920 | A word of length two as do... |
| wrdl2exs2 14921 | A word of length two is a ... |
| pfx2 14922 | A prefix of length two. (... |
| wrd3tpop 14923 | A word of length three rep... |
| wrdlen3s3 14924 | A word of length three as ... |
| repsw2 14925 | The "repeated symbol word"... |
| repsw3 14926 | The "repeated symbol word"... |
| swrd2lsw 14927 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14928 | Two words of length at lea... |
| ccatw2s1ccatws2 14929 | The concatenation of a wor... |
| ccat2s1fvwALT 14930 | Alternate proof of ~ ccat2... |
| wwlktovf 14931 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14932 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14933 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14934 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14935 | There is a bijection betwe... |
| eqwrds3 14936 | A word is equal with a len... |
| wrdl3s3 14937 | A word of length 3 is a le... |
| s3sndisj 14938 | The singletons consisting ... |
| s3iunsndisj 14939 | The union of singletons co... |
| ofccat 14940 | Letterwise operations on w... |
| ofs1 14941 | Letterwise operations on a... |
| ofs2 14942 | Letterwise operations on a... |
| coss12d 14943 | Subset deduction for compo... |
| trrelssd 14944 | The composition of subclas... |
| xpcogend 14945 | The most interesting case ... |
| xpcoidgend 14946 | If two classes are not dis... |
| cotr2g 14947 | Two ways of saying that th... |
| cotr2 14948 | Two ways of saying a relat... |
| cotr3 14949 | Two ways of saying a relat... |
| coemptyd 14950 | Deduction about compositio... |
| xptrrel 14951 | The cross product is alway... |
| 0trrel 14952 | The empty class is a trans... |
| cleq1lem 14953 | Equality implies bijection... |
| cleq1 14954 | Equality of relations impl... |
| clsslem 14955 | The closure of a subclass ... |
| trcleq1 14960 | Equality of relations impl... |
| trclsslem 14961 | The transitive closure (as... |
| trcleq2lem 14962 | Equality implies bijection... |
| cvbtrcl 14963 | Change of bound variable i... |
| trcleq12lem 14964 | Equality implies bijection... |
| trclexlem 14965 | Existence of relation impl... |
| trclublem 14966 | If a relation exists then ... |
| trclubi 14967 | The Cartesian product of t... |
| trclubgi 14968 | The union with the Cartesi... |
| trclub 14969 | The Cartesian product of t... |
| trclubg 14970 | The union with the Cartesi... |
| trclfv 14971 | The transitive closure of ... |
| brintclab 14972 | Two ways to express a bina... |
| brtrclfv 14973 | Two ways of expressing the... |
| brcnvtrclfv 14974 | Two ways of expressing the... |
| brtrclfvcnv 14975 | Two ways of expressing the... |
| brcnvtrclfvcnv 14976 | Two ways of expressing the... |
| trclfvss 14977 | The transitive closure (as... |
| trclfvub 14978 | The transitive closure of ... |
| trclfvlb 14979 | The transitive closure of ... |
| trclfvcotr 14980 | The transitive closure of ... |
| trclfvlb2 14981 | The transitive closure of ... |
| trclfvlb3 14982 | The transitive closure of ... |
| cotrtrclfv 14983 | The transitive closure of ... |
| trclidm 14984 | The transitive closure of ... |
| trclun 14985 | Transitive closure of a un... |
| trclfvg 14986 | The value of the transitiv... |
| trclfvcotrg 14987 | The value of the transitiv... |
| reltrclfv 14988 | The transitive closure of ... |
| dmtrclfv 14989 | The domain of the transiti... |
| reldmrelexp 14992 | The domain of the repeated... |
| relexp0g 14993 | A relation composed zero t... |
| relexp0 14994 | A relation composed zero t... |
| relexp0d 14995 | A relation composed zero t... |
| relexpsucnnr 14996 | A reduction for relation e... |
| relexp1g 14997 | A relation composed once i... |
| dfid5 14998 | Identity relation is equal... |
| dfid6 14999 | Identity relation expresse... |
| relexp1d 15000 | A relation composed once i... |
| relexpsucnnl 15001 | A reduction for relation e... |
| relexpsucl 15002 | A reduction for relation e... |
| relexpsucr 15003 | A reduction for relation e... |
| relexpsucrd 15004 | A reduction for relation e... |
| relexpsucld 15005 | A reduction for relation e... |
| relexpcnv 15006 | Commutation of converse an... |
| relexpcnvd 15007 | Commutation of converse an... |
| relexp0rel 15008 | The exponentiation of a cl... |
| relexprelg 15009 | The exponentiation of a cl... |
| relexprel 15010 | The exponentiation of a re... |
| relexpreld 15011 | The exponentiation of a re... |
| relexpnndm 15012 | The domain of an exponenti... |
| relexpdmg 15013 | The domain of an exponenti... |
| relexpdm 15014 | The domain of an exponenti... |
| relexpdmd 15015 | The domain of an exponenti... |
| relexpnnrn 15016 | The range of an exponentia... |
| relexprng 15017 | The range of an exponentia... |
| relexprn 15018 | The range of an exponentia... |
| relexprnd 15019 | The range of an exponentia... |
| relexpfld 15020 | The field of an exponentia... |
| relexpfldd 15021 | The field of an exponentia... |
| relexpaddnn 15022 | Relation composition becom... |
| relexpuzrel 15023 | The exponentiation of a cl... |
| relexpaddg 15024 | Relation composition becom... |
| relexpaddd 15025 | Relation composition becom... |
| rtrclreclem1 15028 | The reflexive, transitive ... |
| dfrtrclrec2 15029 | If two elements are connec... |
| rtrclreclem2 15030 | The reflexive, transitive ... |
| rtrclreclem3 15031 | The reflexive, transitive ... |
| rtrclreclem4 15032 | The reflexive, transitive ... |
| dfrtrcl2 15033 | The two definitions ` t* `... |
| relexpindlem 15034 | Principle of transitive in... |
| relexpind 15035 | Principle of transitive in... |
| rtrclind 15036 | Principle of transitive in... |
| shftlem 15039 | Two ways to write a shifte... |
| shftuz 15040 | A shift of the upper integ... |
| shftfval 15041 | The value of the sequence ... |
| shftdm 15042 | Domain of a relation shift... |
| shftfib 15043 | Value of a fiber of the re... |
| shftfn 15044 | Functionality and domain o... |
| shftval 15045 | Value of a sequence shifte... |
| shftval2 15046 | Value of a sequence shifte... |
| shftval3 15047 | Value of a sequence shifte... |
| shftval4 15048 | Value of a sequence shifte... |
| shftval5 15049 | Value of a shifted sequenc... |
| shftf 15050 | Functionality of a shifted... |
| 2shfti 15051 | Composite shift operations... |
| shftidt2 15052 | Identity law for the shift... |
| shftidt 15053 | Identity law for the shift... |
| shftcan1 15054 | Cancellation law for the s... |
| shftcan2 15055 | Cancellation law for the s... |
| seqshft 15056 | Shifting the index set of ... |
| sgnval 15059 | Value of the signum functi... |
| sgn0 15060 | The signum of 0 is 0. (Co... |
| sgnp 15061 | The signum of a positive e... |
| sgnrrp 15062 | The signum of a positive r... |
| sgn1 15063 | The signum of 1 is 1. (Co... |
| sgnpnf 15064 | The signum of ` +oo ` is 1... |
| sgnn 15065 | The signum of a negative e... |
| sgnmnf 15066 | The signum of ` -oo ` is -... |
| cjval 15073 | The value of the conjugate... |
| cjth 15074 | The defining property of t... |
| cjf 15075 | Domain and codomain of the... |
| cjcl 15076 | The conjugate of a complex... |
| reval 15077 | The value of the real part... |
| imval 15078 | The value of the imaginary... |
| imre 15079 | The imaginary part of a co... |
| reim 15080 | The real part of a complex... |
| recl 15081 | The real part of a complex... |
| imcl 15082 | The imaginary part of a co... |
| ref 15083 | Domain and codomain of the... |
| imf 15084 | Domain and codomain of the... |
| crre 15085 | The real part of a complex... |
| crim 15086 | The real part of a complex... |
| replim 15087 | Reconstruct a complex numb... |
| remim 15088 | Value of the conjugate of ... |
| reim0 15089 | The imaginary part of a re... |
| reim0b 15090 | A number is real iff its i... |
| rereb 15091 | A number is real iff it eq... |
| mulre 15092 | A product with a nonzero r... |
| rere 15093 | A real number equals its r... |
| cjreb 15094 | A number is real iff it eq... |
| recj 15095 | Real part of a complex con... |
| reneg 15096 | Real part of negative. (C... |
| readd 15097 | Real part distributes over... |
| resub 15098 | Real part distributes over... |
| remullem 15099 | Lemma for ~ remul , ~ immu... |
| remul 15100 | Real part of a product. (... |
| remul2 15101 | Real part of a product. (... |
| rediv 15102 | Real part of a division. ... |
| imcj 15103 | Imaginary part of a comple... |
| imneg 15104 | The imaginary part of a ne... |
| imadd 15105 | Imaginary part distributes... |
| imsub 15106 | Imaginary part distributes... |
| immul 15107 | Imaginary part of a produc... |
| immul2 15108 | Imaginary part of a produc... |
| imdiv 15109 | Imaginary part of a divisi... |
| cjre 15110 | A real number equals its c... |
| cjcj 15111 | The conjugate of the conju... |
| cjadd 15112 | Complex conjugate distribu... |
| cjmul 15113 | Complex conjugate distribu... |
| ipcnval 15114 | Standard inner product on ... |
| cjmulrcl 15115 | A complex number times its... |
| cjmulval 15116 | A complex number times its... |
| cjmulge0 15117 | A complex number times its... |
| cjneg 15118 | Complex conjugate of negat... |
| addcj 15119 | A number plus its conjugat... |
| cjsub 15120 | Complex conjugate distribu... |
| cjexp 15121 | Complex conjugate of posit... |
| imval2 15122 | The imaginary part of a nu... |
| re0 15123 | The real part of zero. (C... |
| im0 15124 | The imaginary part of zero... |
| re1 15125 | The real part of one. (Co... |
| im1 15126 | The imaginary part of one.... |
| rei 15127 | The real part of ` _i ` . ... |
| imi 15128 | The imaginary part of ` _i... |
| cj0 15129 | The conjugate of zero. (C... |
| cji 15130 | The complex conjugate of t... |
| cjreim 15131 | The conjugate of a represe... |
| cjreim2 15132 | The conjugate of the repre... |
| cj11 15133 | Complex conjugate is a one... |
| cjne0 15134 | A number is nonzero iff it... |
| cjdiv 15135 | Complex conjugate distribu... |
| cnrecnv 15136 | The inverse to the canonic... |
| sqeqd 15137 | A deduction for showing tw... |
| recli 15138 | The real part of a complex... |
| imcli 15139 | The imaginary part of a co... |
| cjcli 15140 | Closure law for complex co... |
| replimi 15141 | Construct a complex number... |
| cjcji 15142 | The conjugate of the conju... |
| reim0bi 15143 | A number is real iff its i... |
| rerebi 15144 | A real number equals its r... |
| cjrebi 15145 | A number is real iff it eq... |
| recji 15146 | Real part of a complex con... |
| imcji 15147 | Imaginary part of a comple... |
| cjmulrcli 15148 | A complex number times its... |
| cjmulvali 15149 | A complex number times its... |
| cjmulge0i 15150 | A complex number times its... |
| renegi 15151 | Real part of negative. (C... |
| imnegi 15152 | Imaginary part of negative... |
| cjnegi 15153 | Complex conjugate of negat... |
| addcji 15154 | A number plus its conjugat... |
| readdi 15155 | Real part distributes over... |
| imaddi 15156 | Imaginary part distributes... |
| remuli 15157 | Real part of a product. (... |
| immuli 15158 | Imaginary part of a produc... |
| cjaddi 15159 | Complex conjugate distribu... |
| cjmuli 15160 | Complex conjugate distribu... |
| ipcni 15161 | Standard inner product on ... |
| cjdivi 15162 | Complex conjugate distribu... |
| crrei 15163 | The real part of a complex... |
| crimi 15164 | The imaginary part of a co... |
| recld 15165 | The real part of a complex... |
| imcld 15166 | The imaginary part of a co... |
| cjcld 15167 | Closure law for complex co... |
| replimd 15168 | Construct a complex number... |
| remimd 15169 | Value of the conjugate of ... |
| cjcjd 15170 | The conjugate of the conju... |
| reim0bd 15171 | A number is real iff its i... |
| rerebd 15172 | A real number equals its r... |
| cjrebd 15173 | A number is real iff it eq... |
| cjne0d 15174 | A number is nonzero iff it... |
| recjd 15175 | Real part of a complex con... |
| imcjd 15176 | Imaginary part of a comple... |
| cjmulrcld 15177 | A complex number times its... |
| cjmulvald 15178 | A complex number times its... |
| cjmulge0d 15179 | A complex number times its... |
| renegd 15180 | Real part of negative. (C... |
| imnegd 15181 | Imaginary part of negative... |
| cjnegd 15182 | Complex conjugate of negat... |
| addcjd 15183 | A number plus its conjugat... |
| cjexpd 15184 | Complex conjugate of posit... |
| readdd 15185 | Real part distributes over... |
| imaddd 15186 | Imaginary part distributes... |
| resubd 15187 | Real part distributes over... |
| imsubd 15188 | Imaginary part distributes... |
| remuld 15189 | Real part of a product. (... |
| immuld 15190 | Imaginary part of a produc... |
| cjaddd 15191 | Complex conjugate distribu... |
| cjmuld 15192 | Complex conjugate distribu... |
| ipcnd 15193 | Standard inner product on ... |
| cjdivd 15194 | Complex conjugate distribu... |
| rered 15195 | A real number equals its r... |
| reim0d 15196 | The imaginary part of a re... |
| cjred 15197 | A real number equals its c... |
| remul2d 15198 | Real part of a product. (... |
| immul2d 15199 | Imaginary part of a produc... |
| redivd 15200 | Real part of a division. ... |
| imdivd 15201 | Imaginary part of a divisi... |
| crred 15202 | The real part of a complex... |
| crimd 15203 | The imaginary part of a co... |
| sqrtval 15208 | Value of square root funct... |
| absval 15209 | The absolute value (modulu... |
| rennim 15210 | A real number does not lie... |
| cnpart 15211 | The specification of restr... |
| sqrt0 15212 | The square root of zero is... |
| 01sqrexlem1 15213 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15214 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15215 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15216 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15217 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15218 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15219 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15220 | Existence of a square root... |
| resqrex 15221 | Existence of a square root... |
| sqrmo 15222 | Uniqueness for the square ... |
| resqreu 15223 | Existence and uniqueness f... |
| resqrtcl 15224 | Closure of the square root... |
| resqrtthlem 15225 | Lemma for ~ resqrtth . (C... |
| resqrtth 15226 | Square root theorem over t... |
| remsqsqrt 15227 | Square of square root. (C... |
| sqrtge0 15228 | The square root function i... |
| sqrtgt0 15229 | The square root function i... |
| sqrtmul 15230 | Square root distributes ov... |
| sqrtle 15231 | Square root is monotonic. ... |
| sqrtlt 15232 | Square root is strictly mo... |
| sqrt11 15233 | The square root function i... |
| sqrt00 15234 | A square root is zero iff ... |
| rpsqrtcl 15235 | The square root of a posit... |
| sqrtdiv 15236 | Square root distributes ov... |
| sqrtneglem 15237 | The square root of a negat... |
| sqrtneg 15238 | The square root of a negat... |
| sqrtsq2 15239 | Relationship between squar... |
| sqrtsq 15240 | Square root of square. (C... |
| sqrtmsq 15241 | Square root of square. (C... |
| sqrt1 15242 | The square root of 1 is 1.... |
| sqrt4 15243 | The square root of 4 is 2.... |
| sqrt9 15244 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15245 | The square root of 2 is bo... |
| sqrtm1 15246 | The imaginary unit is the ... |
| nn0sqeq1 15247 | A natural number with squa... |
| absneg 15248 | Absolute value of the nega... |
| abscl 15249 | Real closure of absolute v... |
| abscj 15250 | The absolute value of a nu... |
| absvalsq 15251 | Square of value of absolut... |
| absvalsq2 15252 | Square of value of absolut... |
| sqabsadd 15253 | Square of absolute value o... |
| sqabssub 15254 | Square of absolute value o... |
| absval2 15255 | Value of absolute value fu... |
| abs0 15256 | The absolute value of 0. ... |
| absi 15257 | The absolute value of the ... |
| absge0 15258 | Absolute value is nonnegat... |
| absrpcl 15259 | The absolute value of a no... |
| abs00 15260 | The absolute value of a nu... |
| abs00ad 15261 | A complex number is zero i... |
| abs00bd 15262 | If a complex number is zer... |
| absreimsq 15263 | Square of the absolute val... |
| absreim 15264 | Absolute value of a number... |
| absmul 15265 | Absolute value distributes... |
| absdiv 15266 | Absolute value distributes... |
| absid 15267 | A nonnegative number is it... |
| abs1 15268 | The absolute value of one ... |
| absnid 15269 | For a negative number, its... |
| leabs 15270 | A real number is less than... |
| absor 15271 | The absolute value of a re... |
| absre 15272 | Absolute value of a real n... |
| absresq 15273 | Square of the absolute val... |
| absmod0 15274 | ` A ` is divisible by ` B ... |
| absexp 15275 | Absolute value of positive... |
| absexpz 15276 | Absolute value of integer ... |
| abssq 15277 | Square can be moved in and... |
| sqabs 15278 | The squares of two reals a... |
| absrele 15279 | The absolute value of a co... |
| absimle 15280 | The absolute value of a co... |
| max0add 15281 | The sum of the positive an... |
| absz 15282 | A real number is an intege... |
| nn0abscl 15283 | The absolute value of an i... |
| zabscl 15284 | The absolute value of an i... |
| abslt 15285 | Absolute value and 'less t... |
| absle 15286 | Absolute value and 'less t... |
| abssubne0 15287 | If the absolute value of a... |
| absdiflt 15288 | The absolute value of a di... |
| absdifle 15289 | The absolute value of a di... |
| elicc4abs 15290 | Membership in a symmetric ... |
| lenegsq 15291 | Comparison to a nonnegativ... |
| releabs 15292 | The real part of a number ... |
| recval 15293 | Reciprocal expressed with ... |
| absidm 15294 | The absolute value functio... |
| absgt0 15295 | The absolute value of a no... |
| nnabscl 15296 | The absolute value of a no... |
| abssub 15297 | Swapping order of subtract... |
| abssubge0 15298 | Absolute value of a nonneg... |
| abssuble0 15299 | Absolute value of a nonpos... |
| absmax 15300 | The maximum of two numbers... |
| abstri 15301 | Triangle inequality for ab... |
| abs3dif 15302 | Absolute value of differen... |
| abs2dif 15303 | Difference of absolute val... |
| abs2dif2 15304 | Difference of absolute val... |
| abs2difabs 15305 | Absolute value of differen... |
| abs1m 15306 | For any complex number, th... |
| recan 15307 | Cancellation law involving... |
| absf 15308 | Mapping domain and codomai... |
| abs3lem 15309 | Lemma involving absolute v... |
| abslem2 15310 | Lemma involving absolute v... |
| rddif 15311 | The difference between a r... |
| absrdbnd 15312 | Bound on the absolute valu... |
| fzomaxdiflem 15313 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15314 | A bound on the separation ... |
| uzin2 15315 | The upper integers are clo... |
| rexanuz 15316 | Combine two different uppe... |
| rexanre 15317 | Combine two different uppe... |
| rexfiuz 15318 | Combine finitely many diff... |
| rexuz3 15319 | Restrict the base of the u... |
| rexanuz2 15320 | Combine two different uppe... |
| r19.29uz 15321 | A version of ~ 19.29 for u... |
| r19.2uz 15322 | A version of ~ r19.2z for ... |
| rexuzre 15323 | Convert an upper real quan... |
| rexico 15324 | Restrict the base of an up... |
| cau3lem 15325 | Lemma for ~ cau3 . (Contr... |
| cau3 15326 | Convert between three-quan... |
| cau4 15327 | Change the base of a Cauch... |
| caubnd2 15328 | A Cauchy sequence of compl... |
| caubnd 15329 | A Cauchy sequence of compl... |
| sqreulem 15330 | Lemma for ~ sqreu : write ... |
| sqreu 15331 | Existence and uniqueness f... |
| sqrtcl 15332 | Closure of the square root... |
| sqrtthlem 15333 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15334 | Mapping domain and codomai... |
| sqrtth 15335 | Square root theorem over t... |
| sqrtrege0 15336 | The square root function m... |
| eqsqrtor 15337 | Solve an equation containi... |
| eqsqrtd 15338 | A deduction for showing th... |
| eqsqrt2d 15339 | A deduction for showing th... |
| amgm2 15340 | Arithmetic-geometric mean ... |
| sqrtthi 15341 | Square root theorem. Theo... |
| sqrtcli 15342 | The square root of a nonne... |
| sqrtgt0i 15343 | The square root of a posit... |
| sqrtmsqi 15344 | Square root of square. (C... |
| sqrtsqi 15345 | Square root of square. (C... |
| sqsqrti 15346 | Square of square root. (C... |
| sqrtge0i 15347 | The square root of a nonne... |
| absidi 15348 | A nonnegative number is it... |
| absnidi 15349 | A negative number is the n... |
| leabsi 15350 | A real number is less than... |
| absori 15351 | The absolute value of a re... |
| absrei 15352 | Absolute value of a real n... |
| sqrtpclii 15353 | The square root of a posit... |
| sqrtgt0ii 15354 | The square root of a posit... |
| sqrt11i 15355 | The square root function i... |
| sqrtmuli 15356 | Square root distributes ov... |
| sqrtmulii 15357 | Square root distributes ov... |
| sqrtmsq2i 15358 | Relationship between squar... |
| sqrtlei 15359 | Square root is monotonic. ... |
| sqrtlti 15360 | Square root is strictly mo... |
| abslti 15361 | Absolute value and 'less t... |
| abslei 15362 | Absolute value and 'less t... |
| cnsqrt00 15363 | A square root of a complex... |
| absvalsqi 15364 | Square of value of absolut... |
| absvalsq2i 15365 | Square of value of absolut... |
| abscli 15366 | Real closure of absolute v... |
| absge0i 15367 | Absolute value is nonnegat... |
| absval2i 15368 | Value of absolute value fu... |
| abs00i 15369 | The absolute value of a nu... |
| absgt0i 15370 | The absolute value of a no... |
| absnegi 15371 | Absolute value of negative... |
| abscji 15372 | The absolute value of a nu... |
| releabsi 15373 | The real part of a number ... |
| abssubi 15374 | Swapping order of subtract... |
| absmuli 15375 | Absolute value distributes... |
| sqabsaddi 15376 | Square of absolute value o... |
| sqabssubi 15377 | Square of absolute value o... |
| absdivzi 15378 | Absolute value distributes... |
| abstrii 15379 | Triangle inequality for ab... |
| abs3difi 15380 | Absolute value of differen... |
| abs3lemi 15381 | Lemma involving absolute v... |
| rpsqrtcld 15382 | The square root of a posit... |
| sqrtgt0d 15383 | The square root of a posit... |
| absnidd 15384 | A negative number is the n... |
| leabsd 15385 | A real number is less than... |
| absord 15386 | The absolute value of a re... |
| absred 15387 | Absolute value of a real n... |
| resqrtcld 15388 | The square root of a nonne... |
| sqrtmsqd 15389 | Square root of square. (C... |
| sqrtsqd 15390 | Square root of square. (C... |
| sqrtge0d 15391 | The square root of a nonne... |
| sqrtnegd 15392 | The square root of a negat... |
| absidd 15393 | A nonnegative number is it... |
| sqrtdivd 15394 | Square root distributes ov... |
| sqrtmuld 15395 | Square root distributes ov... |
| sqrtsq2d 15396 | Relationship between squar... |
| sqrtled 15397 | Square root is monotonic. ... |
| sqrtltd 15398 | Square root is strictly mo... |
| sqr11d 15399 | The square root function i... |
| absltd 15400 | Absolute value and 'less t... |
| absled 15401 | Absolute value and 'less t... |
| abssubge0d 15402 | Absolute value of a nonneg... |
| abssuble0d 15403 | Absolute value of a nonpos... |
| absdifltd 15404 | The absolute value of a di... |
| absdifled 15405 | The absolute value of a di... |
| icodiamlt 15406 | Two elements in a half-ope... |
| abscld 15407 | Real closure of absolute v... |
| sqrtcld 15408 | Closure of the square root... |
| sqrtrege0d 15409 | The real part of the squar... |
| sqsqrtd 15410 | Square root theorem. Theo... |
| msqsqrtd 15411 | Square root theorem. Theo... |
| sqr00d 15412 | A square root is zero iff ... |
| absvalsqd 15413 | Square of value of absolut... |
| absvalsq2d 15414 | Square of value of absolut... |
| absge0d 15415 | Absolute value is nonnegat... |
| absval2d 15416 | Value of absolute value fu... |
| abs00d 15417 | The absolute value of a nu... |
| absne0d 15418 | The absolute value of a nu... |
| absrpcld 15419 | The absolute value of a no... |
| absnegd 15420 | Absolute value of negative... |
| abscjd 15421 | The absolute value of a nu... |
| releabsd 15422 | The real part of a number ... |
| absexpd 15423 | Absolute value of positive... |
| abssubd 15424 | Swapping order of subtract... |
| absmuld 15425 | Absolute value distributes... |
| absdivd 15426 | Absolute value distributes... |
| abstrid 15427 | Triangle inequality for ab... |
| abs2difd 15428 | Difference of absolute val... |
| abs2dif2d 15429 | Difference of absolute val... |
| abs2difabsd 15430 | Absolute value of differen... |
| abs3difd 15431 | Absolute value of differen... |
| abs3lemd 15432 | Lemma involving absolute v... |
| reusq0 15433 | A complex number is the sq... |
| bhmafibid1cn 15434 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15435 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15436 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15437 | The Brahmagupta-Fibonacci ... |
| limsupgord 15440 | Ordering property of the s... |
| limsupcl 15441 | Closure of the superior li... |
| limsupval 15442 | The superior limit of an i... |
| limsupgf 15443 | Closure of the superior li... |
| limsupgval 15444 | Value of the superior limi... |
| limsupgle 15445 | The defining property of t... |
| limsuple 15446 | The defining property of t... |
| limsuplt 15447 | The defining property of t... |
| limsupval2 15448 | The superior limit, relati... |
| limsupgre 15449 | If a sequence of real numb... |
| limsupbnd1 15450 | If a sequence is eventuall... |
| limsupbnd2 15451 | If a sequence is eventuall... |
| climrel 15460 | The limit relation is a re... |
| rlimrel 15461 | The limit relation is a re... |
| clim 15462 | Express the predicate: Th... |
| rlim 15463 | Express the predicate: Th... |
| rlim2 15464 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15465 | Use strictly less-than in ... |
| rlim3 15466 | Restrict the range of the ... |
| climcl 15467 | Closure of the limit of a ... |
| rlimpm 15468 | Closure of a function with... |
| rlimf 15469 | Closure of a function with... |
| rlimss 15470 | Domain closure of a functi... |
| rlimcl 15471 | Closure of the limit of a ... |
| clim2 15472 | Express the predicate: Th... |
| clim2c 15473 | Express the predicate ` F ... |
| clim0 15474 | Express the predicate ` F ... |
| clim0c 15475 | Express the predicate ` F ... |
| rlim0 15476 | Express the predicate ` B ... |
| rlim0lt 15477 | Use strictly less-than in ... |
| climi 15478 | Convergence of a sequence ... |
| climi2 15479 | Convergence of a sequence ... |
| climi0 15480 | Convergence of a sequence ... |
| rlimi 15481 | Convergence at infinity of... |
| rlimi2 15482 | Convergence at infinity of... |
| ello1 15483 | Elementhood in the set of ... |
| ello12 15484 | Elementhood in the set of ... |
| ello12r 15485 | Sufficient condition for e... |
| lo1f 15486 | An eventually upper bounde... |
| lo1dm 15487 | An eventually upper bounde... |
| lo1bdd 15488 | The defining property of a... |
| ello1mpt 15489 | Elementhood in the set of ... |
| ello1mpt2 15490 | Elementhood in the set of ... |
| ello1d 15491 | Sufficient condition for e... |
| lo1bdd2 15492 | If an eventually bounded f... |
| lo1bddrp 15493 | Refine ~ o1bdd2 to give a ... |
| elo1 15494 | Elementhood in the set of ... |
| elo12 15495 | Elementhood in the set of ... |
| elo12r 15496 | Sufficient condition for e... |
| o1f 15497 | An eventually bounded func... |
| o1dm 15498 | An eventually bounded func... |
| o1bdd 15499 | The defining property of a... |
| lo1o1 15500 | A function is eventually b... |
| lo1o12 15501 | A function is eventually b... |
| elo1mpt 15502 | Elementhood in the set of ... |
| elo1mpt2 15503 | Elementhood in the set of ... |
| elo1d 15504 | Sufficient condition for e... |
| o1lo1 15505 | A real function is eventua... |
| o1lo12 15506 | A lower bounded real funct... |
| o1lo1d 15507 | A real eventually bounded ... |
| icco1 15508 | Derive eventual boundednes... |
| o1bdd2 15509 | If an eventually bounded f... |
| o1bddrp 15510 | Refine ~ o1bdd2 to give a ... |
| climconst 15511 | An (eventually) constant s... |
| rlimconst 15512 | A constant sequence conver... |
| rlimclim1 15513 | Forward direction of ~ rli... |
| rlimclim 15514 | A sequence on an upper int... |
| climrlim2 15515 | Produce a real limit from ... |
| climconst2 15516 | A constant sequence conver... |
| climz 15517 | The zero sequence converge... |
| rlimuni 15518 | A real function whose doma... |
| rlimdm 15519 | Two ways to express that a... |
| climuni 15520 | An infinite sequence of co... |
| fclim 15521 | The limit relation is func... |
| climdm 15522 | Two ways to express that a... |
| climeu 15523 | An infinite sequence of co... |
| climreu 15524 | An infinite sequence of co... |
| climmo 15525 | An infinite sequence of co... |
| rlimres 15526 | The restriction of a funct... |
| lo1res 15527 | The restriction of an even... |
| o1res 15528 | The restriction of an even... |
| rlimres2 15529 | The restriction of a funct... |
| lo1res2 15530 | The restriction of a funct... |
| o1res2 15531 | The restriction of a funct... |
| lo1resb 15532 | The restriction of a funct... |
| rlimresb 15533 | The restriction of a funct... |
| o1resb 15534 | The restriction of a funct... |
| climeq 15535 | Two functions that are eve... |
| lo1eq 15536 | Two functions that are eve... |
| rlimeq 15537 | Two functions that are eve... |
| o1eq 15538 | Two functions that are eve... |
| climmpt 15539 | Exhibit a function ` G ` w... |
| 2clim 15540 | If two sequences converge ... |
| climmpt2 15541 | Relate an integer limit on... |
| climshftlem 15542 | A shifted function converg... |
| climres 15543 | A function restricted to u... |
| climshft 15544 | A shifted function converg... |
| serclim0 15545 | The zero series converges ... |
| rlimcld2 15546 | If ` D ` is a closed set i... |
| rlimrege0 15547 | The limit of a sequence of... |
| rlimrecl 15548 | The limit of a real sequen... |
| rlimge0 15549 | The limit of a sequence of... |
| climshft2 15550 | A shifted function converg... |
| climrecl 15551 | The limit of a convergent ... |
| climge0 15552 | A nonnegative sequence con... |
| climabs0 15553 | Convergence to zero of the... |
| o1co 15554 | Sufficient condition for t... |
| o1compt 15555 | Sufficient condition for t... |
| rlimcn1 15556 | Image of a limit under a c... |
| rlimcn1b 15557 | Image of a limit under a c... |
| rlimcn3 15558 | Image of a limit under a c... |
| rlimcn2 15559 | Image of a limit under a c... |
| climcn1 15560 | Image of a limit under a c... |
| climcn2 15561 | Image of a limit under a c... |
| addcn2 15562 | Complex number addition is... |
| subcn2 15563 | Complex number subtraction... |
| mulcn2 15564 | Complex number multiplicat... |
| reccn2 15565 | The reciprocal function is... |
| cn1lem 15566 | A sufficient condition for... |
| abscn2 15567 | The absolute value functio... |
| cjcn2 15568 | The complex conjugate func... |
| recn2 15569 | The real part function is ... |
| imcn2 15570 | The imaginary part functio... |
| climcn1lem 15571 | The limit of a continuous ... |
| climabs 15572 | Limit of the absolute valu... |
| climcj 15573 | Limit of the complex conju... |
| climre 15574 | Limit of the real part of ... |
| climim 15575 | Limit of the imaginary par... |
| rlimmptrcl 15576 | Reverse closure for a real... |
| rlimabs 15577 | Limit of the absolute valu... |
| rlimcj 15578 | Limit of the complex conju... |
| rlimre 15579 | Limit of the real part of ... |
| rlimim 15580 | Limit of the imaginary par... |
| o1of2 15581 | Show that a binary operati... |
| o1add 15582 | The sum of two eventually ... |
| o1mul 15583 | The product of two eventua... |
| o1sub 15584 | The difference of two even... |
| rlimo1 15585 | Any function with a finite... |
| rlimdmo1 15586 | A convergent function is e... |
| o1rlimmul 15587 | The product of an eventual... |
| o1const 15588 | A constant function is eve... |
| lo1const 15589 | A constant function is eve... |
| lo1mptrcl 15590 | Reverse closure for an eve... |
| o1mptrcl 15591 | Reverse closure for an eve... |
| o1add2 15592 | The sum of two eventually ... |
| o1mul2 15593 | The product of two eventua... |
| o1sub2 15594 | The product of two eventua... |
| lo1add 15595 | The sum of two eventually ... |
| lo1mul 15596 | The product of an eventual... |
| lo1mul2 15597 | The product of an eventual... |
| o1dif 15598 | If the difference of two f... |
| lo1sub 15599 | The difference of an event... |
| climadd 15600 | Limit of the sum of two co... |
| climmul 15601 | Limit of the product of tw... |
| climsub 15602 | Limit of the difference of... |
| climaddc1 15603 | Limit of a constant ` C ` ... |
| climaddc2 15604 | Limit of a constant ` C ` ... |
| climmulc2 15605 | Limit of a sequence multip... |
| climsubc1 15606 | Limit of a constant ` C ` ... |
| climsubc2 15607 | Limit of a constant ` C ` ... |
| climle 15608 | Comparison of the limits o... |
| climsqz 15609 | Convergence of a sequence ... |
| climsqz2 15610 | Convergence of a sequence ... |
| rlimadd 15611 | Limit of the sum of two co... |
| rlimaddOLD 15612 | Obsolete version of ~ rlim... |
| rlimsub 15613 | Limit of the difference of... |
| rlimmul 15614 | Limit of the product of tw... |
| rlimmulOLD 15615 | Obsolete version of ~ rlim... |
| rlimdiv 15616 | Limit of the quotient of t... |
| rlimneg 15617 | Limit of the negative of a... |
| rlimle 15618 | Comparison of the limits o... |
| rlimsqzlem 15619 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15620 | Convergence of a sequence ... |
| rlimsqz2 15621 | Convergence of a sequence ... |
| lo1le 15622 | Transfer eventual upper bo... |
| o1le 15623 | Transfer eventual boundedn... |
| rlimno1 15624 | A function whose inverse c... |
| clim2ser 15625 | The limit of an infinite s... |
| clim2ser2 15626 | The limit of an infinite s... |
| iserex 15627 | An infinite series converg... |
| isermulc2 15628 | Multiplication of an infin... |
| climlec2 15629 | Comparison of a constant t... |
| iserle 15630 | Comparison of the limits o... |
| iserge0 15631 | The limit of an infinite s... |
| climub 15632 | The limit of a monotonic s... |
| climserle 15633 | The partial sums of a conv... |
| isershft 15634 | Index shift of the limit o... |
| isercolllem1 15635 | Lemma for ~ isercoll . (C... |
| isercolllem2 15636 | Lemma for ~ isercoll . (C... |
| isercolllem3 15637 | Lemma for ~ isercoll . (C... |
| isercoll 15638 | Rearrange an infinite seri... |
| isercoll2 15639 | Generalize ~ isercoll so t... |
| climsup 15640 | A bounded monotonic sequen... |
| climcau 15641 | A converging sequence of c... |
| climbdd 15642 | A converging sequence of c... |
| caucvgrlem 15643 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15644 | A Cauchy sequence of real ... |
| caucvgrlem2 15645 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15646 | A Cauchy sequence of compl... |
| caurcvg 15647 | A Cauchy sequence of real ... |
| caurcvg2 15648 | A Cauchy sequence of real ... |
| caucvg 15649 | A Cauchy sequence of compl... |
| caucvgb 15650 | A function is convergent i... |
| serf0 15651 | If an infinite series conv... |
| iseraltlem1 15652 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15653 | Lemma for ~ iseralt . The... |
| iseraltlem3 15654 | Lemma for ~ iseralt . Fro... |
| iseralt 15655 | The alternating series tes... |
| sumex 15658 | A sum is a set. (Contribu... |
| sumeq1 15659 | Equality theorem for a sum... |
| nfsum1 15660 | Bound-variable hypothesis ... |
| nfsum 15661 | Bound-variable hypothesis ... |
| sumeq2w 15662 | Equality theorem for sum, ... |
| sumeq2ii 15663 | Equality theorem for sum, ... |
| sumeq2 15664 | Equality theorem for sum. ... |
| cbvsum 15665 | Change bound variable in a... |
| cbvsumv 15666 | Change bound variable in a... |
| cbvsumi 15667 | Change bound variable in a... |
| sumeq1i 15668 | Equality inference for sum... |
| sumeq2i 15669 | Equality inference for sum... |
| sumeq12i 15670 | Equality inference for sum... |
| sumeq1d 15671 | Equality deduction for sum... |
| sumeq2d 15672 | Equality deduction for sum... |
| sumeq2dv 15673 | Equality deduction for sum... |
| sumeq2sdv 15674 | Equality deduction for sum... |
| 2sumeq2dv 15675 | Equality deduction for dou... |
| sumeq12dv 15676 | Equality deduction for sum... |
| sumeq12rdv 15677 | Equality deduction for sum... |
| sum2id 15678 | The second class argument ... |
| sumfc 15679 | A lemma to facilitate conv... |
| fz1f1o 15680 | A lemma for working with f... |
| sumrblem 15681 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15682 | The sequence of partial su... |
| sumrb 15683 | Rebase the starting point ... |
| summolem3 15684 | Lemma for ~ summo . (Cont... |
| summolem2a 15685 | Lemma for ~ summo . (Cont... |
| summolem2 15686 | Lemma for ~ summo . (Cont... |
| summo 15687 | A sum has at most one limi... |
| zsum 15688 | Series sum with index set ... |
| isum 15689 | Series sum with an upper i... |
| fsum 15690 | The value of a sum over a ... |
| sum0 15691 | Any sum over the empty set... |
| sumz 15692 | Any sum of zero over a sum... |
| fsumf1o 15693 | Re-index a finite sum usin... |
| sumss 15694 | Change the index set to a ... |
| fsumss 15695 | Change the index set to a ... |
| sumss2 15696 | Change the index set of a ... |
| fsumcvg2 15697 | The sequence of partial su... |
| fsumsers 15698 | Special case of series sum... |
| fsumcvg3 15699 | A finite sum is convergent... |
| fsumser 15700 | A finite sum expressed in ... |
| fsumcl2lem 15701 | - Lemma for finite sum clo... |
| fsumcllem 15702 | - Lemma for finite sum clo... |
| fsumcl 15703 | Closure of a finite sum of... |
| fsumrecl 15704 | Closure of a finite sum of... |
| fsumzcl 15705 | Closure of a finite sum of... |
| fsumnn0cl 15706 | Closure of a finite sum of... |
| fsumrpcl 15707 | Closure of a finite sum of... |
| fsumclf 15708 | Closure of a finite sum of... |
| fsumzcl2 15709 | A finite sum with integer ... |
| fsumadd 15710 | The sum of two finite sums... |
| fsumsplit 15711 | Split a sum into two parts... |
| fsumsplitf 15712 | Split a sum into two parts... |
| sumsnf 15713 | A sum of a singleton is th... |
| fsumsplitsn 15714 | Separate out a term in a f... |
| fsumsplit1 15715 | Separate out a term in a f... |
| sumsn 15716 | A sum of a singleton is th... |
| fsum1 15717 | The finite sum of ` A ( k ... |
| sumpr 15718 | A sum over a pair is the s... |
| sumtp 15719 | A sum over a triple is the... |
| sumsns 15720 | A sum of a singleton is th... |
| fsumm1 15721 | Separate out the last term... |
| fzosump1 15722 | Separate out the last term... |
| fsum1p 15723 | Separate out the first ter... |
| fsummsnunz 15724 | A finite sum all of whose ... |
| fsumsplitsnun 15725 | Separate out a term in a f... |
| fsump1 15726 | The addition of the next t... |
| isumclim 15727 | An infinite sum equals the... |
| isumclim2 15728 | A converging series conver... |
| isumclim3 15729 | The sequence of partial fi... |
| sumnul 15730 | The sum of a non-convergen... |
| isumcl 15731 | The sum of a converging in... |
| isummulc2 15732 | An infinite sum multiplied... |
| isummulc1 15733 | An infinite sum multiplied... |
| isumdivc 15734 | An infinite sum divided by... |
| isumrecl 15735 | The sum of a converging in... |
| isumge0 15736 | An infinite sum of nonnega... |
| isumadd 15737 | Addition of infinite sums.... |
| sumsplit 15738 | Split a sum into two parts... |
| fsump1i 15739 | Optimized version of ~ fsu... |
| fsum2dlem 15740 | Lemma for ~ fsum2d - induc... |
| fsum2d 15741 | Write a double sum as a su... |
| fsumxp 15742 | Combine two sums into a si... |
| fsumcnv 15743 | Transform a region of summ... |
| fsumcom2 15744 | Interchange order of summa... |
| fsumcom 15745 | Interchange order of summa... |
| fsum0diaglem 15746 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15747 | Two ways to express "the s... |
| mptfzshft 15748 | 1-1 onto function in maps-... |
| fsumrev 15749 | Reversal of a finite sum. ... |
| fsumshft 15750 | Index shift of a finite su... |
| fsumshftm 15751 | Negative index shift of a ... |
| fsumrev2 15752 | Reversal of a finite sum. ... |
| fsum0diag2 15753 | Two ways to express "the s... |
| fsummulc2 15754 | A finite sum multiplied by... |
| fsummulc1 15755 | A finite sum multiplied by... |
| fsumdivc 15756 | A finite sum divided by a ... |
| fsumneg 15757 | Negation of a finite sum. ... |
| fsumsub 15758 | Split a finite sum over a ... |
| fsum2mul 15759 | Separate the nested sum of... |
| fsumconst 15760 | The sum of constant terms ... |
| fsumdifsnconst 15761 | The sum of constant terms ... |
| modfsummodslem1 15762 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15763 | Induction step for ~ modfs... |
| modfsummod 15764 | A finite sum modulo a posi... |
| fsumge0 15765 | If all of the terms of a f... |
| fsumless 15766 | A shorter sum of nonnegati... |
| fsumge1 15767 | A sum of nonnegative numbe... |
| fsum00 15768 | A sum of nonnegative numbe... |
| fsumle 15769 | If all of the terms of fin... |
| fsumlt 15770 | If every term in one finit... |
| fsumabs 15771 | Generalized triangle inequ... |
| telfsumo 15772 | Sum of a telescoping serie... |
| telfsumo2 15773 | Sum of a telescoping serie... |
| telfsum 15774 | Sum of a telescoping serie... |
| telfsum2 15775 | Sum of a telescoping serie... |
| fsumparts 15776 | Summation by parts. (Cont... |
| fsumrelem 15777 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15778 | The real part of a sum. (... |
| fsumim 15779 | The imaginary part of a su... |
| fsumcj 15780 | The complex conjugate of a... |
| fsumrlim 15781 | Limit of a finite sum of c... |
| fsumo1 15782 | The finite sum of eventual... |
| o1fsum 15783 | If ` A ( k ) ` is O(1), th... |
| seqabs 15784 | Generalized triangle inequ... |
| iserabs 15785 | Generalized triangle inequ... |
| cvgcmp 15786 | A comparison test for conv... |
| cvgcmpub 15787 | An upper bound for the lim... |
| cvgcmpce 15788 | A comparison test for conv... |
| abscvgcvg 15789 | An absolutely convergent s... |
| climfsum 15790 | Limit of a finite sum of c... |
| fsumiun 15791 | Sum over a disjoint indexe... |
| hashiun 15792 | The cardinality of a disjo... |
| hash2iun 15793 | The cardinality of a neste... |
| hash2iun1dif1 15794 | The cardinality of a neste... |
| hashrabrex 15795 | The number of elements in ... |
| hashuni 15796 | The cardinality of a disjo... |
| qshash 15797 | The cardinality of a set w... |
| ackbijnn 15798 | Translate the Ackermann bi... |
| binomlem 15799 | Lemma for ~ binom (binomia... |
| binom 15800 | The binomial theorem: ` ( ... |
| binom1p 15801 | Special case of the binomi... |
| binom11 15802 | Special case of the binomi... |
| binom1dif 15803 | A summation for the differ... |
| bcxmaslem1 15804 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15805 | Parallel summation (Christ... |
| incexclem 15806 | Lemma for ~ incexc . (Con... |
| incexc 15807 | The inclusion/exclusion pr... |
| incexc2 15808 | The inclusion/exclusion pr... |
| isumshft 15809 | Index shift of an infinite... |
| isumsplit 15810 | Split off the first ` N ` ... |
| isum1p 15811 | The infinite sum of a conv... |
| isumnn0nn 15812 | Sum from 0 to infinity in ... |
| isumrpcl 15813 | The infinite sum of positi... |
| isumle 15814 | Comparison of two infinite... |
| isumless 15815 | A finite sum of nonnegativ... |
| isumsup2 15816 | An infinite sum of nonnega... |
| isumsup 15817 | An infinite sum of nonnega... |
| isumltss 15818 | A partial sum of a series ... |
| climcndslem1 15819 | Lemma for ~ climcnds : bou... |
| climcndslem2 15820 | Lemma for ~ climcnds : bou... |
| climcnds 15821 | The Cauchy condensation te... |
| divrcnv 15822 | The sequence of reciprocal... |
| divcnv 15823 | The sequence of reciprocal... |
| flo1 15824 | The floor function satisfi... |
| divcnvshft 15825 | Limit of a ratio function.... |
| supcvg 15826 | Extract a sequence ` f ` i... |
| infcvgaux1i 15827 | Auxiliary theorem for appl... |
| infcvgaux2i 15828 | Auxiliary theorem for appl... |
| harmonic 15829 | The harmonic series ` H ` ... |
| arisum 15830 | Arithmetic series sum of t... |
| arisum2 15831 | Arithmetic series sum of t... |
| trireciplem 15832 | Lemma for ~ trirecip . Sh... |
| trirecip 15833 | The sum of the reciprocals... |
| expcnv 15834 | A sequence of powers of a ... |
| explecnv 15835 | A sequence of terms conver... |
| geoserg 15836 | The value of the finite ge... |
| geoser 15837 | The value of the finite ge... |
| pwdif 15838 | The difference of two numb... |
| pwm1geoser 15839 | The n-th power of a number... |
| geolim 15840 | The partial sums in the in... |
| geolim2 15841 | The partial sums in the ge... |
| georeclim 15842 | The limit of a geometric s... |
| geo2sum 15843 | The value of the finite ge... |
| geo2sum2 15844 | The value of the finite ge... |
| geo2lim 15845 | The value of the infinite ... |
| geomulcvg 15846 | The geometric series conve... |
| geoisum 15847 | The infinite sum of ` 1 + ... |
| geoisumr 15848 | The infinite sum of recipr... |
| geoisum1 15849 | The infinite sum of ` A ^ ... |
| geoisum1c 15850 | The infinite sum of ` A x.... |
| 0.999... 15851 | The recurring decimal 0.99... |
| geoihalfsum 15852 | Prove that the infinite ge... |
| cvgrat 15853 | Ratio test for convergence... |
| mertenslem1 15854 | Lemma for ~ mertens . (Co... |
| mertenslem2 15855 | Lemma for ~ mertens . (Co... |
| mertens 15856 | Mertens' theorem. If ` A ... |
| prodf 15857 | An infinite product of com... |
| clim2prod 15858 | The limit of an infinite p... |
| clim2div 15859 | The limit of an infinite p... |
| prodfmul 15860 | The product of two infinit... |
| prodf1 15861 | The value of the partial p... |
| prodf1f 15862 | A one-valued infinite prod... |
| prodfclim1 15863 | The constant one product c... |
| prodfn0 15864 | No term of a nonzero infin... |
| prodfrec 15865 | The reciprocal of an infin... |
| prodfdiv 15866 | The quotient of two infini... |
| ntrivcvg 15867 | A non-trivially converging... |
| ntrivcvgn0 15868 | A product that converges t... |
| ntrivcvgfvn0 15869 | Any value of a product seq... |
| ntrivcvgtail 15870 | A tail of a non-trivially ... |
| ntrivcvgmullem 15871 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15872 | The product of two non-tri... |
| prodex 15875 | A product is a set. (Cont... |
| prodeq1f 15876 | Equality theorem for a pro... |
| prodeq1 15877 | Equality theorem for a pro... |
| nfcprod1 15878 | Bound-variable hypothesis ... |
| nfcprod 15879 | Bound-variable hypothesis ... |
| prodeq2w 15880 | Equality theorem for produ... |
| prodeq2ii 15881 | Equality theorem for produ... |
| prodeq2 15882 | Equality theorem for produ... |
| cbvprod 15883 | Change bound variable in a... |
| cbvprodv 15884 | Change bound variable in a... |
| cbvprodi 15885 | Change bound variable in a... |
| prodeq1i 15886 | Equality inference for pro... |
| prodeq2i 15887 | Equality inference for pro... |
| prodeq12i 15888 | Equality inference for pro... |
| prodeq1d 15889 | Equality deduction for pro... |
| prodeq2d 15890 | Equality deduction for pro... |
| prodeq2dv 15891 | Equality deduction for pro... |
| prodeq2sdv 15892 | Equality deduction for pro... |
| 2cprodeq2dv 15893 | Equality deduction for dou... |
| prodeq12dv 15894 | Equality deduction for pro... |
| prodeq12rdv 15895 | Equality deduction for pro... |
| prod2id 15896 | The second class argument ... |
| prodrblem 15897 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15898 | The sequence of partial pr... |
| prodrblem2 15899 | Lemma for ~ prodrb . (Con... |
| prodrb 15900 | Rebase the starting point ... |
| prodmolem3 15901 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15902 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15903 | Lemma for ~ prodmo . (Con... |
| prodmo 15904 | A product has at most one ... |
| zprod 15905 | Series product with index ... |
| iprod 15906 | Series product with an upp... |
| zprodn0 15907 | Nonzero series product wit... |
| iprodn0 15908 | Nonzero series product wit... |
| fprod 15909 | The value of a product ove... |
| fprodntriv 15910 | A non-triviality lemma for... |
| prod0 15911 | A product over the empty s... |
| prod1 15912 | Any product of one over a ... |
| prodfc 15913 | A lemma to facilitate conv... |
| fprodf1o 15914 | Re-index a finite product ... |
| prodss 15915 | Change the index set to a ... |
| fprodss 15916 | Change the index set to a ... |
| fprodser 15917 | A finite product expressed... |
| fprodcl2lem 15918 | Finite product closure lem... |
| fprodcllem 15919 | Finite product closure lem... |
| fprodcl 15920 | Closure of a finite produc... |
| fprodrecl 15921 | Closure of a finite produc... |
| fprodzcl 15922 | Closure of a finite produc... |
| fprodnncl 15923 | Closure of a finite produc... |
| fprodrpcl 15924 | Closure of a finite produc... |
| fprodnn0cl 15925 | Closure of a finite produc... |
| fprodcllemf 15926 | Finite product closure lem... |
| fprodreclf 15927 | Closure of a finite produc... |
| fprodmul 15928 | The product of two finite ... |
| fproddiv 15929 | The quotient of two finite... |
| prodsn 15930 | A product of a singleton i... |
| fprod1 15931 | A finite product of only o... |
| prodsnf 15932 | A product of a singleton i... |
| climprod1 15933 | The limit of a product ove... |
| fprodsplit 15934 | Split a finite product int... |
| fprodm1 15935 | Separate out the last term... |
| fprod1p 15936 | Separate out the first ter... |
| fprodp1 15937 | Multiply in the last term ... |
| fprodm1s 15938 | Separate out the last term... |
| fprodp1s 15939 | Multiply in the last term ... |
| prodsns 15940 | A product of the singleton... |
| fprodfac 15941 | Factorial using product no... |
| fprodabs 15942 | The absolute value of a fi... |
| fprodeq0 15943 | Any finite product contain... |
| fprodshft 15944 | Shift the index of a finit... |
| fprodrev 15945 | Reversal of a finite produ... |
| fprodconst 15946 | The product of constant te... |
| fprodn0 15947 | A finite product of nonzer... |
| fprod2dlem 15948 | Lemma for ~ fprod2d - indu... |
| fprod2d 15949 | Write a double product as ... |
| fprodxp 15950 | Combine two products into ... |
| fprodcnv 15951 | Transform a product region... |
| fprodcom2 15952 | Interchange order of multi... |
| fprodcom 15953 | Interchange product order.... |
| fprod0diag 15954 | Two ways to express "the p... |
| fproddivf 15955 | The quotient of two finite... |
| fprodsplitf 15956 | Split a finite product int... |
| fprodsplitsn 15957 | Separate out a term in a f... |
| fprodsplit1f 15958 | Separate out a term in a f... |
| fprodn0f 15959 | A finite product of nonzer... |
| fprodclf 15960 | Closure of a finite produc... |
| fprodge0 15961 | If all the terms of a fini... |
| fprodeq0g 15962 | Any finite product contain... |
| fprodge1 15963 | If all of the terms of a f... |
| fprodle 15964 | If all the terms of two fi... |
| fprodmodd 15965 | If all factors of two fini... |
| iprodclim 15966 | An infinite product equals... |
| iprodclim2 15967 | A converging product conve... |
| iprodclim3 15968 | The sequence of partial fi... |
| iprodcl 15969 | The product of a non-trivi... |
| iprodrecl 15970 | The product of a non-trivi... |
| iprodmul 15971 | Multiplication of infinite... |
| risefacval 15976 | The value of the rising fa... |
| fallfacval 15977 | The value of the falling f... |
| risefacval2 15978 | One-based value of rising ... |
| fallfacval2 15979 | One-based value of falling... |
| fallfacval3 15980 | A product representation o... |
| risefaccllem 15981 | Lemma for rising factorial... |
| fallfaccllem 15982 | Lemma for falling factoria... |
| risefaccl 15983 | Closure law for rising fac... |
| fallfaccl 15984 | Closure law for falling fa... |
| rerisefaccl 15985 | Closure law for rising fac... |
| refallfaccl 15986 | Closure law for falling fa... |
| nnrisefaccl 15987 | Closure law for rising fac... |
| zrisefaccl 15988 | Closure law for rising fac... |
| zfallfaccl 15989 | Closure law for falling fa... |
| nn0risefaccl 15990 | Closure law for rising fac... |
| rprisefaccl 15991 | Closure law for rising fac... |
| risefallfac 15992 | A relationship between ris... |
| fallrisefac 15993 | A relationship between fal... |
| risefall0lem 15994 | Lemma for ~ risefac0 and ~... |
| risefac0 15995 | The value of the rising fa... |
| fallfac0 15996 | The value of the falling f... |
| risefacp1 15997 | The value of the rising fa... |
| fallfacp1 15998 | The value of the falling f... |
| risefacp1d 15999 | The value of the rising fa... |
| fallfacp1d 16000 | The value of the falling f... |
| risefac1 16001 | The value of rising factor... |
| fallfac1 16002 | The value of falling facto... |
| risefacfac 16003 | Relate rising factorial to... |
| fallfacfwd 16004 | The forward difference of ... |
| 0fallfac 16005 | The value of the zero fall... |
| 0risefac 16006 | The value of the zero risi... |
| binomfallfaclem1 16007 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 16008 | Lemma for ~ binomfallfac .... |
| binomfallfac 16009 | A version of the binomial ... |
| binomrisefac 16010 | A version of the binomial ... |
| fallfacval4 16011 | Represent the falling fact... |
| bcfallfac 16012 | Binomial coefficient in te... |
| fallfacfac 16013 | Relate falling factorial t... |
| bpolylem 16016 | Lemma for ~ bpolyval . (C... |
| bpolyval 16017 | The value of the Bernoulli... |
| bpoly0 16018 | The value of the Bernoulli... |
| bpoly1 16019 | The value of the Bernoulli... |
| bpolycl 16020 | Closure law for Bernoulli ... |
| bpolysum 16021 | A sum for Bernoulli polyno... |
| bpolydiflem 16022 | Lemma for ~ bpolydif . (C... |
| bpolydif 16023 | Calculate the difference b... |
| fsumkthpow 16024 | A closed-form expression f... |
| bpoly2 16025 | The Bernoulli polynomials ... |
| bpoly3 16026 | The Bernoulli polynomials ... |
| bpoly4 16027 | The Bernoulli polynomials ... |
| fsumcube 16028 | Express the sum of cubes i... |
| eftcl 16041 | Closure of a term in the s... |
| reeftcl 16042 | The terms of the series ex... |
| eftabs 16043 | The absolute value of a te... |
| eftval 16044 | The value of a term in the... |
| efcllem 16045 | Lemma for ~ efcl . The se... |
| ef0lem 16046 | The series defining the ex... |
| efval 16047 | Value of the exponential f... |
| esum 16048 | Value of Euler's constant ... |
| eff 16049 | Domain and codomain of the... |
| efcl 16050 | Closure law for the expone... |
| efcld 16051 | Closure law for the expone... |
| efval2 16052 | Value of the exponential f... |
| efcvg 16053 | The series that defines th... |
| efcvgfsum 16054 | Exponential function conve... |
| reefcl 16055 | The exponential function i... |
| reefcld 16056 | The exponential function i... |
| ere 16057 | Euler's constant ` _e ` = ... |
| ege2le3 16058 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16059 | Value of the exponential f... |
| efcj 16060 | The exponential of a compl... |
| efaddlem 16061 | Lemma for ~ efadd (exponen... |
| efadd 16062 | Sum of exponents law for e... |
| fprodefsum 16063 | Move the exponential funct... |
| efcan 16064 | Cancellation law for expon... |
| efne0 16065 | The exponential of a compl... |
| efneg 16066 | The exponential of the opp... |
| eff2 16067 | The exponential function m... |
| efsub 16068 | Difference of exponents la... |
| efexp 16069 | The exponential of an inte... |
| efzval 16070 | Value of the exponential f... |
| efgt0 16071 | The exponential of a real ... |
| rpefcl 16072 | The exponential of a real ... |
| rpefcld 16073 | The exponential of a real ... |
| eftlcvg 16074 | The tail series of the exp... |
| eftlcl 16075 | Closure of the sum of an i... |
| reeftlcl 16076 | Closure of the sum of an i... |
| eftlub 16077 | An upper bound on the abso... |
| efsep 16078 | Separate out the next term... |
| effsumlt 16079 | The partial sums of the se... |
| eft0val 16080 | The value of the first ter... |
| ef4p 16081 | Separate out the first fou... |
| efgt1p2 16082 | The exponential of a posit... |
| efgt1p 16083 | The exponential of a posit... |
| efgt1 16084 | The exponential of a posit... |
| eflt 16085 | The exponential function o... |
| efle 16086 | The exponential function o... |
| reef11 16087 | The exponential function o... |
| reeff1 16088 | The exponential function m... |
| eflegeo 16089 | The exponential function o... |
| sinval 16090 | Value of the sine function... |
| cosval 16091 | Value of the cosine functi... |
| sinf 16092 | Domain and codomain of the... |
| cosf 16093 | Domain and codomain of the... |
| sincl 16094 | Closure of the sine functi... |
| coscl 16095 | Closure of the cosine func... |
| tanval 16096 | Value of the tangent funct... |
| tancl 16097 | The closure of the tangent... |
| sincld 16098 | Closure of the sine functi... |
| coscld 16099 | Closure of the cosine func... |
| tancld 16100 | Closure of the tangent fun... |
| tanval2 16101 | Express the tangent functi... |
| tanval3 16102 | Express the tangent functi... |
| resinval 16103 | The sine of a real number ... |
| recosval 16104 | The cosine of a real numbe... |
| efi4p 16105 | Separate out the first fou... |
| resin4p 16106 | Separate out the first fou... |
| recos4p 16107 | Separate out the first fou... |
| resincl 16108 | The sine of a real number ... |
| recoscl 16109 | The cosine of a real numbe... |
| retancl 16110 | The closure of the tangent... |
| resincld 16111 | Closure of the sine functi... |
| recoscld 16112 | Closure of the cosine func... |
| retancld 16113 | Closure of the tangent fun... |
| sinneg 16114 | The sine of a negative is ... |
| cosneg 16115 | The cosines of a number an... |
| tanneg 16116 | The tangent of a negative ... |
| sin0 16117 | Value of the sine function... |
| cos0 16118 | Value of the cosine functi... |
| tan0 16119 | The value of the tangent f... |
| efival 16120 | The exponential function i... |
| efmival 16121 | The exponential function i... |
| sinhval 16122 | Value of the hyperbolic si... |
| coshval 16123 | Value of the hyperbolic co... |
| resinhcl 16124 | The hyperbolic sine of a r... |
| rpcoshcl 16125 | The hyperbolic cosine of a... |
| recoshcl 16126 | The hyperbolic cosine of a... |
| retanhcl 16127 | The hyperbolic tangent of ... |
| tanhlt1 16128 | The hyperbolic tangent of ... |
| tanhbnd 16129 | The hyperbolic tangent of ... |
| efeul 16130 | Eulerian representation of... |
| efieq 16131 | The exponentials of two im... |
| sinadd 16132 | Addition formula for sine.... |
| cosadd 16133 | Addition formula for cosin... |
| tanaddlem 16134 | A useful intermediate step... |
| tanadd 16135 | Addition formula for tange... |
| sinsub 16136 | Sine of difference. (Cont... |
| cossub 16137 | Cosine of difference. (Co... |
| addsin 16138 | Sum of sines. (Contribute... |
| subsin 16139 | Difference of sines. (Con... |
| sinmul 16140 | Product of sines can be re... |
| cosmul 16141 | Product of cosines can be ... |
| addcos 16142 | Sum of cosines. (Contribu... |
| subcos 16143 | Difference of cosines. (C... |
| sincossq 16144 | Sine squared plus cosine s... |
| sin2t 16145 | Double-angle formula for s... |
| cos2t 16146 | Double-angle formula for c... |
| cos2tsin 16147 | Double-angle formula for c... |
| sinbnd 16148 | The sine of a real number ... |
| cosbnd 16149 | The cosine of a real numbe... |
| sinbnd2 16150 | The sine of a real number ... |
| cosbnd2 16151 | The cosine of a real numbe... |
| ef01bndlem 16152 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16153 | Bounds on the sine of a po... |
| cos01bnd 16154 | Bounds on the cosine of a ... |
| cos1bnd 16155 | Bounds on the cosine of 1.... |
| cos2bnd 16156 | Bounds on the cosine of 2.... |
| sinltx 16157 | The sine of a positive rea... |
| sin01gt0 16158 | The sine of a positive rea... |
| cos01gt0 16159 | The cosine of a positive r... |
| sin02gt0 16160 | The sine of a positive rea... |
| sincos1sgn 16161 | The signs of the sine and ... |
| sincos2sgn 16162 | The signs of the sine and ... |
| sin4lt0 16163 | The sine of 4 is negative.... |
| absefi 16164 | The absolute value of the ... |
| absef 16165 | The absolute value of the ... |
| absefib 16166 | A complex number is real i... |
| efieq1re 16167 | A number whose imaginary e... |
| demoivre 16168 | De Moivre's Formula. Proo... |
| demoivreALT 16169 | Alternate proof of ~ demoi... |
| eirrlem 16172 | Lemma for ~ eirr . (Contr... |
| eirr 16173 | ` _e ` is irrational. (Co... |
| egt2lt3 16174 | Euler's constant ` _e ` = ... |
| epos 16175 | Euler's constant ` _e ` is... |
| epr 16176 | Euler's constant ` _e ` is... |
| ene0 16177 | ` _e ` is not 0. (Contrib... |
| ene1 16178 | ` _e ` is not 1. (Contrib... |
| xpnnen 16179 | The Cartesian product of t... |
| znnen 16180 | The set of integers and th... |
| qnnen 16181 | The rational numbers are c... |
| rpnnen2lem1 16182 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16183 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16184 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16185 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16186 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16187 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16188 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16189 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16190 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16191 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16192 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16193 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16194 | The other half of ~ rpnnen... |
| rpnnen 16195 | The cardinality of the con... |
| rexpen 16196 | The real numbers are equin... |
| cpnnen 16197 | The complex numbers are eq... |
| rucALT 16198 | Alternate proof of ~ ruc .... |
| ruclem1 16199 | Lemma for ~ ruc (the reals... |
| ruclem2 16200 | Lemma for ~ ruc . Orderin... |
| ruclem3 16201 | Lemma for ~ ruc . The con... |
| ruclem4 16202 | Lemma for ~ ruc . Initial... |
| ruclem6 16203 | Lemma for ~ ruc . Domain ... |
| ruclem7 16204 | Lemma for ~ ruc . Success... |
| ruclem8 16205 | Lemma for ~ ruc . The int... |
| ruclem9 16206 | Lemma for ~ ruc . The fir... |
| ruclem10 16207 | Lemma for ~ ruc . Every f... |
| ruclem11 16208 | Lemma for ~ ruc . Closure... |
| ruclem12 16209 | Lemma for ~ ruc . The sup... |
| ruclem13 16210 | Lemma for ~ ruc . There i... |
| ruc 16211 | The set of positive intege... |
| resdomq 16212 | The set of rationals is st... |
| aleph1re 16213 | There are at least aleph-o... |
| aleph1irr 16214 | There are at least aleph-o... |
| cnso 16215 | The complex numbers can be... |
| sqrt2irrlem 16216 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16217 | The square root of 2 is ir... |
| sqrt2re 16218 | The square root of 2 exist... |
| sqrt2irr0 16219 | The square root of 2 is an... |
| nthruc 16220 | The sequence ` NN ` , ` ZZ... |
| nthruz 16221 | The sequence ` NN ` , ` NN... |
| divides 16224 | Define the divides relatio... |
| dvdsval2 16225 | One nonzero integer divide... |
| dvdsval3 16226 | One nonzero integer divide... |
| dvdszrcl 16227 | Reverse closure for the di... |
| dvdsmod0 16228 | If a positive integer divi... |
| p1modz1 16229 | If a number greater than 1... |
| dvdsmodexp 16230 | If a positive integer divi... |
| nndivdvds 16231 | Strong form of ~ dvdsval2 ... |
| nndivides 16232 | Definition of the divides ... |
| moddvds 16233 | Two ways to say ` A == B `... |
| modm1div 16234 | An integer greater than on... |
| dvds0lem 16235 | A lemma to assist theorems... |
| dvds1lem 16236 | A lemma to assist theorems... |
| dvds2lem 16237 | A lemma to assist theorems... |
| iddvds 16238 | An integer divides itself.... |
| 1dvds 16239 | 1 divides any integer. Th... |
| dvds0 16240 | Any integer divides 0. Th... |
| negdvdsb 16241 | An integer divides another... |
| dvdsnegb 16242 | An integer divides another... |
| absdvdsb 16243 | An integer divides another... |
| dvdsabsb 16244 | An integer divides another... |
| 0dvds 16245 | Only 0 is divisible by 0. ... |
| dvdsmul1 16246 | An integer divides a multi... |
| dvdsmul2 16247 | An integer divides a multi... |
| iddvdsexp 16248 | An integer divides a posit... |
| muldvds1 16249 | If a product divides an in... |
| muldvds2 16250 | If a product divides an in... |
| dvdscmul 16251 | Multiplication by a consta... |
| dvdsmulc 16252 | Multiplication by a consta... |
| dvdscmulr 16253 | Cancellation law for the d... |
| dvdsmulcr 16254 | Cancellation law for the d... |
| summodnegmod 16255 | The sum of two integers mo... |
| modmulconst 16256 | Constant multiplication in... |
| dvds2ln 16257 | If an integer divides each... |
| dvds2add 16258 | If an integer divides each... |
| dvds2sub 16259 | If an integer divides each... |
| dvds2addd 16260 | Deduction form of ~ dvds2a... |
| dvds2subd 16261 | Deduction form of ~ dvds2s... |
| dvdstr 16262 | The divides relation is tr... |
| dvdstrd 16263 | The divides relation is tr... |
| dvdsmultr1 16264 | If an integer divides anot... |
| dvdsmultr1d 16265 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16266 | If an integer divides anot... |
| dvdsmultr2d 16267 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16268 | If an integer divides eith... |
| dvdssub2 16269 | If an integer divides a di... |
| dvdsadd 16270 | An integer divides another... |
| dvdsaddr 16271 | An integer divides another... |
| dvdssub 16272 | An integer divides another... |
| dvdssubr 16273 | An integer divides another... |
| dvdsadd2b 16274 | Adding a multiple of the b... |
| dvdsaddre2b 16275 | Adding a multiple of the b... |
| fsumdvds 16276 | If every term in a sum is ... |
| dvdslelem 16277 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16278 | The divisors of a positive... |
| dvdsleabs 16279 | The divisors of a nonzero ... |
| dvdsleabs2 16280 | Transfer divisibility to a... |
| dvdsabseq 16281 | If two integers divide eac... |
| dvdseq 16282 | If two nonnegative integer... |
| divconjdvds 16283 | If a nonzero integer ` M `... |
| dvdsdivcl 16284 | The complement of a diviso... |
| dvdsflip 16285 | An involution of the divis... |
| dvdsssfz1 16286 | The set of divisors of a n... |
| dvds1 16287 | The only nonnegative integ... |
| alzdvds 16288 | Only 0 is divisible by all... |
| dvdsext 16289 | Poset extensionality for d... |
| fzm1ndvds 16290 | No number between ` 1 ` an... |
| fzo0dvdseq 16291 | Zero is the only one of th... |
| fzocongeq 16292 | Two different elements of ... |
| addmodlteqALT 16293 | Two nonnegative integers l... |
| dvdsfac 16294 | A positive integer divides... |
| dvdsexp2im 16295 | If an integer divides anot... |
| dvdsexp 16296 | A power divides a power wi... |
| dvdsmod 16297 | Any number ` K ` whose mod... |
| mulmoddvds 16298 | If an integer is divisible... |
| 3dvds 16299 | A rule for divisibility by... |
| 3dvdsdec 16300 | A decimal number is divisi... |
| 3dvds2dec 16301 | A decimal number is divisi... |
| fprodfvdvdsd 16302 | A finite product of intege... |
| fproddvdsd 16303 | A finite product of intege... |
| evenelz 16304 | An even number is an integ... |
| zeo3 16305 | An integer is even or odd.... |
| zeo4 16306 | An integer is even or odd ... |
| zeneo 16307 | No even integer equals an ... |
| odd2np1lem 16308 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16309 | An integer is odd iff it i... |
| even2n 16310 | An integer is even iff it ... |
| oddm1even 16311 | An integer is odd iff its ... |
| oddp1even 16312 | An integer is odd iff its ... |
| oexpneg 16313 | The exponential of the neg... |
| mod2eq0even 16314 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16315 | An integer is 1 modulo 2 i... |
| oddnn02np1 16316 | A nonnegative integer is o... |
| oddge22np1 16317 | An integer greater than on... |
| evennn02n 16318 | A nonnegative integer is e... |
| evennn2n 16319 | A positive integer is even... |
| 2tp1odd 16320 | A number which is twice an... |
| mulsucdiv2z 16321 | An integer multiplied with... |
| sqoddm1div8z 16322 | A squared odd number minus... |
| 2teven 16323 | A number which is twice an... |
| zeo5 16324 | An integer is either even ... |
| evend2 16325 | An integer is even iff its... |
| oddp1d2 16326 | An integer is odd iff its ... |
| zob 16327 | Alternate characterization... |
| oddm1d2 16328 | An integer is odd iff its ... |
| ltoddhalfle 16329 | An integer is less than ha... |
| halfleoddlt 16330 | An integer is greater than... |
| opoe 16331 | The sum of two odds is eve... |
| omoe 16332 | The difference of two odds... |
| opeo 16333 | The sum of an odd and an e... |
| omeo 16334 | The difference of an odd a... |
| z0even 16335 | 2 divides 0. That means 0... |
| n2dvds1 16336 | 2 does not divide 1. That... |
| n2dvdsm1 16337 | 2 does not divide -1. Tha... |
| z2even 16338 | 2 divides 2. That means 2... |
| n2dvds3 16339 | 2 does not divide 3. That... |
| z4even 16340 | 2 divides 4. That means 4... |
| 4dvdseven 16341 | An integer which is divisi... |
| m1expe 16342 | Exponentiation of -1 by an... |
| m1expo 16343 | Exponentiation of -1 by an... |
| m1exp1 16344 | Exponentiation of negative... |
| nn0enne 16345 | A positive integer is an e... |
| nn0ehalf 16346 | The half of an even nonneg... |
| nnehalf 16347 | The half of an even positi... |
| nn0onn 16348 | An odd nonnegative integer... |
| nn0o1gt2 16349 | An odd nonnegative integer... |
| nno 16350 | An alternate characterizat... |
| nn0o 16351 | An alternate characterizat... |
| nn0ob 16352 | Alternate characterization... |
| nn0oddm1d2 16353 | A positive integer is odd ... |
| nnoddm1d2 16354 | A positive integer is odd ... |
| sumeven 16355 | If every term in a sum is ... |
| sumodd 16356 | If every term in a sum is ... |
| evensumodd 16357 | If every term in a sum wit... |
| oddsumodd 16358 | If every term in a sum wit... |
| pwp1fsum 16359 | The n-th power of a number... |
| oddpwp1fsum 16360 | An odd power of a number i... |
| divalglem0 16361 | Lemma for ~ divalg . (Con... |
| divalglem1 16362 | Lemma for ~ divalg . (Con... |
| divalglem2 16363 | Lemma for ~ divalg . (Con... |
| divalglem4 16364 | Lemma for ~ divalg . (Con... |
| divalglem5 16365 | Lemma for ~ divalg . (Con... |
| divalglem6 16366 | Lemma for ~ divalg . (Con... |
| divalglem7 16367 | Lemma for ~ divalg . (Con... |
| divalglem8 16368 | Lemma for ~ divalg . (Con... |
| divalglem9 16369 | Lemma for ~ divalg . (Con... |
| divalglem10 16370 | Lemma for ~ divalg . (Con... |
| divalg 16371 | The division algorithm (th... |
| divalgb 16372 | Express the division algor... |
| divalg2 16373 | The division algorithm (th... |
| divalgmod 16374 | The result of the ` mod ` ... |
| divalgmodcl 16375 | The result of the ` mod ` ... |
| modremain 16376 | The result of the modulo o... |
| ndvdssub 16377 | Corollary of the division ... |
| ndvdsadd 16378 | Corollary of the division ... |
| ndvdsp1 16379 | Special case of ~ ndvdsadd... |
| ndvdsi 16380 | A quick test for non-divis... |
| flodddiv4 16381 | The floor of an odd intege... |
| fldivndvdslt 16382 | The floor of an integer di... |
| flodddiv4lt 16383 | The floor of an odd number... |
| flodddiv4t2lthalf 16384 | The floor of an odd number... |
| bitsfval 16389 | Expand the definition of t... |
| bitsval 16390 | Expand the definition of t... |
| bitsval2 16391 | Expand the definition of t... |
| bitsss 16392 | The set of bits of an inte... |
| bitsf 16393 | The ` bits ` function is a... |
| bits0 16394 | Value of the zeroth bit. ... |
| bits0e 16395 | The zeroth bit of an even ... |
| bits0o 16396 | The zeroth bit of an odd n... |
| bitsp1 16397 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16398 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16399 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16400 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16401 | The bits of a number are a... |
| bitsmod 16402 | Truncating the bit sequenc... |
| bitsfi 16403 | Every number is associated... |
| bitscmp 16404 | The bit complement of ` N ... |
| 0bits 16405 | The bits of zero. (Contri... |
| m1bits 16406 | The bits of negative one. ... |
| bitsinv1lem 16407 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16408 | There is an explicit inver... |
| bitsinv2 16409 | There is an explicit inver... |
| bitsf1ocnv 16410 | The ` bits ` function rest... |
| bitsf1o 16411 | The ` bits ` function rest... |
| bitsf1 16412 | The ` bits ` function is a... |
| 2ebits 16413 | The bits of a power of two... |
| bitsinv 16414 | The inverse of the ` bits ... |
| bitsinvp1 16415 | Recursive definition of th... |
| sadadd2lem2 16416 | The core of the proof of ~... |
| sadfval 16418 | Define the addition of two... |
| sadcf 16419 | The carry sequence is a se... |
| sadc0 16420 | The initial element of the... |
| sadcp1 16421 | The carry sequence (which ... |
| sadval 16422 | The full adder sequence is... |
| sadcaddlem 16423 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16424 | Non-recursive definition o... |
| sadadd2lem 16425 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16426 | Sum of initial segments of... |
| sadadd3 16427 | Sum of initial segments of... |
| sadcl 16428 | The sum of two sequences i... |
| sadcom 16429 | The adder sequence functio... |
| saddisjlem 16430 | Lemma for ~ sadadd . (Con... |
| saddisj 16431 | The sum of disjoint sequen... |
| sadaddlem 16432 | Lemma for ~ sadadd . (Con... |
| sadadd 16433 | For sequences that corresp... |
| sadid1 16434 | The adder sequence functio... |
| sadid2 16435 | The adder sequence functio... |
| sadasslem 16436 | Lemma for ~ sadass . (Con... |
| sadass 16437 | Sequence addition is assoc... |
| sadeq 16438 | Any element of a sequence ... |
| bitsres 16439 | Restrict the bits of a num... |
| bitsuz 16440 | The bits of a number are a... |
| bitsshft 16441 | Shifting a bit sequence to... |
| smufval 16443 | The multiplication of two ... |
| smupf 16444 | The sequence of partial su... |
| smup0 16445 | The initial element of the... |
| smupp1 16446 | The initial element of the... |
| smuval 16447 | Define the addition of two... |
| smuval2 16448 | The partial sum sequence s... |
| smupvallem 16449 | If ` A ` only has elements... |
| smucl 16450 | The product of two sequenc... |
| smu01lem 16451 | Lemma for ~ smu01 and ~ sm... |
| smu01 16452 | Multiplication of a sequen... |
| smu02 16453 | Multiplication of a sequen... |
| smupval 16454 | Rewrite the elements of th... |
| smup1 16455 | Rewrite ~ smupp1 using onl... |
| smueqlem 16456 | Any element of a sequence ... |
| smueq 16457 | Any element of a sequence ... |
| smumullem 16458 | Lemma for ~ smumul . (Con... |
| smumul 16459 | For sequences that corresp... |
| gcdval 16462 | The value of the ` gcd ` o... |
| gcd0val 16463 | The value, by convention, ... |
| gcdn0val 16464 | The value of the ` gcd ` o... |
| gcdcllem1 16465 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16466 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16467 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16468 | Closure of the ` gcd ` ope... |
| gcddvds 16469 | The gcd of two integers di... |
| dvdslegcd 16470 | An integer which divides b... |
| nndvdslegcd 16471 | A positive integer which d... |
| gcdcl 16472 | Closure of the ` gcd ` ope... |
| gcdnncl 16473 | Closure of the ` gcd ` ope... |
| gcdcld 16474 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16475 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16476 | An integer is the product ... |
| divgcdz 16477 | An integer divided by the ... |
| gcdf 16478 | Domain and codomain of the... |
| gcdcom 16479 | The ` gcd ` operator is co... |
| gcdcomd 16480 | The ` gcd ` operator is co... |
| divgcdnn 16481 | A positive integer divided... |
| divgcdnnr 16482 | A positive integer divided... |
| gcdeq0 16483 | The gcd of two integers is... |
| gcdn0gt0 16484 | The gcd of two integers is... |
| gcd0id 16485 | The gcd of 0 and an intege... |
| gcdid0 16486 | The gcd of an integer and ... |
| nn0gcdid0 16487 | The gcd of a nonnegative i... |
| gcdneg 16488 | Negating one operand of th... |
| neggcd 16489 | Negating one operand of th... |
| gcdaddmlem 16490 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16491 | Adding a multiple of one o... |
| gcdadd 16492 | The GCD of two numbers is ... |
| gcdid 16493 | The gcd of a number and it... |
| gcd1 16494 | The gcd of a number with 1... |
| gcdabs1 16495 | ` gcd ` of the absolute va... |
| gcdabs2 16496 | ` gcd ` of the absolute va... |
| gcdabs 16497 | The gcd of two integers is... |
| gcdabsOLD 16498 | Obsolete version of ~ gcda... |
| modgcd 16499 | The gcd remains unchanged ... |
| 1gcd 16500 | The GCD of one and an inte... |
| gcdmultipled 16501 | The greatest common diviso... |
| gcdmultiplez 16502 | The GCD of a multiple of a... |
| gcdmultiple 16503 | The GCD of a multiple of a... |
| dvdsgcdidd 16504 | The greatest common diviso... |
| 6gcd4e2 16505 | The greatest common diviso... |
| bezoutlem1 16506 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16507 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16508 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16509 | Lemma for ~ bezout . (Con... |
| bezout 16510 | Bézout's identity: ... |
| dvdsgcd 16511 | An integer which divides e... |
| dvdsgcdb 16512 | Biconditional form of ~ dv... |
| dfgcd2 16513 | Alternate definition of th... |
| gcdass 16514 | Associative law for ` gcd ... |
| mulgcd 16515 | Distribute multiplication ... |
| absmulgcd 16516 | Distribute absolute value ... |
| mulgcdr 16517 | Reverse distribution law f... |
| gcddiv 16518 | Division law for GCD. (Con... |
| gcdzeq 16519 | A positive integer ` A ` i... |
| gcdeq 16520 | ` A ` is equal to its gcd ... |
| dvdssqim 16521 | Unidirectional form of ~ d... |
| dvdsmulgcd 16522 | A divisibility equivalent ... |
| rpmulgcd 16523 | If ` K ` and ` M ` are rel... |
| rplpwr 16524 | If ` A ` and ` B ` are rel... |
| rprpwr 16525 | If ` A ` and ` B ` are rel... |
| rppwr 16526 | If ` A ` and ` B ` are rel... |
| sqgcd 16527 | Square distributes over gc... |
| dvdssqlem 16528 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16529 | Two numbers are divisible ... |
| bezoutr 16530 | Partial converse to ~ bezo... |
| bezoutr1 16531 | Converse of ~ bezout for w... |
| nn0seqcvgd 16532 | A strictly-decreasing nonn... |
| seq1st 16533 | A sequence whose iteration... |
| algr0 16534 | The value of the algorithm... |
| algrf 16535 | An algorithm is a step fun... |
| algrp1 16536 | The value of the algorithm... |
| alginv 16537 | If ` I ` is an invariant o... |
| algcvg 16538 | One way to prove that an a... |
| algcvgblem 16539 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16540 | Two ways of expressing tha... |
| algcvga 16541 | The countdown function ` C... |
| algfx 16542 | If ` F ` reaches a fixed p... |
| eucalgval2 16543 | The value of the step func... |
| eucalgval 16544 | Euclid's Algorithm ~ eucal... |
| eucalgf 16545 | Domain and codomain of the... |
| eucalginv 16546 | The invariant of the step ... |
| eucalglt 16547 | The second member of the s... |
| eucalgcvga 16548 | Once Euclid's Algorithm ha... |
| eucalg 16549 | Euclid's Algorithm compute... |
| lcmval 16554 | Value of the ` lcm ` opera... |
| lcmcom 16555 | The ` lcm ` operator is co... |
| lcm0val 16556 | The value, by convention, ... |
| lcmn0val 16557 | The value of the ` lcm ` o... |
| lcmcllem 16558 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16559 | Closure of the ` lcm ` ope... |
| dvdslcm 16560 | The lcm of two integers is... |
| lcmledvds 16561 | A positive integer which b... |
| lcmeq0 16562 | The lcm of two integers is... |
| lcmcl 16563 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16564 | The greatest common diviso... |
| lcmneg 16565 | Negating one operand of th... |
| neglcm 16566 | Negating one operand of th... |
| lcmabs 16567 | The lcm of two integers is... |
| lcmgcdlem 16568 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16569 | The product of two numbers... |
| lcmdvds 16570 | The lcm of two integers di... |
| lcmid 16571 | The lcm of an integer and ... |
| lcm1 16572 | The lcm of an integer and ... |
| lcmgcdnn 16573 | The product of two positiv... |
| lcmgcdeq 16574 | Two integers' absolute val... |
| lcmdvdsb 16575 | Biconditional form of ~ lc... |
| lcmass 16576 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16577 | The least common multiple ... |
| 6lcm4e12 16578 | The least common multiple ... |
| absproddvds 16579 | The absolute value of the ... |
| absprodnn 16580 | The absolute value of the ... |
| fissn0dvds 16581 | For each finite subset of ... |
| fissn0dvdsn0 16582 | For each finite subset of ... |
| lcmfval 16583 | Value of the ` _lcm ` func... |
| lcmf0val 16584 | The value, by convention, ... |
| lcmfn0val 16585 | The value of the ` _lcm ` ... |
| lcmfnnval 16586 | The value of the ` _lcm ` ... |
| lcmfcllem 16587 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16588 | Closure of the ` _lcm ` fu... |
| lcmfpr 16589 | The value of the ` _lcm ` ... |
| lcmfcl 16590 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16591 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16592 | The least common multiple ... |
| dvdslcmf 16593 | The least common multiple ... |
| lcmfledvds 16594 | A positive integer which i... |
| lcmf 16595 | Characterization of the le... |
| lcmf0 16596 | The least common multiple ... |
| lcmfsn 16597 | The least common multiple ... |
| lcmftp 16598 | The least common multiple ... |
| lcmfunsnlem1 16599 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16600 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16601 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16602 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16603 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16604 | The least common multiple ... |
| lcmfdvdsb 16605 | Biconditional form of ~ lc... |
| lcmfunsn 16606 | The ` _lcm ` function for ... |
| lcmfun 16607 | The ` _lcm ` function for ... |
| lcmfass 16608 | Associative law for the ` ... |
| lcmf2a3a4e12 16609 | The least common multiple ... |
| lcmflefac 16610 | The least common multiple ... |
| coprmgcdb 16611 | Two positive integers are ... |
| ncoprmgcdne1b 16612 | Two positive integers are ... |
| ncoprmgcdgt1b 16613 | Two positive integers are ... |
| coprmdvds1 16614 | If two positive integers a... |
| coprmdvds 16615 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16616 | If an integer is divisible... |
| mulgcddvds 16617 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16618 | If ` M ` is relatively pri... |
| qredeq 16619 | Two equal reduced fraction... |
| qredeu 16620 | Every rational number has ... |
| rpmul 16621 | If ` K ` is relatively pri... |
| rpdvds 16622 | If ` K ` is relatively pri... |
| coprmprod 16623 | The product of the element... |
| coprmproddvdslem 16624 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16625 | If a positive integer is d... |
| congr 16626 | Definition of congruence b... |
| divgcdcoprm0 16627 | Integers divided by gcd ar... |
| divgcdcoprmex 16628 | Integers divided by gcd ar... |
| cncongr1 16629 | One direction of the bicon... |
| cncongr2 16630 | The other direction of the... |
| cncongr 16631 | Cancellability of Congruen... |
| cncongrcoprm 16632 | Corollary 1 of Cancellabil... |
| isprm 16635 | The predicate "is a prime ... |
| prmnn 16636 | A prime number is a positi... |
| prmz 16637 | A prime number is an integ... |
| prmssnn 16638 | The prime numbers are a su... |
| prmex 16639 | The set of prime numbers e... |
| 0nprm 16640 | 0 is not a prime number. ... |
| 1nprm 16641 | 1 is not a prime number. ... |
| 1idssfct 16642 | The positive divisors of a... |
| isprm2lem 16643 | Lemma for ~ isprm2 . (Con... |
| isprm2 16644 | The predicate "is a prime ... |
| isprm3 16645 | The predicate "is a prime ... |
| isprm4 16646 | The predicate "is a prime ... |
| prmind2 16647 | A variation on ~ prmind as... |
| prmind 16648 | Perform induction over the... |
| dvdsprime 16649 | If ` M ` divides a prime, ... |
| nprm 16650 | A product of two integers ... |
| nprmi 16651 | An inference for composite... |
| dvdsnprmd 16652 | If a number is divisible b... |
| prm2orodd 16653 | A prime number is either 2... |
| 2prm 16654 | 2 is a prime number. (Con... |
| 2mulprm 16655 | A multiple of two is prime... |
| 3prm 16656 | 3 is a prime number. (Con... |
| 4nprm 16657 | 4 is not a prime number. ... |
| prmuz2 16658 | A prime number is an integ... |
| prmgt1 16659 | A prime number is an integ... |
| prmm2nn0 16660 | Subtracting 2 from a prime... |
| oddprmgt2 16661 | An odd prime is greater th... |
| oddprmge3 16662 | An odd prime is greater th... |
| ge2nprmge4 16663 | A composite integer greate... |
| sqnprm 16664 | A square is never prime. ... |
| dvdsprm 16665 | An integer greater than or... |
| exprmfct 16666 | Every integer greater than... |
| prmdvdsfz 16667 | Each integer greater than ... |
| nprmdvds1 16668 | No prime number divides 1.... |
| isprm5 16669 | One need only check prime ... |
| isprm7 16670 | One need only check prime ... |
| maxprmfct 16671 | The set of prime factors o... |
| divgcdodd 16672 | Either ` A / ( A gcd B ) `... |
| coprm 16673 | A prime number either divi... |
| prmrp 16674 | Unequal prime numbers are ... |
| euclemma 16675 | Euclid's lemma. A prime n... |
| isprm6 16676 | A number is prime iff it s... |
| prmdvdsexp 16677 | A prime divides a positive... |
| prmdvdsexpb 16678 | A prime divides a positive... |
| prmdvdsexpr 16679 | If a prime divides a nonne... |
| prmdvdssq 16680 | Condition for a prime divi... |
| prmdvdssqOLD 16681 | Obsolete version of ~ prmd... |
| prmexpb 16682 | Two positive prime powers ... |
| prmfac1 16683 | The factorial of a number ... |
| dvdszzq 16684 | Divisibility for an intege... |
| rpexp 16685 | If two numbers ` A ` and `... |
| rpexp1i 16686 | Relative primality passes ... |
| rpexp12i 16687 | Relative primality passes ... |
| prmndvdsfaclt 16688 | A prime number does not di... |
| prmdvdsbc 16689 | Condition for a prime numb... |
| prmdvdsncoprmbd 16690 | Two positive integers are ... |
| ncoprmlnprm 16691 | If two positive integers a... |
| cncongrprm 16692 | Corollary 2 of Cancellabil... |
| isevengcd2 16693 | The predicate "is an even ... |
| isoddgcd1 16694 | The predicate "is an odd n... |
| 3lcm2e6 16695 | The least common multiple ... |
| qnumval 16700 | Value of the canonical num... |
| qdenval 16701 | Value of the canonical den... |
| qnumdencl 16702 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16703 | The canonical numerator of... |
| qdencl 16704 | The canonical denominator ... |
| fnum 16705 | Canonical numerator define... |
| fden 16706 | Canonical denominator defi... |
| qnumdenbi 16707 | Two numbers are the canoni... |
| qnumdencoprm 16708 | The canonical representati... |
| qeqnumdivden 16709 | Recover a rational number ... |
| qmuldeneqnum 16710 | Multiplying a rational by ... |
| divnumden 16711 | Calculate the reduced form... |
| divdenle 16712 | Reducing a quotient never ... |
| qnumgt0 16713 | A rational is positive iff... |
| qgt0numnn 16714 | A rational is positive iff... |
| nn0gcdsq 16715 | Squaring commutes with GCD... |
| zgcdsq 16716 | ~ nn0gcdsq extended to int... |
| numdensq 16717 | Squaring a rational square... |
| numsq 16718 | Square commutes with canon... |
| densq 16719 | Square commutes with canon... |
| qden1elz 16720 | A rational is an integer i... |
| zsqrtelqelz 16721 | If an integer has a ration... |
| nonsq 16722 | Any integer strictly betwe... |
| phival 16727 | Value of the Euler ` phi `... |
| phicl2 16728 | Bounds and closure for the... |
| phicl 16729 | Closure for the value of t... |
| phibndlem 16730 | Lemma for ~ phibnd . (Con... |
| phibnd 16731 | A slightly tighter bound o... |
| phicld 16732 | Closure for the value of t... |
| phi1 16733 | Value of the Euler ` phi `... |
| dfphi2 16734 | Alternate definition of th... |
| hashdvds 16735 | The number of numbers in a... |
| phiprmpw 16736 | Value of the Euler ` phi `... |
| phiprm 16737 | Value of the Euler ` phi `... |
| crth 16738 | The Chinese Remainder Theo... |
| phimullem 16739 | Lemma for ~ phimul . (Con... |
| phimul 16740 | The Euler ` phi ` function... |
| eulerthlem1 16741 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16742 | Lemma for ~ eulerth . (Co... |
| eulerth 16743 | Euler's theorem, a general... |
| fermltl 16744 | Fermat's little theorem. ... |
| prmdiv 16745 | Show an explicit expressio... |
| prmdiveq 16746 | The modular inverse of ` A... |
| prmdivdiv 16747 | The (modular) inverse of t... |
| hashgcdlem 16748 | A correspondence between e... |
| hashgcdeq 16749 | Number of initial positive... |
| phisum 16750 | The divisor sum identity o... |
| odzval 16751 | Value of the order functio... |
| odzcllem 16752 | - Lemma for ~ odzcl , show... |
| odzcl 16753 | The order of a group eleme... |
| odzid 16754 | Any element raised to the ... |
| odzdvds 16755 | The only powers of ` A ` t... |
| odzphi 16756 | The order of any group ele... |
| modprm1div 16757 | A prime number divides an ... |
| m1dvdsndvds 16758 | If an integer minus 1 is d... |
| modprminv 16759 | Show an explicit expressio... |
| modprminveq 16760 | The modular inverse of ` A... |
| vfermltl 16761 | Variant of Fermat's little... |
| vfermltlALT 16762 | Alternate proof of ~ vferm... |
| powm2modprm 16763 | If an integer minus 1 is d... |
| reumodprminv 16764 | For any prime number and f... |
| modprm0 16765 | For two positive integers ... |
| nnnn0modprm0 16766 | For a positive integer and... |
| modprmn0modprm0 16767 | For an integer not being 0... |
| coprimeprodsq 16768 | If three numbers are copri... |
| coprimeprodsq2 16769 | If three numbers are copri... |
| oddprm 16770 | A prime not equal to ` 2 `... |
| nnoddn2prm 16771 | A prime not equal to ` 2 `... |
| oddn2prm 16772 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16773 | A number is a prime number... |
| prm23lt5 16774 | A prime less than 5 is eit... |
| prm23ge5 16775 | A prime is either 2 or 3 o... |
| pythagtriplem1 16776 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16777 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16778 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16779 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16780 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16781 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16782 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16783 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16784 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16785 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16786 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16787 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16788 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16789 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16790 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16791 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16792 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16793 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16794 | Parameterize the Pythagore... |
| iserodd 16795 | Collect the odd terms in a... |
| pclem 16798 | - Lemma for the prime powe... |
| pcprecl 16799 | Closure of the prime power... |
| pcprendvds 16800 | Non-divisibility property ... |
| pcprendvds2 16801 | Non-divisibility property ... |
| pcpre1 16802 | Value of the prime power p... |
| pcpremul 16803 | Multiplicative property of... |
| pcval 16804 | The value of the prime pow... |
| pceulem 16805 | Lemma for ~ pceu . (Contr... |
| pceu 16806 | Uniqueness for the prime p... |
| pczpre 16807 | Connect the prime count pr... |
| pczcl 16808 | Closure of the prime power... |
| pccl 16809 | Closure of the prime power... |
| pccld 16810 | Closure of the prime power... |
| pcmul 16811 | Multiplication property of... |
| pcdiv 16812 | Division property of the p... |
| pcqmul 16813 | Multiplication property of... |
| pc0 16814 | The value of the prime pow... |
| pc1 16815 | Value of the prime count f... |
| pcqcl 16816 | Closure of the general pri... |
| pcqdiv 16817 | Division property of the p... |
| pcrec 16818 | Prime power of a reciproca... |
| pcexp 16819 | Prime power of an exponent... |
| pcxnn0cl 16820 | Extended nonnegative integ... |
| pcxcl 16821 | Extended real closure of t... |
| pcge0 16822 | The prime count of an inte... |
| pczdvds 16823 | Defining property of the p... |
| pcdvds 16824 | Defining property of the p... |
| pczndvds 16825 | Defining property of the p... |
| pcndvds 16826 | Defining property of the p... |
| pczndvds2 16827 | The remainder after dividi... |
| pcndvds2 16828 | The remainder after dividi... |
| pcdvdsb 16829 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16830 | There are a positive numbe... |
| pceq0 16831 | There are zero powers of a... |
| pcidlem 16832 | The prime count of a prime... |
| pcid 16833 | The prime count of a prime... |
| pcneg 16834 | The prime count of a negat... |
| pcabs 16835 | The prime count of an abso... |
| pcdvdstr 16836 | The prime count increases ... |
| pcgcd1 16837 | The prime count of a GCD i... |
| pcgcd 16838 | The prime count of a GCD i... |
| pc2dvds 16839 | A characterization of divi... |
| pc11 16840 | The prime count function, ... |
| pcz 16841 | The prime count function c... |
| pcprmpw2 16842 | Self-referential expressio... |
| pcprmpw 16843 | Self-referential expressio... |
| dvdsprmpweq 16844 | If a positive integer divi... |
| dvdsprmpweqnn 16845 | If an integer greater than... |
| dvdsprmpweqle 16846 | If a positive integer divi... |
| difsqpwdvds 16847 | If the difference of two s... |
| pcaddlem 16848 | Lemma for ~ pcadd . The o... |
| pcadd 16849 | An inequality for the prim... |
| pcadd2 16850 | The inequality of ~ pcadd ... |
| pcmptcl 16851 | Closure for the prime powe... |
| pcmpt 16852 | Construct a function with ... |
| pcmpt2 16853 | Dividing two prime count m... |
| pcmptdvds 16854 | The partial products of th... |
| pcprod 16855 | The product of the primes ... |
| sumhash 16856 | The sum of 1 over a set is... |
| fldivp1 16857 | The difference between the... |
| pcfaclem 16858 | Lemma for ~ pcfac . (Cont... |
| pcfac 16859 | Calculate the prime count ... |
| pcbc 16860 | Calculate the prime count ... |
| qexpz 16861 | If a power of a rational n... |
| expnprm 16862 | A second or higher power o... |
| oddprmdvds 16863 | Every positive integer whi... |
| prmpwdvds 16864 | A relation involving divis... |
| pockthlem 16865 | Lemma for ~ pockthg . (Co... |
| pockthg 16866 | The generalized Pocklingto... |
| pockthi 16867 | Pocklington's theorem, whi... |
| unbenlem 16868 | Lemma for ~ unben . (Cont... |
| unben 16869 | An unbounded set of positi... |
| infpnlem1 16870 | Lemma for ~ infpn . The s... |
| infpnlem2 16871 | Lemma for ~ infpn . For a... |
| infpn 16872 | There exist infinitely man... |
| infpn2 16873 | There exist infinitely man... |
| prmunb 16874 | The primes are unbounded. ... |
| prminf 16875 | There are an infinite numb... |
| prmreclem1 16876 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16877 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16878 | Lemma for ~ prmrec . The ... |
| prmreclem4 16879 | Lemma for ~ prmrec . Show... |
| prmreclem5 16880 | Lemma for ~ prmrec . Here... |
| prmreclem6 16881 | Lemma for ~ prmrec . If t... |
| prmrec 16882 | The sum of the reciprocals... |
| 1arithlem1 16883 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16884 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16885 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16886 | Lemma for ~ 1arith . (Con... |
| 1arith 16887 | Fundamental theorem of ari... |
| 1arith2 16888 | Fundamental theorem of ari... |
| elgz 16891 | Elementhood in the gaussia... |
| gzcn 16892 | A gaussian integer is a co... |
| zgz 16893 | An integer is a gaussian i... |
| igz 16894 | ` _i ` is a gaussian integ... |
| gznegcl 16895 | The gaussian integers are ... |
| gzcjcl 16896 | The gaussian integers are ... |
| gzaddcl 16897 | The gaussian integers are ... |
| gzmulcl 16898 | The gaussian integers are ... |
| gzreim 16899 | Construct a gaussian integ... |
| gzsubcl 16900 | The gaussian integers are ... |
| gzabssqcl 16901 | The squared norm of a gaus... |
| 4sqlem5 16902 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16903 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16904 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16905 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16906 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16907 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16908 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16909 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16910 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16911 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16912 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16913 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16914 | Euler's four-square identi... |
| 4sqlem11 16915 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16916 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16917 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16918 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16919 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16920 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16921 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16922 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16923 | Lemma for ~ 4sq . The pro... |
| 4sq 16924 | Lagrange's four-square the... |
| vdwapfval 16931 | Define the arithmetic prog... |
| vdwapf 16932 | The arithmetic progression... |
| vdwapval 16933 | Value of the arithmetic pr... |
| vdwapun 16934 | Remove the first element o... |
| vdwapid1 16935 | The first element of an ar... |
| vdwap0 16936 | Value of a length-1 arithm... |
| vdwap1 16937 | Value of a length-1 arithm... |
| vdwmc 16938 | The predicate " The ` <. R... |
| vdwmc2 16939 | Expand out the definition ... |
| vdwpc 16940 | The predicate " The colori... |
| vdwlem1 16941 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16942 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16943 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16944 | Lemma for ~ vdw . (Contri... |
| vdwlem5 16945 | Lemma for ~ vdw . (Contri... |
| vdwlem6 16946 | Lemma for ~ vdw . (Contri... |
| vdwlem7 16947 | Lemma for ~ vdw . (Contri... |
| vdwlem8 16948 | Lemma for ~ vdw . (Contri... |
| vdwlem9 16949 | Lemma for ~ vdw . (Contri... |
| vdwlem10 16950 | Lemma for ~ vdw . Set up ... |
| vdwlem11 16951 | Lemma for ~ vdw . (Contri... |
| vdwlem12 16952 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 16953 | Lemma for ~ vdw . Main in... |
| vdw 16954 | Van der Waerden's theorem.... |
| vdwnnlem1 16955 | Corollary of ~ vdw , and l... |
| vdwnnlem2 16956 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 16957 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 16958 | Van der Waerden's theorem,... |
| ramtlecl 16960 | The set ` T ` of numbers w... |
| hashbcval 16962 | Value of the "binomial set... |
| hashbccl 16963 | The binomial set is a fini... |
| hashbcss 16964 | Subset relation for the bi... |
| hashbc0 16965 | The set of subsets of size... |
| hashbc2 16966 | The size of the binomial s... |
| 0hashbc 16967 | There are no subsets of th... |
| ramval 16968 | The value of the Ramsey nu... |
| ramcl2lem 16969 | Lemma for extended real cl... |
| ramtcl 16970 | The Ramsey number has the ... |
| ramtcl2 16971 | The Ramsey number is an in... |
| ramtub 16972 | The Ramsey number is a low... |
| ramub 16973 | The Ramsey number is a low... |
| ramub2 16974 | It is sufficient to check ... |
| rami 16975 | The defining property of a... |
| ramcl2 16976 | The Ramsey number is eithe... |
| ramxrcl 16977 | The Ramsey number is an ex... |
| ramubcl 16978 | If the Ramsey number is up... |
| ramlb 16979 | Establish a lower bound on... |
| 0ram 16980 | The Ramsey number when ` M... |
| 0ram2 16981 | The Ramsey number when ` M... |
| ram0 16982 | The Ramsey number when ` R... |
| 0ramcl 16983 | Lemma for ~ ramcl : Exist... |
| ramz2 16984 | The Ramsey number when ` F... |
| ramz 16985 | The Ramsey number when ` F... |
| ramub1lem1 16986 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 16987 | Lemma for ~ ramub1 . (Con... |
| ramub1 16988 | Inductive step for Ramsey'... |
| ramcl 16989 | Ramsey's theorem: the Rams... |
| ramsey 16990 | Ramsey's theorem with the ... |
| prmoval 16993 | Value of the primorial fun... |
| prmocl 16994 | Closure of the primorial f... |
| prmone0 16995 | The primorial function is ... |
| prmo0 16996 | The primorial of 0. (Cont... |
| prmo1 16997 | The primorial of 1. (Cont... |
| prmop1 16998 | The primorial of a success... |
| prmonn2 16999 | Value of the primorial fun... |
| prmo2 17000 | The primorial of 2. (Cont... |
| prmo3 17001 | The primorial of 3. (Cont... |
| prmdvdsprmo 17002 | The primorial of a number ... |
| prmdvdsprmop 17003 | The primorial of a number ... |
| fvprmselelfz 17004 | The value of the prime sel... |
| fvprmselgcd1 17005 | The greatest common diviso... |
| prmolefac 17006 | The primorial of a positiv... |
| prmodvdslcmf 17007 | The primorial of a nonnega... |
| prmolelcmf 17008 | The primorial of a positiv... |
| prmgaplem1 17009 | Lemma for ~ prmgap : The ... |
| prmgaplem2 17010 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 17011 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 17012 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 17013 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 17014 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 17015 | Lemma for ~ prmgap : for e... |
| prmgaplem6 17016 | Lemma for ~ prmgap : for e... |
| prmgaplem7 17017 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 17018 | Lemma for ~ prmgap . (Con... |
| prmgap 17019 | The prime gap theorem: for... |
| prmgaplcm 17020 | Alternate proof of ~ prmga... |
| prmgapprmolem 17021 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 17022 | Alternate proof of ~ prmga... |
| dec2dvds 17023 | Divisibility by two is obv... |
| dec5dvds 17024 | Divisibility by five is ob... |
| dec5dvds2 17025 | Divisibility by five is ob... |
| dec5nprm 17026 | Divisibility by five is ob... |
| dec2nprm 17027 | Divisibility by two is obv... |
| modxai 17028 | Add exponents in a power m... |
| mod2xi 17029 | Double exponents in a powe... |
| modxp1i 17030 | Add one to an exponent in ... |
| mod2xnegi 17031 | Version of ~ mod2xi with a... |
| modsubi 17032 | Subtract from within a mod... |
| gcdi 17033 | Calculate a GCD via Euclid... |
| gcdmodi 17034 | Calculate a GCD via Euclid... |
| decexp2 17035 | Calculate a power of two. ... |
| numexp0 17036 | Calculate an integer power... |
| numexp1 17037 | Calculate an integer power... |
| numexpp1 17038 | Calculate an integer power... |
| numexp2x 17039 | Double an integer power. ... |
| decsplit0b 17040 | Split a decimal number int... |
| decsplit0 17041 | Split a decimal number int... |
| decsplit1 17042 | Split a decimal number int... |
| decsplit 17043 | Split a decimal number int... |
| karatsuba 17044 | The Karatsuba multiplicati... |
| 2exp4 17045 | Two to the fourth power is... |
| 2exp5 17046 | Two to the fifth power is ... |
| 2exp6 17047 | Two to the sixth power is ... |
| 2exp7 17048 | Two to the seventh power i... |
| 2exp8 17049 | Two to the eighth power is... |
| 2exp11 17050 | Two to the eleventh power ... |
| 2exp16 17051 | Two to the sixteenth power... |
| 3exp3 17052 | Three to the third power i... |
| 2expltfac 17053 | The factorial grows faster... |
| cshwsidrepsw 17054 | If cyclically shifting a w... |
| cshwsidrepswmod0 17055 | If cyclically shifting a w... |
| cshwshashlem1 17056 | If cyclically shifting a w... |
| cshwshashlem2 17057 | If cyclically shifting a w... |
| cshwshashlem3 17058 | If cyclically shifting a w... |
| cshwsdisj 17059 | The singletons resulting b... |
| cshwsiun 17060 | The set of (different!) wo... |
| cshwsex 17061 | The class of (different!) ... |
| cshws0 17062 | The size of the set of (di... |
| cshwrepswhash1 17063 | The size of the set of (di... |
| cshwshashnsame 17064 | If a word (not consisting ... |
| cshwshash 17065 | If a word has a length bei... |
| prmlem0 17066 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 17067 | A quick proof skeleton to ... |
| prmlem1 17068 | A quick proof skeleton to ... |
| 5prm 17069 | 5 is a prime number. (Con... |
| 6nprm 17070 | 6 is not a prime number. ... |
| 7prm 17071 | 7 is a prime number. (Con... |
| 8nprm 17072 | 8 is not a prime number. ... |
| 9nprm 17073 | 9 is not a prime number. ... |
| 10nprm 17074 | 10 is not a prime number. ... |
| 11prm 17075 | 11 is a prime number. (Co... |
| 13prm 17076 | 13 is a prime number. (Co... |
| 17prm 17077 | 17 is a prime number. (Co... |
| 19prm 17078 | 19 is a prime number. (Co... |
| 23prm 17079 | 23 is a prime number. (Co... |
| prmlem2 17080 | Our last proving session g... |
| 37prm 17081 | 37 is a prime number. (Co... |
| 43prm 17082 | 43 is a prime number. (Co... |
| 83prm 17083 | 83 is a prime number. (Co... |
| 139prm 17084 | 139 is a prime number. (C... |
| 163prm 17085 | 163 is a prime number. (C... |
| 317prm 17086 | 317 is a prime number. (C... |
| 631prm 17087 | 631 is a prime number. (C... |
| prmo4 17088 | The primorial of 4. (Cont... |
| prmo5 17089 | The primorial of 5. (Cont... |
| prmo6 17090 | The primorial of 6. (Cont... |
| 1259lem1 17091 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17092 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17093 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17094 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17095 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17096 | 1259 is a prime number. (... |
| 2503lem1 17097 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17098 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17099 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17100 | 2503 is a prime number. (... |
| 4001lem1 17101 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17102 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17103 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17104 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17105 | 4001 is a prime number. (... |
| brstruct 17108 | The structure relation is ... |
| isstruct2 17109 | The property of being a st... |
| structex 17110 | A structure is a set. (Co... |
| structn0fun 17111 | A structure without the em... |
| isstruct 17112 | The property of being a st... |
| structcnvcnv 17113 | Two ways to express the re... |
| structfung 17114 | The converse of the conver... |
| structfun 17115 | Convert between two kinds ... |
| structfn 17116 | Convert between two kinds ... |
| strleun 17117 | Combine two structures int... |
| strle1 17118 | Make a structure from a si... |
| strle2 17119 | Make a structure from a pa... |
| strle3 17120 | Make a structure from a tr... |
| sbcie2s 17121 | A special version of class... |
| sbcie3s 17122 | A special version of class... |
| reldmsets 17125 | The structure override ope... |
| setsvalg 17126 | Value of the structure rep... |
| setsval 17127 | Value of the structure rep... |
| fvsetsid 17128 | The value of the structure... |
| fsets 17129 | The structure replacement ... |
| setsdm 17130 | The domain of a structure ... |
| setsfun 17131 | A structure with replaceme... |
| setsfun0 17132 | A structure with replaceme... |
| setsn0fun 17133 | The value of the structure... |
| setsstruct2 17134 | An extensible structure wi... |
| setsexstruct2 17135 | An extensible structure wi... |
| setsstruct 17136 | An extensible structure wi... |
| wunsets 17137 | Closure of structure repla... |
| setsres 17138 | The structure replacement ... |
| setsabs 17139 | Replacing the same compone... |
| setscom 17140 | Different components can b... |
| sloteq 17143 | Equality theorem for the `... |
| slotfn 17144 | A slot is a function on se... |
| strfvnd 17145 | Deduction version of ~ str... |
| strfvn 17146 | Value of a structure compo... |
| strfvss 17147 | A structure component extr... |
| wunstr 17148 | Closure of a structure ind... |
| str0 17149 | All components of the empt... |
| strfvi 17150 | Structure slot extractors ... |
| fveqprc 17151 | Lemma for showing the equa... |
| oveqprc 17152 | Lemma for showing the equa... |
| wunndx 17155 | Closure of the index extra... |
| ndxarg 17156 | Get the numeric argument f... |
| ndxid 17157 | A structure component extr... |
| strndxid 17158 | The value of a structure c... |
| setsidvald 17159 | Value of the structure rep... |
| setsidvaldOLD 17160 | Obsolete version of ~ sets... |
| strfvd 17161 | Deduction version of ~ str... |
| strfv2d 17162 | Deduction version of ~ str... |
| strfv2 17163 | A variation on ~ strfv to ... |
| strfv 17164 | Extract a structure compon... |
| strfv3 17165 | Variant on ~ strfv for lar... |
| strssd 17166 | Deduction version of ~ str... |
| strss 17167 | Propagate component extrac... |
| setsid 17168 | Value of the structure rep... |
| setsnid 17169 | Value of the structure rep... |
| setsnidOLD 17170 | Obsolete proof of ~ setsni... |
| baseval 17173 | Value of the base set extr... |
| baseid 17174 | Utility theorem: index-ind... |
| basfn 17175 | The base set extractor is ... |
| base0 17176 | The base set of the empty ... |
| elbasfv 17177 | Utility theorem: reverse c... |
| elbasov 17178 | Utility theorem: reverse c... |
| strov2rcl 17179 | Partial reverse closure fo... |
| basendx 17180 | Index value of the base se... |
| basendxnn 17181 | The index value of the bas... |
| basendxnnOLD 17182 | Obsolete proof of ~ basend... |
| basndxelwund 17183 | The index of the base set ... |
| basprssdmsets 17184 | The pair of the base index... |
| opelstrbas 17185 | The base set of a structur... |
| 1strstr 17186 | A constructed one-slot str... |
| 1strstr1 17187 | A constructed one-slot str... |
| 1strbas 17188 | The base set of a construc... |
| 1strbasOLD 17189 | Obsolete proof of ~ 1strba... |
| 1strwunbndx 17190 | A constructed one-slot str... |
| 1strwun 17191 | A constructed one-slot str... |
| 1strwunOLD 17192 | Obsolete version of ~ 1str... |
| 2strstr 17193 | A constructed two-slot str... |
| 2strbas 17194 | The base set of a construc... |
| 2strop 17195 | The other slot of a constr... |
| 2strstr1 17196 | A constructed two-slot str... |
| 2strstr1OLD 17197 | Obsolete version of ~ 2str... |
| 2strbas1 17198 | The base set of a construc... |
| 2strop1 17199 | The other slot of a constr... |
| reldmress 17202 | The structure restriction ... |
| ressval 17203 | Value of structure restric... |
| ressid2 17204 | General behavior of trivia... |
| ressval2 17205 | Value of nontrivial struct... |
| ressbas 17206 | Base set of a structure re... |
| ressbasOLD 17207 | Obsolete proof of ~ ressba... |
| ressbasssg 17208 | The base set of a restrict... |
| ressbas2 17209 | Base set of a structure re... |
| ressbasss 17210 | The base set of a restrict... |
| ressbasssOLD 17211 | Obsolete proof of ~ ressba... |
| ressbasss2 17212 | The base set of a restrict... |
| resseqnbas 17213 | The components of an exten... |
| resslemOLD 17214 | Obsolete version of ~ ress... |
| ress0 17215 | All restrictions of the nu... |
| ressid 17216 | Behavior of trivial restri... |
| ressinbas 17217 | Restriction only cares abo... |
| ressval3d 17218 | Value of structure restric... |
| ressval3dOLD 17219 | Obsolete version of ~ ress... |
| ressress 17220 | Restriction composition la... |
| ressabs 17221 | Restriction absorption law... |
| wunress 17222 | Closure of structure restr... |
| wunressOLD 17223 | Obsolete proof of ~ wunres... |
| plusgndx 17250 | Index value of the ~ df-pl... |
| plusgid 17251 | Utility theorem: index-ind... |
| plusgndxnn 17252 | The index of the slot for ... |
| basendxltplusgndx 17253 | The index of the slot for ... |
| basendxnplusgndx 17254 | The slot for the base set ... |
| basendxnplusgndxOLD 17255 | Obsolete version of ~ base... |
| grpstr 17256 | A constructed group is a s... |
| grpstrndx 17257 | A constructed group is a s... |
| grpbase 17258 | The base set of a construc... |
| grpbaseOLD 17259 | Obsolete version of ~ grpb... |
| grpplusg 17260 | The operation of a constru... |
| grpplusgOLD 17261 | Obsolete version of ~ grpp... |
| ressplusg 17262 | ` +g ` is unaffected by re... |
| grpbasex 17263 | The base of an explicitly ... |
| grpplusgx 17264 | The operation of an explic... |
| mulrndx 17265 | Index value of the ~ df-mu... |
| mulridx 17266 | Utility theorem: index-ind... |
| basendxnmulrndx 17267 | The slot for the base set ... |
| basendxnmulrndxOLD 17268 | Obsolete proof of ~ basend... |
| plusgndxnmulrndx 17269 | The slot for the group (ad... |
| rngstr 17270 | A constructed ring is a st... |
| rngbase 17271 | The base set of a construc... |
| rngplusg 17272 | The additive operation of ... |
| rngmulr 17273 | The multiplicative operati... |
| starvndx 17274 | Index value of the ~ df-st... |
| starvid 17275 | Utility theorem: index-ind... |
| starvndxnbasendx 17276 | The slot for the involutio... |
| starvndxnplusgndx 17277 | The slot for the involutio... |
| starvndxnmulrndx 17278 | The slot for the involutio... |
| ressmulr 17279 | ` .r ` is unaffected by re... |
| ressstarv 17280 | ` *r ` is unaffected by re... |
| srngstr 17281 | A constructed star ring is... |
| srngbase 17282 | The base set of a construc... |
| srngplusg 17283 | The addition operation of ... |
| srngmulr 17284 | The multiplication operati... |
| srnginvl 17285 | The involution function of... |
| scandx 17286 | Index value of the ~ df-sc... |
| scaid 17287 | Utility theorem: index-ind... |
| scandxnbasendx 17288 | The slot for the scalar is... |
| scandxnplusgndx 17289 | The slot for the scalar fi... |
| scandxnmulrndx 17290 | The slot for the scalar fi... |
| vscandx 17291 | Index value of the ~ df-vs... |
| vscaid 17292 | Utility theorem: index-ind... |
| vscandxnbasendx 17293 | The slot for the scalar pr... |
| vscandxnplusgndx 17294 | The slot for the scalar pr... |
| vscandxnmulrndx 17295 | The slot for the scalar pr... |
| vscandxnscandx 17296 | The slot for the scalar pr... |
| lmodstr 17297 | A constructed left module ... |
| lmodbase 17298 | The base set of a construc... |
| lmodplusg 17299 | The additive operation of ... |
| lmodsca 17300 | The set of scalars of a co... |
| lmodvsca 17301 | The scalar product operati... |
| ipndx 17302 | Index value of the ~ df-ip... |
| ipid 17303 | Utility theorem: index-ind... |
| ipndxnbasendx 17304 | The slot for the inner pro... |
| ipndxnplusgndx 17305 | The slot for the inner pro... |
| ipndxnmulrndx 17306 | The slot for the inner pro... |
| slotsdifipndx 17307 | The slot for the scalar is... |
| ipsstr 17308 | Lemma to shorten proofs of... |
| ipsbase 17309 | The base set of a construc... |
| ipsaddg 17310 | The additive operation of ... |
| ipsmulr 17311 | The multiplicative operati... |
| ipssca 17312 | The set of scalars of a co... |
| ipsvsca 17313 | The scalar product operati... |
| ipsip 17314 | The multiplicative operati... |
| resssca 17315 | ` Scalar ` is unaffected b... |
| ressvsca 17316 | ` .s ` is unaffected by re... |
| ressip 17317 | The inner product is unaff... |
| phlstr 17318 | A constructed pre-Hilbert ... |
| phlbase 17319 | The base set of a construc... |
| phlplusg 17320 | The additive operation of ... |
| phlsca 17321 | The ring of scalars of a c... |
| phlvsca 17322 | The scalar product operati... |
| phlip 17323 | The inner product (Hermiti... |
| tsetndx 17324 | Index value of the ~ df-ts... |
| tsetid 17325 | Utility theorem: index-ind... |
| tsetndxnn 17326 | The index of the slot for ... |
| basendxlttsetndx 17327 | The index of the slot for ... |
| tsetndxnbasendx 17328 | The slot for the topology ... |
| tsetndxnplusgndx 17329 | The slot for the topology ... |
| tsetndxnmulrndx 17330 | The slot for the topology ... |
| tsetndxnstarvndx 17331 | The slot for the topology ... |
| slotstnscsi 17332 | The slots ` Scalar ` , ` .... |
| topgrpstr 17333 | A constructed topological ... |
| topgrpbas 17334 | The base set of a construc... |
| topgrpplusg 17335 | The additive operation of ... |
| topgrptset 17336 | The topology of a construc... |
| resstset 17337 | ` TopSet ` is unaffected b... |
| plendx 17338 | Index value of the ~ df-pl... |
| pleid 17339 | Utility theorem: self-refe... |
| plendxnn 17340 | The index value of the ord... |
| basendxltplendx 17341 | The index value of the ` B... |
| plendxnbasendx 17342 | The slot for the order is ... |
| plendxnplusgndx 17343 | The slot for the "less tha... |
| plendxnmulrndx 17344 | The slot for the "less tha... |
| plendxnscandx 17345 | The slot for the "less tha... |
| plendxnvscandx 17346 | The slot for the "less tha... |
| slotsdifplendx 17347 | The index of the slot for ... |
| otpsstr 17348 | Functionality of a topolog... |
| otpsbas 17349 | The base set of a topologi... |
| otpstset 17350 | The open sets of a topolog... |
| otpsle 17351 | The order of a topological... |
| ressle 17352 | ` le ` is unaffected by re... |
| ocndx 17353 | Index value of the ~ df-oc... |
| ocid 17354 | Utility theorem: index-ind... |
| basendxnocndx 17355 | The slot for the orthocomp... |
| plendxnocndx 17356 | The slot for the orthocomp... |
| dsndx 17357 | Index value of the ~ df-ds... |
| dsid 17358 | Utility theorem: index-ind... |
| dsndxnn 17359 | The index of the slot for ... |
| basendxltdsndx 17360 | The index of the slot for ... |
| dsndxnbasendx 17361 | The slot for the distance ... |
| dsndxnplusgndx 17362 | The slot for the distance ... |
| dsndxnmulrndx 17363 | The slot for the distance ... |
| slotsdnscsi 17364 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17365 | The slot for the distance ... |
| slotsdifdsndx 17366 | The index of the slot for ... |
| unifndx 17367 | Index value of the ~ df-un... |
| unifid 17368 | Utility theorem: index-ind... |
| unifndxnn 17369 | The index of the slot for ... |
| basendxltunifndx 17370 | The index of the slot for ... |
| unifndxnbasendx 17371 | The slot for the uniform s... |
| unifndxntsetndx 17372 | The slot for the uniform s... |
| slotsdifunifndx 17373 | The index of the slot for ... |
| ressunif 17374 | ` UnifSet ` is unaffected ... |
| odrngstr 17375 | Functionality of an ordere... |
| odrngbas 17376 | The base set of an ordered... |
| odrngplusg 17377 | The addition operation of ... |
| odrngmulr 17378 | The multiplication operati... |
| odrngtset 17379 | The open sets of an ordere... |
| odrngle 17380 | The order of an ordered me... |
| odrngds 17381 | The metric of an ordered m... |
| ressds 17382 | ` dist ` is unaffected by ... |
| homndx 17383 | Index value of the ~ df-ho... |
| homid 17384 | Utility theorem: index-ind... |
| ccondx 17385 | Index value of the ~ df-cc... |
| ccoid 17386 | Utility theorem: index-ind... |
| slotsbhcdif 17387 | The slots ` Base ` , ` Hom... |
| slotsbhcdifOLD 17388 | Obsolete proof of ~ slotsb... |
| slotsdifplendx2 17389 | The index of the slot for ... |
| slotsdifocndx 17390 | The index of the slot for ... |
| resshom 17391 | ` Hom ` is unaffected by r... |
| ressco 17392 | ` comp ` is unaffected by ... |
| restfn 17397 | The subspace topology oper... |
| topnfn 17398 | The topology extractor fun... |
| restval 17399 | The subspace topology indu... |
| elrest 17400 | The predicate "is an open ... |
| elrestr 17401 | Sufficient condition for b... |
| 0rest 17402 | Value of the structure res... |
| restid2 17403 | The subspace topology over... |
| restsspw 17404 | The subspace topology is a... |
| firest 17405 | The finite intersections o... |
| restid 17406 | The subspace topology of t... |
| topnval 17407 | Value of the topology extr... |
| topnid 17408 | Value of the topology extr... |
| topnpropd 17409 | The topology extractor fun... |
| reldmprds 17421 | The structure product is a... |
| prdsbasex 17423 | Lemma for structure produc... |
| imasvalstr 17424 | An image structure value i... |
| prdsvalstr 17425 | Structure product value is... |
| prdsbaslem 17426 | Lemma for ~ prdsbas and si... |
| prdsvallem 17427 | Lemma for ~ prdsval . (Co... |
| prdsval 17428 | Value of the structure pro... |
| prdssca 17429 | Scalar ring of a structure... |
| prdsbas 17430 | Base set of a structure pr... |
| prdsplusg 17431 | Addition in a structure pr... |
| prdsmulr 17432 | Multiplication in a struct... |
| prdsvsca 17433 | Scalar multiplication in a... |
| prdsip 17434 | Inner product in a structu... |
| prdsle 17435 | Structure product weak ord... |
| prdsless 17436 | Closure of the order relat... |
| prdsds 17437 | Structure product distance... |
| prdsdsfn 17438 | Structure product distance... |
| prdstset 17439 | Structure product topology... |
| prdshom 17440 | Structure product hom-sets... |
| prdsco 17441 | Structure product composit... |
| prdsbas2 17442 | The base set of a structur... |
| prdsbasmpt 17443 | A constructed tuple is a p... |
| prdsbasfn 17444 | Points in the structure pr... |
| prdsbasprj 17445 | Each point in a structure ... |
| prdsplusgval 17446 | Value of a componentwise s... |
| prdsplusgfval 17447 | Value of a structure produ... |
| prdsmulrval 17448 | Value of a componentwise r... |
| prdsmulrfval 17449 | Value of a structure produ... |
| prdsleval 17450 | Value of the product order... |
| prdsdsval 17451 | Value of the metric in a s... |
| prdsvscaval 17452 | Scalar multiplication in a... |
| prdsvscafval 17453 | Scalar multiplication of a... |
| prdsbas3 17454 | The base set of an indexed... |
| prdsbasmpt2 17455 | A constructed tuple is a p... |
| prdsbascl 17456 | An element of the base has... |
| prdsdsval2 17457 | Value of the metric in a s... |
| prdsdsval3 17458 | Value of the metric in a s... |
| pwsval 17459 | Value of a structure power... |
| pwsbas 17460 | Base set of a structure po... |
| pwselbasb 17461 | Membership in the base set... |
| pwselbas 17462 | An element of a structure ... |
| pwsplusgval 17463 | Value of addition in a str... |
| pwsmulrval 17464 | Value of multiplication in... |
| pwsle 17465 | Ordering in a structure po... |
| pwsleval 17466 | Ordering in a structure po... |
| pwsvscafval 17467 | Scalar multiplication in a... |
| pwsvscaval 17468 | Scalar multiplication of a... |
| pwssca 17469 | The ring of scalars of a s... |
| pwsdiagel 17470 | Membership of diagonal ele... |
| pwssnf1o 17471 | Triviality of singleton po... |
| imasval 17484 | Value of an image structur... |
| imasbas 17485 | The base set of an image s... |
| imasds 17486 | The distance function of a... |
| imasdsfn 17487 | The distance function is a... |
| imasdsval 17488 | The distance function of a... |
| imasdsval2 17489 | The distance function of a... |
| imasplusg 17490 | The group operation in an ... |
| imasmulr 17491 | The ring multiplication in... |
| imassca 17492 | The scalar field of an ima... |
| imasvsca 17493 | The scalar multiplication ... |
| imasip 17494 | The inner product of an im... |
| imastset 17495 | The topology of an image s... |
| imasle 17496 | The ordering of an image s... |
| f1ocpbllem 17497 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17498 | An injection is compatible... |
| f1ovscpbl 17499 | An injection is compatible... |
| f1olecpbl 17500 | An injection is compatible... |
| imasaddfnlem 17501 | The image structure operat... |
| imasaddvallem 17502 | The operation of an image ... |
| imasaddflem 17503 | The image set operations a... |
| imasaddfn 17504 | The image structure's grou... |
| imasaddval 17505 | The value of an image stru... |
| imasaddf 17506 | The image structure's grou... |
| imasmulfn 17507 | The image structure's ring... |
| imasmulval 17508 | The value of an image stru... |
| imasmulf 17509 | The image structure's ring... |
| imasvscafn 17510 | The image structure's scal... |
| imasvscaval 17511 | The value of an image stru... |
| imasvscaf 17512 | The image structure's scal... |
| imasless 17513 | The order relation defined... |
| imasleval 17514 | The value of the image str... |
| qusval 17515 | Value of a quotient struct... |
| quslem 17516 | The function in ~ qusval i... |
| qusin 17517 | Restrict the equivalence r... |
| qusbas 17518 | Base set of a quotient str... |
| quss 17519 | The scalar field of a quot... |
| divsfval 17520 | Value of the function in ~... |
| ercpbllem 17521 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17522 | Translate the function com... |
| erlecpbl 17523 | Translate the relation com... |
| qusaddvallem 17524 | Value of an operation defi... |
| qusaddflem 17525 | The operation of a quotien... |
| qusaddval 17526 | The addition in a quotient... |
| qusaddf 17527 | The addition in a quotient... |
| qusmulval 17528 | The multiplication in a qu... |
| qusmulf 17529 | The multiplication in a qu... |
| fnpr2o 17530 | Function with a domain of ... |
| fnpr2ob 17531 | Biconditional version of ~... |
| fvpr0o 17532 | The value of a function wi... |
| fvpr1o 17533 | The value of a function wi... |
| fvprif 17534 | The value of the pair func... |
| xpsfrnel 17535 | Elementhood in the target ... |
| xpsfeq 17536 | A function on ` 2o ` is de... |
| xpsfrnel2 17537 | Elementhood in the target ... |
| xpscf 17538 | Equivalent condition for t... |
| xpsfval 17539 | The value of the function ... |
| xpsff1o 17540 | The function appearing in ... |
| xpsfrn 17541 | A short expression for the... |
| xpsff1o2 17542 | The function appearing in ... |
| xpsval 17543 | Value of the binary struct... |
| xpsrnbas 17544 | The indexed structure prod... |
| xpsbas 17545 | The base set of the binary... |
| xpsaddlem 17546 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17547 | Value of the addition oper... |
| xpsmul 17548 | Value of the multiplicatio... |
| xpssca 17549 | Value of the scalar field ... |
| xpsvsca 17550 | Value of the scalar multip... |
| xpsless 17551 | Closure of the ordering in... |
| xpsle 17552 | Value of the ordering in a... |
| ismre 17561 | Property of being a Moore ... |
| fnmre 17562 | The Moore collection gener... |
| mresspw 17563 | A Moore collection is a su... |
| mress 17564 | A Moore-closed subset is a... |
| mre1cl 17565 | In any Moore collection th... |
| mreintcl 17566 | A nonempty collection of c... |
| mreiincl 17567 | A nonempty indexed interse... |
| mrerintcl 17568 | The relative intersection ... |
| mreriincl 17569 | The relative intersection ... |
| mreincl 17570 | Two closed sets have a clo... |
| mreuni 17571 | Since the entire base set ... |
| mreunirn 17572 | Two ways to express the no... |
| ismred 17573 | Properties that determine ... |
| ismred2 17574 | Properties that determine ... |
| mremre 17575 | The Moore collections of s... |
| submre 17576 | The subcollection of a clo... |
| mrcflem 17577 | The domain and codomain of... |
| fnmrc 17578 | Moore-closure is a well-be... |
| mrcfval 17579 | Value of the function expr... |
| mrcf 17580 | The Moore closure is a fun... |
| mrcval 17581 | Evaluation of the Moore cl... |
| mrccl 17582 | The Moore closure of a set... |
| mrcsncl 17583 | The Moore closure of a sin... |
| mrcid 17584 | The closure of a closed se... |
| mrcssv 17585 | The closure of a set is a ... |
| mrcidb 17586 | A set is closed iff it is ... |
| mrcss 17587 | Closure preserves subset o... |
| mrcssid 17588 | The closure of a set is a ... |
| mrcidb2 17589 | A set is closed iff it con... |
| mrcidm 17590 | The closure operation is i... |
| mrcsscl 17591 | The closure is the minimal... |
| mrcuni 17592 | Idempotence of closure und... |
| mrcun 17593 | Idempotence of closure und... |
| mrcssvd 17594 | The Moore closure of a set... |
| mrcssd 17595 | Moore closure preserves su... |
| mrcssidd 17596 | A set is contained in its ... |
| mrcidmd 17597 | Moore closure is idempoten... |
| mressmrcd 17598 | In a Moore system, if a se... |
| submrc 17599 | In a closure system which ... |
| mrieqvlemd 17600 | In a Moore system, if ` Y ... |
| mrisval 17601 | Value of the set of indepe... |
| ismri 17602 | Criterion for a set to be ... |
| ismri2 17603 | Criterion for a subset of ... |
| ismri2d 17604 | Criterion for a subset of ... |
| ismri2dd 17605 | Definition of independence... |
| mriss 17606 | An independent set of a Mo... |
| mrissd 17607 | An independent set of a Mo... |
| ismri2dad 17608 | Consequence of a set in a ... |
| mrieqvd 17609 | In a Moore system, a set i... |
| mrieqv2d 17610 | In a Moore system, a set i... |
| mrissmrcd 17611 | In a Moore system, if an i... |
| mrissmrid 17612 | In a Moore system, subsets... |
| mreexd 17613 | In a Moore system, the clo... |
| mreexmrid 17614 | In a Moore system whose cl... |
| mreexexlemd 17615 | This lemma is used to gene... |
| mreexexlem2d 17616 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17617 | Base case of the induction... |
| mreexexlem4d 17618 | Induction step of the indu... |
| mreexexd 17619 | Exchange-type theorem. In... |
| mreexdomd 17620 | In a Moore system whose cl... |
| mreexfidimd 17621 | In a Moore system whose cl... |
| isacs 17622 | A set is an algebraic clos... |
| acsmre 17623 | Algebraic closure systems ... |
| isacs2 17624 | In the definition of an al... |
| acsfiel 17625 | A set is closed in an alge... |
| acsfiel2 17626 | A set is closed in an alge... |
| acsmred 17627 | An algebraic closure syste... |
| isacs1i 17628 | A closure system determine... |
| mreacs 17629 | Algebraicity is a composab... |
| acsfn 17630 | Algebraicity of a conditio... |
| acsfn0 17631 | Algebraicity of a point cl... |
| acsfn1 17632 | Algebraicity of a one-argu... |
| acsfn1c 17633 | Algebraicity of a one-argu... |
| acsfn2 17634 | Algebraicity of a two-argu... |
| iscat 17643 | The predicate "is a catego... |
| iscatd 17644 | Properties that determine ... |
| catidex 17645 | Each object in a category ... |
| catideu 17646 | Each object in a category ... |
| cidfval 17647 | Each object in a category ... |
| cidval 17648 | Each object in a category ... |
| cidffn 17649 | The identity arrow constru... |
| cidfn 17650 | The identity arrow operato... |
| catidd 17651 | Deduce the identity arrow ... |
| iscatd2 17652 | Version of ~ iscatd with a... |
| catidcl 17653 | Each object in a category ... |
| catlid 17654 | Left identity property of ... |
| catrid 17655 | Right identity property of... |
| catcocl 17656 | Closure of a composition a... |
| catass 17657 | Associativity of compositi... |
| catcone0 17658 | Composition of non-empty h... |
| 0catg 17659 | Any structure with an empt... |
| 0cat 17660 | The empty set is a categor... |
| homffval 17661 | Value of the functionalize... |
| fnhomeqhomf 17662 | If the Hom-set operation i... |
| homfval 17663 | Value of the functionalize... |
| homffn 17664 | The functionalized Hom-set... |
| homfeq 17665 | Condition for two categori... |
| homfeqd 17666 | If two structures have the... |
| homfeqbas 17667 | Deduce equality of base se... |
| homfeqval 17668 | Value of the functionalize... |
| comfffval 17669 | Value of the functionalize... |
| comffval 17670 | Value of the functionalize... |
| comfval 17671 | Value of the functionalize... |
| comfffval2 17672 | Value of the functionalize... |
| comffval2 17673 | Value of the functionalize... |
| comfval2 17674 | Value of the functionalize... |
| comfffn 17675 | The functionalized composi... |
| comffn 17676 | The functionalized composi... |
| comfeq 17677 | Condition for two categori... |
| comfeqd 17678 | Condition for two categori... |
| comfeqval 17679 | Equality of two compositio... |
| catpropd 17680 | Two structures with the sa... |
| cidpropd 17681 | Two structures with the sa... |
| oppcval 17684 | Value of the opposite cate... |
| oppchomfval 17685 | Hom-sets of the opposite c... |
| oppchomfvalOLD 17686 | Obsolete proof of ~ oppcho... |
| oppchom 17687 | Hom-sets of the opposite c... |
| oppccofval 17688 | Composition in the opposit... |
| oppcco 17689 | Composition in the opposit... |
| oppcbas 17690 | Base set of an opposite ca... |
| oppcbasOLD 17691 | Obsolete version of ~ oppc... |
| oppccatid 17692 | Lemma for ~ oppccat . (Co... |
| oppchomf 17693 | Hom-sets of the opposite c... |
| oppcid 17694 | Identity function of an op... |
| oppccat 17695 | An opposite category is a ... |
| 2oppcbas 17696 | The double opposite catego... |
| 2oppchomf 17697 | The double opposite catego... |
| 2oppccomf 17698 | The double opposite catego... |
| oppchomfpropd 17699 | If two categories have the... |
| oppccomfpropd 17700 | If two categories have the... |
| oppccatf 17701 | ` oppCat ` restricted to `... |
| monfval 17706 | Definition of a monomorphi... |
| ismon 17707 | Definition of a monomorphi... |
| ismon2 17708 | Write out the monomorphism... |
| monhom 17709 | A monomorphism is a morphi... |
| moni 17710 | Property of a monomorphism... |
| monpropd 17711 | If two categories have the... |
| oppcmon 17712 | A monomorphism in the oppo... |
| oppcepi 17713 | An epimorphism in the oppo... |
| isepi 17714 | Definition of an epimorphi... |
| isepi2 17715 | Write out the epimorphism ... |
| epihom 17716 | An epimorphism is a morphi... |
| epii 17717 | Property of an epimorphism... |
| sectffval 17724 | Value of the section opera... |
| sectfval 17725 | Value of the section relat... |
| sectss 17726 | The section relation is a ... |
| issect 17727 | The property " ` F ` is a ... |
| issect2 17728 | Property of being a sectio... |
| sectcan 17729 | If ` G ` is a section of `... |
| sectco 17730 | Composition of two section... |
| isofval 17731 | Function value of the func... |
| invffval 17732 | Value of the inverse relat... |
| invfval 17733 | Value of the inverse relat... |
| isinv 17734 | Value of the inverse relat... |
| invss 17735 | The inverse relation is a ... |
| invsym 17736 | The inverse relation is sy... |
| invsym2 17737 | The inverse relation is sy... |
| invfun 17738 | The inverse relation is a ... |
| isoval 17739 | The isomorphisms are the d... |
| inviso1 17740 | If ` G ` is an inverse to ... |
| inviso2 17741 | If ` G ` is an inverse to ... |
| invf 17742 | The inverse relation is a ... |
| invf1o 17743 | The inverse relation is a ... |
| invinv 17744 | The inverse of the inverse... |
| invco 17745 | The composition of two iso... |
| dfiso2 17746 | Alternate definition of an... |
| dfiso3 17747 | Alternate definition of an... |
| inveq 17748 | If there are two inverses ... |
| isofn 17749 | The function value of the ... |
| isohom 17750 | An isomorphism is a homomo... |
| isoco 17751 | The composition of two iso... |
| oppcsect 17752 | A section in the opposite ... |
| oppcsect2 17753 | A section in the opposite ... |
| oppcinv 17754 | An inverse in the opposite... |
| oppciso 17755 | An isomorphism in the oppo... |
| sectmon 17756 | If ` F ` is a section of `... |
| monsect 17757 | If ` F ` is a monomorphism... |
| sectepi 17758 | If ` F ` is a section of `... |
| episect 17759 | If ` F ` is an epimorphism... |
| sectid 17760 | The identity is a section ... |
| invid 17761 | The inverse of the identit... |
| idiso 17762 | The identity is an isomorp... |
| idinv 17763 | The inverse of the identit... |
| invisoinvl 17764 | The inverse of an isomorph... |
| invisoinvr 17765 | The inverse of an isomorph... |
| invcoisoid 17766 | The inverse of an isomorph... |
| isocoinvid 17767 | The inverse of an isomorph... |
| rcaninv 17768 | Right cancellation of an i... |
| cicfval 17771 | The set of isomorphic obje... |
| brcic 17772 | The relation "is isomorphi... |
| cic 17773 | Objects ` X ` and ` Y ` in... |
| brcici 17774 | Prove that two objects are... |
| cicref 17775 | Isomorphism is reflexive. ... |
| ciclcl 17776 | Isomorphism implies the le... |
| cicrcl 17777 | Isomorphism implies the ri... |
| cicsym 17778 | Isomorphism is symmetric. ... |
| cictr 17779 | Isomorphism is transitive.... |
| cicer 17780 | Isomorphism is an equivale... |
| sscrel 17787 | The subcategory subset rel... |
| brssc 17788 | The subcategory subset rel... |
| sscpwex 17789 | An analogue of ~ pwex for ... |
| subcrcl 17790 | Reverse closure for the su... |
| sscfn1 17791 | The subcategory subset rel... |
| sscfn2 17792 | The subcategory subset rel... |
| ssclem 17793 | Lemma for ~ ssc1 and simil... |
| isssc 17794 | Value of the subcategory s... |
| ssc1 17795 | Infer subset relation on o... |
| ssc2 17796 | Infer subset relation on m... |
| sscres 17797 | Any function restricted to... |
| sscid 17798 | The subcategory subset rel... |
| ssctr 17799 | The subcategory subset rel... |
| ssceq 17800 | The subcategory subset rel... |
| rescval 17801 | Value of the category rest... |
| rescval2 17802 | Value of the category rest... |
| rescbas 17803 | Base set of the category r... |
| rescbasOLD 17804 | Obsolete version of ~ resc... |
| reschom 17805 | Hom-sets of the category r... |
| reschomf 17806 | Hom-sets of the category r... |
| rescco 17807 | Composition in the categor... |
| resccoOLD 17808 | Obsolete proof of ~ rescco... |
| rescabs 17809 | Restriction absorption law... |
| rescabsOLD 17810 | Obsolete proof of ~ seqp1d... |
| rescabs2 17811 | Restriction absorption law... |
| issubc 17812 | Elementhood in the set of ... |
| issubc2 17813 | Elementhood in the set of ... |
| 0ssc 17814 | For any category ` C ` , t... |
| 0subcat 17815 | For any category ` C ` , t... |
| catsubcat 17816 | For any category ` C ` , `... |
| subcssc 17817 | An element in the set of s... |
| subcfn 17818 | An element in the set of s... |
| subcss1 17819 | The objects of a subcatego... |
| subcss2 17820 | The morphisms of a subcate... |
| subcidcl 17821 | The identity of the origin... |
| subccocl 17822 | A subcategory is closed un... |
| subccatid 17823 | A subcategory is a categor... |
| subcid 17824 | The identity in a subcateg... |
| subccat 17825 | A subcategory is a categor... |
| issubc3 17826 | Alternate definition of a ... |
| fullsubc 17827 | The full subcategory gener... |
| fullresc 17828 | The category formed by str... |
| resscat 17829 | A category restricted to a... |
| subsubc 17830 | A subcategory of a subcate... |
| relfunc 17839 | The set of functors is a r... |
| funcrcl 17840 | Reverse closure for a func... |
| isfunc 17841 | Value of the set of functo... |
| isfuncd 17842 | Deduce that an operation i... |
| funcf1 17843 | The object part of a funct... |
| funcixp 17844 | The morphism part of a fun... |
| funcf2 17845 | The morphism part of a fun... |
| funcfn2 17846 | The morphism part of a fun... |
| funcid 17847 | A functor maps each identi... |
| funcco 17848 | A functor maps composition... |
| funcsect 17849 | The image of a section und... |
| funcinv 17850 | The image of an inverse un... |
| funciso 17851 | The image of an isomorphis... |
| funcoppc 17852 | A functor on categories yi... |
| idfuval 17853 | Value of the identity func... |
| idfu2nd 17854 | Value of the morphism part... |
| idfu2 17855 | Value of the morphism part... |
| idfu1st 17856 | Value of the object part o... |
| idfu1 17857 | Value of the object part o... |
| idfucl 17858 | The identity functor is a ... |
| cofuval 17859 | Value of the composition o... |
| cofu1st 17860 | Value of the object part o... |
| cofu1 17861 | Value of the object part o... |
| cofu2nd 17862 | Value of the morphism part... |
| cofu2 17863 | Value of the morphism part... |
| cofuval2 17864 | Value of the composition o... |
| cofucl 17865 | The composition of two fun... |
| cofuass 17866 | Functor composition is ass... |
| cofulid 17867 | The identity functor is a ... |
| cofurid 17868 | The identity functor is a ... |
| resfval 17869 | Value of the functor restr... |
| resfval2 17870 | Value of the functor restr... |
| resf1st 17871 | Value of the functor restr... |
| resf2nd 17872 | Value of the functor restr... |
| funcres 17873 | A functor restricted to a ... |
| funcres2b 17874 | Condition for a functor to... |
| funcres2 17875 | A functor into a restricte... |
| idfusubc0 17876 | The identity functor for a... |
| idfusubc 17877 | The identity functor for a... |
| wunfunc 17878 | A weak universe is closed ... |
| wunfuncOLD 17879 | Obsolete proof of ~ wunfun... |
| funcpropd 17880 | If two categories have the... |
| funcres2c 17881 | Condition for a functor to... |
| fullfunc 17886 | A full functor is a functo... |
| fthfunc 17887 | A faithful functor is a fu... |
| relfull 17888 | The set of full functors i... |
| relfth 17889 | The set of faithful functo... |
| isfull 17890 | Value of the set of full f... |
| isfull2 17891 | Equivalent condition for a... |
| fullfo 17892 | The morphism map of a full... |
| fulli 17893 | The morphism map of a full... |
| isfth 17894 | Value of the set of faithf... |
| isfth2 17895 | Equivalent condition for a... |
| isffth2 17896 | A fully faithful functor i... |
| fthf1 17897 | The morphism map of a fait... |
| fthi 17898 | The morphism map of a fait... |
| ffthf1o 17899 | The morphism map of a full... |
| fullpropd 17900 | If two categories have the... |
| fthpropd 17901 | If two categories have the... |
| fulloppc 17902 | The opposite functor of a ... |
| fthoppc 17903 | The opposite functor of a ... |
| ffthoppc 17904 | The opposite functor of a ... |
| fthsect 17905 | A faithful functor reflect... |
| fthinv 17906 | A faithful functor reflect... |
| fthmon 17907 | A faithful functor reflect... |
| fthepi 17908 | A faithful functor reflect... |
| ffthiso 17909 | A fully faithful functor r... |
| fthres2b 17910 | Condition for a faithful f... |
| fthres2c 17911 | Condition for a faithful f... |
| fthres2 17912 | A faithful functor into a ... |
| idffth 17913 | The identity functor is a ... |
| cofull 17914 | The composition of two ful... |
| cofth 17915 | The composition of two fai... |
| coffth 17916 | The composition of two ful... |
| rescfth 17917 | The inclusion functor from... |
| ressffth 17918 | The inclusion functor from... |
| fullres2c 17919 | Condition for a full funct... |
| ffthres2c 17920 | Condition for a fully fait... |
| inclfusubc 17921 | The "inclusion functor" fr... |
| fnfuc 17926 | The ` FuncCat ` operation ... |
| natfval 17927 | Value of the function givi... |
| isnat 17928 | Property of being a natura... |
| isnat2 17929 | Property of being a natura... |
| natffn 17930 | The natural transformation... |
| natrcl 17931 | Reverse closure for a natu... |
| nat1st2nd 17932 | Rewrite the natural transf... |
| natixp 17933 | A natural transformation i... |
| natcl 17934 | A component of a natural t... |
| natfn 17935 | A natural transformation i... |
| nati 17936 | Naturality property of a n... |
| wunnat 17937 | A weak universe is closed ... |
| wunnatOLD 17938 | Obsolete proof of ~ wunnat... |
| catstr 17939 | A category structure is a ... |
| fucval 17940 | Value of the functor categ... |
| fuccofval 17941 | Value of the functor categ... |
| fucbas 17942 | The objects of the functor... |
| fuchom 17943 | The morphisms in the funct... |
| fuchomOLD 17944 | Obsolete proof of ~ fuchom... |
| fucco 17945 | Value of the composition o... |
| fuccoval 17946 | Value of the functor categ... |
| fuccocl 17947 | The composition of two nat... |
| fucidcl 17948 | The identity natural trans... |
| fuclid 17949 | Left identity of natural t... |
| fucrid 17950 | Right identity of natural ... |
| fucass 17951 | Associativity of natural t... |
| fuccatid 17952 | The functor category is a ... |
| fuccat 17953 | The functor category is a ... |
| fucid 17954 | The identity morphism in t... |
| fucsect 17955 | Two natural transformation... |
| fucinv 17956 | Two natural transformation... |
| invfuc 17957 | If ` V ( x ) ` is an inver... |
| fuciso 17958 | A natural transformation i... |
| natpropd 17959 | If two categories have the... |
| fucpropd 17960 | If two categories have the... |
| initofn 17967 | ` InitO ` is a function on... |
| termofn 17968 | ` TermO ` is a function on... |
| zeroofn 17969 | ` ZeroO ` is a function on... |
| initorcl 17970 | Reverse closure for an ini... |
| termorcl 17971 | Reverse closure for a term... |
| zeroorcl 17972 | Reverse closure for a zero... |
| initoval 17973 | The value of the initial o... |
| termoval 17974 | The value of the terminal ... |
| zerooval 17975 | The value of the zero obje... |
| isinito 17976 | The predicate "is an initi... |
| istermo 17977 | The predicate "is a termin... |
| iszeroo 17978 | The predicate "is a zero o... |
| isinitoi 17979 | Implication of a class bei... |
| istermoi 17980 | Implication of a class bei... |
| initoid 17981 | For an initial object, the... |
| termoid 17982 | For a terminal object, the... |
| dfinito2 17983 | An initial object is a ter... |
| dftermo2 17984 | A terminal object is an in... |
| dfinito3 17985 | An alternate definition of... |
| dftermo3 17986 | An alternate definition of... |
| initoo 17987 | An initial object is an ob... |
| termoo 17988 | A terminal object is an ob... |
| iszeroi 17989 | Implication of a class bei... |
| 2initoinv 17990 | Morphisms between two init... |
| initoeu1 17991 | Initial objects are essent... |
| initoeu1w 17992 | Initial objects are essent... |
| initoeu2lem0 17993 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 17994 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 17995 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 17996 | Initial objects are essent... |
| 2termoinv 17997 | Morphisms between two term... |
| termoeu1 17998 | Terminal objects are essen... |
| termoeu1w 17999 | Terminal objects are essen... |
| homarcl 18008 | Reverse closure for an arr... |
| homafval 18009 | Value of the disjointified... |
| homaf 18010 | Functionality of the disjo... |
| homaval 18011 | Value of the disjointified... |
| elhoma 18012 | Value of the disjointified... |
| elhomai 18013 | Produce an arrow from a mo... |
| elhomai2 18014 | Produce an arrow from a mo... |
| homarcl2 18015 | Reverse closure for the do... |
| homarel 18016 | An arrow is an ordered pai... |
| homa1 18017 | The first component of an ... |
| homahom2 18018 | The second component of an... |
| homahom 18019 | The second component of an... |
| homadm 18020 | The domain of an arrow wit... |
| homacd 18021 | The codomain of an arrow w... |
| homadmcd 18022 | Decompose an arrow into do... |
| arwval 18023 | The set of arrows is the u... |
| arwrcl 18024 | The first component of an ... |
| arwhoma 18025 | An arrow is contained in t... |
| homarw 18026 | A hom-set is a subset of t... |
| arwdm 18027 | The domain of an arrow is ... |
| arwcd 18028 | The codomain of an arrow i... |
| dmaf 18029 | The domain function is a f... |
| cdaf 18030 | The codomain function is a... |
| arwhom 18031 | The second component of an... |
| arwdmcd 18032 | Decompose an arrow into do... |
| idafval 18037 | Value of the identity arro... |
| idaval 18038 | Value of the identity arro... |
| ida2 18039 | Morphism part of the ident... |
| idahom 18040 | Domain and codomain of the... |
| idadm 18041 | Domain of the identity arr... |
| idacd 18042 | Codomain of the identity a... |
| idaf 18043 | The identity arrow functio... |
| coafval 18044 | The value of the compositi... |
| eldmcoa 18045 | A pair ` <. G , F >. ` is ... |
| dmcoass 18046 | The domain of composition ... |
| homdmcoa 18047 | If ` F : X --> Y ` and ` G... |
| coaval 18048 | Value of composition for c... |
| coa2 18049 | The morphism part of arrow... |
| coahom 18050 | The composition of two com... |
| coapm 18051 | Composition of arrows is a... |
| arwlid 18052 | Left identity of a categor... |
| arwrid 18053 | Right identity of a catego... |
| arwass 18054 | Associativity of compositi... |
| setcval 18057 | Value of the category of s... |
| setcbas 18058 | Set of objects of the cate... |
| setchomfval 18059 | Set of arrows of the categ... |
| setchom 18060 | Set of arrows of the categ... |
| elsetchom 18061 | A morphism of sets is a fu... |
| setccofval 18062 | Composition in the categor... |
| setcco 18063 | Composition in the categor... |
| setccatid 18064 | Lemma for ~ setccat . (Co... |
| setccat 18065 | The category of sets is a ... |
| setcid 18066 | The identity arrow in the ... |
| setcmon 18067 | A monomorphism of sets is ... |
| setcepi 18068 | An epimorphism of sets is ... |
| setcsect 18069 | A section in the category ... |
| setcinv 18070 | An inverse in the category... |
| setciso 18071 | An isomorphism in the cate... |
| resssetc 18072 | The restriction of the cat... |
| funcsetcres2 18073 | A functor into a smaller c... |
| setc2obas 18074 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18075 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18076 | The category of sets in a ... |
| cat1 18077 | The definition of category... |
| catcval 18080 | Value of the category of c... |
| catcbas 18081 | Set of objects of the cate... |
| catchomfval 18082 | Set of arrows of the categ... |
| catchom 18083 | Set of arrows of the categ... |
| catccofval 18084 | Composition in the categor... |
| catcco 18085 | Composition in the categor... |
| catccatid 18086 | Lemma for ~ catccat . (Co... |
| catcid 18087 | The identity arrow in the ... |
| catccat 18088 | The category of categories... |
| resscatc 18089 | The restriction of the cat... |
| catcisolem 18090 | Lemma for ~ catciso . (Co... |
| catciso 18091 | A functor is an isomorphis... |
| catcbascl 18092 | An element of the base set... |
| catcslotelcl 18093 | A slot entry of an element... |
| catcbaselcl 18094 | The base set of an element... |
| catchomcl 18095 | The Hom-set of an element ... |
| catcccocl 18096 | The composition operation ... |
| catcoppccl 18097 | The category of categories... |
| catcoppcclOLD 18098 | Obsolete proof of ~ catcop... |
| catcfuccl 18099 | The category of categories... |
| catcfucclOLD 18100 | Obsolete proof of ~ catcfu... |
| fncnvimaeqv 18101 | The inverse images of the ... |
| bascnvimaeqv 18102 | The inverse image of the u... |
| estrcval 18105 | Value of the category of e... |
| estrcbas 18106 | Set of objects of the cate... |
| estrchomfval 18107 | Set of morphisms ("arrows"... |
| estrchom 18108 | The morphisms between exte... |
| elestrchom 18109 | A morphism between extensi... |
| estrccofval 18110 | Composition in the categor... |
| estrcco 18111 | Composition in the categor... |
| estrcbasbas 18112 | An element of the base set... |
| estrccatid 18113 | Lemma for ~ estrccat . (C... |
| estrccat 18114 | The category of extensible... |
| estrcid 18115 | The identity arrow in the ... |
| estrchomfn 18116 | The Hom-set operation in t... |
| estrchomfeqhom 18117 | The functionalized Hom-set... |
| estrreslem1 18118 | Lemma 1 for ~ estrres . (... |
| estrreslem1OLD 18119 | Obsolete version of ~ estr... |
| estrreslem2 18120 | Lemma 2 for ~ estrres . (... |
| estrres 18121 | Any restriction of a categ... |
| funcestrcsetclem1 18122 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18123 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18124 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18125 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18126 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18127 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18128 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18129 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18130 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18131 | The "natural forgetful fun... |
| fthestrcsetc 18132 | The "natural forgetful fun... |
| fullestrcsetc 18133 | The "natural forgetful fun... |
| equivestrcsetc 18134 | The "natural forgetful fun... |
| setc1strwun 18135 | A constructed one-slot str... |
| funcsetcestrclem1 18136 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18137 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18138 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18139 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18140 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18141 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18142 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18143 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18144 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18145 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18146 | The "embedding functor" fr... |
| fthsetcestrc 18147 | The "embedding functor" fr... |
| fullsetcestrc 18148 | The "embedding functor" fr... |
| embedsetcestrc 18149 | The "embedding functor" fr... |
| fnxpc 18158 | The binary product of cate... |
| xpcval 18159 | Value of the binary produc... |
| xpcbas 18160 | Set of objects of the bina... |
| xpchomfval 18161 | Set of morphisms of the bi... |
| xpchom 18162 | Set of morphisms of the bi... |
| relxpchom 18163 | A hom-set in the binary pr... |
| xpccofval 18164 | Value of composition in th... |
| xpcco 18165 | Value of composition in th... |
| xpcco1st 18166 | Value of composition in th... |
| xpcco2nd 18167 | Value of composition in th... |
| xpchom2 18168 | Value of the set of morphi... |
| xpcco2 18169 | Value of composition in th... |
| xpccatid 18170 | The product of two categor... |
| xpcid 18171 | The identity morphism in t... |
| xpccat 18172 | The product of two categor... |
| 1stfval 18173 | Value of the first project... |
| 1stf1 18174 | Value of the first project... |
| 1stf2 18175 | Value of the first project... |
| 2ndfval 18176 | Value of the first project... |
| 2ndf1 18177 | Value of the first project... |
| 2ndf2 18178 | Value of the first project... |
| 1stfcl 18179 | The first projection funct... |
| 2ndfcl 18180 | The second projection func... |
| prfval 18181 | Value of the pairing funct... |
| prf1 18182 | Value of the pairing funct... |
| prf2fval 18183 | Value of the pairing funct... |
| prf2 18184 | Value of the pairing funct... |
| prfcl 18185 | The pairing of functors ` ... |
| prf1st 18186 | Cancellation of pairing wi... |
| prf2nd 18187 | Cancellation of pairing wi... |
| 1st2ndprf 18188 | Break a functor into a pro... |
| catcxpccl 18189 | The category of categories... |
| catcxpcclOLD 18190 | Obsolete proof of ~ catcxp... |
| xpcpropd 18191 | If two categories have the... |
| evlfval 18200 | Value of the evaluation fu... |
| evlf2 18201 | Value of the evaluation fu... |
| evlf2val 18202 | Value of the evaluation na... |
| evlf1 18203 | Value of the evaluation fu... |
| evlfcllem 18204 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18205 | The evaluation functor is ... |
| curfval 18206 | Value of the curry functor... |
| curf1fval 18207 | Value of the object part o... |
| curf1 18208 | Value of the object part o... |
| curf11 18209 | Value of the double evalua... |
| curf12 18210 | The partially evaluated cu... |
| curf1cl 18211 | The partially evaluated cu... |
| curf2 18212 | Value of the curry functor... |
| curf2val 18213 | Value of a component of th... |
| curf2cl 18214 | The curry functor at a mor... |
| curfcl 18215 | The curry functor of a fun... |
| curfpropd 18216 | If two categories have the... |
| uncfval 18217 | Value of the uncurry funct... |
| uncfcl 18218 | The uncurry operation take... |
| uncf1 18219 | Value of the uncurry funct... |
| uncf2 18220 | Value of the uncurry funct... |
| curfuncf 18221 | Cancellation of curry with... |
| uncfcurf 18222 | Cancellation of uncurry wi... |
| diagval 18223 | Define the diagonal functo... |
| diagcl 18224 | The diagonal functor is a ... |
| diag1cl 18225 | The constant functor of ` ... |
| diag11 18226 | Value of the constant func... |
| diag12 18227 | Value of the constant func... |
| diag2 18228 | Value of the diagonal func... |
| diag2cl 18229 | The diagonal functor at a ... |
| curf2ndf 18230 | As shown in ~ diagval , th... |
| hofval 18235 | Value of the Hom functor, ... |
| hof1fval 18236 | The object part of the Hom... |
| hof1 18237 | The object part of the Hom... |
| hof2fval 18238 | The morphism part of the H... |
| hof2val 18239 | The morphism part of the H... |
| hof2 18240 | The morphism part of the H... |
| hofcllem 18241 | Lemma for ~ hofcl . (Cont... |
| hofcl 18242 | Closure of the Hom functor... |
| oppchofcl 18243 | Closure of the opposite Ho... |
| yonval 18244 | Value of the Yoneda embedd... |
| yoncl 18245 | The Yoneda embedding is a ... |
| yon1cl 18246 | The Yoneda embedding at an... |
| yon11 18247 | Value of the Yoneda embedd... |
| yon12 18248 | Value of the Yoneda embedd... |
| yon2 18249 | Value of the Yoneda embedd... |
| hofpropd 18250 | If two categories have the... |
| yonpropd 18251 | If two categories have the... |
| oppcyon 18252 | Value of the opposite Yone... |
| oyoncl 18253 | The opposite Yoneda embedd... |
| oyon1cl 18254 | The opposite Yoneda embedd... |
| yonedalem1 18255 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18256 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18257 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18258 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18259 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18260 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18261 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18262 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18263 | Lemma for ~ yoneda . (Con... |
| yonedainv 18264 | The Yoneda Lemma with expl... |
| yonffthlem 18265 | Lemma for ~ yonffth . (Co... |
| yoneda 18266 | The Yoneda Lemma. There i... |
| yonffth 18267 | The Yoneda Lemma. The Yon... |
| yoniso 18268 | If the codomain is recover... |
| oduval 18271 | Value of an order dual str... |
| oduleval 18272 | Value of the less-equal re... |
| oduleg 18273 | Truth of the less-equal re... |
| odubas 18274 | Base set of an order dual ... |
| odubasOLD 18275 | Obsolete proof of ~ odubas... |
| isprs 18280 | Property of being a preord... |
| prslem 18281 | Lemma for ~ prsref and ~ p... |
| prsref 18282 | "Less than or equal to" is... |
| prstr 18283 | "Less than or equal to" is... |
| isdrs 18284 | Property of being a direct... |
| drsdir 18285 | Direction of a directed se... |
| drsprs 18286 | A directed set is a proset... |
| drsbn0 18287 | The base of a directed set... |
| drsdirfi 18288 | Any _finite_ number of ele... |
| isdrs2 18289 | Directed sets may be defin... |
| ispos 18297 | The predicate "is a poset"... |
| ispos2 18298 | A poset is an antisymmetri... |
| posprs 18299 | A poset is a proset. (Con... |
| posi 18300 | Lemma for poset properties... |
| posref 18301 | A poset ordering is reflex... |
| posasymb 18302 | A poset ordering is asymme... |
| postr 18303 | A poset ordering is transi... |
| 0pos 18304 | Technical lemma to simplif... |
| 0posOLD 18305 | Obsolete proof of ~ 0pos a... |
| isposd 18306 | Properties that determine ... |
| isposi 18307 | Properties that determine ... |
| isposix 18308 | Properties that determine ... |
| isposixOLD 18309 | Obsolete proof of ~ isposi... |
| pospropd 18310 | Posethood is determined on... |
| odupos 18311 | Being a poset is a self-du... |
| oduposb 18312 | Being a poset is a self-du... |
| pltfval 18314 | Value of the less-than rel... |
| pltval 18315 | Less-than relation. ( ~ d... |
| pltle 18316 | "Less than" implies "less ... |
| pltne 18317 | The "less than" relation i... |
| pltirr 18318 | The "less than" relation i... |
| pleval2i 18319 | One direction of ~ pleval2... |
| pleval2 18320 | "Less than or equal to" in... |
| pltnle 18321 | "Less than" implies not co... |
| pltval3 18322 | Alternate expression for t... |
| pltnlt 18323 | The less-than relation imp... |
| pltn2lp 18324 | The less-than relation has... |
| plttr 18325 | The less-than relation is ... |
| pltletr 18326 | Transitive law for chained... |
| plelttr 18327 | Transitive law for chained... |
| pospo 18328 | Write a poset structure in... |
| lubfval 18333 | Value of the least upper b... |
| lubdm 18334 | Domain of the least upper ... |
| lubfun 18335 | The LUB is a function. (C... |
| lubeldm 18336 | Member of the domain of th... |
| lubelss 18337 | A member of the domain of ... |
| lubeu 18338 | Unique existence proper of... |
| lubval 18339 | Value of the least upper b... |
| lubcl 18340 | The least upper bound func... |
| lubprop 18341 | Properties of greatest low... |
| luble 18342 | The greatest lower bound i... |
| lublecllem 18343 | Lemma for ~ lublecl and ~ ... |
| lublecl 18344 | The set of all elements le... |
| lubid 18345 | The LUB of elements less t... |
| glbfval 18346 | Value of the greatest lowe... |
| glbdm 18347 | Domain of the greatest low... |
| glbfun 18348 | The GLB is a function. (C... |
| glbeldm 18349 | Member of the domain of th... |
| glbelss 18350 | A member of the domain of ... |
| glbeu 18351 | Unique existence proper of... |
| glbval 18352 | Value of the greatest lowe... |
| glbcl 18353 | The least upper bound func... |
| glbprop 18354 | Properties of greatest low... |
| glble 18355 | The greatest lower bound i... |
| joinfval 18356 | Value of join function for... |
| joinfval2 18357 | Value of join function for... |
| joindm 18358 | Domain of join function fo... |
| joindef 18359 | Two ways to say that a joi... |
| joinval 18360 | Join value. Since both si... |
| joincl 18361 | Closure of join of element... |
| joindmss 18362 | Subset property of domain ... |
| joinval2lem 18363 | Lemma for ~ joinval2 and ~... |
| joinval2 18364 | Value of join for a poset ... |
| joineu 18365 | Uniqueness of join of elem... |
| joinlem 18366 | Lemma for join properties.... |
| lejoin1 18367 | A join's first argument is... |
| lejoin2 18368 | A join's second argument i... |
| joinle 18369 | A join is less than or equ... |
| meetfval 18370 | Value of meet function for... |
| meetfval2 18371 | Value of meet function for... |
| meetdm 18372 | Domain of meet function fo... |
| meetdef 18373 | Two ways to say that a mee... |
| meetval 18374 | Meet value. Since both si... |
| meetcl 18375 | Closure of meet of element... |
| meetdmss 18376 | Subset property of domain ... |
| meetval2lem 18377 | Lemma for ~ meetval2 and ~... |
| meetval2 18378 | Value of meet for a poset ... |
| meeteu 18379 | Uniqueness of meet of elem... |
| meetlem 18380 | Lemma for meet properties.... |
| lemeet1 18381 | A meet's first argument is... |
| lemeet2 18382 | A meet's second argument i... |
| meetle 18383 | A meet is less than or equ... |
| joincomALT 18384 | The join of a poset is com... |
| joincom 18385 | The join of a poset is com... |
| meetcomALT 18386 | The meet of a poset is com... |
| meetcom 18387 | The meet of a poset is com... |
| join0 18388 | Lemma for ~ odumeet . (Co... |
| meet0 18389 | Lemma for ~ odujoin . (Co... |
| odulub 18390 | Least upper bounds in a du... |
| odujoin 18391 | Joins in a dual order are ... |
| oduglb 18392 | Greatest lower bounds in a... |
| odumeet 18393 | Meets in a dual order are ... |
| poslubmo 18394 | Least upper bounds in a po... |
| posglbmo 18395 | Greatest lower bounds in a... |
| poslubd 18396 | Properties which determine... |
| poslubdg 18397 | Properties which determine... |
| posglbdg 18398 | Properties which determine... |
| istos 18401 | The predicate "is a toset"... |
| tosso 18402 | Write the totally ordered ... |
| tospos 18403 | A Toset is a Poset. (Cont... |
| tleile 18404 | In a Toset, any two elemen... |
| tltnle 18405 | In a Toset, "less than" is... |
| p0val 18410 | Value of poset zero. (Con... |
| p1val 18411 | Value of poset zero. (Con... |
| p0le 18412 | Any element is less than o... |
| ple1 18413 | Any element is less than o... |
| islat 18416 | The predicate "is a lattic... |
| odulatb 18417 | Being a lattice is self-du... |
| odulat 18418 | Being a lattice is self-du... |
| latcl2 18419 | The join and meet of any t... |
| latlem 18420 | Lemma for lattice properti... |
| latpos 18421 | A lattice is a poset. (Co... |
| latjcl 18422 | Closure of join operation ... |
| latmcl 18423 | Closure of meet operation ... |
| latref 18424 | A lattice ordering is refl... |
| latasymb 18425 | A lattice ordering is asym... |
| latasym 18426 | A lattice ordering is asym... |
| lattr 18427 | A lattice ordering is tran... |
| latasymd 18428 | Deduce equality from latti... |
| lattrd 18429 | A lattice ordering is tran... |
| latjcom 18430 | The join of a lattice comm... |
| latlej1 18431 | A join's first argument is... |
| latlej2 18432 | A join's second argument i... |
| latjle12 18433 | A join is less than or equ... |
| latleeqj1 18434 | "Less than or equal to" in... |
| latleeqj2 18435 | "Less than or equal to" in... |
| latjlej1 18436 | Add join to both sides of ... |
| latjlej2 18437 | Add join to both sides of ... |
| latjlej12 18438 | Add join to both sides of ... |
| latnlej 18439 | An idiom to express that a... |
| latnlej1l 18440 | An idiom to express that a... |
| latnlej1r 18441 | An idiom to express that a... |
| latnlej2 18442 | An idiom to express that a... |
| latnlej2l 18443 | An idiom to express that a... |
| latnlej2r 18444 | An idiom to express that a... |
| latjidm 18445 | Lattice join is idempotent... |
| latmcom 18446 | The join of a lattice comm... |
| latmle1 18447 | A meet is less than or equ... |
| latmle2 18448 | A meet is less than or equ... |
| latlem12 18449 | An element is less than or... |
| latleeqm1 18450 | "Less than or equal to" in... |
| latleeqm2 18451 | "Less than or equal to" in... |
| latmlem1 18452 | Add meet to both sides of ... |
| latmlem2 18453 | Add meet to both sides of ... |
| latmlem12 18454 | Add join to both sides of ... |
| latnlemlt 18455 | Negation of "less than or ... |
| latnle 18456 | Equivalent expressions for... |
| latmidm 18457 | Lattice meet is idempotent... |
| latabs1 18458 | Lattice absorption law. F... |
| latabs2 18459 | Lattice absorption law. F... |
| latledi 18460 | An ortholattice is distrib... |
| latmlej11 18461 | Ordering of a meet and joi... |
| latmlej12 18462 | Ordering of a meet and joi... |
| latmlej21 18463 | Ordering of a meet and joi... |
| latmlej22 18464 | Ordering of a meet and joi... |
| lubsn 18465 | The least upper bound of a... |
| latjass 18466 | Lattice join is associativ... |
| latj12 18467 | Swap 1st and 2nd members o... |
| latj32 18468 | Swap 2nd and 3rd members o... |
| latj13 18469 | Swap 1st and 3rd members o... |
| latj31 18470 | Swap 2nd and 3rd members o... |
| latjrot 18471 | Rotate lattice join of 3 c... |
| latj4 18472 | Rearrangement of lattice j... |
| latj4rot 18473 | Rotate lattice join of 4 c... |
| latjjdi 18474 | Lattice join distributes o... |
| latjjdir 18475 | Lattice join distributes o... |
| mod1ile 18476 | The weak direction of the ... |
| mod2ile 18477 | The weak direction of the ... |
| latmass 18478 | Lattice meet is associativ... |
| latdisdlem 18479 | Lemma for ~ latdisd . (Co... |
| latdisd 18480 | In a lattice, joins distri... |
| isclat 18483 | The predicate "is a comple... |
| clatpos 18484 | A complete lattice is a po... |
| clatlem 18485 | Lemma for properties of a ... |
| clatlubcl 18486 | Any subset of the base set... |
| clatlubcl2 18487 | Any subset of the base set... |
| clatglbcl 18488 | Any subset of the base set... |
| clatglbcl2 18489 | Any subset of the base set... |
| oduclatb 18490 | Being a complete lattice i... |
| clatl 18491 | A complete lattice is a la... |
| isglbd 18492 | Properties that determine ... |
| lublem 18493 | Lemma for the least upper ... |
| lubub 18494 | The LUB of a complete latt... |
| lubl 18495 | The LUB of a complete latt... |
| lubss 18496 | Subset law for least upper... |
| lubel 18497 | An element of a set is les... |
| lubun 18498 | The LUB of a union. (Cont... |
| clatglb 18499 | Properties of greatest low... |
| clatglble 18500 | The greatest lower bound i... |
| clatleglb 18501 | Two ways of expressing "le... |
| clatglbss 18502 | Subset law for greatest lo... |
| isdlat 18505 | Property of being a distri... |
| dlatmjdi 18506 | In a distributive lattice,... |
| dlatl 18507 | A distributive lattice is ... |
| odudlatb 18508 | The dual of a distributive... |
| dlatjmdi 18509 | In a distributive lattice,... |
| ipostr 18512 | The structure of ~ df-ipo ... |
| ipoval 18513 | Value of the inclusion pos... |
| ipobas 18514 | Base set of the inclusion ... |
| ipolerval 18515 | Relation of the inclusion ... |
| ipotset 18516 | Topology of the inclusion ... |
| ipole 18517 | Weak order condition of th... |
| ipolt 18518 | Strict order condition of ... |
| ipopos 18519 | The inclusion poset on a f... |
| isipodrs 18520 | Condition for a family of ... |
| ipodrscl 18521 | Direction by inclusion as ... |
| ipodrsfi 18522 | Finite upper bound propert... |
| fpwipodrs 18523 | The finite subsets of any ... |
| ipodrsima 18524 | The monotone image of a di... |
| isacs3lem 18525 | An algebraic closure syste... |
| acsdrsel 18526 | An algebraic closure syste... |
| isacs4lem 18527 | In a closure system in whi... |
| isacs5lem 18528 | If closure commutes with d... |
| acsdrscl 18529 | In an algebraic closure sy... |
| acsficl 18530 | A closure in an algebraic ... |
| isacs5 18531 | A closure system is algebr... |
| isacs4 18532 | A closure system is algebr... |
| isacs3 18533 | A closure system is algebr... |
| acsficld 18534 | In an algebraic closure sy... |
| acsficl2d 18535 | In an algebraic closure sy... |
| acsfiindd 18536 | In an algebraic closure sy... |
| acsmapd 18537 | In an algebraic closure sy... |
| acsmap2d 18538 | In an algebraic closure sy... |
| acsinfd 18539 | In an algebraic closure sy... |
| acsdomd 18540 | In an algebraic closure sy... |
| acsinfdimd 18541 | In an algebraic closure sy... |
| acsexdimd 18542 | In an algebraic closure sy... |
| mrelatglb 18543 | Greatest lower bounds in a... |
| mrelatglb0 18544 | The empty intersection in ... |
| mrelatlub 18545 | Least upper bounds in a Mo... |
| mreclatBAD 18546 | A Moore space is a complet... |
| isps 18551 | The predicate "is a poset"... |
| psrel 18552 | A poset is a relation. (C... |
| psref2 18553 | A poset is antisymmetric a... |
| pstr2 18554 | A poset is transitive. (C... |
| pslem 18555 | Lemma for ~ psref and othe... |
| psdmrn 18556 | The domain and range of a ... |
| psref 18557 | A poset is reflexive. (Co... |
| psrn 18558 | The range of a poset equal... |
| psasym 18559 | A poset is antisymmetric. ... |
| pstr 18560 | A poset is transitive. (C... |
| cnvps 18561 | The converse of a poset is... |
| cnvpsb 18562 | The converse of a poset is... |
| psss 18563 | Any subset of a partially ... |
| psssdm2 18564 | Field of a subposet. (Con... |
| psssdm 18565 | Field of a subposet. (Con... |
| istsr 18566 | The predicate is a toset. ... |
| istsr2 18567 | The predicate is a toset. ... |
| tsrlin 18568 | A toset is a linear order.... |
| tsrlemax 18569 | Two ways of saying a numbe... |
| tsrps 18570 | A toset is a poset. (Cont... |
| cnvtsr 18571 | The converse of a toset is... |
| tsrss 18572 | Any subset of a totally or... |
| ledm 18573 | The domain of ` <_ ` is ` ... |
| lern 18574 | The range of ` <_ ` is ` R... |
| lefld 18575 | The field of the 'less or ... |
| letsr 18576 | The "less than or equal to... |
| isdir 18581 | A condition for a relation... |
| reldir 18582 | A direction is a relation.... |
| dirdm 18583 | A direction's domain is eq... |
| dirref 18584 | A direction is reflexive. ... |
| dirtr 18585 | A direction is transitive.... |
| dirge 18586 | For any two elements of a ... |
| tsrdir 18587 | A totally ordered set is a... |
| ismgm 18592 | The predicate "is a magma"... |
| ismgmn0 18593 | The predicate "is a magma"... |
| mgmcl 18594 | Closure of the operation o... |
| isnmgm 18595 | A condition for a structur... |
| mgmsscl 18596 | If the base set of a magma... |
| plusffval 18597 | The group addition operati... |
| plusfval 18598 | The group addition operati... |
| plusfeq 18599 | If the addition operation ... |
| plusffn 18600 | The group addition operati... |
| mgmplusf 18601 | The group addition functio... |
| mgmpropd 18602 | If two structures have the... |
| ismgmd 18603 | Deduce a magma from its pr... |
| issstrmgm 18604 | Characterize a substructur... |
| intopsn 18605 | The internal operation for... |
| mgmb1mgm1 18606 | The only magma with a base... |
| mgm0 18607 | Any set with an empty base... |
| mgm0b 18608 | The structure with an empt... |
| mgm1 18609 | The structure with one ele... |
| opifismgm 18610 | A structure with a group a... |
| mgmidmo 18611 | A two-sided identity eleme... |
| grpidval 18612 | The value of the identity ... |
| grpidpropd 18613 | If two structures have the... |
| fn0g 18614 | The group zero extractor i... |
| 0g0 18615 | The identity element funct... |
| ismgmid 18616 | The identity element of a ... |
| mgmidcl 18617 | The identity element of a ... |
| mgmlrid 18618 | The identity element of a ... |
| ismgmid2 18619 | Show that a given element ... |
| lidrideqd 18620 | If there is a left and rig... |
| lidrididd 18621 | If there is a left and rig... |
| grpidd 18622 | Deduce the identity elemen... |
| mgmidsssn0 18623 | Property of the set of ide... |
| grpinvalem 18624 | Lemma for ~ grpinva . (Co... |
| grpinva 18625 | Deduce right inverse from ... |
| grprida 18626 | Deduce right identity from... |
| gsumvalx 18627 | Expand out the substitutio... |
| gsumval 18628 | Expand out the substitutio... |
| gsumpropd 18629 | The group sum depends only... |
| gsumpropd2lem 18630 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18631 | A stronger version of ~ gs... |
| gsummgmpropd 18632 | A stronger version of ~ gs... |
| gsumress 18633 | The group sum in a substru... |
| gsumval1 18634 | Value of the group sum ope... |
| gsum0 18635 | Value of the empty group s... |
| gsumval2a 18636 | Value of the group sum ope... |
| gsumval2 18637 | Value of the group sum ope... |
| gsumsplit1r 18638 | Splitting off the rightmos... |
| gsumprval 18639 | Value of the group sum ope... |
| gsumpr12val 18640 | Value of the group sum ope... |
| mgmhmrcl 18645 | Reverse closure of a magma... |
| submgmrcl 18646 | Reverse closure for submag... |
| ismgmhm 18647 | Property of a magma homomo... |
| mgmhmf 18648 | A magma homomorphism is a ... |
| mgmhmpropd 18649 | Magma homomorphism depends... |
| mgmhmlin 18650 | A magma homomorphism prese... |
| mgmhmf1o 18651 | A magma homomorphism is bi... |
| idmgmhm 18652 | The identity homomorphism ... |
| issubmgm 18653 | Expand definition of a sub... |
| issubmgm2 18654 | Submagmas are subsets that... |
| rabsubmgmd 18655 | Deduction for proving that... |
| submgmss 18656 | Submagmas are subsets of t... |
| submgmid 18657 | Every magma is trivially a... |
| submgmcl 18658 | Submagmas are closed under... |
| submgmmgm 18659 | Submagmas are themselves m... |
| submgmbas 18660 | The base set of a submagma... |
| subsubmgm 18661 | A submagma of a submagma i... |
| resmgmhm 18662 | Restriction of a magma hom... |
| resmgmhm2 18663 | One direction of ~ resmgmh... |
| resmgmhm2b 18664 | Restriction of the codomai... |
| mgmhmco 18665 | The composition of magma h... |
| mgmhmima 18666 | The homomorphic image of a... |
| mgmhmeql 18667 | The equalizer of two magma... |
| submgmacs 18668 | Submagmas are an algebraic... |
| issgrp 18671 | The predicate "is a semigr... |
| issgrpv 18672 | The predicate "is a semigr... |
| issgrpn0 18673 | The predicate "is a semigr... |
| isnsgrp 18674 | A condition for a structur... |
| sgrpmgm 18675 | A semigroup is a magma. (... |
| sgrpass 18676 | A semigroup operation is a... |
| sgrpcl 18677 | Closure of the operation o... |
| sgrp0 18678 | Any set with an empty base... |
| sgrp0b 18679 | The structure with an empt... |
| sgrp1 18680 | The structure with one ele... |
| issgrpd 18681 | Deduce a semigroup from it... |
| sgrppropd 18682 | If two structures are sets... |
| prdsplusgsgrpcl 18683 | Structure product pointwis... |
| prdssgrpd 18684 | The product of a family of... |
| ismnddef 18687 | The predicate "is a monoid... |
| ismnd 18688 | The predicate "is a monoid... |
| isnmnd 18689 | A condition for a structur... |
| sgrpidmnd 18690 | A semigroup with an identi... |
| mndsgrp 18691 | A monoid is a semigroup. ... |
| mndmgm 18692 | A monoid is a magma. (Con... |
| mndcl 18693 | Closure of the operation o... |
| mndass 18694 | A monoid operation is asso... |
| mndid 18695 | A monoid has a two-sided i... |
| mndideu 18696 | The two-sided identity ele... |
| mnd32g 18697 | Commutative/associative la... |
| mnd12g 18698 | Commutative/associative la... |
| mnd4g 18699 | Commutative/associative la... |
| mndidcl 18700 | The identity element of a ... |
| mndbn0 18701 | The base set of a monoid i... |
| hashfinmndnn 18702 | A finite monoid has positi... |
| mndplusf 18703 | The group addition operati... |
| mndlrid 18704 | A monoid's identity elemen... |
| mndlid 18705 | The identity element of a ... |
| mndrid 18706 | The identity element of a ... |
| ismndd 18707 | Deduce a monoid from its p... |
| mndpfo 18708 | The addition operation of ... |
| mndfo 18709 | The addition operation of ... |
| mndpropd 18710 | If two structures have the... |
| mndprop 18711 | If two structures have the... |
| issubmnd 18712 | Characterize a submonoid b... |
| ress0g 18713 | ` 0g ` is unaffected by re... |
| submnd0 18714 | The zero of a submonoid is... |
| mndinvmod 18715 | Uniqueness of an inverse e... |
| prdsplusgcl 18716 | Structure product pointwis... |
| prdsidlem 18717 | Characterization of identi... |
| prdsmndd 18718 | The product of a family of... |
| prds0g 18719 | Zero in a product of monoi... |
| pwsmnd 18720 | The structure power of a m... |
| pws0g 18721 | Zero in a structure power ... |
| imasmnd2 18722 | The image structure of a m... |
| imasmnd 18723 | The image structure of a m... |
| imasmndf1 18724 | The image of a monoid unde... |
| xpsmnd 18725 | The binary product of mono... |
| xpsmnd0 18726 | The identity element of a ... |
| mnd1 18727 | The (smallest) structure r... |
| mnd1id 18728 | The singleton element of a... |
| ismhm 18733 | Property of a monoid homom... |
| ismhmd 18734 | Deduction version of ~ ism... |
| mhmrcl1 18735 | Reverse closure of a monoi... |
| mhmrcl2 18736 | Reverse closure of a monoi... |
| mhmf 18737 | A monoid homomorphism is a... |
| ismhm0 18738 | Property of a monoid homom... |
| mhmismgmhm 18739 | Each monoid homomorphism i... |
| mhmpropd 18740 | Monoid homomorphism depend... |
| mhmlin 18741 | A monoid homomorphism comm... |
| mhm0 18742 | A monoid homomorphism pres... |
| idmhm 18743 | The identity homomorphism ... |
| mhmf1o 18744 | A monoid homomorphism is b... |
| submrcl 18745 | Reverse closure for submon... |
| issubm 18746 | Expand definition of a sub... |
| issubm2 18747 | Submonoids are subsets tha... |
| issubmndb 18748 | The submonoid predicate. ... |
| issubmd 18749 | Deduction for proving a su... |
| mndissubm 18750 | If the base set of a monoi... |
| resmndismnd 18751 | If the base set of a monoi... |
| submss 18752 | Submonoids are subsets of ... |
| submid 18753 | Every monoid is trivially ... |
| subm0cl 18754 | Submonoids contain zero. ... |
| submcl 18755 | Submonoids are closed unde... |
| submmnd 18756 | Submonoids are themselves ... |
| submbas 18757 | The base set of a submonoi... |
| subm0 18758 | Submonoids have the same i... |
| subsubm 18759 | A submonoid of a submonoid... |
| 0subm 18760 | The zero submonoid of an a... |
| insubm 18761 | The intersection of two su... |
| 0mhm 18762 | The constant zero linear f... |
| resmhm 18763 | Restriction of a monoid ho... |
| resmhm2 18764 | One direction of ~ resmhm2... |
| resmhm2b 18765 | Restriction of the codomai... |
| mhmco 18766 | The composition of monoid ... |
| mhmimalem 18767 | Lemma for ~ mhmima and sim... |
| mhmima 18768 | The homomorphic image of a... |
| mhmeql 18769 | The equalizer of two monoi... |
| submacs 18770 | Submonoids are an algebrai... |
| mndind 18771 | Induction in a monoid. In... |
| prdspjmhm 18772 | A projection from a produc... |
| pwspjmhm 18773 | A projection from a struct... |
| pwsdiagmhm 18774 | Diagonal monoid homomorphi... |
| pwsco1mhm 18775 | Right composition with a f... |
| pwsco2mhm 18776 | Left composition with a mo... |
| gsumvallem2 18777 | Lemma for properties of th... |
| gsumsubm 18778 | Evaluate a group sum in a ... |
| gsumz 18779 | Value of a group sum over ... |
| gsumwsubmcl 18780 | Closure of the composite i... |
| gsumws1 18781 | A singleton composite reco... |
| gsumwcl 18782 | Closure of the composite o... |
| gsumsgrpccat 18783 | Homomorphic property of no... |
| gsumccat 18784 | Homomorphic property of co... |
| gsumws2 18785 | Valuation of a pair in a m... |
| gsumccatsn 18786 | Homomorphic property of co... |
| gsumspl 18787 | The primary purpose of the... |
| gsumwmhm 18788 | Behavior of homomorphisms ... |
| gsumwspan 18789 | The submonoid generated by... |
| frmdval 18794 | Value of the free monoid c... |
| frmdbas 18795 | The base set of a free mon... |
| frmdelbas 18796 | An element of the base set... |
| frmdplusg 18797 | The monoid operation of a ... |
| frmdadd 18798 | Value of the monoid operat... |
| vrmdfval 18799 | The canonical injection fr... |
| vrmdval 18800 | The value of the generatin... |
| vrmdf 18801 | The mapping from the index... |
| frmdmnd 18802 | A free monoid is a monoid.... |
| frmd0 18803 | The identity of the free m... |
| frmdsssubm 18804 | The set of words taking va... |
| frmdgsum 18805 | Any word in a free monoid ... |
| frmdss2 18806 | A subset of generators is ... |
| frmdup1 18807 | Any assignment of the gene... |
| frmdup2 18808 | The evaluation map has the... |
| frmdup3lem 18809 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18810 | Universal property of the ... |
| efmnd 18813 | The monoid of endofunction... |
| efmndbas 18814 | The base set of the monoid... |
| efmndbasabf 18815 | The base set of the monoid... |
| elefmndbas 18816 | Two ways of saying a funct... |
| elefmndbas2 18817 | Two ways of saying a funct... |
| efmndbasf 18818 | Elements in the monoid of ... |
| efmndhash 18819 | The monoid of endofunction... |
| efmndbasfi 18820 | The monoid of endofunction... |
| efmndfv 18821 | The function value of an e... |
| efmndtset 18822 | The topology of the monoid... |
| efmndplusg 18823 | The group operation of a m... |
| efmndov 18824 | The value of the group ope... |
| efmndcl 18825 | The group operation of the... |
| efmndtopn 18826 | The topology of the monoid... |
| symggrplem 18827 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18828 | The monoid of endofunction... |
| efmndsgrp 18829 | The monoid of endofunction... |
| ielefmnd 18830 | The identity function rest... |
| efmndid 18831 | The identity function rest... |
| efmndmnd 18832 | The monoid of endofunction... |
| efmnd0nmnd 18833 | Even the monoid of endofun... |
| efmndbas0 18834 | The base set of the monoid... |
| efmnd1hash 18835 | The monoid of endofunction... |
| efmnd1bas 18836 | The monoid of endofunction... |
| efmnd2hash 18837 | The monoid of endofunction... |
| submefmnd 18838 | If the base set of a monoi... |
| sursubmefmnd 18839 | The set of surjective endo... |
| injsubmefmnd 18840 | The set of injective endof... |
| idressubmefmnd 18841 | The singleton containing o... |
| idresefmnd 18842 | The structure with the sin... |
| smndex1ibas 18843 | The modulo function ` I ` ... |
| smndex1iidm 18844 | The modulo function ` I ` ... |
| smndex1gbas 18845 | The constant functions ` (... |
| smndex1gid 18846 | The composition of a const... |
| smndex1igid 18847 | The composition of the mod... |
| smndex1basss 18848 | The modulo function ` I ` ... |
| smndex1bas 18849 | The base set of the monoid... |
| smndex1mgm 18850 | The monoid of endofunction... |
| smndex1sgrp 18851 | The monoid of endofunction... |
| smndex1mndlem 18852 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18853 | The monoid of endofunction... |
| smndex1id 18854 | The modulo function ` I ` ... |
| smndex1n0mnd 18855 | The identity of the monoid... |
| nsmndex1 18856 | The base set ` B ` of the ... |
| smndex2dbas 18857 | The doubling function ` D ... |
| smndex2dnrinv 18858 | The doubling function ` D ... |
| smndex2hbas 18859 | The halving functions ` H ... |
| smndex2dlinvh 18860 | The halving functions ` H ... |
| mgm2nsgrplem1 18861 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18862 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18863 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18864 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18865 | A small magma (with two el... |
| sgrp2nmndlem1 18866 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18867 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18868 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18869 | A small semigroup (with tw... |
| sgrp2rid2ex 18870 | A small semigroup (with tw... |
| sgrp2nmndlem4 18871 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18872 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18873 | A small semigroup (with tw... |
| mgmnsgrpex 18874 | There is a magma which is ... |
| sgrpnmndex 18875 | There is a semigroup which... |
| sgrpssmgm 18876 | The class of all semigroup... |
| mndsssgrp 18877 | The class of all monoids i... |
| pwmndgplus 18878 | The operation of the monoi... |
| pwmndid 18879 | The identity of the monoid... |
| pwmnd 18880 | The power set of a class `... |
| isgrp 18887 | The predicate "is a group"... |
| grpmnd 18888 | A group is a monoid. (Con... |
| grpcl 18889 | Closure of the operation o... |
| grpass 18890 | A group operation is assoc... |
| grpinvex 18891 | Every member of a group ha... |
| grpideu 18892 | The two-sided identity ele... |
| grpassd 18893 | A group operation is assoc... |
| grpmndd 18894 | A group is a monoid. (Con... |
| grpcld 18895 | Closure of the operation o... |
| grpplusf 18896 | The group addition operati... |
| grpplusfo 18897 | The group addition operati... |
| resgrpplusfrn 18898 | The underlying set of a gr... |
| grppropd 18899 | If two structures have the... |
| grpprop 18900 | If two structures have the... |
| grppropstr 18901 | Generalize a specific 2-el... |
| grpss 18902 | Show that a structure exte... |
| isgrpd2e 18903 | Deduce a group from its pr... |
| isgrpd2 18904 | Deduce a group from its pr... |
| isgrpde 18905 | Deduce a group from its pr... |
| isgrpd 18906 | Deduce a group from its pr... |
| isgrpi 18907 | Properties that determine ... |
| grpsgrp 18908 | A group is a semigroup. (... |
| grpmgmd 18909 | A group is a magma, deduct... |
| dfgrp2 18910 | Alternate definition of a ... |
| dfgrp2e 18911 | Alternate definition of a ... |
| isgrpix 18912 | Properties that determine ... |
| grpidcl 18913 | The identity element of a ... |
| grpbn0 18914 | The base set of a group is... |
| grplid 18915 | The identity element of a ... |
| grprid 18916 | The identity element of a ... |
| grplidd 18917 | The identity element of a ... |
| grpridd 18918 | The identity element of a ... |
| grpn0 18919 | A group is not empty. (Co... |
| hashfingrpnn 18920 | A finite group has positiv... |
| grprcan 18921 | Right cancellation law for... |
| grpinveu 18922 | The left inverse element o... |
| grpid 18923 | Two ways of saying that an... |
| isgrpid2 18924 | Properties showing that an... |
| grpidd2 18925 | Deduce the identity elemen... |
| grpinvfval 18926 | The inverse function of a ... |
| grpinvfvalALT 18927 | Shorter proof of ~ grpinvf... |
| grpinvval 18928 | The inverse of a group ele... |
| grpinvfn 18929 | Functionality of the group... |
| grpinvfvi 18930 | The group inverse function... |
| grpsubfval 18931 | Group subtraction (divisio... |
| grpsubfvalALT 18932 | Shorter proof of ~ grpsubf... |
| grpsubval 18933 | Group subtraction (divisio... |
| grpinvf 18934 | The group inversion operat... |
| grpinvcl 18935 | A group element's inverse ... |
| grpinvcld 18936 | A group element's inverse ... |
| grplinv 18937 | The left inverse of a grou... |
| grprinv 18938 | The right inverse of a gro... |
| grpinvid1 18939 | The inverse of a group ele... |
| grpinvid2 18940 | The inverse of a group ele... |
| isgrpinv 18941 | Properties showing that a ... |
| grplinvd 18942 | The left inverse of a grou... |
| grprinvd 18943 | The right inverse of a gro... |
| grplrinv 18944 | In a group, every member h... |
| grpidinv2 18945 | A group's properties using... |
| grpidinv 18946 | A group has a left and rig... |
| grpinvid 18947 | The inverse of the identit... |
| grplcan 18948 | Left cancellation law for ... |
| grpasscan1 18949 | An associative cancellatio... |
| grpasscan2 18950 | An associative cancellatio... |
| grpidrcan 18951 | If right adding an element... |
| grpidlcan 18952 | If left adding an element ... |
| grpinvinv 18953 | Double inverse law for gro... |
| grpinvcnv 18954 | The group inverse is its o... |
| grpinv11 18955 | The group inverse is one-t... |
| grpinvf1o 18956 | The group inverse is a one... |
| grpinvnz 18957 | The inverse of a nonzero g... |
| grpinvnzcl 18958 | The inverse of a nonzero g... |
| grpsubinv 18959 | Subtraction of an inverse.... |
| grplmulf1o 18960 | Left multiplication by a g... |
| grpraddf1o 18961 | Right addition by a group ... |
| grpinvpropd 18962 | If two structures have the... |
| grpidssd 18963 | If the base set of a group... |
| grpinvssd 18964 | If the base set of a group... |
| grpinvadd 18965 | The inverse of the group o... |
| grpsubf 18966 | Functionality of group sub... |
| grpsubcl 18967 | Closure of group subtracti... |
| grpsubrcan 18968 | Right cancellation law for... |
| grpinvsub 18969 | Inverse of a group subtrac... |
| grpinvval2 18970 | A ~ df-neg -like equation ... |
| grpsubid 18971 | Subtraction of a group ele... |
| grpsubid1 18972 | Subtraction of the identit... |
| grpsubeq0 18973 | If the difference between ... |
| grpsubadd0sub 18974 | Subtraction expressed as a... |
| grpsubadd 18975 | Relationship between group... |
| grpsubsub 18976 | Double group subtraction. ... |
| grpaddsubass 18977 | Associative-type law for g... |
| grppncan 18978 | Cancellation law for subtr... |
| grpnpcan 18979 | Cancellation law for subtr... |
| grpsubsub4 18980 | Double group subtraction (... |
| grppnpcan2 18981 | Cancellation law for mixed... |
| grpnpncan 18982 | Cancellation law for group... |
| grpnpncan0 18983 | Cancellation law for group... |
| grpnnncan2 18984 | Cancellation law for group... |
| dfgrp3lem 18985 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 18986 | Alternate definition of a ... |
| dfgrp3e 18987 | Alternate definition of a ... |
| grplactfval 18988 | The left group action of e... |
| grplactval 18989 | The value of the left grou... |
| grplactcnv 18990 | The left group action of e... |
| grplactf1o 18991 | The left group action of e... |
| grpsubpropd 18992 | Weak property deduction fo... |
| grpsubpropd2 18993 | Strong property deduction ... |
| grp1 18994 | The (smallest) structure r... |
| grp1inv 18995 | The inverse function of th... |
| prdsinvlem 18996 | Characterization of invers... |
| prdsgrpd 18997 | The product of a family of... |
| prdsinvgd 18998 | Negation in a product of g... |
| pwsgrp 18999 | A structure power of a gro... |
| pwsinvg 19000 | Negation in a group power.... |
| pwssub 19001 | Subtraction in a group pow... |
| imasgrp2 19002 | The image structure of a g... |
| imasgrp 19003 | The image structure of a g... |
| imasgrpf1 19004 | The image of a group under... |
| qusgrp2 19005 | Prove that a quotient stru... |
| xpsgrp 19006 | The binary product of grou... |
| xpsinv 19007 | Value of the negation oper... |
| xpsgrpsub 19008 | Value of the subtraction o... |
| mhmlem 19009 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 19010 | A surjective monoid morphi... |
| mhmmnd 19011 | The image of a monoid ` G ... |
| mhmfmhm 19012 | The function fulfilling th... |
| ghmgrp 19013 | The image of a group ` G `... |
| mulgfval 19016 | Group multiple (exponentia... |
| mulgfvalALT 19017 | Shorter proof of ~ mulgfva... |
| mulgval 19018 | Value of the group multipl... |
| mulgfn 19019 | Functionality of the group... |
| mulgfvi 19020 | The group multiple operati... |
| mulg0 19021 | Group multiple (exponentia... |
| mulgnn 19022 | Group multiple (exponentia... |
| ressmulgnn 19023 | Values for the group multi... |
| ressmulgnn0 19024 | Values for the group multi... |
| mulgnngsum 19025 | Group multiple (exponentia... |
| mulgnn0gsum 19026 | Group multiple (exponentia... |
| mulg1 19027 | Group multiple (exponentia... |
| mulgnnp1 19028 | Group multiple (exponentia... |
| mulg2 19029 | Group multiple (exponentia... |
| mulgnegnn 19030 | Group multiple (exponentia... |
| mulgnn0p1 19031 | Group multiple (exponentia... |
| mulgnnsubcl 19032 | Closure of the group multi... |
| mulgnn0subcl 19033 | Closure of the group multi... |
| mulgsubcl 19034 | Closure of the group multi... |
| mulgnncl 19035 | Closure of the group multi... |
| mulgnn0cl 19036 | Closure of the group multi... |
| mulgcl 19037 | Closure of the group multi... |
| mulgneg 19038 | Group multiple (exponentia... |
| mulgnegneg 19039 | The inverse of a negative ... |
| mulgm1 19040 | Group multiple (exponentia... |
| mulgnn0cld 19041 | Closure of the group multi... |
| mulgcld 19042 | Deduction associated with ... |
| mulgaddcomlem 19043 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19044 | The group multiple operato... |
| mulginvcom 19045 | The group multiple operato... |
| mulginvinv 19046 | The group multiple operato... |
| mulgnn0z 19047 | A group multiple of the id... |
| mulgz 19048 | A group multiple of the id... |
| mulgnndir 19049 | Sum of group multiples, fo... |
| mulgnn0dir 19050 | Sum of group multiples, ge... |
| mulgdirlem 19051 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19052 | Sum of group multiples, ge... |
| mulgp1 19053 | Group multiple (exponentia... |
| mulgneg2 19054 | Group multiple (exponentia... |
| mulgnnass 19055 | Product of group multiples... |
| mulgnn0ass 19056 | Product of group multiples... |
| mulgass 19057 | Product of group multiples... |
| mulgassr 19058 | Reversed product of group ... |
| mulgmodid 19059 | Casting out multiples of t... |
| mulgsubdir 19060 | Distribution of group mult... |
| mhmmulg 19061 | A homomorphism of monoids ... |
| mulgpropd 19062 | Two structures with the sa... |
| submmulgcl 19063 | Closure of the group multi... |
| submmulg 19064 | A group multiple is the sa... |
| pwsmulg 19065 | Value of a group multiple ... |
| issubg 19072 | The subgroup predicate. (... |
| subgss 19073 | A subgroup is a subset. (... |
| subgid 19074 | A group is a subgroup of i... |
| subggrp 19075 | A subgroup is a group. (C... |
| subgbas 19076 | The base of the restricted... |
| subgrcl 19077 | Reverse closure for the su... |
| subg0 19078 | A subgroup of a group must... |
| subginv 19079 | The inverse of an element ... |
| subg0cl 19080 | The group identity is an e... |
| subginvcl 19081 | The inverse of an element ... |
| subgcl 19082 | A subgroup is closed under... |
| subgsubcl 19083 | A subgroup is closed under... |
| subgsub 19084 | The subtraction of element... |
| subgmulgcl 19085 | Closure of the group multi... |
| subgmulg 19086 | A group multiple is the sa... |
| issubg2 19087 | Characterize the subgroups... |
| issubgrpd2 19088 | Prove a subgroup by closur... |
| issubgrpd 19089 | Prove a subgroup by closur... |
| issubg3 19090 | A subgroup is a symmetric ... |
| issubg4 19091 | A subgroup is a nonempty s... |
| grpissubg 19092 | If the base set of a group... |
| resgrpisgrp 19093 | If the base set of a group... |
| subgsubm 19094 | A subgroup is a submonoid.... |
| subsubg 19095 | A subgroup of a subgroup i... |
| subgint 19096 | The intersection of a none... |
| 0subg 19097 | The zero subgroup of an ar... |
| 0subgOLD 19098 | Obsolete version of ~ 0sub... |
| trivsubgd 19099 | The only subgroup of a tri... |
| trivsubgsnd 19100 | The only subgroup of a tri... |
| isnsg 19101 | Property of being a normal... |
| isnsg2 19102 | Weaken the condition of ~ ... |
| nsgbi 19103 | Defining property of a nor... |
| nsgsubg 19104 | A normal subgroup is a sub... |
| nsgconj 19105 | The conjugation of an elem... |
| isnsg3 19106 | A subgroup is normal iff t... |
| subgacs 19107 | Subgroups are an algebraic... |
| nsgacs 19108 | Normal subgroups form an a... |
| elnmz 19109 | Elementhood in the normali... |
| nmzbi 19110 | Defining property of the n... |
| nmzsubg 19111 | The normalizer N_G(S) of a... |
| ssnmz 19112 | A subgroup is a subset of ... |
| isnsg4 19113 | A subgroup is normal iff i... |
| nmznsg 19114 | Any subgroup is a normal s... |
| 0nsg 19115 | The zero subgroup is norma... |
| nsgid 19116 | The whole group is a norma... |
| 0idnsgd 19117 | The whole group and the ze... |
| trivnsgd 19118 | The only normal subgroup o... |
| triv1nsgd 19119 | A trivial group has exactl... |
| 1nsgtrivd 19120 | A group with exactly one n... |
| releqg 19121 | The left coset equivalence... |
| eqgfval 19122 | Value of the subgroup left... |
| eqgval 19123 | Value of the subgroup left... |
| eqger 19124 | The subgroup coset equival... |
| eqglact 19125 | A left coset can be expres... |
| eqgid 19126 | The left coset containing ... |
| eqgen 19127 | Each coset is equipotent t... |
| eqgcpbl 19128 | The subgroup coset equival... |
| eqg0el 19129 | Equivalence class of a quo... |
| quselbas 19130 | Membership in the base set... |
| quseccl0 19131 | Closure of the quotient ma... |
| qusgrp 19132 | If ` Y ` is a normal subgr... |
| quseccl 19133 | Closure of the quotient ma... |
| qusadd 19134 | Value of the group operati... |
| qus0 19135 | Value of the group identit... |
| qusinv 19136 | Value of the group inverse... |
| qussub 19137 | Value of the group subtrac... |
| ecqusaddd 19138 | Addition of equivalence cl... |
| ecqusaddcl 19139 | Closure of the addition in... |
| lagsubg2 19140 | Lagrange's theorem for fin... |
| lagsubg 19141 | Lagrange's theorem for Gro... |
| eqg0subg 19142 | The coset equivalence rela... |
| eqg0subgecsn 19143 | The equivalence classes mo... |
| qus0subgbas 19144 | The base set of a quotient... |
| qus0subgadd 19145 | The addition in a quotient... |
| cycsubmel 19146 | Characterization of an ele... |
| cycsubmcl 19147 | The set of nonnegative int... |
| cycsubm 19148 | The set of nonnegative int... |
| cyccom 19149 | Condition for an operation... |
| cycsubmcom 19150 | The operation of a monoid ... |
| cycsubggend 19151 | The cyclic subgroup genera... |
| cycsubgcl 19152 | The set of integer powers ... |
| cycsubgss 19153 | The cyclic subgroup genera... |
| cycsubg 19154 | The cyclic group generated... |
| cycsubgcld 19155 | The cyclic subgroup genera... |
| cycsubg2 19156 | The subgroup generated by ... |
| cycsubg2cl 19157 | Any multiple of an element... |
| reldmghm 19160 | Lemma for group homomorphi... |
| isghm 19161 | Property of being a homomo... |
| isghm3 19162 | Property of a group homomo... |
| ghmgrp1 19163 | A group homomorphism is on... |
| ghmgrp2 19164 | A group homomorphism is on... |
| ghmf 19165 | A group homomorphism is a ... |
| ghmlin 19166 | A homomorphism of groups i... |
| ghmid 19167 | A homomorphism of groups p... |
| ghminv 19168 | A homomorphism of groups p... |
| ghmsub 19169 | Linearity of subtraction t... |
| isghmd 19170 | Deduction for a group homo... |
| ghmmhm 19171 | A group homomorphism is a ... |
| ghmmhmb 19172 | Group homomorphisms and mo... |
| ghmmulg 19173 | A homomorphism of monoids ... |
| ghmrn 19174 | The range of a homomorphis... |
| 0ghm 19175 | The constant zero linear f... |
| idghm 19176 | The identity homomorphism ... |
| resghm 19177 | Restriction of a homomorph... |
| resghm2 19178 | One direction of ~ resghm2... |
| resghm2b 19179 | Restriction of the codomai... |
| ghmghmrn 19180 | A group homomorphism from ... |
| ghmco 19181 | The composition of group h... |
| ghmima 19182 | The image of a subgroup un... |
| ghmpreima 19183 | The inverse image of a sub... |
| ghmeql 19184 | The equalizer of two group... |
| ghmnsgima 19185 | The image of a normal subg... |
| ghmnsgpreima 19186 | The inverse image of a nor... |
| ghmker 19187 | The kernel of a homomorphi... |
| ghmeqker 19188 | Two source points map to t... |
| pwsdiagghm 19189 | Diagonal homomorphism into... |
| f1ghm0to0 19190 | If a group homomorphism ` ... |
| ghmf1 19191 | Two ways of saying a group... |
| kerf1ghm 19192 | A group homomorphism ` F `... |
| ghmf1o 19193 | A bijective group homomorp... |
| conjghm 19194 | Conjugation is an automorp... |
| conjsubg 19195 | A conjugated subgroup is a... |
| conjsubgen 19196 | A conjugated subgroup is e... |
| conjnmz 19197 | A subgroup is unchanged un... |
| conjnmzb 19198 | Alternative condition for ... |
| conjnsg 19199 | A normal subgroup is uncha... |
| qusghm 19200 | If ` Y ` is a normal subgr... |
| ghmpropd 19201 | Group homomorphism depends... |
| gimfn 19206 | The group isomorphism func... |
| isgim 19207 | An isomorphism of groups i... |
| gimf1o 19208 | An isomorphism of groups i... |
| gimghm 19209 | An isomorphism of groups i... |
| isgim2 19210 | A group isomorphism is a h... |
| subggim 19211 | Behavior of subgroups unde... |
| gimcnv 19212 | The converse of a group is... |
| gimco 19213 | The composition of group i... |
| gim0to0 19214 | A group isomorphism maps t... |
| brgic 19215 | The relation "is isomorphi... |
| brgici 19216 | Prove isomorphic by an exp... |
| gicref 19217 | Isomorphism is reflexive. ... |
| giclcl 19218 | Isomorphism implies the le... |
| gicrcl 19219 | Isomorphism implies the ri... |
| gicsym 19220 | Isomorphism is symmetric. ... |
| gictr 19221 | Isomorphism is transitive.... |
| gicer 19222 | Isomorphism is an equivale... |
| gicen 19223 | Isomorphic groups have equ... |
| gicsubgen 19224 | A less trivial example of ... |
| ghmquskerlem1 19225 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19226 | In the case of theorem ~ g... |
| ghmquskerlem2 19227 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19228 | The mapping ` H ` induced ... |
| ghmqusker 19229 | A surjective group homomor... |
| gicqusker 19230 | The image ` H ` of a group... |
| isga 19233 | The predicate "is a (left)... |
| gagrp 19234 | The left argument of a gro... |
| gaset 19235 | The right argument of a gr... |
| gagrpid 19236 | The identity of the group ... |
| gaf 19237 | The mapping of the group a... |
| gafo 19238 | A group action is onto its... |
| gaass 19239 | An "associative" property ... |
| ga0 19240 | The action of a group on t... |
| gaid 19241 | The trivial action of a gr... |
| subgga 19242 | A subgroup acts on its par... |
| gass 19243 | A subset of a group action... |
| gasubg 19244 | The restriction of a group... |
| gaid2 19245 | A group operation is a lef... |
| galcan 19246 | The action of a particular... |
| gacan 19247 | Group inverses cancel in a... |
| gapm 19248 | The action of a particular... |
| gaorb 19249 | The orbit equivalence rela... |
| gaorber 19250 | The orbit equivalence rela... |
| gastacl 19251 | The stabilizer subgroup in... |
| gastacos 19252 | Write the coset relation f... |
| orbstafun 19253 | Existence and uniqueness f... |
| orbstaval 19254 | Value of the function at a... |
| orbsta 19255 | The Orbit-Stabilizer theor... |
| orbsta2 19256 | Relation between the size ... |
| cntrval 19261 | Substitute definition of t... |
| cntzfval 19262 | First level substitution f... |
| cntzval 19263 | Definition substitution fo... |
| elcntz 19264 | Elementhood in the central... |
| cntzel 19265 | Membership in a centralize... |
| cntzsnval 19266 | Special substitution for t... |
| elcntzsn 19267 | Value of the centralizer o... |
| sscntz 19268 | A centralizer expression f... |
| cntzrcl 19269 | Reverse closure for elemen... |
| cntzssv 19270 | The centralizer is uncondi... |
| cntzi 19271 | Membership in a centralize... |
| elcntr 19272 | Elementhood in the center ... |
| cntrss 19273 | The center is a subset of ... |
| cntri 19274 | Defining property of the c... |
| resscntz 19275 | Centralizer in a substruct... |
| cntzsgrpcl 19276 | Centralizers are closed un... |
| cntz2ss 19277 | Centralizers reverse the s... |
| cntzrec 19278 | Reciprocity relationship f... |
| cntziinsn 19279 | Express any centralizer as... |
| cntzsubm 19280 | Centralizers in a monoid a... |
| cntzsubg 19281 | Centralizers in a group ar... |
| cntzidss 19282 | If the elements of ` S ` c... |
| cntzmhm 19283 | Centralizers in a monoid a... |
| cntzmhm2 19284 | Centralizers in a monoid a... |
| cntrsubgnsg 19285 | A central subgroup is norm... |
| cntrnsg 19286 | The center of a group is a... |
| oppgval 19289 | Value of the opposite grou... |
| oppgplusfval 19290 | Value of the addition oper... |
| oppgplus 19291 | Value of the addition oper... |
| setsplusg 19292 | The other components of an... |
| oppglemOLD 19293 | Obsolete version of ~ sets... |
| oppgbas 19294 | Base set of an opposite gr... |
| oppgbasOLD 19295 | Obsolete version of ~ oppg... |
| oppgtset 19296 | Topology of an opposite gr... |
| oppgtsetOLD 19297 | Obsolete version of ~ oppg... |
| oppgtopn 19298 | Topology of an opposite gr... |
| oppgmnd 19299 | The opposite of a monoid i... |
| oppgmndb 19300 | Bidirectional form of ~ op... |
| oppgid 19301 | Zero in a monoid is a symm... |
| oppggrp 19302 | The opposite of a group is... |
| oppggrpb 19303 | Bidirectional form of ~ op... |
| oppginv 19304 | Inverses in a group are a ... |
| invoppggim 19305 | The inverse is an antiauto... |
| oppggic 19306 | Every group is (naturally)... |
| oppgsubm 19307 | Being a submonoid is a sym... |
| oppgsubg 19308 | Being a subgroup is a symm... |
| oppgcntz 19309 | A centralizer in a group i... |
| oppgcntr 19310 | The center of a group is t... |
| gsumwrev 19311 | A sum in an opposite monoi... |
| symgval 19314 | The value of the symmetric... |
| permsetexOLD 19315 | Obsolete version of ~ f1os... |
| symgbas 19316 | The base set of the symmet... |
| symgbasexOLD 19317 | Obsolete as of 8-Aug-2024.... |
| elsymgbas2 19318 | Two ways of saying a funct... |
| elsymgbas 19319 | Two ways of saying a funct... |
| symgbasf1o 19320 | Elements in the symmetric ... |
| symgbasf 19321 | A permutation (element of ... |
| symgbasmap 19322 | A permutation (element of ... |
| symghash 19323 | The symmetric group on ` n... |
| symgbasfi 19324 | The symmetric group on a f... |
| symgfv 19325 | The function value of a pe... |
| symgfvne 19326 | The function values of a p... |
| symgressbas 19327 | The symmetric group on ` A... |
| symgplusg 19328 | The group operation of a s... |
| symgov 19329 | The value of the group ope... |
| symgcl 19330 | The group operation of the... |
| idresperm 19331 | The identity function rest... |
| symgmov1 19332 | For a permutation of a set... |
| symgmov2 19333 | For a permutation of a set... |
| symgbas0 19334 | The base set of the symmet... |
| symg1hash 19335 | The symmetric group on a s... |
| symg1bas 19336 | The symmetric group on a s... |
| symg2hash 19337 | The symmetric group on a (... |
| symg2bas 19338 | The symmetric group on a p... |
| 0symgefmndeq 19339 | The symmetric group on the... |
| snsymgefmndeq 19340 | The symmetric group on a s... |
| symgpssefmnd 19341 | For a set ` A ` with more ... |
| symgvalstruct 19342 | The value of the symmetric... |
| symgvalstructOLD 19343 | Obsolete proof of ~ symgva... |
| symgsubmefmnd 19344 | The symmetric group on a s... |
| symgtset 19345 | The topology of the symmet... |
| symggrp 19346 | The symmetric group on a s... |
| symgid 19347 | The group identity element... |
| symginv 19348 | The group inverse in the s... |
| symgsubmefmndALT 19349 | The symmetric group on a s... |
| galactghm 19350 | The currying of a group ac... |
| lactghmga 19351 | The converse of ~ galactgh... |
| symgtopn 19352 | The topology of the symmet... |
| symgga 19353 | The symmetric group induce... |
| pgrpsubgsymgbi 19354 | Every permutation group is... |
| pgrpsubgsymg 19355 | Every permutation group is... |
| idressubgsymg 19356 | The singleton containing o... |
| idrespermg 19357 | The structure with the sin... |
| cayleylem1 19358 | Lemma for ~ cayley . (Con... |
| cayleylem2 19359 | Lemma for ~ cayley . (Con... |
| cayley 19360 | Cayley's Theorem (construc... |
| cayleyth 19361 | Cayley's Theorem (existenc... |
| symgfix2 19362 | If a permutation does not ... |
| symgextf 19363 | The extension of a permuta... |
| symgextfv 19364 | The function value of the ... |
| symgextfve 19365 | The function value of the ... |
| symgextf1lem 19366 | Lemma for ~ symgextf1 . (... |
| symgextf1 19367 | The extension of a permuta... |
| symgextfo 19368 | The extension of a permuta... |
| symgextf1o 19369 | The extension of a permuta... |
| symgextsymg 19370 | The extension of a permuta... |
| symgextres 19371 | The restriction of the ext... |
| gsumccatsymgsn 19372 | Homomorphic property of co... |
| gsmsymgrfixlem1 19373 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19374 | The composition of permuta... |
| fvcosymgeq 19375 | The values of two composit... |
| gsmsymgreqlem1 19376 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19377 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19378 | Two combination of permuta... |
| symgfixelq 19379 | A permutation of a set fix... |
| symgfixels 19380 | The restriction of a permu... |
| symgfixelsi 19381 | The restriction of a permu... |
| symgfixf 19382 | The mapping of a permutati... |
| symgfixf1 19383 | The mapping of a permutati... |
| symgfixfolem1 19384 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19385 | The mapping of a permutati... |
| symgfixf1o 19386 | The mapping of a permutati... |
| f1omvdmvd 19389 | A permutation of any class... |
| f1omvdcnv 19390 | A permutation and its inve... |
| mvdco 19391 | Composing two permutations... |
| f1omvdconj 19392 | Conjugation of a permutati... |
| f1otrspeq 19393 | A transposition is charact... |
| f1omvdco2 19394 | If exactly one of two perm... |
| f1omvdco3 19395 | If a point is moved by exa... |
| pmtrfval 19396 | The function generating tr... |
| pmtrval 19397 | A generated transposition,... |
| pmtrfv 19398 | General value of mapping a... |
| pmtrprfv 19399 | In a transposition of two ... |
| pmtrprfv3 19400 | In a transposition of two ... |
| pmtrf 19401 | Functionality of a transpo... |
| pmtrmvd 19402 | A transposition moves prec... |
| pmtrrn 19403 | Transposing two points giv... |
| pmtrfrn 19404 | A transposition (as a kind... |
| pmtrffv 19405 | Mapping of a point under a... |
| pmtrrn2 19406 | For any transposition ther... |
| pmtrfinv 19407 | A transposition function i... |
| pmtrfmvdn0 19408 | A transposition moves at l... |
| pmtrff1o 19409 | A transposition function i... |
| pmtrfcnv 19410 | A transposition function i... |
| pmtrfb 19411 | An intrinsic characterizat... |
| pmtrfconj 19412 | Any conjugate of a transpo... |
| symgsssg 19413 | The symmetric group has su... |
| symgfisg 19414 | The symmetric group has a ... |
| symgtrf 19415 | Transpositions are element... |
| symggen 19416 | The span of the transposit... |
| symggen2 19417 | A finite permutation group... |
| symgtrinv 19418 | To invert a permutation re... |
| pmtr3ncomlem1 19419 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19420 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19421 | Transpositions over sets w... |
| pmtrdifellem1 19422 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19423 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19424 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19425 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19426 | A transposition of element... |
| pmtrdifwrdellem1 19427 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19428 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19429 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19430 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19431 | A sequence of transpositio... |
| pmtrdifwrdel2 19432 | A sequence of transpositio... |
| pmtrprfval 19433 | The transpositions on a pa... |
| pmtrprfvalrn 19434 | The range of the transposi... |
| psgnunilem1 19439 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19440 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19441 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19442 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19443 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19444 | Addition and subtraction o... |
| psgnuni 19445 | If the same permutation ca... |
| psgnfval 19446 | Function definition of the... |
| psgnfn 19447 | Functionality and domain o... |
| psgndmsubg 19448 | The finitary permutations ... |
| psgneldm 19449 | Property of being a finita... |
| psgneldm2 19450 | The finitary permutations ... |
| psgneldm2i 19451 | A sequence of transpositio... |
| psgneu 19452 | A finitary permutation has... |
| psgnval 19453 | Value of the permutation s... |
| psgnvali 19454 | A finitary permutation has... |
| psgnvalii 19455 | Any representation of a pe... |
| psgnpmtr 19456 | All transpositions are odd... |
| psgn0fv0 19457 | The permutation sign funct... |
| sygbasnfpfi 19458 | The class of non-fixed poi... |
| psgnfvalfi 19459 | Function definition of the... |
| psgnvalfi 19460 | Value of the permutation s... |
| psgnran 19461 | The range of the permutati... |
| gsmtrcl 19462 | The group sum of transposi... |
| psgnfitr 19463 | A permutation of a finite ... |
| psgnfieu 19464 | A permutation of a finite ... |
| pmtrsn 19465 | The value of the transposi... |
| psgnsn 19466 | The permutation sign funct... |
| psgnprfval 19467 | The permutation sign funct... |
| psgnprfval1 19468 | The permutation sign of th... |
| psgnprfval2 19469 | The permutation sign of th... |
| odfval 19478 | Value of the order functio... |
| odfvalALT 19479 | Shorter proof of ~ odfval ... |
| odval 19480 | Second substitution for th... |
| odlem1 19481 | The group element order is... |
| odcl 19482 | The order of a group eleme... |
| odf 19483 | Functionality of the group... |
| odid 19484 | Any element to the power o... |
| odlem2 19485 | Any positive annihilator o... |
| odmodnn0 19486 | Reduce the argument of a g... |
| mndodconglem 19487 | Lemma for ~ mndodcong . (... |
| mndodcong 19488 | If two multipliers are con... |
| mndodcongi 19489 | If two multipliers are con... |
| oddvdsnn0 19490 | The only multiples of ` A ... |
| odnncl 19491 | If a nonzero multiple of a... |
| odmod 19492 | Reduce the argument of a g... |
| oddvds 19493 | The only multiples of ` A ... |
| oddvdsi 19494 | Any group element is annih... |
| odcong 19495 | If two multipliers are con... |
| odeq 19496 | The ~ oddvds property uniq... |
| odval2 19497 | A non-conditional definiti... |
| odcld 19498 | The order of a group eleme... |
| odm1inv 19499 | The (order-1)th multiple o... |
| odmulgid 19500 | A relationship between the... |
| odmulg2 19501 | The order of a multiple di... |
| odmulg 19502 | Relationship between the o... |
| odmulgeq 19503 | A multiple of a point of f... |
| odbezout 19504 | If ` N ` is coprime to the... |
| od1 19505 | The order of the group ide... |
| odeq1 19506 | The group identity is the ... |
| odinv 19507 | The order of the inverse o... |
| odf1 19508 | The multiples of an elemen... |
| odinf 19509 | The multiples of an elemen... |
| dfod2 19510 | An alternative definition ... |
| odcl2 19511 | The order of an element of... |
| oddvds2 19512 | The order of an element of... |
| finodsubmsubg 19513 | A submonoid whose elements... |
| 0subgALT 19514 | A shorter proof of ~ 0subg... |
| submod 19515 | The order of an element is... |
| subgod 19516 | The order of an element is... |
| odsubdvds 19517 | The order of an element of... |
| odf1o1 19518 | An element with zero order... |
| odf1o2 19519 | An element with nonzero or... |
| odhash 19520 | An element of zero order g... |
| odhash2 19521 | If an element has nonzero ... |
| odhash3 19522 | An element which generates... |
| odngen 19523 | A cyclic subgroup of size ... |
| gexval 19524 | Value of the exponent of a... |
| gexlem1 19525 | The group element order is... |
| gexcl 19526 | The exponent of a group is... |
| gexid 19527 | Any element to the power o... |
| gexlem2 19528 | Any positive annihilator o... |
| gexdvdsi 19529 | Any group element is annih... |
| gexdvds 19530 | The only ` N ` that annihi... |
| gexdvds2 19531 | An integer divides the gro... |
| gexod 19532 | Any group element is annih... |
| gexcl3 19533 | If the order of every grou... |
| gexnnod 19534 | Every group element has fi... |
| gexcl2 19535 | The exponent of a finite g... |
| gexdvds3 19536 | The exponent of a finite g... |
| gex1 19537 | A group or monoid has expo... |
| ispgp 19538 | A group is a ` P ` -group ... |
| pgpprm 19539 | Reverse closure for the fi... |
| pgpgrp 19540 | Reverse closure for the se... |
| pgpfi1 19541 | A finite group with order ... |
| pgp0 19542 | The identity subgroup is a... |
| subgpgp 19543 | A subgroup of a p-group is... |
| sylow1lem1 19544 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19545 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19546 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19547 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19548 | Lemma for ~ sylow1 . Usin... |
| sylow1 19549 | Sylow's first theorem. If... |
| odcau 19550 | Cauchy's theorem for the o... |
| pgpfi 19551 | The converse to ~ pgpfi1 .... |
| pgpfi2 19552 | Alternate version of ~ pgp... |
| pgphash 19553 | The order of a p-group. (... |
| isslw 19554 | The property of being a Sy... |
| slwprm 19555 | Reverse closure for the fi... |
| slwsubg 19556 | A Sylow ` P ` -subgroup is... |
| slwispgp 19557 | Defining property of a Syl... |
| slwpss 19558 | A proper superset of a Syl... |
| slwpgp 19559 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19560 | Every ` P ` -subgroup is c... |
| slwn0 19561 | Every finite group contain... |
| subgslw 19562 | A Sylow subgroup that is c... |
| sylow2alem1 19563 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19564 | Lemma for ~ sylow2a . All... |
| sylow2a 19565 | A named lemma of Sylow's s... |
| sylow2blem1 19566 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19567 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19568 | Sylow's second theorem. P... |
| sylow2b 19569 | Sylow's second theorem. A... |
| slwhash 19570 | A sylow subgroup has cardi... |
| fislw 19571 | The sylow subgroups of a f... |
| sylow2 19572 | Sylow's second theorem. S... |
| sylow3lem1 19573 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19574 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19575 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19576 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19577 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19578 | Lemma for ~ sylow3 , secon... |
| sylow3 19579 | Sylow's third theorem. Th... |
| lsmfval 19584 | The subgroup sum function ... |
| lsmvalx 19585 | Subspace sum value (for a ... |
| lsmelvalx 19586 | Subspace sum membership (f... |
| lsmelvalix 19587 | Subspace sum membership (f... |
| oppglsm 19588 | The subspace sum operation... |
| lsmssv 19589 | Subgroup sum is a subset o... |
| lsmless1x 19590 | Subset implies subgroup su... |
| lsmless2x 19591 | Subset implies subgroup su... |
| lsmub1x 19592 | Subgroup sum is an upper b... |
| lsmub2x 19593 | Subgroup sum is an upper b... |
| lsmval 19594 | Subgroup sum value (for a ... |
| lsmelval 19595 | Subgroup sum membership (f... |
| lsmelvali 19596 | Subgroup sum membership (f... |
| lsmelvalm 19597 | Subgroup sum membership an... |
| lsmelvalmi 19598 | Membership of vector subtr... |
| lsmsubm 19599 | The sum of two commuting s... |
| lsmsubg 19600 | The sum of two commuting s... |
| lsmcom2 19601 | Subgroup sum commutes. (C... |
| smndlsmidm 19602 | The direct product is idem... |
| lsmub1 19603 | Subgroup sum is an upper b... |
| lsmub2 19604 | Subgroup sum is an upper b... |
| lsmunss 19605 | Union of subgroups is a su... |
| lsmless1 19606 | Subset implies subgroup su... |
| lsmless2 19607 | Subset implies subgroup su... |
| lsmless12 19608 | Subset implies subgroup su... |
| lsmidm 19609 | Subgroup sum is idempotent... |
| lsmlub 19610 | The least upper bound prop... |
| lsmss1 19611 | Subgroup sum with a subset... |
| lsmss1b 19612 | Subgroup sum with a subset... |
| lsmss2 19613 | Subgroup sum with a subset... |
| lsmss2b 19614 | Subgroup sum with a subset... |
| lsmass 19615 | Subgroup sum is associativ... |
| mndlsmidm 19616 | Subgroup sum is idempotent... |
| lsm01 19617 | Subgroup sum with the zero... |
| lsm02 19618 | Subgroup sum with the zero... |
| subglsm 19619 | The subgroup sum evaluated... |
| lssnle 19620 | Equivalent expressions for... |
| lsmmod 19621 | The modular law holds for ... |
| lsmmod2 19622 | Modular law dual for subgr... |
| lsmpropd 19623 | If two structures have the... |
| cntzrecd 19624 | Commute the "subgroups com... |
| lsmcntz 19625 | The "subgroups commute" pr... |
| lsmcntzr 19626 | The "subgroups commute" pr... |
| lsmdisj 19627 | Disjointness from a subgro... |
| lsmdisj2 19628 | Association of the disjoin... |
| lsmdisj3 19629 | Association of the disjoin... |
| lsmdisjr 19630 | Disjointness from a subgro... |
| lsmdisj2r 19631 | Association of the disjoin... |
| lsmdisj3r 19632 | Association of the disjoin... |
| lsmdisj2a 19633 | Association of the disjoin... |
| lsmdisj2b 19634 | Association of the disjoin... |
| lsmdisj3a 19635 | Association of the disjoin... |
| lsmdisj3b 19636 | Association of the disjoin... |
| subgdisj1 19637 | Vectors belonging to disjo... |
| subgdisj2 19638 | Vectors belonging to disjo... |
| subgdisjb 19639 | Vectors belonging to disjo... |
| pj1fval 19640 | The left projection functi... |
| pj1val 19641 | The left projection functi... |
| pj1eu 19642 | Uniqueness of a left proje... |
| pj1f 19643 | The left projection functi... |
| pj2f 19644 | The right projection funct... |
| pj1id 19645 | Any element of a direct su... |
| pj1eq 19646 | Any element of a direct su... |
| pj1lid 19647 | The left projection functi... |
| pj1rid 19648 | The left projection functi... |
| pj1ghm 19649 | The left projection functi... |
| pj1ghm2 19650 | The left projection functi... |
| lsmhash 19651 | The order of the direct pr... |
| efgmval 19658 | Value of the formal invers... |
| efgmf 19659 | The formal inverse operati... |
| efgmnvl 19660 | The inversion function on ... |
| efgrcl 19661 | Lemma for ~ efgval . (Con... |
| efglem 19662 | Lemma for ~ efgval . (Con... |
| efgval 19663 | Value of the free group co... |
| efger 19664 | Value of the free group co... |
| efgi 19665 | Value of the free group co... |
| efgi0 19666 | Value of the free group co... |
| efgi1 19667 | Value of the free group co... |
| efgtf 19668 | Value of the free group co... |
| efgtval 19669 | Value of the extension fun... |
| efgval2 19670 | Value of the free group co... |
| efgi2 19671 | Value of the free group co... |
| efgtlen 19672 | Value of the free group co... |
| efginvrel2 19673 | The inverse of the reverse... |
| efginvrel1 19674 | The inverse of the reverse... |
| efgsf 19675 | Value of the auxiliary fun... |
| efgsdm 19676 | Elementhood in the domain ... |
| efgsval 19677 | Value of the auxiliary fun... |
| efgsdmi 19678 | Property of the last link ... |
| efgsval2 19679 | Value of the auxiliary fun... |
| efgsrel 19680 | The start and end of any e... |
| efgs1 19681 | A singleton of an irreduci... |
| efgs1b 19682 | Every extension sequence e... |
| efgsp1 19683 | If ` F ` is an extension s... |
| efgsres 19684 | An initial segment of an e... |
| efgsfo 19685 | For any word, there is a s... |
| efgredlema 19686 | The reduced word that form... |
| efgredlemf 19687 | Lemma for ~ efgredleme . ... |
| efgredlemg 19688 | Lemma for ~ efgred . (Con... |
| efgredleme 19689 | Lemma for ~ efgred . (Con... |
| efgredlemd 19690 | The reduced word that form... |
| efgredlemc 19691 | The reduced word that form... |
| efgredlemb 19692 | The reduced word that form... |
| efgredlem 19693 | The reduced word that form... |
| efgred 19694 | The reduced word that form... |
| efgrelexlema 19695 | If two words ` A , B ` are... |
| efgrelexlemb 19696 | If two words ` A , B ` are... |
| efgrelex 19697 | If two words ` A , B ` are... |
| efgredeu 19698 | There is a unique reduced ... |
| efgred2 19699 | Two extension sequences ha... |
| efgcpbllema 19700 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19701 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19702 | Two extension sequences ha... |
| efgcpbl2 19703 | Two extension sequences ha... |
| frgpval 19704 | Value of the free group co... |
| frgpcpbl 19705 | Compatibility of the group... |
| frgp0 19706 | The free group is a group.... |
| frgpeccl 19707 | Closure of the quotient ma... |
| frgpgrp 19708 | The free group is a group.... |
| frgpadd 19709 | Addition in the free group... |
| frgpinv 19710 | The inverse of an element ... |
| frgpmhm 19711 | The "natural map" from wor... |
| vrgpfval 19712 | The canonical injection fr... |
| vrgpval 19713 | The value of the generatin... |
| vrgpf 19714 | The mapping from the index... |
| vrgpinv 19715 | The inverse of a generatin... |
| frgpuptf 19716 | Any assignment of the gene... |
| frgpuptinv 19717 | Any assignment of the gene... |
| frgpuplem 19718 | Any assignment of the gene... |
| frgpupf 19719 | Any assignment of the gene... |
| frgpupval 19720 | Any assignment of the gene... |
| frgpup1 19721 | Any assignment of the gene... |
| frgpup2 19722 | The evaluation map has the... |
| frgpup3lem 19723 | The evaluation map has the... |
| frgpup3 19724 | Universal property of the ... |
| 0frgp 19725 | The free group on zero gen... |
| isabl 19730 | The predicate "is an Abeli... |
| ablgrp 19731 | An Abelian group is a grou... |
| ablgrpd 19732 | An Abelian group is a grou... |
| ablcmn 19733 | An Abelian group is a comm... |
| ablcmnd 19734 | An Abelian group is a comm... |
| iscmn 19735 | The predicate "is a commut... |
| isabl2 19736 | The predicate "is an Abeli... |
| cmnpropd 19737 | If two structures have the... |
| ablpropd 19738 | If two structures have the... |
| ablprop 19739 | If two structures have the... |
| iscmnd 19740 | Properties that determine ... |
| isabld 19741 | Properties that determine ... |
| isabli 19742 | Properties that determine ... |
| cmnmnd 19743 | A commutative monoid is a ... |
| cmncom 19744 | A commutative monoid is co... |
| ablcom 19745 | An Abelian group operation... |
| cmn32 19746 | Commutative/associative la... |
| cmn4 19747 | Commutative/associative la... |
| cmn12 19748 | Commutative/associative la... |
| abl32 19749 | Commutative/associative la... |
| cmnmndd 19750 | A commutative monoid is a ... |
| cmnbascntr 19751 | The base set of a commutat... |
| rinvmod 19752 | Uniqueness of a right inve... |
| ablinvadd 19753 | The inverse of an Abelian ... |
| ablsub2inv 19754 | Abelian group subtraction ... |
| ablsubadd 19755 | Relationship between Abeli... |
| ablsub4 19756 | Commutative/associative su... |
| abladdsub4 19757 | Abelian group addition/sub... |
| abladdsub 19758 | Associative-type law for g... |
| ablsubadd23 19759 | Commutative/associative la... |
| ablsubaddsub 19760 | Double subtraction and add... |
| ablpncan2 19761 | Cancellation law for subtr... |
| ablpncan3 19762 | A cancellation law for Abe... |
| ablsubsub 19763 | Law for double subtraction... |
| ablsubsub4 19764 | Law for double subtraction... |
| ablpnpcan 19765 | Cancellation law for mixed... |
| ablnncan 19766 | Cancellation law for group... |
| ablsub32 19767 | Swap the second and third ... |
| ablnnncan 19768 | Cancellation law for group... |
| ablnnncan1 19769 | Cancellation law for group... |
| ablsubsub23 19770 | Swap subtrahend and result... |
| mulgnn0di 19771 | Group multiple of a sum, f... |
| mulgdi 19772 | Group multiple of a sum. ... |
| mulgmhm 19773 | The map from ` x ` to ` n ... |
| mulgghm 19774 | The map from ` x ` to ` n ... |
| mulgsubdi 19775 | Group multiple of a differ... |
| ghmfghm 19776 | The function fulfilling th... |
| ghmcmn 19777 | The image of a commutative... |
| ghmabl 19778 | The image of an abelian gr... |
| invghm 19779 | The inversion map is a gro... |
| eqgabl 19780 | Value of the subgroup cose... |
| qusecsub 19781 | Two subgroup cosets are eq... |
| subgabl 19782 | A subgroup of an abelian g... |
| subcmn 19783 | A submonoid of a commutati... |
| submcmn 19784 | A submonoid of a commutati... |
| submcmn2 19785 | A submonoid is commutative... |
| cntzcmn 19786 | The centralizer of any sub... |
| cntzcmnss 19787 | Any subset in a commutativ... |
| cntrcmnd 19788 | The center of a monoid is ... |
| cntrabl 19789 | The center of a group is a... |
| cntzspan 19790 | If the generators commute,... |
| cntzcmnf 19791 | Discharge the centralizer ... |
| ghmplusg 19792 | The pointwise sum of two l... |
| ablnsg 19793 | Every subgroup of an abeli... |
| odadd1 19794 | The order of a product in ... |
| odadd2 19795 | The order of a product in ... |
| odadd 19796 | The order of a product is ... |
| gex2abl 19797 | A group with exponent 2 (o... |
| gexexlem 19798 | Lemma for ~ gexex . (Cont... |
| gexex 19799 | In an abelian group with f... |
| torsubg 19800 | The set of all elements of... |
| oddvdssubg 19801 | The set of all elements wh... |
| lsmcomx 19802 | Subgroup sum commutes (ext... |
| ablcntzd 19803 | All subgroups in an abelia... |
| lsmcom 19804 | Subgroup sum commutes. (C... |
| lsmsubg2 19805 | The sum of two subgroups i... |
| lsm4 19806 | Commutative/associative la... |
| prdscmnd 19807 | The product of a family of... |
| prdsabld 19808 | The product of a family of... |
| pwscmn 19809 | The structure power on a c... |
| pwsabl 19810 | The structure power on an ... |
| qusabl 19811 | If ` Y ` is a subgroup of ... |
| abl1 19812 | The (smallest) structure r... |
| abln0 19813 | Abelian groups (and theref... |
| cnaddablx 19814 | The complex numbers are an... |
| cnaddabl 19815 | The complex numbers are an... |
| cnaddid 19816 | The group identity element... |
| cnaddinv 19817 | Value of the group inverse... |
| zaddablx 19818 | The integers are an Abelia... |
| frgpnabllem1 19819 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19820 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19821 | The free group on two or m... |
| imasabl 19822 | The image structure of an ... |
| iscyg 19825 | Definition of a cyclic gro... |
| iscyggen 19826 | The property of being a cy... |
| iscyggen2 19827 | The property of being a cy... |
| iscyg2 19828 | A cyclic group is a group ... |
| cyggeninv 19829 | The inverse of a cyclic ge... |
| cyggenod 19830 | An element is the generato... |
| cyggenod2 19831 | In an infinite cyclic grou... |
| iscyg3 19832 | Definition of a cyclic gro... |
| iscygd 19833 | Definition of a cyclic gro... |
| iscygodd 19834 | Show that a group with an ... |
| cycsubmcmn 19835 | The set of nonnegative int... |
| cyggrp 19836 | A cyclic group is a group.... |
| cygabl 19837 | A cyclic group is abelian.... |
| cygctb 19838 | A cyclic group is countabl... |
| 0cyg 19839 | The trivial group is cycli... |
| prmcyg 19840 | A group with prime order i... |
| lt6abl 19841 | A group with fewer than ` ... |
| ghmcyg 19842 | The image of a cyclic grou... |
| cyggex2 19843 | The exponent of a cyclic g... |
| cyggex 19844 | The exponent of a finite c... |
| cyggexb 19845 | A finite abelian group is ... |
| giccyg 19846 | Cyclicity is a group prope... |
| cycsubgcyg 19847 | The cyclic subgroup genera... |
| cycsubgcyg2 19848 | The cyclic subgroup genera... |
| gsumval3a 19849 | Value of the group sum ope... |
| gsumval3eu 19850 | The group sum as defined i... |
| gsumval3lem1 19851 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19852 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19853 | Value of the group sum ope... |
| gsumcllem 19854 | Lemma for ~ gsumcl and rel... |
| gsumzres 19855 | Extend a finite group sum ... |
| gsumzcl2 19856 | Closure of a finite group ... |
| gsumzcl 19857 | Closure of a finite group ... |
| gsumzf1o 19858 | Re-index a finite group su... |
| gsumres 19859 | Extend a finite group sum ... |
| gsumcl2 19860 | Closure of a finite group ... |
| gsumcl 19861 | Closure of a finite group ... |
| gsumf1o 19862 | Re-index a finite group su... |
| gsumreidx 19863 | Re-index a finite group su... |
| gsumzsubmcl 19864 | Closure of a group sum in ... |
| gsumsubmcl 19865 | Closure of a group sum in ... |
| gsumsubgcl 19866 | Closure of a group sum in ... |
| gsumzaddlem 19867 | The sum of two group sums.... |
| gsumzadd 19868 | The sum of two group sums.... |
| gsumadd 19869 | The sum of two group sums.... |
| gsummptfsadd 19870 | The sum of two group sums ... |
| gsummptfidmadd 19871 | The sum of two group sums ... |
| gsummptfidmadd2 19872 | The sum of two group sums ... |
| gsumzsplit 19873 | Split a group sum into two... |
| gsumsplit 19874 | Split a group sum into two... |
| gsumsplit2 19875 | Split a group sum into two... |
| gsummptfidmsplit 19876 | Split a group sum expresse... |
| gsummptfidmsplitres 19877 | Split a group sum expresse... |
| gsummptfzsplit 19878 | Split a group sum expresse... |
| gsummptfzsplitl 19879 | Split a group sum expresse... |
| gsumconst 19880 | Sum of a constant series. ... |
| gsumconstf 19881 | Sum of a constant series. ... |
| gsummptshft 19882 | Index shift of a finite gr... |
| gsumzmhm 19883 | Apply a group homomorphism... |
| gsummhm 19884 | Apply a group homomorphism... |
| gsummhm2 19885 | Apply a group homomorphism... |
| gsummptmhm 19886 | Apply a group homomorphism... |
| gsummulglem 19887 | Lemma for ~ gsummulg and ~... |
| gsummulg 19888 | Nonnegative multiple of a ... |
| gsummulgz 19889 | Integer multiple of a grou... |
| gsumzoppg 19890 | The opposite of a group su... |
| gsumzinv 19891 | Inverse of a group sum. (... |
| gsuminv 19892 | Inverse of a group sum. (... |
| gsummptfidminv 19893 | Inverse of a group sum exp... |
| gsumsub 19894 | The difference of two grou... |
| gsummptfssub 19895 | The difference of two grou... |
| gsummptfidmsub 19896 | The difference of two grou... |
| gsumsnfd 19897 | Group sum of a singleton, ... |
| gsumsnd 19898 | Group sum of a singleton, ... |
| gsumsnf 19899 | Group sum of a singleton, ... |
| gsumsn 19900 | Group sum of a singleton. ... |
| gsumpr 19901 | Group sum of a pair. (Con... |
| gsumzunsnd 19902 | Append an element to a fin... |
| gsumunsnfd 19903 | Append an element to a fin... |
| gsumunsnd 19904 | Append an element to a fin... |
| gsumunsnf 19905 | Append an element to a fin... |
| gsumunsn 19906 | Append an element to a fin... |
| gsumdifsnd 19907 | Extract a summand from a f... |
| gsumpt 19908 | Sum of a family that is no... |
| gsummptf1o 19909 | Re-index a finite group su... |
| gsummptun 19910 | Group sum of a disjoint un... |
| gsummpt1n0 19911 | If only one summand in a f... |
| gsummptif1n0 19912 | If only one summand in a f... |
| gsummptcl 19913 | Closure of a finite group ... |
| gsummptfif1o 19914 | Re-index a finite group su... |
| gsummptfzcl 19915 | Closure of a finite group ... |
| gsum2dlem1 19916 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19917 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19918 | Write a sum over a two-dim... |
| gsum2d2lem 19919 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19920 | Write a group sum over a t... |
| gsumcom2 19921 | Two-dimensional commutatio... |
| gsumxp 19922 | Write a group sum over a c... |
| gsumcom 19923 | Commute the arguments of a... |
| gsumcom3 19924 | A commutative law for fini... |
| gsumcom3fi 19925 | A commutative law for fini... |
| gsumxp2 19926 | Write a group sum over a c... |
| prdsgsum 19927 | Finite commutative sums in... |
| pwsgsum 19928 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19929 | Replacing a finitely suppo... |
| nn0gsumfz 19930 | Replacing a finitely suppo... |
| nn0gsumfz0 19931 | Replacing a finitely suppo... |
| gsummptnn0fz 19932 | A final group sum over a f... |
| gsummptnn0fzfv 19933 | A final group sum over a f... |
| telgsumfzslem 19934 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19935 | Telescoping group sum rang... |
| telgsumfz 19936 | Telescoping group sum rang... |
| telgsumfz0s 19937 | Telescoping finite group s... |
| telgsumfz0 19938 | Telescoping finite group s... |
| telgsums 19939 | Telescoping finitely suppo... |
| telgsum 19940 | Telescoping finitely suppo... |
| reldmdprd 19945 | The domain of the internal... |
| dmdprd 19946 | The domain of definition o... |
| dmdprdd 19947 | Show that a given family i... |
| dprddomprc 19948 | A family of subgroups inde... |
| dprddomcld 19949 | If a family of subgroups i... |
| dprdval0prc 19950 | The internal direct produc... |
| dprdval 19951 | The value of the internal ... |
| eldprd 19952 | A class ` A ` is an intern... |
| dprdgrp 19953 | Reverse closure for the in... |
| dprdf 19954 | The function ` S ` is a fa... |
| dprdf2 19955 | The function ` S ` is a fa... |
| dprdcntz 19956 | The function ` S ` is a fa... |
| dprddisj 19957 | The function ` S ` is a fa... |
| dprdw 19958 | The property of being a fi... |
| dprdwd 19959 | A mapping being a finitely... |
| dprdff 19960 | A finitely supported funct... |
| dprdfcl 19961 | A finitely supported funct... |
| dprdffsupp 19962 | A finitely supported funct... |
| dprdfcntz 19963 | A function on the elements... |
| dprdssv 19964 | The internal direct produc... |
| dprdfid 19965 | A function mapping all but... |
| eldprdi 19966 | The domain of definition o... |
| dprdfinv 19967 | Take the inverse of a grou... |
| dprdfadd 19968 | Take the sum of group sums... |
| dprdfsub 19969 | Take the difference of gro... |
| dprdfeq0 19970 | The zero function is the o... |
| dprdf11 19971 | Two group sums over a dire... |
| dprdsubg 19972 | The internal direct produc... |
| dprdub 19973 | Each factor is a subset of... |
| dprdlub 19974 | The direct product is smal... |
| dprdspan 19975 | The direct product is the ... |
| dprdres 19976 | Restriction of a direct pr... |
| dprdss 19977 | Create a direct product by... |
| dprdz 19978 | A family consisting entire... |
| dprd0 19979 | The empty family is an int... |
| dprdf1o 19980 | Rearrange the index set of... |
| dprdf1 19981 | Rearrange the index set of... |
| subgdmdprd 19982 | A direct product in a subg... |
| subgdprd 19983 | A direct product in a subg... |
| dprdsn 19984 | A singleton family is an i... |
| dmdprdsplitlem 19985 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 19986 | The function ` S ` is a fa... |
| dprddisj2 19987 | The function ` S ` is a fa... |
| dprd2dlem2 19988 | The direct product of a co... |
| dprd2dlem1 19989 | The direct product of a co... |
| dprd2da 19990 | The direct product of a co... |
| dprd2db 19991 | The direct product of a co... |
| dprd2d2 19992 | The direct product of a co... |
| dmdprdsplit2lem 19993 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 19994 | The direct product splits ... |
| dmdprdsplit 19995 | The direct product splits ... |
| dprdsplit 19996 | The direct product is the ... |
| dmdprdpr 19997 | A singleton family is an i... |
| dprdpr 19998 | A singleton family is an i... |
| dpjlem 19999 | Lemma for theorems about d... |
| dpjcntz 20000 | The two subgroups that app... |
| dpjdisj 20001 | The two subgroups that app... |
| dpjlsm 20002 | The two subgroups that app... |
| dpjfval 20003 | Value of the direct produc... |
| dpjval 20004 | Value of the direct produc... |
| dpjf 20005 | The ` X ` -th index projec... |
| dpjidcl 20006 | The key property of projec... |
| dpjeq 20007 | Decompose a group sum into... |
| dpjid 20008 | The key property of projec... |
| dpjlid 20009 | The ` X ` -th index projec... |
| dpjrid 20010 | The ` Y ` -th index projec... |
| dpjghm 20011 | The direct product is the ... |
| dpjghm2 20012 | The direct product is the ... |
| ablfacrplem 20013 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 20014 | A finite abelian group who... |
| ablfacrp2 20015 | The factors ` K , L ` of ~... |
| ablfac1lem 20016 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 20017 | The factors of ~ ablfac1b ... |
| ablfac1b 20018 | Any abelian group is the d... |
| ablfac1c 20019 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20020 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20021 | The factorization of ~ abl... |
| pgpfac1lem1 20022 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20023 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20024 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20025 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20026 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20027 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20028 | Factorization of a finite ... |
| pgpfaclem1 20029 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20030 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20031 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20032 | Full factorization of a fi... |
| ablfaclem1 20033 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20034 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20035 | Lemma for ~ ablfac . (Con... |
| ablfac 20036 | The Fundamental Theorem of... |
| ablfac2 20037 | Choose generators for each... |
| issimpg 20040 | The predicate "is a simple... |
| issimpgd 20041 | Deduce a simple group from... |
| simpggrp 20042 | A simple group is a group.... |
| simpggrpd 20043 | A simple group is a group.... |
| simpg2nsg 20044 | A simple group has two nor... |
| trivnsimpgd 20045 | Trivial groups are not sim... |
| simpgntrivd 20046 | Simple groups are nontrivi... |
| simpgnideld 20047 | A simple group contains a ... |
| simpgnsgd 20048 | The only normal subgroups ... |
| simpgnsgeqd 20049 | A normal subgroup of a sim... |
| 2nsgsimpgd 20050 | If any normal subgroup of ... |
| simpgnsgbid 20051 | A nontrivial group is simp... |
| ablsimpnosubgd 20052 | A subgroup of an abelian s... |
| ablsimpg1gend 20053 | An abelian simple group is... |
| ablsimpgcygd 20054 | An abelian simple group is... |
| ablsimpgfindlem1 20055 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20056 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20057 | Relationship between the o... |
| ablsimpgfind 20058 | An abelian simple group is... |
| fincygsubgd 20059 | The subgroup referenced in... |
| fincygsubgodd 20060 | Calculate the order of a s... |
| fincygsubgodexd 20061 | A finite cyclic group has ... |
| prmgrpsimpgd 20062 | A group of prime order is ... |
| ablsimpgprmd 20063 | An abelian simple group ha... |
| ablsimpgd 20064 | An abelian group is simple... |
| fnmgp 20067 | The multiplicative group o... |
| mgpval 20068 | Value of the multiplicatio... |
| mgpplusg 20069 | Value of the group operati... |
| mgplemOLD 20070 | Obsolete version of ~ sets... |
| mgpbas 20071 | Base set of the multiplica... |
| mgpbasOLD 20072 | Obsolete version of ~ mgpb... |
| mgpsca 20073 | The multiplication monoid ... |
| mgpscaOLD 20074 | Obsolete version of ~ mgps... |
| mgptset 20075 | Topology component of the ... |
| mgptsetOLD 20076 | Obsolete version of ~ mgpt... |
| mgptopn 20077 | Topology of the multiplica... |
| mgpds 20078 | Distance function of the m... |
| mgpdsOLD 20079 | Obsolete version of ~ mgpd... |
| mgpress 20080 | Subgroup commutes with the... |
| mgpressOLD 20081 | Obsolete version of ~ mgpr... |
| prdsmgp 20082 | The multiplicative monoid ... |
| isrng 20085 | The predicate "is a non-un... |
| rngabl 20086 | A non-unital ring is an (a... |
| rngmgp 20087 | A non-unital ring is a sem... |
| rngmgpf 20088 | Restricted functionality o... |
| rnggrp 20089 | A non-unital ring is a (ad... |
| rngass 20090 | Associative law for the mu... |
| rngdi 20091 | Distributive law for the m... |
| rngdir 20092 | Distributive law for the m... |
| rngacl 20093 | Closure of the addition op... |
| rng0cl 20094 | The zero element of a non-... |
| rngcl 20095 | Closure of the multiplicat... |
| rnglz 20096 | The zero of a non-unital r... |
| rngrz 20097 | The zero of a non-unital r... |
| rngmneg1 20098 | Negation of a product in a... |
| rngmneg2 20099 | Negation of a product in a... |
| rngm2neg 20100 | Double negation of a produ... |
| rngansg 20101 | Every additive subgroup of... |
| rngsubdi 20102 | Ring multiplication distri... |
| rngsubdir 20103 | Ring multiplication distri... |
| isrngd 20104 | Properties that determine ... |
| rngpropd 20105 | If two structures have the... |
| prdsmulrngcl 20106 | Closure of the multiplicat... |
| prdsrngd 20107 | A product of non-unital ri... |
| imasrng 20108 | The image structure of a n... |
| imasrngf1 20109 | The image of a non-unital ... |
| xpsrngd 20110 | A product of two non-unita... |
| qusrng 20111 | The quotient structure of ... |
| ringidval 20114 | The value of the unity ele... |
| dfur2 20115 | The multiplicative identit... |
| ringurd 20116 | Deduce the unity element o... |
| issrg 20119 | The predicate "is a semiri... |
| srgcmn 20120 | A semiring is a commutativ... |
| srgmnd 20121 | A semiring is a monoid. (... |
| srgmgp 20122 | A semiring is a monoid und... |
| srgdilem 20123 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20124 | Closure of the multiplicat... |
| srgass 20125 | Associative law for the mu... |
| srgideu 20126 | The unity element of a sem... |
| srgfcl 20127 | Functionality of the multi... |
| srgdi 20128 | Distributive law for the m... |
| srgdir 20129 | Distributive law for the m... |
| srgidcl 20130 | The unity element of a sem... |
| srg0cl 20131 | The zero element of a semi... |
| srgidmlem 20132 | Lemma for ~ srglidm and ~ ... |
| srglidm 20133 | The unity element of a sem... |
| srgridm 20134 | The unity element of a sem... |
| issrgid 20135 | Properties showing that an... |
| srgacl 20136 | Closure of the addition op... |
| srgcom 20137 | Commutativity of the addit... |
| srgrz 20138 | The zero of a semiring is ... |
| srglz 20139 | The zero of a semiring is ... |
| srgisid 20140 | In a semiring, the only le... |
| o2timesd 20141 | An element of a ring-like ... |
| rglcom4d 20142 | Restricted commutativity o... |
| srgo2times 20143 | A semiring element plus it... |
| srgcom4lem 20144 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20145 | Restricted commutativity o... |
| srg1zr 20146 | The only semiring with a b... |
| srgen1zr 20147 | The only semiring with one... |
| srgmulgass 20148 | An associative property be... |
| srgpcomp 20149 | If two elements of a semir... |
| srgpcompp 20150 | If two elements of a semir... |
| srgpcomppsc 20151 | If two elements of a semir... |
| srglmhm 20152 | Left-multiplication in a s... |
| srgrmhm 20153 | Right-multiplication in a ... |
| srgsummulcr 20154 | A finite semiring sum mult... |
| sgsummulcl 20155 | A finite semiring sum mult... |
| srg1expzeq1 20156 | The exponentiation (by a n... |
| srgbinomlem1 20157 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20158 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20159 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20160 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20161 | Lemma for ~ srgbinom . In... |
| srgbinom 20162 | The binomial theorem for c... |
| csrgbinom 20163 | The binomial theorem for c... |
| isring 20168 | The predicate "is a (unita... |
| ringgrp 20169 | A ring is a group. (Contr... |
| ringmgp 20170 | A ring is a monoid under m... |
| iscrng 20171 | A commutative ring is a ri... |
| crngmgp 20172 | A commutative ring's multi... |
| ringgrpd 20173 | A ring is a group. (Contr... |
| ringmnd 20174 | A ring is a monoid under a... |
| ringmgm 20175 | A ring is a magma. (Contr... |
| crngring 20176 | A commutative ring is a ri... |
| crngringd 20177 | A commutative ring is a ri... |
| crnggrpd 20178 | A commutative ring is a gr... |
| mgpf 20179 | Restricted functionality o... |
| ringdilem 20180 | Properties of a unital rin... |
| ringcl 20181 | Closure of the multiplicat... |
| crngcom 20182 | A commutative ring's multi... |
| iscrng2 20183 | A commutative ring is a ri... |
| ringass 20184 | Associative law for multip... |
| ringideu 20185 | The unity element of a rin... |
| crngbascntr 20186 | The base set of a commutat... |
| ringassd 20187 | Associative law for multip... |
| ringcld 20188 | Closure of the multiplicat... |
| ringdi 20189 | Distributive law for the m... |
| ringdir 20190 | Distributive law for the m... |
| ringidcl 20191 | The unity element of a rin... |
| ring0cl 20192 | The zero element of a ring... |
| ringidmlem 20193 | Lemma for ~ ringlidm and ~... |
| ringlidm 20194 | The unity element of a rin... |
| ringridm 20195 | The unity element of a rin... |
| isringid 20196 | Properties showing that an... |
| ringlidmd 20197 | The unity element of a rin... |
| ringridmd 20198 | The unity element of a rin... |
| ringid 20199 | The multiplication operati... |
| ringo2times 20200 | A ring element plus itself... |
| ringadd2 20201 | A ring element plus itself... |
| ringidss 20202 | A subset of the multiplica... |
| ringacl 20203 | Closure of the addition op... |
| ringcomlem 20204 | Lemma for ~ ringcom . Thi... |
| ringcom 20205 | Commutativity of the addit... |
| ringabl 20206 | A ring is an Abelian group... |
| ringcmn 20207 | A ring is a commutative mo... |
| ringabld 20208 | A ring is an Abelian group... |
| ringcmnd 20209 | A ring is a commutative mo... |
| ringrng 20210 | A unital ring is a non-uni... |
| ringssrng 20211 | The unital rings are non-u... |
| isringrng 20212 | The predicate "is a unital... |
| ringpropd 20213 | If two structures have the... |
| crngpropd 20214 | If two structures have the... |
| ringprop 20215 | If two structures have the... |
| isringd 20216 | Properties that determine ... |
| iscrngd 20217 | Properties that determine ... |
| ringlz 20218 | The zero of a unital ring ... |
| ringrz 20219 | The zero of a unital ring ... |
| ringlzd 20220 | The zero of a unital ring ... |
| ringrzd 20221 | The zero of a unital ring ... |
| ringsrg 20222 | Any ring is also a semirin... |
| ring1eq0 20223 | If one and zero are equal,... |
| ring1ne0 20224 | If a ring has at least two... |
| ringinvnz1ne0 20225 | In a unital ring, a left i... |
| ringinvnzdiv 20226 | In a unital ring, a left i... |
| ringnegl 20227 | Negation in a ring is the ... |
| ringnegr 20228 | Negation in a ring is the ... |
| ringmneg1 20229 | Negation of a product in a... |
| ringmneg2 20230 | Negation of a product in a... |
| ringm2neg 20231 | Double negation of a produ... |
| ringsubdi 20232 | Ring multiplication distri... |
| ringsubdir 20233 | Ring multiplication distri... |
| mulgass2 20234 | An associative property be... |
| ring1 20235 | The (smallest) structure r... |
| ringn0 20236 | Rings exist. (Contributed... |
| ringlghm 20237 | Left-multiplication in a r... |
| ringrghm 20238 | Right-multiplication in a ... |
| gsummulc1OLD 20239 | Obsolete version of ~ gsum... |
| gsummulc2OLD 20240 | Obsolete version of ~ gsum... |
| gsummulc1 20241 | A finite ring sum multipli... |
| gsummulc2 20242 | A finite ring sum multipli... |
| gsummgp0 20243 | If one factor in a finite ... |
| gsumdixp 20244 | Distribute a binary produc... |
| prdsmulrcl 20245 | A structure product of rin... |
| prdsringd 20246 | A product of rings is a ri... |
| prdscrngd 20247 | A product of commutative r... |
| prds1 20248 | Value of the ring unity in... |
| pwsring 20249 | A structure power of a rin... |
| pws1 20250 | Value of the ring unity in... |
| pwscrng 20251 | A structure power of a com... |
| pwsmgp 20252 | The multiplicative group o... |
| pwspjmhmmgpd 20253 | The projection given by ~ ... |
| pwsexpg 20254 | Value of a group exponenti... |
| imasring 20255 | The image structure of a r... |
| imasringf1 20256 | The image of a ring under ... |
| xpsringd 20257 | A product of two rings is ... |
| xpsring1d 20258 | The multiplicative identit... |
| qusring2 20259 | The quotient structure of ... |
| crngbinom 20260 | The binomial theorem for c... |
| opprval 20263 | Value of the opposite ring... |
| opprmulfval 20264 | Value of the multiplicatio... |
| opprmul 20265 | Value of the multiplicatio... |
| crngoppr 20266 | In a commutative ring, the... |
| opprlem 20267 | Lemma for ~ opprbas and ~ ... |
| opprlemOLD 20268 | Obsolete version of ~ oppr... |
| opprbas 20269 | Base set of an opposite ri... |
| opprbasOLD 20270 | Obsolete proof of ~ opprba... |
| oppradd 20271 | Addition operation of an o... |
| oppraddOLD 20272 | Obsolete proof of ~ opprba... |
| opprrng 20273 | An opposite non-unital rin... |
| opprrngb 20274 | A class is a non-unital ri... |
| opprring 20275 | An opposite ring is a ring... |
| opprringb 20276 | Bidirectional form of ~ op... |
| oppr0 20277 | Additive identity of an op... |
| oppr1 20278 | Multiplicative identity of... |
| opprneg 20279 | The negative function in a... |
| opprsubg 20280 | Being a subgroup is a symm... |
| mulgass3 20281 | An associative property be... |
| reldvdsr 20288 | The divides relation is a ... |
| dvdsrval 20289 | Value of the divides relat... |
| dvdsr 20290 | Value of the divides relat... |
| dvdsr2 20291 | Value of the divides relat... |
| dvdsrmul 20292 | A left-multiple of ` X ` i... |
| dvdsrcl 20293 | Closure of a dividing elem... |
| dvdsrcl2 20294 | Closure of a dividing elem... |
| dvdsrid 20295 | An element in a (unital) r... |
| dvdsrtr 20296 | Divisibility is transitive... |
| dvdsrmul1 20297 | The divisibility relation ... |
| dvdsrneg 20298 | An element divides its neg... |
| dvdsr01 20299 | In a ring, zero is divisib... |
| dvdsr02 20300 | Only zero is divisible by ... |
| isunit 20301 | Property of being a unit o... |
| 1unit 20302 | The multiplicative identit... |
| unitcl 20303 | A unit is an element of th... |
| unitss 20304 | The set of units is contai... |
| opprunit 20305 | Being a unit is a symmetri... |
| crngunit 20306 | Property of being a unit i... |
| dvdsunit 20307 | A divisor of a unit is a u... |
| unitmulcl 20308 | The product of units is a ... |
| unitmulclb 20309 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20310 | The base set of the group ... |
| unitgrp 20311 | The group of units is a gr... |
| unitabl 20312 | The group of units of a co... |
| unitgrpid 20313 | The identity of the group ... |
| unitsubm 20314 | The group of units is a su... |
| invrfval 20317 | Multiplicative inverse fun... |
| unitinvcl 20318 | The inverse of a unit exis... |
| unitinvinv 20319 | The inverse of the inverse... |
| ringinvcl 20320 | The inverse of a unit is a... |
| unitlinv 20321 | A unit times its inverse i... |
| unitrinv 20322 | A unit times its inverse i... |
| 1rinv 20323 | The inverse of the ring un... |
| 0unit 20324 | The additive identity is a... |
| unitnegcl 20325 | The negative of a unit is ... |
| ringunitnzdiv 20326 | In a unitary ring, a unit ... |
| ring1nzdiv 20327 | In a unitary ring, the rin... |
| dvrfval 20330 | Division operation in a ri... |
| dvrval 20331 | Division operation in a ri... |
| dvrcl 20332 | Closure of division operat... |
| unitdvcl 20333 | The units are closed under... |
| dvrid 20334 | A ring element divided by ... |
| dvr1 20335 | A ring element divided by ... |
| dvrass 20336 | An associative law for div... |
| dvrcan1 20337 | A cancellation law for div... |
| dvrcan3 20338 | A cancellation law for div... |
| dvreq1 20339 | Equality in terms of ratio... |
| dvrdir 20340 | Distributive law for the d... |
| rdivmuldivd 20341 | Multiplication of two rati... |
| ringinvdv 20342 | Write the inverse function... |
| rngidpropd 20343 | The ring unity depends onl... |
| dvdsrpropd 20344 | The divisibility relation ... |
| unitpropd 20345 | The set of units depends o... |
| invrpropd 20346 | The ring inverse function ... |
| isirred 20347 | An irreducible element of ... |
| isnirred 20348 | The property of being a no... |
| isirred2 20349 | Expand out the class diffe... |
| opprirred 20350 | Irreducibility is symmetri... |
| irredn0 20351 | The additive identity is n... |
| irredcl 20352 | An irreducible element is ... |
| irrednu 20353 | An irreducible element is ... |
| irredn1 20354 | The multiplicative identit... |
| irredrmul 20355 | The product of an irreduci... |
| irredlmul 20356 | The product of a unit and ... |
| irredmul 20357 | If product of two elements... |
| irredneg 20358 | The negative of an irreduc... |
| irrednegb 20359 | An element is irreducible ... |
| rnghmrcl 20366 | Reverse closure of a non-u... |
| rnghmfn 20367 | The mapping of two non-uni... |
| rnghmval 20368 | The set of the non-unital ... |
| isrnghm 20369 | A function is a non-unital... |
| isrnghmmul 20370 | A function is a non-unital... |
| rnghmmgmhm 20371 | A non-unital ring homomorp... |
| rnghmval2 20372 | The non-unital ring homomo... |
| isrngim 20373 | An isomorphism of non-unit... |
| rngimrcl 20374 | Reverse closure for an iso... |
| rnghmghm 20375 | A non-unital ring homomorp... |
| rnghmf 20376 | A ring homomorphism is a f... |
| rnghmmul 20377 | A homomorphism of non-unit... |
| isrnghm2d 20378 | Demonstration of non-unita... |
| isrnghmd 20379 | Demonstration of non-unita... |
| rnghmf1o 20380 | A non-unital ring homomorp... |
| isrngim2 20381 | An isomorphism of non-unit... |
| rngimf1o 20382 | An isomorphism of non-unit... |
| rngimrnghm 20383 | An isomorphism of non-unit... |
| rngimcnv 20384 | The converse of an isomorp... |
| rnghmco 20385 | The composition of non-uni... |
| idrnghm 20386 | The identity homomorphism ... |
| c0mgm 20387 | The constant mapping to ze... |
| c0mhm 20388 | The constant mapping to ze... |
| c0ghm 20389 | The constant mapping to ze... |
| c0snmgmhm 20390 | The constant mapping to ze... |
| c0snmhm 20391 | The constant mapping to ze... |
| c0snghm 20392 | The constant mapping to ze... |
| rngisomfv1 20393 | If there is a non-unital r... |
| rngisom1 20394 | If there is a non-unital r... |
| rngisomring 20395 | If there is a non-unital r... |
| rngisomring1 20396 | If there is a non-unital r... |
| dfrhm2 20402 | The property of a ring hom... |
| rhmrcl1 20404 | Reverse closure of a ring ... |
| rhmrcl2 20405 | Reverse closure of a ring ... |
| isrhm 20406 | A function is a ring homom... |
| rhmmhm 20407 | A ring homomorphism is a h... |
| rhmisrnghm 20408 | Each unital ring homomorph... |
| isrim0OLD 20409 | Obsolete version of ~ isri... |
| rimrcl 20410 | Reverse closure for an iso... |
| isrim0 20411 | A ring isomorphism is a ho... |
| rhmghm 20412 | A ring homomorphism is an ... |
| rhmf 20413 | A ring homomorphism is a f... |
| rhmmul 20414 | A homomorphism of rings pr... |
| isrhm2d 20415 | Demonstration of ring homo... |
| isrhmd 20416 | Demonstration of ring homo... |
| rhm1 20417 | Ring homomorphisms are req... |
| idrhm 20418 | The identity homomorphism ... |
| rhmf1o 20419 | A ring homomorphism is bij... |
| isrim 20420 | An isomorphism of rings is... |
| isrimOLD 20421 | Obsolete version of ~ isri... |
| rimf1o 20422 | An isomorphism of rings is... |
| rimrhmOLD 20423 | Obsolete version of ~ rimr... |
| rimrhm 20424 | A ring isomorphism is a ho... |
| rimgim 20425 | An isomorphism of rings is... |
| rimisrngim 20426 | Each unital ring isomorphi... |
| rhmfn 20427 | The mapping of two rings t... |
| rhmval 20428 | The ring homomorphisms bet... |
| rhmco 20429 | The composition of ring ho... |
| pwsco1rhm 20430 | Right composition with a f... |
| pwsco2rhm 20431 | Left composition with a ri... |
| brric 20432 | The relation "is isomorphi... |
| brrici 20433 | Prove isomorphic by an exp... |
| brric2 20434 | The relation "is isomorphi... |
| ricgic 20435 | If two rings are (ring) is... |
| rhmdvdsr 20436 | A ring homomorphism preser... |
| rhmopp 20437 | A ring homomorphism is als... |
| elrhmunit 20438 | Ring homomorphisms preserv... |
| rhmunitinv 20439 | Ring homomorphisms preserv... |
| isnzr 20442 | Property of a nonzero ring... |
| nzrnz 20443 | One and zero are different... |
| nzrring 20444 | A nonzero ring is a ring. ... |
| nzrringOLD 20445 | Obsolete version of ~ nzrr... |
| isnzr2 20446 | Equivalent characterizatio... |
| isnzr2hash 20447 | Equivalent characterizatio... |
| opprnzr 20448 | The opposite of a nonzero ... |
| ringelnzr 20449 | A ring is nonzero if it ha... |
| nzrunit 20450 | A unit is nonzero in any n... |
| 0ringnnzr 20451 | A ring is a zero ring iff ... |
| 0ring 20452 | If a ring has only one ele... |
| 0ringdif 20453 | A zero ring is a ring whic... |
| 0ringbas 20454 | The base set of a zero rin... |
| 0ring01eq 20455 | In a ring with only one el... |
| 01eq0ring 20456 | If the zero and the identi... |
| 01eq0ringOLD 20457 | Obsolete version of ~ 01eq... |
| 0ring01eqbi 20458 | In a unital ring the zero ... |
| 0ring1eq0 20459 | In a zero ring, a ring whi... |
| c0rhm 20460 | The constant mapping to ze... |
| c0rnghm 20461 | The constant mapping to ze... |
| zrrnghm 20462 | The constant mapping to ze... |
| nrhmzr 20463 | There is no ring homomorph... |
| islring 20466 | The predicate "is a local ... |
| lringnzr 20467 | A local ring is a nonzero ... |
| lringring 20468 | A local ring is a ring. (... |
| lringnz 20469 | A local ring is a nonzero ... |
| lringuplu 20470 | If the sum of two elements... |
| issubrng 20473 | The subring of non-unital ... |
| subrngss 20474 | A subring is a subset. (C... |
| subrngid 20475 | Every non-unital ring is a... |
| subrngrng 20476 | A subring is a non-unital ... |
| subrngrcl 20477 | Reverse closure for a subr... |
| subrngsubg 20478 | A subring is a subgroup. ... |
| subrngringnsg 20479 | A subring is a normal subg... |
| subrngbas 20480 | Base set of a subring stru... |
| subrng0 20481 | A subring always has the s... |
| subrngacl 20482 | A subring is closed under ... |
| subrngmcl 20483 | A subgroup is closed under... |
| issubrng2 20484 | Characterize the subrings ... |
| opprsubrng 20485 | Being a subring is a symme... |
| subrngint 20486 | The intersection of a none... |
| subrngin 20487 | The intersection of two su... |
| subrngmre 20488 | The subrings of a non-unit... |
| subsubrng 20489 | A subring of a subring is ... |
| subsubrng2 20490 | The set of subrings of a s... |
| rhmimasubrnglem 20491 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20492 | The homomorphic image of a... |
| cntzsubrng 20493 | Centralizers in a non-unit... |
| subrngpropd 20494 | If two structures have the... |
| issubrg 20499 | The subring predicate. (C... |
| subrgss 20500 | A subring is a subset. (C... |
| subrgid 20501 | Every ring is a subring of... |
| subrgring 20502 | A subring is a ring. (Con... |
| subrgcrng 20503 | A subring of a commutative... |
| subrgrcl 20504 | Reverse closure for a subr... |
| subrgsubg 20505 | A subring is a subgroup. ... |
| subrgsubrng 20506 | A subring of a unital ring... |
| subrg0 20507 | A subring always has the s... |
| subrg1cl 20508 | A subring contains the mul... |
| subrgbas 20509 | Base set of a subring stru... |
| subrg1 20510 | A subring always has the s... |
| subrgacl 20511 | A subring is closed under ... |
| subrgmcl 20512 | A subgroup is closed under... |
| subrgsubm 20513 | A subring is a submonoid o... |
| subrgdvds 20514 | If an element divides anot... |
| subrguss 20515 | A unit of a subring is a u... |
| subrginv 20516 | A subring always has the s... |
| subrgdv 20517 | A subring always has the s... |
| subrgunit 20518 | An element of a ring is a ... |
| subrgugrp 20519 | The units of a subring for... |
| issubrg2 20520 | Characterize the subrings ... |
| opprsubrg 20521 | Being a subring is a symme... |
| subrgnzr 20522 | A subring of a nonzero rin... |
| subrgint 20523 | The intersection of a none... |
| subrgin 20524 | The intersection of two su... |
| subrgmre 20525 | The subrings of a ring are... |
| subsubrg 20526 | A subring of a subring is ... |
| subsubrg2 20527 | The set of subrings of a s... |
| issubrg3 20528 | A subring is an additive s... |
| resrhm 20529 | Restriction of a ring homo... |
| resrhm2b 20530 | Restriction of the codomai... |
| rhmeql 20531 | The equalizer of two ring ... |
| rhmima 20532 | The homomorphic image of a... |
| rnrhmsubrg 20533 | The range of a ring homomo... |
| cntzsubr 20534 | Centralizers in a ring are... |
| pwsdiagrhm 20535 | Diagonal homomorphism into... |
| subrgpropd 20536 | If two structures have the... |
| rhmpropd 20537 | Ring homomorphism depends ... |
| rngcval 20540 | Value of the category of n... |
| rnghmresfn 20541 | The class of non-unital ri... |
| rnghmresel 20542 | An element of the non-unit... |
| rngcbas 20543 | Set of objects of the cate... |
| rngchomfval 20544 | Set of arrows of the categ... |
| rngchom 20545 | Set of arrows of the categ... |
| elrngchom 20546 | A morphism of non-unital r... |
| rngchomfeqhom 20547 | The functionalized Hom-set... |
| rngccofval 20548 | Composition in the categor... |
| rngcco 20549 | Composition in the categor... |
| dfrngc2 20550 | Alternate definition of th... |
| rnghmsscmap2 20551 | The non-unital ring homomo... |
| rnghmsscmap 20552 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20553 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20554 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20555 | The non-unital ring homomo... |
| rngccat 20556 | The category of non-unital... |
| rngcid 20557 | The identity arrow in the ... |
| rngcsect 20558 | A section in the category ... |
| rngcinv 20559 | An inverse in the category... |
| rngciso 20560 | An isomorphism in the cate... |
| rngcifuestrc 20561 | The "inclusion functor" fr... |
| funcrngcsetc 20562 | The "natural forgetful fun... |
| funcrngcsetcALT 20563 | Alternate proof of ~ funcr... |
| zrinitorngc 20564 | The zero ring is an initia... |
| zrtermorngc 20565 | The zero ring is a termina... |
| zrzeroorngc 20566 | The zero ring is a zero ob... |
| ringcval 20569 | Value of the category of u... |
| rhmresfn 20570 | The class of unital ring h... |
| rhmresel 20571 | An element of the unital r... |
| ringcbas 20572 | Set of objects of the cate... |
| ringchomfval 20573 | Set of arrows of the categ... |
| ringchom 20574 | Set of arrows of the categ... |
| elringchom 20575 | A morphism of unital rings... |
| ringchomfeqhom 20576 | The functionalized Hom-set... |
| ringccofval 20577 | Composition in the categor... |
| ringcco 20578 | Composition in the categor... |
| dfringc2 20579 | Alternate definition of th... |
| rhmsscmap2 20580 | The unital ring homomorphi... |
| rhmsscmap 20581 | The unital ring homomorphi... |
| rhmsubcsetclem1 20582 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20583 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20584 | The unital ring homomorphi... |
| ringccat 20585 | The category of unital rin... |
| ringcid 20586 | The identity arrow in the ... |
| rhmsscrnghm 20587 | The unital ring homomorphi... |
| rhmsubcrngclem1 20588 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20589 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20590 | The unital ring homomorphi... |
| rngcresringcat 20591 | The restriction of the cat... |
| ringcsect 20592 | A section in the category ... |
| ringcinv 20593 | An inverse in the category... |
| ringciso 20594 | An isomorphism in the cate... |
| ringcbasbas 20595 | An element of the base set... |
| funcringcsetc 20596 | The "natural forgetful fun... |
| zrtermoringc 20597 | The zero ring is a termina... |
| zrninitoringc 20598 | The zero ring is not an in... |
| srhmsubclem1 20599 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20600 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20601 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20602 | According to ~ df-subc , t... |
| sringcat 20603 | The restriction of the cat... |
| crhmsubc 20604 | According to ~ df-subc , t... |
| cringcat 20605 | The restriction of the cat... |
| rngcrescrhm 20606 | The category of non-unital... |
| rhmsubclem1 20607 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20608 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20609 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20610 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20611 | According to ~ df-subc , t... |
| rhmsubccat 20612 | The restriction of the cat... |
| isdrng 20617 | The predicate "is a divisi... |
| drngunit 20618 | Elementhood in the set of ... |
| drngui 20619 | The set of units of a divi... |
| drngring 20620 | A division ring is a ring.... |
| drngringd 20621 | A division ring is a ring.... |
| drnggrpd 20622 | A division ring is a group... |
| drnggrp 20623 | A division ring is a group... |
| isfld 20624 | A field is a commutative d... |
| flddrngd 20625 | A field is a division ring... |
| fldcrngd 20626 | A field is a commutative r... |
| isdrng2 20627 | A division ring can equiva... |
| drngprop 20628 | If two structures have the... |
| drngmgp 20629 | A division ring contains a... |
| drngmcl 20630 | The product of two nonzero... |
| drngid 20631 | A division ring's unity is... |
| drngunz 20632 | A division ring's unity is... |
| drngnzr 20633 | All division rings are non... |
| drngid2 20634 | Properties showing that an... |
| drnginvrcl 20635 | Closure of the multiplicat... |
| drnginvrn0 20636 | The multiplicative inverse... |
| drnginvrcld 20637 | Closure of the multiplicat... |
| drnginvrl 20638 | Property of the multiplica... |
| drnginvrr 20639 | Property of the multiplica... |
| drnginvrld 20640 | Property of the multiplica... |
| drnginvrrd 20641 | Property of the multiplica... |
| drngmul0or 20642 | A product is zero iff one ... |
| drngmulne0 20643 | A product is nonzero iff b... |
| drngmuleq0 20644 | An element is zero iff its... |
| opprdrng 20645 | The opposite of a division... |
| isdrngd 20646 | Properties that characteri... |
| isdrngrd 20647 | Properties that characteri... |
| isdrngdOLD 20648 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20649 | Obsolete version of ~ isdr... |
| drngpropd 20650 | If two structures have the... |
| fldpropd 20651 | If two structures have the... |
| rng1nnzr 20652 | The (smallest) structure r... |
| ring1zr 20653 | The only (unital) ring wit... |
| rngen1zr 20654 | The only (unital) ring wit... |
| ringen1zr 20655 | The only unital ring with ... |
| rng1nfld 20656 | The zero ring is not a fie... |
| issubdrg 20657 | Characterize the subfields... |
| drhmsubc 20658 | According to ~ df-subc , t... |
| drngcat 20659 | The restriction of the cat... |
| fldcat 20660 | The restriction of the cat... |
| fldc 20661 | The restriction of the cat... |
| fldhmsubc 20662 | According to ~ df-subc , t... |
| issdrg 20665 | Property of a division sub... |
| sdrgrcl 20666 | Reverse closure for a sub-... |
| sdrgdrng 20667 | A sub-division-ring is a d... |
| sdrgsubrg 20668 | A sub-division-ring is a s... |
| sdrgid 20669 | Every division ring is a d... |
| sdrgss 20670 | A division subring is a su... |
| sdrgbas 20671 | Base set of a sub-division... |
| issdrg2 20672 | Property of a division sub... |
| sdrgunit 20673 | A unit of a sub-division-r... |
| imadrhmcl 20674 | The image of a (nontrivial... |
| fldsdrgfld 20675 | A sub-division-ring of a f... |
| acsfn1p 20676 | Construction of a closure ... |
| subrgacs 20677 | Closure property of subrin... |
| sdrgacs 20678 | Closure property of divisi... |
| cntzsdrg 20679 | Centralizers in division r... |
| subdrgint 20680 | The intersection of a none... |
| sdrgint 20681 | The intersection of a none... |
| primefld 20682 | The smallest sub division ... |
| primefld0cl 20683 | The prime field contains t... |
| primefld1cl 20684 | The prime field contains t... |
| abvfval 20687 | Value of the set of absolu... |
| isabv 20688 | Elementhood in the set of ... |
| isabvd 20689 | Properties that determine ... |
| abvrcl 20690 | Reverse closure for the ab... |
| abvfge0 20691 | An absolute value is a fun... |
| abvf 20692 | An absolute value is a fun... |
| abvcl 20693 | An absolute value is a fun... |
| abvge0 20694 | The absolute value of a nu... |
| abveq0 20695 | The value of an absolute v... |
| abvne0 20696 | The absolute value of a no... |
| abvgt0 20697 | The absolute value of a no... |
| abvmul 20698 | An absolute value distribu... |
| abvtri 20699 | An absolute value satisfie... |
| abv0 20700 | The absolute value of zero... |
| abv1z 20701 | The absolute value of one ... |
| abv1 20702 | The absolute value of one ... |
| abvneg 20703 | The absolute value of a ne... |
| abvsubtri 20704 | An absolute value satisfie... |
| abvrec 20705 | The absolute value distrib... |
| abvdiv 20706 | The absolute value distrib... |
| abvdom 20707 | Any ring with an absolute ... |
| abvres 20708 | The restriction of an abso... |
| abvtrivd 20709 | The trivial absolute value... |
| abvtriv 20710 | The trivial absolute value... |
| abvpropd 20711 | If two structures have the... |
| staffval 20716 | The functionalization of t... |
| stafval 20717 | The functionalization of t... |
| staffn 20718 | The functionalization is e... |
| issrng 20719 | The predicate "is a star r... |
| srngrhm 20720 | The involution function in... |
| srngring 20721 | A star ring is a ring. (C... |
| srngcnv 20722 | The involution function in... |
| srngf1o 20723 | The involution function in... |
| srngcl 20724 | The involution function in... |
| srngnvl 20725 | The involution function in... |
| srngadd 20726 | The involution function in... |
| srngmul 20727 | The involution function in... |
| srng1 20728 | The conjugate of the ring ... |
| srng0 20729 | The conjugate of the ring ... |
| issrngd 20730 | Properties that determine ... |
| idsrngd 20731 | A commutative ring is a st... |
| islmod 20736 | The predicate "is a left m... |
| lmodlema 20737 | Lemma for properties of a ... |
| islmodd 20738 | Properties that determine ... |
| lmodgrp 20739 | A left module is a group. ... |
| lmodring 20740 | The scalar component of a ... |
| lmodfgrp 20741 | The scalar component of a ... |
| lmodgrpd 20742 | A left module is a group. ... |
| lmodbn0 20743 | The base set of a left mod... |
| lmodacl 20744 | Closure of ring addition f... |
| lmodmcl 20745 | Closure of ring multiplica... |
| lmodsn0 20746 | The set of scalars in a le... |
| lmodvacl 20747 | Closure of vector addition... |
| lmodass 20748 | Left module vector sum is ... |
| lmodlcan 20749 | Left cancellation law for ... |
| lmodvscl 20750 | Closure of scalar product ... |
| lmodvscld 20751 | Closure of scalar product ... |
| scaffval 20752 | The scalar multiplication ... |
| scafval 20753 | The scalar multiplication ... |
| scafeq 20754 | If the scalar multiplicati... |
| scaffn 20755 | The scalar multiplication ... |
| lmodscaf 20756 | The scalar multiplication ... |
| lmodvsdi 20757 | Distributive law for scala... |
| lmodvsdir 20758 | Distributive law for scala... |
| lmodvsass 20759 | Associative law for scalar... |
| lmod0cl 20760 | The ring zero in a left mo... |
| lmod1cl 20761 | The ring unity in a left m... |
| lmodvs1 20762 | Scalar product with the ri... |
| lmod0vcl 20763 | The zero vector is a vecto... |
| lmod0vlid 20764 | Left identity law for the ... |
| lmod0vrid 20765 | Right identity law for the... |
| lmod0vid 20766 | Identity equivalent to the... |
| lmod0vs 20767 | Zero times a vector is the... |
| lmodvs0 20768 | Anything times the zero ve... |
| lmodvsmmulgdi 20769 | Distributive law for a gro... |
| lmodfopnelem1 20770 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20771 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20772 | The (functionalized) opera... |
| lcomf 20773 | A linear-combination sum i... |
| lcomfsupp 20774 | A linear-combination sum i... |
| lmodvnegcl 20775 | Closure of vector negative... |
| lmodvnegid 20776 | Addition of a vector with ... |
| lmodvneg1 20777 | Minus 1 times a vector is ... |
| lmodvsneg 20778 | Multiplication of a vector... |
| lmodvsubcl 20779 | Closure of vector subtract... |
| lmodcom 20780 | Left module vector sum is ... |
| lmodabl 20781 | A left module is an abelia... |
| lmodcmn 20782 | A left module is a commuta... |
| lmodnegadd 20783 | Distribute negation throug... |
| lmod4 20784 | Commutative/associative la... |
| lmodvsubadd 20785 | Relationship between vecto... |
| lmodvaddsub4 20786 | Vector addition/subtractio... |
| lmodvpncan 20787 | Addition/subtraction cance... |
| lmodvnpcan 20788 | Cancellation law for vecto... |
| lmodvsubval2 20789 | Value of vector subtractio... |
| lmodsubvs 20790 | Subtraction of a scalar pr... |
| lmodsubdi 20791 | Scalar multiplication dist... |
| lmodsubdir 20792 | Scalar multiplication dist... |
| lmodsubeq0 20793 | If the difference between ... |
| lmodsubid 20794 | Subtraction of a vector fr... |
| lmodvsghm 20795 | Scalar multiplication of t... |
| lmodprop2d 20796 | If two structures have the... |
| lmodpropd 20797 | If two structures have the... |
| gsumvsmul 20798 | Pull a scalar multiplicati... |
| mptscmfsupp0 20799 | A mapping to a scalar prod... |
| mptscmfsuppd 20800 | A function mapping to a sc... |
| rmodislmodlem 20801 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20802 | The right module ` R ` ind... |
| rmodislmodOLD 20803 | Obsolete version of ~ rmod... |
| lssset 20806 | The set of all (not necess... |
| islss 20807 | The predicate "is a subspa... |
| islssd 20808 | Properties that determine ... |
| lssss 20809 | A subspace is a set of vec... |
| lssel 20810 | A subspace member is a vec... |
| lss1 20811 | The set of vectors in a le... |
| lssuni 20812 | The union of all subspaces... |
| lssn0 20813 | A subspace is not empty. ... |
| 00lss 20814 | The empty structure has no... |
| lsscl 20815 | Closure property of a subs... |
| lssvacl 20816 | Closure of vector addition... |
| lssvsubcl 20817 | Closure of vector subtract... |
| lssvancl1 20818 | Non-closure: if one vector... |
| lssvancl2 20819 | Non-closure: if one vector... |
| lss0cl 20820 | The zero vector belongs to... |
| lsssn0 20821 | The singleton of the zero ... |
| lss0ss 20822 | The zero subspace is inclu... |
| lssle0 20823 | No subspace is smaller tha... |
| lssne0 20824 | A nonzero subspace has a n... |
| lssvneln0 20825 | A vector ` X ` which doesn... |
| lssneln0 20826 | A vector ` X ` which doesn... |
| lssssr 20827 | Conclude subspace ordering... |
| lssvscl 20828 | Closure of scalar product ... |
| lssvnegcl 20829 | Closure of negative vector... |
| lsssubg 20830 | All subspaces are subgroup... |
| lsssssubg 20831 | All subspaces are subgroup... |
| islss3 20832 | A linear subspace of a mod... |
| lsslmod 20833 | A submodule is a module. ... |
| lsslss 20834 | The subspaces of a subspac... |
| islss4 20835 | A linear subspace is a sub... |
| lss1d 20836 | One-dimensional subspace (... |
| lssintcl 20837 | The intersection of a none... |
| lssincl 20838 | The intersection of two su... |
| lssmre 20839 | The subspaces of a module ... |
| lssacs 20840 | Submodules are an algebrai... |
| prdsvscacl 20841 | Pointwise scalar multiplic... |
| prdslmodd 20842 | The product of a family of... |
| pwslmod 20843 | A structure power of a lef... |
| lspfval 20846 | The span function for a le... |
| lspf 20847 | The span function on a lef... |
| lspval 20848 | The span of a set of vecto... |
| lspcl 20849 | The span of a set of vecto... |
| lspsncl 20850 | The span of a singleton is... |
| lspprcl 20851 | The span of a pair is a su... |
| lsptpcl 20852 | The span of an unordered t... |
| lspsnsubg 20853 | The span of a singleton is... |
| 00lsp 20854 | ~ fvco4i lemma for linear ... |
| lspid 20855 | The span of a subspace is ... |
| lspssv 20856 | A span is a set of vectors... |
| lspss 20857 | Span preserves subset orde... |
| lspssid 20858 | A set of vectors is a subs... |
| lspidm 20859 | The span of a set of vecto... |
| lspun 20860 | The span of union is the s... |
| lspssp 20861 | If a set of vectors is a s... |
| mrclsp 20862 | Moore closure generalizes ... |
| lspsnss 20863 | The span of the singleton ... |
| lspsnel3 20864 | A member of the span of th... |
| lspprss 20865 | The span of a pair of vect... |
| lspsnid 20866 | A vector belongs to the sp... |
| lspsnel6 20867 | Relationship between a vec... |
| lspsnel5 20868 | Relationship between a vec... |
| lspsnel5a 20869 | Relationship between a vec... |
| lspprid1 20870 | A member of a pair of vect... |
| lspprid2 20871 | A member of a pair of vect... |
| lspprvacl 20872 | The sum of two vectors bel... |
| lssats2 20873 | A way to express atomistic... |
| lspsneli 20874 | A scalar product with a ve... |
| lspsn 20875 | Span of the singleton of a... |
| lspsnel 20876 | Member of span of the sing... |
| lspsnvsi 20877 | Span of a scalar product o... |
| lspsnss2 20878 | Comparable spans of single... |
| lspsnneg 20879 | Negation does not change t... |
| lspsnsub 20880 | Swapping subtraction order... |
| lspsn0 20881 | Span of the singleton of t... |
| lsp0 20882 | Span of the empty set. (C... |
| lspuni0 20883 | Union of the span of the e... |
| lspun0 20884 | The span of a union with t... |
| lspsneq0 20885 | Span of the singleton is t... |
| lspsneq0b 20886 | Equal singleton spans impl... |
| lmodindp1 20887 | Two independent (non-colin... |
| lsslsp 20888 | Spans in submodules corres... |
| lsslspOLD 20889 | Obsolete version of ~ lssl... |
| lss0v 20890 | The zero vector in a submo... |
| lsspropd 20891 | If two structures have the... |
| lsppropd 20892 | If two structures have the... |
| reldmlmhm 20899 | Lemma for module homomorph... |
| lmimfn 20900 | Lemma for module isomorphi... |
| islmhm 20901 | Property of being a homomo... |
| islmhm3 20902 | Property of a module homom... |
| lmhmlem 20903 | Non-quantified consequence... |
| lmhmsca 20904 | A homomorphism of left mod... |
| lmghm 20905 | A homomorphism of left mod... |
| lmhmlmod2 20906 | A homomorphism of left mod... |
| lmhmlmod1 20907 | A homomorphism of left mod... |
| lmhmf 20908 | A homomorphism of left mod... |
| lmhmlin 20909 | A homomorphism of left mod... |
| lmodvsinv 20910 | Multiplication of a vector... |
| lmodvsinv2 20911 | Multiplying a negated vect... |
| islmhm2 20912 | A one-equation proof of li... |
| islmhmd 20913 | Deduction for a module hom... |
| 0lmhm 20914 | The constant zero linear f... |
| idlmhm 20915 | The identity function on a... |
| invlmhm 20916 | The negative function on a... |
| lmhmco 20917 | The composition of two mod... |
| lmhmplusg 20918 | The pointwise sum of two l... |
| lmhmvsca 20919 | The pointwise scalar produ... |
| lmhmf1o 20920 | A bijective module homomor... |
| lmhmima 20921 | The image of a subspace un... |
| lmhmpreima 20922 | The inverse image of a sub... |
| lmhmlsp 20923 | Homomorphisms preserve spa... |
| lmhmrnlss 20924 | The range of a homomorphis... |
| lmhmkerlss 20925 | The kernel of a homomorphi... |
| reslmhm 20926 | Restriction of a homomorph... |
| reslmhm2 20927 | Expansion of the codomain ... |
| reslmhm2b 20928 | Expansion of the codomain ... |
| lmhmeql 20929 | The equalizer of two modul... |
| lspextmo 20930 | A linear function is compl... |
| pwsdiaglmhm 20931 | Diagonal homomorphism into... |
| pwssplit0 20932 | Splitting for structure po... |
| pwssplit1 20933 | Splitting for structure po... |
| pwssplit2 20934 | Splitting for structure po... |
| pwssplit3 20935 | Splitting for structure po... |
| islmim 20936 | An isomorphism of left mod... |
| lmimf1o 20937 | An isomorphism of left mod... |
| lmimlmhm 20938 | An isomorphism of modules ... |
| lmimgim 20939 | An isomorphism of modules ... |
| islmim2 20940 | An isomorphism of left mod... |
| lmimcnv 20941 | The converse of a bijectiv... |
| brlmic 20942 | The relation "is isomorphi... |
| brlmici 20943 | Prove isomorphic by an exp... |
| lmiclcl 20944 | Isomorphism implies the le... |
| lmicrcl 20945 | Isomorphism implies the ri... |
| lmicsym 20946 | Module isomorphism is symm... |
| lmhmpropd 20947 | Module homomorphism depend... |
| islbs 20950 | The predicate " ` B ` is a... |
| lbsss 20951 | A basis is a set of vector... |
| lbsel 20952 | An element of a basis is a... |
| lbssp 20953 | The span of a basis is the... |
| lbsind 20954 | A basis is linearly indepe... |
| lbsind2 20955 | A basis is linearly indepe... |
| lbspss 20956 | No proper subset of a basi... |
| lsmcl 20957 | The sum of two subspaces i... |
| lsmspsn 20958 | Member of subspace sum of ... |
| lsmelval2 20959 | Subspace sum membership in... |
| lsmsp 20960 | Subspace sum in terms of s... |
| lsmsp2 20961 | Subspace sum of spans of s... |
| lsmssspx 20962 | Subspace sum (in its exten... |
| lsmpr 20963 | The span of a pair of vect... |
| lsppreli 20964 | A vector expressed as a su... |
| lsmelpr 20965 | Two ways to say that a vec... |
| lsppr0 20966 | The span of a vector paire... |
| lsppr 20967 | Span of a pair of vectors.... |
| lspprel 20968 | Member of the span of a pa... |
| lspprabs 20969 | Absorption of vector sum i... |
| lspvadd 20970 | The span of a vector sum i... |
| lspsntri 20971 | Triangle-type inequality f... |
| lspsntrim 20972 | Triangle-type inequality f... |
| lbspropd 20973 | If two structures have the... |
| pj1lmhm 20974 | The left projection functi... |
| pj1lmhm2 20975 | The left projection functi... |
| islvec 20978 | The predicate "is a left v... |
| lvecdrng 20979 | The set of scalars of a le... |
| lveclmod 20980 | A left vector space is a l... |
| lveclmodd 20981 | A vector space is a left m... |
| lvecgrpd 20982 | A vector space is a group.... |
| lsslvec 20983 | A vector subspace is a vec... |
| lmhmlvec 20984 | The property for modules t... |
| lvecvs0or 20985 | If a scalar product is zer... |
| lvecvsn0 20986 | A scalar product is nonzer... |
| lssvs0or 20987 | If a scalar product belong... |
| lvecvscan 20988 | Cancellation law for scala... |
| lvecvscan2 20989 | Cancellation law for scala... |
| lvecinv 20990 | Invert coefficient of scal... |
| lspsnvs 20991 | A nonzero scalar product d... |
| lspsneleq 20992 | Membership relation that i... |
| lspsncmp 20993 | Comparable spans of nonzer... |
| lspsnne1 20994 | Two ways to express that v... |
| lspsnne2 20995 | Two ways to express that v... |
| lspsnnecom 20996 | Swap two vectors with diff... |
| lspabs2 20997 | Absorption law for span of... |
| lspabs3 20998 | Absorption law for span of... |
| lspsneq 20999 | Equal spans of singletons ... |
| lspsneu 21000 | Nonzero vectors with equal... |
| lspsnel4 21001 | A member of the span of th... |
| lspdisj 21002 | The span of a vector not i... |
| lspdisjb 21003 | A nonzero vector is not in... |
| lspdisj2 21004 | Unequal spans are disjoint... |
| lspfixed 21005 | Show membership in the spa... |
| lspexch 21006 | Exchange property for span... |
| lspexchn1 21007 | Exchange property for span... |
| lspexchn2 21008 | Exchange property for span... |
| lspindpi 21009 | Partial independence prope... |
| lspindp1 21010 | Alternate way to say 3 vec... |
| lspindp2l 21011 | Alternate way to say 3 vec... |
| lspindp2 21012 | Alternate way to say 3 vec... |
| lspindp3 21013 | Independence of 2 vectors ... |
| lspindp4 21014 | (Partial) independence of ... |
| lvecindp 21015 | Compute the ` X ` coeffici... |
| lvecindp2 21016 | Sums of independent vector... |
| lspsnsubn0 21017 | Unequal singleton spans im... |
| lsmcv 21018 | Subspace sum has the cover... |
| lspsolvlem 21019 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21020 | If ` X ` is in the span of... |
| lssacsex 21021 | In a vector space, subspac... |
| lspsnat 21022 | There is no subspace stric... |
| lspsncv0 21023 | The span of a singleton co... |
| lsppratlem1 21024 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21025 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21026 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21027 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21028 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21029 | Lemma for ~ lspprat . Neg... |
| lspprat 21030 | A proper subspace of the s... |
| islbs2 21031 | An equivalent formulation ... |
| islbs3 21032 | An equivalent formulation ... |
| lbsacsbs 21033 | Being a basis in a vector ... |
| lvecdim 21034 | The dimension theorem for ... |
| lbsextlem1 21035 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21036 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21037 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21038 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21039 | For any linearly independe... |
| lbsext 21040 | For any linearly independe... |
| lbsexg 21041 | Every vector space has a b... |
| lbsex 21042 | Every vector space has a b... |
| lvecprop2d 21043 | If two structures have the... |
| lvecpropd 21044 | If two structures have the... |
| sraval 21049 | Lemma for ~ srabase throug... |
| sralem 21050 | Lemma for ~ srabase and si... |
| sralemOLD 21051 | Obsolete version of ~ sral... |
| srabase 21052 | Base set of a subring alge... |
| srabaseOLD 21053 | Obsolete proof of ~ srabas... |
| sraaddg 21054 | Additive operation of a su... |
| sraaddgOLD 21055 | Obsolete proof of ~ sraadd... |
| sramulr 21056 | Multiplicative operation o... |
| sramulrOLD 21057 | Obsolete proof of ~ sramul... |
| srasca 21058 | The set of scalars of a su... |
| srascaOLD 21059 | Obsolete proof of ~ srasca... |
| sravsca 21060 | The scalar product operati... |
| sravscaOLD 21061 | Obsolete proof of ~ sravsc... |
| sraip 21062 | The inner product operatio... |
| sratset 21063 | Topology component of a su... |
| sratsetOLD 21064 | Obsolete proof of ~ sratse... |
| sratopn 21065 | Topology component of a su... |
| srads 21066 | Distance function of a sub... |
| sradsOLD 21067 | Obsolete proof of ~ srads ... |
| sraring 21068 | Condition for a subring al... |
| sralmod 21069 | The subring algebra is a l... |
| sralmod0 21070 | The subring module inherit... |
| issubrgd 21071 | Prove a subring by closure... |
| rlmfn 21072 | ` ringLMod ` is a function... |
| rlmval 21073 | Value of the ring module. ... |
| rlmval2 21074 | Value of the ring module e... |
| rlmbas 21075 | Base set of the ring modul... |
| rlmplusg 21076 | Vector addition in the rin... |
| rlm0 21077 | Zero vector in the ring mo... |
| rlmsub 21078 | Subtraction in the ring mo... |
| rlmmulr 21079 | Ring multiplication in the... |
| rlmsca 21080 | Scalars in the ring module... |
| rlmsca2 21081 | Scalars in the ring module... |
| rlmvsca 21082 | Scalar multiplication in t... |
| rlmtopn 21083 | Topology component of the ... |
| rlmds 21084 | Metric component of the ri... |
| rlmlmod 21085 | The ring module is a modul... |
| rlmlvec 21086 | The ring module over a div... |
| rlmlsm 21087 | Subgroup sum of the ring m... |
| rlmvneg 21088 | Vector negation in the rin... |
| rlmscaf 21089 | Functionalized scalar mult... |
| ixpsnbasval 21090 | The value of an infinite C... |
| lidlval 21095 | Value of the set of ring i... |
| rspval 21096 | Value of the ring span fun... |
| lidlss 21097 | An ideal is a subset of th... |
| lidlssbas 21098 | The base set of the restri... |
| lidlbas 21099 | A (left) ideal of a ring i... |
| islidl 21100 | Predicate of being a (left... |
| rnglidlmcl 21101 | A (left) ideal containing ... |
| rngridlmcl 21102 | A right ideal (which is a ... |
| dflidl2rng 21103 | Alternate (the usual textb... |
| isridlrng 21104 | A right ideal is a left id... |
| lidl0cl 21105 | An ideal contains 0. (Con... |
| lidlacl 21106 | An ideal is closed under a... |
| lidlnegcl 21107 | An ideal contains negative... |
| lidlsubg 21108 | An ideal is a subgroup of ... |
| lidlsubcl 21109 | An ideal is closed under s... |
| lidlmcl 21110 | An ideal is closed under l... |
| lidl1el 21111 | An ideal contains 1 iff it... |
| dflidl2 21112 | Alternate (the usual textb... |
| lidl0ALT 21113 | Alternate proof for ~ lidl... |
| rnglidl0 21114 | Every non-unital ring cont... |
| lidl0 21115 | Every ring contains a zero... |
| lidl1ALT 21116 | Alternate proof for ~ lidl... |
| rnglidl1 21117 | The base set of every non-... |
| lidl1 21118 | Every ring contains a unit... |
| lidlacs 21119 | The ideal system is an alg... |
| rspcl 21120 | The span of a set of ring ... |
| rspssid 21121 | The span of a set of ring ... |
| rsp1 21122 | The span of the identity e... |
| rsp0 21123 | The span of the zero eleme... |
| rspssp 21124 | The ideal span of a set of... |
| mrcrsp 21125 | Moore closure generalizes ... |
| lidlnz 21126 | A nonzero ideal contains a... |
| drngnidl 21127 | A division ring has only t... |
| lidlrsppropd 21128 | The left ideals and ring s... |
| rnglidlmmgm 21129 | The multiplicative group o... |
| rnglidlmsgrp 21130 | The multiplicative group o... |
| rnglidlrng 21131 | A (left) ideal of a non-un... |
| 2idlval 21134 | Definition of a two-sided ... |
| isridl 21135 | A right ideal is a left id... |
| 2idlelb 21136 | Membership in a two-sided ... |
| 2idllidld 21137 | A two-sided ideal is a lef... |
| 2idlridld 21138 | A two-sided ideal is a rig... |
| df2idl2rng 21139 | Alternate (the usual textb... |
| df2idl2 21140 | Alternate (the usual textb... |
| ridl0 21141 | Every ring contains a zero... |
| ridl1 21142 | Every ring contains a unit... |
| 2idl0 21143 | Every ring contains a zero... |
| 2idl1 21144 | Every ring contains a unit... |
| 2idlss 21145 | A two-sided ideal is a sub... |
| 2idlbas 21146 | The base set of a two-side... |
| 2idlelbas 21147 | The base set of a two-side... |
| rng2idlsubrng 21148 | A two-sided ideal of a non... |
| rng2idlnsg 21149 | A two-sided ideal of a non... |
| rng2idl0 21150 | The zero (additive identit... |
| rng2idlsubgsubrng 21151 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21152 | A two-sided ideal of a non... |
| rng2idlsubg0 21153 | The zero (additive identit... |
| 2idlcpblrng 21154 | The coset equivalence rela... |
| 2idlcpbl 21155 | The coset equivalence rela... |
| qus2idrng 21156 | The quotient of a non-unit... |
| qus1 21157 | The multiplicative identit... |
| qusring 21158 | If ` S ` is a two-sided id... |
| qusrhm 21159 | If ` S ` is a two-sided id... |
| qusmul2 21160 | Value of the ring operatio... |
| crngridl 21161 | In a commutative ring, the... |
| crng2idl 21162 | In a commutative ring, a t... |
| qusmulrng 21163 | Value of the multiplicatio... |
| quscrng 21164 | The quotient of a commutat... |
| rngqiprng1elbas 21165 | The ring unity of a two-si... |
| rngqiprngghmlem1 21166 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21167 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21168 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21169 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21170 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21171 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21172 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21173 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21174 | The base set of the produc... |
| rngqiprng 21175 | The product of the quotien... |
| rngqiprngimf 21176 | ` F ` is a function from (... |
| rngqiprngimfv 21177 | The value of the function ... |
| rngqiprngghm 21178 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21179 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21180 | ` F ` is a function from (... |
| rngqiprnglin 21181 | ` F ` is linear with respe... |
| rngqiprngho 21182 | ` F ` is a homomorphism of... |
| rngqiprngim 21183 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21184 | The unity of a two-sided i... |
| rngringbdlem1 21185 | In a unital ring, the quot... |
| rngringbdlem2 21186 | A non-unital ring is unita... |
| rngringbd 21187 | A non-unital ring is unita... |
| ring2idlqus 21188 | For every unital ring ther... |
| ring2idlqusb 21189 | A non-unital ring is unita... |
| rngqiprngfulem1 21190 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21191 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21192 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21193 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21194 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21195 | The ring unity of the prod... |
| rngqiprngfu 21196 | The function value of ` F ... |
| rngqiprngu 21197 | If a non-unital ring has a... |
| ring2idlqus1 21198 | If a non-unital ring has a... |
| lpival 21203 | Value of the set of princi... |
| islpidl 21204 | Property of being a princi... |
| lpi0 21205 | The zero ideal is always p... |
| lpi1 21206 | The unit ideal is always p... |
| islpir 21207 | Principal ideal rings are ... |
| lpiss 21208 | Principal ideals are a sub... |
| islpir2 21209 | Principal ideal rings are ... |
| lpirring 21210 | Principal ideal rings are ... |
| drnglpir 21211 | Division rings are princip... |
| rspsn 21212 | Membership in principal id... |
| lidldvgen 21213 | An element generates an id... |
| lpigen 21214 | An ideal is principal iff ... |
| rrgval 21223 | Value of the set or left-r... |
| isrrg 21224 | Membership in the set of l... |
| rrgeq0i 21225 | Property of a left-regular... |
| rrgeq0 21226 | Left-multiplication by a l... |
| rrgsupp 21227 | Left multiplication by a l... |
| rrgss 21228 | Left-regular elements are ... |
| unitrrg 21229 | Units are regular elements... |
| isdomn 21230 | Expand definition of a dom... |
| domnnzr 21231 | A domain is a nonzero ring... |
| domnring 21232 | A domain is a ring. (Cont... |
| domneq0 21233 | In a domain, a product is ... |
| domnmuln0 21234 | In a domain, a product of ... |
| isdomn2 21235 | A ring is a domain iff all... |
| domnrrg 21236 | In a domain, any nonzero e... |
| isdomn5 21237 | The right conjunct in the ... |
| isdomn4 21238 | A ring is a domain iff it ... |
| opprdomn 21239 | The opposite of a domain i... |
| abvn0b 21240 | Another characterization o... |
| drngdomn 21241 | A division ring is a domai... |
| isidom 21242 | An integral domain is a co... |
| idomdomd 21243 | An integral domain is a do... |
| idomringd 21244 | An integral domain is a ri... |
| fldidom 21245 | A field is an integral dom... |
| fldidomOLD 21246 | Obsolete version of ~ fldi... |
| fidomndrnglem 21247 | Lemma for ~ fidomndrng . ... |
| fidomndrng 21248 | A finite domain is a divis... |
| fiidomfld 21249 | A finite integral domain i... |
| cnfldstr 21268 | The field of complex numbe... |
| cnfldex 21269 | The field of complex numbe... |
| cnfldbas 21270 | The base set of the field ... |
| mpocnfldadd 21271 | The addition operation of ... |
| cnfldadd 21272 | The addition operation of ... |
| mpocnfldmul 21273 | The multiplication operati... |
| cnfldmul 21274 | The multiplication operati... |
| cnfldcj 21275 | The conjugation operation ... |
| cnfldtset 21276 | The topology component of ... |
| cnfldle 21277 | The ordering of the field ... |
| cnfldds 21278 | The metric of the field of... |
| cnfldunif 21279 | The uniform structure comp... |
| cnfldfun 21280 | The field of complex numbe... |
| cnfldfunALT 21281 | The field of complex numbe... |
| dfcnfldOLD 21282 | Obsolete version of ~ df-c... |
| cnfldstrOLD 21283 | Obsolete version of ~ cnfl... |
| cnfldexOLD 21284 | Obsolete version of ~ cnfl... |
| cnfldbasOLD 21285 | Obsolete version of ~ cnfl... |
| cnfldaddOLD 21286 | Obsolete version of ~ cnfl... |
| cnfldmulOLD 21287 | Obsolete version of ~ cnfl... |
| cnfldcjOLD 21288 | Obsolete version of ~ cnfl... |
| cnfldtsetOLD 21289 | Obsolete version of ~ cnfl... |
| cnfldleOLD 21290 | Obsolete version of ~ cnfl... |
| cnflddsOLD 21291 | Obsolete version of ~ cnfl... |
| cnfldunifOLD 21292 | Obsolete version of ~ cnfl... |
| cnfldfunOLD 21293 | Obsolete version of ~ cnfl... |
| cnfldfunALTOLD 21294 | Obsolete version of ~ cnfl... |
| cnfldfunALTOLDOLD 21295 | Obsolete proof of ~ cnfldf... |
| xrsstr 21296 | The extended real structur... |
| xrsex 21297 | The extended real structur... |
| xrsbas 21298 | The base set of the extend... |
| xrsadd 21299 | The addition operation of ... |
| xrsmul 21300 | The multiplication operati... |
| xrstset 21301 | The topology component of ... |
| xrsle 21302 | The ordering of the extend... |
| cncrng 21303 | The complex numbers form a... |
| cncrngOLD 21304 | Obsolete version of ~ cncr... |
| cnring 21305 | The complex numbers form a... |
| xrsmcmn 21306 | The "multiplicative group"... |
| cnfld0 21307 | Zero is the zero element o... |
| cnfld1 21308 | One is the unity element o... |
| cnfld1OLD 21309 | Obsolete version of ~ cnfl... |
| cnfldneg 21310 | The additive inverse in th... |
| cnfldplusf 21311 | The functionalized additio... |
| cnfldsub 21312 | The subtraction operator i... |
| cndrng 21313 | The complex numbers form a... |
| cndrngOLD 21314 | Obsolete version of ~ cndr... |
| cnflddiv 21315 | The division operation in ... |
| cnflddivOLD 21316 | Obsolete version of ~ cnfl... |
| cnfldinv 21317 | The multiplicative inverse... |
| cnfldmulg 21318 | The group multiple functio... |
| cnfldexp 21319 | The exponentiation operato... |
| cnsrng 21320 | The complex numbers form a... |
| xrsmgm 21321 | The "additive group" of th... |
| xrsnsgrp 21322 | The "additive group" of th... |
| xrsmgmdifsgrp 21323 | The "additive group" of th... |
| xrs1mnd 21324 | The extended real numbers,... |
| xrs10 21325 | The zero of the extended r... |
| xrs1cmn 21326 | The extended real numbers ... |
| xrge0subm 21327 | The nonnegative extended r... |
| xrge0cmn 21328 | The nonnegative extended r... |
| xrsds 21329 | The metric of the extended... |
| xrsdsval 21330 | The metric of the extended... |
| xrsdsreval 21331 | The metric of the extended... |
| xrsdsreclblem 21332 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21333 | The metric of the extended... |
| cnsubmlem 21334 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21335 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21336 | Lemma for ~ resubdrg and f... |
| cnsubrglemOLD 21337 | Obsolete version of ~ cnsu... |
| cnsubdrglem 21338 | Lemma for ~ resubdrg and f... |
| qsubdrg 21339 | The rational numbers form ... |
| zsubrg 21340 | The integers form a subrin... |
| gzsubrg 21341 | The gaussian integers form... |
| nn0subm 21342 | The nonnegative integers f... |
| rege0subm 21343 | The nonnegative reals form... |
| absabv 21344 | The regular absolute value... |
| zsssubrg 21345 | The integers are a subset ... |
| qsssubdrg 21346 | The rational numbers are a... |
| cnsubrg 21347 | There are no subrings of t... |
| cnmgpabl 21348 | The unit group of the comp... |
| cnmgpid 21349 | The group identity element... |
| cnmsubglem 21350 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21351 | The positive reals form a ... |
| gzrngunitlem 21352 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21353 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21354 | Relate a group sum on ` CC... |
| regsumfsum 21355 | Relate a group sum on ` ( ... |
| expmhm 21356 | Exponentiation is a monoid... |
| nn0srg 21357 | The nonnegative integers f... |
| rge0srg 21358 | The nonnegative real numbe... |
| zringcrng 21361 | The ring of integers is a ... |
| zringring 21362 | The ring of integers is a ... |
| zringrng 21363 | The ring of integers is a ... |
| zringabl 21364 | The ring of integers is an... |
| zringgrp 21365 | The ring of integers is an... |
| zringbas 21366 | The integers are the base ... |
| zringplusg 21367 | The addition operation of ... |
| zringsub 21368 | The subtraction of element... |
| zringmulg 21369 | The multiplication (group ... |
| zringmulr 21370 | The multiplication operati... |
| zring0 21371 | The zero element of the ri... |
| zring1 21372 | The unity element of the r... |
| zringnzr 21373 | The ring of integers is a ... |
| dvdsrzring 21374 | Ring divisibility in the r... |
| zringlpirlem1 21375 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21376 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21377 | Lemma for ~ zringlpir . A... |
| zringinvg 21378 | The additive inverse of an... |
| zringunit 21379 | The units of ` ZZ ` are th... |
| zringlpir 21380 | The integers are a princip... |
| zringndrg 21381 | The integers are not a div... |
| zringcyg 21382 | The integers are a cyclic ... |
| zringsubgval 21383 | Subtraction in the ring of... |
| zringmpg 21384 | The multiplicative group o... |
| prmirredlem 21385 | A positive integer is irre... |
| dfprm2 21386 | The positive irreducible e... |
| prmirred 21387 | The irreducible elements o... |
| expghm 21388 | Exponentiation is a group ... |
| mulgghm2 21389 | The powers of a group elem... |
| mulgrhm 21390 | The powers of the element ... |
| mulgrhm2 21391 | The powers of the element ... |
| irinitoringc 21392 | The ring of integers is an... |
| nzerooringczr 21393 | There is no zero object in... |
| pzriprnglem1 21394 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21395 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21396 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21397 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21398 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21399 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21400 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21401 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21402 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21403 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21404 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21405 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21406 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21407 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21408 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21409 | The ring unity of the ring... |
| pzriprng 21410 | The non-unital ring ` ( ZZ... |
| pzriprng1 21411 | The ring unity of the ring... |
| zrhval 21420 | Define the unique homomorp... |
| zrhval2 21421 | Alternate value of the ` Z... |
| zrhmulg 21422 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21423 | The ` ZRHom ` homomorphism... |
| zrhrhm 21424 | The ` ZRHom ` homomorphism... |
| zrh1 21425 | Interpretation of 1 in a r... |
| zrh0 21426 | Interpretation of 0 in a r... |
| zrhpropd 21427 | The ` ZZ ` ring homomorphi... |
| zlmval 21428 | Augment an abelian group w... |
| zlmlem 21429 | Lemma for ~ zlmbas and ~ z... |
| zlmlemOLD 21430 | Obsolete version of ~ zlml... |
| zlmbas 21431 | Base set of a ` ZZ ` -modu... |
| zlmbasOLD 21432 | Obsolete version of ~ zlmb... |
| zlmplusg 21433 | Group operation of a ` ZZ ... |
| zlmplusgOLD 21434 | Obsolete version of ~ zlmb... |
| zlmmulr 21435 | Ring operation of a ` ZZ `... |
| zlmmulrOLD 21436 | Obsolete version of ~ zlmb... |
| zlmsca 21437 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21438 | Scalar multiplication oper... |
| zlmlmod 21439 | The ` ZZ ` -module operati... |
| chrval 21440 | Definition substitution of... |
| chrcl 21441 | Closure of the characteris... |
| chrid 21442 | The canonical ` ZZ ` ring ... |
| chrdvds 21443 | The ` ZZ ` ring homomorphi... |
| chrcong 21444 | If two integers are congru... |
| dvdschrmulg 21445 | In a ring, any multiple of... |
| fermltlchr 21446 | A generalization of Fermat... |
| chrnzr 21447 | Nonzero rings are precisel... |
| chrrhm 21448 | The characteristic restric... |
| domnchr 21449 | The characteristic of a do... |
| znlidl 21450 | The set ` n ZZ ` is an ide... |
| zncrng2 21451 | The value of the ` Z/nZ ` ... |
| znval 21452 | The value of the ` Z/nZ ` ... |
| znle 21453 | The value of the ` Z/nZ ` ... |
| znval2 21454 | Self-referential expressio... |
| znbaslem 21455 | Lemma for ~ znbas . (Cont... |
| znbaslemOLD 21456 | Obsolete version of ~ znba... |
| znbas2 21457 | The base set of ` Z/nZ ` i... |
| znbas2OLD 21458 | Obsolete version of ~ znba... |
| znadd 21459 | The additive structure of ... |
| znaddOLD 21460 | Obsolete version of ~ znad... |
| znmul 21461 | The multiplicative structu... |
| znmulOLD 21462 | Obsolete version of ~ znad... |
| znzrh 21463 | The ` ZZ ` ring homomorphi... |
| znbas 21464 | The base set of ` Z/nZ ` s... |
| zncrng 21465 | ` Z/nZ ` is a commutative ... |
| znzrh2 21466 | The ` ZZ ` ring homomorphi... |
| znzrhval 21467 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21468 | The ` ZZ ` ring homomorphi... |
| zncyg 21469 | The group ` ZZ / n ZZ ` is... |
| zndvds 21470 | Express equality of equiva... |
| zndvds0 21471 | Special case of ~ zndvds w... |
| znf1o 21472 | The function ` F ` enumera... |
| zzngim 21473 | The ` ZZ ` ring homomorphi... |
| znle2 21474 | The ordering of the ` Z/nZ... |
| znleval 21475 | The ordering of the ` Z/nZ... |
| znleval2 21476 | The ordering of the ` Z/nZ... |
| zntoslem 21477 | Lemma for ~ zntos . (Cont... |
| zntos 21478 | The ` Z/nZ ` structure is ... |
| znhash 21479 | The ` Z/nZ ` structure has... |
| znfi 21480 | The ` Z/nZ ` structure is ... |
| znfld 21481 | The ` Z/nZ ` structure is ... |
| znidomb 21482 | The ` Z/nZ ` structure is ... |
| znchr 21483 | Cyclic rings are defined b... |
| znunit 21484 | The units of ` Z/nZ ` are ... |
| znunithash 21485 | The size of the unit group... |
| znrrg 21486 | The regular elements of ` ... |
| cygznlem1 21487 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21488 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21489 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21490 | A cyclic group with ` n ` ... |
| cygzn 21491 | A cyclic group with ` n ` ... |
| cygth 21492 | The "fundamental theorem o... |
| cyggic 21493 | Cyclic groups are isomorph... |
| frgpcyg 21494 | A free group is cyclic iff... |
| freshmansdream 21495 | For a prime number ` P ` ,... |
| cnmsgnsubg 21496 | The signs form a multiplic... |
| cnmsgnbas 21497 | The base set of the sign s... |
| cnmsgngrp 21498 | The group of signs under m... |
| psgnghm 21499 | The sign is a homomorphism... |
| psgnghm2 21500 | The sign is a homomorphism... |
| psgninv 21501 | The sign of a permutation ... |
| psgnco 21502 | Multiplicativity of the pe... |
| zrhpsgnmhm 21503 | Embedding of permutation s... |
| zrhpsgninv 21504 | The embedded sign of a per... |
| evpmss 21505 | Even permutations are perm... |
| psgnevpmb 21506 | A class is an even permuta... |
| psgnodpm 21507 | A permutation which is odd... |
| psgnevpm 21508 | A permutation which is eve... |
| psgnodpmr 21509 | If a permutation has sign ... |
| zrhpsgnevpm 21510 | The sign of an even permut... |
| zrhpsgnodpm 21511 | The sign of an odd permuta... |
| cofipsgn 21512 | Composition of any class `... |
| zrhpsgnelbas 21513 | Embedding of permutation s... |
| zrhcopsgnelbas 21514 | Embedding of permutation s... |
| evpmodpmf1o 21515 | The function for performin... |
| pmtrodpm 21516 | A transposition is an odd ... |
| psgnfix1 21517 | A permutation of a finite ... |
| psgnfix2 21518 | A permutation of a finite ... |
| psgndiflemB 21519 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21520 | Lemma 2 for ~ psgndif . (... |
| psgndif 21521 | Embedding of permutation s... |
| copsgndif 21522 | Embedding of permutation s... |
| rebase 21525 | The base of the field of r... |
| remulg 21526 | The multiplication (group ... |
| resubdrg 21527 | The real numbers form a di... |
| resubgval 21528 | Subtraction in the field o... |
| replusg 21529 | The addition operation of ... |
| remulr 21530 | The multiplication operati... |
| re0g 21531 | The zero element of the fi... |
| re1r 21532 | The unity element of the f... |
| rele2 21533 | The ordering relation of t... |
| relt 21534 | The ordering relation of t... |
| reds 21535 | The distance of the field ... |
| redvr 21536 | The division operation of ... |
| retos 21537 | The real numbers are a tot... |
| refld 21538 | The real numbers form a fi... |
| refldcj 21539 | The conjugation operation ... |
| resrng 21540 | The real numbers form a st... |
| regsumsupp 21541 | The group sum over the rea... |
| rzgrp 21542 | The quotient group ` RR / ... |
| isphl 21547 | The predicate "is a genera... |
| phllvec 21548 | A pre-Hilbert space is a l... |
| phllmod 21549 | A pre-Hilbert space is a l... |
| phlsrng 21550 | The scalar ring of a pre-H... |
| phllmhm 21551 | The inner product of a pre... |
| ipcl 21552 | Closure of the inner produ... |
| ipcj 21553 | Conjugate of an inner prod... |
| iporthcom 21554 | Orthogonality (meaning inn... |
| ip0l 21555 | Inner product with a zero ... |
| ip0r 21556 | Inner product with a zero ... |
| ipeq0 21557 | The inner product of a vec... |
| ipdir 21558 | Distributive law for inner... |
| ipdi 21559 | Distributive law for inner... |
| ip2di 21560 | Distributive law for inner... |
| ipsubdir 21561 | Distributive law for inner... |
| ipsubdi 21562 | Distributive law for inner... |
| ip2subdi 21563 | Distributive law for inner... |
| ipass 21564 | Associative law for inner ... |
| ipassr 21565 | "Associative" law for seco... |
| ipassr2 21566 | "Associative" law for inne... |
| ipffval 21567 | The inner product operatio... |
| ipfval 21568 | The inner product operatio... |
| ipfeq 21569 | If the inner product opera... |
| ipffn 21570 | The inner product operatio... |
| phlipf 21571 | The inner product operatio... |
| ip2eq 21572 | Two vectors are equal iff ... |
| isphld 21573 | Properties that determine ... |
| phlpropd 21574 | If two structures have the... |
| ssipeq 21575 | The inner product on a sub... |
| phssipval 21576 | The inner product on a sub... |
| phssip 21577 | The inner product (as a fu... |
| phlssphl 21578 | A subspace of an inner pro... |
| ocvfval 21585 | The orthocomplement operat... |
| ocvval 21586 | Value of the orthocompleme... |
| elocv 21587 | Elementhood in the orthoco... |
| ocvi 21588 | Property of a member of th... |
| ocvss 21589 | The orthocomplement of a s... |
| ocvocv 21590 | A set is contained in its ... |
| ocvlss 21591 | The orthocomplement of a s... |
| ocv2ss 21592 | Orthocomplements reverse s... |
| ocvin 21593 | An orthocomplement has tri... |
| ocvsscon 21594 | Two ways to say that ` S `... |
| ocvlsp 21595 | The orthocomplement of a l... |
| ocv0 21596 | The orthocomplement of the... |
| ocvz 21597 | The orthocomplement of the... |
| ocv1 21598 | The orthocomplement of the... |
| unocv 21599 | The orthocomplement of a u... |
| iunocv 21600 | The orthocomplement of an ... |
| cssval 21601 | The set of closed subspace... |
| iscss 21602 | The predicate "is a closed... |
| cssi 21603 | Property of a closed subsp... |
| cssss 21604 | A closed subspace is a sub... |
| iscss2 21605 | It is sufficient to prove ... |
| ocvcss 21606 | The orthocomplement of any... |
| cssincl 21607 | The zero subspace is a clo... |
| css0 21608 | The zero subspace is a clo... |
| css1 21609 | The whole space is a close... |
| csslss 21610 | A closed subspace of a pre... |
| lsmcss 21611 | A subset of a pre-Hilbert ... |
| cssmre 21612 | The closed subspaces of a ... |
| mrccss 21613 | The Moore closure correspo... |
| thlval 21614 | Value of the Hilbert latti... |
| thlbas 21615 | Base set of the Hilbert la... |
| thlbasOLD 21616 | Obsolete proof of ~ thlbas... |
| thlle 21617 | Ordering on the Hilbert la... |
| thlleOLD 21618 | Obsolete proof of ~ thlle ... |
| thlleval 21619 | Ordering on the Hilbert la... |
| thloc 21620 | Orthocomplement on the Hil... |
| pjfval 21627 | The value of the projectio... |
| pjdm 21628 | A subspace is in the domai... |
| pjpm 21629 | The projection map is a pa... |
| pjfval2 21630 | Value of the projection ma... |
| pjval 21631 | Value of the projection ma... |
| pjdm2 21632 | A subspace is in the domai... |
| pjff 21633 | A projection is a linear o... |
| pjf 21634 | A projection is a function... |
| pjf2 21635 | A projection is a function... |
| pjfo 21636 | A projection is a surjecti... |
| pjcss 21637 | A projection subspace is a... |
| ocvpj 21638 | The orthocomplement of a p... |
| ishil 21639 | The predicate "is a Hilber... |
| ishil2 21640 | The predicate "is a Hilber... |
| isobs 21641 | The predicate "is an ortho... |
| obsip 21642 | The inner product of two e... |
| obsipid 21643 | A basis element has length... |
| obsrcl 21644 | Reverse closure for an ort... |
| obsss 21645 | An orthonormal basis is a ... |
| obsne0 21646 | A basis element is nonzero... |
| obsocv 21647 | An orthonormal basis has t... |
| obs2ocv 21648 | The double orthocomplement... |
| obselocv 21649 | A basis element is in the ... |
| obs2ss 21650 | A basis has no proper subs... |
| obslbs 21651 | An orthogonal basis is a l... |
| reldmdsmm 21654 | The direct sum is a well-b... |
| dsmmval 21655 | Value of the module direct... |
| dsmmbase 21656 | Base set of the module dir... |
| dsmmval2 21657 | Self-referential definitio... |
| dsmmbas2 21658 | Base set of the direct sum... |
| dsmmfi 21659 | For finite products, the d... |
| dsmmelbas 21660 | Membership in the finitely... |
| dsmm0cl 21661 | The all-zero vector is con... |
| dsmmacl 21662 | The finite hull is closed ... |
| prdsinvgd2 21663 | Negation of a single coord... |
| dsmmsubg 21664 | The finite hull of a produ... |
| dsmmlss 21665 | The finite hull of a produ... |
| dsmmlmod 21666 | The direct sum of a family... |
| frlmval 21669 | Value of the "free module"... |
| frlmlmod 21670 | The free module is a modul... |
| frlmpws 21671 | The free module as a restr... |
| frlmlss 21672 | The base set of the free m... |
| frlmpwsfi 21673 | The finite free module is ... |
| frlmsca 21674 | The ring of scalars of a f... |
| frlm0 21675 | Zero in a free module (rin... |
| frlmbas 21676 | Base set of the free modul... |
| frlmelbas 21677 | Membership in the base set... |
| frlmrcl 21678 | If a free module is inhabi... |
| frlmbasfsupp 21679 | Elements of the free modul... |
| frlmbasmap 21680 | Elements of the free modul... |
| frlmbasf 21681 | Elements of the free modul... |
| frlmlvec 21682 | The free module over a div... |
| frlmfibas 21683 | The base set of the finite... |
| elfrlmbasn0 21684 | If the dimension of a free... |
| frlmplusgval 21685 | Addition in a free module.... |
| frlmsubgval 21686 | Subtraction in a free modu... |
| frlmvscafval 21687 | Scalar multiplication in a... |
| frlmvplusgvalc 21688 | Coordinates of a sum with ... |
| frlmvscaval 21689 | Coordinates of a scalar mu... |
| frlmplusgvalb 21690 | Addition in a free module ... |
| frlmvscavalb 21691 | Scalar multiplication in a... |
| frlmvplusgscavalb 21692 | Addition combined with sca... |
| frlmgsum 21693 | Finite commutative sums in... |
| frlmsplit2 21694 | Restriction is homomorphic... |
| frlmsslss 21695 | A subset of a free module ... |
| frlmsslss2 21696 | A subset of a free module ... |
| frlmbas3 21697 | An element of the base set... |
| mpofrlmd 21698 | Elements of the free modul... |
| frlmip 21699 | The inner product of a fre... |
| frlmipval 21700 | The inner product of a fre... |
| frlmphllem 21701 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21702 | Conditions for a free modu... |
| uvcfval 21705 | Value of the unit-vector g... |
| uvcval 21706 | Value of a single unit vec... |
| uvcvval 21707 | Value of a unit vector coo... |
| uvcvvcl 21708 | A coordinate of a unit vec... |
| uvcvvcl2 21709 | A unit vector coordinate i... |
| uvcvv1 21710 | The unit vector is one at ... |
| uvcvv0 21711 | The unit vector is zero at... |
| uvcff 21712 | Domain and codomain of the... |
| uvcf1 21713 | In a nonzero ring, each un... |
| uvcresum 21714 | Any element of a free modu... |
| frlmssuvc1 21715 | A scalar multiple of a uni... |
| frlmssuvc2 21716 | A nonzero scalar multiple ... |
| frlmsslsp 21717 | A subset of a free module ... |
| frlmlbs 21718 | The unit vectors comprise ... |
| frlmup1 21719 | Any assignment of unit vec... |
| frlmup2 21720 | The evaluation map has the... |
| frlmup3 21721 | The range of such an evalu... |
| frlmup4 21722 | Universal property of the ... |
| ellspd 21723 | The elements of the span o... |
| elfilspd 21724 | Simplified version of ~ el... |
| rellindf 21729 | The independent-family pre... |
| islinds 21730 | Property of an independent... |
| linds1 21731 | An independent set of vect... |
| linds2 21732 | An independent set of vect... |
| islindf 21733 | Property of an independent... |
| islinds2 21734 | Expanded property of an in... |
| islindf2 21735 | Property of an independent... |
| lindff 21736 | Functional property of a l... |
| lindfind 21737 | A linearly independent fam... |
| lindsind 21738 | A linearly independent set... |
| lindfind2 21739 | In a linearly independent ... |
| lindsind2 21740 | In a linearly independent ... |
| lindff1 21741 | A linearly independent fam... |
| lindfrn 21742 | The range of an independen... |
| f1lindf 21743 | Rearranging and deleting e... |
| lindfres 21744 | Any restriction of an inde... |
| lindsss 21745 | Any subset of an independe... |
| f1linds 21746 | A family constructed from ... |
| islindf3 21747 | In a nonzero ring, indepen... |
| lindfmm 21748 | Linear independence of a f... |
| lindsmm 21749 | Linear independence of a s... |
| lindsmm2 21750 | The monomorphic image of a... |
| lsslindf 21751 | Linear independence is unc... |
| lsslinds 21752 | Linear independence is unc... |
| islbs4 21753 | A basis is an independent ... |
| lbslinds 21754 | A basis is independent. (... |
| islinds3 21755 | A subset is linearly indep... |
| islinds4 21756 | A set is independent in a ... |
| lmimlbs 21757 | The isomorphic image of a ... |
| lmiclbs 21758 | Having a basis is an isomo... |
| islindf4 21759 | A family is independent if... |
| islindf5 21760 | A family is independent if... |
| indlcim 21761 | An independent, spanning f... |
| lbslcic 21762 | A module with a basis is i... |
| lmisfree 21763 | A module has a basis iff i... |
| lvecisfrlm 21764 | Every vector space is isom... |
| lmimco 21765 | The composition of two iso... |
| lmictra 21766 | Module isomorphism is tran... |
| uvcf1o 21767 | In a nonzero ring, the map... |
| uvcendim 21768 | In a nonzero ring, the num... |
| frlmisfrlm 21769 | A free module is isomorphi... |
| frlmiscvec 21770 | Every free module is isomo... |
| isassa 21777 | The properties of an assoc... |
| assalem 21778 | The properties of an assoc... |
| assaass 21779 | Left-associative property ... |
| assaassr 21780 | Right-associative property... |
| assalmod 21781 | An associative algebra is ... |
| assaring 21782 | An associative algebra is ... |
| assasca 21783 | The scalars of an associat... |
| assa2ass 21784 | Left- and right-associativ... |
| isassad 21785 | Sufficient condition for b... |
| issubassa3 21786 | A subring that is also a s... |
| issubassa 21787 | The subalgebras of an asso... |
| sraassab 21788 | A subring algebra is an as... |
| sraassa 21789 | The subring algebra over a... |
| sraassaOLD 21790 | Obsolete version of ~ sraa... |
| rlmassa 21791 | The ring module over a com... |
| assapropd 21792 | If two structures have the... |
| aspval 21793 | Value of the algebraic clo... |
| asplss 21794 | The algebraic span of a se... |
| aspid 21795 | The algebraic span of a su... |
| aspsubrg 21796 | The algebraic span of a se... |
| aspss 21797 | Span preserves subset orde... |
| aspssid 21798 | A set of vectors is a subs... |
| asclfval 21799 | Function value of the alge... |
| asclval 21800 | Value of a mapped algebra ... |
| asclfn 21801 | Unconditional functionalit... |
| asclf 21802 | The algebra scalars functi... |
| asclghm 21803 | The algebra scalars functi... |
| ascl0 21804 | The scalar 0 embedded into... |
| ascl1 21805 | The scalar 1 embedded into... |
| asclmul1 21806 | Left multiplication by a l... |
| asclmul2 21807 | Right multiplication by a ... |
| ascldimul 21808 | The algebra scalars functi... |
| asclinvg 21809 | The group inverse (negatio... |
| asclrhm 21810 | The scalar injection is a ... |
| rnascl 21811 | The set of injected scalar... |
| issubassa2 21812 | A subring of a unital alge... |
| rnasclsubrg 21813 | The scalar multiples of th... |
| rnasclmulcl 21814 | (Vector) multiplication is... |
| rnasclassa 21815 | The scalar multiples of th... |
| ressascl 21816 | The injection of scalars i... |
| asclpropd 21817 | If two structures have the... |
| aspval2 21818 | The algebraic closure is t... |
| assamulgscmlem1 21819 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21820 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21821 | Exponentiation of a scalar... |
| asclmulg 21822 | Apply group multiplication... |
| zlmassa 21823 | The ` ZZ ` -module operati... |
| reldmpsr 21834 | The multivariate power ser... |
| psrval 21835 | Value of the multivariate ... |
| psrvalstr 21836 | The multivariate power ser... |
| psrbag 21837 | Elementhood in the set of ... |
| psrbagf 21838 | A finite bag is a function... |
| psrbagfOLD 21839 | Obsolete version of ~ psrb... |
| psrbagfsupp 21840 | Finite bags have finite su... |
| psrbagfsuppOLD 21841 | Obsolete version of ~ psrb... |
| snifpsrbag 21842 | A bag containing one eleme... |
| fczpsrbag 21843 | The constant function equa... |
| psrbaglesupp 21844 | The support of a dominated... |
| psrbaglesuppOLD 21845 | Obsolete version of ~ psrb... |
| psrbaglecl 21846 | The set of finite bags is ... |
| psrbagleclOLD 21847 | Obsolete version of ~ psrb... |
| psrbagaddcl 21848 | The sum of two finite bags... |
| psrbagaddclOLD 21849 | Obsolete version of ~ psrb... |
| psrbagcon 21850 | The analogue of the statem... |
| psrbagconOLD 21851 | Obsolete version of ~ psrb... |
| psrbaglefi 21852 | There are finitely many ba... |
| psrbaglefiOLD 21853 | Obsolete version of ~ psrb... |
| psrbagconcl 21854 | The complement of a bag is... |
| psrbagconclOLD 21855 | Obsolete version of ~ psrb... |
| psrbagleadd1 21856 | The analogue of " ` X <_ F... |
| psrbagconf1o 21857 | Bag complementation is a b... |
| psrbagconf1oOLD 21858 | Obsolete version of ~ psrb... |
| gsumbagdiaglemOLD 21859 | Obsolete version of ~ gsum... |
| gsumbagdiagOLD 21860 | Obsolete version of ~ gsum... |
| psrass1lemOLD 21861 | Obsolete version of ~ psra... |
| gsumbagdiaglem 21862 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21863 | Two-dimensional commutatio... |
| psrass1lem 21864 | A group sum commutation us... |
| psrbas 21865 | The base set of the multiv... |
| psrelbas 21866 | An element of the set of p... |
| psrelbasfun 21867 | An element of the set of p... |
| psrplusg 21868 | The addition operation of ... |
| psradd 21869 | The addition operation of ... |
| psraddcl 21870 | Closure of the power serie... |
| psraddclOLD 21871 | Obsolete version of ~ psra... |
| psrmulr 21872 | The multiplication operati... |
| psrmulfval 21873 | The multiplication operati... |
| psrmulval 21874 | The multiplication operati... |
| psrmulcllem 21875 | Closure of the power serie... |
| psrmulcl 21876 | Closure of the power serie... |
| psrsca 21877 | The scalar field of the mu... |
| psrvscafval 21878 | The scalar multiplication ... |
| psrvsca 21879 | The scalar multiplication ... |
| psrvscaval 21880 | The scalar multiplication ... |
| psrvscacl 21881 | Closure of the power serie... |
| psr0cl 21882 | The zero element of the ri... |
| psr0lid 21883 | The zero element of the ri... |
| psrnegcl 21884 | The negative function in t... |
| psrlinv 21885 | The negative function in t... |
| psrgrp 21886 | The ring of power series i... |
| psrgrpOLD 21887 | Obsolete proof of ~ psrgrp... |
| psr0 21888 | The zero element of the ri... |
| psrneg 21889 | The negative function of t... |
| psrlmod 21890 | The ring of power series i... |
| psr1cl 21891 | The identity element of th... |
| psrlidm 21892 | The identity element of th... |
| psrridm 21893 | The identity element of th... |
| psrass1 21894 | Associative identity for t... |
| psrdi 21895 | Distributive law for the r... |
| psrdir 21896 | Distributive law for the r... |
| psrass23l 21897 | Associative identity for t... |
| psrcom 21898 | Commutative law for the ri... |
| psrass23 21899 | Associative identities for... |
| psrring 21900 | The ring of power series i... |
| psr1 21901 | The identity element of th... |
| psrcrng 21902 | The ring of power series i... |
| psrassa 21903 | The ring of power series i... |
| resspsrbas 21904 | A restricted power series ... |
| resspsradd 21905 | A restricted power series ... |
| resspsrmul 21906 | A restricted power series ... |
| resspsrvsca 21907 | A restricted power series ... |
| subrgpsr 21908 | A subring of the base ring... |
| psrasclcl 21909 | A scalar is lifted into a ... |
| mvrfval 21910 | Value of the generating el... |
| mvrval 21911 | Value of the generating el... |
| mvrval2 21912 | Value of the generating el... |
| mvrid 21913 | The ` X i ` -th coefficien... |
| mvrf 21914 | The power series variable ... |
| mvrf1 21915 | The power series variable ... |
| mvrcl2 21916 | A power series variable is... |
| reldmmpl 21917 | The multivariate polynomia... |
| mplval 21918 | Value of the set of multiv... |
| mplbas 21919 | Base set of the set of mul... |
| mplelbas 21920 | Property of being a polyno... |
| mvrcl 21921 | A power series variable is... |
| mvrf2 21922 | The power series/polynomia... |
| mplrcl 21923 | Reverse closure for the po... |
| mplelsfi 21924 | A polynomial treated as a ... |
| mplval2 21925 | Self-referential expressio... |
| mplbasss 21926 | The set of polynomials is ... |
| mplelf 21927 | A polynomial is defined as... |
| mplsubglem 21928 | If ` A ` is an ideal of se... |
| mpllsslem 21929 | If ` A ` is an ideal of su... |
| mplsubglem2 21930 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 21931 | The set of polynomials is ... |
| mpllss 21932 | The set of polynomials is ... |
| mplsubrglem 21933 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 21934 | The set of polynomials is ... |
| mpl0 21935 | The zero polynomial. (Con... |
| mplplusg 21936 | Value of addition in a pol... |
| mplmulr 21937 | Value of multiplication in... |
| mpladd 21938 | The addition operation on ... |
| mplneg 21939 | The negative function on m... |
| mplmul 21940 | The multiplication operati... |
| mpl1 21941 | The identity element of th... |
| mplsca 21942 | The scalar field of a mult... |
| mplvsca2 21943 | The scalar multiplication ... |
| mplvsca 21944 | The scalar multiplication ... |
| mplvscaval 21945 | The scalar multiplication ... |
| mplgrp 21946 | The polynomial ring is a g... |
| mpllmod 21947 | The polynomial ring is a l... |
| mplring 21948 | The polynomial ring is a r... |
| mpllvec 21949 | The polynomial ring is a v... |
| mplcrng 21950 | The polynomial ring is a c... |
| mplassa 21951 | The polynomial ring is an ... |
| ressmplbas2 21952 | The base set of a restrict... |
| ressmplbas 21953 | A restricted polynomial al... |
| ressmpladd 21954 | A restricted polynomial al... |
| ressmplmul 21955 | A restricted polynomial al... |
| ressmplvsca 21956 | A restricted power series ... |
| subrgmpl 21957 | A subring of the base ring... |
| subrgmvr 21958 | The variables in a subring... |
| subrgmvrf 21959 | The variables in a polynom... |
| mplmon 21960 | A monomial is a polynomial... |
| mplmonmul 21961 | The product of two monomia... |
| mplcoe1 21962 | Decompose a polynomial int... |
| mplcoe3 21963 | Decompose a monomial in on... |
| mplcoe5lem 21964 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 21965 | Decompose a monomial into ... |
| mplcoe2 21966 | Decompose a monomial into ... |
| mplbas2 21967 | An alternative expression ... |
| ltbval 21968 | Value of the well-order on... |
| ltbwe 21969 | The finite bag order is a ... |
| reldmopsr 21970 | Lemma for ordered power se... |
| opsrval 21971 | The value of the "ordered ... |
| opsrle 21972 | An alternative expression ... |
| opsrval2 21973 | Self-referential expressio... |
| opsrbaslem 21974 | Get a component of the ord... |
| opsrbaslemOLD 21975 | Obsolete version of ~ opsr... |
| opsrbas 21976 | The base set of the ordere... |
| opsrbasOLD 21977 | Obsolete version of ~ opsr... |
| opsrplusg 21978 | The addition operation of ... |
| opsrplusgOLD 21979 | Obsolete version of ~ opsr... |
| opsrmulr 21980 | The multiplication operati... |
| opsrmulrOLD 21981 | Obsolete version of ~ opsr... |
| opsrvsca 21982 | The scalar product operati... |
| opsrvscaOLD 21983 | Obsolete version of ~ opsr... |
| opsrsca 21984 | The scalar ring of the ord... |
| opsrscaOLD 21985 | Obsolete version of ~ opsr... |
| opsrtoslem1 21986 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 21987 | Lemma for ~ opsrtos . (Co... |
| opsrtos 21988 | The ordered power series s... |
| opsrso 21989 | The ordered power series s... |
| opsrcrng 21990 | The ring of ordered power ... |
| opsrassa 21991 | The ring of ordered power ... |
| mplmon2 21992 | Express a scaled monomial.... |
| psrbag0 21993 | The empty bag is a bag. (... |
| psrbagsn 21994 | A singleton bag is a bag. ... |
| mplascl 21995 | Value of the scalar inject... |
| mplasclf 21996 | The scalar injection is a ... |
| subrgascl 21997 | The scalar injection funct... |
| subrgasclcl 21998 | The scalars in a polynomia... |
| mplmon2cl 21999 | A scaled monomial is a pol... |
| mplmon2mul 22000 | Product of scaled monomial... |
| mplind 22001 | Prove a property of polyno... |
| mplcoe4 22002 | Decompose a polynomial int... |
| evlslem4 22007 | The support of a tensor pr... |
| psrbagev1 22008 | A bag of multipliers provi... |
| psrbagev1OLD 22009 | Obsolete version of ~ psrb... |
| psrbagev2 22010 | Closure of a sum using a b... |
| psrbagev2OLD 22011 | Obsolete version of ~ psrb... |
| evlslem2 22012 | A linear function on the p... |
| evlslem3 22013 | Lemma for ~ evlseu . Poly... |
| evlslem6 22014 | Lemma for ~ evlseu . Fini... |
| evlslem1 22015 | Lemma for ~ evlseu , give ... |
| evlseu 22016 | For a given interpretation... |
| reldmevls 22017 | Well-behaved binary operat... |
| mpfrcl 22018 | Reverse closure for the se... |
| evlsval 22019 | Value of the polynomial ev... |
| evlsval2 22020 | Characterizing properties ... |
| evlsrhm 22021 | Polynomial evaluation is a... |
| evlssca 22022 | Polynomial evaluation maps... |
| evlsvar 22023 | Polynomial evaluation maps... |
| evlsgsumadd 22024 | Polynomial evaluation maps... |
| evlsgsummul 22025 | Polynomial evaluation maps... |
| evlspw 22026 | Polynomial evaluation for ... |
| evlsvarpw 22027 | Polynomial evaluation for ... |
| evlval 22028 | Value of the simple/same r... |
| evlrhm 22029 | The simple evaluation map ... |
| evlsscasrng 22030 | The evaluation of a scalar... |
| evlsca 22031 | Simple polynomial evaluati... |
| evlsvarsrng 22032 | The evaluation of the vari... |
| evlvar 22033 | Simple polynomial evaluati... |
| mpfconst 22034 | Constants are multivariate... |
| mpfproj 22035 | Projections are multivaria... |
| mpfsubrg 22036 | Polynomial functions are a... |
| mpff 22037 | Polynomial functions are f... |
| mpfaddcl 22038 | The sum of multivariate po... |
| mpfmulcl 22039 | The product of multivariat... |
| mpfind 22040 | Prove a property of polyno... |
| selvffval 22046 | Value of the "variable sel... |
| selvfval 22047 | Value of the "variable sel... |
| selvval 22048 | Value of the "variable sel... |
| mhpfval 22050 | Value of the "homogeneous ... |
| mhpval 22051 | Value of the "homogeneous ... |
| ismhp 22052 | Property of being a homoge... |
| ismhp2 22053 | Deduce a homogeneous polyn... |
| ismhp3 22054 | A polynomial is homogeneou... |
| mhpmpl 22055 | A homogeneous polynomial i... |
| mhpdeg 22056 | All nonzero terms of a hom... |
| mhp0cl 22057 | The zero polynomial is hom... |
| mhpsclcl 22058 | A scalar (or constant) pol... |
| mhpvarcl 22059 | A power series variable is... |
| mhpmulcl 22060 | A product of homogeneous p... |
| mhppwdeg 22061 | Degree of a homogeneous po... |
| mhpaddcl 22062 | Homogeneous polynomials ar... |
| mhpinvcl 22063 | Homogeneous polynomials ar... |
| mhpsubg 22064 | Homogeneous polynomials fo... |
| mhpvscacl 22065 | Homogeneous polynomials ar... |
| mhplss 22066 | Homogeneous polynomials fo... |
| psdffval 22068 | Value of the power series ... |
| psdfval 22069 | Give a map between power s... |
| psdval 22070 | Evaluate the partial deriv... |
| psdcoef 22071 | Coefficient of a term of t... |
| psdcl 22072 | The derivative of a power ... |
| psdmplcl 22073 | The derivative of a polyno... |
| psdadd 22074 | The derivative of a sum is... |
| psdvsca 22075 | The derivative of a scaled... |
| psdmullem 22076 | Lemma for ~ psdmul . Tran... |
| psdmul 22077 | Product rule for power ser... |
| psd1 22078 | The derivative of one is z... |
| psdascl 22079 | The derivative of a consta... |
| psr1baslem 22091 | The set of finite bags on ... |
| psr1val 22092 | Value of the ring of univa... |
| psr1crng 22093 | The ring of univariate pow... |
| psr1assa 22094 | The ring of univariate pow... |
| psr1tos 22095 | The ordered power series s... |
| psr1bas2 22096 | The base set of the ring o... |
| psr1bas 22097 | The base set of the ring o... |
| vr1val 22098 | The value of the generator... |
| vr1cl2 22099 | The variable ` X ` is a me... |
| ply1val 22100 | The value of the set of un... |
| ply1bas 22101 | The value of the base set ... |
| ply1lss 22102 | Univariate polynomials for... |
| ply1subrg 22103 | Univariate polynomials for... |
| ply1crng 22104 | The ring of univariate pol... |
| ply1assa 22105 | The ring of univariate pol... |
| psr1bascl 22106 | A univariate power series ... |
| psr1basf 22107 | Univariate power series ba... |
| ply1basf 22108 | Univariate polynomial base... |
| ply1bascl 22109 | A univariate polynomial is... |
| ply1bascl2 22110 | A univariate polynomial is... |
| coe1fval 22111 | Value of the univariate po... |
| coe1fv 22112 | Value of an evaluated coef... |
| fvcoe1 22113 | Value of a multivariate co... |
| coe1fval3 22114 | Univariate power series co... |
| coe1f2 22115 | Functionality of univariat... |
| coe1fval2 22116 | Univariate polynomial coef... |
| coe1f 22117 | Functionality of univariat... |
| coe1fvalcl 22118 | A coefficient of a univari... |
| coe1sfi 22119 | Finite support of univaria... |
| coe1fsupp 22120 | The coefficient vector of ... |
| mptcoe1fsupp 22121 | A mapping involving coeffi... |
| coe1ae0 22122 | The coefficient vector of ... |
| vr1cl 22123 | The generator of a univari... |
| opsr0 22124 | Zero in the ordered power ... |
| opsr1 22125 | One in the ordered power s... |
| psr1plusg 22126 | Value of addition in a uni... |
| psr1vsca 22127 | Value of scalar multiplica... |
| psr1mulr 22128 | Value of multiplication in... |
| ply1plusg 22129 | Value of addition in a uni... |
| ply1vsca 22130 | Value of scalar multiplica... |
| ply1mulr 22131 | Value of multiplication in... |
| ply1ass23l 22132 | Associative identity with ... |
| ressply1bas2 22133 | The base set of a restrict... |
| ressply1bas 22134 | A restricted polynomial al... |
| ressply1add 22135 | A restricted polynomial al... |
| ressply1mul 22136 | A restricted polynomial al... |
| ressply1vsca 22137 | A restricted power series ... |
| subrgply1 22138 | A subring of the base ring... |
| gsumply1subr 22139 | Evaluate a group sum in a ... |
| psrbaspropd 22140 | Property deduction for pow... |
| psrplusgpropd 22141 | Property deduction for pow... |
| mplbaspropd 22142 | Property deduction for pol... |
| psropprmul 22143 | Reversing multiplication i... |
| ply1opprmul 22144 | Reversing multiplication i... |
| 00ply1bas 22145 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22146 | Protection compatibility o... |
| ply1plusgfvi 22147 | Protection compatibility o... |
| ply1baspropd 22148 | Property deduction for uni... |
| ply1plusgpropd 22149 | Property deduction for uni... |
| opsrring 22150 | Ordered power series form ... |
| opsrlmod 22151 | Ordered power series form ... |
| psr1ring 22152 | Univariate power series fo... |
| ply1ring 22153 | Univariate polynomials for... |
| psr1lmod 22154 | Univariate power series fo... |
| psr1sca 22155 | Scalars of a univariate po... |
| psr1sca2 22156 | Scalars of a univariate po... |
| ply1lmod 22157 | Univariate polynomials for... |
| ply1sca 22158 | Scalars of a univariate po... |
| ply1sca2 22159 | Scalars of a univariate po... |
| ply1ascl0 22160 | The zero scalar as a polyn... |
| ply1mpl0 22161 | The univariate polynomial ... |
| ply10s0 22162 | Zero times a univariate po... |
| ply1mpl1 22163 | The univariate polynomial ... |
| ply1ascl 22164 | The univariate polynomial ... |
| subrg1ascl 22165 | The scalar injection funct... |
| subrg1asclcl 22166 | The scalars in a polynomia... |
| subrgvr1 22167 | The variables in a subring... |
| subrgvr1cl 22168 | The variables in a polynom... |
| coe1z 22169 | The coefficient vector of ... |
| coe1add 22170 | The coefficient vector of ... |
| coe1addfv 22171 | A particular coefficient o... |
| coe1subfv 22172 | A particular coefficient o... |
| coe1mul2lem1 22173 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22174 | An equivalence for ~ coe1m... |
| coe1mul2 22175 | The coefficient vector of ... |
| coe1mul 22176 | The coefficient vector of ... |
| ply1moncl 22177 | Closure of the expression ... |
| ply1tmcl 22178 | Closure of the expression ... |
| coe1tm 22179 | Coefficient vector of a po... |
| coe1tmfv1 22180 | Nonzero coefficient of a p... |
| coe1tmfv2 22181 | Zero coefficient of a poly... |
| coe1tmmul2 22182 | Coefficient vector of a po... |
| coe1tmmul 22183 | Coefficient vector of a po... |
| coe1tmmul2fv 22184 | Function value of a right-... |
| coe1pwmul 22185 | Coefficient vector of a po... |
| coe1pwmulfv 22186 | Function value of a right-... |
| ply1scltm 22187 | A scalar is a term with ze... |
| coe1sclmul 22188 | Coefficient vector of a po... |
| coe1sclmulfv 22189 | A single coefficient of a ... |
| coe1sclmul2 22190 | Coefficient vector of a po... |
| ply1sclf 22191 | A scalar polynomial is a p... |
| ply1sclcl 22192 | The value of the algebra s... |
| coe1scl 22193 | Coefficient vector of a sc... |
| ply1sclid 22194 | Recover the base scalar fr... |
| ply1sclf1 22195 | The polynomial scalar func... |
| ply1scl0 22196 | The zero scalar is zero. ... |
| ply1scl0OLD 22197 | Obsolete version of ~ ply1... |
| ply1scln0 22198 | Nonzero scalars create non... |
| ply1scl1 22199 | The one scalar is the unit... |
| ply1scl1OLD 22200 | Obsolete version of ~ ply1... |
| ply1idvr1 22201 | The identity of a polynomi... |
| cply1mul 22202 | The product of two constan... |
| ply1coefsupp 22203 | The decomposition of a uni... |
| ply1coe 22204 | Decompose a univariate pol... |
| eqcoe1ply1eq 22205 | Two polynomials over the s... |
| ply1coe1eq 22206 | Two polynomials over the s... |
| cply1coe0 22207 | All but the first coeffici... |
| cply1coe0bi 22208 | A polynomial is constant (... |
| coe1fzgsumdlem 22209 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22210 | Value of an evaluated coef... |
| ply1scleq 22211 | Equality of a constant pol... |
| ply1chr 22212 | The characteristic of a po... |
| gsumsmonply1 22213 | A finite group sum of scal... |
| gsummoncoe1 22214 | A coefficient of the polyn... |
| gsumply1eq 22215 | Two univariate polynomials... |
| lply1binom 22216 | The binomial theorem for l... |
| lply1binomsc 22217 | The binomial theorem for l... |
| ply1fermltlchr 22218 | Fermat's little theorem fo... |
| reldmevls1 22223 | Well-behaved binary operat... |
| ply1frcl 22224 | Reverse closure for the se... |
| evls1fval 22225 | Value of the univariate po... |
| evls1val 22226 | Value of the univariate po... |
| evls1rhmlem 22227 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22228 | Polynomial evaluation is a... |
| evls1sca 22229 | Univariate polynomial eval... |
| evls1gsumadd 22230 | Univariate polynomial eval... |
| evls1gsummul 22231 | Univariate polynomial eval... |
| evls1pw 22232 | Univariate polynomial eval... |
| evls1varpw 22233 | Univariate polynomial eval... |
| evl1fval 22234 | Value of the simple/same r... |
| evl1val 22235 | Value of the simple/same r... |
| evl1fval1lem 22236 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22237 | Value of the simple/same r... |
| evl1rhm 22238 | Polynomial evaluation is a... |
| fveval1fvcl 22239 | The function value of the ... |
| evl1sca 22240 | Polynomial evaluation maps... |
| evl1scad 22241 | Polynomial evaluation buil... |
| evl1var 22242 | Polynomial evaluation maps... |
| evl1vard 22243 | Polynomial evaluation buil... |
| evls1var 22244 | Univariate polynomial eval... |
| evls1scasrng 22245 | The evaluation of a scalar... |
| evls1varsrng 22246 | The evaluation of the vari... |
| evl1addd 22247 | Polynomial evaluation buil... |
| evl1subd 22248 | Polynomial evaluation buil... |
| evl1muld 22249 | Polynomial evaluation buil... |
| evl1vsd 22250 | Polynomial evaluation buil... |
| evl1expd 22251 | Polynomial evaluation buil... |
| pf1const 22252 | Constants are polynomial f... |
| pf1id 22253 | The identity is a polynomi... |
| pf1subrg 22254 | Polynomial functions are a... |
| pf1rcl 22255 | Reverse closure for the se... |
| pf1f 22256 | Polynomial functions are f... |
| mpfpf1 22257 | Convert a multivariate pol... |
| pf1mpf 22258 | Convert a univariate polyn... |
| pf1addcl 22259 | The sum of multivariate po... |
| pf1mulcl 22260 | The product of multivariat... |
| pf1ind 22261 | Prove a property of polyno... |
| evl1gsumdlem 22262 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22263 | Polynomial evaluation buil... |
| evl1gsumadd 22264 | Univariate polynomial eval... |
| evl1gsumaddval 22265 | Value of a univariate poly... |
| evl1gsummul 22266 | Univariate polynomial eval... |
| evl1varpw 22267 | Univariate polynomial eval... |
| evl1varpwval 22268 | Value of a univariate poly... |
| evl1scvarpw 22269 | Univariate polynomial eval... |
| evl1scvarpwval 22270 | Value of a univariate poly... |
| evl1gsummon 22271 | Value of a univariate poly... |
| mamufval 22274 | Functional value of the ma... |
| mamuval 22275 | Multiplication of two matr... |
| mamufv 22276 | A cell in the multiplicati... |
| mamudm 22277 | The domain of the matrix m... |
| mamufacex 22278 | Every solution of the equa... |
| mamures 22279 | Rows in a matrix product a... |
| mndvcl 22280 | Tuple-wise additive closur... |
| mndvass 22281 | Tuple-wise associativity i... |
| mndvlid 22282 | Tuple-wise left identity i... |
| mndvrid 22283 | Tuple-wise right identity ... |
| grpvlinv 22284 | Tuple-wise left inverse in... |
| grpvrinv 22285 | Tuple-wise right inverse i... |
| mhmvlin 22286 | Tuple extension of monoid ... |
| ringvcl 22287 | Tuple-wise multiplication ... |
| mamucl 22288 | Operation closure of matri... |
| mamuass 22289 | Matrix multiplication is a... |
| mamudi 22290 | Matrix multiplication dist... |
| mamudir 22291 | Matrix multiplication dist... |
| mamuvs1 22292 | Matrix multiplication dist... |
| mamuvs2 22293 | Matrix multiplication dist... |
| matbas0pc 22296 | There is no matrix with a ... |
| matbas0 22297 | There is no matrix for a n... |
| matval 22298 | Value of the matrix algebr... |
| matrcl 22299 | Reverse closure for the ma... |
| matbas 22300 | The matrix ring has the sa... |
| matplusg 22301 | The matrix ring has the sa... |
| matsca 22302 | The matrix ring has the sa... |
| matscaOLD 22303 | Obsolete proof of ~ matsca... |
| matvsca 22304 | The matrix ring has the sa... |
| matvscaOLD 22305 | Obsolete proof of ~ matvsc... |
| mat0 22306 | The matrix ring has the sa... |
| matinvg 22307 | The matrix ring has the sa... |
| mat0op 22308 | Value of a zero matrix as ... |
| matsca2 22309 | The scalars of the matrix ... |
| matbas2 22310 | The base set of the matrix... |
| matbas2i 22311 | A matrix is a function. (... |
| matbas2d 22312 | The base set of the matrix... |
| eqmat 22313 | Two square matrices of the... |
| matecl 22314 | Each entry (according to W... |
| matecld 22315 | Each entry (according to W... |
| matplusg2 22316 | Addition in the matrix rin... |
| matvsca2 22317 | Scalar multiplication in t... |
| matlmod 22318 | The matrix ring is a linea... |
| matgrp 22319 | The matrix ring is a group... |
| matvscl 22320 | Closure of the scalar mult... |
| matsubg 22321 | The matrix ring has the sa... |
| matplusgcell 22322 | Addition in the matrix rin... |
| matsubgcell 22323 | Subtraction in the matrix ... |
| matinvgcell 22324 | Additive inversion in the ... |
| matvscacell 22325 | Scalar multiplication in t... |
| matgsum 22326 | Finite commutative sums in... |
| matmulr 22327 | Multiplication in the matr... |
| mamumat1cl 22328 | The identity matrix (as op... |
| mat1comp 22329 | The components of the iden... |
| mamulid 22330 | The identity matrix (as op... |
| mamurid 22331 | The identity matrix (as op... |
| matring 22332 | Existence of the matrix ri... |
| matassa 22333 | Existence of the matrix al... |
| matmulcell 22334 | Multiplication in the matr... |
| mpomatmul 22335 | Multiplication of two N x ... |
| mat1 22336 | Value of an identity matri... |
| mat1ov 22337 | Entries of an identity mat... |
| mat1bas 22338 | The identity matrix is a m... |
| matsc 22339 | The identity matrix multip... |
| ofco2 22340 | Distribution law for the f... |
| oftpos 22341 | The transposition of the v... |
| mattposcl 22342 | The transpose of a square ... |
| mattpostpos 22343 | The transpose of the trans... |
| mattposvs 22344 | The transposition of a mat... |
| mattpos1 22345 | The transposition of the i... |
| tposmap 22346 | The transposition of an I ... |
| mamutpos 22347 | Behavior of transposes in ... |
| mattposm 22348 | Multiplying two transposed... |
| matgsumcl 22349 | Closure of a group sum ove... |
| madetsumid 22350 | The identity summand in th... |
| matepmcl 22351 | Each entry of a matrix wit... |
| matepm2cl 22352 | Each entry of a matrix wit... |
| madetsmelbas 22353 | A summand of the determina... |
| madetsmelbas2 22354 | A summand of the determina... |
| mat0dimbas0 22355 | The empty set is the one a... |
| mat0dim0 22356 | The zero of the algebra of... |
| mat0dimid 22357 | The identity of the algebr... |
| mat0dimscm 22358 | The scalar multiplication ... |
| mat0dimcrng 22359 | The algebra of matrices wi... |
| mat1dimelbas 22360 | A matrix with dimension 1 ... |
| mat1dimbas 22361 | A matrix with dimension 1 ... |
| mat1dim0 22362 | The zero of the algebra of... |
| mat1dimid 22363 | The identity of the algebr... |
| mat1dimscm 22364 | The scalar multiplication ... |
| mat1dimmul 22365 | The ring multiplication in... |
| mat1dimcrng 22366 | The algebra of matrices wi... |
| mat1f1o 22367 | There is a 1-1 function fr... |
| mat1rhmval 22368 | The value of the ring homo... |
| mat1rhmelval 22369 | The value of the ring homo... |
| mat1rhmcl 22370 | The value of the ring homo... |
| mat1f 22371 | There is a function from a... |
| mat1ghm 22372 | There is a group homomorph... |
| mat1mhm 22373 | There is a monoid homomorp... |
| mat1rhm 22374 | There is a ring homomorphi... |
| mat1rngiso 22375 | There is a ring isomorphis... |
| mat1ric 22376 | A ring is isomorphic to th... |
| dmatval 22381 | The set of ` N ` x ` N ` d... |
| dmatel 22382 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22383 | An ` N ` x ` N ` diagonal ... |
| dmatid 22384 | The identity matrix is a d... |
| dmatelnd 22385 | An extradiagonal entry of ... |
| dmatmul 22386 | The product of two diagona... |
| dmatsubcl 22387 | The difference of two diag... |
| dmatsgrp 22388 | The set of diagonal matric... |
| dmatmulcl 22389 | The product of two diagona... |
| dmatsrng 22390 | The set of diagonal matric... |
| dmatcrng 22391 | The subring of diagonal ma... |
| dmatscmcl 22392 | The multiplication of a di... |
| scmatval 22393 | The set of ` N ` x ` N ` s... |
| scmatel 22394 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22395 | A scalar matrix can be exp... |
| scmatscmide 22396 | An entry of a scalar matri... |
| scmatscmiddistr 22397 | Distributive law for scala... |
| scmatmat 22398 | An ` N ` x ` N ` scalar ma... |
| scmate 22399 | An entry of an ` N ` x ` N... |
| scmatmats 22400 | The set of an ` N ` x ` N ... |
| scmateALT 22401 | Alternate proof of ~ scmat... |
| scmatscm 22402 | The multiplication of a ma... |
| scmatid 22403 | The identity matrix is a s... |
| scmatdmat 22404 | A scalar matrix is a diago... |
| scmataddcl 22405 | The sum of two scalar matr... |
| scmatsubcl 22406 | The difference of two scal... |
| scmatmulcl 22407 | The product of two scalar ... |
| scmatsgrp 22408 | The set of scalar matrices... |
| scmatsrng 22409 | The set of scalar matrices... |
| scmatcrng 22410 | The subring of scalar matr... |
| scmatsgrp1 22411 | The set of scalar matrices... |
| scmatsrng1 22412 | The set of scalar matrices... |
| smatvscl 22413 | Closure of the scalar mult... |
| scmatlss 22414 | The set of scalar matrices... |
| scmatstrbas 22415 | The set of scalar matrices... |
| scmatrhmval 22416 | The value of the ring homo... |
| scmatrhmcl 22417 | The value of the ring homo... |
| scmatf 22418 | There is a function from a... |
| scmatfo 22419 | There is a function from a... |
| scmatf1 22420 | There is a 1-1 function fr... |
| scmatf1o 22421 | There is a bijection betwe... |
| scmatghm 22422 | There is a group homomorph... |
| scmatmhm 22423 | There is a monoid homomorp... |
| scmatrhm 22424 | There is a ring homomorphi... |
| scmatrngiso 22425 | There is a ring isomorphis... |
| scmatric 22426 | A ring is isomorphic to ev... |
| mat0scmat 22427 | The empty matrix over a ri... |
| mat1scmat 22428 | A 1-dimensional matrix ove... |
| mvmulfval 22431 | Functional value of the ma... |
| mvmulval 22432 | Multiplication of a vector... |
| mvmulfv 22433 | A cell/element in the vect... |
| mavmulval 22434 | Multiplication of a vector... |
| mavmulfv 22435 | A cell/element in the vect... |
| mavmulcl 22436 | Multiplication of an NxN m... |
| 1mavmul 22437 | Multiplication of the iden... |
| mavmulass 22438 | Associativity of the multi... |
| mavmuldm 22439 | The domain of the matrix v... |
| mavmulsolcl 22440 | Every solution of the equa... |
| mavmul0 22441 | Multiplication of a 0-dime... |
| mavmul0g 22442 | The result of the 0-dimens... |
| mvmumamul1 22443 | The multiplication of an M... |
| mavmumamul1 22444 | The multiplication of an N... |
| marrepfval 22449 | First substitution for the... |
| marrepval0 22450 | Second substitution for th... |
| marrepval 22451 | Third substitution for the... |
| marrepeval 22452 | An entry of a matrix with ... |
| marrepcl 22453 | Closure of the row replace... |
| marepvfval 22454 | First substitution for the... |
| marepvval0 22455 | Second substitution for th... |
| marepvval 22456 | Third substitution for the... |
| marepveval 22457 | An entry of a matrix with ... |
| marepvcl 22458 | Closure of the column repl... |
| ma1repvcl 22459 | Closure of the column repl... |
| ma1repveval 22460 | An entry of an identity ma... |
| mulmarep1el 22461 | Element by element multipl... |
| mulmarep1gsum1 22462 | The sum of element by elem... |
| mulmarep1gsum2 22463 | The sum of element by elem... |
| 1marepvmarrepid 22464 | Replacing the ith row by 0... |
| submabas 22467 | Any subset of the index se... |
| submafval 22468 | First substitution for a s... |
| submaval0 22469 | Second substitution for a ... |
| submaval 22470 | Third substitution for a s... |
| submaeval 22471 | An entry of a submatrix of... |
| 1marepvsma1 22472 | The submatrix of the ident... |
| mdetfval 22475 | First substitution for the... |
| mdetleib 22476 | Full substitution of our d... |
| mdetleib2 22477 | Leibniz' formula can also ... |
| nfimdetndef 22478 | The determinant is not def... |
| mdetfval1 22479 | First substitution of an a... |
| mdetleib1 22480 | Full substitution of an al... |
| mdet0pr 22481 | The determinant function f... |
| mdet0f1o 22482 | The determinant function f... |
| mdet0fv0 22483 | The determinant of the emp... |
| mdetf 22484 | Functionality of the deter... |
| mdetcl 22485 | The determinant evaluates ... |
| m1detdiag 22486 | The determinant of a 1-dim... |
| mdetdiaglem 22487 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22488 | The determinant of a diago... |
| mdetdiagid 22489 | The determinant of a diago... |
| mdet1 22490 | The determinant of the ide... |
| mdetrlin 22491 | The determinant function i... |
| mdetrsca 22492 | The determinant function i... |
| mdetrsca2 22493 | The determinant function i... |
| mdetr0 22494 | The determinant of a matri... |
| mdet0 22495 | The determinant of the zer... |
| mdetrlin2 22496 | The determinant function i... |
| mdetralt 22497 | The determinant function i... |
| mdetralt2 22498 | The determinant function i... |
| mdetero 22499 | The determinant function i... |
| mdettpos 22500 | Determinant is invariant u... |
| mdetunilem1 22501 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22502 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22503 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22504 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22505 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22506 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22507 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22508 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22509 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22510 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22511 | According to the definitio... |
| mdetmul 22512 | Multiplicativity of the de... |
| m2detleiblem1 22513 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22514 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22515 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22516 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22517 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22518 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22519 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22520 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22525 | Lemma for ~ maducoeval2 . ... |
| madufval 22526 | First substitution for the... |
| maduval 22527 | Second substitution for th... |
| maducoeval 22528 | An entry of the adjunct (c... |
| maducoeval2 22529 | An entry of the adjunct (c... |
| maduf 22530 | Creating the adjunct of ma... |
| madutpos 22531 | The adjuct of a transposed... |
| madugsum 22532 | The determinant of a matri... |
| madurid 22533 | Multiplying a matrix with ... |
| madulid 22534 | Multiplying the adjunct of... |
| minmar1fval 22535 | First substitution for the... |
| minmar1val0 22536 | Second substitution for th... |
| minmar1val 22537 | Third substitution for the... |
| minmar1eval 22538 | An entry of a matrix for a... |
| minmar1marrep 22539 | The minor matrix is a spec... |
| minmar1cl 22540 | Closure of the row replace... |
| maducoevalmin1 22541 | The coefficients of an adj... |
| symgmatr01lem 22542 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22543 | Applying a permutation tha... |
| gsummatr01lem1 22544 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22545 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22546 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22547 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22548 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22549 | Replacing a row of a squar... |
| smadiadetlem0 22550 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22551 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22552 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22553 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22554 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22555 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22556 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22557 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22558 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22559 | The determinant of a subma... |
| smadiadetglem1 22560 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22561 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22562 | The determinant of a squar... |
| smadiadetg0 22563 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22564 | The determinant of a squar... |
| invrvald 22565 | If a matrix multiplied wit... |
| matinv 22566 | The inverse of a matrix is... |
| matunit 22567 | A matrix is a unit in the ... |
| slesolvec 22568 | Every solution of a system... |
| slesolinv 22569 | The solution of a system o... |
| slesolinvbi 22570 | The solution of a system o... |
| slesolex 22571 | Every system of linear equ... |
| cramerimplem1 22572 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22573 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22574 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22575 | One direction of Cramer's ... |
| cramerlem1 22576 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22577 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22578 | Lemma 3 for ~ cramer . (C... |
| cramer0 22579 | Special case of Cramer's r... |
| cramer 22580 | Cramer's rule. According ... |
| pmatring 22581 | The set of polynomial matr... |
| pmatlmod 22582 | The set of polynomial matr... |
| pmatassa 22583 | The set of polynomial matr... |
| pmat0op 22584 | The zero polynomial matrix... |
| pmat1op 22585 | The identity polynomial ma... |
| pmat1ovd 22586 | Entries of the identity po... |
| pmat0opsc 22587 | The zero polynomial matrix... |
| pmat1opsc 22588 | The identity polynomial ma... |
| pmat1ovscd 22589 | Entries of the identity po... |
| pmatcoe1fsupp 22590 | For a polynomial matrix th... |
| 1pmatscmul 22591 | The scalar product of the ... |
| cpmat 22598 | Value of the constructor o... |
| cpmatpmat 22599 | A constant polynomial matr... |
| cpmatel 22600 | Property of a constant pol... |
| cpmatelimp 22601 | Implication of a set being... |
| cpmatel2 22602 | Another property of a cons... |
| cpmatelimp2 22603 | Another implication of a s... |
| 1elcpmat 22604 | The identity of the ring o... |
| cpmatacl 22605 | The set of all constant po... |
| cpmatinvcl 22606 | The set of all constant po... |
| cpmatmcllem 22607 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22608 | The set of all constant po... |
| cpmatsubgpmat 22609 | The set of all constant po... |
| cpmatsrgpmat 22610 | The set of all constant po... |
| 0elcpmat 22611 | The zero of the ring of al... |
| mat2pmatfval 22612 | Value of the matrix transf... |
| mat2pmatval 22613 | The result of a matrix tra... |
| mat2pmatvalel 22614 | A (matrix) element of the ... |
| mat2pmatbas 22615 | The result of a matrix tra... |
| mat2pmatbas0 22616 | The result of a matrix tra... |
| mat2pmatf 22617 | The matrix transformation ... |
| mat2pmatf1 22618 | The matrix transformation ... |
| mat2pmatghm 22619 | The transformation of matr... |
| mat2pmatmul 22620 | The transformation of matr... |
| mat2pmat1 22621 | The transformation of the ... |
| mat2pmatmhm 22622 | The transformation of matr... |
| mat2pmatrhm 22623 | The transformation of matr... |
| mat2pmatlin 22624 | The transformation of matr... |
| 0mat2pmat 22625 | The transformed zero matri... |
| idmatidpmat 22626 | The transformed identity m... |
| d0mat2pmat 22627 | The transformed empty set ... |
| d1mat2pmat 22628 | The transformation of a ma... |
| mat2pmatscmxcl 22629 | A transformed matrix multi... |
| m2cpm 22630 | The result of a matrix tra... |
| m2cpmf 22631 | The matrix transformation ... |
| m2cpmf1 22632 | The matrix transformation ... |
| m2cpmghm 22633 | The transformation of matr... |
| m2cpmmhm 22634 | The transformation of matr... |
| m2cpmrhm 22635 | The transformation of matr... |
| m2pmfzmap 22636 | The transformed values of ... |
| m2pmfzgsumcl 22637 | Closure of the sum of scal... |
| cpm2mfval 22638 | Value of the inverse matri... |
| cpm2mval 22639 | The result of an inverse m... |
| cpm2mvalel 22640 | A (matrix) element of the ... |
| cpm2mf 22641 | The inverse matrix transfo... |
| m2cpminvid 22642 | The inverse transformation... |
| m2cpminvid2lem 22643 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22644 | The transformation applied... |
| m2cpmfo 22645 | The matrix transformation ... |
| m2cpmf1o 22646 | The matrix transformation ... |
| m2cpmrngiso 22647 | The transformation of matr... |
| matcpmric 22648 | The ring of matrices over ... |
| m2cpminv 22649 | The inverse matrix transfo... |
| m2cpminv0 22650 | The inverse matrix transfo... |
| decpmatval0 22653 | The matrix consisting of t... |
| decpmatval 22654 | The matrix consisting of t... |
| decpmate 22655 | An entry of the matrix con... |
| decpmatcl 22656 | Closure of the decompositi... |
| decpmataa0 22657 | The matrix consisting of t... |
| decpmatfsupp 22658 | The mapping to the matrice... |
| decpmatid 22659 | The matrix consisting of t... |
| decpmatmullem 22660 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22661 | The matrix consisting of t... |
| decpmatmulsumfsupp 22662 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22663 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22664 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22665 | Write a polynomial matrix ... |
| pmatcollpw2lem 22666 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22667 | Write a polynomial matrix ... |
| monmatcollpw 22668 | The matrix consisting of t... |
| pmatcollpwlem 22669 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22670 | Write a polynomial matrix ... |
| pmatcollpwfi 22671 | Write a polynomial matrix ... |
| pmatcollpw3lem 22672 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22673 | Write a polynomial matrix ... |
| pmatcollpw3fi 22674 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22675 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22676 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22677 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22678 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22679 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22680 | Write a scalar matrix over... |
| pm2mpf1lem 22683 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22684 | Value of the transformatio... |
| pm2mpfval 22685 | A polynomial matrix transf... |
| pm2mpcl 22686 | The transformation of poly... |
| pm2mpf 22687 | The transformation of poly... |
| pm2mpf1 22688 | The transformation of poly... |
| pm2mpcoe1 22689 | A coefficient of the polyn... |
| idpm2idmp 22690 | The transformation of the ... |
| mptcoe1matfsupp 22691 | The mapping extracting the... |
| mply1topmatcllem 22692 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22693 | A polynomial over matrices... |
| mply1topmatcl 22694 | A polynomial over matrices... |
| mp2pm2mplem1 22695 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22696 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22697 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22698 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22699 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22700 | A polynomial over matrices... |
| pm2mpghmlem2 22701 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22702 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22703 | The transformation of poly... |
| pm2mpf1o 22704 | The transformation of poly... |
| pm2mpghm 22705 | The transformation of poly... |
| pm2mpgrpiso 22706 | The transformation of poly... |
| pm2mpmhmlem1 22707 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22708 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22709 | The transformation of poly... |
| pm2mprhm 22710 | The transformation of poly... |
| pm2mprngiso 22711 | The transformation of poly... |
| pmmpric 22712 | The ring of polynomial mat... |
| monmat2matmon 22713 | The transformation of a po... |
| pm2mp 22714 | The transformation of a su... |
| chmatcl 22717 | Closure of the characteris... |
| chmatval 22718 | The entries of the charact... |
| chpmatfval 22719 | Value of the characteristi... |
| chpmatval 22720 | The characteristic polynom... |
| chpmatply1 22721 | The characteristic polynom... |
| chpmatval2 22722 | The characteristic polynom... |
| chpmat0d 22723 | The characteristic polynom... |
| chpmat1dlem 22724 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22725 | The characteristic polynom... |
| chpdmatlem0 22726 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22727 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22728 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22729 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22730 | The characteristic polynom... |
| chpscmat 22731 | The characteristic polynom... |
| chpscmat0 22732 | The characteristic polynom... |
| chpscmatgsumbin 22733 | The characteristic polynom... |
| chpscmatgsummon 22734 | The characteristic polynom... |
| chp0mat 22735 | The characteristic polynom... |
| chpidmat 22736 | The characteristic polynom... |
| chmaidscmat 22737 | The characteristic polynom... |
| fvmptnn04if 22738 | The function values of a m... |
| fvmptnn04ifa 22739 | The function value of a ma... |
| fvmptnn04ifb 22740 | The function value of a ma... |
| fvmptnn04ifc 22741 | The function value of a ma... |
| fvmptnn04ifd 22742 | The function value of a ma... |
| chfacfisf 22743 | The "characteristic factor... |
| chfacfisfcpmat 22744 | The "characteristic factor... |
| chfacffsupp 22745 | The "characteristic factor... |
| chfacfscmulcl 22746 | Closure of a scaled value ... |
| chfacfscmul0 22747 | A scaled value of the "cha... |
| chfacfscmulfsupp 22748 | A mapping of scaled values... |
| chfacfscmulgsum 22749 | Breaking up a sum of value... |
| chfacfpmmulcl 22750 | Closure of the value of th... |
| chfacfpmmul0 22751 | The value of the "characte... |
| chfacfpmmulfsupp 22752 | A mapping of values of the... |
| chfacfpmmulgsum 22753 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22754 | Breaking up a sum of value... |
| cayhamlem1 22755 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22756 | The right-hand fundamental... |
| cpmidgsum 22757 | Representation of the iden... |
| cpmidgsumm2pm 22758 | Representation of the iden... |
| cpmidpmatlem1 22759 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22760 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22761 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22762 | Representation of the iden... |
| cpmadugsumlemB 22763 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22764 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22765 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22766 | The product of the charact... |
| cpmadugsum 22767 | The product of the charact... |
| cpmidgsum2 22768 | Representation of the iden... |
| cpmidg2sum 22769 | Equality of two sums repre... |
| cpmadumatpolylem1 22770 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22771 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22772 | The product of the charact... |
| cayhamlem2 22773 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22774 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22775 | The coefficients of the ch... |
| cayhamlem3 22776 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22777 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22778 | The Cayley-Hamilton theore... |
| cayleyhamilton 22779 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22780 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22781 | The Cayley-Hamilton theore... |
| istopg 22784 | Express the predicate " ` ... |
| istop2g 22785 | Express the predicate " ` ... |
| uniopn 22786 | The union of a subset of a... |
| iunopn 22787 | The indexed union of a sub... |
| inopn 22788 | The intersection of two op... |
| fitop 22789 | A topology is closed under... |
| fiinopn 22790 | The intersection of a none... |
| iinopn 22791 | The intersection of a none... |
| unopn 22792 | The union of two open sets... |
| 0opn 22793 | The empty set is an open s... |
| 0ntop 22794 | The empty set is not a top... |
| topopn 22795 | The underlying set of a to... |
| eltopss 22796 | A member of a topology is ... |
| riinopn 22797 | A finite indexed relative ... |
| rintopn 22798 | A finite relative intersec... |
| istopon 22801 | Property of being a topolo... |
| topontop 22802 | A topology on a given base... |
| toponuni 22803 | The base set of a topology... |
| topontopi 22804 | A topology on a given base... |
| toponunii 22805 | The base set of a topology... |
| toptopon 22806 | Alternative definition of ... |
| toptopon2 22807 | A topology is the same thi... |
| topontopon 22808 | A topology on a set is a t... |
| funtopon 22809 | The class ` TopOn ` is a f... |
| toponrestid 22810 | Given a topology on a set,... |
| toponsspwpw 22811 | The set of topologies on a... |
| dmtopon 22812 | The domain of ` TopOn ` is... |
| fntopon 22813 | The class ` TopOn ` is a f... |
| toprntopon 22814 | A topology is the same thi... |
| toponmax 22815 | The base set of a topology... |
| toponss 22816 | A member of a topology is ... |
| toponcom 22817 | If ` K ` is a topology on ... |
| toponcomb 22818 | Biconditional form of ~ to... |
| topgele 22819 | The topologies over the sa... |
| topsn 22820 | The only topology on a sin... |
| istps 22823 | Express the predicate "is ... |
| istps2 22824 | Express the predicate "is ... |
| tpsuni 22825 | The base set of a topologi... |
| tpstop 22826 | The topology extractor on ... |
| tpspropd 22827 | A topological space depend... |
| tpsprop2d 22828 | A topological space depend... |
| topontopn 22829 | Express the predicate "is ... |
| tsettps 22830 | If the topology component ... |
| istpsi 22831 | Properties that determine ... |
| eltpsg 22832 | Properties that determine ... |
| eltpsgOLD 22833 | Obsolete version of ~ eltp... |
| eltpsi 22834 | Properties that determine ... |
| isbasisg 22837 | Express the predicate "the... |
| isbasis2g 22838 | Express the predicate "the... |
| isbasis3g 22839 | Express the predicate "the... |
| basis1 22840 | Property of a basis. (Con... |
| basis2 22841 | Property of a basis. (Con... |
| fiinbas 22842 | If a set is closed under f... |
| basdif0 22843 | A basis is not affected by... |
| baspartn 22844 | A disjoint system of sets ... |
| tgval 22845 | The topology generated by ... |
| tgval2 22846 | Definition of a topology g... |
| eltg 22847 | Membership in a topology g... |
| eltg2 22848 | Membership in a topology g... |
| eltg2b 22849 | Membership in a topology g... |
| eltg4i 22850 | An open set in a topology ... |
| eltg3i 22851 | The union of a set of basi... |
| eltg3 22852 | Membership in a topology g... |
| tgval3 22853 | Alternate expression for t... |
| tg1 22854 | Property of a member of a ... |
| tg2 22855 | Property of a member of a ... |
| bastg 22856 | A member of a basis is a s... |
| unitg 22857 | The topology generated by ... |
| tgss 22858 | Subset relation for genera... |
| tgcl 22859 | Show that a basis generate... |
| tgclb 22860 | The property ~ tgcl can be... |
| tgtopon 22861 | A basis generates a topolo... |
| topbas 22862 | A topology is its own basi... |
| tgtop 22863 | A topology is its own basi... |
| eltop 22864 | Membership in a topology, ... |
| eltop2 22865 | Membership in a topology. ... |
| eltop3 22866 | Membership in a topology. ... |
| fibas 22867 | A collection of finite int... |
| tgdom 22868 | A space has no more open s... |
| tgiun 22869 | The indexed union of a set... |
| tgidm 22870 | The topology generator fun... |
| bastop 22871 | Two ways to express that a... |
| tgtop11 22872 | The topology generation fu... |
| 0top 22873 | The singleton of the empty... |
| en1top 22874 | ` { (/) } ` is the only to... |
| en2top 22875 | If a topology has two elem... |
| tgss3 22876 | A criterion for determinin... |
| tgss2 22877 | A criterion for determinin... |
| basgen 22878 | Given a topology ` J ` , s... |
| basgen2 22879 | Given a topology ` J ` , s... |
| 2basgen 22880 | Conditions that determine ... |
| tgfiss 22881 | If a subbase is included i... |
| tgdif0 22882 | A generated topology is no... |
| bastop1 22883 | A subset of a topology is ... |
| bastop2 22884 | A version of ~ bastop1 tha... |
| distop 22885 | The discrete topology on a... |
| topnex 22886 | The class of all topologie... |
| distopon 22887 | The discrete topology on a... |
| sn0topon 22888 | The singleton of the empty... |
| sn0top 22889 | The singleton of the empty... |
| indislem 22890 | A lemma to eliminate some ... |
| indistopon 22891 | The indiscrete topology on... |
| indistop 22892 | The indiscrete topology on... |
| indisuni 22893 | The base set of the indisc... |
| fctop 22894 | The finite complement topo... |
| fctop2 22895 | The finite complement topo... |
| cctop 22896 | The countable complement t... |
| ppttop 22897 | The particular point topol... |
| pptbas 22898 | The particular point topol... |
| epttop 22899 | The excluded point topolog... |
| indistpsx 22900 | The indiscrete topology on... |
| indistps 22901 | The indiscrete topology on... |
| indistps2 22902 | The indiscrete topology on... |
| indistpsALT 22903 | The indiscrete topology on... |
| indistpsALTOLD 22904 | Obsolete version of ~ indi... |
| indistps2ALT 22905 | The indiscrete topology on... |
| distps 22906 | The discrete topology on a... |
| fncld 22913 | The closed-set generator i... |
| cldval 22914 | The set of closed sets of ... |
| ntrfval 22915 | The interior function on t... |
| clsfval 22916 | The closure function on th... |
| cldrcl 22917 | Reverse closure of the clo... |
| iscld 22918 | The predicate "the class `... |
| iscld2 22919 | A subset of the underlying... |
| cldss 22920 | A closed set is a subset o... |
| cldss2 22921 | The set of closed sets is ... |
| cldopn 22922 | The complement of a closed... |
| isopn2 22923 | A subset of the underlying... |
| opncld 22924 | The complement of an open ... |
| difopn 22925 | The difference of a closed... |
| topcld 22926 | The underlying set of a to... |
| ntrval 22927 | The interior of a subset o... |
| clsval 22928 | The closure of a subset of... |
| 0cld 22929 | The empty set is closed. ... |
| iincld 22930 | The indexed intersection o... |
| intcld 22931 | The intersection of a set ... |
| uncld 22932 | The union of two closed se... |
| cldcls 22933 | A closed subset equals its... |
| incld 22934 | The intersection of two cl... |
| riincld 22935 | An indexed relative inters... |
| iuncld 22936 | A finite indexed union of ... |
| unicld 22937 | A finite union of closed s... |
| clscld 22938 | The closure of a subset of... |
| clsf 22939 | The closure function is a ... |
| ntropn 22940 | The interior of a subset o... |
| clsval2 22941 | Express closure in terms o... |
| ntrval2 22942 | Interior expressed in term... |
| ntrdif 22943 | An interior of a complemen... |
| clsdif 22944 | A closure of a complement ... |
| clsss 22945 | Subset relationship for cl... |
| ntrss 22946 | Subset relationship for in... |
| sscls 22947 | A subset of a topology's u... |
| ntrss2 22948 | A subset includes its inte... |
| ssntr 22949 | An open subset of a set is... |
| clsss3 22950 | The closure of a subset of... |
| ntrss3 22951 | The interior of a subset o... |
| ntrin 22952 | A pairwise intersection of... |
| cmclsopn 22953 | The complement of a closur... |
| cmntrcld 22954 | The complement of an inter... |
| iscld3 22955 | A subset is closed iff it ... |
| iscld4 22956 | A subset is closed iff it ... |
| isopn3 22957 | A subset is open iff it eq... |
| clsidm 22958 | The closure operation is i... |
| ntridm 22959 | The interior operation is ... |
| clstop 22960 | The closure of a topology'... |
| ntrtop 22961 | The interior of a topology... |
| 0ntr 22962 | A subset with an empty int... |
| clsss2 22963 | If a subset is included in... |
| elcls 22964 | Membership in a closure. ... |
| elcls2 22965 | Membership in a closure. ... |
| clsndisj 22966 | Any open set containing a ... |
| ntrcls0 22967 | A subset whose closure has... |
| ntreq0 22968 | Two ways to say that a sub... |
| cldmre 22969 | The closed sets of a topol... |
| mrccls 22970 | Moore closure generalizes ... |
| cls0 22971 | The closure of the empty s... |
| ntr0 22972 | The interior of the empty ... |
| isopn3i 22973 | An open subset equals its ... |
| elcls3 22974 | Membership in a closure in... |
| opncldf1 22975 | A bijection useful for con... |
| opncldf2 22976 | The values of the open-clo... |
| opncldf3 22977 | The values of the converse... |
| isclo 22978 | A set ` A ` is clopen iff ... |
| isclo2 22979 | A set ` A ` is clopen iff ... |
| discld 22980 | The open sets of a discret... |
| sn0cld 22981 | The closed sets of the top... |
| indiscld 22982 | The closed sets of an indi... |
| mretopd 22983 | A Moore collection which i... |
| toponmre 22984 | The topologies over a give... |
| cldmreon 22985 | The closed sets of a topol... |
| iscldtop 22986 | A family is the closed set... |
| mreclatdemoBAD 22987 | The closed subspaces of a ... |
| neifval 22990 | Value of the neighborhood ... |
| neif 22991 | The neighborhood function ... |
| neiss2 22992 | A set with a neighborhood ... |
| neival 22993 | Value of the set of neighb... |
| isnei 22994 | The predicate "the class `... |
| neiint 22995 | An intuitive definition of... |
| isneip 22996 | The predicate "the class `... |
| neii1 22997 | A neighborhood is included... |
| neisspw 22998 | The neighborhoods of any s... |
| neii2 22999 | Property of a neighborhood... |
| neiss 23000 | Any neighborhood of a set ... |
| ssnei 23001 | A set is included in any o... |
| elnei 23002 | A point belongs to any of ... |
| 0nnei 23003 | The empty set is not a nei... |
| neips 23004 | A neighborhood of a set is... |
| opnneissb 23005 | An open set is a neighborh... |
| opnssneib 23006 | Any superset of an open se... |
| ssnei2 23007 | Any subset ` M ` of ` X ` ... |
| neindisj 23008 | Any neighborhood of an ele... |
| opnneiss 23009 | An open set is a neighborh... |
| opnneip 23010 | An open set is a neighborh... |
| opnnei 23011 | A set is open iff it is a ... |
| tpnei 23012 | The underlying set of a to... |
| neiuni 23013 | The union of the neighborh... |
| neindisj2 23014 | A point ` P ` belongs to t... |
| topssnei 23015 | A finer topology has more ... |
| innei 23016 | The intersection of two ne... |
| opnneiid 23017 | Only an open set is a neig... |
| neissex 23018 | For any neighborhood ` N `... |
| 0nei 23019 | The empty set is a neighbo... |
| neipeltop 23020 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23021 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23022 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23023 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23024 | If, to each element ` P ` ... |
| lpfval 23029 | The limit point function o... |
| lpval 23030 | The set of limit points of... |
| islp 23031 | The predicate "the class `... |
| lpsscls 23032 | The limit points of a subs... |
| lpss 23033 | The limit points of a subs... |
| lpdifsn 23034 | ` P ` is a limit point of ... |
| lpss3 23035 | Subset relationship for li... |
| islp2 23036 | The predicate " ` P ` is a... |
| islp3 23037 | The predicate " ` P ` is a... |
| maxlp 23038 | A point is a limit point o... |
| clslp 23039 | The closure of a subset of... |
| islpi 23040 | A point belonging to a set... |
| cldlp 23041 | A subset of a topological ... |
| isperf 23042 | Definition of a perfect sp... |
| isperf2 23043 | Definition of a perfect sp... |
| isperf3 23044 | A perfect space is a topol... |
| perflp 23045 | The limit points of a perf... |
| perfi 23046 | Property of a perfect spac... |
| perftop 23047 | A perfect space is a topol... |
| restrcl 23048 | Reverse closure for the su... |
| restbas 23049 | A subspace topology basis ... |
| tgrest 23050 | A subspace can be generate... |
| resttop 23051 | A subspace topology is a t... |
| resttopon 23052 | A subspace topology is a t... |
| restuni 23053 | The underlying set of a su... |
| stoig 23054 | The topological space buil... |
| restco 23055 | Composition of subspaces. ... |
| restabs 23056 | Equivalence of being a sub... |
| restin 23057 | When the subspace region i... |
| restuni2 23058 | The underlying set of a su... |
| resttopon2 23059 | The underlying set of a su... |
| rest0 23060 | The subspace topology indu... |
| restsn 23061 | The only subspace topology... |
| restsn2 23062 | The subspace topology indu... |
| restcld 23063 | A closed set of a subspace... |
| restcldi 23064 | A closed set is closed in ... |
| restcldr 23065 | A set which is closed in t... |
| restopnb 23066 | If ` B ` is an open subset... |
| ssrest 23067 | If ` K ` is a finer topolo... |
| restopn2 23068 | If ` A ` is open, then ` B... |
| restdis 23069 | A subspace of a discrete t... |
| restfpw 23070 | The restriction of the set... |
| neitr 23071 | The neighborhood of a trac... |
| restcls 23072 | A closure in a subspace to... |
| restntr 23073 | An interior in a subspace ... |
| restlp 23074 | The limit points of a subs... |
| restperf 23075 | Perfection of a subspace. ... |
| perfopn 23076 | An open subset of a perfec... |
| resstopn 23077 | The topology of a restrict... |
| resstps 23078 | A restricted topological s... |
| ordtbaslem 23079 | Lemma for ~ ordtbas . In ... |
| ordtval 23080 | Value of the order topolog... |
| ordtuni 23081 | Value of the order topolog... |
| ordtbas2 23082 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23083 | In a total order, the fini... |
| ordttopon 23084 | Value of the order topolog... |
| ordtopn1 23085 | An upward ray ` ( P , +oo ... |
| ordtopn2 23086 | A downward ray ` ( -oo , P... |
| ordtopn3 23087 | An open interval ` ( A , B... |
| ordtcld1 23088 | A downward ray ` ( -oo , P... |
| ordtcld2 23089 | An upward ray ` [ P , +oo ... |
| ordtcld3 23090 | A closed interval ` [ A , ... |
| ordttop 23091 | The order topology is a to... |
| ordtcnv 23092 | The order dual generates t... |
| ordtrest 23093 | The subspace topology of a... |
| ordtrest2lem 23094 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23095 | An interval-closed set ` A... |
| letopon 23096 | The topology of the extend... |
| letop 23097 | The topology of the extend... |
| letopuni 23098 | The topology of the extend... |
| xrstopn 23099 | The topology component of ... |
| xrstps 23100 | The extended real number s... |
| leordtvallem1 23101 | Lemma for ~ leordtval . (... |
| leordtvallem2 23102 | Lemma for ~ leordtval . (... |
| leordtval2 23103 | The topology of the extend... |
| leordtval 23104 | The topology of the extend... |
| iccordt 23105 | A closed interval is close... |
| iocpnfordt 23106 | An unbounded above open in... |
| icomnfordt 23107 | An unbounded above open in... |
| iooordt 23108 | An open interval is open i... |
| reordt 23109 | The real numbers are an op... |
| lecldbas 23110 | The set of closed interval... |
| pnfnei 23111 | A neighborhood of ` +oo ` ... |
| mnfnei 23112 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23113 | The restriction of the les... |
| ordtresticc 23114 | The restriction of the les... |
| lmrel 23121 | The topological space conv... |
| lmrcl 23122 | Reverse closure for the co... |
| lmfval 23123 | The relation "sequence ` f... |
| cnfval 23124 | The set of all continuous ... |
| cnpfval 23125 | The function mapping the p... |
| iscn 23126 | The predicate "the class `... |
| cnpval 23127 | The set of all functions f... |
| iscnp 23128 | The predicate "the class `... |
| iscn2 23129 | The predicate "the class `... |
| iscnp2 23130 | The predicate "the class `... |
| cntop1 23131 | Reverse closure for a cont... |
| cntop2 23132 | Reverse closure for a cont... |
| cnptop1 23133 | Reverse closure for a func... |
| cnptop2 23134 | Reverse closure for a func... |
| iscnp3 23135 | The predicate "the class `... |
| cnprcl 23136 | Reverse closure for a func... |
| cnf 23137 | A continuous function is a... |
| cnpf 23138 | A continuous function at p... |
| cnpcl 23139 | The value of a continuous ... |
| cnf2 23140 | A continuous function is a... |
| cnpf2 23141 | A continuous function at p... |
| cnprcl2 23142 | Reverse closure for a func... |
| tgcn 23143 | The continuity predicate w... |
| tgcnp 23144 | The "continuous at a point... |
| subbascn 23145 | The continuity predicate w... |
| ssidcn 23146 | The identity function is a... |
| cnpimaex 23147 | Property of a function con... |
| idcn 23148 | A restricted identity func... |
| lmbr 23149 | Express the binary relatio... |
| lmbr2 23150 | Express the binary relatio... |
| lmbrf 23151 | Express the binary relatio... |
| lmconst 23152 | A constant sequence conver... |
| lmcvg 23153 | Convergence property of a ... |
| iscnp4 23154 | The predicate "the class `... |
| cnpnei 23155 | A condition for continuity... |
| cnima 23156 | An open subset of the codo... |
| cnco 23157 | The composition of two con... |
| cnpco 23158 | The composition of a funct... |
| cnclima 23159 | A closed subset of the cod... |
| iscncl 23160 | A characterization of a co... |
| cncls2i 23161 | Property of the preimage o... |
| cnntri 23162 | Property of the preimage o... |
| cnclsi 23163 | Property of the image of a... |
| cncls2 23164 | Continuity in terms of clo... |
| cncls 23165 | Continuity in terms of clo... |
| cnntr 23166 | Continuity in terms of int... |
| cnss1 23167 | If the topology ` K ` is f... |
| cnss2 23168 | If the topology ` K ` is f... |
| cncnpi 23169 | A continuous function is c... |
| cnsscnp 23170 | The set of continuous func... |
| cncnp 23171 | A continuous function is c... |
| cncnp2 23172 | A continuous function is c... |
| cnnei 23173 | Continuity in terms of nei... |
| cnconst2 23174 | A constant function is con... |
| cnconst 23175 | A constant function is con... |
| cnrest 23176 | Continuity of a restrictio... |
| cnrest2 23177 | Equivalence of continuity ... |
| cnrest2r 23178 | Equivalence of continuity ... |
| cnpresti 23179 | One direction of ~ cnprest... |
| cnprest 23180 | Equivalence of continuity ... |
| cnprest2 23181 | Equivalence of point-conti... |
| cndis 23182 | Every function is continuo... |
| cnindis 23183 | Every function is continuo... |
| cnpdis 23184 | If ` A ` is an isolated po... |
| paste 23185 | Pasting lemma. If ` A ` a... |
| lmfpm 23186 | If ` F ` converges, then `... |
| lmfss 23187 | Inclusion of a function ha... |
| lmcl 23188 | Closure of a limit. (Cont... |
| lmss 23189 | Limit on a subspace. (Con... |
| sslm 23190 | A finer topology has fewer... |
| lmres 23191 | A function converges iff i... |
| lmff 23192 | If ` F ` converges, there ... |
| lmcls 23193 | Any convergent sequence of... |
| lmcld 23194 | Any convergent sequence of... |
| lmcnp 23195 | The image of a convergent ... |
| lmcn 23196 | The image of a convergent ... |
| ist0 23211 | The predicate "is a T_0 sp... |
| ist1 23212 | The predicate "is a T_1 sp... |
| ishaus 23213 | The predicate "is a Hausdo... |
| iscnrm 23214 | The property of being comp... |
| t0sep 23215 | Any two topologically indi... |
| t0dist 23216 | Any two distinct points in... |
| t1sncld 23217 | In a T_1 space, singletons... |
| t1ficld 23218 | In a T_1 space, finite set... |
| hausnei 23219 | Neighborhood property of a... |
| t0top 23220 | A T_0 space is a topologic... |
| t1top 23221 | A T_1 space is a topologic... |
| haustop 23222 | A Hausdorff space is a top... |
| isreg 23223 | The predicate "is a regula... |
| regtop 23224 | A regular space is a topol... |
| regsep 23225 | In a regular space, every ... |
| isnrm 23226 | The predicate "is a normal... |
| nrmtop 23227 | A normal space is a topolo... |
| cnrmtop 23228 | A completely normal space ... |
| iscnrm2 23229 | The property of being comp... |
| ispnrm 23230 | The property of being perf... |
| pnrmnrm 23231 | A perfectly normal space i... |
| pnrmtop 23232 | A perfectly normal space i... |
| pnrmcld 23233 | A closed set in a perfectl... |
| pnrmopn 23234 | An open set in a perfectly... |
| ist0-2 23235 | The predicate "is a T_0 sp... |
| ist0-3 23236 | The predicate "is a T_0 sp... |
| cnt0 23237 | The preimage of a T_0 topo... |
| ist1-2 23238 | An alternate characterizat... |
| t1t0 23239 | A T_1 space is a T_0 space... |
| ist1-3 23240 | A space is T_1 iff every p... |
| cnt1 23241 | The preimage of a T_1 topo... |
| ishaus2 23242 | Express the predicate " ` ... |
| haust1 23243 | A Hausdorff space is a T_1... |
| hausnei2 23244 | The Hausdorff condition st... |
| cnhaus 23245 | The preimage of a Hausdorf... |
| nrmsep3 23246 | In a normal space, given a... |
| nrmsep2 23247 | In a normal space, any two... |
| nrmsep 23248 | In a normal space, disjoin... |
| isnrm2 23249 | An alternate characterizat... |
| isnrm3 23250 | A topological space is nor... |
| cnrmi 23251 | A subspace of a completely... |
| cnrmnrm 23252 | A completely normal space ... |
| restcnrm 23253 | A subspace of a completely... |
| resthauslem 23254 | Lemma for ~ resthaus and s... |
| lpcls 23255 | The limit points of the cl... |
| perfcls 23256 | A subset of a perfect spac... |
| restt0 23257 | A subspace of a T_0 topolo... |
| restt1 23258 | A subspace of a T_1 topolo... |
| resthaus 23259 | A subspace of a Hausdorff ... |
| t1sep2 23260 | Any two points in a T_1 sp... |
| t1sep 23261 | Any two distinct points in... |
| sncld 23262 | A singleton is closed in a... |
| sshauslem 23263 | Lemma for ~ sshaus and sim... |
| sst0 23264 | A topology finer than a T_... |
| sst1 23265 | A topology finer than a T_... |
| sshaus 23266 | A topology finer than a Ha... |
| regsep2 23267 | In a regular space, a clos... |
| isreg2 23268 | A topological space is reg... |
| dnsconst 23269 | If a continuous mapping to... |
| ordtt1 23270 | The order topology is T_1 ... |
| lmmo 23271 | A sequence in a Hausdorff ... |
| lmfun 23272 | The convergence relation i... |
| dishaus 23273 | A discrete topology is Hau... |
| ordthauslem 23274 | Lemma for ~ ordthaus . (C... |
| ordthaus 23275 | The order topology of a to... |
| xrhaus 23276 | The topology of the extend... |
| iscmp 23279 | The predicate "is a compac... |
| cmpcov 23280 | An open cover of a compact... |
| cmpcov2 23281 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23282 | Combine ~ cmpcov with ~ ac... |
| cncmp 23283 | Compactness is respected b... |
| fincmp 23284 | A finite topology is compa... |
| 0cmp 23285 | The singleton of the empty... |
| cmptop 23286 | A compact topology is a to... |
| rncmp 23287 | The image of a compact set... |
| imacmp 23288 | The image of a compact set... |
| discmp 23289 | A discrete topology is com... |
| cmpsublem 23290 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23291 | Two equivalent ways of des... |
| tgcmp 23292 | A topology generated by a ... |
| cmpcld 23293 | A closed subset of a compa... |
| uncmp 23294 | The union of two compact s... |
| fiuncmp 23295 | A finite union of compact ... |
| sscmp 23296 | A subset of a compact topo... |
| hauscmplem 23297 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23298 | A compact subspace of a T2... |
| cmpfi 23299 | If a topology is compact a... |
| cmpfii 23300 | In a compact topology, a s... |
| bwth 23301 | The glorious Bolzano-Weier... |
| isconn 23304 | The predicate ` J ` is a c... |
| isconn2 23305 | The predicate ` J ` is a c... |
| connclo 23306 | The only nonempty clopen s... |
| conndisj 23307 | If a topology is connected... |
| conntop 23308 | A connected topology is a ... |
| indisconn 23309 | The indiscrete topology (o... |
| dfconn2 23310 | An alternate definition of... |
| connsuba 23311 | Connectedness for a subspa... |
| connsub 23312 | Two equivalent ways of say... |
| cnconn 23313 | Connectedness is respected... |
| nconnsubb 23314 | Disconnectedness for a sub... |
| connsubclo 23315 | If a clopen set meets a co... |
| connima 23316 | The image of a connected s... |
| conncn 23317 | A continuous function from... |
| iunconnlem 23318 | Lemma for ~ iunconn . (Co... |
| iunconn 23319 | The indexed union of conne... |
| unconn 23320 | The union of two connected... |
| clsconn 23321 | The closure of a connected... |
| conncompid 23322 | The connected component co... |
| conncompconn 23323 | The connected component co... |
| conncompss 23324 | The connected component co... |
| conncompcld 23325 | The connected component co... |
| conncompclo 23326 | The connected component co... |
| t1connperf 23327 | A connected T_1 space is p... |
| is1stc 23332 | The predicate "is a first-... |
| is1stc2 23333 | An equivalent way of sayin... |
| 1stctop 23334 | A first-countable topology... |
| 1stcclb 23335 | A property of points in a ... |
| 1stcfb 23336 | For any point ` A ` in a f... |
| is2ndc 23337 | The property of being seco... |
| 2ndctop 23338 | A second-countable topolog... |
| 2ndci 23339 | A countable basis generate... |
| 2ndcsb 23340 | Having a countable subbase... |
| 2ndcredom 23341 | A second-countable space h... |
| 2ndc1stc 23342 | A second-countable space i... |
| 1stcrestlem 23343 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23344 | A subspace of a first-coun... |
| 2ndcrest 23345 | A subspace of a second-cou... |
| 2ndcctbss 23346 | If a topology is second-co... |
| 2ndcdisj 23347 | Any disjoint family of ope... |
| 2ndcdisj2 23348 | Any disjoint collection of... |
| 2ndcomap 23349 | A surjective continuous op... |
| 2ndcsep 23350 | A second-countable topolog... |
| dis2ndc 23351 | A discrete space is second... |
| 1stcelcls 23352 | A point belongs to the clo... |
| 1stccnp 23353 | A mapping is continuous at... |
| 1stccn 23354 | A mapping ` X --> Y ` , wh... |
| islly 23359 | The property of being a lo... |
| isnlly 23360 | The property of being an n... |
| llyeq 23361 | Equality theorem for the `... |
| nllyeq 23362 | Equality theorem for the `... |
| llytop 23363 | A locally ` A ` space is a... |
| nllytop 23364 | A locally ` A ` space is a... |
| llyi 23365 | The property of a locally ... |
| nllyi 23366 | The property of an n-local... |
| nlly2i 23367 | Eliminate the neighborhood... |
| llynlly 23368 | A locally ` A ` space is n... |
| llyssnlly 23369 | A locally ` A ` space is n... |
| llyss 23370 | The "locally" predicate re... |
| nllyss 23371 | The "n-locally" predicate ... |
| subislly 23372 | The property of a subspace... |
| restnlly 23373 | If the property ` A ` pass... |
| restlly 23374 | If the property ` A ` pass... |
| islly2 23375 | An alternative expression ... |
| llyrest 23376 | An open subspace of a loca... |
| nllyrest 23377 | An open subspace of an n-l... |
| loclly 23378 | If ` A ` is a local proper... |
| llyidm 23379 | Idempotence of the "locall... |
| nllyidm 23380 | Idempotence of the "n-loca... |
| toplly 23381 | A topology is locally a to... |
| topnlly 23382 | A topology is n-locally a ... |
| hauslly 23383 | A Hausdorff space is local... |
| hausnlly 23384 | A Hausdorff space is n-loc... |
| hausllycmp 23385 | A compact Hausdorff space ... |
| cldllycmp 23386 | A closed subspace of a loc... |
| lly1stc 23387 | First-countability is a lo... |
| dislly 23388 | The discrete space ` ~P X ... |
| disllycmp 23389 | A discrete space is locall... |
| dis1stc 23390 | A discrete space is first-... |
| hausmapdom 23391 | If ` X ` is a first-counta... |
| hauspwdom 23392 | Simplify the cardinal ` A ... |
| refrel 23399 | Refinement is a relation. ... |
| isref 23400 | The property of being a re... |
| refbas 23401 | A refinement covers the sa... |
| refssex 23402 | Every set in a refinement ... |
| ssref 23403 | A subcover is a refinement... |
| refref 23404 | Reflexivity of refinement.... |
| reftr 23405 | Refinement is transitive. ... |
| refun0 23406 | Adding the empty set prese... |
| isptfin 23407 | The statement "is a point-... |
| islocfin 23408 | The statement "is a locall... |
| finptfin 23409 | A finite cover is a point-... |
| ptfinfin 23410 | A point covered by a point... |
| finlocfin 23411 | A finite cover of a topolo... |
| locfintop 23412 | A locally finite cover cov... |
| locfinbas 23413 | A locally finite cover mus... |
| locfinnei 23414 | A point covered by a local... |
| lfinpfin 23415 | A locally finite cover is ... |
| lfinun 23416 | Adding a finite set preser... |
| locfincmp 23417 | For a compact space, the l... |
| unisngl 23418 | Taking the union of the se... |
| dissnref 23419 | The set of singletons is a... |
| dissnlocfin 23420 | The set of singletons is l... |
| locfindis 23421 | The locally finite covers ... |
| locfincf 23422 | A locally finite cover in ... |
| comppfsc 23423 | A space where every open c... |
| kgenval 23426 | Value of the compact gener... |
| elkgen 23427 | Value of the compact gener... |
| kgeni 23428 | Property of the open sets ... |
| kgentopon 23429 | The compact generator gene... |
| kgenuni 23430 | The base set of the compac... |
| kgenftop 23431 | The compact generator gene... |
| kgenf 23432 | The compact generator is a... |
| kgentop 23433 | A compactly generated spac... |
| kgenss 23434 | The compact generator gene... |
| kgenhaus 23435 | The compact generator gene... |
| kgencmp 23436 | The compact generator topo... |
| kgencmp2 23437 | The compact generator topo... |
| kgenidm 23438 | The compact generator is i... |
| iskgen2 23439 | A space is compactly gener... |
| iskgen3 23440 | Derive the usual definitio... |
| llycmpkgen2 23441 | A locally compact space is... |
| cmpkgen 23442 | A compact space is compact... |
| llycmpkgen 23443 | A locally compact space is... |
| 1stckgenlem 23444 | The one-point compactifica... |
| 1stckgen 23445 | A first-countable space is... |
| kgen2ss 23446 | The compact generator pres... |
| kgencn 23447 | A function from a compactl... |
| kgencn2 23448 | A function ` F : J --> K `... |
| kgencn3 23449 | The set of continuous func... |
| kgen2cn 23450 | A continuous function is a... |
| txval 23455 | Value of the binary topolo... |
| txuni2 23456 | The underlying set of the ... |
| txbasex 23457 | The basis for the product ... |
| txbas 23458 | The set of Cartesian produ... |
| eltx 23459 | A set in a product is open... |
| txtop 23460 | The product of two topolog... |
| ptval 23461 | The value of the product t... |
| ptpjpre1 23462 | The preimage of a projecti... |
| elpt 23463 | Elementhood in the bases o... |
| elptr 23464 | A basic open set in the pr... |
| elptr2 23465 | A basic open set in the pr... |
| ptbasid 23466 | The base set of the produc... |
| ptuni2 23467 | The base set for the produ... |
| ptbasin 23468 | The basis for a product to... |
| ptbasin2 23469 | The basis for a product to... |
| ptbas 23470 | The basis for a product to... |
| ptpjpre2 23471 | The basis for a product to... |
| ptbasfi 23472 | The basis for the product ... |
| pttop 23473 | The product topology is a ... |
| ptopn 23474 | A basic open set in the pr... |
| ptopn2 23475 | A sub-basic open set in th... |
| xkotf 23476 | Functionality of function ... |
| xkobval 23477 | Alternative expression for... |
| xkoval 23478 | Value of the compact-open ... |
| xkotop 23479 | The compact-open topology ... |
| xkoopn 23480 | A basic open set of the co... |
| txtopi 23481 | The product of two topolog... |
| txtopon 23482 | The underlying set of the ... |
| txuni 23483 | The underlying set of the ... |
| txunii 23484 | The underlying set of the ... |
| ptuni 23485 | The base set for the produ... |
| ptunimpt 23486 | Base set of a product topo... |
| pttopon 23487 | The base set for the produ... |
| pttoponconst 23488 | The base set for a product... |
| ptuniconst 23489 | The base set for a product... |
| xkouni 23490 | The base set of the compac... |
| xkotopon 23491 | The base set of the compac... |
| ptval2 23492 | The value of the product t... |
| txopn 23493 | The product of two open se... |
| txcld 23494 | The product of two closed ... |
| txcls 23495 | Closure of a rectangle in ... |
| txss12 23496 | Subset property of the top... |
| txbasval 23497 | It is sufficient to consid... |
| neitx 23498 | The Cartesian product of t... |
| txcnpi 23499 | Continuity of a two-argume... |
| tx1cn 23500 | Continuity of the first pr... |
| tx2cn 23501 | Continuity of the second p... |
| ptpjcn 23502 | Continuity of a projection... |
| ptpjopn 23503 | The projection map is an o... |
| ptcld 23504 | A closed box in the produc... |
| ptcldmpt 23505 | A closed box in the produc... |
| ptclsg 23506 | The closure of a box in th... |
| ptcls 23507 | The closure of a box in th... |
| dfac14lem 23508 | Lemma for ~ dfac14 . By e... |
| dfac14 23509 | Theorem ~ ptcls is an equi... |
| xkoccn 23510 | The "constant function" fu... |
| txcnp 23511 | If two functions are conti... |
| ptcnplem 23512 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23513 | If every projection of a f... |
| upxp 23514 | Universal property of the ... |
| txcnmpt 23515 | A map into the product of ... |
| uptx 23516 | Universal property of the ... |
| txcn 23517 | A map into the product of ... |
| ptcn 23518 | If every projection of a f... |
| prdstopn 23519 | Topology of a structure pr... |
| prdstps 23520 | A structure product of top... |
| pwstps 23521 | A structure power of a top... |
| txrest 23522 | The subspace of a topologi... |
| txdis 23523 | The topological product of... |
| txindislem 23524 | Lemma for ~ txindis . (Co... |
| txindis 23525 | The topological product of... |
| txdis1cn 23526 | A function is jointly cont... |
| txlly 23527 | If the property ` A ` is p... |
| txnlly 23528 | If the property ` A ` is p... |
| pthaus 23529 | The product of a collectio... |
| ptrescn 23530 | Restriction is a continuou... |
| txtube 23531 | The "tube lemma". If ` X ... |
| txcmplem1 23532 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23533 | Lemma for ~ txcmp . (Cont... |
| txcmp 23534 | The topological product of... |
| txcmpb 23535 | The topological product of... |
| hausdiag 23536 | A topology is Hausdorff if... |
| hauseqlcld 23537 | In a Hausdorff topology, t... |
| txhaus 23538 | The topological product of... |
| txlm 23539 | Two sequences converge iff... |
| lmcn2 23540 | The image of a convergent ... |
| tx1stc 23541 | The topological product of... |
| tx2ndc 23542 | The topological product of... |
| txkgen 23543 | The topological product of... |
| xkohaus 23544 | If the codomain space is H... |
| xkoptsub 23545 | The compact-open topology ... |
| xkopt 23546 | The compact-open topology ... |
| xkopjcn 23547 | Continuity of a projection... |
| xkoco1cn 23548 | If ` F ` is a continuous f... |
| xkoco2cn 23549 | If ` F ` is a continuous f... |
| xkococnlem 23550 | Continuity of the composit... |
| xkococn 23551 | Continuity of the composit... |
| cnmptid 23552 | The identity function is c... |
| cnmptc 23553 | A constant function is con... |
| cnmpt11 23554 | The composition of continu... |
| cnmpt11f 23555 | The composition of continu... |
| cnmpt1t 23556 | The composition of continu... |
| cnmpt12f 23557 | The composition of continu... |
| cnmpt12 23558 | The composition of continu... |
| cnmpt1st 23559 | The projection onto the fi... |
| cnmpt2nd 23560 | The projection onto the se... |
| cnmpt2c 23561 | A constant function is con... |
| cnmpt21 23562 | The composition of continu... |
| cnmpt21f 23563 | The composition of continu... |
| cnmpt2t 23564 | The composition of continu... |
| cnmpt22 23565 | The composition of continu... |
| cnmpt22f 23566 | The composition of continu... |
| cnmpt1res 23567 | The restriction of a conti... |
| cnmpt2res 23568 | The restriction of a conti... |
| cnmptcom 23569 | The argument converse of a... |
| cnmptkc 23570 | The curried first projecti... |
| cnmptkp 23571 | The evaluation of the inne... |
| cnmptk1 23572 | The composition of a curri... |
| cnmpt1k 23573 | The composition of a one-a... |
| cnmptkk 23574 | The composition of two cur... |
| xkofvcn 23575 | Joint continuity of the fu... |
| cnmptk1p 23576 | The evaluation of a currie... |
| cnmptk2 23577 | The uncurrying of a currie... |
| xkoinjcn 23578 | Continuity of "injection",... |
| cnmpt2k 23579 | The currying of a two-argu... |
| txconn 23580 | The topological product of... |
| imasnopn 23581 | If a relation graph is ope... |
| imasncld 23582 | If a relation graph is clo... |
| imasncls 23583 | If a relation graph is clo... |
| qtopval 23586 | Value of the quotient topo... |
| qtopval2 23587 | Value of the quotient topo... |
| elqtop 23588 | Value of the quotient topo... |
| qtopres 23589 | The quotient topology is u... |
| qtoptop2 23590 | The quotient topology is a... |
| qtoptop 23591 | The quotient topology is a... |
| elqtop2 23592 | Value of the quotient topo... |
| qtopuni 23593 | The base set of the quotie... |
| elqtop3 23594 | Value of the quotient topo... |
| qtoptopon 23595 | The base set of the quotie... |
| qtopid 23596 | A quotient map is a contin... |
| idqtop 23597 | The quotient topology indu... |
| qtopcmplem 23598 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23599 | A quotient of a compact sp... |
| qtopconn 23600 | A quotient of a connected ... |
| qtopkgen 23601 | A quotient of a compactly ... |
| basqtop 23602 | An injection maps bases to... |
| tgqtop 23603 | An injection maps generate... |
| qtopcld 23604 | The property of being a cl... |
| qtopcn 23605 | Universal property of a qu... |
| qtopss 23606 | A surjective continuous fu... |
| qtopeu 23607 | Universal property of the ... |
| qtoprest 23608 | If ` A ` is a saturated op... |
| qtopomap 23609 | If ` F ` is a surjective c... |
| qtopcmap 23610 | If ` F ` is a surjective c... |
| imastopn 23611 | The topology of an image s... |
| imastps 23612 | The image of a topological... |
| qustps 23613 | A quotient structure is a ... |
| kqfval 23614 | Value of the function appe... |
| kqfeq 23615 | Two points in the Kolmogor... |
| kqffn 23616 | The topological indistingu... |
| kqval 23617 | Value of the quotient topo... |
| kqtopon 23618 | The Kolmogorov quotient is... |
| kqid 23619 | The topological indistingu... |
| ist0-4 23620 | The topological indistingu... |
| kqfvima 23621 | When the image set is open... |
| kqsat 23622 | Any open set is saturated ... |
| kqdisj 23623 | A version of ~ imain for t... |
| kqcldsat 23624 | Any closed set is saturate... |
| kqopn 23625 | The topological indistingu... |
| kqcld 23626 | The topological indistingu... |
| kqt0lem 23627 | Lemma for ~ kqt0 . (Contr... |
| isr0 23628 | The property " ` J ` is an... |
| r0cld 23629 | The analogue of the T_1 ax... |
| regr1lem 23630 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23631 | A Kolmogorov quotient of a... |
| kqreglem1 23632 | A Kolmogorov quotient of a... |
| kqreglem2 23633 | If the Kolmogorov quotient... |
| kqnrmlem1 23634 | A Kolmogorov quotient of a... |
| kqnrmlem2 23635 | If the Kolmogorov quotient... |
| kqtop 23636 | The Kolmogorov quotient is... |
| kqt0 23637 | The Kolmogorov quotient is... |
| kqf 23638 | The Kolmogorov quotient is... |
| r0sep 23639 | The separation property of... |
| nrmr0reg 23640 | A normal R_0 space is also... |
| regr1 23641 | A regular space is R_1, wh... |
| kqreg 23642 | The Kolmogorov quotient of... |
| kqnrm 23643 | The Kolmogorov quotient of... |
| hmeofn 23648 | The set of homeomorphisms ... |
| hmeofval 23649 | The set of all the homeomo... |
| ishmeo 23650 | The predicate F is a homeo... |
| hmeocn 23651 | A homeomorphism is continu... |
| hmeocnvcn 23652 | The converse of a homeomor... |
| hmeocnv 23653 | The converse of a homeomor... |
| hmeof1o2 23654 | A homeomorphism is a 1-1-o... |
| hmeof1o 23655 | A homeomorphism is a 1-1-o... |
| hmeoima 23656 | The image of an open set b... |
| hmeoopn 23657 | Homeomorphisms preserve op... |
| hmeocld 23658 | Homeomorphisms preserve cl... |
| hmeocls 23659 | Homeomorphisms preserve cl... |
| hmeontr 23660 | Homeomorphisms preserve in... |
| hmeoimaf1o 23661 | The function mapping open ... |
| hmeores 23662 | The restriction of a homeo... |
| hmeoco 23663 | The composite of two homeo... |
| idhmeo 23664 | The identity function is a... |
| hmeocnvb 23665 | The converse of a homeomor... |
| hmeoqtop 23666 | A homeomorphism is a quoti... |
| hmph 23667 | Express the predicate ` J ... |
| hmphi 23668 | If there is a homeomorphis... |
| hmphtop 23669 | Reverse closure for the ho... |
| hmphtop1 23670 | The relation "being homeom... |
| hmphtop2 23671 | The relation "being homeom... |
| hmphref 23672 | "Is homeomorphic to" is re... |
| hmphsym 23673 | "Is homeomorphic to" is sy... |
| hmphtr 23674 | "Is homeomorphic to" is tr... |
| hmpher 23675 | "Is homeomorphic to" is an... |
| hmphen 23676 | Homeomorphisms preserve th... |
| hmphsymb 23677 | "Is homeomorphic to" is sy... |
| haushmphlem 23678 | Lemma for ~ haushmph and s... |
| cmphmph 23679 | Compactness is a topologic... |
| connhmph 23680 | Connectedness is a topolog... |
| t0hmph 23681 | T_0 is a topological prope... |
| t1hmph 23682 | T_1 is a topological prope... |
| haushmph 23683 | Hausdorff-ness is a topolo... |
| reghmph 23684 | Regularity is a topologica... |
| nrmhmph 23685 | Normality is a topological... |
| hmph0 23686 | A topology homeomorphic to... |
| hmphdis 23687 | Homeomorphisms preserve to... |
| hmphindis 23688 | Homeomorphisms preserve to... |
| indishmph 23689 | Equinumerous sets equipped... |
| hmphen2 23690 | Homeomorphisms preserve th... |
| cmphaushmeo 23691 | A continuous bijection fro... |
| ordthmeolem 23692 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23693 | An order isomorphism is a ... |
| txhmeo 23694 | Lift a pair of homeomorphi... |
| txswaphmeolem 23695 | Show inverse for the "swap... |
| txswaphmeo 23696 | There is a homeomorphism f... |
| pt1hmeo 23697 | The canonical homeomorphis... |
| ptuncnv 23698 | Exhibit the converse funct... |
| ptunhmeo 23699 | Define a homeomorphism fro... |
| xpstopnlem1 23700 | The function ` F ` used in... |
| xpstps 23701 | A binary product of topolo... |
| xpstopnlem2 23702 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23703 | The topology on a binary p... |
| ptcmpfi 23704 | A topological product of f... |
| xkocnv 23705 | The inverse of the "curryi... |
| xkohmeo 23706 | The Exponential Law for to... |
| qtopf1 23707 | If a quotient map is injec... |
| qtophmeo 23708 | If two functions on a base... |
| t0kq 23709 | A topological space is T_0... |
| kqhmph 23710 | A topological space is T_0... |
| ist1-5lem 23711 | Lemma for ~ ist1-5 and sim... |
| t1r0 23712 | A T_1 space is R_0. That ... |
| ist1-5 23713 | A topological space is T_1... |
| ishaus3 23714 | A topological space is Hau... |
| nrmreg 23715 | A normal T_1 space is regu... |
| reghaus 23716 | A regular T_0 space is Hau... |
| nrmhaus 23717 | A T_1 normal space is Haus... |
| elmptrab 23718 | Membership in a one-parame... |
| elmptrab2 23719 | Membership in a one-parame... |
| isfbas 23720 | The predicate " ` F ` is a... |
| fbasne0 23721 | There are no empty filter ... |
| 0nelfb 23722 | No filter base contains th... |
| fbsspw 23723 | A filter base on a set is ... |
| fbelss 23724 | An element of the filter b... |
| fbdmn0 23725 | The domain of a filter bas... |
| isfbas2 23726 | The predicate " ` F ` is a... |
| fbasssin 23727 | A filter base contains sub... |
| fbssfi 23728 | A filter base contains sub... |
| fbssint 23729 | A filter base contains sub... |
| fbncp 23730 | A filter base does not con... |
| fbun 23731 | A necessary and sufficient... |
| fbfinnfr 23732 | No filter base containing ... |
| opnfbas 23733 | The collection of open sup... |
| trfbas2 23734 | Conditions for the trace o... |
| trfbas 23735 | Conditions for the trace o... |
| isfil 23738 | The predicate "is a filter... |
| filfbas 23739 | A filter is a filter base.... |
| 0nelfil 23740 | The empty set doesn't belo... |
| fileln0 23741 | An element of a filter is ... |
| filsspw 23742 | A filter is a subset of th... |
| filelss 23743 | An element of a filter is ... |
| filss 23744 | A filter is closed under t... |
| filin 23745 | A filter is closed under t... |
| filtop 23746 | The underlying set belongs... |
| isfil2 23747 | Derive the standard axioms... |
| isfildlem 23748 | Lemma for ~ isfild . (Con... |
| isfild 23749 | Sufficient condition for a... |
| filfi 23750 | A filter is closed under t... |
| filinn0 23751 | The intersection of two el... |
| filintn0 23752 | A filter has the finite in... |
| filn0 23753 | The empty set is not a fil... |
| infil 23754 | The intersection of two fi... |
| snfil 23755 | A singleton is a filter. ... |
| fbasweak 23756 | A filter base on any set i... |
| snfbas 23757 | Condition for a singleton ... |
| fsubbas 23758 | A condition for a set to g... |
| fbasfip 23759 | A filter base has the fini... |
| fbunfip 23760 | A helpful lemma for showin... |
| fgval 23761 | The filter generating clas... |
| elfg 23762 | A condition for elements o... |
| ssfg 23763 | A filter base is a subset ... |
| fgss 23764 | A bigger base generates a ... |
| fgss2 23765 | A condition for a filter t... |
| fgfil 23766 | A filter generates itself.... |
| elfilss 23767 | An element belongs to a fi... |
| filfinnfr 23768 | No filter containing a fin... |
| fgcl 23769 | A generated filter is a fi... |
| fgabs 23770 | Absorption law for filter ... |
| neifil 23771 | The neighborhoods of a non... |
| filunibas 23772 | Recover the base set from ... |
| filunirn 23773 | Two ways to express a filt... |
| filconn 23774 | A filter gives rise to a c... |
| fbasrn 23775 | Given a filter on a domain... |
| filuni 23776 | The union of a nonempty se... |
| trfil1 23777 | Conditions for the trace o... |
| trfil2 23778 | Conditions for the trace o... |
| trfil3 23779 | Conditions for the trace o... |
| trfilss 23780 | If ` A ` is a member of th... |
| fgtr 23781 | If ` A ` is a member of th... |
| trfg 23782 | The trace operation and th... |
| trnei 23783 | The trace, over a set ` A ... |
| cfinfil 23784 | Relative complements of th... |
| csdfil 23785 | The set of all elements wh... |
| supfil 23786 | The supersets of a nonempt... |
| zfbas 23787 | The set of upper sets of i... |
| uzrest 23788 | The restriction of the set... |
| uzfbas 23789 | The set of upper sets of i... |
| isufil 23794 | The property of being an u... |
| ufilfil 23795 | An ultrafilter is a filter... |
| ufilss 23796 | For any subset of the base... |
| ufilb 23797 | The complement is in an ul... |
| ufilmax 23798 | Any filter finer than an u... |
| isufil2 23799 | The maximal property of an... |
| ufprim 23800 | An ultrafilter is a prime ... |
| trufil 23801 | Conditions for the trace o... |
| filssufilg 23802 | A filter is contained in s... |
| filssufil 23803 | A filter is contained in s... |
| isufl 23804 | Define the (strong) ultraf... |
| ufli 23805 | Property of a set that sat... |
| numufl 23806 | Consequence of ~ filssufil... |
| fiufl 23807 | A finite set satisfies the... |
| acufl 23808 | The axiom of choice implie... |
| ssufl 23809 | If ` Y ` is a subset of ` ... |
| ufileu 23810 | If the ultrafilter contain... |
| filufint 23811 | A filter is equal to the i... |
| uffix 23812 | Lemma for ~ fixufil and ~ ... |
| fixufil 23813 | The condition describing a... |
| uffixfr 23814 | An ultrafilter is either f... |
| uffix2 23815 | A classification of fixed ... |
| uffixsn 23816 | The singleton of the gener... |
| ufildom1 23817 | An ultrafilter is generate... |
| uffinfix 23818 | An ultrafilter containing ... |
| cfinufil 23819 | An ultrafilter is free iff... |
| ufinffr 23820 | An infinite subset is cont... |
| ufilen 23821 | Any infinite set has an ul... |
| ufildr 23822 | An ultrafilter gives rise ... |
| fin1aufil 23823 | There are no definable fre... |
| fmval 23834 | Introduce a function that ... |
| fmfil 23835 | A mapping filter is a filt... |
| fmf 23836 | Pushing-forward via a func... |
| fmss 23837 | A finer filter produces a ... |
| elfm 23838 | An element of a mapping fi... |
| elfm2 23839 | An element of a mapping fi... |
| fmfg 23840 | The image filter of a filt... |
| elfm3 23841 | An alternate formulation o... |
| imaelfm 23842 | An image of a filter eleme... |
| rnelfmlem 23843 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23844 | A condition for a filter t... |
| fmfnfmlem1 23845 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23846 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23847 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23848 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23849 | A filter finer than an ima... |
| fmufil 23850 | An image filter of an ultr... |
| fmid 23851 | The filter map applied to ... |
| fmco 23852 | Composition of image filte... |
| ufldom 23853 | The ultrafilter lemma prop... |
| flimval 23854 | The set of limit points of... |
| elflim2 23855 | The predicate "is a limit ... |
| flimtop 23856 | Reverse closure for the li... |
| flimneiss 23857 | A filter contains the neig... |
| flimnei 23858 | A filter contains all of t... |
| flimelbas 23859 | A limit point of a filter ... |
| flimfil 23860 | Reverse closure for the li... |
| flimtopon 23861 | Reverse closure for the li... |
| elflim 23862 | The predicate "is a limit ... |
| flimss2 23863 | A limit point of a filter ... |
| flimss1 23864 | A limit point of a filter ... |
| neiflim 23865 | A point is a limit point o... |
| flimopn 23866 | The condition for being a ... |
| fbflim 23867 | A condition for a filter t... |
| fbflim2 23868 | A condition for a filter b... |
| flimclsi 23869 | The convergent points of a... |
| hausflimlem 23870 | If ` A ` and ` B ` are bot... |
| hausflimi 23871 | One direction of ~ hausfli... |
| hausflim 23872 | A condition for a topology... |
| flimcf 23873 | Fineness is properly chara... |
| flimrest 23874 | The set of limit points in... |
| flimclslem 23875 | Lemma for ~ flimcls . (Co... |
| flimcls 23876 | Closure in terms of filter... |
| flimsncls 23877 | If ` A ` is a limit point ... |
| hauspwpwf1 23878 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 23879 | If ` X ` is a Hausdorff sp... |
| flffval 23880 | Given a topology and a fil... |
| flfval 23881 | Given a function from a fi... |
| flfnei 23882 | The property of being a li... |
| flfneii 23883 | A neighborhood of a limit ... |
| isflf 23884 | The property of being a li... |
| flfelbas 23885 | A limit point of a functio... |
| flffbas 23886 | Limit points of a function... |
| flftg 23887 | Limit points of a function... |
| hausflf 23888 | If a function has its valu... |
| hausflf2 23889 | If a convergent function h... |
| cnpflfi 23890 | Forward direction of ~ cnp... |
| cnpflf2 23891 | ` F ` is continuous at poi... |
| cnpflf 23892 | Continuity of a function a... |
| cnflf 23893 | A function is continuous i... |
| cnflf2 23894 | A function is continuous i... |
| flfcnp 23895 | A continuous function pres... |
| lmflf 23896 | The topological limit rela... |
| txflf 23897 | Two sequences converge in ... |
| flfcnp2 23898 | The image of a convergent ... |
| fclsval 23899 | The set of all cluster poi... |
| isfcls 23900 | A cluster point of a filte... |
| fclsfil 23901 | Reverse closure for the cl... |
| fclstop 23902 | Reverse closure for the cl... |
| fclstopon 23903 | Reverse closure for the cl... |
| isfcls2 23904 | A cluster point of a filte... |
| fclsopn 23905 | Write the cluster point co... |
| fclsopni 23906 | An open neighborhood of a ... |
| fclselbas 23907 | A cluster point is in the ... |
| fclsneii 23908 | A neighborhood of a cluste... |
| fclssscls 23909 | The set of cluster points ... |
| fclsnei 23910 | Cluster points in terms of... |
| supnfcls 23911 | The filter of supersets of... |
| fclsbas 23912 | Cluster points in terms of... |
| fclsss1 23913 | A finer topology has fewer... |
| fclsss2 23914 | A finer filter has fewer c... |
| fclsrest 23915 | The set of cluster points ... |
| fclscf 23916 | Characterization of finene... |
| flimfcls 23917 | A limit point is a cluster... |
| fclsfnflim 23918 | A filter clusters at a poi... |
| flimfnfcls 23919 | A filter converges to a po... |
| fclscmpi 23920 | Forward direction of ~ fcl... |
| fclscmp 23921 | A space is compact iff eve... |
| uffclsflim 23922 | The cluster points of an u... |
| ufilcmp 23923 | A space is compact iff eve... |
| fcfval 23924 | The set of cluster points ... |
| isfcf 23925 | The property of being a cl... |
| fcfnei 23926 | The property of being a cl... |
| fcfelbas 23927 | A cluster point of a funct... |
| fcfneii 23928 | A neighborhood of a cluste... |
| flfssfcf 23929 | A limit point of a functio... |
| uffcfflf 23930 | If the domain filter is an... |
| cnpfcfi 23931 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 23932 | A function ` F ` is contin... |
| cnfcf 23933 | Continuity of a function i... |
| flfcntr 23934 | A continuous function's va... |
| alexsublem 23935 | Lemma for ~ alexsub . (Co... |
| alexsub 23936 | The Alexander Subbase Theo... |
| alexsubb 23937 | Biconditional form of the ... |
| alexsubALTlem1 23938 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 23939 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 23940 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 23941 | Lemma for ~ alexsubALT . ... |
| alexsubALT 23942 | The Alexander Subbase Theo... |
| ptcmplem1 23943 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 23944 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 23945 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 23946 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 23947 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 23948 | Tychonoff's theorem: The ... |
| ptcmp 23949 | Tychonoff's theorem: The ... |
| cnextval 23952 | The function applying cont... |
| cnextfval 23953 | The continuous extension o... |
| cnextrel 23954 | In the general case, a con... |
| cnextfun 23955 | If the target space is Hau... |
| cnextfvval 23956 | The value of the continuou... |
| cnextf 23957 | Extension by continuity. ... |
| cnextcn 23958 | Extension by continuity. ... |
| cnextfres1 23959 | ` F ` and its extension by... |
| cnextfres 23960 | ` F ` and its extension by... |
| istmd 23965 | The predicate "is a topolo... |
| tmdmnd 23966 | A topological monoid is a ... |
| tmdtps 23967 | A topological monoid is a ... |
| istgp 23968 | The predicate "is a topolo... |
| tgpgrp 23969 | A topological group is a g... |
| tgptmd 23970 | A topological group is a t... |
| tgptps 23971 | A topological group is a t... |
| tmdtopon 23972 | The topology of a topologi... |
| tgptopon 23973 | The topology of a topologi... |
| tmdcn 23974 | In a topological monoid, t... |
| tgpcn 23975 | In a topological group, th... |
| tgpinv 23976 | In a topological group, th... |
| grpinvhmeo 23977 | The inverse function in a ... |
| cnmpt1plusg 23978 | Continuity of the group su... |
| cnmpt2plusg 23979 | Continuity of the group su... |
| tmdcn2 23980 | Write out the definition o... |
| tgpsubcn 23981 | In a topological group, th... |
| istgp2 23982 | A group with a topology is... |
| tmdmulg 23983 | In a topological monoid, t... |
| tgpmulg 23984 | In a topological group, th... |
| tgpmulg2 23985 | In a topological monoid, t... |
| tmdgsum 23986 | In a topological monoid, t... |
| tmdgsum2 23987 | For any neighborhood ` U `... |
| oppgtmd 23988 | The opposite of a topologi... |
| oppgtgp 23989 | The opposite of a topologi... |
| distgp 23990 | Any group equipped with th... |
| indistgp 23991 | Any group equipped with th... |
| efmndtmd 23992 | The monoid of endofunction... |
| tmdlactcn 23993 | The left group action of e... |
| tgplacthmeo 23994 | The left group action of e... |
| submtmd 23995 | A submonoid of a topologic... |
| subgtgp 23996 | A subgroup of a topologica... |
| symgtgp 23997 | The symmetric group is a t... |
| subgntr 23998 | A subgroup of a topologica... |
| opnsubg 23999 | An open subgroup of a topo... |
| clssubg 24000 | The closure of a subgroup ... |
| clsnsg 24001 | The closure of a normal su... |
| cldsubg 24002 | A subgroup of finite index... |
| tgpconncompeqg 24003 | The connected component co... |
| tgpconncomp 24004 | The identity component, th... |
| tgpconncompss 24005 | The identity component is ... |
| ghmcnp 24006 | A group homomorphism on to... |
| snclseqg 24007 | The coset of the closure o... |
| tgphaus 24008 | A topological group is Hau... |
| tgpt1 24009 | Hausdorff and T1 are equiv... |
| tgpt0 24010 | Hausdorff and T0 are equiv... |
| qustgpopn 24011 | A quotient map in a topolo... |
| qustgplem 24012 | Lemma for ~ qustgp . (Con... |
| qustgp 24013 | The quotient of a topologi... |
| qustgphaus 24014 | The quotient of a topologi... |
| prdstmdd 24015 | The product of a family of... |
| prdstgpd 24016 | The product of a family of... |
| tsmsfbas 24019 | The collection of all sets... |
| tsmslem1 24020 | The finite partial sums of... |
| tsmsval2 24021 | Definition of the topologi... |
| tsmsval 24022 | Definition of the topologi... |
| tsmspropd 24023 | The group sum depends only... |
| eltsms 24024 | The property of being a su... |
| tsmsi 24025 | The property of being a su... |
| tsmscl 24026 | A sum in a topological gro... |
| haustsms 24027 | In a Hausdorff topological... |
| haustsms2 24028 | In a Hausdorff topological... |
| tsmscls 24029 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24030 | The convergent points of a... |
| tsmsid 24031 | If a sum is finite, the us... |
| haustsmsid 24032 | In a Hausdorff topological... |
| tsms0 24033 | The sum of zero is zero. ... |
| tsmssubm 24034 | Evaluate an infinite group... |
| tsmsres 24035 | Extend an infinite group s... |
| tsmsf1o 24036 | Re-index an infinite group... |
| tsmsmhm 24037 | Apply a continuous group h... |
| tsmsadd 24038 | The sum of two infinite gr... |
| tsmsinv 24039 | Inverse of an infinite gro... |
| tsmssub 24040 | The difference of two infi... |
| tgptsmscls 24041 | A sum in a topological gro... |
| tgptsmscld 24042 | The set of limit points to... |
| tsmssplit 24043 | Split a topological group ... |
| tsmsxplem1 24044 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24045 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24046 | Write a sum over a two-dim... |
| istrg 24055 | Express the predicate " ` ... |
| trgtmd 24056 | The multiplicative monoid ... |
| istdrg 24057 | Express the predicate " ` ... |
| tdrgunit 24058 | The unit group of a topolo... |
| trgtgp 24059 | A topological ring is a to... |
| trgtmd2 24060 | A topological ring is a to... |
| trgtps 24061 | A topological ring is a to... |
| trgring 24062 | A topological ring is a ri... |
| trggrp 24063 | A topological ring is a gr... |
| tdrgtrg 24064 | A topological division rin... |
| tdrgdrng 24065 | A topological division rin... |
| tdrgring 24066 | A topological division rin... |
| tdrgtmd 24067 | A topological division rin... |
| tdrgtps 24068 | A topological division rin... |
| istdrg2 24069 | A topological-ring divisio... |
| mulrcn 24070 | The functionalization of t... |
| invrcn2 24071 | The multiplicative inverse... |
| invrcn 24072 | The multiplicative inverse... |
| cnmpt1mulr 24073 | Continuity of ring multipl... |
| cnmpt2mulr 24074 | Continuity of ring multipl... |
| dvrcn 24075 | The division function is c... |
| istlm 24076 | The predicate " ` W ` is a... |
| vscacn 24077 | The scalar multiplication ... |
| tlmtmd 24078 | A topological module is a ... |
| tlmtps 24079 | A topological module is a ... |
| tlmlmod 24080 | A topological module is a ... |
| tlmtrg 24081 | The scalar ring of a topol... |
| tlmscatps 24082 | The scalar ring of a topol... |
| istvc 24083 | A topological vector space... |
| tvctdrg 24084 | The scalar field of a topo... |
| cnmpt1vsca 24085 | Continuity of scalar multi... |
| cnmpt2vsca 24086 | Continuity of scalar multi... |
| tlmtgp 24087 | A topological vector space... |
| tvctlm 24088 | A topological vector space... |
| tvclmod 24089 | A topological vector space... |
| tvclvec 24090 | A topological vector space... |
| ustfn 24093 | The defined uniform struct... |
| ustval 24094 | The class of all uniform s... |
| isust 24095 | The predicate " ` U ` is a... |
| ustssxp 24096 | Entourages are subsets of ... |
| ustssel 24097 | A uniform structure is upw... |
| ustbasel 24098 | The full set is always an ... |
| ustincl 24099 | A uniform structure is clo... |
| ustdiag 24100 | The diagonal set is includ... |
| ustinvel 24101 | If ` V ` is an entourage, ... |
| ustexhalf 24102 | For each entourage ` V ` t... |
| ustrel 24103 | The elements of uniform st... |
| ustfilxp 24104 | A uniform structure on a n... |
| ustne0 24105 | A uniform structure cannot... |
| ustssco 24106 | In an uniform structure, a... |
| ustexsym 24107 | In an uniform structure, f... |
| ustex2sym 24108 | In an uniform structure, f... |
| ustex3sym 24109 | In an uniform structure, f... |
| ustref 24110 | Any element of the base se... |
| ust0 24111 | The unique uniform structu... |
| ustn0 24112 | The empty set is not an un... |
| ustund 24113 | If two intersecting sets `... |
| ustelimasn 24114 | Any point ` A ` is near en... |
| ustneism 24115 | For a point ` A ` in ` X `... |
| elrnustOLD 24116 | Obsolete version of ~ elfv... |
| ustbas2 24117 | Second direction for ~ ust... |
| ustuni 24118 | The set union of a uniform... |
| ustbas 24119 | Recover the base of an uni... |
| ustimasn 24120 | Lemma for ~ ustuqtop . (C... |
| trust 24121 | The trace of a uniform str... |
| utopval 24124 | The topology induced by a ... |
| elutop 24125 | Open sets in the topology ... |
| utoptop 24126 | The topology induced by a ... |
| utopbas 24127 | The base of the topology i... |
| utoptopon 24128 | Topology induced by a unif... |
| restutop 24129 | Restriction of a topology ... |
| restutopopn 24130 | The restriction of the top... |
| ustuqtoplem 24131 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24132 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24133 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24134 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24135 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24136 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24137 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24138 | For a given uniform struct... |
| utopsnneiplem 24139 | The neighborhoods of a poi... |
| utopsnneip 24140 | The neighborhoods of a poi... |
| utopsnnei 24141 | Images of singletons by en... |
| utop2nei 24142 | For any symmetrical entour... |
| utop3cls 24143 | Relation between a topolog... |
| utopreg 24144 | All Hausdorff uniform spac... |
| ussval 24151 | The uniform structure on u... |
| ussid 24152 | In case the base of the ` ... |
| isusp 24153 | The predicate ` W ` is a u... |
| ressuss 24154 | Value of the uniform struc... |
| ressust 24155 | The uniform structure of a... |
| ressusp 24156 | The restriction of a unifo... |
| tusval 24157 | The value of the uniform s... |
| tuslem 24158 | Lemma for ~ tusbas , ~ tus... |
| tuslemOLD 24159 | Obsolete proof of ~ tuslem... |
| tusbas 24160 | The base set of a construc... |
| tusunif 24161 | The uniform structure of a... |
| tususs 24162 | The uniform structure of a... |
| tustopn 24163 | The topology induced by a ... |
| tususp 24164 | A constructed uniform spac... |
| tustps 24165 | A constructed uniform spac... |
| uspreg 24166 | If a uniform space is Haus... |
| ucnval 24169 | The set of all uniformly c... |
| isucn 24170 | The predicate " ` F ` is a... |
| isucn2 24171 | The predicate " ` F ` is a... |
| ucnimalem 24172 | Reformulate the ` G ` func... |
| ucnima 24173 | An equivalent statement of... |
| ucnprima 24174 | The preimage by a uniforml... |
| iducn 24175 | The identity is uniformly ... |
| cstucnd 24176 | A constant function is uni... |
| ucncn 24177 | Uniform continuity implies... |
| iscfilu 24180 | The predicate " ` F ` is a... |
| cfilufbas 24181 | A Cauchy filter base is a ... |
| cfiluexsm 24182 | For a Cauchy filter base a... |
| fmucndlem 24183 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24184 | The image of a Cauchy filt... |
| cfilufg 24185 | The filter generated by a ... |
| trcfilu 24186 | Condition for the trace of... |
| cfiluweak 24187 | A Cauchy filter base is al... |
| neipcfilu 24188 | In an uniform space, a nei... |
| iscusp 24191 | The predicate " ` W ` is a... |
| cuspusp 24192 | A complete uniform space i... |
| cuspcvg 24193 | In a complete uniform spac... |
| iscusp2 24194 | The predicate " ` W ` is a... |
| cnextucn 24195 | Extension by continuity. ... |
| ucnextcn 24196 | Extension by continuity. ... |
| ispsmet 24197 | Express the predicate " ` ... |
| psmetdmdm 24198 | Recover the base set from ... |
| psmetf 24199 | The distance function of a... |
| psmetcl 24200 | Closure of the distance fu... |
| psmet0 24201 | The distance function of a... |
| psmettri2 24202 | Triangle inequality for th... |
| psmetsym 24203 | The distance function of a... |
| psmettri 24204 | Triangle inequality for th... |
| psmetge0 24205 | The distance function of a... |
| psmetxrge0 24206 | The distance function of a... |
| psmetres2 24207 | Restriction of a pseudomet... |
| psmetlecl 24208 | Real closure of an extende... |
| distspace 24209 | A set ` X ` together with ... |
| ismet 24216 | Express the predicate " ` ... |
| isxmet 24217 | Express the predicate " ` ... |
| ismeti 24218 | Properties that determine ... |
| isxmetd 24219 | Properties that determine ... |
| isxmet2d 24220 | It is safe to only require... |
| metflem 24221 | Lemma for ~ metf and other... |
| xmetf 24222 | Mapping of the distance fu... |
| metf 24223 | Mapping of the distance fu... |
| xmetcl 24224 | Closure of the distance fu... |
| metcl 24225 | Closure of the distance fu... |
| ismet2 24226 | An extended metric is a me... |
| metxmet 24227 | A metric is an extended me... |
| xmetdmdm 24228 | Recover the base set from ... |
| metdmdm 24229 | Recover the base set from ... |
| xmetunirn 24230 | Two ways to express an ext... |
| xmeteq0 24231 | The value of an extended m... |
| meteq0 24232 | The value of a metric is z... |
| xmettri2 24233 | Triangle inequality for th... |
| mettri2 24234 | Triangle inequality for th... |
| xmet0 24235 | The distance function of a... |
| met0 24236 | The distance function of a... |
| xmetge0 24237 | The distance function of a... |
| metge0 24238 | The distance function of a... |
| xmetlecl 24239 | Real closure of an extende... |
| xmetsym 24240 | The distance function of a... |
| xmetpsmet 24241 | An extended metric is a ps... |
| xmettpos 24242 | The distance function of a... |
| metsym 24243 | The distance function of a... |
| xmettri 24244 | Triangle inequality for th... |
| mettri 24245 | Triangle inequality for th... |
| xmettri3 24246 | Triangle inequality for th... |
| mettri3 24247 | Triangle inequality for th... |
| xmetrtri 24248 | One half of the reverse tr... |
| xmetrtri2 24249 | The reverse triangle inequ... |
| metrtri 24250 | Reverse triangle inequalit... |
| xmetgt0 24251 | The distance function of a... |
| metgt0 24252 | The distance function of a... |
| metn0 24253 | A metric space is nonempty... |
| xmetres2 24254 | Restriction of an extended... |
| metreslem 24255 | Lemma for ~ metres . (Con... |
| metres2 24256 | Lemma for ~ metres . (Con... |
| xmetres 24257 | A restriction of an extend... |
| metres 24258 | A restriction of a metric ... |
| 0met 24259 | The empty metric. (Contri... |
| prdsdsf 24260 | The product metric is a fu... |
| prdsxmetlem 24261 | The product metric is an e... |
| prdsxmet 24262 | The product metric is an e... |
| prdsmet 24263 | The product metric is a me... |
| ressprdsds 24264 | Restriction of a product m... |
| resspwsds 24265 | Restriction of a power met... |
| imasdsf1olem 24266 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24267 | The distance function is t... |
| imasf1oxmet 24268 | The image of an extended m... |
| imasf1omet 24269 | The image of a metric is a... |
| xpsdsfn 24270 | Closure of the metric in a... |
| xpsdsfn2 24271 | Closure of the metric in a... |
| xpsxmetlem 24272 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24273 | A product metric of extend... |
| xpsdsval 24274 | Value of the metric in a b... |
| xpsmet 24275 | The direct product of two ... |
| blfvalps 24276 | The value of the ball func... |
| blfval 24277 | The value of the ball func... |
| blvalps 24278 | The ball around a point ` ... |
| blval 24279 | The ball around a point ` ... |
| elblps 24280 | Membership in a ball. (Co... |
| elbl 24281 | Membership in a ball. (Co... |
| elbl2ps 24282 | Membership in a ball. (Co... |
| elbl2 24283 | Membership in a ball. (Co... |
| elbl3ps 24284 | Membership in a ball, with... |
| elbl3 24285 | Membership in a ball, with... |
| blcomps 24286 | Commute the arguments to t... |
| blcom 24287 | Commute the arguments to t... |
| xblpnfps 24288 | The infinity ball in an ex... |
| xblpnf 24289 | The infinity ball in an ex... |
| blpnf 24290 | The infinity ball in a sta... |
| bldisj 24291 | Two balls are disjoint if ... |
| blgt0 24292 | A nonempty ball implies th... |
| bl2in 24293 | Two balls are disjoint if ... |
| xblss2ps 24294 | One ball is contained in a... |
| xblss2 24295 | One ball is contained in a... |
| blss2ps 24296 | One ball is contained in a... |
| blss2 24297 | One ball is contained in a... |
| blhalf 24298 | A ball of radius ` R / 2 `... |
| blfps 24299 | Mapping of a ball. (Contr... |
| blf 24300 | Mapping of a ball. (Contr... |
| blrnps 24301 | Membership in the range of... |
| blrn 24302 | Membership in the range of... |
| xblcntrps 24303 | A ball contains its center... |
| xblcntr 24304 | A ball contains its center... |
| blcntrps 24305 | A ball contains its center... |
| blcntr 24306 | A ball contains its center... |
| xbln0 24307 | A ball is nonempty iff the... |
| bln0 24308 | A ball is not empty. (Con... |
| blelrnps 24309 | A ball belongs to the set ... |
| blelrn 24310 | A ball belongs to the set ... |
| blssm 24311 | A ball is a subset of the ... |
| unirnblps 24312 | The union of the set of ba... |
| unirnbl 24313 | The union of the set of ba... |
| blin 24314 | The intersection of two ba... |
| ssblps 24315 | The size of a ball increas... |
| ssbl 24316 | The size of a ball increas... |
| blssps 24317 | Any point ` P ` in a ball ... |
| blss 24318 | Any point ` P ` in a ball ... |
| blssexps 24319 | Two ways to express the ex... |
| blssex 24320 | Two ways to express the ex... |
| ssblex 24321 | A nested ball exists whose... |
| blin2 24322 | Given any two balls and a ... |
| blbas 24323 | The balls of a metric spac... |
| blres 24324 | A ball in a restricted met... |
| xmeterval 24325 | Value of the "finitely sep... |
| xmeter 24326 | The "finitely separated" r... |
| xmetec 24327 | The equivalence classes un... |
| blssec 24328 | A ball centered at ` P ` i... |
| blpnfctr 24329 | The infinity ball in an ex... |
| xmetresbl 24330 | An extended metric restric... |
| mopnval 24331 | An open set is a subset of... |
| mopntopon 24332 | The set of open sets of a ... |
| mopntop 24333 | The set of open sets of a ... |
| mopnuni 24334 | The union of all open sets... |
| elmopn 24335 | The defining property of a... |
| mopnfss 24336 | The family of open sets of... |
| mopnm 24337 | The base set of a metric s... |
| elmopn2 24338 | A defining property of an ... |
| mopnss 24339 | An open set of a metric sp... |
| isxms 24340 | Express the predicate " ` ... |
| isxms2 24341 | Express the predicate " ` ... |
| isms 24342 | Express the predicate " ` ... |
| isms2 24343 | Express the predicate " ` ... |
| xmstopn 24344 | The topology component of ... |
| mstopn 24345 | The topology component of ... |
| xmstps 24346 | An extended metric space i... |
| msxms 24347 | A metric space is an exten... |
| mstps 24348 | A metric space is a topolo... |
| xmsxmet 24349 | The distance function, sui... |
| msmet 24350 | The distance function, sui... |
| msf 24351 | The distance function of a... |
| xmsxmet2 24352 | The distance function, sui... |
| msmet2 24353 | The distance function, sui... |
| mscl 24354 | Closure of the distance fu... |
| xmscl 24355 | Closure of the distance fu... |
| xmsge0 24356 | The distance function in a... |
| xmseq0 24357 | The distance between two p... |
| xmssym 24358 | The distance function in a... |
| xmstri2 24359 | Triangle inequality for th... |
| mstri2 24360 | Triangle inequality for th... |
| xmstri 24361 | Triangle inequality for th... |
| mstri 24362 | Triangle inequality for th... |
| xmstri3 24363 | Triangle inequality for th... |
| mstri3 24364 | Triangle inequality for th... |
| msrtri 24365 | Reverse triangle inequalit... |
| xmspropd 24366 | Property deduction for an ... |
| mspropd 24367 | Property deduction for a m... |
| setsmsbas 24368 | The base set of a construc... |
| setsmsbasOLD 24369 | Obsolete proof of ~ setsms... |
| setsmsds 24370 | The distance function of a... |
| setsmsdsOLD 24371 | Obsolete proof of ~ setsms... |
| setsmstset 24372 | The topology of a construc... |
| setsmstopn 24373 | The topology of a construc... |
| setsxms 24374 | The constructed metric spa... |
| setsms 24375 | The constructed metric spa... |
| tmsval 24376 | For any metric there is an... |
| tmslem 24377 | Lemma for ~ tmsbas , ~ tms... |
| tmslemOLD 24378 | Obsolete version of ~ tmsl... |
| tmsbas 24379 | The base set of a construc... |
| tmsds 24380 | The metric of a constructe... |
| tmstopn 24381 | The topology of a construc... |
| tmsxms 24382 | The constructed metric spa... |
| tmsms 24383 | The constructed metric spa... |
| imasf1obl 24384 | The image of a metric spac... |
| imasf1oxms 24385 | The image of a metric spac... |
| imasf1oms 24386 | The image of a metric spac... |
| prdsbl 24387 | A ball in the product metr... |
| mopni 24388 | An open set of a metric sp... |
| mopni2 24389 | An open set of a metric sp... |
| mopni3 24390 | An open set of a metric sp... |
| blssopn 24391 | The balls of a metric spac... |
| unimopn 24392 | The union of a collection ... |
| mopnin 24393 | The intersection of two op... |
| mopn0 24394 | The empty set is an open s... |
| rnblopn 24395 | A ball of a metric space i... |
| blopn 24396 | A ball of a metric space i... |
| neibl 24397 | The neighborhoods around a... |
| blnei 24398 | A ball around a point is a... |
| lpbl 24399 | Every ball around a limit ... |
| blsscls2 24400 | A smaller closed ball is c... |
| blcld 24401 | A "closed ball" in a metri... |
| blcls 24402 | The closure of an open bal... |
| blsscls 24403 | If two concentric balls ha... |
| metss 24404 | Two ways of saying that me... |
| metequiv 24405 | Two ways of saying that tw... |
| metequiv2 24406 | If there is a sequence of ... |
| metss2lem 24407 | Lemma for ~ metss2 . (Con... |
| metss2 24408 | If the metric ` D ` is "st... |
| comet 24409 | The composition of an exte... |
| stdbdmetval 24410 | Value of the standard boun... |
| stdbdxmet 24411 | The standard bounded metri... |
| stdbdmet 24412 | The standard bounded metri... |
| stdbdbl 24413 | The standard bounded metri... |
| stdbdmopn 24414 | The standard bounded metri... |
| mopnex 24415 | The topology generated by ... |
| methaus 24416 | The topology generated by ... |
| met1stc 24417 | The topology generated by ... |
| met2ndci 24418 | A separable metric space (... |
| met2ndc 24419 | A metric space is second-c... |
| metrest 24420 | Two alternate formulations... |
| ressxms 24421 | The restriction of a metri... |
| ressms 24422 | The restriction of a metri... |
| prdsmslem1 24423 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24424 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24425 | Lemma for ~ prdsxms . The... |
| prdsxms 24426 | The indexed product struct... |
| prdsms 24427 | The indexed product struct... |
| pwsxms 24428 | A power of an extended met... |
| pwsms 24429 | A power of a metric space ... |
| xpsxms 24430 | A binary product of metric... |
| xpsms 24431 | A binary product of metric... |
| tmsxps 24432 | Express the product of two... |
| tmsxpsmopn 24433 | Express the product of two... |
| tmsxpsval 24434 | Value of the product of tw... |
| tmsxpsval2 24435 | Value of the product of tw... |
| metcnp3 24436 | Two ways to express that `... |
| metcnp 24437 | Two ways to say a mapping ... |
| metcnp2 24438 | Two ways to say a mapping ... |
| metcn 24439 | Two ways to say a mapping ... |
| metcnpi 24440 | Epsilon-delta property of ... |
| metcnpi2 24441 | Epsilon-delta property of ... |
| metcnpi3 24442 | Epsilon-delta property of ... |
| txmetcnp 24443 | Continuity of a binary ope... |
| txmetcn 24444 | Continuity of a binary ope... |
| metuval 24445 | Value of the uniform struc... |
| metustel 24446 | Define a filter base ` F `... |
| metustss 24447 | Range of the elements of t... |
| metustrel 24448 | Elements of the filter bas... |
| metustto 24449 | Any two elements of the fi... |
| metustid 24450 | The identity diagonal is i... |
| metustsym 24451 | Elements of the filter bas... |
| metustexhalf 24452 | For any element ` A ` of t... |
| metustfbas 24453 | The filter base generated ... |
| metust 24454 | The uniform structure gene... |
| cfilucfil 24455 | Given a metric ` D ` and a... |
| metuust 24456 | The uniform structure gene... |
| cfilucfil2 24457 | Given a metric ` D ` and a... |
| blval2 24458 | The ball around a point ` ... |
| elbl4 24459 | Membership in a ball, alte... |
| metuel 24460 | Elementhood in the uniform... |
| metuel2 24461 | Elementhood in the uniform... |
| metustbl 24462 | The "section" image of an ... |
| psmetutop 24463 | The topology induced by a ... |
| xmetutop 24464 | The topology induced by a ... |
| xmsusp 24465 | If the uniform set of a me... |
| restmetu 24466 | The uniform structure gene... |
| metucn 24467 | Uniform continuity in metr... |
| dscmet 24468 | The discrete metric on any... |
| dscopn 24469 | The discrete metric genera... |
| nrmmetd 24470 | Show that a group norm gen... |
| abvmet 24471 | An absolute value ` F ` ge... |
| nmfval 24484 | The value of the norm func... |
| nmval 24485 | The value of the norm as t... |
| nmfval0 24486 | The value of the norm func... |
| nmfval2 24487 | The value of the norm func... |
| nmval2 24488 | The value of the norm on a... |
| nmf2 24489 | The norm on a metric group... |
| nmpropd 24490 | Weak property deduction fo... |
| nmpropd2 24491 | Strong property deduction ... |
| isngp 24492 | The property of being a no... |
| isngp2 24493 | The property of being a no... |
| isngp3 24494 | The property of being a no... |
| ngpgrp 24495 | A normed group is a group.... |
| ngpms 24496 | A normed group is a metric... |
| ngpxms 24497 | A normed group is an exten... |
| ngptps 24498 | A normed group is a topolo... |
| ngpmet 24499 | The (induced) metric of a ... |
| ngpds 24500 | Value of the distance func... |
| ngpdsr 24501 | Value of the distance func... |
| ngpds2 24502 | Write the distance between... |
| ngpds2r 24503 | Write the distance between... |
| ngpds3 24504 | Write the distance between... |
| ngpds3r 24505 | Write the distance between... |
| ngprcan 24506 | Cancel right addition insi... |
| ngplcan 24507 | Cancel left addition insid... |
| isngp4 24508 | Express the property of be... |
| ngpinvds 24509 | Two elements are the same ... |
| ngpsubcan 24510 | Cancel right subtraction i... |
| nmf 24511 | The norm on a normed group... |
| nmcl 24512 | The norm of a normed group... |
| nmge0 24513 | The norm of a normed group... |
| nmeq0 24514 | The identity is the only e... |
| nmne0 24515 | The norm of a nonzero elem... |
| nmrpcl 24516 | The norm of a nonzero elem... |
| nminv 24517 | The norm of a negated elem... |
| nmmtri 24518 | The triangle inequality fo... |
| nmsub 24519 | The norm of the difference... |
| nmrtri 24520 | Reverse triangle inequalit... |
| nm2dif 24521 | Inequality for the differe... |
| nmtri 24522 | The triangle inequality fo... |
| nmtri2 24523 | Triangle inequality for th... |
| ngpi 24524 | The properties of a normed... |
| nm0 24525 | Norm of the identity eleme... |
| nmgt0 24526 | The norm of a nonzero elem... |
| sgrim 24527 | The induced metric on a su... |
| sgrimval 24528 | The induced metric on a su... |
| subgnm 24529 | The norm in a subgroup. (... |
| subgnm2 24530 | A substructure assigns the... |
| subgngp 24531 | A normed group restricted ... |
| ngptgp 24532 | A normed abelian group is ... |
| ngppropd 24533 | Property deduction for a n... |
| reldmtng 24534 | The function ` toNrmGrp ` ... |
| tngval 24535 | Value of the function whic... |
| tnglem 24536 | Lemma for ~ tngbas and sim... |
| tnglemOLD 24537 | Obsolete version of ~ tngl... |
| tngbas 24538 | The base set of a structur... |
| tngbasOLD 24539 | Obsolete proof of ~ tngbas... |
| tngplusg 24540 | The group addition of a st... |
| tngplusgOLD 24541 | Obsolete proof of ~ tngplu... |
| tng0 24542 | The group identity of a st... |
| tngmulr 24543 | The ring multiplication of... |
| tngmulrOLD 24544 | Obsolete proof of ~ tngmul... |
| tngsca 24545 | The scalar ring of a struc... |
| tngscaOLD 24546 | Obsolete proof of ~ tngsca... |
| tngvsca 24547 | The scalar multiplication ... |
| tngvscaOLD 24548 | Obsolete proof of ~ tngvsc... |
| tngip 24549 | The inner product operatio... |
| tngipOLD 24550 | Obsolete proof of ~ tngip ... |
| tngds 24551 | The metric function of a s... |
| tngdsOLD 24552 | Obsolete proof of ~ tngds ... |
| tngtset 24553 | The topology generated by ... |
| tngtopn 24554 | The topology generated by ... |
| tngnm 24555 | The topology generated by ... |
| tngngp2 24556 | A norm turns a group into ... |
| tngngpd 24557 | Derive the axioms for a no... |
| tngngp 24558 | Derive the axioms for a no... |
| tnggrpr 24559 | If a structure equipped wi... |
| tngngp3 24560 | Alternate definition of a ... |
| nrmtngdist 24561 | The augmentation of a norm... |
| nrmtngnrm 24562 | The augmentation of a norm... |
| tngngpim 24563 | The induced metric of a no... |
| isnrg 24564 | A normed ring is a ring wi... |
| nrgabv 24565 | The norm of a normed ring ... |
| nrgngp 24566 | A normed ring is a normed ... |
| nrgring 24567 | A normed ring is a ring. ... |
| nmmul 24568 | The norm of a product in a... |
| nrgdsdi 24569 | Distribute a distance calc... |
| nrgdsdir 24570 | Distribute a distance calc... |
| nm1 24571 | The norm of one in a nonze... |
| unitnmn0 24572 | The norm of a unit is nonz... |
| nminvr 24573 | The norm of an inverse in ... |
| nmdvr 24574 | The norm of a division in ... |
| nrgdomn 24575 | A nonzero normed ring is a... |
| nrgtgp 24576 | A normed ring is a topolog... |
| subrgnrg 24577 | A normed ring restricted t... |
| tngnrg 24578 | Given any absolute value o... |
| isnlm 24579 | A normed (left) module is ... |
| nmvs 24580 | Defining property of a nor... |
| nlmngp 24581 | A normed module is a norme... |
| nlmlmod 24582 | A normed module is a left ... |
| nlmnrg 24583 | The scalar component of a ... |
| nlmngp2 24584 | The scalar component of a ... |
| nlmdsdi 24585 | Distribute a distance calc... |
| nlmdsdir 24586 | Distribute a distance calc... |
| nlmmul0or 24587 | If a scalar product is zer... |
| sranlm 24588 | The subring algebra over a... |
| nlmvscnlem2 24589 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24590 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24591 | The scalar multiplication ... |
| rlmnlm 24592 | The ring module over a nor... |
| rlmnm 24593 | The norm function in the r... |
| nrgtrg 24594 | A normed ring is a topolog... |
| nrginvrcnlem 24595 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24596 | The ring inverse function ... |
| nrgtdrg 24597 | A normed division ring is ... |
| nlmtlm 24598 | A normed module is a topol... |
| isnvc 24599 | A normed vector space is j... |
| nvcnlm 24600 | A normed vector space is a... |
| nvclvec 24601 | A normed vector space is a... |
| nvclmod 24602 | A normed vector space is a... |
| isnvc2 24603 | A normed vector space is j... |
| nvctvc 24604 | A normed vector space is a... |
| lssnlm 24605 | A subspace of a normed mod... |
| lssnvc 24606 | A subspace of a normed vec... |
| rlmnvc 24607 | The ring module over a nor... |
| ngpocelbl 24608 | Membership of an off-cente... |
| nmoffn 24615 | The function producing ope... |
| reldmnghm 24616 | Lemma for normed group hom... |
| reldmnmhm 24617 | Lemma for module homomorph... |
| nmofval 24618 | Value of the operator norm... |
| nmoval 24619 | Value of the operator norm... |
| nmogelb 24620 | Property of the operator n... |
| nmolb 24621 | Any upper bound on the val... |
| nmolb2d 24622 | Any upper bound on the val... |
| nmof 24623 | The operator norm is a fun... |
| nmocl 24624 | The operator norm of an op... |
| nmoge0 24625 | The operator norm of an op... |
| nghmfval 24626 | A normed group homomorphis... |
| isnghm 24627 | A normed group homomorphis... |
| isnghm2 24628 | A normed group homomorphis... |
| isnghm3 24629 | A normed group homomorphis... |
| bddnghm 24630 | A bounded group homomorphi... |
| nghmcl 24631 | A normed group homomorphis... |
| nmoi 24632 | The operator norm achieves... |
| nmoix 24633 | The operator norm is a bou... |
| nmoi2 24634 | The operator norm is a bou... |
| nmoleub 24635 | The operator norm, defined... |
| nghmrcl1 24636 | Reverse closure for a norm... |
| nghmrcl2 24637 | Reverse closure for a norm... |
| nghmghm 24638 | A normed group homomorphis... |
| nmo0 24639 | The operator norm of the z... |
| nmoeq0 24640 | The operator norm is zero ... |
| nmoco 24641 | An upper bound on the oper... |
| nghmco 24642 | The composition of normed ... |
| nmotri 24643 | Triangle inequality for th... |
| nghmplusg 24644 | The sum of two bounded lin... |
| 0nghm 24645 | The zero operator is a nor... |
| nmoid 24646 | The operator norm of the i... |
| idnghm 24647 | The identity operator is a... |
| nmods 24648 | Upper bound for the distan... |
| nghmcn 24649 | A normed group homomorphis... |
| isnmhm 24650 | A normed module homomorphi... |
| nmhmrcl1 24651 | Reverse closure for a norm... |
| nmhmrcl2 24652 | Reverse closure for a norm... |
| nmhmlmhm 24653 | A normed module homomorphi... |
| nmhmnghm 24654 | A normed module homomorphi... |
| nmhmghm 24655 | A normed module homomorphi... |
| isnmhm2 24656 | A normed module homomorphi... |
| nmhmcl 24657 | A normed module homomorphi... |
| idnmhm 24658 | The identity operator is a... |
| 0nmhm 24659 | The zero operator is a bou... |
| nmhmco 24660 | The composition of bounded... |
| nmhmplusg 24661 | The sum of two bounded lin... |
| qtopbaslem 24662 | The set of open intervals ... |
| qtopbas 24663 | The set of open intervals ... |
| retopbas 24664 | A basis for the standard t... |
| retop 24665 | The standard topology on t... |
| uniretop 24666 | The underlying set of the ... |
| retopon 24667 | The standard topology on t... |
| retps 24668 | The standard topological s... |
| iooretop 24669 | Open intervals are open se... |
| icccld 24670 | Closed intervals are close... |
| icopnfcld 24671 | Right-unbounded closed int... |
| iocmnfcld 24672 | Left-unbounded closed inte... |
| qdensere 24673 | ` QQ ` is dense in the sta... |
| cnmetdval 24674 | Value of the distance func... |
| cnmet 24675 | The absolute value metric ... |
| cnxmet 24676 | The absolute value metric ... |
| cnbl0 24677 | Two ways to write the open... |
| cnblcld 24678 | Two ways to write the clos... |
| cnfldms 24679 | The complex number field i... |
| cnfldxms 24680 | The complex number field i... |
| cnfldtps 24681 | The complex number field i... |
| cnfldnm 24682 | The norm of the field of c... |
| cnngp 24683 | The complex numbers form a... |
| cnnrg 24684 | The complex numbers form a... |
| cnfldtopn 24685 | The topology of the comple... |
| cnfldtopon 24686 | The topology of the comple... |
| cnfldtop 24687 | The topology of the comple... |
| cnfldhaus 24688 | The topology of the comple... |
| unicntop 24689 | The underlying set of the ... |
| cnopn 24690 | The set of complex numbers... |
| zringnrg 24691 | The ring of integers is a ... |
| remetdval 24692 | Value of the distance func... |
| remet 24693 | The absolute value metric ... |
| rexmet 24694 | The absolute value metric ... |
| bl2ioo 24695 | A ball in terms of an open... |
| ioo2bl 24696 | An open interval of reals ... |
| ioo2blex 24697 | An open interval of reals ... |
| blssioo 24698 | The balls of the standard ... |
| tgioo 24699 | The topology generated by ... |
| qdensere2 24700 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24701 | An open ball in the comple... |
| rehaus 24702 | The standard topology on t... |
| tgqioo 24703 | The topology generated by ... |
| re2ndc 24704 | The standard topology on t... |
| resubmet 24705 | The subspace topology indu... |
| tgioo2 24706 | The standard topology on t... |
| rerest 24707 | The subspace topology indu... |
| tgioo3 24708 | The standard topology on t... |
| xrtgioo 24709 | The topology on the extend... |
| xrrest 24710 | The subspace topology indu... |
| xrrest2 24711 | The subspace topology indu... |
| xrsxmet 24712 | The metric on the extended... |
| xrsdsre 24713 | The metric on the extended... |
| xrsblre 24714 | Any ball of the metric of ... |
| xrsmopn 24715 | The metric on the extended... |
| zcld 24716 | The integers are a closed ... |
| recld2 24717 | The real numbers are a clo... |
| zcld2 24718 | The integers are a closed ... |
| zdis 24719 | The integers are a discret... |
| sszcld 24720 | Every subset of the intege... |
| reperflem 24721 | A subset of the real numbe... |
| reperf 24722 | The real numbers are a per... |
| cnperf 24723 | The complex numbers are a ... |
| iccntr 24724 | The interior of a closed i... |
| icccmplem1 24725 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24726 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24727 | Lemma for ~ icccmp . (Con... |
| icccmp 24728 | A closed interval in ` RR ... |
| reconnlem1 24729 | Lemma for ~ reconn . Conn... |
| reconnlem2 24730 | Lemma for ~ reconn . (Con... |
| reconn 24731 | A subset of the reals is c... |
| retopconn 24732 | Corollary of ~ reconn . T... |
| iccconn 24733 | A closed interval is conne... |
| opnreen 24734 | Every nonempty open set is... |
| rectbntr0 24735 | A countable subset of the ... |
| xrge0gsumle 24736 | A finite sum in the nonneg... |
| xrge0tsms 24737 | Any finite or infinite sum... |
| xrge0tsms2 24738 | Any finite or infinite sum... |
| metdcnlem 24739 | The metric function of a m... |
| xmetdcn2 24740 | The metric function of an ... |
| xmetdcn 24741 | The metric function of an ... |
| metdcn2 24742 | The metric function of a m... |
| metdcn 24743 | The metric function of a m... |
| msdcn 24744 | The metric function of a m... |
| cnmpt1ds 24745 | Continuity of the metric f... |
| cnmpt2ds 24746 | Continuity of the metric f... |
| nmcn 24747 | The norm of a normed group... |
| ngnmcncn 24748 | The norm of a normed group... |
| abscn 24749 | The absolute value functio... |
| metdsval 24750 | Value of the "distance to ... |
| metdsf 24751 | The distance from a point ... |
| metdsge 24752 | The distance from the poin... |
| metds0 24753 | If a point is in a set, it... |
| metdstri 24754 | A generalization of the tr... |
| metdsle 24755 | The distance from a point ... |
| metdsre 24756 | The distance from a point ... |
| metdseq0 24757 | The distance from a point ... |
| metdscnlem 24758 | Lemma for ~ metdscn . (Co... |
| metdscn 24759 | The function ` F ` which g... |
| metdscn2 24760 | The function ` F ` which g... |
| metnrmlem1a 24761 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24762 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24763 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24764 | Lemma for ~ metnrm . (Con... |
| metnrm 24765 | A metric space is normal. ... |
| metreg 24766 | A metric space is regular.... |
| addcnlem 24767 | Lemma for ~ addcn , ~ subc... |
| addcn 24768 | Complex number addition is... |
| subcn 24769 | Complex number subtraction... |
| mulcn 24770 | Complex number multiplicat... |
| divcnOLD 24771 | Obsolete version of ~ divc... |
| mpomulcn 24772 | Complex number multiplicat... |
| divcn 24773 | Complex number division is... |
| cnfldtgp 24774 | The complex numbers form a... |
| fsumcn 24775 | A finite sum of functions ... |
| fsum2cn 24776 | Version of ~ fsumcn for tw... |
| expcn 24777 | The power function on comp... |
| divccn 24778 | Division by a nonzero cons... |
| expcnOLD 24779 | Obsolete version of ~ expc... |
| divccnOLD 24780 | Obsolete version of ~ divc... |
| sqcn 24781 | The square function on com... |
| iitopon 24786 | The unit interval is a top... |
| iitop 24787 | The unit interval is a top... |
| iiuni 24788 | The base set of the unit i... |
| dfii2 24789 | Alternate definition of th... |
| dfii3 24790 | Alternate definition of th... |
| dfii4 24791 | Alternate definition of th... |
| dfii5 24792 | The unit interval expresse... |
| iicmp 24793 | The unit interval is compa... |
| iiconn 24794 | The unit interval is conne... |
| cncfval 24795 | The value of the continuou... |
| elcncf 24796 | Membership in the set of c... |
| elcncf2 24797 | Version of ~ elcncf with a... |
| cncfrss 24798 | Reverse closure of the con... |
| cncfrss2 24799 | Reverse closure of the con... |
| cncff 24800 | A continuous complex funct... |
| cncfi 24801 | Defining property of a con... |
| elcncf1di 24802 | Membership in the set of c... |
| elcncf1ii 24803 | Membership in the set of c... |
| rescncf 24804 | A continuous complex funct... |
| cncfcdm 24805 | Change the codomain of a c... |
| cncfss 24806 | The set of continuous func... |
| climcncf 24807 | Image of a limit under a c... |
| abscncf 24808 | Absolute value is continuo... |
| recncf 24809 | Real part is continuous. ... |
| imcncf 24810 | Imaginary part is continuo... |
| cjcncf 24811 | Complex conjugate is conti... |
| mulc1cncf 24812 | Multiplication by a consta... |
| divccncf 24813 | Division by a constant is ... |
| cncfco 24814 | The composition of two con... |
| cncfcompt2 24815 | Composition of continuous ... |
| cncfmet 24816 | Relate complex function co... |
| cncfcn 24817 | Relate complex function co... |
| cncfcn1 24818 | Relate complex function co... |
| cncfmptc 24819 | A constant function is a c... |
| cncfmptid 24820 | The identity function is a... |
| cncfmpt1f 24821 | Composition of continuous ... |
| cncfmpt2f 24822 | Composition of continuous ... |
| cncfmpt2ss 24823 | Composition of continuous ... |
| addccncf 24824 | Adding a constant is a con... |
| idcncf 24825 | The identity function is a... |
| sub1cncf 24826 | Subtracting a constant is ... |
| sub2cncf 24827 | Subtraction from a constan... |
| cdivcncf 24828 | Division with a constant n... |
| negcncf 24829 | The negative function is c... |
| negcncfOLD 24830 | Obsolete version of ~ negc... |
| negfcncf 24831 | The negative of a continuo... |
| abscncfALT 24832 | Absolute value is continuo... |
| cncfcnvcn 24833 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24834 | The power function on comp... |
| cnmptre 24835 | Lemma for ~ iirevcn and re... |
| cnmpopc 24836 | Piecewise definition of a ... |
| iirev 24837 | Reverse the unit interval.... |
| iirevcn 24838 | The reversion function is ... |
| iihalf1 24839 | Map the first half of ` II... |
| iihalf1cn 24840 | The first half function is... |
| iihalf1cnOLD 24841 | Obsolete version of ~ iiha... |
| iihalf2 24842 | Map the second half of ` I... |
| iihalf2cn 24843 | The second half function i... |
| iihalf2cnOLD 24844 | Obsolete version of ~ iiha... |
| elii1 24845 | Divide the unit interval i... |
| elii2 24846 | Divide the unit interval i... |
| iimulcl 24847 | The unit interval is close... |
| iimulcn 24848 | Multiplication is a contin... |
| iimulcnOLD 24849 | Obsolete version of ~ iimu... |
| icoopnst 24850 | A half-open interval start... |
| iocopnst 24851 | A half-open interval endin... |
| icchmeo 24852 | The natural bijection from... |
| icchmeoOLD 24853 | Obsolete version of ~ icch... |
| icopnfcnv 24854 | Define a bijection from ` ... |
| icopnfhmeo 24855 | The defined bijection from... |
| iccpnfcnv 24856 | Define a bijection from ` ... |
| iccpnfhmeo 24857 | The defined bijection from... |
| xrhmeo 24858 | The bijection from ` [ -u ... |
| xrhmph 24859 | The extended reals are hom... |
| xrcmp 24860 | The topology of the extend... |
| xrconn 24861 | The topology of the extend... |
| icccvx 24862 | A linear combination of tw... |
| oprpiece1res1 24863 | Restriction to the first p... |
| oprpiece1res2 24864 | Restriction to the second ... |
| cnrehmeo 24865 | The canonical bijection fr... |
| cnrehmeoOLD 24866 | Obsolete version of ~ cnre... |
| cnheiborlem 24867 | Lemma for ~ cnheibor . (C... |
| cnheibor 24868 | Heine-Borel theorem for co... |
| cnllycmp 24869 | The topology on the comple... |
| rellycmp 24870 | The topology on the reals ... |
| bndth 24871 | The Boundedness Theorem. ... |
| evth 24872 | The Extreme Value Theorem.... |
| evth2 24873 | The Extreme Value Theorem,... |
| lebnumlem1 24874 | Lemma for ~ lebnum . The ... |
| lebnumlem2 24875 | Lemma for ~ lebnum . As a... |
| lebnumlem3 24876 | Lemma for ~ lebnum . By t... |
| lebnum 24877 | The Lebesgue number lemma,... |
| xlebnum 24878 | Generalize ~ lebnum to ext... |
| lebnumii 24879 | Specialize the Lebesgue nu... |
| ishtpy 24885 | Membership in the class of... |
| htpycn 24886 | A homotopy is a continuous... |
| htpyi 24887 | A homotopy evaluated at it... |
| ishtpyd 24888 | Deduction for membership i... |
| htpycom 24889 | Given a homotopy from ` F ... |
| htpyid 24890 | A homotopy from a function... |
| htpyco1 24891 | Compose a homotopy with a ... |
| htpyco2 24892 | Compose a homotopy with a ... |
| htpycc 24893 | Concatenate two homotopies... |
| isphtpy 24894 | Membership in the class of... |
| phtpyhtpy 24895 | A path homotopy is a homot... |
| phtpycn 24896 | A path homotopy is a conti... |
| phtpyi 24897 | Membership in the class of... |
| phtpy01 24898 | Two path-homotopic paths h... |
| isphtpyd 24899 | Deduction for membership i... |
| isphtpy2d 24900 | Deduction for membership i... |
| phtpycom 24901 | Given a homotopy from ` F ... |
| phtpyid 24902 | A homotopy from a path to ... |
| phtpyco2 24903 | Compose a path homotopy wi... |
| phtpycc 24904 | Concatenate two path homot... |
| phtpcrel 24906 | The path homotopy relation... |
| isphtpc 24907 | The relation "is path homo... |
| phtpcer 24908 | Path homotopy is an equiva... |
| phtpc01 24909 | Path homotopic paths have ... |
| reparphti 24910 | Lemma for ~ reparpht . (C... |
| reparphtiOLD 24911 | Obsolete version of ~ repa... |
| reparpht 24912 | Reparametrization lemma. ... |
| phtpcco2 24913 | Compose a path homotopy wi... |
| pcofval 24924 | The value of the path conc... |
| pcoval 24925 | The concatenation of two p... |
| pcovalg 24926 | Evaluate the concatenation... |
| pcoval1 24927 | Evaluate the concatenation... |
| pco0 24928 | The starting point of a pa... |
| pco1 24929 | The ending point of a path... |
| pcoval2 24930 | Evaluate the concatenation... |
| pcocn 24931 | The concatenation of two p... |
| copco 24932 | The composition of a conca... |
| pcohtpylem 24933 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 24934 | Homotopy invariance of pat... |
| pcoptcl 24935 | A constant function is a p... |
| pcopt 24936 | Concatenation with a point... |
| pcopt2 24937 | Concatenation with a point... |
| pcoass 24938 | Order of concatenation doe... |
| pcorevcl 24939 | Closure for a reversed pat... |
| pcorevlem 24940 | Lemma for ~ pcorev . Prov... |
| pcorev 24941 | Concatenation with the rev... |
| pcorev2 24942 | Concatenation with the rev... |
| pcophtb 24943 | The path homotopy equivale... |
| om1val 24944 | The definition of the loop... |
| om1bas 24945 | The base set of the loop s... |
| om1elbas 24946 | Elementhood in the base se... |
| om1addcl 24947 | Closure of the group opera... |
| om1plusg 24948 | The group operation (which... |
| om1tset 24949 | The topology of the loop s... |
| om1opn 24950 | The topology of the loop s... |
| pi1val 24951 | The definition of the fund... |
| pi1bas 24952 | The base set of the fundam... |
| pi1blem 24953 | Lemma for ~ pi1buni . (Co... |
| pi1buni 24954 | Another way to write the l... |
| pi1bas2 24955 | The base set of the fundam... |
| pi1eluni 24956 | Elementhood in the base se... |
| pi1bas3 24957 | The base set of the fundam... |
| pi1cpbl 24958 | The group operation, loop ... |
| elpi1 24959 | The elements of the fundam... |
| elpi1i 24960 | The elements of the fundam... |
| pi1addf 24961 | The group operation of ` p... |
| pi1addval 24962 | The concatenation of two p... |
| pi1grplem 24963 | Lemma for ~ pi1grp . (Con... |
| pi1grp 24964 | The fundamental group is a... |
| pi1id 24965 | The identity element of th... |
| pi1inv 24966 | An inverse in the fundamen... |
| pi1xfrf 24967 | Functionality of the loop ... |
| pi1xfrval 24968 | The value of the loop tran... |
| pi1xfr 24969 | Given a path ` F ` and its... |
| pi1xfrcnvlem 24970 | Given a path ` F ` between... |
| pi1xfrcnv 24971 | Given a path ` F ` between... |
| pi1xfrgim 24972 | The mapping ` G ` between ... |
| pi1cof 24973 | Functionality of the loop ... |
| pi1coval 24974 | The value of the loop tran... |
| pi1coghm 24975 | The mapping ` G ` between ... |
| isclm 24978 | A subcomplex module is a l... |
| clmsca 24979 | The ring of scalars ` F ` ... |
| clmsubrg 24980 | The base set of the ring o... |
| clmlmod 24981 | A subcomplex module is a l... |
| clmgrp 24982 | A subcomplex module is an ... |
| clmabl 24983 | A subcomplex module is an ... |
| clmring 24984 | The scalar ring of a subco... |
| clmfgrp 24985 | The scalar ring of a subco... |
| clm0 24986 | The zero of the scalar rin... |
| clm1 24987 | The identity of the scalar... |
| clmadd 24988 | The addition of the scalar... |
| clmmul 24989 | The multiplication of the ... |
| clmcj 24990 | The conjugation of the sca... |
| isclmi 24991 | Reverse direction of ~ isc... |
| clmzss 24992 | The scalar ring of a subco... |
| clmsscn 24993 | The scalar ring of a subco... |
| clmsub 24994 | Subtraction in the scalar ... |
| clmneg 24995 | Negation in the scalar rin... |
| clmneg1 24996 | Minus one is in the scalar... |
| clmabs 24997 | Norm in the scalar ring of... |
| clmacl 24998 | Closure of ring addition f... |
| clmmcl 24999 | Closure of ring multiplica... |
| clmsubcl 25000 | Closure of ring subtractio... |
| lmhmclm 25001 | The domain of a linear ope... |
| clmvscl 25002 | Closure of scalar product ... |
| clmvsass 25003 | Associative law for scalar... |
| clmvscom 25004 | Commutative law for the sc... |
| clmvsdir 25005 | Distributive law for scala... |
| clmvsdi 25006 | Distributive law for scala... |
| clmvs1 25007 | Scalar product with ring u... |
| clmvs2 25008 | A vector plus itself is tw... |
| clm0vs 25009 | Zero times a vector is the... |
| clmopfne 25010 | The (functionalized) opera... |
| isclmp 25011 | The predicate "is a subcom... |
| isclmi0 25012 | Properties that determine ... |
| clmvneg1 25013 | Minus 1 times a vector is ... |
| clmvsneg 25014 | Multiplication of a vector... |
| clmmulg 25015 | The group multiple functio... |
| clmsubdir 25016 | Scalar multiplication dist... |
| clmpm1dir 25017 | Subtractive distributive l... |
| clmnegneg 25018 | Double negative of a vecto... |
| clmnegsubdi2 25019 | Distribution of negative o... |
| clmsub4 25020 | Rearrangement of 4 terms i... |
| clmvsrinv 25021 | A vector minus itself. (C... |
| clmvslinv 25022 | Minus a vector plus itself... |
| clmvsubval 25023 | Value of vector subtractio... |
| clmvsubval2 25024 | Value of vector subtractio... |
| clmvz 25025 | Two ways to express the ne... |
| zlmclm 25026 | The ` ZZ ` -module operati... |
| clmzlmvsca 25027 | The scalar product of a su... |
| nmoleub2lem 25028 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25029 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25030 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25031 | The operator norm is the s... |
| nmoleub2b 25032 | The operator norm is the s... |
| nmoleub3 25033 | The operator norm is the s... |
| nmhmcn 25034 | A linear operator over a n... |
| cmodscexp 25035 | The powers of ` _i ` belon... |
| cmodscmulexp 25036 | The scalar product of a ve... |
| cvslvec 25039 | A subcomplex vector space ... |
| cvsclm 25040 | A subcomplex vector space ... |
| iscvs 25041 | A subcomplex vector space ... |
| iscvsp 25042 | The predicate "is a subcom... |
| iscvsi 25043 | Properties that determine ... |
| cvsi 25044 | The properties of a subcom... |
| cvsunit 25045 | Unit group of the scalar r... |
| cvsdiv 25046 | Division of the scalar rin... |
| cvsdivcl 25047 | The scalar field of a subc... |
| cvsmuleqdivd 25048 | An equality involving rati... |
| cvsdiveqd 25049 | An equality involving rati... |
| cnlmodlem1 25050 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25051 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25052 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25053 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25054 | The set of complex numbers... |
| cnstrcvs 25055 | The set of complex numbers... |
| cnrbas 25056 | The set of complex numbers... |
| cnrlmod 25057 | The complex left module of... |
| cnrlvec 25058 | The complex left module of... |
| cncvs 25059 | The complex left module of... |
| recvs 25060 | The field of the real numb... |
| recvsOLD 25061 | Obsolete version of ~ recv... |
| qcvs 25062 | The field of rational numb... |
| zclmncvs 25063 | The ring of integers as le... |
| isncvsngp 25064 | A normed subcomplex vector... |
| isncvsngpd 25065 | Properties that determine ... |
| ncvsi 25066 | The properties of a normed... |
| ncvsprp 25067 | Proportionality property o... |
| ncvsge0 25068 | The norm of a scalar produ... |
| ncvsm1 25069 | The norm of the opposite o... |
| ncvsdif 25070 | The norm of the difference... |
| ncvspi 25071 | The norm of a vector plus ... |
| ncvs1 25072 | From any nonzero vector of... |
| cnrnvc 25073 | The module of complex numb... |
| cnncvs 25074 | The module of complex numb... |
| cnnm 25075 | The norm of the normed sub... |
| ncvspds 25076 | Value of the distance func... |
| cnindmet 25077 | The metric induced on the ... |
| cnncvsaddassdemo 25078 | Derive the associative law... |
| cnncvsmulassdemo 25079 | Derive the associative law... |
| cnncvsabsnegdemo 25080 | Derive the absolute value ... |
| iscph 25085 | A subcomplex pre-Hilbert s... |
| cphphl 25086 | A subcomplex pre-Hilbert s... |
| cphnlm 25087 | A subcomplex pre-Hilbert s... |
| cphngp 25088 | A subcomplex pre-Hilbert s... |
| cphlmod 25089 | A subcomplex pre-Hilbert s... |
| cphlvec 25090 | A subcomplex pre-Hilbert s... |
| cphnvc 25091 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25092 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25093 | Lemma for ~ cphreccl . (C... |
| cphsca 25094 | A subcomplex pre-Hilbert s... |
| cphsubrg 25095 | The scalar field of a subc... |
| cphreccl 25096 | The scalar field of a subc... |
| cphdivcl 25097 | The scalar field of a subc... |
| cphcjcl 25098 | The scalar field of a subc... |
| cphsqrtcl 25099 | The scalar field of a subc... |
| cphabscl 25100 | The scalar field of a subc... |
| cphsqrtcl2 25101 | The scalar field of a subc... |
| cphsqrtcl3 25102 | If the scalar field of a s... |
| cphqss 25103 | The scalar field of a subc... |
| cphclm 25104 | A subcomplex pre-Hilbert s... |
| cphnmvs 25105 | Norm of a scalar product. ... |
| cphipcl 25106 | An inner product is a memb... |
| cphnmfval 25107 | The value of the norm in a... |
| cphnm 25108 | The square of the norm is ... |
| nmsq 25109 | The square of the norm is ... |
| cphnmf 25110 | The norm of a vector is a ... |
| cphnmcl 25111 | The norm of a vector is a ... |
| reipcl 25112 | An inner product of an ele... |
| ipge0 25113 | The inner product in a sub... |
| cphipcj 25114 | Conjugate of an inner prod... |
| cphipipcj 25115 | An inner product times its... |
| cphorthcom 25116 | Orthogonality (meaning inn... |
| cphip0l 25117 | Inner product with a zero ... |
| cphip0r 25118 | Inner product with a zero ... |
| cphipeq0 25119 | The inner product of a vec... |
| cphdir 25120 | Distributive law for inner... |
| cphdi 25121 | Distributive law for inner... |
| cph2di 25122 | Distributive law for inner... |
| cphsubdir 25123 | Distributive law for inner... |
| cphsubdi 25124 | Distributive law for inner... |
| cph2subdi 25125 | Distributive law for inner... |
| cphass 25126 | Associative law for inner ... |
| cphassr 25127 | "Associative" law for seco... |
| cph2ass 25128 | Move scalar multiplication... |
| cphassi 25129 | Associative law for the fi... |
| cphassir 25130 | "Associative" law for the ... |
| cphpyth 25131 | The pythagorean theorem fo... |
| tcphex 25132 | Lemma for ~ tcphbas and si... |
| tcphval 25133 | Define a function to augme... |
| tcphbas 25134 | The base set of a subcompl... |
| tchplusg 25135 | The addition operation of ... |
| tcphsub 25136 | The subtraction operation ... |
| tcphmulr 25137 | The ring operation of a su... |
| tcphsca 25138 | The scalar field of a subc... |
| tcphvsca 25139 | The scalar multiplication ... |
| tcphip 25140 | The inner product of a sub... |
| tcphtopn 25141 | The topology of a subcompl... |
| tcphphl 25142 | Augmentation of a subcompl... |
| tchnmfval 25143 | The norm of a subcomplex p... |
| tcphnmval 25144 | The norm of a subcomplex p... |
| cphtcphnm 25145 | The norm of a norm-augment... |
| tcphds 25146 | The distance of a pre-Hilb... |
| phclm 25147 | A pre-Hilbert space whose ... |
| tcphcphlem3 25148 | Lemma for ~ tcphcph : real... |
| ipcau2 25149 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25150 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25151 | Lemma for ~ tcphcph : homo... |
| tcphcph 25152 | The standard definition of... |
| ipcau 25153 | The Cauchy-Schwarz inequal... |
| nmparlem 25154 | Lemma for ~ nmpar . (Cont... |
| nmpar 25155 | A subcomplex pre-Hilbert s... |
| cphipval2 25156 | Value of the inner product... |
| 4cphipval2 25157 | Four times the inner produ... |
| cphipval 25158 | Value of the inner product... |
| ipcnlem2 25159 | The inner product operatio... |
| ipcnlem1 25160 | The inner product operatio... |
| ipcn 25161 | The inner product operatio... |
| cnmpt1ip 25162 | Continuity of inner produc... |
| cnmpt2ip 25163 | Continuity of inner produc... |
| csscld 25164 | A "closed subspace" in a s... |
| clsocv 25165 | The orthogonal complement ... |
| cphsscph 25166 | A subspace of a subcomplex... |
| lmmbr 25173 | Express the binary relatio... |
| lmmbr2 25174 | Express the binary relatio... |
| lmmbr3 25175 | Express the binary relatio... |
| lmmcvg 25176 | Convergence property of a ... |
| lmmbrf 25177 | Express the binary relatio... |
| lmnn 25178 | A condition that implies c... |
| cfilfval 25179 | The set of Cauchy filters ... |
| iscfil 25180 | The property of being a Ca... |
| iscfil2 25181 | The property of being a Ca... |
| cfilfil 25182 | A Cauchy filter is a filte... |
| cfili 25183 | Property of a Cauchy filte... |
| cfil3i 25184 | A Cauchy filter contains b... |
| cfilss 25185 | A filter finer than a Cauc... |
| fgcfil 25186 | The Cauchy filter conditio... |
| fmcfil 25187 | The Cauchy filter conditio... |
| iscfil3 25188 | A filter is Cauchy iff it ... |
| cfilfcls 25189 | Similar to ultrafilters ( ... |
| caufval 25190 | The set of Cauchy sequence... |
| iscau 25191 | Express the property " ` F... |
| iscau2 25192 | Express the property " ` F... |
| iscau3 25193 | Express the Cauchy sequenc... |
| iscau4 25194 | Express the property " ` F... |
| iscauf 25195 | Express the property " ` F... |
| caun0 25196 | A metric with a Cauchy seq... |
| caufpm 25197 | Inclusion of a Cauchy sequ... |
| caucfil 25198 | A Cauchy sequence predicat... |
| iscmet 25199 | The property " ` D ` is a ... |
| cmetcvg 25200 | The convergence of a Cauch... |
| cmetmet 25201 | A complete metric space is... |
| cmetmeti 25202 | A complete metric space is... |
| cmetcaulem 25203 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25204 | The convergence of a Cauch... |
| iscmet3lem3 25205 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25206 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25207 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25208 | The property " ` D ` is a ... |
| iscmet2 25209 | A metric ` D ` is complete... |
| cfilresi 25210 | A Cauchy filter on a metri... |
| cfilres 25211 | Cauchy filter on a metric ... |
| caussi 25212 | Cauchy sequence on a metri... |
| causs 25213 | Cauchy sequence on a metri... |
| equivcfil 25214 | If the metric ` D ` is "st... |
| equivcau 25215 | If the metric ` D ` is "st... |
| lmle 25216 | If the distance from each ... |
| nglmle 25217 | If the norm of each member... |
| lmclim 25218 | Relate a limit on the metr... |
| lmclimf 25219 | Relate a limit on the metr... |
| metelcls 25220 | A point belongs to the clo... |
| metcld 25221 | A subset of a metric space... |
| metcld2 25222 | A subset of a metric space... |
| caubl 25223 | Sufficient condition to en... |
| caublcls 25224 | The convergent point of a ... |
| metcnp4 25225 | Two ways to say a mapping ... |
| metcn4 25226 | Two ways to say a mapping ... |
| iscmet3i 25227 | Properties that determine ... |
| lmcau 25228 | Every convergent sequence ... |
| flimcfil 25229 | Every convergent filter in... |
| metsscmetcld 25230 | A complete subspace of a m... |
| cmetss 25231 | A subspace of a complete m... |
| equivcmet 25232 | If two metrics are strongl... |
| relcmpcmet 25233 | If ` D ` is a metric space... |
| cmpcmet 25234 | A compact metric space is ... |
| cfilucfil3 25235 | Given a metric ` D ` and a... |
| cfilucfil4 25236 | Given a metric ` D ` and a... |
| cncmet 25237 | The set of complex numbers... |
| recmet 25238 | The real numbers are a com... |
| bcthlem1 25239 | Lemma for ~ bcth . Substi... |
| bcthlem2 25240 | Lemma for ~ bcth . The ba... |
| bcthlem3 25241 | Lemma for ~ bcth . The li... |
| bcthlem4 25242 | Lemma for ~ bcth . Given ... |
| bcthlem5 25243 | Lemma for ~ bcth . The pr... |
| bcth 25244 | Baire's Category Theorem. ... |
| bcth2 25245 | Baire's Category Theorem, ... |
| bcth3 25246 | Baire's Category Theorem, ... |
| isbn 25253 | A Banach space is a normed... |
| bnsca 25254 | The scalar field of a Bana... |
| bnnvc 25255 | A Banach space is a normed... |
| bnnlm 25256 | A Banach space is a normed... |
| bnngp 25257 | A Banach space is a normed... |
| bnlmod 25258 | A Banach space is a left m... |
| bncms 25259 | A Banach space is a comple... |
| iscms 25260 | A complete metric space is... |
| cmscmet 25261 | The induced metric on a co... |
| bncmet 25262 | The induced metric on Bana... |
| cmsms 25263 | A complete metric space is... |
| cmspropd 25264 | Property deduction for a c... |
| cmssmscld 25265 | The restriction of a metri... |
| cmsss 25266 | The restriction of a compl... |
| lssbn 25267 | A subspace of a Banach spa... |
| cmetcusp1 25268 | If the uniform set of a co... |
| cmetcusp 25269 | The uniform space generate... |
| cncms 25270 | The field of complex numbe... |
| cnflduss 25271 | The uniform structure of t... |
| cnfldcusp 25272 | The field of complex numbe... |
| resscdrg 25273 | The real numbers are a sub... |
| cncdrg 25274 | The only complete subfield... |
| srabn 25275 | The subring algebra over a... |
| rlmbn 25276 | The ring module over a com... |
| ishl 25277 | The predicate "is a subcom... |
| hlbn 25278 | Every subcomplex Hilbert s... |
| hlcph 25279 | Every subcomplex Hilbert s... |
| hlphl 25280 | Every subcomplex Hilbert s... |
| hlcms 25281 | Every subcomplex Hilbert s... |
| hlprlem 25282 | Lemma for ~ hlpr . (Contr... |
| hlress 25283 | The scalar field of a subc... |
| hlpr 25284 | The scalar field of a subc... |
| ishl2 25285 | A Hilbert space is a compl... |
| cphssphl 25286 | A Banach subspace of a sub... |
| cmslssbn 25287 | A complete linear subspace... |
| cmscsscms 25288 | A closed subspace of a com... |
| bncssbn 25289 | A closed subspace of a Ban... |
| cssbn 25290 | A complete subspace of a n... |
| csschl 25291 | A complete subspace of a c... |
| cmslsschl 25292 | A complete linear subspace... |
| chlcsschl 25293 | A closed subspace of a sub... |
| retopn 25294 | The topology of the real n... |
| recms 25295 | The real numbers form a co... |
| reust 25296 | The Uniform structure of t... |
| recusp 25297 | The real numbers form a co... |
| rrxval 25302 | Value of the generalized E... |
| rrxbase 25303 | The base of the generalize... |
| rrxprds 25304 | Expand the definition of t... |
| rrxip 25305 | The inner product of the g... |
| rrxnm 25306 | The norm of the generalize... |
| rrxcph 25307 | Generalized Euclidean real... |
| rrxds 25308 | The distance over generali... |
| rrxvsca 25309 | The scalar product over ge... |
| rrxplusgvscavalb 25310 | The result of the addition... |
| rrxsca 25311 | The field of real numbers ... |
| rrx0 25312 | The zero ("origin") in a g... |
| rrx0el 25313 | The zero ("origin") in a g... |
| csbren 25314 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25315 | Triangle inequality in R^n... |
| rrxf 25316 | Euclidean vectors as funct... |
| rrxfsupp 25317 | Euclidean vectors are of f... |
| rrxsuppss 25318 | Support of Euclidean vecto... |
| rrxmvallem 25319 | Support of the function us... |
| rrxmval 25320 | The value of the Euclidean... |
| rrxmfval 25321 | The value of the Euclidean... |
| rrxmetlem 25322 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25323 | Euclidean space is a metri... |
| rrxdstprj1 25324 | The distance between two p... |
| rrxbasefi 25325 | The base of the generalize... |
| rrxdsfi 25326 | The distance over generali... |
| rrxmetfi 25327 | Euclidean space is a metri... |
| rrxdsfival 25328 | The value of the Euclidean... |
| ehlval 25329 | Value of the Euclidean spa... |
| ehlbase 25330 | The base of the Euclidean ... |
| ehl0base 25331 | The base of the Euclidean ... |
| ehl0 25332 | The Euclidean space of dim... |
| ehleudis 25333 | The Euclidean distance fun... |
| ehleudisval 25334 | The value of the Euclidean... |
| ehl1eudis 25335 | The Euclidean distance fun... |
| ehl1eudisval 25336 | The value of the Euclidean... |
| ehl2eudis 25337 | The Euclidean distance fun... |
| ehl2eudisval 25338 | The value of the Euclidean... |
| minveclem1 25339 | Lemma for ~ minvec . The ... |
| minveclem4c 25340 | Lemma for ~ minvec . The ... |
| minveclem2 25341 | Lemma for ~ minvec . Any ... |
| minveclem3a 25342 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25343 | Lemma for ~ minvec . The ... |
| minveclem3 25344 | Lemma for ~ minvec . The ... |
| minveclem4a 25345 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25346 | Lemma for ~ minvec . The ... |
| minveclem4 25347 | Lemma for ~ minvec . The ... |
| minveclem5 25348 | Lemma for ~ minvec . Disc... |
| minveclem6 25349 | Lemma for ~ minvec . Any ... |
| minveclem7 25350 | Lemma for ~ minvec . Sinc... |
| minvec 25351 | Minimizing vector theorem,... |
| pjthlem1 25352 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25353 | Lemma for ~ pjth . (Contr... |
| pjth 25354 | Projection Theorem: Any H... |
| pjth2 25355 | Projection Theorem with ab... |
| cldcss 25356 | Corollary of the Projectio... |
| cldcss2 25357 | Corollary of the Projectio... |
| hlhil 25358 | Corollary of the Projectio... |
| addcncf 25359 | The addition of two contin... |
| subcncf 25360 | The addition of two contin... |
| mulcncf 25361 | The multiplication of two ... |
| mulcncfOLD 25362 | Obsolete version of ~ mulc... |
| divcncf 25363 | The quotient of two contin... |
| pmltpclem1 25364 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25365 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25366 | Any function on the reals ... |
| ivthlem1 25367 | Lemma for ~ ivth . The se... |
| ivthlem2 25368 | Lemma for ~ ivth . Show t... |
| ivthlem3 25369 | Lemma for ~ ivth , the int... |
| ivth 25370 | The intermediate value the... |
| ivth2 25371 | The intermediate value the... |
| ivthle 25372 | The intermediate value the... |
| ivthle2 25373 | The intermediate value the... |
| ivthicc 25374 | The interval between any t... |
| evthicc 25375 | Specialization of the Extr... |
| evthicc2 25376 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25377 | A continuous function on a... |
| ovolfcl 25382 | Closure for the interval e... |
| ovolfioo 25383 | Unpack the interval coveri... |
| ovolficc 25384 | Unpack the interval coveri... |
| ovolficcss 25385 | Any (closed) interval cove... |
| ovolfsval 25386 | The value of the interval ... |
| ovolfsf 25387 | Closure for the interval l... |
| ovolsf 25388 | Closure for the partial su... |
| ovolval 25389 | The value of the outer mea... |
| elovolmlem 25390 | Lemma for ~ elovolm and re... |
| elovolm 25391 | Elementhood in the set ` M... |
| elovolmr 25392 | Sufficient condition for e... |
| ovolmge0 25393 | The set ` M ` is composed ... |
| ovolcl 25394 | The volume of a set is an ... |
| ovollb 25395 | The outer volume is a lowe... |
| ovolgelb 25396 | The outer volume is the gr... |
| ovolge0 25397 | The volume of a set is alw... |
| ovolf 25398 | The domain and codomain of... |
| ovollecl 25399 | If an outer volume is boun... |
| ovolsslem 25400 | Lemma for ~ ovolss . (Con... |
| ovolss 25401 | The volume of a set is mon... |
| ovolsscl 25402 | If a set is contained in a... |
| ovolssnul 25403 | A subset of a nullset is n... |
| ovollb2lem 25404 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25405 | It is often more convenien... |
| ovolctb 25406 | The volume of a denumerabl... |
| ovolq 25407 | The rational numbers have ... |
| ovolctb2 25408 | The volume of a countable ... |
| ovol0 25409 | The empty set has 0 outer ... |
| ovolfi 25410 | A finite set has 0 outer L... |
| ovolsn 25411 | A singleton has 0 outer Le... |
| ovolunlem1a 25412 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25413 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25414 | Lemma for ~ ovolun . (Con... |
| ovolun 25415 | The Lebesgue outer measure... |
| ovolunnul 25416 | Adding a nullset does not ... |
| ovolfiniun 25417 | The Lebesgue outer measure... |
| ovoliunlem1 25418 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25419 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25420 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25421 | The Lebesgue outer measure... |
| ovoliun2 25422 | The Lebesgue outer measure... |
| ovoliunnul 25423 | A countable union of nulls... |
| shft2rab 25424 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25425 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25426 | Lemma for ~ ovolshft . (C... |
| ovolshft 25427 | The Lebesgue outer measure... |
| sca2rab 25428 | If ` B ` is a scale of ` A... |
| ovolscalem1 25429 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25430 | Lemma for ~ ovolshft . (C... |
| ovolsca 25431 | The Lebesgue outer measure... |
| ovolicc1 25432 | The measure of a closed in... |
| ovolicc2lem1 25433 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25434 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25435 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25436 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25437 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25438 | The measure of a closed in... |
| ovolicc 25439 | The measure of a closed in... |
| ovolicopnf 25440 | The measure of a right-unb... |
| ovolre 25441 | The measure of the real nu... |
| ismbl 25442 | The predicate " ` A ` is L... |
| ismbl2 25443 | From ~ ovolun , it suffice... |
| volres 25444 | A self-referencing abbrevi... |
| volf 25445 | The domain and codomain of... |
| mblvol 25446 | The volume of a measurable... |
| mblss 25447 | A measurable set is a subs... |
| mblsplit 25448 | The defining property of m... |
| volss 25449 | The Lebesgue measure is mo... |
| cmmbl 25450 | The complement of a measur... |
| nulmbl 25451 | A nullset is measurable. ... |
| nulmbl2 25452 | A set of outer measure zer... |
| unmbl 25453 | A union of measurable sets... |
| shftmbl 25454 | A shift of a measurable se... |
| 0mbl 25455 | The empty set is measurabl... |
| rembl 25456 | The set of all real number... |
| unidmvol 25457 | The union of the Lebesgue ... |
| inmbl 25458 | An intersection of measura... |
| difmbl 25459 | A difference of measurable... |
| finiunmbl 25460 | A finite union of measurab... |
| volun 25461 | The Lebesgue measure funct... |
| volinun 25462 | Addition of non-disjoint s... |
| volfiniun 25463 | The volume of a disjoint f... |
| iundisj 25464 | Rewrite a countable union ... |
| iundisj2 25465 | A disjoint union is disjoi... |
| voliunlem1 25466 | Lemma for ~ voliun . (Con... |
| voliunlem2 25467 | Lemma for ~ voliun . (Con... |
| voliunlem3 25468 | Lemma for ~ voliun . (Con... |
| iunmbl 25469 | The measurable sets are cl... |
| voliun 25470 | The Lebesgue measure funct... |
| volsuplem 25471 | Lemma for ~ volsup . (Con... |
| volsup 25472 | The volume of the limit of... |
| iunmbl2 25473 | The measurable sets are cl... |
| ioombl1lem1 25474 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25475 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25476 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25477 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25478 | An open right-unbounded in... |
| icombl1 25479 | A closed unbounded-above i... |
| icombl 25480 | A closed-below, open-above... |
| ioombl 25481 | An open real interval is m... |
| iccmbl 25482 | A closed real interval is ... |
| iccvolcl 25483 | A closed real interval has... |
| ovolioo 25484 | The measure of an open int... |
| volioo 25485 | The measure of an open int... |
| ioovolcl 25486 | An open real interval has ... |
| ovolfs2 25487 | Alternative expression for... |
| ioorcl2 25488 | An open interval with fini... |
| ioorf 25489 | Define a function from ope... |
| ioorval 25490 | Define a function from ope... |
| ioorinv2 25491 | The function ` F ` is an "... |
| ioorinv 25492 | The function ` F ` is an "... |
| ioorcl 25493 | The function ` F ` does no... |
| uniiccdif 25494 | A union of closed interval... |
| uniioovol 25495 | A disjoint union of open i... |
| uniiccvol 25496 | An almost-disjoint union o... |
| uniioombllem1 25497 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25498 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25499 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25500 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25501 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25502 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25503 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25504 | Lemma for ~ uniioombl . (... |
| uniioombl 25505 | A disjoint union of open i... |
| uniiccmbl 25506 | An almost-disjoint union o... |
| dyadf 25507 | The function ` F ` returns... |
| dyadval 25508 | Value of the dyadic ration... |
| dyadovol 25509 | Volume of a dyadic rationa... |
| dyadss 25510 | Two closed dyadic rational... |
| dyaddisjlem 25511 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25512 | Two closed dyadic rational... |
| dyadmaxlem 25513 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25514 | Any nonempty set of dyadic... |
| dyadmbllem 25515 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25516 | Any union of dyadic ration... |
| opnmbllem 25517 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25518 | All open sets are measurab... |
| opnmblALT 25519 | All open sets are measurab... |
| subopnmbl 25520 | Sets which are open in a m... |
| volsup2 25521 | The volume of ` A ` is the... |
| volcn 25522 | The function formed by res... |
| volivth 25523 | The Intermediate Value The... |
| vitalilem1 25524 | Lemma for ~ vitali . (Con... |
| vitalilem2 25525 | Lemma for ~ vitali . (Con... |
| vitalilem3 25526 | Lemma for ~ vitali . (Con... |
| vitalilem4 25527 | Lemma for ~ vitali . (Con... |
| vitalilem5 25528 | Lemma for ~ vitali . (Con... |
| vitali 25529 | If the reals can be well-o... |
| ismbf1 25540 | The predicate " ` F ` is a... |
| mbff 25541 | A measurable function is a... |
| mbfdm 25542 | The domain of a measurable... |
| mbfconstlem 25543 | Lemma for ~ mbfconst and r... |
| ismbf 25544 | The predicate " ` F ` is a... |
| ismbfcn 25545 | A complex function is meas... |
| mbfima 25546 | Definitional property of a... |
| mbfimaicc 25547 | The preimage of any closed... |
| mbfimasn 25548 | The preimage of a point un... |
| mbfconst 25549 | A constant function is mea... |
| mbf0 25550 | The empty function is meas... |
| mbfid 25551 | The identity function is m... |
| mbfmptcl 25552 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25553 | The domain of a measurable... |
| ismbfcn2 25554 | A complex function is meas... |
| ismbfd 25555 | Deduction to prove measura... |
| ismbf2d 25556 | Deduction to prove measura... |
| mbfeqalem1 25557 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25558 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25559 | If two functions are equal... |
| mbfres 25560 | The restriction of a measu... |
| mbfres2 25561 | Measurability of a piecewi... |
| mbfss 25562 | Change the domain of a mea... |
| mbfmulc2lem 25563 | Multiplication by a consta... |
| mbfmulc2re 25564 | Multiplication by a consta... |
| mbfmax 25565 | The maximum of two functio... |
| mbfneg 25566 | The negative of a measurab... |
| mbfpos 25567 | The positive part of a mea... |
| mbfposr 25568 | Converse to ~ mbfpos . (C... |
| mbfposb 25569 | A function is measurable i... |
| ismbf3d 25570 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25571 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25572 | The preimage of any open s... |
| mbfimaopn2 25573 | The preimage of any set op... |
| cncombf 25574 | The composition of a conti... |
| cnmbf 25575 | A continuous function is m... |
| mbfaddlem 25576 | The sum of two measurable ... |
| mbfadd 25577 | The sum of two measurable ... |
| mbfsub 25578 | The difference of two meas... |
| mbfmulc2 25579 | A complex constant times a... |
| mbfsup 25580 | The supremum of a sequence... |
| mbfinf 25581 | The infimum of a sequence ... |
| mbflimsup 25582 | The limit supremum of a se... |
| mbflimlem 25583 | The pointwise limit of a s... |
| mbflim 25584 | The pointwise limit of a s... |
| 0pval 25587 | The zero function evaluate... |
| 0plef 25588 | Two ways to say that the f... |
| 0pledm 25589 | Adjust the domain of the l... |
| isi1f 25590 | The predicate " ` F ` is a... |
| i1fmbf 25591 | Simple functions are measu... |
| i1ff 25592 | A simple function is a fun... |
| i1frn 25593 | A simple function has fini... |
| i1fima 25594 | Any preimage of a simple f... |
| i1fima2 25595 | Any preimage of a simple f... |
| i1fima2sn 25596 | Preimage of a singleton. ... |
| i1fd 25597 | A simplified set of assump... |
| i1f0rn 25598 | Any simple function takes ... |
| itg1val 25599 | The value of the integral ... |
| itg1val2 25600 | The value of the integral ... |
| itg1cl 25601 | Closure of the integral on... |
| itg1ge0 25602 | Closure of the integral on... |
| i1f0 25603 | The zero function is simpl... |
| itg10 25604 | The zero function has zero... |
| i1f1lem 25605 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25606 | Base case simple functions... |
| itg11 25607 | The integral of an indicat... |
| itg1addlem1 25608 | Decompose a preimage, whic... |
| i1faddlem 25609 | Decompose the preimage of ... |
| i1fmullem 25610 | Decompose the preimage of ... |
| i1fadd 25611 | The sum of two simple func... |
| i1fmul 25612 | The pointwise product of t... |
| itg1addlem2 25613 | Lemma for ~ itg1add . The... |
| itg1addlem3 25614 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25615 | Lemma for ~ itg1add . (Co... |
| itg1addlem4OLD 25616 | Obsolete version of ~ itg1... |
| itg1addlem5 25617 | Lemma for ~ itg1add . (Co... |
| itg1add 25618 | The integral of a sum of s... |
| i1fmulclem 25619 | Decompose the preimage of ... |
| i1fmulc 25620 | A nonnegative constant tim... |
| itg1mulc 25621 | The integral of a constant... |
| i1fres 25622 | The "restriction" of a sim... |
| i1fpos 25623 | The positive part of a sim... |
| i1fposd 25624 | Deduction form of ~ i1fpos... |
| i1fsub 25625 | The difference of two simp... |
| itg1sub 25626 | The integral of a differen... |
| itg10a 25627 | The integral of a simple f... |
| itg1ge0a 25628 | The integral of an almost ... |
| itg1lea 25629 | Approximate version of ~ i... |
| itg1le 25630 | If one simple function dom... |
| itg1climres 25631 | Restricting the simple fun... |
| mbfi1fseqlem1 25632 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25633 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25634 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25635 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25636 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25637 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25638 | A characterization of meas... |
| mbfi1flimlem 25639 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25640 | Any real measurable functi... |
| mbfmullem2 25641 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25642 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25643 | The product of two measura... |
| itg2lcl 25644 | The set of lower sums is a... |
| itg2val 25645 | Value of the integral on n... |
| itg2l 25646 | Elementhood in the set ` L... |
| itg2lr 25647 | Sufficient condition for e... |
| xrge0f 25648 | A real function is a nonne... |
| itg2cl 25649 | The integral of a nonnegat... |
| itg2ub 25650 | The integral of a nonnegat... |
| itg2leub 25651 | Any upper bound on the int... |
| itg2ge0 25652 | The integral of a nonnegat... |
| itg2itg1 25653 | The integral of a nonnegat... |
| itg20 25654 | The integral of the zero f... |
| itg2lecl 25655 | If an ` S.2 ` integral is ... |
| itg2le 25656 | If one function dominates ... |
| itg2const 25657 | Integral of a constant fun... |
| itg2const2 25658 | When the base set of a con... |
| itg2seq 25659 | Definitional property of t... |
| itg2uba 25660 | Approximate version of ~ i... |
| itg2lea 25661 | Approximate version of ~ i... |
| itg2eqa 25662 | Approximate equality of in... |
| itg2mulclem 25663 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25664 | The integral of a nonnegat... |
| itg2splitlem 25665 | Lemma for ~ itg2split . (... |
| itg2split 25666 | The ` S.2 ` integral split... |
| itg2monolem1 25667 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25668 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25669 | Lemma for ~ itg2mono . (C... |
| itg2mono 25670 | The Monotone Convergence T... |
| itg2i1fseqle 25671 | Subject to the conditions ... |
| itg2i1fseq 25672 | Subject to the conditions ... |
| itg2i1fseq2 25673 | In an extension to the res... |
| itg2i1fseq3 25674 | Special case of ~ itg2i1fs... |
| itg2addlem 25675 | Lemma for ~ itg2add . (Co... |
| itg2add 25676 | The ` S.2 ` integral is li... |
| itg2gt0 25677 | If the function ` F ` is s... |
| itg2cnlem1 25678 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25679 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25680 | A sort of absolute continu... |
| ibllem 25681 | Conditioned equality theor... |
| isibl 25682 | The predicate " ` F ` is i... |
| isibl2 25683 | The predicate " ` F ` is i... |
| iblmbf 25684 | An integrable function is ... |
| iblitg 25685 | If a function is integrabl... |
| dfitg 25686 | Evaluate the class substit... |
| itgex 25687 | An integral is a set. (Co... |
| itgeq1f 25688 | Equality theorem for an in... |
| itgeq1 25689 | Equality theorem for an in... |
| nfitg1 25690 | Bound-variable hypothesis ... |
| nfitg 25691 | Bound-variable hypothesis ... |
| cbvitg 25692 | Change bound variable in a... |
| cbvitgv 25693 | Change bound variable in a... |
| itgeq2 25694 | Equality theorem for an in... |
| itgresr 25695 | The domain of an integral ... |
| itg0 25696 | The integral of anything o... |
| itgz 25697 | The integral of zero on an... |
| itgeq2dv 25698 | Equality theorem for an in... |
| itgmpt 25699 | Change bound variable in a... |
| itgcl 25700 | The integral of an integra... |
| itgvallem 25701 | Substitution lemma. (Cont... |
| itgvallem3 25702 | Lemma for ~ itgposval and ... |
| ibl0 25703 | The zero function is integ... |
| iblcnlem1 25704 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25705 | Expand out the universal q... |
| itgcnlem 25706 | Expand out the sum in ~ df... |
| iblrelem 25707 | Integrability of a real fu... |
| iblposlem 25708 | Lemma for ~ iblpos . (Con... |
| iblpos 25709 | Integrability of a nonnega... |
| iblre 25710 | Integrability of a real fu... |
| itgrevallem1 25711 | Lemma for ~ itgposval and ... |
| itgposval 25712 | The integral of a nonnegat... |
| itgreval 25713 | Decompose the integral of ... |
| itgrecl 25714 | Real closure of an integra... |
| iblcn 25715 | Integrability of a complex... |
| itgcnval 25716 | Decompose the integral of ... |
| itgre 25717 | Real part of an integral. ... |
| itgim 25718 | Imaginary part of an integ... |
| iblneg 25719 | The negative of an integra... |
| itgneg 25720 | Negation of an integral. ... |
| iblss 25721 | A subset of an integrable ... |
| iblss2 25722 | Change the domain of an in... |
| itgitg2 25723 | Transfer an integral using... |
| i1fibl 25724 | A simple function is integ... |
| itgitg1 25725 | Transfer an integral using... |
| itgle 25726 | Monotonicity of an integra... |
| itgge0 25727 | The integral of a positive... |
| itgss 25728 | Expand the set of an integ... |
| itgss2 25729 | Expand the set of an integ... |
| itgeqa 25730 | Approximate equality of in... |
| itgss3 25731 | Expand the set of an integ... |
| itgioo 25732 | Equality of integrals on o... |
| itgless 25733 | Expand the integral of a n... |
| iblconst 25734 | A constant function is int... |
| itgconst 25735 | Integral of a constant fun... |
| ibladdlem 25736 | Lemma for ~ ibladd . (Con... |
| ibladd 25737 | Add two integrals over the... |
| iblsub 25738 | Subtract two integrals ove... |
| itgaddlem1 25739 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25740 | Lemma for ~ itgadd . (Con... |
| itgadd 25741 | Add two integrals over the... |
| itgsub 25742 | Subtract two integrals ove... |
| itgfsum 25743 | Take a finite sum of integ... |
| iblabslem 25744 | Lemma for ~ iblabs . (Con... |
| iblabs 25745 | The absolute value of an i... |
| iblabsr 25746 | A measurable function is i... |
| iblmulc2 25747 | Multiply an integral by a ... |
| itgmulc2lem1 25748 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25749 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25750 | Multiply an integral by a ... |
| itgabs 25751 | The triangle inequality fo... |
| itgsplit 25752 | The ` S. ` integral splits... |
| itgspliticc 25753 | The ` S. ` integral splits... |
| itgsplitioo 25754 | The ` S. ` integral splits... |
| bddmulibl 25755 | A bounded function times a... |
| bddibl 25756 | A bounded function is inte... |
| cniccibl 25757 | A continuous function on a... |
| bddiblnc 25758 | Choice-free proof of ~ bdd... |
| cnicciblnc 25759 | Choice-free proof of ~ cni... |
| itggt0 25760 | The integral of a strictly... |
| itgcn 25761 | Transfer ~ itg2cn to the f... |
| ditgeq1 25764 | Equality theorem for the d... |
| ditgeq2 25765 | Equality theorem for the d... |
| ditgeq3 25766 | Equality theorem for the d... |
| ditgeq3dv 25767 | Equality theorem for the d... |
| ditgex 25768 | A directed integral is a s... |
| ditg0 25769 | Value of the directed inte... |
| cbvditg 25770 | Change bound variable in a... |
| cbvditgv 25771 | Change bound variable in a... |
| ditgpos 25772 | Value of the directed inte... |
| ditgneg 25773 | Value of the directed inte... |
| ditgcl 25774 | Closure of a directed inte... |
| ditgswap 25775 | Reverse a directed integra... |
| ditgsplitlem 25776 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25777 | This theorem is the raison... |
| reldv 25786 | The derivative function is... |
| limcvallem 25787 | Lemma for ~ ellimc . (Con... |
| limcfval 25788 | Value and set bounds on th... |
| ellimc 25789 | Value of the limit predica... |
| limcrcl 25790 | Reverse closure for the li... |
| limccl 25791 | Closure of the limit opera... |
| limcdif 25792 | It suffices to consider fu... |
| ellimc2 25793 | Write the definition of a ... |
| limcnlp 25794 | If ` B ` is not a limit po... |
| ellimc3 25795 | Write the epsilon-delta de... |
| limcflflem 25796 | Lemma for ~ limcflf . (Co... |
| limcflf 25797 | The limit operator can be ... |
| limcmo 25798 | If ` B ` is a limit point ... |
| limcmpt 25799 | Express the limit operator... |
| limcmpt2 25800 | Express the limit operator... |
| limcresi 25801 | Any limit of ` F ` is also... |
| limcres 25802 | If ` B ` is an interior po... |
| cnplimc 25803 | A function is continuous a... |
| cnlimc 25804 | ` F ` is a continuous func... |
| cnlimci 25805 | If ` F ` is a continuous f... |
| cnmptlimc 25806 | If ` F ` is a continuous f... |
| limccnp 25807 | If the limit of ` F ` at `... |
| limccnp2 25808 | The image of a convergent ... |
| limcco 25809 | Composition of two limits.... |
| limciun 25810 | A point is a limit of ` F ... |
| limcun 25811 | A point is a limit of ` F ... |
| dvlem 25812 | Closure for a difference q... |
| dvfval 25813 | Value and set bounds on th... |
| eldv 25814 | The differentiable predica... |
| dvcl 25815 | The derivative function ta... |
| dvbssntr 25816 | The set of differentiable ... |
| dvbss 25817 | The set of differentiable ... |
| dvbsss 25818 | The set of differentiable ... |
| perfdvf 25819 | The derivative is a functi... |
| recnprss 25820 | Both ` RR ` and ` CC ` are... |
| recnperf 25821 | Both ` RR ` and ` CC ` are... |
| dvfg 25822 | Explicitly write out the f... |
| dvf 25823 | The derivative is a functi... |
| dvfcn 25824 | The derivative is a functi... |
| dvreslem 25825 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25826 | Lemma for ~ dvres2 . (Con... |
| dvres 25827 | Restriction of a derivativ... |
| dvres2 25828 | Restriction of the base se... |
| dvres3 25829 | Restriction of a complex d... |
| dvres3a 25830 | Restriction of a complex d... |
| dvidlem 25831 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25832 | Derivative of a function r... |
| dvconst 25833 | Derivative of a constant f... |
| dvid 25834 | Derivative of the identity... |
| dvcnp 25835 | The difference quotient is... |
| dvcnp2 25836 | A function is continuous a... |
| dvcnp2OLD 25837 | Obsolete version of ~ dvcn... |
| dvcn 25838 | A differentiable function ... |
| dvnfval 25839 | Value of the iterated deri... |
| dvnff 25840 | The iterated derivative is... |
| dvn0 25841 | Zero times iterated deriva... |
| dvnp1 25842 | Successor iterated derivat... |
| dvn1 25843 | One times iterated derivat... |
| dvnf 25844 | The N-times derivative is ... |
| dvnbss 25845 | The set of N-times differe... |
| dvnadd 25846 | The ` N ` -th derivative o... |
| dvn2bss 25847 | An N-times differentiable ... |
| dvnres 25848 | Multiple derivative versio... |
| cpnfval 25849 | Condition for n-times cont... |
| fncpn 25850 | The ` C^n ` object is a fu... |
| elcpn 25851 | Condition for n-times cont... |
| cpnord 25852 | ` C^n ` conditions are ord... |
| cpncn 25853 | A ` C^n ` function is cont... |
| cpnres 25854 | The restriction of a ` C^n... |
| dvaddbr 25855 | The sum rule for derivativ... |
| dvmulbr 25856 | The product rule for deriv... |
| dvmulbrOLD 25857 | Obsolete version of ~ dvmu... |
| dvadd 25858 | The sum rule for derivativ... |
| dvmul 25859 | The product rule for deriv... |
| dvaddf 25860 | The sum rule for everywher... |
| dvmulf 25861 | The product rule for every... |
| dvcmul 25862 | The product rule when one ... |
| dvcmulf 25863 | The product rule when one ... |
| dvcobr 25864 | The chain rule for derivat... |
| dvcobrOLD 25865 | Obsolete version of ~ dvco... |
| dvco 25866 | The chain rule for derivat... |
| dvcof 25867 | The chain rule for everywh... |
| dvcjbr 25868 | The derivative of the conj... |
| dvcj 25869 | The derivative of the conj... |
| dvfre 25870 | The derivative of a real f... |
| dvnfre 25871 | The ` N ` -th derivative o... |
| dvexp 25872 | Derivative of a power func... |
| dvexp2 25873 | Derivative of an exponenti... |
| dvrec 25874 | Derivative of the reciproc... |
| dvmptres3 25875 | Function-builder for deriv... |
| dvmptid 25876 | Function-builder for deriv... |
| dvmptc 25877 | Function-builder for deriv... |
| dvmptcl 25878 | Closure lemma for ~ dvmptc... |
| dvmptadd 25879 | Function-builder for deriv... |
| dvmptmul 25880 | Function-builder for deriv... |
| dvmptres2 25881 | Function-builder for deriv... |
| dvmptres 25882 | Function-builder for deriv... |
| dvmptcmul 25883 | Function-builder for deriv... |
| dvmptdivc 25884 | Function-builder for deriv... |
| dvmptneg 25885 | Function-builder for deriv... |
| dvmptsub 25886 | Function-builder for deriv... |
| dvmptcj 25887 | Function-builder for deriv... |
| dvmptre 25888 | Function-builder for deriv... |
| dvmptim 25889 | Function-builder for deriv... |
| dvmptntr 25890 | Function-builder for deriv... |
| dvmptco 25891 | Function-builder for deriv... |
| dvrecg 25892 | Derivative of the reciproc... |
| dvmptdiv 25893 | Function-builder for deriv... |
| dvmptfsum 25894 | Function-builder for deriv... |
| dvcnvlem 25895 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 25896 | A weak version of ~ dvcnvr... |
| dvexp3 25897 | Derivative of an exponenti... |
| dveflem 25898 | Derivative of the exponent... |
| dvef 25899 | Derivative of the exponent... |
| dvsincos 25900 | Derivative of the sine and... |
| dvsin 25901 | Derivative of the sine fun... |
| dvcos 25902 | Derivative of the cosine f... |
| dvferm1lem 25903 | Lemma for ~ dvferm . (Con... |
| dvferm1 25904 | One-sided version of ~ dvf... |
| dvferm2lem 25905 | Lemma for ~ dvferm . (Con... |
| dvferm2 25906 | One-sided version of ~ dvf... |
| dvferm 25907 | Fermat's theorem on statio... |
| rollelem 25908 | Lemma for ~ rolle . (Cont... |
| rolle 25909 | Rolle's theorem. If ` F `... |
| cmvth 25910 | Cauchy's Mean Value Theore... |
| cmvthOLD 25911 | Obsolete version of ~ cmvt... |
| mvth 25912 | The Mean Value Theorem. I... |
| dvlip 25913 | A function with derivative... |
| dvlipcn 25914 | A complex function with de... |
| dvlip2 25915 | Combine the results of ~ d... |
| c1liplem1 25916 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 25917 | C^1 functions are Lipschit... |
| c1lip2 25918 | C^1 functions are Lipschit... |
| c1lip3 25919 | C^1 functions are Lipschit... |
| dveq0 25920 | If a continuous function h... |
| dv11cn 25921 | Two functions defined on a... |
| dvgt0lem1 25922 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 25923 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 25924 | A function on a closed int... |
| dvlt0 25925 | A function on a closed int... |
| dvge0 25926 | A function on a closed int... |
| dvle 25927 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 25928 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 25929 | Lemma for ~ dvivth . (Con... |
| dvivth 25930 | Darboux' theorem, or the i... |
| dvne0 25931 | A function on a closed int... |
| dvne0f1 25932 | A function on a closed int... |
| lhop1lem 25933 | Lemma for ~ lhop1 . (Cont... |
| lhop1 25934 | L'Hôpital's Rule for... |
| lhop2 25935 | L'Hôpital's Rule for... |
| lhop 25936 | L'Hôpital's Rule. I... |
| dvcnvrelem1 25937 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 25938 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 25939 | The derivative rule for in... |
| dvcvx 25940 | A real function with stric... |
| dvfsumle 25941 | Compare a finite sum to an... |
| dvfsumleOLD 25942 | Obsolete version of ~ dvfs... |
| dvfsumge 25943 | Compare a finite sum to an... |
| dvfsumabs 25944 | Compare a finite sum to an... |
| dvmptrecl 25945 | Real closure of a derivati... |
| dvfsumrlimf 25946 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 25947 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 25948 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2OLD 25949 | Obsolete version of ~ dvfs... |
| dvfsumlem3 25950 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 25951 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 25952 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 25953 | Compare a finite sum to an... |
| dvfsumrlim2 25954 | Compare a finite sum to an... |
| dvfsumrlim3 25955 | Conjoin the statements of ... |
| dvfsum2 25956 | The reverse of ~ dvfsumrli... |
| ftc1lem1 25957 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 25958 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 25959 | The Fundamental Theorem of... |
| ftc1lem3 25960 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 25961 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 25962 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 25963 | Lemma for ~ ftc1 . (Contr... |
| ftc1 25964 | The Fundamental Theorem of... |
| ftc1cn 25965 | Strengthen the assumptions... |
| ftc2 25966 | The Fundamental Theorem of... |
| ftc2ditglem 25967 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 25968 | Directed integral analogue... |
| itgparts 25969 | Integration by parts. If ... |
| itgsubstlem 25970 | Lemma for ~ itgsubst . (C... |
| itgsubst 25971 | Integration by ` u ` -subs... |
| itgpowd 25972 | The integral of a monomial... |
| reldmmdeg 25977 | Multivariate degree is a b... |
| tdeglem1 25978 | Functionality of the total... |
| tdeglem1OLD 25979 | Obsolete version of ~ tdeg... |
| tdeglem3 25980 | Additivity of the total de... |
| tdeglem3OLD 25981 | Obsolete version of ~ tdeg... |
| tdeglem4 25982 | There is only one multi-in... |
| tdeglem4OLD 25983 | Obsolete version of ~ tdeg... |
| tdeglem2 25984 | Simplification of total de... |
| mdegfval 25985 | Value of the multivariate ... |
| mdegval 25986 | Value of the multivariate ... |
| mdegleb 25987 | Property of being of limit... |
| mdeglt 25988 | If there is an upper limit... |
| mdegldg 25989 | A nonzero polynomial has s... |
| mdegxrcl 25990 | Closure of polynomial degr... |
| mdegxrf 25991 | Functionality of polynomia... |
| mdegcl 25992 | Sharp closure for multivar... |
| mdeg0 25993 | Degree of the zero polynom... |
| mdegnn0cl 25994 | Degree of a nonzero polyno... |
| degltlem1 25995 | Theorem on arithmetic of e... |
| degltp1le 25996 | Theorem on arithmetic of e... |
| mdegaddle 25997 | The degree of a sum is at ... |
| mdegvscale 25998 | The degree of a scalar mul... |
| mdegvsca 25999 | The degree of a scalar mul... |
| mdegle0 26000 | A polynomial has nonpositi... |
| mdegmullem 26001 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 26002 | The multivariate degree of... |
| deg1fval 26003 | Relate univariate polynomi... |
| deg1xrf 26004 | Functionality of univariat... |
| deg1xrcl 26005 | Closure of univariate poly... |
| deg1cl 26006 | Sharp closure of univariat... |
| mdegpropd 26007 | Property deduction for pol... |
| deg1fvi 26008 | Univariate polynomial degr... |
| deg1propd 26009 | Property deduction for pol... |
| deg1z 26010 | Degree of the zero univari... |
| deg1nn0cl 26011 | Degree of a nonzero univar... |
| deg1n0ima 26012 | Degree image of a set of p... |
| deg1nn0clb 26013 | A polynomial is nonzero if... |
| deg1lt0 26014 | A polynomial is zero iff i... |
| deg1ldg 26015 | A nonzero univariate polyn... |
| deg1ldgn 26016 | An index at which a polyno... |
| deg1ldgdomn 26017 | A nonzero univariate polyn... |
| deg1leb 26018 | Property of being of limit... |
| deg1val 26019 | Value of the univariate de... |
| deg1lt 26020 | If the degree of a univari... |
| deg1ge 26021 | Conversely, a nonzero coef... |
| coe1mul3 26022 | The coefficient vector of ... |
| coe1mul4 26023 | Value of the "leading" coe... |
| deg1addle 26024 | The degree of a sum is at ... |
| deg1addle2 26025 | If both factors have degre... |
| deg1add 26026 | Exact degree of a sum of t... |
| deg1vscale 26027 | The degree of a scalar tim... |
| deg1vsca 26028 | The degree of a scalar tim... |
| deg1invg 26029 | The degree of the negated ... |
| deg1suble 26030 | The degree of a difference... |
| deg1sub 26031 | Exact degree of a differen... |
| deg1mulle2 26032 | Produce a bound on the pro... |
| deg1sublt 26033 | Subtraction of two polynom... |
| deg1le0 26034 | A polynomial has nonpositi... |
| deg1sclle 26035 | A scalar polynomial has no... |
| deg1scl 26036 | A nonzero scalar polynomia... |
| deg1mul2 26037 | Degree of multiplication o... |
| deg1mul3 26038 | Degree of multiplication o... |
| deg1mul3le 26039 | Degree of multiplication o... |
| deg1tmle 26040 | Limiting degree of a polyn... |
| deg1tm 26041 | Exact degree of a polynomi... |
| deg1pwle 26042 | Limiting degree of a varia... |
| deg1pw 26043 | Exact degree of a variable... |
| ply1nz 26044 | Univariate polynomials ove... |
| ply1nzb 26045 | Univariate polynomials are... |
| ply1domn 26046 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26047 | The ring of univariate pol... |
| ply1divmo 26058 | Uniqueness of a quotient i... |
| ply1divex 26059 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26060 | The division algorithm for... |
| ply1divalg2 26061 | Reverse the order of multi... |
| uc1pval 26062 | Value of the set of unitic... |
| isuc1p 26063 | Being a unitic polynomial.... |
| mon1pval 26064 | Value of the set of monic ... |
| ismon1p 26065 | Being a monic polynomial. ... |
| uc1pcl 26066 | Unitic polynomials are pol... |
| mon1pcl 26067 | Monic polynomials are poly... |
| uc1pn0 26068 | Unitic polynomials are not... |
| mon1pn0 26069 | Monic polynomials are not ... |
| uc1pdeg 26070 | Unitic polynomials have no... |
| uc1pldg 26071 | Unitic polynomials have un... |
| mon1pldg 26072 | Unitic polynomials have on... |
| mon1puc1p 26073 | Monic polynomials are unit... |
| uc1pmon1p 26074 | Make a unitic polynomial m... |
| deg1submon1p 26075 | The difference of two moni... |
| mon1pid 26076 | Monicity and degree of the... |
| q1pval 26077 | Value of the univariate po... |
| q1peqb 26078 | Characterizing property of... |
| q1pcl 26079 | Closure of the quotient by... |
| r1pval 26080 | Value of the polynomial re... |
| r1pcl 26081 | Closure of remainder follo... |
| r1pdeglt 26082 | The remainder has a degree... |
| r1pid 26083 | Express the original polyn... |
| dvdsq1p 26084 | Divisibility in a polynomi... |
| dvdsr1p 26085 | Divisibility in a polynomi... |
| ply1remlem 26086 | A term of the form ` x - N... |
| ply1rem 26087 | The polynomial remainder t... |
| facth1 26088 | The factor theorem and its... |
| fta1glem1 26089 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26090 | Lemma for ~ fta1g . (Cont... |
| fta1g 26091 | The one-sided fundamental ... |
| fta1blem 26092 | Lemma for ~ fta1b . (Cont... |
| fta1b 26093 | The assumption that ` R ` ... |
| idomrootle 26094 | No element of an integral ... |
| drnguc1p 26095 | Over a division ring, all ... |
| ig1peu 26096 | There is a unique monic po... |
| ig1pval 26097 | Substitutions for the poly... |
| ig1pval2 26098 | Generator of the zero idea... |
| ig1pval3 26099 | Characterizing properties ... |
| ig1pcl 26100 | The monic generator of an ... |
| ig1pdvds 26101 | The monic generator of an ... |
| ig1prsp 26102 | Any ideal of polynomials o... |
| ply1lpir 26103 | The ring of polynomials ov... |
| ply1pid 26104 | The polynomials over a fie... |
| plyco0 26113 | Two ways to say that a fun... |
| plyval 26114 | Value of the polynomial se... |
| plybss 26115 | Reverse closure of the par... |
| elply 26116 | Definition of a polynomial... |
| elply2 26117 | The coefficient function c... |
| plyun0 26118 | The set of polynomials is ... |
| plyf 26119 | The polynomial is a functi... |
| plyss 26120 | The polynomial set functio... |
| plyssc 26121 | Every polynomial ring is c... |
| elplyr 26122 | Sufficient condition for e... |
| elplyd 26123 | Sufficient condition for e... |
| ply1termlem 26124 | Lemma for ~ ply1term . (C... |
| ply1term 26125 | A one-term polynomial. (C... |
| plypow 26126 | A power is a polynomial. ... |
| plyconst 26127 | A constant function is a p... |
| ne0p 26128 | A test to show that a poly... |
| ply0 26129 | The zero function is a pol... |
| plyid 26130 | The identity function is a... |
| plyeq0lem 26131 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26132 | If a polynomial is zero at... |
| plypf1 26133 | Write the set of complex p... |
| plyaddlem1 26134 | Derive the coefficient fun... |
| plymullem1 26135 | Derive the coefficient fun... |
| plyaddlem 26136 | Lemma for ~ plyadd . (Con... |
| plymullem 26137 | Lemma for ~ plymul . (Con... |
| plyadd 26138 | The sum of two polynomials... |
| plymul 26139 | The product of two polynom... |
| plysub 26140 | The difference of two poly... |
| plyaddcl 26141 | The sum of two polynomials... |
| plymulcl 26142 | The product of two polynom... |
| plysubcl 26143 | The difference of two poly... |
| coeval 26144 | Value of the coefficient f... |
| coeeulem 26145 | Lemma for ~ coeeu . (Cont... |
| coeeu 26146 | Uniqueness of the coeffici... |
| coelem 26147 | Lemma for properties of th... |
| coeeq 26148 | If ` A ` satisfies the pro... |
| dgrval 26149 | Value of the degree functi... |
| dgrlem 26150 | Lemma for ~ dgrcl and simi... |
| coef 26151 | The domain and codomain of... |
| coef2 26152 | The domain and codomain of... |
| coef3 26153 | The domain and codomain of... |
| dgrcl 26154 | The degree of any polynomi... |
| dgrub 26155 | If the ` M ` -th coefficie... |
| dgrub2 26156 | All the coefficients above... |
| dgrlb 26157 | If all the coefficients ab... |
| coeidlem 26158 | Lemma for ~ coeid . (Cont... |
| coeid 26159 | Reconstruct a polynomial a... |
| coeid2 26160 | Reconstruct a polynomial a... |
| coeid3 26161 | Reconstruct a polynomial a... |
| plyco 26162 | The composition of two pol... |
| coeeq2 26163 | Compute the coefficient fu... |
| dgrle 26164 | Given an explicit expressi... |
| dgreq 26165 | If the highest term in a p... |
| 0dgr 26166 | A constant function has de... |
| 0dgrb 26167 | A function has degree zero... |
| dgrnznn 26168 | A nonzero polynomial with ... |
| coefv0 26169 | The result of evaluating a... |
| coeaddlem 26170 | Lemma for ~ coeadd and ~ d... |
| coemullem 26171 | Lemma for ~ coemul and ~ d... |
| coeadd 26172 | The coefficient function o... |
| coemul 26173 | A coefficient of a product... |
| coe11 26174 | The coefficient function i... |
| coemulhi 26175 | The leading coefficient of... |
| coemulc 26176 | The coefficient function i... |
| coe0 26177 | The coefficients of the ze... |
| coesub 26178 | The coefficient function o... |
| coe1termlem 26179 | The coefficient function o... |
| coe1term 26180 | The coefficient function o... |
| dgr1term 26181 | The degree of a monomial. ... |
| plycn 26182 | A polynomial is a continuo... |
| plycnOLD 26183 | Obsolete version of ~ plyc... |
| dgr0 26184 | The degree of the zero pol... |
| coeidp 26185 | The coefficients of the id... |
| dgrid 26186 | The degree of the identity... |
| dgreq0 26187 | The leading coefficient of... |
| dgrlt 26188 | Two ways to say that the d... |
| dgradd 26189 | The degree of a sum of pol... |
| dgradd2 26190 | The degree of a sum of pol... |
| dgrmul2 26191 | The degree of a product of... |
| dgrmul 26192 | The degree of a product of... |
| dgrmulc 26193 | Scalar multiplication by a... |
| dgrsub 26194 | The degree of a difference... |
| dgrcolem1 26195 | The degree of a compositio... |
| dgrcolem2 26196 | Lemma for ~ dgrco . (Cont... |
| dgrco 26197 | The degree of a compositio... |
| plycjlem 26198 | Lemma for ~ plycj and ~ co... |
| plycj 26199 | The double conjugation of ... |
| coecj 26200 | Double conjugation of a po... |
| plyrecj 26201 | A polynomial with real coe... |
| plymul0or 26202 | Polynomial multiplication ... |
| ofmulrt 26203 | The set of roots of a prod... |
| plyreres 26204 | Real-coefficient polynomia... |
| dvply1 26205 | Derivative of a polynomial... |
| dvply2g 26206 | The derivative of a polyno... |
| dvply2gOLD 26207 | Obsolete version of ~ dvpl... |
| dvply2 26208 | The derivative of a polyno... |
| dvnply2 26209 | Polynomials have polynomia... |
| dvnply 26210 | Polynomials have polynomia... |
| plycpn 26211 | Polynomials are smooth. (... |
| quotval 26214 | Value of the quotient func... |
| plydivlem1 26215 | Lemma for ~ plydivalg . (... |
| plydivlem2 26216 | Lemma for ~ plydivalg . (... |
| plydivlem3 26217 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26218 | Lemma for ~ plydivex . In... |
| plydivex 26219 | Lemma for ~ plydivalg . (... |
| plydiveu 26220 | Lemma for ~ plydivalg . (... |
| plydivalg 26221 | The division algorithm on ... |
| quotlem 26222 | Lemma for properties of th... |
| quotcl 26223 | The quotient of two polyno... |
| quotcl2 26224 | Closure of the quotient fu... |
| quotdgr 26225 | Remainder property of the ... |
| plyremlem 26226 | Closure of a linear factor... |
| plyrem 26227 | The polynomial remainder t... |
| facth 26228 | The factor theorem. If a ... |
| fta1lem 26229 | Lemma for ~ fta1 . (Contr... |
| fta1 26230 | The easy direction of the ... |
| quotcan 26231 | Exact division with a mult... |
| vieta1lem1 26232 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26233 | Lemma for ~ vieta1 : induc... |
| vieta1 26234 | The first-order Vieta's fo... |
| plyexmo 26235 | An infinite set of values ... |
| elaa 26238 | Elementhood in the set of ... |
| aacn 26239 | An algebraic number is a c... |
| aasscn 26240 | The algebraic numbers are ... |
| elqaalem1 26241 | Lemma for ~ elqaa . The f... |
| elqaalem2 26242 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26243 | Lemma for ~ elqaa . (Cont... |
| elqaa 26244 | The set of numbers generat... |
| qaa 26245 | Every rational number is a... |
| qssaa 26246 | The rational numbers are c... |
| iaa 26247 | The imaginary unit is alge... |
| aareccl 26248 | The reciprocal of an algeb... |
| aacjcl 26249 | The conjugate of an algebr... |
| aannenlem1 26250 | Lemma for ~ aannen . (Con... |
| aannenlem2 26251 | Lemma for ~ aannen . (Con... |
| aannenlem3 26252 | The algebraic numbers are ... |
| aannen 26253 | The algebraic numbers are ... |
| aalioulem1 26254 | Lemma for ~ aaliou . An i... |
| aalioulem2 26255 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26256 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26257 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26258 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26259 | Lemma for ~ aaliou . (Con... |
| aaliou 26260 | Liouville's theorem on dio... |
| geolim3 26261 | Geometric series convergen... |
| aaliou2 26262 | Liouville's approximation ... |
| aaliou2b 26263 | Liouville's approximation ... |
| aaliou3lem1 26264 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26265 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26266 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26267 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26268 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26269 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26270 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26271 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26272 | Example of a "Liouville nu... |
| aaliou3 26273 | Example of a "Liouville nu... |
| taylfvallem1 26278 | Lemma for ~ taylfval . (C... |
| taylfvallem 26279 | Lemma for ~ taylfval . (C... |
| taylfval 26280 | Define the Taylor polynomi... |
| eltayl 26281 | Value of the Taylor series... |
| taylf 26282 | The Taylor series defines ... |
| tayl0 26283 | The Taylor series is alway... |
| taylplem1 26284 | Lemma for ~ taylpfval and ... |
| taylplem2 26285 | Lemma for ~ taylpfval and ... |
| taylpfval 26286 | Define the Taylor polynomi... |
| taylpf 26287 | The Taylor polynomial is a... |
| taylpval 26288 | Value of the Taylor polyno... |
| taylply2 26289 | The Taylor polynomial is a... |
| taylply2OLD 26290 | Obsolete version of ~ tayl... |
| taylply 26291 | The Taylor polynomial is a... |
| dvtaylp 26292 | The derivative of the Tayl... |
| dvntaylp 26293 | The ` M ` -th derivative o... |
| dvntaylp0 26294 | The first ` N ` derivative... |
| taylthlem1 26295 | Lemma for ~ taylth . This... |
| taylthlem2 26296 | Lemma for ~ taylth . (Con... |
| taylthlem2OLD 26297 | Obsolete version of ~ tayl... |
| taylth 26298 | Taylor's theorem. The Tay... |
| ulmrel 26301 | The uniform limit relation... |
| ulmscl 26302 | Closure of the base set in... |
| ulmval 26303 | Express the predicate: Th... |
| ulmcl 26304 | Closure of a uniform limit... |
| ulmf 26305 | Closure of a uniform limit... |
| ulmpm 26306 | Closure of a uniform limit... |
| ulmf2 26307 | Closure of a uniform limit... |
| ulm2 26308 | Simplify ~ ulmval when ` F... |
| ulmi 26309 | The uniform limit property... |
| ulmclm 26310 | A uniform limit of functio... |
| ulmres 26311 | A sequence of functions co... |
| ulmshftlem 26312 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26313 | A sequence of functions co... |
| ulm0 26314 | Every function converges u... |
| ulmuni 26315 | A sequence of functions un... |
| ulmdm 26316 | Two ways to express that a... |
| ulmcaulem 26317 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26318 | A sequence of functions co... |
| ulmcau2 26319 | A sequence of functions co... |
| ulmss 26320 | A uniform limit of functio... |
| ulmbdd 26321 | A uniform limit of bounded... |
| ulmcn 26322 | A uniform limit of continu... |
| ulmdvlem1 26323 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26324 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26325 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26326 | If ` F ` is a sequence of ... |
| mtest 26327 | The Weierstrass M-test. I... |
| mtestbdd 26328 | Given the hypotheses of th... |
| mbfulm 26329 | A uniform limit of measura... |
| iblulm 26330 | A uniform limit of integra... |
| itgulm 26331 | A uniform limit of integra... |
| itgulm2 26332 | A uniform limit of integra... |
| pserval 26333 | Value of the function ` G ... |
| pserval2 26334 | Value of the function ` G ... |
| psergf 26335 | The sequence of terms in t... |
| radcnvlem1 26336 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26337 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26338 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26339 | Zero is always a convergen... |
| radcnvcl 26340 | The radius of convergence ... |
| radcnvlt1 26341 | If ` X ` is within the ope... |
| radcnvlt2 26342 | If ` X ` is within the ope... |
| radcnvle 26343 | If ` X ` is a convergent p... |
| dvradcnv 26344 | The radius of convergence ... |
| pserulm 26345 | If ` S ` is a region conta... |
| psercn2 26346 | Since by ~ pserulm the ser... |
| psercn2OLD 26347 | Obsolete version of ~ pser... |
| psercnlem2 26348 | Lemma for ~ psercn . (Con... |
| psercnlem1 26349 | Lemma for ~ psercn . (Con... |
| psercn 26350 | An infinite series converg... |
| pserdvlem1 26351 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26352 | Lemma for ~ pserdv . (Con... |
| pserdv 26353 | The derivative of a power ... |
| pserdv2 26354 | The derivative of a power ... |
| abelthlem1 26355 | Lemma for ~ abelth . (Con... |
| abelthlem2 26356 | Lemma for ~ abelth . The ... |
| abelthlem3 26357 | Lemma for ~ abelth . (Con... |
| abelthlem4 26358 | Lemma for ~ abelth . (Con... |
| abelthlem5 26359 | Lemma for ~ abelth . (Con... |
| abelthlem6 26360 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26361 | Lemma for ~ abelth . (Con... |
| abelthlem7 26362 | Lemma for ~ abelth . (Con... |
| abelthlem8 26363 | Lemma for ~ abelth . (Con... |
| abelthlem9 26364 | Lemma for ~ abelth . By a... |
| abelth 26365 | Abel's theorem. If the po... |
| abelth2 26366 | Abel's theorem, restricted... |
| efcn 26367 | The exponential function i... |
| sincn 26368 | Sine is continuous. (Cont... |
| coscn 26369 | Cosine is continuous. (Co... |
| reeff1olem 26370 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26371 | The real exponential funct... |
| reefiso 26372 | The exponential function o... |
| efcvx 26373 | The exponential function o... |
| reefgim 26374 | The exponential function i... |
| pilem1 26375 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26376 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26377 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26378 | ` _pi ` is between 2 and 4... |
| sinpi 26379 | The sine of ` _pi ` is 0. ... |
| pire 26380 | ` _pi ` is a real number. ... |
| picn 26381 | ` _pi ` is a complex numbe... |
| pipos 26382 | ` _pi ` is positive. (Con... |
| pirp 26383 | ` _pi ` is a positive real... |
| negpicn 26384 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 26385 | Lemma for ~ sinhalfpi and ... |
| halfpire 26386 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26387 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26388 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26389 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26390 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26391 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26392 | The cosine of ` -u _pi / 2... |
| efhalfpi 26393 | The exponential of ` _i _p... |
| cospi 26394 | The cosine of ` _pi ` is `... |
| efipi 26395 | The exponential of ` _i x.... |
| eulerid 26396 | Euler's identity. (Contri... |
| sin2pi 26397 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26398 | The cosine of ` 2 _pi ` is... |
| ef2pi 26399 | The exponential of ` 2 _pi... |
| ef2kpi 26400 | If ` K ` is an integer, th... |
| efper 26401 | The exponential function i... |
| sinperlem 26402 | Lemma for ~ sinper and ~ c... |
| sinper 26403 | The sine function is perio... |
| cosper 26404 | The cosine function is per... |
| sin2kpi 26405 | If ` K ` is an integer, th... |
| cos2kpi 26406 | If ` K ` is an integer, th... |
| sin2pim 26407 | Sine of a number subtracte... |
| cos2pim 26408 | Cosine of a number subtrac... |
| sinmpi 26409 | Sine of a number less ` _p... |
| cosmpi 26410 | Cosine of a number less ` ... |
| sinppi 26411 | Sine of a number plus ` _p... |
| cosppi 26412 | Cosine of a number plus ` ... |
| efimpi 26413 | The exponential function a... |
| sinhalfpip 26414 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26415 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26416 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26417 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26418 | Ptolemy's Theorem. This t... |
| sincosq1lem 26419 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26420 | The signs of the sine and ... |
| sincosq2sgn 26421 | The signs of the sine and ... |
| sincosq3sgn 26422 | The signs of the sine and ... |
| sincosq4sgn 26423 | The signs of the sine and ... |
| coseq00topi 26424 | Location of the zeroes of ... |
| coseq0negpitopi 26425 | Location of the zeroes of ... |
| tanrpcl 26426 | Positive real closure of t... |
| tangtx 26427 | The tangent function is gr... |
| tanabsge 26428 | The tangent function is gr... |
| sinq12gt0 26429 | The sine of a number stric... |
| sinq12ge0 26430 | The sine of a number betwe... |
| sinq34lt0t 26431 | The sine of a number stric... |
| cosq14gt0 26432 | The cosine of a number str... |
| cosq14ge0 26433 | The cosine of a number bet... |
| sincosq1eq 26434 | Complementarity of the sin... |
| sincos4thpi 26435 | The sine and cosine of ` _... |
| tan4thpi 26436 | The tangent of ` _pi / 4 `... |
| sincos6thpi 26437 | The sine and cosine of ` _... |
| sincos3rdpi 26438 | The sine and cosine of ` _... |
| pigt3 26439 | ` _pi ` is greater than 3.... |
| pige3 26440 | ` _pi ` is greater than or... |
| pige3ALT 26441 | Alternate proof of ~ pige3... |
| abssinper 26442 | The absolute value of sine... |
| sinkpi 26443 | The sine of an integer mul... |
| coskpi 26444 | The absolute value of the ... |
| sineq0 26445 | A complex number whose sin... |
| coseq1 26446 | A complex number whose cos... |
| cos02pilt1 26447 | Cosine is less than one be... |
| cosq34lt1 26448 | Cosine is less than one in... |
| efeq1 26449 | A complex number whose exp... |
| cosne0 26450 | The cosine function has no... |
| cosordlem 26451 | Lemma for ~ cosord . (Con... |
| cosord 26452 | Cosine is decreasing over ... |
| cos0pilt1 26453 | Cosine is between minus on... |
| cos11 26454 | Cosine is one-to-one over ... |
| sinord 26455 | Sine is increasing over th... |
| recosf1o 26456 | The cosine function is a b... |
| resinf1o 26457 | The sine function is a bij... |
| tanord1 26458 | The tangent function is st... |
| tanord 26459 | The tangent function is st... |
| tanregt0 26460 | The real part of the tange... |
| negpitopissre 26461 | The interval ` ( -u _pi (,... |
| efgh 26462 | The exponential function o... |
| efif1olem1 26463 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26464 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26465 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26466 | The exponential function o... |
| efif1o 26467 | The exponential function o... |
| efifo 26468 | The exponential function o... |
| eff1olem 26469 | The exponential function m... |
| eff1o 26470 | The exponential function m... |
| efabl 26471 | The image of a subgroup of... |
| efsubm 26472 | The image of a subgroup of... |
| circgrp 26473 | The circle group ` T ` is ... |
| circsubm 26474 | The circle group ` T ` is ... |
| logrn 26479 | The range of the natural l... |
| ellogrn 26480 | Write out the property ` A... |
| dflog2 26481 | The natural logarithm func... |
| relogrn 26482 | The range of the natural l... |
| logrncn 26483 | The range of the natural l... |
| eff1o2 26484 | The exponential function r... |
| logf1o 26485 | The natural logarithm func... |
| dfrelog 26486 | The natural logarithm func... |
| relogf1o 26487 | The natural logarithm func... |
| logrncl 26488 | Closure of the natural log... |
| logcl 26489 | Closure of the natural log... |
| logimcl 26490 | Closure of the imaginary p... |
| logcld 26491 | The logarithm of a nonzero... |
| logimcld 26492 | The imaginary part of the ... |
| logimclad 26493 | The imaginary part of the ... |
| abslogimle 26494 | The imaginary part of the ... |
| logrnaddcl 26495 | The range of the natural l... |
| relogcl 26496 | Closure of the natural log... |
| eflog 26497 | Relationship between the n... |
| logeq0im1 26498 | If the logarithm of a numb... |
| logccne0 26499 | The logarithm isn't 0 if i... |
| logne0 26500 | Logarithm of a non-1 posit... |
| reeflog 26501 | Relationship between the n... |
| logef 26502 | Relationship between the n... |
| relogef 26503 | Relationship between the n... |
| logeftb 26504 | Relationship between the n... |
| relogeftb 26505 | Relationship between the n... |
| log1 26506 | The natural logarithm of `... |
| loge 26507 | The natural logarithm of `... |
| logi 26508 | The natural logarithm of `... |
| logneg 26509 | The natural logarithm of a... |
| logm1 26510 | The natural logarithm of n... |
| lognegb 26511 | If a number has imaginary ... |
| relogoprlem 26512 | Lemma for ~ relogmul and ~... |
| relogmul 26513 | The natural logarithm of t... |
| relogdiv 26514 | The natural logarithm of t... |
| explog 26515 | Exponentiation of a nonzer... |
| reexplog 26516 | Exponentiation of a positi... |
| relogexp 26517 | The natural logarithm of p... |
| relog 26518 | Real part of a logarithm. ... |
| relogiso 26519 | The natural logarithm func... |
| reloggim 26520 | The natural logarithm is a... |
| logltb 26521 | The natural logarithm func... |
| logfac 26522 | The logarithm of a factori... |
| eflogeq 26523 | Solve an equation involvin... |
| logleb 26524 | Natural logarithm preserve... |
| rplogcl 26525 | Closure of the logarithm f... |
| logge0 26526 | The logarithm of a number ... |
| logcj 26527 | The natural logarithm dist... |
| efiarg 26528 | The exponential of the "ar... |
| cosargd 26529 | The cosine of the argument... |
| cosarg0d 26530 | The cosine of the argument... |
| argregt0 26531 | Closure of the argument of... |
| argrege0 26532 | Closure of the argument of... |
| argimgt0 26533 | Closure of the argument of... |
| argimlt0 26534 | Closure of the argument of... |
| logimul 26535 | Multiplying a number by ` ... |
| logneg2 26536 | The logarithm of the negat... |
| logmul2 26537 | Generalization of ~ relogm... |
| logdiv2 26538 | Generalization of ~ relogd... |
| abslogle 26539 | Bound on the magnitude of ... |
| tanarg 26540 | The basic relation between... |
| logdivlti 26541 | The ` log x / x ` function... |
| logdivlt 26542 | The ` log x / x ` function... |
| logdivle 26543 | The ` log x / x ` function... |
| relogcld 26544 | Closure of the natural log... |
| reeflogd 26545 | Relationship between the n... |
| relogmuld 26546 | The natural logarithm of t... |
| relogdivd 26547 | The natural logarithm of t... |
| logled 26548 | Natural logarithm preserve... |
| relogefd 26549 | Relationship between the n... |
| rplogcld 26550 | Closure of the logarithm f... |
| logge0d 26551 | The logarithm of a number ... |
| logge0b 26552 | The logarithm of a number ... |
| loggt0b 26553 | The logarithm of a number ... |
| logle1b 26554 | The logarithm of a number ... |
| loglt1b 26555 | The logarithm of a number ... |
| divlogrlim 26556 | The inverse logarithm func... |
| logno1 26557 | The logarithm function is ... |
| dvrelog 26558 | The derivative of the real... |
| relogcn 26559 | The real logarithm functio... |
| ellogdm 26560 | Elementhood in the "contin... |
| logdmn0 26561 | A number in the continuous... |
| logdmnrp 26562 | A number in the continuous... |
| logdmss 26563 | The continuity domain of `... |
| logcnlem2 26564 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26565 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26566 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26567 | Lemma for ~ logcn . (Cont... |
| logcn 26568 | The logarithm function is ... |
| dvloglem 26569 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26570 | The "continuous domain" of... |
| logf1o2 26571 | The logarithm maps its con... |
| dvlog 26572 | The derivative of the comp... |
| dvlog2lem 26573 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26574 | The derivative of the comp... |
| advlog 26575 | The antiderivative of the ... |
| advlogexp 26576 | The antiderivative of a po... |
| efopnlem1 26577 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26578 | Lemma for ~ efopn . (Cont... |
| efopn 26579 | The exponential map is an ... |
| logtayllem 26580 | Lemma for ~ logtayl . (Co... |
| logtayl 26581 | The Taylor series for ` -u... |
| logtaylsum 26582 | The Taylor series for ` -u... |
| logtayl2 26583 | Power series expression fo... |
| logccv 26584 | The natural logarithm func... |
| cxpval 26585 | Value of the complex power... |
| cxpef 26586 | Value of the complex power... |
| 0cxp 26587 | Value of the complex power... |
| cxpexpz 26588 | Relate the complex power f... |
| cxpexp 26589 | Relate the complex power f... |
| logcxp 26590 | Logarithm of a complex pow... |
| cxp0 26591 | Value of the complex power... |
| cxp1 26592 | Value of the complex power... |
| 1cxp 26593 | Value of the complex power... |
| ecxp 26594 | Write the exponential func... |
| cxpcl 26595 | Closure of the complex pow... |
| recxpcl 26596 | Real closure of the comple... |
| rpcxpcl 26597 | Positive real closure of t... |
| cxpne0 26598 | Complex exponentiation is ... |
| cxpeq0 26599 | Complex exponentiation is ... |
| cxpadd 26600 | Sum of exponents law for c... |
| cxpp1 26601 | Value of a nonzero complex... |
| cxpneg 26602 | Value of a complex number ... |
| cxpsub 26603 | Exponent subtraction law f... |
| cxpge0 26604 | Nonnegative exponentiation... |
| mulcxplem 26605 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26606 | Complex exponentiation of ... |
| cxprec 26607 | Complex exponentiation of ... |
| divcxp 26608 | Complex exponentiation of ... |
| cxpmul 26609 | Product of exponents law f... |
| cxpmul2 26610 | Product of exponents law f... |
| cxproot 26611 | The complex power function... |
| cxpmul2z 26612 | Generalize ~ cxpmul2 to ne... |
| abscxp 26613 | Absolute value of a power,... |
| abscxp2 26614 | Absolute value of a power,... |
| cxplt 26615 | Ordering property for comp... |
| cxple 26616 | Ordering property for comp... |
| cxplea 26617 | Ordering property for comp... |
| cxple2 26618 | Ordering property for comp... |
| cxplt2 26619 | Ordering property for comp... |
| cxple2a 26620 | Ordering property for comp... |
| cxplt3 26621 | Ordering property for comp... |
| cxple3 26622 | Ordering property for comp... |
| cxpsqrtlem 26623 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26624 | The complex exponential fu... |
| logsqrt 26625 | Logarithm of a square root... |
| cxp0d 26626 | Value of the complex power... |
| cxp1d 26627 | Value of the complex power... |
| 1cxpd 26628 | Value of the complex power... |
| cxpcld 26629 | Closure of the complex pow... |
| cxpmul2d 26630 | Product of exponents law f... |
| 0cxpd 26631 | Value of the complex power... |
| cxpexpzd 26632 | Relate the complex power f... |
| cxpefd 26633 | Value of the complex power... |
| cxpne0d 26634 | Complex exponentiation is ... |
| cxpp1d 26635 | Value of a nonzero complex... |
| cxpnegd 26636 | Value of a complex number ... |
| cxpmul2zd 26637 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26638 | Sum of exponents law for c... |
| cxpsubd 26639 | Exponent subtraction law f... |
| cxpltd 26640 | Ordering property for comp... |
| cxpled 26641 | Ordering property for comp... |
| cxplead 26642 | Ordering property for comp... |
| divcxpd 26643 | Complex exponentiation of ... |
| recxpcld 26644 | Positive real closure of t... |
| cxpge0d 26645 | Nonnegative exponentiation... |
| cxple2ad 26646 | Ordering property for comp... |
| cxplt2d 26647 | Ordering property for comp... |
| cxple2d 26648 | Ordering property for comp... |
| mulcxpd 26649 | Complex exponentiation of ... |
| recxpf1lem 26650 | Complex exponentiation on ... |
| cxpsqrtth 26651 | Square root theorem over t... |
| 2irrexpq 26652 | There exist irrational num... |
| cxprecd 26653 | Complex exponentiation of ... |
| rpcxpcld 26654 | Positive real closure of t... |
| logcxpd 26655 | Logarithm of a complex pow... |
| cxplt3d 26656 | Ordering property for comp... |
| cxple3d 26657 | Ordering property for comp... |
| cxpmuld 26658 | Product of exponents law f... |
| cxpgt0d 26659 | A positive real raised to ... |
| cxpcom 26660 | Commutative law for real e... |
| dvcxp1 26661 | The derivative of a comple... |
| dvcxp2 26662 | The derivative of a comple... |
| dvsqrt 26663 | The derivative of the real... |
| dvcncxp1 26664 | Derivative of complex powe... |
| dvcnsqrt 26665 | Derivative of square root ... |
| cxpcn 26666 | Domain of continuity of th... |
| cxpcnOLD 26667 | Obsolete version of ~ cxpc... |
| cxpcn2 26668 | Continuity of the complex ... |
| cxpcn3lem 26669 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26670 | Extend continuity of the c... |
| resqrtcn 26671 | Continuity of the real squ... |
| sqrtcn 26672 | Continuity of the square r... |
| cxpaddlelem 26673 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26674 | Ordering property for comp... |
| abscxpbnd 26675 | Bound on the absolute valu... |
| root1id 26676 | Property of an ` N ` -th r... |
| root1eq1 26677 | The only powers of an ` N ... |
| root1cj 26678 | Within the ` N ` -th roots... |
| cxpeq 26679 | Solve an equation involvin... |
| loglesqrt 26680 | An upper bound on the loga... |
| logreclem 26681 | Symmetry of the natural lo... |
| logrec 26682 | Logarithm of a reciprocal ... |
| logbval 26685 | Define the value of the ` ... |
| logbcl 26686 | General logarithm closure.... |
| logbid1 26687 | General logarithm is 1 whe... |
| logb1 26688 | The logarithm of ` 1 ` to ... |
| elogb 26689 | The general logarithm of a... |
| logbchbase 26690 | Change of base for logarit... |
| relogbval 26691 | Value of the general logar... |
| relogbcl 26692 | Closure of the general log... |
| relogbzcl 26693 | Closure of the general log... |
| relogbreexp 26694 | Power law for the general ... |
| relogbzexp 26695 | Power law for the general ... |
| relogbmul 26696 | The logarithm of the produ... |
| relogbmulexp 26697 | The logarithm of the produ... |
| relogbdiv 26698 | The logarithm of the quoti... |
| relogbexp 26699 | Identity law for general l... |
| nnlogbexp 26700 | Identity law for general l... |
| logbrec 26701 | Logarithm of a reciprocal ... |
| logbleb 26702 | The general logarithm func... |
| logblt 26703 | The general logarithm func... |
| relogbcxp 26704 | Identity law for the gener... |
| cxplogb 26705 | Identity law for the gener... |
| relogbcxpb 26706 | The logarithm is the inver... |
| logbmpt 26707 | The general logarithm to a... |
| logbf 26708 | The general logarithm to a... |
| logbfval 26709 | The general logarithm of a... |
| relogbf 26710 | The general logarithm to a... |
| logblog 26711 | The general logarithm to t... |
| logbgt0b 26712 | The logarithm of a positiv... |
| logbgcd1irr 26713 | The logarithm of an intege... |
| 2logb9irr 26714 | Example for ~ logbgcd1irr ... |
| logbprmirr 26715 | The logarithm of a prime t... |
| 2logb3irr 26716 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26717 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26718 | The square root of two to ... |
| 2irrexpqALT 26719 | Alternate proof of ~ 2irre... |
| angval 26720 | Define the angle function,... |
| angcan 26721 | Cancel a constant multipli... |
| angneg 26722 | Cancel a negative sign in ... |
| angvald 26723 | The (signed) angle between... |
| angcld 26724 | The (signed) angle between... |
| angrteqvd 26725 | Two vectors are at a right... |
| cosangneg2d 26726 | The cosine of the angle be... |
| angrtmuld 26727 | Perpendicularity of two ve... |
| ang180lem1 26728 | Lemma for ~ ang180 . Show... |
| ang180lem2 26729 | Lemma for ~ ang180 . Show... |
| ang180lem3 26730 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26731 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26732 | Lemma for ~ ang180 : Redu... |
| ang180 26733 | The sum of angles ` m A B ... |
| lawcoslem1 26734 | Lemma for ~ lawcos . Here... |
| lawcos 26735 | Law of cosines (also known... |
| pythag 26736 | Pythagorean theorem. Give... |
| isosctrlem1 26737 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26738 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26739 | Lemma for ~ isosctr . Cor... |
| isosctr 26740 | Isosceles triangle theorem... |
| ssscongptld 26741 | If two triangles have equa... |
| affineequiv 26742 | Equivalence between two wa... |
| affineequiv2 26743 | Equivalence between two wa... |
| affineequiv3 26744 | Equivalence between two wa... |
| affineequiv4 26745 | Equivalence between two wa... |
| affineequivne 26746 | Equivalence between two wa... |
| angpieqvdlem 26747 | Equivalence used in the pr... |
| angpieqvdlem2 26748 | Equivalence used in ~ angp... |
| angpined 26749 | If the angle at ABC is ` _... |
| angpieqvd 26750 | The angle ABC is ` _pi ` i... |
| chordthmlem 26751 | If ` M ` is the midpoint o... |
| chordthmlem2 26752 | If M is the midpoint of AB... |
| chordthmlem3 26753 | If M is the midpoint of AB... |
| chordthmlem4 26754 | If P is on the segment AB ... |
| chordthmlem5 26755 | If P is on the segment AB ... |
| chordthm 26756 | The intersecting chords th... |
| heron 26757 | Heron's formula gives the ... |
| quad2 26758 | The quadratic equation, wi... |
| quad 26759 | The quadratic equation. (... |
| 1cubrlem 26760 | The cube roots of unity. ... |
| 1cubr 26761 | The cube roots of unity. ... |
| dcubic1lem 26762 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26763 | Reverse direction of ~ dcu... |
| dcubic1 26764 | Forward direction of ~ dcu... |
| dcubic 26765 | Solutions to the depressed... |
| mcubic 26766 | Solutions to a monic cubic... |
| cubic2 26767 | The solution to the genera... |
| cubic 26768 | The cubic equation, which ... |
| binom4 26769 | Work out a quartic binomia... |
| dquartlem1 26770 | Lemma for ~ dquart . (Con... |
| dquartlem2 26771 | Lemma for ~ dquart . (Con... |
| dquart 26772 | Solve a depressed quartic ... |
| quart1cl 26773 | Closure lemmas for ~ quart... |
| quart1lem 26774 | Lemma for ~ quart1 . (Con... |
| quart1 26775 | Depress a quartic equation... |
| quartlem1 26776 | Lemma for ~ quart . (Cont... |
| quartlem2 26777 | Closure lemmas for ~ quart... |
| quartlem3 26778 | Closure lemmas for ~ quart... |
| quartlem4 26779 | Closure lemmas for ~ quart... |
| quart 26780 | The quartic equation, writ... |
| asinlem 26787 | The argument to the logari... |
| asinlem2 26788 | The argument to the logari... |
| asinlem3a 26789 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26790 | The argument to the logari... |
| asinf 26791 | Domain and codomain of the... |
| asincl 26792 | Closure for the arcsin fun... |
| acosf 26793 | Domain and codoamin of the... |
| acoscl 26794 | Closure for the arccos fun... |
| atandm 26795 | Since the property is a li... |
| atandm2 26796 | This form of ~ atandm is a... |
| atandm3 26797 | A compact form of ~ atandm... |
| atandm4 26798 | A compact form of ~ atandm... |
| atanf 26799 | Domain and codoamin of the... |
| atancl 26800 | Closure for the arctan fun... |
| asinval 26801 | Value of the arcsin functi... |
| acosval 26802 | Value of the arccos functi... |
| atanval 26803 | Value of the arctan functi... |
| atanre 26804 | A real number is in the do... |
| asinneg 26805 | The arcsine function is od... |
| acosneg 26806 | The negative symmetry rela... |
| efiasin 26807 | The exponential of the arc... |
| sinasin 26808 | The arcsine function is an... |
| cosacos 26809 | The arccosine function is ... |
| asinsinlem 26810 | Lemma for ~ asinsin . (Co... |
| asinsin 26811 | The arcsine function compo... |
| acoscos 26812 | The arccosine function is ... |
| asin1 26813 | The arcsine of ` 1 ` is ` ... |
| acos1 26814 | The arccosine of ` 1 ` is ... |
| reasinsin 26815 | The arcsine function compo... |
| asinsinb 26816 | Relationship between sine ... |
| acoscosb 26817 | Relationship between cosin... |
| asinbnd 26818 | The arcsine function has r... |
| acosbnd 26819 | The arccosine function has... |
| asinrebnd 26820 | Bounds on the arcsine func... |
| asinrecl 26821 | The arcsine function is re... |
| acosrecl 26822 | The arccosine function is ... |
| cosasin 26823 | The cosine of the arcsine ... |
| sinacos 26824 | The sine of the arccosine ... |
| atandmneg 26825 | The domain of the arctange... |
| atanneg 26826 | The arctangent function is... |
| atan0 26827 | The arctangent of zero is ... |
| atandmcj 26828 | The arctangent function di... |
| atancj 26829 | The arctangent function di... |
| atanrecl 26830 | The arctangent function is... |
| efiatan 26831 | Value of the exponential o... |
| atanlogaddlem 26832 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26833 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26834 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26835 | A variation on ~ atanlogad... |
| efiatan2 26836 | Value of the exponential o... |
| 2efiatan 26837 | Value of the exponential o... |
| tanatan 26838 | The arctangent function is... |
| atandmtan 26839 | The tangent function has r... |
| cosatan 26840 | The cosine of an arctangen... |
| cosatanne0 26841 | The arctangent function ha... |
| atantan 26842 | The arctangent function is... |
| atantanb 26843 | Relationship between tange... |
| atanbndlem 26844 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26845 | The arctangent function is... |
| atanord 26846 | The arctangent function is... |
| atan1 26847 | The arctangent of ` 1 ` is... |
| bndatandm 26848 | A point in the open unit d... |
| atans 26849 | The "domain of continuity"... |
| atans2 26850 | It suffices to show that `... |
| atansopn 26851 | The domain of continuity o... |
| atansssdm 26852 | The domain of continuity o... |
| ressatans 26853 | The real number line is a ... |
| dvatan 26854 | The derivative of the arct... |
| atancn 26855 | The arctangent is a contin... |
| atantayl 26856 | The Taylor series for ` ar... |
| atantayl2 26857 | The Taylor series for ` ar... |
| atantayl3 26858 | The Taylor series for ` ar... |
| leibpilem1 26859 | Lemma for ~ leibpi . (Con... |
| leibpilem2 26860 | The Leibniz formula for ` ... |
| leibpi 26861 | The Leibniz formula for ` ... |
| leibpisum 26862 | The Leibniz formula for ` ... |
| log2cnv 26863 | Using the Taylor series fo... |
| log2tlbnd 26864 | Bound the error term in th... |
| log2ublem1 26865 | Lemma for ~ log2ub . The ... |
| log2ublem2 26866 | Lemma for ~ log2ub . (Con... |
| log2ublem3 26867 | Lemma for ~ log2ub . In d... |
| log2ub 26868 | ` log 2 ` is less than ` 2... |
| log2le1 26869 | ` log 2 ` is less than ` 1... |
| birthdaylem1 26870 | Lemma for ~ birthday . (C... |
| birthdaylem2 26871 | For general ` N ` and ` K ... |
| birthdaylem3 26872 | For general ` N ` and ` K ... |
| birthday 26873 | The Birthday Problem. The... |
| dmarea 26876 | The domain of the area fun... |
| areambl 26877 | The fibers of a measurable... |
| areass 26878 | A measurable region is a s... |
| dfarea 26879 | Rewrite ~ df-area self-ref... |
| areaf 26880 | Area measurement is a func... |
| areacl 26881 | The area of a measurable r... |
| areage0 26882 | The area of a measurable r... |
| areaval 26883 | The area of a measurable r... |
| rlimcnp 26884 | Relate a limit of a real-v... |
| rlimcnp2 26885 | Relate a limit of a real-v... |
| rlimcnp3 26886 | Relate a limit of a real-v... |
| xrlimcnp 26887 | Relate a limit of a real-v... |
| efrlim 26888 | The limit of the sequence ... |
| efrlimOLD 26889 | Obsolete version of ~ efrl... |
| dfef2 26890 | The limit of the sequence ... |
| cxplim 26891 | A power to a negative expo... |
| sqrtlim 26892 | The inverse square root fu... |
| rlimcxp 26893 | Any power to a positive ex... |
| o1cxp 26894 | An eventually bounded func... |
| cxp2limlem 26895 | A linear factor grows slow... |
| cxp2lim 26896 | Any power grows slower tha... |
| cxploglim 26897 | The logarithm grows slower... |
| cxploglim2 26898 | Every power of the logarit... |
| divsqrtsumlem 26899 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 26900 | The function ` F ` used in... |
| divsqrsum 26901 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 26902 | A bound on the distance of... |
| divsqrtsumo1 26903 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 26904 | Closure of a 0-1 linear co... |
| scvxcvx 26905 | A strictly convex function... |
| jensenlem1 26906 | Lemma for ~ jensen . (Con... |
| jensenlem2 26907 | Lemma for ~ jensen . (Con... |
| jensen 26908 | Jensen's inequality, a fin... |
| amgmlem 26909 | Lemma for ~ amgm . (Contr... |
| amgm 26910 | Inequality of arithmetic a... |
| logdifbnd 26913 | Bound on the difference of... |
| logdiflbnd 26914 | Lower bound on the differe... |
| emcllem1 26915 | Lemma for ~ emcl . The se... |
| emcllem2 26916 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 26917 | Lemma for ~ emcl . The fu... |
| emcllem4 26918 | Lemma for ~ emcl . The di... |
| emcllem5 26919 | Lemma for ~ emcl . The pa... |
| emcllem6 26920 | Lemma for ~ emcl . By the... |
| emcllem7 26921 | Lemma for ~ emcl and ~ har... |
| emcl 26922 | Closure and bounds for the... |
| harmonicbnd 26923 | A bound on the harmonic se... |
| harmonicbnd2 26924 | A bound on the harmonic se... |
| emre 26925 | The Euler-Mascheroni const... |
| emgt0 26926 | The Euler-Mascheroni const... |
| harmonicbnd3 26927 | A bound on the harmonic se... |
| harmoniclbnd 26928 | A bound on the harmonic se... |
| harmonicubnd 26929 | A bound on the harmonic se... |
| harmonicbnd4 26930 | The asymptotic behavior of... |
| fsumharmonic 26931 | Bound a finite sum based o... |
| zetacvg 26934 | The zeta series is converg... |
| eldmgm 26941 | Elementhood in the set of ... |
| dmgmaddn0 26942 | If ` A ` is not a nonposit... |
| dmlogdmgm 26943 | If ` A ` is in the continu... |
| rpdmgm 26944 | A positive real number is ... |
| dmgmn0 26945 | If ` A ` is not a nonposit... |
| dmgmaddnn0 26946 | If ` A ` is not a nonposit... |
| dmgmdivn0 26947 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 26948 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 26949 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 26950 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 26951 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 26952 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 26953 | The series ` G ` is unifor... |
| lgamgulm 26954 | The series ` G ` is unifor... |
| lgamgulm2 26955 | Rewrite the limit of the s... |
| lgambdd 26956 | The log-Gamma function is ... |
| lgamucov 26957 | The ` U ` regions used in ... |
| lgamucov2 26958 | The ` U ` regions used in ... |
| lgamcvglem 26959 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 26960 | The log-Gamma function is ... |
| lgamf 26961 | The log-Gamma function is ... |
| gamf 26962 | The Gamma function is a co... |
| gamcl 26963 | The exponential of the log... |
| eflgam 26964 | The exponential of the log... |
| gamne0 26965 | The Gamma function is neve... |
| igamval 26966 | Value of the inverse Gamma... |
| igamz 26967 | Value of the inverse Gamma... |
| igamgam 26968 | Value of the inverse Gamma... |
| igamlgam 26969 | Value of the inverse Gamma... |
| igamf 26970 | Closure of the inverse Gam... |
| igamcl 26971 | Closure of the inverse Gam... |
| gamigam 26972 | The Gamma function is the ... |
| lgamcvg 26973 | The series ` G ` converges... |
| lgamcvg2 26974 | The series ` G ` converges... |
| gamcvg 26975 | The pointwise exponential ... |
| lgamp1 26976 | The functional equation of... |
| gamp1 26977 | The functional equation of... |
| gamcvg2lem 26978 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 26979 | An infinite product expres... |
| regamcl 26980 | The Gamma function is real... |
| relgamcl 26981 | The log-Gamma function is ... |
| rpgamcl 26982 | The log-Gamma function is ... |
| lgam1 26983 | The log-Gamma function at ... |
| gam1 26984 | The log-Gamma function at ... |
| facgam 26985 | The Gamma function general... |
| gamfac 26986 | The Gamma function general... |
| wilthlem1 26987 | The only elements that are... |
| wilthlem2 26988 | Lemma for ~ wilth : induct... |
| wilthlem3 26989 | Lemma for ~ wilth . Here ... |
| wilth 26990 | Wilson's theorem. A numbe... |
| wilthimp 26991 | The forward implication of... |
| ftalem1 26992 | Lemma for ~ fta : "growth... |
| ftalem2 26993 | Lemma for ~ fta . There e... |
| ftalem3 26994 | Lemma for ~ fta . There e... |
| ftalem4 26995 | Lemma for ~ fta : Closure... |
| ftalem5 26996 | Lemma for ~ fta : Main pr... |
| ftalem6 26997 | Lemma for ~ fta : Dischar... |
| ftalem7 26998 | Lemma for ~ fta . Shift t... |
| fta 26999 | The Fundamental Theorem of... |
| basellem1 27000 | Lemma for ~ basel . Closu... |
| basellem2 27001 | Lemma for ~ basel . Show ... |
| basellem3 27002 | Lemma for ~ basel . Using... |
| basellem4 27003 | Lemma for ~ basel . By ~ ... |
| basellem5 27004 | Lemma for ~ basel . Using... |
| basellem6 27005 | Lemma for ~ basel . The f... |
| basellem7 27006 | Lemma for ~ basel . The f... |
| basellem8 27007 | Lemma for ~ basel . The f... |
| basellem9 27008 | Lemma for ~ basel . Since... |
| basel 27009 | The sum of the inverse squ... |
| efnnfsumcl 27022 | Finite sum closure in the ... |
| ppisval 27023 | The set of primes less tha... |
| ppisval2 27024 | The set of primes less tha... |
| ppifi 27025 | The set of primes less tha... |
| prmdvdsfi 27026 | The set of prime divisors ... |
| chtf 27027 | Domain and codoamin of the... |
| chtcl 27028 | Real closure of the Chebys... |
| chtval 27029 | Value of the Chebyshev fun... |
| efchtcl 27030 | The Chebyshev function is ... |
| chtge0 27031 | The Chebyshev function is ... |
| vmaval 27032 | Value of the von Mangoldt ... |
| isppw 27033 | Two ways to say that ` A `... |
| isppw2 27034 | Two ways to say that ` A `... |
| vmappw 27035 | Value of the von Mangoldt ... |
| vmaprm 27036 | Value of the von Mangoldt ... |
| vmacl 27037 | Closure for the von Mangol... |
| vmaf 27038 | Functionality of the von M... |
| efvmacl 27039 | The von Mangoldt is closed... |
| vmage0 27040 | The von Mangoldt function ... |
| chpval 27041 | Value of the second Chebys... |
| chpf 27042 | Functionality of the secon... |
| chpcl 27043 | Closure for the second Che... |
| efchpcl 27044 | The second Chebyshev funct... |
| chpge0 27045 | The second Chebyshev funct... |
| ppival 27046 | Value of the prime-countin... |
| ppival2 27047 | Value of the prime-countin... |
| ppival2g 27048 | Value of the prime-countin... |
| ppif 27049 | Domain and codomain of the... |
| ppicl 27050 | Real closure of the prime-... |
| muval 27051 | The value of the Möbi... |
| muval1 27052 | The value of the Möbi... |
| muval2 27053 | The value of the Möbi... |
| isnsqf 27054 | Two ways to say that a num... |
| issqf 27055 | Two ways to say that a num... |
| sqfpc 27056 | The prime count of a squar... |
| dvdssqf 27057 | A divisor of a squarefree ... |
| sqf11 27058 | A squarefree number is com... |
| muf 27059 | The Möbius function i... |
| mucl 27060 | Closure of the Möbius... |
| sgmval 27061 | The value of the divisor f... |
| sgmval2 27062 | The value of the divisor f... |
| 0sgm 27063 | The value of the sum-of-di... |
| sgmf 27064 | The divisor function is a ... |
| sgmcl 27065 | Closure of the divisor fun... |
| sgmnncl 27066 | Closure of the divisor fun... |
| mule1 27067 | The Möbius function t... |
| chtfl 27068 | The Chebyshev function doe... |
| chpfl 27069 | The second Chebyshev funct... |
| ppiprm 27070 | The prime-counting functio... |
| ppinprm 27071 | The prime-counting functio... |
| chtprm 27072 | The Chebyshev function at ... |
| chtnprm 27073 | The Chebyshev function at ... |
| chpp1 27074 | The second Chebyshev funct... |
| chtwordi 27075 | The Chebyshev function is ... |
| chpwordi 27076 | The second Chebyshev funct... |
| chtdif 27077 | The difference of the Cheb... |
| efchtdvds 27078 | The exponentiated Chebyshe... |
| ppifl 27079 | The prime-counting functio... |
| ppip1le 27080 | The prime-counting functio... |
| ppiwordi 27081 | The prime-counting functio... |
| ppidif 27082 | The difference of the prim... |
| ppi1 27083 | The prime-counting functio... |
| cht1 27084 | The Chebyshev function at ... |
| vma1 27085 | The von Mangoldt function ... |
| chp1 27086 | The second Chebyshev funct... |
| ppi1i 27087 | Inference form of ~ ppiprm... |
| ppi2i 27088 | Inference form of ~ ppinpr... |
| ppi2 27089 | The prime-counting functio... |
| ppi3 27090 | The prime-counting functio... |
| cht2 27091 | The Chebyshev function at ... |
| cht3 27092 | The Chebyshev function at ... |
| ppinncl 27093 | Closure of the prime-count... |
| chtrpcl 27094 | Closure of the Chebyshev f... |
| ppieq0 27095 | The prime-counting functio... |
| ppiltx 27096 | The prime-counting functio... |
| prmorcht 27097 | Relate the primorial (prod... |
| mumullem1 27098 | Lemma for ~ mumul . A mul... |
| mumullem2 27099 | Lemma for ~ mumul . The p... |
| mumul 27100 | The Möbius function i... |
| sqff1o 27101 | There is a bijection from ... |
| fsumdvdsdiaglem 27102 | A "diagonal commutation" o... |
| fsumdvdsdiag 27103 | A "diagonal commutation" o... |
| fsumdvdscom 27104 | A double commutation of di... |
| dvdsppwf1o 27105 | A bijection from the divis... |
| dvdsflf1o 27106 | A bijection from the numbe... |
| dvdsflsumcom 27107 | A sum commutation from ` s... |
| fsumfldivdiaglem 27108 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27109 | The right-hand side of ~ d... |
| musum 27110 | The sum of the Möbius... |
| musumsum 27111 | Evaluate a collapsing sum ... |
| muinv 27112 | The Möbius inversion ... |
| mpodvdsmulf1o 27113 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27114 | Product of two divisor sum... |
| dvdsmulf1o 27115 | If ` M ` and ` N ` are two... |
| fsumdvdsmulOLD 27116 | Obsolete version of ~ fsum... |
| sgmppw 27117 | The value of the divisor f... |
| 0sgmppw 27118 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27119 | The sum of divisors for a ... |
| 1sgm2ppw 27120 | The sum of the divisors of... |
| sgmmul 27121 | The divisor function for f... |
| ppiublem1 27122 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27123 | A prime greater than ` 3 `... |
| ppiub 27124 | An upper bound on the prim... |
| vmalelog 27125 | The von Mangoldt function ... |
| chtlepsi 27126 | The first Chebyshev functi... |
| chprpcl 27127 | Closure of the second Cheb... |
| chpeq0 27128 | The second Chebyshev funct... |
| chteq0 27129 | The first Chebyshev functi... |
| chtleppi 27130 | Upper bound on the ` theta... |
| chtublem 27131 | Lemma for ~ chtub . (Cont... |
| chtub 27132 | An upper bound on the Cheb... |
| fsumvma 27133 | Rewrite a sum over the von... |
| fsumvma2 27134 | Apply ~ fsumvma for the co... |
| pclogsum 27135 | The logarithmic analogue o... |
| vmasum 27136 | The sum of the von Mangold... |
| logfac2 27137 | Another expression for the... |
| chpval2 27138 | Express the second Chebysh... |
| chpchtsum 27139 | The second Chebyshev funct... |
| chpub 27140 | An upper bound on the seco... |
| logfacubnd 27141 | A simple upper bound on th... |
| logfaclbnd 27142 | A lower bound on the logar... |
| logfacbnd3 27143 | Show the stronger statemen... |
| logfacrlim 27144 | Combine the estimates ~ lo... |
| logexprlim 27145 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27146 | Write out ~ logfacrlim as ... |
| mersenne 27147 | A Mersenne prime is a prim... |
| perfect1 27148 | Euclid's contribution to t... |
| perfectlem1 27149 | Lemma for ~ perfect . (Co... |
| perfectlem2 27150 | Lemma for ~ perfect . (Co... |
| perfect 27151 | The Euclid-Euler theorem, ... |
| dchrval 27154 | Value of the group of Diri... |
| dchrbas 27155 | Base set of the group of D... |
| dchrelbas 27156 | A Dirichlet character is a... |
| dchrelbas2 27157 | A Dirichlet character is a... |
| dchrelbas3 27158 | A Dirichlet character is a... |
| dchrelbasd 27159 | A Dirichlet character is a... |
| dchrrcl 27160 | Reverse closure for a Diri... |
| dchrmhm 27161 | A Dirichlet character is a... |
| dchrf 27162 | A Dirichlet character is a... |
| dchrelbas4 27163 | A Dirichlet character is a... |
| dchrzrh1 27164 | Value of a Dirichlet chara... |
| dchrzrhcl 27165 | A Dirichlet character take... |
| dchrzrhmul 27166 | A Dirichlet character is c... |
| dchrplusg 27167 | Group operation on the gro... |
| dchrmul 27168 | Group operation on the gro... |
| dchrmulcl 27169 | Closure of the group opera... |
| dchrn0 27170 | A Dirichlet character is n... |
| dchr1cl 27171 | Closure of the principal D... |
| dchrmullid 27172 | Left identity for the prin... |
| dchrinvcl 27173 | Closure of the group inver... |
| dchrabl 27174 | The set of Dirichlet chara... |
| dchrfi 27175 | The group of Dirichlet cha... |
| dchrghm 27176 | A Dirichlet character rest... |
| dchr1 27177 | Value of the principal Dir... |
| dchreq 27178 | A Dirichlet character is d... |
| dchrresb 27179 | A Dirichlet character is d... |
| dchrabs 27180 | A Dirichlet character take... |
| dchrinv 27181 | The inverse of a Dirichlet... |
| dchrabs2 27182 | A Dirichlet character take... |
| dchr1re 27183 | The principal Dirichlet ch... |
| dchrptlem1 27184 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27185 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27186 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27187 | For any element other than... |
| dchrsum2 27188 | An orthogonality relation ... |
| dchrsum 27189 | An orthogonality relation ... |
| sumdchr2 27190 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27191 | There are exactly ` phi ( ... |
| sumdchr 27192 | An orthogonality relation ... |
| dchr2sum 27193 | An orthogonality relation ... |
| sum2dchr 27194 | An orthogonality relation ... |
| bcctr 27195 | Value of the central binom... |
| pcbcctr 27196 | Prime count of a central b... |
| bcmono 27197 | The binomial coefficient i... |
| bcmax 27198 | The binomial coefficient t... |
| bcp1ctr 27199 | Ratio of two central binom... |
| bclbnd 27200 | A bound on the binomial co... |
| efexple 27201 | Convert a bound on a power... |
| bpos1lem 27202 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27203 | Bertrand's postulate, chec... |
| bposlem1 27204 | An upper bound on the prim... |
| bposlem2 27205 | There are no odd primes in... |
| bposlem3 27206 | Lemma for ~ bpos . Since ... |
| bposlem4 27207 | Lemma for ~ bpos . (Contr... |
| bposlem5 27208 | Lemma for ~ bpos . Bound ... |
| bposlem6 27209 | Lemma for ~ bpos . By usi... |
| bposlem7 27210 | Lemma for ~ bpos . The fu... |
| bposlem8 27211 | Lemma for ~ bpos . Evalua... |
| bposlem9 27212 | Lemma for ~ bpos . Derive... |
| bpos 27213 | Bertrand's postulate: ther... |
| zabsle1 27216 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27217 | When ` a ` is coprime to t... |
| lgslem2 27218 | The set ` Z ` of all integ... |
| lgslem3 27219 | The set ` Z ` of all integ... |
| lgslem4 27220 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27221 | Value of the Legendre symb... |
| lgsfval 27222 | Value of the function ` F ... |
| lgsfcl2 27223 | The function ` F ` is clos... |
| lgscllem 27224 | The Legendre symbol is an ... |
| lgsfcl 27225 | Closure of the function ` ... |
| lgsfle1 27226 | The function ` F ` has mag... |
| lgsval2lem 27227 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27228 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27229 | The Legendre symbol is an ... |
| lgs0 27230 | The Legendre symbol when t... |
| lgscl 27231 | The Legendre symbol is an ... |
| lgsle1 27232 | The Legendre symbol has ab... |
| lgsval2 27233 | The Legendre symbol at a p... |
| lgs2 27234 | The Legendre symbol at ` 2... |
| lgsval3 27235 | The Legendre symbol at an ... |
| lgsvalmod 27236 | The Legendre symbol is equ... |
| lgsval4 27237 | Restate ~ lgsval for nonze... |
| lgsfcl3 27238 | Closure of the function ` ... |
| lgsval4a 27239 | Same as ~ lgsval4 for posi... |
| lgscl1 27240 | The value of the Legendre ... |
| lgsneg 27241 | The Legendre symbol is eit... |
| lgsneg1 27242 | The Legendre symbol for no... |
| lgsmod 27243 | The Legendre (Jacobi) symb... |
| lgsdilem 27244 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27245 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27246 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27247 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27248 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27249 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27250 | The Legendre symbol is com... |
| lgsdirprm 27251 | The Legendre symbol is com... |
| lgsdir 27252 | The Legendre symbol is com... |
| lgsdilem2 27253 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27254 | The Legendre symbol is com... |
| lgsne0 27255 | The Legendre symbol is non... |
| lgsabs1 27256 | The Legendre symbol is non... |
| lgssq 27257 | The Legendre symbol at a s... |
| lgssq2 27258 | The Legendre symbol at a s... |
| lgsprme0 27259 | The Legendre symbol at any... |
| 1lgs 27260 | The Legendre symbol at ` 1... |
| lgs1 27261 | The Legendre symbol at ` 1... |
| lgsmodeq 27262 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27263 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27264 | Variation on ~ lgsdir vali... |
| lgsdinn0 27265 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27266 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27267 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27268 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27269 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27270 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27271 | The Legendre symbol for od... |
| lgsqrmod 27272 | If the Legendre symbol of ... |
| lgsqrmodndvds 27273 | If the Legendre symbol of ... |
| lgsdchrval 27274 | The Legendre symbol functi... |
| lgsdchr 27275 | The Legendre symbol functi... |
| gausslemma2dlem0a 27276 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27277 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27278 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27279 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27280 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27281 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27282 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27283 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27284 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27285 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27286 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27287 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27288 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27289 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27290 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27291 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27292 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27293 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27294 | Gauss' Lemma (see also the... |
| lgseisenlem1 27295 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27296 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27297 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27298 | Lemma for ~ lgseisen . Th... |
| lgseisen 27299 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27300 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27301 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27302 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27303 | The Law of Quadratic Recip... |
| lgsquad2lem1 27304 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27305 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27306 | Extend ~ lgsquad to coprim... |
| lgsquad3 27307 | Extend ~ lgsquad2 to integ... |
| m1lgs 27308 | The first supplement to th... |
| 2lgslem1a1 27309 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27310 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27311 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27312 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27313 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27314 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27315 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27316 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27317 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27318 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27319 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27320 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27321 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27322 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27323 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27324 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27325 | The Legendre symbol for ` ... |
| 2lgslem4 27326 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27327 | The second supplement to t... |
| 2lgsoddprmlem1 27328 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27329 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27330 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27331 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27332 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27333 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27334 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27335 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27336 | The second supplement to t... |
| 2sqlem1 27337 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27338 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27339 | Fibonacci's identity (actu... |
| 2sqlem3 27340 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27341 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27342 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27343 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27344 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27345 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27346 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27347 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27348 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27349 | Lemma for ~ 2sq . (Contri... |
| 2sq 27350 | All primes of the form ` 4... |
| 2sqblem 27351 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27352 | The converse to ~ 2sq . (... |
| 2sq2 27353 | ` 2 ` is the sum of square... |
| 2sqn0 27354 | If the sum of two squares ... |
| 2sqcoprm 27355 | If the sum of two squares ... |
| 2sqmod 27356 | Given two decompositions o... |
| 2sqmo 27357 | There exists at most one d... |
| 2sqnn0 27358 | All primes of the form ` 4... |
| 2sqnn 27359 | All primes of the form ` 4... |
| addsq2reu 27360 | For each complex number ` ... |
| addsqn2reu 27361 | For each complex number ` ... |
| addsqrexnreu 27362 | For each complex number, t... |
| addsqnreup 27363 | There is no unique decompo... |
| addsq2nreurex 27364 | For each complex number ` ... |
| addsqn2reurex2 27365 | For each complex number ` ... |
| 2sqreulem1 27366 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27367 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27368 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27369 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27370 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27371 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27372 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27373 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27374 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27375 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27376 | There exists a unique deco... |
| 2sqreunn 27377 | There exists a unique deco... |
| 2sqreult 27378 | There exists a unique deco... |
| 2sqreultb 27379 | There exists a unique deco... |
| 2sqreunnlt 27380 | There exists a unique deco... |
| 2sqreunnltb 27381 | There exists a unique deco... |
| 2sqreuop 27382 | There exists a unique deco... |
| 2sqreuopnn 27383 | There exists a unique deco... |
| 2sqreuoplt 27384 | There exists a unique deco... |
| 2sqreuopltb 27385 | There exists a unique deco... |
| 2sqreuopnnlt 27386 | There exists a unique deco... |
| 2sqreuopnnltb 27387 | There exists a unique deco... |
| 2sqreuopb 27388 | There exists a unique deco... |
| chebbnd1lem1 27389 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27390 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27391 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27392 | The Chebyshev bound: The ... |
| chtppilimlem1 27393 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27394 | Lemma for ~ chtppilim . (... |
| chtppilim 27395 | The ` theta ` function is ... |
| chto1ub 27396 | The ` theta ` function is ... |
| chebbnd2 27397 | The Chebyshev bound, part ... |
| chto1lb 27398 | The ` theta ` function is ... |
| chpchtlim 27399 | The ` psi ` and ` theta ` ... |
| chpo1ub 27400 | The ` psi ` function is up... |
| chpo1ubb 27401 | The ` psi ` function is up... |
| vmadivsum 27402 | The sum of the von Mangold... |
| vmadivsumb 27403 | Give a total bound on the ... |
| rplogsumlem1 27404 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27405 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27406 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27407 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27408 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27409 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27410 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27411 | Lemma for ~ dchrisum . Le... |
| dchrisum 27412 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27413 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27414 | The sum of the Möbius... |
| dchrvmasumlem1 27415 | An alternative expression ... |
| dchrvmasum2lem 27416 | Give an expression for ` l... |
| dchrvmasum2if 27417 | Combine the results of ~ d... |
| dchrvmasumlem2 27418 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27419 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27420 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27421 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27422 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27423 | An asymptotic approximatio... |
| dchrvmaeq0 27424 | The set ` W ` is the colle... |
| dchrisum0fval 27425 | Value of the function ` F ... |
| dchrisum0fmul 27426 | The function ` F ` , the d... |
| dchrisum0ff 27427 | The function ` F ` is a re... |
| dchrisum0flblem1 27428 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27429 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27430 | The divisor sum of a real ... |
| dchrisum0fno1 27431 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27432 | A partial result along the... |
| dchrisum0re 27433 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27434 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27435 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27436 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27437 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27438 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27439 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27440 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27441 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27442 | The sum of the Möbius... |
| dchrvmasumlem 27443 | The sum of the Möbius... |
| dchrmusum 27444 | The sum of the Möbius... |
| dchrvmasum 27445 | The sum of the von Mangold... |
| rpvmasum 27446 | The sum of the von Mangold... |
| rplogsum 27447 | The sum of ` log p / p ` o... |
| dirith2 27448 | Dirichlet's theorem: there... |
| dirith 27449 | Dirichlet's theorem: there... |
| mudivsum 27450 | Asymptotic formula for ` s... |
| mulogsumlem 27451 | Lemma for ~ mulogsum . (C... |
| mulogsum 27452 | Asymptotic formula for ... |
| logdivsum 27453 | Asymptotic analysis of ... |
| mulog2sumlem1 27454 | Asymptotic formula for ... |
| mulog2sumlem2 27455 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27456 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27457 | Asymptotic formula for ... |
| vmalogdivsum2 27458 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27459 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27460 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27461 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27462 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27463 | The Möbius inverse of... |
| log2sumbnd 27464 | Bound on the difference be... |
| selberglem1 27465 | Lemma for ~ selberg . Est... |
| selberglem2 27466 | Lemma for ~ selberg . (Co... |
| selberglem3 27467 | Lemma for ~ selberg . Est... |
| selberg 27468 | Selberg's symmetry formula... |
| selbergb 27469 | Convert eventual boundedne... |
| selberg2lem 27470 | Lemma for ~ selberg2 . Eq... |
| selberg2 27471 | Selberg's symmetry formula... |
| selberg2b 27472 | Convert eventual boundedne... |
| chpdifbndlem1 27473 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27474 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27475 | A bound on the difference ... |
| logdivbnd 27476 | A bound on a sum of logs, ... |
| selberg3lem1 27477 | Introduce a log weighting ... |
| selberg3lem2 27478 | Lemma for ~ selberg3 . Eq... |
| selberg3 27479 | Introduce a log weighting ... |
| selberg4lem1 27480 | Lemma for ~ selberg4 . Eq... |
| selberg4 27481 | The Selberg symmetry formu... |
| pntrval 27482 | Define the residual of the... |
| pntrf 27483 | Functionality of the resid... |
| pntrmax 27484 | There is a bound on the re... |
| pntrsumo1 27485 | A bound on a sum over ` R ... |
| pntrsumbnd 27486 | A bound on a sum over ` R ... |
| pntrsumbnd2 27487 | A bound on a sum over ` R ... |
| selbergr 27488 | Selberg's symmetry formula... |
| selberg3r 27489 | Selberg's symmetry formula... |
| selberg4r 27490 | Selberg's symmetry formula... |
| selberg34r 27491 | The sum of ~ selberg3r and... |
| pntsval 27492 | Define the "Selberg functi... |
| pntsf 27493 | Functionality of the Selbe... |
| selbergs 27494 | Selberg's symmetry formula... |
| selbergsb 27495 | Selberg's symmetry formula... |
| pntsval2 27496 | The Selberg function can b... |
| pntrlog2bndlem1 27497 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27498 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27499 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27500 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27501 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27502 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27503 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27504 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27505 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27506 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27507 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27508 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27509 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27510 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27511 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27512 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27513 | Lemma for ~ pnt . Establi... |
| pntlemd 27514 | Lemma for ~ pnt . Closure... |
| pntlemc 27515 | Lemma for ~ pnt . Closure... |
| pntlema 27516 | Lemma for ~ pnt . Closure... |
| pntlemb 27517 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27518 | Lemma for ~ pnt . Closure... |
| pntlemh 27519 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27520 | Lemma for ~ pnt . The "na... |
| pntlemq 27521 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27522 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27523 | Lemma for ~ pnt . The ind... |
| pntlemi 27524 | Lemma for ~ pnt . Elimina... |
| pntlemf 27525 | Lemma for ~ pnt . Add up ... |
| pntlemk 27526 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27527 | Lemma for ~ pnt . Combine... |
| pntleme 27528 | Lemma for ~ pnt . Package... |
| pntlem3 27529 | Lemma for ~ pnt . Equatio... |
| pntlemp 27530 | Lemma for ~ pnt . Wrappin... |
| pntleml 27531 | Lemma for ~ pnt . Equatio... |
| pnt3 27532 | The Prime Number Theorem, ... |
| pnt2 27533 | The Prime Number Theorem, ... |
| pnt 27534 | The Prime Number Theorem: ... |
| abvcxp 27535 | Raising an absolute value ... |
| padicfval 27536 | Value of the p-adic absolu... |
| padicval 27537 | Value of the p-adic absolu... |
| ostth2lem1 27538 | Lemma for ~ ostth2 , altho... |
| qrngbas 27539 | The base set of the field ... |
| qdrng 27540 | The rationals form a divis... |
| qrng0 27541 | The zero element of the fi... |
| qrng1 27542 | The unity element of the f... |
| qrngneg 27543 | The additive inverse in th... |
| qrngdiv 27544 | The division operation in ... |
| qabvle 27545 | By using induction on ` N ... |
| qabvexp 27546 | Induct the product rule ~ ... |
| ostthlem1 27547 | Lemma for ~ ostth . If tw... |
| ostthlem2 27548 | Lemma for ~ ostth . Refin... |
| qabsabv 27549 | The regular absolute value... |
| padicabv 27550 | The p-adic absolute value ... |
| padicabvf 27551 | The p-adic absolute value ... |
| padicabvcxp 27552 | All positive powers of the... |
| ostth1 27553 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27554 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27555 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27556 | Lemma for ~ ostth2 . (Con... |
| ostth2 27557 | - Lemma for ~ ostth : regu... |
| ostth3 27558 | - Lemma for ~ ostth : p-ad... |
| ostth 27559 | Ostrowski's theorem, which... |
| elno 27566 | Membership in the surreals... |
| sltval 27567 | The value of the surreal l... |
| bdayval 27568 | The value of the birthday ... |
| nofun 27569 | A surreal is a function. ... |
| nodmon 27570 | The domain of a surreal is... |
| norn 27571 | The range of a surreal is ... |
| nofnbday 27572 | A surreal is a function ov... |
| nodmord 27573 | The domain of a surreal ha... |
| elno2 27574 | An alternative condition f... |
| elno3 27575 | Another condition for memb... |
| sltval2 27576 | Alternate expression for s... |
| nofv 27577 | The function value of a su... |
| nosgnn0 27578 | ` (/) ` is not a surreal s... |
| nosgnn0i 27579 | If ` X ` is a surreal sign... |
| noreson 27580 | The restriction of a surre... |
| sltintdifex 27581 |
If ` A |
| sltres 27582 | If the restrictions of two... |
| noxp1o 27583 | The Cartesian product of a... |
| noseponlem 27584 | Lemma for ~ nosepon . Con... |
| nosepon 27585 | Given two unequal surreals... |
| noextend 27586 | Extending a surreal by one... |
| noextendseq 27587 | Extend a surreal by a sequ... |
| noextenddif 27588 | Calculate the place where ... |
| noextendlt 27589 | Extending a surreal with a... |
| noextendgt 27590 | Extending a surreal with a... |
| nolesgn2o 27591 | Given ` A ` less-than or e... |
| nolesgn2ores 27592 | Given ` A ` less-than or e... |
| nogesgn1o 27593 | Given ` A ` greater than o... |
| nogesgn1ores 27594 | Given ` A ` greater than o... |
| sltsolem1 27595 | Lemma for ~ sltso . The "... |
| sltso 27596 | Less-than totally orders t... |
| bdayfo 27597 | The birthday function maps... |
| fvnobday 27598 | The value of a surreal at ... |
| nosepnelem 27599 | Lemma for ~ nosepne . (Co... |
| nosepne 27600 | The value of two non-equal... |
| nosep1o 27601 | If the value of a surreal ... |
| nosep2o 27602 | If the value of a surreal ... |
| nosepdmlem 27603 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27604 | The first place two surrea... |
| nosepeq 27605 | The values of two surreals... |
| nosepssdm 27606 | Given two non-equal surrea... |
| nodenselem4 27607 | Lemma for ~ nodense . Sho... |
| nodenselem5 27608 | Lemma for ~ nodense . If ... |
| nodenselem6 27609 | The restriction of a surre... |
| nodenselem7 27610 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27611 | Lemma for ~ nodense . Giv... |
| nodense 27612 | Given two distinct surreal... |
| bdayimaon 27613 | Lemma for full-eta propert... |
| nolt02olem 27614 | Lemma for ~ nolt02o . If ... |
| nolt02o 27615 | Given ` A ` less-than ` B ... |
| nogt01o 27616 | Given ` A ` greater than `... |
| noresle 27617 | Restriction law for surrea... |
| nomaxmo 27618 | A class of surreals has at... |
| nominmo 27619 | A class of surreals has at... |
| nosupprefixmo 27620 | In any class of surreals, ... |
| noinfprefixmo 27621 | In any class of surreals, ... |
| nosupcbv 27622 | Lemma to change bound vari... |
| nosupno 27623 | The next several theorems ... |
| nosupdm 27624 | The domain of the surreal ... |
| nosupbday 27625 | Birthday bounding law for ... |
| nosupfv 27626 | The value of surreal supre... |
| nosupres 27627 | A restriction law for surr... |
| nosupbnd1lem1 27628 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27629 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27630 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27631 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27632 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27633 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27634 | Bounding law from below fo... |
| nosupbnd2lem1 27635 | Bounding law from above wh... |
| nosupbnd2 27636 | Bounding law from above fo... |
| noinfcbv 27637 | Change bound variables for... |
| noinfno 27638 | The next several theorems ... |
| noinfdm 27639 | Next, we calculate the dom... |
| noinfbday 27640 | Birthday bounding law for ... |
| noinffv 27641 | The value of surreal infim... |
| noinfres 27642 | The restriction of surreal... |
| noinfbnd1lem1 27643 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27644 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27645 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27646 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27647 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27648 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27649 | Bounding law from above fo... |
| noinfbnd2lem1 27650 | Bounding law from below wh... |
| noinfbnd2 27651 | Bounding law from below fo... |
| nosupinfsep 27652 | Given two sets of surreals... |
| noetasuplem1 27653 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27654 | Lemma for ~ noeta . The r... |
| noetasuplem3 27655 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27656 | Lemma for ~ noeta . When ... |
| noetainflem1 27657 | Lemma for ~ noeta . Estab... |
| noetainflem2 27658 | Lemma for ~ noeta . The r... |
| noetainflem3 27659 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27660 | Lemma for ~ noeta . If ` ... |
| noetalem1 27661 | Lemma for ~ noeta . Eithe... |
| noetalem2 27662 | Lemma for ~ noeta . The f... |
| noeta 27663 | The full-eta axiom for the... |
| sltirr 27666 | Surreal less-than is irref... |
| slttr 27667 | Surreal less-than is trans... |
| sltasym 27668 | Surreal less-than is asymm... |
| sltlin 27669 | Surreal less-than obeys tr... |
| slttrieq2 27670 | Trichotomy law for surreal... |
| slttrine 27671 | Trichotomy law for surreal... |
| slenlt 27672 | Surreal less-than or equal... |
| sltnle 27673 | Surreal less-than in terms... |
| sleloe 27674 | Surreal less-than or equal... |
| sletri3 27675 | Trichotomy law for surreal... |
| sltletr 27676 | Surreal transitive law. (... |
| slelttr 27677 | Surreal transitive law. (... |
| sletr 27678 | Surreal transitive law. (... |
| slttrd 27679 | Surreal less-than is trans... |
| sltletrd 27680 | Surreal less-than is trans... |
| slelttrd 27681 | Surreal less-than is trans... |
| sletrd 27682 | Surreal less-than or equal... |
| slerflex 27683 | Surreal less-than or equal... |
| sletric 27684 | Surreal trichotomy law. (... |
| maxs1 27685 | A surreal is less than or ... |
| maxs2 27686 | A surreal is less than or ... |
| mins1 27687 | The minimum of two surreal... |
| mins2 27688 | The minimum of two surreal... |
| sltled 27689 | Surreal less-than implies ... |
| sltne 27690 | Surreal less-than implies ... |
| sltlend 27691 | Surreal less-than in terms... |
| bdayfun 27692 | The birthday function is a... |
| bdayfn 27693 | The birthday function is a... |
| bdaydm 27694 | The birthday function's do... |
| bdayrn 27695 | The birthday function's ra... |
| bdayelon 27696 | The value of the birthday ... |
| nocvxminlem 27697 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27698 | Given a nonempty convex cl... |
| noprc 27699 | The surreal numbers are a ... |
| noeta2 27704 | A version of ~ noeta with ... |
| brsslt 27705 | Binary relation form of th... |
| ssltex1 27706 | The first argument of surr... |
| ssltex2 27707 | The second argument of sur... |
| ssltss1 27708 | The first argument of surr... |
| ssltss2 27709 | The second argument of sur... |
| ssltsep 27710 | The separation property of... |
| ssltd 27711 | Deduce surreal set less-th... |
| ssltsn 27712 | Surreal set less-than of t... |
| ssltsepc 27713 | Two elements of separated ... |
| ssltsepcd 27714 | Two elements of separated ... |
| sssslt1 27715 | Relation between surreal s... |
| sssslt2 27716 | Relation between surreal s... |
| nulsslt 27717 | The empty set is less-than... |
| nulssgt 27718 | The empty set is greater t... |
| conway 27719 | Conway's Simplicity Theore... |
| scutval 27720 | The value of the surreal c... |
| scutcut 27721 | Cut properties of the surr... |
| scutcl 27722 | Closure law for surreal cu... |
| scutcld 27723 | Closure law for surreal cu... |
| scutbday 27724 | The birthday of the surrea... |
| eqscut 27725 | Condition for equality to ... |
| eqscut2 27726 | Condition for equality to ... |
| sslttr 27727 | Transitive law for surreal... |
| ssltun1 27728 | Union law for surreal set ... |
| ssltun2 27729 | Union law for surreal set ... |
| scutun12 27730 | Union law for surreal cuts... |
| dmscut 27731 | The domain of the surreal ... |
| scutf 27732 | Functionality statement fo... |
| etasslt 27733 | A restatement of ~ noeta u... |
| etasslt2 27734 | A version of ~ etasslt wit... |
| scutbdaybnd 27735 | An upper bound on the birt... |
| scutbdaybnd2 27736 | An upper bound on the birt... |
| scutbdaybnd2lim 27737 | An upper bound on the birt... |
| scutbdaylt 27738 | If a surreal lies in a gap... |
| slerec 27739 | A comparison law for surre... |
| sltrec 27740 | A comparison law for surre... |
| ssltdisj 27741 | If ` A ` preceeds ` B ` , ... |
| 0sno 27746 | Surreal zero is a surreal.... |
| 1sno 27747 | Surreal one is a surreal. ... |
| bday0s 27748 | Calculate the birthday of ... |
| 0slt1s 27749 | Surreal zero is less than ... |
| bday0b 27750 | The only surreal with birt... |
| bday1s 27751 | The birthday of surreal on... |
| cuteq0 27752 | Condition for a surreal cu... |
| cuteq1 27753 | Condition for a surreal cu... |
| sgt0ne0 27754 | A positive surreal is not ... |
| sgt0ne0d 27755 | A positive surreal is not ... |
| madeval 27766 | The value of the made by f... |
| madeval2 27767 | Alternative characterizati... |
| oldval 27768 | The value of the old optio... |
| newval 27769 | The value of the new optio... |
| madef 27770 | The made function is a fun... |
| oldf 27771 | The older function is a fu... |
| newf 27772 | The new function is a func... |
| old0 27773 | No surreal is older than `... |
| madessno 27774 | Made sets are surreals. (... |
| oldssno 27775 | Old sets are surreals. (C... |
| newssno 27776 | New sets are surreals. (C... |
| leftval 27777 | The value of the left opti... |
| rightval 27778 | The value of the right opt... |
| leftf 27779 | The functionality of the l... |
| rightf 27780 | The functionality of the r... |
| elmade 27781 | Membership in the made fun... |
| elmade2 27782 | Membership in the made fun... |
| elold 27783 | Membership in an old set. ... |
| ssltleft 27784 | A surreal is greater than ... |
| ssltright 27785 | A surreal is less than its... |
| lltropt 27786 | The left options of a surr... |
| made0 27787 | The only surreal made on d... |
| new0 27788 | The only surreal new on da... |
| old1 27789 | The only surreal older tha... |
| madess 27790 | If ` A ` is less than or e... |
| oldssmade 27791 | The older-than set is a su... |
| leftssold 27792 | The left options are a sub... |
| rightssold 27793 | The right options are a su... |
| leftssno 27794 | The left set of a surreal ... |
| rightssno 27795 | The right set of a surreal... |
| madecut 27796 | Given a section that is a ... |
| madeun 27797 | The made set is the union ... |
| madeoldsuc 27798 | The made set is the old se... |
| oldsuc 27799 | The value of the old set a... |
| oldlim 27800 | The value of the old set a... |
| madebdayim 27801 | If a surreal is a member o... |
| oldbdayim 27802 | If ` X ` is in the old set... |
| oldirr 27803 | No surreal is a member of ... |
| leftirr 27804 | No surreal is a member of ... |
| rightirr 27805 | No surreal is a member of ... |
| left0s 27806 | The left set of ` 0s ` is ... |
| right0s 27807 | The right set of ` 0s ` is... |
| left1s 27808 | The left set of ` 1s ` is ... |
| right1s 27809 | The right set of ` 1s ` is... |
| lrold 27810 | The union of the left and ... |
| madebdaylemold 27811 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27812 | Lemma for ~ madebday . If... |
| madebday 27813 | A surreal is part of the s... |
| oldbday 27814 | A surreal is part of the s... |
| newbday 27815 | A surreal is an element of... |
| lrcut 27816 | A surreal is equal to the ... |
| scutfo 27817 | The surreal cut function i... |
| sltn0 27818 | If ` X ` is less than ` Y ... |
| lruneq 27819 | If two surreals share a bi... |
| sltlpss 27820 | If two surreals share a bi... |
| slelss 27821 | If two surreals ` A ` and ... |
| 0elold 27822 | Zero is in the old set of ... |
| 0elleft 27823 | Zero is in the left set of... |
| 0elright 27824 | Zero is in the right set o... |
| cofsslt 27825 | If every element of ` A ` ... |
| coinitsslt 27826 | If ` B ` is coinitial with... |
| cofcut1 27827 | If ` C ` is cofinal with `... |
| cofcut1d 27828 | If ` C ` is cofinal with `... |
| cofcut2 27829 | If ` A ` and ` C ` are mut... |
| cofcut2d 27830 | If ` A ` and ` C ` are mut... |
| cofcutr 27831 | If ` X ` is the cut of ` A... |
| cofcutr1d 27832 | If ` X ` is the cut of ` A... |
| cofcutr2d 27833 | If ` X ` is the cut of ` A... |
| cofcutrtime 27834 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 27835 | If ` X ` is a timely cut o... |
| cofcutrtime2d 27836 | If ` X ` is a timely cut o... |
| cofss 27837 | Cofinality for a subset. ... |
| coiniss 27838 | Coinitiality for a subset.... |
| cutlt 27839 | Eliminating all elements b... |
| cutpos 27840 | Reduce the elements of a c... |
| lrrecval 27843 | The next step in the devel... |
| lrrecval2 27844 | Next, we establish an alte... |
| lrrecpo 27845 | Now, we establish that ` R... |
| lrrecse 27846 | Next, we show that ` R ` i... |
| lrrecfr 27847 | Now we show that ` R ` is ... |
| lrrecpred 27848 | Finally, we calculate the ... |
| noinds 27849 | Induction principle for a ... |
| norecfn 27850 | Surreal recursion over one... |
| norecov 27851 | Calculate the value of the... |
| noxpordpo 27854 | To get through most of the... |
| noxpordfr 27855 | Next we establish the foun... |
| noxpordse 27856 | Next we establish the set-... |
| noxpordpred 27857 | Next we calculate the pred... |
| no2indslem 27858 | Double induction on surrea... |
| no2inds 27859 | Double induction on surrea... |
| norec2fn 27860 | The double-recursion opera... |
| norec2ov 27861 | The value of the double-re... |
| no3inds 27862 | Triple induction over surr... |
| addsfn 27865 | Surreal addition is a func... |
| addsval 27866 | The value of surreal addit... |
| addsval2 27867 | The value of surreal addit... |
| addsrid 27868 | Surreal addition to zero i... |
| addsridd 27869 | Surreal addition to zero i... |
| addscom 27870 | Surreal addition commutes.... |
| addscomd 27871 | Surreal addition commutes.... |
| addslid 27872 | Surreal addition to zero i... |
| addsproplem1 27873 | Lemma for surreal addition... |
| addsproplem2 27874 | Lemma for surreal addition... |
| addsproplem3 27875 | Lemma for surreal addition... |
| addsproplem4 27876 | Lemma for surreal addition... |
| addsproplem5 27877 | Lemma for surreal addition... |
| addsproplem6 27878 | Lemma for surreal addition... |
| addsproplem7 27879 | Lemma for surreal addition... |
| addsprop 27880 | Inductively show that surr... |
| addscutlem 27881 | Lemma for ~ addscut . Sho... |
| addscut 27882 | Demonstrate the cut proper... |
| addscut2 27883 | Show that the cut involved... |
| addscld 27884 | Surreal numbers are closed... |
| addscl 27885 | Surreal numbers are closed... |
| addsf 27886 | Function statement for sur... |
| addsfo 27887 | Surreal addition is onto. ... |
| peano2no 27888 | A theorem for surreals tha... |
| sltadd1im 27889 | Surreal less-than is prese... |
| sltadd2im 27890 | Surreal less-than is prese... |
| sleadd1im 27891 | Surreal less-than or equal... |
| sleadd2im 27892 | Surreal less-than or equal... |
| sleadd1 27893 | Addition to both sides of ... |
| sleadd2 27894 | Addition to both sides of ... |
| sltadd2 27895 | Addition to both sides of ... |
| sltadd1 27896 | Addition to both sides of ... |
| addscan2 27897 | Cancellation law for surre... |
| addscan1 27898 | Cancellation law for surre... |
| sleadd1d 27899 | Addition to both sides of ... |
| sleadd2d 27900 | Addition to both sides of ... |
| sltadd2d 27901 | Addition to both sides of ... |
| sltadd1d 27902 | Addition to both sides of ... |
| addscan2d 27903 | Cancellation law for surre... |
| addscan1d 27904 | Cancellation law for surre... |
| addsuniflem 27905 | Lemma for ~ addsunif . St... |
| addsunif 27906 | Uniformity theorem for sur... |
| addsasslem1 27907 | Lemma for addition associa... |
| addsasslem2 27908 | Lemma for addition associa... |
| addsass 27909 | Surreal addition is associ... |
| addsassd 27910 | Surreal addition is associ... |
| adds32d 27911 | Commutative/associative la... |
| adds12d 27912 | Commutative/associative la... |
| adds4d 27913 | Rearrangement of four term... |
| adds42d 27914 | Rearrangement of four term... |
| sltaddpos1d 27915 | Addition of a positive num... |
| sltaddpos2d 27916 | Addition of a positive num... |
| slt2addd 27917 | Adding both sides of two s... |
| addsgt0d 27918 | The sum of two positive su... |
| negsfn 27923 | Surreal negation is a func... |
| subsfn 27924 | Surreal subtraction is a f... |
| negsval 27925 | The value of the surreal n... |
| negs0s 27926 | Negative surreal zero is s... |
| negsproplem1 27927 | Lemma for surreal negation... |
| negsproplem2 27928 | Lemma for surreal negation... |
| negsproplem3 27929 | Lemma for surreal negation... |
| negsproplem4 27930 | Lemma for surreal negation... |
| negsproplem5 27931 | Lemma for surreal negation... |
| negsproplem6 27932 | Lemma for surreal negation... |
| negsproplem7 27933 | Lemma for surreal negation... |
| negsprop 27934 | Show closure and ordering ... |
| negscl 27935 | The surreals are closed un... |
| negscld 27936 | The surreals are closed un... |
| sltnegim 27937 | The forward direction of t... |
| negscut 27938 | The cut properties of surr... |
| negscut2 27939 | The cut that defines surre... |
| negsid 27940 | Surreal addition of a numb... |
| negsidd 27941 | Surreal addition of a numb... |
| negsex 27942 | Every surreal has a negati... |
| negnegs 27943 | A surreal is equal to the ... |
| sltneg 27944 | Negative of both sides of ... |
| sleneg 27945 | Negative of both sides of ... |
| sltnegd 27946 | Negative of both sides of ... |
| slenegd 27947 | Negative of both sides of ... |
| negs11 27948 | Surreal negation is one-to... |
| negsdi 27949 | Distribution of surreal ne... |
| slt0neg2d 27950 | Comparison of a surreal an... |
| negsf 27951 | Function statement for sur... |
| negsfo 27952 | Function statement for sur... |
| negsf1o 27953 | Surreal negation is a bije... |
| negsunif 27954 | Uniformity property for su... |
| negsbdaylem 27955 | Lemma for ~ negsbday . Bo... |
| negsbday 27956 | Negation of a surreal numb... |
| subsval 27957 | The value of surreal subtr... |
| subsvald 27958 | The value of surreal subtr... |
| subscl 27959 | Closure law for surreal su... |
| subscld 27960 | Closure law for surreal su... |
| negsval2 27961 | Surreal negation in terms ... |
| negsval2d 27962 | Surreal negation in terms ... |
| subsid1 27963 | Identity law for subtracti... |
| subsid 27964 | Subtraction of a surreal f... |
| subadds 27965 | Relationship between addit... |
| subaddsd 27966 | Relationship between addit... |
| pncans 27967 | Cancellation law for surre... |
| pncan3s 27968 | Subtraction and addition o... |
| pncan2s 27969 | Cancellation law for surre... |
| npcans 27970 | Cancellation law for surre... |
| sltsub1 27971 | Subtraction from both side... |
| sltsub2 27972 | Subtraction from both side... |
| sltsub1d 27973 | Subtraction from both side... |
| sltsub2d 27974 | Subtraction from both side... |
| negsubsdi2d 27975 | Distribution of negative o... |
| addsubsassd 27976 | Associative-type law for s... |
| addsubsd 27977 | Law for surreal addition a... |
| sltsubsubbd 27978 | Equivalence for the surrea... |
| sltsubsub2bd 27979 | Equivalence for the surrea... |
| sltsubsub3bd 27980 | Equivalence for the surrea... |
| slesubsubbd 27981 | Equivalence for the surrea... |
| slesubsub2bd 27982 | Equivalence for the surrea... |
| slesubsub3bd 27983 | Equivalence for the surrea... |
| sltsubaddd 27984 | Surreal less-than relation... |
| sltsubadd2d 27985 | Surreal less-than relation... |
| sltaddsubd 27986 | Surreal less-than relation... |
| sltaddsub2d 27987 | Surreal less-than relation... |
| subsubs4d 27988 | Law for double surreal sub... |
| subsubs2d 27989 | Law for double surreal sub... |
| nncansd 27990 | Cancellation law for surre... |
| posdifsd 27991 | Comparison of two surreals... |
| sltsubposd 27992 | Subtraction of a positive ... |
| mulsfn 27995 | Surreal multiplication is ... |
| mulsval 27996 | The value of surreal multi... |
| mulsval2lem 27997 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 27998 | The value of surreal multi... |
| muls01 27999 | Surreal multiplication by ... |
| mulsrid 28000 | Surreal one is a right ide... |
| mulsridd 28001 | Surreal one is a right ide... |
| mulsproplemcbv 28002 | Lemma for surreal multipli... |
| mulsproplem1 28003 | Lemma for surreal multipli... |
| mulsproplem2 28004 | Lemma for surreal multipli... |
| mulsproplem3 28005 | Lemma for surreal multipli... |
| mulsproplem4 28006 | Lemma for surreal multipli... |
| mulsproplem5 28007 | Lemma for surreal multipli... |
| mulsproplem6 28008 | Lemma for surreal multipli... |
| mulsproplem7 28009 | Lemma for surreal multipli... |
| mulsproplem8 28010 | Lemma for surreal multipli... |
| mulsproplem9 28011 | Lemma for surreal multipli... |
| mulsproplem10 28012 | Lemma for surreal multipli... |
| mulsproplem11 28013 | Lemma for surreal multipli... |
| mulsproplem12 28014 | Lemma for surreal multipli... |
| mulsproplem13 28015 | Lemma for surreal multipli... |
| mulsproplem14 28016 | Lemma for surreal multipli... |
| mulsprop 28017 | Surreals are closed under ... |
| mulscutlem 28018 | Lemma for ~ mulscut . Sta... |
| mulscut 28019 | Show the cut properties of... |
| mulscut2 28020 | Show that the cut involved... |
| mulscl 28021 | The surreals are closed un... |
| mulscld 28022 | The surreals are closed un... |
| sltmul 28023 | An ordering relationship f... |
| sltmuld 28024 | An ordering relationship f... |
| slemuld 28025 | An ordering relationship f... |
| mulscom 28026 | Surreal multiplication com... |
| mulscomd 28027 | Surreal multiplication com... |
| muls02 28028 | Surreal multiplication by ... |
| mulslid 28029 | Surreal one is a left iden... |
| mulslidd 28030 | Surreal one is a left iden... |
| mulsgt0 28031 | The product of two positiv... |
| mulsgt0d 28032 | The product of two positiv... |
| mulsge0d 28033 | The product of two non-neg... |
| ssltmul1 28034 | One surreal set less-than ... |
| ssltmul2 28035 | One surreal set less-than ... |
| mulsuniflem 28036 | Lemma for ~ mulsunif . St... |
| mulsunif 28037 | Surreal multiplication has... |
| addsdilem1 28038 | Lemma for surreal distribu... |
| addsdilem2 28039 | Lemma for surreal distribu... |
| addsdilem3 28040 | Lemma for ~ addsdi . Show... |
| addsdilem4 28041 | Lemma for ~ addsdi . Show... |
| addsdi 28042 | Distributive law for surre... |
| addsdid 28043 | Distributive law for surre... |
| addsdird 28044 | Distributive law for surre... |
| subsdid 28045 | Distribution of surreal mu... |
| subsdird 28046 | Distribution of surreal mu... |
| mulnegs1d 28047 | Product with negative is n... |
| mulnegs2d 28048 | Product with negative is n... |
| mul2negsd 28049 | Surreal product of two neg... |
| mulsasslem1 28050 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28051 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28052 | Lemma for ~ mulsass . Dem... |
| mulsass 28053 | Associative law for surrea... |
| mulsassd 28054 | Associative law for surrea... |
| muls4d 28055 | Rearrangement of four surr... |
| mulsunif2lem 28056 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28057 | Alternate expression for s... |
| sltmul2 28058 | Multiplication of both sid... |
| sltmul2d 28059 | Multiplication of both sid... |
| sltmul1d 28060 | Multiplication of both sid... |
| slemul2d 28061 | Multiplication of both sid... |
| slemul1d 28062 | Multiplication of both sid... |
| sltmulneg1d 28063 | Multiplication of both sid... |
| sltmulneg2d 28064 | Multiplication of both sid... |
| mulscan2dlem 28065 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28066 | Cancellation of surreal mu... |
| mulscan1d 28067 | Cancellation of surreal mu... |
| muls12d 28068 | Commutative/associative la... |
| slemul1ad 28069 | Multiplication of both sid... |
| sltmul12ad 28070 | Comparison of the product ... |
| divsmo 28071 | Uniqueness of surreal inve... |
| muls0ord 28072 | If a surreal product is ze... |
| mulsne0bd 28073 | The product of two non-zer... |
| divsval 28076 | The value of surreal divis... |
| norecdiv 28077 | If a surreal has a recipro... |
| noreceuw 28078 | If a surreal has a recipro... |
| divsmulw 28079 | Relationship between surre... |
| divsmulwd 28080 | Relationship between surre... |
| divsclw 28081 | Weak division closure law.... |
| divsclwd 28082 | Weak division closure law.... |
| divscan2wd 28083 | A weak cancellation law fo... |
| divscan1wd 28084 | A weak cancellation law fo... |
| sltdivmulwd 28085 | Surreal less-than relation... |
| sltdivmul2wd 28086 | Surreal less-than relation... |
| sltmuldivwd 28087 | Surreal less-than relation... |
| sltmuldiv2wd 28088 | Surreal less-than relation... |
| divsasswd 28089 | An associative law for sur... |
| divs1 28090 | A surreal divided by one i... |
| precsexlemcbv 28091 | Lemma for surreal reciproc... |
| precsexlem1 28092 | Lemma for surreal reciproc... |
| precsexlem2 28093 | Lemma for surreal reciproc... |
| precsexlem3 28094 | Lemma for surreal reciproc... |
| precsexlem4 28095 | Lemma for surreal reciproc... |
| precsexlem5 28096 | Lemma for surreal reciproc... |
| precsexlem6 28097 | Lemma for surreal reciproc... |
| precsexlem7 28098 | Lemma for surreal reciproc... |
| precsexlem8 28099 | Lemma for surreal reciproc... |
| precsexlem9 28100 | Lemma for surreal reciproc... |
| precsexlem10 28101 | Lemma for surreal reciproc... |
| precsexlem11 28102 | Lemma for surreal reciproc... |
| precsex 28103 | Every positive surreal has... |
| recsex 28104 | A non-zero surreal has a r... |
| recsexd 28105 | A non-zero surreal has a r... |
| divsmul 28106 | Relationship between surre... |
| divsmuld 28107 | Relationship between surre... |
| divscl 28108 | Surreal division closure l... |
| divscld 28109 | Surreal division closure l... |
| divscan2d 28110 | A cancellation law for sur... |
| divscan1d 28111 | A cancellation law for sur... |
| sltdivmuld 28112 | Surreal less-than relation... |
| sltdivmul2d 28113 | Surreal less-than relation... |
| sltmuldivd 28114 | Surreal less-than relation... |
| sltmuldiv2d 28115 | Surreal less-than relation... |
| divsassd 28116 | An associative law for sur... |
| divmuldivsd 28117 | Multiplication of two surr... |
| abssval 28120 | The value of surreal absol... |
| absscl 28121 | Closure law for surreal ab... |
| abssid 28122 | The absolute value of a no... |
| abs0s 28123 | The absolute value of surr... |
| abssnid 28124 | For a negative surreal, it... |
| absmuls 28125 | Surreal absolute value dis... |
| abssge0 28126 | The absolute value of a su... |
| abssor 28127 | The absolute value of a su... |
| abssneg 28128 | Surreal absolute value of ... |
| sleabs 28129 | A surreal is less than or ... |
| absslt 28130 | Surreal absolute value and... |
| elons 28133 | Membership in the class of... |
| onssno 28134 | The surreal ordinals are a... |
| onsno 28135 | A surreal ordinal is a sur... |
| 0ons 28136 | Surreal zero is a surreal ... |
| 1ons 28137 | Surreal one is a surreal o... |
| elons2 28138 | A surreal is ordinal iff i... |
| elons2d 28139 | The cut of any set of surr... |
| sltonold 28140 | The class of ordinals less... |
| sltonex 28141 | The class of ordinals less... |
| onscutleft 28142 | A surreal ordinal is equal... |
| seqsex 28145 | Existence of the surreal s... |
| seqseq123d 28146 | Equality deduction for the... |
| nfseqs 28147 | Hypothesis builder for the... |
| seqsval 28148 | The value of the surreal s... |
| noseqex 28149 | The next several theorems ... |
| noseq0 28150 | The surreal ` A ` is a mem... |
| noseqp1 28151 | One plus an element of ` Z... |
| noseqind 28152 | Peano's inductive postulat... |
| noseqinds 28153 | Induction schema for surre... |
| noseqssno 28154 | A surreal sequence is a su... |
| noseqno 28155 | An element of a surreal se... |
| om2noseq0 28156 | The mapping ` G ` is a one... |
| om2noseqsuc 28157 | The value of ` G ` at a su... |
| om2noseqfo 28158 | Function statement for ` G... |
| om2noseqlt 28159 | Surreal less-than relation... |
| om2noseqlt2 28160 | The mapping ` G ` preserve... |
| om2noseqf1o 28161 | ` G ` is a bijection. (Co... |
| om2noseqiso 28162 | ` G ` is an isomorphism fr... |
| om2noseqoi 28163 | An alternative definition ... |
| om2noseqrdg 28164 | A helper lemma for the val... |
| noseqrdglem 28165 | A helper lemma for the val... |
| noseqrdgfn 28166 | The recursive definition g... |
| noseqrdg0 28167 | Initial value of a recursi... |
| noseqrdgsuc 28168 | Successor value of a recur... |
| seqsfn 28169 | The surreal sequence build... |
| seqs1 28170 | The value of the surreal s... |
| seqsp1 28171 | The value of the surreal s... |
| n0sex 28176 | The set of all non-negativ... |
| nnsex 28177 | The set of all positive su... |
| peano5n0s 28178 | Peano's inductive postulat... |
| n0ssno 28179 | The non-negative surreal i... |
| nnssn0s 28180 | The positive surreal integ... |
| nnssno 28181 | The positive surreal integ... |
| n0sno 28182 | A non-negative surreal int... |
| nnsno 28183 | A positive surreal integer... |
| n0snod 28184 | A non-negative surreal int... |
| nnsnod 28185 | A positive surreal integer... |
| 0n0s 28186 | Peano postulate: ` 0s ` is... |
| peano2n0s 28187 | Peano postulate: the succe... |
| dfn0s2 28188 | Alternate definition of th... |
| n0sind 28189 | Principle of Mathematical ... |
| n0scut 28190 | A cut form for surreal nat... |
| n0ons 28191 | A surreal natural is a sur... |
| nnne0s 28192 | A surreal positive integer... |
| n0sge0 28193 | A non-negative integer is ... |
| nnsgt0 28194 | A positive integer is grea... |
| elnns 28195 | Membership in the positive... |
| elnns2 28196 | A positive surreal integer... |
| n0addscl 28197 | The non-negative surreal i... |
| n0mulscl 28198 | The non-negative surreal i... |
| nnaddscl 28199 | The positive surreal integ... |
| nnmulscl 28200 | The positive surreal integ... |
| 1n0s 28201 | Surreal one is a non-negat... |
| 1nns 28202 | Surreal one is a positive ... |
| peano2nns 28203 | Peano postulate for positi... |
| n0sbday 28204 | A non-negative surreal int... |
| n0ssold 28205 | The non-negative surreal i... |
| nnsrecgt0d 28206 | The reciprocal of a positi... |
| seqn0sfn 28207 | The surreal sequence build... |
| elreno 28210 | Membership in the set of s... |
| recut 28211 | The cut involved in defini... |
| 0reno 28212 | Surreal zero is a surreal ... |
| renegscl 28213 | The surreal reals are clos... |
| readdscl 28214 | The surreal reals are clos... |
| remulscllem1 28215 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28216 | Lemma for ~ remulscl . Bo... |
| remulscl 28217 | The surreal reals are clos... |
| itvndx 28228 | Index value of the Interva... |
| lngndx 28229 | Index value of the "line" ... |
| itvid 28230 | Utility theorem: index-ind... |
| lngid 28231 | Utility theorem: index-ind... |
| slotsinbpsd 28232 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28233 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28234 | The slot for the line is n... |
| trkgstr 28235 | Functionality of a Tarski ... |
| trkgbas 28236 | The base set of a Tarski g... |
| trkgdist 28237 | The measure of a distance ... |
| trkgitv 28238 | The congruence relation in... |
| istrkgc 28245 | Property of being a Tarski... |
| istrkgb 28246 | Property of being a Tarski... |
| istrkgcb 28247 | Property of being a Tarski... |
| istrkge 28248 | Property of fulfilling Euc... |
| istrkgl 28249 | Building lines from the se... |
| istrkgld 28250 | Property of fulfilling the... |
| istrkg2ld 28251 | Property of fulfilling the... |
| istrkg3ld 28252 | Property of fulfilling the... |
| axtgcgrrflx 28253 | Axiom of reflexivity of co... |
| axtgcgrid 28254 | Axiom of identity of congr... |
| axtgsegcon 28255 | Axiom of segment construct... |
| axtg5seg 28256 | Five segments axiom, Axiom... |
| axtgbtwnid 28257 | Identity of Betweenness. ... |
| axtgpasch 28258 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28259 | Axiom of Continuity. Axio... |
| axtgcont 28260 | Axiom of Continuity. Axio... |
| axtglowdim2 28261 | Lower dimension axiom for ... |
| axtgupdim2 28262 | Upper dimension axiom for ... |
| axtgeucl 28263 | Euclid's Axiom. Axiom A10... |
| tgjustf 28264 | Given any function ` F ` ,... |
| tgjustr 28265 | Given any equivalence rela... |
| tgjustc1 28266 | A justification for using ... |
| tgjustc2 28267 | A justification for using ... |
| tgcgrcomimp 28268 | Congruence commutes on the... |
| tgcgrcomr 28269 | Congruence commutes on the... |
| tgcgrcoml 28270 | Congruence commutes on the... |
| tgcgrcomlr 28271 | Congruence commutes on bot... |
| tgcgreqb 28272 | Congruence and equality. ... |
| tgcgreq 28273 | Congruence and equality. ... |
| tgcgrneq 28274 | Congruence and equality. ... |
| tgcgrtriv 28275 | Degenerate segments are co... |
| tgcgrextend 28276 | Link congruence over a pai... |
| tgsegconeq 28277 | Two points that satisfy th... |
| tgbtwntriv2 28278 | Betweenness always holds f... |
| tgbtwncom 28279 | Betweenness commutes. The... |
| tgbtwncomb 28280 | Betweenness commutes, bico... |
| tgbtwnne 28281 | Betweenness and inequality... |
| tgbtwntriv1 28282 | Betweenness always holds f... |
| tgbtwnswapid 28283 | If you can swap the first ... |
| tgbtwnintr 28284 | Inner transitivity law for... |
| tgbtwnexch3 28285 | Exchange the first endpoin... |
| tgbtwnouttr2 28286 | Outer transitivity law for... |
| tgbtwnexch2 28287 | Exchange the outer point o... |
| tgbtwnouttr 28288 | Outer transitivity law for... |
| tgbtwnexch 28289 | Outer transitivity law for... |
| tgtrisegint 28290 | A line segment between two... |
| tglowdim1 28291 | Lower dimension axiom for ... |
| tglowdim1i 28292 | Lower dimension axiom for ... |
| tgldimor 28293 | Excluded-middle like state... |
| tgldim0eq 28294 | In dimension zero, any two... |
| tgldim0itv 28295 | In dimension zero, any two... |
| tgldim0cgr 28296 | In dimension zero, any two... |
| tgbtwndiff 28297 | There is always a ` c ` di... |
| tgdim01 28298 | In geometries of dimension... |
| tgifscgr 28299 | Inner five segment congrue... |
| tgcgrsub 28300 | Removing identical parts f... |
| iscgrg 28303 | The congruence property fo... |
| iscgrgd 28304 | The property for two seque... |
| iscgrglt 28305 | The property for two seque... |
| trgcgrg 28306 | The property for two trian... |
| trgcgr 28307 | Triangle congruence. (Con... |
| ercgrg 28308 | The shape congruence relat... |
| tgcgrxfr 28309 | A line segment can be divi... |
| cgr3id 28310 | Reflexivity law for three-... |
| cgr3simp1 28311 | Deduce segment congruence ... |
| cgr3simp2 28312 | Deduce segment congruence ... |
| cgr3simp3 28313 | Deduce segment congruence ... |
| cgr3swap12 28314 | Permutation law for three-... |
| cgr3swap23 28315 | Permutation law for three-... |
| cgr3swap13 28316 | Permutation law for three-... |
| cgr3rotr 28317 | Permutation law for three-... |
| cgr3rotl 28318 | Permutation law for three-... |
| trgcgrcom 28319 | Commutative law for three-... |
| cgr3tr 28320 | Transitivity law for three... |
| tgbtwnxfr 28321 | A condition for extending ... |
| tgcgr4 28322 | Two quadrilaterals to be c... |
| isismt 28325 | Property of being an isome... |
| ismot 28326 | Property of being an isome... |
| motcgr 28327 | Property of a motion: dist... |
| idmot 28328 | The identity is a motion. ... |
| motf1o 28329 | Motions are bijections. (... |
| motcl 28330 | Closure of motions. (Cont... |
| motco 28331 | The composition of two mot... |
| cnvmot 28332 | The converse of a motion i... |
| motplusg 28333 | The operation for motions ... |
| motgrp 28334 | The motions of a geometry ... |
| motcgrg 28335 | Property of a motion: dist... |
| motcgr3 28336 | Property of a motion: dist... |
| tglng 28337 | Lines of a Tarski Geometry... |
| tglnfn 28338 | Lines as functions. (Cont... |
| tglnunirn 28339 | Lines are sets of points. ... |
| tglnpt 28340 | Lines are sets of points. ... |
| tglngne 28341 | It takes two different poi... |
| tglngval 28342 | The line going through poi... |
| tglnssp 28343 | Lines are subset of the ge... |
| tgellng 28344 | Property of lying on the l... |
| tgcolg 28345 | We choose the notation ` (... |
| btwncolg1 28346 | Betweenness implies coline... |
| btwncolg2 28347 | Betweenness implies coline... |
| btwncolg3 28348 | Betweenness implies coline... |
| colcom 28349 | Swapping the points defini... |
| colrot1 28350 | Rotating the points defini... |
| colrot2 28351 | Rotating the points defini... |
| ncolcom 28352 | Swapping non-colinear poin... |
| ncolrot1 28353 | Rotating non-colinear poin... |
| ncolrot2 28354 | Rotating non-colinear poin... |
| tgdim01ln 28355 | In geometries of dimension... |
| ncoltgdim2 28356 | If there are three non-col... |
| lnxfr 28357 | Transfer law for colineari... |
| lnext 28358 | Extend a line with a missi... |
| tgfscgr 28359 | Congruence law for the gen... |
| lncgr 28360 | Congruence rule for lines.... |
| lnid 28361 | Identity law for points on... |
| tgidinside 28362 | Law for finding a point in... |
| tgbtwnconn1lem1 28363 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 28364 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 28365 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 28366 | Connectivity law for betwe... |
| tgbtwnconn2 28367 | Another connectivity law f... |
| tgbtwnconn3 28368 | Inner connectivity law for... |
| tgbtwnconnln3 28369 | Derive colinearity from be... |
| tgbtwnconn22 28370 | Double connectivity law fo... |
| tgbtwnconnln1 28371 | Derive colinearity from be... |
| tgbtwnconnln2 28372 | Derive colinearity from be... |
| legval 28375 | Value of the less-than rel... |
| legov 28376 | Value of the less-than rel... |
| legov2 28377 | An equivalent definition o... |
| legid 28378 | Reflexivity of the less-th... |
| btwnleg 28379 | Betweenness implies less-t... |
| legtrd 28380 | Transitivity of the less-t... |
| legtri3 28381 | Equality from the less-tha... |
| legtrid 28382 | Trichotomy law for the les... |
| leg0 28383 | Degenerated (zero-length) ... |
| legeq 28384 | Deduce equality from "less... |
| legbtwn 28385 | Deduce betweenness from "l... |
| tgcgrsub2 28386 | Removing identical parts f... |
| ltgseg 28387 | The set ` E ` denotes the ... |
| ltgov 28388 | Strict "shorter than" geom... |
| legov3 28389 | An equivalent definition o... |
| legso 28390 | The "shorter than" relatio... |
| ishlg 28393 | Rays : Definition 6.1 of ... |
| hlcomb 28394 | The half-line relation com... |
| hlcomd 28395 | The half-line relation com... |
| hlne1 28396 | The half-line relation imp... |
| hlne2 28397 | The half-line relation imp... |
| hlln 28398 | The half-line relation imp... |
| hleqnid 28399 | The endpoint does not belo... |
| hlid 28400 | The half-line relation is ... |
| hltr 28401 | The half-line relation is ... |
| hlbtwn 28402 | Betweenness is a sufficien... |
| btwnhl1 28403 | Deduce half-line from betw... |
| btwnhl2 28404 | Deduce half-line from betw... |
| btwnhl 28405 | Swap betweenness for a hal... |
| lnhl 28406 | Either a point ` C ` on th... |
| hlcgrex 28407 | Construct a point on a hal... |
| hlcgreulem 28408 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 28409 | The point constructed in ~... |
| btwnlng1 28410 | Betweenness implies coline... |
| btwnlng2 28411 | Betweenness implies coline... |
| btwnlng3 28412 | Betweenness implies coline... |
| lncom 28413 | Swapping the points defini... |
| lnrot1 28414 | Rotating the points defini... |
| lnrot2 28415 | Rotating the points defini... |
| ncolne1 28416 | Non-colinear points are di... |
| ncolne2 28417 | Non-colinear points are di... |
| tgisline 28418 | The property of being a pr... |
| tglnne 28419 | It takes two different poi... |
| tglndim0 28420 | There are no lines in dime... |
| tgelrnln 28421 | The property of being a pr... |
| tglineeltr 28422 | Transitivity law for lines... |
| tglineelsb2 28423 | If ` S ` lies on PQ , then... |
| tglinerflx1 28424 | Reflexivity law for line m... |
| tglinerflx2 28425 | Reflexivity law for line m... |
| tglinecom 28426 | Commutativity law for line... |
| tglinethru 28427 | If ` A ` is a line contain... |
| tghilberti1 28428 | There is a line through an... |
| tghilberti2 28429 | There is at most one line ... |
| tglinethrueu 28430 | There is a unique line goi... |
| tglnne0 28431 | A line ` A ` has at least ... |
| tglnpt2 28432 | Find a second point on a l... |
| tglineintmo 28433 | Two distinct lines interse... |
| tglineineq 28434 | Two distinct lines interse... |
| tglineneq 28435 | Given three non-colinear p... |
| tglineinteq 28436 | Two distinct lines interse... |
| ncolncol 28437 | Deduce non-colinearity fro... |
| coltr 28438 | A transitivity law for col... |
| coltr3 28439 | A transitivity law for col... |
| colline 28440 | Three points are colinear ... |
| tglowdim2l 28441 | Reformulation of the lower... |
| tglowdim2ln 28442 | There is always one point ... |
| mirreu3 28445 | Existential uniqueness of ... |
| mirval 28446 | Value of the point inversi... |
| mirfv 28447 | Value of the point inversi... |
| mircgr 28448 | Property of the image by t... |
| mirbtwn 28449 | Property of the image by t... |
| ismir 28450 | Property of the image by t... |
| mirf 28451 | Point inversion as functio... |
| mircl 28452 | Closure of the point inver... |
| mirmir 28453 | The point inversion functi... |
| mircom 28454 | Variation on ~ mirmir . (... |
| mirreu 28455 | Any point has a unique ant... |
| mireq 28456 | Equality deduction for poi... |
| mirinv 28457 | The only invariant point o... |
| mirne 28458 | Mirror of non-center point... |
| mircinv 28459 | The center point is invari... |
| mirf1o 28460 | The point inversion functi... |
| miriso 28461 | The point inversion functi... |
| mirbtwni 28462 | Point inversion preserves ... |
| mirbtwnb 28463 | Point inversion preserves ... |
| mircgrs 28464 | Point inversion preserves ... |
| mirmir2 28465 | Point inversion of a point... |
| mirmot 28466 | Point investion is a motio... |
| mirln 28467 | If two points are on the s... |
| mirln2 28468 | If a point and its mirror ... |
| mirconn 28469 | Point inversion of connect... |
| mirhl 28470 | If two points ` X ` and ` ... |
| mirbtwnhl 28471 | If the center of the point... |
| mirhl2 28472 | Deduce half-line relation ... |
| mircgrextend 28473 | Link congruence over a pai... |
| mirtrcgr 28474 | Point inversion of one poi... |
| mirauto 28475 | Point inversion preserves ... |
| miduniq 28476 | Uniqueness of the middle p... |
| miduniq1 28477 | Uniqueness of the middle p... |
| miduniq2 28478 | If two point inversions co... |
| colmid 28479 | Colinearity and equidistan... |
| symquadlem 28480 | Lemma of the symetrial qua... |
| krippenlem 28481 | Lemma for ~ krippen . We ... |
| krippen 28482 | Krippenlemma (German for c... |
| midexlem 28483 | Lemma for the existence of... |
| israg 28488 | Property for 3 points A, B... |
| ragcom 28489 | Commutative rule for right... |
| ragcol 28490 | The right angle property i... |
| ragmir 28491 | Right angle property is pr... |
| mirrag 28492 | Right angle is conserved b... |
| ragtrivb 28493 | Trivial right angle. Theo... |
| ragflat2 28494 | Deduce equality from two r... |
| ragflat 28495 | Deduce equality from two r... |
| ragtriva 28496 | Trivial right angle. Theo... |
| ragflat3 28497 | Right angle and colinearit... |
| ragcgr 28498 | Right angle and colinearit... |
| motrag 28499 | Right angles are preserved... |
| ragncol 28500 | Right angle implies non-co... |
| perpln1 28501 | Derive a line from perpend... |
| perpln2 28502 | Derive a line from perpend... |
| isperp 28503 | Property for 2 lines A, B ... |
| perpcom 28504 | The "perpendicular" relati... |
| perpneq 28505 | Two perpendicular lines ar... |
| isperp2 28506 | Property for 2 lines A, B,... |
| isperp2d 28507 | One direction of ~ isperp2... |
| ragperp 28508 | Deduce that two lines are ... |
| footexALT 28509 | Alternative version of ~ f... |
| footexlem1 28510 | Lemma for ~ footex . (Con... |
| footexlem2 28511 | Lemma for ~ footex . (Con... |
| footex 28512 | From a point ` C ` outside... |
| foot 28513 | From a point ` C ` outside... |
| footne 28514 | Uniqueness of the foot poi... |
| footeq 28515 | Uniqueness of the foot poi... |
| hlperpnel 28516 | A point on a half-line whi... |
| perprag 28517 | Deduce a right angle from ... |
| perpdragALT 28518 | Deduce a right angle from ... |
| perpdrag 28519 | Deduce a right angle from ... |
| colperp 28520 | Deduce a perpendicularity ... |
| colperpexlem1 28521 | Lemma for ~ colperp . Fir... |
| colperpexlem2 28522 | Lemma for ~ colperpex . S... |
| colperpexlem3 28523 | Lemma for ~ colperpex . C... |
| colperpex 28524 | In dimension 2 and above, ... |
| mideulem2 28525 | Lemma for ~ opphllem , whi... |
| opphllem 28526 | Lemma 8.24 of [Schwabhause... |
| mideulem 28527 | Lemma for ~ mideu . We ca... |
| midex 28528 | Existence of the midpoint,... |
| mideu 28529 | Existence and uniqueness o... |
| islnopp 28530 | The property for two point... |
| islnoppd 28531 | Deduce that ` A ` and ` B ... |
| oppne1 28532 | Points lying on opposite s... |
| oppne2 28533 | Points lying on opposite s... |
| oppne3 28534 | Points lying on opposite s... |
| oppcom 28535 | Commutativity rule for "op... |
| opptgdim2 28536 | If two points opposite to ... |
| oppnid 28537 | The "opposite to a line" r... |
| opphllem1 28538 | Lemma for ~ opphl . (Cont... |
| opphllem2 28539 | Lemma for ~ opphl . Lemma... |
| opphllem3 28540 | Lemma for ~ opphl : We as... |
| opphllem4 28541 | Lemma for ~ opphl . (Cont... |
| opphllem5 28542 | Second part of Lemma 9.4 o... |
| opphllem6 28543 | First part of Lemma 9.4 of... |
| oppperpex 28544 | Restating ~ colperpex usin... |
| opphl 28545 | If two points ` A ` and ` ... |
| outpasch 28546 | Axiom of Pasch, outer form... |
| hlpasch 28547 | An application of the axio... |
| ishpg 28550 | Value of the half-plane re... |
| hpgbr 28551 | Half-planes : property for... |
| hpgne1 28552 | Points on the open half pl... |
| hpgne2 28553 | Points on the open half pl... |
| lnopp2hpgb 28554 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 28555 | If two points lie on the o... |
| hpgerlem 28556 | Lemma for the proof that t... |
| hpgid 28557 | The half-plane relation is... |
| hpgcom 28558 | The half-plane relation co... |
| hpgtr 28559 | The half-plane relation is... |
| colopp 28560 | Opposite sides of a line f... |
| colhp 28561 | Half-plane relation for co... |
| hphl 28562 | If two points are on the s... |
| midf 28567 | Midpoint as a function. (... |
| midcl 28568 | Closure of the midpoint. ... |
| ismidb 28569 | Property of the midpoint. ... |
| midbtwn 28570 | Betweenness of midpoint. ... |
| midcgr 28571 | Congruence of midpoint. (... |
| midid 28572 | Midpoint of a null segment... |
| midcom 28573 | Commutativity rule for the... |
| mirmid 28574 | Point inversion preserves ... |
| lmieu 28575 | Uniqueness of the line mir... |
| lmif 28576 | Line mirror as a function.... |
| lmicl 28577 | Closure of the line mirror... |
| islmib 28578 | Property of the line mirro... |
| lmicom 28579 | The line mirroring functio... |
| lmilmi 28580 | Line mirroring is an invol... |
| lmireu 28581 | Any point has a unique ant... |
| lmieq 28582 | Equality deduction for lin... |
| lmiinv 28583 | The invariants of the line... |
| lmicinv 28584 | The mirroring line is an i... |
| lmimid 28585 | If we have a right angle, ... |
| lmif1o 28586 | The line mirroring functio... |
| lmiisolem 28587 | Lemma for ~ lmiiso . (Con... |
| lmiiso 28588 | The line mirroring functio... |
| lmimot 28589 | Line mirroring is a motion... |
| hypcgrlem1 28590 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 28591 | Lemma for ~ hypcgr , case ... |
| hypcgr 28592 | If the catheti of two righ... |
| lmiopp 28593 | Line mirroring produces po... |
| lnperpex 28594 | Existence of a perpendicul... |
| trgcopy 28595 | Triangle construction: a c... |
| trgcopyeulem 28596 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 28597 | Triangle construction: a c... |
| iscgra 28600 | Property for two angles AB... |
| iscgra1 28601 | A special version of ~ isc... |
| iscgrad 28602 | Sufficient conditions for ... |
| cgrane1 28603 | Angles imply inequality. ... |
| cgrane2 28604 | Angles imply inequality. ... |
| cgrane3 28605 | Angles imply inequality. ... |
| cgrane4 28606 | Angles imply inequality. ... |
| cgrahl1 28607 | Angle congruence is indepe... |
| cgrahl2 28608 | Angle congruence is indepe... |
| cgracgr 28609 | First direction of proposi... |
| cgraid 28610 | Angle congruence is reflex... |
| cgraswap 28611 | Swap rays in a congruence ... |
| cgrcgra 28612 | Triangle congruence implie... |
| cgracom 28613 | Angle congruence commutes.... |
| cgratr 28614 | Angle congruence is transi... |
| flatcgra 28615 | Flat angles are congruent.... |
| cgraswaplr 28616 | Swap both side of angle co... |
| cgrabtwn 28617 | Angle congruence preserves... |
| cgrahl 28618 | Angle congruence preserves... |
| cgracol 28619 | Angle congruence preserves... |
| cgrancol 28620 | Angle congruence preserves... |
| dfcgra2 28621 | This is the full statement... |
| sacgr 28622 | Supplementary angles of co... |
| oacgr 28623 | Vertical angle theorem. V... |
| acopy 28624 | Angle construction. Theor... |
| acopyeu 28625 | Angle construction. Theor... |
| isinag 28629 | Property for point ` X ` t... |
| isinagd 28630 | Sufficient conditions for ... |
| inagflat 28631 | Any point lies in a flat a... |
| inagswap 28632 | Swap the order of the half... |
| inagne1 28633 | Deduce inequality from the... |
| inagne2 28634 | Deduce inequality from the... |
| inagne3 28635 | Deduce inequality from the... |
| inaghl 28636 | The "point lie in angle" r... |
| isleag 28638 | Geometrical "less than" pr... |
| isleagd 28639 | Sufficient condition for "... |
| leagne1 28640 | Deduce inequality from the... |
| leagne2 28641 | Deduce inequality from the... |
| leagne3 28642 | Deduce inequality from the... |
| leagne4 28643 | Deduce inequality from the... |
| cgrg3col4 28644 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 28645 | First congruence theorem: ... |
| tgsas 28646 | First congruence theorem: ... |
| tgsas2 28647 | First congruence theorem: ... |
| tgsas3 28648 | First congruence theorem: ... |
| tgasa1 28649 | Second congruence theorem:... |
| tgasa 28650 | Second congruence theorem:... |
| tgsss1 28651 | Third congruence theorem: ... |
| tgsss2 28652 | Third congruence theorem: ... |
| tgsss3 28653 | Third congruence theorem: ... |
| dfcgrg2 28654 | Congruence for two triangl... |
| isoas 28655 | Congruence theorem for iso... |
| iseqlg 28658 | Property of a triangle bei... |
| iseqlgd 28659 | Condition for a triangle t... |
| f1otrgds 28660 | Convenient lemma for ~ f1o... |
| f1otrgitv 28661 | Convenient lemma for ~ f1o... |
| f1otrg 28662 | A bijection between bases ... |
| f1otrge 28663 | A bijection between bases ... |
| ttgval 28666 | Define a function to augme... |
| ttgvalOLD 28667 | Obsolete proof of ~ ttgval... |
| ttglem 28668 | Lemma for ~ ttgbas , ~ ttg... |
| ttglemOLD 28669 | Obsolete version of ~ ttgl... |
| ttgbas 28670 | The base set of a subcompl... |
| ttgbasOLD 28671 | Obsolete proof of ~ ttgbas... |
| ttgplusg 28672 | The addition operation of ... |
| ttgplusgOLD 28673 | Obsolete proof of ~ ttgplu... |
| ttgsub 28674 | The subtraction operation ... |
| ttgvsca 28675 | The scalar product of a su... |
| ttgvscaOLD 28676 | Obsolete proof of ~ ttgvsc... |
| ttgds 28677 | The metric of a subcomplex... |
| ttgdsOLD 28678 | Obsolete proof of ~ ttgds ... |
| ttgitvval 28679 | Betweenness for a subcompl... |
| ttgelitv 28680 | Betweenness for a subcompl... |
| ttgbtwnid 28681 | Any subcomplex module equi... |
| ttgcontlem1 28682 | Lemma for % ttgcont . (Co... |
| xmstrkgc 28683 | Any metric space fulfills ... |
| cchhllem 28684 | Lemma for chlbas and chlvs... |
| cchhllemOLD 28685 | Obsolete version of ~ cchh... |
| elee 28692 | Membership in a Euclidean ... |
| mptelee 28693 | A condition for a mapping ... |
| eleenn 28694 | If ` A ` is in ` ( EE `` N... |
| eleei 28695 | The forward direction of ~... |
| eedimeq 28696 | A point belongs to at most... |
| brbtwn 28697 | The binary relation form o... |
| brcgr 28698 | The binary relation form o... |
| fveere 28699 | The function value of a po... |
| fveecn 28700 | The function value of a po... |
| eqeefv 28701 | Two points are equal iff t... |
| eqeelen 28702 | Two points are equal iff t... |
| brbtwn2 28703 | Alternate characterization... |
| colinearalglem1 28704 | Lemma for ~ colinearalg . ... |
| colinearalglem2 28705 | Lemma for ~ colinearalg . ... |
| colinearalglem3 28706 | Lemma for ~ colinearalg . ... |
| colinearalglem4 28707 | Lemma for ~ colinearalg . ... |
| colinearalg 28708 | An algebraic characterizat... |
| eleesub 28709 | Membership of a subtractio... |
| eleesubd 28710 | Membership of a subtractio... |
| axdimuniq 28711 | The unique dimension axiom... |
| axcgrrflx 28712 | ` A ` is as far from ` B `... |
| axcgrtr 28713 | Congruence is transitive. ... |
| axcgrid 28714 | If there is no distance be... |
| axsegconlem1 28715 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 28716 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 28717 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 28718 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 28719 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 28720 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 28721 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 28722 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 28723 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 28724 | Lemma for ~ axsegcon . Sh... |
| axsegcon 28725 | Any segment ` A B ` can be... |
| ax5seglem1 28726 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 28727 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 28728 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 28729 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 28730 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 28731 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 28732 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 28733 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 28734 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 28735 | Lemma for ~ ax5seg . Take... |
| ax5seg 28736 | The five segment axiom. T... |
| axbtwnid 28737 | Points are indivisible. T... |
| axpaschlem 28738 | Lemma for ~ axpasch . Set... |
| axpasch 28739 | The inner Pasch axiom. Ta... |
| axlowdimlem1 28740 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 28741 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 28742 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 28743 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 28744 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 28745 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 28746 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 28747 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 28748 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 28749 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 28750 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 28751 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 28752 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 28753 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 28754 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 28755 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 28756 | Lemma for ~ axlowdim . Es... |
| axlowdim1 28757 | The lower dimension axiom ... |
| axlowdim2 28758 | The lower two-dimensional ... |
| axlowdim 28759 | The general lower dimensio... |
| axeuclidlem 28760 | Lemma for ~ axeuclid . Ha... |
| axeuclid 28761 | Euclid's axiom. Take an a... |
| axcontlem1 28762 | Lemma for ~ axcont . Chan... |
| axcontlem2 28763 | Lemma for ~ axcont . The ... |
| axcontlem3 28764 | Lemma for ~ axcont . Give... |
| axcontlem4 28765 | Lemma for ~ axcont . Give... |
| axcontlem5 28766 | Lemma for ~ axcont . Comp... |
| axcontlem6 28767 | Lemma for ~ axcont . Stat... |
| axcontlem7 28768 | Lemma for ~ axcont . Give... |
| axcontlem8 28769 | Lemma for ~ axcont . A po... |
| axcontlem9 28770 | Lemma for ~ axcont . Give... |
| axcontlem10 28771 | Lemma for ~ axcont . Give... |
| axcontlem11 28772 | Lemma for ~ axcont . Elim... |
| axcontlem12 28773 | Lemma for ~ axcont . Elim... |
| axcont 28774 | The axiom of continuity. ... |
| eengv 28777 | The value of the Euclidean... |
| eengstr 28778 | The Euclidean geometry as ... |
| eengbas 28779 | The Base of the Euclidean ... |
| ebtwntg 28780 | The betweenness relation u... |
| ecgrtg 28781 | The congruence relation us... |
| elntg 28782 | The line definition in the... |
| elntg2 28783 | The line definition in the... |
| eengtrkg 28784 | The geometry structure for... |
| eengtrkge 28785 | The geometry structure for... |
| edgfid 28788 | Utility theorem: index-ind... |
| edgfndx 28789 | Index value of the ~ df-ed... |
| edgfndxnn 28790 | The index value of the edg... |
| edgfndxid 28791 | The value of the edge func... |
| edgfndxidOLD 28792 | Obsolete version of ~ edgf... |
| basendxltedgfndx 28793 | The index value of the ` B... |
| baseltedgfOLD 28794 | Obsolete proof of ~ basend... |
| basendxnedgfndx 28795 | The slots ` Base ` and ` .... |
| vtxval 28800 | The set of vertices of a g... |
| iedgval 28801 | The set of indexed edges o... |
| 1vgrex 28802 | A graph with at least one ... |
| opvtxval 28803 | The set of vertices of a g... |
| opvtxfv 28804 | The set of vertices of a g... |
| opvtxov 28805 | The set of vertices of a g... |
| opiedgval 28806 | The set of indexed edges o... |
| opiedgfv 28807 | The set of indexed edges o... |
| opiedgov 28808 | The set of indexed edges o... |
| opvtxfvi 28809 | The set of vertices of a g... |
| opiedgfvi 28810 | The set of indexed edges o... |
| funvtxdmge2val 28811 | The set of vertices of an ... |
| funiedgdmge2val 28812 | The set of indexed edges o... |
| funvtxdm2val 28813 | The set of vertices of an ... |
| funiedgdm2val 28814 | The set of indexed edges o... |
| funvtxval0 28815 | The set of vertices of an ... |
| basvtxval 28816 | The set of vertices of a g... |
| edgfiedgval 28817 | The set of indexed edges o... |
| funvtxval 28818 | The set of vertices of a g... |
| funiedgval 28819 | The set of indexed edges o... |
| structvtxvallem 28820 | Lemma for ~ structvtxval a... |
| structvtxval 28821 | The set of vertices of an ... |
| structiedg0val 28822 | The set of indexed edges o... |
| structgrssvtxlem 28823 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 28824 | The set of vertices of a g... |
| structgrssiedg 28825 | The set of indexed edges o... |
| struct2grstr 28826 | A graph represented as an ... |
| struct2grvtx 28827 | The set of vertices of a g... |
| struct2griedg 28828 | The set of indexed edges o... |
| graop 28829 | Any representation of a gr... |
| grastruct 28830 | Any representation of a gr... |
| gropd 28831 | If any representation of a... |
| grstructd 28832 | If any representation of a... |
| gropeld 28833 | If any representation of a... |
| grstructeld 28834 | If any representation of a... |
| setsvtx 28835 | The vertices of a structur... |
| setsiedg 28836 | The (indexed) edges of a s... |
| snstrvtxval 28837 | The set of vertices of a g... |
| snstriedgval 28838 | The set of indexed edges o... |
| vtxval0 28839 | Degenerated case 1 for ver... |
| iedgval0 28840 | Degenerated case 1 for edg... |
| vtxvalsnop 28841 | Degenerated case 2 for ver... |
| iedgvalsnop 28842 | Degenerated case 2 for edg... |
| vtxval3sn 28843 | Degenerated case 3 for ver... |
| iedgval3sn 28844 | Degenerated case 3 for edg... |
| vtxvalprc 28845 | Degenerated case 4 for ver... |
| iedgvalprc 28846 | Degenerated case 4 for edg... |
| edgval 28849 | The edges of a graph. (Co... |
| iedgedg 28850 | An indexed edge is an edge... |
| edgopval 28851 | The edges of a graph repre... |
| edgov 28852 | The edges of a graph repre... |
| edgstruct 28853 | The edges of a graph repre... |
| edgiedgb 28854 | A set is an edge iff it is... |
| edg0iedg0 28855 | There is no edge in a grap... |
| isuhgr 28860 | The predicate "is an undir... |
| isushgr 28861 | The predicate "is an undir... |
| uhgrf 28862 | The edge function of an un... |
| ushgrf 28863 | The edge function of an un... |
| uhgrss 28864 | An edge is a subset of ver... |
| uhgreq12g 28865 | If two sets have the same ... |
| uhgrfun 28866 | The edge function of an un... |
| uhgrn0 28867 | An edge is a nonempty subs... |
| lpvtx 28868 | The endpoints of a loop (w... |
| ushgruhgr 28869 | An undirected simple hyper... |
| isuhgrop 28870 | The property of being an u... |
| uhgr0e 28871 | The empty graph, with vert... |
| uhgr0vb 28872 | The null graph, with no ve... |
| uhgr0 28873 | The null graph represented... |
| uhgrun 28874 | The union ` U ` of two (un... |
| uhgrunop 28875 | The union of two (undirect... |
| ushgrun 28876 | The union ` U ` of two (un... |
| ushgrunop 28877 | The union of two (undirect... |
| uhgrstrrepe 28878 | Replacing (or adding) the ... |
| incistruhgr 28879 | An _incidence structure_ `... |
| isupgr 28884 | The property of being an u... |
| wrdupgr 28885 | The property of being an u... |
| upgrf 28886 | The edge function of an un... |
| upgrfn 28887 | The edge function of an un... |
| upgrss 28888 | An edge is a subset of ver... |
| upgrn0 28889 | An edge is a nonempty subs... |
| upgrle 28890 | An edge of an undirected p... |
| upgrfi 28891 | An edge is a finite subset... |
| upgrex 28892 | An edge is an unordered pa... |
| upgrbi 28893 | Show that an unordered pai... |
| upgrop 28894 | A pseudograph represented ... |
| isumgr 28895 | The property of being an u... |
| isumgrs 28896 | The simplified property of... |
| wrdumgr 28897 | The property of being an u... |
| umgrf 28898 | The edge function of an un... |
| umgrfn 28899 | The edge function of an un... |
| umgredg2 28900 | An edge of a multigraph ha... |
| umgrbi 28901 | Show that an unordered pai... |
| upgruhgr 28902 | An undirected pseudograph ... |
| umgrupgr 28903 | An undirected multigraph i... |
| umgruhgr 28904 | An undirected multigraph i... |
| upgrle2 28905 | An edge of an undirected p... |
| umgrnloopv 28906 | In a multigraph, there is ... |
| umgredgprv 28907 | In a multigraph, an edge i... |
| umgrnloop 28908 | In a multigraph, there is ... |
| umgrnloop0 28909 | A multigraph has no loops.... |
| umgr0e 28910 | The empty graph, with vert... |
| upgr0e 28911 | The empty graph, with vert... |
| upgr1elem 28912 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 28913 | A pseudograph with one edg... |
| upgr0eop 28914 | The empty graph, with vert... |
| upgr1eop 28915 | A pseudograph with one edg... |
| upgr0eopALT 28916 | Alternate proof of ~ upgr0... |
| upgr1eopALT 28917 | Alternate proof of ~ upgr1... |
| upgrun 28918 | The union ` U ` of two pse... |
| upgrunop 28919 | The union of two pseudogra... |
| umgrun 28920 | The union ` U ` of two mul... |
| umgrunop 28921 | The union of two multigrap... |
| umgrislfupgrlem 28922 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 28923 | A multigraph is a loop-fre... |
| lfgredgge2 28924 | An edge of a loop-free gra... |
| lfgrnloop 28925 | A loop-free graph has no l... |
| uhgredgiedgb 28926 | In a hypergraph, a set is ... |
| uhgriedg0edg0 28927 | A hypergraph has no edges ... |
| uhgredgn0 28928 | An edge of a hypergraph is... |
| edguhgr 28929 | An edge of a hypergraph is... |
| uhgredgrnv 28930 | An edge of a hypergraph co... |
| uhgredgss 28931 | The set of edges of a hype... |
| upgredgss 28932 | The set of edges of a pseu... |
| umgredgss 28933 | The set of edges of a mult... |
| edgupgr 28934 | Properties of an edge of a... |
| edgumgr 28935 | Properties of an edge of a... |
| uhgrvtxedgiedgb 28936 | In a hypergraph, a vertex ... |
| upgredg 28937 | For each edge in a pseudog... |
| umgredg 28938 | For each edge in a multigr... |
| upgrpredgv 28939 | An edge of a pseudograph a... |
| umgrpredgv 28940 | An edge of a multigraph al... |
| upgredg2vtx 28941 | For a vertex incident to a... |
| upgredgpr 28942 | If a proper pair (of verti... |
| edglnl 28943 | The edges incident with a ... |
| numedglnl 28944 | The number of edges incide... |
| umgredgne 28945 | An edge of a multigraph al... |
| umgrnloop2 28946 | A multigraph has no loops.... |
| umgredgnlp 28947 | An edge of a multigraph is... |
| isuspgr 28952 | The property of being a si... |
| isusgr 28953 | The property of being a si... |
| uspgrf 28954 | The edge function of a sim... |
| usgrf 28955 | The edge function of a sim... |
| isusgrs 28956 | The property of being a si... |
| usgrfs 28957 | The edge function of a sim... |
| usgrfun 28958 | The edge function of a sim... |
| usgredgss 28959 | The set of edges of a simp... |
| edgusgr 28960 | An edge of a simple graph ... |
| isuspgrop 28961 | The property of being an u... |
| isusgrop 28962 | The property of being an u... |
| usgrop 28963 | A simple graph represented... |
| isausgr 28964 | The property of an unorder... |
| ausgrusgrb 28965 | The equivalence of the def... |
| usgrausgri 28966 | A simple graph represented... |
| ausgrumgri 28967 | If an alternatively define... |
| ausgrusgri 28968 | The equivalence of the def... |
| usgrausgrb 28969 | The equivalence of the def... |
| usgredgop 28970 | An edge of a simple graph ... |
| usgrf1o 28971 | The edge function of a sim... |
| usgrf1 28972 | The edge function of a sim... |
| uspgrf1oedg 28973 | The edge function of a sim... |
| usgrss 28974 | An edge is a subset of ver... |
| uspgredgiedg 28975 | In a simple pseudograph, f... |
| uspgriedgedg 28976 | In a simple pseudograph, f... |
| uspgrushgr 28977 | A simple pseudograph is an... |
| uspgrupgr 28978 | A simple pseudograph is an... |
| uspgrupgrushgr 28979 | A graph is a simple pseudo... |
| usgruspgr 28980 | A simple graph is a simple... |
| usgrumgr 28981 | A simple graph is an undir... |
| usgrumgruspgr 28982 | A graph is a simple graph ... |
| usgruspgrb 28983 | A class is a simple graph ... |
| uspgruhgr 28984 | An undirected simple pseud... |
| usgrupgr 28985 | A simple graph is an undir... |
| usgruhgr 28986 | A simple graph is an undir... |
| usgrislfuspgr 28987 | A simple graph is a loop-f... |
| uspgrun 28988 | The union ` U ` of two sim... |
| uspgrunop 28989 | The union of two simple ps... |
| usgrun 28990 | The union ` U ` of two sim... |
| usgrunop 28991 | The union of two simple gr... |
| usgredg2 28992 | The value of the "edge fun... |
| usgredg2ALT 28993 | Alternate proof of ~ usgre... |
| usgredgprv 28994 | In a simple graph, an edge... |
| usgredgprvALT 28995 | Alternate proof of ~ usgre... |
| usgredgppr 28996 | An edge of a simple graph ... |
| usgrpredgv 28997 | An edge of a simple graph ... |
| edgssv2 28998 | An edge of a simple graph ... |
| usgredg 28999 | For each edge in a simple ... |
| usgrnloopv 29000 | In a simple graph, there i... |
| usgrnloopvALT 29001 | Alternate proof of ~ usgrn... |
| usgrnloop 29002 | In a simple graph, there i... |
| usgrnloopALT 29003 | Alternate proof of ~ usgrn... |
| usgrnloop0 29004 | A simple graph has no loop... |
| usgrnloop0ALT 29005 | Alternate proof of ~ usgrn... |
| usgredgne 29006 | An edge of a simple graph ... |
| usgrf1oedg 29007 | The edge function of a sim... |
| uhgr2edg 29008 | If a vertex is adjacent to... |
| umgr2edg 29009 | If a vertex is adjacent to... |
| usgr2edg 29010 | If a vertex is adjacent to... |
| umgr2edg1 29011 | If a vertex is adjacent to... |
| usgr2edg1 29012 | If a vertex is adjacent to... |
| umgrvad2edg 29013 | If a vertex is adjacent to... |
| umgr2edgneu 29014 | If a vertex is adjacent to... |
| usgrsizedg 29015 | In a simple graph, the siz... |
| usgredg3 29016 | The value of the "edge fun... |
| usgredg4 29017 | For a vertex incident to a... |
| usgredgreu 29018 | For a vertex incident to a... |
| usgredg2vtx 29019 | For a vertex incident to a... |
| uspgredg2vtxeu 29020 | For a vertex incident to a... |
| usgredg2vtxeu 29021 | For a vertex incident to a... |
| usgredg2vtxeuALT 29022 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29023 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29024 | In a simple pseudograph, t... |
| usgredg2vlem1 29025 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29026 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29027 | In a simple graph, the map... |
| usgriedgleord 29028 | Alternate version of ~ usg... |
| ushgredgedg 29029 | In a simple hypergraph the... |
| usgredgedg 29030 | In a simple graph there is... |
| ushgredgedgloop 29031 | In a simple hypergraph the... |
| uspgredgleord 29032 | In a simple pseudograph th... |
| usgredgleord 29033 | In a simple graph the numb... |
| usgredgleordALT 29034 | Alternate proof for ~ usgr... |
| usgrstrrepe 29035 | Replacing (or adding) the ... |
| usgr0e 29036 | The empty graph, with vert... |
| usgr0vb 29037 | The null graph, with no ve... |
| uhgr0v0e 29038 | The null graph, with no ve... |
| uhgr0vsize0 29039 | The size of a hypergraph w... |
| uhgr0edgfi 29040 | A graph of order 0 (i.e. w... |
| usgr0v 29041 | The null graph, with no ve... |
| uhgr0vusgr 29042 | The null graph, with no ve... |
| usgr0 29043 | The null graph represented... |
| uspgr1e 29044 | A simple pseudograph with ... |
| usgr1e 29045 | A simple graph with one ed... |
| usgr0eop 29046 | The empty graph, with vert... |
| uspgr1eop 29047 | A simple pseudograph with ... |
| uspgr1ewop 29048 | A simple pseudograph with ... |
| uspgr1v1eop 29049 | A simple pseudograph with ... |
| usgr1eop 29050 | A simple graph with (at le... |
| uspgr2v1e2w 29051 | A simple pseudograph with ... |
| usgr2v1e2w 29052 | A simple graph with two ve... |
| edg0usgr 29053 | A class without edges is a... |
| lfuhgr1v0e 29054 | A loop-free hypergraph wit... |
| usgr1vr 29055 | A simple graph with one ve... |
| usgr1v 29056 | A class with one (or no) v... |
| usgr1v0edg 29057 | A class with one (or no) v... |
| usgrexmpldifpr 29058 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29059 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29060 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29061 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29062 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29063 | ` G ` is a simple graph of... |
| griedg0prc 29064 | The class of empty graphs ... |
| griedg0ssusgr 29065 | The class of all simple gr... |
| usgrprc 29066 | The class of simple graphs... |
| relsubgr 29069 | The class of the subgraph ... |
| subgrv 29070 | If a class is a subgraph o... |
| issubgr 29071 | The property of a set to b... |
| issubgr2 29072 | The property of a set to b... |
| subgrprop 29073 | The properties of a subgra... |
| subgrprop2 29074 | The properties of a subgra... |
| uhgrissubgr 29075 | The property of a hypergra... |
| subgrprop3 29076 | The properties of a subgra... |
| egrsubgr 29077 | An empty graph consisting ... |
| 0grsubgr 29078 | The null graph (represente... |
| 0uhgrsubgr 29079 | The null graph (as hypergr... |
| uhgrsubgrself 29080 | A hypergraph is a subgraph... |
| subgrfun 29081 | The edge function of a sub... |
| subgruhgrfun 29082 | The edge function of a sub... |
| subgreldmiedg 29083 | An element of the domain o... |
| subgruhgredgd 29084 | An edge of a subgraph of a... |
| subumgredg2 29085 | An edge of a subgraph of a... |
| subuhgr 29086 | A subgraph of a hypergraph... |
| subupgr 29087 | A subgraph of a pseudograp... |
| subumgr 29088 | A subgraph of a multigraph... |
| subusgr 29089 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29090 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29091 | A spanning subgraph ` S ` ... |
| uhgrspan 29092 | A spanning subgraph ` S ` ... |
| upgrspan 29093 | A spanning subgraph ` S ` ... |
| umgrspan 29094 | A spanning subgraph ` S ` ... |
| usgrspan 29095 | A spanning subgraph ` S ` ... |
| uhgrspanop 29096 | A spanning subgraph of a h... |
| upgrspanop 29097 | A spanning subgraph of a p... |
| umgrspanop 29098 | A spanning subgraph of a m... |
| usgrspanop 29099 | A spanning subgraph of a s... |
| uhgrspan1lem1 29100 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29101 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29102 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29103 | The induced subgraph ` S `... |
| upgrreslem 29104 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29105 | Lemma for ~ umgrres and ~ ... |
| upgrres 29106 | A subgraph obtained by rem... |
| umgrres 29107 | A subgraph obtained by rem... |
| usgrres 29108 | A subgraph obtained by rem... |
| upgrres1lem1 29109 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29110 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29111 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29112 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29113 | A pseudograph obtained by ... |
| umgrres1 29114 | A multigraph obtained by r... |
| usgrres1 29115 | Restricting a simple graph... |
| isfusgr 29118 | The property of being a fi... |
| fusgrvtxfi 29119 | A finite simple graph has ... |
| isfusgrf1 29120 | The property of being a fi... |
| isfusgrcl 29121 | The property of being a fi... |
| fusgrusgr 29122 | A finite simple graph is a... |
| opfusgr 29123 | A finite simple graph repr... |
| usgredgffibi 29124 | The number of edges in a s... |
| fusgredgfi 29125 | In a finite simple graph t... |
| usgr1v0e 29126 | The size of a (finite) sim... |
| usgrfilem 29127 | In a finite simple graph, ... |
| fusgrfisbase 29128 | Induction base for ~ fusgr... |
| fusgrfisstep 29129 | Induction step in ~ fusgrf... |
| fusgrfis 29130 | A finite simple graph is o... |
| fusgrfupgrfs 29131 | A finite simple graph is a... |
| nbgrprc0 29134 | The set of neighbors is em... |
| nbgrcl 29135 | If a class ` X ` has at le... |
| nbgrval 29136 | The set of neighbors of a ... |
| dfnbgr2 29137 | Alternate definition of th... |
| dfnbgr3 29138 | Alternate definition of th... |
| nbgrnvtx0 29139 | If a class ` X ` is not a ... |
| nbgrel 29140 | Characterization of a neig... |
| nbgrisvtx 29141 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29142 | The neighbors of a vertex ... |
| nbuhgr 29143 | The set of neighbors of a ... |
| nbupgr 29144 | The set of neighbors of a ... |
| nbupgrel 29145 | A neighbor of a vertex in ... |
| nbumgrvtx 29146 | The set of neighbors of a ... |
| nbumgr 29147 | The set of neighbors of an... |
| nbusgrvtx 29148 | The set of neighbors of a ... |
| nbusgr 29149 | The set of neighbors of an... |
| nbgr2vtx1edg 29150 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29151 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29152 | If a hypergraph has two ve... |
| nbusgreledg 29153 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29154 | A vertex which is not endp... |
| nbgr0vtxlem 29155 | Lemma for ~ nbgr0vtx and ~... |
| nbgr0vtx 29156 | In a null graph (with no v... |
| nbgr0edg 29157 | In an empty graph (with no... |
| nbgr1vtx 29158 | In a graph with one vertex... |
| nbgrnself 29159 | A vertex in a graph is not... |
| nbgrnself2 29160 | A class ` X ` is not a nei... |
| nbgrssovtx 29161 | The neighbors of a vertex ... |
| nbgrssvwo2 29162 | The neighbors of a vertex ... |
| nbgrsym 29163 | In a graph, the neighborho... |
| nbupgrres 29164 | The neighborhood of a vert... |
| usgrnbcnvfv 29165 | Applying the edge function... |
| nbusgredgeu 29166 | For each neighbor of a ver... |
| edgnbusgreu 29167 | For each edge incident to ... |
| nbusgredgeu0 29168 | For each neighbor of a ver... |
| nbusgrf1o0 29169 | The mapping of neighbors o... |
| nbusgrf1o1 29170 | The set of neighbors of a ... |
| nbusgrf1o 29171 | The set of neighbors of a ... |
| nbedgusgr 29172 | The number of neighbors of... |
| edgusgrnbfin 29173 | The number of neighbors of... |
| nbusgrfi 29174 | The class of neighbors of ... |
| nbfiusgrfi 29175 | The class of neighbors of ... |
| hashnbusgrnn0 29176 | The number of neighbors of... |
| nbfusgrlevtxm1 29177 | The number of neighbors of... |
| nbfusgrlevtxm2 29178 | If there is a vertex which... |
| nbusgrvtxm1 29179 | If the number of neighbors... |
| nb3grprlem1 29180 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29181 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29182 | The neighbors of a vertex ... |
| nb3grpr2 29183 | The neighbors of a vertex ... |
| nb3gr2nb 29184 | If the neighbors of two ve... |
| uvtxval 29187 | The set of all universal v... |
| uvtxel 29188 | A universal vertex, i.e. a... |
| uvtxisvtx 29189 | A universal vertex is a ve... |
| uvtxssvtx 29190 | The set of the universal v... |
| vtxnbuvtx 29191 | A universal vertex has all... |
| uvtxnbgrss 29192 | A universal vertex has all... |
| uvtxnbgrvtx 29193 | A universal vertex is neig... |
| uvtx0 29194 | There is no universal vert... |
| isuvtx 29195 | The set of all universal v... |
| uvtxel1 29196 | Characterization of a univ... |
| uvtx01vtx 29197 | If a graph/class has no ed... |
| uvtx2vtx1edg 29198 | If a graph has two vertice... |
| uvtx2vtx1edgb 29199 | If a hypergraph has two ve... |
| uvtxnbgr 29200 | A universal vertex has all... |
| uvtxnbgrb 29201 | A vertex is universal iff ... |
| uvtxusgr 29202 | The set of all universal v... |
| uvtxusgrel 29203 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 29204 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 29205 | If the number of neighbors... |
| uvtxnbvtxm1 29206 | A universal vertex has ` n... |
| nbupgruvtxres 29207 | The neighborhood of a univ... |
| uvtxupgrres 29208 | A universal vertex is univ... |
| cplgruvtxb 29213 | A graph ` G ` is complete ... |
| prcliscplgr 29214 | A proper class (representi... |
| iscplgr 29215 | The property of being a co... |
| iscplgrnb 29216 | A graph is complete iff al... |
| iscplgredg 29217 | A graph ` G ` is complete ... |
| iscusgr 29218 | The property of being a co... |
| cusgrusgr 29219 | A complete simple graph is... |
| cusgrcplgr 29220 | A complete simple graph is... |
| iscusgrvtx 29221 | A simple graph is complete... |
| cusgruvtxb 29222 | A simple graph is complete... |
| iscusgredg 29223 | A simple graph is complete... |
| cusgredg 29224 | In a complete simple graph... |
| cplgr0 29225 | The null graph (with no ve... |
| cusgr0 29226 | The null graph (with no ve... |
| cplgr0v 29227 | A null graph (with no vert... |
| cusgr0v 29228 | A graph with no vertices a... |
| cplgr1vlem 29229 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 29230 | A graph with one vertex is... |
| cusgr1v 29231 | A graph with one vertex an... |
| cplgr2v 29232 | An undirected hypergraph w... |
| cplgr2vpr 29233 | An undirected hypergraph w... |
| nbcplgr 29234 | In a complete graph, each ... |
| cplgr3v 29235 | A pseudograph with three (... |
| cusgr3vnbpr 29236 | The neighbors of a vertex ... |
| cplgrop 29237 | A complete graph represent... |
| cusgrop 29238 | A complete simple graph re... |
| cusgrexilem1 29239 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 29240 | Lemma for ~ usgrexi . (Co... |
| usgrexi 29241 | An arbitrary set regarded ... |
| cusgrexilem2 29242 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 29243 | An arbitrary set ` V ` reg... |
| cusgrexg 29244 | For each set there is a se... |
| structtousgr 29245 | Any (extensible) structure... |
| structtocusgr 29246 | Any (extensible) structure... |
| cffldtocusgr 29247 | The field of complex numbe... |
| cffldtocusgrOLD 29248 | Obsolete version of ~ cffl... |
| cusgrres 29249 | Restricting a complete sim... |
| cusgrsizeindb0 29250 | Base case of the induction... |
| cusgrsizeindb1 29251 | Base case of the induction... |
| cusgrsizeindslem 29252 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 29253 | Part 1 of induction step i... |
| cusgrsize2inds 29254 | Induction step in ~ cusgrs... |
| cusgrsize 29255 | The size of a finite compl... |
| cusgrfilem1 29256 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 29257 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 29258 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 29259 | If the size of a complete ... |
| usgredgsscusgredg 29260 | A simple graph is a subgra... |
| usgrsscusgr 29261 | A simple graph is a subgra... |
| sizusglecusglem1 29262 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 29263 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 29264 | The size of a simple graph... |
| fusgrmaxsize 29265 | The maximum size of a fini... |
| vtxdgfval 29268 | The value of the vertex de... |
| vtxdgval 29269 | The degree of a vertex. (... |
| vtxdgfival 29270 | The degree of a vertex for... |
| vtxdgop 29271 | The vertex degree expresse... |
| vtxdgf 29272 | The vertex degree function... |
| vtxdgelxnn0 29273 | The degree of a vertex is ... |
| vtxdg0v 29274 | The degree of a vertex in ... |
| vtxdg0e 29275 | The degree of a vertex in ... |
| vtxdgfisnn0 29276 | The degree of a vertex in ... |
| vtxdgfisf 29277 | The vertex degree function... |
| vtxdeqd 29278 | Equality theorem for the v... |
| vtxduhgr0e 29279 | The degree of a vertex in ... |
| vtxdlfuhgr1v 29280 | The degree of the vertex i... |
| vdumgr0 29281 | A vertex in a multigraph h... |
| vtxdun 29282 | The degree of a vertex in ... |
| vtxdfiun 29283 | The degree of a vertex in ... |
| vtxduhgrun 29284 | The degree of a vertex in ... |
| vtxduhgrfiun 29285 | The degree of a vertex in ... |
| vtxdlfgrval 29286 | The value of the vertex de... |
| vtxdumgrval 29287 | The value of the vertex de... |
| vtxdusgrval 29288 | The value of the vertex de... |
| vtxd0nedgb 29289 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 29290 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 29291 | The value of the vertex de... |
| vtxdusgrfvedg 29292 | The value of the vertex de... |
| vtxduhgr0nedg 29293 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 29294 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 29295 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 29296 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 29297 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 29298 | The vertex degree function... |
| vtxdgfusgr 29299 | In a finite simple graph, ... |
| fusgrn0degnn0 29300 | In a nonempty, finite grap... |
| 1loopgruspgr 29301 | A graph with one edge whic... |
| 1loopgredg 29302 | The set of edges in a grap... |
| 1loopgrnb0 29303 | In a graph (simple pseudog... |
| 1loopgrvd2 29304 | The vertex degree of a one... |
| 1loopgrvd0 29305 | The vertex degree of a one... |
| 1hevtxdg0 29306 | The vertex degree of verte... |
| 1hevtxdg1 29307 | The vertex degree of verte... |
| 1hegrvtxdg1 29308 | The vertex degree of a gra... |
| 1hegrvtxdg1r 29309 | The vertex degree of a gra... |
| 1egrvtxdg1 29310 | The vertex degree of a one... |
| 1egrvtxdg1r 29311 | The vertex degree of a one... |
| 1egrvtxdg0 29312 | The vertex degree of a one... |
| p1evtxdeqlem 29313 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 29314 | If an edge ` E ` which doe... |
| p1evtxdp1 29315 | If an edge ` E ` (not bein... |
| uspgrloopvtx 29316 | The set of vertices in a g... |
| uspgrloopvtxel 29317 | A vertex in a graph (simpl... |
| uspgrloopiedg 29318 | The set of edges in a grap... |
| uspgrloopedg 29319 | The set of edges in a grap... |
| uspgrloopnb0 29320 | In a graph (simple pseudog... |
| uspgrloopvd2 29321 | The vertex degree of a one... |
| umgr2v2evtx 29322 | The set of vertices in a m... |
| umgr2v2evtxel 29323 | A vertex in a multigraph w... |
| umgr2v2eiedg 29324 | The edge function in a mul... |
| umgr2v2eedg 29325 | The set of edges in a mult... |
| umgr2v2e 29326 | A multigraph with two edge... |
| umgr2v2enb1 29327 | In a multigraph with two e... |
| umgr2v2evd2 29328 | In a multigraph with two e... |
| hashnbusgrvd 29329 | In a simple graph, the num... |
| usgruvtxvdb 29330 | In a finite simple graph w... |
| vdiscusgrb 29331 | A finite simple graph with... |
| vdiscusgr 29332 | In a finite complete simpl... |
| vtxdusgradjvtx 29333 | The degree of a vertex in ... |
| usgrvd0nedg 29334 | If a vertex in a simple gr... |
| uhgrvd00 29335 | If every vertex in a hyper... |
| usgrvd00 29336 | If every vertex in a simpl... |
| vdegp1ai 29337 | The induction step for a v... |
| vdegp1bi 29338 | The induction step for a v... |
| vdegp1ci 29339 | The induction step for a v... |
| vtxdginducedm1lem1 29340 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 29341 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 29342 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 29343 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 29344 | The degree of a vertex ` v... |
| vtxdginducedm1fi 29345 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 29346 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 29347 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 29348 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 29349 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 29350 | Induction step of ~ finsum... |
| finsumvtxdg2size 29351 | The sum of the degrees of ... |
| fusgr1th 29352 | The sum of the degrees of ... |
| finsumvtxdgeven 29353 | The sum of the degrees of ... |
| vtxdgoddnumeven 29354 | The number of vertices of ... |
| fusgrvtxdgonume 29355 | The number of vertices of ... |
| isrgr 29360 | The property of a class be... |
| rgrprop 29361 | The properties of a k-regu... |
| isrusgr 29362 | The property of being a k-... |
| rusgrprop 29363 | The properties of a k-regu... |
| rusgrrgr 29364 | A k-regular simple graph i... |
| rusgrusgr 29365 | A k-regular simple graph i... |
| finrusgrfusgr 29366 | A finite regular simple gr... |
| isrusgr0 29367 | The property of being a k-... |
| rusgrprop0 29368 | The properties of a k-regu... |
| usgreqdrusgr 29369 | If all vertices in a simpl... |
| fusgrregdegfi 29370 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 29371 | If all vertices in a nonem... |
| frusgrnn0 29372 | In a nonempty finite k-reg... |
| 0edg0rgr 29373 | A graph is 0-regular if it... |
| uhgr0edg0rgr 29374 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 29375 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 29376 | A simple graph is 0-regula... |
| 0vtxrgr 29377 | A null graph (with no vert... |
| 0vtxrusgr 29378 | A graph with no vertices a... |
| 0uhgrrusgr 29379 | The null graph as hypergra... |
| 0grrusgr 29380 | The null graph represented... |
| 0grrgr 29381 | The null graph represented... |
| cusgrrusgr 29382 | A complete simple graph wi... |
| cusgrm1rusgr 29383 | A finite simple graph with... |
| rusgrpropnb 29384 | The properties of a k-regu... |
| rusgrpropedg 29385 | The properties of a k-regu... |
| rusgrpropadjvtx 29386 | The properties of a k-regu... |
| rusgrnumwrdl2 29387 | In a k-regular simple grap... |
| rusgr1vtxlem 29388 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 29389 | If a k-regular simple grap... |
| rgrusgrprc 29390 | The class of 0-regular sim... |
| rusgrprc 29391 | The class of 0-regular sim... |
| rgrprc 29392 | The class of 0-regular gra... |
| rgrprcx 29393 | The class of 0-regular gra... |
| rgrx0ndm 29394 | 0 is not in the domain of ... |
| rgrx0nd 29395 | The potentially alternativ... |
| ewlksfval 29402 | The set of s-walks of edge... |
| isewlk 29403 | Conditions for a function ... |
| ewlkprop 29404 | Properties of an s-walk of... |
| ewlkinedg 29405 | The intersection (common v... |
| ewlkle 29406 | An s-walk of edges is also... |
| upgrewlkle2 29407 | In a pseudograph, there is... |
| wkslem1 29408 | Lemma 1 for walks to subst... |
| wkslem2 29409 | Lemma 2 for walks to subst... |
| wksfval 29410 | The set of walks (in an un... |
| iswlk 29411 | Properties of a pair of fu... |
| wlkprop 29412 | Properties of a walk. (Co... |
| wlkv 29413 | The classes involved in a ... |
| iswlkg 29414 | Generalization of ~ iswlk ... |
| wlkf 29415 | The mapping enumerating th... |
| wlkcl 29416 | A walk has length ` # ( F ... |
| wlkp 29417 | The mapping enumerating th... |
| wlkpwrd 29418 | The sequence of vertices o... |
| wlklenvp1 29419 | The number of vertices of ... |
| wksv 29420 | The class of walks is a se... |
| wksvOLD 29421 | Obsolete version of ~ wksv... |
| wlkn0 29422 | The sequence of vertices o... |
| wlklenvm1 29423 | The number of edges of a w... |
| ifpsnprss 29424 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 29425 | Each pair of adjacent vert... |
| wlkvtxiedg 29426 | The vertices of a walk are... |
| relwlk 29427 | The set ` ( Walks `` G ) `... |
| wlkvv 29428 | If there is at least one w... |
| wlkop 29429 | A walk is an ordered pair.... |
| wlkcpr 29430 | A walk as class with two c... |
| wlk2f 29431 | If there is a walk ` W ` t... |
| wlkcomp 29432 | A walk expressed by proper... |
| wlkcompim 29433 | Implications for the prope... |
| wlkelwrd 29434 | The components of a walk a... |
| wlkeq 29435 | Conditions for two walks (... |
| edginwlk 29436 | The value of the edge func... |
| upgredginwlk 29437 | The value of the edge func... |
| iedginwlk 29438 | The value of the edge func... |
| wlkl1loop 29439 | A walk of length 1 from a ... |
| wlk1walk 29440 | A walk is a 1-walk "on the... |
| wlk1ewlk 29441 | A walk is an s-walk "on th... |
| upgriswlk 29442 | Properties of a pair of fu... |
| upgrwlkedg 29443 | The edges of a walk in a p... |
| upgrwlkcompim 29444 | Implications for the prope... |
| wlkvtxedg 29445 | The vertices of a walk are... |
| upgrwlkvtxedg 29446 | The pairs of connected ver... |
| uspgr2wlkeq 29447 | Conditions for two walks w... |
| uspgr2wlkeq2 29448 | Conditions for two walks w... |
| uspgr2wlkeqi 29449 | Conditions for two walks w... |
| umgrwlknloop 29450 | In a multigraph, each walk... |
| wlkResOLD 29451 | Obsolete version of ~ opab... |
| wlkv0 29452 | If there is a walk in the ... |
| g0wlk0 29453 | There is no walk in a null... |
| 0wlk0 29454 | There is no walk for the e... |
| wlk0prc 29455 | There is no walk in a null... |
| wlklenvclwlk 29456 | The number of vertices in ... |
| wlkson 29457 | The set of walks between t... |
| iswlkon 29458 | Properties of a pair of fu... |
| wlkonprop 29459 | Properties of a walk betwe... |
| wlkpvtx 29460 | A walk connects vertices. ... |
| wlkepvtx 29461 | The endpoints of a walk ar... |
| wlkoniswlk 29462 | A walk between two vertice... |
| wlkonwlk 29463 | A walk is a walk between i... |
| wlkonwlk1l 29464 | A walk is a walk from its ... |
| wlksoneq1eq2 29465 | Two walks with identical s... |
| wlkonl1iedg 29466 | If there is a walk between... |
| wlkon2n0 29467 | The length of a walk betwe... |
| 2wlklem 29468 | Lemma for theorems for wal... |
| upgr2wlk 29469 | Properties of a pair of fu... |
| wlkreslem 29470 | Lemma for ~ wlkres . (Con... |
| wlkres 29471 | The restriction ` <. H , Q... |
| redwlklem 29472 | Lemma for ~ redwlk . (Con... |
| redwlk 29473 | A walk ending at the last ... |
| wlkp1lem1 29474 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 29475 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 29476 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 29477 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 29478 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 29479 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 29480 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 29481 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 29482 | Append one path segment (e... |
| wlkdlem1 29483 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 29484 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 29485 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 29486 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 29487 | Two words representing a w... |
| lfgrwlkprop 29488 | Two adjacent vertices in a... |
| lfgriswlk 29489 | Conditions for a pair of f... |
| lfgrwlknloop 29490 | In a loop-free graph, each... |
| reltrls 29495 | The set ` ( Trails `` G ) ... |
| trlsfval 29496 | The set of trails (in an u... |
| istrl 29497 | Conditions for a pair of c... |
| trliswlk 29498 | A trail is a walk. (Contr... |
| trlf1 29499 | The enumeration ` F ` of a... |
| trlreslem 29500 | Lemma for ~ trlres . Form... |
| trlres 29501 | The restriction ` <. H , Q... |
| upgrtrls 29502 | The set of trails in a pse... |
| upgristrl 29503 | Properties of a pair of fu... |
| upgrf1istrl 29504 | Properties of a pair of a ... |
| wksonproplem 29505 | Lemma for theorems for pro... |
| wksonproplemOLD 29506 | Obsolete version of ~ wkso... |
| trlsonfval 29507 | The set of trails between ... |
| istrlson 29508 | Properties of a pair of fu... |
| trlsonprop 29509 | Properties of a trail betw... |
| trlsonistrl 29510 | A trail between two vertic... |
| trlsonwlkon 29511 | A trail between two vertic... |
| trlontrl 29512 | A trail is a trail between... |
| relpths 29521 | The set ` ( Paths `` G ) `... |
| pthsfval 29522 | The set of paths (in an un... |
| spthsfval 29523 | The set of simple paths (i... |
| ispth 29524 | Conditions for a pair of c... |
| isspth 29525 | Conditions for a pair of c... |
| pthistrl 29526 | A path is a trail (in an u... |
| spthispth 29527 | A simple path is a path (i... |
| pthiswlk 29528 | A path is a walk (in an un... |
| spthiswlk 29529 | A simple path is a walk (i... |
| pthdivtx 29530 | The inner vertices of a pa... |
| pthdadjvtx 29531 | The adjacent vertices of a... |
| 2pthnloop 29532 | A path of length at least ... |
| upgr2pthnlp 29533 | A path of length at least ... |
| spthdifv 29534 | The vertices of a simple p... |
| spthdep 29535 | A simple path (at least of... |
| pthdepisspth 29536 | A path with different star... |
| upgrwlkdvdelem 29537 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 29538 | In a pseudograph, all edge... |
| upgrspthswlk 29539 | The set of simple paths in... |
| upgrwlkdvspth 29540 | A walk consisting of diffe... |
| pthsonfval 29541 | The set of paths between t... |
| spthson 29542 | The set of simple paths be... |
| ispthson 29543 | Properties of a pair of fu... |
| isspthson 29544 | Properties of a pair of fu... |
| pthsonprop 29545 | Properties of a path betwe... |
| spthonprop 29546 | Properties of a simple pat... |
| pthonispth 29547 | A path between two vertice... |
| pthontrlon 29548 | A path between two vertice... |
| pthonpth 29549 | A path is a path between i... |
| isspthonpth 29550 | A pair of functions is a s... |
| spthonisspth 29551 | A simple path between to v... |
| spthonpthon 29552 | A simple path between two ... |
| spthonepeq 29553 | The endpoints of a simple ... |
| uhgrwkspthlem1 29554 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 29555 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 29556 | Any walk of length 1 betwe... |
| usgr2wlkneq 29557 | The vertices and edges are... |
| usgr2wlkspthlem1 29558 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 29559 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 29560 | In a simple graph, any wal... |
| usgr2trlncl 29561 | In a simple graph, any tra... |
| usgr2trlspth 29562 | In a simple graph, any tra... |
| usgr2pthspth 29563 | In a simple graph, any pat... |
| usgr2pthlem 29564 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 29565 | In a simple graph, there i... |
| usgr2pth0 29566 | In a simply graph, there i... |
| pthdlem1 29567 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 29568 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 29569 | Lemma 2 for ~ pthd . (Con... |
| pthd 29570 | Two words representing a t... |
| clwlks 29573 | The set of closed walks (i... |
| isclwlk 29574 | A pair of functions repres... |
| clwlkiswlk 29575 | A closed walk is a walk (i... |
| clwlkwlk 29576 | Closed walks are walks (in... |
| clwlkswks 29577 | Closed walks are walks (in... |
| isclwlke 29578 | Properties of a pair of fu... |
| isclwlkupgr 29579 | Properties of a pair of fu... |
| clwlkcomp 29580 | A closed walk expressed by... |
| clwlkcompim 29581 | Implications for the prope... |
| upgrclwlkcompim 29582 | Implications for the prope... |
| clwlkcompbp 29583 | Basic properties of the co... |
| clwlkl1loop 29584 | A closed walk of length 1 ... |
| crcts 29589 | The set of circuits (in an... |
| cycls 29590 | The set of cycles (in an u... |
| iscrct 29591 | Sufficient and necessary c... |
| iscycl 29592 | Sufficient and necessary c... |
| crctprop 29593 | The properties of a circui... |
| cyclprop 29594 | The properties of a cycle:... |
| crctisclwlk 29595 | A circuit is a closed walk... |
| crctistrl 29596 | A circuit is a trail. (Co... |
| crctiswlk 29597 | A circuit is a walk. (Con... |
| cyclispth 29598 | A cycle is a path. (Contr... |
| cycliswlk 29599 | A cycle is a walk. (Contr... |
| cycliscrct 29600 | A cycle is a circuit. (Co... |
| cyclnspth 29601 | A (non-trivial) cycle is n... |
| cyclispthon 29602 | A cycle is a path starting... |
| lfgrn1cycl 29603 | In a loop-free graph there... |
| usgr2trlncrct 29604 | In a simple graph, any tra... |
| umgrn1cycl 29605 | In a multigraph graph (wit... |
| uspgrn2crct 29606 | In a simple pseudograph th... |
| usgrn2cycl 29607 | In a simple graph there ar... |
| crctcshwlkn0lem1 29608 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 29609 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 29610 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 29611 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 29612 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 29613 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 29614 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 29615 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 29616 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 29617 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 29618 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 29619 | Cyclically shifting the in... |
| crctcshwlk 29620 | Cyclically shifting the in... |
| crctcshtrl 29621 | Cyclically shifting the in... |
| crctcsh 29622 | Cyclically shifting the in... |
| wwlks 29633 | The set of walks (in an un... |
| iswwlks 29634 | A word over the set of ver... |
| wwlksn 29635 | The set of walks (in an un... |
| iswwlksn 29636 | A word over the set of ver... |
| wwlksnprcl 29637 | Derivation of the length o... |
| iswwlksnx 29638 | Properties of a word to re... |
| wwlkbp 29639 | Basic properties of a walk... |
| wwlknbp 29640 | Basic properties of a walk... |
| wwlknp 29641 | Properties of a set being ... |
| wwlknbp1 29642 | Other basic properties of ... |
| wwlknvtx 29643 | The symbols of a word ` W ... |
| wwlknllvtx 29644 | If a word ` W ` represents... |
| wwlknlsw 29645 | If a word represents a wal... |
| wspthsn 29646 | The set of simple paths of... |
| iswspthn 29647 | An element of the set of s... |
| wspthnp 29648 | Properties of a set being ... |
| wwlksnon 29649 | The set of walks of a fixe... |
| wspthsnon 29650 | The set of simple paths of... |
| iswwlksnon 29651 | The set of walks of a fixe... |
| wwlksnon0 29652 | Sufficient conditions for ... |
| wwlksonvtx 29653 | If a word ` W ` represents... |
| iswspthsnon 29654 | The set of simple paths of... |
| wwlknon 29655 | An element of the set of w... |
| wspthnon 29656 | An element of the set of s... |
| wspthnonp 29657 | Properties of a set being ... |
| wspthneq1eq2 29658 | Two simple paths with iden... |
| wwlksn0s 29659 | The set of all walks as wo... |
| wwlkssswrd 29660 | Walks (represented by word... |
| wwlksn0 29661 | A walk of length 0 is repr... |
| 0enwwlksnge1 29662 | In graphs without edges, t... |
| wwlkswwlksn 29663 | A walk of a fixed length a... |
| wwlkssswwlksn 29664 | The walks of a fixed lengt... |
| wlkiswwlks1 29665 | The sequence of vertices i... |
| wlklnwwlkln1 29666 | The sequence of vertices i... |
| wlkiswwlks2lem1 29667 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 29668 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 29669 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 29670 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 29671 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 29672 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 29673 | A walk as word corresponds... |
| wlkiswwlks 29674 | A walk as word corresponds... |
| wlkiswwlksupgr2 29675 | A walk as word corresponds... |
| wlkiswwlkupgr 29676 | A walk as word corresponds... |
| wlkswwlksf1o 29677 | The mapping of (ordinary) ... |
| wlkswwlksen 29678 | The set of walks as words ... |
| wwlksm1edg 29679 | Removing the trailing edge... |
| wlklnwwlkln2lem 29680 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 29681 | A walk of length ` N ` as ... |
| wlklnwwlkn 29682 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 29683 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 29684 | A walk of length ` N ` as ... |
| wlknewwlksn 29685 | If a walk in a pseudograph... |
| wlknwwlksnbij 29686 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 29687 | In a simple pseudograph, t... |
| wlknwwlksneqs 29688 | The set of walks of a fixe... |
| wwlkseq 29689 | Equality of two walks (as ... |
| wwlksnred 29690 | Reduction of a walk (as wo... |
| wwlksnext 29691 | Extension of a walk (as wo... |
| wwlksnextbi 29692 | Extension of a walk (as wo... |
| wwlksnredwwlkn 29693 | For each walk (as word) of... |
| wwlksnredwwlkn0 29694 | For each walk (as word) of... |
| wwlksnextwrd 29695 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 29696 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 29697 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 29698 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 29699 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 29700 | There is a bijection betwe... |
| wwlksnexthasheq 29701 | The number of the extensio... |
| disjxwwlksn 29702 | Sets of walks (as words) e... |
| wwlksnndef 29703 | Conditions for ` WWalksN `... |
| wwlksnfi 29704 | The number of walks repres... |
| wlksnfi 29705 | The number of walks of fix... |
| wlksnwwlknvbij 29706 | There is a bijection betwe... |
| wwlksnextproplem1 29707 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 29708 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 29709 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 29710 | Adding additional properti... |
| disjxwwlkn 29711 | Sets of walks (as words) e... |
| hashwwlksnext 29712 | Number of walks (as words)... |
| wwlksnwwlksnon 29713 | A walk of fixed length is ... |
| wspthsnwspthsnon 29714 | A simple path of fixed len... |
| wspthsnonn0vne 29715 | If the set of simple paths... |
| wspthsswwlkn 29716 | The set of simple paths of... |
| wspthnfi 29717 | In a finite graph, the set... |
| wwlksnonfi 29718 | In a finite graph, the set... |
| wspthsswwlknon 29719 | The set of simple paths of... |
| wspthnonfi 29720 | In a finite graph, the set... |
| wspniunwspnon 29721 | The set of nonempty simple... |
| wspn0 29722 | If there are no vertices, ... |
| 2wlkdlem1 29723 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 29724 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 29725 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 29726 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 29727 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 29728 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 29729 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 29730 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 29731 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 29732 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 29733 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 29734 | Construction of a walk fro... |
| 2wlkond 29735 | A walk of length 2 from on... |
| 2trld 29736 | Construction of a trail fr... |
| 2trlond 29737 | A trail of length 2 from o... |
| 2pthd 29738 | A path of length 2 from on... |
| 2spthd 29739 | A simple path of length 2 ... |
| 2pthond 29740 | A simple path of length 2 ... |
| 2pthon3v 29741 | For a vertex adjacent to t... |
| umgr2adedgwlklem 29742 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 29743 | In a multigraph, two adjac... |
| umgr2adedgwlkon 29744 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 29745 | Alternate proof for ~ umgr... |
| umgr2adedgspth 29746 | In a multigraph, two adjac... |
| umgr2wlk 29747 | In a multigraph, there is ... |
| umgr2wlkon 29748 | For each pair of adjacent ... |
| elwwlks2s3 29749 | A walk of length 2 as word... |
| midwwlks2s3 29750 | There is a vertex between ... |
| wwlks2onv 29751 | If a length 3 string repre... |
| elwwlks2ons3im 29752 | A walk as word of length 2... |
| elwwlks2ons3 29753 | For each walk of length 2 ... |
| s3wwlks2on 29754 | A length 3 string which re... |
| umgrwwlks2on 29755 | A walk of length 2 between... |
| wwlks2onsym 29756 | There is a walk of length ... |
| elwwlks2on 29757 | A walk of length 2 between... |
| elwspths2on 29758 | A simple path of length 2 ... |
| wpthswwlks2on 29759 | For two different vertices... |
| 2wspdisj 29760 | All simple paths of length... |
| 2wspiundisj 29761 | All simple paths of length... |
| usgr2wspthons3 29762 | A simple path of length 2 ... |
| usgr2wspthon 29763 | A simple path of length 2 ... |
| elwwlks2 29764 | A walk of length 2 between... |
| elwspths2spth 29765 | A simple path of length 2 ... |
| rusgrnumwwlkl1 29766 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 29767 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 29768 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 29769 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 29770 | Induction base 1 for ~ rus... |
| rusgr0edg 29771 | Special case for graphs wi... |
| rusgrnumwwlks 29772 | Induction step for ~ rusgr... |
| rusgrnumwwlk 29773 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 29774 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 29775 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 29776 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 29777 | In a ` K `-regular graph, ... |
| clwwlk 29780 | The set of closed walks (i... |
| isclwwlk 29781 | Properties of a word to re... |
| clwwlkbp 29782 | Basic properties of a clos... |
| clwwlkgt0 29783 | There is no empty closed w... |
| clwwlksswrd 29784 | Closed walks (represented ... |
| clwwlk1loop 29785 | A closed walk of length 1 ... |
| clwwlkccatlem 29786 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 29787 | The concatenation of two w... |
| umgrclwwlkge2 29788 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 29789 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 29790 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 29791 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 29792 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 29793 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 29794 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 29795 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 29796 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 29797 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 29798 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 29799 | A closed walk as word of l... |
| clwlkclwwlk2 29800 | A closed walk corresponds ... |
| clwlkclwwlkflem 29801 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 29802 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 29803 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 29804 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 29805 | ` F ` is a function from t... |
| clwlkclwwlkfo 29806 | ` F ` is a function from t... |
| clwlkclwwlkf1 29807 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 29808 | ` F ` is a bijection betwe... |
| clwlkclwwlken 29809 | The set of the nonempty cl... |
| clwwisshclwwslemlem 29810 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 29811 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 29812 | Cyclically shifting a clos... |
| clwwisshclwwsn 29813 | Cyclically shifting a clos... |
| erclwwlkrel 29814 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 29815 | Two classes are equivalent... |
| erclwwlkeqlen 29816 | If two classes are equival... |
| erclwwlkref 29817 | ` .~ ` is a reflexive rela... |
| erclwwlksym 29818 | ` .~ ` is a symmetric rela... |
| erclwwlktr 29819 | ` .~ ` is a transitive rel... |
| erclwwlk 29820 | ` .~ ` is an equivalence r... |
| clwwlkn 29823 | The set of closed walks of... |
| isclwwlkn 29824 | A word over the set of ver... |
| clwwlkn0 29825 | There is no closed walk of... |
| clwwlkneq0 29826 | Sufficient conditions for ... |
| clwwlkclwwlkn 29827 | A closed walk of a fixed l... |
| clwwlksclwwlkn 29828 | The closed walks of a fixe... |
| clwwlknlen 29829 | The length of a word repre... |
| clwwlknnn 29830 | The length of a closed wal... |
| clwwlknwrd 29831 | A closed walk of a fixed l... |
| clwwlknbp 29832 | Basic properties of a clos... |
| isclwwlknx 29833 | Characterization of a word... |
| clwwlknp 29834 | Properties of a set being ... |
| clwwlknwwlksn 29835 | A word representing a clos... |
| clwwlknlbonbgr1 29836 | The last but one vertex in... |
| clwwlkinwwlk 29837 | If the initial vertex of a... |
| clwwlkn1 29838 | A closed walk of length 1 ... |
| loopclwwlkn1b 29839 | The singleton word consist... |
| clwwlkn1loopb 29840 | A word represents a closed... |
| clwwlkn2 29841 | A closed walk of length 2 ... |
| clwwlknfi 29842 | If there is only a finite ... |
| clwwlkel 29843 | Obtaining a closed walk (a... |
| clwwlkf 29844 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 29845 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 29846 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 29847 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 29848 | F is a 1-1 onto function, ... |
| clwwlken 29849 | The set of closed walks of... |
| clwwlknwwlkncl 29850 | Obtaining a closed walk (a... |
| clwwlkwwlksb 29851 | A nonempty word over verti... |
| clwwlknwwlksnb 29852 | A word over vertices repre... |
| clwwlkext2edg 29853 | If a word concatenated wit... |
| wwlksext2clwwlk 29854 | If a word represents a wal... |
| wwlksubclwwlk 29855 | Any prefix of a word repre... |
| clwwnisshclwwsn 29856 | Cyclically shifting a clos... |
| eleclclwwlknlem1 29857 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 29858 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 29859 | The set of cyclical shifts... |
| clwwlknccat 29860 | The concatenation of two w... |
| umgr2cwwk2dif 29861 | If a word represents a clo... |
| umgr2cwwkdifex 29862 | If a word represents a clo... |
| erclwwlknrel 29863 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 29864 | Two classes are equivalent... |
| erclwwlkneqlen 29865 | If two classes are equival... |
| erclwwlknref 29866 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 29867 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 29868 | ` .~ ` is a transitive rel... |
| erclwwlkn 29869 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 29870 | The quotient set of the se... |
| hashclwwlkn0 29871 | The number of closed walks... |
| eclclwwlkn1 29872 | An equivalence class accor... |
| eleclclwwlkn 29873 | A member of an equivalence... |
| hashecclwwlkn1 29874 | The size of every equivale... |
| umgrhashecclwwlk 29875 | The size of every equivale... |
| fusgrhashclwwlkn 29876 | The size of the set of clo... |
| clwwlkndivn 29877 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 29878 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 29879 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 29880 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 29881 | There is a one-to-one onto... |
| clwlkssizeeq 29882 | The size of the set of clo... |
| clwlksndivn 29883 | The size of the set of clo... |
| clwwlknonmpo 29886 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 29887 | The set of closed walks on... |
| isclwwlknon 29888 | A word over the set of ver... |
| clwwlk0on0 29889 | There is no word over the ... |
| clwwlknon0 29890 | Sufficient conditions for ... |
| clwwlknonfin 29891 | In a finite graph ` G ` , ... |
| clwwlknonel 29892 | Characterization of a word... |
| clwwlknonccat 29893 | The concatenation of two w... |
| clwwlknon1 29894 | The set of closed walks on... |
| clwwlknon1loop 29895 | If there is a loop at vert... |
| clwwlknon1nloop 29896 | If there is no loop at ver... |
| clwwlknon1sn 29897 | The set of (closed) walks ... |
| clwwlknon1le1 29898 | There is at most one (clos... |
| clwwlknon2 29899 | The set of closed walks on... |
| clwwlknon2x 29900 | The set of closed walks on... |
| s2elclwwlknon2 29901 | Sufficient conditions of a... |
| clwwlknon2num 29902 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 29903 | A word over vertices repre... |
| clwwlknonex2lem1 29904 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 29905 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 29906 | Extending a closed walk ` ... |
| clwwlknonex2e 29907 | Extending a closed walk ` ... |
| clwwlknondisj 29908 | The sets of closed walks o... |
| clwwlknun 29909 | The set of closed walks of... |
| clwwlkvbij 29910 | There is a bijection betwe... |
| 0ewlk 29911 | The empty set (empty seque... |
| 1ewlk 29912 | A sequence of 1 edge is an... |
| 0wlk 29913 | A pair of an empty set (of... |
| is0wlk 29914 | A pair of an empty set (of... |
| 0wlkonlem1 29915 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 29916 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 29917 | A walk of length 0 from a ... |
| 0wlkons1 29918 | A walk of length 0 from a ... |
| 0trl 29919 | A pair of an empty set (of... |
| is0trl 29920 | A pair of an empty set (of... |
| 0trlon 29921 | A trail of length 0 from a... |
| 0pth 29922 | A pair of an empty set (of... |
| 0spth 29923 | A pair of an empty set (of... |
| 0pthon 29924 | A path of length 0 from a ... |
| 0pthon1 29925 | A path of length 0 from a ... |
| 0pthonv 29926 | For each vertex there is a... |
| 0clwlk 29927 | A pair of an empty set (of... |
| 0clwlkv 29928 | Any vertex (more precisely... |
| 0clwlk0 29929 | There is no closed walk in... |
| 0crct 29930 | A pair of an empty set (of... |
| 0cycl 29931 | A pair of an empty set (of... |
| 1pthdlem1 29932 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 29933 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 29934 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 29935 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 29936 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 29937 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 29938 | In a graph with two vertic... |
| 1trld 29939 | In a graph with two vertic... |
| 1pthd 29940 | In a graph with two vertic... |
| 1pthond 29941 | In a graph with two vertic... |
| upgr1wlkdlem1 29942 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 29943 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 29944 | In a pseudograph with two ... |
| upgr1trld 29945 | In a pseudograph with two ... |
| upgr1pthd 29946 | In a pseudograph with two ... |
| upgr1pthond 29947 | In a pseudograph with two ... |
| lppthon 29948 | A loop (which is an edge a... |
| lp1cycl 29949 | A loop (which is an edge a... |
| 1pthon2v 29950 | For each pair of adjacent ... |
| 1pthon2ve 29951 | For each pair of adjacent ... |
| wlk2v2elem1 29952 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 29953 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 29954 | In a graph with two vertic... |
| ntrl2v2e 29955 | A walk which is not a trai... |
| 3wlkdlem1 29956 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 29957 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 29958 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 29959 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 29960 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 29961 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 29962 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 29963 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 29964 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 29965 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 29966 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 29967 | Construction of a walk fro... |
| 3wlkond 29968 | A walk of length 3 from on... |
| 3trld 29969 | Construction of a trail fr... |
| 3trlond 29970 | A trail of length 3 from o... |
| 3pthd 29971 | A path of length 3 from on... |
| 3pthond 29972 | A path of length 3 from on... |
| 3spthd 29973 | A simple path of length 3 ... |
| 3spthond 29974 | A simple path of length 3 ... |
| 3cycld 29975 | Construction of a 3-cycle ... |
| 3cyclpd 29976 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 29977 | If there is a cycle of len... |
| uhgr3cyclexlem 29978 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 29979 | If there are three differe... |
| umgr3cyclex 29980 | If there are three (differ... |
| umgr3v3e3cycl 29981 | If and only if there is a ... |
| upgr4cycl4dv4e 29982 | If there is a cycle of len... |
| dfconngr1 29985 | Alternative definition of ... |
| isconngr 29986 | The property of being a co... |
| isconngr1 29987 | The property of being a co... |
| cusconngr 29988 | A complete hypergraph is c... |
| 0conngr 29989 | A graph without vertices i... |
| 0vconngr 29990 | A graph without vertices i... |
| 1conngr 29991 | A graph with (at most) one... |
| conngrv2edg 29992 | A vertex in a connected gr... |
| vdn0conngrumgrv2 29993 | A vertex in a connected mu... |
| releupth 29996 | The set ` ( EulerPaths `` ... |
| eupths 29997 | The Eulerian paths on the ... |
| iseupth 29998 | The property " ` <. F , P ... |
| iseupthf1o 29999 | The property " ` <. F , P ... |
| eupthi 30000 | Properties of an Eulerian ... |
| eupthf1o 30001 | The ` F ` function in an E... |
| eupthfi 30002 | Any graph with an Eulerian... |
| eupthseg 30003 | The ` N ` -th edge in an e... |
| upgriseupth 30004 | The property " ` <. F , P ... |
| upgreupthi 30005 | Properties of an Eulerian ... |
| upgreupthseg 30006 | The ` N ` -th edge in an e... |
| eupthcl 30007 | An Eulerian path has lengt... |
| eupthistrl 30008 | An Eulerian path is a trai... |
| eupthiswlk 30009 | An Eulerian path is a walk... |
| eupthpf 30010 | The ` P ` function in an E... |
| eupth0 30011 | There is an Eulerian path ... |
| eupthres 30012 | The restriction ` <. H , Q... |
| eupthp1 30013 | Append one path segment to... |
| eupth2eucrct 30014 | Append one path segment to... |
| eupth2lem1 30015 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30016 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30017 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30018 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30019 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30020 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30021 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30022 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30023 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30024 | Formerly part of proof of ... |
| eupth2lem3lem1 30025 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30026 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30027 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30028 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30029 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30030 | Formerly part of proof of ... |
| eupth2lem3lem7 30031 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30032 | Formerly part of proof of ... |
| eupth2lem3 30033 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30034 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30035 | Lemma for ~ eupth2 (induct... |
| eupth2 30036 | The only vertices of odd d... |
| eulerpathpr 30037 | A graph with an Eulerian p... |
| eulerpath 30038 | A pseudograph with an Eule... |
| eulercrct 30039 | A pseudograph with an Eule... |
| eucrctshift 30040 | Cyclically shifting the in... |
| eucrct2eupth1 30041 | Removing one edge ` ( I ``... |
| eucrct2eupth 30042 | Removing one edge ` ( I ``... |
| konigsbergvtx 30043 | The set of vertices of the... |
| konigsbergiedg 30044 | The indexed edges of the K... |
| konigsbergiedgw 30045 | The indexed edges of the K... |
| konigsbergssiedgwpr 30046 | Each subset of the indexed... |
| konigsbergssiedgw 30047 | Each subset of the indexed... |
| konigsbergumgr 30048 | The Königsberg graph ... |
| konigsberglem1 30049 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30050 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30051 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30052 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30053 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30054 | The Königsberg Bridge... |
| isfrgr 30057 | The property of being a fr... |
| frgrusgr 30058 | A friendship graph is a si... |
| frgr0v 30059 | Any null graph (set with n... |
| frgr0vb 30060 | Any null graph (without ve... |
| frgruhgr0v 30061 | Any null graph (without ve... |
| frgr0 30062 | The null graph (graph with... |
| frcond1 30063 | The friendship condition: ... |
| frcond2 30064 | The friendship condition: ... |
| frgreu 30065 | Variant of ~ frcond2 : An... |
| frcond3 30066 | The friendship condition, ... |
| frcond4 30067 | The friendship condition, ... |
| frgr1v 30068 | Any graph with (at most) o... |
| nfrgr2v 30069 | Any graph with two (differ... |
| frgr3vlem1 30070 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30071 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30072 | Any graph with three verti... |
| 1vwmgr 30073 | Every graph with one verte... |
| 3vfriswmgrlem 30074 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30075 | Every friendship graph wit... |
| 1to2vfriswmgr 30076 | Every friendship graph wit... |
| 1to3vfriswmgr 30077 | Every friendship graph wit... |
| 1to3vfriendship 30078 | The friendship theorem for... |
| 2pthfrgrrn 30079 | Between any two (different... |
| 2pthfrgrrn2 30080 | Between any two (different... |
| 2pthfrgr 30081 | Between any two (different... |
| 3cyclfrgrrn1 30082 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30083 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30084 | Every vertex in a friendsh... |
| 3cyclfrgr 30085 | Every vertex in a friendsh... |
| 4cycl2v2nb 30086 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30087 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30088 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30089 | A graph with a 4-cycle is ... |
| frgrnbnb 30090 | If two neighbors ` U ` and... |
| frgrconngr 30091 | A friendship graph is conn... |
| vdgn0frgrv2 30092 | A vertex in a friendship g... |
| vdgn1frgrv2 30093 | Any vertex in a friendship... |
| vdgn1frgrv3 30094 | Any vertex in a friendship... |
| vdgfrgrgt2 30095 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30096 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30097 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30098 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30099 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30100 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30101 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30102 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30103 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30104 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30105 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30106 | In a friendship graph, two... |
| frgrwopreglem4a 30107 | In a friendship graph any ... |
| frgrwopreglem5a 30108 | If a friendship graph has ... |
| frgrwopreglem1 30109 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30110 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30111 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30112 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30113 | According to statement 5 i... |
| frgrwopregbsn 30114 | According to statement 5 i... |
| frgrwopreg1 30115 | According to statement 5 i... |
| frgrwopreg2 30116 | According to statement 5 i... |
| frgrwopreglem5lem 30117 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30118 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30119 | Alternate direct proof of ... |
| frgrwopreg 30120 | In a friendship graph ther... |
| frgrregorufr0 30121 | In a friendship graph ther... |
| frgrregorufr 30122 | If there is a vertex havin... |
| frgrregorufrg 30123 | If there is a vertex havin... |
| frgr2wwlkeu 30124 | For two different vertices... |
| frgr2wwlkn0 30125 | In a friendship graph, the... |
| frgr2wwlk1 30126 | In a friendship graph, the... |
| frgr2wsp1 30127 | In a friendship graph, the... |
| frgr2wwlkeqm 30128 | If there is a (simple) pat... |
| frgrhash2wsp 30129 | The number of simple paths... |
| fusgreg2wsplem 30130 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30131 | The set of paths of length... |
| fusgreghash2wspv 30132 | According to statement 7 i... |
| fusgreg2wsp 30133 | In a finite simple graph, ... |
| 2wspmdisj 30134 | The sets of paths of lengt... |
| fusgreghash2wsp 30135 | In a finite k-regular grap... |
| frrusgrord0lem 30136 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30137 | If a nonempty finite frien... |
| frrusgrord 30138 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30139 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30140 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30141 | If the initial vertex of a... |
| clwwnonrepclwwnon 30142 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30143 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30144 | Value of operation ` C ` ,... |
| 2clwwlk2 30145 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30146 | Characterization of an ele... |
| 2clwwlk2clwwlk 30147 | An element of the value of... |
| numclwwlk1lem2foalem 30148 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30149 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30150 | Characterization of an ele... |
| numclwwlk1lem2foa 30151 | Going forth and back from ... |
| numclwwlk1lem2f 30152 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30153 | Value of the function ` T ... |
| numclwwlk1lem2f1 30154 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30155 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30156 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30157 | The set of double loops of... |
| numclwwlk1 30158 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30159 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30160 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30161 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30162 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30163 | The sets of the two repres... |
| wlkl0 30164 | There is exactly one walk ... |
| clwlknon2num 30165 | There are k walks of lengt... |
| numclwlk1lem1 30166 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 30167 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 30168 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 30169 | Value of operation ` H ` ,... |
| numclwwlkovh 30170 | Value of operation ` H ` ,... |
| numclwwlkovq 30171 | Value of operation ` Q ` ,... |
| numclwwlkqhash 30172 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 30173 | In a friendship graph, for... |
| numclwlk2lem2f 30174 | ` R ` is a function mappin... |
| numclwlk2lem2fv 30175 | Value of the function ` R ... |
| numclwlk2lem2f1o 30176 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 30177 | In a friendship graph, the... |
| numclwwlk2 30178 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 30179 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 30180 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 30181 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 30182 | Statement 12 in [Huneke] p... |
| numclwwlk4 30183 | The total number of closed... |
| numclwwlk5lem 30184 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 30185 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 30186 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 30187 | For a prime divisor ` P ` ... |
| numclwwlk7 30188 | Statement 14 in [Huneke] p... |
| numclwwlk8 30189 | The size of the set of clo... |
| frgrreggt1 30190 | If a finite nonempty frien... |
| frgrreg 30191 | If a finite nonempty frien... |
| frgrregord013 30192 | If a finite friendship gra... |
| frgrregord13 30193 | If a nonempty finite frien... |
| frgrogt3nreg 30194 | If a finite friendship gra... |
| friendshipgt3 30195 | The friendship theorem for... |
| friendship 30196 | The friendship theorem: I... |
| conventions 30197 |
H... |
| conventions-labels 30198 |
... |
| conventions-comments 30199 |
... |
| natded 30200 | Here are typical n... |
| ex-natded5.2 30201 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 30202 | A more efficient proof of ... |
| ex-natded5.2i 30203 | The same as ~ ex-natded5.2... |
| ex-natded5.3 30204 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 30205 | A more efficient proof of ... |
| ex-natded5.3i 30206 | The same as ~ ex-natded5.3... |
| ex-natded5.5 30207 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 30208 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 30209 | A more efficient proof of ... |
| ex-natded5.8 30210 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 30211 | A more efficient proof of ... |
| ex-natded5.13 30212 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 30213 | A more efficient proof of ... |
| ex-natded9.20 30214 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 30215 | A more efficient proof of ... |
| ex-natded9.26 30216 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 30217 | A more efficient proof of ... |
| ex-or 30218 | Example for ~ df-or . Exa... |
| ex-an 30219 | Example for ~ df-an . Exa... |
| ex-dif 30220 | Example for ~ df-dif . Ex... |
| ex-un 30221 | Example for ~ df-un . Exa... |
| ex-in 30222 | Example for ~ df-in . Exa... |
| ex-uni 30223 | Example for ~ df-uni . Ex... |
| ex-ss 30224 | Example for ~ df-ss . Exa... |
| ex-pss 30225 | Example for ~ df-pss . Ex... |
| ex-pw 30226 | Example for ~ df-pw . Exa... |
| ex-pr 30227 | Example for ~ df-pr . (Co... |
| ex-br 30228 | Example for ~ df-br . Exa... |
| ex-opab 30229 | Example for ~ df-opab . E... |
| ex-eprel 30230 | Example for ~ df-eprel . ... |
| ex-id 30231 | Example for ~ df-id . Exa... |
| ex-po 30232 | Example for ~ df-po . Exa... |
| ex-xp 30233 | Example for ~ df-xp . Exa... |
| ex-cnv 30234 | Example for ~ df-cnv . Ex... |
| ex-co 30235 | Example for ~ df-co . Exa... |
| ex-dm 30236 | Example for ~ df-dm . Exa... |
| ex-rn 30237 | Example for ~ df-rn . Exa... |
| ex-res 30238 | Example for ~ df-res . Ex... |
| ex-ima 30239 | Example for ~ df-ima . Ex... |
| ex-fv 30240 | Example for ~ df-fv . Exa... |
| ex-1st 30241 | Example for ~ df-1st . Ex... |
| ex-2nd 30242 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 30243 | Example for ~ df-dec , 100... |
| ex-fl 30244 | Example for ~ df-fl . Exa... |
| ex-ceil 30245 | Example for ~ df-ceil . (... |
| ex-mod 30246 | Example for ~ df-mod . (C... |
| ex-exp 30247 | Example for ~ df-exp . (C... |
| ex-fac 30248 | Example for ~ df-fac . (C... |
| ex-bc 30249 | Example for ~ df-bc . (Co... |
| ex-hash 30250 | Example for ~ df-hash . (... |
| ex-sqrt 30251 | Example for ~ df-sqrt . (... |
| ex-abs 30252 | Example for ~ df-abs . (C... |
| ex-dvds 30253 | Example for ~ df-dvds : 3 ... |
| ex-gcd 30254 | Example for ~ df-gcd . (C... |
| ex-lcm 30255 | Example for ~ df-lcm . (C... |
| ex-prmo 30256 | Example for ~ df-prmo : ` ... |
| aevdemo 30257 | Proof illustrating the com... |
| ex-ind-dvds 30258 | Example of a proof by indu... |
| ex-fpar 30259 | Formalized example provide... |
| avril1 30260 | Poisson d'Avril's Theorem.... |
| 2bornot2b 30261 | The law of excluded middle... |
| helloworld 30262 | The classic "Hello world" ... |
| 1p1e2apr1 30263 | One plus one equals two. ... |
| eqid1 30264 | Law of identity (reflexivi... |
| 1div0apr 30265 | Division by zero is forbid... |
| topnfbey 30266 | Nothing seems to be imposs... |
| 9p10ne21 30267 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 30268 | 9 + 10 equals 21. This as... |
| nrt2irr 30270 | The ` N ` -th root of 2 is... |
| isplig 30273 | The predicate "is a planar... |
| ispligb 30274 | The predicate "is a planar... |
| tncp 30275 | In any planar incidence ge... |
| l2p 30276 | For any line in a planar i... |
| lpni 30277 | For any line in a planar i... |
| nsnlplig 30278 | There is no "one-point lin... |
| nsnlpligALT 30279 | Alternate version of ~ nsn... |
| n0lplig 30280 | There is no "empty line" i... |
| n0lpligALT 30281 | Alternate version of ~ n0l... |
| eulplig 30282 | Through two distinct point... |
| pliguhgr 30283 | Any planar incidence geome... |
| dummylink 30284 | Alias for ~ a1ii that may ... |
| id1 30285 | Alias for ~ idALT that may... |
| isgrpo 30294 | The predicate "is a group ... |
| isgrpoi 30295 | Properties that determine ... |
| grpofo 30296 | A group operation maps ont... |
| grpocl 30297 | Closure law for a group op... |
| grpolidinv 30298 | A group has a left identit... |
| grpon0 30299 | The base set of a group is... |
| grpoass 30300 | A group operation is assoc... |
| grpoidinvlem1 30301 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 30302 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 30303 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 30304 | Lemma for ~ grpoidinv . (... |
| grpoidinv 30305 | A group has a left and rig... |
| grpoideu 30306 | The left identity element ... |
| grporndm 30307 | A group's range in terms o... |
| 0ngrp 30308 | The empty set is not a gro... |
| gidval 30309 | The value of the identity ... |
| grpoidval 30310 | Lemma for ~ grpoidcl and o... |
| grpoidcl 30311 | The identity element of a ... |
| grpoidinv2 30312 | A group's properties using... |
| grpolid 30313 | The identity element of a ... |
| grporid 30314 | The identity element of a ... |
| grporcan 30315 | Right cancellation law for... |
| grpoinveu 30316 | The left inverse element o... |
| grpoid 30317 | Two ways of saying that an... |
| grporn 30318 | The range of a group opera... |
| grpoinvfval 30319 | The inverse function of a ... |
| grpoinvval 30320 | The inverse of a group ele... |
| grpoinvcl 30321 | A group element's inverse ... |
| grpoinv 30322 | The properties of a group ... |
| grpolinv 30323 | The left inverse of a grou... |
| grporinv 30324 | The right inverse of a gro... |
| grpoinvid1 30325 | The inverse of a group ele... |
| grpoinvid2 30326 | The inverse of a group ele... |
| grpolcan 30327 | Left cancellation law for ... |
| grpo2inv 30328 | Double inverse law for gro... |
| grpoinvf 30329 | Mapping of the inverse fun... |
| grpoinvop 30330 | The inverse of the group o... |
| grpodivfval 30331 | Group division (or subtrac... |
| grpodivval 30332 | Group division (or subtrac... |
| grpodivinv 30333 | Group division by an inver... |
| grpoinvdiv 30334 | Inverse of a group divisio... |
| grpodivf 30335 | Mapping for group division... |
| grpodivcl 30336 | Closure of group division ... |
| grpodivdiv 30337 | Double group division. (C... |
| grpomuldivass 30338 | Associative-type law for m... |
| grpodivid 30339 | Division of a group member... |
| grponpcan 30340 | Cancellation law for group... |
| isablo 30343 | The predicate "is an Abeli... |
| ablogrpo 30344 | An Abelian group operation... |
| ablocom 30345 | An Abelian group operation... |
| ablo32 30346 | Commutative/associative la... |
| ablo4 30347 | Commutative/associative la... |
| isabloi 30348 | Properties that determine ... |
| ablomuldiv 30349 | Law for group multiplicati... |
| ablodivdiv 30350 | Law for double group divis... |
| ablodivdiv4 30351 | Law for double group divis... |
| ablodiv32 30352 | Swap the second and third ... |
| ablonncan 30353 | Cancellation law for group... |
| ablonnncan1 30354 | Cancellation law for group... |
| vcrel 30357 | The class of all complex v... |
| vciOLD 30358 | Obsolete version of ~ cvsi... |
| vcsm 30359 | Functionality of th scalar... |
| vccl 30360 | Closure of the scalar prod... |
| vcidOLD 30361 | Identity element for the s... |
| vcdi 30362 | Distributive law for the s... |
| vcdir 30363 | Distributive law for the s... |
| vcass 30364 | Associative law for the sc... |
| vc2OLD 30365 | A vector plus itself is tw... |
| vcablo 30366 | Vector addition is an Abel... |
| vcgrp 30367 | Vector addition is a group... |
| vclcan 30368 | Left cancellation law for ... |
| vczcl 30369 | The zero vector is a vecto... |
| vc0rid 30370 | The zero vector is a right... |
| vc0 30371 | Zero times a vector is the... |
| vcz 30372 | Anything times the zero ve... |
| vcm 30373 | Minus 1 times a vector is ... |
| isvclem 30374 | Lemma for ~ isvcOLD . (Co... |
| vcex 30375 | The components of a comple... |
| isvcOLD 30376 | The predicate "is a comple... |
| isvciOLD 30377 | Properties that determine ... |
| cnaddabloOLD 30378 | Obsolete version of ~ cnad... |
| cnidOLD 30379 | Obsolete version of ~ cnad... |
| cncvcOLD 30380 | Obsolete version of ~ cncv... |
| nvss 30390 | Structure of the class of ... |
| nvvcop 30391 | A normed complex vector sp... |
| nvrel 30399 | The class of all normed co... |
| vafval 30400 | Value of the function for ... |
| bafval 30401 | Value of the function for ... |
| smfval 30402 | Value of the function for ... |
| 0vfval 30403 | Value of the function for ... |
| nmcvfval 30404 | Value of the norm function... |
| nvop2 30405 | A normed complex vector sp... |
| nvvop 30406 | The vector space component... |
| isnvlem 30407 | Lemma for ~ isnv . (Contr... |
| nvex 30408 | The components of a normed... |
| isnv 30409 | The predicate "is a normed... |
| isnvi 30410 | Properties that determine ... |
| nvi 30411 | The properties of a normed... |
| nvvc 30412 | The vector space component... |
| nvablo 30413 | The vector addition operat... |
| nvgrp 30414 | The vector addition operat... |
| nvgf 30415 | Mapping for the vector add... |
| nvsf 30416 | Mapping for the scalar mul... |
| nvgcl 30417 | Closure law for the vector... |
| nvcom 30418 | The vector addition (group... |
| nvass 30419 | The vector addition (group... |
| nvadd32 30420 | Commutative/associative la... |
| nvrcan 30421 | Right cancellation law for... |
| nvadd4 30422 | Rearrangement of 4 terms i... |
| nvscl 30423 | Closure law for the scalar... |
| nvsid 30424 | Identity element for the s... |
| nvsass 30425 | Associative law for the sc... |
| nvscom 30426 | Commutative law for the sc... |
| nvdi 30427 | Distributive law for the s... |
| nvdir 30428 | Distributive law for the s... |
| nv2 30429 | A vector plus itself is tw... |
| vsfval 30430 | Value of the function for ... |
| nvzcl 30431 | Closure law for the zero v... |
| nv0rid 30432 | The zero vector is a right... |
| nv0lid 30433 | The zero vector is a left ... |
| nv0 30434 | Zero times a vector is the... |
| nvsz 30435 | Anything times the zero ve... |
| nvinv 30436 | Minus 1 times a vector is ... |
| nvinvfval 30437 | Function for the negative ... |
| nvm 30438 | Vector subtraction in term... |
| nvmval 30439 | Value of vector subtractio... |
| nvmval2 30440 | Value of vector subtractio... |
| nvmfval 30441 | Value of the function for ... |
| nvmf 30442 | Mapping for the vector sub... |
| nvmcl 30443 | Closure law for the vector... |
| nvnnncan1 30444 | Cancellation law for vecto... |
| nvmdi 30445 | Distributive law for scala... |
| nvnegneg 30446 | Double negative of a vecto... |
| nvmul0or 30447 | If a scalar product is zer... |
| nvrinv 30448 | A vector minus itself. (C... |
| nvlinv 30449 | Minus a vector plus itself... |
| nvpncan2 30450 | Cancellation law for vecto... |
| nvpncan 30451 | Cancellation law for vecto... |
| nvaddsub 30452 | Commutative/associative la... |
| nvnpcan 30453 | Cancellation law for a nor... |
| nvaddsub4 30454 | Rearrangement of 4 terms i... |
| nvmeq0 30455 | The difference between two... |
| nvmid 30456 | A vector minus itself is t... |
| nvf 30457 | Mapping for the norm funct... |
| nvcl 30458 | The norm of a normed compl... |
| nvcli 30459 | The norm of a normed compl... |
| nvs 30460 | Proportionality property o... |
| nvsge0 30461 | The norm of a scalar produ... |
| nvm1 30462 | The norm of the negative o... |
| nvdif 30463 | The norm of the difference... |
| nvpi 30464 | The norm of a vector plus ... |
| nvz0 30465 | The norm of a zero vector ... |
| nvz 30466 | The norm of a vector is ze... |
| nvtri 30467 | Triangle inequality for th... |
| nvmtri 30468 | Triangle inequality for th... |
| nvabs 30469 | Norm difference property o... |
| nvge0 30470 | The norm of a normed compl... |
| nvgt0 30471 | A nonzero norm is positive... |
| nv1 30472 | From any nonzero vector, c... |
| nvop 30473 | A complex inner product sp... |
| cnnv 30474 | The set of complex numbers... |
| cnnvg 30475 | The vector addition (group... |
| cnnvba 30476 | The base set of the normed... |
| cnnvs 30477 | The scalar product operati... |
| cnnvnm 30478 | The norm operation of the ... |
| cnnvm 30479 | The vector subtraction ope... |
| elimnv 30480 | Hypothesis elimination lem... |
| elimnvu 30481 | Hypothesis elimination lem... |
| imsval 30482 | Value of the induced metri... |
| imsdval 30483 | Value of the induced metri... |
| imsdval2 30484 | Value of the distance func... |
| nvnd 30485 | The norm of a normed compl... |
| imsdf 30486 | Mapping for the induced me... |
| imsmetlem 30487 | Lemma for ~ imsmet . (Con... |
| imsmet 30488 | The induced metric of a no... |
| imsxmet 30489 | The induced metric of a no... |
| cnims 30490 | The metric induced on the ... |
| vacn 30491 | Vector addition is jointly... |
| nmcvcn 30492 | The norm of a normed compl... |
| nmcnc 30493 | The norm of a normed compl... |
| smcnlem 30494 | Lemma for ~ smcn . (Contr... |
| smcn 30495 | Scalar multiplication is j... |
| vmcn 30496 | Vector subtraction is join... |
| dipfval 30499 | The inner product function... |
| ipval 30500 | Value of the inner product... |
| ipval2lem2 30501 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 30502 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 30503 | Lemma for ~ ipval3 . (Con... |
| ipval2 30504 | Expansion of the inner pro... |
| 4ipval2 30505 | Four times the inner produ... |
| ipval3 30506 | Expansion of the inner pro... |
| ipidsq 30507 | The inner product of a vec... |
| ipnm 30508 | Norm expressed in terms of... |
| dipcl 30509 | An inner product is a comp... |
| ipf 30510 | Mapping for the inner prod... |
| dipcj 30511 | The complex conjugate of a... |
| ipipcj 30512 | An inner product times its... |
| diporthcom 30513 | Orthogonality (meaning inn... |
| dip0r 30514 | Inner product with a zero ... |
| dip0l 30515 | Inner product with a zero ... |
| ipz 30516 | The inner product of a vec... |
| dipcn 30517 | Inner product is jointly c... |
| sspval 30520 | The set of all subspaces o... |
| isssp 30521 | The predicate "is a subspa... |
| sspid 30522 | A normed complex vector sp... |
| sspnv 30523 | A subspace is a normed com... |
| sspba 30524 | The base set of a subspace... |
| sspg 30525 | Vector addition on a subsp... |
| sspgval 30526 | Vector addition on a subsp... |
| ssps 30527 | Scalar multiplication on a... |
| sspsval 30528 | Scalar multiplication on a... |
| sspmlem 30529 | Lemma for ~ sspm and other... |
| sspmval 30530 | Vector addition on a subsp... |
| sspm 30531 | Vector subtraction on a su... |
| sspz 30532 | The zero vector of a subsp... |
| sspn 30533 | The norm on a subspace is ... |
| sspnval 30534 | The norm on a subspace in ... |
| sspimsval 30535 | The induced metric on a su... |
| sspims 30536 | The induced metric on a su... |
| lnoval 30549 | The set of linear operator... |
| islno 30550 | The predicate "is a linear... |
| lnolin 30551 | Basic linearity property o... |
| lnof 30552 | A linear operator is a map... |
| lno0 30553 | The value of a linear oper... |
| lnocoi 30554 | The composition of two lin... |
| lnoadd 30555 | Addition property of a lin... |
| lnosub 30556 | Subtraction property of a ... |
| lnomul 30557 | Scalar multiplication prop... |
| nvo00 30558 | Two ways to express a zero... |
| nmoofval 30559 | The operator norm function... |
| nmooval 30560 | The operator norm function... |
| nmosetre 30561 | The set in the supremum of... |
| nmosetn0 30562 | The set in the supremum of... |
| nmoxr 30563 | The norm of an operator is... |
| nmooge0 30564 | The norm of an operator is... |
| nmorepnf 30565 | The norm of an operator is... |
| nmoreltpnf 30566 | The norm of any operator i... |
| nmogtmnf 30567 | The norm of an operator is... |
| nmoolb 30568 | A lower bound for an opera... |
| nmoubi 30569 | An upper bound for an oper... |
| nmoub3i 30570 | An upper bound for an oper... |
| nmoub2i 30571 | An upper bound for an oper... |
| nmobndi 30572 | Two ways to express that a... |
| nmounbi 30573 | Two ways two express that ... |
| nmounbseqi 30574 | An unbounded operator dete... |
| nmounbseqiALT 30575 | Alternate shorter proof of... |
| nmobndseqi 30576 | A bounded sequence determi... |
| nmobndseqiALT 30577 | Alternate shorter proof of... |
| bloval 30578 | The class of bounded linea... |
| isblo 30579 | The predicate "is a bounde... |
| isblo2 30580 | The predicate "is a bounde... |
| bloln 30581 | A bounded operator is a li... |
| blof 30582 | A bounded operator is an o... |
| nmblore 30583 | The norm of a bounded oper... |
| 0ofval 30584 | The zero operator between ... |
| 0oval 30585 | Value of the zero operator... |
| 0oo 30586 | The zero operator is an op... |
| 0lno 30587 | The zero operator is linea... |
| nmoo0 30588 | The operator norm of the z... |
| 0blo 30589 | The zero operator is a bou... |
| nmlno0lem 30590 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 30591 | The norm of a linear opera... |
| nmlno0 30592 | The norm of a linear opera... |
| nmlnoubi 30593 | An upper bound for the ope... |
| nmlnogt0 30594 | The norm of a nonzero line... |
| lnon0 30595 | The domain of a nonzero li... |
| nmblolbii 30596 | A lower bound for the norm... |
| nmblolbi 30597 | A lower bound for the norm... |
| isblo3i 30598 | The predicate "is a bounde... |
| blo3i 30599 | Properties that determine ... |
| blometi 30600 | Upper bound for the distan... |
| blocnilem 30601 | Lemma for ~ blocni and ~ l... |
| blocni 30602 | A linear operator is conti... |
| lnocni 30603 | If a linear operator is co... |
| blocn 30604 | A linear operator is conti... |
| blocn2 30605 | A bounded linear operator ... |
| ajfval 30606 | The adjoint function. (Co... |
| hmoval 30607 | The set of Hermitian (self... |
| ishmo 30608 | The predicate "is a hermit... |
| phnv 30611 | Every complex inner produc... |
| phrel 30612 | The class of all complex i... |
| phnvi 30613 | Every complex inner produc... |
| isphg 30614 | The predicate "is a comple... |
| phop 30615 | A complex inner product sp... |
| cncph 30616 | The set of complex numbers... |
| elimph 30617 | Hypothesis elimination lem... |
| elimphu 30618 | Hypothesis elimination lem... |
| isph 30619 | The predicate "is an inner... |
| phpar2 30620 | The parallelogram law for ... |
| phpar 30621 | The parallelogram law for ... |
| ip0i 30622 | A slight variant of Equati... |
| ip1ilem 30623 | Lemma for ~ ip1i . (Contr... |
| ip1i 30624 | Equation 6.47 of [Ponnusam... |
| ip2i 30625 | Equation 6.48 of [Ponnusam... |
| ipdirilem 30626 | Lemma for ~ ipdiri . (Con... |
| ipdiri 30627 | Distributive law for inner... |
| ipasslem1 30628 | Lemma for ~ ipassi . Show... |
| ipasslem2 30629 | Lemma for ~ ipassi . Show... |
| ipasslem3 30630 | Lemma for ~ ipassi . Show... |
| ipasslem4 30631 | Lemma for ~ ipassi . Show... |
| ipasslem5 30632 | Lemma for ~ ipassi . Show... |
| ipasslem7 30633 | Lemma for ~ ipassi . Show... |
| ipasslem8 30634 | Lemma for ~ ipassi . By ~... |
| ipasslem9 30635 | Lemma for ~ ipassi . Conc... |
| ipasslem10 30636 | Lemma for ~ ipassi . Show... |
| ipasslem11 30637 | Lemma for ~ ipassi . Show... |
| ipassi 30638 | Associative law for inner ... |
| dipdir 30639 | Distributive law for inner... |
| dipdi 30640 | Distributive law for inner... |
| ip2dii 30641 | Inner product of two sums.... |
| dipass 30642 | Associative law for inner ... |
| dipassr 30643 | "Associative" law for seco... |
| dipassr2 30644 | "Associative" law for inne... |
| dipsubdir 30645 | Distributive law for inner... |
| dipsubdi 30646 | Distributive law for inner... |
| pythi 30647 | The Pythagorean theorem fo... |
| siilem1 30648 | Lemma for ~ sii . (Contri... |
| siilem2 30649 | Lemma for ~ sii . (Contri... |
| siii 30650 | Inference from ~ sii . (C... |
| sii 30651 | Obsolete version of ~ ipca... |
| ipblnfi 30652 | A function ` F ` generated... |
| ip2eqi 30653 | Two vectors are equal iff ... |
| phoeqi 30654 | A condition implying that ... |
| ajmoi 30655 | Every operator has at most... |
| ajfuni 30656 | The adjoint function is a ... |
| ajfun 30657 | The adjoint function is a ... |
| ajval 30658 | Value of the adjoint funct... |
| iscbn 30661 | A complex Banach space is ... |
| cbncms 30662 | The induced metric on comp... |
| bnnv 30663 | Every complex Banach space... |
| bnrel 30664 | The class of all complex B... |
| bnsscmcl 30665 | A subspace of a Banach spa... |
| cnbn 30666 | The set of complex numbers... |
| ubthlem1 30667 | Lemma for ~ ubth . The fu... |
| ubthlem2 30668 | Lemma for ~ ubth . Given ... |
| ubthlem3 30669 | Lemma for ~ ubth . Prove ... |
| ubth 30670 | Uniform Boundedness Theore... |
| minvecolem1 30671 | Lemma for ~ minveco . The... |
| minvecolem2 30672 | Lemma for ~ minveco . Any... |
| minvecolem3 30673 | Lemma for ~ minveco . The... |
| minvecolem4a 30674 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 30675 | Lemma for ~ minveco . The... |
| minvecolem4c 30676 | Lemma for ~ minveco . The... |
| minvecolem4 30677 | Lemma for ~ minveco . The... |
| minvecolem5 30678 | Lemma for ~ minveco . Dis... |
| minvecolem6 30679 | Lemma for ~ minveco . Any... |
| minvecolem7 30680 | Lemma for ~ minveco . Sin... |
| minveco 30681 | Minimizing vector theorem,... |
| ishlo 30684 | The predicate "is a comple... |
| hlobn 30685 | Every complex Hilbert spac... |
| hlph 30686 | Every complex Hilbert spac... |
| hlrel 30687 | The class of all complex H... |
| hlnv 30688 | Every complex Hilbert spac... |
| hlnvi 30689 | Every complex Hilbert spac... |
| hlvc 30690 | Every complex Hilbert spac... |
| hlcmet 30691 | The induced metric on a co... |
| hlmet 30692 | The induced metric on a co... |
| hlpar2 30693 | The parallelogram law sati... |
| hlpar 30694 | The parallelogram law sati... |
| hlex 30695 | The base set of a Hilbert ... |
| hladdf 30696 | Mapping for Hilbert space ... |
| hlcom 30697 | Hilbert space vector addit... |
| hlass 30698 | Hilbert space vector addit... |
| hl0cl 30699 | The Hilbert space zero vec... |
| hladdid 30700 | Hilbert space addition wit... |
| hlmulf 30701 | Mapping for Hilbert space ... |
| hlmulid 30702 | Hilbert space scalar multi... |
| hlmulass 30703 | Hilbert space scalar multi... |
| hldi 30704 | Hilbert space scalar multi... |
| hldir 30705 | Hilbert space scalar multi... |
| hlmul0 30706 | Hilbert space scalar multi... |
| hlipf 30707 | Mapping for Hilbert space ... |
| hlipcj 30708 | Conjugate law for Hilbert ... |
| hlipdir 30709 | Distributive law for Hilbe... |
| hlipass 30710 | Associative law for Hilber... |
| hlipgt0 30711 | The inner product of a Hil... |
| hlcompl 30712 | Completeness of a Hilbert ... |
| cnchl 30713 | The set of complex numbers... |
| htthlem 30714 | Lemma for ~ htth . The co... |
| htth 30715 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 30771 | The group (addition) opera... |
| h2hsm 30772 | The scalar product operati... |
| h2hnm 30773 | The norm function of Hilbe... |
| h2hvs 30774 | The vector subtraction ope... |
| h2hmetdval 30775 | Value of the distance func... |
| h2hcau 30776 | The Cauchy sequences of Hi... |
| h2hlm 30777 | The limit sequences of Hil... |
| axhilex-zf 30778 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 30779 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 30780 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 30781 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 30782 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 30783 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 30784 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 30785 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 30786 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 30787 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 30788 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 30789 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 30790 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 30791 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 30792 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 30793 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 30794 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 30795 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 30808 | The Hilbert space scalar p... |
| hvaddcl 30809 | Closure of vector addition... |
| hvmulcl 30810 | Closure of scalar multipli... |
| hvmulcli 30811 | Closure inference for scal... |
| hvsubf 30812 | Mapping domain and codomai... |
| hvsubval 30813 | Value of vector subtractio... |
| hvsubcl 30814 | Closure of vector subtract... |
| hvaddcli 30815 | Closure of vector addition... |
| hvcomi 30816 | Commutation of vector addi... |
| hvsubvali 30817 | Value of vector subtractio... |
| hvsubcli 30818 | Closure of vector subtract... |
| ifhvhv0 30819 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 30820 | Addition with the zero vec... |
| hvmul0 30821 | Scalar multiplication with... |
| hvmul0or 30822 | If a scalar product is zer... |
| hvsubid 30823 | Subtraction of a vector fr... |
| hvnegid 30824 | Addition of negative of a ... |
| hv2neg 30825 | Two ways to express the ne... |
| hvaddlidi 30826 | Addition with the zero vec... |
| hvnegidi 30827 | Addition of negative of a ... |
| hv2negi 30828 | Two ways to express the ne... |
| hvm1neg 30829 | Convert minus one times a ... |
| hvaddsubval 30830 | Value of vector addition i... |
| hvadd32 30831 | Commutative/associative la... |
| hvadd12 30832 | Commutative/associative la... |
| hvadd4 30833 | Hilbert vector space addit... |
| hvsub4 30834 | Hilbert vector space addit... |
| hvaddsub12 30835 | Commutative/associative la... |
| hvpncan 30836 | Addition/subtraction cance... |
| hvpncan2 30837 | Addition/subtraction cance... |
| hvaddsubass 30838 | Associativity of sum and d... |
| hvpncan3 30839 | Subtraction and addition o... |
| hvmulcom 30840 | Scalar multiplication comm... |
| hvsubass 30841 | Hilbert vector space assoc... |
| hvsub32 30842 | Hilbert vector space commu... |
| hvmulassi 30843 | Scalar multiplication asso... |
| hvmulcomi 30844 | Scalar multiplication comm... |
| hvmul2negi 30845 | Double negative in scalar ... |
| hvsubdistr1 30846 | Scalar multiplication dist... |
| hvsubdistr2 30847 | Scalar multiplication dist... |
| hvdistr1i 30848 | Scalar multiplication dist... |
| hvsubdistr1i 30849 | Scalar multiplication dist... |
| hvassi 30850 | Hilbert vector space assoc... |
| hvadd32i 30851 | Hilbert vector space commu... |
| hvsubassi 30852 | Hilbert vector space assoc... |
| hvsub32i 30853 | Hilbert vector space commu... |
| hvadd12i 30854 | Hilbert vector space commu... |
| hvadd4i 30855 | Hilbert vector space addit... |
| hvsubsub4i 30856 | Hilbert vector space addit... |
| hvsubsub4 30857 | Hilbert vector space addit... |
| hv2times 30858 | Two times a vector. (Cont... |
| hvnegdii 30859 | Distribution of negative o... |
| hvsubeq0i 30860 | If the difference between ... |
| hvsubcan2i 30861 | Vector cancellation law. ... |
| hvaddcani 30862 | Cancellation law for vecto... |
| hvsubaddi 30863 | Relationship between vecto... |
| hvnegdi 30864 | Distribution of negative o... |
| hvsubeq0 30865 | If the difference between ... |
| hvaddeq0 30866 | If the sum of two vectors ... |
| hvaddcan 30867 | Cancellation law for vecto... |
| hvaddcan2 30868 | Cancellation law for vecto... |
| hvmulcan 30869 | Cancellation law for scala... |
| hvmulcan2 30870 | Cancellation law for scala... |
| hvsubcan 30871 | Cancellation law for vecto... |
| hvsubcan2 30872 | Cancellation law for vecto... |
| hvsub0 30873 | Subtraction of a zero vect... |
| hvsubadd 30874 | Relationship between vecto... |
| hvaddsub4 30875 | Hilbert vector space addit... |
| hicl 30877 | Closure of inner product. ... |
| hicli 30878 | Closure inference for inne... |
| his5 30883 | Associative law for inner ... |
| his52 30884 | Associative law for inner ... |
| his35 30885 | Move scalar multiplication... |
| his35i 30886 | Move scalar multiplication... |
| his7 30887 | Distributive law for inner... |
| hiassdi 30888 | Distributive/associative l... |
| his2sub 30889 | Distributive law for inner... |
| his2sub2 30890 | Distributive law for inner... |
| hire 30891 | A necessary and sufficient... |
| hiidrcl 30892 | Real closure of inner prod... |
| hi01 30893 | Inner product with the 0 v... |
| hi02 30894 | Inner product with the 0 v... |
| hiidge0 30895 | Inner product with self is... |
| his6 30896 | Zero inner product with se... |
| his1i 30897 | Conjugate law for inner pr... |
| abshicom 30898 | Commuted inner products ha... |
| hial0 30899 | A vector whose inner produ... |
| hial02 30900 | A vector whose inner produ... |
| hisubcomi 30901 | Two vector subtractions si... |
| hi2eq 30902 | Lemma used to prove equali... |
| hial2eq 30903 | Two vectors whose inner pr... |
| hial2eq2 30904 | Two vectors whose inner pr... |
| orthcom 30905 | Orthogonality commutes. (... |
| normlem0 30906 | Lemma used to derive prope... |
| normlem1 30907 | Lemma used to derive prope... |
| normlem2 30908 | Lemma used to derive prope... |
| normlem3 30909 | Lemma used to derive prope... |
| normlem4 30910 | Lemma used to derive prope... |
| normlem5 30911 | Lemma used to derive prope... |
| normlem6 30912 | Lemma used to derive prope... |
| normlem7 30913 | Lemma used to derive prope... |
| normlem8 30914 | Lemma used to derive prope... |
| normlem9 30915 | Lemma used to derive prope... |
| normlem7tALT 30916 | Lemma used to derive prope... |
| bcseqi 30917 | Equality case of Bunjakova... |
| normlem9at 30918 | Lemma used to derive prope... |
| dfhnorm2 30919 | Alternate definition of th... |
| normf 30920 | The norm function maps fro... |
| normval 30921 | The value of the norm of a... |
| normcl 30922 | Real closure of the norm o... |
| normge0 30923 | The norm of a vector is no... |
| normgt0 30924 | The norm of nonzero vector... |
| norm0 30925 | The norm of a zero vector.... |
| norm-i 30926 | Theorem 3.3(i) of [Beran] ... |
| normne0 30927 | A norm is nonzero iff its ... |
| normcli 30928 | Real closure of the norm o... |
| normsqi 30929 | The square of a norm. (Co... |
| norm-i-i 30930 | Theorem 3.3(i) of [Beran] ... |
| normsq 30931 | The square of a norm. (Co... |
| normsub0i 30932 | Two vectors are equal iff ... |
| normsub0 30933 | Two vectors are equal iff ... |
| norm-ii-i 30934 | Triangle inequality for no... |
| norm-ii 30935 | Triangle inequality for no... |
| norm-iii-i 30936 | Theorem 3.3(iii) of [Beran... |
| norm-iii 30937 | Theorem 3.3(iii) of [Beran... |
| normsubi 30938 | Negative doesn't change th... |
| normpythi 30939 | Analogy to Pythagorean the... |
| normsub 30940 | Swapping order of subtract... |
| normneg 30941 | The norm of a vector equal... |
| normpyth 30942 | Analogy to Pythagorean the... |
| normpyc 30943 | Corollary to Pythagorean t... |
| norm3difi 30944 | Norm of differences around... |
| norm3adifii 30945 | Norm of differences around... |
| norm3lem 30946 | Lemma involving norm of di... |
| norm3dif 30947 | Norm of differences around... |
| norm3dif2 30948 | Norm of differences around... |
| norm3lemt 30949 | Lemma involving norm of di... |
| norm3adifi 30950 | Norm of differences around... |
| normpari 30951 | Parallelogram law for norm... |
| normpar 30952 | Parallelogram law for norm... |
| normpar2i 30953 | Corollary of parallelogram... |
| polid2i 30954 | Generalized polarization i... |
| polidi 30955 | Polarization identity. Re... |
| polid 30956 | Polarization identity. Re... |
| hilablo 30957 | Hilbert space vector addit... |
| hilid 30958 | The group identity element... |
| hilvc 30959 | Hilbert space is a complex... |
| hilnormi 30960 | Hilbert space norm in term... |
| hilhhi 30961 | Deduce the structure of Hi... |
| hhnv 30962 | Hilbert space is a normed ... |
| hhva 30963 | The group (addition) opera... |
| hhba 30964 | The base set of Hilbert sp... |
| hh0v 30965 | The zero vector of Hilbert... |
| hhsm 30966 | The scalar product operati... |
| hhvs 30967 | The vector subtraction ope... |
| hhnm 30968 | The norm function of Hilbe... |
| hhims 30969 | The induced metric of Hilb... |
| hhims2 30970 | Hilbert space distance met... |
| hhmet 30971 | The induced metric of Hilb... |
| hhxmet 30972 | The induced metric of Hilb... |
| hhmetdval 30973 | Value of the distance func... |
| hhip 30974 | The inner product operatio... |
| hhph 30975 | The Hilbert space of the H... |
| bcsiALT 30976 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 30977 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 30978 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 30979 | Corollary of the Bunjakova... |
| bcs3 30980 | Corollary of the Bunjakova... |
| hcau 30981 | Member of the set of Cauch... |
| hcauseq 30982 | A Cauchy sequences on a Hi... |
| hcaucvg 30983 | A Cauchy sequence on a Hil... |
| seq1hcau 30984 | A sequence on a Hilbert sp... |
| hlimi 30985 | Express the predicate: Th... |
| hlimseqi 30986 | A sequence with a limit on... |
| hlimveci 30987 | Closure of the limit of a ... |
| hlimconvi 30988 | Convergence of a sequence ... |
| hlim2 30989 | The limit of a sequence on... |
| hlimadd 30990 | Limit of the sum of two se... |
| hilmet 30991 | The Hilbert space norm det... |
| hilxmet 30992 | The Hilbert space norm det... |
| hilmetdval 30993 | Value of the distance func... |
| hilims 30994 | Hilbert space distance met... |
| hhcau 30995 | The Cauchy sequences of Hi... |
| hhlm 30996 | The limit sequences of Hil... |
| hhcmpl 30997 | Lemma used for derivation ... |
| hilcompl 30998 | Lemma used for derivation ... |
| hhcms 31000 | The Hilbert space induced ... |
| hhhl 31001 | The Hilbert space structur... |
| hilcms 31002 | The Hilbert space norm det... |
| hilhl 31003 | The Hilbert space of the H... |
| issh 31005 | Subspace ` H ` of a Hilber... |
| issh2 31006 | Subspace ` H ` of a Hilber... |
| shss 31007 | A subspace is a subset of ... |
| shel 31008 | A member of a subspace of ... |
| shex 31009 | The set of subspaces of a ... |
| shssii 31010 | A closed subspace of a Hil... |
| sheli 31011 | A member of a subspace of ... |
| shelii 31012 | A member of a subspace of ... |
| sh0 31013 | The zero vector belongs to... |
| shaddcl 31014 | Closure of vector addition... |
| shmulcl 31015 | Closure of vector scalar m... |
| issh3 31016 | Subspace ` H ` of a Hilber... |
| shsubcl 31017 | Closure of vector subtract... |
| isch 31019 | Closed subspace ` H ` of a... |
| isch2 31020 | Closed subspace ` H ` of a... |
| chsh 31021 | A closed subspace is a sub... |
| chsssh 31022 | Closed subspaces are subsp... |
| chex 31023 | The set of closed subspace... |
| chshii 31024 | A closed subspace is a sub... |
| ch0 31025 | The zero vector belongs to... |
| chss 31026 | A closed subspace of a Hil... |
| chel 31027 | A member of a closed subsp... |
| chssii 31028 | A closed subspace of a Hil... |
| cheli 31029 | A member of a closed subsp... |
| chelii 31030 | A member of a closed subsp... |
| chlimi 31031 | The limit property of a cl... |
| hlim0 31032 | The zero sequence in Hilbe... |
| hlimcaui 31033 | If a sequence in Hilbert s... |
| hlimf 31034 | Function-like behavior of ... |
| hlimuni 31035 | A Hilbert space sequence c... |
| hlimreui 31036 | The limit of a Hilbert spa... |
| hlimeui 31037 | The limit of a Hilbert spa... |
| isch3 31038 | A Hilbert subspace is clos... |
| chcompl 31039 | Completeness of a closed s... |
| helch 31040 | The Hilbert lattice one (w... |
| ifchhv 31041 | Prove ` if ( A e. CH , A ,... |
| helsh 31042 | Hilbert space is a subspac... |
| shsspwh 31043 | Subspaces are subsets of H... |
| chsspwh 31044 | Closed subspaces are subse... |
| hsn0elch 31045 | The zero subspace belongs ... |
| norm1 31046 | From any nonzero Hilbert s... |
| norm1exi 31047 | A normalized vector exists... |
| norm1hex 31048 | A normalized vector can ex... |
| elch0 31051 | Membership in zero for clo... |
| h0elch 31052 | The zero subspace is a clo... |
| h0elsh 31053 | The zero subspace is a sub... |
| hhssva 31054 | The vector addition operat... |
| hhsssm 31055 | The scalar multiplication ... |
| hhssnm 31056 | The norm operation on a su... |
| issubgoilem 31057 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31058 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31059 | Abelian group property of ... |
| hhssablo 31060 | Abelian group property of ... |
| hhssnv 31061 | Normed complex vector spac... |
| hhssnvt 31062 | Normed complex vector spac... |
| hhsst 31063 | A member of ` SH ` is a su... |
| hhshsslem1 31064 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31065 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31066 | The predicate " ` H ` is a... |
| hhsssh2 31067 | The predicate " ` H ` is a... |
| hhssba 31068 | The base set of a subspace... |
| hhssvs 31069 | The vector subtraction ope... |
| hhssvsf 31070 | Mapping of the vector subt... |
| hhssims 31071 | Induced metric of a subspa... |
| hhssims2 31072 | Induced metric of a subspa... |
| hhssmet 31073 | Induced metric of a subspa... |
| hhssmetdval 31074 | Value of the distance func... |
| hhsscms 31075 | The induced metric of a cl... |
| hhssbnOLD 31076 | Obsolete version of ~ cssb... |
| ocval 31077 | Value of orthogonal comple... |
| ocel 31078 | Membership in orthogonal c... |
| shocel 31079 | Membership in orthogonal c... |
| ocsh 31080 | The orthogonal complement ... |
| shocsh 31081 | The orthogonal complement ... |
| ocss 31082 | An orthogonal complement i... |
| shocss 31083 | An orthogonal complement i... |
| occon 31084 | Contraposition law for ort... |
| occon2 31085 | Double contraposition for ... |
| occon2i 31086 | Double contraposition for ... |
| oc0 31087 | The zero vector belongs to... |
| ocorth 31088 | Members of a subset and it... |
| shocorth 31089 | Members of a subspace and ... |
| ococss 31090 | Inclusion in complement of... |
| shococss 31091 | Inclusion in complement of... |
| shorth 31092 | Members of orthogonal subs... |
| ocin 31093 | Intersection of a Hilbert ... |
| occon3 31094 | Hilbert lattice contraposi... |
| ocnel 31095 | A nonzero vector in the co... |
| chocvali 31096 | Value of the orthogonal co... |
| shuni 31097 | Two subspaces with trivial... |
| chocunii 31098 | Lemma for uniqueness part ... |
| pjhthmo 31099 | Projection Theorem, unique... |
| occllem 31100 | Lemma for ~ occl . (Contr... |
| occl 31101 | Closure of complement of H... |
| shoccl 31102 | Closure of complement of H... |
| choccl 31103 | Closure of complement of H... |
| choccli 31104 | Closure of ` CH ` orthocom... |
| shsval 31109 | Value of subspace sum of t... |
| shsss 31110 | The subspace sum is a subs... |
| shsel 31111 | Membership in the subspace... |
| shsel3 31112 | Membership in the subspace... |
| shseli 31113 | Membership in subspace sum... |
| shscli 31114 | Closure of subspace sum. ... |
| shscl 31115 | Closure of subspace sum. ... |
| shscom 31116 | Commutative law for subspa... |
| shsva 31117 | Vector sum belongs to subs... |
| shsel1 31118 | A subspace sum contains a ... |
| shsel2 31119 | A subspace sum contains a ... |
| shsvs 31120 | Vector subtraction belongs... |
| shsub1 31121 | Subspace sum is an upper b... |
| shsub2 31122 | Subspace sum is an upper b... |
| choc0 31123 | The orthocomplement of the... |
| choc1 31124 | The orthocomplement of the... |
| chocnul 31125 | Orthogonal complement of t... |
| shintcli 31126 | Closure of intersection of... |
| shintcl 31127 | The intersection of a none... |
| chintcli 31128 | The intersection of a none... |
| chintcl 31129 | The intersection (infimum)... |
| spanval 31130 | Value of the linear span o... |
| hsupval 31131 | Value of supremum of set o... |
| chsupval 31132 | The value of the supremum ... |
| spancl 31133 | The span of a subset of Hi... |
| elspancl 31134 | A member of a span is a ve... |
| shsupcl 31135 | Closure of the subspace su... |
| hsupcl 31136 | Closure of supremum of set... |
| chsupcl 31137 | Closure of supremum of sub... |
| hsupss 31138 | Subset relation for suprem... |
| chsupss 31139 | Subset relation for suprem... |
| hsupunss 31140 | The union of a set of Hilb... |
| chsupunss 31141 | The union of a set of clos... |
| spanss2 31142 | A subset of Hilbert space ... |
| shsupunss 31143 | The union of a set of subs... |
| spanid 31144 | A subspace of Hilbert spac... |
| spanss 31145 | Ordering relationship for ... |
| spanssoc 31146 | The span of a subset of Hi... |
| sshjval 31147 | Value of join for subsets ... |
| shjval 31148 | Value of join in ` SH ` . ... |
| chjval 31149 | Value of join in ` CH ` . ... |
| chjvali 31150 | Value of join in ` CH ` . ... |
| sshjval3 31151 | Value of join for subsets ... |
| sshjcl 31152 | Closure of join for subset... |
| shjcl 31153 | Closure of join in ` SH ` ... |
| chjcl 31154 | Closure of join in ` CH ` ... |
| shjcom 31155 | Commutative law for Hilber... |
| shless 31156 | Subset implies subset of s... |
| shlej1 31157 | Add disjunct to both sides... |
| shlej2 31158 | Add disjunct to both sides... |
| shincli 31159 | Closure of intersection of... |
| shscomi 31160 | Commutative law for subspa... |
| shsvai 31161 | Vector sum belongs to subs... |
| shsel1i 31162 | A subspace sum contains a ... |
| shsel2i 31163 | A subspace sum contains a ... |
| shsvsi 31164 | Vector subtraction belongs... |
| shunssi 31165 | Union is smaller than subs... |
| shunssji 31166 | Union is smaller than Hilb... |
| shsleji 31167 | Subspace sum is smaller th... |
| shjcomi 31168 | Commutative law for join i... |
| shsub1i 31169 | Subspace sum is an upper b... |
| shsub2i 31170 | Subspace sum is an upper b... |
| shub1i 31171 | Hilbert lattice join is an... |
| shjcli 31172 | Closure of ` CH ` join. (... |
| shjshcli 31173 | ` SH ` closure of join. (... |
| shlessi 31174 | Subset implies subset of s... |
| shlej1i 31175 | Add disjunct to both sides... |
| shlej2i 31176 | Add disjunct to both sides... |
| shslej 31177 | Subspace sum is smaller th... |
| shincl 31178 | Closure of intersection of... |
| shub1 31179 | Hilbert lattice join is an... |
| shub2 31180 | A subspace is a subset of ... |
| shsidmi 31181 | Idempotent law for Hilbert... |
| shslubi 31182 | The least upper bound law ... |
| shlesb1i 31183 | Hilbert lattice ordering i... |
| shsval2i 31184 | An alternate way to expres... |
| shsval3i 31185 | An alternate way to expres... |
| shmodsi 31186 | The modular law holds for ... |
| shmodi 31187 | The modular law is implied... |
| pjhthlem1 31188 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 31189 | Lemma for ~ pjhth . (Cont... |
| pjhth 31190 | Projection Theorem: Any H... |
| pjhtheu 31191 | Projection Theorem: Any H... |
| pjhfval 31193 | The value of the projectio... |
| pjhval 31194 | Value of a projection. (C... |
| pjpreeq 31195 | Equality with a projection... |
| pjeq 31196 | Equality with a projection... |
| axpjcl 31197 | Closure of a projection in... |
| pjhcl 31198 | Closure of a projection in... |
| omlsilem 31199 | Lemma for orthomodular law... |
| omlsii 31200 | Subspace inference form of... |
| omlsi 31201 | Subspace form of orthomodu... |
| ococi 31202 | Complement of complement o... |
| ococ 31203 | Complement of complement o... |
| dfch2 31204 | Alternate definition of th... |
| ococin 31205 | The double complement is t... |
| hsupval2 31206 | Alternate definition of su... |
| chsupval2 31207 | The value of the supremum ... |
| sshjval2 31208 | Value of join in the set o... |
| chsupid 31209 | A subspace is the supremum... |
| chsupsn 31210 | Value of supremum of subse... |
| shlub 31211 | Hilbert lattice join is th... |
| shlubi 31212 | Hilbert lattice join is th... |
| pjhtheu2 31213 | Uniqueness of ` y ` for th... |
| pjcli 31214 | Closure of a projection in... |
| pjhcli 31215 | Closure of a projection in... |
| pjpjpre 31216 | Decomposition of a vector ... |
| axpjpj 31217 | Decomposition of a vector ... |
| pjclii 31218 | Closure of a projection in... |
| pjhclii 31219 | Closure of a projection in... |
| pjpj0i 31220 | Decomposition of a vector ... |
| pjpji 31221 | Decomposition of a vector ... |
| pjpjhth 31222 | Projection Theorem: Any H... |
| pjpjhthi 31223 | Projection Theorem: Any H... |
| pjop 31224 | Orthocomplement projection... |
| pjpo 31225 | Projection in terms of ort... |
| pjopi 31226 | Orthocomplement projection... |
| pjpoi 31227 | Projection in terms of ort... |
| pjoc1i 31228 | Projection of a vector in ... |
| pjchi 31229 | Projection of a vector in ... |
| pjoccl 31230 | The part of a vector that ... |
| pjoc1 31231 | Projection of a vector in ... |
| pjomli 31232 | Subspace form of orthomodu... |
| pjoml 31233 | Subspace form of orthomodu... |
| pjococi 31234 | Proof of orthocomplement t... |
| pjoc2i 31235 | Projection of a vector in ... |
| pjoc2 31236 | Projection of a vector in ... |
| sh0le 31237 | The zero subspace is the s... |
| ch0le 31238 | The zero subspace is the s... |
| shle0 31239 | No subspace is smaller tha... |
| chle0 31240 | No Hilbert lattice element... |
| chnlen0 31241 | A Hilbert lattice element ... |
| ch0pss 31242 | The zero subspace is a pro... |
| orthin 31243 | The intersection of orthog... |
| ssjo 31244 | The lattice join of a subs... |
| shne0i 31245 | A nonzero subspace has a n... |
| shs0i 31246 | Hilbert subspace sum with ... |
| shs00i 31247 | Two subspaces are zero iff... |
| ch0lei 31248 | The closed subspace zero i... |
| chle0i 31249 | No Hilbert closed subspace... |
| chne0i 31250 | A nonzero closed subspace ... |
| chocini 31251 | Intersection of a closed s... |
| chj0i 31252 | Join with lattice zero in ... |
| chm1i 31253 | Meet with lattice one in `... |
| chjcli 31254 | Closure of ` CH ` join. (... |
| chsleji 31255 | Subspace sum is smaller th... |
| chseli 31256 | Membership in subspace sum... |
| chincli 31257 | Closure of Hilbert lattice... |
| chsscon3i 31258 | Hilbert lattice contraposi... |
| chsscon1i 31259 | Hilbert lattice contraposi... |
| chsscon2i 31260 | Hilbert lattice contraposi... |
| chcon2i 31261 | Hilbert lattice contraposi... |
| chcon1i 31262 | Hilbert lattice contraposi... |
| chcon3i 31263 | Hilbert lattice contraposi... |
| chunssji 31264 | Union is smaller than ` CH... |
| chjcomi 31265 | Commutative law for join i... |
| chub1i 31266 | ` CH ` join is an upper bo... |
| chub2i 31267 | ` CH ` join is an upper bo... |
| chlubi 31268 | Hilbert lattice join is th... |
| chlubii 31269 | Hilbert lattice join is th... |
| chlej1i 31270 | Add join to both sides of ... |
| chlej2i 31271 | Add join to both sides of ... |
| chlej12i 31272 | Add join to both sides of ... |
| chlejb1i 31273 | Hilbert lattice ordering i... |
| chdmm1i 31274 | De Morgan's law for meet i... |
| chdmm2i 31275 | De Morgan's law for meet i... |
| chdmm3i 31276 | De Morgan's law for meet i... |
| chdmm4i 31277 | De Morgan's law for meet i... |
| chdmj1i 31278 | De Morgan's law for join i... |
| chdmj2i 31279 | De Morgan's law for join i... |
| chdmj3i 31280 | De Morgan's law for join i... |
| chdmj4i 31281 | De Morgan's law for join i... |
| chnlei 31282 | Equivalent expressions for... |
| chjassi 31283 | Associative law for Hilber... |
| chj00i 31284 | Two Hilbert lattice elemen... |
| chjoi 31285 | The join of a closed subsp... |
| chj1i 31286 | Join with Hilbert lattice ... |
| chm0i 31287 | Meet with Hilbert lattice ... |
| chm0 31288 | Meet with Hilbert lattice ... |
| shjshsi 31289 | Hilbert lattice join equal... |
| shjshseli 31290 | A closed subspace sum equa... |
| chne0 31291 | A nonzero closed subspace ... |
| chocin 31292 | Intersection of a closed s... |
| chssoc 31293 | A closed subspace less tha... |
| chj0 31294 | Join with Hilbert lattice ... |
| chslej 31295 | Subspace sum is smaller th... |
| chincl 31296 | Closure of Hilbert lattice... |
| chsscon3 31297 | Hilbert lattice contraposi... |
| chsscon1 31298 | Hilbert lattice contraposi... |
| chsscon2 31299 | Hilbert lattice contraposi... |
| chpsscon3 31300 | Hilbert lattice contraposi... |
| chpsscon1 31301 | Hilbert lattice contraposi... |
| chpsscon2 31302 | Hilbert lattice contraposi... |
| chjcom 31303 | Commutative law for Hilber... |
| chub1 31304 | Hilbert lattice join is gr... |
| chub2 31305 | Hilbert lattice join is gr... |
| chlub 31306 | Hilbert lattice join is th... |
| chlej1 31307 | Add join to both sides of ... |
| chlej2 31308 | Add join to both sides of ... |
| chlejb1 31309 | Hilbert lattice ordering i... |
| chlejb2 31310 | Hilbert lattice ordering i... |
| chnle 31311 | Equivalent expressions for... |
| chjo 31312 | The join of a closed subsp... |
| chabs1 31313 | Hilbert lattice absorption... |
| chabs2 31314 | Hilbert lattice absorption... |
| chabs1i 31315 | Hilbert lattice absorption... |
| chabs2i 31316 | Hilbert lattice absorption... |
| chjidm 31317 | Idempotent law for Hilbert... |
| chjidmi 31318 | Idempotent law for Hilbert... |
| chj12i 31319 | A rearrangement of Hilbert... |
| chj4i 31320 | Rearrangement of the join ... |
| chjjdiri 31321 | Hilbert lattice join distr... |
| chdmm1 31322 | De Morgan's law for meet i... |
| chdmm2 31323 | De Morgan's law for meet i... |
| chdmm3 31324 | De Morgan's law for meet i... |
| chdmm4 31325 | De Morgan's law for meet i... |
| chdmj1 31326 | De Morgan's law for join i... |
| chdmj2 31327 | De Morgan's law for join i... |
| chdmj3 31328 | De Morgan's law for join i... |
| chdmj4 31329 | De Morgan's law for join i... |
| chjass 31330 | Associative law for Hilber... |
| chj12 31331 | A rearrangement of Hilbert... |
| chj4 31332 | Rearrangement of the join ... |
| ledii 31333 | An ortholattice is distrib... |
| lediri 31334 | An ortholattice is distrib... |
| lejdii 31335 | An ortholattice is distrib... |
| lejdiri 31336 | An ortholattice is distrib... |
| ledi 31337 | An ortholattice is distrib... |
| spansn0 31338 | The span of the singleton ... |
| span0 31339 | The span of the empty set ... |
| elspani 31340 | Membership in the span of ... |
| spanuni 31341 | The span of a union is the... |
| spanun 31342 | The span of a union is the... |
| sshhococi 31343 | The join of two Hilbert sp... |
| hne0 31344 | Hilbert space has a nonzer... |
| chsup0 31345 | The supremum of the empty ... |
| h1deoi 31346 | Membership in orthocomplem... |
| h1dei 31347 | Membership in 1-dimensiona... |
| h1did 31348 | A generating vector belong... |
| h1dn0 31349 | A nonzero vector generates... |
| h1de2i 31350 | Membership in 1-dimensiona... |
| h1de2bi 31351 | Membership in 1-dimensiona... |
| h1de2ctlem 31352 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 31353 | Membership in 1-dimensiona... |
| spansni 31354 | The span of a singleton in... |
| elspansni 31355 | Membership in the span of ... |
| spansn 31356 | The span of a singleton in... |
| spansnch 31357 | The span of a Hilbert spac... |
| spansnsh 31358 | The span of a Hilbert spac... |
| spansnchi 31359 | The span of a singleton in... |
| spansnid 31360 | A vector belongs to the sp... |
| spansnmul 31361 | A scalar product with a ve... |
| elspansncl 31362 | A member of a span of a si... |
| elspansn 31363 | Membership in the span of ... |
| elspansn2 31364 | Membership in the span of ... |
| spansncol 31365 | The singletons of collinea... |
| spansneleqi 31366 | Membership relation implie... |
| spansneleq 31367 | Membership relation that i... |
| spansnss 31368 | The span of the singleton ... |
| elspansn3 31369 | A member of the span of th... |
| elspansn4 31370 | A span membership conditio... |
| elspansn5 31371 | A vector belonging to both... |
| spansnss2 31372 | The span of the singleton ... |
| normcan 31373 | Cancellation-type law that... |
| pjspansn 31374 | A projection on the span o... |
| spansnpji 31375 | A subset of Hilbert space ... |
| spanunsni 31376 | The span of the union of a... |
| spanpr 31377 | The span of a pair of vect... |
| h1datomi 31378 | A 1-dimensional subspace i... |
| h1datom 31379 | A 1-dimensional subspace i... |
| cmbr 31381 | Binary relation expressing... |
| pjoml2i 31382 | Variation of orthomodular ... |
| pjoml3i 31383 | Variation of orthomodular ... |
| pjoml4i 31384 | Variation of orthomodular ... |
| pjoml5i 31385 | The orthomodular law. Rem... |
| pjoml6i 31386 | An equivalent of the ortho... |
| cmbri 31387 | Binary relation expressing... |
| cmcmlem 31388 | Commutation is symmetric. ... |
| cmcmi 31389 | Commutation is symmetric. ... |
| cmcm2i 31390 | Commutation with orthocomp... |
| cmcm3i 31391 | Commutation with orthocomp... |
| cmcm4i 31392 | Commutation with orthocomp... |
| cmbr2i 31393 | Alternate definition of th... |
| cmcmii 31394 | Commutation is symmetric. ... |
| cmcm2ii 31395 | Commutation with orthocomp... |
| cmcm3ii 31396 | Commutation with orthocomp... |
| cmbr3i 31397 | Alternate definition for t... |
| cmbr4i 31398 | Alternate definition for t... |
| lecmi 31399 | Comparable Hilbert lattice... |
| lecmii 31400 | Comparable Hilbert lattice... |
| cmj1i 31401 | A Hilbert lattice element ... |
| cmj2i 31402 | A Hilbert lattice element ... |
| cmm1i 31403 | A Hilbert lattice element ... |
| cmm2i 31404 | A Hilbert lattice element ... |
| cmbr3 31405 | Alternate definition for t... |
| cm0 31406 | The zero Hilbert lattice e... |
| cmidi 31407 | The commutes relation is r... |
| pjoml2 31408 | Variation of orthomodular ... |
| pjoml3 31409 | Variation of orthomodular ... |
| pjoml5 31410 | The orthomodular law. Rem... |
| cmcm 31411 | Commutation is symmetric. ... |
| cmcm3 31412 | Commutation with orthocomp... |
| cmcm2 31413 | Commutation with orthocomp... |
| lecm 31414 | Comparable Hilbert lattice... |
| fh1 31415 | Foulis-Holland Theorem. I... |
| fh2 31416 | Foulis-Holland Theorem. I... |
| cm2j 31417 | A lattice element that com... |
| fh1i 31418 | Foulis-Holland Theorem. I... |
| fh2i 31419 | Foulis-Holland Theorem. I... |
| fh3i 31420 | Variation of the Foulis-Ho... |
| fh4i 31421 | Variation of the Foulis-Ho... |
| cm2ji 31422 | A lattice element that com... |
| cm2mi 31423 | A lattice element that com... |
| qlax1i 31424 | One of the equations showi... |
| qlax2i 31425 | One of the equations showi... |
| qlax3i 31426 | One of the equations showi... |
| qlax4i 31427 | One of the equations showi... |
| qlax5i 31428 | One of the equations showi... |
| qlaxr1i 31429 | One of the conditions show... |
| qlaxr2i 31430 | One of the conditions show... |
| qlaxr4i 31431 | One of the conditions show... |
| qlaxr5i 31432 | One of the conditions show... |
| qlaxr3i 31433 | A variation of the orthomo... |
| chscllem1 31434 | Lemma for ~ chscl . (Cont... |
| chscllem2 31435 | Lemma for ~ chscl . (Cont... |
| chscllem3 31436 | Lemma for ~ chscl . (Cont... |
| chscllem4 31437 | Lemma for ~ chscl . (Cont... |
| chscl 31438 | The subspace sum of two cl... |
| osumi 31439 | If two closed subspaces of... |
| osumcori 31440 | Corollary of ~ osumi . (C... |
| osumcor2i 31441 | Corollary of ~ osumi , sho... |
| osum 31442 | If two closed subspaces of... |
| spansnji 31443 | The subspace sum of a clos... |
| spansnj 31444 | The subspace sum of a clos... |
| spansnscl 31445 | The subspace sum of a clos... |
| sumspansn 31446 | The sum of two vectors bel... |
| spansnm0i 31447 | The meet of different one-... |
| nonbooli 31448 | A Hilbert lattice with two... |
| spansncvi 31449 | Hilbert space has the cove... |
| spansncv 31450 | Hilbert space has the cove... |
| 5oalem1 31451 | Lemma for orthoarguesian l... |
| 5oalem2 31452 | Lemma for orthoarguesian l... |
| 5oalem3 31453 | Lemma for orthoarguesian l... |
| 5oalem4 31454 | Lemma for orthoarguesian l... |
| 5oalem5 31455 | Lemma for orthoarguesian l... |
| 5oalem6 31456 | Lemma for orthoarguesian l... |
| 5oalem7 31457 | Lemma for orthoarguesian l... |
| 5oai 31458 | Orthoarguesian law 5OA. Th... |
| 3oalem1 31459 | Lemma for 3OA (weak) ortho... |
| 3oalem2 31460 | Lemma for 3OA (weak) ortho... |
| 3oalem3 31461 | Lemma for 3OA (weak) ortho... |
| 3oalem4 31462 | Lemma for 3OA (weak) ortho... |
| 3oalem5 31463 | Lemma for 3OA (weak) ortho... |
| 3oalem6 31464 | Lemma for 3OA (weak) ortho... |
| 3oai 31465 | 3OA (weak) orthoarguesian ... |
| pjorthi 31466 | Projection components on o... |
| pjch1 31467 | Property of identity proje... |
| pjo 31468 | The orthogonal projection.... |
| pjcompi 31469 | Component of a projection.... |
| pjidmi 31470 | A projection is idempotent... |
| pjadjii 31471 | A projection is self-adjoi... |
| pjaddii 31472 | Projection of vector sum i... |
| pjinormii 31473 | The inner product of a pro... |
| pjmulii 31474 | Projection of (scalar) pro... |
| pjsubii 31475 | Projection of vector diffe... |
| pjsslem 31476 | Lemma for subset relations... |
| pjss2i 31477 | Subset relationship for pr... |
| pjssmii 31478 | Projection meet property. ... |
| pjssge0ii 31479 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 31480 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 31481 | The projection on a subspa... |
| pjadji 31482 | A projection is self-adjoi... |
| pjaddi 31483 | Projection of vector sum i... |
| pjinormi 31484 | The inner product of a pro... |
| pjsubi 31485 | Projection of vector diffe... |
| pjmuli 31486 | Projection of scalar produ... |
| pjige0i 31487 | The inner product of a pro... |
| pjige0 31488 | The inner product of a pro... |
| pjcjt2 31489 | The projection on a subspa... |
| pj0i 31490 | The projection of the zero... |
| pjch 31491 | Projection of a vector in ... |
| pjid 31492 | The projection of a vector... |
| pjvec 31493 | The set of vectors belongi... |
| pjocvec 31494 | The set of vectors belongi... |
| pjocini 31495 | Membership of projection i... |
| pjini 31496 | Membership of projection i... |
| pjjsi 31497 | A sufficient condition for... |
| pjfni 31498 | Functionality of a project... |
| pjrni 31499 | The range of a projection.... |
| pjfoi 31500 | A projection maps onto its... |
| pjfi 31501 | The mapping of a projectio... |
| pjvi 31502 | The value of a projection ... |
| pjhfo 31503 | A projection maps onto its... |
| pjrn 31504 | The range of a projection.... |
| pjhf 31505 | The mapping of a projectio... |
| pjfn 31506 | Functionality of a project... |
| pjsumi 31507 | The projection on a subspa... |
| pj11i 31508 | One-to-one correspondence ... |
| pjdsi 31509 | Vector decomposition into ... |
| pjds3i 31510 | Vector decomposition into ... |
| pj11 31511 | One-to-one correspondence ... |
| pjmfn 31512 | Functionality of the proje... |
| pjmf1 31513 | The projector function map... |
| pjoi0 31514 | The inner product of proje... |
| pjoi0i 31515 | The inner product of proje... |
| pjopythi 31516 | Pythagorean theorem for pr... |
| pjopyth 31517 | Pythagorean theorem for pr... |
| pjnormi 31518 | The norm of the projection... |
| pjpythi 31519 | Pythagorean theorem for pr... |
| pjneli 31520 | If a vector does not belon... |
| pjnorm 31521 | The norm of the projection... |
| pjpyth 31522 | Pythagorean theorem for pr... |
| pjnel 31523 | If a vector does not belon... |
| pjnorm2 31524 | A vector belongs to the su... |
| mayete3i 31525 | Mayet's equation E_3. Par... |
| mayetes3i 31526 | Mayet's equation E^*_3, de... |
| hosmval 31532 | Value of the sum of two Hi... |
| hommval 31533 | Value of the scalar produc... |
| hodmval 31534 | Value of the difference of... |
| hfsmval 31535 | Value of the sum of two Hi... |
| hfmmval 31536 | Value of the scalar produc... |
| hosval 31537 | Value of the sum of two Hi... |
| homval 31538 | Value of the scalar produc... |
| hodval 31539 | Value of the difference of... |
| hfsval 31540 | Value of the sum of two Hi... |
| hfmval 31541 | Value of the scalar produc... |
| hoscl 31542 | Closure of the sum of two ... |
| homcl 31543 | Closure of the scalar prod... |
| hodcl 31544 | Closure of the difference ... |
| ho0val 31547 | Value of the zero Hilbert ... |
| ho0f 31548 | Functionality of the zero ... |
| df0op2 31549 | Alternate definition of Hi... |
| dfiop2 31550 | Alternate definition of Hi... |
| hoif 31551 | Functionality of the Hilbe... |
| hoival 31552 | The value of the Hilbert s... |
| hoico1 31553 | Composition with the Hilbe... |
| hoico2 31554 | Composition with the Hilbe... |
| hoaddcl 31555 | The sum of Hilbert space o... |
| homulcl 31556 | The scalar product of a Hi... |
| hoeq 31557 | Equality of Hilbert space ... |
| hoeqi 31558 | Equality of Hilbert space ... |
| hoscli 31559 | Closure of Hilbert space o... |
| hodcli 31560 | Closure of Hilbert space o... |
| hocoi 31561 | Composition of Hilbert spa... |
| hococli 31562 | Closure of composition of ... |
| hocofi 31563 | Mapping of composition of ... |
| hocofni 31564 | Functionality of compositi... |
| hoaddcli 31565 | Mapping of sum of Hilbert ... |
| hosubcli 31566 | Mapping of difference of H... |
| hoaddfni 31567 | Functionality of sum of Hi... |
| hosubfni 31568 | Functionality of differenc... |
| hoaddcomi 31569 | Commutativity of sum of Hi... |
| hosubcl 31570 | Mapping of difference of H... |
| hoaddcom 31571 | Commutativity of sum of Hi... |
| hodsi 31572 | Relationship between Hilbe... |
| hoaddassi 31573 | Associativity of sum of Hi... |
| hoadd12i 31574 | Commutative/associative la... |
| hoadd32i 31575 | Commutative/associative la... |
| hocadddiri 31576 | Distributive law for Hilbe... |
| hocsubdiri 31577 | Distributive law for Hilbe... |
| ho2coi 31578 | Double composition of Hilb... |
| hoaddass 31579 | Associativity of sum of Hi... |
| hoadd32 31580 | Commutative/associative la... |
| hoadd4 31581 | Rearrangement of 4 terms i... |
| hocsubdir 31582 | Distributive law for Hilbe... |
| hoaddridi 31583 | Sum of a Hilbert space ope... |
| hodidi 31584 | Difference of a Hilbert sp... |
| ho0coi 31585 | Composition of the zero op... |
| hoid1i 31586 | Composition of Hilbert spa... |
| hoid1ri 31587 | Composition of Hilbert spa... |
| hoaddrid 31588 | Sum of a Hilbert space ope... |
| hodid 31589 | Difference of a Hilbert sp... |
| hon0 31590 | A Hilbert space operator i... |
| hodseqi 31591 | Subtraction and addition o... |
| ho0subi 31592 | Subtraction of Hilbert spa... |
| honegsubi 31593 | Relationship between Hilbe... |
| ho0sub 31594 | Subtraction of Hilbert spa... |
| hosubid1 31595 | The zero operator subtract... |
| honegsub 31596 | Relationship between Hilbe... |
| homullid 31597 | An operator equals its sca... |
| homco1 31598 | Associative law for scalar... |
| homulass 31599 | Scalar product associative... |
| hoadddi 31600 | Scalar product distributiv... |
| hoadddir 31601 | Scalar product reverse dis... |
| homul12 31602 | Swap first and second fact... |
| honegneg 31603 | Double negative of a Hilbe... |
| hosubneg 31604 | Relationship between opera... |
| hosubdi 31605 | Scalar product distributiv... |
| honegdi 31606 | Distribution of negative o... |
| honegsubdi 31607 | Distribution of negative o... |
| honegsubdi2 31608 | Distribution of negative o... |
| hosubsub2 31609 | Law for double subtraction... |
| hosub4 31610 | Rearrangement of 4 terms i... |
| hosubadd4 31611 | Rearrangement of 4 terms i... |
| hoaddsubass 31612 | Associative-type law for a... |
| hoaddsub 31613 | Law for operator addition ... |
| hosubsub 31614 | Law for double subtraction... |
| hosubsub4 31615 | Law for double subtraction... |
| ho2times 31616 | Two times a Hilbert space ... |
| hoaddsubassi 31617 | Associativity of sum and d... |
| hoaddsubi 31618 | Law for sum and difference... |
| hosd1i 31619 | Hilbert space operator sum... |
| hosd2i 31620 | Hilbert space operator sum... |
| hopncani 31621 | Hilbert space operator can... |
| honpcani 31622 | Hilbert space operator can... |
| hosubeq0i 31623 | If the difference between ... |
| honpncani 31624 | Hilbert space operator can... |
| ho01i 31625 | A condition implying that ... |
| ho02i 31626 | A condition implying that ... |
| hoeq1 31627 | A condition implying that ... |
| hoeq2 31628 | A condition implying that ... |
| adjmo 31629 | Every Hilbert space operat... |
| adjsym 31630 | Symmetry property of an ad... |
| eigrei 31631 | A necessary and sufficient... |
| eigre 31632 | A necessary and sufficient... |
| eigposi 31633 | A sufficient condition (fi... |
| eigorthi 31634 | A necessary and sufficient... |
| eigorth 31635 | A necessary and sufficient... |
| nmopval 31653 | Value of the norm of a Hil... |
| elcnop 31654 | Property defining a contin... |
| ellnop 31655 | Property defining a linear... |
| lnopf 31656 | A linear Hilbert space ope... |
| elbdop 31657 | Property defining a bounde... |
| bdopln 31658 | A bounded linear Hilbert s... |
| bdopf 31659 | A bounded linear Hilbert s... |
| nmopsetretALT 31660 | The set in the supremum of... |
| nmopsetretHIL 31661 | The set in the supremum of... |
| nmopsetn0 31662 | The set in the supremum of... |
| nmopxr 31663 | The norm of a Hilbert spac... |
| nmoprepnf 31664 | The norm of a Hilbert spac... |
| nmopgtmnf 31665 | The norm of a Hilbert spac... |
| nmopreltpnf 31666 | The norm of a Hilbert spac... |
| nmopre 31667 | The norm of a bounded oper... |
| elbdop2 31668 | Property defining a bounde... |
| elunop 31669 | Property defining a unitar... |
| elhmop 31670 | Property defining a Hermit... |
| hmopf 31671 | A Hermitian operator is a ... |
| hmopex 31672 | The class of Hermitian ope... |
| nmfnval 31673 | Value of the norm of a Hil... |
| nmfnsetre 31674 | The set in the supremum of... |
| nmfnsetn0 31675 | The set in the supremum of... |
| nmfnxr 31676 | The norm of any Hilbert sp... |
| nmfnrepnf 31677 | The norm of a Hilbert spac... |
| nlfnval 31678 | Value of the null space of... |
| elcnfn 31679 | Property defining a contin... |
| ellnfn 31680 | Property defining a linear... |
| lnfnf 31681 | A linear Hilbert space fun... |
| dfadj2 31682 | Alternate definition of th... |
| funadj 31683 | Functionality of the adjoi... |
| dmadjss 31684 | The domain of the adjoint ... |
| dmadjop 31685 | A member of the domain of ... |
| adjeu 31686 | Elementhood in the domain ... |
| adjval 31687 | Value of the adjoint funct... |
| adjval2 31688 | Value of the adjoint funct... |
| cnvadj 31689 | The adjoint function equal... |
| funcnvadj 31690 | The converse of the adjoin... |
| adj1o 31691 | The adjoint function maps ... |
| dmadjrn 31692 | The adjoint of an operator... |
| eigvecval 31693 | The set of eigenvectors of... |
| eigvalfval 31694 | The eigenvalues of eigenve... |
| specval 31695 | The value of the spectrum ... |
| speccl 31696 | The spectrum of an operato... |
| hhlnoi 31697 | The linear operators of Hi... |
| hhnmoi 31698 | The norm of an operator in... |
| hhbloi 31699 | A bounded linear operator ... |
| hh0oi 31700 | The zero operator in Hilbe... |
| hhcno 31701 | The continuous operators o... |
| hhcnf 31702 | The continuous functionals... |
| dmadjrnb 31703 | The adjoint of an operator... |
| nmoplb 31704 | A lower bound for an opera... |
| nmopub 31705 | An upper bound for an oper... |
| nmopub2tALT 31706 | An upper bound for an oper... |
| nmopub2tHIL 31707 | An upper bound for an oper... |
| nmopge0 31708 | The norm of any Hilbert sp... |
| nmopgt0 31709 | A linear Hilbert space ope... |
| cnopc 31710 | Basic continuity property ... |
| lnopl 31711 | Basic linearity property o... |
| unop 31712 | Basic inner product proper... |
| unopf1o 31713 | A unitary operator in Hilb... |
| unopnorm 31714 | A unitary operator is idem... |
| cnvunop 31715 | The inverse (converse) of ... |
| unopadj 31716 | The inverse (converse) of ... |
| unoplin 31717 | A unitary operator is line... |
| counop 31718 | The composition of two uni... |
| hmop 31719 | Basic inner product proper... |
| hmopre 31720 | The inner product of the v... |
| nmfnlb 31721 | A lower bound for a functi... |
| nmfnleub 31722 | An upper bound for the nor... |
| nmfnleub2 31723 | An upper bound for the nor... |
| nmfnge0 31724 | The norm of any Hilbert sp... |
| elnlfn 31725 | Membership in the null spa... |
| elnlfn2 31726 | Membership in the null spa... |
| cnfnc 31727 | Basic continuity property ... |
| lnfnl 31728 | Basic linearity property o... |
| adjcl 31729 | Closure of the adjoint of ... |
| adj1 31730 | Property of an adjoint Hil... |
| adj2 31731 | Property of an adjoint Hil... |
| adjeq 31732 | A property that determines... |
| adjadj 31733 | Double adjoint. Theorem 3... |
| adjvalval 31734 | Value of the value of the ... |
| unopadj2 31735 | The adjoint of a unitary o... |
| hmopadj 31736 | A Hermitian operator is se... |
| hmdmadj 31737 | Every Hermitian operator h... |
| hmopadj2 31738 | An operator is Hermitian i... |
| hmoplin 31739 | A Hermitian operator is li... |
| brafval 31740 | The bra of a vector, expre... |
| braval 31741 | A bra-ket juxtaposition, e... |
| braadd 31742 | Linearity property of bra ... |
| bramul 31743 | Linearity property of bra ... |
| brafn 31744 | The bra function is a func... |
| bralnfn 31745 | The Dirac bra function is ... |
| bracl 31746 | Closure of the bra functio... |
| bra0 31747 | The Dirac bra of the zero ... |
| brafnmul 31748 | Anti-linearity property of... |
| kbfval 31749 | The outer product of two v... |
| kbop 31750 | The outer product of two v... |
| kbval 31751 | The value of the operator ... |
| kbmul 31752 | Multiplication property of... |
| kbpj 31753 | If a vector ` A ` has norm... |
| eleigvec 31754 | Membership in the set of e... |
| eleigvec2 31755 | Membership in the set of e... |
| eleigveccl 31756 | Closure of an eigenvector ... |
| eigvalval 31757 | The eigenvalue of an eigen... |
| eigvalcl 31758 | An eigenvalue is a complex... |
| eigvec1 31759 | Property of an eigenvector... |
| eighmre 31760 | The eigenvalues of a Hermi... |
| eighmorth 31761 | Eigenvectors of a Hermitia... |
| nmopnegi 31762 | Value of the norm of the n... |
| lnop0 31763 | The value of a linear Hilb... |
| lnopmul 31764 | Multiplicative property of... |
| lnopli 31765 | Basic scalar product prope... |
| lnopfi 31766 | A linear Hilbert space ope... |
| lnop0i 31767 | The value of a linear Hilb... |
| lnopaddi 31768 | Additive property of a lin... |
| lnopmuli 31769 | Multiplicative property of... |
| lnopaddmuli 31770 | Sum/product property of a ... |
| lnopsubi 31771 | Subtraction property for a... |
| lnopsubmuli 31772 | Subtraction/product proper... |
| lnopmulsubi 31773 | Product/subtraction proper... |
| homco2 31774 | Move a scalar product out ... |
| idunop 31775 | The identity function (res... |
| 0cnop 31776 | The identically zero funct... |
| 0cnfn 31777 | The identically zero funct... |
| idcnop 31778 | The identity function (res... |
| idhmop 31779 | The Hilbert space identity... |
| 0hmop 31780 | The identically zero funct... |
| 0lnop 31781 | The identically zero funct... |
| 0lnfn 31782 | The identically zero funct... |
| nmop0 31783 | The norm of the zero opera... |
| nmfn0 31784 | The norm of the identicall... |
| hmopbdoptHIL 31785 | A Hermitian operator is a ... |
| hoddii 31786 | Distributive law for Hilbe... |
| hoddi 31787 | Distributive law for Hilbe... |
| nmop0h 31788 | The norm of any operator o... |
| idlnop 31789 | The identity function (res... |
| 0bdop 31790 | The identically zero opera... |
| adj0 31791 | Adjoint of the zero operat... |
| nmlnop0iALT 31792 | A linear operator with a z... |
| nmlnop0iHIL 31793 | A linear operator with a z... |
| nmlnopgt0i 31794 | A linear Hilbert space ope... |
| nmlnop0 31795 | A linear operator with a z... |
| nmlnopne0 31796 | A linear operator with a n... |
| lnopmi 31797 | The scalar product of a li... |
| lnophsi 31798 | The sum of two linear oper... |
| lnophdi 31799 | The difference of two line... |
| lnopcoi 31800 | The composition of two lin... |
| lnopco0i 31801 | The composition of a linea... |
| lnopeq0lem1 31802 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 31803 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 31804 | A condition implying that ... |
| lnopeqi 31805 | Two linear Hilbert space o... |
| lnopeq 31806 | Two linear Hilbert space o... |
| lnopunilem1 31807 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 31808 | Lemma for ~ lnopunii . (C... |
| lnopunii 31809 | If a linear operator (whos... |
| elunop2 31810 | An operator is unitary iff... |
| nmopun 31811 | Norm of a unitary Hilbert ... |
| unopbd 31812 | A unitary operator is a bo... |
| lnophmlem1 31813 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 31814 | Lemma for ~ lnophmi . (Co... |
| lnophmi 31815 | A linear operator is Hermi... |
| lnophm 31816 | A linear operator is Hermi... |
| hmops 31817 | The sum of two Hermitian o... |
| hmopm 31818 | The scalar product of a He... |
| hmopd 31819 | The difference of two Herm... |
| hmopco 31820 | The composition of two com... |
| nmbdoplbi 31821 | A lower bound for the norm... |
| nmbdoplb 31822 | A lower bound for the norm... |
| nmcexi 31823 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 31824 | The norm of a continuous l... |
| nmcoplbi 31825 | A lower bound for the norm... |
| nmcopex 31826 | The norm of a continuous l... |
| nmcoplb 31827 | A lower bound for the norm... |
| nmophmi 31828 | The norm of the scalar pro... |
| bdophmi 31829 | The scalar product of a bo... |
| lnconi 31830 | Lemma for ~ lnopconi and ~... |
| lnopconi 31831 | A condition equivalent to ... |
| lnopcon 31832 | A condition equivalent to ... |
| lnopcnbd 31833 | A linear operator is conti... |
| lncnopbd 31834 | A continuous linear operat... |
| lncnbd 31835 | A continuous linear operat... |
| lnopcnre 31836 | A linear operator is conti... |
| lnfnli 31837 | Basic property of a linear... |
| lnfnfi 31838 | A linear Hilbert space fun... |
| lnfn0i 31839 | The value of a linear Hilb... |
| lnfnaddi 31840 | Additive property of a lin... |
| lnfnmuli 31841 | Multiplicative property of... |
| lnfnaddmuli 31842 | Sum/product property of a ... |
| lnfnsubi 31843 | Subtraction property for a... |
| lnfn0 31844 | The value of a linear Hilb... |
| lnfnmul 31845 | Multiplicative property of... |
| nmbdfnlbi 31846 | A lower bound for the norm... |
| nmbdfnlb 31847 | A lower bound for the norm... |
| nmcfnexi 31848 | The norm of a continuous l... |
| nmcfnlbi 31849 | A lower bound for the norm... |
| nmcfnex 31850 | The norm of a continuous l... |
| nmcfnlb 31851 | A lower bound of the norm ... |
| lnfnconi 31852 | A condition equivalent to ... |
| lnfncon 31853 | A condition equivalent to ... |
| lnfncnbd 31854 | A linear functional is con... |
| imaelshi 31855 | The image of a subspace un... |
| rnelshi 31856 | The range of a linear oper... |
| nlelshi 31857 | The null space of a linear... |
| nlelchi 31858 | The null space of a contin... |
| riesz3i 31859 | A continuous linear functi... |
| riesz4i 31860 | A continuous linear functi... |
| riesz4 31861 | A continuous linear functi... |
| riesz1 31862 | Part 1 of the Riesz repres... |
| riesz2 31863 | Part 2 of the Riesz repres... |
| cnlnadjlem1 31864 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 31865 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 31866 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 31867 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 31868 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 31869 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 31870 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 31871 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 31872 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 31873 | Every continuous linear op... |
| cnlnadjeui 31874 | Every continuous linear op... |
| cnlnadjeu 31875 | Every continuous linear op... |
| cnlnadj 31876 | Every continuous linear op... |
| cnlnssadj 31877 | Every continuous linear Hi... |
| bdopssadj 31878 | Every bounded linear Hilbe... |
| bdopadj 31879 | Every bounded linear Hilbe... |
| adjbdln 31880 | The adjoint of a bounded l... |
| adjbdlnb 31881 | An operator is bounded and... |
| adjbd1o 31882 | The mapping of adjoints of... |
| adjlnop 31883 | The adjoint of an operator... |
| adjsslnop 31884 | Every operator with an adj... |
| nmopadjlei 31885 | Property of the norm of an... |
| nmopadjlem 31886 | Lemma for ~ nmopadji . (C... |
| nmopadji 31887 | Property of the norm of an... |
| adjeq0 31888 | An operator is zero iff it... |
| adjmul 31889 | The adjoint of the scalar ... |
| adjadd 31890 | The adjoint of the sum of ... |
| nmoptrii 31891 | Triangle inequality for th... |
| nmopcoi 31892 | Upper bound for the norm o... |
| bdophsi 31893 | The sum of two bounded lin... |
| bdophdi 31894 | The difference between two... |
| bdopcoi 31895 | The composition of two bou... |
| nmoptri2i 31896 | Triangle-type inequality f... |
| adjcoi 31897 | The adjoint of a compositi... |
| nmopcoadji 31898 | The norm of an operator co... |
| nmopcoadj2i 31899 | The norm of an operator co... |
| nmopcoadj0i 31900 | An operator composed with ... |
| unierri 31901 | If we approximate a chain ... |
| branmfn 31902 | The norm of the bra functi... |
| brabn 31903 | The bra of a vector is a b... |
| rnbra 31904 | The set of bras equals the... |
| bra11 31905 | The bra function maps vect... |
| bracnln 31906 | A bra is a continuous line... |
| cnvbraval 31907 | Value of the converse of t... |
| cnvbracl 31908 | Closure of the converse of... |
| cnvbrabra 31909 | The converse bra of the br... |
| bracnvbra 31910 | The bra of the converse br... |
| bracnlnval 31911 | The vector that a continuo... |
| cnvbramul 31912 | Multiplication property of... |
| kbass1 31913 | Dirac bra-ket associative ... |
| kbass2 31914 | Dirac bra-ket associative ... |
| kbass3 31915 | Dirac bra-ket associative ... |
| kbass4 31916 | Dirac bra-ket associative ... |
| kbass5 31917 | Dirac bra-ket associative ... |
| kbass6 31918 | Dirac bra-ket associative ... |
| leopg 31919 | Ordering relation for posi... |
| leop 31920 | Ordering relation for oper... |
| leop2 31921 | Ordering relation for oper... |
| leop3 31922 | Operator ordering in terms... |
| leoppos 31923 | Binary relation defining a... |
| leoprf2 31924 | The ordering relation for ... |
| leoprf 31925 | The ordering relation for ... |
| leopsq 31926 | The square of a Hermitian ... |
| 0leop 31927 | The zero operator is a pos... |
| idleop 31928 | The identity operator is a... |
| leopadd 31929 | The sum of two positive op... |
| leopmuli 31930 | The scalar product of a no... |
| leopmul 31931 | The scalar product of a po... |
| leopmul2i 31932 | Scalar product applied to ... |
| leoptri 31933 | The positive operator orde... |
| leoptr 31934 | The positive operator orde... |
| leopnmid 31935 | A bounded Hermitian operat... |
| nmopleid 31936 | A nonzero, bounded Hermiti... |
| opsqrlem1 31937 | Lemma for opsqri . (Contr... |
| opsqrlem2 31938 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 31939 | Lemma for opsqri . (Contr... |
| opsqrlem4 31940 | Lemma for opsqri . (Contr... |
| opsqrlem5 31941 | Lemma for opsqri . (Contr... |
| opsqrlem6 31942 | Lemma for opsqri . (Contr... |
| pjhmopi 31943 | A projector is a Hermitian... |
| pjlnopi 31944 | A projector is a linear op... |
| pjnmopi 31945 | The operator norm of a pro... |
| pjbdlni 31946 | A projector is a bounded l... |
| pjhmop 31947 | A projection is a Hermitia... |
| hmopidmchi 31948 | An idempotent Hermitian op... |
| hmopidmpji 31949 | An idempotent Hermitian op... |
| hmopidmch 31950 | An idempotent Hermitian op... |
| hmopidmpj 31951 | An idempotent Hermitian op... |
| pjsdii 31952 | Distributive law for Hilbe... |
| pjddii 31953 | Distributive law for Hilbe... |
| pjsdi2i 31954 | Chained distributive law f... |
| pjcoi 31955 | Composition of projections... |
| pjcocli 31956 | Closure of composition of ... |
| pjcohcli 31957 | Closure of composition of ... |
| pjadjcoi 31958 | Adjoint of composition of ... |
| pjcofni 31959 | Functionality of compositi... |
| pjss1coi 31960 | Subset relationship for pr... |
| pjss2coi 31961 | Subset relationship for pr... |
| pjssmi 31962 | Projection meet property. ... |
| pjssge0i 31963 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 31964 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 31965 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 31966 | Composition of projections... |
| pjscji 31967 | The projection of orthogon... |
| pjssumi 31968 | The projection on a subspa... |
| pjssposi 31969 | Projector ordering can be ... |
| pjordi 31970 | The definition of projecto... |
| pjssdif2i 31971 | The projection subspace of... |
| pjssdif1i 31972 | A necessary and sufficient... |
| pjimai 31973 | The image of a projection.... |
| pjidmcoi 31974 | A projection is idempotent... |
| pjoccoi 31975 | Composition of projections... |
| pjtoi 31976 | Subspace sum of projection... |
| pjoci 31977 | Projection of orthocomplem... |
| pjidmco 31978 | A projection operator is i... |
| dfpjop 31979 | Definition of projection o... |
| pjhmopidm 31980 | Two ways to express the se... |
| elpjidm 31981 | A projection operator is i... |
| elpjhmop 31982 | A projection operator is H... |
| 0leopj 31983 | A projector is a positive ... |
| pjadj2 31984 | A projector is self-adjoin... |
| pjadj3 31985 | A projector is self-adjoin... |
| elpjch 31986 | Reconstruction of the subs... |
| elpjrn 31987 | Reconstruction of the subs... |
| pjinvari 31988 | A closed subspace ` H ` wi... |
| pjin1i 31989 | Lemma for Theorem 1.22 of ... |
| pjin2i 31990 | Lemma for Theorem 1.22 of ... |
| pjin3i 31991 | Lemma for Theorem 1.22 of ... |
| pjclem1 31992 | Lemma for projection commu... |
| pjclem2 31993 | Lemma for projection commu... |
| pjclem3 31994 | Lemma for projection commu... |
| pjclem4a 31995 | Lemma for projection commu... |
| pjclem4 31996 | Lemma for projection commu... |
| pjci 31997 | Two subspaces commute iff ... |
| pjcmul1i 31998 | A necessary and sufficient... |
| pjcmul2i 31999 | The projection subspace of... |
| pjcohocli 32000 | Closure of composition of ... |
| pjadj2coi 32001 | Adjoint of double composit... |
| pj2cocli 32002 | Closure of double composit... |
| pj3lem1 32003 | Lemma for projection tripl... |
| pj3si 32004 | Stronger projection triple... |
| pj3i 32005 | Projection triplet theorem... |
| pj3cor1i 32006 | Projection triplet corolla... |
| pjs14i 32007 | Theorem S-14 of Watanabe, ... |
| isst 32010 | Property of a state. (Con... |
| ishst 32011 | Property of a complex Hilb... |
| sticl 32012 | ` [ 0 , 1 ] ` closure of t... |
| stcl 32013 | Real closure of the value ... |
| hstcl 32014 | Closure of the value of a ... |
| hst1a 32015 | Unit value of a Hilbert-sp... |
| hstel2 32016 | Properties of a Hilbert-sp... |
| hstorth 32017 | Orthogonality property of ... |
| hstosum 32018 | Orthogonal sum property of... |
| hstoc 32019 | Sum of a Hilbert-space-val... |
| hstnmoc 32020 | Sum of norms of a Hilbert-... |
| stge0 32021 | The value of a state is no... |
| stle1 32022 | The value of a state is le... |
| hstle1 32023 | The norm of the value of a... |
| hst1h 32024 | The norm of a Hilbert-spac... |
| hst0h 32025 | The norm of a Hilbert-spac... |
| hstpyth 32026 | Pythagorean property of a ... |
| hstle 32027 | Ordering property of a Hil... |
| hstles 32028 | Ordering property of a Hil... |
| hstoh 32029 | A Hilbert-space-valued sta... |
| hst0 32030 | A Hilbert-space-valued sta... |
| sthil 32031 | The value of a state at th... |
| stj 32032 | The value of a state on a ... |
| sto1i 32033 | The state of a subspace pl... |
| sto2i 32034 | The state of the orthocomp... |
| stge1i 32035 | If a state is greater than... |
| stle0i 32036 | If a state is less than or... |
| stlei 32037 | Ordering law for states. ... |
| stlesi 32038 | Ordering law for states. ... |
| stji1i 32039 | Join of components of Sasa... |
| stm1i 32040 | State of component of unit... |
| stm1ri 32041 | State of component of unit... |
| stm1addi 32042 | Sum of states whose meet i... |
| staddi 32043 | If the sum of 2 states is ... |
| stm1add3i 32044 | Sum of states whose meet i... |
| stadd3i 32045 | If the sum of 3 states is ... |
| st0 32046 | The state of the zero subs... |
| strlem1 32047 | Lemma for strong state the... |
| strlem2 32048 | Lemma for strong state the... |
| strlem3a 32049 | Lemma for strong state the... |
| strlem3 32050 | Lemma for strong state the... |
| strlem4 32051 | Lemma for strong state the... |
| strlem5 32052 | Lemma for strong state the... |
| strlem6 32053 | Lemma for strong state the... |
| stri 32054 | Strong state theorem. The... |
| strb 32055 | Strong state theorem (bidi... |
| hstrlem2 32056 | Lemma for strong set of CH... |
| hstrlem3a 32057 | Lemma for strong set of CH... |
| hstrlem3 32058 | Lemma for strong set of CH... |
| hstrlem4 32059 | Lemma for strong set of CH... |
| hstrlem5 32060 | Lemma for strong set of CH... |
| hstrlem6 32061 | Lemma for strong set of CH... |
| hstri 32062 | Hilbert space admits a str... |
| hstrbi 32063 | Strong CH-state theorem (b... |
| largei 32064 | A Hilbert lattice admits a... |
| jplem1 32065 | Lemma for Jauch-Piron theo... |
| jplem2 32066 | Lemma for Jauch-Piron theo... |
| jpi 32067 | The function ` S ` , that ... |
| golem1 32068 | Lemma for Godowski's equat... |
| golem2 32069 | Lemma for Godowski's equat... |
| goeqi 32070 | Godowski's equation, shown... |
| stcltr1i 32071 | Property of a strong class... |
| stcltr2i 32072 | Property of a strong class... |
| stcltrlem1 32073 | Lemma for strong classical... |
| stcltrlem2 32074 | Lemma for strong classical... |
| stcltrthi 32075 | Theorem for classically st... |
| cvbr 32079 | Binary relation expressing... |
| cvbr2 32080 | Binary relation expressing... |
| cvcon3 32081 | Contraposition law for the... |
| cvpss 32082 | The covers relation implie... |
| cvnbtwn 32083 | The covers relation implie... |
| cvnbtwn2 32084 | The covers relation implie... |
| cvnbtwn3 32085 | The covers relation implie... |
| cvnbtwn4 32086 | The covers relation implie... |
| cvnsym 32087 | The covers relation is not... |
| cvnref 32088 | The covers relation is not... |
| cvntr 32089 | The covers relation is not... |
| spansncv2 32090 | Hilbert space has the cove... |
| mdbr 32091 | Binary relation expressing... |
| mdi 32092 | Consequence of the modular... |
| mdbr2 32093 | Binary relation expressing... |
| mdbr3 32094 | Binary relation expressing... |
| mdbr4 32095 | Binary relation expressing... |
| dmdbr 32096 | Binary relation expressing... |
| dmdmd 32097 | The dual modular pair prop... |
| mddmd 32098 | The modular pair property ... |
| dmdi 32099 | Consequence of the dual mo... |
| dmdbr2 32100 | Binary relation expressing... |
| dmdi2 32101 | Consequence of the dual mo... |
| dmdbr3 32102 | Binary relation expressing... |
| dmdbr4 32103 | Binary relation expressing... |
| dmdi4 32104 | Consequence of the dual mo... |
| dmdbr5 32105 | Binary relation expressing... |
| mddmd2 32106 | Relationship between modul... |
| mdsl0 32107 | A sublattice condition tha... |
| ssmd1 32108 | Ordering implies the modul... |
| ssmd2 32109 | Ordering implies the modul... |
| ssdmd1 32110 | Ordering implies the dual ... |
| ssdmd2 32111 | Ordering implies the dual ... |
| dmdsl3 32112 | Sublattice mapping for a d... |
| mdsl3 32113 | Sublattice mapping for a m... |
| mdslle1i 32114 | Order preservation of the ... |
| mdslle2i 32115 | Order preservation of the ... |
| mdslj1i 32116 | Join preservation of the o... |
| mdslj2i 32117 | Meet preservation of the r... |
| mdsl1i 32118 | If the modular pair proper... |
| mdsl2i 32119 | If the modular pair proper... |
| mdsl2bi 32120 | If the modular pair proper... |
| cvmdi 32121 | The covering property impl... |
| mdslmd1lem1 32122 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 32123 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 32124 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 32125 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 32126 | Preservation of the modula... |
| mdslmd2i 32127 | Preservation of the modula... |
| mdsldmd1i 32128 | Preservation of the dual m... |
| mdslmd3i 32129 | Modular pair conditions th... |
| mdslmd4i 32130 | Modular pair condition tha... |
| csmdsymi 32131 | Cross-symmetry implies M-s... |
| mdexchi 32132 | An exchange lemma for modu... |
| cvmd 32133 | The covering property impl... |
| cvdmd 32134 | The covering property impl... |
| ela 32136 | Atoms in a Hilbert lattice... |
| elat2 32137 | Expanded membership relati... |
| elatcv0 32138 | A Hilbert lattice element ... |
| atcv0 32139 | An atom covers the zero su... |
| atssch 32140 | Atoms are a subset of the ... |
| atelch 32141 | An atom is a Hilbert latti... |
| atne0 32142 | An atom is not the Hilbert... |
| atss 32143 | A lattice element smaller ... |
| atsseq 32144 | Two atoms in a subset rela... |
| atcveq0 32145 | A Hilbert lattice element ... |
| h1da 32146 | A 1-dimensional subspace i... |
| spansna 32147 | The span of the singleton ... |
| sh1dle 32148 | A 1-dimensional subspace i... |
| ch1dle 32149 | A 1-dimensional subspace i... |
| atom1d 32150 | The 1-dimensional subspace... |
| superpos 32151 | Superposition Principle. ... |
| chcv1 32152 | The Hilbert lattice has th... |
| chcv2 32153 | The Hilbert lattice has th... |
| chjatom 32154 | The join of a closed subsp... |
| shatomici 32155 | The lattice of Hilbert sub... |
| hatomici 32156 | The Hilbert lattice is ato... |
| hatomic 32157 | A Hilbert lattice is atomi... |
| shatomistici 32158 | The lattice of Hilbert sub... |
| hatomistici 32159 | ` CH ` is atomistic, i.e. ... |
| chpssati 32160 | Two Hilbert lattice elemen... |
| chrelati 32161 | The Hilbert lattice is rel... |
| chrelat2i 32162 | A consequence of relative ... |
| cvati 32163 | If a Hilbert lattice eleme... |
| cvbr4i 32164 | An alternate way to expres... |
| cvexchlem 32165 | Lemma for ~ cvexchi . (Co... |
| cvexchi 32166 | The Hilbert lattice satisf... |
| chrelat2 32167 | A consequence of relative ... |
| chrelat3 32168 | A consequence of relative ... |
| chrelat3i 32169 | A consequence of the relat... |
| chrelat4i 32170 | A consequence of relative ... |
| cvexch 32171 | The Hilbert lattice satisf... |
| cvp 32172 | The Hilbert lattice satisf... |
| atnssm0 32173 | The meet of a Hilbert latt... |
| atnemeq0 32174 | The meet of distinct atoms... |
| atssma 32175 | The meet with an atom's su... |
| atcv0eq 32176 | Two atoms covering the zer... |
| atcv1 32177 | Two atoms covering the zer... |
| atexch 32178 | The Hilbert lattice satisf... |
| atomli 32179 | An assertion holding in at... |
| atoml2i 32180 | An assertion holding in at... |
| atordi 32181 | An ordering law for a Hilb... |
| atcvatlem 32182 | Lemma for ~ atcvati . (Co... |
| atcvati 32183 | A nonzero Hilbert lattice ... |
| atcvat2i 32184 | A Hilbert lattice element ... |
| atord 32185 | An ordering law for a Hilb... |
| atcvat2 32186 | A Hilbert lattice element ... |
| chirredlem1 32187 | Lemma for ~ chirredi . (C... |
| chirredlem2 32188 | Lemma for ~ chirredi . (C... |
| chirredlem3 32189 | Lemma for ~ chirredi . (C... |
| chirredlem4 32190 | Lemma for ~ chirredi . (C... |
| chirredi 32191 | The Hilbert lattice is irr... |
| chirred 32192 | The Hilbert lattice is irr... |
| atcvat3i 32193 | A condition implying that ... |
| atcvat4i 32194 | A condition implying exist... |
| atdmd 32195 | Two Hilbert lattice elemen... |
| atmd 32196 | Two Hilbert lattice elemen... |
| atmd2 32197 | Two Hilbert lattice elemen... |
| atabsi 32198 | Absorption of an incompara... |
| atabs2i 32199 | Absorption of an incompara... |
| mdsymlem1 32200 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 32201 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 32202 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 32203 | Lemma for ~ mdsymi . This... |
| mdsymlem5 32204 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 32205 | Lemma for ~ mdsymi . This... |
| mdsymlem7 32206 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 32207 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 32208 | M-symmetry of the Hilbert ... |
| mdsym 32209 | M-symmetry of the Hilbert ... |
| dmdsym 32210 | Dual M-symmetry of the Hil... |
| atdmd2 32211 | Two Hilbert lattice elemen... |
| sumdmdii 32212 | If the subspace sum of two... |
| cmmdi 32213 | Commuting subspaces form a... |
| cmdmdi 32214 | Commuting subspaces form a... |
| sumdmdlem 32215 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 32216 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 32217 | The subspace sum of two Hi... |
| dmdbr4ati 32218 | Dual modular pair property... |
| dmdbr5ati 32219 | Dual modular pair property... |
| dmdbr6ati 32220 | Dual modular pair property... |
| dmdbr7ati 32221 | Dual modular pair property... |
| mdoc1i 32222 | Orthocomplements form a mo... |
| mdoc2i 32223 | Orthocomplements form a mo... |
| dmdoc1i 32224 | Orthocomplements form a du... |
| dmdoc2i 32225 | Orthocomplements form a du... |
| mdcompli 32226 | A condition equivalent to ... |
| dmdcompli 32227 | A condition equivalent to ... |
| mddmdin0i 32228 | If dual modular implies mo... |
| cdjreui 32229 | A member of the sum of dis... |
| cdj1i 32230 | Two ways to express " ` A ... |
| cdj3lem1 32231 | A property of " ` A ` and ... |
| cdj3lem2 32232 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 32233 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 32234 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 32235 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 32236 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 32237 | Lemma for ~ cdj3i . The s... |
| cdj3i 32238 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 32239 | (_This theorem is a dummy ... |
| sa-abvi 32240 | A theorem about the univer... |
| xfree 32241 | A partial converse to ~ 19... |
| xfree2 32242 | A partial converse to ~ 19... |
| addltmulALT 32243 | A proof readability experi... |
| bian1d 32244 | Adding a superfluous conju... |
| or3di 32245 | Distributive law for disju... |
| or3dir 32246 | Distributive law for disju... |
| 3o1cs 32247 | Deduction eliminating disj... |
| 3o2cs 32248 | Deduction eliminating disj... |
| 3o3cs 32249 | Deduction eliminating disj... |
| 13an22anass 32250 | Associative law for four c... |
| sbc2iedf 32251 | Conversion of implicit sub... |
| rspc2daf 32252 | Double restricted speciali... |
| ralcom4f 32253 | Commutation of restricted ... |
| rexcom4f 32254 | Commutation of restricted ... |
| 19.9d2rf 32255 | A deduction version of one... |
| 19.9d2r 32256 | A deduction version of one... |
| r19.29ffa 32257 | A commonly used pattern ba... |
| eqtrb 32258 | A transposition of equalit... |
| eqelbid 32259 | A variable elimination law... |
| opsbc2ie 32260 | Conversion of implicit sub... |
| opreu2reuALT 32261 | Correspondence between uni... |
| 2reucom 32264 | Double restricted existent... |
| 2reu2rex1 32265 | Double restricted existent... |
| 2reureurex 32266 | Double restricted existent... |
| 2reu2reu2 32267 | Double restricted existent... |
| opreu2reu1 32268 | Equivalent definition of t... |
| sq2reunnltb 32269 | There exists a unique deco... |
| addsqnot2reu 32270 | For each complex number ` ... |
| sbceqbidf 32271 | Equality theorem for class... |
| sbcies 32272 | A special version of class... |
| mo5f 32273 | Alternate definition of "a... |
| nmo 32274 | Negation of "at most one".... |
| reuxfrdf 32275 | Transfer existential uniqu... |
| rexunirn 32276 | Restricted existential qua... |
| rmoxfrd 32277 | Transfer "at most one" res... |
| rmoun 32278 | "At most one" restricted e... |
| rmounid 32279 | A case where an "at most o... |
| riotaeqbidva 32280 | Equivalent wff's yield equ... |
| dmrab 32281 | Domain of a restricted cla... |
| difrab2 32282 | Difference of two restrict... |
| rabexgfGS 32283 | Separation Scheme in terms... |
| rabsnel 32284 | Truth implied by equality ... |
| eqrrabd 32285 | Deduce equality with a res... |
| foresf1o 32286 | From a surjective function... |
| rabfodom 32287 | Domination relation for re... |
| abrexdomjm 32288 | An indexed set is dominate... |
| abrexdom2jm 32289 | An indexed set is dominate... |
| abrexexd 32290 | Existence of a class abstr... |
| elabreximd 32291 | Class substitution in an i... |
| elabreximdv 32292 | Class substitution in an i... |
| abrexss 32293 | A necessary condition for ... |
| elunsn 32294 | Elementhood to a union wit... |
| nelun 32295 | Negated membership for a u... |
| snsssng 32296 | If a singleton is a subset... |
| inin 32297 | Intersection with an inter... |
| inindif 32298 | See ~ inundif . (Contribu... |
| difininv 32299 | Condition for the intersec... |
| difeq 32300 | Rewriting an equation with... |
| eqdif 32301 | If both set differences of... |
| indifbi 32302 | Two ways to express equali... |
| diffib 32303 | Case where ~ diffi is a bi... |
| difxp1ss 32304 | Difference law for Cartesi... |
| difxp2ss 32305 | Difference law for Cartesi... |
| indifundif 32306 | A remarkable equation with... |
| elpwincl1 32307 | Closure of intersection wi... |
| elpwdifcl 32308 | Closure of class differenc... |
| elpwiuncl 32309 | Closure of indexed union w... |
| eqsnd 32310 | Deduce that a set is a sin... |
| elpreq 32311 | Equality wihin a pair. (C... |
| nelpr 32312 | A set ` A ` not in a pair ... |
| inpr0 32313 | Rewrite an empty intersect... |
| neldifpr1 32314 | The first element of a pai... |
| neldifpr2 32315 | The second element of a pa... |
| unidifsnel 32316 | The other element of a pai... |
| unidifsnne 32317 | The other element of a pai... |
| ifeqeqx 32318 | An equality theorem tailor... |
| elimifd 32319 | Elimination of a condition... |
| elim2if 32320 | Elimination of two conditi... |
| elim2ifim 32321 | Elimination of two conditi... |
| ifeq3da 32322 | Given an expression ` C ` ... |
| ifnetrue 32323 | Deduce truth from a condit... |
| ifnefals 32324 | Deduce falsehood from a co... |
| ifnebib 32325 | The converse of ~ ifbi hol... |
| uniinn0 32326 | Sufficient and necessary c... |
| uniin1 32327 | Union of intersection. Ge... |
| uniin2 32328 | Union of intersection. Ge... |
| difuncomp 32329 | Express a class difference... |
| elpwunicl 32330 | Closure of a set union wit... |
| cbviunf 32331 | Rule used to change the bo... |
| iuneq12daf 32332 | Equality deduction for ind... |
| iunin1f 32333 | Indexed union of intersect... |
| ssiun3 32334 | Subset equivalence for an ... |
| ssiun2sf 32335 | Subset relationship for an... |
| iuninc 32336 | The union of an increasing... |
| iundifdifd 32337 | The intersection of a set ... |
| iundifdif 32338 | The intersection of a set ... |
| iunrdx 32339 | Re-index an indexed union.... |
| iunpreima 32340 | Preimage of an indexed uni... |
| iunrnmptss 32341 | A subset relation for an i... |
| iunxunsn 32342 | Appending a set to an inde... |
| iunxunpr 32343 | Appending two sets to an i... |
| iinabrex 32344 | Rewriting an indexed inter... |
| disjnf 32345 | In case ` x ` is not free ... |
| cbvdisjf 32346 | Change bound variables in ... |
| disjss1f 32347 | A subset of a disjoint col... |
| disjeq1f 32348 | Equality theorem for disjo... |
| disjxun0 32349 | Simplify a disjoint union.... |
| disjdifprg 32350 | A trivial partition into a... |
| disjdifprg2 32351 | A trivial partition of a s... |
| disji2f 32352 | Property of a disjoint col... |
| disjif 32353 | Property of a disjoint col... |
| disjorf 32354 | Two ways to say that a col... |
| disjorsf 32355 | Two ways to say that a col... |
| disjif2 32356 | Property of a disjoint col... |
| disjabrex 32357 | Rewriting a disjoint colle... |
| disjabrexf 32358 | Rewriting a disjoint colle... |
| disjpreima 32359 | A preimage of a disjoint s... |
| disjrnmpt 32360 | Rewriting a disjoint colle... |
| disjin 32361 | If a collection is disjoin... |
| disjin2 32362 | If a collection is disjoin... |
| disjxpin 32363 | Derive a disjunction over ... |
| iundisjf 32364 | Rewrite a countable union ... |
| iundisj2f 32365 | A disjoint union is disjoi... |
| disjrdx 32366 | Re-index a disjunct collec... |
| disjex 32367 | Two ways to say that two c... |
| disjexc 32368 | A variant of ~ disjex , ap... |
| disjunsn 32369 | Append an element to a dis... |
| disjun0 32370 | Adding the empty element p... |
| disjiunel 32371 | A set of elements B of a d... |
| disjuniel 32372 | A set of elements B of a d... |
| xpdisjres 32373 | Restriction of a constant ... |
| opeldifid 32374 | Ordered pair elementhood o... |
| difres 32375 | Case when class difference... |
| imadifxp 32376 | Image of the difference wi... |
| relfi 32377 | A relation (set) is finite... |
| 0res 32378 | Restriction of the empty f... |
| fcoinver 32379 | Build an equivalence relat... |
| fcoinvbr 32380 | Binary relation for the eq... |
| brabgaf 32381 | The law of concretion for ... |
| brelg 32382 | Two things in a binary rel... |
| br8d 32383 | Substitution for an eight-... |
| opabdm 32384 | Domain of an ordered-pair ... |
| opabrn 32385 | Range of an ordered-pair c... |
| opabssi 32386 | Sufficient condition for a... |
| opabid2ss 32387 | One direction of ~ opabid2... |
| ssrelf 32388 | A subclass relationship de... |
| eqrelrd2 32389 | A version of ~ eqrelrdv2 w... |
| erbr3b 32390 | Biconditional for equivale... |
| iunsnima 32391 | Image of a singleton by an... |
| iunsnima2 32392 | Version of ~ iunsnima with... |
| ac6sf2 32393 | Alternate version of ~ ac6... |
| fnresin 32394 | Restriction of a function ... |
| f1o3d 32395 | Describe an implicit one-t... |
| eldmne0 32396 | A function of nonempty dom... |
| f1rnen 32397 | Equinumerosity of the rang... |
| rinvf1o 32398 | Sufficient conditions for ... |
| fresf1o 32399 | Conditions for a restricti... |
| nfpconfp 32400 | The set of fixed points of... |
| fmptco1f1o 32401 | The action of composing (t... |
| cofmpt2 32402 | Express composition of a m... |
| f1mptrn 32403 | Express injection for a ma... |
| dfimafnf 32404 | Alternate definition of th... |
| funimass4f 32405 | Membership relation for th... |
| elimampt 32406 | Membership in the image of... |
| suppss2f 32407 | Show that the support of a... |
| ofrn 32408 | The range of the function ... |
| ofrn2 32409 | The range of the function ... |
| off2 32410 | The function operation pro... |
| ofresid 32411 | Applying an operation rest... |
| fimarab 32412 | Expressing the image of a ... |
| unipreima 32413 | Preimage of a class union.... |
| opfv 32414 | Value of a function produc... |
| xppreima 32415 | The preimage of a Cartesia... |
| 2ndimaxp 32416 | Image of a cartesian produ... |
| djussxp2 32417 | Stronger version of ~ djus... |
| 2ndresdju 32418 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 32419 | The ` 2nd ` function restr... |
| xppreima2 32420 | The preimage of a Cartesia... |
| abfmpunirn 32421 | Membership in a union of a... |
| rabfmpunirn 32422 | Membership in a union of a... |
| abfmpeld 32423 | Membership in an element o... |
| abfmpel 32424 | Membership in an element o... |
| fmptdF 32425 | Domain and codomain of the... |
| fmptcof2 32426 | Composition of two functio... |
| fcomptf 32427 | Express composition of two... |
| acunirnmpt 32428 | Axiom of choice for the un... |
| acunirnmpt2 32429 | Axiom of choice for the un... |
| acunirnmpt2f 32430 | Axiom of choice for the un... |
| aciunf1lem 32431 | Choice in an index union. ... |
| aciunf1 32432 | Choice in an index union. ... |
| ofoprabco 32433 | Function operation as a co... |
| ofpreima 32434 | Express the preimage of a ... |
| ofpreima2 32435 | Express the preimage of a ... |
| funcnvmpt 32436 | Condition for a function i... |
| funcnv5mpt 32437 | Two ways to say that a fun... |
| funcnv4mpt 32438 | Two ways to say that a fun... |
| preimane 32439 | Different elements have di... |
| fnpreimac 32440 | Choose a set ` x ` contain... |
| fgreu 32441 | Exactly one point of a fun... |
| fcnvgreu 32442 | If the converse of a relat... |
| rnmposs 32443 | The range of an operation ... |
| mptssALT 32444 | Deduce subset relation of ... |
| dfcnv2 32445 | Alternative definition of ... |
| fnimatp 32446 | The image of an unordered ... |
| mpomptxf 32447 | Express a two-argument fun... |
| suppovss 32448 | A bound for the support of... |
| fvdifsupp 32449 | Function value is zero out... |
| suppiniseg 32450 | Relation between the suppo... |
| fsuppinisegfi 32451 | The initial segment ` ( ``... |
| fressupp 32452 | The restriction of a funct... |
| fdifsuppconst 32453 | A function is a zero const... |
| ressupprn 32454 | The range of a function re... |
| supppreima 32455 | Express the support of a f... |
| fsupprnfi 32456 | Finite support implies fin... |
| mptiffisupp 32457 | Conditions for a mapping f... |
| cosnopne 32458 | Composition of two ordered... |
| cosnop 32459 | Composition of two ordered... |
| cnvprop 32460 | Converse of a pair of orde... |
| brprop 32461 | Binary relation for a pair... |
| mptprop 32462 | Rewrite pairs of ordered p... |
| coprprop 32463 | Composition of two pairs o... |
| gtiso 32464 | Two ways to write a strict... |
| isoun 32465 | Infer an isomorphism from ... |
| disjdsct 32466 | A disjoint collection is d... |
| df1stres 32467 | Definition for a restricti... |
| df2ndres 32468 | Definition for a restricti... |
| 1stpreimas 32469 | The preimage of a singleto... |
| 1stpreima 32470 | The preimage by ` 1st ` is... |
| 2ndpreima 32471 | The preimage by ` 2nd ` is... |
| curry2ima 32472 | The image of a curried fun... |
| preiman0 32473 | The preimage of a nonempty... |
| intimafv 32474 | The intersection of an ima... |
| supssd 32475 | Inequality deduction for s... |
| infssd 32476 | Inequality deduction for i... |
| imafi2 32477 | The image by a finite set ... |
| unifi3 32478 | If a union is finite, then... |
| snct 32479 | A singleton is countable. ... |
| prct 32480 | An unordered pair is count... |
| mpocti 32481 | An operation is countable ... |
| abrexct 32482 | An image set of a countabl... |
| mptctf 32483 | A countable mapping set is... |
| abrexctf 32484 | An image set of a countabl... |
| padct 32485 | Index a countable set with... |
| cnvoprabOLD 32486 | The converse of a class ab... |
| f1od2 32487 | Sufficient condition for a... |
| fcobij 32488 | Composing functions with a... |
| fcobijfs 32489 | Composing finitely support... |
| suppss3 32490 | Deduce a function's suppor... |
| fsuppcurry1 32491 | Finite support of a currie... |
| fsuppcurry2 32492 | Finite support of a currie... |
| offinsupp1 32493 | Finite support for a funct... |
| ffs2 32494 | Rewrite a function's suppo... |
| ffsrn 32495 | The range of a finitely su... |
| resf1o 32496 | Restriction of functions t... |
| maprnin 32497 | Restricting the range of t... |
| fpwrelmapffslem 32498 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 32499 | Define a canonical mapping... |
| fpwrelmapffs 32500 | Define a canonical mapping... |
| creq0 32501 | The real representation of... |
| 1nei 32502 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 32503 | An integer unit times itse... |
| nnmulge 32504 | Multiplying by a positive ... |
| lt2addrd 32505 | If the right-hand side of ... |
| xrlelttric 32506 | Trichotomy law for extende... |
| xaddeq0 32507 | Two extended reals which a... |
| xrinfm 32508 | The extended real numbers ... |
| le2halvesd 32509 | A sum is less than the who... |
| xraddge02 32510 | A number is less than or e... |
| xrge0addge 32511 | A number is less than or e... |
| xlt2addrd 32512 | If the right-hand side of ... |
| xrsupssd 32513 | Inequality deduction for s... |
| xrge0infss 32514 | Any subset of nonnegative ... |
| xrge0infssd 32515 | Inequality deduction for i... |
| xrge0addcld 32516 | Nonnegative extended reals... |
| xrge0subcld 32517 | Condition for closure of n... |
| infxrge0lb 32518 | A member of a set of nonne... |
| infxrge0glb 32519 | The infimum of a set of no... |
| infxrge0gelb 32520 | The infimum of a set of no... |
| xrofsup 32521 | The supremum is preserved ... |
| supxrnemnf 32522 | The supremum of a nonempty... |
| xnn0gt0 32523 | Nonzero extended nonnegati... |
| xnn01gt 32524 | An extended nonnegative in... |
| nn0xmulclb 32525 | Finite multiplication in t... |
| joiniooico 32526 | Disjoint joining an open i... |
| ubico 32527 | A right-open interval does... |
| xeqlelt 32528 | Equality in terms of 'less... |
| eliccelico 32529 | Relate elementhood to a cl... |
| elicoelioo 32530 | Relate elementhood to a cl... |
| iocinioc2 32531 | Intersection between two o... |
| xrdifh 32532 | Class difference of a half... |
| iocinif 32533 | Relate intersection of two... |
| difioo 32534 | The difference between two... |
| difico 32535 | The difference between two... |
| uzssico 32536 | Upper integer sets are a s... |
| fz2ssnn0 32537 | A finite set of sequential... |
| nndiffz1 32538 | Upper set of the positive ... |
| ssnnssfz 32539 | For any finite subset of `... |
| fzne1 32540 | Elementhood in a finite se... |
| fzm1ne1 32541 | Elementhood of an integer ... |
| fzspl 32542 | Split the last element of ... |
| fzdif2 32543 | Split the last element of ... |
| fzodif2 32544 | Split the last element of ... |
| fzodif1 32545 | Set difference of two half... |
| fzsplit3 32546 | Split a finite interval of... |
| bcm1n 32547 | The proportion of one bino... |
| iundisjfi 32548 | Rewrite a countable union ... |
| iundisj2fi 32549 | A disjoint union is disjoi... |
| iundisjcnt 32550 | Rewrite a countable union ... |
| iundisj2cnt 32551 | A countable disjoint union... |
| fzone1 32552 | Elementhood in a half-open... |
| fzom1ne1 32553 | Elementhood in a half-open... |
| f1ocnt 32554 | Given a countable set ` A ... |
| fz1nnct 32555 | NN and integer ranges star... |
| fz1nntr 32556 | NN and integer ranges star... |
| nn0difffzod 32557 | A nonnegative integer that... |
| suppssnn0 32558 | Show that the support of a... |
| hashunif 32559 | The cardinality of a disjo... |
| hashxpe 32560 | The size of the Cartesian ... |
| hashgt1 32561 | Restate "set contains at l... |
| numdenneg 32562 | Numerator and denominator ... |
| divnumden2 32563 | Calculate the reduced form... |
| nnindf 32564 | Principle of Mathematical ... |
| nn0min 32565 | Extracting the minimum pos... |
| subne0nn 32566 | A nonnegative difference i... |
| ltesubnnd 32567 | Subtracting an integer num... |
| fprodeq02 32568 | If one of the factors is z... |
| pr01ssre 32569 | The range of the indicator... |
| fprodex01 32570 | A product of factors equal... |
| prodpr 32571 | A product over a pair is t... |
| prodtp 32572 | A product over a triple is... |
| fsumub 32573 | An upper bound for a term ... |
| fsumiunle 32574 | Upper bound for a sum of n... |
| dfdec100 32575 | Split the hundreds from a ... |
| dp2eq1 32578 | Equality theorem for the d... |
| dp2eq2 32579 | Equality theorem for the d... |
| dp2eq1i 32580 | Equality theorem for the d... |
| dp2eq2i 32581 | Equality theorem for the d... |
| dp2eq12i 32582 | Equality theorem for the d... |
| dp20u 32583 | Add a zero in the tenths (... |
| dp20h 32584 | Add a zero in the unit pla... |
| dp2cl 32585 | Closure for the decimal fr... |
| dp2clq 32586 | Closure for a decimal frac... |
| rpdp2cl 32587 | Closure for a decimal frac... |
| rpdp2cl2 32588 | Closure for a decimal frac... |
| dp2lt10 32589 | Decimal fraction builds re... |
| dp2lt 32590 | Comparing two decimal frac... |
| dp2ltsuc 32591 | Comparing a decimal fracti... |
| dp2ltc 32592 | Comparing two decimal expa... |
| dpval 32595 | Define the value of the de... |
| dpcl 32596 | Prove that the closure of ... |
| dpfrac1 32597 | Prove a simple equivalence... |
| dpval2 32598 | Value of the decimal point... |
| dpval3 32599 | Value of the decimal point... |
| dpmul10 32600 | Multiply by 10 a decimal e... |
| decdiv10 32601 | Divide a decimal number by... |
| dpmul100 32602 | Multiply by 100 a decimal ... |
| dp3mul10 32603 | Multiply by 10 a decimal e... |
| dpmul1000 32604 | Multiply by 1000 a decimal... |
| dpval3rp 32605 | Value of the decimal point... |
| dp0u 32606 | Add a zero in the tenths p... |
| dp0h 32607 | Remove a zero in the units... |
| rpdpcl 32608 | Closure of the decimal poi... |
| dplt 32609 | Comparing two decimal expa... |
| dplti 32610 | Comparing a decimal expans... |
| dpgti 32611 | Comparing a decimal expans... |
| dpltc 32612 | Comparing two decimal inte... |
| dpexpp1 32613 | Add one zero to the mantis... |
| 0dp2dp 32614 | Multiply by 10 a decimal e... |
| dpadd2 32615 | Addition with one decimal,... |
| dpadd 32616 | Addition with one decimal.... |
| dpadd3 32617 | Addition with two decimals... |
| dpmul 32618 | Multiplication with one de... |
| dpmul4 32619 | An upper bound to multipli... |
| threehalves 32620 | Example theorem demonstrat... |
| 1mhdrd 32621 | Example theorem demonstrat... |
| xdivval 32624 | Value of division: the (un... |
| xrecex 32625 | Existence of reciprocal of... |
| xmulcand 32626 | Cancellation law for exten... |
| xreceu 32627 | Existential uniqueness of ... |
| xdivcld 32628 | Closure law for the extend... |
| xdivcl 32629 | Closure law for the extend... |
| xdivmul 32630 | Relationship between divis... |
| rexdiv 32631 | The extended real division... |
| xdivrec 32632 | Relationship between divis... |
| xdivid 32633 | A number divided by itself... |
| xdiv0 32634 | Division into zero is zero... |
| xdiv0rp 32635 | Division into zero is zero... |
| eliccioo 32636 | Membership in a closed int... |
| elxrge02 32637 | Elementhood in the set of ... |
| xdivpnfrp 32638 | Plus infinity divided by a... |
| rpxdivcld 32639 | Closure law for extended d... |
| xrpxdivcld 32640 | Closure law for extended d... |
| wrdfd 32641 | A word is a zero-based seq... |
| wrdres 32642 | Condition for the restrict... |
| wrdsplex 32643 | Existence of a split of a ... |
| pfx1s2 32644 | The prefix of length 1 of ... |
| pfxrn2 32645 | The range of a prefix of a... |
| pfxrn3 32646 | Express the range of a pre... |
| pfxf1 32647 | Condition for a prefix to ... |
| s1f1 32648 | Conditions for a length 1 ... |
| s2rn 32649 | Range of a length 2 string... |
| s2f1 32650 | Conditions for a length 2 ... |
| s3rn 32651 | Range of a length 3 string... |
| s3f1 32652 | Conditions for a length 3 ... |
| s3clhash 32653 | Closure of the words of le... |
| ccatf1 32654 | Conditions for a concatena... |
| pfxlsw2ccat 32655 | Reconstruct a word from it... |
| wrdt2ind 32656 | Perform an induction over ... |
| swrdrn2 32657 | The range of a subword is ... |
| swrdrn3 32658 | Express the range of a sub... |
| swrdf1 32659 | Condition for a subword to... |
| swrdrndisj 32660 | Condition for the range of... |
| splfv3 32661 | Symbols to the right of a ... |
| 1cshid 32662 | Cyclically shifting a sing... |
| cshw1s2 32663 | Cyclically shifting a leng... |
| cshwrnid 32664 | Cyclically shifting a word... |
| cshf1o 32665 | Condition for the cyclic s... |
| ressplusf 32666 | The group operation functi... |
| ressnm 32667 | The norm in a restricted s... |
| abvpropd2 32668 | Weaker version of ~ abvpro... |
| oppgle 32669 | less-than relation of an o... |
| oppgleOLD 32670 | Obsolete version of ~ oppg... |
| oppglt 32671 | less-than relation of an o... |
| ressprs 32672 | The restriction of a prose... |
| oduprs 32673 | Being a proset is a self-d... |
| posrasymb 32674 | A poset ordering is asymet... |
| resspos 32675 | The restriction of a Poset... |
| resstos 32676 | The restriction of a Toset... |
| odutos 32677 | Being a toset is a self-du... |
| tlt2 32678 | In a Toset, two elements m... |
| tlt3 32679 | In a Toset, two elements m... |
| trleile 32680 | In a Toset, two elements m... |
| toslublem 32681 | Lemma for ~ toslub and ~ x... |
| toslub 32682 | In a toset, the lowest upp... |
| tosglblem 32683 | Lemma for ~ tosglb and ~ x... |
| tosglb 32684 | Same theorem as ~ toslub ,... |
| clatp0cl 32685 | The poset zero of a comple... |
| clatp1cl 32686 | The poset one of a complet... |
| mntoval 32691 | Operation value of the mon... |
| ismnt 32692 | Express the statement " ` ... |
| ismntd 32693 | Property of being a monoto... |
| mntf 32694 | A monotone function is a f... |
| mgcoval 32695 | Operation value of the mon... |
| mgcval 32696 | Monotone Galois connection... |
| mgcf1 32697 | The lower adjoint ` F ` of... |
| mgcf2 32698 | The upper adjoint ` G ` of... |
| mgccole1 32699 | An inequality for the kern... |
| mgccole2 32700 | Inequality for the closure... |
| mgcmnt1 32701 | The lower adjoint ` F ` of... |
| mgcmnt2 32702 | The upper adjoint ` G ` of... |
| mgcmntco 32703 | A Galois connection like s... |
| dfmgc2lem 32704 | Lemma for dfmgc2, backward... |
| dfmgc2 32705 | Alternate definition of th... |
| mgcmnt1d 32706 | Galois connection implies ... |
| mgcmnt2d 32707 | Galois connection implies ... |
| mgccnv 32708 | The inverse Galois connect... |
| pwrssmgc 32709 | Given a function ` F ` , e... |
| mgcf1olem1 32710 | Property of a Galois conne... |
| mgcf1olem2 32711 | Property of a Galois conne... |
| mgcf1o 32712 | Given a Galois connection,... |
| xrs0 32715 | The zero of the extended r... |
| xrslt 32716 | The "strictly less than" r... |
| xrsinvgval 32717 | The inversion operation in... |
| xrsmulgzz 32718 | The "multiple" function in... |
| xrstos 32719 | The extended real numbers ... |
| xrsclat 32720 | The extended real numbers ... |
| xrsp0 32721 | The poset 0 of the extende... |
| xrsp1 32722 | The poset 1 of the extende... |
| xrge0base 32723 | The base of the extended n... |
| xrge00 32724 | The zero of the extended n... |
| xrge0plusg 32725 | The additive law of the ex... |
| xrge0le 32726 | The "less than or equal to... |
| xrge0mulgnn0 32727 | The group multiple functio... |
| xrge0addass 32728 | Associativity of extended ... |
| xrge0addgt0 32729 | The sum of nonnegative and... |
| xrge0adddir 32730 | Right-distributivity of ex... |
| xrge0adddi 32731 | Left-distributivity of ext... |
| xrge0npcan 32732 | Extended nonnegative real ... |
| fsumrp0cl 32733 | Closure of a finite sum of... |
| abliso 32734 | The image of an Abelian gr... |
| lmhmghmd 32735 | A module homomorphism is a... |
| mhmimasplusg 32736 | Value of the operation of ... |
| lmhmimasvsca 32737 | Value of the scalar produc... |
| gsumsubg 32738 | The group sum in a subgrou... |
| gsumsra 32739 | The group sum in a subring... |
| gsummpt2co 32740 | Split a finite sum into a ... |
| gsummpt2d 32741 | Express a finite sum over ... |
| lmodvslmhm 32742 | Scalar multiplication in a... |
| gsumvsmul1 32743 | Pull a scalar multiplicati... |
| gsummptres 32744 | Extend a finite group sum ... |
| gsummptres2 32745 | Extend a finite group sum ... |
| gsumzresunsn 32746 | Append an element to a fin... |
| gsumpart 32747 | Express a group sum as a d... |
| gsumhashmul 32748 | Express a group sum by gro... |
| xrge0tsmsd 32749 | Any finite or infinite sum... |
| xrge0tsmsbi 32750 | Any limit of a finite or i... |
| xrge0tsmseq 32751 | Any limit of a finite or i... |
| cntzun 32752 | The centralizer of a union... |
| cntzsnid 32753 | The centralizer of the ide... |
| cntrcrng 32754 | The center of a ring is a ... |
| isomnd 32759 | A (left) ordered monoid is... |
| isogrp 32760 | A (left-)ordered group is ... |
| ogrpgrp 32761 | A left-ordered group is a ... |
| omndmnd 32762 | A left-ordered monoid is a... |
| omndtos 32763 | A left-ordered monoid is a... |
| omndadd 32764 | In an ordered monoid, the ... |
| omndaddr 32765 | In a right ordered monoid,... |
| omndadd2d 32766 | In a commutative left orde... |
| omndadd2rd 32767 | In a left- and right- orde... |
| submomnd 32768 | A submonoid of an ordered ... |
| xrge0omnd 32769 | The nonnegative extended r... |
| omndmul2 32770 | In an ordered monoid, the ... |
| omndmul3 32771 | In an ordered monoid, the ... |
| omndmul 32772 | In a commutative ordered m... |
| ogrpinv0le 32773 | In an ordered group, the o... |
| ogrpsub 32774 | In an ordered group, the o... |
| ogrpaddlt 32775 | In an ordered group, stric... |
| ogrpaddltbi 32776 | In a right ordered group, ... |
| ogrpaddltrd 32777 | In a right ordered group, ... |
| ogrpaddltrbid 32778 | In a right ordered group, ... |
| ogrpsublt 32779 | In an ordered group, stric... |
| ogrpinv0lt 32780 | In an ordered group, the o... |
| ogrpinvlt 32781 | In an ordered group, the o... |
| gsumle 32782 | A finite sum in an ordered... |
| symgfcoeu 32783 | Uniqueness property of per... |
| symgcom 32784 | Two permutations ` X ` and... |
| symgcom2 32785 | Two permutations ` X ` and... |
| symgcntz 32786 | All elements of a (finite)... |
| odpmco 32787 | The composition of two odd... |
| symgsubg 32788 | The value of the group sub... |
| pmtrprfv2 32789 | In a transposition of two ... |
| pmtrcnel 32790 | Composing a permutation ` ... |
| pmtrcnel2 32791 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 32792 | Composing a permutation ` ... |
| pmtridf1o 32793 | Transpositions of ` X ` an... |
| pmtridfv1 32794 | Value at X of the transpos... |
| pmtridfv2 32795 | Value at Y of the transpos... |
| psgnid 32796 | Permutation sign of the id... |
| psgndmfi 32797 | For a finite base set, the... |
| pmtrto1cl 32798 | Useful lemma for the follo... |
| psgnfzto1stlem 32799 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 32800 | Value of our permutation `... |
| fzto1st1 32801 | Special case where the per... |
| fzto1st 32802 | The function moving one el... |
| fzto1stinvn 32803 | Value of the inverse of ou... |
| psgnfzto1st 32804 | The permutation sign for m... |
| tocycval 32807 | Value of the cycle builder... |
| tocycfv 32808 | Function value of a permut... |
| tocycfvres1 32809 | A cyclic permutation is a ... |
| tocycfvres2 32810 | A cyclic permutation is th... |
| cycpmfvlem 32811 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 32812 | Value of a cycle function ... |
| cycpmfv2 32813 | Value of a cycle function ... |
| cycpmfv3 32814 | Values outside of the orbi... |
| cycpmcl 32815 | Cyclic permutations are pe... |
| tocycf 32816 | The permutation cycle buil... |
| tocyc01 32817 | Permutation cycles built f... |
| cycpm2tr 32818 | A cyclic permutation of 2 ... |
| cycpm2cl 32819 | Closure for the 2-cycles. ... |
| cyc2fv1 32820 | Function value of a 2-cycl... |
| cyc2fv2 32821 | Function value of a 2-cycl... |
| trsp2cyc 32822 | Exhibit the word a transpo... |
| cycpmco2f1 32823 | The word U used in ~ cycpm... |
| cycpmco2rn 32824 | The orbit of the compositi... |
| cycpmco2lem1 32825 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 32826 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 32827 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 32828 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 32829 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 32830 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 32831 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 32832 | The composition of a cycli... |
| cyc2fvx 32833 | Function value of a 2-cycl... |
| cycpm3cl 32834 | Closure of the 3-cycles in... |
| cycpm3cl2 32835 | Closure of the 3-cycles in... |
| cyc3fv1 32836 | Function value of a 3-cycl... |
| cyc3fv2 32837 | Function value of a 3-cycl... |
| cyc3fv3 32838 | Function value of a 3-cycl... |
| cyc3co2 32839 | Represent a 3-cycle as a c... |
| cycpmconjvlem 32840 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 32841 | A formula for computing co... |
| cycpmrn 32842 | The range of the word used... |
| tocyccntz 32843 | All elements of a (finite)... |
| evpmval 32844 | Value of the set of even p... |
| cnmsgn0g 32845 | The neutral element of the... |
| evpmsubg 32846 | The alternating group is a... |
| evpmid 32847 | The identity is an even pe... |
| altgnsg 32848 | The alternating group ` ( ... |
| cyc3evpm 32849 | 3-Cycles are even permutat... |
| cyc3genpmlem 32850 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 32851 | The alternating group ` A ... |
| cycpmgcl 32852 | Cyclic permutations are pe... |
| cycpmconjslem1 32853 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 32854 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 32855 | All cycles of the same len... |
| cyc3conja 32856 | All 3-cycles are conjugate... |
| sgnsv 32859 | The sign mapping. (Contri... |
| sgnsval 32860 | The sign value. (Contribu... |
| sgnsf 32861 | The sign function. (Contr... |
| inftmrel 32866 | The infinitesimal relation... |
| isinftm 32867 | Express ` x ` is infinites... |
| isarchi 32868 | Express the predicate " ` ... |
| pnfinf 32869 | Plus infinity is an infini... |
| xrnarchi 32870 | The completed real line is... |
| isarchi2 32871 | Alternative way to express... |
| submarchi 32872 | A submonoid is archimedean... |
| isarchi3 32873 | This is the usual definiti... |
| archirng 32874 | Property of Archimedean or... |
| archirngz 32875 | Property of Archimedean le... |
| archiexdiv 32876 | In an Archimedean group, g... |
| archiabllem1a 32877 | Lemma for ~ archiabl : In... |
| archiabllem1b 32878 | Lemma for ~ archiabl . (C... |
| archiabllem1 32879 | Archimedean ordered groups... |
| archiabllem2a 32880 | Lemma for ~ archiabl , whi... |
| archiabllem2c 32881 | Lemma for ~ archiabl . (C... |
| archiabllem2b 32882 | Lemma for ~ archiabl . (C... |
| archiabllem2 32883 | Archimedean ordered groups... |
| archiabl 32884 | Archimedean left- and righ... |
| isslmd 32887 | The predicate "is a semimo... |
| slmdlema 32888 | Lemma for properties of a ... |
| lmodslmd 32889 | Left semimodules generaliz... |
| slmdcmn 32890 | A semimodule is a commutat... |
| slmdmnd 32891 | A semimodule is a monoid. ... |
| slmdsrg 32892 | The scalar component of a ... |
| slmdbn0 32893 | The base set of a semimodu... |
| slmdacl 32894 | Closure of ring addition f... |
| slmdmcl 32895 | Closure of ring multiplica... |
| slmdsn0 32896 | The set of scalars in a se... |
| slmdvacl 32897 | Closure of vector addition... |
| slmdass 32898 | Semiring left module vecto... |
| slmdvscl 32899 | Closure of scalar product ... |
| slmdvsdi 32900 | Distributive law for scala... |
| slmdvsdir 32901 | Distributive law for scala... |
| slmdvsass 32902 | Associative law for scalar... |
| slmd0cl 32903 | The ring zero in a semimod... |
| slmd1cl 32904 | The ring unity in a semiri... |
| slmdvs1 32905 | Scalar product with ring u... |
| slmd0vcl 32906 | The zero vector is a vecto... |
| slmd0vlid 32907 | Left identity law for the ... |
| slmd0vrid 32908 | Right identity law for the... |
| slmd0vs 32909 | Zero times a vector is the... |
| slmdvs0 32910 | Anything times the zero ve... |
| gsumvsca1 32911 | Scalar product of a finite... |
| gsumvsca2 32912 | Scalar product of a finite... |
| prmsimpcyc 32913 | A group of prime order is ... |
| domnlcan 32914 | Left-cancellation law for ... |
| idomrcan 32915 | Right-cancellation law for... |
| urpropd 32916 | Sufficient condition for r... |
| 0ringsubrg 32917 | A subring of a zero ring i... |
| frobrhm 32918 | In a commutative ring with... |
| ress1r 32919 | ` 1r ` is unaffected by re... |
| ringinvval 32920 | The ring inverse expressed... |
| dvrcan5 32921 | Cancellation law for commo... |
| subrgchr 32922 | If ` A ` is a subring of `... |
| rmfsupp2 32923 | A mapping of a multiplicat... |
| unitnz 32924 | In a nonzero ring, a unit ... |
| eufndx 32927 | Index value of the Euclide... |
| eufid 32928 | Utility theorem: index-ind... |
| ringinveu 32931 | If a ring unit element ` X... |
| isdrng4 32932 | A division ring is a ring ... |
| rndrhmcl 32933 | The image of a division ri... |
| sdrgdvcl 32934 | A sub-division-ring is clo... |
| sdrginvcl 32935 | A sub-division-ring is clo... |
| primefldchr 32936 | The characteristic of a pr... |
| fldgenval 32939 | Value of the field generat... |
| fldgenssid 32940 | The field generated by a s... |
| fldgensdrg 32941 | A generated subfield is a ... |
| fldgenssv 32942 | A generated subfield is a ... |
| fldgenss 32943 | Generated subfields preser... |
| fldgenidfld 32944 | The subfield generated by ... |
| fldgenssp 32945 | The field generated by a s... |
| fldgenid 32946 | The subfield of a field ` ... |
| fldgenfld 32947 | A generated subfield is a ... |
| primefldgen1 32948 | The prime field of a divis... |
| 1fldgenq 32949 | The field of rational numb... |
| isorng 32954 | An ordered ring is a ring ... |
| orngring 32955 | An ordered ring is a ring.... |
| orngogrp 32956 | An ordered ring is an orde... |
| isofld 32957 | An ordered field is a fiel... |
| orngmul 32958 | In an ordered ring, the or... |
| orngsqr 32959 | In an ordered ring, all sq... |
| ornglmulle 32960 | In an ordered ring, multip... |
| orngrmulle 32961 | In an ordered ring, multip... |
| ornglmullt 32962 | In an ordered ring, multip... |
| orngrmullt 32963 | In an ordered ring, multip... |
| orngmullt 32964 | In an ordered ring, the st... |
| ofldfld 32965 | An ordered field is a fiel... |
| ofldtos 32966 | An ordered field is a tota... |
| orng0le1 32967 | In an ordered ring, the ri... |
| ofldlt1 32968 | In an ordered field, the r... |
| ofldchr 32969 | The characteristic of an o... |
| suborng 32970 | Every subring of an ordere... |
| subofld 32971 | Every subfield of an order... |
| isarchiofld 32972 | Axiom of Archimedes : a ch... |
| rhmdvd 32973 | A ring homomorphism preser... |
| kerunit 32974 | If a unit element lies in ... |
| reldmresv 32977 | The scalar restriction is ... |
| resvval 32978 | Value of structure restric... |
| resvid2 32979 | General behavior of trivia... |
| resvval2 32980 | Value of nontrivial struct... |
| resvsca 32981 | Base set of a structure re... |
| resvlem 32982 | Other elements of a scalar... |
| resvlemOLD 32983 | Obsolete version of ~ resv... |
| resvbas 32984 | ` Base ` is unaffected by ... |
| resvbasOLD 32985 | Obsolete proof of ~ resvba... |
| resvplusg 32986 | ` +g ` is unaffected by sc... |
| resvplusgOLD 32987 | Obsolete proof of ~ resvpl... |
| resvvsca 32988 | ` .s ` is unaffected by sc... |
| resvvscaOLD 32989 | Obsolete proof of ~ resvvs... |
| resvmulr 32990 | ` .r ` is unaffected by sc... |
| resvmulrOLD 32991 | Obsolete proof of ~ resvmu... |
| resv0g 32992 | ` 0g ` is unaffected by sc... |
| resv1r 32993 | ` 1r ` is unaffected by sc... |
| resvcmn 32994 | Scalar restriction preserv... |
| gzcrng 32995 | The gaussian integers form... |
| reofld 32996 | The real numbers form an o... |
| nn0omnd 32997 | The nonnegative integers f... |
| rearchi 32998 | The field of the real numb... |
| nn0archi 32999 | The monoid of the nonnegat... |
| xrge0slmod 33000 | The extended nonnegative r... |
| qusker 33001 | The kernel of a quotient m... |
| eqgvscpbl 33002 | The left coset equivalence... |
| qusvscpbl 33003 | The quotient map distribut... |
| qusvsval 33004 | Value of the scalar multip... |
| imaslmod 33005 | The image structure of a l... |
| imasmhm 33006 | Given a function ` F ` wit... |
| imasghm 33007 | Given a function ` F ` wit... |
| imasrhm 33008 | Given a function ` F ` wit... |
| imaslmhm 33009 | Given a function ` F ` wit... |
| quslmod 33010 | If ` G ` is a submodule in... |
| quslmhm 33011 | If ` G ` is a submodule of... |
| quslvec 33012 | If ` S ` is a vector subsp... |
| ecxpid 33013 | The equivalence class of a... |
| qsxpid 33014 | The quotient set of a cart... |
| qusxpid 33015 | The Group quotient equival... |
| qustriv 33016 | The quotient of a group ` ... |
| qustrivr 33017 | Converse of ~ qustriv . (... |
| znfermltl 33018 | Fermat's little theorem in... |
| islinds5 33019 | A set is linearly independ... |
| ellspds 33020 | Variation on ~ ellspd . (... |
| 0ellsp 33021 | Zero is in all spans. (Co... |
| 0nellinds 33022 | The group identity cannot ... |
| rspsnel 33023 | Membership in a principal ... |
| rspsnid 33024 | A principal ideal contains... |
| elrsp 33025 | Write the elements of a ri... |
| rspidlid 33026 | The ideal span of an ideal... |
| pidlnz 33027 | A principal ideal generate... |
| dvdsruassoi 33028 | If two elements ` X ` and ... |
| dvdsruasso 33029 | Two elements ` X ` and ` Y... |
| dvdsrspss 33030 | In a ring, an element ` X ... |
| rspsnasso 33031 | Two elements ` X ` and ` Y... |
| lbslsp 33032 | Any element of a left modu... |
| lindssn 33033 | Any singleton of a nonzero... |
| lindflbs 33034 | Conditions for an independ... |
| islbs5 33035 | An equivalent formulation ... |
| linds2eq 33036 | Deduce equality of element... |
| lindfpropd 33037 | Property deduction for lin... |
| lindspropd 33038 | Property deduction for lin... |
| elgrplsmsn 33039 | Membership in a sumset wit... |
| lsmsnorb 33040 | The sumset of a group with... |
| lsmsnorb2 33041 | The sumset of a single ele... |
| elringlsm 33042 | Membership in a product of... |
| elringlsmd 33043 | Membership in a product of... |
| ringlsmss 33044 | Closure of the product of ... |
| ringlsmss1 33045 | The product of an ideal ` ... |
| ringlsmss2 33046 | The product with an ideal ... |
| lsmsnpridl 33047 | The product of the ring wi... |
| lsmsnidl 33048 | The product of the ring wi... |
| lsmidllsp 33049 | The sum of two ideals is t... |
| lsmidl 33050 | The sum of two ideals is a... |
| lsmssass 33051 | Group sum is associative, ... |
| grplsm0l 33052 | Sumset with the identity s... |
| grplsmid 33053 | The direct sum of an eleme... |
| qusmul 33054 | Value of the ring operatio... |
| quslsm 33055 | Express the image by the q... |
| qusbas2 33056 | Alternate definition of th... |
| qus0g 33057 | The identity element of a ... |
| qusima 33058 | The image of a subgroup by... |
| qusrn 33059 | The natural map from eleme... |
| nsgqus0 33060 | A normal subgroup ` N ` is... |
| nsgmgclem 33061 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 33062 | There is a monotone Galois... |
| nsgqusf1olem1 33063 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 33064 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 33065 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 33066 | The canonical projection h... |
| lmhmqusker 33067 | A surjective module homomo... |
| lmicqusker 33068 | The image ` H ` of a modul... |
| intlidl 33069 | The intersection of a none... |
| rhmpreimaidl 33070 | The preimage of an ideal b... |
| kerlidl 33071 | The kernel of a ring homom... |
| 0ringidl 33072 | The zero ideal is the only... |
| pidlnzb 33073 | A principal ideal is nonze... |
| lidlunitel 33074 | If an ideal ` I ` contains... |
| unitpidl1 33075 | The ideal ` I ` generated ... |
| rhmquskerlem 33076 | The mapping ` J ` induced ... |
| rhmqusker 33077 | A surjective ring homomorp... |
| ricqusker 33078 | The image ` H ` of a ring ... |
| elrspunidl 33079 | Elementhood in the span of... |
| elrspunsn 33080 | Membership to the span of ... |
| lidlincl 33081 | Ideals are closed under in... |
| idlinsubrg 33082 | The intersection between a... |
| rhmimaidl 33083 | The image of an ideal ` I ... |
| drngidl 33084 | A nonzero ring is a divisi... |
| drngidlhash 33085 | A ring is a division ring ... |
| prmidlval 33088 | The class of prime ideals ... |
| isprmidl 33089 | The predicate "is a prime ... |
| prmidlnr 33090 | A prime ideal is a proper ... |
| prmidl 33091 | The main property of a pri... |
| prmidl2 33092 | A condition that shows an ... |
| idlmulssprm 33093 | Let ` P ` be a prime ideal... |
| pridln1 33094 | A proper ideal cannot cont... |
| prmidlidl 33095 | A prime ideal is an ideal.... |
| prmidlssidl 33096 | Prime ideals as a subset o... |
| lidlnsg 33097 | An ideal is a normal subgr... |
| cringm4 33098 | Commutative/associative la... |
| isprmidlc 33099 | The predicate "is prime id... |
| prmidlc 33100 | Property of a prime ideal ... |
| 0ringprmidl 33101 | The trivial ring does not ... |
| prmidl0 33102 | The zero ideal of a commut... |
| rhmpreimaprmidl 33103 | The preimage of a prime id... |
| qsidomlem1 33104 | If the quotient ring of a ... |
| qsidomlem2 33105 | A quotient by a prime idea... |
| qsidom 33106 | An ideal ` I ` in the comm... |
| qsnzr 33107 | A quotient of a non-zero r... |
| mxidlval 33110 | The set of maximal ideals ... |
| ismxidl 33111 | The predicate "is a maxima... |
| mxidlidl 33112 | A maximal ideal is an idea... |
| mxidlnr 33113 | A maximal ideal is proper.... |
| mxidlmax 33114 | A maximal ideal is a maxim... |
| mxidln1 33115 | One is not contained in an... |
| mxidlnzr 33116 | A ring with a maximal idea... |
| mxidlmaxv 33117 | An ideal ` I ` strictly co... |
| crngmxidl 33118 | In a commutative ring, max... |
| mxidlprm 33119 | Every maximal ideal is pri... |
| mxidlirredi 33120 | In an integral domain, the... |
| mxidlirred 33121 | In a principal ideal domai... |
| ssmxidllem 33122 | The set ` P ` used in the ... |
| ssmxidl 33123 | Let ` R ` be a ring, and l... |
| drnglidl1ne0 33124 | In a nonzero ring, the zer... |
| drng0mxidl 33125 | In a division ring, the ze... |
| drngmxidl 33126 | The zero ideal is the only... |
| krull 33127 | Krull's theorem: Any nonz... |
| mxidlnzrb 33128 | A ring is nonzero if and o... |
| opprabs 33129 | The opposite ring of the o... |
| oppreqg 33130 | Group coset equivalence re... |
| opprnsg 33131 | Normal subgroups of the op... |
| opprlidlabs 33132 | The ideals of the opposite... |
| oppr2idl 33133 | Two sided ideal of the opp... |
| opprmxidlabs 33134 | The maximal ideal of the o... |
| opprqusbas 33135 | The base of the quotient o... |
| opprqusplusg 33136 | The group operation of the... |
| opprqus0g 33137 | The group identity element... |
| opprqusmulr 33138 | The multiplication operati... |
| opprqus1r 33139 | The ring unity of the quot... |
| opprqusdrng 33140 | The quotient of the opposi... |
| qsdrngilem 33141 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 33142 | A quotient by a maximal le... |
| qsdrnglem2 33143 | Lemma for ~ qsdrng . (Con... |
| qsdrng 33144 | An ideal ` M ` is both lef... |
| qsfld 33145 | An ideal ` M ` in the comm... |
| mxidlprmALT 33146 | Every maximal ideal is pri... |
| idlsrgstr 33149 | A constructed semiring of ... |
| idlsrgval 33150 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 33151 | Base of the ideals of a ri... |
| idlsrgplusg 33152 | Additive operation of the ... |
| idlsrg0g 33153 | The zero ideal is the addi... |
| idlsrgmulr 33154 | Multiplicative operation o... |
| idlsrgtset 33155 | Topology component of the ... |
| idlsrgmulrval 33156 | Value of the ring multipli... |
| idlsrgmulrcl 33157 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 33158 | In a commutative ring, the... |
| idlsrgmulrss2 33159 | The product of two ideals ... |
| idlsrgmulrssin 33160 | In a commutative ring, the... |
| idlsrgmnd 33161 | The ideals of a ring form ... |
| idlsrgcmnd 33162 | The ideals of a ring form ... |
| isufd 33165 | The property of being a Un... |
| rprmval 33166 | The prime elements of a ri... |
| isrprm 33167 | Property for ` P ` to be a... |
| 0ringmon1p 33168 | There are no monic polynom... |
| fply1 33169 | Conditions for a function ... |
| ply1lvec 33170 | In a division ring, the un... |
| evls1fn 33171 | Functionality of the subri... |
| evls1dm 33172 | The domain of the subring ... |
| evls1fvf 33173 | The subring evaluation fun... |
| evls1scafv 33174 | Value of the univariate po... |
| evls1expd 33175 | Univariate polynomial eval... |
| evls1varpwval 33176 | Univariate polynomial eval... |
| ressdeg1 33177 | The degree of a univariate... |
| ressply10g 33178 | A restricted polynomial al... |
| ressply1mon1p 33179 | The monic polynomials of a... |
| ressply1invg 33180 | An element of a restricted... |
| ressply1sub 33181 | A restricted polynomial al... |
| evls1fpws 33182 | Evaluation of a univariate... |
| ressply1evl 33183 | Evaluation of a univariate... |
| evls1addd 33184 | Univariate polynomial eval... |
| evls1subd 33185 | Univariate polynomial eval... |
| evls1muld 33186 | Univariate polynomial eval... |
| evls1vsca 33187 | Univariate polynomial eval... |
| ply1ascl1 33188 | The multiplicative unit sc... |
| deg1le0eq0 33189 | A polynomial with nonposit... |
| ply1asclunit 33190 | A non-zero scalar polynomi... |
| ply1unit 33191 | In a field ` F ` , a polyn... |
| m1pmeq 33192 | If two monic polynomials `... |
| asclply1subcl 33193 | Closure of the algebra sca... |
| ply1fermltl 33194 | Fermat's little theorem fo... |
| coe1mon 33195 | Coefficient vector of a mo... |
| ply1moneq 33196 | Two monomials are equal if... |
| ply1degltel 33197 | Characterize elementhood i... |
| ply1degleel 33198 | Characterize elementhood i... |
| ply1degltlss 33199 | The space ` S ` of the uni... |
| gsummoncoe1fzo 33200 | A coefficient of the polyn... |
| ply1gsumz 33201 | If a polynomial given as a... |
| deg1addlt 33202 | If both factors have degre... |
| ig1pnunit 33203 | The polynomial ideal gener... |
| ig1pmindeg 33204 | The polynomial ideal gener... |
| q1pdir 33205 | Distribution of univariate... |
| q1pvsca 33206 | Scalar multiplication prop... |
| r1pvsca 33207 | Scalar multiplication prop... |
| r1p0 33208 | Polynomial remainder opera... |
| r1pcyc 33209 | The polynomial remainder o... |
| r1padd1 33210 | Addition property of the p... |
| r1pid2 33211 | Identity law for polynomia... |
| r1plmhm 33212 | The univariate polynomial ... |
| r1pquslmic 33213 | The univariate polynomial ... |
| sra1r 33214 | The unity element of a sub... |
| sradrng 33215 | Condition for a subring al... |
| srasubrg 33216 | A subring of the original ... |
| sralvec 33217 | Given a sub division ring ... |
| srafldlvec 33218 | Given a subfield ` F ` of ... |
| resssra 33219 | The subring algebra of a r... |
| lsssra 33220 | A subring is a subspace of... |
| drgext0g 33221 | The additive neutral eleme... |
| drgextvsca 33222 | The scalar multiplication ... |
| drgext0gsca 33223 | The additive neutral eleme... |
| drgextsubrg 33224 | The scalar field is a subr... |
| drgextlsp 33225 | The scalar field is a subs... |
| drgextgsum 33226 | Group sum in a division ri... |
| lvecdimfi 33227 | Finite version of ~ lvecdi... |
| dimval 33230 | The dimension of a vector ... |
| dimvalfi 33231 | The dimension of a vector ... |
| dimcl 33232 | Closure of the vector spac... |
| lmimdim 33233 | Module isomorphisms preser... |
| lmicdim 33234 | Module isomorphisms preser... |
| lvecdim0i 33235 | A vector space of dimensio... |
| lvecdim0 33236 | A vector space of dimensio... |
| lssdimle 33237 | The dimension of a linear ... |
| dimpropd 33238 | If two structures have the... |
| rlmdim 33239 | The left vector space indu... |
| rgmoddimOLD 33240 | Obsolete version of ~ rlmd... |
| frlmdim 33241 | Dimension of a free left m... |
| tnglvec 33242 | Augmenting a structure wit... |
| tngdim 33243 | Dimension of a left vector... |
| rrxdim 33244 | Dimension of the generaliz... |
| matdim 33245 | Dimension of the space of ... |
| lbslsat 33246 | A nonzero vector ` X ` is ... |
| lsatdim 33247 | A line, spanned by a nonze... |
| drngdimgt0 33248 | The dimension of a vector ... |
| lmhmlvec2 33249 | A homomorphism of left vec... |
| kerlmhm 33250 | The kernel of a vector spa... |
| imlmhm 33251 | The image of a vector spac... |
| ply1degltdimlem 33252 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 33253 | The space ` S ` of the uni... |
| lindsunlem 33254 | Lemma for ~ lindsun . (Co... |
| lindsun 33255 | Condition for the union of... |
| lbsdiflsp0 33256 | The linear spans of two di... |
| dimkerim 33257 | Given a linear map ` F ` b... |
| qusdimsum 33258 | Let ` W ` be a vector spac... |
| fedgmullem1 33259 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 33260 | Lemma for ~ fedgmul . (Co... |
| fedgmul 33261 | The multiplicativity formu... |
| relfldext 33270 | The field extension is a r... |
| brfldext 33271 | The field extension relati... |
| ccfldextrr 33272 | The field of the complex n... |
| fldextfld1 33273 | A field extension is only ... |
| fldextfld2 33274 | A field extension is only ... |
| fldextsubrg 33275 | Field extension implies a ... |
| fldextress 33276 | Field extension implies a ... |
| brfinext 33277 | The finite field extension... |
| extdgval 33278 | Value of the field extensi... |
| fldextsralvec 33279 | The subring algebra associ... |
| extdgcl 33280 | Closure of the field exten... |
| extdggt0 33281 | Degrees of field extension... |
| fldexttr 33282 | Field extension is a trans... |
| fldextid 33283 | The field extension relati... |
| extdgid 33284 | A trivial field extension ... |
| extdgmul 33285 | The multiplicativity formu... |
| finexttrb 33286 | The extension ` E ` of ` K... |
| extdg1id 33287 | If the degree of the exten... |
| extdg1b 33288 | The degree of the extensio... |
| fldextchr 33289 | The characteristic of a su... |
| evls1fldgencl 33290 | Closure of the subring pol... |
| ccfldsrarelvec 33291 | The subring algebra of the... |
| ccfldextdgrr 33292 | The degree of the field ex... |
| irngval 33295 | The elements of a field ` ... |
| elirng 33296 | Property for an element ` ... |
| irngss 33297 | All elements of a subring ... |
| irngssv 33298 | An integral element is an ... |
| 0ringirng 33299 | A zero ring ` R ` has no i... |
| irngnzply1lem 33300 | In the case of a field ` E... |
| irngnzply1 33301 | In the case of a field ` E... |
| evls1fvcl 33304 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 33305 | The function ` F ` mapping... |
| evls1maplmhm 33306 | The function ` F ` mapping... |
| evls1maprnss 33307 | The function ` F ` mapping... |
| ply1annidllem 33308 | Write the set ` Q ` of pol... |
| ply1annidl 33309 | The set ` Q ` of polynomia... |
| ply1annnr 33310 | The set ` Q ` of polynomia... |
| ply1annig1p 33311 | The ideal ` Q ` of polynom... |
| minplyval 33312 | Expand the value of the mi... |
| minplycl 33313 | The minimal polynomial is ... |
| ply1annprmidl 33314 | The set ` Q ` of polynomia... |
| minplyann 33315 | The minimal polynomial for... |
| minplyirredlem 33316 | Lemma for ~ minplyirred . ... |
| minplyirred 33317 | A nonzero minimal polynomi... |
| irngnminplynz 33318 | Integral elements have non... |
| minplym1p 33319 | A minimal polynomial is mo... |
| irredminply 33320 | An irreductible, monic, an... |
| algextdeglem1 33321 | Lemma for ~ algextdeg . (... |
| algextdeglem2 33322 | Lemma for ~ algextdeg . B... |
| algextdeglem3 33323 | Lemma for ~ algextdeg . T... |
| algextdeglem4 33324 | Lemma for ~ algextdeg . B... |
| algextdeglem5 33325 | Lemma for ~ algextdeg . T... |
| algextdeglem6 33326 | Lemma for ~ algextdeg . B... |
| algextdeglem7 33327 | Lemma for ~ algextdeg . T... |
| algextdeglem8 33328 | Lemma for ~ algextdeg . T... |
| algextdeg 33329 | The degree of an algebraic... |
| smatfval 33332 | Value of the submatrix. (... |
| smatrcl 33333 | Closure of the rectangular... |
| smatlem 33334 | Lemma for the next theorem... |
| smattl 33335 | Entries of a submatrix, to... |
| smattr 33336 | Entries of a submatrix, to... |
| smatbl 33337 | Entries of a submatrix, bo... |
| smatbr 33338 | Entries of a submatrix, bo... |
| smatcl 33339 | Closure of the square subm... |
| matmpo 33340 | Write a square matrix as a... |
| 1smat1 33341 | The submatrix of the ident... |
| submat1n 33342 | One case where the submatr... |
| submatres 33343 | Special case where the sub... |
| submateqlem1 33344 | Lemma for ~ submateq . (C... |
| submateqlem2 33345 | Lemma for ~ submateq . (C... |
| submateq 33346 | Sufficient condition for t... |
| submatminr1 33347 | If we take a submatrix by ... |
| lmatval 33350 | Value of the literal matri... |
| lmatfval 33351 | Entries of a literal matri... |
| lmatfvlem 33352 | Useful lemma to extract li... |
| lmatcl 33353 | Closure of the literal mat... |
| lmat22lem 33354 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 33355 | Entry of a 2x2 literal mat... |
| lmat22e12 33356 | Entry of a 2x2 literal mat... |
| lmat22e21 33357 | Entry of a 2x2 literal mat... |
| lmat22e22 33358 | Entry of a 2x2 literal mat... |
| lmat22det 33359 | The determinant of a liter... |
| mdetpmtr1 33360 | The determinant of a matri... |
| mdetpmtr2 33361 | The determinant of a matri... |
| mdetpmtr12 33362 | The determinant of a matri... |
| mdetlap1 33363 | A Laplace expansion of the... |
| madjusmdetlem1 33364 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 33365 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 33366 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 33367 | Lemma for ~ madjusmdet . ... |
| madjusmdet 33368 | Express the cofactor of th... |
| mdetlap 33369 | Laplace expansion of the d... |
| ist0cld 33370 | The predicate "is a T_0 sp... |
| txomap 33371 | Given two open maps ` F ` ... |
| qtopt1 33372 | If every equivalence class... |
| qtophaus 33373 | If an open map's graph in ... |
| circtopn 33374 | The topology of the unit c... |
| circcn 33375 | The function gluing the re... |
| reff 33376 | For any cover refinement, ... |
| locfinreflem 33377 | A locally finite refinemen... |
| locfinref 33378 | A locally finite refinemen... |
| iscref 33381 | The property that every op... |
| crefeq 33382 | Equality theorem for the "... |
| creftop 33383 | A space where every open c... |
| crefi 33384 | The property that every op... |
| crefdf 33385 | A formulation of ~ crefi e... |
| crefss 33386 | The "every open cover has ... |
| cmpcref 33387 | Equivalent definition of c... |
| cmpfiref 33388 | Every open cover of a Comp... |
| ldlfcntref 33391 | Every open cover of a Lind... |
| ispcmp 33394 | The predicate "is a paraco... |
| cmppcmp 33395 | Every compact space is par... |
| dispcmp 33396 | Every discrete space is pa... |
| pcmplfin 33397 | Given a paracompact topolo... |
| pcmplfinf 33398 | Given a paracompact topolo... |
| rspecval 33401 | Value of the spectrum of t... |
| rspecbas 33402 | The prime ideals form the ... |
| rspectset 33403 | Topology component of the ... |
| rspectopn 33404 | The topology component of ... |
| zarcls0 33405 | The closure of the identit... |
| zarcls1 33406 | The unit ideal ` B ` is th... |
| zarclsun 33407 | The union of two closed se... |
| zarclsiin 33408 | In a Zariski topology, the... |
| zarclsint 33409 | The intersection of a fami... |
| zarclssn 33410 | The closed points of Zaris... |
| zarcls 33411 | The open sets of the Zaris... |
| zartopn 33412 | The Zariski topology is a ... |
| zartop 33413 | The Zariski topology is a ... |
| zartopon 33414 | The points of the Zariski ... |
| zar0ring 33415 | The Zariski Topology of th... |
| zart0 33416 | The Zariski topology is T_... |
| zarmxt1 33417 | The Zariski topology restr... |
| zarcmplem 33418 | Lemma for ~ zarcmp . (Con... |
| zarcmp 33419 | The Zariski topology is co... |
| rspectps 33420 | The spectrum of a ring ` R... |
| rhmpreimacnlem 33421 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 33422 | The function mapping a pri... |
| metidval 33427 | Value of the metric identi... |
| metidss 33428 | As a relation, the metric ... |
| metidv 33429 | ` A ` and ` B ` identify b... |
| metideq 33430 | Basic property of the metr... |
| metider 33431 | The metric identification ... |
| pstmval 33432 | Value of the metric induce... |
| pstmfval 33433 | Function value of the metr... |
| pstmxmet 33434 | The metric induced by a ps... |
| hauseqcn 33435 | In a Hausdorff topology, t... |
| elunitge0 33436 | An element of the closed u... |
| unitssxrge0 33437 | The closed unit interval i... |
| unitdivcld 33438 | Necessary conditions for a... |
| iistmd 33439 | The closed unit interval f... |
| unicls 33440 | The union of the closed se... |
| tpr2tp 33441 | The usual topology on ` ( ... |
| tpr2uni 33442 | The usual topology on ` ( ... |
| xpinpreima 33443 | Rewrite the cartesian prod... |
| xpinpreima2 33444 | Rewrite the cartesian prod... |
| sqsscirc1 33445 | The complex square of side... |
| sqsscirc2 33446 | The complex square of side... |
| cnre2csqlem 33447 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 33448 | Image of a centered square... |
| tpr2rico 33449 | For any point of an open s... |
| cnvordtrestixx 33450 | The restriction of the 'gr... |
| prsdm 33451 | Domain of the relation of ... |
| prsrn 33452 | Range of the relation of a... |
| prsss 33453 | Relation of a subproset. ... |
| prsssdm 33454 | Domain of a subproset rela... |
| ordtprsval 33455 | Value of the order topolog... |
| ordtprsuni 33456 | Value of the order topolog... |
| ordtcnvNEW 33457 | The order dual generates t... |
| ordtrestNEW 33458 | The subspace topology of a... |
| ordtrest2NEWlem 33459 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 33460 | An interval-closed set ` A... |
| ordtconnlem1 33461 | Connectedness in the order... |
| ordtconn 33462 | Connectedness in the order... |
| mndpluscn 33463 | A mapping that is both a h... |
| mhmhmeotmd 33464 | Deduce a Topological Monoi... |
| rmulccn 33465 | Multiplication by a real c... |
| raddcn 33466 | Addition in the real numbe... |
| xrmulc1cn 33467 | The operation multiplying ... |
| fmcncfil 33468 | The image of a Cauchy filt... |
| xrge0hmph 33469 | The extended nonnegative r... |
| xrge0iifcnv 33470 | Define a bijection from ` ... |
| xrge0iifcv 33471 | The defined function's val... |
| xrge0iifiso 33472 | The defined bijection from... |
| xrge0iifhmeo 33473 | Expose a homeomorphism fro... |
| xrge0iifhom 33474 | The defined function from ... |
| xrge0iif1 33475 | Condition for the defined ... |
| xrge0iifmhm 33476 | The defined function from ... |
| xrge0pluscn 33477 | The addition operation of ... |
| xrge0mulc1cn 33478 | The operation multiplying ... |
| xrge0tps 33479 | The extended nonnegative r... |
| xrge0topn 33480 | The topology of the extend... |
| xrge0haus 33481 | The topology of the extend... |
| xrge0tmd 33482 | The extended nonnegative r... |
| xrge0tmdALT 33483 | Alternate proof of ~ xrge0... |
| lmlim 33484 | Relate a limit in a given ... |
| lmlimxrge0 33485 | Relate a limit in the nonn... |
| rge0scvg 33486 | Implication of convergence... |
| fsumcvg4 33487 | A serie with finite suppor... |
| pnfneige0 33488 | A neighborhood of ` +oo ` ... |
| lmxrge0 33489 | Express "sequence ` F ` co... |
| lmdvg 33490 | If a monotonic sequence of... |
| lmdvglim 33491 | If a monotonic real number... |
| pl1cn 33492 | A univariate polynomial is... |
| zringnm 33495 | The norm (function) for a ... |
| zzsnm 33496 | The norm of the ring of th... |
| zlm0 33497 | Zero of a ` ZZ ` -module. ... |
| zlm1 33498 | Unity element of a ` ZZ ` ... |
| zlmds 33499 | Distance in a ` ZZ ` -modu... |
| zlmdsOLD 33500 | Obsolete proof of ~ zlmds ... |
| zlmtset 33501 | Topology in a ` ZZ ` -modu... |
| zlmtsetOLD 33502 | Obsolete proof of ~ zlmtse... |
| zlmnm 33503 | Norm of a ` ZZ ` -module (... |
| zhmnrg 33504 | The ` ZZ ` -module built f... |
| nmmulg 33505 | The norm of a group produc... |
| zrhnm 33506 | The norm of the image by `... |
| cnzh 33507 | The ` ZZ ` -module of ` CC... |
| rezh 33508 | The ` ZZ ` -module of ` RR... |
| qqhval 33511 | Value of the canonical hom... |
| zrhf1ker 33512 | The kernel of the homomorp... |
| zrhchr 33513 | The kernel of the homomorp... |
| zrhker 33514 | The kernel of the homomorp... |
| zrhunitpreima 33515 | The preimage by ` ZRHom ` ... |
| elzrhunit 33516 | Condition for the image by... |
| elzdif0 33517 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 33518 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 33519 | Value of the canonical hom... |
| qqhvval 33520 | Value of the canonical hom... |
| qqh0 33521 | The image of ` 0 ` by the ... |
| qqh1 33522 | The image of ` 1 ` by the ... |
| qqhf 33523 | ` QQHom ` as a function. ... |
| qqhvq 33524 | The image of a quotient by... |
| qqhghm 33525 | The ` QQHom ` homomorphism... |
| qqhrhm 33526 | The ` QQHom ` homomorphism... |
| qqhnm 33527 | The norm of the image by `... |
| qqhcn 33528 | The ` QQHom ` homomorphism... |
| qqhucn 33529 | The ` QQHom ` homomorphism... |
| rrhval 33533 | Value of the canonical hom... |
| rrhcn 33534 | If the topology of ` R ` i... |
| rrhf 33535 | If the topology of ` R ` i... |
| isrrext 33537 | Express the property " ` R... |
| rrextnrg 33538 | An extension of ` RR ` is ... |
| rrextdrg 33539 | An extension of ` RR ` is ... |
| rrextnlm 33540 | The norm of an extension o... |
| rrextchr 33541 | The ring characteristic of... |
| rrextcusp 33542 | An extension of ` RR ` is ... |
| rrexttps 33543 | An extension of ` RR ` is ... |
| rrexthaus 33544 | The topology of an extensi... |
| rrextust 33545 | The uniformity of an exten... |
| rerrext 33546 | The field of the real numb... |
| cnrrext 33547 | The field of the complex n... |
| qqtopn 33548 | The topology of the field ... |
| rrhfe 33549 | If ` R ` is an extension o... |
| rrhcne 33550 | If ` R ` is an extension o... |
| rrhqima 33551 | The ` RRHom ` homomorphism... |
| rrh0 33552 | The image of ` 0 ` by the ... |
| xrhval 33555 | The value of the embedding... |
| zrhre 33556 | The ` ZRHom ` homomorphism... |
| qqhre 33557 | The ` QQHom ` homomorphism... |
| rrhre 33558 | The ` RRHom ` homomorphism... |
| relmntop 33561 | Manifold is a relation. (... |
| ismntoplly 33562 | Property of being a manifo... |
| ismntop 33563 | Property of being a manifo... |
| nexple 33564 | A lower bound for an expon... |
| indv 33567 | Value of the indicator fun... |
| indval 33568 | Value of the indicator fun... |
| indval2 33569 | Alternate value of the ind... |
| indf 33570 | An indicator function as a... |
| indfval 33571 | Value of the indicator fun... |
| ind1 33572 | Value of the indicator fun... |
| ind0 33573 | Value of the indicator fun... |
| ind1a 33574 | Value of the indicator fun... |
| indpi1 33575 | Preimage of the singleton ... |
| indsum 33576 | Finite sum of a product wi... |
| indsumin 33577 | Finite sum of a product wi... |
| prodindf 33578 | The product of indicators ... |
| indf1o 33579 | The bijection between a po... |
| indpreima 33580 | A function with range ` { ... |
| indf1ofs 33581 | The bijection between fini... |
| esumex 33584 | An extended sum is a set b... |
| esumcl 33585 | Closure for extended sum i... |
| esumeq12dvaf 33586 | Equality deduction for ext... |
| esumeq12dva 33587 | Equality deduction for ext... |
| esumeq12d 33588 | Equality deduction for ext... |
| esumeq1 33589 | Equality theorem for an ex... |
| esumeq1d 33590 | Equality theorem for an ex... |
| esumeq2 33591 | Equality theorem for exten... |
| esumeq2d 33592 | Equality deduction for ext... |
| esumeq2dv 33593 | Equality deduction for ext... |
| esumeq2sdv 33594 | Equality deduction for ext... |
| nfesum1 33595 | Bound-variable hypothesis ... |
| nfesum2 33596 | Bound-variable hypothesis ... |
| cbvesum 33597 | Change bound variable in a... |
| cbvesumv 33598 | Change bound variable in a... |
| esumid 33599 | Identify the extended sum ... |
| esumgsum 33600 | A finite extended sum is t... |
| esumval 33601 | Develop the value of the e... |
| esumel 33602 | The extended sum is a limi... |
| esumnul 33603 | Extended sum over the empt... |
| esum0 33604 | Extended sum of zero. (Co... |
| esumf1o 33605 | Re-index an extended sum u... |
| esumc 33606 | Convert from the collectio... |
| esumrnmpt 33607 | Rewrite an extended sum in... |
| esumsplit 33608 | Split an extended sum into... |
| esummono 33609 | Extended sum is monotonic.... |
| esumpad 33610 | Extend an extended sum by ... |
| esumpad2 33611 | Remove zeroes from an exte... |
| esumadd 33612 | Addition of infinite sums.... |
| esumle 33613 | If all of the terms of an ... |
| gsumesum 33614 | Relate a group sum on ` ( ... |
| esumlub 33615 | The extended sum is the lo... |
| esumaddf 33616 | Addition of infinite sums.... |
| esumlef 33617 | If all of the terms of an ... |
| esumcst 33618 | The extended sum of a cons... |
| esumsnf 33619 | The extended sum of a sing... |
| esumsn 33620 | The extended sum of a sing... |
| esumpr 33621 | Extended sum over a pair. ... |
| esumpr2 33622 | Extended sum over a pair, ... |
| esumrnmpt2 33623 | Rewrite an extended sum in... |
| esumfzf 33624 | Formulating a partial exte... |
| esumfsup 33625 | Formulating an extended su... |
| esumfsupre 33626 | Formulating an extended su... |
| esumss 33627 | Change the index set to a ... |
| esumpinfval 33628 | The value of the extended ... |
| esumpfinvallem 33629 | Lemma for ~ esumpfinval . ... |
| esumpfinval 33630 | The value of the extended ... |
| esumpfinvalf 33631 | Same as ~ esumpfinval , mi... |
| esumpinfsum 33632 | The value of the extended ... |
| esumpcvgval 33633 | The value of the extended ... |
| esumpmono 33634 | The partial sums in an ext... |
| esumcocn 33635 | Lemma for ~ esummulc2 and ... |
| esummulc1 33636 | An extended sum multiplied... |
| esummulc2 33637 | An extended sum multiplied... |
| esumdivc 33638 | An extended sum divided by... |
| hashf2 33639 | Lemma for ~ hasheuni . (C... |
| hasheuni 33640 | The cardinality of a disjo... |
| esumcvg 33641 | The sequence of partial su... |
| esumcvg2 33642 | Simpler version of ~ esumc... |
| esumcvgsum 33643 | The value of the extended ... |
| esumsup 33644 | Express an extended sum as... |
| esumgect 33645 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 33646 | All terms of a converging ... |
| esum2dlem 33647 | Lemma for ~ esum2d (finite... |
| esum2d 33648 | Write a double extended su... |
| esumiun 33649 | Sum over a nonnecessarily ... |
| ofceq 33652 | Equality theorem for funct... |
| ofcfval 33653 | Value of an operation appl... |
| ofcval 33654 | Evaluate a function/consta... |
| ofcfn 33655 | The function operation pro... |
| ofcfeqd2 33656 | Equality theorem for funct... |
| ofcfval3 33657 | General value of ` ( F oFC... |
| ofcf 33658 | The function/constant oper... |
| ofcfval2 33659 | The function operation exp... |
| ofcfval4 33660 | The function/constant oper... |
| ofcc 33661 | Left operation by a consta... |
| ofcof 33662 | Relate function operation ... |
| sigaex 33665 | Lemma for ~ issiga and ~ i... |
| sigaval 33666 | The set of sigma-algebra w... |
| issiga 33667 | An alternative definition ... |
| isrnsiga 33668 | The property of being a si... |
| 0elsiga 33669 | A sigma-algebra contains t... |
| baselsiga 33670 | A sigma-algebra contains i... |
| sigasspw 33671 | A sigma-algebra is a set o... |
| sigaclcu 33672 | A sigma-algebra is closed ... |
| sigaclcuni 33673 | A sigma-algebra is closed ... |
| sigaclfu 33674 | A sigma-algebra is closed ... |
| sigaclcu2 33675 | A sigma-algebra is closed ... |
| sigaclfu2 33676 | A sigma-algebra is closed ... |
| sigaclcu3 33677 | A sigma-algebra is closed ... |
| issgon 33678 | Property of being a sigma-... |
| sgon 33679 | A sigma-algebra is a sigma... |
| elsigass 33680 | An element of a sigma-alge... |
| elrnsiga 33681 | Dropping the base informat... |
| isrnsigau 33682 | The property of being a si... |
| unielsiga 33683 | A sigma-algebra contains i... |
| dmvlsiga 33684 | Lebesgue-measurable subset... |
| pwsiga 33685 | Any power set forms a sigm... |
| prsiga 33686 | The smallest possible sigm... |
| sigaclci 33687 | A sigma-algebra is closed ... |
| difelsiga 33688 | A sigma-algebra is closed ... |
| unelsiga 33689 | A sigma-algebra is closed ... |
| inelsiga 33690 | A sigma-algebra is closed ... |
| sigainb 33691 | Building a sigma-algebra f... |
| insiga 33692 | The intersection of a coll... |
| sigagenval 33695 | Value of the generated sig... |
| sigagensiga 33696 | A generated sigma-algebra ... |
| sgsiga 33697 | A generated sigma-algebra ... |
| unisg 33698 | The sigma-algebra generate... |
| dmsigagen 33699 | A sigma-algebra can be gen... |
| sssigagen 33700 | A set is a subset of the s... |
| sssigagen2 33701 | A subset of the generating... |
| elsigagen 33702 | Any element of a set is al... |
| elsigagen2 33703 | Any countable union of ele... |
| sigagenss 33704 | The generated sigma-algebr... |
| sigagenss2 33705 | Sufficient condition for i... |
| sigagenid 33706 | The sigma-algebra generate... |
| ispisys 33707 | The property of being a pi... |
| ispisys2 33708 | The property of being a pi... |
| inelpisys 33709 | Pi-systems are closed unde... |
| sigapisys 33710 | All sigma-algebras are pi-... |
| isldsys 33711 | The property of being a la... |
| pwldsys 33712 | The power set of the unive... |
| unelldsys 33713 | Lambda-systems are closed ... |
| sigaldsys 33714 | All sigma-algebras are lam... |
| ldsysgenld 33715 | The intersection of all la... |
| sigapildsyslem 33716 | Lemma for ~ sigapildsys . ... |
| sigapildsys 33717 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 33718 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 33719 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 33720 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 33721 | The lambda system ` E ` ge... |
| dynkin 33722 | Dynkin's lambda-pi theorem... |
| isros 33723 | The property of being a ri... |
| rossspw 33724 | A ring of sets is a collec... |
| 0elros 33725 | A ring of sets contains th... |
| unelros 33726 | A ring of sets is closed u... |
| difelros 33727 | A ring of sets is closed u... |
| inelros 33728 | A ring of sets is closed u... |
| fiunelros 33729 | A ring of sets is closed u... |
| issros 33730 | The property of being a se... |
| srossspw 33731 | A semiring of sets is a co... |
| 0elsros 33732 | A semiring of sets contain... |
| inelsros 33733 | A semiring of sets is clos... |
| diffiunisros 33734 | In semiring of sets, compl... |
| rossros 33735 | Rings of sets are semiring... |
| brsiga 33738 | The Borel Algebra on real ... |
| brsigarn 33739 | The Borel Algebra is a sig... |
| brsigasspwrn 33740 | The Borel Algebra is a set... |
| unibrsiga 33741 | The union of the Borel Alg... |
| cldssbrsiga 33742 | A Borel Algebra contains a... |
| sxval 33745 | Value of the product sigma... |
| sxsiga 33746 | A product sigma-algebra is... |
| sxsigon 33747 | A product sigma-algebra is... |
| sxuni 33748 | The base set of a product ... |
| elsx 33749 | The cartesian product of t... |
| measbase 33752 | The base set of a measure ... |
| measval 33753 | The value of the ` measure... |
| ismeas 33754 | The property of being a me... |
| isrnmeas 33755 | The property of being a me... |
| dmmeas 33756 | The domain of a measure is... |
| measbasedom 33757 | The base set of a measure ... |
| measfrge0 33758 | A measure is a function ov... |
| measfn 33759 | A measure is a function on... |
| measvxrge0 33760 | The values of a measure ar... |
| measvnul 33761 | The measure of the empty s... |
| measge0 33762 | A measure is nonnegative. ... |
| measle0 33763 | If the measure of a given ... |
| measvun 33764 | The measure of a countable... |
| measxun2 33765 | The measure the union of t... |
| measun 33766 | The measure the union of t... |
| measvunilem 33767 | Lemma for ~ measvuni . (C... |
| measvunilem0 33768 | Lemma for ~ measvuni . (C... |
| measvuni 33769 | The measure of a countable... |
| measssd 33770 | A measure is monotone with... |
| measunl 33771 | A measure is sub-additive ... |
| measiuns 33772 | The measure of the union o... |
| measiun 33773 | A measure is sub-additive.... |
| meascnbl 33774 | A measure is continuous fr... |
| measinblem 33775 | Lemma for ~ measinb . (Co... |
| measinb 33776 | Building a measure restric... |
| measres 33777 | Building a measure restric... |
| measinb2 33778 | Building a measure restric... |
| measdivcst 33779 | Division of a measure by a... |
| measdivcstALTV 33780 | Alternate version of ~ mea... |
| cntmeas 33781 | The Counting measure is a ... |
| pwcntmeas 33782 | The counting measure is a ... |
| cntnevol 33783 | Counting and Lebesgue meas... |
| voliune 33784 | The Lebesgue measure funct... |
| volfiniune 33785 | The Lebesgue measure funct... |
| volmeas 33786 | The Lebesgue measure is a ... |
| ddeval1 33789 | Value of the delta measure... |
| ddeval0 33790 | Value of the delta measure... |
| ddemeas 33791 | The Dirac delta measure is... |
| relae 33795 | 'almost everywhere' is a r... |
| brae 33796 | 'almost everywhere' relati... |
| braew 33797 | 'almost everywhere' relati... |
| truae 33798 | A truth holds almost every... |
| aean 33799 | A conjunction holds almost... |
| faeval 33801 | Value of the 'almost every... |
| relfae 33802 | The 'almost everywhere' bu... |
| brfae 33803 | 'almost everywhere' relati... |
| ismbfm 33806 | The predicate " ` F ` is a... |
| elunirnmbfm 33807 | The property of being a me... |
| mbfmfun 33808 | A measurable function is a... |
| mbfmf 33809 | A measurable function as a... |
| isanmbfmOLD 33810 | Obsolete version of ~ isan... |
| mbfmcnvima 33811 | The preimage by a measurab... |
| isanmbfm 33812 | The predicate to be a meas... |
| mbfmbfmOLD 33813 | A measurable function to a... |
| mbfmbfm 33814 | A measurable function to a... |
| mbfmcst 33815 | A constant function is mea... |
| 1stmbfm 33816 | The first projection map i... |
| 2ndmbfm 33817 | The second projection map ... |
| imambfm 33818 | If the sigma-algebra in th... |
| cnmbfm 33819 | A continuous function is m... |
| mbfmco 33820 | The composition of two mea... |
| mbfmco2 33821 | The pair building of two m... |
| mbfmvolf 33822 | Measurable functions with ... |
| elmbfmvol2 33823 | Measurable functions with ... |
| mbfmcnt 33824 | All functions are measurab... |
| br2base 33825 | The base set for the gener... |
| dya2ub 33826 | An upper bound for a dyadi... |
| sxbrsigalem0 33827 | The closed half-spaces of ... |
| sxbrsigalem3 33828 | The sigma-algebra generate... |
| dya2iocival 33829 | The function ` I ` returns... |
| dya2iocress 33830 | Dyadic intervals are subse... |
| dya2iocbrsiga 33831 | Dyadic intervals are Borel... |
| dya2icobrsiga 33832 | Dyadic intervals are Borel... |
| dya2icoseg 33833 | For any point and any clos... |
| dya2icoseg2 33834 | For any point and any open... |
| dya2iocrfn 33835 | The function returning dya... |
| dya2iocct 33836 | The dyadic rectangle set i... |
| dya2iocnrect 33837 | For any point of an open r... |
| dya2iocnei 33838 | For any point of an open s... |
| dya2iocuni 33839 | Every open set of ` ( RR X... |
| dya2iocucvr 33840 | The dyadic rectangular set... |
| sxbrsigalem1 33841 | The Borel algebra on ` ( R... |
| sxbrsigalem2 33842 | The sigma-algebra generate... |
| sxbrsigalem4 33843 | The Borel algebra on ` ( R... |
| sxbrsigalem5 33844 | First direction for ~ sxbr... |
| sxbrsigalem6 33845 | First direction for ~ sxbr... |
| sxbrsiga 33846 | The product sigma-algebra ... |
| omsval 33849 | Value of the function mapp... |
| omsfval 33850 | Value of the outer measure... |
| omscl 33851 | A closure lemma for the co... |
| omsf 33852 | A constructed outer measur... |
| oms0 33853 | A constructed outer measur... |
| omsmon 33854 | A constructed outer measur... |
| omssubaddlem 33855 | For any small margin ` E `... |
| omssubadd 33856 | A constructed outer measur... |
| carsgval 33859 | Value of the Caratheodory ... |
| carsgcl 33860 | Closure of the Caratheodor... |
| elcarsg 33861 | Property of being a Carath... |
| baselcarsg 33862 | The universe set, ` O ` , ... |
| 0elcarsg 33863 | The empty set is Caratheod... |
| carsguni 33864 | The union of all Caratheod... |
| elcarsgss 33865 | Caratheodory measurable se... |
| difelcarsg 33866 | The Caratheodory measurabl... |
| inelcarsg 33867 | The Caratheodory measurabl... |
| unelcarsg 33868 | The Caratheodory-measurabl... |
| difelcarsg2 33869 | The Caratheodory-measurabl... |
| carsgmon 33870 | Utility lemma: Apply mono... |
| carsgsigalem 33871 | Lemma for the following th... |
| fiunelcarsg 33872 | The Caratheodory measurabl... |
| carsgclctunlem1 33873 | Lemma for ~ carsgclctun . ... |
| carsggect 33874 | The outer measure is count... |
| carsgclctunlem2 33875 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 33876 | Lemma for ~ carsgclctun . ... |
| carsgclctun 33877 | The Caratheodory measurabl... |
| carsgsiga 33878 | The Caratheodory measurabl... |
| omsmeas 33879 | The restriction of a const... |
| pmeasmono 33880 | This theorem's hypotheses ... |
| pmeasadd 33881 | A premeasure on a ring of ... |
| itgeq12dv 33882 | Equality theorem for an in... |
| sitgval 33888 | Value of the simple functi... |
| issibf 33889 | The predicate " ` F ` is a... |
| sibf0 33890 | The constant zero function... |
| sibfmbl 33891 | A simple function is measu... |
| sibff 33892 | A simple function is a fun... |
| sibfrn 33893 | A simple function has fini... |
| sibfima 33894 | Any preimage of a singleto... |
| sibfinima 33895 | The measure of the interse... |
| sibfof 33896 | Applying function operatio... |
| sitgfval 33897 | Value of the Bochner integ... |
| sitgclg 33898 | Closure of the Bochner int... |
| sitgclbn 33899 | Closure of the Bochner int... |
| sitgclcn 33900 | Closure of the Bochner int... |
| sitgclre 33901 | Closure of the Bochner int... |
| sitg0 33902 | The integral of the consta... |
| sitgf 33903 | The integral for simple fu... |
| sitgaddlemb 33904 | Lemma for * sitgadd . (Co... |
| sitmval 33905 | Value of the simple functi... |
| sitmfval 33906 | Value of the integral dist... |
| sitmcl 33907 | Closure of the integral di... |
| sitmf 33908 | The integral metric as a f... |
| oddpwdc 33910 | Lemma for ~ eulerpart . T... |
| oddpwdcv 33911 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 33912 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 33913 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 33914 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 33915 | Lemma for ~ eulerpart . (... |
| eulerpartlems 33916 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 33917 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 33918 | Lemma for ~ eulerpart . (... |
| eulerpartleme 33919 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 33920 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 33921 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 33922 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 33923 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 33924 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 33925 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 33926 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 33927 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 33928 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 33929 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 33930 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 33931 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 33932 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 33933 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 33934 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 33935 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 33936 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 33937 | Lemma for ~ eulerpart . (... |
| eulerpart 33938 | Euler's theorem on partiti... |
| subiwrd 33941 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 33942 | Length of a subword of an ... |
| iwrdsplit 33943 | Lemma for ~ sseqp1 . (Con... |
| sseqval 33944 | Value of the strong sequen... |
| sseqfv1 33945 | Value of the strong sequen... |
| sseqfn 33946 | A strong recursive sequenc... |
| sseqmw 33947 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 33948 | A strong recursive sequenc... |
| sseqfres 33949 | The first elements in the ... |
| sseqfv2 33950 | Value of the strong sequen... |
| sseqp1 33951 | Value of the strong sequen... |
| fiblem 33954 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 33955 | Value of the Fibonacci seq... |
| fib1 33956 | Value of the Fibonacci seq... |
| fibp1 33957 | Value of the Fibonacci seq... |
| fib2 33958 | Value of the Fibonacci seq... |
| fib3 33959 | Value of the Fibonacci seq... |
| fib4 33960 | Value of the Fibonacci seq... |
| fib5 33961 | Value of the Fibonacci seq... |
| fib6 33962 | Value of the Fibonacci seq... |
| elprob 33965 | The property of being a pr... |
| domprobmeas 33966 | A probability measure is a... |
| domprobsiga 33967 | The domain of a probabilit... |
| probtot 33968 | The probability of the uni... |
| prob01 33969 | A probability is an elemen... |
| probnul 33970 | The probability of the emp... |
| unveldomd 33971 | The universe is an element... |
| unveldom 33972 | The universe is an element... |
| nuleldmp 33973 | The empty set is an elemen... |
| probcun 33974 | The probability of the uni... |
| probun 33975 | The probability of the uni... |
| probdif 33976 | The probability of the dif... |
| probinc 33977 | A probability law is incre... |
| probdsb 33978 | The probability of the com... |
| probmeasd 33979 | A probability measure is a... |
| probvalrnd 33980 | The value of a probability... |
| probtotrnd 33981 | The probability of the uni... |
| totprobd 33982 | Law of total probability, ... |
| totprob 33983 | Law of total probability. ... |
| probfinmeasb 33984 | Build a probability measur... |
| probfinmeasbALTV 33985 | Alternate version of ~ pro... |
| probmeasb 33986 | Build a probability from a... |
| cndprobval 33989 | The value of the condition... |
| cndprobin 33990 | An identity linking condit... |
| cndprob01 33991 | The conditional probabilit... |
| cndprobtot 33992 | The conditional probabilit... |
| cndprobnul 33993 | The conditional probabilit... |
| cndprobprob 33994 | The conditional probabilit... |
| bayesth 33995 | Bayes Theorem. (Contribut... |
| rrvmbfm 33998 | A real-valued random varia... |
| isrrvv 33999 | Elementhood to the set of ... |
| rrvvf 34000 | A real-valued random varia... |
| rrvfn 34001 | A real-valued random varia... |
| rrvdm 34002 | The domain of a random var... |
| rrvrnss 34003 | The range of a random vari... |
| rrvf2 34004 | A real-valued random varia... |
| rrvdmss 34005 | The domain of a random var... |
| rrvfinvima 34006 | For a real-value random va... |
| 0rrv 34007 | The constant function equa... |
| rrvadd 34008 | The sum of two random vari... |
| rrvmulc 34009 | A random variable multipli... |
| rrvsum 34010 | An indexed sum of random v... |
| orvcval 34013 | Value of the preimage mapp... |
| orvcval2 34014 | Another way to express the... |
| elorvc 34015 | Elementhood of a preimage.... |
| orvcval4 34016 | The value of the preimage ... |
| orvcoel 34017 | If the relation produces o... |
| orvccel 34018 | If the relation produces c... |
| elorrvc 34019 | Elementhood of a preimage ... |
| orrvcval4 34020 | The value of the preimage ... |
| orrvcoel 34021 | If the relation produces o... |
| orrvccel 34022 | If the relation produces c... |
| orvcgteel 34023 | Preimage maps produced by ... |
| orvcelval 34024 | Preimage maps produced by ... |
| orvcelel 34025 | Preimage maps produced by ... |
| dstrvval 34026 | The value of the distribut... |
| dstrvprob 34027 | The distribution of a rand... |
| orvclteel 34028 | Preimage maps produced by ... |
| dstfrvel 34029 | Elementhood of preimage ma... |
| dstfrvunirn 34030 | The limit of all preimage ... |
| orvclteinc 34031 | Preimage maps produced by ... |
| dstfrvinc 34032 | A cumulative distribution ... |
| dstfrvclim1 34033 | The limit of the cumulativ... |
| coinfliplem 34034 | Division in the extended r... |
| coinflipprob 34035 | The ` P ` we defined for c... |
| coinflipspace 34036 | The space of our coin-flip... |
| coinflipuniv 34037 | The universe of our coin-f... |
| coinfliprv 34038 | The ` X ` we defined for c... |
| coinflippv 34039 | The probability of heads i... |
| coinflippvt 34040 | The probability of tails i... |
| ballotlemoex 34041 | ` O ` is a set. (Contribu... |
| ballotlem1 34042 | The size of the universe i... |
| ballotlemelo 34043 | Elementhood in ` O ` . (C... |
| ballotlem2 34044 | The probability that the f... |
| ballotlemfval 34045 | The value of ` F ` . (Con... |
| ballotlemfelz 34046 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 34047 | If the ` J ` th ballot is ... |
| ballotlemfc0 34048 | ` F ` takes value 0 betwee... |
| ballotlemfcc 34049 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 34050 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 34051 | ` ( F `` C ) ` always star... |
| ballotleme 34052 | Elements of ` E ` . (Cont... |
| ballotlemodife 34053 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 34054 | If the first pick is a vot... |
| ballotlem5 34055 | If A is not ahead througho... |
| ballotlemi 34056 | Value of ` I ` for a given... |
| ballotlemiex 34057 | Properties of ` ( I `` C )... |
| ballotlemi1 34058 | The first tie cannot be re... |
| ballotlemii 34059 | The first tie cannot be re... |
| ballotlemsup 34060 | The set of zeroes of ` F `... |
| ballotlemimin 34061 | ` ( I `` C ) ` is the firs... |
| ballotlemic 34062 | If the first vote is for B... |
| ballotlem1c 34063 | If the first vote is for A... |
| ballotlemsval 34064 | Value of ` S ` . (Contrib... |
| ballotlemsv 34065 | Value of ` S ` evaluated a... |
| ballotlemsgt1 34066 | ` S ` maps values less tha... |
| ballotlemsdom 34067 | Domain of ` S ` for a give... |
| ballotlemsel1i 34068 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 34069 | The defined ` S ` is a bij... |
| ballotlemsi 34070 | The image by ` S ` of the ... |
| ballotlemsima 34071 | The image by ` S ` of an i... |
| ballotlemieq 34072 | If two countings share the... |
| ballotlemrval 34073 | Value of ` R ` . (Contrib... |
| ballotlemscr 34074 | The image of ` ( R `` C ) ... |
| ballotlemrv 34075 | Value of ` R ` evaluated a... |
| ballotlemrv1 34076 | Value of ` R ` before the ... |
| ballotlemrv2 34077 | Value of ` R ` after the t... |
| ballotlemro 34078 | Range of ` R ` is included... |
| ballotlemgval 34079 | Expand the value of ` .^ `... |
| ballotlemgun 34080 | A property of the defined ... |
| ballotlemfg 34081 | Express the value of ` ( F... |
| ballotlemfrc 34082 | Express the value of ` ( F... |
| ballotlemfrci 34083 | Reverse counting preserves... |
| ballotlemfrceq 34084 | Value of ` F ` for a rever... |
| ballotlemfrcn0 34085 | Value of ` F ` for a rever... |
| ballotlemrc 34086 | Range of ` R ` . (Contrib... |
| ballotlemirc 34087 | Applying ` R ` does not ch... |
| ballotlemrinv0 34088 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 34089 | ` R ` is its own inverse :... |
| ballotlem1ri 34090 | When the vote on the first... |
| ballotlem7 34091 | ` R ` is a bijection betwe... |
| ballotlem8 34092 | There are as many counting... |
| ballotth 34093 | Bertrand's ballot problem ... |
| sgncl 34094 | Closure of the signum. (C... |
| sgnclre 34095 | Closure of the signum. (C... |
| sgnneg 34096 | Negation of the signum. (... |
| sgn3da 34097 | A conditional containing a... |
| sgnmul 34098 | Signum of a product. (Con... |
| sgnmulrp2 34099 | Multiplication by a positi... |
| sgnsub 34100 | Subtraction of a number of... |
| sgnnbi 34101 | Negative signum. (Contrib... |
| sgnpbi 34102 | Positive signum. (Contrib... |
| sgn0bi 34103 | Zero signum. (Contributed... |
| sgnsgn 34104 | Signum is idempotent. (Co... |
| sgnmulsgn 34105 | If two real numbers are of... |
| sgnmulsgp 34106 | If two real numbers are of... |
| fzssfzo 34107 | Condition for an integer i... |
| gsumncl 34108 | Closure of a group sum in ... |
| gsumnunsn 34109 | Closure of a group sum in ... |
| ccatmulgnn0dir 34110 | Concatenation of words fol... |
| ofcccat 34111 | Letterwise operations on w... |
| ofcs1 34112 | Letterwise operations on a... |
| ofcs2 34113 | Letterwise operations on a... |
| plymul02 34114 | Product of a polynomial wi... |
| plymulx0 34115 | Coefficients of a polynomi... |
| plymulx 34116 | Coefficients of a polynomi... |
| plyrecld 34117 | Closure of a polynomial wi... |
| signsplypnf 34118 | The quotient of a polynomi... |
| signsply0 34119 | Lemma for the rule of sign... |
| signspval 34120 | The value of the skipping ... |
| signsw0glem 34121 | Neutral element property o... |
| signswbase 34122 | The base of ` W ` is the u... |
| signswplusg 34123 | The operation of ` W ` . ... |
| signsw0g 34124 | The neutral element of ` W... |
| signswmnd 34125 | ` W ` is a monoid structur... |
| signswrid 34126 | The zero-skipping operatio... |
| signswlid 34127 | The zero-skipping operatio... |
| signswn0 34128 | The zero-skipping operatio... |
| signswch 34129 | The zero-skipping operatio... |
| signslema 34130 | Computational part of ~~? ... |
| signstfv 34131 | Value of the zero-skipping... |
| signstfval 34132 | Value of the zero-skipping... |
| signstcl 34133 | Closure of the zero skippi... |
| signstf 34134 | The zero skipping sign wor... |
| signstlen 34135 | Length of the zero skippin... |
| signstf0 34136 | Sign of a single letter wo... |
| signstfvn 34137 | Zero-skipping sign in a wo... |
| signsvtn0 34138 | If the last letter is nonz... |
| signstfvp 34139 | Zero-skipping sign in a wo... |
| signstfvneq0 34140 | In case the first letter i... |
| signstfvcl 34141 | Closure of the zero skippi... |
| signstfvc 34142 | Zero-skipping sign in a wo... |
| signstres 34143 | Restriction of a zero skip... |
| signstfveq0a 34144 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 34145 | In case the last letter is... |
| signsvvfval 34146 | The value of ` V ` , which... |
| signsvvf 34147 | ` V ` is a function. (Con... |
| signsvf0 34148 | There is no change of sign... |
| signsvf1 34149 | In a single-letter word, w... |
| signsvfn 34150 | Number of changes in a wor... |
| signsvtp 34151 | Adding a letter of the sam... |
| signsvtn 34152 | Adding a letter of a diffe... |
| signsvfpn 34153 | Adding a letter of the sam... |
| signsvfnn 34154 | Adding a letter of a diffe... |
| signlem0 34155 | Adding a zero as the highe... |
| signshf 34156 | ` H ` , corresponding to t... |
| signshwrd 34157 | ` H ` , corresponding to t... |
| signshlen 34158 | Length of ` H ` , correspo... |
| signshnz 34159 | ` H ` is not the empty wor... |
| iblidicc 34160 | The identity function is i... |
| rpsqrtcn 34161 | Continuity of the real pos... |
| divsqrtid 34162 | A real number divided by i... |
| cxpcncf1 34163 | The power function on comp... |
| efmul2picn 34164 | Multiplying by ` ( _i x. (... |
| fct2relem 34165 | Lemma for ~ ftc2re . (Con... |
| ftc2re 34166 | The Fundamental Theorem of... |
| fdvposlt 34167 | Functions with a positive ... |
| fdvneggt 34168 | Functions with a negative ... |
| fdvposle 34169 | Functions with a nonnegati... |
| fdvnegge 34170 | Functions with a nonpositi... |
| prodfzo03 34171 | A product of three factors... |
| actfunsnf1o 34172 | The action ` F ` of extend... |
| actfunsnrndisj 34173 | The action ` F ` of extend... |
| itgexpif 34174 | The basis for the circle m... |
| fsum2dsub 34175 | Lemma for ~ breprexp - Re-... |
| reprval 34178 | Value of the representatio... |
| repr0 34179 | There is exactly one repre... |
| reprf 34180 | Members of the representat... |
| reprsum 34181 | Sums of values of the memb... |
| reprle 34182 | Upper bound to the terms i... |
| reprsuc 34183 | Express the representation... |
| reprfi 34184 | Bounded representations ar... |
| reprss 34185 | Representations with terms... |
| reprinrn 34186 | Representations with term ... |
| reprlt 34187 | There are no representatio... |
| hashreprin 34188 | Express a sum of represent... |
| reprgt 34189 | There are no representatio... |
| reprinfz1 34190 | For the representation of ... |
| reprfi2 34191 | Corollary of ~ reprinfz1 .... |
| reprfz1 34192 | Corollary of ~ reprinfz1 .... |
| hashrepr 34193 | Develop the number of repr... |
| reprpmtf1o 34194 | Transposing ` 0 ` and ` X ... |
| reprdifc 34195 | Express the representation... |
| chpvalz 34196 | Value of the second Chebys... |
| chtvalz 34197 | Value of the Chebyshev fun... |
| breprexplema 34198 | Lemma for ~ breprexp (indu... |
| breprexplemb 34199 | Lemma for ~ breprexp (clos... |
| breprexplemc 34200 | Lemma for ~ breprexp (indu... |
| breprexp 34201 | Express the ` S ` th power... |
| breprexpnat 34202 | Express the ` S ` th power... |
| vtsval 34205 | Value of the Vinogradov tr... |
| vtscl 34206 | Closure of the Vinogradov ... |
| vtsprod 34207 | Express the Vinogradov tri... |
| circlemeth 34208 | The Hardy, Littlewood and ... |
| circlemethnat 34209 | The Hardy, Littlewood and ... |
| circlevma 34210 | The Circle Method, where t... |
| circlemethhgt 34211 | The circle method, where t... |
| hgt750lemc 34215 | An upper bound to the summ... |
| hgt750lemd 34216 | An upper bound to the summ... |
| hgt749d 34217 | A deduction version of ~ a... |
| logdivsqrle 34218 | Conditions for ` ( ( log `... |
| hgt750lem 34219 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 34220 | Decimal multiplication gal... |
| hgt750lemf 34221 | Lemma for the statement 7.... |
| hgt750lemg 34222 | Lemma for the statement 7.... |
| oddprm2 34223 | Two ways to write the set ... |
| hgt750lemb 34224 | An upper bound on the cont... |
| hgt750lema 34225 | An upper bound on the cont... |
| hgt750leme 34226 | An upper bound on the cont... |
| tgoldbachgnn 34227 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 34228 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 34229 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 34230 | Odd integers greater than ... |
| tgoldbachgt 34231 | Odd integers greater than ... |
| istrkg2d 34234 | Property of fulfilling dim... |
| axtglowdim2ALTV 34235 | Alternate version of ~ axt... |
| axtgupdim2ALTV 34236 | Alternate version of ~ axt... |
| afsval 34239 | Value of the AFS relation ... |
| brafs 34240 | Binary relation form of th... |
| tg5segofs 34241 | Rephrase ~ axtg5seg using ... |
| lpadval 34244 | Value of the ` leftpad ` f... |
| lpadlem1 34245 | Lemma for the ` leftpad ` ... |
| lpadlem3 34246 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 34247 | Length of a left-padded wo... |
| lpadlem2 34248 | Lemma for the ` leftpad ` ... |
| lpadlen2 34249 | Length of a left-padded wo... |
| lpadmax 34250 | Length of a left-padded wo... |
| lpadleft 34251 | The contents of prefix of ... |
| lpadright 34252 | The suffix of a left-padde... |
| bnj170 34265 | ` /\ ` -manipulation. (Co... |
| bnj240 34266 | ` /\ ` -manipulation. (Co... |
| bnj248 34267 | ` /\ ` -manipulation. (Co... |
| bnj250 34268 | ` /\ ` -manipulation. (Co... |
| bnj251 34269 | ` /\ ` -manipulation. (Co... |
| bnj252 34270 | ` /\ ` -manipulation. (Co... |
| bnj253 34271 | ` /\ ` -manipulation. (Co... |
| bnj255 34272 | ` /\ ` -manipulation. (Co... |
| bnj256 34273 | ` /\ ` -manipulation. (Co... |
| bnj257 34274 | ` /\ ` -manipulation. (Co... |
| bnj258 34275 | ` /\ ` -manipulation. (Co... |
| bnj268 34276 | ` /\ ` -manipulation. (Co... |
| bnj290 34277 | ` /\ ` -manipulation. (Co... |
| bnj291 34278 | ` /\ ` -manipulation. (Co... |
| bnj312 34279 | ` /\ ` -manipulation. (Co... |
| bnj334 34280 | ` /\ ` -manipulation. (Co... |
| bnj345 34281 | ` /\ ` -manipulation. (Co... |
| bnj422 34282 | ` /\ ` -manipulation. (Co... |
| bnj432 34283 | ` /\ ` -manipulation. (Co... |
| bnj446 34284 | ` /\ ` -manipulation. (Co... |
| bnj23 34285 | First-order logic and set ... |
| bnj31 34286 | First-order logic and set ... |
| bnj62 34287 | First-order logic and set ... |
| bnj89 34288 | First-order logic and set ... |
| bnj90 34289 | First-order logic and set ... |
| bnj101 34290 | First-order logic and set ... |
| bnj105 34291 | First-order logic and set ... |
| bnj115 34292 | First-order logic and set ... |
| bnj132 34293 | First-order logic and set ... |
| bnj133 34294 | First-order logic and set ... |
| bnj156 34295 | First-order logic and set ... |
| bnj158 34296 | First-order logic and set ... |
| bnj168 34297 | First-order logic and set ... |
| bnj206 34298 | First-order logic and set ... |
| bnj216 34299 | First-order logic and set ... |
| bnj219 34300 | First-order logic and set ... |
| bnj226 34301 | First-order logic and set ... |
| bnj228 34302 | First-order logic and set ... |
| bnj519 34303 | First-order logic and set ... |
| bnj524 34304 | First-order logic and set ... |
| bnj525 34305 | First-order logic and set ... |
| bnj534 34306 | First-order logic and set ... |
| bnj538 34307 | First-order logic and set ... |
| bnj529 34308 | First-order logic and set ... |
| bnj551 34309 | First-order logic and set ... |
| bnj563 34310 | First-order logic and set ... |
| bnj564 34311 | First-order logic and set ... |
| bnj593 34312 | First-order logic and set ... |
| bnj596 34313 | First-order logic and set ... |
| bnj610 34314 | Pass from equality ( ` x =... |
| bnj642 34315 | ` /\ ` -manipulation. (Co... |
| bnj643 34316 | ` /\ ` -manipulation. (Co... |
| bnj645 34317 | ` /\ ` -manipulation. (Co... |
| bnj658 34318 | ` /\ ` -manipulation. (Co... |
| bnj667 34319 | ` /\ ` -manipulation. (Co... |
| bnj705 34320 | ` /\ ` -manipulation. (Co... |
| bnj706 34321 | ` /\ ` -manipulation. (Co... |
| bnj707 34322 | ` /\ ` -manipulation. (Co... |
| bnj708 34323 | ` /\ ` -manipulation. (Co... |
| bnj721 34324 | ` /\ ` -manipulation. (Co... |
| bnj832 34325 | ` /\ ` -manipulation. (Co... |
| bnj835 34326 | ` /\ ` -manipulation. (Co... |
| bnj836 34327 | ` /\ ` -manipulation. (Co... |
| bnj837 34328 | ` /\ ` -manipulation. (Co... |
| bnj769 34329 | ` /\ ` -manipulation. (Co... |
| bnj770 34330 | ` /\ ` -manipulation. (Co... |
| bnj771 34331 | ` /\ ` -manipulation. (Co... |
| bnj887 34332 | ` /\ ` -manipulation. (Co... |
| bnj918 34333 | First-order logic and set ... |
| bnj919 34334 | First-order logic and set ... |
| bnj923 34335 | First-order logic and set ... |
| bnj927 34336 | First-order logic and set ... |
| bnj931 34337 | First-order logic and set ... |
| bnj937 34338 | First-order logic and set ... |
| bnj941 34339 | First-order logic and set ... |
| bnj945 34340 | Technical lemma for ~ bnj6... |
| bnj946 34341 | First-order logic and set ... |
| bnj951 34342 | ` /\ ` -manipulation. (Co... |
| bnj956 34343 | First-order logic and set ... |
| bnj976 34344 | First-order logic and set ... |
| bnj982 34345 | First-order logic and set ... |
| bnj1019 34346 | First-order logic and set ... |
| bnj1023 34347 | First-order logic and set ... |
| bnj1095 34348 | First-order logic and set ... |
| bnj1096 34349 | First-order logic and set ... |
| bnj1098 34350 | First-order logic and set ... |
| bnj1101 34351 | First-order logic and set ... |
| bnj1113 34352 | First-order logic and set ... |
| bnj1109 34353 | First-order logic and set ... |
| bnj1131 34354 | First-order logic and set ... |
| bnj1138 34355 | First-order logic and set ... |
| bnj1142 34356 | First-order logic and set ... |
| bnj1143 34357 | First-order logic and set ... |
| bnj1146 34358 | First-order logic and set ... |
| bnj1149 34359 | First-order logic and set ... |
| bnj1185 34360 | First-order logic and set ... |
| bnj1196 34361 | First-order logic and set ... |
| bnj1198 34362 | First-order logic and set ... |
| bnj1209 34363 | First-order logic and set ... |
| bnj1211 34364 | First-order logic and set ... |
| bnj1213 34365 | First-order logic and set ... |
| bnj1212 34366 | First-order logic and set ... |
| bnj1219 34367 | First-order logic and set ... |
| bnj1224 34368 | First-order logic and set ... |
| bnj1230 34369 | First-order logic and set ... |
| bnj1232 34370 | First-order logic and set ... |
| bnj1235 34371 | First-order logic and set ... |
| bnj1239 34372 | First-order logic and set ... |
| bnj1238 34373 | First-order logic and set ... |
| bnj1241 34374 | First-order logic and set ... |
| bnj1247 34375 | First-order logic and set ... |
| bnj1254 34376 | First-order logic and set ... |
| bnj1262 34377 | First-order logic and set ... |
| bnj1266 34378 | First-order logic and set ... |
| bnj1265 34379 | First-order logic and set ... |
| bnj1275 34380 | First-order logic and set ... |
| bnj1276 34381 | First-order logic and set ... |
| bnj1292 34382 | First-order logic and set ... |
| bnj1293 34383 | First-order logic and set ... |
| bnj1294 34384 | First-order logic and set ... |
| bnj1299 34385 | First-order logic and set ... |
| bnj1304 34386 | First-order logic and set ... |
| bnj1316 34387 | First-order logic and set ... |
| bnj1317 34388 | First-order logic and set ... |
| bnj1322 34389 | First-order logic and set ... |
| bnj1340 34390 | First-order logic and set ... |
| bnj1345 34391 | First-order logic and set ... |
| bnj1350 34392 | First-order logic and set ... |
| bnj1351 34393 | First-order logic and set ... |
| bnj1352 34394 | First-order logic and set ... |
| bnj1361 34395 | First-order logic and set ... |
| bnj1366 34396 | First-order logic and set ... |
| bnj1379 34397 | First-order logic and set ... |
| bnj1383 34398 | First-order logic and set ... |
| bnj1385 34399 | First-order logic and set ... |
| bnj1386 34400 | First-order logic and set ... |
| bnj1397 34401 | First-order logic and set ... |
| bnj1400 34402 | First-order logic and set ... |
| bnj1405 34403 | First-order logic and set ... |
| bnj1422 34404 | First-order logic and set ... |
| bnj1424 34405 | First-order logic and set ... |
| bnj1436 34406 | First-order logic and set ... |
| bnj1441 34407 | First-order logic and set ... |
| bnj1441g 34408 | First-order logic and set ... |
| bnj1454 34409 | First-order logic and set ... |
| bnj1459 34410 | First-order logic and set ... |
| bnj1464 34411 | Conversion of implicit sub... |
| bnj1465 34412 | First-order logic and set ... |
| bnj1468 34413 | Conversion of implicit sub... |
| bnj1476 34414 | First-order logic and set ... |
| bnj1502 34415 | First-order logic and set ... |
| bnj1503 34416 | First-order logic and set ... |
| bnj1517 34417 | First-order logic and set ... |
| bnj1521 34418 | First-order logic and set ... |
| bnj1533 34419 | First-order logic and set ... |
| bnj1534 34420 | First-order logic and set ... |
| bnj1536 34421 | First-order logic and set ... |
| bnj1538 34422 | First-order logic and set ... |
| bnj1541 34423 | First-order logic and set ... |
| bnj1542 34424 | First-order logic and set ... |
| bnj110 34425 | Well-founded induction res... |
| bnj157 34426 | Well-founded induction res... |
| bnj66 34427 | Technical lemma for ~ bnj6... |
| bnj91 34428 | First-order logic and set ... |
| bnj92 34429 | First-order logic and set ... |
| bnj93 34430 | Technical lemma for ~ bnj9... |
| bnj95 34431 | Technical lemma for ~ bnj1... |
| bnj96 34432 | Technical lemma for ~ bnj1... |
| bnj97 34433 | Technical lemma for ~ bnj1... |
| bnj98 34434 | Technical lemma for ~ bnj1... |
| bnj106 34435 | First-order logic and set ... |
| bnj118 34436 | First-order logic and set ... |
| bnj121 34437 | First-order logic and set ... |
| bnj124 34438 | Technical lemma for ~ bnj1... |
| bnj125 34439 | Technical lemma for ~ bnj1... |
| bnj126 34440 | Technical lemma for ~ bnj1... |
| bnj130 34441 | Technical lemma for ~ bnj1... |
| bnj149 34442 | Technical lemma for ~ bnj1... |
| bnj150 34443 | Technical lemma for ~ bnj1... |
| bnj151 34444 | Technical lemma for ~ bnj1... |
| bnj154 34445 | Technical lemma for ~ bnj1... |
| bnj155 34446 | Technical lemma for ~ bnj1... |
| bnj153 34447 | Technical lemma for ~ bnj8... |
| bnj207 34448 | Technical lemma for ~ bnj8... |
| bnj213 34449 | First-order logic and set ... |
| bnj222 34450 | Technical lemma for ~ bnj2... |
| bnj229 34451 | Technical lemma for ~ bnj5... |
| bnj517 34452 | Technical lemma for ~ bnj5... |
| bnj518 34453 | Technical lemma for ~ bnj8... |
| bnj523 34454 | Technical lemma for ~ bnj8... |
| bnj526 34455 | Technical lemma for ~ bnj8... |
| bnj528 34456 | Technical lemma for ~ bnj8... |
| bnj535 34457 | Technical lemma for ~ bnj8... |
| bnj539 34458 | Technical lemma for ~ bnj8... |
| bnj540 34459 | Technical lemma for ~ bnj8... |
| bnj543 34460 | Technical lemma for ~ bnj8... |
| bnj544 34461 | Technical lemma for ~ bnj8... |
| bnj545 34462 | Technical lemma for ~ bnj8... |
| bnj546 34463 | Technical lemma for ~ bnj8... |
| bnj548 34464 | Technical lemma for ~ bnj8... |
| bnj553 34465 | Technical lemma for ~ bnj8... |
| bnj554 34466 | Technical lemma for ~ bnj8... |
| bnj556 34467 | Technical lemma for ~ bnj8... |
| bnj557 34468 | Technical lemma for ~ bnj8... |
| bnj558 34469 | Technical lemma for ~ bnj8... |
| bnj561 34470 | Technical lemma for ~ bnj8... |
| bnj562 34471 | Technical lemma for ~ bnj8... |
| bnj570 34472 | Technical lemma for ~ bnj8... |
| bnj571 34473 | Technical lemma for ~ bnj8... |
| bnj605 34474 | Technical lemma. This lem... |
| bnj581 34475 | Technical lemma for ~ bnj5... |
| bnj589 34476 | Technical lemma for ~ bnj8... |
| bnj590 34477 | Technical lemma for ~ bnj8... |
| bnj591 34478 | Technical lemma for ~ bnj8... |
| bnj594 34479 | Technical lemma for ~ bnj8... |
| bnj580 34480 | Technical lemma for ~ bnj5... |
| bnj579 34481 | Technical lemma for ~ bnj8... |
| bnj602 34482 | Equality theorem for the `... |
| bnj607 34483 | Technical lemma for ~ bnj8... |
| bnj609 34484 | Technical lemma for ~ bnj8... |
| bnj611 34485 | Technical lemma for ~ bnj8... |
| bnj600 34486 | Technical lemma for ~ bnj8... |
| bnj601 34487 | Technical lemma for ~ bnj8... |
| bnj852 34488 | Technical lemma for ~ bnj6... |
| bnj864 34489 | Technical lemma for ~ bnj6... |
| bnj865 34490 | Technical lemma for ~ bnj6... |
| bnj873 34491 | Technical lemma for ~ bnj6... |
| bnj849 34492 | Technical lemma for ~ bnj6... |
| bnj882 34493 | Definition (using hypothes... |
| bnj18eq1 34494 | Equality theorem for trans... |
| bnj893 34495 | Property of ` _trCl ` . U... |
| bnj900 34496 | Technical lemma for ~ bnj6... |
| bnj906 34497 | Property of ` _trCl ` . (... |
| bnj908 34498 | Technical lemma for ~ bnj6... |
| bnj911 34499 | Technical lemma for ~ bnj6... |
| bnj916 34500 | Technical lemma for ~ bnj6... |
| bnj917 34501 | Technical lemma for ~ bnj6... |
| bnj934 34502 | Technical lemma for ~ bnj6... |
| bnj929 34503 | Technical lemma for ~ bnj6... |
| bnj938 34504 | Technical lemma for ~ bnj6... |
| bnj944 34505 | Technical lemma for ~ bnj6... |
| bnj953 34506 | Technical lemma for ~ bnj6... |
| bnj958 34507 | Technical lemma for ~ bnj6... |
| bnj1000 34508 | Technical lemma for ~ bnj8... |
| bnj965 34509 | Technical lemma for ~ bnj8... |
| bnj964 34510 | Technical lemma for ~ bnj6... |
| bnj966 34511 | Technical lemma for ~ bnj6... |
| bnj967 34512 | Technical lemma for ~ bnj6... |
| bnj969 34513 | Technical lemma for ~ bnj6... |
| bnj970 34514 | Technical lemma for ~ bnj6... |
| bnj910 34515 | Technical lemma for ~ bnj6... |
| bnj978 34516 | Technical lemma for ~ bnj6... |
| bnj981 34517 | Technical lemma for ~ bnj6... |
| bnj983 34518 | Technical lemma for ~ bnj6... |
| bnj984 34519 | Technical lemma for ~ bnj6... |
| bnj985v 34520 | Version of ~ bnj985 with a... |
| bnj985 34521 | Technical lemma for ~ bnj6... |
| bnj986 34522 | Technical lemma for ~ bnj6... |
| bnj996 34523 | Technical lemma for ~ bnj6... |
| bnj998 34524 | Technical lemma for ~ bnj6... |
| bnj999 34525 | Technical lemma for ~ bnj6... |
| bnj1001 34526 | Technical lemma for ~ bnj6... |
| bnj1006 34527 | Technical lemma for ~ bnj6... |
| bnj1014 34528 | Technical lemma for ~ bnj6... |
| bnj1015 34529 | Technical lemma for ~ bnj6... |
| bnj1018g 34530 | Version of ~ bnj1018 with ... |
| bnj1018 34531 | Technical lemma for ~ bnj6... |
| bnj1020 34532 | Technical lemma for ~ bnj6... |
| bnj1021 34533 | Technical lemma for ~ bnj6... |
| bnj907 34534 | Technical lemma for ~ bnj6... |
| bnj1029 34535 | Property of ` _trCl ` . (... |
| bnj1033 34536 | Technical lemma for ~ bnj6... |
| bnj1034 34537 | Technical lemma for ~ bnj6... |
| bnj1039 34538 | Technical lemma for ~ bnj6... |
| bnj1040 34539 | Technical lemma for ~ bnj6... |
| bnj1047 34540 | Technical lemma for ~ bnj6... |
| bnj1049 34541 | Technical lemma for ~ bnj6... |
| bnj1052 34542 | Technical lemma for ~ bnj6... |
| bnj1053 34543 | Technical lemma for ~ bnj6... |
| bnj1071 34544 | Technical lemma for ~ bnj6... |
| bnj1083 34545 | Technical lemma for ~ bnj6... |
| bnj1090 34546 | Technical lemma for ~ bnj6... |
| bnj1093 34547 | Technical lemma for ~ bnj6... |
| bnj1097 34548 | Technical lemma for ~ bnj6... |
| bnj1110 34549 | Technical lemma for ~ bnj6... |
| bnj1112 34550 | Technical lemma for ~ bnj6... |
| bnj1118 34551 | Technical lemma for ~ bnj6... |
| bnj1121 34552 | Technical lemma for ~ bnj6... |
| bnj1123 34553 | Technical lemma for ~ bnj6... |
| bnj1030 34554 | Technical lemma for ~ bnj6... |
| bnj1124 34555 | Property of ` _trCl ` . (... |
| bnj1133 34556 | Technical lemma for ~ bnj6... |
| bnj1128 34557 | Technical lemma for ~ bnj6... |
| bnj1127 34558 | Property of ` _trCl ` . (... |
| bnj1125 34559 | Property of ` _trCl ` . (... |
| bnj1145 34560 | Technical lemma for ~ bnj6... |
| bnj1147 34561 | Property of ` _trCl ` . (... |
| bnj1137 34562 | Property of ` _trCl ` . (... |
| bnj1148 34563 | Property of ` _pred ` . (... |
| bnj1136 34564 | Technical lemma for ~ bnj6... |
| bnj1152 34565 | Technical lemma for ~ bnj6... |
| bnj1154 34566 | Property of ` Fr ` . (Con... |
| bnj1171 34567 | Technical lemma for ~ bnj6... |
| bnj1172 34568 | Technical lemma for ~ bnj6... |
| bnj1173 34569 | Technical lemma for ~ bnj6... |
| bnj1174 34570 | Technical lemma for ~ bnj6... |
| bnj1175 34571 | Technical lemma for ~ bnj6... |
| bnj1176 34572 | Technical lemma for ~ bnj6... |
| bnj1177 34573 | Technical lemma for ~ bnj6... |
| bnj1186 34574 | Technical lemma for ~ bnj6... |
| bnj1190 34575 | Technical lemma for ~ bnj6... |
| bnj1189 34576 | Technical lemma for ~ bnj6... |
| bnj69 34577 | Existence of a minimal ele... |
| bnj1228 34578 | Existence of a minimal ele... |
| bnj1204 34579 | Well-founded induction. T... |
| bnj1234 34580 | Technical lemma for ~ bnj6... |
| bnj1245 34581 | Technical lemma for ~ bnj6... |
| bnj1256 34582 | Technical lemma for ~ bnj6... |
| bnj1259 34583 | Technical lemma for ~ bnj6... |
| bnj1253 34584 | Technical lemma for ~ bnj6... |
| bnj1279 34585 | Technical lemma for ~ bnj6... |
| bnj1286 34586 | Technical lemma for ~ bnj6... |
| bnj1280 34587 | Technical lemma for ~ bnj6... |
| bnj1296 34588 | Technical lemma for ~ bnj6... |
| bnj1309 34589 | Technical lemma for ~ bnj6... |
| bnj1307 34590 | Technical lemma for ~ bnj6... |
| bnj1311 34591 | Technical lemma for ~ bnj6... |
| bnj1318 34592 | Technical lemma for ~ bnj6... |
| bnj1326 34593 | Technical lemma for ~ bnj6... |
| bnj1321 34594 | Technical lemma for ~ bnj6... |
| bnj1364 34595 | Property of ` _FrSe ` . (... |
| bnj1371 34596 | Technical lemma for ~ bnj6... |
| bnj1373 34597 | Technical lemma for ~ bnj6... |
| bnj1374 34598 | Technical lemma for ~ bnj6... |
| bnj1384 34599 | Technical lemma for ~ bnj6... |
| bnj1388 34600 | Technical lemma for ~ bnj6... |
| bnj1398 34601 | Technical lemma for ~ bnj6... |
| bnj1413 34602 | Property of ` _trCl ` . (... |
| bnj1408 34603 | Technical lemma for ~ bnj1... |
| bnj1414 34604 | Property of ` _trCl ` . (... |
| bnj1415 34605 | Technical lemma for ~ bnj6... |
| bnj1416 34606 | Technical lemma for ~ bnj6... |
| bnj1418 34607 | Property of ` _pred ` . (... |
| bnj1417 34608 | Technical lemma for ~ bnj6... |
| bnj1421 34609 | Technical lemma for ~ bnj6... |
| bnj1444 34610 | Technical lemma for ~ bnj6... |
| bnj1445 34611 | Technical lemma for ~ bnj6... |
| bnj1446 34612 | Technical lemma for ~ bnj6... |
| bnj1447 34613 | Technical lemma for ~ bnj6... |
| bnj1448 34614 | Technical lemma for ~ bnj6... |
| bnj1449 34615 | Technical lemma for ~ bnj6... |
| bnj1442 34616 | Technical lemma for ~ bnj6... |
| bnj1450 34617 | Technical lemma for ~ bnj6... |
| bnj1423 34618 | Technical lemma for ~ bnj6... |
| bnj1452 34619 | Technical lemma for ~ bnj6... |
| bnj1466 34620 | Technical lemma for ~ bnj6... |
| bnj1467 34621 | Technical lemma for ~ bnj6... |
| bnj1463 34622 | Technical lemma for ~ bnj6... |
| bnj1489 34623 | Technical lemma for ~ bnj6... |
| bnj1491 34624 | Technical lemma for ~ bnj6... |
| bnj1312 34625 | Technical lemma for ~ bnj6... |
| bnj1493 34626 | Technical lemma for ~ bnj6... |
| bnj1497 34627 | Technical lemma for ~ bnj6... |
| bnj1498 34628 | Technical lemma for ~ bnj6... |
| bnj60 34629 | Well-founded recursion, pa... |
| bnj1514 34630 | Technical lemma for ~ bnj1... |
| bnj1518 34631 | Technical lemma for ~ bnj1... |
| bnj1519 34632 | Technical lemma for ~ bnj1... |
| bnj1520 34633 | Technical lemma for ~ bnj1... |
| bnj1501 34634 | Technical lemma for ~ bnj1... |
| bnj1500 34635 | Well-founded recursion, pa... |
| bnj1525 34636 | Technical lemma for ~ bnj1... |
| bnj1529 34637 | Technical lemma for ~ bnj1... |
| bnj1523 34638 | Technical lemma for ~ bnj1... |
| bnj1522 34639 | Well-founded recursion, pa... |
| exdifsn 34640 | There exists an element in... |
| srcmpltd 34641 | If a statement is true for... |
| prsrcmpltd 34642 | If a statement is true for... |
| dff15 34643 | A one-to-one function in t... |
| f1resveqaeq 34644 | If a function restricted t... |
| f1resrcmplf1dlem 34645 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 34646 | If a function's restrictio... |
| funen1cnv 34647 | If a function is equinumer... |
| fnrelpredd 34648 | A function that preserves ... |
| cardpred 34649 | The cardinality function p... |
| nummin 34650 | Every nonempty class of nu... |
| fineqvrep 34651 | If the Axiom of Infinity i... |
| fineqvpow 34652 | If the Axiom of Infinity i... |
| fineqvac 34653 | If the Axiom of Infinity i... |
| fineqvacALT 34654 | Shorter proof of ~ fineqva... |
| zltp1ne 34655 | Integer ordering relation.... |
| nnltp1ne 34656 | Positive integer ordering ... |
| nn0ltp1ne 34657 | Nonnegative integer orderi... |
| 0nn0m1nnn0 34658 | A number is zero if and on... |
| f1resfz0f1d 34659 | If a function with a seque... |
| fisshasheq 34660 | A finite set is equal to i... |
| revpfxsfxrev 34661 | The reverse of a prefix of... |
| swrdrevpfx 34662 | A subword expressed in ter... |
| lfuhgr 34663 | A hypergraph is loop-free ... |
| lfuhgr2 34664 | A hypergraph is loop-free ... |
| lfuhgr3 34665 | A hypergraph is loop-free ... |
| cplgredgex 34666 | Any two (distinct) vertice... |
| cusgredgex 34667 | Any two (distinct) vertice... |
| cusgredgex2 34668 | Any two distinct vertices ... |
| pfxwlk 34669 | A prefix of a walk is a wa... |
| revwlk 34670 | The reverse of a walk is a... |
| revwlkb 34671 | Two words represent a walk... |
| swrdwlk 34672 | Two matching subwords of a... |
| pthhashvtx 34673 | A graph containing a path ... |
| pthisspthorcycl 34674 | A path is either a simple ... |
| spthcycl 34675 | A walk is a trivial path i... |
| usgrgt2cycl 34676 | A non-trivial cycle in a s... |
| usgrcyclgt2v 34677 | A simple graph with a non-... |
| subgrwlk 34678 | If a walk exists in a subg... |
| subgrtrl 34679 | If a trail exists in a sub... |
| subgrpth 34680 | If a path exists in a subg... |
| subgrcycl 34681 | If a cycle exists in a sub... |
| cusgr3cyclex 34682 | Every complete simple grap... |
| loop1cycl 34683 | A hypergraph has a cycle o... |
| 2cycld 34684 | Construction of a 2-cycle ... |
| 2cycl2d 34685 | Construction of a 2-cycle ... |
| umgr2cycllem 34686 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 34687 | A multigraph with two dist... |
| dfacycgr1 34690 | An alternate definition of... |
| isacycgr 34691 | The property of being an a... |
| isacycgr1 34692 | The property of being an a... |
| acycgrcycl 34693 | Any cycle in an acyclic gr... |
| acycgr0v 34694 | A null graph (with no vert... |
| acycgr1v 34695 | A multigraph with one vert... |
| acycgr2v 34696 | A simple graph with two ve... |
| prclisacycgr 34697 | A proper class (representi... |
| acycgrislfgr 34698 | An acyclic hypergraph is a... |
| upgracycumgr 34699 | An acyclic pseudograph is ... |
| umgracycusgr 34700 | An acyclic multigraph is a... |
| upgracycusgr 34701 | An acyclic pseudograph is ... |
| cusgracyclt3v 34702 | A complete simple graph is... |
| pthacycspth 34703 | A path in an acyclic graph... |
| acycgrsubgr 34704 | The subgraph of an acyclic... |
| quartfull 34711 | The quartic equation, writ... |
| deranglem 34712 | Lemma for derangements. (... |
| derangval 34713 | Define the derangement fun... |
| derangf 34714 | The derangement number is ... |
| derang0 34715 | The derangement number of ... |
| derangsn 34716 | The derangement number of ... |
| derangenlem 34717 | One half of ~ derangen . ... |
| derangen 34718 | The derangement number is ... |
| subfacval 34719 | The subfactorial is define... |
| derangen2 34720 | Write the derangement numb... |
| subfacf 34721 | The subfactorial is a func... |
| subfaclefac 34722 | The subfactorial is less t... |
| subfac0 34723 | The subfactorial at zero. ... |
| subfac1 34724 | The subfactorial at one. ... |
| subfacp1lem1 34725 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 34726 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 34727 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 34728 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 34729 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 34730 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 34731 | Lemma for ~ subfacp1 . By... |
| subfacp1 34732 | A two-term recurrence for ... |
| subfacval2 34733 | A closed-form expression f... |
| subfaclim 34734 | The subfactorial converges... |
| subfacval3 34735 | Another closed form expres... |
| derangfmla 34736 | The derangements formula, ... |
| erdszelem1 34737 | Lemma for ~ erdsze . (Con... |
| erdszelem2 34738 | Lemma for ~ erdsze . (Con... |
| erdszelem3 34739 | Lemma for ~ erdsze . (Con... |
| erdszelem4 34740 | Lemma for ~ erdsze . (Con... |
| erdszelem5 34741 | Lemma for ~ erdsze . (Con... |
| erdszelem6 34742 | Lemma for ~ erdsze . (Con... |
| erdszelem7 34743 | Lemma for ~ erdsze . (Con... |
| erdszelem8 34744 | Lemma for ~ erdsze . (Con... |
| erdszelem9 34745 | Lemma for ~ erdsze . (Con... |
| erdszelem10 34746 | Lemma for ~ erdsze . (Con... |
| erdszelem11 34747 | Lemma for ~ erdsze . (Con... |
| erdsze 34748 | The Erdős-Szekeres th... |
| erdsze2lem1 34749 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 34750 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 34751 | Generalize the statement o... |
| kur14lem1 34752 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 34753 | Lemma for ~ kur14 . Write... |
| kur14lem3 34754 | Lemma for ~ kur14 . A clo... |
| kur14lem4 34755 | Lemma for ~ kur14 . Compl... |
| kur14lem5 34756 | Lemma for ~ kur14 . Closu... |
| kur14lem6 34757 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 34758 | Lemma for ~ kur14 : main p... |
| kur14lem8 34759 | Lemma for ~ kur14 . Show ... |
| kur14lem9 34760 | Lemma for ~ kur14 . Since... |
| kur14lem10 34761 | Lemma for ~ kur14 . Disch... |
| kur14 34762 | Kuratowski's closure-compl... |
| ispconn 34769 | The property of being a pa... |
| pconncn 34770 | The property of being a pa... |
| pconntop 34771 | A simply connected space i... |
| issconn 34772 | The property of being a si... |
| sconnpconn 34773 | A simply connected space i... |
| sconntop 34774 | A simply connected space i... |
| sconnpht 34775 | A closed path in a simply ... |
| cnpconn 34776 | An image of a path-connect... |
| pconnconn 34777 | A path-connected space is ... |
| txpconn 34778 | The topological product of... |
| ptpconn 34779 | The topological product of... |
| indispconn 34780 | The indiscrete topology (o... |
| connpconn 34781 | A connected and locally pa... |
| qtoppconn 34782 | A quotient of a path-conne... |
| pconnpi1 34783 | All fundamental groups in ... |
| sconnpht2 34784 | Any two paths in a simply ... |
| sconnpi1 34785 | A path-connected topologic... |
| txsconnlem 34786 | Lemma for ~ txsconn . (Co... |
| txsconn 34787 | The topological product of... |
| cvxpconn 34788 | A convex subset of the com... |
| cvxsconn 34789 | A convex subset of the com... |
| blsconn 34790 | An open ball in the comple... |
| cnllysconn 34791 | The topology of the comple... |
| resconn 34792 | A subset of ` RR ` is simp... |
| ioosconn 34793 | An open interval is simply... |
| iccsconn 34794 | A closed interval is simpl... |
| retopsconn 34795 | The real numbers are simpl... |
| iccllysconn 34796 | A closed interval is local... |
| rellysconn 34797 | The real numbers are local... |
| iisconn 34798 | The unit interval is simpl... |
| iillysconn 34799 | The unit interval is local... |
| iinllyconn 34800 | The unit interval is local... |
| fncvm 34803 | Lemma for covering maps. ... |
| cvmscbv 34804 | Change bound variables in ... |
| iscvm 34805 | The property of being a co... |
| cvmtop1 34806 | Reverse closure for a cove... |
| cvmtop2 34807 | Reverse closure for a cove... |
| cvmcn 34808 | A covering map is a contin... |
| cvmcov 34809 | Property of a covering map... |
| cvmsrcl 34810 | Reverse closure for an eve... |
| cvmsi 34811 | One direction of ~ cvmsval... |
| cvmsval 34812 | Elementhood in the set ` S... |
| cvmsss 34813 | An even covering is a subs... |
| cvmsn0 34814 | An even covering is nonemp... |
| cvmsuni 34815 | An even covering of ` U ` ... |
| cvmsdisj 34816 | An even covering of ` U ` ... |
| cvmshmeo 34817 | Every element of an even c... |
| cvmsf1o 34818 | ` F ` , localized to an el... |
| cvmscld 34819 | The sets of an even coveri... |
| cvmsss2 34820 | An open subset of an evenl... |
| cvmcov2 34821 | The covering map property ... |
| cvmseu 34822 | Every element in ` U. T ` ... |
| cvmsiota 34823 | Identify the unique elemen... |
| cvmopnlem 34824 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 34825 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 34826 | A covering map is an open ... |
| cvmliftmolem1 34827 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 34828 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 34829 | A lift of a continuous fun... |
| cvmliftmo 34830 | A lift of a continuous fun... |
| cvmliftlem1 34831 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 34832 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 34833 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 34834 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 34835 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 34836 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 34837 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 34838 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 34839 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 34840 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 34841 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 34842 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 34843 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 34844 | Lemma for ~ cvmlift . Dis... |
| cvmlift 34845 | One of the important prope... |
| cvmfo 34846 | A covering map is an onto ... |
| cvmliftiota 34847 | Write out a function ` H `... |
| cvmlift2lem1 34848 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 34849 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 34850 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 34851 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 34852 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 34853 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 34854 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 34855 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 34856 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 34857 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 34858 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 34859 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 34860 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 34861 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 34862 | A two-dimensional version ... |
| cvmliftphtlem 34863 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 34864 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 34865 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 34866 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 34867 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 34868 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 34869 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 34870 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 34871 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 34872 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 34873 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 34874 | A general version of ~ cvm... |
| snmlff 34875 | The function ` F ` from ~ ... |
| snmlfval 34876 | The function ` F ` from ~ ... |
| snmlval 34877 | The property " ` A ` is si... |
| snmlflim 34878 | If ` A ` is simply normal,... |
| goel 34893 | A "Godel-set of membership... |
| goelel3xp 34894 | A "Godel-set of membership... |
| goeleq12bg 34895 | Two "Godel-set of membersh... |
| gonafv 34896 | The "Godel-set for the She... |
| goaleq12d 34897 | Equality of the "Godel-set... |
| gonanegoal 34898 | The Godel-set for the Shef... |
| satf 34899 | The satisfaction predicate... |
| satfsucom 34900 | The satisfaction predicate... |
| satfn 34901 | The satisfaction predicate... |
| satom 34902 | The satisfaction predicate... |
| satfvsucom 34903 | The satisfaction predicate... |
| satfv0 34904 | The value of the satisfact... |
| satfvsuclem1 34905 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 34906 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 34907 | The value of the satisfact... |
| satfv1lem 34908 | Lemma for ~ satfv1 . (Con... |
| satfv1 34909 | The value of the satisfact... |
| satfsschain 34910 | The binary relation of a s... |
| satfvsucsuc 34911 | The satisfaction predicate... |
| satfbrsuc 34912 | The binary relation of a s... |
| satfrel 34913 | The value of the satisfact... |
| satfdmlem 34914 | Lemma for ~ satfdm . (Con... |
| satfdm 34915 | The domain of the satisfac... |
| satfrnmapom 34916 | The range of the satisfact... |
| satfv0fun 34917 | The value of the satisfact... |
| satf0 34918 | The satisfaction predicate... |
| satf0sucom 34919 | The satisfaction predicate... |
| satf00 34920 | The value of the satisfact... |
| satf0suclem 34921 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 34922 | The value of the satisfact... |
| satf0op 34923 | An element of a value of t... |
| satf0n0 34924 | The value of the satisfact... |
| sat1el2xp 34925 | The first component of an ... |
| fmlafv 34926 | The valid Godel formulas o... |
| fmla 34927 | The set of all valid Godel... |
| fmla0 34928 | The valid Godel formulas o... |
| fmla0xp 34929 | The valid Godel formulas o... |
| fmlasuc0 34930 | The valid Godel formulas o... |
| fmlafvel 34931 | A class is a valid Godel f... |
| fmlasuc 34932 | The valid Godel formulas o... |
| fmla1 34933 | The valid Godel formulas o... |
| isfmlasuc 34934 | The characterization of a ... |
| fmlasssuc 34935 | The Godel formulas of heig... |
| fmlaomn0 34936 | The empty set is not a God... |
| fmlan0 34937 | The empty set is not a God... |
| gonan0 34938 | The "Godel-set of NAND" is... |
| goaln0 34939 | The "Godel-set of universa... |
| gonarlem 34940 | Lemma for ~ gonar (inducti... |
| gonar 34941 | If the "Godel-set of NAND"... |
| goalrlem 34942 | Lemma for ~ goalr (inducti... |
| goalr 34943 | If the "Godel-set of unive... |
| fmla0disjsuc 34944 | The set of valid Godel for... |
| fmlasucdisj 34945 | The valid Godel formulas o... |
| satfdmfmla 34946 | The domain of the satisfac... |
| satffunlem 34947 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 34948 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 34949 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 34950 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 34951 | The domain of an ordered p... |
| satffunlem2lem2 34952 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 34953 | Lemma 1 for ~ satffun : in... |
| satffunlem2 34954 | Lemma 2 for ~ satffun : in... |
| satffun 34955 | The value of the satisfact... |
| satff 34956 | The satisfaction predicate... |
| satfun 34957 | The satisfaction predicate... |
| satfvel 34958 | An element of the value of... |
| satfv0fvfmla0 34959 | The value of the satisfact... |
| satefv 34960 | The simplified satisfactio... |
| sate0 34961 | The simplified satisfactio... |
| satef 34962 | The simplified satisfactio... |
| sate0fv0 34963 | A simplified satisfaction ... |
| satefvfmla0 34964 | The simplified satisfactio... |
| sategoelfvb 34965 | Characterization of a valu... |
| sategoelfv 34966 | Condition of a valuation `... |
| ex-sategoelel 34967 | Example of a valuation of ... |
| ex-sategoel 34968 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 34969 | The value of the satisfact... |
| 2goelgoanfmla1 34970 | Two Godel-sets of membersh... |
| satefvfmla1 34971 | The simplified satisfactio... |
| ex-sategoelelomsuc 34972 | Example of a valuation of ... |
| ex-sategoelel12 34973 | Example of a valuation of ... |
| prv 34974 | The "proves" relation on a... |
| elnanelprv 34975 | The wff ` ( A e. B -/\ B e... |
| prv0 34976 | Every wff encoded as ` U `... |
| prv1n 34977 | No wff encoded as a Godel-... |
| mvtval 35046 | The set of variable typeco... |
| mrexval 35047 | The set of "raw expression... |
| mexval 35048 | The set of expressions, wh... |
| mexval2 35049 | The set of expressions, wh... |
| mdvval 35050 | The set of disjoint variab... |
| mvrsval 35051 | The set of variables in an... |
| mvrsfpw 35052 | The set of variables in an... |
| mrsubffval 35053 | The substitution of some v... |
| mrsubfval 35054 | The substitution of some v... |
| mrsubval 35055 | The substitution of some v... |
| mrsubcv 35056 | The value of a substituted... |
| mrsubvr 35057 | The value of a substituted... |
| mrsubff 35058 | A substitution is a functi... |
| mrsubrn 35059 | Although it is defined for... |
| mrsubff1 35060 | When restricted to complet... |
| mrsubff1o 35061 | When restricted to complet... |
| mrsub0 35062 | The value of the substitut... |
| mrsubf 35063 | A substitution is a functi... |
| mrsubccat 35064 | Substitution distributes o... |
| mrsubcn 35065 | A substitution does not ch... |
| elmrsubrn 35066 | Characterization of the su... |
| mrsubco 35067 | The composition of two sub... |
| mrsubvrs 35068 | The set of variables in a ... |
| msubffval 35069 | A substitution applied to ... |
| msubfval 35070 | A substitution applied to ... |
| msubval 35071 | A substitution applied to ... |
| msubrsub 35072 | A substitution applied to ... |
| msubty 35073 | The type of a substituted ... |
| elmsubrn 35074 | Characterization of substi... |
| msubrn 35075 | Although it is defined for... |
| msubff 35076 | A substitution is a functi... |
| msubco 35077 | The composition of two sub... |
| msubf 35078 | A substitution is a functi... |
| mvhfval 35079 | Value of the function mapp... |
| mvhval 35080 | Value of the function mapp... |
| mpstval 35081 | A pre-statement is an orde... |
| elmpst 35082 | Property of being a pre-st... |
| msrfval 35083 | Value of the reduct of a p... |
| msrval 35084 | Value of the reduct of a p... |
| mpstssv 35085 | A pre-statement is an orde... |
| mpst123 35086 | Decompose a pre-statement ... |
| mpstrcl 35087 | The elements of a pre-stat... |
| msrf 35088 | The reduct of a pre-statem... |
| msrrcl 35089 | If ` X ` and ` Y ` have th... |
| mstaval 35090 | Value of the set of statem... |
| msrid 35091 | The reduct of a statement ... |
| msrfo 35092 | The reduct of a pre-statem... |
| mstapst 35093 | A statement is a pre-state... |
| elmsta 35094 | Property of being a statem... |
| ismfs 35095 | A formal system is a tuple... |
| mfsdisj 35096 | The constants and variable... |
| mtyf2 35097 | The type function maps var... |
| mtyf 35098 | The type function maps var... |
| mvtss 35099 | The set of variable typeco... |
| maxsta 35100 | An axiom is a statement. ... |
| mvtinf 35101 | Each variable typecode has... |
| msubff1 35102 | When restricted to complet... |
| msubff1o 35103 | When restricted to complet... |
| mvhf 35104 | The function mapping varia... |
| mvhf1 35105 | The function mapping varia... |
| msubvrs 35106 | The set of variables in a ... |
| mclsrcl 35107 | Reverse closure for the cl... |
| mclsssvlem 35108 | Lemma for ~ mclsssv . (Co... |
| mclsval 35109 | The function mapping varia... |
| mclsssv 35110 | The closure of a set of ex... |
| ssmclslem 35111 | Lemma for ~ ssmcls . (Con... |
| vhmcls 35112 | All variable hypotheses ar... |
| ssmcls 35113 | The original expressions a... |
| ss2mcls 35114 | The closure is monotonic u... |
| mclsax 35115 | The closure is closed unde... |
| mclsind 35116 | Induction theorem for clos... |
| mppspstlem 35117 | Lemma for ~ mppspst . (Co... |
| mppsval 35118 | Definition of a provable p... |
| elmpps 35119 | Definition of a provable p... |
| mppspst 35120 | A provable pre-statement i... |
| mthmval 35121 | A theorem is a pre-stateme... |
| elmthm 35122 | A theorem is a pre-stateme... |
| mthmi 35123 | A statement whose reduct i... |
| mthmsta 35124 | A theorem is a pre-stateme... |
| mppsthm 35125 | A provable pre-statement i... |
| mthmblem 35126 | Lemma for ~ mthmb . (Cont... |
| mthmb 35127 | If two statements have the... |
| mthmpps 35128 | Given a theorem, there is ... |
| mclsppslem 35129 | The closure is closed unde... |
| mclspps 35130 | The closure is closed unde... |
| problem1 35205 | Practice problem 1. Clues... |
| problem2 35206 | Practice problem 2. Clues... |
| problem3 35207 | Practice problem 3. Clues... |
| problem4 35208 | Practice problem 4. Clues... |
| problem5 35209 | Practice problem 5. Clues... |
| quad3 35210 | Variant of quadratic equat... |
| climuzcnv 35211 | Utility lemma to convert b... |
| sinccvglem 35212 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 35213 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 35214 | The circumference of a cir... |
| elfzm12 35215 | Membership in a curtailed ... |
| nn0seqcvg 35216 | A strictly-decreasing nonn... |
| lediv2aALT 35217 | Division of both sides of ... |
| abs2sqlei 35218 | The absolute values of two... |
| abs2sqlti 35219 | The absolute values of two... |
| abs2sqle 35220 | The absolute values of two... |
| abs2sqlt 35221 | The absolute values of two... |
| abs2difi 35222 | Difference of absolute val... |
| abs2difabsi 35223 | Absolute value of differen... |
| currybi 35224 | Biconditional version of C... |
| axextprim 35231 | ~ ax-ext without distinct ... |
| axrepprim 35232 | ~ ax-rep without distinct ... |
| axunprim 35233 | ~ ax-un without distinct v... |
| axpowprim 35234 | ~ ax-pow without distinct ... |
| axregprim 35235 | ~ ax-reg without distinct ... |
| axinfprim 35236 | ~ ax-inf without distinct ... |
| axacprim 35237 | ~ ax-ac without distinct v... |
| untelirr 35238 | We call a class "untanged"... |
| untuni 35239 | The union of a class is un... |
| untsucf 35240 | If a class is untangled, t... |
| unt0 35241 | The null set is untangled.... |
| untint 35242 | If there is an untangled e... |
| efrunt 35243 | If ` A ` is well-founded b... |
| untangtr 35244 | A transitive class is unta... |
| 3jaodd 35245 | Double deduction form of ~... |
| 3orit 35246 | Closed form of ~ 3ori . (... |
| biimpexp 35247 | A biconditional in the ant... |
| nepss 35248 | Two classes are unequal if... |
| 3ccased 35249 | Triple disjunction form of... |
| dfso3 35250 | Expansion of the definitio... |
| brtpid1 35251 | A binary relation involvin... |
| brtpid2 35252 | A binary relation involvin... |
| brtpid3 35253 | A binary relation involvin... |
| iota5f 35254 | A method for computing iot... |
| jath 35255 | Closed form of ~ ja . Pro... |
| xpab 35256 | Cartesian product of two c... |
| nnuni 35257 | The union of a finite ordi... |
| sqdivzi 35258 | Distribution of square ove... |
| supfz 35259 | The supremum of a finite s... |
| inffz 35260 | The infimum of a finite se... |
| fz0n 35261 | The sequence ` ( 0 ... ( N... |
| shftvalg 35262 | Value of a sequence shifte... |
| divcnvlin 35263 | Limit of the ratio of two ... |
| climlec3 35264 | Comparison of a constant t... |
| iexpire 35265 | ` _i ` raised to itself is... |
| bcneg1 35266 | The binomial coefficent ov... |
| bcm1nt 35267 | The proportion of one bion... |
| bcprod 35268 | A product identity for bin... |
| bccolsum 35269 | A column-sum rule for bino... |
| iprodefisumlem 35270 | Lemma for ~ iprodefisum . ... |
| iprodefisum 35271 | Applying the exponential f... |
| iprodgam 35272 | An infinite product versio... |
| faclimlem1 35273 | Lemma for ~ faclim . Clos... |
| faclimlem2 35274 | Lemma for ~ faclim . Show... |
| faclimlem3 35275 | Lemma for ~ faclim . Alge... |
| faclim 35276 | An infinite product expres... |
| iprodfac 35277 | An infinite product expres... |
| faclim2 35278 | Another factorial limit du... |
| gcd32 35279 | Swap the second and third ... |
| gcdabsorb 35280 | Absorption law for gcd. (... |
| dftr6 35281 | A potential definition of ... |
| coep 35282 | Composition with the membe... |
| coepr 35283 | Composition with the conve... |
| dffr5 35284 | A quantifier-free definiti... |
| dfso2 35285 | Quantifier-free definition... |
| br8 35286 | Substitution for an eight-... |
| br6 35287 | Substitution for a six-pla... |
| br4 35288 | Substitution for a four-pl... |
| cnvco1 35289 | Another distributive law o... |
| cnvco2 35290 | Another distributive law o... |
| eldm3 35291 | Quantifier-free definition... |
| elrn3 35292 | Quantifier-free definition... |
| pocnv 35293 | The converse of a partial ... |
| socnv 35294 | The converse of a strict o... |
| sotrd 35295 | Transitivity law for stric... |
| elintfv 35296 | Membership in an intersect... |
| funpsstri 35297 | A condition for subset tri... |
| fundmpss 35298 | If a class ` F ` is a prop... |
| funsseq 35299 | Given two functions with e... |
| fununiq 35300 | The uniqueness condition o... |
| funbreq 35301 | An equality condition for ... |
| br1steq 35302 | Uniqueness condition for t... |
| br2ndeq 35303 | Uniqueness condition for t... |
| dfdm5 35304 | Definition of domain in te... |
| dfrn5 35305 | Definition of range in ter... |
| opelco3 35306 | Alternate way of saying th... |
| elima4 35307 | Quantifier-free expression... |
| fv1stcnv 35308 | The value of the converse ... |
| fv2ndcnv 35309 | The value of the converse ... |
| setinds 35310 | Principle of set induction... |
| setinds2f 35311 | ` _E ` induction schema, u... |
| setinds2 35312 | ` _E ` induction schema, u... |
| elpotr 35313 | A class of transitive sets... |
| dford5reg 35314 | Given ~ ax-reg , an ordina... |
| dfon2lem1 35315 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 35316 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 35317 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 35318 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 35319 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 35320 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 35321 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 35322 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 35323 | Lemma for ~ dfon2 . A cla... |
| dfon2 35324 | ` On ` consists of all set... |
| rdgprc0 35325 | The value of the recursive... |
| rdgprc 35326 | The value of the recursive... |
| dfrdg2 35327 | Alternate definition of th... |
| dfrdg3 35328 | Generalization of ~ dfrdg2... |
| axextdfeq 35329 | A version of ~ ax-ext for ... |
| ax8dfeq 35330 | A version of ~ ax-8 for us... |
| axextdist 35331 | ~ ax-ext with distinctors ... |
| axextbdist 35332 | ~ axextb with distinctors ... |
| 19.12b 35333 | Version of ~ 19.12vv with ... |
| exnel 35334 | There is always a set not ... |
| distel 35335 | Distinctors in terms of me... |
| axextndbi 35336 | ~ axextnd as a bicondition... |
| hbntg 35337 | A more general form of ~ h... |
| hbimtg 35338 | A more general and closed ... |
| hbaltg 35339 | A more general and closed ... |
| hbng 35340 | A more general form of ~ h... |
| hbimg 35341 | A more general form of ~ h... |
| wsuceq123 35346 | Equality theorem for well-... |
| wsuceq1 35347 | Equality theorem for well-... |
| wsuceq2 35348 | Equality theorem for well-... |
| wsuceq3 35349 | Equality theorem for well-... |
| nfwsuc 35350 | Bound-variable hypothesis ... |
| wlimeq12 35351 | Equality theorem for the l... |
| wlimeq1 35352 | Equality theorem for the l... |
| wlimeq2 35353 | Equality theorem for the l... |
| nfwlim 35354 | Bound-variable hypothesis ... |
| elwlim 35355 | Membership in the limit cl... |
| wzel 35356 | The zero of a well-founded... |
| wsuclem 35357 | Lemma for the supremum pro... |
| wsucex 35358 | Existence theorem for well... |
| wsuccl 35359 | If ` X ` is a set with an ... |
| wsuclb 35360 | A well-founded successor i... |
| wlimss 35361 | The class of limit points ... |
| txpss3v 35410 | A tail Cartesian product i... |
| txprel 35411 | A tail Cartesian product i... |
| brtxp 35412 | Characterize a ternary rel... |
| brtxp2 35413 | The binary relation over a... |
| dfpprod2 35414 | Expanded definition of par... |
| pprodcnveq 35415 | A converse law for paralle... |
| pprodss4v 35416 | The parallel product is a ... |
| brpprod 35417 | Characterize a quaternary ... |
| brpprod3a 35418 | Condition for parallel pro... |
| brpprod3b 35419 | Condition for parallel pro... |
| relsset 35420 | The subset class is a bina... |
| brsset 35421 | For sets, the ` SSet ` bin... |
| idsset 35422 | ` _I ` is equal to the int... |
| eltrans 35423 | Membership in the class of... |
| dfon3 35424 | A quantifier-free definiti... |
| dfon4 35425 | Another quantifier-free de... |
| brtxpsd 35426 | Expansion of a common form... |
| brtxpsd2 35427 | Another common abbreviatio... |
| brtxpsd3 35428 | A third common abbreviatio... |
| relbigcup 35429 | The ` Bigcup ` relationshi... |
| brbigcup 35430 | Binary relation over ` Big... |
| dfbigcup2 35431 | ` Bigcup ` using maps-to n... |
| fobigcup 35432 | ` Bigcup ` maps the univer... |
| fnbigcup 35433 | ` Bigcup ` is a function o... |
| fvbigcup 35434 | For sets, ` Bigcup ` yield... |
| elfix 35435 | Membership in the fixpoint... |
| elfix2 35436 | Alternative membership in ... |
| dffix2 35437 | The fixpoints of a class i... |
| fixssdm 35438 | The fixpoints of a class a... |
| fixssrn 35439 | The fixpoints of a class a... |
| fixcnv 35440 | The fixpoints of a class a... |
| fixun 35441 | The fixpoint operator dist... |
| ellimits 35442 | Membership in the class of... |
| limitssson 35443 | The class of all limit ord... |
| dfom5b 35444 | A quantifier-free definiti... |
| sscoid 35445 | A condition for subset and... |
| dffun10 35446 | Another potential definiti... |
| elfuns 35447 | Membership in the class of... |
| elfunsg 35448 | Closed form of ~ elfuns . ... |
| brsingle 35449 | The binary relation form o... |
| elsingles 35450 | Membership in the class of... |
| fnsingle 35451 | The singleton relationship... |
| fvsingle 35452 | The value of the singleton... |
| dfsingles2 35453 | Alternate definition of th... |
| snelsingles 35454 | A singleton is a member of... |
| dfiota3 35455 | A definition of iota using... |
| dffv5 35456 | Another quantifier-free de... |
| unisnif 35457 | Express union of singleton... |
| brimage 35458 | Binary relation form of th... |
| brimageg 35459 | Closed form of ~ brimage .... |
| funimage 35460 | ` Image A ` is a function.... |
| fnimage 35461 | ` Image R ` is a function ... |
| imageval 35462 | The image functor in maps-... |
| fvimage 35463 | Value of the image functor... |
| brcart 35464 | Binary relation form of th... |
| brdomain 35465 | Binary relation form of th... |
| brrange 35466 | Binary relation form of th... |
| brdomaing 35467 | Closed form of ~ brdomain ... |
| brrangeg 35468 | Closed form of ~ brrange .... |
| brimg 35469 | Binary relation form of th... |
| brapply 35470 | Binary relation form of th... |
| brcup 35471 | Binary relation form of th... |
| brcap 35472 | Binary relation form of th... |
| brsuccf 35473 | Binary relation form of th... |
| funpartlem 35474 | Lemma for ~ funpartfun . ... |
| funpartfun 35475 | The functional part of ` F... |
| funpartss 35476 | The functional part of ` F... |
| funpartfv 35477 | The function value of the ... |
| fullfunfnv 35478 | The full functional part o... |
| fullfunfv 35479 | The function value of the ... |
| brfullfun 35480 | A binary relation form con... |
| brrestrict 35481 | Binary relation form of th... |
| dfrecs2 35482 | A quantifier-free definiti... |
| dfrdg4 35483 | A quantifier-free definiti... |
| dfint3 35484 | Quantifier-free definition... |
| imagesset 35485 | The Image functor applied ... |
| brub 35486 | Binary relation form of th... |
| brlb 35487 | Binary relation form of th... |
| altopex 35492 | Alternative ordered pairs ... |
| altopthsn 35493 | Two alternate ordered pair... |
| altopeq12 35494 | Equality for alternate ord... |
| altopeq1 35495 | Equality for alternate ord... |
| altopeq2 35496 | Equality for alternate ord... |
| altopth1 35497 | Equality of the first memb... |
| altopth2 35498 | Equality of the second mem... |
| altopthg 35499 | Alternate ordered pair the... |
| altopthbg 35500 | Alternate ordered pair the... |
| altopth 35501 | The alternate ordered pair... |
| altopthb 35502 | Alternate ordered pair the... |
| altopthc 35503 | Alternate ordered pair the... |
| altopthd 35504 | Alternate ordered pair the... |
| altxpeq1 35505 | Equality for alternate Car... |
| altxpeq2 35506 | Equality for alternate Car... |
| elaltxp 35507 | Membership in alternate Ca... |
| altopelaltxp 35508 | Alternate ordered pair mem... |
| altxpsspw 35509 | An inclusion rule for alte... |
| altxpexg 35510 | The alternate Cartesian pr... |
| rankaltopb 35511 | Compute the rank of an alt... |
| nfaltop 35512 | Bound-variable hypothesis ... |
| sbcaltop 35513 | Distribution of class subs... |
| cgrrflx2d 35516 | Deduction form of ~ axcgrr... |
| cgrtr4d 35517 | Deduction form of ~ axcgrt... |
| cgrtr4and 35518 | Deduction form of ~ axcgrt... |
| cgrrflx 35519 | Reflexivity law for congru... |
| cgrrflxd 35520 | Deduction form of ~ cgrrfl... |
| cgrcomim 35521 | Congruence commutes on the... |
| cgrcom 35522 | Congruence commutes betwee... |
| cgrcomand 35523 | Deduction form of ~ cgrcom... |
| cgrtr 35524 | Transitivity law for congr... |
| cgrtrand 35525 | Deduction form of ~ cgrtr ... |
| cgrtr3 35526 | Transitivity law for congr... |
| cgrtr3and 35527 | Deduction form of ~ cgrtr3... |
| cgrcoml 35528 | Congruence commutes on the... |
| cgrcomr 35529 | Congruence commutes on the... |
| cgrcomlr 35530 | Congruence commutes on bot... |
| cgrcomland 35531 | Deduction form of ~ cgrcom... |
| cgrcomrand 35532 | Deduction form of ~ cgrcom... |
| cgrcomlrand 35533 | Deduction form of ~ cgrcom... |
| cgrtriv 35534 | Degenerate segments are co... |
| cgrid2 35535 | Identity law for congruenc... |
| cgrdegen 35536 | Two congruent segments are... |
| brofs 35537 | Binary relation form of th... |
| 5segofs 35538 | Rephrase ~ ax5seg using th... |
| ofscom 35539 | The outer five segment pre... |
| cgrextend 35540 | Link congruence over a pai... |
| cgrextendand 35541 | Deduction form of ~ cgrext... |
| segconeq 35542 | Two points that satisfy th... |
| segconeu 35543 | Existential uniqueness ver... |
| btwntriv2 35544 | Betweenness always holds f... |
| btwncomim 35545 | Betweenness commutes. Imp... |
| btwncom 35546 | Betweenness commutes. (Co... |
| btwncomand 35547 | Deduction form of ~ btwnco... |
| btwntriv1 35548 | Betweenness always holds f... |
| btwnswapid 35549 | If you can swap the first ... |
| btwnswapid2 35550 | If you can swap arguments ... |
| btwnintr 35551 | Inner transitivity law for... |
| btwnexch3 35552 | Exchange the first endpoin... |
| btwnexch3and 35553 | Deduction form of ~ btwnex... |
| btwnouttr2 35554 | Outer transitivity law for... |
| btwnexch2 35555 | Exchange the outer point o... |
| btwnouttr 35556 | Outer transitivity law for... |
| btwnexch 35557 | Outer transitivity law for... |
| btwnexchand 35558 | Deduction form of ~ btwnex... |
| btwndiff 35559 | There is always a ` c ` di... |
| trisegint 35560 | A line segment between two... |
| funtransport 35563 | The ` TransportTo ` relati... |
| fvtransport 35564 | Calculate the value of the... |
| transportcl 35565 | Closure law for segment tr... |
| transportprops 35566 | Calculate the defining pro... |
| brifs 35575 | Binary relation form of th... |
| ifscgr 35576 | Inner five segment congrue... |
| cgrsub 35577 | Removing identical parts f... |
| brcgr3 35578 | Binary relation form of th... |
| cgr3permute3 35579 | Permutation law for three-... |
| cgr3permute1 35580 | Permutation law for three-... |
| cgr3permute2 35581 | Permutation law for three-... |
| cgr3permute4 35582 | Permutation law for three-... |
| cgr3permute5 35583 | Permutation law for three-... |
| cgr3tr4 35584 | Transitivity law for three... |
| cgr3com 35585 | Commutativity law for thre... |
| cgr3rflx 35586 | Identity law for three-pla... |
| cgrxfr 35587 | A line segment can be divi... |
| btwnxfr 35588 | A condition for extending ... |
| colinrel 35589 | Colinearity is a relations... |
| brcolinear2 35590 | Alternate colinearity bina... |
| brcolinear 35591 | The binary relation form o... |
| colinearex 35592 | The colinear predicate exi... |
| colineardim1 35593 | If ` A ` is colinear with ... |
| colinearperm1 35594 | Permutation law for coline... |
| colinearperm3 35595 | Permutation law for coline... |
| colinearperm2 35596 | Permutation law for coline... |
| colinearperm4 35597 | Permutation law for coline... |
| colinearperm5 35598 | Permutation law for coline... |
| colineartriv1 35599 | Trivial case of colinearit... |
| colineartriv2 35600 | Trivial case of colinearit... |
| btwncolinear1 35601 | Betweenness implies coline... |
| btwncolinear2 35602 | Betweenness implies coline... |
| btwncolinear3 35603 | Betweenness implies coline... |
| btwncolinear4 35604 | Betweenness implies coline... |
| btwncolinear5 35605 | Betweenness implies coline... |
| btwncolinear6 35606 | Betweenness implies coline... |
| colinearxfr 35607 | Transfer law for colineari... |
| lineext 35608 | Extend a line with a missi... |
| brofs2 35609 | Change some conditions for... |
| brifs2 35610 | Change some conditions for... |
| brfs 35611 | Binary relation form of th... |
| fscgr 35612 | Congruence law for the gen... |
| linecgr 35613 | Congruence rule for lines.... |
| linecgrand 35614 | Deduction form of ~ linecg... |
| lineid 35615 | Identity law for points on... |
| idinside 35616 | Law for finding a point in... |
| endofsegid 35617 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 35618 | Deduction form of ~ endofs... |
| btwnconn1lem1 35619 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 35620 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 35621 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 35622 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 35623 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 35624 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 35625 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 35626 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 35627 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 35628 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 35629 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 35630 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 35631 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 35632 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 35633 | Connectitivy law for betwe... |
| btwnconn2 35634 | Another connectivity law f... |
| btwnconn3 35635 | Inner connectivity law for... |
| midofsegid 35636 | If two points fall in the ... |
| segcon2 35637 | Generalization of ~ axsegc... |
| brsegle 35640 | Binary relation form of th... |
| brsegle2 35641 | Alternate characterization... |
| seglecgr12im 35642 | Substitution law for segme... |
| seglecgr12 35643 | Substitution law for segme... |
| seglerflx 35644 | Segment comparison is refl... |
| seglemin 35645 | Any segment is at least as... |
| segletr 35646 | Segment less than is trans... |
| segleantisym 35647 | Antisymmetry law for segme... |
| seglelin 35648 | Linearity law for segment ... |
| btwnsegle 35649 | If ` B ` falls between ` A... |
| colinbtwnle 35650 | Given three colinear point... |
| broutsideof 35653 | Binary relation form of ` ... |
| broutsideof2 35654 | Alternate form of ` Outsid... |
| outsidene1 35655 | Outsideness implies inequa... |
| outsidene2 35656 | Outsideness implies inequa... |
| btwnoutside 35657 | A principle linking outsid... |
| broutsideof3 35658 | Characterization of outsid... |
| outsideofrflx 35659 | Reflexivity of outsideness... |
| outsideofcom 35660 | Commutativity law for outs... |
| outsideoftr 35661 | Transitivity law for outsi... |
| outsideofeq 35662 | Uniqueness law for ` Outsi... |
| outsideofeu 35663 | Given a nondegenerate ray,... |
| outsidele 35664 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 35665 | Outside of implies colinea... |
| funray 35672 | Show that the ` Ray ` rela... |
| fvray 35673 | Calculate the value of the... |
| funline 35674 | Show that the ` Line ` rel... |
| linedegen 35675 | When ` Line ` is applied w... |
| fvline 35676 | Calculate the value of the... |
| liness 35677 | A line is a subset of the ... |
| fvline2 35678 | Alternate definition of a ... |
| lineunray 35679 | A line is composed of a po... |
| lineelsb2 35680 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 35681 | Reflexivity law for line m... |
| linecom 35682 | Commutativity law for line... |
| linerflx2 35683 | Reflexivity law for line m... |
| ellines 35684 | Membership in the set of a... |
| linethru 35685 | If ` A ` is a line contain... |
| hilbert1.1 35686 | There is a line through an... |
| hilbert1.2 35687 | There is at most one line ... |
| linethrueu 35688 | There is a unique line goi... |
| lineintmo 35689 | Two distinct lines interse... |
| fwddifval 35694 | Calculate the value of the... |
| fwddifnval 35695 | The value of the forward d... |
| fwddifn0 35696 | The value of the n-iterate... |
| fwddifnp1 35697 | The value of the n-iterate... |
| rankung 35698 | The rank of the union of t... |
| ranksng 35699 | The rank of a singleton. ... |
| rankelg 35700 | The membership relation is... |
| rankpwg 35701 | The rank of a power set. ... |
| rank0 35702 | The rank of the empty set ... |
| rankeq1o 35703 | The only set with rank ` 1... |
| elhf 35706 | Membership in the heredita... |
| elhf2 35707 | Alternate form of membersh... |
| elhf2g 35708 | Hereditarily finiteness vi... |
| 0hf 35709 | The empty set is a heredit... |
| hfun 35710 | The union of two HF sets i... |
| hfsn 35711 | The singleton of an HF set... |
| hfadj 35712 | Adjoining one HF element t... |
| hfelhf 35713 | Any member of an HF set is... |
| hftr 35714 | The class of all hereditar... |
| hfext 35715 | Extensionality for HF sets... |
| hfuni 35716 | The union of an HF set is ... |
| hfpw 35717 | The power class of an HF s... |
| hfninf 35718 | ` _om ` is not hereditaril... |
| mpomulnzcnf 35719 | Multiplication maps nonzer... |
| a1i14 35720 | Add two antecedents to a w... |
| a1i24 35721 | Add two antecedents to a w... |
| exp5d 35722 | An exportation inference. ... |
| exp5g 35723 | An exportation inference. ... |
| exp5k 35724 | An exportation inference. ... |
| exp56 35725 | An exportation inference. ... |
| exp58 35726 | An exportation inference. ... |
| exp510 35727 | An exportation inference. ... |
| exp511 35728 | An exportation inference. ... |
| exp512 35729 | An exportation inference. ... |
| 3com12d 35730 | Commutation in consequent.... |
| imp5p 35731 | A triple importation infer... |
| imp5q 35732 | A triple importation infer... |
| ecase13d 35733 | Deduction for elimination ... |
| subtr 35734 | Transitivity of implicit s... |
| subtr2 35735 | Transitivity of implicit s... |
| trer 35736 | A relation intersected wit... |
| elicc3 35737 | An equivalent membership c... |
| finminlem 35738 | A useful lemma about finit... |
| gtinf 35739 | Any number greater than an... |
| opnrebl 35740 | A set is open in the stand... |
| opnrebl2 35741 | A set is open in the stand... |
| nn0prpwlem 35742 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 35743 | Two nonnegative integers a... |
| topbnd 35744 | Two equivalent expressions... |
| opnbnd 35745 | A set is open iff it is di... |
| cldbnd 35746 | A set is closed iff it con... |
| ntruni 35747 | A union of interiors is a ... |
| clsun 35748 | A pairwise union of closur... |
| clsint2 35749 | The closure of an intersec... |
| opnregcld 35750 | A set is regularly closed ... |
| cldregopn 35751 | A set if regularly open if... |
| neiin 35752 | Two neighborhoods intersec... |
| hmeoclda 35753 | Homeomorphisms preserve cl... |
| hmeocldb 35754 | Homeomorphisms preserve cl... |
| ivthALT 35755 | An alternate proof of the ... |
| fnerel 35758 | Fineness is a relation. (... |
| isfne 35759 | The predicate " ` B ` is f... |
| isfne4 35760 | The predicate " ` B ` is f... |
| isfne4b 35761 | A condition for a topology... |
| isfne2 35762 | The predicate " ` B ` is f... |
| isfne3 35763 | The predicate " ` B ` is f... |
| fnebas 35764 | A finer cover covers the s... |
| fnetg 35765 | A finer cover generates a ... |
| fnessex 35766 | If ` B ` is finer than ` A... |
| fneuni 35767 | If ` B ` is finer than ` A... |
| fneint 35768 | If a cover is finer than a... |
| fness 35769 | A cover is finer than its ... |
| fneref 35770 | Reflexivity of the finenes... |
| fnetr 35771 | Transitivity of the finene... |
| fneval 35772 | Two covers are finer than ... |
| fneer 35773 | Fineness intersected with ... |
| topfne 35774 | Fineness for covers corres... |
| topfneec 35775 | A cover is equivalent to a... |
| topfneec2 35776 | A topology is precisely id... |
| fnessref 35777 | A cover is finer iff it ha... |
| refssfne 35778 | A cover is a refinement if... |
| neibastop1 35779 | A collection of neighborho... |
| neibastop2lem 35780 | Lemma for ~ neibastop2 . ... |
| neibastop2 35781 | In the topology generated ... |
| neibastop3 35782 | The topology generated by ... |
| topmtcl 35783 | The meet of a collection o... |
| topmeet 35784 | Two equivalent formulation... |
| topjoin 35785 | Two equivalent formulation... |
| fnemeet1 35786 | The meet of a collection o... |
| fnemeet2 35787 | The meet of equivalence cl... |
| fnejoin1 35788 | Join of equivalence classe... |
| fnejoin2 35789 | Join of equivalence classe... |
| fgmin 35790 | Minimality property of a g... |
| neifg 35791 | The neighborhood filter of... |
| tailfval 35792 | The tail function for a di... |
| tailval 35793 | The tail of an element in ... |
| eltail 35794 | An element of a tail. (Co... |
| tailf 35795 | The tail function of a dir... |
| tailini 35796 | A tail contains its initia... |
| tailfb 35797 | The collection of tails of... |
| filnetlem1 35798 | Lemma for ~ filnet . Chan... |
| filnetlem2 35799 | Lemma for ~ filnet . The ... |
| filnetlem3 35800 | Lemma for ~ filnet . (Con... |
| filnetlem4 35801 | Lemma for ~ filnet . (Con... |
| filnet 35802 | A filter has the same conv... |
| tb-ax1 35803 | The first of three axioms ... |
| tb-ax2 35804 | The second of three axioms... |
| tb-ax3 35805 | The third of three axioms ... |
| tbsyl 35806 | The weak syllogism from Ta... |
| re1ax2lem 35807 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 35808 | ~ ax-2 rederived from the ... |
| naim1 35809 | Constructor theorem for ` ... |
| naim2 35810 | Constructor theorem for ` ... |
| naim1i 35811 | Constructor rule for ` -/\... |
| naim2i 35812 | Constructor rule for ` -/\... |
| naim12i 35813 | Constructor rule for ` -/\... |
| nabi1i 35814 | Constructor rule for ` -/\... |
| nabi2i 35815 | Constructor rule for ` -/\... |
| nabi12i 35816 | Constructor rule for ` -/\... |
| df3nandALT1 35819 | The double nand expressed ... |
| df3nandALT2 35820 | The double nand expressed ... |
| andnand1 35821 | Double and in terms of dou... |
| imnand2 35822 | An ` -> ` nand relation. ... |
| nalfal 35823 | Not all sets hold ` F. ` a... |
| nexntru 35824 | There does not exist a set... |
| nexfal 35825 | There does not exist a set... |
| neufal 35826 | There does not exist exact... |
| neutru 35827 | There does not exist exact... |
| nmotru 35828 | There does not exist at mo... |
| mofal 35829 | There exist at most one se... |
| nrmo 35830 | "At most one" restricted e... |
| meran1 35831 | A single axiom for proposi... |
| meran2 35832 | A single axiom for proposi... |
| meran3 35833 | A single axiom for proposi... |
| waj-ax 35834 | A single axiom for proposi... |
| lukshef-ax2 35835 | A single axiom for proposi... |
| arg-ax 35836 | A single axiom for proposi... |
| negsym1 35837 | In the paper "On Variable ... |
| imsym1 35838 | A symmetry with ` -> ` . ... |
| bisym1 35839 | A symmetry with ` <-> ` . ... |
| consym1 35840 | A symmetry with ` /\ ` . ... |
| dissym1 35841 | A symmetry with ` \/ ` . ... |
| nandsym1 35842 | A symmetry with ` -/\ ` . ... |
| unisym1 35843 | A symmetry with ` A. ` . ... |
| exisym1 35844 | A symmetry with ` E. ` . ... |
| unqsym1 35845 | A symmetry with ` E! ` . ... |
| amosym1 35846 | A symmetry with ` E* ` . ... |
| subsym1 35847 | A symmetry with ` [ x / y ... |
| ontopbas 35848 | An ordinal number is a top... |
| onsstopbas 35849 | The class of ordinal numbe... |
| onpsstopbas 35850 | The class of ordinal numbe... |
| ontgval 35851 | The topology generated fro... |
| ontgsucval 35852 | The topology generated fro... |
| onsuctop 35853 | A successor ordinal number... |
| onsuctopon 35854 | One of the topologies on a... |
| ordtoplem 35855 | Membership of the class of... |
| ordtop 35856 | An ordinal is a topology i... |
| onsucconni 35857 | A successor ordinal number... |
| onsucconn 35858 | A successor ordinal number... |
| ordtopconn 35859 | An ordinal topology is con... |
| onintopssconn 35860 | An ordinal topology is con... |
| onsuct0 35861 | A successor ordinal number... |
| ordtopt0 35862 | An ordinal topology is T_0... |
| onsucsuccmpi 35863 | The successor of a success... |
| onsucsuccmp 35864 | The successor of a success... |
| limsucncmpi 35865 | The successor of a limit o... |
| limsucncmp 35866 | The successor of a limit o... |
| ordcmp 35867 | An ordinal topology is com... |
| ssoninhaus 35868 | The ordinal topologies ` 1... |
| onint1 35869 | The ordinal T_1 spaces are... |
| oninhaus 35870 | The ordinal Hausdorff spac... |
| fveleq 35871 | Please add description her... |
| findfvcl 35872 | Please add description her... |
| findreccl 35873 | Please add description her... |
| findabrcl 35874 | Please add description her... |
| nnssi2 35875 | Convert a theorem for real... |
| nnssi3 35876 | Convert a theorem for real... |
| nndivsub 35877 | Please add description her... |
| nndivlub 35878 | A factor of a positive int... |
| ee7.2aOLD 35881 | Lemma for Euclid's Element... |
| dnival 35882 | Value of the "distance to ... |
| dnicld1 35883 | Closure theorem for the "d... |
| dnicld2 35884 | Closure theorem for the "d... |
| dnif 35885 | The "distance to nearest i... |
| dnizeq0 35886 | The distance to nearest in... |
| dnizphlfeqhlf 35887 | The distance to nearest in... |
| rddif2 35888 | Variant of ~ rddif . (Con... |
| dnibndlem1 35889 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 35890 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 35891 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 35892 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 35893 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 35894 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 35895 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 35896 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 35897 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 35898 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 35899 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 35900 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 35901 | Lemma for ~ dnibnd . (Con... |
| dnibnd 35902 | The "distance to nearest i... |
| dnicn 35903 | The "distance to nearest i... |
| knoppcnlem1 35904 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 35905 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 35906 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 35907 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 35908 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 35909 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 35910 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 35911 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 35912 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 35913 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 35914 | Lemma for ~ knoppcn . (Co... |
| knoppcn 35915 | The continuous nowhere dif... |
| knoppcld 35916 | Closure theorem for Knopp'... |
| unblimceq0lem 35917 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 35918 | If ` F ` is unbounded near... |
| unbdqndv1 35919 | If the difference quotient... |
| unbdqndv2lem1 35920 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 35921 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 35922 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 35923 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 35924 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 35925 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 35926 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 35927 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 35928 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 35929 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 35930 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 35931 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 35932 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 35933 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 35934 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 35935 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 35936 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 35937 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 35938 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 35939 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 35940 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 35941 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 35942 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 35943 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 35944 | Lemma for ~ knoppndv . (C... |
| knoppndv 35945 | The continuous nowhere dif... |
| knoppf 35946 | Knopp's function is a func... |
| knoppcn2 35947 | Variant of ~ knoppcn with ... |
| cnndvlem1 35948 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 35949 | Lemma for ~ cnndv . (Cont... |
| cnndv 35950 | There exists a continuous ... |
| bj-mp2c 35951 | A double modus ponens infe... |
| bj-mp2d 35952 | A double modus ponens infe... |
| bj-0 35953 | A syntactic theorem. See ... |
| bj-1 35954 | In this proof, the use of ... |
| bj-a1k 35955 | Weakening of ~ ax-1 . As ... |
| bj-poni 35956 | Inference associated with ... |
| bj-nnclav 35957 | When ` F. ` is substituted... |
| bj-nnclavi 35958 | Inference associated with ... |
| bj-nnclavc 35959 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 35960 | Inference associated with ... |
| bj-jarrii 35961 | Inference associated with ... |
| bj-imim21 35962 | The propositional function... |
| bj-imim21i 35963 | Inference associated with ... |
| bj-peircestab 35964 | Over minimal implicational... |
| bj-stabpeirce 35965 | This minimal implicational... |
| bj-syl66ib 35966 | A mixed syllogism inferenc... |
| bj-orim2 35967 | Proof of ~ orim2 from the ... |
| bj-currypeirce 35968 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 35969 | Peirce's axiom ~ peirce im... |
| bj-animbi 35970 | Conjunction in terms of im... |
| bj-currypara 35971 | Curry's paradox. Note tha... |
| bj-con2com 35972 | A commuted form of the con... |
| bj-con2comi 35973 | Inference associated with ... |
| bj-pm2.01i 35974 | Inference associated with ... |
| bj-nimn 35975 | If a formula is true, then... |
| bj-nimni 35976 | Inference associated with ... |
| bj-peircei 35977 | Inference associated with ... |
| bj-looinvi 35978 | Inference associated with ... |
| bj-looinvii 35979 | Inference associated with ... |
| bj-mt2bi 35980 | Version of ~ mt2 where the... |
| bj-ntrufal 35981 | The negation of a theorem ... |
| bj-fal 35982 | Shortening of ~ fal using ... |
| bj-jaoi1 35983 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 35984 | Shortens ~ consensus (110>... |
| bj-dfbi4 35985 | Alternate definition of th... |
| bj-dfbi5 35986 | Alternate definition of th... |
| bj-dfbi6 35987 | Alternate definition of th... |
| bj-bijust0ALT 35988 | Alternate proof of ~ bijus... |
| bj-bijust00 35989 | A self-implication does no... |
| bj-consensus 35990 | Version of ~ consensus exp... |
| bj-consensusALT 35991 | Alternate proof of ~ bj-co... |
| bj-df-ifc 35992 | Candidate definition for t... |
| bj-dfif 35993 | Alternate definition of th... |
| bj-ififc 35994 | A biconditional connecting... |
| bj-imbi12 35995 | Uncurried (imported) form ... |
| bj-biorfi 35996 | This should be labeled "bi... |
| bj-falor 35997 | Dual of ~ truan (which has... |
| bj-falor2 35998 | Dual of ~ truan . (Contri... |
| bj-bibibi 35999 | A property of the bicondit... |
| bj-imn3ani 36000 | Duplication of ~ bnj1224 .... |
| bj-andnotim 36001 | Two ways of expressing a c... |
| bj-bi3ant 36002 | This used to be in the mai... |
| bj-bisym 36003 | This used to be in the mai... |
| bj-bixor 36004 | Equivalence of two ternary... |
| bj-axdd2 36005 | This implication, proved u... |
| bj-axd2d 36006 | This implication, proved u... |
| bj-axtd 36007 | This implication, proved f... |
| bj-gl4 36008 | In a normal modal logic, t... |
| bj-axc4 36009 | Over minimal calculus, the... |
| prvlem1 36014 | An elementary property of ... |
| prvlem2 36015 | An elementary property of ... |
| bj-babygodel 36016 | See the section header com... |
| bj-babylob 36017 | See the section header com... |
| bj-godellob 36018 | Proof of Gödel's theo... |
| bj-genr 36019 | Generalization rule on the... |
| bj-genl 36020 | Generalization rule on the... |
| bj-genan 36021 | Generalization rule on a c... |
| bj-mpgs 36022 | From a closed form theorem... |
| bj-2alim 36023 | Closed form of ~ 2alimi . ... |
| bj-2exim 36024 | Closed form of ~ 2eximi . ... |
| bj-alanim 36025 | Closed form of ~ alanimi .... |
| bj-2albi 36026 | Closed form of ~ 2albii . ... |
| bj-notalbii 36027 | Equivalence of universal q... |
| bj-2exbi 36028 | Closed form of ~ 2exbii . ... |
| bj-3exbi 36029 | Closed form of ~ 3exbii . ... |
| bj-sylgt2 36030 | Uncurried (imported) form ... |
| bj-alrimg 36031 | The general form of the *a... |
| bj-alrimd 36032 | A slightly more general ~ ... |
| bj-sylget 36033 | Dual statement of ~ sylgt ... |
| bj-sylget2 36034 | Uncurried (imported) form ... |
| bj-exlimg 36035 | The general form of the *e... |
| bj-sylge 36036 | Dual statement of ~ sylg (... |
| bj-exlimd 36037 | A slightly more general ~ ... |
| bj-nfimexal 36038 | A weak from of nonfreeness... |
| bj-alexim 36039 | Closed form of ~ aleximi .... |
| bj-nexdh 36040 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 36041 | Uncurried (imported) form ... |
| bj-hbxfrbi 36042 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 36043 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 36044 | Distribute quantifiers ove... |
| bj-exalimi 36045 | An inference for distribut... |
| bj-exalims 36046 | Distributing quantifiers o... |
| bj-exalimsi 36047 | An inference for distribut... |
| bj-ax12ig 36048 | A lemma used to prove a we... |
| bj-ax12i 36049 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 36050 | Closed form of ~ nfim and ... |
| bj-cbvalimt 36051 | A lemma in closed form use... |
| bj-cbveximt 36052 | A lemma in closed form use... |
| bj-eximALT 36053 | Alternate proof of ~ exim ... |
| bj-aleximiALT 36054 | Alternate proof of ~ alexi... |
| bj-eximcom 36055 | A commuted form of ~ exim ... |
| bj-ax12wlem 36056 | A lemma used to prove a we... |
| bj-cbvalim 36057 | A lemma used to prove ~ bj... |
| bj-cbvexim 36058 | A lemma used to prove ~ bj... |
| bj-cbvalimi 36059 | An equality-free general i... |
| bj-cbveximi 36060 | An equality-free general i... |
| bj-cbval 36061 | Changing a bound variable ... |
| bj-cbvex 36062 | Changing a bound variable ... |
| bj-ssbeq 36065 | Substitution in an equalit... |
| bj-ssblem1 36066 | A lemma for the definiens ... |
| bj-ssblem2 36067 | An instance of ~ ax-11 pro... |
| bj-ax12v 36068 | A weaker form of ~ ax-12 a... |
| bj-ax12 36069 | Remove a DV condition from... |
| bj-ax12ssb 36070 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 36071 | Special case of ~ 19.41 pr... |
| bj-equsexval 36072 | Special case of ~ equsexv ... |
| bj-subst 36073 | Proof of ~ sbalex from cor... |
| bj-ssbid2 36074 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 36075 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 36076 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 36077 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 36078 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 36079 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 36080 | Proof of ~ ax6e (hence ~ a... |
| bj-spimvwt 36081 | Closed form of ~ spimvw . ... |
| bj-spnfw 36082 | Theorem close to a closed ... |
| bj-cbvexiw 36083 | Change bound variable. Th... |
| bj-cbvexivw 36084 | Change bound variable. Th... |
| bj-modald 36085 | A short form of the axiom ... |
| bj-denot 36086 | A weakening of ~ ax-6 and ... |
| bj-eqs 36087 | A lemma for substitutions,... |
| bj-cbvexw 36088 | Change bound variable. Th... |
| bj-ax12w 36089 | The general statement that... |
| bj-ax89 36090 | A theorem which could be u... |
| bj-elequ12 36091 | An identity law for the no... |
| bj-cleljusti 36092 | One direction of ~ cleljus... |
| bj-alcomexcom 36093 | Commutation of two existen... |
| bj-hbalt 36094 | Closed form of ~ hbal . W... |
| axc11n11 36095 | Proof of ~ axc11n from { ~... |
| axc11n11r 36096 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 36097 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 36098 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 36099 | Alternate proof of ~ bj-ax... |
| bj-sb 36100 | A weak variant of ~ sbid2 ... |
| bj-modalbe 36101 | The predicate-calculus ver... |
| bj-spst 36102 | Closed form of ~ sps . On... |
| bj-19.21bit 36103 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 36104 | Closed form of ~ 19.23bi .... |
| bj-nexrt 36105 | Closed form of ~ nexr . C... |
| bj-alrim 36106 | Closed form of ~ alrimi . ... |
| bj-alrim2 36107 | Uncurried (imported) form ... |
| bj-nfdt0 36108 | A theorem close to a close... |
| bj-nfdt 36109 | Closed form of ~ nf5d and ... |
| bj-nexdt 36110 | Closed form of ~ nexd . (... |
| bj-nexdvt 36111 | Closed form of ~ nexdv . ... |
| bj-alexbiex 36112 | Adding a second quantifier... |
| bj-exexbiex 36113 | Adding a second quantifier... |
| bj-alalbial 36114 | Adding a second quantifier... |
| bj-exalbial 36115 | Adding a second quantifier... |
| bj-19.9htbi 36116 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 36117 | Strengthening ~ hbnt by re... |
| bj-biexal1 36118 | A general FOL biconditiona... |
| bj-biexal2 36119 | When ` ph ` is substituted... |
| bj-biexal3 36120 | When ` ph ` is substituted... |
| bj-bialal 36121 | When ` ph ` is substituted... |
| bj-biexex 36122 | When ` ph ` is substituted... |
| bj-hbext 36123 | Closed form of ~ hbex . (... |
| bj-nfalt 36124 | Closed form of ~ nfal . (... |
| bj-nfext 36125 | Closed form of ~ nfex . (... |
| bj-eeanvw 36126 | Version of ~ exdistrv with... |
| bj-modal4 36127 | First-order logic form of ... |
| bj-modal4e 36128 | First-order logic form of ... |
| bj-modalb 36129 | A short form of the axiom ... |
| bj-wnf1 36130 | When ` ph ` is substituted... |
| bj-wnf2 36131 | When ` ph ` is substituted... |
| bj-wnfanf 36132 | When ` ph ` is substituted... |
| bj-wnfenf 36133 | When ` ph ` is substituted... |
| bj-substax12 36134 | Equivalent form of the axi... |
| bj-substw 36135 | Weak form of the LHS of ~ ... |
| bj-nnfbi 36138 | If two formulas are equiva... |
| bj-nnfbd 36139 | If two formulas are equiva... |
| bj-nnfbii 36140 | If two formulas are equiva... |
| bj-nnfa 36141 | Nonfreeness implies the eq... |
| bj-nnfad 36142 | Nonfreeness implies the eq... |
| bj-nnfai 36143 | Nonfreeness implies the eq... |
| bj-nnfe 36144 | Nonfreeness implies the eq... |
| bj-nnfed 36145 | Nonfreeness implies the eq... |
| bj-nnfei 36146 | Nonfreeness implies the eq... |
| bj-nnfea 36147 | Nonfreeness implies the eq... |
| bj-nnfead 36148 | Nonfreeness implies the eq... |
| bj-nnfeai 36149 | Nonfreeness implies the eq... |
| bj-dfnnf2 36150 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 36151 | New nonfreeness implies ol... |
| bj-wnfnf 36152 | When ` ph ` is substituted... |
| bj-nnfnt 36153 | A variable is nonfree in a... |
| bj-nnftht 36154 | A variable is nonfree in a... |
| bj-nnfth 36155 | A variable is nonfree in a... |
| bj-nnfnth 36156 | A variable is nonfree in t... |
| bj-nnfim1 36157 | A consequence of nonfreene... |
| bj-nnfim2 36158 | A consequence of nonfreene... |
| bj-nnfim 36159 | Nonfreeness in the anteced... |
| bj-nnfimd 36160 | Nonfreeness in the anteced... |
| bj-nnfan 36161 | Nonfreeness in both conjun... |
| bj-nnfand 36162 | Nonfreeness in both conjun... |
| bj-nnfor 36163 | Nonfreeness in both disjun... |
| bj-nnford 36164 | Nonfreeness in both disjun... |
| bj-nnfbit 36165 | Nonfreeness in both sides ... |
| bj-nnfbid 36166 | Nonfreeness in both sides ... |
| bj-nnfv 36167 | A non-occurring variable i... |
| bj-nnf-alrim 36168 | Proof of the closed form o... |
| bj-nnf-exlim 36169 | Proof of the closed form o... |
| bj-dfnnf3 36170 | Alternate definition of no... |
| bj-nfnnfTEMP 36171 | New nonfreeness is equival... |
| bj-nnfa1 36172 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 36173 | See ~ nfe1 . (Contributed... |
| bj-19.12 36174 | See ~ 19.12 . Could be la... |
| bj-nnflemaa 36175 | One of four lemmas for non... |
| bj-nnflemee 36176 | One of four lemmas for non... |
| bj-nnflemae 36177 | One of four lemmas for non... |
| bj-nnflemea 36178 | One of four lemmas for non... |
| bj-nnfalt 36179 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 36180 | See ~ nfex and ~ bj-nfext ... |
| bj-stdpc5t 36181 | Alias of ~ bj-nnf-alrim fo... |
| bj-19.21t 36182 | Statement ~ 19.21t proved ... |
| bj-19.23t 36183 | Statement ~ 19.23t proved ... |
| bj-19.36im 36184 | One direction of ~ 19.36 f... |
| bj-19.37im 36185 | One direction of ~ 19.37 f... |
| bj-19.42t 36186 | Closed form of ~ 19.42 fro... |
| bj-19.41t 36187 | Closed form of ~ 19.41 fro... |
| bj-sbft 36188 | Version of ~ sbft using ` ... |
| bj-pm11.53vw 36189 | Version of ~ pm11.53v with... |
| bj-pm11.53v 36190 | Version of ~ pm11.53v with... |
| bj-pm11.53a 36191 | A variant of ~ pm11.53v . ... |
| bj-equsvt 36192 | A variant of ~ equsv . (C... |
| bj-equsalvwd 36193 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 36194 | Variant of ~ equsexvw . (... |
| bj-sbievwd 36195 | Variant of ~ sbievw . (Co... |
| bj-axc10 36196 | Alternate proof of ~ axc10... |
| bj-alequex 36197 | A fol lemma. See ~ aleque... |
| bj-spimt2 36198 | A step in the proof of ~ s... |
| bj-cbv3ta 36199 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 36200 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 36201 | A theorem close to a close... |
| bj-hbsb3 36202 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 36203 | A theorem close to a close... |
| bj-nfs1t2 36204 | A theorem close to a close... |
| bj-nfs1 36205 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 36206 | Version of ~ axc10 with a ... |
| bj-spimtv 36207 | Version of ~ spimt with a ... |
| bj-cbv3hv2 36208 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 36209 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 36210 | Version of ~ cbv2h with a ... |
| bj-cbv2v 36211 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 36212 | Version of ~ cbvald with a... |
| bj-cbvexdv 36213 | Version of ~ cbvexd with a... |
| bj-cbval2vv 36214 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 36215 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 36216 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 36217 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 36218 | Version of ~ cbvex4v with ... |
| bj-equsalhv 36219 | Version of ~ equsalh with ... |
| bj-axc11nv 36220 | Version of ~ axc11n with a... |
| bj-aecomsv 36221 | Version of ~ aecoms with a... |
| bj-axc11v 36222 | Version of ~ axc11 with a ... |
| bj-drnf2v 36223 | Version of ~ drnf2 with a ... |
| bj-equs45fv 36224 | Version of ~ equs45f with ... |
| bj-hbs1 36225 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 36226 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 36227 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 36228 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 36229 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 36230 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 36231 | Biconditional version of a... |
| bj-hbaeb 36232 | Biconditional version of ~... |
| bj-hbnaeb 36233 | Biconditional version of ~... |
| bj-dvv 36234 | A special instance of ~ bj... |
| bj-equsal1t 36235 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 36236 | Inference associated with ... |
| bj-equsal1 36237 | One direction of ~ equsal ... |
| bj-equsal2 36238 | One direction of ~ equsal ... |
| bj-equsal 36239 | Shorter proof of ~ equsal ... |
| stdpc5t 36240 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 36241 | More direct proof of ~ std... |
| 2stdpc5 36242 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 36243 | Proof of ~ 19.21t from ~ s... |
| exlimii 36244 | Inference associated with ... |
| ax11-pm 36245 | Proof of ~ ax-11 similar t... |
| ax6er 36246 | Commuted form of ~ ax6e . ... |
| exlimiieq1 36247 | Inferring a theorem when i... |
| exlimiieq2 36248 | Inferring a theorem when i... |
| ax11-pm2 36249 | Proof of ~ ax-11 from the ... |
| bj-sbsb 36250 | Biconditional showing two ... |
| bj-dfsb2 36251 | Alternate (dual) definitio... |
| bj-sbf3 36252 | Substitution has no effect... |
| bj-sbf4 36253 | Substitution has no effect... |
| bj-eu3f 36254 | Version of ~ eu3v where th... |
| bj-sblem1 36255 | Lemma for substitution. (... |
| bj-sblem2 36256 | Lemma for substitution. (... |
| bj-sblem 36257 | Lemma for substitution. (... |
| bj-sbievw1 36258 | Lemma for substitution. (... |
| bj-sbievw2 36259 | Lemma for substitution. (... |
| bj-sbievw 36260 | Lemma for substitution. C... |
| bj-sbievv 36261 | Version of ~ sbie with a s... |
| bj-moeub 36262 | Uniqueness is equivalent t... |
| bj-sbidmOLD 36263 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 36264 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 36265 | Curried (exported) form of... |
| bj-dvelimv 36266 | A version of ~ dvelim usin... |
| bj-nfeel2 36267 | Nonfreeness in a membershi... |
| bj-axc14nf 36268 | Proof of a version of ~ ax... |
| bj-axc14 36269 | Alternate proof of ~ axc14... |
| mobidvALT 36270 | Alternate proof of ~ mobid... |
| sbn1ALT 36271 | Alternate proof of ~ sbn1 ... |
| eliminable1 36272 | A theorem used to prove th... |
| eliminable2a 36273 | A theorem used to prove th... |
| eliminable2b 36274 | A theorem used to prove th... |
| eliminable2c 36275 | A theorem used to prove th... |
| eliminable3a 36276 | A theorem used to prove th... |
| eliminable3b 36277 | A theorem used to prove th... |
| eliminable-velab 36278 | A theorem used to prove th... |
| eliminable-veqab 36279 | A theorem used to prove th... |
| eliminable-abeqv 36280 | A theorem used to prove th... |
| eliminable-abeqab 36281 | A theorem used to prove th... |
| eliminable-abelv 36282 | A theorem used to prove th... |
| eliminable-abelab 36283 | A theorem used to prove th... |
| bj-denoteslem 36284 | Lemma for ~ bj-denotes . ... |
| bj-denotes 36285 | This would be the justific... |
| bj-issettru 36286 | Weak version of ~ isset wi... |
| bj-elabtru 36287 | This is as close as we can... |
| bj-issetwt 36288 | Closed form of ~ bj-issetw... |
| bj-issetw 36289 | The closest one can get to... |
| bj-elissetALT 36290 | Alternate proof of ~ eliss... |
| bj-issetiv 36291 | Version of ~ bj-isseti wit... |
| bj-isseti 36292 | Version of ~ isseti with a... |
| bj-ralvw 36293 | A weak version of ~ ralv n... |
| bj-rexvw 36294 | A weak version of ~ rexv n... |
| bj-rababw 36295 | A weak version of ~ rabab ... |
| bj-rexcom4bv 36296 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 36297 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 36298 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 36299 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 36300 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 36301 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 36302 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 36303 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 36304 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 36305 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 36306 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 36307 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 36308 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 36309 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 36310 | Remove from ~ spcimdv depe... |
| elelb 36311 | Equivalence between two co... |
| bj-pwvrelb 36312 | Characterization of the el... |
| bj-nfcsym 36313 | The nonfreeness quantifier... |
| bj-sbeqALT 36314 | Substitution in an equalit... |
| bj-sbeq 36315 | Distribute proper substitu... |
| bj-sbceqgALT 36316 | Distribute proper substitu... |
| bj-csbsnlem 36317 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 36318 | Substitution in a singleto... |
| bj-sbel1 36319 | Version of ~ sbcel1g when ... |
| bj-abv 36320 | The class of sets verifyin... |
| bj-abvALT 36321 | Alternate version of ~ bj-... |
| bj-ab0 36322 | The class of sets verifyin... |
| bj-abf 36323 | Shorter proof of ~ abf (wh... |
| bj-csbprc 36324 | More direct proof of ~ csb... |
| bj-exlimvmpi 36325 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 36326 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 36327 | Lemma for theorems of the ... |
| bj-exlimmpbir 36328 | Lemma for theorems of the ... |
| bj-vtoclf 36329 | Remove dependency on ~ ax-... |
| bj-vtocl 36330 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 36331 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 36332 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 36333 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 36334 | A version of ~ vtoclg with... |
| bj-rabeqbid 36335 | Version of ~ rabeqbidv wit... |
| bj-seex 36336 | Version of ~ seex with a d... |
| bj-nfcf 36337 | Version of ~ df-nfc with a... |
| bj-zfauscl 36338 | General version of ~ zfaus... |
| bj-elabd2ALT 36339 | Alternate proof of ~ elabd... |
| bj-unrab 36340 | Generalization of ~ unrab ... |
| bj-inrab 36341 | Generalization of ~ inrab ... |
| bj-inrab2 36342 | Shorter proof of ~ inrab .... |
| bj-inrab3 36343 | Generalization of ~ dfrab3... |
| bj-rabtr 36344 | Restricted class abstracti... |
| bj-rabtrALT 36345 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 36346 | Proof of ~ bj-rabtr found ... |
| bj-gabss 36349 | Inclusion of generalized c... |
| bj-gabssd 36350 | Inclusion of generalized c... |
| bj-gabeqd 36351 | Equality of generalized cl... |
| bj-gabeqis 36352 | Equality of generalized cl... |
| bj-elgab 36353 | Elements of a generalized ... |
| bj-gabima 36354 | Generalized class abstract... |
| bj-ru0 36357 | The FOL part of Russell's ... |
| bj-ru1 36358 | A version of Russell's par... |
| bj-ru 36359 | Remove dependency on ~ ax-... |
| currysetlem 36360 | Lemma for ~ currysetlem , ... |
| curryset 36361 | Curry's paradox in set the... |
| currysetlem1 36362 | Lemma for ~ currysetALT . ... |
| currysetlem2 36363 | Lemma for ~ currysetALT . ... |
| currysetlem3 36364 | Lemma for ~ currysetALT . ... |
| currysetALT 36365 | Alternate proof of ~ curry... |
| bj-n0i 36366 | Inference associated with ... |
| bj-disjsn01 36367 | Disjointness of the single... |
| bj-0nel1 36368 | The empty set does not bel... |
| bj-1nel0 36369 | ` 1o ` does not belong to ... |
| bj-xpimasn 36370 | The image of a singleton, ... |
| bj-xpima1sn 36371 | The image of a singleton b... |
| bj-xpima1snALT 36372 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 36373 | The image of a singleton b... |
| bj-xpnzex 36374 | If the first factor of a p... |
| bj-xpexg2 36375 | Curried (exported) form of... |
| bj-xpnzexb 36376 | If the first factor of a p... |
| bj-cleq 36377 | Substitution property for ... |
| bj-snsetex 36378 | The class of sets "whose s... |
| bj-clexab 36379 | Sethood of certain classes... |
| bj-sngleq 36382 | Substitution property for ... |
| bj-elsngl 36383 | Characterization of the el... |
| bj-snglc 36384 | Characterization of the el... |
| bj-snglss 36385 | The singletonization of a ... |
| bj-0nelsngl 36386 | The empty set is not a mem... |
| bj-snglinv 36387 | Inverse of singletonizatio... |
| bj-snglex 36388 | A class is a set if and on... |
| bj-tageq 36391 | Substitution property for ... |
| bj-eltag 36392 | Characterization of the el... |
| bj-0eltag 36393 | The empty set belongs to t... |
| bj-tagn0 36394 | The tagging of a class is ... |
| bj-tagss 36395 | The tagging of a class is ... |
| bj-snglsstag 36396 | The singletonization is in... |
| bj-sngltagi 36397 | The singletonization is in... |
| bj-sngltag 36398 | The singletonization and t... |
| bj-tagci 36399 | Characterization of the el... |
| bj-tagcg 36400 | Characterization of the el... |
| bj-taginv 36401 | Inverse of tagging. (Cont... |
| bj-tagex 36402 | A class is a set if and on... |
| bj-xtageq 36403 | The products of a given cl... |
| bj-xtagex 36404 | The product of a set and t... |
| bj-projeq 36407 | Substitution property for ... |
| bj-projeq2 36408 | Substitution property for ... |
| bj-projun 36409 | The class projection on a ... |
| bj-projex 36410 | Sethood of the class proje... |
| bj-projval 36411 | Value of the class project... |
| bj-1upleq 36414 | Substitution property for ... |
| bj-pr1eq 36417 | Substitution property for ... |
| bj-pr1un 36418 | The first projection prese... |
| bj-pr1val 36419 | Value of the first project... |
| bj-pr11val 36420 | Value of the first project... |
| bj-pr1ex 36421 | Sethood of the first proje... |
| bj-1uplth 36422 | The characteristic propert... |
| bj-1uplex 36423 | A monuple is a set if and ... |
| bj-1upln0 36424 | A monuple is nonempty. (C... |
| bj-2upleq 36427 | Substitution property for ... |
| bj-pr21val 36428 | Value of the first project... |
| bj-pr2eq 36431 | Substitution property for ... |
| bj-pr2un 36432 | The second projection pres... |
| bj-pr2val 36433 | Value of the second projec... |
| bj-pr22val 36434 | Value of the second projec... |
| bj-pr2ex 36435 | Sethood of the second proj... |
| bj-2uplth 36436 | The characteristic propert... |
| bj-2uplex 36437 | A couple is a set if and o... |
| bj-2upln0 36438 | A couple is nonempty. (Co... |
| bj-2upln1upl 36439 | A couple is never equal to... |
| bj-rcleqf 36440 | Relative version of ~ cleq... |
| bj-rcleq 36441 | Relative version of ~ dfcl... |
| bj-reabeq 36442 | Relative form of ~ eqabb .... |
| bj-disj2r 36443 | Relative version of ~ ssdi... |
| bj-sscon 36444 | Contraposition law for rel... |
| bj-abex 36445 | Two ways of stating that t... |
| bj-clex 36446 | Two ways of stating that a... |
| bj-axsn 36447 | Two ways of stating the ax... |
| bj-snexg 36449 | A singleton built on a set... |
| bj-snex 36450 | A singleton is a set. See... |
| bj-axbun 36451 | Two ways of stating the ax... |
| bj-unexg 36453 | Existence of binary unions... |
| bj-prexg 36454 | Existence of unordered pai... |
| bj-prex 36455 | Existence of unordered pai... |
| bj-axadj 36456 | Two ways of stating the ax... |
| bj-adjg1 36458 | Existence of the result of... |
| bj-snfromadj 36459 | Singleton from adjunction ... |
| bj-prfromadj 36460 | Unordered pair from adjunc... |
| bj-adjfrombun 36461 | Adjunction from singleton ... |
| eleq2w2ALT 36462 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 36463 | Alternate proof of ~ clel3... |
| bj-pw0ALT 36464 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 36465 | Quantitative version of ~ ... |
| bj-unirel 36466 | Quantitative version of ~ ... |
| bj-elpwg 36467 | If the intersection of two... |
| bj-velpwALT 36468 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 36469 | Alternate proof of ~ elpwg... |
| bj-vjust 36470 | Justification theorem for ... |
| bj-nul 36471 | Two formulations of the ax... |
| bj-nuliota 36472 | Definition of the empty se... |
| bj-nuliotaALT 36473 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 36474 | Alternate proof of ~ vtocl... |
| bj-elsn12g 36475 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 36476 | Biconditional version of ~... |
| bj-pwcfsdom 36477 | Remove hypothesis from ~ p... |
| bj-grur1 36478 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 36479 | The extension of a predica... |
| bj-dfid2ALT 36480 | Alternate version of ~ dfi... |
| bj-0nelopab 36481 | The empty set is never an ... |
| bj-brrelex12ALT 36482 | Two classes related by a b... |
| bj-epelg 36483 | The membership relation an... |
| bj-epelb 36484 | Two classes are related by... |
| bj-nsnid 36485 | A set does not contain the... |
| bj-rdg0gALT 36486 | Alternate proof of ~ rdg0g... |
| bj-evaleq 36487 | Equality theorem for the `... |
| bj-evalfun 36488 | The evaluation at a class ... |
| bj-evalfn 36489 | The evaluation at a class ... |
| bj-evalval 36490 | Value of the evaluation at... |
| bj-evalid 36491 | The evaluation at a set of... |
| bj-ndxarg 36492 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 36493 | Closed general form of ~ s... |
| bj-rest00 36496 | An elementwise intersectio... |
| bj-restsn 36497 | An elementwise intersectio... |
| bj-restsnss 36498 | Special case of ~ bj-rests... |
| bj-restsnss2 36499 | Special case of ~ bj-rests... |
| bj-restsn0 36500 | An elementwise intersectio... |
| bj-restsn10 36501 | Special case of ~ bj-rests... |
| bj-restsnid 36502 | The elementwise intersecti... |
| bj-rest10 36503 | An elementwise intersectio... |
| bj-rest10b 36504 | Alternate version of ~ bj-... |
| bj-restn0 36505 | An elementwise intersectio... |
| bj-restn0b 36506 | Alternate version of ~ bj-... |
| bj-restpw 36507 | The elementwise intersecti... |
| bj-rest0 36508 | An elementwise intersectio... |
| bj-restb 36509 | An elementwise intersectio... |
| bj-restv 36510 | An elementwise intersectio... |
| bj-resta 36511 | An elementwise intersectio... |
| bj-restuni 36512 | The union of an elementwis... |
| bj-restuni2 36513 | The union of an elementwis... |
| bj-restreg 36514 | A reformulation of the axi... |
| bj-raldifsn 36515 | All elements in a set sati... |
| bj-0int 36516 | If ` A ` is a collection o... |
| bj-mooreset 36517 | A Moore collection is a se... |
| bj-ismoore 36520 | Characterization of Moore ... |
| bj-ismoored0 36521 | Necessary condition to be ... |
| bj-ismoored 36522 | Necessary condition to be ... |
| bj-ismoored2 36523 | Necessary condition to be ... |
| bj-ismooredr 36524 | Sufficient condition to be... |
| bj-ismooredr2 36525 | Sufficient condition to be... |
| bj-discrmoore 36526 | The powerclass ` ~P A ` is... |
| bj-0nmoore 36527 | The empty set is not a Moo... |
| bj-snmoore 36528 | A singleton is a Moore col... |
| bj-snmooreb 36529 | A singleton is a Moore col... |
| bj-prmoore 36530 | A pair formed of two neste... |
| bj-0nelmpt 36531 | The empty set is not an el... |
| bj-mptval 36532 | Value of a function given ... |
| bj-dfmpoa 36533 | An equivalent definition o... |
| bj-mpomptALT 36534 | Alternate proof of ~ mpomp... |
| setsstrset 36551 | Relation between ~ df-sets... |
| bj-nfald 36552 | Variant of ~ nfald . (Con... |
| bj-nfexd 36553 | Variant of ~ nfexd . (Con... |
| copsex2d 36554 | Implicit substitution dedu... |
| copsex2b 36555 | Biconditional form of ~ co... |
| opelopabd 36556 | Membership of an ordere pa... |
| opelopabb 36557 | Membership of an ordered p... |
| opelopabbv 36558 | Membership of an ordered p... |
| bj-opelrelex 36559 | The coordinates of an orde... |
| bj-opelresdm 36560 | If an ordered pair is in a... |
| bj-brresdm 36561 | If two classes are related... |
| brabd0 36562 | Expressing that two sets a... |
| brabd 36563 | Expressing that two sets a... |
| bj-brab2a1 36564 | "Unbounded" version of ~ b... |
| bj-opabssvv 36565 | A variant of ~ relopabiv (... |
| bj-funidres 36566 | The restricted identity re... |
| bj-opelidb 36567 | Characterization of the or... |
| bj-opelidb1 36568 | Characterization of the or... |
| bj-inexeqex 36569 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 36570 | If the intersection of two... |
| bj-opelid 36571 | Characterization of the or... |
| bj-ideqg 36572 | Characterization of the cl... |
| bj-ideqgALT 36573 | Alternate proof of ~ bj-id... |
| bj-ideqb 36574 | Characterization of classe... |
| bj-idres 36575 | Alternate expression for t... |
| bj-opelidres 36576 | Characterization of the or... |
| bj-idreseq 36577 | Sufficient condition for t... |
| bj-idreseqb 36578 | Characterization for two c... |
| bj-ideqg1 36579 | For sets, the identity rel... |
| bj-ideqg1ALT 36580 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 36581 | Characterization of the co... |
| bj-elid3 36582 | Characterization of the co... |
| bj-elid4 36583 | Characterization of the el... |
| bj-elid5 36584 | Characterization of the el... |
| bj-elid6 36585 | Characterization of the el... |
| bj-elid7 36586 | Characterization of the el... |
| bj-diagval 36589 | Value of the functionalize... |
| bj-diagval2 36590 | Value of the functionalize... |
| bj-eldiag 36591 | Characterization of the el... |
| bj-eldiag2 36592 | Characterization of the el... |
| bj-imdirvallem 36595 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 36596 | Value of the functionalize... |
| bj-imdirval2lem 36597 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 36598 | Value of the functionalize... |
| bj-imdirval3 36599 | Value of the functionalize... |
| bj-imdiridlem 36600 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 36601 | Functorial property of the... |
| bj-opelopabid 36602 | Membership in an ordered-p... |
| bj-opabco 36603 | Composition of ordered-pai... |
| bj-xpcossxp 36604 | The composition of two Car... |
| bj-imdirco 36605 | Functorial property of the... |
| bj-iminvval 36608 | Value of the functionalize... |
| bj-iminvval2 36609 | Value of the functionalize... |
| bj-iminvid 36610 | Functorial property of the... |
| bj-inftyexpitaufo 36617 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 36620 | An element of the circle a... |
| bj-inftyexpiinv 36623 | Utility theorem for the in... |
| bj-inftyexpiinj 36624 | Injectivity of the paramet... |
| bj-inftyexpidisj 36625 | An element of the circle a... |
| bj-ccinftydisj 36628 | The circle at infinity is ... |
| bj-elccinfty 36629 | A lemma for infinite exten... |
| bj-ccssccbar 36632 | Complex numbers are extend... |
| bj-ccinftyssccbar 36633 | Infinite extended complex ... |
| bj-pinftyccb 36636 | The class ` pinfty ` is an... |
| bj-pinftynrr 36637 | The extended complex numbe... |
| bj-minftyccb 36640 | The class ` minfty ` is an... |
| bj-minftynrr 36641 | The extended complex numbe... |
| bj-pinftynminfty 36642 | The extended complex numbe... |
| bj-rrhatsscchat 36651 | The real projective line i... |
| bj-imafv 36666 | If the direct image of a s... |
| bj-funun 36667 | Value of a function expres... |
| bj-fununsn1 36668 | Value of a function expres... |
| bj-fununsn2 36669 | Value of a function expres... |
| bj-fvsnun1 36670 | The value of a function wi... |
| bj-fvsnun2 36671 | The value of a function wi... |
| bj-fvmptunsn1 36672 | Value of a function expres... |
| bj-fvmptunsn2 36673 | Value of a function expres... |
| bj-iomnnom 36674 | The canonical bijection fr... |
| bj-smgrpssmgm 36683 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 36684 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 36685 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 36686 | Monoids are semigroups (el... |
| bj-cmnssmnd 36687 | Commutative monoids are mo... |
| bj-cmnssmndel 36688 | Commutative monoids are mo... |
| bj-grpssmnd 36689 | Groups are monoids. (Cont... |
| bj-grpssmndel 36690 | Groups are monoids (elemen... |
| bj-ablssgrp 36691 | Abelian groups are groups.... |
| bj-ablssgrpel 36692 | Abelian groups are groups ... |
| bj-ablsscmn 36693 | Abelian groups are commuta... |
| bj-ablsscmnel 36694 | Abelian groups are commuta... |
| bj-modssabl 36695 | (The additive groups of) m... |
| bj-vecssmod 36696 | Vector spaces are modules.... |
| bj-vecssmodel 36697 | Vector spaces are modules ... |
| bj-finsumval0 36700 | Value of a finite sum. (C... |
| bj-fvimacnv0 36701 | Variant of ~ fvimacnv wher... |
| bj-isvec 36702 | The predicate "is a vector... |
| bj-fldssdrng 36703 | Fields are division rings.... |
| bj-flddrng 36704 | Fields are division rings ... |
| bj-rrdrg 36705 | The field of real numbers ... |
| bj-isclm 36706 | The predicate "is a subcom... |
| bj-isrvec 36709 | The predicate "is a real v... |
| bj-rvecmod 36710 | Real vector spaces are mod... |
| bj-rvecssmod 36711 | Real vector spaces are mod... |
| bj-rvecrr 36712 | The field of scalars of a ... |
| bj-isrvecd 36713 | The predicate "is a real v... |
| bj-rvecvec 36714 | Real vector spaces are vec... |
| bj-isrvec2 36715 | The predicate "is a real v... |
| bj-rvecssvec 36716 | Real vector spaces are vec... |
| bj-rveccmod 36717 | Real vector spaces are sub... |
| bj-rvecsscmod 36718 | Real vector spaces are sub... |
| bj-rvecsscvec 36719 | Real vector spaces are sub... |
| bj-rveccvec 36720 | Real vector spaces are sub... |
| bj-rvecssabl 36721 | (The additive groups of) r... |
| bj-rvecabl 36722 | (The additive groups of) r... |
| bj-subcom 36723 | A consequence of commutati... |
| bj-lineqi 36724 | Solution of a (scalar) lin... |
| bj-bary1lem 36725 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 36726 | Lemma for bj-bary1: comput... |
| bj-bary1 36727 | Barycentric coordinates in... |
| bj-endval 36730 | Value of the monoid of end... |
| bj-endbase 36731 | Base set of the monoid of ... |
| bj-endcomp 36732 | Composition law of the mon... |
| bj-endmnd 36733 | The monoid of endomorphism... |
| taupilem3 36734 | Lemma for tau-related theo... |
| taupilemrplb 36735 | A set of positive reals ha... |
| taupilem1 36736 | Lemma for ~ taupi . A pos... |
| taupilem2 36737 | Lemma for ~ taupi . The s... |
| taupi 36738 | Relationship between ` _ta... |
| dfgcd3 36739 | Alternate definition of th... |
| irrdifflemf 36740 | Lemma for ~ irrdiff . The... |
| irrdiff 36741 | The irrationals are exactl... |
| iccioo01 36742 | The closed unit interval i... |
| csbrecsg 36743 | Move class substitution in... |
| csbrdgg 36744 | Move class substitution in... |
| csboprabg 36745 | Move class substitution in... |
| csbmpo123 36746 | Move class substitution in... |
| con1bii2 36747 | A contraposition inference... |
| con2bii2 36748 | A contraposition inference... |
| vtoclefex 36749 | Implicit substitution of a... |
| rnmptsn 36750 | The range of a function ma... |
| f1omptsnlem 36751 | This is the core of the pr... |
| f1omptsn 36752 | A function mapping to sing... |
| mptsnunlem 36753 | This is the core of the pr... |
| mptsnun 36754 | A class ` B ` is equal to ... |
| dissneqlem 36755 | This is the core of the pr... |
| dissneq 36756 | Any topology that contains... |
| exlimim 36757 | Closed form of ~ exlimimd ... |
| exlimimd 36758 | Existential elimination ru... |
| exellim 36759 | Closed form of ~ exellimdd... |
| exellimddv 36760 | Eliminate an antecedent wh... |
| topdifinfindis 36761 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 36762 | This is the core of the pr... |
| topdifinffin 36763 | Part of Exercise 3 of [Mun... |
| topdifinf 36764 | Part of Exercise 3 of [Mun... |
| topdifinfeq 36765 | Two different ways of defi... |
| icorempo 36766 | Closed-below, open-above i... |
| icoreresf 36767 | Closed-below, open-above i... |
| icoreval 36768 | Value of the closed-below,... |
| icoreelrnab 36769 | Elementhood in the set of ... |
| isbasisrelowllem1 36770 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 36771 | Lemma for ~ isbasisrelowl ... |
| icoreclin 36772 | The set of closed-below, o... |
| isbasisrelowl 36773 | The set of all closed-belo... |
| icoreunrn 36774 | The union of all closed-be... |
| istoprelowl 36775 | The set of all closed-belo... |
| icoreelrn 36776 | A class abstraction which ... |
| iooelexlt 36777 | An element of an open inte... |
| relowlssretop 36778 | The lower limit topology o... |
| relowlpssretop 36779 | The lower limit topology o... |
| sucneqond 36780 | Inequality of an ordinal s... |
| sucneqoni 36781 | Inequality of an ordinal s... |
| onsucuni3 36782 | If an ordinal number has a... |
| 1oequni2o 36783 | The ordinal number ` 1o ` ... |
| rdgsucuni 36784 | If an ordinal number has a... |
| rdgeqoa 36785 | If a recursive function wi... |
| elxp8 36786 | Membership in a Cartesian ... |
| cbveud 36787 | Deduction used to change b... |
| cbvreud 36788 | Deduction used to change b... |
| difunieq 36789 | The difference of unions i... |
| inunissunidif 36790 | Theorem about subsets of t... |
| rdgellim 36791 | Elementhood in a recursive... |
| rdglimss 36792 | A recursive definition at ... |
| rdgssun 36793 | In a recursive definition ... |
| exrecfnlem 36794 | Lemma for ~ exrecfn . (Co... |
| exrecfn 36795 | Theorem about the existenc... |
| exrecfnpw 36796 | For any base set, a set wh... |
| finorwe 36797 | If the Axiom of Infinity i... |
| dffinxpf 36800 | This theorem is the same a... |
| finxpeq1 36801 | Equality theorem for Carte... |
| finxpeq2 36802 | Equality theorem for Carte... |
| csbfinxpg 36803 | Distribute proper substitu... |
| finxpreclem1 36804 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 36805 | Lemma for ` ^^ ` recursion... |
| finxp0 36806 | The value of Cartesian exp... |
| finxp1o 36807 | The value of Cartesian exp... |
| finxpreclem3 36808 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 36809 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 36810 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 36811 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 36812 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 36813 | The value of Cartesian exp... |
| finxp2o 36814 | The value of Cartesian exp... |
| finxp3o 36815 | The value of Cartesian exp... |
| finxpnom 36816 | Cartesian exponentiation w... |
| finxp00 36817 | Cartesian exponentiation o... |
| iunctb2 36818 | Using the axiom of countab... |
| domalom 36819 | A class which dominates ev... |
| isinf2 36820 | The converse of ~ isinf . ... |
| ctbssinf 36821 | Using the axiom of choice,... |
| ralssiun 36822 | The index set of an indexe... |
| nlpineqsn 36823 | For every point ` p ` of a... |
| nlpfvineqsn 36824 | Given a subset ` A ` of ` ... |
| fvineqsnf1 36825 | A theorem about functions ... |
| fvineqsneu 36826 | A theorem about functions ... |
| fvineqsneq 36827 | A theorem about functions ... |
| pibp16 36828 | Property P000016 of pi-bas... |
| pibp19 36829 | Property P000019 of pi-bas... |
| pibp21 36830 | Property P000021 of pi-bas... |
| pibt1 36831 | Theorem T000001 of pi-base... |
| pibt2 36832 | Theorem T000002 of pi-base... |
| wl-section-prop 36833 | Intuitionistic logic is no... |
| wl-section-boot 36837 | In this section, I provide... |
| wl-luk-imim1i 36838 | Inference adding common co... |
| wl-luk-syl 36839 | An inference version of th... |
| wl-luk-imtrid 36840 | A syllogism rule of infere... |
| wl-luk-pm2.18d 36841 | Deduction based on reducti... |
| wl-luk-con4i 36842 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 36843 | Inference rule. Copy of ~... |
| wl-luk-a1i 36844 | Inference rule. Copy of ~... |
| wl-luk-mpi 36845 | A nested modus ponens infe... |
| wl-luk-imim2i 36846 | Inference adding common an... |
| wl-luk-imtrdi 36847 | A syllogism rule of infere... |
| wl-luk-ax3 36848 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 36849 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 36850 | This theorem, called "Asse... |
| wl-luk-com12 36851 | Inference that swaps (comm... |
| wl-luk-pm2.21 36852 | From a wff and its negatio... |
| wl-luk-con1i 36853 | A contraposition inference... |
| wl-luk-ja 36854 | Inference joining the ante... |
| wl-luk-imim2 36855 | A closed form of syllogism... |
| wl-luk-a1d 36856 | Deduction introducing an e... |
| wl-luk-ax2 36857 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 36858 | Principle of identity. Th... |
| wl-luk-notnotr 36859 | Converse of double negatio... |
| wl-luk-pm2.04 36860 | Swap antecedents. Theorem... |
| wl-section-impchain 36861 | An implication like ` ( ps... |
| wl-impchain-mp-x 36862 | This series of theorems pr... |
| wl-impchain-mp-0 36863 | This theorem is the start ... |
| wl-impchain-mp-1 36864 | This theorem is in fact a ... |
| wl-impchain-mp-2 36865 | This theorem is in fact a ... |
| wl-impchain-com-1.x 36866 | It is often convenient to ... |
| wl-impchain-com-1.1 36867 | A degenerate form of antec... |
| wl-impchain-com-1.2 36868 | This theorem is in fact a ... |
| wl-impchain-com-1.3 36869 | This theorem is in fact a ... |
| wl-impchain-com-1.4 36870 | This theorem is in fact a ... |
| wl-impchain-com-n.m 36871 | This series of theorems al... |
| wl-impchain-com-2.3 36872 | This theorem is in fact a ... |
| wl-impchain-com-2.4 36873 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 36874 | This theorem is in fact a ... |
| wl-impchain-a1-x 36875 | If an implication chain is... |
| wl-impchain-a1-1 36876 | Inference rule, a copy of ... |
| wl-impchain-a1-2 36877 | Inference rule, a copy of ... |
| wl-impchain-a1-3 36878 | Inference rule, a copy of ... |
| wl-ifp-ncond1 36879 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 36880 | If one case of an ` if- ` ... |
| wl-ifpimpr 36881 | If one case of an ` if- ` ... |
| wl-ifp4impr 36882 | If one case of an ` if- ` ... |
| wl-df-3xor 36883 | Alternative definition of ... |
| wl-df3xor2 36884 | Alternative definition of ... |
| wl-df3xor3 36885 | Alternative form of ~ wl-d... |
| wl-3xortru 36886 | If the first input is true... |
| wl-3xorfal 36887 | If the first input is fals... |
| wl-3xorbi 36888 | Triple xor can be replaced... |
| wl-3xorbi2 36889 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 36890 | Equivalence theorem for tr... |
| wl-3xorbi123i 36891 | Equivalence theorem for tr... |
| wl-3xorrot 36892 | Rotation law for triple xo... |
| wl-3xorcoma 36893 | Commutative law for triple... |
| wl-3xorcomb 36894 | Commutative law for triple... |
| wl-3xornot1 36895 | Flipping the first input f... |
| wl-3xornot 36896 | Triple xor distributes ove... |
| wl-1xor 36897 | In the recursive scheme ... |
| wl-2xor 36898 | In the recursive scheme ... |
| wl-df-3mintru2 36899 | Alternative definition of ... |
| wl-df2-3mintru2 36900 | The adder carry in disjunc... |
| wl-df3-3mintru2 36901 | The adder carry in conjunc... |
| wl-df4-3mintru2 36902 | An alternative definition ... |
| wl-1mintru1 36903 | Using the recursion formul... |
| wl-1mintru2 36904 | Using the recursion formul... |
| wl-2mintru1 36905 | Using the recursion formul... |
| wl-2mintru2 36906 | Using the recursion formul... |
| wl-df3maxtru1 36907 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 36909 | A version of ~ ax-wl-13v w... |
| wl-mps 36910 | Replacing a nested consequ... |
| wl-syls1 36911 | Replacing a nested consequ... |
| wl-syls2 36912 | Replacing a nested anteced... |
| wl-embant 36913 | A true wff can always be a... |
| wl-orel12 36914 | In a conjunctive normal fo... |
| wl-cases2-dnf 36915 | A particular instance of ~... |
| wl-cbvmotv 36916 | Change bound variable. Us... |
| wl-moteq 36917 | Change bound variable. Us... |
| wl-motae 36918 | Change bound variable. Us... |
| wl-moae 36919 | Two ways to express "at mo... |
| wl-euae 36920 | Two ways to express "exact... |
| wl-nax6im 36921 | The following series of th... |
| wl-hbae1 36922 | This specialization of ~ h... |
| wl-naevhba1v 36923 | An instance of ~ hbn1w app... |
| wl-spae 36924 | Prove an instance of ~ sp ... |
| wl-speqv 36925 | Under the assumption ` -. ... |
| wl-19.8eqv 36926 | Under the assumption ` -. ... |
| wl-19.2reqv 36927 | Under the assumption ` -. ... |
| wl-nfalv 36928 | If ` x ` is not present in... |
| wl-nfimf1 36929 | An antecedent is irrelevan... |
| wl-nfae1 36930 | Unlike ~ nfae , this speci... |
| wl-nfnae1 36931 | Unlike ~ nfnae , this spec... |
| wl-aetr 36932 | A transitive law for varia... |
| wl-axc11r 36933 | Same as ~ axc11r , but usi... |
| wl-dral1d 36934 | A version of ~ dral1 with ... |
| wl-cbvalnaed 36935 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 36936 | A more general version of ... |
| wl-exeq 36937 | The semantics of ` E. x y ... |
| wl-aleq 36938 | The semantics of ` A. x y ... |
| wl-nfeqfb 36939 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 36940 | If ` y ` is not free in ` ... |
| wl-equsalvw 36941 | Version of ~ equsalv with ... |
| wl-equsald 36942 | Deduction version of ~ equ... |
| wl-equsal 36943 | A useful equivalence relat... |
| wl-equsal1t 36944 | The expression ` x = y ` i... |
| wl-equsalcom 36945 | This simple equivalence ea... |
| wl-equsal1i 36946 | The antecedent ` x = y ` i... |
| wl-sb6rft 36947 | A specialization of ~ wl-e... |
| wl-cbvalsbi 36948 | Change bounded variables i... |
| wl-sbrimt 36949 | Substitution with a variab... |
| wl-sblimt 36950 | Substitution with a variab... |
| wl-sb9v 36951 | Commutation of quantificat... |
| wl-sb8ft 36952 | Substitution of variable i... |
| wl-sb8eft 36953 | Substitution of variable i... |
| wl-sb8t 36954 | Substitution of variable i... |
| wl-sb8et 36955 | Substitution of variable i... |
| wl-sbhbt 36956 | Closed form of ~ sbhb . C... |
| wl-sbnf1 36957 | Two ways expressing that `... |
| wl-equsb3 36958 | ~ equsb3 with a distinctor... |
| wl-equsb4 36959 | Substitution applied to an... |
| wl-2sb6d 36960 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 36961 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 36962 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 36963 | Version of ~ sbcom2 with a... |
| wl-sbalnae 36964 | A theorem used in eliminat... |
| wl-sbal1 36965 | A theorem used in eliminat... |
| wl-sbal2 36966 | Move quantifier in and out... |
| wl-2spsbbi 36967 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 36968 | This theorem provides a ba... |
| wl-lem-nexmo 36969 | This theorem provides a ba... |
| wl-lem-moexsb 36970 | The antecedent ` A. x ( ph... |
| wl-alanbii 36971 | This theorem extends ~ ala... |
| wl-mo2df 36972 | Version of ~ mof with a co... |
| wl-mo2tf 36973 | Closed form of ~ mof with ... |
| wl-eudf 36974 | Version of ~ eu6 with a co... |
| wl-eutf 36975 | Closed form of ~ eu6 with ... |
| wl-euequf 36976 | ~ euequ proved with a dist... |
| wl-mo2t 36977 | Closed form of ~ mof . (C... |
| wl-mo3t 36978 | Closed form of ~ mo3 . (C... |
| wl-nfsbtv 36979 | Closed form of ~ nfsbv . ... |
| wl-sb8eut 36980 | Substitution of variable i... |
| wl-sb8eutv 36981 | Substitution of variable i... |
| wl-sb8mot 36982 | Substitution of variable i... |
| wl-sb8motv 36983 | Substitution of variable i... |
| wl-issetft 36984 | A closed form of ~ issetf ... |
| wl-axc11rc11 36985 | Proving ~ axc11r from ~ ax... |
| wl-ax11-lem1 36987 | A transitive law for varia... |
| wl-ax11-lem2 36988 | Lemma. (Contributed by Wo... |
| wl-ax11-lem3 36989 | Lemma. (Contributed by Wo... |
| wl-ax11-lem4 36990 | Lemma. (Contributed by Wo... |
| wl-ax11-lem5 36991 | Lemma. (Contributed by Wo... |
| wl-ax11-lem6 36992 | Lemma. (Contributed by Wo... |
| wl-ax11-lem7 36993 | Lemma. (Contributed by Wo... |
| wl-ax11-lem8 36994 | Lemma. (Contributed by Wo... |
| wl-ax11-lem9 36995 | The easy part when ` x ` c... |
| wl-ax11-lem10 36996 | We now have prepared every... |
| wl-clabv 36997 | Variant of ~ df-clab , whe... |
| wl-dfclab 36998 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 36999 | Using class abstraction in... |
| wl-clabt 37000 | Using class abstraction in... |
| rabiun 37001 | Abstraction restricted to ... |
| iundif1 37002 | Indexed union of class dif... |
| imadifss 37003 | The difference of images i... |
| cureq 37004 | Equality theorem for curry... |
| unceq 37005 | Equality theorem for uncur... |
| curf 37006 | Functional property of cur... |
| uncf 37007 | Functional property of unc... |
| curfv 37008 | Value of currying. (Contr... |
| uncov 37009 | Value of uncurrying. (Con... |
| curunc 37010 | Currying of uncurrying. (... |
| unccur 37011 | Uncurrying of currying. (... |
| phpreu 37012 | Theorem related to pigeonh... |
| finixpnum 37013 | A finite Cartesian product... |
| fin2solem 37014 | Lemma for ~ fin2so . (Con... |
| fin2so 37015 | Any totally ordered Tarski... |
| ltflcei 37016 | Theorem to move the floor ... |
| leceifl 37017 | Theorem to move the floor ... |
| sin2h 37018 | Half-angle rule for sine. ... |
| cos2h 37019 | Half-angle rule for cosine... |
| tan2h 37020 | Half-angle rule for tangen... |
| lindsadd 37021 | In a vector space, the uni... |
| lindsdom 37022 | A linearly independent set... |
| lindsenlbs 37023 | A maximal linearly indepen... |
| matunitlindflem1 37024 | One direction of ~ matunit... |
| matunitlindflem2 37025 | One direction of ~ matunit... |
| matunitlindf 37026 | A matrix over a field is i... |
| ptrest 37027 | Expressing a restriction o... |
| ptrecube 37028 | Any point in an open set o... |
| poimirlem1 37029 | Lemma for ~ poimir - the v... |
| poimirlem2 37030 | Lemma for ~ poimir - conse... |
| poimirlem3 37031 | Lemma for ~ poimir to add ... |
| poimirlem4 37032 | Lemma for ~ poimir connect... |
| poimirlem5 37033 | Lemma for ~ poimir to esta... |
| poimirlem6 37034 | Lemma for ~ poimir establi... |
| poimirlem7 37035 | Lemma for ~ poimir , simil... |
| poimirlem8 37036 | Lemma for ~ poimir , estab... |
| poimirlem9 37037 | Lemma for ~ poimir , estab... |
| poimirlem10 37038 | Lemma for ~ poimir establi... |
| poimirlem11 37039 | Lemma for ~ poimir connect... |
| poimirlem12 37040 | Lemma for ~ poimir connect... |
| poimirlem13 37041 | Lemma for ~ poimir - for a... |
| poimirlem14 37042 | Lemma for ~ poimir - for a... |
| poimirlem15 37043 | Lemma for ~ poimir , that ... |
| poimirlem16 37044 | Lemma for ~ poimir establi... |
| poimirlem17 37045 | Lemma for ~ poimir establi... |
| poimirlem18 37046 | Lemma for ~ poimir stating... |
| poimirlem19 37047 | Lemma for ~ poimir establi... |
| poimirlem20 37048 | Lemma for ~ poimir establi... |
| poimirlem21 37049 | Lemma for ~ poimir stating... |
| poimirlem22 37050 | Lemma for ~ poimir , that ... |
| poimirlem23 37051 | Lemma for ~ poimir , two w... |
| poimirlem24 37052 | Lemma for ~ poimir , two w... |
| poimirlem25 37053 | Lemma for ~ poimir stating... |
| poimirlem26 37054 | Lemma for ~ poimir showing... |
| poimirlem27 37055 | Lemma for ~ poimir showing... |
| poimirlem28 37056 | Lemma for ~ poimir , a var... |
| poimirlem29 37057 | Lemma for ~ poimir connect... |
| poimirlem30 37058 | Lemma for ~ poimir combini... |
| poimirlem31 37059 | Lemma for ~ poimir , assig... |
| poimirlem32 37060 | Lemma for ~ poimir , combi... |
| poimir 37061 | Poincare-Miranda theorem. ... |
| broucube 37062 | Brouwer - or as Kulpa call... |
| heicant 37063 | Heine-Cantor theorem: a co... |
| opnmbllem0 37064 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 37065 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 37066 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 37067 | The difference between two... |
| mblfinlem4 37068 | Backward direction of ~ is... |
| ismblfin 37069 | Measurability in terms of ... |
| ovoliunnfl 37070 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 37071 | Demonstration of ~ ovoliun... |
| voliunnfl 37072 | ~ voliun is incompatible w... |
| volsupnfl 37073 | ~ volsup is incompatible w... |
| mbfresfi 37074 | Measurability of a piecewi... |
| mbfposadd 37075 | If the sum of two measurab... |
| cnambfre 37076 | A real-valued, a.e. contin... |
| dvtanlem 37077 | Lemma for ~ dvtan - the do... |
| dvtan 37078 | Derivative of tangent. (C... |
| itg2addnclem 37079 | An alternate expression fo... |
| itg2addnclem2 37080 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 37081 | Lemma incomprehensible in ... |
| itg2addnc 37082 | Alternate proof of ~ itg2a... |
| itg2gt0cn 37083 | ~ itg2gt0 holds on functio... |
| ibladdnclem 37084 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 37085 | Choice-free analogue of ~ ... |
| itgaddnclem1 37086 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 37087 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 37088 | Choice-free analogue of ~ ... |
| iblsubnc 37089 | Choice-free analogue of ~ ... |
| itgsubnc 37090 | Choice-free analogue of ~ ... |
| iblabsnclem 37091 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 37092 | Choice-free analogue of ~ ... |
| iblmulc2nc 37093 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 37094 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 37095 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 37096 | Choice-free analogue of ~ ... |
| itgabsnc 37097 | Choice-free analogue of ~ ... |
| itggt0cn 37098 | ~ itggt0 holds for continu... |
| ftc1cnnclem 37099 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 37100 | Choice-free proof of ~ ftc... |
| ftc1anclem1 37101 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 37102 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 37103 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 37104 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 37105 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 37106 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 37107 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 37108 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 37109 | ~ ftc1a holds for function... |
| ftc2nc 37110 | Choice-free proof of ~ ftc... |
| asindmre 37111 | Real part of domain of dif... |
| dvasin 37112 | Derivative of arcsine. (C... |
| dvacos 37113 | Derivative of arccosine. ... |
| dvreasin 37114 | Real derivative of arcsine... |
| dvreacos 37115 | Real derivative of arccosi... |
| areacirclem1 37116 | Antiderivative of cross-se... |
| areacirclem2 37117 | Endpoint-inclusive continu... |
| areacirclem3 37118 | Integrability of cross-sec... |
| areacirclem4 37119 | Endpoint-inclusive continu... |
| areacirclem5 37120 | Finding the cross-section ... |
| areacirc 37121 | The area of a circle of ra... |
| unirep 37122 | Define a quantity whose de... |
| cover2 37123 | Two ways of expressing the... |
| cover2g 37124 | Two ways of expressing the... |
| brabg2 37125 | Relation by a binary relat... |
| opelopab3 37126 | Ordered pair membership in... |
| cocanfo 37127 | Cancellation of a surjecti... |
| brresi2 37128 | Restriction of a binary re... |
| fnopabeqd 37129 | Equality deduction for fun... |
| fvopabf4g 37130 | Function value of an opera... |
| fnopabco 37131 | Composition of a function ... |
| opropabco 37132 | Composition of an operator... |
| cocnv 37133 | Composition with a functio... |
| f1ocan1fv 37134 | Cancel a composition by a ... |
| f1ocan2fv 37135 | Cancel a composition by th... |
| inixp 37136 | Intersection of Cartesian ... |
| upixp 37137 | Universal property of the ... |
| abrexdom 37138 | An indexed set is dominate... |
| abrexdom2 37139 | An indexed set is dominate... |
| ac6gf 37140 | Axiom of Choice. (Contrib... |
| indexa 37141 | If for every element of an... |
| indexdom 37142 | If for every element of an... |
| frinfm 37143 | A subset of a well-founded... |
| welb 37144 | A nonempty subset of a wel... |
| supex2g 37145 | Existence of supremum. (C... |
| supclt 37146 | Closure of supremum. (Con... |
| supubt 37147 | Upper bound property of su... |
| filbcmb 37148 | Combine a finite set of lo... |
| fzmul 37149 | Membership of a product in... |
| sdclem2 37150 | Lemma for ~ sdc . (Contri... |
| sdclem1 37151 | Lemma for ~ sdc . (Contri... |
| sdc 37152 | Strong dependent choice. ... |
| fdc 37153 | Finite version of dependen... |
| fdc1 37154 | Variant of ~ fdc with no s... |
| seqpo 37155 | Two ways to say that a seq... |
| incsequz 37156 | An increasing sequence of ... |
| incsequz2 37157 | An increasing sequence of ... |
| nnubfi 37158 | A bounded above set of pos... |
| nninfnub 37159 | An infinite set of positiv... |
| subspopn 37160 | An open set is open in the... |
| neificl 37161 | Neighborhoods are closed u... |
| lpss2 37162 | Limit points of a subset a... |
| metf1o 37163 | Use a bijection with a met... |
| blssp 37164 | A ball in the subspace met... |
| mettrifi 37165 | Generalized triangle inequ... |
| lmclim2 37166 | A sequence in a metric spa... |
| geomcau 37167 | If the distance between co... |
| caures 37168 | The restriction of a Cauch... |
| caushft 37169 | A shifted Cauchy sequence ... |
| constcncf 37170 | A constant function is a c... |
| cnres2 37171 | The restriction of a conti... |
| cnresima 37172 | A continuous function is c... |
| cncfres 37173 | A continuous function on c... |
| istotbnd 37177 | The predicate "is a totall... |
| istotbnd2 37178 | The predicate "is a totall... |
| istotbnd3 37179 | A metric space is totally ... |
| totbndmet 37180 | The predicate "totally bou... |
| 0totbnd 37181 | The metric (there is only ... |
| sstotbnd2 37182 | Condition for a subset of ... |
| sstotbnd 37183 | Condition for a subset of ... |
| sstotbnd3 37184 | Use a net that is not nece... |
| totbndss 37185 | A subset of a totally boun... |
| equivtotbnd 37186 | If the metric ` M ` is "st... |
| isbnd 37188 | The predicate "is a bounde... |
| bndmet 37189 | A bounded metric space is ... |
| isbndx 37190 | A "bounded extended metric... |
| isbnd2 37191 | The predicate "is a bounde... |
| isbnd3 37192 | A metric space is bounded ... |
| isbnd3b 37193 | A metric space is bounded ... |
| bndss 37194 | A subset of a bounded metr... |
| blbnd 37195 | A ball is bounded. (Contr... |
| ssbnd 37196 | A subset of a metric space... |
| totbndbnd 37197 | A totally bounded metric s... |
| equivbnd 37198 | If the metric ` M ` is "st... |
| bnd2lem 37199 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 37200 | If balls are totally bound... |
| prdsbnd 37201 | The product metric over fi... |
| prdstotbnd 37202 | The product metric over fi... |
| prdsbnd2 37203 | If balls are totally bound... |
| cntotbnd 37204 | A subset of the complex nu... |
| cnpwstotbnd 37205 | A subset of ` A ^ I ` , wh... |
| ismtyval 37208 | The set of isometries betw... |
| isismty 37209 | The condition "is an isome... |
| ismtycnv 37210 | The inverse of an isometry... |
| ismtyima 37211 | The image of a ball under ... |
| ismtyhmeolem 37212 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 37213 | An isometry is a homeomorp... |
| ismtybndlem 37214 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 37215 | Isometries preserve bounde... |
| ismtyres 37216 | A restriction of an isomet... |
| heibor1lem 37217 | Lemma for ~ heibor1 . A c... |
| heibor1 37218 | One half of ~ heibor , tha... |
| heiborlem1 37219 | Lemma for ~ heibor . We w... |
| heiborlem2 37220 | Lemma for ~ heibor . Subs... |
| heiborlem3 37221 | Lemma for ~ heibor . Usin... |
| heiborlem4 37222 | Lemma for ~ heibor . Usin... |
| heiborlem5 37223 | Lemma for ~ heibor . The ... |
| heiborlem6 37224 | Lemma for ~ heibor . Sinc... |
| heiborlem7 37225 | Lemma for ~ heibor . Sinc... |
| heiborlem8 37226 | Lemma for ~ heibor . The ... |
| heiborlem9 37227 | Lemma for ~ heibor . Disc... |
| heiborlem10 37228 | Lemma for ~ heibor . The ... |
| heibor 37229 | Generalized Heine-Borel Th... |
| bfplem1 37230 | Lemma for ~ bfp . The seq... |
| bfplem2 37231 | Lemma for ~ bfp . Using t... |
| bfp 37232 | Banach fixed point theorem... |
| rrnval 37235 | The n-dimensional Euclidea... |
| rrnmval 37236 | The value of the Euclidean... |
| rrnmet 37237 | Euclidean space is a metri... |
| rrndstprj1 37238 | The distance between two p... |
| rrndstprj2 37239 | Bound on the distance betw... |
| rrncmslem 37240 | Lemma for ~ rrncms . (Con... |
| rrncms 37241 | Euclidean space is complet... |
| repwsmet 37242 | The supremum metric on ` R... |
| rrnequiv 37243 | The supremum metric on ` R... |
| rrntotbnd 37244 | A set in Euclidean space i... |
| rrnheibor 37245 | Heine-Borel theorem for Eu... |
| ismrer1 37246 | An isometry between ` RR `... |
| reheibor 37247 | Heine-Borel theorem for re... |
| iccbnd 37248 | A closed interval in ` RR ... |
| icccmpALT 37249 | A closed interval in ` RR ... |
| isass 37254 | The predicate "is an assoc... |
| isexid 37255 | The predicate ` G ` has a ... |
| ismgmOLD 37258 | Obsolete version of ~ ismg... |
| clmgmOLD 37259 | Obsolete version of ~ mgmc... |
| opidonOLD 37260 | Obsolete version of ~ mndp... |
| rngopidOLD 37261 | Obsolete version of ~ mndp... |
| opidon2OLD 37262 | Obsolete version of ~ mndp... |
| isexid2 37263 | If ` G e. ( Magma i^i ExId... |
| exidu1 37264 | Uniqueness of the left and... |
| idrval 37265 | The value of the identity ... |
| iorlid 37266 | A magma right and left ide... |
| cmpidelt 37267 | A magma right and left ide... |
| smgrpismgmOLD 37270 | Obsolete version of ~ sgrp... |
| issmgrpOLD 37271 | Obsolete version of ~ issg... |
| smgrpmgm 37272 | A semigroup is a magma. (... |
| smgrpassOLD 37273 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 37276 | Obsolete version of ~ mnds... |
| mndoisexid 37277 | A monoid has an identity e... |
| mndoismgmOLD 37278 | Obsolete version of ~ mndm... |
| mndomgmid 37279 | A monoid is a magma with a... |
| ismndo 37280 | The predicate "is a monoid... |
| ismndo1 37281 | The predicate "is a monoid... |
| ismndo2 37282 | The predicate "is a monoid... |
| grpomndo 37283 | A group is a monoid. (Con... |
| exidcl 37284 | Closure of the binary oper... |
| exidreslem 37285 | Lemma for ~ exidres and ~ ... |
| exidres 37286 | The restriction of a binar... |
| exidresid 37287 | The restriction of a binar... |
| ablo4pnp 37288 | A commutative/associative ... |
| grpoeqdivid 37289 | Two group elements are equ... |
| grposnOLD 37290 | The group operation for th... |
| elghomlem1OLD 37293 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 37294 | Obsolete as of 15-Mar-2020... |
| elghomOLD 37295 | Obsolete version of ~ isgh... |
| ghomlinOLD 37296 | Obsolete version of ~ ghml... |
| ghomidOLD 37297 | Obsolete version of ~ ghmi... |
| ghomf 37298 | Mapping property of a grou... |
| ghomco 37299 | The composition of two gro... |
| ghomdiv 37300 | Group homomorphisms preser... |
| grpokerinj 37301 | A group homomorphism is in... |
| relrngo 37304 | The class of all unital ri... |
| isrngo 37305 | The predicate "is a (unita... |
| isrngod 37306 | Conditions that determine ... |
| rngoi 37307 | The properties of a unital... |
| rngosm 37308 | Functionality of the multi... |
| rngocl 37309 | Closure of the multiplicat... |
| rngoid 37310 | The multiplication operati... |
| rngoideu 37311 | The unity element of a rin... |
| rngodi 37312 | Distributive law for the m... |
| rngodir 37313 | Distributive law for the m... |
| rngoass 37314 | Associative law for the mu... |
| rngo2 37315 | A ring element plus itself... |
| rngoablo 37316 | A ring's addition operatio... |
| rngoablo2 37317 | In a unital ring the addit... |
| rngogrpo 37318 | A ring's addition operatio... |
| rngone0 37319 | The base set of a ring is ... |
| rngogcl 37320 | Closure law for the additi... |
| rngocom 37321 | The addition operation of ... |
| rngoaass 37322 | The addition operation of ... |
| rngoa32 37323 | The addition operation of ... |
| rngoa4 37324 | Rearrangement of 4 terms i... |
| rngorcan 37325 | Right cancellation law for... |
| rngolcan 37326 | Left cancellation law for ... |
| rngo0cl 37327 | A ring has an additive ide... |
| rngo0rid 37328 | The additive identity of a... |
| rngo0lid 37329 | The additive identity of a... |
| rngolz 37330 | The zero of a unital ring ... |
| rngorz 37331 | The zero of a unital ring ... |
| rngosn3 37332 | Obsolete as of 25-Jan-2020... |
| rngosn4 37333 | Obsolete as of 25-Jan-2020... |
| rngosn6 37334 | Obsolete as of 25-Jan-2020... |
| rngonegcl 37335 | A ring is closed under neg... |
| rngoaddneg1 37336 | Adding the negative in a r... |
| rngoaddneg2 37337 | Adding the negative in a r... |
| rngosub 37338 | Subtraction in a ring, in ... |
| rngmgmbs4 37339 | The range of an internal o... |
| rngodm1dm2 37340 | In a unital ring the domai... |
| rngorn1 37341 | In a unital ring the range... |
| rngorn1eq 37342 | In a unital ring the range... |
| rngomndo 37343 | In a unital ring the multi... |
| rngoidmlem 37344 | The unity element of a rin... |
| rngolidm 37345 | The unity element of a rin... |
| rngoridm 37346 | The unity element of a rin... |
| rngo1cl 37347 | The unity element of a rin... |
| rngoueqz 37348 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 37349 | Negation in a ring is the ... |
| rngonegmn1r 37350 | Negation in a ring is the ... |
| rngoneglmul 37351 | Negation of a product in a... |
| rngonegrmul 37352 | Negation of a product in a... |
| rngosubdi 37353 | Ring multiplication distri... |
| rngosubdir 37354 | Ring multiplication distri... |
| zerdivemp1x 37355 | In a unital ring a left in... |
| isdivrngo 37358 | The predicate "is a divisi... |
| drngoi 37359 | The properties of a divisi... |
| gidsn 37360 | Obsolete as of 23-Jan-2020... |
| zrdivrng 37361 | The zero ring is not a div... |
| dvrunz 37362 | In a division ring the rin... |
| isgrpda 37363 | Properties that determine ... |
| isdrngo1 37364 | The predicate "is a divisi... |
| divrngcl 37365 | The product of two nonzero... |
| isdrngo2 37366 | A division ring is a ring ... |
| isdrngo3 37367 | A division ring is a ring ... |
| rngohomval 37372 | The set of ring homomorphi... |
| isrngohom 37373 | The predicate "is a ring h... |
| rngohomf 37374 | A ring homomorphism is a f... |
| rngohomcl 37375 | Closure law for a ring hom... |
| rngohom1 37376 | A ring homomorphism preser... |
| rngohomadd 37377 | Ring homomorphisms preserv... |
| rngohommul 37378 | Ring homomorphisms preserv... |
| rngogrphom 37379 | A ring homomorphism is a g... |
| rngohom0 37380 | A ring homomorphism preser... |
| rngohomsub 37381 | Ring homomorphisms preserv... |
| rngohomco 37382 | The composition of two rin... |
| rngokerinj 37383 | A ring homomorphism is inj... |
| rngoisoval 37385 | The set of ring isomorphis... |
| isrngoiso 37386 | The predicate "is a ring i... |
| rngoiso1o 37387 | A ring isomorphism is a bi... |
| rngoisohom 37388 | A ring isomorphism is a ri... |
| rngoisocnv 37389 | The inverse of a ring isom... |
| rngoisoco 37390 | The composition of two rin... |
| isriscg 37392 | The ring isomorphism relat... |
| isrisc 37393 | The ring isomorphism relat... |
| risc 37394 | The ring isomorphism relat... |
| risci 37395 | Determine that two rings a... |
| riscer 37396 | Ring isomorphism is an equ... |
| iscom2 37403 | A device to add commutativ... |
| iscrngo 37404 | The predicate "is a commut... |
| iscrngo2 37405 | The predicate "is a commut... |
| iscringd 37406 | Conditions that determine ... |
| flddivrng 37407 | A field is a division ring... |
| crngorngo 37408 | A commutative ring is a ri... |
| crngocom 37409 | The multiplication operati... |
| crngm23 37410 | Commutative/associative la... |
| crngm4 37411 | Commutative/associative la... |
| fldcrngo 37412 | A field is a commutative r... |
| isfld2 37413 | The predicate "is a field"... |
| crngohomfo 37414 | The image of a homomorphis... |
| idlval 37421 | The class of ideals of a r... |
| isidl 37422 | The predicate "is an ideal... |
| isidlc 37423 | The predicate "is an ideal... |
| idlss 37424 | An ideal of ` R ` is a sub... |
| idlcl 37425 | An element of an ideal is ... |
| idl0cl 37426 | An ideal contains ` 0 ` . ... |
| idladdcl 37427 | An ideal is closed under a... |
| idllmulcl 37428 | An ideal is closed under m... |
| idlrmulcl 37429 | An ideal is closed under m... |
| idlnegcl 37430 | An ideal is closed under n... |
| idlsubcl 37431 | An ideal is closed under s... |
| rngoidl 37432 | A ring ` R ` is an ` R ` i... |
| 0idl 37433 | The set containing only ` ... |
| 1idl 37434 | Two ways of expressing the... |
| 0rngo 37435 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 37436 | The only ideals in a divis... |
| intidl 37437 | The intersection of a none... |
| inidl 37438 | The intersection of two id... |
| unichnidl 37439 | The union of a nonempty ch... |
| keridl 37440 | The kernel of a ring homom... |
| pridlval 37441 | The class of prime ideals ... |
| ispridl 37442 | The predicate "is a prime ... |
| pridlidl 37443 | A prime ideal is an ideal.... |
| pridlnr 37444 | A prime ideal is a proper ... |
| pridl 37445 | The main property of a pri... |
| ispridl2 37446 | A condition that shows an ... |
| maxidlval 37447 | The set of maximal ideals ... |
| ismaxidl 37448 | The predicate "is a maxima... |
| maxidlidl 37449 | A maximal ideal is an idea... |
| maxidlnr 37450 | A maximal ideal is proper.... |
| maxidlmax 37451 | A maximal ideal is a maxim... |
| maxidln1 37452 | One is not contained in an... |
| maxidln0 37453 | A ring with a maximal idea... |
| isprrngo 37458 | The predicate "is a prime ... |
| prrngorngo 37459 | A prime ring is a ring. (... |
| smprngopr 37460 | A simple ring (one whose o... |
| divrngpr 37461 | A division ring is a prime... |
| isdmn 37462 | The predicate "is a domain... |
| isdmn2 37463 | The predicate "is a domain... |
| dmncrng 37464 | A domain is a commutative ... |
| dmnrngo 37465 | A domain is a ring. (Cont... |
| flddmn 37466 | A field is a domain. (Con... |
| igenval 37469 | The ideal generated by a s... |
| igenss 37470 | A set is a subset of the i... |
| igenidl 37471 | The ideal generated by a s... |
| igenmin 37472 | The ideal generated by a s... |
| igenidl2 37473 | The ideal generated by an ... |
| igenval2 37474 | The ideal generated by a s... |
| prnc 37475 | A principal ideal (an idea... |
| isfldidl 37476 | Determine if a ring is a f... |
| isfldidl2 37477 | Determine if a ring is a f... |
| ispridlc 37478 | The predicate "is a prime ... |
| pridlc 37479 | Property of a prime ideal ... |
| pridlc2 37480 | Property of a prime ideal ... |
| pridlc3 37481 | Property of a prime ideal ... |
| isdmn3 37482 | The predicate "is a domain... |
| dmnnzd 37483 | A domain has no zero-divis... |
| dmncan1 37484 | Cancellation law for domai... |
| dmncan2 37485 | Cancellation law for domai... |
| efald2 37486 | A proof by contradiction. ... |
| notbinot1 37487 | Simplification rule of neg... |
| bicontr 37488 | Biconditional of its own n... |
| impor 37489 | An equivalent formula for ... |
| orfa 37490 | The falsum ` F. ` can be r... |
| notbinot2 37491 | Commutation rule between n... |
| biimpor 37492 | A rewriting rule for bicon... |
| orfa1 37493 | Add a contradicting disjun... |
| orfa2 37494 | Remove a contradicting dis... |
| bifald 37495 | Infer the equivalence to a... |
| orsild 37496 | A lemma for not-or-not eli... |
| orsird 37497 | A lemma for not-or-not eli... |
| cnf1dd 37498 | A lemma for Conjunctive No... |
| cnf2dd 37499 | A lemma for Conjunctive No... |
| cnfn1dd 37500 | A lemma for Conjunctive No... |
| cnfn2dd 37501 | A lemma for Conjunctive No... |
| or32dd 37502 | A rearrangement of disjunc... |
| notornotel1 37503 | A lemma for not-or-not eli... |
| notornotel2 37504 | A lemma for not-or-not eli... |
| contrd 37505 | A proof by contradiction, ... |
| an12i 37506 | An inference from commutin... |
| exmid2 37507 | An excluded middle law. (... |
| selconj 37508 | An inference for selecting... |
| truconj 37509 | Add true as a conjunct. (... |
| orel 37510 | An inference for disjuncti... |
| negel 37511 | An inference for negation ... |
| botel 37512 | An inference for bottom el... |
| tradd 37513 | Add top ad a conjunct. (C... |
| gm-sbtru 37514 | Substitution does not chan... |
| sbfal 37515 | Substitution does not chan... |
| sbcani 37516 | Distribution of class subs... |
| sbcori 37517 | Distribution of class subs... |
| sbcimi 37518 | Distribution of class subs... |
| sbcni 37519 | Move class substitution in... |
| sbali 37520 | Discard class substitution... |
| sbexi 37521 | Discard class substitution... |
| sbcalf 37522 | Move universal quantifier ... |
| sbcexf 37523 | Move existential quantifie... |
| sbcalfi 37524 | Move universal quantifier ... |
| sbcexfi 37525 | Move existential quantifie... |
| spsbcdi 37526 | A lemma for eliminating a ... |
| alrimii 37527 | A lemma for introducing a ... |
| spesbcdi 37528 | A lemma for introducing an... |
| exlimddvf 37529 | A lemma for eliminating an... |
| exlimddvfi 37530 | A lemma for eliminating an... |
| sbceq1ddi 37531 | A lemma for eliminating in... |
| sbccom2lem 37532 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 37533 | Commutative law for double... |
| sbccom2f 37534 | Commutative law for double... |
| sbccom2fi 37535 | Commutative law for double... |
| csbcom2fi 37536 | Commutative law for double... |
| fald 37537 | Refutation of falsity, in ... |
| tsim1 37538 | A Tseitin axiom for logica... |
| tsim2 37539 | A Tseitin axiom for logica... |
| tsim3 37540 | A Tseitin axiom for logica... |
| tsbi1 37541 | A Tseitin axiom for logica... |
| tsbi2 37542 | A Tseitin axiom for logica... |
| tsbi3 37543 | A Tseitin axiom for logica... |
| tsbi4 37544 | A Tseitin axiom for logica... |
| tsxo1 37545 | A Tseitin axiom for logica... |
| tsxo2 37546 | A Tseitin axiom for logica... |
| tsxo3 37547 | A Tseitin axiom for logica... |
| tsxo4 37548 | A Tseitin axiom for logica... |
| tsan1 37549 | A Tseitin axiom for logica... |
| tsan2 37550 | A Tseitin axiom for logica... |
| tsan3 37551 | A Tseitin axiom for logica... |
| tsna1 37552 | A Tseitin axiom for logica... |
| tsna2 37553 | A Tseitin axiom for logica... |
| tsna3 37554 | A Tseitin axiom for logica... |
| tsor1 37555 | A Tseitin axiom for logica... |
| tsor2 37556 | A Tseitin axiom for logica... |
| tsor3 37557 | A Tseitin axiom for logica... |
| ts3an1 37558 | A Tseitin axiom for triple... |
| ts3an2 37559 | A Tseitin axiom for triple... |
| ts3an3 37560 | A Tseitin axiom for triple... |
| ts3or1 37561 | A Tseitin axiom for triple... |
| ts3or2 37562 | A Tseitin axiom for triple... |
| ts3or3 37563 | A Tseitin axiom for triple... |
| iuneq2f 37564 | Equality deduction for ind... |
| rabeq12f 37565 | Equality deduction for res... |
| csbeq12 37566 | Equality deduction for sub... |
| sbeqi 37567 | Equality deduction for sub... |
| ralbi12f 37568 | Equality deduction for res... |
| oprabbi 37569 | Equality deduction for cla... |
| mpobi123f 37570 | Equality deduction for map... |
| iuneq12f 37571 | Equality deduction for ind... |
| iineq12f 37572 | Equality deduction for ind... |
| opabbi 37573 | Equality deduction for cla... |
| mptbi12f 37574 | Equality deduction for map... |
| orcomdd 37575 | Commutativity of logic dis... |
| scottexf 37576 | A version of ~ scottex wit... |
| scott0f 37577 | A version of ~ scott0 with... |
| scottn0f 37578 | A version of ~ scott0f wit... |
| ac6s3f 37579 | Generalization of the Axio... |
| ac6s6 37580 | Generalization of the Axio... |
| ac6s6f 37581 | Generalization of the Axio... |
| el2v1 37625 | New way ( ~ elv , and the ... |
| el3v 37626 | New way ( ~ elv , and the ... |
| el3v1 37627 | New way ( ~ elv , and the ... |
| el3v2 37628 | New way ( ~ elv , and the ... |
| el3v3 37629 | New way ( ~ elv , and the ... |
| el3v12 37630 | New way ( ~ elv , and the ... |
| el3v13 37631 | New way ( ~ elv , and the ... |
| el3v23 37632 | New way ( ~ elv , and the ... |
| anan 37633 | Multiple commutations in c... |
| triantru3 37634 | A wff is equivalent to its... |
| bianim 37635 | Exchanging conjunction in ... |
| biorfd 37636 | A wff is equivalent to its... |
| eqbrtr 37637 | Substitution of equal clas... |
| eqbrb 37638 | Substitution of equal clas... |
| eqeltr 37639 | Substitution of equal clas... |
| eqelb 37640 | Substitution of equal clas... |
| eqeqan2d 37641 | Implication of introducing... |
| suceqsneq 37642 | One-to-one relationship be... |
| sucdifsn2 37643 | Absorption of union with a... |
| sucdifsn 37644 | The difference between the... |
| disjresin 37645 | The restriction to a disjo... |
| disjresdisj 37646 | The intersection of restri... |
| disjresdif 37647 | The difference between res... |
| disjresundif 37648 | Lemma for ~ ressucdifsn2 .... |
| ressucdifsn2 37649 | The difference between res... |
| ressucdifsn 37650 | The difference between res... |
| inres2 37651 | Two ways of expressing the... |
| coideq 37652 | Equality theorem for compo... |
| nexmo1 37653 | If there is no case where ... |
| ralin 37654 | Restricted universal quant... |
| r2alan 37655 | Double restricted universa... |
| ssrabi 37656 | Inference of restricted ab... |
| rabbieq 37657 | Equivalent wff's correspon... |
| rabimbieq 37658 | Restricted equivalent wff'... |
| abeqin 37659 | Intersection with class ab... |
| abeqinbi 37660 | Intersection with class ab... |
| rabeqel 37661 | Class element of a restric... |
| eqrelf 37662 | The equality connective be... |
| br1cnvinxp 37663 | Binary relation on the con... |
| releleccnv 37664 | Elementhood in a converse ... |
| releccnveq 37665 | Equality of converse ` R `... |
| opelvvdif 37666 | Negated elementhood of ord... |
| vvdifopab 37667 | Ordered-pair class abstrac... |
| brvdif 37668 | Binary relation with unive... |
| brvdif2 37669 | Binary relation with unive... |
| brvvdif 37670 | Binary relation with the c... |
| brvbrvvdif 37671 | Binary relation with the c... |
| brcnvep 37672 | The converse of the binary... |
| elecALTV 37673 | Elementhood in the ` R ` -... |
| brcnvepres 37674 | Restricted converse epsilo... |
| brres2 37675 | Binary relation on a restr... |
| br1cnvres 37676 | Binary relation on the con... |
| eldmres 37677 | Elementhood in the domain ... |
| elrnres 37678 | Element of the range of a ... |
| eldmressnALTV 37679 | Element of the domain of a... |
| elrnressn 37680 | Element of the range of a ... |
| eldm4 37681 | Elementhood in a domain. ... |
| eldmres2 37682 | Elementhood in the domain ... |
| eceq1i 37683 | Equality theorem for ` C `... |
| elecres 37684 | Elementhood in the restric... |
| ecres 37685 | Restricted coset of ` B ` ... |
| ecres2 37686 | The restricted coset of ` ... |
| eccnvepres 37687 | Restricted converse epsilo... |
| eleccnvep 37688 | Elementhood in the convers... |
| eccnvep 37689 | The converse epsilon coset... |
| extep 37690 | Property of epsilon relati... |
| disjeccnvep 37691 | Property of the epsilon re... |
| eccnvepres2 37692 | The restricted converse ep... |
| eccnvepres3 37693 | Condition for a restricted... |
| eldmqsres 37694 | Elementhood in a restricte... |
| eldmqsres2 37695 | Elementhood in a restricte... |
| qsss1 37696 | Subclass theorem for quoti... |
| qseq1i 37697 | Equality theorem for quoti... |
| qseq1d 37698 | Equality theorem for quoti... |
| brinxprnres 37699 | Binary relation on a restr... |
| inxprnres 37700 | Restriction of a class as ... |
| dfres4 37701 | Alternate definition of th... |
| exan3 37702 | Equivalent expressions wit... |
| exanres 37703 | Equivalent expressions wit... |
| exanres3 37704 | Equivalent expressions wit... |
| exanres2 37705 | Equivalent expressions wit... |
| cnvepres 37706 | Restricted converse epsilo... |
| eqrel2 37707 | Equality of relations. (C... |
| rncnv 37708 | Range of converse is the d... |
| dfdm6 37709 | Alternate definition of do... |
| dfrn6 37710 | Alternate definition of ra... |
| rncnvepres 37711 | The range of the restricte... |
| dmecd 37712 | Equality of the coset of `... |
| dmec2d 37713 | Equality of the coset of `... |
| brid 37714 | Property of the identity b... |
| ideq2 37715 | For sets, the identity bin... |
| idresssidinxp 37716 | Condition for the identity... |
| idreseqidinxp 37717 | Condition for the identity... |
| extid 37718 | Property of identity relat... |
| inxpss 37719 | Two ways to say that an in... |
| idinxpss 37720 | Two ways to say that an in... |
| ref5 37721 | Two ways to say that an in... |
| inxpss3 37722 | Two ways to say that an in... |
| inxpss2 37723 | Two ways to say that inter... |
| inxpssidinxp 37724 | Two ways to say that inter... |
| idinxpssinxp 37725 | Two ways to say that inter... |
| idinxpssinxp2 37726 | Identity intersection with... |
| idinxpssinxp3 37727 | Identity intersection with... |
| idinxpssinxp4 37728 | Identity intersection with... |
| relcnveq3 37729 | Two ways of saying a relat... |
| relcnveq 37730 | Two ways of saying a relat... |
| relcnveq2 37731 | Two ways of saying a relat... |
| relcnveq4 37732 | Two ways of saying a relat... |
| qsresid 37733 | Simplification of a specia... |
| n0elqs 37734 | Two ways of expressing tha... |
| n0elqs2 37735 | Two ways of expressing tha... |
| ecex2 37736 | Condition for a coset to b... |
| uniqsALTV 37737 | The union of a quotient se... |
| imaexALTV 37738 | Existence of an image of a... |
| ecexALTV 37739 | Existence of a coset, like... |
| rnresequniqs 37740 | The range of a restriction... |
| n0el2 37741 | Two ways of expressing tha... |
| cnvepresex 37742 | Sethood condition for the ... |
| eccnvepex 37743 | The converse epsilon coset... |
| cnvepimaex 37744 | The image of converse epsi... |
| cnvepima 37745 | The image of converse epsi... |
| inex3 37746 | Sufficient condition for t... |
| inxpex 37747 | Sufficient condition for a... |
| eqres 37748 | Converting a class constan... |
| brrabga 37749 | The law of concretion for ... |
| brcnvrabga 37750 | The law of concretion for ... |
| opideq 37751 | Equality conditions for or... |
| iss2 37752 | A subclass of the identity... |
| eldmcnv 37753 | Elementhood in a domain of... |
| dfrel5 37754 | Alternate definition of th... |
| dfrel6 37755 | Alternate definition of th... |
| cnvresrn 37756 | Converse restricted to ran... |
| relssinxpdmrn 37757 | Subset of restriction, spe... |
| cnvref4 37758 | Two ways to say that a rel... |
| cnvref5 37759 | Two ways to say that a rel... |
| ecin0 37760 | Two ways of saying that th... |
| ecinn0 37761 | Two ways of saying that th... |
| ineleq 37762 | Equivalence of restricted ... |
| inecmo 37763 | Equivalence of a double re... |
| inecmo2 37764 | Equivalence of a double re... |
| ineccnvmo 37765 | Equivalence of a double re... |
| alrmomorn 37766 | Equivalence of an "at most... |
| alrmomodm 37767 | Equivalence of an "at most... |
| ineccnvmo2 37768 | Equivalence of a double un... |
| inecmo3 37769 | Equivalence of a double un... |
| moeu2 37770 | Uniqueness is equivalent t... |
| mopickr 37771 | "At most one" picks a vari... |
| moantr 37772 | Sufficient condition for t... |
| brabidgaw 37773 | The law of concretion for ... |
| brabidga 37774 | The law of concretion for ... |
| inxp2 37775 | Intersection with a Cartes... |
| opabf 37776 | A class abstraction of a c... |
| ec0 37777 | The empty-coset of a class... |
| 0qs 37778 | Quotient set with the empt... |
| brcnvin 37779 | Intersection with a conver... |
| xrnss3v 37781 | A range Cartesian product ... |
| xrnrel 37782 | A range Cartesian product ... |
| brxrn 37783 | Characterize a ternary rel... |
| brxrn2 37784 | A characterization of the ... |
| dfxrn2 37785 | Alternate definition of th... |
| xrneq1 37786 | Equality theorem for the r... |
| xrneq1i 37787 | Equality theorem for the r... |
| xrneq1d 37788 | Equality theorem for the r... |
| xrneq2 37789 | Equality theorem for the r... |
| xrneq2i 37790 | Equality theorem for the r... |
| xrneq2d 37791 | Equality theorem for the r... |
| xrneq12 37792 | Equality theorem for the r... |
| xrneq12i 37793 | Equality theorem for the r... |
| xrneq12d 37794 | Equality theorem for the r... |
| elecxrn 37795 | Elementhood in the ` ( R |... |
| ecxrn 37796 | The ` ( R |X. S ) ` -coset... |
| disjressuc2 37797 | Double restricted quantifi... |
| disjecxrn 37798 | Two ways of saying that ` ... |
| disjecxrncnvep 37799 | Two ways of saying that co... |
| disjsuc2 37800 | Double restricted quantifi... |
| xrninxp 37801 | Intersection of a range Ca... |
| xrninxp2 37802 | Intersection of a range Ca... |
| xrninxpex 37803 | Sufficient condition for t... |
| inxpxrn 37804 | Two ways to express the in... |
| br1cnvxrn2 37805 | The converse of a binary r... |
| elec1cnvxrn2 37806 | Elementhood in the convers... |
| rnxrn 37807 | Range of the range Cartesi... |
| rnxrnres 37808 | Range of a range Cartesian... |
| rnxrncnvepres 37809 | Range of a range Cartesian... |
| rnxrnidres 37810 | Range of a range Cartesian... |
| xrnres 37811 | Two ways to express restri... |
| xrnres2 37812 | Two ways to express restri... |
| xrnres3 37813 | Two ways to express restri... |
| xrnres4 37814 | Two ways to express restri... |
| xrnresex 37815 | Sufficient condition for a... |
| xrnidresex 37816 | Sufficient condition for a... |
| xrncnvepresex 37817 | Sufficient condition for a... |
| brin2 37818 | Binary relation on an inte... |
| brin3 37819 | Binary relation on an inte... |
| dfcoss2 37822 | Alternate definition of th... |
| dfcoss3 37823 | Alternate definition of th... |
| dfcoss4 37824 | Alternate definition of th... |
| cosscnv 37825 | Class of cosets by the con... |
| coss1cnvres 37826 | Class of cosets by the con... |
| coss2cnvepres 37827 | Special case of ~ coss1cnv... |
| cossex 37828 | If ` A ` is a set then the... |
| cosscnvex 37829 | If ` A ` is a set then the... |
| 1cosscnvepresex 37830 | Sufficient condition for a... |
| 1cossxrncnvepresex 37831 | Sufficient condition for a... |
| relcoss 37832 | Cosets by ` R ` is a relat... |
| relcoels 37833 | Coelements on ` A ` is a r... |
| cossss 37834 | Subclass theorem for the c... |
| cosseq 37835 | Equality theorem for the c... |
| cosseqi 37836 | Equality theorem for the c... |
| cosseqd 37837 | Equality theorem for the c... |
| 1cossres 37838 | The class of cosets by a r... |
| dfcoels 37839 | Alternate definition of th... |
| brcoss 37840 | ` A ` and ` B ` are cosets... |
| brcoss2 37841 | Alternate form of the ` A ... |
| brcoss3 37842 | Alternate form of the ` A ... |
| brcosscnvcoss 37843 | For sets, the ` A ` and ` ... |
| brcoels 37844 | ` B ` and ` C ` are coelem... |
| cocossss 37845 | Two ways of saying that co... |
| cnvcosseq 37846 | The converse of cosets by ... |
| br2coss 37847 | Cosets by ` ,~ R ` binary ... |
| br1cossres 37848 | ` B ` and ` C ` are cosets... |
| br1cossres2 37849 | ` B ` and ` C ` are cosets... |
| brressn 37850 | Binary relation on a restr... |
| ressn2 37851 | A class ' R ' restricted t... |
| refressn 37852 | Any class ' R ' restricted... |
| antisymressn 37853 | Every class ' R ' restrict... |
| trressn 37854 | Any class ' R ' restricted... |
| relbrcoss 37855 | ` A ` and ` B ` are cosets... |
| br1cossinres 37856 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 37857 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 37858 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 37859 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 37860 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 37861 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 37862 | The domain of cosets is th... |
| dmcoss2 37863 | The domain of cosets is th... |
| rncossdmcoss 37864 | The range of cosets is the... |
| dm1cosscnvepres 37865 | The domain of cosets of th... |
| dmcoels 37866 | The domain of coelements i... |
| eldmcoss 37867 | Elementhood in the domain ... |
| eldmcoss2 37868 | Elementhood in the domain ... |
| eldm1cossres 37869 | Elementhood in the domain ... |
| eldm1cossres2 37870 | Elementhood in the domain ... |
| refrelcosslem 37871 | Lemma for the left side of... |
| refrelcoss3 37872 | The class of cosets by ` R... |
| refrelcoss2 37873 | The class of cosets by ` R... |
| symrelcoss3 37874 | The class of cosets by ` R... |
| symrelcoss2 37875 | The class of cosets by ` R... |
| cossssid 37876 | Equivalent expressions for... |
| cossssid2 37877 | Equivalent expressions for... |
| cossssid3 37878 | Equivalent expressions for... |
| cossssid4 37879 | Equivalent expressions for... |
| cossssid5 37880 | Equivalent expressions for... |
| brcosscnv 37881 | ` A ` and ` B ` are cosets... |
| brcosscnv2 37882 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 37883 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 37884 | Cosets by the converse ran... |
| cosscnvssid3 37885 | Equivalent expressions for... |
| cosscnvssid4 37886 | Equivalent expressions for... |
| cosscnvssid5 37887 | Equivalent expressions for... |
| coss0 37888 | Cosets by the empty set ar... |
| cossid 37889 | Cosets by the identity rel... |
| cosscnvid 37890 | Cosets by the converse ide... |
| trcoss 37891 | Sufficient condition for t... |
| eleccossin 37892 | Two ways of saying that th... |
| trcoss2 37893 | Equivalent expressions for... |
| elrels2 37895 | The element of the relatio... |
| elrelsrel 37896 | The element of the relatio... |
| elrelsrelim 37897 | The element of the relatio... |
| elrels5 37898 | Equivalent expressions for... |
| elrels6 37899 | Equivalent expressions for... |
| elrelscnveq3 37900 | Two ways of saying a relat... |
| elrelscnveq 37901 | Two ways of saying a relat... |
| elrelscnveq2 37902 | Two ways of saying a relat... |
| elrelscnveq4 37903 | Two ways of saying a relat... |
| cnvelrels 37904 | The converse of a set is a... |
| cosselrels 37905 | Cosets of sets are element... |
| cosscnvelrels 37906 | Cosets of converse sets ar... |
| dfssr2 37908 | Alternate definition of th... |
| relssr 37909 | The subset relation is a r... |
| brssr 37910 | The subset relation and su... |
| brssrid 37911 | Any set is a subset of its... |
| issetssr 37912 | Two ways of expressing set... |
| brssrres 37913 | Restricted subset binary r... |
| br1cnvssrres 37914 | Restricted converse subset... |
| brcnvssr 37915 | The converse of a subset r... |
| brcnvssrid 37916 | Any set is a converse subs... |
| br1cossxrncnvssrres 37917 | ` <. B , C >. ` and ` <. D... |
| extssr 37918 | Property of subset relatio... |
| dfrefrels2 37922 | Alternate definition of th... |
| dfrefrels3 37923 | Alternate definition of th... |
| dfrefrel2 37924 | Alternate definition of th... |
| dfrefrel3 37925 | Alternate definition of th... |
| dfrefrel5 37926 | Alternate definition of th... |
| elrefrels2 37927 | Element of the class of re... |
| elrefrels3 37928 | Element of the class of re... |
| elrefrelsrel 37929 | For sets, being an element... |
| refreleq 37930 | Equality theorem for refle... |
| refrelid 37931 | Identity relation is refle... |
| refrelcoss 37932 | The class of cosets by ` R... |
| refrelressn 37933 | Any class ' R ' restricted... |
| dfcnvrefrels2 37937 | Alternate definition of th... |
| dfcnvrefrels3 37938 | Alternate definition of th... |
| dfcnvrefrel2 37939 | Alternate definition of th... |
| dfcnvrefrel3 37940 | Alternate definition of th... |
| dfcnvrefrel4 37941 | Alternate definition of th... |
| dfcnvrefrel5 37942 | Alternate definition of th... |
| elcnvrefrels2 37943 | Element of the class of co... |
| elcnvrefrels3 37944 | Element of the class of co... |
| elcnvrefrelsrel 37945 | For sets, being an element... |
| cnvrefrelcoss2 37946 | Necessary and sufficient c... |
| cosselcnvrefrels2 37947 | Necessary and sufficient c... |
| cosselcnvrefrels3 37948 | Necessary and sufficient c... |
| cosselcnvrefrels4 37949 | Necessary and sufficient c... |
| cosselcnvrefrels5 37950 | Necessary and sufficient c... |
| dfsymrels2 37954 | Alternate definition of th... |
| dfsymrels3 37955 | Alternate definition of th... |
| dfsymrels4 37956 | Alternate definition of th... |
| dfsymrels5 37957 | Alternate definition of th... |
| dfsymrel2 37958 | Alternate definition of th... |
| dfsymrel3 37959 | Alternate definition of th... |
| dfsymrel4 37960 | Alternate definition of th... |
| dfsymrel5 37961 | Alternate definition of th... |
| elsymrels2 37962 | Element of the class of sy... |
| elsymrels3 37963 | Element of the class of sy... |
| elsymrels4 37964 | Element of the class of sy... |
| elsymrels5 37965 | Element of the class of sy... |
| elsymrelsrel 37966 | For sets, being an element... |
| symreleq 37967 | Equality theorem for symme... |
| symrelim 37968 | Symmetric relation implies... |
| symrelcoss 37969 | The class of cosets by ` R... |
| idsymrel 37970 | The identity relation is s... |
| epnsymrel 37971 | The membership (epsilon) r... |
| symrefref2 37972 | Symmetry is a sufficient c... |
| symrefref3 37973 | Symmetry is a sufficient c... |
| refsymrels2 37974 | Elements of the class of r... |
| refsymrels3 37975 | Elements of the class of r... |
| refsymrel2 37976 | A relation which is reflex... |
| refsymrel3 37977 | A relation which is reflex... |
| elrefsymrels2 37978 | Elements of the class of r... |
| elrefsymrels3 37979 | Elements of the class of r... |
| elrefsymrelsrel 37980 | For sets, being an element... |
| dftrrels2 37984 | Alternate definition of th... |
| dftrrels3 37985 | Alternate definition of th... |
| dftrrel2 37986 | Alternate definition of th... |
| dftrrel3 37987 | Alternate definition of th... |
| eltrrels2 37988 | Element of the class of tr... |
| eltrrels3 37989 | Element of the class of tr... |
| eltrrelsrel 37990 | For sets, being an element... |
| trreleq 37991 | Equality theorem for the t... |
| trrelressn 37992 | Any class ' R ' restricted... |
| dfeqvrels2 37997 | Alternate definition of th... |
| dfeqvrels3 37998 | Alternate definition of th... |
| dfeqvrel2 37999 | Alternate definition of th... |
| dfeqvrel3 38000 | Alternate definition of th... |
| eleqvrels2 38001 | Element of the class of eq... |
| eleqvrels3 38002 | Element of the class of eq... |
| eleqvrelsrel 38003 | For sets, being an element... |
| elcoeleqvrels 38004 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 38005 | For sets, being an element... |
| eqvrelrel 38006 | An equivalence relation is... |
| eqvrelrefrel 38007 | An equivalence relation is... |
| eqvrelsymrel 38008 | An equivalence relation is... |
| eqvreltrrel 38009 | An equivalence relation is... |
| eqvrelim 38010 | Equivalence relation impli... |
| eqvreleq 38011 | Equality theorem for equiv... |
| eqvreleqi 38012 | Equality theorem for equiv... |
| eqvreleqd 38013 | Equality theorem for equiv... |
| eqvrelsym 38014 | An equivalence relation is... |
| eqvrelsymb 38015 | An equivalence relation is... |
| eqvreltr 38016 | An equivalence relation is... |
| eqvreltrd 38017 | A transitivity relation fo... |
| eqvreltr4d 38018 | A transitivity relation fo... |
| eqvrelref 38019 | An equivalence relation is... |
| eqvrelth 38020 | Basic property of equivale... |
| eqvrelcl 38021 | Elementhood in the field o... |
| eqvrelthi 38022 | Basic property of equivale... |
| eqvreldisj 38023 | Equivalence classes do not... |
| qsdisjALTV 38024 | Elements of a quotient set... |
| eqvrelqsel 38025 | If an element of a quotien... |
| eqvrelcoss 38026 | Two ways to express equiva... |
| eqvrelcoss3 38027 | Two ways to express equiva... |
| eqvrelcoss2 38028 | Two ways to express equiva... |
| eqvrelcoss4 38029 | Two ways to express equiva... |
| dfcoeleqvrels 38030 | Alternate definition of th... |
| dfcoeleqvrel 38031 | Alternate definition of th... |
| brredunds 38035 | Binary relation on the cla... |
| brredundsredund 38036 | For sets, binary relation ... |
| redundss3 38037 | Implication of redundancy ... |
| redundeq1 38038 | Equivalence of redundancy ... |
| redundpim3 38039 | Implication of redundancy ... |
| redundpbi1 38040 | Equivalence of redundancy ... |
| refrelsredund4 38041 | The naive version of the c... |
| refrelsredund2 38042 | The naive version of the c... |
| refrelsredund3 38043 | The naive version of the c... |
| refrelredund4 38044 | The naive version of the d... |
| refrelredund2 38045 | The naive version of the d... |
| refrelredund3 38046 | The naive version of the d... |
| dmqseq 38049 | Equality theorem for domai... |
| dmqseqi 38050 | Equality theorem for domai... |
| dmqseqd 38051 | Equality theorem for domai... |
| dmqseqeq1 38052 | Equality theorem for domai... |
| dmqseqeq1i 38053 | Equality theorem for domai... |
| dmqseqeq1d 38054 | Equality theorem for domai... |
| brdmqss 38055 | The domain quotient binary... |
| brdmqssqs 38056 | If ` A ` and ` R ` are set... |
| n0eldmqs 38057 | The empty set is not an el... |
| n0eldmqseq 38058 | The empty set is not an el... |
| n0elim 38059 | Implication of that the em... |
| n0el3 38060 | Two ways of expressing tha... |
| cnvepresdmqss 38061 | The domain quotient binary... |
| cnvepresdmqs 38062 | The domain quotient predic... |
| unidmqs 38063 | The range of a relation is... |
| unidmqseq 38064 | The union of the domain qu... |
| dmqseqim 38065 | If the domain quotient of ... |
| dmqseqim2 38066 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 38067 | Elementhood in the domain ... |
| eldmqs1cossres 38068 | Elementhood in the domain ... |
| releldmqscoss 38069 | Elementhood in the domain ... |
| dmqscoelseq 38070 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 38071 | Two ways to express the eq... |
| brers 38076 | Binary equivalence relatio... |
| dferALTV2 38077 | Equivalence relation with ... |
| erALTVeq1 38078 | Equality theorem for equiv... |
| erALTVeq1i 38079 | Equality theorem for equiv... |
| erALTVeq1d 38080 | Equality theorem for equiv... |
| dfcomember 38081 | Alternate definition of th... |
| dfcomember2 38082 | Alternate definition of th... |
| dfcomember3 38083 | Alternate definition of th... |
| eqvreldmqs 38084 | Two ways to express comemb... |
| eqvreldmqs2 38085 | Two ways to express comemb... |
| brerser 38086 | Binary equivalence relatio... |
| erimeq2 38087 | Equivalence relation on it... |
| erimeq 38088 | Equivalence relation on it... |
| dffunsALTV 38092 | Alternate definition of th... |
| dffunsALTV2 38093 | Alternate definition of th... |
| dffunsALTV3 38094 | Alternate definition of th... |
| dffunsALTV4 38095 | Alternate definition of th... |
| dffunsALTV5 38096 | Alternate definition of th... |
| dffunALTV2 38097 | Alternate definition of th... |
| dffunALTV3 38098 | Alternate definition of th... |
| dffunALTV4 38099 | Alternate definition of th... |
| dffunALTV5 38100 | Alternate definition of th... |
| elfunsALTV 38101 | Elementhood in the class o... |
| elfunsALTV2 38102 | Elementhood in the class o... |
| elfunsALTV3 38103 | Elementhood in the class o... |
| elfunsALTV4 38104 | Elementhood in the class o... |
| elfunsALTV5 38105 | Elementhood in the class o... |
| elfunsALTVfunALTV 38106 | The element of the class o... |
| funALTVfun 38107 | Our definition of the func... |
| funALTVss 38108 | Subclass theorem for funct... |
| funALTVeq 38109 | Equality theorem for funct... |
| funALTVeqi 38110 | Equality inference for the... |
| funALTVeqd 38111 | Equality deduction for the... |
| dfdisjs 38117 | Alternate definition of th... |
| dfdisjs2 38118 | Alternate definition of th... |
| dfdisjs3 38119 | Alternate definition of th... |
| dfdisjs4 38120 | Alternate definition of th... |
| dfdisjs5 38121 | Alternate definition of th... |
| dfdisjALTV 38122 | Alternate definition of th... |
| dfdisjALTV2 38123 | Alternate definition of th... |
| dfdisjALTV3 38124 | Alternate definition of th... |
| dfdisjALTV4 38125 | Alternate definition of th... |
| dfdisjALTV5 38126 | Alternate definition of th... |
| dfeldisj2 38127 | Alternate definition of th... |
| dfeldisj3 38128 | Alternate definition of th... |
| dfeldisj4 38129 | Alternate definition of th... |
| dfeldisj5 38130 | Alternate definition of th... |
| eldisjs 38131 | Elementhood in the class o... |
| eldisjs2 38132 | Elementhood in the class o... |
| eldisjs3 38133 | Elementhood in the class o... |
| eldisjs4 38134 | Elementhood in the class o... |
| eldisjs5 38135 | Elementhood in the class o... |
| eldisjsdisj 38136 | The element of the class o... |
| eleldisjs 38137 | Elementhood in the disjoin... |
| eleldisjseldisj 38138 | The element of the disjoin... |
| disjrel 38139 | Disjoint relation is a rel... |
| disjss 38140 | Subclass theorem for disjo... |
| disjssi 38141 | Subclass theorem for disjo... |
| disjssd 38142 | Subclass theorem for disjo... |
| disjeq 38143 | Equality theorem for disjo... |
| disjeqi 38144 | Equality theorem for disjo... |
| disjeqd 38145 | Equality theorem for disjo... |
| disjdmqseqeq1 38146 | Lemma for the equality the... |
| eldisjss 38147 | Subclass theorem for disjo... |
| eldisjssi 38148 | Subclass theorem for disjo... |
| eldisjssd 38149 | Subclass theorem for disjo... |
| eldisjeq 38150 | Equality theorem for disjo... |
| eldisjeqi 38151 | Equality theorem for disjo... |
| eldisjeqd 38152 | Equality theorem for disjo... |
| disjres 38153 | Disjoint restriction. (Co... |
| eldisjn0elb 38154 | Two forms of disjoint elem... |
| disjxrn 38155 | Two ways of saying that a ... |
| disjxrnres5 38156 | Disjoint range Cartesian p... |
| disjorimxrn 38157 | Disjointness condition for... |
| disjimxrn 38158 | Disjointness condition for... |
| disjimres 38159 | Disjointness condition for... |
| disjimin 38160 | Disjointness condition for... |
| disjiminres 38161 | Disjointness condition for... |
| disjimxrnres 38162 | Disjointness condition for... |
| disjALTV0 38163 | The null class is disjoint... |
| disjALTVid 38164 | The class of identity rela... |
| disjALTVidres 38165 | The class of identity rela... |
| disjALTVinidres 38166 | The intersection with rest... |
| disjALTVxrnidres 38167 | The class of range Cartesi... |
| disjsuc 38168 | Disjoint range Cartesian p... |
| dfantisymrel4 38170 | Alternate definition of th... |
| dfantisymrel5 38171 | Alternate definition of th... |
| antisymrelres 38172 | (Contributed by Peter Mazs... |
| antisymrelressn 38173 | (Contributed by Peter Mazs... |
| dfpart2 38178 | Alternate definition of th... |
| dfmembpart2 38179 | Alternate definition of th... |
| brparts 38180 | Binary partitions relation... |
| brparts2 38181 | Binary partitions relation... |
| brpartspart 38182 | Binary partition and the p... |
| parteq1 38183 | Equality theorem for parti... |
| parteq2 38184 | Equality theorem for parti... |
| parteq12 38185 | Equality theorem for parti... |
| parteq1i 38186 | Equality theorem for parti... |
| parteq1d 38187 | Equality theorem for parti... |
| partsuc2 38188 | Property of the partition.... |
| partsuc 38189 | Property of the partition.... |
| disjim 38190 | The "Divide et Aequivalere... |
| disjimi 38191 | Every disjoint relation ge... |
| detlem 38192 | If a relation is disjoint,... |
| eldisjim 38193 | If the elements of ` A ` a... |
| eldisjim2 38194 | Alternate form of ~ eldisj... |
| eqvrel0 38195 | The null class is an equiv... |
| det0 38196 | The cosets by the null cla... |
| eqvrelcoss0 38197 | The cosets by the null cla... |
| eqvrelid 38198 | The identity relation is a... |
| eqvrel1cossidres 38199 | The cosets by a restricted... |
| eqvrel1cossinidres 38200 | The cosets by an intersect... |
| eqvrel1cossxrnidres 38201 | The cosets by a range Cart... |
| detid 38202 | The cosets by the identity... |
| eqvrelcossid 38203 | The cosets by the identity... |
| detidres 38204 | The cosets by the restrict... |
| detinidres 38205 | The cosets by the intersec... |
| detxrnidres 38206 | The cosets by the range Ca... |
| disjlem14 38207 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 38208 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 38209 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 38210 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 38211 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 38212 | Lemma for ~ disjdmqseq via... |
| disjdmqs 38213 | If a relation is disjoint,... |
| disjdmqseq 38214 | If a relation is disjoint,... |
| eldisjn0el 38215 | Special case of ~ disjdmqs... |
| partim2 38216 | Disjoint relation on its n... |
| partim 38217 | Partition implies equivale... |
| partimeq 38218 | Partition implies that the... |
| eldisjlem19 38219 | Special case of ~ disjlem1... |
| membpartlem19 38220 | Together with ~ disjlem19 ... |
| petlem 38221 | If you can prove that the ... |
| petlemi 38222 | If you can prove disjointn... |
| pet02 38223 | Class ` A ` is a partition... |
| pet0 38224 | Class ` A ` is a partition... |
| petid2 38225 | Class ` A ` is a partition... |
| petid 38226 | A class is a partition by ... |
| petidres2 38227 | Class ` A ` is a partition... |
| petidres 38228 | A class is a partition by ... |
| petinidres2 38229 | Class ` A ` is a partition... |
| petinidres 38230 | A class is a partition by ... |
| petxrnidres2 38231 | Class ` A ` is a partition... |
| petxrnidres 38232 | A class is a partition by ... |
| eqvreldisj1 38233 | The elements of the quotie... |
| eqvreldisj2 38234 | The elements of the quotie... |
| eqvreldisj3 38235 | The elements of the quotie... |
| eqvreldisj4 38236 | Intersection with the conv... |
| eqvreldisj5 38237 | Range Cartesian product wi... |
| eqvrelqseqdisj2 38238 | Implication of ~ eqvreldis... |
| fences3 38239 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 38240 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 38241 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 38242 | Lemma for the Partition-Eq... |
| mainer 38243 | The Main Theorem of Equiva... |
| partimcomember 38244 | Partition with general ` R... |
| mpet3 38245 | Member Partition-Equivalen... |
| cpet2 38246 | The conventional form of t... |
| cpet 38247 | The conventional form of M... |
| mpet 38248 | Member Partition-Equivalen... |
| mpet2 38249 | Member Partition-Equivalen... |
| mpets2 38250 | Member Partition-Equivalen... |
| mpets 38251 | Member Partition-Equivalen... |
| mainpart 38252 | Partition with general ` R... |
| fences 38253 | The Theorem of Fences by E... |
| fences2 38254 | The Theorem of Fences by E... |
| mainer2 38255 | The Main Theorem of Equiva... |
| mainerim 38256 | Every equivalence relation... |
| petincnvepres2 38257 | A partition-equivalence th... |
| petincnvepres 38258 | The shortest form of a par... |
| pet2 38259 | Partition-Equivalence Theo... |
| pet 38260 | Partition-Equivalence Theo... |
| pets 38261 | Partition-Equivalence Theo... |
| prtlem60 38262 | Lemma for ~ prter3 . (Con... |
| bicomdd 38263 | Commute two sides of a bic... |
| jca2r 38264 | Inference conjoining the c... |
| jca3 38265 | Inference conjoining the c... |
| prtlem70 38266 | Lemma for ~ prter3 : a rea... |
| ibdr 38267 | Reverse of ~ ibd . (Contr... |
| prtlem100 38268 | Lemma for ~ prter3 . (Con... |
| prtlem5 38269 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 38270 | Lemma for ~ prter2 . (Con... |
| brabsb2 38271 | A closed form of ~ brabsb ... |
| eqbrrdv2 38272 | Other version of ~ eqbrrdi... |
| prtlem9 38273 | Lemma for ~ prter3 . (Con... |
| prtlem10 38274 | Lemma for ~ prter3 . (Con... |
| prtlem11 38275 | Lemma for ~ prter2 . (Con... |
| prtlem12 38276 | Lemma for ~ prtex and ~ pr... |
| prtlem13 38277 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 38278 | Lemma for ~ prtex , ~ prte... |
| prtlem400 38279 | Lemma for ~ prter2 and als... |
| erprt 38282 | The quotient set of an equ... |
| prtlem14 38283 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 38284 | Lemma for ~ prter1 and ~ p... |
| prtlem17 38285 | Lemma for ~ prter2 . (Con... |
| prtlem18 38286 | Lemma for ~ prter2 . (Con... |
| prtlem19 38287 | Lemma for ~ prter2 . (Con... |
| prter1 38288 | Every partition generates ... |
| prtex 38289 | The equivalence relation g... |
| prter2 38290 | The quotient set of the eq... |
| prter3 38291 | For every partition there ... |
| axc5 38302 | This theorem repeats ~ sp ... |
| ax4fromc4 38303 | Rederivation of Axiom ~ ax... |
| ax10fromc7 38304 | Rederivation of Axiom ~ ax... |
| ax6fromc10 38305 | Rederivation of Axiom ~ ax... |
| hba1-o 38306 | The setvar ` x ` is not fr... |
| axc4i-o 38307 | Inference version of ~ ax-... |
| equid1 38308 | Proof of ~ equid from our ... |
| equcomi1 38309 | Proof of ~ equcomi from ~ ... |
| aecom-o 38310 | Commutation law for identi... |
| aecoms-o 38311 | A commutation rule for ide... |
| hbae-o 38312 | All variables are effectiv... |
| dral1-o 38313 | Formula-building lemma for... |
| ax12fromc15 38314 | Rederivation of Axiom ~ ax... |
| ax13fromc9 38315 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 38316 | Axiom to quantify a variab... |
| sps-o 38317 | Generalization of antecede... |
| hbequid 38318 | Bound-variable hypothesis ... |
| nfequid-o 38319 | Bound-variable hypothesis ... |
| axc5c7 38320 | Proof of a single axiom th... |
| axc5c7toc5 38321 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 38322 | Rederivation of ~ ax-c7 fr... |
| axc711 38323 | Proof of a single axiom th... |
| nfa1-o 38324 | ` x ` is not free in ` A. ... |
| axc711toc7 38325 | Rederivation of ~ ax-c7 fr... |
| axc711to11 38326 | Rederivation of ~ ax-11 fr... |
| axc5c711 38327 | Proof of a single axiom th... |
| axc5c711toc5 38328 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 38329 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 38330 | Rederivation of ~ ax-11 fr... |
| equidqe 38331 | ~ equid with existential q... |
| axc5sp1 38332 | A special case of ~ ax-c5 ... |
| equidq 38333 | ~ equid with universal qua... |
| equid1ALT 38334 | Alternate proof of ~ equid... |
| axc11nfromc11 38335 | Rederivation of ~ ax-c11n ... |
| naecoms-o 38336 | A commutation rule for dis... |
| hbnae-o 38337 | All variables are effectiv... |
| dvelimf-o 38338 | Proof of ~ dvelimh that us... |
| dral2-o 38339 | Formula-building lemma for... |
| aev-o 38340 | A "distinctor elimination"... |
| ax5eq 38341 | Theorem to add distinct qu... |
| dveeq2-o 38342 | Quantifier introduction wh... |
| axc16g-o 38343 | A generalization of Axiom ... |
| dveeq1-o 38344 | Quantifier introduction wh... |
| dveeq1-o16 38345 | Version of ~ dveeq1 using ... |
| ax5el 38346 | Theorem to add distinct qu... |
| axc11n-16 38347 | This theorem shows that, g... |
| dveel2ALT 38348 | Alternate proof of ~ dveel... |
| ax12f 38349 | Basis step for constructin... |
| ax12eq 38350 | Basis step for constructin... |
| ax12el 38351 | Basis step for constructin... |
| ax12indn 38352 | Induction step for constru... |
| ax12indi 38353 | Induction step for constru... |
| ax12indalem 38354 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 38355 | Alternate proof of ~ ax12i... |
| ax12inda2 38356 | Induction step for constru... |
| ax12inda 38357 | Induction step for constru... |
| ax12v2-o 38358 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 38359 | Derive ~ ax-c15 from a hyp... |
| axc11-o 38360 | Show that ~ ax-c11 can be ... |
| fsumshftd 38361 | Index shift of a finite su... |
| riotaclbgBAD 38363 | Closure of restricted iota... |
| riotaclbBAD 38364 | Closure of restricted iota... |
| riotasvd 38365 | Deduction version of ~ rio... |
| riotasv2d 38366 | Value of description binde... |
| riotasv2s 38367 | The value of description b... |
| riotasv 38368 | Value of description binde... |
| riotasv3d 38369 | A property ` ch ` holding ... |
| elimhyps 38370 | A version of ~ elimhyp usi... |
| dedths 38371 | A version of weak deductio... |
| renegclALT 38372 | Closure law for negative o... |
| elimhyps2 38373 | Generalization of ~ elimhy... |
| dedths2 38374 | Generalization of ~ dedths... |
| nfcxfrdf 38375 | A utility lemma to transfe... |
| nfded 38376 | A deduction theorem that c... |
| nfded2 38377 | A deduction theorem that c... |
| nfunidALT2 38378 | Deduction version of ~ nfu... |
| nfunidALT 38379 | Deduction version of ~ nfu... |
| nfopdALT 38380 | Deduction version of bound... |
| cnaddcom 38381 | Recover the commutative la... |
| toycom 38382 | Show the commutative law f... |
| lshpset 38387 | The set of all hyperplanes... |
| islshp 38388 | The predicate "is a hyperp... |
| islshpsm 38389 | Hyperplane properties expr... |
| lshplss 38390 | A hyperplane is a subspace... |
| lshpne 38391 | A hyperplane is not equal ... |
| lshpnel 38392 | A hyperplane's generating ... |
| lshpnelb 38393 | The subspace sum of a hype... |
| lshpnel2N 38394 | Condition that determines ... |
| lshpne0 38395 | The member of the span in ... |
| lshpdisj 38396 | A hyperplane and the span ... |
| lshpcmp 38397 | If two hyperplanes are com... |
| lshpinN 38398 | The intersection of two di... |
| lsatset 38399 | The set of all 1-dim subsp... |
| islsat 38400 | The predicate "is a 1-dim ... |
| lsatlspsn2 38401 | The span of a nonzero sing... |
| lsatlspsn 38402 | The span of a nonzero sing... |
| islsati 38403 | A 1-dim subspace (atom) (o... |
| lsateln0 38404 | A 1-dim subspace (atom) (o... |
| lsatlss 38405 | The set of 1-dim subspaces... |
| lsatlssel 38406 | An atom is a subspace. (C... |
| lsatssv 38407 | An atom is a set of vector... |
| lsatn0 38408 | A 1-dim subspace (atom) of... |
| lsatspn0 38409 | The span of a vector is an... |
| lsator0sp 38410 | The span of a vector is ei... |
| lsatssn0 38411 | A subspace (or any class) ... |
| lsatcmp 38412 | If two atoms are comparabl... |
| lsatcmp2 38413 | If an atom is included in ... |
| lsatel 38414 | A nonzero vector in an ato... |
| lsatelbN 38415 | A nonzero vector in an ato... |
| lsat2el 38416 | Two atoms sharing a nonzer... |
| lsmsat 38417 | Convert comparison of atom... |
| lsatfixedN 38418 | Show equality with the spa... |
| lsmsatcv 38419 | Subspace sum has the cover... |
| lssatomic 38420 | The lattice of subspaces i... |
| lssats 38421 | The lattice of subspaces i... |
| lpssat 38422 | Two subspaces in a proper ... |
| lrelat 38423 | Subspaces are relatively a... |
| lssatle 38424 | The ordering of two subspa... |
| lssat 38425 | Two subspaces in a proper ... |
| islshpat 38426 | Hyperplane properties expr... |
| lcvfbr 38429 | The covers relation for a ... |
| lcvbr 38430 | The covers relation for a ... |
| lcvbr2 38431 | The covers relation for a ... |
| lcvbr3 38432 | The covers relation for a ... |
| lcvpss 38433 | The covers relation implie... |
| lcvnbtwn 38434 | The covers relation implie... |
| lcvntr 38435 | The covers relation is not... |
| lcvnbtwn2 38436 | The covers relation implie... |
| lcvnbtwn3 38437 | The covers relation implie... |
| lsmcv2 38438 | Subspace sum has the cover... |
| lcvat 38439 | If a subspace covers anoth... |
| lsatcv0 38440 | An atom covers the zero su... |
| lsatcveq0 38441 | A subspace covered by an a... |
| lsat0cv 38442 | A subspace is an atom iff ... |
| lcvexchlem1 38443 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 38444 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 38445 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 38446 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 38447 | Lemma for ~ lcvexch . (Co... |
| lcvexch 38448 | Subspaces satisfy the exch... |
| lcvp 38449 | Covering property of Defin... |
| lcv1 38450 | Covering property of a sub... |
| lcv2 38451 | Covering property of a sub... |
| lsatexch 38452 | The atom exchange property... |
| lsatnle 38453 | The meet of a subspace and... |
| lsatnem0 38454 | The meet of distinct atoms... |
| lsatexch1 38455 | The atom exch1ange propert... |
| lsatcv0eq 38456 | If the sum of two atoms co... |
| lsatcv1 38457 | Two atoms covering the zer... |
| lsatcvatlem 38458 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 38459 | A nonzero subspace less th... |
| lsatcvat2 38460 | A subspace covered by the ... |
| lsatcvat3 38461 | A condition implying that ... |
| islshpcv 38462 | Hyperplane properties expr... |
| l1cvpat 38463 | A subspace covered by the ... |
| l1cvat 38464 | Create an atom under an el... |
| lshpat 38465 | Create an atom under a hyp... |
| lflset 38468 | The set of linear function... |
| islfl 38469 | The predicate "is a linear... |
| lfli 38470 | Property of a linear funct... |
| islfld 38471 | Properties that determine ... |
| lflf 38472 | A linear functional is a f... |
| lflcl 38473 | A linear functional value ... |
| lfl0 38474 | A linear functional is zer... |
| lfladd 38475 | Property of a linear funct... |
| lflsub 38476 | Property of a linear funct... |
| lflmul 38477 | Property of a linear funct... |
| lfl0f 38478 | The zero function is a fun... |
| lfl1 38479 | A nonzero functional has a... |
| lfladdcl 38480 | Closure of addition of two... |
| lfladdcom 38481 | Commutativity of functiona... |
| lfladdass 38482 | Associativity of functiona... |
| lfladd0l 38483 | Functional addition with t... |
| lflnegcl 38484 | Closure of the negative of... |
| lflnegl 38485 | A functional plus its nega... |
| lflvscl 38486 | Closure of a scalar produc... |
| lflvsdi1 38487 | Distributive law for (righ... |
| lflvsdi2 38488 | Reverse distributive law f... |
| lflvsdi2a 38489 | Reverse distributive law f... |
| lflvsass 38490 | Associative law for (right... |
| lfl0sc 38491 | The (right vector space) s... |
| lflsc0N 38492 | The scalar product with th... |
| lfl1sc 38493 | The (right vector space) s... |
| lkrfval 38496 | The kernel of a functional... |
| lkrval 38497 | Value of the kernel of a f... |
| ellkr 38498 | Membership in the kernel o... |
| lkrval2 38499 | Value of the kernel of a f... |
| ellkr2 38500 | Membership in the kernel o... |
| lkrcl 38501 | A member of the kernel of ... |
| lkrf0 38502 | The value of a functional ... |
| lkr0f 38503 | The kernel of the zero fun... |
| lkrlss 38504 | The kernel of a linear fun... |
| lkrssv 38505 | The kernel of a linear fun... |
| lkrsc 38506 | The kernel of a nonzero sc... |
| lkrscss 38507 | The kernel of a scalar pro... |
| eqlkr 38508 | Two functionals with the s... |
| eqlkr2 38509 | Two functionals with the s... |
| eqlkr3 38510 | Two functionals with the s... |
| lkrlsp 38511 | The subspace sum of a kern... |
| lkrlsp2 38512 | The subspace sum of a kern... |
| lkrlsp3 38513 | The subspace sum of a kern... |
| lkrshp 38514 | The kernel of a nonzero fu... |
| lkrshp3 38515 | The kernels of nonzero fun... |
| lkrshpor 38516 | The kernel of a functional... |
| lkrshp4 38517 | A kernel is a hyperplane i... |
| lshpsmreu 38518 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 38519 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 38520 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 38521 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 38522 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 38523 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 38524 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 38525 | The set ` G ` defined by h... |
| lshpkr 38526 | The kernel of functional `... |
| lshpkrex 38527 | There exists a functional ... |
| lshpset2N 38528 | The set of all hyperplanes... |
| islshpkrN 38529 | The predicate "is a hyperp... |
| lfl1dim 38530 | Equivalent expressions for... |
| lfl1dim2N 38531 | Equivalent expressions for... |
| ldualset 38534 | Define the (left) dual of ... |
| ldualvbase 38535 | The vectors of a dual spac... |
| ldualelvbase 38536 | Utility theorem for conver... |
| ldualfvadd 38537 | Vector addition in the dua... |
| ldualvadd 38538 | Vector addition in the dua... |
| ldualvaddcl 38539 | The value of vector additi... |
| ldualvaddval 38540 | The value of the value of ... |
| ldualsca 38541 | The ring of scalars of the... |
| ldualsbase 38542 | Base set of scalar ring fo... |
| ldualsaddN 38543 | Scalar addition for the du... |
| ldualsmul 38544 | Scalar multiplication for ... |
| ldualfvs 38545 | Scalar product operation f... |
| ldualvs 38546 | Scalar product operation v... |
| ldualvsval 38547 | Value of scalar product op... |
| ldualvscl 38548 | The scalar product operati... |
| ldualvaddcom 38549 | Commutative law for vector... |
| ldualvsass 38550 | Associative law for scalar... |
| ldualvsass2 38551 | Associative law for scalar... |
| ldualvsdi1 38552 | Distributive law for scala... |
| ldualvsdi2 38553 | Reverse distributive law f... |
| ldualgrplem 38554 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 38555 | The dual of a vector space... |
| ldual0 38556 | The zero scalar of the dua... |
| ldual1 38557 | The unit scalar of the dua... |
| ldualneg 38558 | The negative of a scalar o... |
| ldual0v 38559 | The zero vector of the dua... |
| ldual0vcl 38560 | The dual zero vector is a ... |
| lduallmodlem 38561 | Lemma for ~ lduallmod . (... |
| lduallmod 38562 | The dual of a left module ... |
| lduallvec 38563 | The dual of a left vector ... |
| ldualvsub 38564 | The value of vector subtra... |
| ldualvsubcl 38565 | Closure of vector subtract... |
| ldualvsubval 38566 | The value of the value of ... |
| ldualssvscl 38567 | Closure of scalar product ... |
| ldualssvsubcl 38568 | Closure of vector subtract... |
| ldual0vs 38569 | Scalar zero times a functi... |
| lkr0f2 38570 | The kernel of the zero fun... |
| lduallkr3 38571 | The kernels of nonzero fun... |
| lkrpssN 38572 | Proper subset relation bet... |
| lkrin 38573 | Intersection of the kernel... |
| eqlkr4 38574 | Two functionals with the s... |
| ldual1dim 38575 | Equivalent expressions for... |
| ldualkrsc 38576 | The kernel of a nonzero sc... |
| lkrss 38577 | The kernel of a scalar pro... |
| lkrss2N 38578 | Two functionals with kerne... |
| lkreqN 38579 | Proportional functionals h... |
| lkrlspeqN 38580 | Condition for colinear fun... |
| isopos 38589 | The predicate "is an ortho... |
| opposet 38590 | Every orthoposet is a pose... |
| oposlem 38591 | Lemma for orthoposet prope... |
| op01dm 38592 | Conditions necessary for z... |
| op0cl 38593 | An orthoposet has a zero e... |
| op1cl 38594 | An orthoposet has a unity ... |
| op0le 38595 | Orthoposet zero is less th... |
| ople0 38596 | An element less than or eq... |
| opnlen0 38597 | An element not less than a... |
| lub0N 38598 | The least upper bound of t... |
| opltn0 38599 | A lattice element greater ... |
| ople1 38600 | Any element is less than t... |
| op1le 38601 | If the orthoposet unity is... |
| glb0N 38602 | The greatest lower bound o... |
| opoccl 38603 | Closure of orthocomplement... |
| opococ 38604 | Double negative law for or... |
| opcon3b 38605 | Contraposition law for ort... |
| opcon2b 38606 | Orthocomplement contraposi... |
| opcon1b 38607 | Orthocomplement contraposi... |
| oplecon3 38608 | Contraposition law for ort... |
| oplecon3b 38609 | Contraposition law for ort... |
| oplecon1b 38610 | Contraposition law for str... |
| opoc1 38611 | Orthocomplement of orthopo... |
| opoc0 38612 | Orthocomplement of orthopo... |
| opltcon3b 38613 | Contraposition law for str... |
| opltcon1b 38614 | Contraposition law for str... |
| opltcon2b 38615 | Contraposition law for str... |
| opexmid 38616 | Law of excluded middle for... |
| opnoncon 38617 | Law of contradiction for o... |
| riotaocN 38618 | The orthocomplement of the... |
| cmtfvalN 38619 | Value of commutes relation... |
| cmtvalN 38620 | Equivalence for commutes r... |
| isolat 38621 | The predicate "is an ortho... |
| ollat 38622 | An ortholattice is a latti... |
| olop 38623 | An ortholattice is an orth... |
| olposN 38624 | An ortholattice is a poset... |
| isolatiN 38625 | Properties that determine ... |
| oldmm1 38626 | De Morgan's law for meet i... |
| oldmm2 38627 | De Morgan's law for meet i... |
| oldmm3N 38628 | De Morgan's law for meet i... |
| oldmm4 38629 | De Morgan's law for meet i... |
| oldmj1 38630 | De Morgan's law for join i... |
| oldmj2 38631 | De Morgan's law for join i... |
| oldmj3 38632 | De Morgan's law for join i... |
| oldmj4 38633 | De Morgan's law for join i... |
| olj01 38634 | An ortholattice element jo... |
| olj02 38635 | An ortholattice element jo... |
| olm11 38636 | The meet of an ortholattic... |
| olm12 38637 | The meet of an ortholattic... |
| latmassOLD 38638 | Ortholattice meet is assoc... |
| latm12 38639 | A rearrangement of lattice... |
| latm32 38640 | A rearrangement of lattice... |
| latmrot 38641 | Rotate lattice meet of 3 c... |
| latm4 38642 | Rearrangement of lattice m... |
| latmmdiN 38643 | Lattice meet distributes o... |
| latmmdir 38644 | Lattice meet distributes o... |
| olm01 38645 | Meet with lattice zero is ... |
| olm02 38646 | Meet with lattice zero is ... |
| isoml 38647 | The predicate "is an ortho... |
| isomliN 38648 | Properties that determine ... |
| omlol 38649 | An orthomodular lattice is... |
| omlop 38650 | An orthomodular lattice is... |
| omllat 38651 | An orthomodular lattice is... |
| omllaw 38652 | The orthomodular law. (Co... |
| omllaw2N 38653 | Variation of orthomodular ... |
| omllaw3 38654 | Orthomodular law equivalen... |
| omllaw4 38655 | Orthomodular law equivalen... |
| omllaw5N 38656 | The orthomodular law. Rem... |
| cmtcomlemN 38657 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 38658 | Commutation is symmetric. ... |
| cmt2N 38659 | Commutation with orthocomp... |
| cmt3N 38660 | Commutation with orthocomp... |
| cmt4N 38661 | Commutation with orthocomp... |
| cmtbr2N 38662 | Alternate definition of th... |
| cmtbr3N 38663 | Alternate definition for t... |
| cmtbr4N 38664 | Alternate definition for t... |
| lecmtN 38665 | Ordered elements commute. ... |
| cmtidN 38666 | Any element commutes with ... |
| omlfh1N 38667 | Foulis-Holland Theorem, pa... |
| omlfh3N 38668 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 38669 | Analogue of modular law ~ ... |
| omlspjN 38670 | Contraction of a Sasaki pr... |
| cvrfval 38677 | Value of covers relation "... |
| cvrval 38678 | Binary relation expressing... |
| cvrlt 38679 | The covers relation implie... |
| cvrnbtwn 38680 | There is no element betwee... |
| ncvr1 38681 | No element covers the latt... |
| cvrletrN 38682 | Property of an element abo... |
| cvrval2 38683 | Binary relation expressing... |
| cvrnbtwn2 38684 | The covers relation implie... |
| cvrnbtwn3 38685 | The covers relation implie... |
| cvrcon3b 38686 | Contraposition law for the... |
| cvrle 38687 | The covers relation implie... |
| cvrnbtwn4 38688 | The covers relation implie... |
| cvrnle 38689 | The covers relation implie... |
| cvrne 38690 | The covers relation implie... |
| cvrnrefN 38691 | The covers relation is not... |
| cvrcmp 38692 | If two lattice elements th... |
| cvrcmp2 38693 | If two lattice elements co... |
| pats 38694 | The set of atoms in a pose... |
| isat 38695 | The predicate "is an atom"... |
| isat2 38696 | The predicate "is an atom"... |
| atcvr0 38697 | An atom covers zero. ( ~ ... |
| atbase 38698 | An atom is a member of the... |
| atssbase 38699 | The set of atoms is a subs... |
| 0ltat 38700 | An atom is greater than ze... |
| leatb 38701 | A poset element less than ... |
| leat 38702 | A poset element less than ... |
| leat2 38703 | A nonzero poset element le... |
| leat3 38704 | A poset element less than ... |
| meetat 38705 | The meet of any element wi... |
| meetat2 38706 | The meet of any element wi... |
| isatl 38708 | The predicate "is an atomi... |
| atllat 38709 | An atomic lattice is a lat... |
| atlpos 38710 | An atomic lattice is a pos... |
| atl0dm 38711 | Condition necessary for ze... |
| atl0cl 38712 | An atomic lattice has a ze... |
| atl0le 38713 | Orthoposet zero is less th... |
| atlle0 38714 | An element less than or eq... |
| atlltn0 38715 | A lattice element greater ... |
| isat3 38716 | The predicate "is an atom"... |
| atn0 38717 | An atom is not zero. ( ~ ... |
| atnle0 38718 | An atom is not less than o... |
| atlen0 38719 | A lattice element is nonze... |
| atcmp 38720 | If two atoms are comparabl... |
| atncmp 38721 | Frequently-used variation ... |
| atnlt 38722 | Two atoms cannot satisfy t... |
| atcvreq0 38723 | An element covered by an a... |
| atncvrN 38724 | Two atoms cannot satisfy t... |
| atlex 38725 | Every nonzero element of a... |
| atnle 38726 | Two ways of expressing "an... |
| atnem0 38727 | The meet of distinct atoms... |
| atlatmstc 38728 | An atomic, complete, ortho... |
| atlatle 38729 | The ordering of two Hilber... |
| atlrelat1 38730 | An atomistic lattice with ... |
| iscvlat 38732 | The predicate "is an atomi... |
| iscvlat2N 38733 | The predicate "is an atomi... |
| cvlatl 38734 | An atomic lattice with the... |
| cvllat 38735 | An atomic lattice with the... |
| cvlposN 38736 | An atomic lattice with the... |
| cvlexch1 38737 | An atomic covering lattice... |
| cvlexch2 38738 | An atomic covering lattice... |
| cvlexchb1 38739 | An atomic covering lattice... |
| cvlexchb2 38740 | An atomic covering lattice... |
| cvlexch3 38741 | An atomic covering lattice... |
| cvlexch4N 38742 | An atomic covering lattice... |
| cvlatexchb1 38743 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 38744 | A version of ~ cvlexchb2 f... |
| cvlatexch1 38745 | Atom exchange property. (... |
| cvlatexch2 38746 | Atom exchange property. (... |
| cvlatexch3 38747 | Atom exchange property. (... |
| cvlcvr1 38748 | The covering property. Pr... |
| cvlcvrp 38749 | A Hilbert lattice satisfie... |
| cvlatcvr1 38750 | An atom is covered by its ... |
| cvlatcvr2 38751 | An atom is covered by its ... |
| cvlsupr2 38752 | Two equivalent ways of exp... |
| cvlsupr3 38753 | Two equivalent ways of exp... |
| cvlsupr4 38754 | Consequence of superpositi... |
| cvlsupr5 38755 | Consequence of superpositi... |
| cvlsupr6 38756 | Consequence of superpositi... |
| cvlsupr7 38757 | Consequence of superpositi... |
| cvlsupr8 38758 | Consequence of superpositi... |
| ishlat1 38761 | The predicate "is a Hilber... |
| ishlat2 38762 | The predicate "is a Hilber... |
| ishlat3N 38763 | The predicate "is a Hilber... |
| ishlatiN 38764 | Properties that determine ... |
| hlomcmcv 38765 | A Hilbert lattice is ortho... |
| hloml 38766 | A Hilbert lattice is ortho... |
| hlclat 38767 | A Hilbert lattice is compl... |
| hlcvl 38768 | A Hilbert lattice is an at... |
| hlatl 38769 | A Hilbert lattice is atomi... |
| hlol 38770 | A Hilbert lattice is an or... |
| hlop 38771 | A Hilbert lattice is an or... |
| hllat 38772 | A Hilbert lattice is a lat... |
| hllatd 38773 | Deduction form of ~ hllat ... |
| hlomcmat 38774 | A Hilbert lattice is ortho... |
| hlpos 38775 | A Hilbert lattice is a pos... |
| hlatjcl 38776 | Closure of join operation.... |
| hlatjcom 38777 | Commutatitivity of join op... |
| hlatjidm 38778 | Idempotence of join operat... |
| hlatjass 38779 | Lattice join is associativ... |
| hlatj12 38780 | Swap 1st and 2nd members o... |
| hlatj32 38781 | Swap 2nd and 3rd members o... |
| hlatjrot 38782 | Rotate lattice join of 3 c... |
| hlatj4 38783 | Rearrangement of lattice j... |
| hlatlej1 38784 | A join's first argument is... |
| hlatlej2 38785 | A join's second argument i... |
| glbconN 38786 | De Morgan's law for GLB an... |
| glbconNOLD 38787 | Obsolete version of ~ glbc... |
| glbconxN 38788 | De Morgan's law for GLB an... |
| atnlej1 38789 | If an atom is not less tha... |
| atnlej2 38790 | If an atom is not less tha... |
| hlsuprexch 38791 | A Hilbert lattice has the ... |
| hlexch1 38792 | A Hilbert lattice has the ... |
| hlexch2 38793 | A Hilbert lattice has the ... |
| hlexchb1 38794 | A Hilbert lattice has the ... |
| hlexchb2 38795 | A Hilbert lattice has the ... |
| hlsupr 38796 | A Hilbert lattice has the ... |
| hlsupr2 38797 | A Hilbert lattice has the ... |
| hlhgt4 38798 | A Hilbert lattice has a he... |
| hlhgt2 38799 | A Hilbert lattice has a he... |
| hl0lt1N 38800 | Lattice 0 is less than lat... |
| hlexch3 38801 | A Hilbert lattice has the ... |
| hlexch4N 38802 | A Hilbert lattice has the ... |
| hlatexchb1 38803 | A version of ~ hlexchb1 fo... |
| hlatexchb2 38804 | A version of ~ hlexchb2 fo... |
| hlatexch1 38805 | Atom exchange property. (... |
| hlatexch2 38806 | Atom exchange property. (... |
| hlatmstcOLDN 38807 | An atomic, complete, ortho... |
| hlatle 38808 | The ordering of two Hilber... |
| hlateq 38809 | The equality of two Hilber... |
| hlrelat1 38810 | An atomistic lattice with ... |
| hlrelat5N 38811 | An atomistic lattice with ... |
| hlrelat 38812 | A Hilbert lattice is relat... |
| hlrelat2 38813 | A consequence of relative ... |
| exatleN 38814 | A condition for an atom to... |
| hl2at 38815 | A Hilbert lattice has at l... |
| atex 38816 | At least one atom exists. ... |
| intnatN 38817 | If the intersection with a... |
| 2llnne2N 38818 | Condition implying that tw... |
| 2llnneN 38819 | Condition implying that tw... |
| cvr1 38820 | A Hilbert lattice has the ... |
| cvr2N 38821 | Less-than and covers equiv... |
| hlrelat3 38822 | The Hilbert lattice is rel... |
| cvrval3 38823 | Binary relation expressing... |
| cvrval4N 38824 | Binary relation expressing... |
| cvrval5 38825 | Binary relation expressing... |
| cvrp 38826 | A Hilbert lattice satisfie... |
| atcvr1 38827 | An atom is covered by its ... |
| atcvr2 38828 | An atom is covered by its ... |
| cvrexchlem 38829 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 38830 | A Hilbert lattice satisfie... |
| cvratlem 38831 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 38832 | A nonzero Hilbert lattice ... |
| ltltncvr 38833 | A chained strong ordering ... |
| ltcvrntr 38834 | Non-transitive condition f... |
| cvrntr 38835 | The covers relation is not... |
| atcvr0eq 38836 | The covers relation is not... |
| lnnat 38837 | A line (the join of two di... |
| atcvrj0 38838 | Two atoms covering the zer... |
| cvrat2 38839 | A Hilbert lattice element ... |
| atcvrneN 38840 | Inequality derived from at... |
| atcvrj1 38841 | Condition for an atom to b... |
| atcvrj2b 38842 | Condition for an atom to b... |
| atcvrj2 38843 | Condition for an atom to b... |
| atleneN 38844 | Inequality derived from at... |
| atltcvr 38845 | An equivalence of less-tha... |
| atle 38846 | Any nonzero element has an... |
| atlt 38847 | Two atoms are unequal iff ... |
| atlelt 38848 | Transfer less-than relatio... |
| 2atlt 38849 | Given an atom less than an... |
| atexchcvrN 38850 | Atom exchange property. V... |
| atexchltN 38851 | Atom exchange property. V... |
| cvrat3 38852 | A condition implying that ... |
| cvrat4 38853 | A condition implying exist... |
| cvrat42 38854 | Commuted version of ~ cvra... |
| 2atjm 38855 | The meet of a line (expres... |
| atbtwn 38856 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 38857 | There exists a 3rd atom ` ... |
| atbtwnex 38858 | Given atoms ` P ` in ` X `... |
| 3noncolr2 38859 | Two ways to express 3 non-... |
| 3noncolr1N 38860 | Two ways to express 3 non-... |
| hlatcon3 38861 | Atom exchange combined wit... |
| hlatcon2 38862 | Atom exchange combined wit... |
| 4noncolr3 38863 | A way to express 4 non-col... |
| 4noncolr2 38864 | A way to express 4 non-col... |
| 4noncolr1 38865 | A way to express 4 non-col... |
| athgt 38866 | A Hilbert lattice, whose h... |
| 3dim0 38867 | There exists a 3-dimension... |
| 3dimlem1 38868 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 38869 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 38870 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 38871 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 38872 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 38873 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 38874 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 38875 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 38876 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 38877 | Construct a 3-dimensional ... |
| 3dim2 38878 | Construct 2 new layers on ... |
| 3dim3 38879 | Construct a new layer on t... |
| 2dim 38880 | Generate a height-3 elemen... |
| 1dimN 38881 | An atom is covered by a he... |
| 1cvrco 38882 | The orthocomplement of an ... |
| 1cvratex 38883 | There exists an atom less ... |
| 1cvratlt 38884 | An atom less than or equal... |
| 1cvrjat 38885 | An element covered by the ... |
| 1cvrat 38886 | Create an atom under an el... |
| ps-1 38887 | The join of two atoms ` R ... |
| ps-2 38888 | Lattice analogue for the p... |
| 2atjlej 38889 | Two atoms are different if... |
| hlatexch3N 38890 | Rearrange join of atoms in... |
| hlatexch4 38891 | Exchange 2 atoms. (Contri... |
| ps-2b 38892 | Variation of projective ge... |
| 3atlem1 38893 | Lemma for ~ 3at . (Contri... |
| 3atlem2 38894 | Lemma for ~ 3at . (Contri... |
| 3atlem3 38895 | Lemma for ~ 3at . (Contri... |
| 3atlem4 38896 | Lemma for ~ 3at . (Contri... |
| 3atlem5 38897 | Lemma for ~ 3at . (Contri... |
| 3atlem6 38898 | Lemma for ~ 3at . (Contri... |
| 3atlem7 38899 | Lemma for ~ 3at . (Contri... |
| 3at 38900 | Any three non-colinear ato... |
| llnset 38915 | The set of lattice lines i... |
| islln 38916 | The predicate "is a lattic... |
| islln4 38917 | The predicate "is a lattic... |
| llni 38918 | Condition implying a latti... |
| llnbase 38919 | A lattice line is a lattic... |
| islln3 38920 | The predicate "is a lattic... |
| islln2 38921 | The predicate "is a lattic... |
| llni2 38922 | The join of two different ... |
| llnnleat 38923 | An atom cannot majorize a ... |
| llnneat 38924 | A lattice line is not an a... |
| 2atneat 38925 | The join of two distinct a... |
| llnn0 38926 | A lattice line is nonzero.... |
| islln2a 38927 | The predicate "is a lattic... |
| llnle 38928 | Any element greater than 0... |
| atcvrlln2 38929 | An atom under a line is co... |
| atcvrlln 38930 | An element covering an ato... |
| llnexatN 38931 | Given an atom on a line, t... |
| llncmp 38932 | If two lattice lines are c... |
| llnnlt 38933 | Two lattice lines cannot s... |
| 2llnmat 38934 | Two intersecting lines int... |
| 2at0mat0 38935 | Special case of ~ 2atmat0 ... |
| 2atmat0 38936 | The meet of two unequal li... |
| 2atm 38937 | An atom majorized by two d... |
| ps-2c 38938 | Variation of projective ge... |
| lplnset 38939 | The set of lattice planes ... |
| islpln 38940 | The predicate "is a lattic... |
| islpln4 38941 | The predicate "is a lattic... |
| lplni 38942 | Condition implying a latti... |
| islpln3 38943 | The predicate "is a lattic... |
| lplnbase 38944 | A lattice plane is a latti... |
| islpln5 38945 | The predicate "is a lattic... |
| islpln2 38946 | The predicate "is a lattic... |
| lplni2 38947 | The join of 3 different at... |
| lvolex3N 38948 | There is an atom outside o... |
| llnmlplnN 38949 | The intersection of a line... |
| lplnle 38950 | Any element greater than 0... |
| lplnnle2at 38951 | A lattice line (or atom) c... |
| lplnnleat 38952 | A lattice plane cannot maj... |
| lplnnlelln 38953 | A lattice plane is not les... |
| 2atnelpln 38954 | The join of two atoms is n... |
| lplnneat 38955 | No lattice plane is an ato... |
| lplnnelln 38956 | No lattice plane is a latt... |
| lplnn0N 38957 | A lattice plane is nonzero... |
| islpln2a 38958 | The predicate "is a lattic... |
| islpln2ah 38959 | The predicate "is a lattic... |
| lplnriaN 38960 | Property of a lattice plan... |
| lplnribN 38961 | Property of a lattice plan... |
| lplnric 38962 | Property of a lattice plan... |
| lplnri1 38963 | Property of a lattice plan... |
| lplnri2N 38964 | Property of a lattice plan... |
| lplnri3N 38965 | Property of a lattice plan... |
| lplnllnneN 38966 | Two lattice lines defined ... |
| llncvrlpln2 38967 | A lattice line under a lat... |
| llncvrlpln 38968 | An element covering a latt... |
| 2lplnmN 38969 | If the join of two lattice... |
| 2llnmj 38970 | The meet of two lattice li... |
| 2atmat 38971 | The meet of two intersecti... |
| lplncmp 38972 | If two lattice planes are ... |
| lplnexatN 38973 | Given a lattice line on a ... |
| lplnexllnN 38974 | Given an atom on a lattice... |
| lplnnlt 38975 | Two lattice planes cannot ... |
| 2llnjaN 38976 | The join of two different ... |
| 2llnjN 38977 | The join of two different ... |
| 2llnm2N 38978 | The meet of two different ... |
| 2llnm3N 38979 | Two lattice lines in a lat... |
| 2llnm4 38980 | Two lattice lines that maj... |
| 2llnmeqat 38981 | An atom equals the interse... |
| lvolset 38982 | The set of 3-dim lattice v... |
| islvol 38983 | The predicate "is a 3-dim ... |
| islvol4 38984 | The predicate "is a 3-dim ... |
| lvoli 38985 | Condition implying a 3-dim... |
| islvol3 38986 | The predicate "is a 3-dim ... |
| lvoli3 38987 | Condition implying a 3-dim... |
| lvolbase 38988 | A 3-dim lattice volume is ... |
| islvol5 38989 | The predicate "is a 3-dim ... |
| islvol2 38990 | The predicate "is a 3-dim ... |
| lvoli2 38991 | The join of 4 different at... |
| lvolnle3at 38992 | A lattice plane (or lattic... |
| lvolnleat 38993 | An atom cannot majorize a ... |
| lvolnlelln 38994 | A lattice line cannot majo... |
| lvolnlelpln 38995 | A lattice plane cannot maj... |
| 3atnelvolN 38996 | The join of 3 atoms is not... |
| 2atnelvolN 38997 | The join of two atoms is n... |
| lvolneatN 38998 | No lattice volume is an at... |
| lvolnelln 38999 | No lattice volume is a lat... |
| lvolnelpln 39000 | No lattice volume is a lat... |
| lvoln0N 39001 | A lattice volume is nonzer... |
| islvol2aN 39002 | The predicate "is a lattic... |
| 4atlem0a 39003 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 39004 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 39005 | Lemma for ~ 4at . (Contri... |
| 4atlem3 39006 | Lemma for ~ 4at . Break i... |
| 4atlem3a 39007 | Lemma for ~ 4at . Break i... |
| 4atlem3b 39008 | Lemma for ~ 4at . Break i... |
| 4atlem4a 39009 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 39010 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 39011 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 39012 | Lemma for ~ 4at . Frequen... |
| 4atlem9 39013 | Lemma for ~ 4at . Substit... |
| 4atlem10a 39014 | Lemma for ~ 4at . Substit... |
| 4atlem10b 39015 | Lemma for ~ 4at . Substit... |
| 4atlem10 39016 | Lemma for ~ 4at . Combine... |
| 4atlem11a 39017 | Lemma for ~ 4at . Substit... |
| 4atlem11b 39018 | Lemma for ~ 4at . Substit... |
| 4atlem11 39019 | Lemma for ~ 4at . Combine... |
| 4atlem12a 39020 | Lemma for ~ 4at . Substit... |
| 4atlem12b 39021 | Lemma for ~ 4at . Substit... |
| 4atlem12 39022 | Lemma for ~ 4at . Combine... |
| 4at 39023 | Four atoms determine a lat... |
| 4at2 39024 | Four atoms determine a lat... |
| lplncvrlvol2 39025 | A lattice line under a lat... |
| lplncvrlvol 39026 | An element covering a latt... |
| lvolcmp 39027 | If two lattice planes are ... |
| lvolnltN 39028 | Two lattice volumes cannot... |
| 2lplnja 39029 | The join of two different ... |
| 2lplnj 39030 | The join of two different ... |
| 2lplnm2N 39031 | The meet of two different ... |
| 2lplnmj 39032 | The meet of two lattice pl... |
| dalemkehl 39033 | Lemma for ~ dath . Freque... |
| dalemkelat 39034 | Lemma for ~ dath . Freque... |
| dalemkeop 39035 | Lemma for ~ dath . Freque... |
| dalempea 39036 | Lemma for ~ dath . Freque... |
| dalemqea 39037 | Lemma for ~ dath . Freque... |
| dalemrea 39038 | Lemma for ~ dath . Freque... |
| dalemsea 39039 | Lemma for ~ dath . Freque... |
| dalemtea 39040 | Lemma for ~ dath . Freque... |
| dalemuea 39041 | Lemma for ~ dath . Freque... |
| dalemyeo 39042 | Lemma for ~ dath . Freque... |
| dalemzeo 39043 | Lemma for ~ dath . Freque... |
| dalemclpjs 39044 | Lemma for ~ dath . Freque... |
| dalemclqjt 39045 | Lemma for ~ dath . Freque... |
| dalemclrju 39046 | Lemma for ~ dath . Freque... |
| dalem-clpjq 39047 | Lemma for ~ dath . Freque... |
| dalemceb 39048 | Lemma for ~ dath . Freque... |
| dalempeb 39049 | Lemma for ~ dath . Freque... |
| dalemqeb 39050 | Lemma for ~ dath . Freque... |
| dalemreb 39051 | Lemma for ~ dath . Freque... |
| dalemseb 39052 | Lemma for ~ dath . Freque... |
| dalemteb 39053 | Lemma for ~ dath . Freque... |
| dalemueb 39054 | Lemma for ~ dath . Freque... |
| dalempjqeb 39055 | Lemma for ~ dath . Freque... |
| dalemsjteb 39056 | Lemma for ~ dath . Freque... |
| dalemtjueb 39057 | Lemma for ~ dath . Freque... |
| dalemqrprot 39058 | Lemma for ~ dath . Freque... |
| dalemyeb 39059 | Lemma for ~ dath . Freque... |
| dalemcnes 39060 | Lemma for ~ dath . Freque... |
| dalempnes 39061 | Lemma for ~ dath . Freque... |
| dalemqnet 39062 | Lemma for ~ dath . Freque... |
| dalempjsen 39063 | Lemma for ~ dath . Freque... |
| dalemply 39064 | Lemma for ~ dath . Freque... |
| dalemsly 39065 | Lemma for ~ dath . Freque... |
| dalemswapyz 39066 | Lemma for ~ dath . Swap t... |
| dalemrot 39067 | Lemma for ~ dath . Rotate... |
| dalemrotyz 39068 | Lemma for ~ dath . Rotate... |
| dalem1 39069 | Lemma for ~ dath . Show t... |
| dalemcea 39070 | Lemma for ~ dath . Freque... |
| dalem2 39071 | Lemma for ~ dath . Show t... |
| dalemdea 39072 | Lemma for ~ dath . Freque... |
| dalemeea 39073 | Lemma for ~ dath . Freque... |
| dalem3 39074 | Lemma for ~ dalemdnee . (... |
| dalem4 39075 | Lemma for ~ dalemdnee . (... |
| dalemdnee 39076 | Lemma for ~ dath . Axis o... |
| dalem5 39077 | Lemma for ~ dath . Atom `... |
| dalem6 39078 | Lemma for ~ dath . Analog... |
| dalem7 39079 | Lemma for ~ dath . Analog... |
| dalem8 39080 | Lemma for ~ dath . Plane ... |
| dalem-cly 39081 | Lemma for ~ dalem9 . Cent... |
| dalem9 39082 | Lemma for ~ dath . Since ... |
| dalem10 39083 | Lemma for ~ dath . Atom `... |
| dalem11 39084 | Lemma for ~ dath . Analog... |
| dalem12 39085 | Lemma for ~ dath . Analog... |
| dalem13 39086 | Lemma for ~ dalem14 . (Co... |
| dalem14 39087 | Lemma for ~ dath . Planes... |
| dalem15 39088 | Lemma for ~ dath . The ax... |
| dalem16 39089 | Lemma for ~ dath . The at... |
| dalem17 39090 | Lemma for ~ dath . When p... |
| dalem18 39091 | Lemma for ~ dath . Show t... |
| dalem19 39092 | Lemma for ~ dath . Show t... |
| dalemccea 39093 | Lemma for ~ dath . Freque... |
| dalemddea 39094 | Lemma for ~ dath . Freque... |
| dalem-ccly 39095 | Lemma for ~ dath . Freque... |
| dalem-ddly 39096 | Lemma for ~ dath . Freque... |
| dalemccnedd 39097 | Lemma for ~ dath . Freque... |
| dalemclccjdd 39098 | Lemma for ~ dath . Freque... |
| dalemcceb 39099 | Lemma for ~ dath . Freque... |
| dalemswapyzps 39100 | Lemma for ~ dath . Swap t... |
| dalemrotps 39101 | Lemma for ~ dath . Rotate... |
| dalemcjden 39102 | Lemma for ~ dath . Show t... |
| dalem20 39103 | Lemma for ~ dath . Show t... |
| dalem21 39104 | Lemma for ~ dath . Show t... |
| dalem22 39105 | Lemma for ~ dath . Show t... |
| dalem23 39106 | Lemma for ~ dath . Show t... |
| dalem24 39107 | Lemma for ~ dath . Show t... |
| dalem25 39108 | Lemma for ~ dath . Show t... |
| dalem27 39109 | Lemma for ~ dath . Show t... |
| dalem28 39110 | Lemma for ~ dath . Lemma ... |
| dalem29 39111 | Lemma for ~ dath . Analog... |
| dalem30 39112 | Lemma for ~ dath . Analog... |
| dalem31N 39113 | Lemma for ~ dath . Analog... |
| dalem32 39114 | Lemma for ~ dath . Analog... |
| dalem33 39115 | Lemma for ~ dath . Analog... |
| dalem34 39116 | Lemma for ~ dath . Analog... |
| dalem35 39117 | Lemma for ~ dath . Analog... |
| dalem36 39118 | Lemma for ~ dath . Analog... |
| dalem37 39119 | Lemma for ~ dath . Analog... |
| dalem38 39120 | Lemma for ~ dath . Plane ... |
| dalem39 39121 | Lemma for ~ dath . Auxili... |
| dalem40 39122 | Lemma for ~ dath . Analog... |
| dalem41 39123 | Lemma for ~ dath . (Contr... |
| dalem42 39124 | Lemma for ~ dath . Auxili... |
| dalem43 39125 | Lemma for ~ dath . Planes... |
| dalem44 39126 | Lemma for ~ dath . Dummy ... |
| dalem45 39127 | Lemma for ~ dath . Dummy ... |
| dalem46 39128 | Lemma for ~ dath . Analog... |
| dalem47 39129 | Lemma for ~ dath . Analog... |
| dalem48 39130 | Lemma for ~ dath . Analog... |
| dalem49 39131 | Lemma for ~ dath . Analog... |
| dalem50 39132 | Lemma for ~ dath . Analog... |
| dalem51 39133 | Lemma for ~ dath . Constr... |
| dalem52 39134 | Lemma for ~ dath . Lines ... |
| dalem53 39135 | Lemma for ~ dath . The au... |
| dalem54 39136 | Lemma for ~ dath . Line `... |
| dalem55 39137 | Lemma for ~ dath . Lines ... |
| dalem56 39138 | Lemma for ~ dath . Analog... |
| dalem57 39139 | Lemma for ~ dath . Axis o... |
| dalem58 39140 | Lemma for ~ dath . Analog... |
| dalem59 39141 | Lemma for ~ dath . Analog... |
| dalem60 39142 | Lemma for ~ dath . ` B ` i... |
| dalem61 39143 | Lemma for ~ dath . Show t... |
| dalem62 39144 | Lemma for ~ dath . Elimin... |
| dalem63 39145 | Lemma for ~ dath . Combin... |
| dath 39146 | Desargues's theorem of pro... |
| dath2 39147 | Version of Desargues's the... |
| lineset 39148 | The set of lines in a Hilb... |
| isline 39149 | The predicate "is a line".... |
| islinei 39150 | Condition implying "is a l... |
| pointsetN 39151 | The set of points in a Hil... |
| ispointN 39152 | The predicate "is a point"... |
| atpointN 39153 | The singleton of an atom i... |
| psubspset 39154 | The set of projective subs... |
| ispsubsp 39155 | The predicate "is a projec... |
| ispsubsp2 39156 | The predicate "is a projec... |
| psubspi 39157 | Property of a projective s... |
| psubspi2N 39158 | Property of a projective s... |
| 0psubN 39159 | The empty set is a project... |
| snatpsubN 39160 | The singleton of an atom i... |
| pointpsubN 39161 | A point (singleton of an a... |
| linepsubN 39162 | A line is a projective sub... |
| atpsubN 39163 | The set of all atoms is a ... |
| psubssat 39164 | A projective subspace cons... |
| psubatN 39165 | A member of a projective s... |
| pmapfval 39166 | The projective map of a Hi... |
| pmapval 39167 | Value of the projective ma... |
| elpmap 39168 | Member of a projective map... |
| pmapssat 39169 | The projective map of a Hi... |
| pmapssbaN 39170 | A weakening of ~ pmapssat ... |
| pmaple 39171 | The projective map of a Hi... |
| pmap11 39172 | The projective map of a Hi... |
| pmapat 39173 | The projective map of an a... |
| elpmapat 39174 | Member of the projective m... |
| pmap0 39175 | Value of the projective ma... |
| pmapeq0 39176 | A projective map value is ... |
| pmap1N 39177 | Value of the projective ma... |
| pmapsub 39178 | The projective map of a Hi... |
| pmapglbx 39179 | The projective map of the ... |
| pmapglb 39180 | The projective map of the ... |
| pmapglb2N 39181 | The projective map of the ... |
| pmapglb2xN 39182 | The projective map of the ... |
| pmapmeet 39183 | The projective map of a me... |
| isline2 39184 | Definition of line in term... |
| linepmap 39185 | A line described with a pr... |
| isline3 39186 | Definition of line in term... |
| isline4N 39187 | Definition of line in term... |
| lneq2at 39188 | A line equals the join of ... |
| lnatexN 39189 | There is an atom in a line... |
| lnjatN 39190 | Given an atom in a line, t... |
| lncvrelatN 39191 | A lattice element covered ... |
| lncvrat 39192 | A line covers the atoms it... |
| lncmp 39193 | If two lines are comparabl... |
| 2lnat 39194 | Two intersecting lines int... |
| 2atm2atN 39195 | Two joins with a common at... |
| 2llnma1b 39196 | Generalization of ~ 2llnma... |
| 2llnma1 39197 | Two different intersecting... |
| 2llnma3r 39198 | Two different intersecting... |
| 2llnma2 39199 | Two different intersecting... |
| 2llnma2rN 39200 | Two different intersecting... |
| cdlema1N 39201 | A condition for required f... |
| cdlema2N 39202 | A condition for required f... |
| cdlemblem 39203 | Lemma for ~ cdlemb . (Con... |
| cdlemb 39204 | Given two atoms not less t... |
| paddfval 39207 | Projective subspace sum op... |
| paddval 39208 | Projective subspace sum op... |
| elpadd 39209 | Member of a projective sub... |
| elpaddn0 39210 | Member of projective subsp... |
| paddvaln0N 39211 | Projective subspace sum op... |
| elpaddri 39212 | Condition implying members... |
| elpaddatriN 39213 | Condition implying members... |
| elpaddat 39214 | Membership in a projective... |
| elpaddatiN 39215 | Consequence of membership ... |
| elpadd2at 39216 | Membership in a projective... |
| elpadd2at2 39217 | Membership in a projective... |
| paddunssN 39218 | Projective subspace sum in... |
| elpadd0 39219 | Member of projective subsp... |
| paddval0 39220 | Projective subspace sum wi... |
| padd01 39221 | Projective subspace sum wi... |
| padd02 39222 | Projective subspace sum wi... |
| paddcom 39223 | Projective subspace sum co... |
| paddssat 39224 | A projective subspace sum ... |
| sspadd1 39225 | A projective subspace sum ... |
| sspadd2 39226 | A projective subspace sum ... |
| paddss1 39227 | Subset law for projective ... |
| paddss2 39228 | Subset law for projective ... |
| paddss12 39229 | Subset law for projective ... |
| paddasslem1 39230 | Lemma for ~ paddass . (Co... |
| paddasslem2 39231 | Lemma for ~ paddass . (Co... |
| paddasslem3 39232 | Lemma for ~ paddass . Res... |
| paddasslem4 39233 | Lemma for ~ paddass . Com... |
| paddasslem5 39234 | Lemma for ~ paddass . Sho... |
| paddasslem6 39235 | Lemma for ~ paddass . (Co... |
| paddasslem7 39236 | Lemma for ~ paddass . Com... |
| paddasslem8 39237 | Lemma for ~ paddass . (Co... |
| paddasslem9 39238 | Lemma for ~ paddass . Com... |
| paddasslem10 39239 | Lemma for ~ paddass . Use... |
| paddasslem11 39240 | Lemma for ~ paddass . The... |
| paddasslem12 39241 | Lemma for ~ paddass . The... |
| paddasslem13 39242 | Lemma for ~ paddass . The... |
| paddasslem14 39243 | Lemma for ~ paddass . Rem... |
| paddasslem15 39244 | Lemma for ~ paddass . Use... |
| paddasslem16 39245 | Lemma for ~ paddass . Use... |
| paddasslem17 39246 | Lemma for ~ paddass . The... |
| paddasslem18 39247 | Lemma for ~ paddass . Com... |
| paddass 39248 | Projective subspace sum is... |
| padd12N 39249 | Commutative/associative la... |
| padd4N 39250 | Rearrangement of 4 terms i... |
| paddidm 39251 | Projective subspace sum is... |
| paddclN 39252 | The projective sum of two ... |
| paddssw1 39253 | Subset law for projective ... |
| paddssw2 39254 | Subset law for projective ... |
| paddss 39255 | Subset law for projective ... |
| pmodlem1 39256 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 39257 | Lemma for ~ pmod1i . (Con... |
| pmod1i 39258 | The modular law holds in a... |
| pmod2iN 39259 | Dual of the modular law. ... |
| pmodN 39260 | The modular law for projec... |
| pmodl42N 39261 | Lemma derived from modular... |
| pmapjoin 39262 | The projective map of the ... |
| pmapjat1 39263 | The projective map of the ... |
| pmapjat2 39264 | The projective map of the ... |
| pmapjlln1 39265 | The projective map of the ... |
| hlmod1i 39266 | A version of the modular l... |
| atmod1i1 39267 | Version of modular law ~ p... |
| atmod1i1m 39268 | Version of modular law ~ p... |
| atmod1i2 39269 | Version of modular law ~ p... |
| llnmod1i2 39270 | Version of modular law ~ p... |
| atmod2i1 39271 | Version of modular law ~ p... |
| atmod2i2 39272 | Version of modular law ~ p... |
| llnmod2i2 39273 | Version of modular law ~ p... |
| atmod3i1 39274 | Version of modular law tha... |
| atmod3i2 39275 | Version of modular law tha... |
| atmod4i1 39276 | Version of modular law tha... |
| atmod4i2 39277 | Version of modular law tha... |
| llnexchb2lem 39278 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 39279 | Line exchange property (co... |
| llnexch2N 39280 | Line exchange property (co... |
| dalawlem1 39281 | Lemma for ~ dalaw . Speci... |
| dalawlem2 39282 | Lemma for ~ dalaw . Utili... |
| dalawlem3 39283 | Lemma for ~ dalaw . First... |
| dalawlem4 39284 | Lemma for ~ dalaw . Secon... |
| dalawlem5 39285 | Lemma for ~ dalaw . Speci... |
| dalawlem6 39286 | Lemma for ~ dalaw . First... |
| dalawlem7 39287 | Lemma for ~ dalaw . Secon... |
| dalawlem8 39288 | Lemma for ~ dalaw . Speci... |
| dalawlem9 39289 | Lemma for ~ dalaw . Speci... |
| dalawlem10 39290 | Lemma for ~ dalaw . Combi... |
| dalawlem11 39291 | Lemma for ~ dalaw . First... |
| dalawlem12 39292 | Lemma for ~ dalaw . Secon... |
| dalawlem13 39293 | Lemma for ~ dalaw . Speci... |
| dalawlem14 39294 | Lemma for ~ dalaw . Combi... |
| dalawlem15 39295 | Lemma for ~ dalaw . Swap ... |
| dalaw 39296 | Desargues's law, derived f... |
| pclfvalN 39299 | The projective subspace cl... |
| pclvalN 39300 | Value of the projective su... |
| pclclN 39301 | Closure of the projective ... |
| elpclN 39302 | Membership in the projecti... |
| elpcliN 39303 | Implication of membership ... |
| pclssN 39304 | Ordering is preserved by s... |
| pclssidN 39305 | A set of atoms is included... |
| pclidN 39306 | The projective subspace cl... |
| pclbtwnN 39307 | A projective subspace sand... |
| pclunN 39308 | The projective subspace cl... |
| pclun2N 39309 | The projective subspace cl... |
| pclfinN 39310 | The projective subspace cl... |
| pclcmpatN 39311 | The set of projective subs... |
| polfvalN 39314 | The projective subspace po... |
| polvalN 39315 | Value of the projective su... |
| polval2N 39316 | Alternate expression for v... |
| polsubN 39317 | The polarity of a set of a... |
| polssatN 39318 | The polarity of a set of a... |
| pol0N 39319 | The polarity of the empty ... |
| pol1N 39320 | The polarity of the whole ... |
| 2pol0N 39321 | The closed subspace closur... |
| polpmapN 39322 | The polarity of a projecti... |
| 2polpmapN 39323 | Double polarity of a proje... |
| 2polvalN 39324 | Value of double polarity. ... |
| 2polssN 39325 | A set of atoms is a subset... |
| 3polN 39326 | Triple polarity cancels to... |
| polcon3N 39327 | Contraposition law for pol... |
| 2polcon4bN 39328 | Contraposition law for pol... |
| polcon2N 39329 | Contraposition law for pol... |
| polcon2bN 39330 | Contraposition law for pol... |
| pclss2polN 39331 | The projective subspace cl... |
| pcl0N 39332 | The projective subspace cl... |
| pcl0bN 39333 | The projective subspace cl... |
| pmaplubN 39334 | The LUB of a projective ma... |
| sspmaplubN 39335 | A set of atoms is a subset... |
| 2pmaplubN 39336 | Double projective map of a... |
| paddunN 39337 | The closure of the project... |
| poldmj1N 39338 | De Morgan's law for polari... |
| pmapj2N 39339 | The projective map of the ... |
| pmapocjN 39340 | The projective map of the ... |
| polatN 39341 | The polarity of the single... |
| 2polatN 39342 | Double polarity of the sin... |
| pnonsingN 39343 | The intersection of a set ... |
| psubclsetN 39346 | The set of closed projecti... |
| ispsubclN 39347 | The predicate "is a closed... |
| psubcliN 39348 | Property of a closed proje... |
| psubcli2N 39349 | Property of a closed proje... |
| psubclsubN 39350 | A closed projective subspa... |
| psubclssatN 39351 | A closed projective subspa... |
| pmapidclN 39352 | Projective map of the LUB ... |
| 0psubclN 39353 | The empty set is a closed ... |
| 1psubclN 39354 | The set of all atoms is a ... |
| atpsubclN 39355 | A point (singleton of an a... |
| pmapsubclN 39356 | A projective map value is ... |
| ispsubcl2N 39357 | Alternate predicate for "i... |
| psubclinN 39358 | The intersection of two cl... |
| paddatclN 39359 | The projective sum of a cl... |
| pclfinclN 39360 | The projective subspace cl... |
| linepsubclN 39361 | A line is a closed project... |
| polsubclN 39362 | A polarity is a closed pro... |
| poml4N 39363 | Orthomodular law for proje... |
| poml5N 39364 | Orthomodular law for proje... |
| poml6N 39365 | Orthomodular law for proje... |
| osumcllem1N 39366 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 39367 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 39368 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 39369 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 39370 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 39371 | Lemma for ~ osumclN . Use... |
| osumcllem7N 39372 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 39373 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 39374 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 39375 | Lemma for ~ osumclN . Con... |
| osumcllem11N 39376 | Lemma for ~ osumclN . (Co... |
| osumclN 39377 | Closure of orthogonal sum.... |
| pmapojoinN 39378 | For orthogonal elements, p... |
| pexmidN 39379 | Excluded middle law for cl... |
| pexmidlem1N 39380 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 39381 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 39382 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 39383 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 39384 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 39385 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 39386 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 39387 | Lemma for ~ pexmidN . The... |
| pexmidALTN 39388 | Excluded middle law for cl... |
| pl42lem1N 39389 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 39390 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 39391 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 39392 | Lemma for ~ pl42N . (Cont... |
| pl42N 39393 | Law holding in a Hilbert l... |
| watfvalN 39402 | The W atoms function. (Co... |
| watvalN 39403 | Value of the W atoms funct... |
| iswatN 39404 | The predicate "is a W atom... |
| lhpset 39405 | The set of co-atoms (latti... |
| islhp 39406 | The predicate "is a co-ato... |
| islhp2 39407 | The predicate "is a co-ato... |
| lhpbase 39408 | A co-atom is a member of t... |
| lhp1cvr 39409 | The lattice unity covers a... |
| lhplt 39410 | An atom under a co-atom is... |
| lhp2lt 39411 | The join of two atoms unde... |
| lhpexlt 39412 | There exists an atom less ... |
| lhp0lt 39413 | A co-atom is greater than ... |
| lhpn0 39414 | A co-atom is nonzero. TOD... |
| lhpexle 39415 | There exists an atom under... |
| lhpexnle 39416 | There exists an atom not u... |
| lhpexle1lem 39417 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 39418 | There exists an atom under... |
| lhpexle2lem 39419 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 39420 | There exists atom under a ... |
| lhpexle3lem 39421 | There exists atom under a ... |
| lhpexle3 39422 | There exists atom under a ... |
| lhpex2leN 39423 | There exist at least two d... |
| lhpoc 39424 | The orthocomplement of a c... |
| lhpoc2N 39425 | The orthocomplement of an ... |
| lhpocnle 39426 | The orthocomplement of a c... |
| lhpocat 39427 | The orthocomplement of a c... |
| lhpocnel 39428 | The orthocomplement of a c... |
| lhpocnel2 39429 | The orthocomplement of a c... |
| lhpjat1 39430 | The join of a co-atom (hyp... |
| lhpjat2 39431 | The join of a co-atom (hyp... |
| lhpj1 39432 | The join of a co-atom (hyp... |
| lhpmcvr 39433 | The meet of a lattice hype... |
| lhpmcvr2 39434 | Alternate way to express t... |
| lhpmcvr3 39435 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 39436 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 39437 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 39438 | Specialization of ~ lhpmcv... |
| lhpm0atN 39439 | If the meet of a lattice h... |
| lhpmat 39440 | An element covered by the ... |
| lhpmatb 39441 | An element covered by the ... |
| lhp2at0 39442 | Join and meet with differe... |
| lhp2atnle 39443 | Inequality for 2 different... |
| lhp2atne 39444 | Inequality for joins with ... |
| lhp2at0nle 39445 | Inequality for 2 different... |
| lhp2at0ne 39446 | Inequality for joins with ... |
| lhpelim 39447 | Eliminate an atom not unde... |
| lhpmod2i2 39448 | Modular law for hyperplane... |
| lhpmod6i1 39449 | Modular law for hyperplane... |
| lhprelat3N 39450 | The Hilbert lattice is rel... |
| cdlemb2 39451 | Given two atoms not under ... |
| lhple 39452 | Property of a lattice elem... |
| lhpat 39453 | Create an atom under a co-... |
| lhpat4N 39454 | Property of an atom under ... |
| lhpat2 39455 | Create an atom under a co-... |
| lhpat3 39456 | There is only one atom und... |
| 4atexlemk 39457 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 39458 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 39459 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 39460 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 39461 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 39462 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 39463 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 39464 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 39465 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 39466 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 39467 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 39468 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 39469 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 39470 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 39471 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 39472 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 39473 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 39474 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 39475 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 39476 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 39477 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 39478 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 39479 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 39480 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 39481 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 39482 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 39483 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 39484 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 39485 | Whenever there are at leas... |
| 4atex 39486 | Whenever there are at leas... |
| 4atex2 39487 | More general version of ~ ... |
| 4atex2-0aOLDN 39488 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 39489 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 39490 | Same as ~ 4atex2 except th... |
| 4atex3 39491 | More general version of ~ ... |
| lautset 39492 | The set of lattice automor... |
| islaut 39493 | The predicate "is a lattic... |
| lautle 39494 | Less-than or equal propert... |
| laut1o 39495 | A lattice automorphism is ... |
| laut11 39496 | One-to-one property of a l... |
| lautcl 39497 | A lattice automorphism val... |
| lautcnvclN 39498 | Reverse closure of a latti... |
| lautcnvle 39499 | Less-than or equal propert... |
| lautcnv 39500 | The converse of a lattice ... |
| lautlt 39501 | Less-than property of a la... |
| lautcvr 39502 | Covering property of a lat... |
| lautj 39503 | Meet property of a lattice... |
| lautm 39504 | Meet property of a lattice... |
| lauteq 39505 | A lattice automorphism arg... |
| idlaut 39506 | The identity function is a... |
| lautco 39507 | The composition of two lat... |
| pautsetN 39508 | The set of projective auto... |
| ispautN 39509 | The predicate "is a projec... |
| ldilfset 39518 | The mapping from fiducial ... |
| ldilset 39519 | The set of lattice dilatio... |
| isldil 39520 | The predicate "is a lattic... |
| ldillaut 39521 | A lattice dilation is an a... |
| ldil1o 39522 | A lattice dilation is a on... |
| ldilval 39523 | Value of a lattice dilatio... |
| idldil 39524 | The identity function is a... |
| ldilcnv 39525 | The converse of a lattice ... |
| ldilco 39526 | The composition of two lat... |
| ltrnfset 39527 | The set of all lattice tra... |
| ltrnset 39528 | The set of lattice transla... |
| isltrn 39529 | The predicate "is a lattic... |
| isltrn2N 39530 | The predicate "is a lattic... |
| ltrnu 39531 | Uniqueness property of a l... |
| ltrnldil 39532 | A lattice translation is a... |
| ltrnlaut 39533 | A lattice translation is a... |
| ltrn1o 39534 | A lattice translation is a... |
| ltrncl 39535 | Closure of a lattice trans... |
| ltrn11 39536 | One-to-one property of a l... |
| ltrncnvnid 39537 | If a translation is differ... |
| ltrncoidN 39538 | Two translations are equal... |
| ltrnle 39539 | Less-than or equal propert... |
| ltrncnvleN 39540 | Less-than or equal propert... |
| ltrnm 39541 | Lattice translation of a m... |
| ltrnj 39542 | Lattice translation of a m... |
| ltrncvr 39543 | Covering property of a lat... |
| ltrnval1 39544 | Value of a lattice transla... |
| ltrnid 39545 | A lattice translation is t... |
| ltrnnid 39546 | If a lattice translation i... |
| ltrnatb 39547 | The lattice translation of... |
| ltrncnvatb 39548 | The converse of the lattic... |
| ltrnel 39549 | The lattice translation of... |
| ltrnat 39550 | The lattice translation of... |
| ltrncnvat 39551 | The converse of the lattic... |
| ltrncnvel 39552 | The converse of the lattic... |
| ltrncoelN 39553 | Composition of lattice tra... |
| ltrncoat 39554 | Composition of lattice tra... |
| ltrncoval 39555 | Two ways to express value ... |
| ltrncnv 39556 | The converse of a lattice ... |
| ltrn11at 39557 | Frequently used one-to-one... |
| ltrneq2 39558 | The equality of two transl... |
| ltrneq 39559 | The equality of two transl... |
| idltrn 39560 | The identity function is a... |
| ltrnmw 39561 | Property of lattice transl... |
| dilfsetN 39562 | The mapping from fiducial ... |
| dilsetN 39563 | The set of dilations for a... |
| isdilN 39564 | The predicate "is a dilati... |
| trnfsetN 39565 | The mapping from fiducial ... |
| trnsetN 39566 | The set of translations fo... |
| istrnN 39567 | The predicate "is a transl... |
| trlfset 39570 | The set of all traces of l... |
| trlset 39571 | The set of traces of latti... |
| trlval 39572 | The value of the trace of ... |
| trlval2 39573 | The value of the trace of ... |
| trlcl 39574 | Closure of the trace of a ... |
| trlcnv 39575 | The trace of the converse ... |
| trljat1 39576 | The value of a translation... |
| trljat2 39577 | The value of a translation... |
| trljat3 39578 | The value of a translation... |
| trlat 39579 | If an atom differs from it... |
| trl0 39580 | If an atom not under the f... |
| trlator0 39581 | The trace of a lattice tra... |
| trlatn0 39582 | The trace of a lattice tra... |
| trlnidat 39583 | The trace of a lattice tra... |
| ltrnnidn 39584 | If a lattice translation i... |
| ltrnideq 39585 | Property of the identity l... |
| trlid0 39586 | The trace of the identity ... |
| trlnidatb 39587 | A lattice translation is n... |
| trlid0b 39588 | A lattice translation is t... |
| trlnid 39589 | Different translations wit... |
| ltrn2ateq 39590 | Property of the equality o... |
| ltrnateq 39591 | If any atom (under ` W ` )... |
| ltrnatneq 39592 | If any atom (under ` W ` )... |
| ltrnatlw 39593 | If the value of an atom eq... |
| trlle 39594 | The trace of a lattice tra... |
| trlne 39595 | The trace of a lattice tra... |
| trlnle 39596 | The atom not under the fid... |
| trlval3 39597 | The value of the trace of ... |
| trlval4 39598 | The value of the trace of ... |
| trlval5 39599 | The value of the trace of ... |
| arglem1N 39600 | Lemma for Desargues's law.... |
| cdlemc1 39601 | Part of proof of Lemma C i... |
| cdlemc2 39602 | Part of proof of Lemma C i... |
| cdlemc3 39603 | Part of proof of Lemma C i... |
| cdlemc4 39604 | Part of proof of Lemma C i... |
| cdlemc5 39605 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 39606 | Lemma for ~ cdlemc . (Con... |
| cdlemc 39607 | Lemma C in [Crawley] p. 11... |
| cdlemd1 39608 | Part of proof of Lemma D i... |
| cdlemd2 39609 | Part of proof of Lemma D i... |
| cdlemd3 39610 | Part of proof of Lemma D i... |
| cdlemd4 39611 | Part of proof of Lemma D i... |
| cdlemd5 39612 | Part of proof of Lemma D i... |
| cdlemd6 39613 | Part of proof of Lemma D i... |
| cdlemd7 39614 | Part of proof of Lemma D i... |
| cdlemd8 39615 | Part of proof of Lemma D i... |
| cdlemd9 39616 | Part of proof of Lemma D i... |
| cdlemd 39617 | If two translations agree ... |
| ltrneq3 39618 | Two translations agree at ... |
| cdleme00a 39619 | Part of proof of Lemma E i... |
| cdleme0aa 39620 | Part of proof of Lemma E i... |
| cdleme0a 39621 | Part of proof of Lemma E i... |
| cdleme0b 39622 | Part of proof of Lemma E i... |
| cdleme0c 39623 | Part of proof of Lemma E i... |
| cdleme0cp 39624 | Part of proof of Lemma E i... |
| cdleme0cq 39625 | Part of proof of Lemma E i... |
| cdleme0dN 39626 | Part of proof of Lemma E i... |
| cdleme0e 39627 | Part of proof of Lemma E i... |
| cdleme0fN 39628 | Part of proof of Lemma E i... |
| cdleme0gN 39629 | Part of proof of Lemma E i... |
| cdlemeulpq 39630 | Part of proof of Lemma E i... |
| cdleme01N 39631 | Part of proof of Lemma E i... |
| cdleme02N 39632 | Part of proof of Lemma E i... |
| cdleme0ex1N 39633 | Part of proof of Lemma E i... |
| cdleme0ex2N 39634 | Part of proof of Lemma E i... |
| cdleme0moN 39635 | Part of proof of Lemma E i... |
| cdleme1b 39636 | Part of proof of Lemma E i... |
| cdleme1 39637 | Part of proof of Lemma E i... |
| cdleme2 39638 | Part of proof of Lemma E i... |
| cdleme3b 39639 | Part of proof of Lemma E i... |
| cdleme3c 39640 | Part of proof of Lemma E i... |
| cdleme3d 39641 | Part of proof of Lemma E i... |
| cdleme3e 39642 | Part of proof of Lemma E i... |
| cdleme3fN 39643 | Part of proof of Lemma E i... |
| cdleme3g 39644 | Part of proof of Lemma E i... |
| cdleme3h 39645 | Part of proof of Lemma E i... |
| cdleme3fa 39646 | Part of proof of Lemma E i... |
| cdleme3 39647 | Part of proof of Lemma E i... |
| cdleme4 39648 | Part of proof of Lemma E i... |
| cdleme4a 39649 | Part of proof of Lemma E i... |
| cdleme5 39650 | Part of proof of Lemma E i... |
| cdleme6 39651 | Part of proof of Lemma E i... |
| cdleme7aa 39652 | Part of proof of Lemma E i... |
| cdleme7a 39653 | Part of proof of Lemma E i... |
| cdleme7b 39654 | Part of proof of Lemma E i... |
| cdleme7c 39655 | Part of proof of Lemma E i... |
| cdleme7d 39656 | Part of proof of Lemma E i... |
| cdleme7e 39657 | Part of proof of Lemma E i... |
| cdleme7ga 39658 | Part of proof of Lemma E i... |
| cdleme7 39659 | Part of proof of Lemma E i... |
| cdleme8 39660 | Part of proof of Lemma E i... |
| cdleme9a 39661 | Part of proof of Lemma E i... |
| cdleme9b 39662 | Utility lemma for Lemma E ... |
| cdleme9 39663 | Part of proof of Lemma E i... |
| cdleme10 39664 | Part of proof of Lemma E i... |
| cdleme8tN 39665 | Part of proof of Lemma E i... |
| cdleme9taN 39666 | Part of proof of Lemma E i... |
| cdleme9tN 39667 | Part of proof of Lemma E i... |
| cdleme10tN 39668 | Part of proof of Lemma E i... |
| cdleme16aN 39669 | Part of proof of Lemma E i... |
| cdleme11a 39670 | Part of proof of Lemma E i... |
| cdleme11c 39671 | Part of proof of Lemma E i... |
| cdleme11dN 39672 | Part of proof of Lemma E i... |
| cdleme11e 39673 | Part of proof of Lemma E i... |
| cdleme11fN 39674 | Part of proof of Lemma E i... |
| cdleme11g 39675 | Part of proof of Lemma E i... |
| cdleme11h 39676 | Part of proof of Lemma E i... |
| cdleme11j 39677 | Part of proof of Lemma E i... |
| cdleme11k 39678 | Part of proof of Lemma E i... |
| cdleme11l 39679 | Part of proof of Lemma E i... |
| cdleme11 39680 | Part of proof of Lemma E i... |
| cdleme12 39681 | Part of proof of Lemma E i... |
| cdleme13 39682 | Part of proof of Lemma E i... |
| cdleme14 39683 | Part of proof of Lemma E i... |
| cdleme15a 39684 | Part of proof of Lemma E i... |
| cdleme15b 39685 | Part of proof of Lemma E i... |
| cdleme15c 39686 | Part of proof of Lemma E i... |
| cdleme15d 39687 | Part of proof of Lemma E i... |
| cdleme15 39688 | Part of proof of Lemma E i... |
| cdleme16b 39689 | Part of proof of Lemma E i... |
| cdleme16c 39690 | Part of proof of Lemma E i... |
| cdleme16d 39691 | Part of proof of Lemma E i... |
| cdleme16e 39692 | Part of proof of Lemma E i... |
| cdleme16f 39693 | Part of proof of Lemma E i... |
| cdleme16g 39694 | Part of proof of Lemma E i... |
| cdleme16 39695 | Part of proof of Lemma E i... |
| cdleme17a 39696 | Part of proof of Lemma E i... |
| cdleme17b 39697 | Lemma leading to ~ cdleme1... |
| cdleme17c 39698 | Part of proof of Lemma E i... |
| cdleme17d1 39699 | Part of proof of Lemma E i... |
| cdleme0nex 39700 | Part of proof of Lemma E i... |
| cdleme18a 39701 | Part of proof of Lemma E i... |
| cdleme18b 39702 | Part of proof of Lemma E i... |
| cdleme18c 39703 | Part of proof of Lemma E i... |
| cdleme22gb 39704 | Utility lemma for Lemma E ... |
| cdleme18d 39705 | Part of proof of Lemma E i... |
| cdlemesner 39706 | Part of proof of Lemma E i... |
| cdlemedb 39707 | Part of proof of Lemma E i... |
| cdlemeda 39708 | Part of proof of Lemma E i... |
| cdlemednpq 39709 | Part of proof of Lemma E i... |
| cdlemednuN 39710 | Part of proof of Lemma E i... |
| cdleme20zN 39711 | Part of proof of Lemma E i... |
| cdleme20y 39712 | Part of proof of Lemma E i... |
| cdleme19a 39713 | Part of proof of Lemma E i... |
| cdleme19b 39714 | Part of proof of Lemma E i... |
| cdleme19c 39715 | Part of proof of Lemma E i... |
| cdleme19d 39716 | Part of proof of Lemma E i... |
| cdleme19e 39717 | Part of proof of Lemma E i... |
| cdleme19f 39718 | Part of proof of Lemma E i... |
| cdleme20aN 39719 | Part of proof of Lemma E i... |
| cdleme20bN 39720 | Part of proof of Lemma E i... |
| cdleme20c 39721 | Part of proof of Lemma E i... |
| cdleme20d 39722 | Part of proof of Lemma E i... |
| cdleme20e 39723 | Part of proof of Lemma E i... |
| cdleme20f 39724 | Part of proof of Lemma E i... |
| cdleme20g 39725 | Part of proof of Lemma E i... |
| cdleme20h 39726 | Part of proof of Lemma E i... |
| cdleme20i 39727 | Part of proof of Lemma E i... |
| cdleme20j 39728 | Part of proof of Lemma E i... |
| cdleme20k 39729 | Part of proof of Lemma E i... |
| cdleme20l1 39730 | Part of proof of Lemma E i... |
| cdleme20l2 39731 | Part of proof of Lemma E i... |
| cdleme20l 39732 | Part of proof of Lemma E i... |
| cdleme20m 39733 | Part of proof of Lemma E i... |
| cdleme20 39734 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 39735 | Part of proof of Lemma E i... |
| cdleme21b 39736 | Part of proof of Lemma E i... |
| cdleme21c 39737 | Part of proof of Lemma E i... |
| cdleme21at 39738 | Part of proof of Lemma E i... |
| cdleme21ct 39739 | Part of proof of Lemma E i... |
| cdleme21d 39740 | Part of proof of Lemma E i... |
| cdleme21e 39741 | Part of proof of Lemma E i... |
| cdleme21f 39742 | Part of proof of Lemma E i... |
| cdleme21g 39743 | Part of proof of Lemma E i... |
| cdleme21h 39744 | Part of proof of Lemma E i... |
| cdleme21i 39745 | Part of proof of Lemma E i... |
| cdleme21j 39746 | Combine ~ cdleme20 and ~ c... |
| cdleme21 39747 | Part of proof of Lemma E i... |
| cdleme21k 39748 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 39749 | Part of proof of Lemma E i... |
| cdleme22a 39750 | Part of proof of Lemma E i... |
| cdleme22b 39751 | Part of proof of Lemma E i... |
| cdleme22cN 39752 | Part of proof of Lemma E i... |
| cdleme22d 39753 | Part of proof of Lemma E i... |
| cdleme22e 39754 | Part of proof of Lemma E i... |
| cdleme22eALTN 39755 | Part of proof of Lemma E i... |
| cdleme22f 39756 | Part of proof of Lemma E i... |
| cdleme22f2 39757 | Part of proof of Lemma E i... |
| cdleme22g 39758 | Part of proof of Lemma E i... |
| cdleme23a 39759 | Part of proof of Lemma E i... |
| cdleme23b 39760 | Part of proof of Lemma E i... |
| cdleme23c 39761 | Part of proof of Lemma E i... |
| cdleme24 39762 | Quantified version of ~ cd... |
| cdleme25a 39763 | Lemma for ~ cdleme25b . (... |
| cdleme25b 39764 | Transform ~ cdleme24 . TO... |
| cdleme25c 39765 | Transform ~ cdleme25b . (... |
| cdleme25dN 39766 | Transform ~ cdleme25c . (... |
| cdleme25cl 39767 | Show closure of the unique... |
| cdleme25cv 39768 | Change bound variables in ... |
| cdleme26e 39769 | Part of proof of Lemma E i... |
| cdleme26ee 39770 | Part of proof of Lemma E i... |
| cdleme26eALTN 39771 | Part of proof of Lemma E i... |
| cdleme26fALTN 39772 | Part of proof of Lemma E i... |
| cdleme26f 39773 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 39774 | Part of proof of Lemma E i... |
| cdleme26f2 39775 | Part of proof of Lemma E i... |
| cdleme27cl 39776 | Part of proof of Lemma E i... |
| cdleme27a 39777 | Part of proof of Lemma E i... |
| cdleme27b 39778 | Lemma for ~ cdleme27N . (... |
| cdleme27N 39779 | Part of proof of Lemma E i... |
| cdleme28a 39780 | Lemma for ~ cdleme25b . T... |
| cdleme28b 39781 | Lemma for ~ cdleme25b . T... |
| cdleme28c 39782 | Part of proof of Lemma E i... |
| cdleme28 39783 | Quantified version of ~ cd... |
| cdleme29ex 39784 | Lemma for ~ cdleme29b . (... |
| cdleme29b 39785 | Transform ~ cdleme28 . (C... |
| cdleme29c 39786 | Transform ~ cdleme28b . (... |
| cdleme29cl 39787 | Show closure of the unique... |
| cdleme30a 39788 | Part of proof of Lemma E i... |
| cdleme31so 39789 | Part of proof of Lemma E i... |
| cdleme31sn 39790 | Part of proof of Lemma E i... |
| cdleme31sn1 39791 | Part of proof of Lemma E i... |
| cdleme31se 39792 | Part of proof of Lemma D i... |
| cdleme31se2 39793 | Part of proof of Lemma D i... |
| cdleme31sc 39794 | Part of proof of Lemma E i... |
| cdleme31sde 39795 | Part of proof of Lemma D i... |
| cdleme31snd 39796 | Part of proof of Lemma D i... |
| cdleme31sdnN 39797 | Part of proof of Lemma E i... |
| cdleme31sn1c 39798 | Part of proof of Lemma E i... |
| cdleme31sn2 39799 | Part of proof of Lemma E i... |
| cdleme31fv 39800 | Part of proof of Lemma E i... |
| cdleme31fv1 39801 | Part of proof of Lemma E i... |
| cdleme31fv1s 39802 | Part of proof of Lemma E i... |
| cdleme31fv2 39803 | Part of proof of Lemma E i... |
| cdleme31id 39804 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 39805 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 39806 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 39807 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 39808 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 39809 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 39810 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 39811 | Part of proof of Lemma E i... |
| cdlemefr29exN 39812 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 39813 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 39814 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 39815 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 39816 | TODO fix comment. (Contri... |
| cdlemefr29clN 39817 | Show closure of the unique... |
| cdleme43frv1snN 39818 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 39819 | Part of proof of Lemma E i... |
| cdlemefr32fva1 39820 | Part of proof of Lemma E i... |
| cdlemefr31fv1 39821 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 39822 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 39823 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 39824 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 39825 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 39826 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 39827 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 39828 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 39829 | Show closure of the unique... |
| cdleme43fsv1snlem 39830 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 39831 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 39832 | Part of proof of Lemma E i... |
| cdlemefs32fva1 39833 | Part of proof of Lemma E i... |
| cdlemefs31fv1 39834 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 39835 | Value of f(r) when r is an... |
| cdlemefs44 39836 | Value of f_s(r) when r is ... |
| cdlemefr45 39837 | Value of f(r) when r is an... |
| cdlemefr45e 39838 | Explicit expansion of ~ cd... |
| cdlemefs45 39839 | Value of f_s(r) when r is ... |
| cdlemefs45ee 39840 | Explicit expansion of ~ cd... |
| cdlemefs45eN 39841 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 39842 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 39843 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 39844 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 39845 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 39846 | Show closure of ` [_ R / s... |
| cdleme32fva 39847 | Part of proof of Lemma D i... |
| cdleme32fva1 39848 | Part of proof of Lemma D i... |
| cdleme32fvaw 39849 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 39850 | Part of proof of Lemma D i... |
| cdleme32a 39851 | Part of proof of Lemma D i... |
| cdleme32b 39852 | Part of proof of Lemma D i... |
| cdleme32c 39853 | Part of proof of Lemma D i... |
| cdleme32d 39854 | Part of proof of Lemma D i... |
| cdleme32e 39855 | Part of proof of Lemma D i... |
| cdleme32f 39856 | Part of proof of Lemma D i... |
| cdleme32le 39857 | Part of proof of Lemma D i... |
| cdleme35a 39858 | Part of proof of Lemma E i... |
| cdleme35fnpq 39859 | Part of proof of Lemma E i... |
| cdleme35b 39860 | Part of proof of Lemma E i... |
| cdleme35c 39861 | Part of proof of Lemma E i... |
| cdleme35d 39862 | Part of proof of Lemma E i... |
| cdleme35e 39863 | Part of proof of Lemma E i... |
| cdleme35f 39864 | Part of proof of Lemma E i... |
| cdleme35g 39865 | Part of proof of Lemma E i... |
| cdleme35h 39866 | Part of proof of Lemma E i... |
| cdleme35h2 39867 | Part of proof of Lemma E i... |
| cdleme35sn2aw 39868 | Part of proof of Lemma E i... |
| cdleme35sn3a 39869 | Part of proof of Lemma E i... |
| cdleme36a 39870 | Part of proof of Lemma E i... |
| cdleme36m 39871 | Part of proof of Lemma E i... |
| cdleme37m 39872 | Part of proof of Lemma E i... |
| cdleme38m 39873 | Part of proof of Lemma E i... |
| cdleme38n 39874 | Part of proof of Lemma E i... |
| cdleme39a 39875 | Part of proof of Lemma E i... |
| cdleme39n 39876 | Part of proof of Lemma E i... |
| cdleme40m 39877 | Part of proof of Lemma E i... |
| cdleme40n 39878 | Part of proof of Lemma E i... |
| cdleme40v 39879 | Part of proof of Lemma E i... |
| cdleme40w 39880 | Part of proof of Lemma E i... |
| cdleme42a 39881 | Part of proof of Lemma E i... |
| cdleme42c 39882 | Part of proof of Lemma E i... |
| cdleme42d 39883 | Part of proof of Lemma E i... |
| cdleme41sn3aw 39884 | Part of proof of Lemma E i... |
| cdleme41sn4aw 39885 | Part of proof of Lemma E i... |
| cdleme41snaw 39886 | Part of proof of Lemma E i... |
| cdleme41fva11 39887 | Part of proof of Lemma E i... |
| cdleme42b 39888 | Part of proof of Lemma E i... |
| cdleme42e 39889 | Part of proof of Lemma E i... |
| cdleme42f 39890 | Part of proof of Lemma E i... |
| cdleme42g 39891 | Part of proof of Lemma E i... |
| cdleme42h 39892 | Part of proof of Lemma E i... |
| cdleme42i 39893 | Part of proof of Lemma E i... |
| cdleme42k 39894 | Part of proof of Lemma E i... |
| cdleme42ke 39895 | Part of proof of Lemma E i... |
| cdleme42keg 39896 | Part of proof of Lemma E i... |
| cdleme42mN 39897 | Part of proof of Lemma E i... |
| cdleme42mgN 39898 | Part of proof of Lemma E i... |
| cdleme43aN 39899 | Part of proof of Lemma E i... |
| cdleme43bN 39900 | Lemma for Lemma E in [Craw... |
| cdleme43cN 39901 | Part of proof of Lemma E i... |
| cdleme43dN 39902 | Part of proof of Lemma E i... |
| cdleme46f2g2 39903 | Conversion for ` G ` to re... |
| cdleme46f2g1 39904 | Conversion for ` G ` to re... |
| cdleme17d2 39905 | Part of proof of Lemma E i... |
| cdleme17d3 39906 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 39907 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 39908 | Part of proof of Lemma E i... |
| cdleme48fv 39909 | Part of proof of Lemma D i... |
| cdleme48fvg 39910 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 39911 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 39912 | TODO: fix comment. TODO: ... |
| cdleme48b 39913 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 39914 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 39915 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 39916 | TODO: fix comment. (Contr... |
| cdleme4gfv 39917 | Part of proof of Lemma D i... |
| cdlemeg47b 39918 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 39919 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 39920 | Value of g_s(r) when r is ... |
| cdlemeg49le 39921 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 39922 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 39923 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 39924 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 39925 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 39926 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 39927 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 39928 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 39929 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 39930 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 39931 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 39932 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 39933 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 39934 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 39935 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 39936 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 39937 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 39938 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 39939 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 39940 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 39941 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 39942 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 39943 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 39944 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 39945 | TODO: fix comment. (Contr... |
| cdleme48gfv1 39946 | TODO: fix comment. (Contr... |
| cdleme48gfv 39947 | TODO: fix comment. (Contr... |
| cdleme48fgv 39948 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 39949 | Part of proof of Lemma D i... |
| cdleme50lebi 39950 | Part of proof of Lemma D i... |
| cdleme50eq 39951 | Part of proof of Lemma D i... |
| cdleme50f 39952 | Part of proof of Lemma D i... |
| cdleme50f1 39953 | Part of proof of Lemma D i... |
| cdleme50rnlem 39954 | Part of proof of Lemma D i... |
| cdleme50rn 39955 | Part of proof of Lemma D i... |
| cdleme50f1o 39956 | Part of proof of Lemma D i... |
| cdleme50laut 39957 | Part of proof of Lemma D i... |
| cdleme50ldil 39958 | Part of proof of Lemma D i... |
| cdleme50trn1 39959 | Part of proof that ` F ` i... |
| cdleme50trn2a 39960 | Part of proof that ` F ` i... |
| cdleme50trn2 39961 | Part of proof that ` F ` i... |
| cdleme50trn12 39962 | Part of proof that ` F ` i... |
| cdleme50trn3 39963 | Part of proof that ` F ` i... |
| cdleme50trn123 39964 | Part of proof that ` F ` i... |
| cdleme51finvfvN 39965 | Part of proof of Lemma E i... |
| cdleme51finvN 39966 | Part of proof of Lemma E i... |
| cdleme50ltrn 39967 | Part of proof of Lemma E i... |
| cdleme51finvtrN 39968 | Part of proof of Lemma E i... |
| cdleme50ex 39969 | Part of Lemma E in [Crawle... |
| cdleme 39970 | Lemma E in [Crawley] p. 11... |
| cdlemf1 39971 | Part of Lemma F in [Crawle... |
| cdlemf2 39972 | Part of Lemma F in [Crawle... |
| cdlemf 39973 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 39974 | ~ cdlemf with additional c... |
| cdlemftr3 39975 | Special case of ~ cdlemf s... |
| cdlemftr2 39976 | Special case of ~ cdlemf s... |
| cdlemftr1 39977 | Part of proof of Lemma G o... |
| cdlemftr0 39978 | Special case of ~ cdlemf s... |
| trlord 39979 | The ordering of two Hilber... |
| cdlemg1a 39980 | Shorter expression for ` G... |
| cdlemg1b2 39981 | This theorem can be used t... |
| cdlemg1idlemN 39982 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 39983 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 39984 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 39985 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 39986 | This theorem can be used t... |
| cdlemg1idN 39987 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 39988 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 39989 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 39990 | Version of ~ cdleme51finvt... |
| ltrniotaval 39991 | Value of the unique transl... |
| ltrniotacnvval 39992 | Converse value of the uniq... |
| ltrniotaidvalN 39993 | Value of the unique transl... |
| ltrniotavalbN 39994 | Value of the unique transl... |
| cdlemeiota 39995 | A translation is uniquely ... |
| cdlemg1ci2 39996 | Any function of the form o... |
| cdlemg1cN 39997 | Any translation belongs to... |
| cdlemg1cex 39998 | Any translation is one of ... |
| cdlemg2cN 39999 | Any translation belongs to... |
| cdlemg2dN 40000 | This theorem can be used t... |
| cdlemg2cex 40001 | Any translation is one of ... |
| cdlemg2ce 40002 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 40003 | Part of proof of Lemma E i... |
| cdlemg2fvlem 40004 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 40005 | ~ cdleme42keg with simpler... |
| cdlemg2idN 40006 | Version of ~ cdleme31id wi... |
| cdlemg3a 40007 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 40008 | TODO: Replace this with ~... |
| cdlemg2fv 40009 | Value of a translation in ... |
| cdlemg2fv2 40010 | Value of a translation in ... |
| cdlemg2k 40011 | ~ cdleme42keg with simpler... |
| cdlemg2kq 40012 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 40013 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 40014 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 40015 | TODO: Is there a simpler ... |
| cdlemb3 40016 | Given two atoms not under ... |
| cdlemg7fvbwN 40017 | Properties of a translatio... |
| cdlemg4a 40018 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 40019 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 40020 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 40021 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 40022 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 40023 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 40024 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 40025 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 40026 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 40027 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 40028 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 40029 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 40030 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 40031 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 40032 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 40033 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 40034 | Value of a translation com... |
| cdlemg7aN 40035 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 40036 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 40037 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 40038 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 40039 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 40040 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 40041 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 40042 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 40043 | The triples ` <. P , ( F `... |
| cdlemg9 40044 | The triples ` <. P , ( F `... |
| cdlemg10b 40045 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 40046 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 40047 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 40048 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 40049 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 40050 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 40051 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 40052 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 40053 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 40054 | The triples ` <. P , ( F `... |
| cdlemg12c 40055 | The triples ` <. P , ( F `... |
| cdlemg12d 40056 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 40057 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 40058 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 40059 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 40060 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 40061 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 40062 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 40063 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 40064 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 40065 | Eliminate the ` ( F `` P )... |
| cdlemg15 40066 | Eliminate the ` ( (... |
| cdlemg16 40067 | Part of proof of Lemma G o... |
| cdlemg16ALTN 40068 | This version of ~ cdlemg16... |
| cdlemg16z 40069 | Eliminate ` ( ( F `... |
| cdlemg16zz 40070 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 40071 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 40072 | Part of proof of Lemma G i... |
| cdlemg17dN 40073 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 40074 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 40075 | TODO: fix comment. (Contr... |
| cdlemg17f 40076 | TODO: fix comment. (Contr... |
| cdlemg17g 40077 | TODO: fix comment. (Contr... |
| cdlemg17h 40078 | TODO: fix comment. (Contr... |
| cdlemg17i 40079 | TODO: fix comment. (Contr... |
| cdlemg17ir 40080 | TODO: fix comment. (Contr... |
| cdlemg17j 40081 | TODO: fix comment. (Contr... |
| cdlemg17pq 40082 | Utility theorem for swappi... |
| cdlemg17bq 40083 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 40084 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 40085 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 40086 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 40087 | Part of Lemma G of [Crawle... |
| cdlemg18a 40088 | Show two lines are differe... |
| cdlemg18b 40089 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 40090 | Show two lines intersect a... |
| cdlemg18d 40091 | Show two lines intersect a... |
| cdlemg18 40092 | Show two lines intersect a... |
| cdlemg19a 40093 | Show two lines intersect a... |
| cdlemg19 40094 | Show two lines intersect a... |
| cdlemg20 40095 | Show two lines intersect a... |
| cdlemg21 40096 | Version of cdlemg19 with `... |
| cdlemg22 40097 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 40098 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 40099 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 40100 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 40101 | ~ cdlemg16zz restated for ... |
| cdlemg27a 40102 | For use with case when ` (... |
| cdlemg28a 40103 | Part of proof of Lemma G o... |
| cdlemg31b0N 40104 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 40105 | TODO: Fix comment. (Cont... |
| cdlemg27b 40106 | TODO: Fix comment. (Cont... |
| cdlemg31a 40107 | TODO: fix comment. (Contr... |
| cdlemg31b 40108 | TODO: fix comment. (Contr... |
| cdlemg31c 40109 | Show that when ` N ` is an... |
| cdlemg31d 40110 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 40111 | TODO: Fix comment. (Cont... |
| cdlemg33c0 40112 | TODO: Fix comment. (Cont... |
| cdlemg28b 40113 | Part of proof of Lemma G o... |
| cdlemg28 40114 | Part of proof of Lemma G o... |
| cdlemg29 40115 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 40116 | TODO: Fix comment. (Cont... |
| cdlemg33b 40117 | TODO: Fix comment. (Cont... |
| cdlemg33c 40118 | TODO: Fix comment. (Cont... |
| cdlemg33d 40119 | TODO: Fix comment. (Cont... |
| cdlemg33e 40120 | TODO: Fix comment. (Cont... |
| cdlemg33 40121 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 40122 | Use cdlemg33 to eliminate ... |
| cdlemg35 40123 | TODO: Fix comment. TODO:... |
| cdlemg36 40124 | Use cdlemg35 to eliminate ... |
| cdlemg38 40125 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 40126 | Eliminate ` =/= ` conditio... |
| cdlemg40 40127 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 40128 | Convert ~ cdlemg40 to func... |
| ltrnco 40129 | The composition of two tra... |
| trlcocnv 40130 | Swap the arguments of the ... |
| trlcoabs 40131 | Absorption into a composit... |
| trlcoabs2N 40132 | Absorption of the trace of... |
| trlcoat 40133 | The trace of a composition... |
| trlcocnvat 40134 | Commonly used special case... |
| trlconid 40135 | The composition of two dif... |
| trlcolem 40136 | Lemma for ~ trlco . (Cont... |
| trlco 40137 | The trace of a composition... |
| trlcone 40138 | If two translations have d... |
| cdlemg42 40139 | Part of proof of Lemma G o... |
| cdlemg43 40140 | Part of proof of Lemma G o... |
| cdlemg44a 40141 | Part of proof of Lemma G o... |
| cdlemg44b 40142 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 40143 | Part of proof of Lemma G o... |
| cdlemg47a 40144 | TODO: fix comment. TODO: ... |
| cdlemg46 40145 | Part of proof of Lemma G o... |
| cdlemg47 40146 | Part of proof of Lemma G o... |
| cdlemg48 40147 | Eliminate ` h ` from ~ cdl... |
| ltrncom 40148 | Composition is commutative... |
| ltrnco4 40149 | Rearrange a composition of... |
| trljco 40150 | Trace joined with trace of... |
| trljco2 40151 | Trace joined with trace of... |
| tgrpfset 40154 | The translation group maps... |
| tgrpset 40155 | The translation group for ... |
| tgrpbase 40156 | The base set of the transl... |
| tgrpopr 40157 | The group operation of the... |
| tgrpov 40158 | The group operation value ... |
| tgrpgrplem 40159 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 40160 | The translation group is a... |
| tgrpabl 40161 | The translation group is a... |
| tendofset 40168 | The set of all trace-prese... |
| tendoset 40169 | The set of trace-preservin... |
| istendo 40170 | The predicate "is a trace-... |
| tendotp 40171 | Trace-preserving property ... |
| istendod 40172 | Deduce the predicate "is a... |
| tendof 40173 | Functionality of a trace-p... |
| tendoeq1 40174 | Condition determining equa... |
| tendovalco 40175 | Value of composition of tr... |
| tendocoval 40176 | Value of composition of en... |
| tendocl 40177 | Closure of a trace-preserv... |
| tendoco2 40178 | Distribution of compositio... |
| tendoidcl 40179 | The identity is a trace-pr... |
| tendo1mul 40180 | Multiplicative identity mu... |
| tendo1mulr 40181 | Multiplicative identity mu... |
| tendococl 40182 | The composition of two tra... |
| tendoid 40183 | The identity value of a tr... |
| tendoeq2 40184 | Condition determining equa... |
| tendoplcbv 40185 | Define sum operation for t... |
| tendopl 40186 | Value of endomorphism sum ... |
| tendopl2 40187 | Value of result of endomor... |
| tendoplcl2 40188 | Value of result of endomor... |
| tendoplco2 40189 | Value of result of endomor... |
| tendopltp 40190 | Trace-preserving property ... |
| tendoplcl 40191 | Endomorphism sum is a trac... |
| tendoplcom 40192 | The endomorphism sum opera... |
| tendoplass 40193 | The endomorphism sum opera... |
| tendodi1 40194 | Endomorphism composition d... |
| tendodi2 40195 | Endomorphism composition d... |
| tendo0cbv 40196 | Define additive identity f... |
| tendo02 40197 | Value of additive identity... |
| tendo0co2 40198 | The additive identity trac... |
| tendo0tp 40199 | Trace-preserving property ... |
| tendo0cl 40200 | The additive identity is a... |
| tendo0pl 40201 | Property of the additive i... |
| tendo0plr 40202 | Property of the additive i... |
| tendoicbv 40203 | Define inverse function fo... |
| tendoi 40204 | Value of inverse endomorph... |
| tendoi2 40205 | Value of additive inverse ... |
| tendoicl 40206 | Closure of the additive in... |
| tendoipl 40207 | Property of the additive i... |
| tendoipl2 40208 | Property of the additive i... |
| erngfset 40209 | The division rings on trac... |
| erngset 40210 | The division ring on trace... |
| erngbase 40211 | The base set of the divisi... |
| erngfplus 40212 | Ring addition operation. ... |
| erngplus 40213 | Ring addition operation. ... |
| erngplus2 40214 | Ring addition operation. ... |
| erngfmul 40215 | Ring multiplication operat... |
| erngmul 40216 | Ring addition operation. ... |
| erngfset-rN 40217 | The division rings on trac... |
| erngset-rN 40218 | The division ring on trace... |
| erngbase-rN 40219 | The base set of the divisi... |
| erngfplus-rN 40220 | Ring addition operation. ... |
| erngplus-rN 40221 | Ring addition operation. ... |
| erngplus2-rN 40222 | Ring addition operation. ... |
| erngfmul-rN 40223 | Ring multiplication operat... |
| erngmul-rN 40224 | Ring addition operation. ... |
| cdlemh1 40225 | Part of proof of Lemma H o... |
| cdlemh2 40226 | Part of proof of Lemma H o... |
| cdlemh 40227 | Lemma H of [Crawley] p. 11... |
| cdlemi1 40228 | Part of proof of Lemma I o... |
| cdlemi2 40229 | Part of proof of Lemma I o... |
| cdlemi 40230 | Lemma I of [Crawley] p. 11... |
| cdlemj1 40231 | Part of proof of Lemma J o... |
| cdlemj2 40232 | Part of proof of Lemma J o... |
| cdlemj3 40233 | Part of proof of Lemma J o... |
| tendocan 40234 | Cancellation law: if the v... |
| tendoid0 40235 | A trace-preserving endomor... |
| tendo0mul 40236 | Additive identity multipli... |
| tendo0mulr 40237 | Additive identity multipli... |
| tendo1ne0 40238 | The identity (unity) is no... |
| tendoconid 40239 | The composition (product) ... |
| tendotr 40240 | The trace of the value of ... |
| cdlemk1 40241 | Part of proof of Lemma K o... |
| cdlemk2 40242 | Part of proof of Lemma K o... |
| cdlemk3 40243 | Part of proof of Lemma K o... |
| cdlemk4 40244 | Part of proof of Lemma K o... |
| cdlemk5a 40245 | Part of proof of Lemma K o... |
| cdlemk5 40246 | Part of proof of Lemma K o... |
| cdlemk6 40247 | Part of proof of Lemma K o... |
| cdlemk8 40248 | Part of proof of Lemma K o... |
| cdlemk9 40249 | Part of proof of Lemma K o... |
| cdlemk9bN 40250 | Part of proof of Lemma K o... |
| cdlemki 40251 | Part of proof of Lemma K o... |
| cdlemkvcl 40252 | Part of proof of Lemma K o... |
| cdlemk10 40253 | Part of proof of Lemma K o... |
| cdlemksv 40254 | Part of proof of Lemma K o... |
| cdlemksel 40255 | Part of proof of Lemma K o... |
| cdlemksat 40256 | Part of proof of Lemma K o... |
| cdlemksv2 40257 | Part of proof of Lemma K o... |
| cdlemk7 40258 | Part of proof of Lemma K o... |
| cdlemk11 40259 | Part of proof of Lemma K o... |
| cdlemk12 40260 | Part of proof of Lemma K o... |
| cdlemkoatnle 40261 | Utility lemma. (Contribut... |
| cdlemk13 40262 | Part of proof of Lemma K o... |
| cdlemkole 40263 | Utility lemma. (Contribut... |
| cdlemk14 40264 | Part of proof of Lemma K o... |
| cdlemk15 40265 | Part of proof of Lemma K o... |
| cdlemk16a 40266 | Part of proof of Lemma K o... |
| cdlemk16 40267 | Part of proof of Lemma K o... |
| cdlemk17 40268 | Part of proof of Lemma K o... |
| cdlemk1u 40269 | Part of proof of Lemma K o... |
| cdlemk5auN 40270 | Part of proof of Lemma K o... |
| cdlemk5u 40271 | Part of proof of Lemma K o... |
| cdlemk6u 40272 | Part of proof of Lemma K o... |
| cdlemkj 40273 | Part of proof of Lemma K o... |
| cdlemkuvN 40274 | Part of proof of Lemma K o... |
| cdlemkuel 40275 | Part of proof of Lemma K o... |
| cdlemkuat 40276 | Part of proof of Lemma K o... |
| cdlemkuv2 40277 | Part of proof of Lemma K o... |
| cdlemk18 40278 | Part of proof of Lemma K o... |
| cdlemk19 40279 | Part of proof of Lemma K o... |
| cdlemk7u 40280 | Part of proof of Lemma K o... |
| cdlemk11u 40281 | Part of proof of Lemma K o... |
| cdlemk12u 40282 | Part of proof of Lemma K o... |
| cdlemk21N 40283 | Part of proof of Lemma K o... |
| cdlemk20 40284 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 40285 | Utility lemma. (Contribut... |
| cdlemk13-2N 40286 | Part of proof of Lemma K o... |
| cdlemkole-2N 40287 | Utility lemma. (Contribut... |
| cdlemk14-2N 40288 | Part of proof of Lemma K o... |
| cdlemk15-2N 40289 | Part of proof of Lemma K o... |
| cdlemk16-2N 40290 | Part of proof of Lemma K o... |
| cdlemk17-2N 40291 | Part of proof of Lemma K o... |
| cdlemkj-2N 40292 | Part of proof of Lemma K o... |
| cdlemkuv-2N 40293 | Part of proof of Lemma K o... |
| cdlemkuel-2N 40294 | Part of proof of Lemma K o... |
| cdlemkuv2-2 40295 | Part of proof of Lemma K o... |
| cdlemk18-2N 40296 | Part of proof of Lemma K o... |
| cdlemk19-2N 40297 | Part of proof of Lemma K o... |
| cdlemk7u-2N 40298 | Part of proof of Lemma K o... |
| cdlemk11u-2N 40299 | Part of proof of Lemma K o... |
| cdlemk12u-2N 40300 | Part of proof of Lemma K o... |
| cdlemk21-2N 40301 | Part of proof of Lemma K o... |
| cdlemk20-2N 40302 | Part of proof of Lemma K o... |
| cdlemk22 40303 | Part of proof of Lemma K o... |
| cdlemk30 40304 | Part of proof of Lemma K o... |
| cdlemkuu 40305 | Convert between function a... |
| cdlemk31 40306 | Part of proof of Lemma K o... |
| cdlemk32 40307 | Part of proof of Lemma K o... |
| cdlemkuel-3 40308 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 40309 | Part of proof of Lemma K o... |
| cdlemk18-3N 40310 | Part of proof of Lemma K o... |
| cdlemk22-3 40311 | Part of proof of Lemma K o... |
| cdlemk23-3 40312 | Part of proof of Lemma K o... |
| cdlemk24-3 40313 | Part of proof of Lemma K o... |
| cdlemk25-3 40314 | Part of proof of Lemma K o... |
| cdlemk26b-3 40315 | Part of proof of Lemma K o... |
| cdlemk26-3 40316 | Part of proof of Lemma K o... |
| cdlemk27-3 40317 | Part of proof of Lemma K o... |
| cdlemk28-3 40318 | Part of proof of Lemma K o... |
| cdlemk33N 40319 | Part of proof of Lemma K o... |
| cdlemk34 40320 | Part of proof of Lemma K o... |
| cdlemk29-3 40321 | Part of proof of Lemma K o... |
| cdlemk35 40322 | Part of proof of Lemma K o... |
| cdlemk36 40323 | Part of proof of Lemma K o... |
| cdlemk37 40324 | Part of proof of Lemma K o... |
| cdlemk38 40325 | Part of proof of Lemma K o... |
| cdlemk39 40326 | Part of proof of Lemma K o... |
| cdlemk40 40327 | TODO: fix comment. (Contr... |
| cdlemk40t 40328 | TODO: fix comment. (Contr... |
| cdlemk40f 40329 | TODO: fix comment. (Contr... |
| cdlemk41 40330 | Part of proof of Lemma K o... |
| cdlemkfid1N 40331 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 40332 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 40333 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 40334 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 40335 | TODO: is this useful or sh... |
| cdlemky 40336 | Part of proof of Lemma K o... |
| cdlemkyu 40337 | Convert between function a... |
| cdlemkyuu 40338 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 40339 | Part of proof of Lemma K o... |
| cdlemk19ylem 40340 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 40341 | Part of proof of Lemma K o... |
| cdlemk19y 40342 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 40343 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 40344 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 40345 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 40346 | The value of the tau funct... |
| cdlemk35s 40347 | Substitution version of ~ ... |
| cdlemk35s-id 40348 | Substitution version of ~ ... |
| cdlemk39s 40349 | Substitution version of ~ ... |
| cdlemk39s-id 40350 | Substitution version of ~ ... |
| cdlemk42 40351 | Part of proof of Lemma K o... |
| cdlemk19xlem 40352 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 40353 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 40354 | Part of proof of Lemma K o... |
| cdlemk11tc 40355 | Part of proof of Lemma K o... |
| cdlemk11t 40356 | Part of proof of Lemma K o... |
| cdlemk45 40357 | Part of proof of Lemma K o... |
| cdlemk46 40358 | Part of proof of Lemma K o... |
| cdlemk47 40359 | Part of proof of Lemma K o... |
| cdlemk48 40360 | Part of proof of Lemma K o... |
| cdlemk49 40361 | Part of proof of Lemma K o... |
| cdlemk50 40362 | Part of proof of Lemma K o... |
| cdlemk51 40363 | Part of proof of Lemma K o... |
| cdlemk52 40364 | Part of proof of Lemma K o... |
| cdlemk53a 40365 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 40366 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 40367 | Part of proof of Lemma K o... |
| cdlemk54 40368 | Part of proof of Lemma K o... |
| cdlemk55a 40369 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 40370 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 40371 | Part of proof of Lemma K o... |
| cdlemkyyN 40372 | Part of proof of Lemma K o... |
| cdlemk43N 40373 | Part of proof of Lemma K o... |
| cdlemk35u 40374 | Substitution version of ~ ... |
| cdlemk55u1 40375 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 40376 | Part of proof of Lemma K o... |
| cdlemk39u1 40377 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 40378 | Part of proof of Lemma K o... |
| cdlemk19u1 40379 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 40380 | Part of Lemma K of [Crawle... |
| cdlemk56 40381 | Part of Lemma K of [Crawle... |
| cdlemk19w 40382 | Use a fixed element to eli... |
| cdlemk56w 40383 | Use a fixed element to eli... |
| cdlemk 40384 | Lemma K of [Crawley] p. 11... |
| tendoex 40385 | Generalization of Lemma K ... |
| cdleml1N 40386 | Part of proof of Lemma L o... |
| cdleml2N 40387 | Part of proof of Lemma L o... |
| cdleml3N 40388 | Part of proof of Lemma L o... |
| cdleml4N 40389 | Part of proof of Lemma L o... |
| cdleml5N 40390 | Part of proof of Lemma L o... |
| cdleml6 40391 | Part of proof of Lemma L o... |
| cdleml7 40392 | Part of proof of Lemma L o... |
| cdleml8 40393 | Part of proof of Lemma L o... |
| cdleml9 40394 | Part of proof of Lemma L o... |
| dva1dim 40395 | Two expressions for the 1-... |
| dvhb1dimN 40396 | Two expressions for the 1-... |
| erng1lem 40397 | Value of the endomorphism ... |
| erngdvlem1 40398 | Lemma for ~ eringring . (... |
| erngdvlem2N 40399 | Lemma for ~ eringring . (... |
| erngdvlem3 40400 | Lemma for ~ eringring . (... |
| erngdvlem4 40401 | Lemma for ~ erngdv . (Con... |
| eringring 40402 | An endomorphism ring is a ... |
| erngdv 40403 | An endomorphism ring is a ... |
| erng0g 40404 | The division ring zero of ... |
| erng1r 40405 | The division ring unity of... |
| erngdvlem1-rN 40406 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 40407 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 40408 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 40409 | Lemma for ~ erngdv . (Con... |
| erngring-rN 40410 | An endomorphism ring is a ... |
| erngdv-rN 40411 | An endomorphism ring is a ... |
| dvafset 40414 | The constructed partial ve... |
| dvaset 40415 | The constructed partial ve... |
| dvasca 40416 | The ring base set of the c... |
| dvabase 40417 | The ring base set of the c... |
| dvafplusg 40418 | Ring addition operation fo... |
| dvaplusg 40419 | Ring addition operation fo... |
| dvaplusgv 40420 | Ring addition operation fo... |
| dvafmulr 40421 | Ring multiplication operat... |
| dvamulr 40422 | Ring multiplication operat... |
| dvavbase 40423 | The vectors (vector base s... |
| dvafvadd 40424 | The vector sum operation f... |
| dvavadd 40425 | Ring addition operation fo... |
| dvafvsca 40426 | Ring addition operation fo... |
| dvavsca 40427 | Ring addition operation fo... |
| tendospcl 40428 | Closure of endomorphism sc... |
| tendospass 40429 | Associative law for endomo... |
| tendospdi1 40430 | Forward distributive law f... |
| tendocnv 40431 | Converse of a trace-preser... |
| tendospdi2 40432 | Reverse distributive law f... |
| tendospcanN 40433 | Cancellation law for trace... |
| dvaabl 40434 | The constructed partial ve... |
| dvalveclem 40435 | Lemma for ~ dvalvec . (Co... |
| dvalvec 40436 | The constructed partial ve... |
| dva0g 40437 | The zero vector of partial... |
| diaffval 40440 | The partial isomorphism A ... |
| diafval 40441 | The partial isomorphism A ... |
| diaval 40442 | The partial isomorphism A ... |
| diaelval 40443 | Member of the partial isom... |
| diafn 40444 | Functionality and domain o... |
| diadm 40445 | Domain of the partial isom... |
| diaeldm 40446 | Member of domain of the pa... |
| diadmclN 40447 | A member of domain of the ... |
| diadmleN 40448 | A member of domain of the ... |
| dian0 40449 | The value of the partial i... |
| dia0eldmN 40450 | The lattice zero belongs t... |
| dia1eldmN 40451 | The fiducial hyperplane (t... |
| diass 40452 | The value of the partial i... |
| diael 40453 | A member of the value of t... |
| diatrl 40454 | Trace of a member of the p... |
| diaelrnN 40455 | Any value of the partial i... |
| dialss 40456 | The value of partial isomo... |
| diaord 40457 | The partial isomorphism A ... |
| dia11N 40458 | The partial isomorphism A ... |
| diaf11N 40459 | The partial isomorphism A ... |
| diaclN 40460 | Closure of partial isomorp... |
| diacnvclN 40461 | Closure of partial isomorp... |
| dia0 40462 | The value of the partial i... |
| dia1N 40463 | The value of the partial i... |
| dia1elN 40464 | The largest subspace in th... |
| diaglbN 40465 | Partial isomorphism A of a... |
| diameetN 40466 | Partial isomorphism A of a... |
| diainN 40467 | Inverse partial isomorphis... |
| diaintclN 40468 | The intersection of partia... |
| diasslssN 40469 | The partial isomorphism A ... |
| diassdvaN 40470 | The partial isomorphism A ... |
| dia1dim 40471 | Two expressions for the 1-... |
| dia1dim2 40472 | Two expressions for a 1-di... |
| dia1dimid 40473 | A vector (translation) bel... |
| dia2dimlem1 40474 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 40475 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 40476 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 40477 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 40478 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 40479 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 40480 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 40481 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 40482 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 40483 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 40484 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 40485 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 40486 | Lemma for ~ dia2dim . Eli... |
| dia2dim 40487 | A two-dimensional subspace... |
| dvhfset 40490 | The constructed full vecto... |
| dvhset 40491 | The constructed full vecto... |
| dvhsca 40492 | The ring of scalars of the... |
| dvhbase 40493 | The ring base set of the c... |
| dvhfplusr 40494 | Ring addition operation fo... |
| dvhfmulr 40495 | Ring multiplication operat... |
| dvhmulr 40496 | Ring multiplication operat... |
| dvhvbase 40497 | The vectors (vector base s... |
| dvhelvbasei 40498 | Vector membership in the c... |
| dvhvaddcbv 40499 | Change bound variables to ... |
| dvhvaddval 40500 | The vector sum operation f... |
| dvhfvadd 40501 | The vector sum operation f... |
| dvhvadd 40502 | The vector sum operation f... |
| dvhopvadd 40503 | The vector sum operation f... |
| dvhopvadd2 40504 | The vector sum operation f... |
| dvhvaddcl 40505 | Closure of the vector sum ... |
| dvhvaddcomN 40506 | Commutativity of vector su... |
| dvhvaddass 40507 | Associativity of vector su... |
| dvhvscacbv 40508 | Change bound variables to ... |
| dvhvscaval 40509 | The scalar product operati... |
| dvhfvsca 40510 | Scalar product operation f... |
| dvhvsca 40511 | Scalar product operation f... |
| dvhopvsca 40512 | Scalar product operation f... |
| dvhvscacl 40513 | Closure of the scalar prod... |
| tendoinvcl 40514 | Closure of multiplicative ... |
| tendolinv 40515 | Left multiplicative invers... |
| tendorinv 40516 | Right multiplicative inver... |
| dvhgrp 40517 | The full vector space ` U ... |
| dvhlveclem 40518 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 40519 | The full vector space ` U ... |
| dvhlmod 40520 | The full vector space ` U ... |
| dvh0g 40521 | The zero vector of vector ... |
| dvheveccl 40522 | Properties of a unit vecto... |
| dvhopclN 40523 | Closure of a ` DVecH ` vec... |
| dvhopaddN 40524 | Sum of ` DVecH ` vectors e... |
| dvhopspN 40525 | Scalar product of ` DVecH ... |
| dvhopN 40526 | Decompose a ` DVecH ` vect... |
| dvhopellsm 40527 | Ordered pair membership in... |
| cdlemm10N 40528 | The image of the map ` G `... |
| docaffvalN 40531 | Subspace orthocomplement f... |
| docafvalN 40532 | Subspace orthocomplement f... |
| docavalN 40533 | Subspace orthocomplement f... |
| docaclN 40534 | Closure of subspace orthoc... |
| diaocN 40535 | Value of partial isomorphi... |
| doca2N 40536 | Double orthocomplement of ... |
| doca3N 40537 | Double orthocomplement of ... |
| dvadiaN 40538 | Any closed subspace is a m... |
| diarnN 40539 | Partial isomorphism A maps... |
| diaf1oN 40540 | The partial isomorphism A ... |
| djaffvalN 40543 | Subspace join for ` DVecA ... |
| djafvalN 40544 | Subspace join for ` DVecA ... |
| djavalN 40545 | Subspace join for ` DVecA ... |
| djaclN 40546 | Closure of subspace join f... |
| djajN 40547 | Transfer lattice join to `... |
| dibffval 40550 | The partial isomorphism B ... |
| dibfval 40551 | The partial isomorphism B ... |
| dibval 40552 | The partial isomorphism B ... |
| dibopelvalN 40553 | Member of the partial isom... |
| dibval2 40554 | Value of the partial isomo... |
| dibopelval2 40555 | Member of the partial isom... |
| dibval3N 40556 | Value of the partial isomo... |
| dibelval3 40557 | Member of the partial isom... |
| dibopelval3 40558 | Member of the partial isom... |
| dibelval1st 40559 | Membership in value of the... |
| dibelval1st1 40560 | Membership in value of the... |
| dibelval1st2N 40561 | Membership in value of the... |
| dibelval2nd 40562 | Membership in value of the... |
| dibn0 40563 | The value of the partial i... |
| dibfna 40564 | Functionality and domain o... |
| dibdiadm 40565 | Domain of the partial isom... |
| dibfnN 40566 | Functionality and domain o... |
| dibdmN 40567 | Domain of the partial isom... |
| dibeldmN 40568 | Member of domain of the pa... |
| dibord 40569 | The isomorphism B for a la... |
| dib11N 40570 | The isomorphism B for a la... |
| dibf11N 40571 | The partial isomorphism A ... |
| dibclN 40572 | Closure of partial isomorp... |
| dibvalrel 40573 | The value of partial isomo... |
| dib0 40574 | The value of partial isomo... |
| dib1dim 40575 | Two expressions for the 1-... |
| dibglbN 40576 | Partial isomorphism B of a... |
| dibintclN 40577 | The intersection of partia... |
| dib1dim2 40578 | Two expressions for a 1-di... |
| dibss 40579 | The partial isomorphism B ... |
| diblss 40580 | The value of partial isomo... |
| diblsmopel 40581 | Membership in subspace sum... |
| dicffval 40584 | The partial isomorphism C ... |
| dicfval 40585 | The partial isomorphism C ... |
| dicval 40586 | The partial isomorphism C ... |
| dicopelval 40587 | Membership in value of the... |
| dicelvalN 40588 | Membership in value of the... |
| dicval2 40589 | The partial isomorphism C ... |
| dicelval3 40590 | Member of the partial isom... |
| dicopelval2 40591 | Membership in value of the... |
| dicelval2N 40592 | Membership in value of the... |
| dicfnN 40593 | Functionality and domain o... |
| dicdmN 40594 | Domain of the partial isom... |
| dicvalrelN 40595 | The value of partial isomo... |
| dicssdvh 40596 | The partial isomorphism C ... |
| dicelval1sta 40597 | Membership in value of the... |
| dicelval1stN 40598 | Membership in value of the... |
| dicelval2nd 40599 | Membership in value of the... |
| dicvaddcl 40600 | Membership in value of the... |
| dicvscacl 40601 | Membership in value of the... |
| dicn0 40602 | The value of the partial i... |
| diclss 40603 | The value of partial isomo... |
| diclspsn 40604 | The value of isomorphism C... |
| cdlemn2 40605 | Part of proof of Lemma N o... |
| cdlemn2a 40606 | Part of proof of Lemma N o... |
| cdlemn3 40607 | Part of proof of Lemma N o... |
| cdlemn4 40608 | Part of proof of Lemma N o... |
| cdlemn4a 40609 | Part of proof of Lemma N o... |
| cdlemn5pre 40610 | Part of proof of Lemma N o... |
| cdlemn5 40611 | Part of proof of Lemma N o... |
| cdlemn6 40612 | Part of proof of Lemma N o... |
| cdlemn7 40613 | Part of proof of Lemma N o... |
| cdlemn8 40614 | Part of proof of Lemma N o... |
| cdlemn9 40615 | Part of proof of Lemma N o... |
| cdlemn10 40616 | Part of proof of Lemma N o... |
| cdlemn11a 40617 | Part of proof of Lemma N o... |
| cdlemn11b 40618 | Part of proof of Lemma N o... |
| cdlemn11c 40619 | Part of proof of Lemma N o... |
| cdlemn11pre 40620 | Part of proof of Lemma N o... |
| cdlemn11 40621 | Part of proof of Lemma N o... |
| cdlemn 40622 | Lemma N of [Crawley] p. 12... |
| dihordlem6 40623 | Part of proof of Lemma N o... |
| dihordlem7 40624 | Part of proof of Lemma N o... |
| dihordlem7b 40625 | Part of proof of Lemma N o... |
| dihjustlem 40626 | Part of proof after Lemma ... |
| dihjust 40627 | Part of proof after Lemma ... |
| dihord1 40628 | Part of proof after Lemma ... |
| dihord2a 40629 | Part of proof after Lemma ... |
| dihord2b 40630 | Part of proof after Lemma ... |
| dihord2cN 40631 | Part of proof after Lemma ... |
| dihord11b 40632 | Part of proof after Lemma ... |
| dihord10 40633 | Part of proof after Lemma ... |
| dihord11c 40634 | Part of proof after Lemma ... |
| dihord2pre 40635 | Part of proof after Lemma ... |
| dihord2pre2 40636 | Part of proof after Lemma ... |
| dihord2 40637 | Part of proof after Lemma ... |
| dihffval 40640 | The isomorphism H for a la... |
| dihfval 40641 | Isomorphism H for a lattic... |
| dihval 40642 | Value of isomorphism H for... |
| dihvalc 40643 | Value of isomorphism H for... |
| dihlsscpre 40644 | Closure of isomorphism H f... |
| dihvalcqpre 40645 | Value of isomorphism H for... |
| dihvalcq 40646 | Value of isomorphism H for... |
| dihvalb 40647 | Value of isomorphism H for... |
| dihopelvalbN 40648 | Ordered pair member of the... |
| dihvalcqat 40649 | Value of isomorphism H for... |
| dih1dimb 40650 | Two expressions for a 1-di... |
| dih1dimb2 40651 | Isomorphism H at an atom u... |
| dih1dimc 40652 | Isomorphism H at an atom n... |
| dib2dim 40653 | Extend ~ dia2dim to partia... |
| dih2dimb 40654 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 40655 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 40656 | Ordered pair member of the... |
| dihvalcq2 40657 | Value of isomorphism H for... |
| dihopelvalcpre 40658 | Member of value of isomorp... |
| dihopelvalc 40659 | Member of value of isomorp... |
| dihlss 40660 | The value of isomorphism H... |
| dihss 40661 | The value of isomorphism H... |
| dihssxp 40662 | An isomorphism H value is ... |
| dihopcl 40663 | Closure of an ordered pair... |
| xihopellsmN 40664 | Ordered pair membership in... |
| dihopellsm 40665 | Ordered pair membership in... |
| dihord6apre 40666 | Part of proof that isomorp... |
| dihord3 40667 | The isomorphism H for a la... |
| dihord4 40668 | The isomorphism H for a la... |
| dihord5b 40669 | Part of proof that isomorp... |
| dihord6b 40670 | Part of proof that isomorp... |
| dihord6a 40671 | Part of proof that isomorp... |
| dihord5apre 40672 | Part of proof that isomorp... |
| dihord5a 40673 | Part of proof that isomorp... |
| dihord 40674 | The isomorphism H is order... |
| dih11 40675 | The isomorphism H is one-t... |
| dihf11lem 40676 | Functionality of the isomo... |
| dihf11 40677 | The isomorphism H for a la... |
| dihfn 40678 | Functionality and domain o... |
| dihdm 40679 | Domain of isomorphism H. (... |
| dihcl 40680 | Closure of isomorphism H. ... |
| dihcnvcl 40681 | Closure of isomorphism H c... |
| dihcnvid1 40682 | The converse isomorphism o... |
| dihcnvid2 40683 | The isomorphism of a conve... |
| dihcnvord 40684 | Ordering property for conv... |
| dihcnv11 40685 | The converse of isomorphis... |
| dihsslss 40686 | The isomorphism H maps to ... |
| dihrnlss 40687 | The isomorphism H maps to ... |
| dihrnss 40688 | The isomorphism H maps to ... |
| dihvalrel 40689 | The value of isomorphism H... |
| dih0 40690 | The value of isomorphism H... |
| dih0bN 40691 | A lattice element is zero ... |
| dih0vbN 40692 | A vector is zero iff its s... |
| dih0cnv 40693 | The isomorphism H converse... |
| dih0rn 40694 | The zero subspace belongs ... |
| dih0sb 40695 | A subspace is zero iff the... |
| dih1 40696 | The value of isomorphism H... |
| dih1rn 40697 | The full vector space belo... |
| dih1cnv 40698 | The isomorphism H converse... |
| dihwN 40699 | Value of isomorphism H at ... |
| dihmeetlem1N 40700 | Isomorphism H of a conjunc... |
| dihglblem5apreN 40701 | A conjunction property of ... |
| dihglblem5aN 40702 | A conjunction property of ... |
| dihglblem2aN 40703 | Lemma for isomorphism H of... |
| dihglblem2N 40704 | The GLB of a set of lattic... |
| dihglblem3N 40705 | Isomorphism H of a lattice... |
| dihglblem3aN 40706 | Isomorphism H of a lattice... |
| dihglblem4 40707 | Isomorphism H of a lattice... |
| dihglblem5 40708 | Isomorphism H of a lattice... |
| dihmeetlem2N 40709 | Isomorphism H of a conjunc... |
| dihglbcpreN 40710 | Isomorphism H of a lattice... |
| dihglbcN 40711 | Isomorphism H of a lattice... |
| dihmeetcN 40712 | Isomorphism H of a lattice... |
| dihmeetbN 40713 | Isomorphism H of a lattice... |
| dihmeetbclemN 40714 | Lemma for isomorphism H of... |
| dihmeetlem3N 40715 | Lemma for isomorphism H of... |
| dihmeetlem4preN 40716 | Lemma for isomorphism H of... |
| dihmeetlem4N 40717 | Lemma for isomorphism H of... |
| dihmeetlem5 40718 | Part of proof that isomorp... |
| dihmeetlem6 40719 | Lemma for isomorphism H of... |
| dihmeetlem7N 40720 | Lemma for isomorphism H of... |
| dihjatc1 40721 | Lemma for isomorphism H of... |
| dihjatc2N 40722 | Isomorphism H of join with... |
| dihjatc3 40723 | Isomorphism H of join with... |
| dihmeetlem8N 40724 | Lemma for isomorphism H of... |
| dihmeetlem9N 40725 | Lemma for isomorphism H of... |
| dihmeetlem10N 40726 | Lemma for isomorphism H of... |
| dihmeetlem11N 40727 | Lemma for isomorphism H of... |
| dihmeetlem12N 40728 | Lemma for isomorphism H of... |
| dihmeetlem13N 40729 | Lemma for isomorphism H of... |
| dihmeetlem14N 40730 | Lemma for isomorphism H of... |
| dihmeetlem15N 40731 | Lemma for isomorphism H of... |
| dihmeetlem16N 40732 | Lemma for isomorphism H of... |
| dihmeetlem17N 40733 | Lemma for isomorphism H of... |
| dihmeetlem18N 40734 | Lemma for isomorphism H of... |
| dihmeetlem19N 40735 | Lemma for isomorphism H of... |
| dihmeetlem20N 40736 | Lemma for isomorphism H of... |
| dihmeetALTN 40737 | Isomorphism H of a lattice... |
| dih1dimatlem0 40738 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 40739 | Lemma for ~ dih1dimat . (... |
| dih1dimat 40740 | Any 1-dimensional subspace... |
| dihlsprn 40741 | The span of a vector belon... |
| dihlspsnssN 40742 | A subspace included in a 1... |
| dihlspsnat 40743 | The inverse isomorphism H ... |
| dihatlat 40744 | The isomorphism H of an at... |
| dihat 40745 | There exists at least one ... |
| dihpN 40746 | The value of isomorphism H... |
| dihlatat 40747 | The reverse isomorphism H ... |
| dihatexv 40748 | There is a nonzero vector ... |
| dihatexv2 40749 | There is a nonzero vector ... |
| dihglblem6 40750 | Isomorphism H of a lattice... |
| dihglb 40751 | Isomorphism H of a lattice... |
| dihglb2 40752 | Isomorphism H of a lattice... |
| dihmeet 40753 | Isomorphism H of a lattice... |
| dihintcl 40754 | The intersection of closed... |
| dihmeetcl 40755 | Closure of closed subspace... |
| dihmeet2 40756 | Reverse isomorphism H of a... |
| dochffval 40759 | Subspace orthocomplement f... |
| dochfval 40760 | Subspace orthocomplement f... |
| dochval 40761 | Subspace orthocomplement f... |
| dochval2 40762 | Subspace orthocomplement f... |
| dochcl 40763 | Closure of subspace orthoc... |
| dochlss 40764 | A subspace orthocomplement... |
| dochssv 40765 | A subspace orthocomplement... |
| dochfN 40766 | Domain and codomain of the... |
| dochvalr 40767 | Orthocomplement of a close... |
| doch0 40768 | Orthocomplement of the zer... |
| doch1 40769 | Orthocomplement of the uni... |
| dochoc0 40770 | The zero subspace is close... |
| dochoc1 40771 | The unit subspace (all vec... |
| dochvalr2 40772 | Orthocomplement of a close... |
| dochvalr3 40773 | Orthocomplement of a close... |
| doch2val2 40774 | Double orthocomplement for... |
| dochss 40775 | Subset law for orthocomple... |
| dochocss 40776 | Double negative law for or... |
| dochoc 40777 | Double negative law for or... |
| dochsscl 40778 | If a set of vectors is inc... |
| dochoccl 40779 | A set of vectors is closed... |
| dochord 40780 | Ordering law for orthocomp... |
| dochord2N 40781 | Ordering law for orthocomp... |
| dochord3 40782 | Ordering law for orthocomp... |
| doch11 40783 | Orthocomplement is one-to-... |
| dochsordN 40784 | Strict ordering law for or... |
| dochn0nv 40785 | An orthocomplement is nonz... |
| dihoml4c 40786 | Version of ~ dihoml4 with ... |
| dihoml4 40787 | Orthomodular law for const... |
| dochspss 40788 | The span of a set of vecto... |
| dochocsp 40789 | The span of an orthocomple... |
| dochspocN 40790 | The span of an orthocomple... |
| dochocsn 40791 | The double orthocomplement... |
| dochsncom 40792 | Swap vectors in an orthoco... |
| dochsat 40793 | The double orthocomplement... |
| dochshpncl 40794 | If a hyperplane is not clo... |
| dochlkr 40795 | Equivalent conditions for ... |
| dochkrshp 40796 | The closure of a kernel is... |
| dochkrshp2 40797 | Properties of the closure ... |
| dochkrshp3 40798 | Properties of the closure ... |
| dochkrshp4 40799 | Properties of the closure ... |
| dochdmj1 40800 | De Morgan-like law for sub... |
| dochnoncon 40801 | Law of noncontradiction. ... |
| dochnel2 40802 | A nonzero member of a subs... |
| dochnel 40803 | A nonzero vector doesn't b... |
| djhffval 40806 | Subspace join for ` DVecH ... |
| djhfval 40807 | Subspace join for ` DVecH ... |
| djhval 40808 | Subspace join for ` DVecH ... |
| djhval2 40809 | Value of subspace join for... |
| djhcl 40810 | Closure of subspace join f... |
| djhlj 40811 | Transfer lattice join to `... |
| djhljjN 40812 | Lattice join in terms of `... |
| djhjlj 40813 | ` DVecH ` vector space clo... |
| djhj 40814 | ` DVecH ` vector space clo... |
| djhcom 40815 | Subspace join commutes. (... |
| djhspss 40816 | Subspace span of union is ... |
| djhsumss 40817 | Subspace sum is a subset o... |
| dihsumssj 40818 | The subspace sum of two is... |
| djhunssN 40819 | Subspace union is a subset... |
| dochdmm1 40820 | De Morgan-like law for clo... |
| djhexmid 40821 | Excluded middle property o... |
| djh01 40822 | Closed subspace join with ... |
| djh02 40823 | Closed subspace join with ... |
| djhlsmcl 40824 | A closed subspace sum equa... |
| djhcvat42 40825 | A covering property. ( ~ ... |
| dihjatb 40826 | Isomorphism H of lattice j... |
| dihjatc 40827 | Isomorphism H of lattice j... |
| dihjatcclem1 40828 | Lemma for isomorphism H of... |
| dihjatcclem2 40829 | Lemma for isomorphism H of... |
| dihjatcclem3 40830 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 40831 | Lemma for isomorphism H of... |
| dihjatcc 40832 | Isomorphism H of lattice j... |
| dihjat 40833 | Isomorphism H of lattice j... |
| dihprrnlem1N 40834 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 40835 | Lemma for ~ dihprrn . (Co... |
| dihprrn 40836 | The span of a vector pair ... |
| djhlsmat 40837 | The sum of two subspace at... |
| dihjat1lem 40838 | Subspace sum of a closed s... |
| dihjat1 40839 | Subspace sum of a closed s... |
| dihsmsprn 40840 | Subspace sum of a closed s... |
| dihjat2 40841 | The subspace sum of a clos... |
| dihjat3 40842 | Isomorphism H of lattice j... |
| dihjat4 40843 | Transfer the subspace sum ... |
| dihjat6 40844 | Transfer the subspace sum ... |
| dihsmsnrn 40845 | The subspace sum of two si... |
| dihsmatrn 40846 | The subspace sum of a clos... |
| dihjat5N 40847 | Transfer lattice join with... |
| dvh4dimat 40848 | There is an atom that is o... |
| dvh3dimatN 40849 | There is an atom that is o... |
| dvh2dimatN 40850 | Given an atom, there exist... |
| dvh1dimat 40851 | There exists an atom. (Co... |
| dvh1dim 40852 | There exists a nonzero vec... |
| dvh4dimlem 40853 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 40854 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 40855 | There is a vector that is ... |
| dvh3dim 40856 | There is a vector that is ... |
| dvh4dimN 40857 | There is a vector that is ... |
| dvh3dim2 40858 | There is a vector that is ... |
| dvh3dim3N 40859 | There is a vector that is ... |
| dochsnnz 40860 | The orthocomplement of a s... |
| dochsatshp 40861 | The orthocomplement of a s... |
| dochsatshpb 40862 | The orthocomplement of a s... |
| dochsnshp 40863 | The orthocomplement of a n... |
| dochshpsat 40864 | A hyperplane is closed iff... |
| dochkrsat 40865 | The orthocomplement of a k... |
| dochkrsat2 40866 | The orthocomplement of a k... |
| dochsat0 40867 | The orthocomplement of a k... |
| dochkrsm 40868 | The subspace sum of a clos... |
| dochexmidat 40869 | Special case of excluded m... |
| dochexmidlem1 40870 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 40871 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 40872 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 40873 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 40874 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 40875 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 40876 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 40877 | Lemma for ~ dochexmid . T... |
| dochexmid 40878 | Excluded middle law for cl... |
| dochsnkrlem1 40879 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 40880 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 40881 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 40882 | A (closed) kernel expresse... |
| dochsnkr2 40883 | Kernel of the explicit fun... |
| dochsnkr2cl 40884 | The ` X ` determining func... |
| dochflcl 40885 | Closure of the explicit fu... |
| dochfl1 40886 | The value of the explicit ... |
| dochfln0 40887 | The value of a functional ... |
| dochkr1 40888 | A nonzero functional has a... |
| dochkr1OLDN 40889 | A nonzero functional has a... |
| lpolsetN 40892 | The set of polarities of a... |
| islpolN 40893 | The predicate "is a polari... |
| islpoldN 40894 | Properties that determine ... |
| lpolfN 40895 | Functionality of a polarit... |
| lpolvN 40896 | The polarity of the whole ... |
| lpolconN 40897 | Contraposition property of... |
| lpolsatN 40898 | The polarity of an atomic ... |
| lpolpolsatN 40899 | Property of a polarity. (... |
| dochpolN 40900 | The subspace orthocompleme... |
| lcfl1lem 40901 | Property of a functional w... |
| lcfl1 40902 | Property of a functional w... |
| lcfl2 40903 | Property of a functional w... |
| lcfl3 40904 | Property of a functional w... |
| lcfl4N 40905 | Property of a functional w... |
| lcfl5 40906 | Property of a functional w... |
| lcfl5a 40907 | Property of a functional w... |
| lcfl6lem 40908 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 40909 | Lemma for ~ lcfl7N . If t... |
| lcfl6 40910 | Property of a functional w... |
| lcfl7N 40911 | Property of a functional w... |
| lcfl8 40912 | Property of a functional w... |
| lcfl8a 40913 | Property of a functional w... |
| lcfl8b 40914 | Property of a nonzero func... |
| lcfl9a 40915 | Property implying that a f... |
| lclkrlem1 40916 | The set of functionals hav... |
| lclkrlem2a 40917 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 40918 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 40919 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 40920 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 40921 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 40922 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 40923 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 40924 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 40925 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 40926 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 40927 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 40928 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 40929 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 40930 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 40931 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 40932 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 40933 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 40934 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 40935 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 40936 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 40937 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 40938 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 40939 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 40940 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 40941 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 40942 | The set of functionals hav... |
| lclkr 40943 | The set of functionals wit... |
| lcfls1lem 40944 | Property of a functional w... |
| lcfls1N 40945 | Property of a functional w... |
| lcfls1c 40946 | Property of a functional w... |
| lclkrslem1 40947 | The set of functionals hav... |
| lclkrslem2 40948 | The set of functionals hav... |
| lclkrs 40949 | The set of functionals hav... |
| lclkrs2 40950 | The set of functionals wit... |
| lcfrvalsnN 40951 | Reconstruction from the du... |
| lcfrlem1 40952 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 40953 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 40954 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 40955 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 40956 | Lemma for ~ lcfr . The se... |
| lcfrlem6 40957 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 40958 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 40959 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 40960 | Lemma for ~ lcf1o . (This... |
| lcf1o 40961 | Define a function ` J ` th... |
| lcfrlem10 40962 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 40963 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 40964 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 40965 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 40966 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 40967 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 40968 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 40969 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 40970 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 40971 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 40972 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 40973 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 40974 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 40975 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 40976 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 40977 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 40978 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 40979 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 40980 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 40981 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 40982 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 40983 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 40984 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 40985 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 40986 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 40987 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 40988 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 40989 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 40990 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 40991 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 40992 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 40993 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 40994 | Lemma for ~ lcfr . Elimin... |
| lcfr 40995 | Reconstruction of a subspa... |
| lcdfval 40998 | Dual vector space of funct... |
| lcdval 40999 | Dual vector space of funct... |
| lcdval2 41000 | Dual vector space of funct... |
| lcdlvec 41001 | The dual vector space of f... |
| lcdlmod 41002 | The dual vector space of f... |
| lcdvbase 41003 | Vector base set of a dual ... |
| lcdvbasess 41004 | The vector base set of the... |
| lcdvbaselfl 41005 | A vector in the base set o... |
| lcdvbasecl 41006 | Closure of the value of a ... |
| lcdvadd 41007 | Vector addition for the cl... |
| lcdvaddval 41008 | The value of the value of ... |
| lcdsca 41009 | The ring of scalars of the... |
| lcdsbase 41010 | Base set of scalar ring fo... |
| lcdsadd 41011 | Scalar addition for the cl... |
| lcdsmul 41012 | Scalar multiplication for ... |
| lcdvs 41013 | Scalar product for the clo... |
| lcdvsval 41014 | Value of scalar product op... |
| lcdvscl 41015 | The scalar product operati... |
| lcdlssvscl 41016 | Closure of scalar product ... |
| lcdvsass 41017 | Associative law for scalar... |
| lcd0 41018 | The zero scalar of the clo... |
| lcd1 41019 | The unit scalar of the clo... |
| lcdneg 41020 | The unit scalar of the clo... |
| lcd0v 41021 | The zero functional in the... |
| lcd0v2 41022 | The zero functional in the... |
| lcd0vvalN 41023 | Value of the zero function... |
| lcd0vcl 41024 | Closure of the zero functi... |
| lcd0vs 41025 | A scalar zero times a func... |
| lcdvs0N 41026 | A scalar times the zero fu... |
| lcdvsub 41027 | The value of vector subtra... |
| lcdvsubval 41028 | The value of the value of ... |
| lcdlss 41029 | Subspaces of a dual vector... |
| lcdlss2N 41030 | Subspaces of a dual vector... |
| lcdlsp 41031 | Span in the set of functio... |
| lcdlkreqN 41032 | Colinear functionals have ... |
| lcdlkreq2N 41033 | Colinear functionals have ... |
| mapdffval 41036 | Projectivity from vector s... |
| mapdfval 41037 | Projectivity from vector s... |
| mapdval 41038 | Value of projectivity from... |
| mapdvalc 41039 | Value of projectivity from... |
| mapdval2N 41040 | Value of projectivity from... |
| mapdval3N 41041 | Value of projectivity from... |
| mapdval4N 41042 | Value of projectivity from... |
| mapdval5N 41043 | Value of projectivity from... |
| mapdordlem1a 41044 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 41045 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 41046 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 41047 | Lemma for ~ mapdord . Ord... |
| mapdord 41048 | Ordering property of the m... |
| mapd11 41049 | The map defined by ~ df-ma... |
| mapddlssN 41050 | The mapping of a subspace ... |
| mapdsn 41051 | Value of the map defined b... |
| mapdsn2 41052 | Value of the map defined b... |
| mapdsn3 41053 | Value of the map defined b... |
| mapd1dim2lem1N 41054 | Value of the map defined b... |
| mapdrvallem2 41055 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 41056 | Lemma for ~ mapdrval . (C... |
| mapdrval 41057 | Given a dual subspace ` R ... |
| mapd1o 41058 | The map defined by ~ df-ma... |
| mapdrn 41059 | Range of the map defined b... |
| mapdunirnN 41060 | Union of the range of the ... |
| mapdrn2 41061 | Range of the map defined b... |
| mapdcnvcl 41062 | Closure of the converse of... |
| mapdcl 41063 | Closure the value of the m... |
| mapdcnvid1N 41064 | Converse of the value of t... |
| mapdsord 41065 | Strong ordering property o... |
| mapdcl2 41066 | The mapping of a subspace ... |
| mapdcnvid2 41067 | Value of the converse of t... |
| mapdcnvordN 41068 | Ordering property of the c... |
| mapdcnv11N 41069 | The converse of the map de... |
| mapdcv 41070 | Covering property of the c... |
| mapdincl 41071 | Closure of dual subspace i... |
| mapdin 41072 | Subspace intersection is p... |
| mapdlsmcl 41073 | Closure of dual subspace s... |
| mapdlsm 41074 | Subspace sum is preserved ... |
| mapd0 41075 | Projectivity map of the ze... |
| mapdcnvatN 41076 | Atoms are preserved by the... |
| mapdat 41077 | Atoms are preserved by the... |
| mapdspex 41078 | The map of a span equals t... |
| mapdn0 41079 | Transfer nonzero property ... |
| mapdncol 41080 | Transfer non-colinearity f... |
| mapdindp 41081 | Transfer (part of) vector ... |
| mapdpglem1 41082 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 41083 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 41084 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 41085 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 41086 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 41087 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 41088 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 41089 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 41090 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 41091 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 41092 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 41093 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 41094 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 41095 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 41096 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 41097 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 41098 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 41099 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 41100 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 41101 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 41102 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 41103 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 41104 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 41105 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 41106 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 41107 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 41108 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 41109 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 41110 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 41111 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 41112 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 41113 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 41114 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 41115 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 41116 | Part 1 of proof of the fir... |
| baerlem3lem1 41117 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 41118 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 41119 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 41120 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 41121 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 41122 | Lemma for ~ baerlem5b . (... |
| baerlem3 41123 | An equality that holds whe... |
| baerlem5a 41124 | An equality that holds whe... |
| baerlem5b 41125 | An equality that holds whe... |
| baerlem5amN 41126 | An equality that holds whe... |
| baerlem5bmN 41127 | An equality that holds whe... |
| baerlem5abmN 41128 | An equality that holds whe... |
| mapdindp0 41129 | Vector independence lemma.... |
| mapdindp1 41130 | Vector independence lemma.... |
| mapdindp2 41131 | Vector independence lemma.... |
| mapdindp3 41132 | Vector independence lemma.... |
| mapdindp4 41133 | Vector independence lemma.... |
| mapdhval 41134 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 41135 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 41136 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 41137 | Lemmma for ~~? mapdh . (C... |
| mapdheq 41138 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 41139 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 41140 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 41141 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 41142 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 41143 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 41144 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 41145 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 41146 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 41147 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 41148 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 41149 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 41150 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 41151 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 41152 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 41153 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 41154 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 41155 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 41156 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 41157 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 41158 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 41159 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 41160 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 41161 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 41162 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 41163 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 41164 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 41165 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 41168 | Map from nonzero vectors t... |
| hvmapfval 41169 | Map from nonzero vectors t... |
| hvmapval 41170 | Value of map from nonzero ... |
| hvmapvalvalN 41171 | Value of value of map (i.e... |
| hvmapidN 41172 | The value of the vector to... |
| hvmap1o 41173 | The vector to functional m... |
| hvmapclN 41174 | Closure of the vector to f... |
| hvmap1o2 41175 | The vector to functional m... |
| hvmapcl2 41176 | Closure of the vector to f... |
| hvmaplfl 41177 | The vector to functional m... |
| hvmaplkr 41178 | Kernel of the vector to fu... |
| mapdhvmap 41179 | Relationship between ` map... |
| lspindp5 41180 | Obtain an independent vect... |
| hdmaplem1 41181 | Lemma to convert a frequen... |
| hdmaplem2N 41182 | Lemma to convert a frequen... |
| hdmaplem3 41183 | Lemma to convert a frequen... |
| hdmaplem4 41184 | Lemma to convert a frequen... |
| mapdh8a 41185 | Part of Part (8) in [Baer]... |
| mapdh8aa 41186 | Part of Part (8) in [Baer]... |
| mapdh8ab 41187 | Part of Part (8) in [Baer]... |
| mapdh8ac 41188 | Part of Part (8) in [Baer]... |
| mapdh8ad 41189 | Part of Part (8) in [Baer]... |
| mapdh8b 41190 | Part of Part (8) in [Baer]... |
| mapdh8c 41191 | Part of Part (8) in [Baer]... |
| mapdh8d0N 41192 | Part of Part (8) in [Baer]... |
| mapdh8d 41193 | Part of Part (8) in [Baer]... |
| mapdh8e 41194 | Part of Part (8) in [Baer]... |
| mapdh8g 41195 | Part of Part (8) in [Baer]... |
| mapdh8i 41196 | Part of Part (8) in [Baer]... |
| mapdh8j 41197 | Part of Part (8) in [Baer]... |
| mapdh8 41198 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 41199 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 41200 | Lemma for part (9) in [Bae... |
| hdmap1ffval 41205 | Preliminary map from vecto... |
| hdmap1fval 41206 | Preliminary map from vecto... |
| hdmap1vallem 41207 | Value of preliminary map f... |
| hdmap1val 41208 | Value of preliminary map f... |
| hdmap1val0 41209 | Value of preliminary map f... |
| hdmap1val2 41210 | Value of preliminary map f... |
| hdmap1eq 41211 | The defining equation for ... |
| hdmap1cbv 41212 | Frequently used lemma to c... |
| hdmap1valc 41213 | Connect the value of the p... |
| hdmap1cl 41214 | Convert closure theorem ~ ... |
| hdmap1eq2 41215 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 41216 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 41217 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 41218 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 41219 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 41220 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 41221 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 41222 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 41223 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 41224 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 41225 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 41226 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 41227 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 41228 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 41229 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 41230 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 41231 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 41232 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 41233 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 41234 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 41235 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 41236 | Map from vectors to functi... |
| hdmapfval 41237 | Map from vectors to functi... |
| hdmapval 41238 | Value of map from vectors ... |
| hdmapfnN 41239 | Functionality of map from ... |
| hdmapcl 41240 | Closure of map from vector... |
| hdmapval2lem 41241 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 41242 | Value of map from vectors ... |
| hdmapval0 41243 | Value of map from vectors ... |
| hdmapeveclem 41244 | Lemma for ~ hdmapevec . T... |
| hdmapevec 41245 | Value of map from vectors ... |
| hdmapevec2 41246 | The inner product of the r... |
| hdmapval3lemN 41247 | Value of map from vectors ... |
| hdmapval3N 41248 | Value of map from vectors ... |
| hdmap10lem 41249 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 41250 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 41251 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 41252 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 41253 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 41254 | Part of proof of part 12 i... |
| hdmapnzcl 41255 | Nonzero vector closure of ... |
| hdmapneg 41256 | Part of proof of part 12 i... |
| hdmapsub 41257 | Part of proof of part 12 i... |
| hdmap11 41258 | Part of proof of part 12 i... |
| hdmaprnlem1N 41259 | Part of proof of part 12 i... |
| hdmaprnlem3N 41260 | Part of proof of part 12 i... |
| hdmaprnlem3uN 41261 | Part of proof of part 12 i... |
| hdmaprnlem4tN 41262 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 41263 | Part of proof of part 12 i... |
| hdmaprnlem6N 41264 | Part of proof of part 12 i... |
| hdmaprnlem7N 41265 | Part of proof of part 12 i... |
| hdmaprnlem8N 41266 | Part of proof of part 12 i... |
| hdmaprnlem9N 41267 | Part of proof of part 12 i... |
| hdmaprnlem3eN 41268 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 41269 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 41270 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 41271 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 41272 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 41273 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 41274 | Part of proof of part 12 i... |
| hdmapf1oN 41275 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 41276 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 41277 | Prior to part 14 in [Baer]... |
| hdmap14lem1 41278 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 41279 | Prior to part 14 in [Baer]... |
| hdmap14lem3 41280 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 41281 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 41282 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 41283 | Case where ` F ` is zero. ... |
| hdmap14lem7 41284 | Combine cases of ` F ` . ... |
| hdmap14lem8 41285 | Part of proof of part 14 i... |
| hdmap14lem9 41286 | Part of proof of part 14 i... |
| hdmap14lem10 41287 | Part of proof of part 14 i... |
| hdmap14lem11 41288 | Part of proof of part 14 i... |
| hdmap14lem12 41289 | Lemma for proof of part 14... |
| hdmap14lem13 41290 | Lemma for proof of part 14... |
| hdmap14lem14 41291 | Part of proof of part 14 i... |
| hdmap14lem15 41292 | Part of proof of part 14 i... |
| hgmapffval 41295 | Map from the scalar divisi... |
| hgmapfval 41296 | Map from the scalar divisi... |
| hgmapval 41297 | Value of map from the scal... |
| hgmapfnN 41298 | Functionality of scalar si... |
| hgmapcl 41299 | Closure of scalar sigma ma... |
| hgmapdcl 41300 | Closure of the vector spac... |
| hgmapvs 41301 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 41302 | Value of the scalar sigma ... |
| hgmapval1 41303 | Value of the scalar sigma ... |
| hgmapadd 41304 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 41305 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 41306 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 41307 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 41308 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 41309 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 41310 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 41311 | Part of proof of part 16 i... |
| hgmap11 41312 | The scalar sigma map is on... |
| hgmapf1oN 41313 | The scalar sigma map is a ... |
| hgmapeq0 41314 | The scalar sigma map is ze... |
| hdmapipcl 41315 | The inner product (Hermiti... |
| hdmapln1 41316 | Linearity property that wi... |
| hdmaplna1 41317 | Additive property of first... |
| hdmaplns1 41318 | Subtraction property of fi... |
| hdmaplnm1 41319 | Multiplicative property of... |
| hdmaplna2 41320 | Additive property of secon... |
| hdmapglnm2 41321 | g-linear property of secon... |
| hdmapgln2 41322 | g-linear property that wil... |
| hdmaplkr 41323 | Kernel of the vector to du... |
| hdmapellkr 41324 | Membership in the kernel (... |
| hdmapip0 41325 | Zero property that will be... |
| hdmapip1 41326 | Construct a proportional v... |
| hdmapip0com 41327 | Commutation property of Ba... |
| hdmapinvlem1 41328 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 41329 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 41330 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 41331 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 41332 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 41333 | Involution property of sca... |
| hgmapvvlem2 41334 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 41335 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 41336 | Value of a double involuti... |
| hdmapglem7a 41337 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 41338 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 41339 | Lemma for ~ hdmapg . Line... |
| hdmapg 41340 | Apply the scalar sigma fun... |
| hdmapoc 41341 | Express our constructed or... |
| hlhilset 41344 | The final Hilbert space co... |
| hlhilsca 41345 | The scalar of the final co... |
| hlhilbase 41346 | The base set of the final ... |
| hlhilplus 41347 | The vector addition for th... |
| hlhilslem 41348 | Lemma for ~ hlhilsbase etc... |
| hlhilslemOLD 41349 | Obsolete version of ~ hlhi... |
| hlhilsbase 41350 | The scalar base set of the... |
| hlhilsbaseOLD 41351 | Obsolete version of ~ hlhi... |
| hlhilsplus 41352 | Scalar addition for the fi... |
| hlhilsplusOLD 41353 | Obsolete version of ~ hlhi... |
| hlhilsmul 41354 | Scalar multiplication for ... |
| hlhilsmulOLD 41355 | Obsolete version of ~ hlhi... |
| hlhilsbase2 41356 | The scalar base set of the... |
| hlhilsplus2 41357 | Scalar addition for the fi... |
| hlhilsmul2 41358 | Scalar multiplication for ... |
| hlhils0 41359 | The scalar ring zero for t... |
| hlhils1N 41360 | The scalar ring unity for ... |
| hlhilvsca 41361 | The scalar product for the... |
| hlhilip 41362 | Inner product operation fo... |
| hlhilipval 41363 | Value of inner product ope... |
| hlhilnvl 41364 | The involution operation o... |
| hlhillvec 41365 | The final constructed Hilb... |
| hlhildrng 41366 | The star division ring for... |
| hlhilsrnglem 41367 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 41368 | The star division ring for... |
| hlhil0 41369 | The zero vector for the fi... |
| hlhillsm 41370 | The vector sum operation f... |
| hlhilocv 41371 | The orthocomplement for th... |
| hlhillcs 41372 | The closed subspaces of th... |
| hlhilphllem 41373 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 41374 | Lemma for ~ hlhil . (Cont... |
| hlathil 41375 | Construction of a Hilbert ... |
| iscsrg 41378 | A commutative semiring is ... |
| leexp1ad 41379 | Weak base ordering relatio... |
| relogbcld 41380 | Closure of the general log... |
| relogbexpd 41381 | Identity law for general l... |
| relogbzexpd 41382 | Power law for the general ... |
| logblebd 41383 | The general logarithm is m... |
| uzindd 41384 | Induction on the upper int... |
| fzadd2d 41385 | Membership of a sum in a f... |
| zltlem1d 41386 | Integer ordering relation,... |
| zltp1led 41387 | Integer ordering relation,... |
| fzne2d 41388 | Elementhood in a finite se... |
| eqfnfv2d2 41389 | Equality of functions is d... |
| fzsplitnd 41390 | Split a finite interval of... |
| fzsplitnr 41391 | Split a finite interval of... |
| addassnni 41392 | Associative law for additi... |
| addcomnni 41393 | Commutative law for additi... |
| mulassnni 41394 | Associative law for multip... |
| mulcomnni 41395 | Commutative law for multip... |
| gcdcomnni 41396 | Commutative law for gcd. ... |
| gcdnegnni 41397 | Negation invariance for gc... |
| neggcdnni 41398 | Negation invariance for gc... |
| bccl2d 41399 | Closure of the binomial co... |
| recbothd 41400 | Take reciprocal on both si... |
| gcdmultiplei 41401 | The GCD of a multiple of a... |
| gcdaddmzz2nni 41402 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 41403 | Adding a multiple of one o... |
| gcdnncli 41404 | Closure of the gcd operato... |
| muldvds1d 41405 | If a product divides an in... |
| muldvds2d 41406 | If a product divides an in... |
| nndivdvdsd 41407 | A positive integer divides... |
| nnproddivdvdsd 41408 | A product of natural numbe... |
| coprmdvds2d 41409 | If an integer is divisible... |
| imadomfi 41410 | An image of a function und... |
| 12gcd5e1 41411 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 41412 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 41413 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 41414 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 41415 | Calculate the least common... |
| 12lcm5e60 41416 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 41417 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 41418 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 41419 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 41420 | Useful equation to calcula... |
| lcm1un 41421 | Least common multiple of n... |
| lcm2un 41422 | Least common multiple of n... |
| lcm3un 41423 | Least common multiple of n... |
| lcm4un 41424 | Least common multiple of n... |
| lcm5un 41425 | Least common multiple of n... |
| lcm6un 41426 | Least common multiple of n... |
| lcm7un 41427 | Least common multiple of n... |
| lcm8un 41428 | Least common multiple of n... |
| 3factsumint1 41429 | Move constants out of inte... |
| 3factsumint2 41430 | Move constants out of inte... |
| 3factsumint3 41431 | Move constants out of inte... |
| 3factsumint4 41432 | Move constants out of inte... |
| 3factsumint 41433 | Helpful equation for lcm i... |
| resopunitintvd 41434 | Restrict continuous functi... |
| resclunitintvd 41435 | Restrict continuous functi... |
| resdvopclptsd 41436 | Restrict derivative on uni... |
| lcmineqlem1 41437 | Part of lcm inequality lem... |
| lcmineqlem2 41438 | Part of lcm inequality lem... |
| lcmineqlem3 41439 | Part of lcm inequality lem... |
| lcmineqlem4 41440 | Part of lcm inequality lem... |
| lcmineqlem5 41441 | Technical lemma for recipr... |
| lcmineqlem6 41442 | Part of lcm inequality lem... |
| lcmineqlem7 41443 | Derivative of 1-x for chai... |
| lcmineqlem8 41444 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 41445 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 41446 | Induction step of ~ lcmine... |
| lcmineqlem11 41447 | Induction step, continuati... |
| lcmineqlem12 41448 | Base case for induction. ... |
| lcmineqlem13 41449 | Induction proof for lcm in... |
| lcmineqlem14 41450 | Technical lemma for inequa... |
| lcmineqlem15 41451 | F times the least common m... |
| lcmineqlem16 41452 | Technical divisibility lem... |
| lcmineqlem17 41453 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 41454 | Technical lemma to shift f... |
| lcmineqlem19 41455 | Dividing implies inequalit... |
| lcmineqlem20 41456 | Inequality for lcm lemma. ... |
| lcmineqlem21 41457 | The lcm inequality lemma w... |
| lcmineqlem22 41458 | The lcm inequality lemma w... |
| lcmineqlem23 41459 | Penultimate step to the lc... |
| lcmineqlem 41460 | The least common multiple ... |
| 3exp7 41461 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 41462 | First inequality in inequa... |
| 3lexlogpow5ineq2 41463 | Second inequality in inequ... |
| 3lexlogpow5ineq4 41464 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 41465 | Combined inequality chain ... |
| 3lexlogpow2ineq1 41466 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 41467 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 41468 | Result for bound in AKS in... |
| intlewftc 41469 | Inequality inference by in... |
| aks4d1lem1 41470 | Technical lemma to reduce ... |
| aks4d1p1p1 41471 | Exponential law for finite... |
| dvrelog2 41472 | The derivative of the loga... |
| dvrelog3 41473 | The derivative of the loga... |
| dvrelog2b 41474 | Derivative of the binary l... |
| 0nonelalab 41475 | Technical lemma for open i... |
| dvrelogpow2b 41476 | Derivative of the power of... |
| aks4d1p1p3 41477 | Bound of a ceiling of the ... |
| aks4d1p1p2 41478 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 41479 | Technical step for inequal... |
| dvle2 41480 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 41481 | Inequality lift to differe... |
| aks4d1p1p7 41482 | Bound of intermediary of i... |
| aks4d1p1p5 41483 | Show inequality for existe... |
| aks4d1p1 41484 | Show inequality for existe... |
| aks4d1p2 41485 | Technical lemma for existe... |
| aks4d1p3 41486 | There exists a small enoug... |
| aks4d1p4 41487 | There exists a small enoug... |
| aks4d1p5 41488 | Show that ` N ` and ` R ` ... |
| aks4d1p6 41489 | The maximal prime power ex... |
| aks4d1p7d1 41490 | Technical step in AKS lemm... |
| aks4d1p7 41491 | Technical step in AKS lemm... |
| aks4d1p8d1 41492 | If a prime divides one num... |
| aks4d1p8d2 41493 | Any prime power dividing a... |
| aks4d1p8d3 41494 | The remainder of a divisio... |
| aks4d1p8 41495 | Show that ` N ` and ` R ` ... |
| aks4d1p9 41496 | Show that the order is bou... |
| aks4d1 41497 | Lemma 4.1 from ~ https://w... |
| fldhmf1 41498 | A field homomorphism is in... |
| isprimroot 41501 | The value of a primitive r... |
| mndmolinv 41502 | An element of a monoid tha... |
| linvh 41503 | If an element has a unique... |
| primrootsunit1 41504 | Primitive roots have left ... |
| primrootsunit 41505 | Primitive roots have left ... |
| primrootscoprmpow 41506 | Coprime powers of primitiv... |
| posbezout 41507 | Bezout's identity restrict... |
| primrootscoprf 41508 | Coprime powers of primitiv... |
| primrootscoprbij 41509 | A bijection between coprim... |
| primrootscoprbij2 41510 | A bijection between coprim... |
| aks6d1c1p1 41511 | Definition of the introspe... |
| aks6d1c1p1rcl 41512 | Reverse closure of the int... |
| aks6d1c1p2 41513 | ` P ` and linear factors a... |
| aks6d1c1p3 41514 | In a field with a Frobeniu... |
| aks6d1c1p4 41515 | The product of polynomials... |
| aks6d1c1p5 41516 | The product of exponents i... |
| aks6d1c1p7 41517 | ` X ` is introspective to ... |
| aks6d1c1p6 41518 | If a polynomials ` F ` is ... |
| aks6d1c1p8 41519 | If a number ` E ` is intro... |
| aks6d1c1 41520 | Claim 1 of Theorem 6.1 ~ h... |
| evl1gprodd 41521 | Polynomial evaluation buil... |
| aks6d1c2p1 41522 | In the AKS-theorem the sub... |
| aks6d1c2p2 41523 | Injective condition for co... |
| hashscontpowcl 41524 | Closure of E for ~ https:/... |
| hashscontpow1 41525 | Helper lemma for to prove ... |
| hashscontpow 41526 | If a set contains all ` N ... |
| aks6d1c3 41527 | Claim 3 of Theorem 6.1 of ... |
| aks6d1c1rh 41528 | Claim 1 of AKS primality p... |
| aks6d1c2lem3 41529 | Lemma for ~ aks6d1c2 to si... |
| aks6d1c2lem4 41530 | Claim 2 of Theorem 6.1 AKS... |
| hashnexinj 41531 | If the number of elements ... |
| hashnexinjle 41532 | If the number of elements ... |
| aks6d1c2 41533 | Claim 2 of Theorem 6.1 of ... |
| rspcsbnea 41534 | Special case related to ~ ... |
| idomnnzpownz 41535 | A non-zero power in an int... |
| idomnnzgmulnz 41536 | A finite product of non-ze... |
| ringexp0nn 41537 | Zero to the power of a pos... |
| aks6d1c5lem0 41538 | Lemma for Claim 5 of Theor... |
| aks6d1c5lem1 41539 | Lemma for claim 5, evaluat... |
| aks6d1c5lem3 41540 | Lemma for Claim 5, polynom... |
| aks6d1c5lem2 41541 | Lemma for Claim 5, contrad... |
| aks6d1c5 41542 | Claim 5 of Theorem 6.1 ~ h... |
| deg1mul 41543 | Degree of multiplication o... |
| deg1gprod 41544 | Degree multiplication is a... |
| deg1pow 41545 | Exact degree of a power of... |
| 5bc2eq10 41546 | The value of 5 choose 2. ... |
| facp2 41547 | The factorial of a success... |
| 2np3bcnp1 41548 | Part of induction step for... |
| 2ap1caineq 41549 | Inequality for Theorem 6.6... |
| sticksstones1 41550 | Different strictly monoton... |
| sticksstones2 41551 | The range function on stri... |
| sticksstones3 41552 | The range function on stri... |
| sticksstones4 41553 | Equinumerosity lemma for s... |
| sticksstones5 41554 | Count the number of strict... |
| sticksstones6 41555 | Function induces an order ... |
| sticksstones7 41556 | Closure property of sticks... |
| sticksstones8 41557 | Establish mapping between ... |
| sticksstones9 41558 | Establish mapping between ... |
| sticksstones10 41559 | Establish mapping between ... |
| sticksstones11 41560 | Establish bijective mappin... |
| sticksstones12a 41561 | Establish bijective mappin... |
| sticksstones12 41562 | Establish bijective mappin... |
| sticksstones13 41563 | Establish bijective mappin... |
| sticksstones14 41564 | Sticks and stones with def... |
| sticksstones15 41565 | Sticks and stones with alm... |
| sticksstones16 41566 | Sticks and stones with col... |
| sticksstones17 41567 | Extend sticks and stones t... |
| sticksstones18 41568 | Extend sticks and stones t... |
| sticksstones19 41569 | Extend sticks and stones t... |
| sticksstones20 41570 | Lift sticks and stones to ... |
| sticksstones21 41571 | Lift sticks and stones to ... |
| sticksstones22 41572 | Non-exhaustive sticks and ... |
| sticksstones23 41573 | Non-exhaustive sticks and ... |
| aks6d1c6lem1 41574 | Lemma for claim 6, deduce ... |
| aks6d1c6lem2 41575 | Every primitive root is ro... |
| aks6d1c6lem3 41576 | Claim 6 of Theorem 6.1 of ... |
| metakunt1 41577 | A is an endomapping. (Con... |
| metakunt2 41578 | A is an endomapping. (Con... |
| metakunt3 41579 | Value of A. (Contributed b... |
| metakunt4 41580 | Value of A. (Contributed b... |
| metakunt5 41581 | C is the left inverse for ... |
| metakunt6 41582 | C is the left inverse for ... |
| metakunt7 41583 | C is the left inverse for ... |
| metakunt8 41584 | C is the left inverse for ... |
| metakunt9 41585 | C is the left inverse for ... |
| metakunt10 41586 | C is the right inverse for... |
| metakunt11 41587 | C is the right inverse for... |
| metakunt12 41588 | C is the right inverse for... |
| metakunt13 41589 | C is the right inverse for... |
| metakunt14 41590 | A is a primitive permutati... |
| metakunt15 41591 | Construction of another pe... |
| metakunt16 41592 | Construction of another pe... |
| metakunt17 41593 | The union of three disjoin... |
| metakunt18 41594 | Disjoint domains and codom... |
| metakunt19 41595 | Domains on restrictions of... |
| metakunt20 41596 | Show that B coincides on t... |
| metakunt21 41597 | Show that B coincides on t... |
| metakunt22 41598 | Show that B coincides on t... |
| metakunt23 41599 | B coincides on the union o... |
| metakunt24 41600 | Technical condition such t... |
| metakunt25 41601 | B is a permutation. (Cont... |
| metakunt26 41602 | Construction of one soluti... |
| metakunt27 41603 | Construction of one soluti... |
| metakunt28 41604 | Construction of one soluti... |
| metakunt29 41605 | Construction of one soluti... |
| metakunt30 41606 | Construction of one soluti... |
| metakunt31 41607 | Construction of one soluti... |
| metakunt32 41608 | Construction of one soluti... |
| metakunt33 41609 | Construction of one soluti... |
| metakunt34 41610 | ` D ` is a permutation. (... |
| fac2xp3 41611 | Factorial of 2x+3, sublemm... |
| prodsplit 41612 | Product split into two fac... |
| 2xp3dxp2ge1d 41613 | 2x+3 is greater than or eq... |
| factwoffsmonot 41614 | A factorial with offset is... |
| ioin9i8 41615 | Miscellaneous inference cr... |
| jaodd 41616 | Double deduction form of ~... |
| syl3an12 41617 | A double syllogism inferen... |
| sbtd 41618 | A true statement is true u... |
| sbor2 41619 | One direction of ~ sbor , ... |
| 19.9dev 41620 | ~ 19.9d in the case of an ... |
| 3rspcedvdw 41621 | Triple application of ~ rs... |
| 3rspcedvd 41622 | Triple application of ~ rs... |
| rabdif 41623 | Move difference in and out... |
| sn-axrep5v 41624 | A condensed form of ~ axre... |
| sn-axprlem3 41625 | ~ axprlem3 using only Tars... |
| sn-exelALT 41626 | Alternate proof of ~ exel ... |
| ss2ab1 41627 | Class abstractions in a su... |
| ssabdv 41628 | Deduction of abstraction s... |
| sn-iotalem 41629 | An unused lemma showing th... |
| sn-iotalemcor 41630 | Corollary of ~ sn-iotalem ... |
| abbi1sn 41631 | Originally part of ~ uniab... |
| brif2 41632 | Move a relation inside and... |
| brif12 41633 | Move a relation inside and... |
| pssexg 41634 | The proper subset of a set... |
| pssn0 41635 | A proper superset is nonem... |
| psspwb 41636 | Classes are proper subclas... |
| xppss12 41637 | Proper subset theorem for ... |
| coexd 41638 | The composition of two set... |
| elpwbi 41639 | Membership in a power set,... |
| imaopab 41640 | The image of a class of or... |
| fnsnbt 41641 | A function's domain is a s... |
| fnimasnd 41642 | The image of a function by... |
| eqresfnbd 41643 | Property of being the rest... |
| f1o2d2 41644 | Sufficient condition for a... |
| fmpocos 41645 | Composition of two functio... |
| ovmpogad 41646 | Value of an operation give... |
| ofun 41647 | A function operation of un... |
| dfqs2 41648 | Alternate definition of qu... |
| dfqs3 41649 | Alternate definition of qu... |
| qseq12d 41650 | Equality theorem for quoti... |
| qsalrel 41651 | The quotient set is equal ... |
| elmapssresd 41652 | A restricted mapping is a ... |
| mapcod 41653 | Compose two mappings. (Co... |
| fzosumm1 41654 | Separate out the last term... |
| ccatcan2d 41655 | Cancellation law for conca... |
| nelsubginvcld 41656 | The inverse of a non-subgr... |
| nelsubgcld 41657 | A non-subgroup-member plus... |
| nelsubgsubcld 41658 | A non-subgroup-member minu... |
| rnasclg 41659 | The set of injected scalar... |
| frlmfielbas 41660 | The vectors of a finite fr... |
| frlmfzwrd 41661 | A vector of a module with ... |
| frlmfzowrd 41662 | A vector of a module with ... |
| frlmfzolen 41663 | The dimension of a vector ... |
| frlmfzowrdb 41664 | The vectors of a module wi... |
| frlmfzoccat 41665 | The concatenation of two v... |
| frlmvscadiccat 41666 | Scalar multiplication dist... |
| grpasscan2d 41667 | An associative cancellatio... |
| grpcominv1 41668 | If two elements commute, t... |
| grpcominv2 41669 | If two elements commute, t... |
| finsubmsubg 41670 | A submonoid of a finite gr... |
| crngcomd 41671 | Multiplication is commutat... |
| crng12d 41672 | Commutative/associative la... |
| imacrhmcl 41673 | The image of a commutative... |
| rimrcl1 41674 | Reverse closure of a ring ... |
| rimrcl2 41675 | Reverse closure of a ring ... |
| rimcnv 41676 | The converse of a ring iso... |
| rimco 41677 | The composition of ring is... |
| ricsym 41678 | Ring isomorphism is symmet... |
| rictr 41679 | Ring isomorphism is transi... |
| riccrng1 41680 | Ring isomorphism preserves... |
| riccrng 41681 | A ring is commutative if a... |
| drnginvrn0d 41682 | A multiplicative inverse i... |
| drngmulcanad 41683 | Cancellation of a nonzero ... |
| drngmulcan2ad 41684 | Cancellation of a nonzero ... |
| drnginvmuld 41685 | Inverse of a nonzero produ... |
| ricdrng1 41686 | A ring isomorphism maps a ... |
| ricdrng 41687 | A ring is a division ring ... |
| ricfld 41688 | A ring is a field if and o... |
| lvecgrp 41689 | A vector space is a group.... |
| lvecring 41690 | The scalar component of a ... |
| frlm0vald 41691 | All coordinates of the zer... |
| frlmsnic 41692 | Given a free module with a... |
| uvccl 41693 | A unit vector is a vector.... |
| uvcn0 41694 | A unit vector is nonzero. ... |
| pwselbasr 41695 | The reverse direction of ~... |
| pwsgprod 41696 | Finite products in a power... |
| psrmnd 41697 | The ring of power series i... |
| psrbagres 41698 | Restrict a bag of variable... |
| mpllmodd 41699 | The polynomial ring is a l... |
| mplringd 41700 | The polynomial ring is a r... |
| mplcrngd 41701 | The polynomial ring is a c... |
| mplsubrgcl 41702 | An element of a polynomial... |
| mhmcompl 41703 | The composition of a monoi... |
| rhmmpllem1 41704 | Lemma for ~ rhmmpl . A su... |
| rhmmpllem2 41705 | Lemma for ~ rhmmpl . A su... |
| mhmcoaddmpl 41706 | Show that the ring homomor... |
| rhmcomulmpl 41707 | Show that the ring homomor... |
| rhmmpl 41708 | Provide a ring homomorphis... |
| mplascl0 41709 | The zero scalar as a polyn... |
| mplascl1 41710 | The one scalar as a polyno... |
| mplmapghm 41711 | The function ` H ` mapping... |
| evl0 41712 | The zero polynomial evalua... |
| evlscl 41713 | A polynomial over the ring... |
| evlsval3 41714 | Give a formula for the pol... |
| evlsvval 41715 | Give a formula for the eva... |
| evlsvvvallem 41716 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 41717 | Lemma for theorems using ~... |
| evlsvvval 41718 | Give a formula for the eva... |
| evlsscaval 41719 | Polynomial evaluation buil... |
| evlsvarval 41720 | Polynomial evaluation buil... |
| evlsbagval 41721 | Polynomial evaluation buil... |
| evlsexpval 41722 | Polynomial evaluation buil... |
| evlsaddval 41723 | Polynomial evaluation buil... |
| evlsmulval 41724 | Polynomial evaluation buil... |
| evlsmaprhm 41725 | The function ` F ` mapping... |
| evlsevl 41726 | Evaluation in a subring is... |
| evlcl 41727 | A polynomial over the ring... |
| evlvvval 41728 | Give a formula for the eva... |
| evlvvvallem 41729 | Lemma for theorems using ~... |
| evladdval 41730 | Polynomial evaluation buil... |
| evlmulval 41731 | Polynomial evaluation buil... |
| selvcllem1 41732 | ` T ` is an associative al... |
| selvcllem2 41733 | ` D ` is a ring homomorphi... |
| selvcllem3 41734 | The third argument passed ... |
| selvcllemh 41735 | Apply the third argument (... |
| selvcllem4 41736 | The fourth argument passed... |
| selvcllem5 41737 | The fifth argument passed ... |
| selvcl 41738 | Closure of the "variable s... |
| selvval2 41739 | Value of the "variable sel... |
| selvvvval 41740 | Recover the original polyn... |
| evlselvlem 41741 | Lemma for ~ evlselv . Use... |
| evlselv 41742 | Evaluating a selection of ... |
| selvadd 41743 | The "variable selection" f... |
| selvmul 41744 | The "variable selection" f... |
| fsuppind 41745 | Induction on functions ` F... |
| fsuppssindlem1 41746 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 41747 | Lemma for ~ fsuppssind . ... |
| fsuppssind 41748 | Induction on functions ` F... |
| mhpind 41749 | The homogeneous polynomial... |
| evlsmhpvvval 41750 | Give a formula for the eva... |
| mhphflem 41751 | Lemma for ~ mhphf . Add s... |
| mhphf 41752 | A homogeneous polynomial d... |
| mhphf2 41753 | A homogeneous polynomial d... |
| mhphf3 41754 | A homogeneous polynomial d... |
| mhphf4 41755 | A homogeneous polynomial d... |
| c0exALT 41756 | Alternate proof of ~ c0ex ... |
| 0cnALT3 41757 | Alternate proof of ~ 0cn u... |
| elre0re 41758 | Specialized version of ~ 0... |
| 1t1e1ALT 41759 | Alternate proof of ~ 1t1e1... |
| remulcan2d 41760 | ~ mulcan2d for real number... |
| readdridaddlidd 41761 | Given some real number ` B... |
| sn-1ne2 41762 | A proof of ~ 1ne2 without ... |
| nnn1suc 41763 | A positive integer that is... |
| nnadd1com 41764 | Addition with 1 is commuta... |
| nnaddcom 41765 | Addition is commutative fo... |
| nnaddcomli 41766 | Version of ~ addcomli for ... |
| nnadddir 41767 | Right-distributivity for n... |
| nnmul1com 41768 | Multiplication with 1 is c... |
| nnmulcom 41769 | Multiplication is commutat... |
| readdrcl2d 41770 | Reverse closure for additi... |
| mvrrsubd 41771 | Move a subtraction in the ... |
| laddrotrd 41772 | Rotate the variables right... |
| raddcom12d 41773 | Swap the first two variabl... |
| lsubrotld 41774 | Rotate the variables left ... |
| lsubcom23d 41775 | Swap the second and third ... |
| addsubeq4com 41776 | Relation between sums and ... |
| sqsumi 41777 | A sum squared. (Contribut... |
| negn0nposznnd 41778 | Lemma for ~ dffltz . (Con... |
| sqmid3api 41779 | Value of the square of the... |
| decaddcom 41780 | Commute ones place in addi... |
| sqn5i 41781 | The square of a number end... |
| sqn5ii 41782 | The square of a number end... |
| decpmulnc 41783 | Partial products algorithm... |
| decpmul 41784 | Partial products algorithm... |
| sqdeccom12 41785 | The square of a number in ... |
| sq3deccom12 41786 | Variant of ~ sqdeccom12 wi... |
| 4t5e20 41787 | 4 times 5 equals 20. (Con... |
| sq9 41788 | The square of 9 is 81. (C... |
| 235t711 41789 | Calculate a product by lon... |
| ex-decpmul 41790 | Example usage of ~ decpmul... |
| fz1sumconst 41791 | The sum of ` N ` constant ... |
| fz1sump1 41792 | Add one more term to a sum... |
| oddnumth 41793 | The Odd Number Theorem. T... |
| nicomachus 41794 | Nicomachus's Theorem. The... |
| sumcubes 41795 | The sum of the first ` N `... |
| pine0 41796 | ` _pi ` is nonzero. (Cont... |
| ine1 41797 | ` _i ` is not 1. (Contrib... |
| 0tie0 41798 | 0 times ` _i ` equals 0. ... |
| it1ei 41799 | ` _i ` times 1 equals ` _i... |
| 1tiei 41800 | 1 times ` _i ` equals ` _i... |
| itrere 41801 | ` _i ` times a real is rea... |
| retire 41802 | A real times ` _i ` is rea... |
| oexpreposd 41803 | Lemma for ~ dffltz . TODO... |
| ltexp1d 41804 | ~ ltmul1d for exponentiati... |
| ltexp1dd 41805 | Raising both sides of 'les... |
| exp11nnd 41806 | ~ sq11d for positive real ... |
| exp11d 41807 | ~ exp11nnd for nonzero int... |
| 0dvds0 41808 | 0 divides 0. (Contributed... |
| absdvdsabsb 41809 | Divisibility is invariant ... |
| dvdsexpim 41810 | ~ dvdssqim generalized to ... |
| gcdnn0id 41811 | The ` gcd ` of a nonnegati... |
| gcdle1d 41812 | The greatest common diviso... |
| gcdle2d 41813 | The greatest common diviso... |
| dvdsexpad 41814 | Deduction associated with ... |
| nn0rppwr 41815 | If ` A ` and ` B ` are rel... |
| expgcd 41816 | Exponentiation distributes... |
| nn0expgcd 41817 | Exponentiation distributes... |
| zexpgcd 41818 | Exponentiation distributes... |
| numdenexp 41819 | ~ numdensq extended to non... |
| numexp 41820 | ~ numsq extended to nonneg... |
| denexp 41821 | ~ densq extended to nonneg... |
| dvdsexpnn 41822 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 41823 | ~ dvdsexpnn generalized to... |
| dvdsexpb 41824 | ~ dvdssq generalized to po... |
| posqsqznn 41825 | When a positive rational s... |
| zrtelqelz 41826 | ~ zsqrtelqelz generalized ... |
| zrtdvds 41827 | A positive integer root di... |
| rtprmirr 41828 | The root of a prime number... |
| zdivgd 41829 | Two ways to express " ` N ... |
| efne0d 41830 | The exponential of a compl... |
| efsubd 41831 | Difference of exponents la... |
| ef11d 41832 | General condition for the ... |
| logccne0d 41833 | The logarithm isn't 0 if i... |
| cxp112d 41834 | General condition for comp... |
| cxp111d 41835 | General condition for comp... |
| cxpi11d 41836 | ` _i ` to the powers of ` ... |
| logne0d 41837 | Deduction form of ~ logne0... |
| rxp112d 41838 | Real exponentiation is one... |
| log11d 41839 | The natural logarithm is o... |
| rplog11d 41840 | The natural logarithm is o... |
| rxp11d 41841 | Real exponentiation is one... |
| resubval 41844 | Value of real subtraction,... |
| renegeulemv 41845 | Lemma for ~ renegeu and si... |
| renegeulem 41846 | Lemma for ~ renegeu and si... |
| renegeu 41847 | Existential uniqueness of ... |
| rernegcl 41848 | Closure law for negative r... |
| renegadd 41849 | Relationship between real ... |
| renegid 41850 | Addition of a real number ... |
| reneg0addlid 41851 | Negative zero is a left ad... |
| resubeulem1 41852 | Lemma for ~ resubeu . A v... |
| resubeulem2 41853 | Lemma for ~ resubeu . A v... |
| resubeu 41854 | Existential uniqueness of ... |
| rersubcl 41855 | Closure for real subtracti... |
| resubadd 41856 | Relation between real subt... |
| resubaddd 41857 | Relationship between subtr... |
| resubf 41858 | Real subtraction is an ope... |
| repncan2 41859 | Addition and subtraction o... |
| repncan3 41860 | Addition and subtraction o... |
| readdsub 41861 | Law for addition and subtr... |
| reladdrsub 41862 | Move LHS of a sum into RHS... |
| reltsub1 41863 | Subtraction from both side... |
| reltsubadd2 41864 | 'Less than' relationship b... |
| resubcan2 41865 | Cancellation law for real ... |
| resubsub4 41866 | Law for double subtraction... |
| rennncan2 41867 | Cancellation law for real ... |
| renpncan3 41868 | Cancellation law for real ... |
| repnpcan 41869 | Cancellation law for addit... |
| reppncan 41870 | Cancellation law for mixed... |
| resubidaddlidlem 41871 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 41872 | Any real number subtracted... |
| resubdi 41873 | Distribution of multiplica... |
| re1m1e0m0 41874 | Equality of two left-addit... |
| sn-00idlem1 41875 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 41876 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 41877 | Lemma for ~ sn-00id . (Co... |
| sn-00id 41878 | ~ 00id proven without ~ ax... |
| re0m0e0 41879 | Real number version of ~ 0... |
| readdlid 41880 | Real number version of ~ a... |
| sn-addlid 41881 | ~ addlid without ~ ax-mulc... |
| remul02 41882 | Real number version of ~ m... |
| sn-0ne2 41883 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 41884 | Real number version of ~ m... |
| resubid 41885 | Subtraction of a real numb... |
| readdrid 41886 | Real number version of ~ a... |
| resubid1 41887 | Real number version of ~ s... |
| renegneg 41888 | A real number is equal to ... |
| readdcan2 41889 | Commuted version of ~ read... |
| renegid2 41890 | Commuted version of ~ rene... |
| remulneg2d 41891 | Product with negative is n... |
| sn-it0e0 41892 | Proof of ~ it0e0 without ~... |
| sn-negex12 41893 | A combination of ~ cnegex ... |
| sn-negex 41894 | Proof of ~ cnegex without ... |
| sn-negex2 41895 | Proof of ~ cnegex2 without... |
| sn-addcand 41896 | ~ addcand without ~ ax-mul... |
| sn-addrid 41897 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 41898 | ~ addcan2d without ~ ax-mu... |
| reixi 41899 | ~ ixi without ~ ax-mulcom ... |
| rei4 41900 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 41901 | A number that sums to itse... |
| sn-mul01 41902 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 41903 | ~ negeu without ~ ax-mulco... |
| sn-subcl 41904 | ~ subcl without ~ ax-mulco... |
| sn-subf 41905 | ~ subf without ~ ax-mulcom... |
| resubeqsub 41906 | Equivalence between real s... |
| subresre 41907 | Subtraction restricted to ... |
| addinvcom 41908 | A number commutes with its... |
| remulinvcom 41909 | A left multiplicative inve... |
| remullid 41910 | Commuted version of ~ ax-1... |
| sn-1ticom 41911 | Lemma for ~ sn-mullid and ... |
| sn-mullid 41912 | ~ mullid without ~ ax-mulc... |
| sn-it1ei 41913 | ~ it1ei without ~ ax-mulco... |
| ipiiie0 41914 | The multiplicative inverse... |
| remulcand 41915 | Commuted version of ~ remu... |
| sn-0tie0 41916 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 41917 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 41918 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 41919 | ~ ltaddneg without ~ ax-mu... |
| reposdif 41920 | Comparison of two numbers ... |
| relt0neg1 41921 | Comparison of a real and i... |
| relt0neg2 41922 | Comparison of a real and i... |
| sn-addlt0d 41923 | The sum of negative number... |
| sn-addgt0d 41924 | The sum of positive number... |
| sn-nnne0 41925 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 41926 | ~ elznn0nn restated using ... |
| nn0addcom 41927 | Addition is commutative fo... |
| zaddcomlem 41928 | Lemma for ~ zaddcom . (Co... |
| zaddcom 41929 | Addition is commutative fo... |
| renegmulnnass 41930 | Move multiplication by a n... |
| nn0mulcom 41931 | Multiplication is commutat... |
| zmulcomlem 41932 | Lemma for ~ zmulcom . (Co... |
| zmulcom 41933 | Multiplication is commutat... |
| mulgt0con1dlem 41934 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 41935 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 41936 | Lemma for ~ mulgt0b2d and ... |
| mulgt0b2d 41937 | Biconditional, deductive f... |
| sn-ltmul2d 41938 | ~ ltmul2d without ~ ax-mul... |
| sn-0lt1 41939 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 41940 | ~ ltp1 without ~ ax-mulcom... |
| reneg1lt0 41941 | Lemma for ~ sn-inelr . (C... |
| sn-inelr 41942 | ~ inelr without ~ ax-mulco... |
| sn-itrere 41943 | ` _i ` times a real is rea... |
| sn-retire 41944 | Commuted version of ~ sn-i... |
| cnreeu 41945 | The reals in the expressio... |
| sn-sup2 41946 | ~ sup2 with exactly the sa... |
| prjspval 41949 | Value of the projective sp... |
| prjsprel 41950 | Utility theorem regarding ... |
| prjspertr 41951 | The relation in ` PrjSp ` ... |
| prjsperref 41952 | The relation in ` PrjSp ` ... |
| prjspersym 41953 | The relation in ` PrjSp ` ... |
| prjsper 41954 | The relation used to defin... |
| prjspreln0 41955 | Two nonzero vectors are eq... |
| prjspvs 41956 | A nonzero multiple of a ve... |
| prjsprellsp 41957 | Two vectors are equivalent... |
| prjspeclsp 41958 | The vectors equivalent to ... |
| prjspval2 41959 | Alternate definition of pr... |
| prjspnval 41962 | Value of the n-dimensional... |
| prjspnerlem 41963 | A lemma showing that the e... |
| prjspnval2 41964 | Value of the n-dimensional... |
| prjspner 41965 | The relation used to defin... |
| prjspnvs 41966 | A nonzero multiple of a ve... |
| prjspnssbas 41967 | A projective point spans a... |
| prjspnn0 41968 | A projective point is none... |
| 0prjspnlem 41969 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 41970 | Any vector is equivalent t... |
| prjspner01 41971 | Any vector is equivalent t... |
| prjspner1 41972 | Two vectors whose zeroth c... |
| 0prjspnrel 41973 | In the zero-dimensional pr... |
| 0prjspn 41974 | A zero-dimensional project... |
| prjcrvfval 41977 | Value of the projective cu... |
| prjcrvval 41978 | Value of the projective cu... |
| prjcrv0 41979 | The "curve" (zero set) cor... |
| dffltz 41980 | Fermat's Last Theorem (FLT... |
| fltmul 41981 | A counterexample to FLT st... |
| fltdiv 41982 | A counterexample to FLT st... |
| flt0 41983 | A counterexample for FLT d... |
| fltdvdsabdvdsc 41984 | Any factor of both ` A ` a... |
| fltabcoprmex 41985 | A counterexample to FLT im... |
| fltaccoprm 41986 | A counterexample to FLT wi... |
| fltbccoprm 41987 | A counterexample to FLT wi... |
| fltabcoprm 41988 | A counterexample to FLT wi... |
| infdesc 41989 | Infinite descent. The hyp... |
| fltne 41990 | If a counterexample to FLT... |
| flt4lem 41991 | Raising a number to the fo... |
| flt4lem1 41992 | Satisfy the antecedent use... |
| flt4lem2 41993 | If ` A ` is even, ` B ` is... |
| flt4lem3 41994 | Equivalent to ~ pythagtrip... |
| flt4lem4 41995 | If the product of two copr... |
| flt4lem5 41996 | In the context of the lemm... |
| flt4lem5elem 41997 | Version of ~ fltaccoprm an... |
| flt4lem5a 41998 | Part 1 of Equation 1 of ... |
| flt4lem5b 41999 | Part 2 of Equation 1 of ... |
| flt4lem5c 42000 | Part 2 of Equation 2 of ... |
| flt4lem5d 42001 | Part 3 of Equation 2 of ... |
| flt4lem5e 42002 | Satisfy the hypotheses of ... |
| flt4lem5f 42003 | Final equation of ~... |
| flt4lem6 42004 | Remove shared factors in a... |
| flt4lem7 42005 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 42006 | Strengthening of Fermat's ... |
| fltltc 42007 | ` ( C ^ N ) ` is the large... |
| fltnltalem 42008 | Lemma for ~ fltnlta . A l... |
| fltnlta 42009 | In a Fermat counterexample... |
| iddii 42010 | Version of ~ a1ii with the... |
| bicomdALT 42011 | Alternate proof of ~ bicom... |
| elabgw 42012 | Membership in a class abst... |
| elab2gw 42013 | Membership in a class abst... |
| elrab2w 42014 | Membership in a restricted... |
| ruvALT 42015 | Alternate proof of ~ ruv w... |
| sn-wcdeq 42016 | Alternative to ~ wcdeq and... |
| sq45 42017 | 45 squared is 2025. (Cont... |
| sum9cubes 42018 | The sum of the first nine ... |
| acos1half 42019 | The arccosine of ` 1 / 2 `... |
| aprilfools2025 42020 | An abuse of notation. (Co... |
| binom2d 42021 | Deduction form of binom2. ... |
| cu3addd 42022 | Cube of sum of three numbe... |
| sqnegd 42023 | The square of the negative... |
| negexpidd 42024 | The sum of a real number t... |
| rexlimdv3d 42025 | An extended version of ~ r... |
| 3cubeslem1 42026 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 42027 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 42028 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 42029 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 42030 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 42031 | Lemma for ~ 3cubes . This... |
| 3cubes 42032 | Every rational number is a... |
| rntrclfvOAI 42033 | The range of the transitiv... |
| moxfr 42034 | Transfer at-most-one betwe... |
| imaiinfv 42035 | Indexed intersection of an... |
| elrfi 42036 | Elementhood in a set of re... |
| elrfirn 42037 | Elementhood in a set of re... |
| elrfirn2 42038 | Elementhood in a set of re... |
| cmpfiiin 42039 | In a compact topology, a s... |
| ismrcd1 42040 | Any function from the subs... |
| ismrcd2 42041 | Second half of ~ ismrcd1 .... |
| istopclsd 42042 | A closure function which s... |
| ismrc 42043 | A function is a Moore clos... |
| isnacs 42046 | Expand definition of Noeth... |
| nacsfg 42047 | In a Noetherian-type closu... |
| isnacs2 42048 | Express Noetherian-type cl... |
| mrefg2 42049 | Slight variation on finite... |
| mrefg3 42050 | Slight variation on finite... |
| nacsacs 42051 | A closure system of Noethe... |
| isnacs3 42052 | A choice-free order equiva... |
| incssnn0 42053 | Transitivity induction of ... |
| nacsfix 42054 | An increasing sequence of ... |
| constmap 42055 | A constant (represented wi... |
| mapco2g 42056 | Renaming indices in a tupl... |
| mapco2 42057 | Post-composition (renaming... |
| mapfzcons 42058 | Extending a one-based mapp... |
| mapfzcons1 42059 | Recover prefix mapping fro... |
| mapfzcons1cl 42060 | A nonempty mapping has a p... |
| mapfzcons2 42061 | Recover added element from... |
| mptfcl 42062 | Interpret range of a maps-... |
| mzpclval 42067 | Substitution lemma for ` m... |
| elmzpcl 42068 | Double substitution lemma ... |
| mzpclall 42069 | The set of all functions w... |
| mzpcln0 42070 | Corollary of ~ mzpclall : ... |
| mzpcl1 42071 | Defining property 1 of a p... |
| mzpcl2 42072 | Defining property 2 of a p... |
| mzpcl34 42073 | Defining properties 3 and ... |
| mzpval 42074 | Value of the ` mzPoly ` fu... |
| dmmzp 42075 | ` mzPoly ` is defined for ... |
| mzpincl 42076 | Polynomial closedness is a... |
| mzpconst 42077 | Constant functions are pol... |
| mzpf 42078 | A polynomial function is a... |
| mzpproj 42079 | A projection function is p... |
| mzpadd 42080 | The pointwise sum of two p... |
| mzpmul 42081 | The pointwise product of t... |
| mzpconstmpt 42082 | A constant function expres... |
| mzpaddmpt 42083 | Sum of polynomial function... |
| mzpmulmpt 42084 | Product of polynomial func... |
| mzpsubmpt 42085 | The difference of two poly... |
| mzpnegmpt 42086 | Negation of a polynomial f... |
| mzpexpmpt 42087 | Raise a polynomial functio... |
| mzpindd 42088 | "Structural" induction to ... |
| mzpmfp 42089 | Relationship between multi... |
| mzpsubst 42090 | Substituting polynomials f... |
| mzprename 42091 | Simplified version of ~ mz... |
| mzpresrename 42092 | A polynomial is a polynomi... |
| mzpcompact2lem 42093 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 42094 | Polynomials are finitary o... |
| coeq0i 42095 | ~ coeq0 but without explic... |
| fzsplit1nn0 42096 | Split a finite 1-based set... |
| eldiophb 42099 | Initial expression of Diop... |
| eldioph 42100 | Condition for a set to be ... |
| diophrw 42101 | Renaming and adding unused... |
| eldioph2lem1 42102 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 42103 | Lemma for ~ eldioph2 . Co... |
| eldioph2 42104 | Construct a Diophantine se... |
| eldioph2b 42105 | While Diophantine sets wer... |
| eldiophelnn0 42106 | Remove antecedent on ` B `... |
| eldioph3b 42107 | Define Diophantine sets in... |
| eldioph3 42108 | Inference version of ~ eld... |
| ellz1 42109 | Membership in a lower set ... |
| lzunuz 42110 | The union of a lower set o... |
| fz1eqin 42111 | Express a one-based finite... |
| lzenom 42112 | Lower integers are countab... |
| elmapresaunres2 42113 | ~ fresaunres2 transposed t... |
| diophin 42114 | If two sets are Diophantin... |
| diophun 42115 | If two sets are Diophantin... |
| eldiophss 42116 | Diophantine sets are sets ... |
| diophrex 42117 | Projecting a Diophantine s... |
| eq0rabdioph 42118 | This is the first of a num... |
| eqrabdioph 42119 | Diophantine set builder fo... |
| 0dioph 42120 | The null set is Diophantin... |
| vdioph 42121 | The "universal" set (as la... |
| anrabdioph 42122 | Diophantine set builder fo... |
| orrabdioph 42123 | Diophantine set builder fo... |
| 3anrabdioph 42124 | Diophantine set builder fo... |
| 3orrabdioph 42125 | Diophantine set builder fo... |
| 2sbcrex 42126 | Exchange an existential qu... |
| sbcrexgOLD 42127 | Interchange class substitu... |
| 2sbcrexOLD 42128 | Exchange an existential qu... |
| sbc2rex 42129 | Exchange a substitution wi... |
| sbc2rexgOLD 42130 | Exchange a substitution wi... |
| sbc4rex 42131 | Exchange a substitution wi... |
| sbc4rexgOLD 42132 | Exchange a substitution wi... |
| sbcrot3 42133 | Rotate a sequence of three... |
| sbcrot5 42134 | Rotate a sequence of five ... |
| sbccomieg 42135 | Commute two explicit subst... |
| rexrabdioph 42136 | Diophantine set builder fo... |
| rexfrabdioph 42137 | Diophantine set builder fo... |
| 2rexfrabdioph 42138 | Diophantine set builder fo... |
| 3rexfrabdioph 42139 | Diophantine set builder fo... |
| 4rexfrabdioph 42140 | Diophantine set builder fo... |
| 6rexfrabdioph 42141 | Diophantine set builder fo... |
| 7rexfrabdioph 42142 | Diophantine set builder fo... |
| rabdiophlem1 42143 | Lemma for arithmetic dioph... |
| rabdiophlem2 42144 | Lemma for arithmetic dioph... |
| elnn0rabdioph 42145 | Diophantine set builder fo... |
| rexzrexnn0 42146 | Rewrite an existential qua... |
| lerabdioph 42147 | Diophantine set builder fo... |
| eluzrabdioph 42148 | Diophantine set builder fo... |
| elnnrabdioph 42149 | Diophantine set builder fo... |
| ltrabdioph 42150 | Diophantine set builder fo... |
| nerabdioph 42151 | Diophantine set builder fo... |
| dvdsrabdioph 42152 | Divisibility is a Diophant... |
| eldioph4b 42153 | Membership in ` Dioph ` ex... |
| eldioph4i 42154 | Forward-only version of ~ ... |
| diophren 42155 | Change variables in a Diop... |
| rabrenfdioph 42156 | Change variable numbers in... |
| rabren3dioph 42157 | Change variable numbers in... |
| fphpd 42158 | Pigeonhole principle expre... |
| fphpdo 42159 | Pigeonhole principle for s... |
| ctbnfien 42160 | An infinite subset of a co... |
| fiphp3d 42161 | Infinite pigeonhole princi... |
| rencldnfilem 42162 | Lemma for ~ rencldnfi . (... |
| rencldnfi 42163 | A set of real numbers whic... |
| irrapxlem1 42164 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 42165 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 42166 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 42167 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 42168 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 42169 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 42170 | Dirichlet's approximation ... |
| pellexlem1 42171 | Lemma for ~ pellex . Arit... |
| pellexlem2 42172 | Lemma for ~ pellex . Arit... |
| pellexlem3 42173 | Lemma for ~ pellex . To e... |
| pellexlem4 42174 | Lemma for ~ pellex . Invo... |
| pellexlem5 42175 | Lemma for ~ pellex . Invo... |
| pellexlem6 42176 | Lemma for ~ pellex . Doin... |
| pellex 42177 | Every Pell equation has a ... |
| pell1qrval 42188 | Value of the set of first-... |
| elpell1qr 42189 | Membership in a first-quad... |
| pell14qrval 42190 | Value of the set of positi... |
| elpell14qr 42191 | Membership in the set of p... |
| pell1234qrval 42192 | Value of the set of genera... |
| elpell1234qr 42193 | Membership in the set of g... |
| pell1234qrre 42194 | General Pell solutions are... |
| pell1234qrne0 42195 | No solution to a Pell equa... |
| pell1234qrreccl 42196 | General solutions of the P... |
| pell1234qrmulcl 42197 | General solutions of the P... |
| pell14qrss1234 42198 | A positive Pell solution i... |
| pell14qrre 42199 | A positive Pell solution i... |
| pell14qrne0 42200 | A positive Pell solution i... |
| pell14qrgt0 42201 | A positive Pell solution i... |
| pell14qrrp 42202 | A positive Pell solution i... |
| pell1234qrdich 42203 | A general Pell solution is... |
| elpell14qr2 42204 | A number is a positive Pel... |
| pell14qrmulcl 42205 | Positive Pell solutions ar... |
| pell14qrreccl 42206 | Positive Pell solutions ar... |
| pell14qrdivcl 42207 | Positive Pell solutions ar... |
| pell14qrexpclnn0 42208 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 42209 | Positive Pell solutions ar... |
| pell1qrss14 42210 | First-quadrant Pell soluti... |
| pell14qrdich 42211 | A positive Pell solution i... |
| pell1qrge1 42212 | A Pell solution in the fir... |
| pell1qr1 42213 | 1 is a Pell solution and i... |
| elpell1qr2 42214 | The first quadrant solutio... |
| pell1qrgaplem 42215 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 42216 | First-quadrant Pell soluti... |
| pell14qrgap 42217 | Positive Pell solutions ar... |
| pell14qrgapw 42218 | Positive Pell solutions ar... |
| pellqrexplicit 42219 | Condition for a calculated... |
| infmrgelbi 42220 | Any lower bound of a nonem... |
| pellqrex 42221 | There is a nontrivial solu... |
| pellfundval 42222 | Value of the fundamental s... |
| pellfundre 42223 | The fundamental solution o... |
| pellfundge 42224 | Lower bound on the fundame... |
| pellfundgt1 42225 | Weak lower bound on the Pe... |
| pellfundlb 42226 | A nontrivial first quadran... |
| pellfundglb 42227 | If a real is larger than t... |
| pellfundex 42228 | The fundamental solution a... |
| pellfund14gap 42229 | There are no solutions bet... |
| pellfundrp 42230 | The fundamental Pell solut... |
| pellfundne1 42231 | The fundamental Pell solut... |
| reglogcl 42232 | General logarithm is a rea... |
| reglogltb 42233 | General logarithm preserve... |
| reglogleb 42234 | General logarithm preserve... |
| reglogmul 42235 | Multiplication law for gen... |
| reglogexp 42236 | Power law for general log.... |
| reglogbas 42237 | General log of the base is... |
| reglog1 42238 | General log of 1 is 0. (C... |
| reglogexpbas 42239 | General log of a power of ... |
| pellfund14 42240 | Every positive Pell soluti... |
| pellfund14b 42241 | The positive Pell solution... |
| rmxfval 42246 | Value of the X sequence. ... |
| rmyfval 42247 | Value of the Y sequence. ... |
| rmspecsqrtnq 42248 | The discriminant used to d... |
| rmspecnonsq 42249 | The discriminant used to d... |
| qirropth 42250 | This lemma implements the ... |
| rmspecfund 42251 | The base of exponent used ... |
| rmxyelqirr 42252 | The solutions used to cons... |
| rmxyelqirrOLD 42253 | Obsolete version of ~ rmxy... |
| rmxypairf1o 42254 | The function used to extra... |
| rmxyelxp 42255 | Lemma for ~ frmx and ~ frm... |
| frmx 42256 | The X sequence is a nonneg... |
| frmy 42257 | The Y sequence is an integ... |
| rmxyval 42258 | Main definition of the X a... |
| rmspecpos 42259 | The discriminant used to d... |
| rmxycomplete 42260 | The X and Y sequences take... |
| rmxynorm 42261 | The X and Y sequences defi... |
| rmbaserp 42262 | The base of exponentiation... |
| rmxyneg 42263 | Negation law for X and Y s... |
| rmxyadd 42264 | Addition formula for X and... |
| rmxy1 42265 | Value of the X and Y seque... |
| rmxy0 42266 | Value of the X and Y seque... |
| rmxneg 42267 | Negation law (even functio... |
| rmx0 42268 | Value of X sequence at 0. ... |
| rmx1 42269 | Value of X sequence at 1. ... |
| rmxadd 42270 | Addition formula for X seq... |
| rmyneg 42271 | Negation formula for Y seq... |
| rmy0 42272 | Value of Y sequence at 0. ... |
| rmy1 42273 | Value of Y sequence at 1. ... |
| rmyadd 42274 | Addition formula for Y seq... |
| rmxp1 42275 | Special addition-of-1 form... |
| rmyp1 42276 | Special addition of 1 form... |
| rmxm1 42277 | Subtraction of 1 formula f... |
| rmym1 42278 | Subtraction of 1 formula f... |
| rmxluc 42279 | The X sequence is a Lucas ... |
| rmyluc 42280 | The Y sequence is a Lucas ... |
| rmyluc2 42281 | Lucas sequence property of... |
| rmxdbl 42282 | "Double-angle formula" for... |
| rmydbl 42283 | "Double-angle formula" for... |
| monotuz 42284 | A function defined on an u... |
| monotoddzzfi 42285 | A function which is odd an... |
| monotoddzz 42286 | A function (given implicit... |
| oddcomabszz 42287 | An odd function which take... |
| 2nn0ind 42288 | Induction on nonnegative i... |
| zindbi 42289 | Inductively transfer a pro... |
| rmxypos 42290 | For all nonnegative indice... |
| ltrmynn0 42291 | The Y-sequence is strictly... |
| ltrmxnn0 42292 | The X-sequence is strictly... |
| lermxnn0 42293 | The X-sequence is monotoni... |
| rmxnn 42294 | The X-sequence is defined ... |
| ltrmy 42295 | The Y-sequence is strictly... |
| rmyeq0 42296 | Y is zero only at zero. (... |
| rmyeq 42297 | Y is one-to-one. (Contrib... |
| lermy 42298 | Y is monotonic (non-strict... |
| rmynn 42299 | ` rmY ` is positive for po... |
| rmynn0 42300 | ` rmY ` is nonnegative for... |
| rmyabs 42301 | ` rmY ` commutes with ` ab... |
| jm2.24nn 42302 | X(n) is strictly greater t... |
| jm2.17a 42303 | First half of lemma 2.17 o... |
| jm2.17b 42304 | Weak form of the second ha... |
| jm2.17c 42305 | Second half of lemma 2.17 ... |
| jm2.24 42306 | Lemma 2.24 of [JonesMatija... |
| rmygeid 42307 | Y(n) increases faster than... |
| congtr 42308 | A wff of the form ` A || (... |
| congadd 42309 | If two pairs of numbers ar... |
| congmul 42310 | If two pairs of numbers ar... |
| congsym 42311 | Congruence mod ` A ` is a ... |
| congneg 42312 | If two integers are congru... |
| congsub 42313 | If two pairs of numbers ar... |
| congid 42314 | Every integer is congruent... |
| mzpcong 42315 | Polynomials commute with c... |
| congrep 42316 | Every integer is congruent... |
| congabseq 42317 | If two integers are congru... |
| acongid 42318 | A wff like that in this th... |
| acongsym 42319 | Symmetry of alternating co... |
| acongneg2 42320 | Negate right side of alter... |
| acongtr 42321 | Transitivity of alternatin... |
| acongeq12d 42322 | Substitution deduction for... |
| acongrep 42323 | Every integer is alternati... |
| fzmaxdif 42324 | Bound on the difference be... |
| fzneg 42325 | Reflection of a finite ran... |
| acongeq 42326 | Two numbers in the fundame... |
| dvdsacongtr 42327 | Alternating congruence pas... |
| coprmdvdsb 42328 | Multiplication by a coprim... |
| modabsdifz 42329 | Divisibility in terms of m... |
| dvdsabsmod0 42330 | Divisibility in terms of m... |
| jm2.18 42331 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 42332 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 42333 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 42334 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 42335 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 42336 | Lemma 2.19 of [JonesMatija... |
| jm2.21 42337 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 42338 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 42339 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 42340 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 42341 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 42342 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 42343 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 42344 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 42345 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 42346 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 42347 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 42348 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 42349 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 42350 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 42351 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 42352 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 42353 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 42354 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 42355 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 42356 | Lemma for ~ rmydioph . Us... |
| rmydioph 42357 | ~ jm2.27 restated in terms... |
| rmxdiophlem 42358 | X can be expressed in term... |
| rmxdioph 42359 | X is a Diophantine functio... |
| jm3.1lem1 42360 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 42361 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 42362 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 42363 | Diophantine expression for... |
| expdiophlem1 42364 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 42365 | Lemma for ~ expdioph . Ex... |
| expdioph 42366 | The exponential function i... |
| setindtr 42367 | Set induction for sets con... |
| setindtrs 42368 | Set induction scheme witho... |
| dford3lem1 42369 | Lemma for ~ dford3 . (Con... |
| dford3lem2 42370 | Lemma for ~ dford3 . (Con... |
| dford3 42371 | Ordinals are precisely the... |
| dford4 42372 | ~ dford3 expressed in prim... |
| wopprc 42373 | Unrelated: Wiener pairs t... |
| rpnnen3lem 42374 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 42375 | Dedekind cut injection of ... |
| axac10 42376 | Characterization of choice... |
| harinf 42377 | The Hartogs number of an i... |
| wdom2d2 42378 | Deduction for weak dominan... |
| ttac 42379 | Tarski's theorem about cho... |
| pw2f1ocnv 42380 | Define a bijection between... |
| pw2f1o2 42381 | Define a bijection between... |
| pw2f1o2val 42382 | Function value of the ~ pw... |
| pw2f1o2val2 42383 | Membership in a mapped set... |
| soeq12d 42384 | Equality deduction for tot... |
| freq12d 42385 | Equality deduction for fou... |
| weeq12d 42386 | Equality deduction for wel... |
| limsuc2 42387 | Limit ordinals in the sens... |
| wepwsolem 42388 | Transfer an ordering on ch... |
| wepwso 42389 | A well-ordering induces a ... |
| dnnumch1 42390 | Define an enumeration of a... |
| dnnumch2 42391 | Define an enumeration (wea... |
| dnnumch3lem 42392 | Value of the ordinal injec... |
| dnnumch3 42393 | Define an injection from a... |
| dnwech 42394 | Define a well-ordering fro... |
| fnwe2val 42395 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 42396 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 42397 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 42398 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 42399 | A well-ordering can be con... |
| aomclem1 42400 | Lemma for ~ dfac11 . This... |
| aomclem2 42401 | Lemma for ~ dfac11 . Succ... |
| aomclem3 42402 | Lemma for ~ dfac11 . Succ... |
| aomclem4 42403 | Lemma for ~ dfac11 . Limi... |
| aomclem5 42404 | Lemma for ~ dfac11 . Comb... |
| aomclem6 42405 | Lemma for ~ dfac11 . Tran... |
| aomclem7 42406 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 42407 | Lemma for ~ dfac11 . Perf... |
| dfac11 42408 | The right-hand side of thi... |
| kelac1 42409 | Kelley's choice, basic for... |
| kelac2lem 42410 | Lemma for ~ kelac2 and ~ d... |
| kelac2 42411 | Kelley's choice, most comm... |
| dfac21 42412 | Tychonoff's theorem is a c... |
| islmodfg 42415 | Property of a finitely gen... |
| islssfg 42416 | Property of a finitely gen... |
| islssfg2 42417 | Property of a finitely gen... |
| islssfgi 42418 | Finitely spanned subspaces... |
| fglmod 42419 | Finitely generated left mo... |
| lsmfgcl 42420 | The sum of two finitely ge... |
| islnm 42423 | Property of being a Noethe... |
| islnm2 42424 | Property of being a Noethe... |
| lnmlmod 42425 | A Noetherian left module i... |
| lnmlssfg 42426 | A submodule of Noetherian ... |
| lnmlsslnm 42427 | All submodules of a Noethe... |
| lnmfg 42428 | A Noetherian left module i... |
| kercvrlsm 42429 | The domain of a linear fun... |
| lmhmfgima 42430 | A homomorphism maps finite... |
| lnmepi 42431 | Epimorphic images of Noeth... |
| lmhmfgsplit 42432 | If the kernel and range of... |
| lmhmlnmsplit 42433 | If the kernel and range of... |
| lnmlmic 42434 | Noetherian is an invariant... |
| pwssplit4 42435 | Splitting for structure po... |
| filnm 42436 | Finite left modules are No... |
| pwslnmlem0 42437 | Zeroeth powers are Noether... |
| pwslnmlem1 42438 | First powers are Noetheria... |
| pwslnmlem2 42439 | A sum of powers is Noether... |
| pwslnm 42440 | Finite powers of Noetheria... |
| unxpwdom3 42441 | Weaker version of ~ unxpwd... |
| pwfi2f1o 42442 | The ~ pw2f1o bijection rel... |
| pwfi2en 42443 | Finitely supported indicat... |
| frlmpwfi 42444 | Formal linear combinations... |
| gicabl 42445 | Being Abelian is a group i... |
| imasgim 42446 | A relabeling of the elemen... |
| isnumbasgrplem1 42447 | A set which is equipollent... |
| harn0 42448 | The Hartogs number of a se... |
| numinfctb 42449 | A numerable infinite set c... |
| isnumbasgrplem2 42450 | If the (to be thought of a... |
| isnumbasgrplem3 42451 | Every nonempty numerable s... |
| isnumbasabl 42452 | A set is numerable iff it ... |
| isnumbasgrp 42453 | A set is numerable iff it ... |
| dfacbasgrp 42454 | A choice equivalent in abs... |
| islnr 42457 | Property of a left-Noether... |
| lnrring 42458 | Left-Noetherian rings are ... |
| lnrlnm 42459 | Left-Noetherian rings have... |
| islnr2 42460 | Property of being a left-N... |
| islnr3 42461 | Relate left-Noetherian rin... |
| lnr2i 42462 | Given an ideal in a left-N... |
| lpirlnr 42463 | Left principal ideal rings... |
| lnrfrlm 42464 | Finite-dimensional free mo... |
| lnrfg 42465 | Finitely-generated modules... |
| lnrfgtr 42466 | A submodule of a finitely ... |
| hbtlem1 42469 | Value of the leading coeff... |
| hbtlem2 42470 | Leading coefficient ideals... |
| hbtlem7 42471 | Functionality of leading c... |
| hbtlem4 42472 | The leading ideal function... |
| hbtlem3 42473 | The leading ideal function... |
| hbtlem5 42474 | The leading ideal function... |
| hbtlem6 42475 | There is a finite set of p... |
| hbt 42476 | The Hilbert Basis Theorem ... |
| dgrsub2 42481 | Subtracting two polynomial... |
| elmnc 42482 | Property of a monic polyno... |
| mncply 42483 | A monic polynomial is a po... |
| mnccoe 42484 | A monic polynomial has lea... |
| mncn0 42485 | A monic polynomial is not ... |
| dgraaval 42490 | Value of the degree functi... |
| dgraalem 42491 | Properties of the degree o... |
| dgraacl 42492 | Closure of the degree func... |
| dgraaf 42493 | Degree function on algebra... |
| dgraaub 42494 | Upper bound on degree of a... |
| dgraa0p 42495 | A rational polynomial of d... |
| mpaaeu 42496 | An algebraic number has ex... |
| mpaaval 42497 | Value of the minimal polyn... |
| mpaalem 42498 | Properties of the minimal ... |
| mpaacl 42499 | Minimal polynomial is a po... |
| mpaadgr 42500 | Minimal polynomial has deg... |
| mpaaroot 42501 | The minimal polynomial of ... |
| mpaamn 42502 | Minimal polynomial is moni... |
| itgoval 42507 | Value of the integral-over... |
| aaitgo 42508 | The standard algebraic num... |
| itgoss 42509 | An integral element is int... |
| itgocn 42510 | All integral elements are ... |
| cnsrexpcl 42511 | Exponentiation is closed i... |
| fsumcnsrcl 42512 | Finite sums are closed in ... |
| cnsrplycl 42513 | Polynomials are closed in ... |
| rgspnval 42514 | Value of the ring-span of ... |
| rgspncl 42515 | The ring-span of a set is ... |
| rgspnssid 42516 | The ring-span of a set con... |
| rgspnmin 42517 | The ring-span is contained... |
| rgspnid 42518 | The span of a subring is i... |
| rngunsnply 42519 | Adjoining one element to a... |
| flcidc 42520 | Finite linear combinations... |
| algstr 42523 | Lemma to shorten proofs of... |
| algbase 42524 | The base set of a construc... |
| algaddg 42525 | The additive operation of ... |
| algmulr 42526 | The multiplicative operati... |
| algsca 42527 | The set of scalars of a co... |
| algvsca 42528 | The scalar product operati... |
| mendval 42529 | Value of the module endomo... |
| mendbas 42530 | Base set of the module end... |
| mendplusgfval 42531 | Addition in the module end... |
| mendplusg 42532 | A specific addition in the... |
| mendmulrfval 42533 | Multiplication in the modu... |
| mendmulr 42534 | A specific multiplication ... |
| mendsca 42535 | The module endomorphism al... |
| mendvscafval 42536 | Scalar multiplication in t... |
| mendvsca 42537 | A specific scalar multipli... |
| mendring 42538 | The module endomorphism al... |
| mendlmod 42539 | The module endomorphism al... |
| mendassa 42540 | The module endomorphism al... |
| idomodle 42541 | Limit on the number of ` N... |
| fiuneneq 42542 | Two finite sets of equal s... |
| idomsubgmo 42543 | The units of an integral d... |
| proot1mul 42544 | Any primitive ` N ` -th ro... |
| proot1hash 42545 | If an integral domain has ... |
| proot1ex 42546 | The complex field has prim... |
| isdomn3 42549 | Nonzero elements form a mu... |
| mon1psubm 42550 | Monic polynomials are a mu... |
| deg1mhm 42551 | Homomorphic property of th... |
| cytpfn 42552 | Functionality of the cyclo... |
| cytpval 42553 | Substitutions for the Nth ... |
| fgraphopab 42554 | Express a function as a su... |
| fgraphxp 42555 | Express a function as a su... |
| hausgraph 42556 | The graph of a continuous ... |
| r1sssucd 42561 | Deductive form of ~ r1sssu... |
| iocunico 42562 | Split an open interval int... |
| iocinico 42563 | The intersection of two se... |
| iocmbl 42564 | An open-below, closed-abov... |
| cnioobibld 42565 | A bounded, continuous func... |
| arearect 42566 | The area of a rectangle wh... |
| areaquad 42567 | The area of a quadrilatera... |
| uniel 42568 | Two ways to say a union is... |
| unielss 42569 | Two ways to say the union ... |
| unielid 42570 | Two ways to say the union ... |
| ssunib 42571 | Two ways to say a class is... |
| rp-intrabeq 42572 | Equality theorem for supre... |
| rp-unirabeq 42573 | Equality theorem for infim... |
| onmaxnelsup 42574 | Two ways to say the maximu... |
| onsupneqmaxlim0 42575 | If the supremum of a class... |
| onsupcl2 42576 | The supremum of a set of o... |
| onuniintrab 42577 | The union of a set of ordi... |
| onintunirab 42578 | The intersection of a non-... |
| onsupnmax 42579 | If the union of a class of... |
| onsupuni 42580 | The supremum of a set of o... |
| onsupuni2 42581 | The supremum of a set of o... |
| onsupintrab 42582 | The supremum of a set of o... |
| onsupintrab2 42583 | The supremum of a set of o... |
| onsupcl3 42584 | The supremum of a set of o... |
| onsupex3 42585 | The supremum of a set of o... |
| onuniintrab2 42586 | The union of a set of ordi... |
| oninfint 42587 | The infimum of a non-empty... |
| oninfunirab 42588 | The infimum of a non-empty... |
| oninfcl2 42589 | The infimum of a non-empty... |
| onsupmaxb 42590 | The union of a class of or... |
| onexgt 42591 | For any ordinal, there is ... |
| onexomgt 42592 | For any ordinal, there is ... |
| omlimcl2 42593 | The product of a limit ord... |
| onexlimgt 42594 | For any ordinal, there is ... |
| onexoegt 42595 | For any ordinal, there is ... |
| oninfex2 42596 | The infimum of a non-empty... |
| onsupeqmax 42597 | Condition when the supremu... |
| onsupeqnmax 42598 | Condition when the supremu... |
| onsuplub 42599 | The supremum of a set of o... |
| onsupnub 42600 | An upper bound of a set of... |
| onfisupcl 42601 | Sufficient condition when ... |
| onelord 42602 | Every element of a ordinal... |
| onepsuc 42603 | Every ordinal is less than... |
| epsoon 42604 | The ordinals are strictly ... |
| epirron 42605 | The strict order on the or... |
| oneptr 42606 | The strict order on the or... |
| oneltr 42607 | The elementhood relation o... |
| oneptri 42608 | The strict, complete (line... |
| oneltri 42609 | The elementhood relation o... |
| ordeldif 42610 | Membership in the differen... |
| ordeldifsucon 42611 | Membership in the differen... |
| ordeldif1o 42612 | Membership in the differen... |
| ordne0gt0 42613 | Ordinal zero is less than ... |
| ondif1i 42614 | Ordinal zero is less than ... |
| onsucelab 42615 | The successor of every ord... |
| dflim6 42616 | A limit ordinal is a non-z... |
| limnsuc 42617 | A limit ordinal is not an ... |
| onsucss 42618 | If one ordinal is less tha... |
| ordnexbtwnsuc 42619 | For any distinct pair of o... |
| orddif0suc 42620 | For any distinct pair of o... |
| onsucf1lem 42621 | For ordinals, the successo... |
| onsucf1olem 42622 | The successor operation is... |
| onsucrn 42623 | The successor operation is... |
| onsucf1o 42624 | The successor operation is... |
| dflim7 42625 | A limit ordinal is a non-z... |
| onov0suclim 42626 | Compactly express rules fo... |
| oa0suclim 42627 | Closed form expression of ... |
| om0suclim 42628 | Closed form expression of ... |
| oe0suclim 42629 | Closed form expression of ... |
| oaomoecl 42630 | The operations of addition... |
| onsupsucismax 42631 | If the union of a set of o... |
| onsssupeqcond 42632 | If for every element of a ... |
| limexissup 42633 | An ordinal which is a limi... |
| limiun 42634 | A limit ordinal is the uni... |
| limexissupab 42635 | An ordinal which is a limi... |
| om1om1r 42636 | Ordinal one is both a left... |
| oe0rif 42637 | Ordinal zero raised to any... |
| oasubex 42638 | While subtraction can't be... |
| nnamecl 42639 | Natural numbers are closed... |
| onsucwordi 42640 | The successor operation pr... |
| oalim2cl 42641 | The ordinal sum of any ord... |
| oaltublim 42642 | Given ` C ` is a limit ord... |
| oaordi3 42643 | Ordinal addition of the sa... |
| oaord3 42644 | When the same ordinal is a... |
| 1oaomeqom 42645 | Ordinal one plus omega is ... |
| oaabsb 42646 | The right addend absorbs t... |
| oaordnrex 42647 | When omega is added on the... |
| oaordnr 42648 | When the same ordinal is a... |
| omge1 42649 | Any non-zero ordinal produ... |
| omge2 42650 | Any non-zero ordinal produ... |
| omlim2 42651 | The non-zero product with ... |
| omord2lim 42652 | Given a limit ordinal, the... |
| omord2i 42653 | Ordinal multiplication of ... |
| omord2com 42654 | When the same non-zero ord... |
| 2omomeqom 42655 | Ordinal two times omega is... |
| omnord1ex 42656 | When omega is multiplied o... |
| omnord1 42657 | When the same non-zero ord... |
| oege1 42658 | Any non-zero ordinal power... |
| oege2 42659 | Any power of an ordinal at... |
| rp-oelim2 42660 | The power of an ordinal at... |
| oeord2lim 42661 | Given a limit ordinal, the... |
| oeord2i 42662 | Ordinal exponentiation of ... |
| oeord2com 42663 | When the same base at leas... |
| nnoeomeqom 42664 | Any natural number at leas... |
| df3o2 42665 | Ordinal 3 is the unordered... |
| df3o3 42666 | Ordinal 3, fully expanded.... |
| oenord1ex 42667 | When ordinals two and thre... |
| oenord1 42668 | When two ordinals (both at... |
| oaomoencom 42669 | Ordinal addition, multipli... |
| oenassex 42670 | Ordinal two raised to two ... |
| oenass 42671 | Ordinal exponentiation is ... |
| cantnftermord 42672 | For terms of the form of a... |
| cantnfub 42673 | Given a finite number of t... |
| cantnfub2 42674 | Given a finite number of t... |
| bropabg 42675 | Equivalence for two classe... |
| cantnfresb 42676 | A Cantor normal form which... |
| cantnf2 42677 | For every ordinal, ` A ` ,... |
| oawordex2 42678 | If ` C ` is between ` A ` ... |
| nnawordexg 42679 | If an ordinal, ` B ` , is ... |
| succlg 42680 | Closure law for ordinal su... |
| dflim5 42681 | A limit ordinal is either ... |
| oacl2g 42682 | Closure law for ordinal ad... |
| onmcl 42683 | If an ordinal is less than... |
| omabs2 42684 | Ordinal multiplication by ... |
| omcl2 42685 | Closure law for ordinal mu... |
| omcl3g 42686 | Closure law for ordinal mu... |
| ordsssucb 42687 | An ordinal number is less ... |
| tfsconcatlem 42688 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 42689 | The concatenation of two t... |
| tfsconcatfn 42690 | The concatenation of two t... |
| tfsconcatfv1 42691 | An early value of the conc... |
| tfsconcatfv2 42692 | A latter value of the conc... |
| tfsconcatfv 42693 | The value of the concatena... |
| tfsconcatrn 42694 | The range of the concatena... |
| tfsconcatfo 42695 | The concatenation of two t... |
| tfsconcatb0 42696 | The concatentation with th... |
| tfsconcat0i 42697 | The concatentation with th... |
| tfsconcat0b 42698 | The concatentation with th... |
| tfsconcat00 42699 | The concatentation of two ... |
| tfsconcatrev 42700 | If the domain of a transfi... |
| tfsconcatrnss12 42701 | The range of the concatena... |
| tfsconcatrnss 42702 | The concatenation of trans... |
| tfsconcatrnsson 42703 | The concatenation of trans... |
| tfsnfin 42704 | A transfinite sequence is ... |
| rp-tfslim 42705 | The limit of a sequence of... |
| ofoafg 42706 | Addition operator for func... |
| ofoaf 42707 | Addition operator for func... |
| ofoafo 42708 | Addition operator for func... |
| ofoacl 42709 | Closure law for component ... |
| ofoaid1 42710 | Identity law for component... |
| ofoaid2 42711 | Identity law for component... |
| ofoaass 42712 | Component-wise addition of... |
| ofoacom 42713 | Component-wise addition of... |
| naddcnff 42714 | Addition operator for Cant... |
| naddcnffn 42715 | Addition operator for Cant... |
| naddcnffo 42716 | Addition of Cantor normal ... |
| naddcnfcl 42717 | Closure law for component-... |
| naddcnfcom 42718 | Component-wise ordinal add... |
| naddcnfid1 42719 | Identity law for component... |
| naddcnfid2 42720 | Identity law for component... |
| naddcnfass 42721 | Component-wise addition of... |
| onsucunifi 42722 | The successor to the union... |
| sucunisn 42723 | The successor to the union... |
| onsucunipr 42724 | The successor to the union... |
| onsucunitp 42725 | The successor to the union... |
| oaun3lem1 42726 | The class of all ordinal s... |
| oaun3lem2 42727 | The class of all ordinal s... |
| oaun3lem3 42728 | The class of all ordinal s... |
| oaun3lem4 42729 | The class of all ordinal s... |
| rp-abid 42730 | Two ways to express a clas... |
| oadif1lem 42731 | Express the set difference... |
| oadif1 42732 | Express the set difference... |
| oaun2 42733 | Ordinal addition as a unio... |
| oaun3 42734 | Ordinal addition as a unio... |
| naddov4 42735 | Alternate expression for n... |
| nadd2rabtr 42736 | The set of ordinals which ... |
| nadd2rabord 42737 | The set of ordinals which ... |
| nadd2rabex 42738 | The class of ordinals whic... |
| nadd2rabon 42739 | The set of ordinals which ... |
| nadd1rabtr 42740 | The set of ordinals which ... |
| nadd1rabord 42741 | The set of ordinals which ... |
| nadd1rabex 42742 | The class of ordinals whic... |
| nadd1rabon 42743 | The set of ordinals which ... |
| nadd1suc 42744 | Natural addition with 1 is... |
| naddsuc2 42745 | Natural addition with succ... |
| naddass1 42746 | Natural addition of ordina... |
| naddgeoa 42747 | Natural addition results i... |
| naddonnn 42748 | Natural addition with a na... |
| naddwordnexlem0 42749 | When ` A ` is the sum of a... |
| naddwordnexlem1 42750 | When ` A ` is the sum of a... |
| naddwordnexlem2 42751 | When ` A ` is the sum of a... |
| naddwordnexlem3 42752 | When ` A ` is the sum of a... |
| oawordex3 42753 | When ` A ` is the sum of a... |
| naddwordnexlem4 42754 | When ` A ` is the sum of a... |
| ordsssucim 42755 | If an ordinal is less than... |
| insucid 42756 | The intersection of a clas... |
| om2 42757 | Two ways to double an ordi... |
| oaltom 42758 | Multiplication eventually ... |
| oe2 42759 | Two ways to square an ordi... |
| omltoe 42760 | Exponentiation eventually ... |
| abeqabi 42761 | Generalized condition for ... |
| abpr 42762 | Condition for a class abst... |
| abtp 42763 | Condition for a class abst... |
| ralopabb 42764 | Restricted universal quant... |
| fpwfvss 42765 | Functions into a powerset ... |
| sdomne0 42766 | A class that strictly domi... |
| sdomne0d 42767 | A class that strictly domi... |
| safesnsupfiss 42768 | If ` B ` is a finite subse... |
| safesnsupfiub 42769 | If ` B ` is a finite subse... |
| safesnsupfidom1o 42770 | If ` B ` is a finite subse... |
| safesnsupfilb 42771 | If ` B ` is a finite subse... |
| isoeq145d 42772 | Equality deduction for iso... |
| resisoeq45d 42773 | Equality deduction for equ... |
| negslem1 42774 | An equivalence between ide... |
| nvocnvb 42775 | Equivalence to saying the ... |
| rp-brsslt 42776 | Binary relation form of a ... |
| nla0002 42777 | Extending a linear order t... |
| nla0003 42778 | Extending a linear order t... |
| nla0001 42779 | Extending a linear order t... |
| faosnf0.11b 42780 | ` B ` is called a non-limi... |
| dfno2 42781 | A surreal number, in the f... |
| onnog 42782 | Every ordinal maps to a su... |
| onnobdayg 42783 | Every ordinal maps to a su... |
| bdaybndex 42784 | Bounds formed from the bir... |
| bdaybndbday 42785 | Bounds formed from the bir... |
| onno 42786 | Every ordinal maps to a su... |
| onnoi 42787 | Every ordinal maps to a su... |
| 0no 42788 | Ordinal zero maps to a sur... |
| 1no 42789 | Ordinal one maps to a surr... |
| 2no 42790 | Ordinal two maps to a surr... |
| 3no 42791 | Ordinal three maps to a su... |
| 4no 42792 | Ordinal four maps to a sur... |
| fnimafnex 42793 | The functional image of a ... |
| nlimsuc 42794 | A successor is not a limit... |
| nlim1NEW 42795 | 1 is not a limit ordinal. ... |
| nlim2NEW 42796 | 2 is not a limit ordinal. ... |
| nlim3 42797 | 3 is not a limit ordinal. ... |
| nlim4 42798 | 4 is not a limit ordinal. ... |
| oa1un 42799 | Given ` A e. On ` , let ` ... |
| oa1cl 42800 | ` A +o 1o ` is in ` On ` .... |
| 0finon 42801 | 0 is a finite ordinal. Se... |
| 1finon 42802 | 1 is a finite ordinal. Se... |
| 2finon 42803 | 2 is a finite ordinal. Se... |
| 3finon 42804 | 3 is a finite ordinal. Se... |
| 4finon 42805 | 4 is a finite ordinal. Se... |
| finona1cl 42806 | The finite ordinals are cl... |
| finonex 42807 | The finite ordinals are a ... |
| fzunt 42808 | Union of two adjacent fini... |
| fzuntd 42809 | Union of two adjacent fini... |
| fzunt1d 42810 | Union of two overlapping f... |
| fzuntgd 42811 | Union of two adjacent or o... |
| ifpan123g 42812 | Conjunction of conditional... |
| ifpan23 42813 | Conjunction of conditional... |
| ifpdfor2 42814 | Define or in terms of cond... |
| ifporcor 42815 | Corollary of commutation o... |
| ifpdfan2 42816 | Define and with conditiona... |
| ifpancor 42817 | Corollary of commutation o... |
| ifpdfor 42818 | Define or in terms of cond... |
| ifpdfan 42819 | Define and with conditiona... |
| ifpbi2 42820 | Equivalence theorem for co... |
| ifpbi3 42821 | Equivalence theorem for co... |
| ifpim1 42822 | Restate implication as con... |
| ifpnot 42823 | Restate negated wff as con... |
| ifpid2 42824 | Restate wff as conditional... |
| ifpim2 42825 | Restate implication as con... |
| ifpbi23 42826 | Equivalence theorem for co... |
| ifpbiidcor 42827 | Restatement of ~ biid . (... |
| ifpbicor 42828 | Corollary of commutation o... |
| ifpxorcor 42829 | Corollary of commutation o... |
| ifpbi1 42830 | Equivalence theorem for co... |
| ifpnot23 42831 | Negation of conditional lo... |
| ifpnotnotb 42832 | Factor conditional logic o... |
| ifpnorcor 42833 | Corollary of commutation o... |
| ifpnancor 42834 | Corollary of commutation o... |
| ifpnot23b 42835 | Negation of conditional lo... |
| ifpbiidcor2 42836 | Restatement of ~ biid . (... |
| ifpnot23c 42837 | Negation of conditional lo... |
| ifpnot23d 42838 | Negation of conditional lo... |
| ifpdfnan 42839 | Define nand as conditional... |
| ifpdfxor 42840 | Define xor as conditional ... |
| ifpbi12 42841 | Equivalence theorem for co... |
| ifpbi13 42842 | Equivalence theorem for co... |
| ifpbi123 42843 | Equivalence theorem for co... |
| ifpidg 42844 | Restate wff as conditional... |
| ifpid3g 42845 | Restate wff as conditional... |
| ifpid2g 42846 | Restate wff as conditional... |
| ifpid1g 42847 | Restate wff as conditional... |
| ifpim23g 42848 | Restate implication as con... |
| ifpim3 42849 | Restate implication as con... |
| ifpnim1 42850 | Restate negated implicatio... |
| ifpim4 42851 | Restate implication as con... |
| ifpnim2 42852 | Restate negated implicatio... |
| ifpim123g 42853 | Implication of conditional... |
| ifpim1g 42854 | Implication of conditional... |
| ifp1bi 42855 | Substitute the first eleme... |
| ifpbi1b 42856 | When the first variable is... |
| ifpimimb 42857 | Factor conditional logic o... |
| ifpororb 42858 | Factor conditional logic o... |
| ifpananb 42859 | Factor conditional logic o... |
| ifpnannanb 42860 | Factor conditional logic o... |
| ifpor123g 42861 | Disjunction of conditional... |
| ifpimim 42862 | Consequnce of implication.... |
| ifpbibib 42863 | Factor conditional logic o... |
| ifpxorxorb 42864 | Factor conditional logic o... |
| rp-fakeimass 42865 | A special case where impli... |
| rp-fakeanorass 42866 | A special case where a mix... |
| rp-fakeoranass 42867 | A special case where a mix... |
| rp-fakeinunass 42868 | A special case where a mix... |
| rp-fakeuninass 42869 | A special case where a mix... |
| rp-isfinite5 42870 | A set is said to be finite... |
| rp-isfinite6 42871 | A set is said to be finite... |
| intabssd 42872 | When for each element ` y ... |
| eu0 42873 | There is only one empty se... |
| epelon2 42874 | Over the ordinal numbers, ... |
| ontric3g 42875 | For all ` x , y e. On ` , ... |
| dfsucon 42876 | ` A ` is called a successo... |
| snen1g 42877 | A singleton is equinumerou... |
| snen1el 42878 | A singleton is equinumerou... |
| sn1dom 42879 | A singleton is dominated b... |
| pr2dom 42880 | An unordered pair is domin... |
| tr3dom 42881 | An unordered triple is dom... |
| ensucne0 42882 | A class equinumerous to a ... |
| ensucne0OLD 42883 | A class equinumerous to a ... |
| dfom6 42884 | Let ` _om ` be defined to ... |
| infordmin 42885 | ` _om ` is the smallest in... |
| iscard4 42886 | Two ways to express the pr... |
| minregex 42887 | Given any cardinal number ... |
| minregex2 42888 | Given any cardinal number ... |
| iscard5 42889 | Two ways to express the pr... |
| elrncard 42890 | Let us define a cardinal n... |
| harval3 42891 | ` ( har `` A ) ` is the le... |
| harval3on 42892 | For any ordinal number ` A... |
| omssrncard 42893 | All natural numbers are ca... |
| 0iscard 42894 | 0 is a cardinal number. (... |
| 1iscard 42895 | 1 is a cardinal number. (... |
| omiscard 42896 | ` _om ` is a cardinal numb... |
| sucomisnotcard 42897 | ` _om +o 1o ` is not a car... |
| nna1iscard 42898 | For any natural number, th... |
| har2o 42899 | The least cardinal greater... |
| en2pr 42900 | A class is equinumerous to... |
| pr2cv 42901 | If an unordered pair is eq... |
| pr2el1 42902 | If an unordered pair is eq... |
| pr2cv1 42903 | If an unordered pair is eq... |
| pr2el2 42904 | If an unordered pair is eq... |
| pr2cv2 42905 | If an unordered pair is eq... |
| pren2 42906 | An unordered pair is equin... |
| pr2eldif1 42907 | If an unordered pair is eq... |
| pr2eldif2 42908 | If an unordered pair is eq... |
| pren2d 42909 | A pair of two distinct set... |
| aleph1min 42910 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 42911 | ` aleph ` is a strictly or... |
| alephiso3 42912 | ` aleph ` is a strictly or... |
| pwelg 42913 | The powerclass is an eleme... |
| pwinfig 42914 | The powerclass of an infin... |
| pwinfi2 42915 | The powerclass of an infin... |
| pwinfi3 42916 | The powerclass of an infin... |
| pwinfi 42917 | The powerclass of an infin... |
| fipjust 42918 | A definition of the finite... |
| cllem0 42919 | The class of all sets with... |
| superficl 42920 | The class of all supersets... |
| superuncl 42921 | The class of all supersets... |
| ssficl 42922 | The class of all subsets o... |
| ssuncl 42923 | The class of all subsets o... |
| ssdifcl 42924 | The class of all subsets o... |
| sssymdifcl 42925 | The class of all subsets o... |
| fiinfi 42926 | If two classes have the fi... |
| rababg 42927 | Condition when restricted ... |
| elinintab 42928 | Two ways of saying a set i... |
| elmapintrab 42929 | Two ways to say a set is a... |
| elinintrab 42930 | Two ways of saying a set i... |
| inintabss 42931 | Upper bound on intersectio... |
| inintabd 42932 | Value of the intersection ... |
| xpinintabd 42933 | Value of the intersection ... |
| relintabex 42934 | If the intersection of a c... |
| elcnvcnvintab 42935 | Two ways of saying a set i... |
| relintab 42936 | Value of the intersection ... |
| nonrel 42937 | A non-relation is equal to... |
| elnonrel 42938 | Only an ordered pair where... |
| cnvssb 42939 | Subclass theorem for conve... |
| relnonrel 42940 | The non-relation part of a... |
| cnvnonrel 42941 | The converse of the non-re... |
| brnonrel 42942 | A non-relation cannot rela... |
| dmnonrel 42943 | The domain of the non-rela... |
| rnnonrel 42944 | The range of the non-relat... |
| resnonrel 42945 | A restriction of the non-r... |
| imanonrel 42946 | An image under the non-rel... |
| cononrel1 42947 | Composition with the non-r... |
| cononrel2 42948 | Composition with the non-r... |
| elmapintab 42949 | Two ways to say a set is a... |
| fvnonrel 42950 | The function value of any ... |
| elinlem 42951 | Two ways to say a set is a... |
| elcnvcnvlem 42952 | Two ways to say a set is a... |
| cnvcnvintabd 42953 | Value of the relationship ... |
| elcnvlem 42954 | Two ways to say a set is a... |
| elcnvintab 42955 | Two ways of saying a set i... |
| cnvintabd 42956 | Value of the converse of t... |
| undmrnresiss 42957 | Two ways of saying the ide... |
| reflexg 42958 | Two ways of saying a relat... |
| cnvssco 42959 | A condition weaker than re... |
| refimssco 42960 | Reflexive relations are su... |
| cleq2lem 42961 | Equality implies bijection... |
| cbvcllem 42962 | Change of bound variable i... |
| clublem 42963 | If a superset ` Y ` of ` X... |
| clss2lem 42964 | The closure of a property ... |
| dfid7 42965 | Definition of identity rel... |
| mptrcllem 42966 | Show two versions of a clo... |
| cotrintab 42967 | The intersection of a clas... |
| rclexi 42968 | The reflexive closure of a... |
| rtrclexlem 42969 | Existence of relation impl... |
| rtrclex 42970 | The reflexive-transitive c... |
| trclubgNEW 42971 | If a relation exists then ... |
| trclubNEW 42972 | If a relation exists then ... |
| trclexi 42973 | The transitive closure of ... |
| rtrclexi 42974 | The reflexive-transitive c... |
| clrellem 42975 | When the property ` ps ` h... |
| clcnvlem 42976 | When ` A ` , an upper boun... |
| cnvtrucl0 42977 | The converse of the trivia... |
| cnvrcl0 42978 | The converse of the reflex... |
| cnvtrcl0 42979 | The converse of the transi... |
| dmtrcl 42980 | The domain of the transiti... |
| rntrcl 42981 | The range of the transitiv... |
| dfrtrcl5 42982 | Definition of reflexive-tr... |
| trcleq2lemRP 42983 | Equality implies bijection... |
| sqrtcvallem1 42984 | Two ways of saying a compl... |
| reabsifneg 42985 | Alternate expression for t... |
| reabsifnpos 42986 | Alternate expression for t... |
| reabsifpos 42987 | Alternate expression for t... |
| reabsifnneg 42988 | Alternate expression for t... |
| reabssgn 42989 | Alternate expression for t... |
| sqrtcvallem2 42990 | Equivalent to saying that ... |
| sqrtcvallem3 42991 | Equivalent to saying that ... |
| sqrtcvallem4 42992 | Equivalent to saying that ... |
| sqrtcvallem5 42993 | Equivalent to saying that ... |
| sqrtcval 42994 | Explicit formula for the c... |
| sqrtcval2 42995 | Explicit formula for the c... |
| resqrtval 42996 | Real part of the complex s... |
| imsqrtval 42997 | Imaginary part of the comp... |
| resqrtvalex 42998 | Example for ~ resqrtval . ... |
| imsqrtvalex 42999 | Example for ~ imsqrtval . ... |
| al3im 43000 | Version of ~ ax-4 for a ne... |
| intima0 43001 | Two ways of expressing the... |
| elimaint 43002 | Element of image of inters... |
| cnviun 43003 | Converse of indexed union.... |
| imaiun1 43004 | The image of an indexed un... |
| coiun1 43005 | Composition with an indexe... |
| elintima 43006 | Element of intersection of... |
| intimass 43007 | The image under the inters... |
| intimass2 43008 | The image under the inters... |
| intimag 43009 | Requirement for the image ... |
| intimasn 43010 | Two ways to express the im... |
| intimasn2 43011 | Two ways to express the im... |
| ss2iundf 43012 | Subclass theorem for index... |
| ss2iundv 43013 | Subclass theorem for index... |
| cbviuneq12df 43014 | Rule used to change the bo... |
| cbviuneq12dv 43015 | Rule used to change the bo... |
| conrel1d 43016 | Deduction about compositio... |
| conrel2d 43017 | Deduction about compositio... |
| trrelind 43018 | The intersection of transi... |
| xpintrreld 43019 | The intersection of a tran... |
| restrreld 43020 | The restriction of a trans... |
| trrelsuperreldg 43021 | Concrete construction of a... |
| trficl 43022 | The class of all transitiv... |
| cnvtrrel 43023 | The converse of a transiti... |
| trrelsuperrel2dg 43024 | Concrete construction of a... |
| dfrcl2 43027 | Reflexive closure of a rel... |
| dfrcl3 43028 | Reflexive closure of a rel... |
| dfrcl4 43029 | Reflexive closure of a rel... |
| relexp2 43030 | A set operated on by the r... |
| relexpnul 43031 | If the domain and range of... |
| eliunov2 43032 | Membership in the indexed ... |
| eltrclrec 43033 | Membership in the indexed ... |
| elrtrclrec 43034 | Membership in the indexed ... |
| briunov2 43035 | Two classes related by the... |
| brmptiunrelexpd 43036 | If two elements are connec... |
| fvmptiunrelexplb0d 43037 | If the indexed union range... |
| fvmptiunrelexplb0da 43038 | If the indexed union range... |
| fvmptiunrelexplb1d 43039 | If the indexed union range... |
| brfvid 43040 | If two elements are connec... |
| brfvidRP 43041 | If two elements are connec... |
| fvilbd 43042 | A set is a subset of its i... |
| fvilbdRP 43043 | A set is a subset of its i... |
| brfvrcld 43044 | If two elements are connec... |
| brfvrcld2 43045 | If two elements are connec... |
| fvrcllb0d 43046 | A restriction of the ident... |
| fvrcllb0da 43047 | A restriction of the ident... |
| fvrcllb1d 43048 | A set is a subset of its i... |
| brtrclrec 43049 | Two classes related by the... |
| brrtrclrec 43050 | Two classes related by the... |
| briunov2uz 43051 | Two classes related by the... |
| eliunov2uz 43052 | Membership in the indexed ... |
| ov2ssiunov2 43053 | Any particular operator va... |
| relexp0eq 43054 | The zeroth power of relati... |
| iunrelexp0 43055 | Simplification of zeroth p... |
| relexpxpnnidm 43056 | Any positive power of a Ca... |
| relexpiidm 43057 | Any power of any restricti... |
| relexpss1d 43058 | The relational power of a ... |
| comptiunov2i 43059 | The composition two indexe... |
| corclrcl 43060 | The reflexive closure is i... |
| iunrelexpmin1 43061 | The indexed union of relat... |
| relexpmulnn 43062 | With exponents limited to ... |
| relexpmulg 43063 | With ordered exponents, th... |
| trclrelexplem 43064 | The union of relational po... |
| iunrelexpmin2 43065 | The indexed union of relat... |
| relexp01min 43066 | With exponents limited to ... |
| relexp1idm 43067 | Repeated raising a relatio... |
| relexp0idm 43068 | Repeated raising a relatio... |
| relexp0a 43069 | Absorption law for zeroth ... |
| relexpxpmin 43070 | The composition of powers ... |
| relexpaddss 43071 | The composition of two pow... |
| iunrelexpuztr 43072 | The indexed union of relat... |
| dftrcl3 43073 | Transitive closure of a re... |
| brfvtrcld 43074 | If two elements are connec... |
| fvtrcllb1d 43075 | A set is a subset of its i... |
| trclfvcom 43076 | The transitive closure of ... |
| cnvtrclfv 43077 | The converse of the transi... |
| cotrcltrcl 43078 | The transitive closure is ... |
| trclimalb2 43079 | Lower bound for image unde... |
| brtrclfv2 43080 | Two ways to indicate two e... |
| trclfvdecomr 43081 | The transitive closure of ... |
| trclfvdecoml 43082 | The transitive closure of ... |
| dmtrclfvRP 43083 | The domain of the transiti... |
| rntrclfvRP 43084 | The range of the transitiv... |
| rntrclfv 43085 | The range of the transitiv... |
| dfrtrcl3 43086 | Reflexive-transitive closu... |
| brfvrtrcld 43087 | If two elements are connec... |
| fvrtrcllb0d 43088 | A restriction of the ident... |
| fvrtrcllb0da 43089 | A restriction of the ident... |
| fvrtrcllb1d 43090 | A set is a subset of its i... |
| dfrtrcl4 43091 | Reflexive-transitive closu... |
| corcltrcl 43092 | The composition of the ref... |
| cortrcltrcl 43093 | Composition with the refle... |
| corclrtrcl 43094 | Composition with the refle... |
| cotrclrcl 43095 | The composition of the ref... |
| cortrclrcl 43096 | Composition with the refle... |
| cotrclrtrcl 43097 | Composition with the refle... |
| cortrclrtrcl 43098 | The reflexive-transitive c... |
| frege77d 43099 | If the images of both ` { ... |
| frege81d 43100 | If the image of ` U ` is a... |
| frege83d 43101 | If the image of the union ... |
| frege96d 43102 | If ` C ` follows ` A ` in ... |
| frege87d 43103 | If the images of both ` { ... |
| frege91d 43104 | If ` B ` follows ` A ` in ... |
| frege97d 43105 | If ` A ` contains all elem... |
| frege98d 43106 | If ` C ` follows ` A ` and... |
| frege102d 43107 | If either ` A ` and ` C ` ... |
| frege106d 43108 | If ` B ` follows ` A ` in ... |
| frege108d 43109 | If either ` A ` and ` C ` ... |
| frege109d 43110 | If ` A ` contains all elem... |
| frege114d 43111 | If either ` R ` relates ` ... |
| frege111d 43112 | If either ` A ` and ` C ` ... |
| frege122d 43113 | If ` F ` is a function, ` ... |
| frege124d 43114 | If ` F ` is a function, ` ... |
| frege126d 43115 | If ` F ` is a function, ` ... |
| frege129d 43116 | If ` F ` is a function and... |
| frege131d 43117 | If ` F ` is a function and... |
| frege133d 43118 | If ` F ` is a function and... |
| dfxor4 43119 | Express exclusive-or in te... |
| dfxor5 43120 | Express exclusive-or in te... |
| df3or2 43121 | Express triple-or in terms... |
| df3an2 43122 | Express triple-and in term... |
| nev 43123 | Express that not every set... |
| 0pssin 43124 | Express that an intersecti... |
| dfhe2 43127 | The property of relation `... |
| dfhe3 43128 | The property of relation `... |
| heeq12 43129 | Equality law for relations... |
| heeq1 43130 | Equality law for relations... |
| heeq2 43131 | Equality law for relations... |
| sbcheg 43132 | Distribute proper substitu... |
| hess 43133 | Subclass law for relations... |
| xphe 43134 | Any Cartesian product is h... |
| 0he 43135 | The empty relation is here... |
| 0heALT 43136 | The empty relation is here... |
| he0 43137 | Any relation is hereditary... |
| unhe1 43138 | The union of two relations... |
| snhesn 43139 | Any singleton is hereditar... |
| idhe 43140 | The identity relation is h... |
| psshepw 43141 | The relation between sets ... |
| sshepw 43142 | The relation between sets ... |
| rp-simp2-frege 43145 | Simplification of triple c... |
| rp-simp2 43146 | Simplification of triple c... |
| rp-frege3g 43147 | Add antecedent to ~ ax-fre... |
| frege3 43148 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 43149 | Double-use of ~ ax-frege2 ... |
| rp-frege24 43150 | Introducing an embedded an... |
| rp-frege4g 43151 | Deduction related to distr... |
| frege4 43152 | Special case of closed for... |
| frege5 43153 | A closed form of ~ syl . ... |
| rp-7frege 43154 | Distribute antecedent and ... |
| rp-4frege 43155 | Elimination of a nested an... |
| rp-6frege 43156 | Elimination of a nested an... |
| rp-8frege 43157 | Eliminate antecedent when ... |
| rp-frege25 43158 | Closed form for ~ a1dd . ... |
| frege6 43159 | A closed form of ~ imim2d ... |
| axfrege8 43160 | Swap antecedents. Identic... |
| frege7 43161 | A closed form of ~ syl6 . ... |
| frege26 43163 | Identical to ~ idd . Prop... |
| frege27 43164 | We cannot (at the same tim... |
| frege9 43165 | Closed form of ~ syl with ... |
| frege12 43166 | A closed form of ~ com23 .... |
| frege11 43167 | Elimination of a nested an... |
| frege24 43168 | Closed form for ~ a1d . D... |
| frege16 43169 | A closed form of ~ com34 .... |
| frege25 43170 | Closed form for ~ a1dd . ... |
| frege18 43171 | Closed form of a syllogism... |
| frege22 43172 | A closed form of ~ com45 .... |
| frege10 43173 | Result commuting anteceden... |
| frege17 43174 | A closed form of ~ com3l .... |
| frege13 43175 | A closed form of ~ com3r .... |
| frege14 43176 | Closed form of a deduction... |
| frege19 43177 | A closed form of ~ syl6 . ... |
| frege23 43178 | Syllogism followed by rota... |
| frege15 43179 | A closed form of ~ com4r .... |
| frege21 43180 | Replace antecedent in ante... |
| frege20 43181 | A closed form of ~ syl8 . ... |
| axfrege28 43182 | Contraposition. Identical... |
| frege29 43184 | Closed form of ~ con3d . ... |
| frege30 43185 | Commuted, closed form of ~... |
| axfrege31 43186 | Identical to ~ notnotr . ... |
| frege32 43188 | Deduce ~ con1 from ~ con3 ... |
| frege33 43189 | If ` ph ` or ` ps ` takes ... |
| frege34 43190 | If as a consequence of the... |
| frege35 43191 | Commuted, closed form of ~... |
| frege36 43192 | The case in which ` ps ` i... |
| frege37 43193 | If ` ch ` is a necessary c... |
| frege38 43194 | Identical to ~ pm2.21 . P... |
| frege39 43195 | Syllogism between ~ pm2.18... |
| frege40 43196 | Anything implies ~ pm2.18 ... |
| axfrege41 43197 | Identical to ~ notnot . A... |
| frege42 43199 | Not not ~ id . Propositio... |
| frege43 43200 | If there is a choice only ... |
| frege44 43201 | Similar to a commuted ~ pm... |
| frege45 43202 | Deduce ~ pm2.6 from ~ con1... |
| frege46 43203 | If ` ps ` holds when ` ph ... |
| frege47 43204 | Deduce consequence follows... |
| frege48 43205 | Closed form of syllogism w... |
| frege49 43206 | Closed form of deduction w... |
| frege50 43207 | Closed form of ~ jaoi . P... |
| frege51 43208 | Compare with ~ jaod . Pro... |
| axfrege52a 43209 | Justification for ~ ax-fre... |
| frege52aid 43211 | The case when the content ... |
| frege53aid 43212 | Specialization of ~ frege5... |
| frege53a 43213 | Lemma for ~ frege55a . Pr... |
| axfrege54a 43214 | Justification for ~ ax-fre... |
| frege54cor0a 43216 | Synonym for logical equiva... |
| frege54cor1a 43217 | Reflexive equality. (Cont... |
| frege55aid 43218 | Lemma for ~ frege57aid . ... |
| frege55lem1a 43219 | Necessary deduction regard... |
| frege55lem2a 43220 | Core proof of Proposition ... |
| frege55a 43221 | Proposition 55 of [Frege18... |
| frege55cor1a 43222 | Proposition 55 of [Frege18... |
| frege56aid 43223 | Lemma for ~ frege57aid . ... |
| frege56a 43224 | Proposition 56 of [Frege18... |
| frege57aid 43225 | This is the all imporant f... |
| frege57a 43226 | Analogue of ~ frege57aid .... |
| axfrege58a 43227 | Identical to ~ anifp . Ju... |
| frege58acor 43229 | Lemma for ~ frege59a . (C... |
| frege59a 43230 | A kind of Aristotelian inf... |
| frege60a 43231 | Swap antecedents of ~ ax-f... |
| frege61a 43232 | Lemma for ~ frege65a . Pr... |
| frege62a 43233 | A kind of Aristotelian inf... |
| frege63a 43234 | Proposition 63 of [Frege18... |
| frege64a 43235 | Lemma for ~ frege65a . Pr... |
| frege65a 43236 | A kind of Aristotelian inf... |
| frege66a 43237 | Swap antecedents of ~ freg... |
| frege67a 43238 | Lemma for ~ frege68a . Pr... |
| frege68a 43239 | Combination of applying a ... |
| axfrege52c 43240 | Justification for ~ ax-fre... |
| frege52b 43242 | The case when the content ... |
| frege53b 43243 | Lemma for frege102 (via ~ ... |
| axfrege54c 43244 | Reflexive equality of clas... |
| frege54b 43246 | Reflexive equality of sets... |
| frege54cor1b 43247 | Reflexive equality. (Cont... |
| frege55lem1b 43248 | Necessary deduction regard... |
| frege55lem2b 43249 | Lemma for ~ frege55b . Co... |
| frege55b 43250 | Lemma for ~ frege57b . Pr... |
| frege56b 43251 | Lemma for ~ frege57b . Pr... |
| frege57b 43252 | Analogue of ~ frege57aid .... |
| axfrege58b 43253 | If ` A. x ph ` is affirmed... |
| frege58bid 43255 | If ` A. x ph ` is affirmed... |
| frege58bcor 43256 | Lemma for ~ frege59b . (C... |
| frege59b 43257 | A kind of Aristotelian inf... |
| frege60b 43258 | Swap antecedents of ~ ax-f... |
| frege61b 43259 | Lemma for ~ frege65b . Pr... |
| frege62b 43260 | A kind of Aristotelian inf... |
| frege63b 43261 | Lemma for ~ frege91 . Pro... |
| frege64b 43262 | Lemma for ~ frege65b . Pr... |
| frege65b 43263 | A kind of Aristotelian inf... |
| frege66b 43264 | Swap antecedents of ~ freg... |
| frege67b 43265 | Lemma for ~ frege68b . Pr... |
| frege68b 43266 | Combination of applying a ... |
| frege53c 43267 | Proposition 53 of [Frege18... |
| frege54cor1c 43268 | Reflexive equality. (Cont... |
| frege55lem1c 43269 | Necessary deduction regard... |
| frege55lem2c 43270 | Core proof of Proposition ... |
| frege55c 43271 | Proposition 55 of [Frege18... |
| frege56c 43272 | Lemma for ~ frege57c . Pr... |
| frege57c 43273 | Swap order of implication ... |
| frege58c 43274 | Principle related to ~ sp ... |
| frege59c 43275 | A kind of Aristotelian inf... |
| frege60c 43276 | Swap antecedents of ~ freg... |
| frege61c 43277 | Lemma for ~ frege65c . Pr... |
| frege62c 43278 | A kind of Aristotelian inf... |
| frege63c 43279 | Analogue of ~ frege63b . ... |
| frege64c 43280 | Lemma for ~ frege65c . Pr... |
| frege65c 43281 | A kind of Aristotelian inf... |
| frege66c 43282 | Swap antecedents of ~ freg... |
| frege67c 43283 | Lemma for ~ frege68c . Pr... |
| frege68c 43284 | Combination of applying a ... |
| dffrege69 43285 | If from the proposition th... |
| frege70 43286 | Lemma for ~ frege72 . Pro... |
| frege71 43287 | Lemma for ~ frege72 . Pro... |
| frege72 43288 | If property ` A ` is hered... |
| frege73 43289 | Lemma for ~ frege87 . Pro... |
| frege74 43290 | If ` X ` has a property ` ... |
| frege75 43291 | If from the proposition th... |
| dffrege76 43292 | If from the two propositio... |
| frege77 43293 | If ` Y ` follows ` X ` in ... |
| frege78 43294 | Commuted form of ~ frege77... |
| frege79 43295 | Distributed form of ~ freg... |
| frege80 43296 | Add additional condition t... |
| frege81 43297 | If ` X ` has a property ` ... |
| frege82 43298 | Closed-form deduction base... |
| frege83 43299 | Apply commuted form of ~ f... |
| frege84 43300 | Commuted form of ~ frege81... |
| frege85 43301 | Commuted form of ~ frege77... |
| frege86 43302 | Conclusion about element o... |
| frege87 43303 | If ` Z ` is a result of an... |
| frege88 43304 | Commuted form of ~ frege87... |
| frege89 43305 | One direction of ~ dffrege... |
| frege90 43306 | Add antecedent to ~ frege8... |
| frege91 43307 | Every result of an applica... |
| frege92 43308 | Inference from ~ frege91 .... |
| frege93 43309 | Necessary condition for tw... |
| frege94 43310 | Looking one past a pair re... |
| frege95 43311 | Looking one past a pair re... |
| frege96 43312 | Every result of an applica... |
| frege97 43313 | The property of following ... |
| frege98 43314 | If ` Y ` follows ` X ` and... |
| dffrege99 43315 | If ` Z ` is identical with... |
| frege100 43316 | One direction of ~ dffrege... |
| frege101 43317 | Lemma for ~ frege102 . Pr... |
| frege102 43318 | If ` Z ` belongs to the ` ... |
| frege103 43319 | Proposition 103 of [Frege1... |
| frege104 43320 | Proposition 104 of [Frege1... |
| frege105 43321 | Proposition 105 of [Frege1... |
| frege106 43322 | Whatever follows ` X ` in ... |
| frege107 43323 | Proposition 107 of [Frege1... |
| frege108 43324 | If ` Y ` belongs to the ` ... |
| frege109 43325 | The property of belonging ... |
| frege110 43326 | Proposition 110 of [Frege1... |
| frege111 43327 | If ` Y ` belongs to the ` ... |
| frege112 43328 | Identity implies belonging... |
| frege113 43329 | Proposition 113 of [Frege1... |
| frege114 43330 | If ` X ` belongs to the ` ... |
| dffrege115 43331 | If from the circumstance t... |
| frege116 43332 | One direction of ~ dffrege... |
| frege117 43333 | Lemma for ~ frege118 . Pr... |
| frege118 43334 | Simplified application of ... |
| frege119 43335 | Lemma for ~ frege120 . Pr... |
| frege120 43336 | Simplified application of ... |
| frege121 43337 | Lemma for ~ frege122 . Pr... |
| frege122 43338 | If ` X ` is a result of an... |
| frege123 43339 | Lemma for ~ frege124 . Pr... |
| frege124 43340 | If ` X ` is a result of an... |
| frege125 43341 | Lemma for ~ frege126 . Pr... |
| frege126 43342 | If ` M ` follows ` Y ` in ... |
| frege127 43343 | Communte antecedents of ~ ... |
| frege128 43344 | Lemma for ~ frege129 . Pr... |
| frege129 43345 | If the procedure ` R ` is ... |
| frege130 43346 | Lemma for ~ frege131 . Pr... |
| frege131 43347 | If the procedure ` R ` is ... |
| frege132 43348 | Lemma for ~ frege133 . Pr... |
| frege133 43349 | If the procedure ` R ` is ... |
| enrelmap 43350 | The set of all possible re... |
| enrelmapr 43351 | The set of all possible re... |
| enmappw 43352 | The set of all mappings fr... |
| enmappwid 43353 | The set of all mappings fr... |
| rfovd 43354 | Value of the operator, ` (... |
| rfovfvd 43355 | Value of the operator, ` (... |
| rfovfvfvd 43356 | Value of the operator, ` (... |
| rfovcnvf1od 43357 | Properties of the operator... |
| rfovcnvd 43358 | Value of the converse of t... |
| rfovf1od 43359 | The value of the operator,... |
| rfovcnvfvd 43360 | Value of the converse of t... |
| fsovd 43361 | Value of the operator, ` (... |
| fsovrfovd 43362 | The operator which gives a... |
| fsovfvd 43363 | Value of the operator, ` (... |
| fsovfvfvd 43364 | Value of the operator, ` (... |
| fsovfd 43365 | The operator, ` ( A O B ) ... |
| fsovcnvlem 43366 | The ` O ` operator, which ... |
| fsovcnvd 43367 | The value of the converse ... |
| fsovcnvfvd 43368 | The value of the converse ... |
| fsovf1od 43369 | The value of ` ( A O B ) `... |
| dssmapfvd 43370 | Value of the duality opera... |
| dssmapfv2d 43371 | Value of the duality opera... |
| dssmapfv3d 43372 | Value of the duality opera... |
| dssmapnvod 43373 | For any base set ` B ` the... |
| dssmapf1od 43374 | For any base set ` B ` the... |
| dssmap2d 43375 | For any base set ` B ` the... |
| or3or 43376 | Decompose disjunction into... |
| andi3or 43377 | Distribute over triple dis... |
| uneqsn 43378 | If a union of classes is e... |
| brfvimex 43379 | If a binary relation holds... |
| brovmptimex 43380 | If a binary relation holds... |
| brovmptimex1 43381 | If a binary relation holds... |
| brovmptimex2 43382 | If a binary relation holds... |
| brcoffn 43383 | Conditions allowing the de... |
| brcofffn 43384 | Conditions allowing the de... |
| brco2f1o 43385 | Conditions allowing the de... |
| brco3f1o 43386 | Conditions allowing the de... |
| ntrclsbex 43387 | If (pseudo-)interior and (... |
| ntrclsrcomplex 43388 | The relative complement of... |
| neik0imk0p 43389 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 43390 | If an interior function is... |
| ntrkbimka 43391 | If the interiors of disjoi... |
| ntrk0kbimka 43392 | If the interiors of disjoi... |
| clsk3nimkb 43393 | If the base set is not emp... |
| clsk1indlem0 43394 | The ansatz closure functio... |
| clsk1indlem2 43395 | The ansatz closure functio... |
| clsk1indlem3 43396 | The ansatz closure functio... |
| clsk1indlem4 43397 | The ansatz closure functio... |
| clsk1indlem1 43398 | The ansatz closure functio... |
| clsk1independent 43399 | For generalized closure fu... |
| neik0pk1imk0 43400 | Kuratowski's K0' and K1 ax... |
| isotone1 43401 | Two different ways to say ... |
| isotone2 43402 | Two different ways to say ... |
| ntrk1k3eqk13 43403 | An interior function is bo... |
| ntrclsf1o 43404 | If (pseudo-)interior and (... |
| ntrclsnvobr 43405 | If (pseudo-)interior and (... |
| ntrclsiex 43406 | If (pseudo-)interior and (... |
| ntrclskex 43407 | If (pseudo-)interior and (... |
| ntrclsfv1 43408 | If (pseudo-)interior and (... |
| ntrclsfv2 43409 | If (pseudo-)interior and (... |
| ntrclselnel1 43410 | If (pseudo-)interior and (... |
| ntrclselnel2 43411 | If (pseudo-)interior and (... |
| ntrclsfv 43412 | The value of the interior ... |
| ntrclsfveq1 43413 | If interior and closure fu... |
| ntrclsfveq2 43414 | If interior and closure fu... |
| ntrclsfveq 43415 | If interior and closure fu... |
| ntrclsss 43416 | If interior and closure fu... |
| ntrclsneine0lem 43417 | If (pseudo-)interior and (... |
| ntrclsneine0 43418 | If (pseudo-)interior and (... |
| ntrclscls00 43419 | If (pseudo-)interior and (... |
| ntrclsiso 43420 | If (pseudo-)interior and (... |
| ntrclsk2 43421 | An interior function is co... |
| ntrclskb 43422 | The interiors of disjoint ... |
| ntrclsk3 43423 | The intersection of interi... |
| ntrclsk13 43424 | The interior of the inters... |
| ntrclsk4 43425 | Idempotence of the interio... |
| ntrneibex 43426 | If (pseudo-)interior and (... |
| ntrneircomplex 43427 | The relative complement of... |
| ntrneif1o 43428 | If (pseudo-)interior and (... |
| ntrneiiex 43429 | If (pseudo-)interior and (... |
| ntrneinex 43430 | If (pseudo-)interior and (... |
| ntrneicnv 43431 | If (pseudo-)interior and (... |
| ntrneifv1 43432 | If (pseudo-)interior and (... |
| ntrneifv2 43433 | If (pseudo-)interior and (... |
| ntrneiel 43434 | If (pseudo-)interior and (... |
| ntrneifv3 43435 | The value of the neighbors... |
| ntrneineine0lem 43436 | If (pseudo-)interior and (... |
| ntrneineine1lem 43437 | If (pseudo-)interior and (... |
| ntrneifv4 43438 | The value of the interior ... |
| ntrneiel2 43439 | Membership in iterated int... |
| ntrneineine0 43440 | If (pseudo-)interior and (... |
| ntrneineine1 43441 | If (pseudo-)interior and (... |
| ntrneicls00 43442 | If (pseudo-)interior and (... |
| ntrneicls11 43443 | If (pseudo-)interior and (... |
| ntrneiiso 43444 | If (pseudo-)interior and (... |
| ntrneik2 43445 | An interior function is co... |
| ntrneix2 43446 | An interior (closure) func... |
| ntrneikb 43447 | The interiors of disjoint ... |
| ntrneixb 43448 | The interiors (closures) o... |
| ntrneik3 43449 | The intersection of interi... |
| ntrneix3 43450 | The closure of the union o... |
| ntrneik13 43451 | The interior of the inters... |
| ntrneix13 43452 | The closure of the union o... |
| ntrneik4w 43453 | Idempotence of the interio... |
| ntrneik4 43454 | Idempotence of the interio... |
| clsneibex 43455 | If (pseudo-)closure and (p... |
| clsneircomplex 43456 | The relative complement of... |
| clsneif1o 43457 | If a (pseudo-)closure func... |
| clsneicnv 43458 | If a (pseudo-)closure func... |
| clsneikex 43459 | If closure and neighborhoo... |
| clsneinex 43460 | If closure and neighborhoo... |
| clsneiel1 43461 | If a (pseudo-)closure func... |
| clsneiel2 43462 | If a (pseudo-)closure func... |
| clsneifv3 43463 | Value of the neighborhoods... |
| clsneifv4 43464 | Value of the closure (inte... |
| neicvgbex 43465 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 43466 | The relative complement of... |
| neicvgf1o 43467 | If neighborhood and conver... |
| neicvgnvo 43468 | If neighborhood and conver... |
| neicvgnvor 43469 | If neighborhood and conver... |
| neicvgmex 43470 | If the neighborhoods and c... |
| neicvgnex 43471 | If the neighborhoods and c... |
| neicvgel1 43472 | A subset being an element ... |
| neicvgel2 43473 | The complement of a subset... |
| neicvgfv 43474 | The value of the neighborh... |
| ntrrn 43475 | The range of the interior ... |
| ntrf 43476 | The interior function of a... |
| ntrf2 43477 | The interior function is a... |
| ntrelmap 43478 | The interior function is a... |
| clsf2 43479 | The closure function is a ... |
| clselmap 43480 | The closure function is a ... |
| dssmapntrcls 43481 | The interior and closure o... |
| dssmapclsntr 43482 | The closure and interior o... |
| gneispa 43483 | Each point ` p ` of the ne... |
| gneispb 43484 | Given a neighborhood ` N `... |
| gneispace2 43485 | The predicate that ` F ` i... |
| gneispace3 43486 | The predicate that ` F ` i... |
| gneispace 43487 | The predicate that ` F ` i... |
| gneispacef 43488 | A generic neighborhood spa... |
| gneispacef2 43489 | A generic neighborhood spa... |
| gneispacefun 43490 | A generic neighborhood spa... |
| gneispacern 43491 | A generic neighborhood spa... |
| gneispacern2 43492 | A generic neighborhood spa... |
| gneispace0nelrn 43493 | A generic neighborhood spa... |
| gneispace0nelrn2 43494 | A generic neighborhood spa... |
| gneispace0nelrn3 43495 | A generic neighborhood spa... |
| gneispaceel 43496 | Every neighborhood of a po... |
| gneispaceel2 43497 | Every neighborhood of a po... |
| gneispacess 43498 | All supersets of a neighbo... |
| gneispacess2 43499 | All supersets of a neighbo... |
| k0004lem1 43500 | Application of ~ ssin to r... |
| k0004lem2 43501 | A mapping with a particula... |
| k0004lem3 43502 | When the value of a mappin... |
| k0004val 43503 | The topological simplex of... |
| k0004ss1 43504 | The topological simplex of... |
| k0004ss2 43505 | The topological simplex of... |
| k0004ss3 43506 | The topological simplex of... |
| k0004val0 43507 | The topological simplex of... |
| inductionexd 43508 | Simple induction example. ... |
| wwlemuld 43509 | Natural deduction form of ... |
| leeq1d 43510 | Specialization of ~ breq1d... |
| leeq2d 43511 | Specialization of ~ breq2d... |
| absmulrposd 43512 | Specialization of absmuld ... |
| imadisjld 43513 | Natural dduction form of o... |
| wnefimgd 43514 | The image of a mapping fro... |
| fco2d 43515 | Natural deduction form of ... |
| wfximgfd 43516 | The value of a function on... |
| extoimad 43517 | If |f(x)| <= C for all x t... |
| imo72b2lem0 43518 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 43519 | Natural deduction form of ... |
| imo72b2lem2 43520 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 43521 | Natural deduction form of ... |
| imo72b2lem1 43522 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 43523 | 'Less than or equal to' re... |
| lemuldiv4d 43524 | 'Less than or equal to' re... |
| imo72b2 43525 | IMO 1972 B2. (14th Intern... |
| int-addcomd 43526 | AdditionCommutativity gene... |
| int-addassocd 43527 | AdditionAssociativity gene... |
| int-addsimpd 43528 | AdditionSimplification gen... |
| int-mulcomd 43529 | MultiplicationCommutativit... |
| int-mulassocd 43530 | MultiplicationAssociativit... |
| int-mulsimpd 43531 | MultiplicationSimplificati... |
| int-leftdistd 43532 | AdditionMultiplicationLeft... |
| int-rightdistd 43533 | AdditionMultiplicationRigh... |
| int-sqdefd 43534 | SquareDefinition generator... |
| int-mul11d 43535 | First MultiplicationOne ge... |
| int-mul12d 43536 | Second MultiplicationOne g... |
| int-add01d 43537 | First AdditionZero generat... |
| int-add02d 43538 | Second AdditionZero genera... |
| int-sqgeq0d 43539 | SquareGEQZero generator ru... |
| int-eqprincd 43540 | PrincipleOfEquality genera... |
| int-eqtransd 43541 | EqualityTransitivity gener... |
| int-eqmvtd 43542 | EquMoveTerm generator rule... |
| int-eqineqd 43543 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 43544 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 43545 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 43546 | SecondPrincipleOfInequalit... |
| int-ineqtransd 43547 | InequalityTransitivity gen... |
| unitadd 43548 | Theorem used in conjunctio... |
| gsumws3 43549 | Valuation of a length 3 wo... |
| gsumws4 43550 | Valuation of a length 4 wo... |
| amgm2d 43551 | Arithmetic-geometric mean ... |
| amgm3d 43552 | Arithmetic-geometric mean ... |
| amgm4d 43553 | Arithmetic-geometric mean ... |
| spALT 43554 | ~ sp can be proven from th... |
| elnelneqd 43555 | Two classes are not equal ... |
| elnelneq2d 43556 | Two classes are not equal ... |
| rr-spce 43557 | Prove an existential. (Co... |
| rexlimdvaacbv 43558 | Unpack a restricted existe... |
| rexlimddvcbvw 43559 | Unpack a restricted existe... |
| rexlimddvcbv 43560 | Unpack a restricted existe... |
| rr-elrnmpt3d 43561 | Elementhood in an image se... |
| finnzfsuppd 43562 | If a function is zero outs... |
| rr-phpd 43563 | Equivalent of ~ php withou... |
| suceqd 43564 | Deduction associated with ... |
| tfindsd 43565 | Deduction associated with ... |
| mnringvald 43568 | Value of the monoid ring f... |
| mnringnmulrd 43569 | Components of a monoid rin... |
| mnringnmulrdOLD 43570 | Obsolete version of ~ mnri... |
| mnringbased 43571 | The base set of a monoid r... |
| mnringbasedOLD 43572 | Obsolete version of ~ mnri... |
| mnringbaserd 43573 | The base set of a monoid r... |
| mnringelbased 43574 | Membership in the base set... |
| mnringbasefd 43575 | Elements of a monoid ring ... |
| mnringbasefsuppd 43576 | Elements of a monoid ring ... |
| mnringaddgd 43577 | The additive operation of ... |
| mnringaddgdOLD 43578 | Obsolete version of ~ mnri... |
| mnring0gd 43579 | The additive identity of a... |
| mnring0g2d 43580 | The additive identity of a... |
| mnringmulrd 43581 | The ring product of a mono... |
| mnringscad 43582 | The scalar ring of a monoi... |
| mnringscadOLD 43583 | Obsolete version of ~ mnri... |
| mnringvscad 43584 | The scalar product of a mo... |
| mnringvscadOLD 43585 | Obsolete version of ~ mnri... |
| mnringlmodd 43586 | Monoid rings are left modu... |
| mnringmulrvald 43587 | Value of multiplication in... |
| mnringmulrcld 43588 | Monoid rings are closed un... |
| gru0eld 43589 | A nonempty Grothendieck un... |
| grusucd 43590 | Grothendieck universes are... |
| r1rankcld 43591 | Any rank of the cumulative... |
| grur1cld 43592 | Grothendieck universes are... |
| grurankcld 43593 | Grothendieck universes are... |
| grurankrcld 43594 | If a Grothendieck universe... |
| scotteqd 43597 | Equality theorem for the S... |
| scotteq 43598 | Closed form of ~ scotteqd ... |
| nfscott 43599 | Bound-variable hypothesis ... |
| scottabf 43600 | Value of the Scott operati... |
| scottab 43601 | Value of the Scott operati... |
| scottabes 43602 | Value of the Scott operati... |
| scottss 43603 | Scott's trick produces a s... |
| elscottab 43604 | An element of the output o... |
| scottex2 43605 | ~ scottex expressed using ... |
| scotteld 43606 | The Scott operation sends ... |
| scottelrankd 43607 | Property of a Scott's tric... |
| scottrankd 43608 | Rank of a nonempty Scott's... |
| gruscottcld 43609 | If a Grothendieck universe... |
| dfcoll2 43612 | Alternate definition of th... |
| colleq12d 43613 | Equality theorem for the c... |
| colleq1 43614 | Equality theorem for the c... |
| colleq2 43615 | Equality theorem for the c... |
| nfcoll 43616 | Bound-variable hypothesis ... |
| collexd 43617 | The output of the collecti... |
| cpcolld 43618 | Property of the collection... |
| cpcoll2d 43619 | ~ cpcolld with an extra ex... |
| grucollcld 43620 | A Grothendieck universe co... |
| ismnu 43621 | The hypothesis of this the... |
| mnuop123d 43622 | Operations of a minimal un... |
| mnussd 43623 | Minimal universes are clos... |
| mnuss2d 43624 | ~ mnussd with arguments pr... |
| mnu0eld 43625 | A nonempty minimal univers... |
| mnuop23d 43626 | Second and third operation... |
| mnupwd 43627 | Minimal universes are clos... |
| mnusnd 43628 | Minimal universes are clos... |
| mnuprssd 43629 | A minimal universe contain... |
| mnuprss2d 43630 | Special case of ~ mnuprssd... |
| mnuop3d 43631 | Third operation of a minim... |
| mnuprdlem1 43632 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 43633 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 43634 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 43635 | Lemma for ~ mnuprd . Gene... |
| mnuprd 43636 | Minimal universes are clos... |
| mnuunid 43637 | Minimal universes are clos... |
| mnuund 43638 | Minimal universes are clos... |
| mnutrcld 43639 | Minimal universes contain ... |
| mnutrd 43640 | Minimal universes are tran... |
| mnurndlem1 43641 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 43642 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 43643 | Minimal universes contain ... |
| mnugrud 43644 | Minimal universes are Grot... |
| grumnudlem 43645 | Lemma for ~ grumnud . (Co... |
| grumnud 43646 | Grothendieck universes are... |
| grumnueq 43647 | The class of Grothendieck ... |
| expandan 43648 | Expand conjunction to prim... |
| expandexn 43649 | Expand an existential quan... |
| expandral 43650 | Expand a restricted univer... |
| expandrexn 43651 | Expand a restricted existe... |
| expandrex 43652 | Expand a restricted existe... |
| expanduniss 43653 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 43654 | Express the predicate on `... |
| rr-grothprimbi 43655 | Express "every set is cont... |
| inagrud 43656 | Inaccessible levels of the... |
| inaex 43657 | Assuming the Tarski-Grothe... |
| gruex 43658 | Assuming the Tarski-Grothe... |
| rr-groth 43659 | An equivalent of ~ ax-grot... |
| rr-grothprim 43660 | An equivalent of ~ ax-grot... |
| ismnushort 43661 | Express the predicate on `... |
| dfuniv2 43662 | Alternative definition of ... |
| rr-grothshortbi 43663 | Express "every set is cont... |
| rr-grothshort 43664 | A shorter equivalent of ~ ... |
| nanorxor 43665 | 'nand' is equivalent to th... |
| undisjrab 43666 | Union of two disjoint rest... |
| iso0 43667 | The empty set is an ` R , ... |
| ssrecnpr 43668 | ` RR ` is a subset of both... |
| seff 43669 | Let set ` S ` be the real ... |
| sblpnf 43670 | The infinity ball in the a... |
| prmunb2 43671 | The primes are unbounded. ... |
| dvgrat 43672 | Ratio test for divergence ... |
| cvgdvgrat 43673 | Ratio test for convergence... |
| radcnvrat 43674 | Let ` L ` be the limit, if... |
| reldvds 43675 | The divides relation is in... |
| nznngen 43676 | All positive integers in t... |
| nzss 43677 | The set of multiples of _m... |
| nzin 43678 | The intersection of the se... |
| nzprmdif 43679 | Subtract one prime's multi... |
| hashnzfz 43680 | Special case of ~ hashdvds... |
| hashnzfz2 43681 | Special case of ~ hashnzfz... |
| hashnzfzclim 43682 | As the upper bound ` K ` o... |
| caofcan 43683 | Transfer a cancellation la... |
| ofsubid 43684 | Function analogue of ~ sub... |
| ofmul12 43685 | Function analogue of ~ mul... |
| ofdivrec 43686 | Function analogue of ~ div... |
| ofdivcan4 43687 | Function analogue of ~ div... |
| ofdivdiv2 43688 | Function analogue of ~ div... |
| lhe4.4ex1a 43689 | Example of the Fundamental... |
| dvsconst 43690 | Derivative of a constant f... |
| dvsid 43691 | Derivative of the identity... |
| dvsef 43692 | Derivative of the exponent... |
| expgrowthi 43693 | Exponential growth and dec... |
| dvconstbi 43694 | The derivative of a functi... |
| expgrowth 43695 | Exponential growth and dec... |
| bccval 43698 | Value of the generalized b... |
| bcccl 43699 | Closure of the generalized... |
| bcc0 43700 | The generalized binomial c... |
| bccp1k 43701 | Generalized binomial coeff... |
| bccm1k 43702 | Generalized binomial coeff... |
| bccn0 43703 | Generalized binomial coeff... |
| bccn1 43704 | Generalized binomial coeff... |
| bccbc 43705 | The binomial coefficient a... |
| uzmptshftfval 43706 | When ` F ` is a maps-to fu... |
| dvradcnv2 43707 | The radius of convergence ... |
| binomcxplemwb 43708 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 43709 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 43710 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 43711 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 43712 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 43713 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 43714 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 43715 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 43716 | Lemma for ~ binomcxp . Wh... |
| binomcxp 43717 | Generalize the binomial th... |
| pm10.12 43718 | Theorem *10.12 in [Whitehe... |
| pm10.14 43719 | Theorem *10.14 in [Whitehe... |
| pm10.251 43720 | Theorem *10.251 in [Whiteh... |
| pm10.252 43721 | Theorem *10.252 in [Whiteh... |
| pm10.253 43722 | Theorem *10.253 in [Whiteh... |
| albitr 43723 | Theorem *10.301 in [Whiteh... |
| pm10.42 43724 | Theorem *10.42 in [Whitehe... |
| pm10.52 43725 | Theorem *10.52 in [Whitehe... |
| pm10.53 43726 | Theorem *10.53 in [Whitehe... |
| pm10.541 43727 | Theorem *10.541 in [Whiteh... |
| pm10.542 43728 | Theorem *10.542 in [Whiteh... |
| pm10.55 43729 | Theorem *10.55 in [Whitehe... |
| pm10.56 43730 | Theorem *10.56 in [Whitehe... |
| pm10.57 43731 | Theorem *10.57 in [Whitehe... |
| 2alanimi 43732 | Removes two universal quan... |
| 2al2imi 43733 | Removes two universal quan... |
| pm11.11 43734 | Theorem *11.11 in [Whitehe... |
| pm11.12 43735 | Theorem *11.12 in [Whitehe... |
| 19.21vv 43736 | Compare Theorem *11.3 in [... |
| 2alim 43737 | Theorem *11.32 in [Whitehe... |
| 2albi 43738 | Theorem *11.33 in [Whitehe... |
| 2exim 43739 | Theorem *11.34 in [Whitehe... |
| 2exbi 43740 | Theorem *11.341 in [Whiteh... |
| spsbce-2 43741 | Theorem *11.36 in [Whitehe... |
| 19.33-2 43742 | Theorem *11.421 in [Whiteh... |
| 19.36vv 43743 | Theorem *11.43 in [Whitehe... |
| 19.31vv 43744 | Theorem *11.44 in [Whitehe... |
| 19.37vv 43745 | Theorem *11.46 in [Whitehe... |
| 19.28vv 43746 | Theorem *11.47 in [Whitehe... |
| pm11.52 43747 | Theorem *11.52 in [Whitehe... |
| aaanv 43748 | Theorem *11.56 in [Whitehe... |
| pm11.57 43749 | Theorem *11.57 in [Whitehe... |
| pm11.58 43750 | Theorem *11.58 in [Whitehe... |
| pm11.59 43751 | Theorem *11.59 in [Whitehe... |
| pm11.6 43752 | Theorem *11.6 in [Whitehea... |
| pm11.61 43753 | Theorem *11.61 in [Whitehe... |
| pm11.62 43754 | Theorem *11.62 in [Whitehe... |
| pm11.63 43755 | Theorem *11.63 in [Whitehe... |
| pm11.7 43756 | Theorem *11.7 in [Whitehea... |
| pm11.71 43757 | Theorem *11.71 in [Whitehe... |
| sbeqal1 43758 | If ` x = y ` always implie... |
| sbeqal1i 43759 | Suppose you know ` x = y `... |
| sbeqal2i 43760 | If ` x = y ` implies ` x =... |
| axc5c4c711 43761 | Proof of a theorem that ca... |
| axc5c4c711toc5 43762 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 43763 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 43764 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 43765 | Rederivation of ~ ax-11 fr... |
| axc11next 43766 | This theorem shows that, g... |
| pm13.13a 43767 | One result of theorem *13.... |
| pm13.13b 43768 | Theorem *13.13 in [Whitehe... |
| pm13.14 43769 | Theorem *13.14 in [Whitehe... |
| pm13.192 43770 | Theorem *13.192 in [Whiteh... |
| pm13.193 43771 | Theorem *13.193 in [Whiteh... |
| pm13.194 43772 | Theorem *13.194 in [Whiteh... |
| pm13.195 43773 | Theorem *13.195 in [Whiteh... |
| pm13.196a 43774 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 43775 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 43776 | Theorem *13.22 in [Whitehe... |
| iotain 43777 | Equivalence between two di... |
| iotaexeu 43778 | The iota class exists. Th... |
| iotasbc 43779 | Definition *14.01 in [Whit... |
| iotasbc2 43780 | Theorem *14.111 in [Whiteh... |
| pm14.12 43781 | Theorem *14.12 in [Whitehe... |
| pm14.122a 43782 | Theorem *14.122 in [Whiteh... |
| pm14.122b 43783 | Theorem *14.122 in [Whiteh... |
| pm14.122c 43784 | Theorem *14.122 in [Whiteh... |
| pm14.123a 43785 | Theorem *14.123 in [Whiteh... |
| pm14.123b 43786 | Theorem *14.123 in [Whiteh... |
| pm14.123c 43787 | Theorem *14.123 in [Whiteh... |
| pm14.18 43788 | Theorem *14.18 in [Whitehe... |
| iotaequ 43789 | Theorem *14.2 in [Whitehea... |
| iotavalb 43790 | Theorem *14.202 in [Whiteh... |
| iotasbc5 43791 | Theorem *14.205 in [Whiteh... |
| pm14.24 43792 | Theorem *14.24 in [Whitehe... |
| iotavalsb 43793 | Theorem *14.242 in [Whiteh... |
| sbiota1 43794 | Theorem *14.25 in [Whitehe... |
| sbaniota 43795 | Theorem *14.26 in [Whitehe... |
| eubiOLD 43796 | Obsolete proof of ~ eubi a... |
| iotasbcq 43797 | Theorem *14.272 in [Whiteh... |
| elnev 43798 | Any set that contains one ... |
| rusbcALT 43799 | A version of Russell's par... |
| compeq 43800 | Equality between two ways ... |
| compne 43801 | The complement of ` A ` is... |
| compab 43802 | Two ways of saying "the co... |
| conss2 43803 | Contrapositive law for sub... |
| conss1 43804 | Contrapositive law for sub... |
| ralbidar 43805 | More general form of ~ ral... |
| rexbidar 43806 | More general form of ~ rex... |
| dropab1 43807 | Theorem to aid use of the ... |
| dropab2 43808 | Theorem to aid use of the ... |
| ipo0 43809 | If the identity relation p... |
| ifr0 43810 | A class that is founded by... |
| ordpss 43811 | ~ ordelpss with an anteced... |
| fvsb 43812 | Explicit substitution of a... |
| fveqsb 43813 | Implicit substitution of a... |
| xpexb 43814 | A Cartesian product exists... |
| trelpss 43815 | An element of a transitive... |
| addcomgi 43816 | Generalization of commutat... |
| addrval 43826 | Value of the operation of ... |
| subrval 43827 | Value of the operation of ... |
| mulvval 43828 | Value of the operation of ... |
| addrfv 43829 | Vector addition at a value... |
| subrfv 43830 | Vector subtraction at a va... |
| mulvfv 43831 | Scalar multiplication at a... |
| addrfn 43832 | Vector addition produces a... |
| subrfn 43833 | Vector subtraction produce... |
| mulvfn 43834 | Scalar multiplication prod... |
| addrcom 43835 | Vector addition is commuta... |
| idiALT 43839 | Placeholder for ~ idi . T... |
| exbir 43840 | Exportation implication al... |
| 3impexpbicom 43841 | Version of ~ 3impexp where... |
| 3impexpbicomi 43842 | Inference associated with ... |
| bi1imp 43843 | Importation inference simi... |
| bi2imp 43844 | Importation inference simi... |
| bi3impb 43845 | Similar to ~ 3impb with im... |
| bi3impa 43846 | Similar to ~ 3impa with im... |
| bi23impib 43847 | ~ 3impib with the inner im... |
| bi13impib 43848 | ~ 3impib with the outer im... |
| bi123impib 43849 | ~ 3impib with the implicat... |
| bi13impia 43850 | ~ 3impia with the outer im... |
| bi123impia 43851 | ~ 3impia with the implicat... |
| bi33imp12 43852 | ~ 3imp with innermost impl... |
| bi23imp13 43853 | ~ 3imp with middle implica... |
| bi13imp23 43854 | ~ 3imp with outermost impl... |
| bi13imp2 43855 | Similar to ~ 3imp except t... |
| bi12imp3 43856 | Similar to ~ 3imp except a... |
| bi23imp1 43857 | Similar to ~ 3imp except a... |
| bi123imp0 43858 | Similar to ~ 3imp except a... |
| 4animp1 43859 | A single hypothesis unific... |
| 4an31 43860 | A rearrangement of conjunc... |
| 4an4132 43861 | A rearrangement of conjunc... |
| expcomdg 43862 | Biconditional form of ~ ex... |
| iidn3 43863 | ~ idn3 without virtual ded... |
| ee222 43864 | ~ e222 without virtual ded... |
| ee3bir 43865 | Right-biconditional form o... |
| ee13 43866 | ~ e13 without virtual dedu... |
| ee121 43867 | ~ e121 without virtual ded... |
| ee122 43868 | ~ e122 without virtual ded... |
| ee333 43869 | ~ e333 without virtual ded... |
| ee323 43870 | ~ e323 without virtual ded... |
| 3ornot23 43871 | If the second and third di... |
| orbi1r 43872 | ~ orbi1 with order of disj... |
| 3orbi123 43873 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 43874 | Closed form of ~ syl5 . D... |
| impexpd 43875 | The following User's Proof... |
| com3rgbi 43876 | The following User's Proof... |
| impexpdcom 43877 | The following User's Proof... |
| ee1111 43878 | Non-virtual deduction form... |
| pm2.43bgbi 43879 | Logical equivalence of a 2... |
| pm2.43cbi 43880 | Logical equivalence of a 3... |
| ee233 43881 | Non-virtual deduction form... |
| imbi13 43882 | Join three logical equival... |
| ee33 43883 | Non-virtual deduction form... |
| con5 43884 | Biconditional contrapositi... |
| con5i 43885 | Inference form of ~ con5 .... |
| exlimexi 43886 | Inference similar to Theor... |
| sb5ALT 43887 | Equivalence for substituti... |
| eexinst01 43888 | ~ exinst01 without virtual... |
| eexinst11 43889 | ~ exinst11 without virtual... |
| vk15.4j 43890 | Excercise 4j of Unit 15 of... |
| notnotrALT 43891 | Converse of double negatio... |
| con3ALT2 43892 | Contraposition. Alternate... |
| ssralv2 43893 | Quantification restricted ... |
| sbc3or 43894 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 43895 | Closed form of ~ alrimi wi... |
| rspsbc2 43896 | ~ rspsbc with two quantify... |
| sbcoreleleq 43897 | Substitution of a setvar v... |
| tratrb 43898 | If a class is transitive a... |
| ordelordALT 43899 | An element of an ordinal c... |
| sbcim2g 43900 | Distribution of class subs... |
| sbcbi 43901 | Implication form of ~ sbcb... |
| trsbc 43902 | Formula-building inference... |
| truniALT 43903 | The union of a class of tr... |
| onfrALTlem5 43904 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 43905 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 43906 | Lemma for ~ onfrALT . (Co... |
| ggen31 43907 | ~ gen31 without virtual de... |
| onfrALTlem2 43908 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 43909 | A theorem pertaining to th... |
| onfrALTlem1 43910 | Lemma for ~ onfrALT . (Co... |
| onfrALT 43911 | The membership relation is... |
| 19.41rg 43912 | Closed form of right-to-le... |
| opelopab4 43913 | Ordered pair membership in... |
| 2pm13.193 43914 | ~ pm13.193 for two variabl... |
| hbntal 43915 | A closed form of ~ hbn . ~... |
| hbimpg 43916 | A closed form of ~ hbim . ... |
| hbalg 43917 | Closed form of ~ hbal . D... |
| hbexg 43918 | Closed form of ~ nfex . D... |
| ax6e2eq 43919 | Alternate form of ~ ax6e f... |
| ax6e2nd 43920 | If at least two sets exist... |
| ax6e2ndeq 43921 | "At least two sets exist" ... |
| 2sb5nd 43922 | Equivalence for double sub... |
| 2uasbanh 43923 | Distribute the unabbreviat... |
| 2uasban 43924 | Distribute the unabbreviat... |
| e2ebind 43925 | Absorption of an existenti... |
| elpwgded 43926 | ~ elpwgdedVD in convention... |
| trelded 43927 | Deduction form of ~ trel .... |
| jaoded 43928 | Deduction form of ~ jao . ... |
| sbtT 43929 | A substitution into a theo... |
| not12an2impnot1 43930 | If a double conjunction is... |
| in1 43933 | Inference form of ~ df-vd1... |
| iin1 43934 | ~ in1 without virtual dedu... |
| dfvd1ir 43935 | Inference form of ~ df-vd1... |
| idn1 43936 | Virtual deduction identity... |
| dfvd1imp 43937 | Left-to-right part of defi... |
| dfvd1impr 43938 | Right-to-left part of defi... |
| dfvd2 43941 | Definition of a 2-hypothes... |
| dfvd2an 43944 | Definition of a 2-hypothes... |
| dfvd2ani 43945 | Inference form of ~ dfvd2a... |
| dfvd2anir 43946 | Right-to-left inference fo... |
| dfvd2i 43947 | Inference form of ~ dfvd2 ... |
| dfvd2ir 43948 | Right-to-left inference fo... |
| dfvd3 43953 | Definition of a 3-hypothes... |
| dfvd3i 43954 | Inference form of ~ dfvd3 ... |
| dfvd3ir 43955 | Right-to-left inference fo... |
| dfvd3an 43956 | Definition of a 3-hypothes... |
| dfvd3ani 43957 | Inference form of ~ dfvd3a... |
| dfvd3anir 43958 | Right-to-left inference fo... |
| vd01 43959 | A virtual hypothesis virtu... |
| vd02 43960 | Two virtual hypotheses vir... |
| vd03 43961 | A theorem is virtually inf... |
| vd12 43962 | A virtual deduction with 1... |
| vd13 43963 | A virtual deduction with 1... |
| vd23 43964 | A virtual deduction with 2... |
| dfvd2imp 43965 | The virtual deduction form... |
| dfvd2impr 43966 | A 2-antecedent nested impl... |
| in2 43967 | The virtual deduction intr... |
| int2 43968 | The virtual deduction intr... |
| iin2 43969 | ~ in2 without virtual dedu... |
| in2an 43970 | The virtual deduction intr... |
| in3 43971 | The virtual deduction intr... |
| iin3 43972 | ~ in3 without virtual dedu... |
| in3an 43973 | The virtual deduction intr... |
| int3 43974 | The virtual deduction intr... |
| idn2 43975 | Virtual deduction identity... |
| iden2 43976 | Virtual deduction identity... |
| idn3 43977 | Virtual deduction identity... |
| gen11 43978 | Virtual deduction generali... |
| gen11nv 43979 | Virtual deduction generali... |
| gen12 43980 | Virtual deduction generali... |
| gen21 43981 | Virtual deduction generali... |
| gen21nv 43982 | Virtual deduction form of ... |
| gen31 43983 | Virtual deduction generali... |
| gen22 43984 | Virtual deduction generali... |
| ggen22 43985 | ~ gen22 without virtual de... |
| exinst 43986 | Existential Instantiation.... |
| exinst01 43987 | Existential Instantiation.... |
| exinst11 43988 | Existential Instantiation.... |
| e1a 43989 | A Virtual deduction elimin... |
| el1 43990 | A Virtual deduction elimin... |
| e1bi 43991 | Biconditional form of ~ e1... |
| e1bir 43992 | Right biconditional form o... |
| e2 43993 | A virtual deduction elimin... |
| e2bi 43994 | Biconditional form of ~ e2... |
| e2bir 43995 | Right biconditional form o... |
| ee223 43996 | ~ e223 without virtual ded... |
| e223 43997 | A virtual deduction elimin... |
| e222 43998 | A virtual deduction elimin... |
| e220 43999 | A virtual deduction elimin... |
| ee220 44000 | ~ e220 without virtual ded... |
| e202 44001 | A virtual deduction elimin... |
| ee202 44002 | ~ e202 without virtual ded... |
| e022 44003 | A virtual deduction elimin... |
| ee022 44004 | ~ e022 without virtual ded... |
| e002 44005 | A virtual deduction elimin... |
| ee002 44006 | ~ e002 without virtual ded... |
| e020 44007 | A virtual deduction elimin... |
| ee020 44008 | ~ e020 without virtual ded... |
| e200 44009 | A virtual deduction elimin... |
| ee200 44010 | ~ e200 without virtual ded... |
| e221 44011 | A virtual deduction elimin... |
| ee221 44012 | ~ e221 without virtual ded... |
| e212 44013 | A virtual deduction elimin... |
| ee212 44014 | ~ e212 without virtual ded... |
| e122 44015 | A virtual deduction elimin... |
| e112 44016 | A virtual deduction elimin... |
| ee112 44017 | ~ e112 without virtual ded... |
| e121 44018 | A virtual deduction elimin... |
| e211 44019 | A virtual deduction elimin... |
| ee211 44020 | ~ e211 without virtual ded... |
| e210 44021 | A virtual deduction elimin... |
| ee210 44022 | ~ e210 without virtual ded... |
| e201 44023 | A virtual deduction elimin... |
| ee201 44024 | ~ e201 without virtual ded... |
| e120 44025 | A virtual deduction elimin... |
| ee120 44026 | Virtual deduction rule ~ e... |
| e021 44027 | A virtual deduction elimin... |
| ee021 44028 | ~ e021 without virtual ded... |
| e012 44029 | A virtual deduction elimin... |
| ee012 44030 | ~ e012 without virtual ded... |
| e102 44031 | A virtual deduction elimin... |
| ee102 44032 | ~ e102 without virtual ded... |
| e22 44033 | A virtual deduction elimin... |
| e22an 44034 | Conjunction form of ~ e22 ... |
| ee22an 44035 | ~ e22an without virtual de... |
| e111 44036 | A virtual deduction elimin... |
| e1111 44037 | A virtual deduction elimin... |
| e110 44038 | A virtual deduction elimin... |
| ee110 44039 | ~ e110 without virtual ded... |
| e101 44040 | A virtual deduction elimin... |
| ee101 44041 | ~ e101 without virtual ded... |
| e011 44042 | A virtual deduction elimin... |
| ee011 44043 | ~ e011 without virtual ded... |
| e100 44044 | A virtual deduction elimin... |
| ee100 44045 | ~ e100 without virtual ded... |
| e010 44046 | A virtual deduction elimin... |
| ee010 44047 | ~ e010 without virtual ded... |
| e001 44048 | A virtual deduction elimin... |
| ee001 44049 | ~ e001 without virtual ded... |
| e11 44050 | A virtual deduction elimin... |
| e11an 44051 | Conjunction form of ~ e11 ... |
| ee11an 44052 | ~ e11an without virtual de... |
| e01 44053 | A virtual deduction elimin... |
| e01an 44054 | Conjunction form of ~ e01 ... |
| ee01an 44055 | ~ e01an without virtual de... |
| e10 44056 | A virtual deduction elimin... |
| e10an 44057 | Conjunction form of ~ e10 ... |
| ee10an 44058 | ~ e10an without virtual de... |
| e02 44059 | A virtual deduction elimin... |
| e02an 44060 | Conjunction form of ~ e02 ... |
| ee02an 44061 | ~ e02an without virtual de... |
| eel021old 44062 | ~ el021old without virtual... |
| el021old 44063 | A virtual deduction elimin... |
| eel132 44064 | ~ syl2an with antecedents ... |
| eel000cT 44065 | An elimination deduction. ... |
| eel0TT 44066 | An elimination deduction. ... |
| eelT00 44067 | An elimination deduction. ... |
| eelTTT 44068 | An elimination deduction. ... |
| eelT11 44069 | An elimination deduction. ... |
| eelT1 44070 | Syllogism inference combin... |
| eelT12 44071 | An elimination deduction. ... |
| eelTT1 44072 | An elimination deduction. ... |
| eelT01 44073 | An elimination deduction. ... |
| eel0T1 44074 | An elimination deduction. ... |
| eel12131 44075 | An elimination deduction. ... |
| eel2131 44076 | ~ syl2an with antecedents ... |
| eel3132 44077 | ~ syl2an with antecedents ... |
| eel0321old 44078 | ~ el0321old without virtua... |
| el0321old 44079 | A virtual deduction elimin... |
| eel2122old 44080 | ~ el2122old without virtua... |
| el2122old 44081 | A virtual deduction elimin... |
| eel0000 44082 | Elimination rule similar t... |
| eel00001 44083 | An elimination deduction. ... |
| eel00000 44084 | Elimination rule similar ~... |
| eel11111 44085 | Five-hypothesis eliminatio... |
| e12 44086 | A virtual deduction elimin... |
| e12an 44087 | Conjunction form of ~ e12 ... |
| el12 44088 | Virtual deduction form of ... |
| e20 44089 | A virtual deduction elimin... |
| e20an 44090 | Conjunction form of ~ e20 ... |
| ee20an 44091 | ~ e20an without virtual de... |
| e21 44092 | A virtual deduction elimin... |
| e21an 44093 | Conjunction form of ~ e21 ... |
| ee21an 44094 | ~ e21an without virtual de... |
| e333 44095 | A virtual deduction elimin... |
| e33 44096 | A virtual deduction elimin... |
| e33an 44097 | Conjunction form of ~ e33 ... |
| ee33an 44098 | ~ e33an without virtual de... |
| e3 44099 | Meta-connective form of ~ ... |
| e3bi 44100 | Biconditional form of ~ e3... |
| e3bir 44101 | Right biconditional form o... |
| e03 44102 | A virtual deduction elimin... |
| ee03 44103 | ~ e03 without virtual dedu... |
| e03an 44104 | Conjunction form of ~ e03 ... |
| ee03an 44105 | Conjunction form of ~ ee03... |
| e30 44106 | A virtual deduction elimin... |
| ee30 44107 | ~ e30 without virtual dedu... |
| e30an 44108 | A virtual deduction elimin... |
| ee30an 44109 | Conjunction form of ~ ee30... |
| e13 44110 | A virtual deduction elimin... |
| e13an 44111 | A virtual deduction elimin... |
| ee13an 44112 | ~ e13an without virtual de... |
| e31 44113 | A virtual deduction elimin... |
| ee31 44114 | ~ e31 without virtual dedu... |
| e31an 44115 | A virtual deduction elimin... |
| ee31an 44116 | ~ e31an without virtual de... |
| e23 44117 | A virtual deduction elimin... |
| e23an 44118 | A virtual deduction elimin... |
| ee23an 44119 | ~ e23an without virtual de... |
| e32 44120 | A virtual deduction elimin... |
| ee32 44121 | ~ e32 without virtual dedu... |
| e32an 44122 | A virtual deduction elimin... |
| ee32an 44123 | ~ e33an without virtual de... |
| e123 44124 | A virtual deduction elimin... |
| ee123 44125 | ~ e123 without virtual ded... |
| el123 44126 | A virtual deduction elimin... |
| e233 44127 | A virtual deduction elimin... |
| e323 44128 | A virtual deduction elimin... |
| e000 44129 | A virtual deduction elimin... |
| e00 44130 | Elimination rule identical... |
| e00an 44131 | Elimination rule identical... |
| eel00cT 44132 | An elimination deduction. ... |
| eelTT 44133 | An elimination deduction. ... |
| e0a 44134 | Elimination rule identical... |
| eelT 44135 | An elimination deduction. ... |
| eel0cT 44136 | An elimination deduction. ... |
| eelT0 44137 | An elimination deduction. ... |
| e0bi 44138 | Elimination rule identical... |
| e0bir 44139 | Elimination rule identical... |
| uun0.1 44140 | Convention notation form o... |
| un0.1 44141 | ` T. ` is the constant tru... |
| uunT1 44142 | A deduction unionizing a n... |
| uunT1p1 44143 | A deduction unionizing a n... |
| uunT21 44144 | A deduction unionizing a n... |
| uun121 44145 | A deduction unionizing a n... |
| uun121p1 44146 | A deduction unionizing a n... |
| uun132 44147 | A deduction unionizing a n... |
| uun132p1 44148 | A deduction unionizing a n... |
| anabss7p1 44149 | A deduction unionizing a n... |
| un10 44150 | A unionizing deduction. (... |
| un01 44151 | A unionizing deduction. (... |
| un2122 44152 | A deduction unionizing a n... |
| uun2131 44153 | A deduction unionizing a n... |
| uun2131p1 44154 | A deduction unionizing a n... |
| uunTT1 44155 | A deduction unionizing a n... |
| uunTT1p1 44156 | A deduction unionizing a n... |
| uunTT1p2 44157 | A deduction unionizing a n... |
| uunT11 44158 | A deduction unionizing a n... |
| uunT11p1 44159 | A deduction unionizing a n... |
| uunT11p2 44160 | A deduction unionizing a n... |
| uunT12 44161 | A deduction unionizing a n... |
| uunT12p1 44162 | A deduction unionizing a n... |
| uunT12p2 44163 | A deduction unionizing a n... |
| uunT12p3 44164 | A deduction unionizing a n... |
| uunT12p4 44165 | A deduction unionizing a n... |
| uunT12p5 44166 | A deduction unionizing a n... |
| uun111 44167 | A deduction unionizing a n... |
| 3anidm12p1 44168 | A deduction unionizing a n... |
| 3anidm12p2 44169 | A deduction unionizing a n... |
| uun123 44170 | A deduction unionizing a n... |
| uun123p1 44171 | A deduction unionizing a n... |
| uun123p2 44172 | A deduction unionizing a n... |
| uun123p3 44173 | A deduction unionizing a n... |
| uun123p4 44174 | A deduction unionizing a n... |
| uun2221 44175 | A deduction unionizing a n... |
| uun2221p1 44176 | A deduction unionizing a n... |
| uun2221p2 44177 | A deduction unionizing a n... |
| 3impdirp1 44178 | A deduction unionizing a n... |
| 3impcombi 44179 | A 1-hypothesis proposition... |
| trsspwALT 44180 | Virtual deduction proof of... |
| trsspwALT2 44181 | Virtual deduction proof of... |
| trsspwALT3 44182 | Short predicate calculus p... |
| sspwtr 44183 | Virtual deduction proof of... |
| sspwtrALT 44184 | Virtual deduction proof of... |
| sspwtrALT2 44185 | Short predicate calculus p... |
| pwtrVD 44186 | Virtual deduction proof of... |
| pwtrrVD 44187 | Virtual deduction proof of... |
| suctrALT 44188 | The successor of a transit... |
| snssiALTVD 44189 | Virtual deduction proof of... |
| snssiALT 44190 | If a class is an element o... |
| snsslVD 44191 | Virtual deduction proof of... |
| snssl 44192 | If a singleton is a subcla... |
| snelpwrVD 44193 | Virtual deduction proof of... |
| unipwrVD 44194 | Virtual deduction proof of... |
| unipwr 44195 | A class is a subclass of t... |
| sstrALT2VD 44196 | Virtual deduction proof of... |
| sstrALT2 44197 | Virtual deduction proof of... |
| suctrALT2VD 44198 | Virtual deduction proof of... |
| suctrALT2 44199 | Virtual deduction proof of... |
| elex2VD 44200 | Virtual deduction proof of... |
| elex22VD 44201 | Virtual deduction proof of... |
| eqsbc2VD 44202 | Virtual deduction proof of... |
| zfregs2VD 44203 | Virtual deduction proof of... |
| tpid3gVD 44204 | Virtual deduction proof of... |
| en3lplem1VD 44205 | Virtual deduction proof of... |
| en3lplem2VD 44206 | Virtual deduction proof of... |
| en3lpVD 44207 | Virtual deduction proof of... |
| simplbi2VD 44208 | Virtual deduction proof of... |
| 3ornot23VD 44209 | Virtual deduction proof of... |
| orbi1rVD 44210 | Virtual deduction proof of... |
| bitr3VD 44211 | Virtual deduction proof of... |
| 3orbi123VD 44212 | Virtual deduction proof of... |
| sbc3orgVD 44213 | Virtual deduction proof of... |
| 19.21a3con13vVD 44214 | Virtual deduction proof of... |
| exbirVD 44215 | Virtual deduction proof of... |
| exbiriVD 44216 | Virtual deduction proof of... |
| rspsbc2VD 44217 | Virtual deduction proof of... |
| 3impexpVD 44218 | Virtual deduction proof of... |
| 3impexpbicomVD 44219 | Virtual deduction proof of... |
| 3impexpbicomiVD 44220 | Virtual deduction proof of... |
| sbcoreleleqVD 44221 | Virtual deduction proof of... |
| hbra2VD 44222 | Virtual deduction proof of... |
| tratrbVD 44223 | Virtual deduction proof of... |
| al2imVD 44224 | Virtual deduction proof of... |
| syl5impVD 44225 | Virtual deduction proof of... |
| idiVD 44226 | Virtual deduction proof of... |
| ancomstVD 44227 | Closed form of ~ ancoms . ... |
| ssralv2VD 44228 | Quantification restricted ... |
| ordelordALTVD 44229 | An element of an ordinal c... |
| equncomVD 44230 | If a class equals the unio... |
| equncomiVD 44231 | Inference form of ~ equnco... |
| sucidALTVD 44232 | A set belongs to its succe... |
| sucidALT 44233 | A set belongs to its succe... |
| sucidVD 44234 | A set belongs to its succe... |
| imbi12VD 44235 | Implication form of ~ imbi... |
| imbi13VD 44236 | Join three logical equival... |
| sbcim2gVD 44237 | Distribution of class subs... |
| sbcbiVD 44238 | Implication form of ~ sbcb... |
| trsbcVD 44239 | Formula-building inference... |
| truniALTVD 44240 | The union of a class of tr... |
| ee33VD 44241 | Non-virtual deduction form... |
| trintALTVD 44242 | The intersection of a clas... |
| trintALT 44243 | The intersection of a clas... |
| undif3VD 44244 | The first equality of Exer... |
| sbcssgVD 44245 | Virtual deduction proof of... |
| csbingVD 44246 | Virtual deduction proof of... |
| onfrALTlem5VD 44247 | Virtual deduction proof of... |
| onfrALTlem4VD 44248 | Virtual deduction proof of... |
| onfrALTlem3VD 44249 | Virtual deduction proof of... |
| simplbi2comtVD 44250 | Virtual deduction proof of... |
| onfrALTlem2VD 44251 | Virtual deduction proof of... |
| onfrALTlem1VD 44252 | Virtual deduction proof of... |
| onfrALTVD 44253 | Virtual deduction proof of... |
| csbeq2gVD 44254 | Virtual deduction proof of... |
| csbsngVD 44255 | Virtual deduction proof of... |
| csbxpgVD 44256 | Virtual deduction proof of... |
| csbresgVD 44257 | Virtual deduction proof of... |
| csbrngVD 44258 | Virtual deduction proof of... |
| csbima12gALTVD 44259 | Virtual deduction proof of... |
| csbunigVD 44260 | Virtual deduction proof of... |
| csbfv12gALTVD 44261 | Virtual deduction proof of... |
| con5VD 44262 | Virtual deduction proof of... |
| relopabVD 44263 | Virtual deduction proof of... |
| 19.41rgVD 44264 | Virtual deduction proof of... |
| 2pm13.193VD 44265 | Virtual deduction proof of... |
| hbimpgVD 44266 | Virtual deduction proof of... |
| hbalgVD 44267 | Virtual deduction proof of... |
| hbexgVD 44268 | Virtual deduction proof of... |
| ax6e2eqVD 44269 | The following User's Proof... |
| ax6e2ndVD 44270 | The following User's Proof... |
| ax6e2ndeqVD 44271 | The following User's Proof... |
| 2sb5ndVD 44272 | The following User's Proof... |
| 2uasbanhVD 44273 | The following User's Proof... |
| e2ebindVD 44274 | The following User's Proof... |
| sb5ALTVD 44275 | The following User's Proof... |
| vk15.4jVD 44276 | The following User's Proof... |
| notnotrALTVD 44277 | The following User's Proof... |
| con3ALTVD 44278 | The following User's Proof... |
| elpwgdedVD 44279 | Membership in a power clas... |
| sspwimp 44280 | If a class is a subclass o... |
| sspwimpVD 44281 | The following User's Proof... |
| sspwimpcf 44282 | If a class is a subclass o... |
| sspwimpcfVD 44283 | The following User's Proof... |
| suctrALTcf 44284 | The sucessor of a transiti... |
| suctrALTcfVD 44285 | The following User's Proof... |
| suctrALT3 44286 | The successor of a transit... |
| sspwimpALT 44287 | If a class is a subclass o... |
| unisnALT 44288 | A set equals the union of ... |
| notnotrALT2 44289 | Converse of double negatio... |
| sspwimpALT2 44290 | If a class is a subclass o... |
| e2ebindALT 44291 | Absorption of an existenti... |
| ax6e2ndALT 44292 | If at least two sets exist... |
| ax6e2ndeqALT 44293 | "At least two sets exist" ... |
| 2sb5ndALT 44294 | Equivalence for double sub... |
| chordthmALT 44295 | The intersecting chords th... |
| isosctrlem1ALT 44296 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 44297 | The indexed union of conne... |
| iunconnALT 44298 | The indexed union of conne... |
| sineq0ALT 44299 | A complex number whose sin... |
| evth2f 44300 | A version of ~ evth2 using... |
| elunif 44301 | A version of ~ eluni using... |
| rzalf 44302 | A version of ~ rzal using ... |
| fvelrnbf 44303 | A version of ~ fvelrnb usi... |
| rfcnpre1 44304 | If F is a continuous funct... |
| ubelsupr 44305 | If U belongs to A and U is... |
| fsumcnf 44306 | A finite sum of functions ... |
| mulltgt0 44307 | The product of a negative ... |
| rspcegf 44308 | A version of ~ rspcev usin... |
| rabexgf 44309 | A version of ~ rabexg usin... |
| fcnre 44310 | A function continuous with... |
| sumsnd 44311 | A sum of a singleton is th... |
| evthf 44312 | A version of ~ evth using ... |
| cnfex 44313 | The class of continuous fu... |
| fnchoice 44314 | For a finite set, a choice... |
| refsumcn 44315 | A finite sum of continuous... |
| rfcnpre2 44316 | If ` F ` is a continuous f... |
| cncmpmax 44317 | When the hypothesis for th... |
| rfcnpre3 44318 | If F is a continuous funct... |
| rfcnpre4 44319 | If F is a continuous funct... |
| sumpair 44320 | Sum of two distinct comple... |
| rfcnnnub 44321 | Given a real continuous fu... |
| refsum2cnlem1 44322 | This is the core Lemma for... |
| refsum2cn 44323 | The sum of two continuus r... |
| adantlllr 44324 | Deduction adding a conjunc... |
| 3adantlr3 44325 | Deduction adding a conjunc... |
| 3adantll2 44326 | Deduction adding a conjunc... |
| 3adantll3 44327 | Deduction adding a conjunc... |
| ssnel 44328 | If not element of a set, t... |
| sncldre 44329 | A singleton is closed w.r.... |
| n0p 44330 | A polynomial with a nonzer... |
| pm2.65ni 44331 | Inference rule for proof b... |
| pwssfi 44332 | Every element of the power... |
| iuneq2df 44333 | Equality deduction for ind... |
| nnfoctb 44334 | There exists a mapping fro... |
| ssinss1d 44335 | Intersection preserves sub... |
| elpwinss 44336 | An element of the powerset... |
| unidmex 44337 | If ` F ` is a set, then ` ... |
| ndisj2 44338 | A non-disjointness conditi... |
| zenom 44339 | The set of integer numbers... |
| uzwo4 44340 | Well-ordering principle: a... |
| unisn0 44341 | The union of the singleton... |
| ssin0 44342 | If two classes are disjoin... |
| inabs3 44343 | Absorption law for interse... |
| pwpwuni 44344 | Relationship between power... |
| disjiun2 44345 | In a disjoint collection, ... |
| 0pwfi 44346 | The empty set is in any po... |
| ssinss2d 44347 | Intersection preserves sub... |
| zct 44348 | The set of integer numbers... |
| pwfin0 44349 | A finite set always belong... |
| uzct 44350 | An upper integer set is co... |
| iunxsnf 44351 | A singleton index picks ou... |
| fiiuncl 44352 | If a set is closed under t... |
| iunp1 44353 | The addition of the next s... |
| fiunicl 44354 | If a set is closed under t... |
| ixpeq2d 44355 | Equality theorem for infin... |
| disjxp1 44356 | The sets of a cartesian pr... |
| disjsnxp 44357 | The sets in the cartesian ... |
| eliind 44358 | Membership in indexed inte... |
| rspcef 44359 | Restricted existential spe... |
| inn0f 44360 | A nonempty intersection. ... |
| ixpssmapc 44361 | An infinite Cartesian prod... |
| inn0 44362 | A nonempty intersection. ... |
| elintd 44363 | Membership in class inters... |
| ssdf 44364 | A sufficient condition for... |
| brneqtrd 44365 | Substitution of equal clas... |
| ssnct 44366 | A set containing an uncoun... |
| ssuniint 44367 | Sufficient condition for b... |
| elintdv 44368 | Membership in class inters... |
| ssd 44369 | A sufficient condition for... |
| ralimralim 44370 | Introducing any antecedent... |
| snelmap 44371 | Membership of the element ... |
| xrnmnfpnf 44372 | An extended real that is n... |
| nelrnmpt 44373 | Non-membership in the rang... |
| iuneq1i 44374 | Equality theorem for index... |
| nssrex 44375 | Negation of subclass relat... |
| ssinc 44376 | Inclusion relation for a m... |
| ssdec 44377 | Inclusion relation for a m... |
| elixpconstg 44378 | Membership in an infinite ... |
| iineq1d 44379 | Equality theorem for index... |
| metpsmet 44380 | A metric is a pseudometric... |
| ixpssixp 44381 | Subclass theorem for infin... |
| ballss3 44382 | A sufficient condition for... |
| iunincfi 44383 | Given a sequence of increa... |
| nsstr 44384 | If it's not a subclass, it... |
| rexanuz3 44385 | Combine two different uppe... |
| cbvmpo2 44386 | Rule to change the second ... |
| cbvmpo1 44387 | Rule to change the first b... |
| eliuniin 44388 | Indexed union of indexed i... |
| ssabf 44389 | Subclass of a class abstra... |
| pssnssi 44390 | A proper subclass does not... |
| rabidim2 44391 | Membership in a restricted... |
| eluni2f 44392 | Membership in class union.... |
| eliin2f 44393 | Membership in indexed inte... |
| nssd 44394 | Negation of subclass relat... |
| iineq12dv 44395 | Equality deduction for ind... |
| supxrcld 44396 | The supremum of an arbitra... |
| elrestd 44397 | A sufficient condition for... |
| eliuniincex 44398 | Counterexample to show tha... |
| eliincex 44399 | Counterexample to show tha... |
| eliinid 44400 | Membership in an indexed i... |
| abssf 44401 | Class abstraction in a sub... |
| supxrubd 44402 | A member of a set of exten... |
| ssrabf 44403 | Subclass of a restricted c... |
| ssrabdf 44404 | Subclass of a restricted c... |
| eliin2 44405 | Membership in indexed inte... |
| ssrab2f 44406 | Subclass relation for a re... |
| restuni3 44407 | The underlying set of a su... |
| rabssf 44408 | Restricted class abstracti... |
| eliuniin2 44409 | Indexed union of indexed i... |
| restuni4 44410 | The underlying set of a su... |
| restuni6 44411 | The underlying set of a su... |
| restuni5 44412 | The underlying set of a su... |
| unirestss 44413 | The union of an elementwis... |
| iniin1 44414 | Indexed intersection of in... |
| iniin2 44415 | Indexed intersection of in... |
| cbvrabv2 44416 | A more general version of ... |
| cbvrabv2w 44417 | A more general version of ... |
| iinssiin 44418 | Subset implication for an ... |
| eliind2 44419 | Membership in indexed inte... |
| iinssd 44420 | Subset implication for an ... |
| rabbida2 44421 | Equivalent wff's yield equ... |
| iinexd 44422 | The existence of an indexe... |
| rabexf 44423 | Separation Scheme in terms... |
| rabbida3 44424 | Equivalent wff's yield equ... |
| r19.36vf 44425 | Restricted quantifier vers... |
| raleqd 44426 | Equality deduction for res... |
| iinssf 44427 | Subset implication for an ... |
| iinssdf 44428 | Subset implication for an ... |
| resabs2i 44429 | Absorption law for restric... |
| ssdf2 44430 | A sufficient condition for... |
| rabssd 44431 | Restricted class abstracti... |
| rexnegd 44432 | Minus a real number. (Con... |
| rexlimd3 44433 | * Inference from Theorem 1... |
| resabs1i 44434 | Absorption law for restric... |
| nel1nelin 44435 | Membership in an intersect... |
| nel2nelin 44436 | Membership in an intersect... |
| nel1nelini 44437 | Membership in an intersect... |
| nel2nelini 44438 | Membership in an intersect... |
| eliunid 44439 | Membership in indexed unio... |
| reximddv3 44440 | Deduction from Theorem 19.... |
| reximdd 44441 | Deduction from Theorem 19.... |
| unfid 44442 | The union of two finite se... |
| inopnd 44443 | The intersection of two op... |
| ss2rabdf 44444 | Deduction of restricted ab... |
| restopn3 44445 | If ` A ` is open, then ` A... |
| restopnssd 44446 | A topology restricted to a... |
| restsubel 44447 | A subset belongs in the sp... |
| toprestsubel 44448 | A subset is open in the to... |
| rabidd 44449 | An "identity" law of concr... |
| iunssdf 44450 | Subset theorem for an inde... |
| iinss2d 44451 | Subset implication for an ... |
| r19.3rzf 44452 | Restricted quantification ... |
| r19.28zf 44453 | Restricted quantifier vers... |
| iindif2f 44454 | Indexed intersection of cl... |
| ralfal 44455 | Two ways of expressing emp... |
| archd 44456 | Archimedean property of re... |
| eliund 44457 | Membership in indexed unio... |
| nimnbi 44458 | If an implication is false... |
| nimnbi2 44459 | If an implication is false... |
| notbicom 44460 | Commutative law for the ne... |
| rexeqif 44461 | Equality inference for res... |
| rspced 44462 | Restricted existential spe... |
| feq1dd 44463 | Equality deduction for fun... |
| fnresdmss 44464 | A function does not change... |
| fmptsnxp 44465 | Maps-to notation and Carte... |
| fvmpt2bd 44466 | Value of a function given ... |
| rnmptfi 44467 | The range of a function wi... |
| fresin2 44468 | Restriction of a function ... |
| ffi 44469 | A function with finite dom... |
| suprnmpt 44470 | An explicit bound for the ... |
| rnffi 44471 | The range of a function wi... |
| mptelpm 44472 | A function in maps-to nota... |
| rnmptpr 44473 | Range of a function define... |
| resmpti 44474 | Restriction of the mapping... |
| founiiun 44475 | Union expressed as an inde... |
| rnresun 44476 | Distribution law for range... |
| elrnmptf 44477 | The range of a function in... |
| rnmptssrn 44478 | Inclusion relation for two... |
| disjf1 44479 | A 1 to 1 mapping built fro... |
| rnsnf 44480 | The range of a function wh... |
| wessf1ornlem 44481 | Given a function ` F ` on ... |
| wessf1orn 44482 | Given a function ` F ` on ... |
| nelrnres 44483 | If ` A ` is not in the ran... |
| disjrnmpt2 44484 | Disjointness of the range ... |
| elrnmpt1sf 44485 | Elementhood in an image se... |
| founiiun0 44486 | Union expressed as an inde... |
| disjf1o 44487 | A bijection built from dis... |
| disjinfi 44488 | Only a finite number of di... |
| fvovco 44489 | Value of the composition o... |
| ssnnf1octb 44490 | There exists a bijection b... |
| nnf1oxpnn 44491 | There is a bijection betwe... |
| rnmptssd 44492 | The range of a function gi... |
| projf1o 44493 | A biijection from a set to... |
| fvmap 44494 | Function value for a membe... |
| fvixp2 44495 | Projection of a factor of ... |
| choicefi 44496 | For a finite set, a choice... |
| mpct 44497 | The exponentiation of a co... |
| cnmetcoval 44498 | Value of the distance func... |
| fcomptss 44499 | Express composition of two... |
| elmapsnd 44500 | Membership in a set expone... |
| mapss2 44501 | Subset inheritance for set... |
| fsneq 44502 | Equality condition for two... |
| difmap 44503 | Difference of two sets exp... |
| unirnmap 44504 | Given a subset of a set ex... |
| inmap 44505 | Intersection of two sets e... |
| fcoss 44506 | Composition of two mapping... |
| fsneqrn 44507 | Equality condition for two... |
| difmapsn 44508 | Difference of two sets exp... |
| mapssbi 44509 | Subset inheritance for set... |
| unirnmapsn 44510 | Equality theorem for a sub... |
| iunmapss 44511 | The indexed union of set e... |
| ssmapsn 44512 | A subset ` C ` of a set ex... |
| iunmapsn 44513 | The indexed union of set e... |
| absfico 44514 | Mapping domain and codomai... |
| icof 44515 | The set of left-closed rig... |
| elpmrn 44516 | The range of a partial fun... |
| imaexi 44517 | The image of a set is a se... |
| axccdom 44518 | Relax the constraint on ax... |
| dmmptdff 44519 | The domain of the mapping ... |
| dmmptdf 44520 | The domain of the mapping ... |
| elpmi2 44521 | The domain of a partial fu... |
| dmrelrnrel 44522 | A relation preserving func... |
| fvcod 44523 | Value of a function compos... |
| elrnmpoid 44524 | Membership in the range of... |
| axccd 44525 | An alternative version of ... |
| axccd2 44526 | An alternative version of ... |
| feqresmptf 44527 | Express a restricted funct... |
| dmresss 44528 | The domain of a restrictio... |
| dmmptssf 44529 | The domain of a mapping is... |
| dmmptdf2 44530 | The domain of the mapping ... |
| dmuz 44531 | Domain of the upper intege... |
| fmptd2f 44532 | Domain and codomain of the... |
| mpteq1df 44533 | An equality theorem for th... |
| mpteq1dfOLD 44534 | Obsolete version of ~ mpte... |
| mptexf 44535 | If the domain of a functio... |
| fvmpt4 44536 | Value of a function given ... |
| fmptf 44537 | Functionality of the mappi... |
| resimass 44538 | The image of a restriction... |
| mptssid 44539 | The mapping operation expr... |
| mptfnd 44540 | The maps-to notation defin... |
| mpteq12daOLD 44541 | Obsolete version of ~ mpte... |
| rnmptlb 44542 | Boundness below of the ran... |
| rnmptbddlem 44543 | Boundness of the range of ... |
| rnmptbdd 44544 | Boundness of the range of ... |
| funimaeq 44545 | Membership relation for th... |
| rnmptssf 44546 | The range of a function gi... |
| rnmptbd2lem 44547 | Boundness below of the ran... |
| rnmptbd2 44548 | Boundness below of the ran... |
| infnsuprnmpt 44549 | The indexed infimum of rea... |
| suprclrnmpt 44550 | Closure of the indexed sup... |
| suprubrnmpt2 44551 | A member of a nonempty ind... |
| suprubrnmpt 44552 | A member of a nonempty ind... |
| rnmptssdf 44553 | The range of a function gi... |
| rnmptbdlem 44554 | Boundness above of the ran... |
| rnmptbd 44555 | Boundness above of the ran... |
| rnmptss2 44556 | The range of a function gi... |
| elmptima 44557 | The image of a function in... |
| ralrnmpt3 44558 | A restricted quantifier ov... |
| fvelima2 44559 | Function value in an image... |
| rnmptssbi 44560 | The range of a function gi... |
| imass2d 44561 | Subset theorem for image. ... |
| imassmpt 44562 | Membership relation for th... |
| fpmd 44563 | A total function is a part... |
| fconst7 44564 | An alternative way to expr... |
| fnmptif 44565 | Functionality and domain o... |
| dmmptif 44566 | Domain of the mapping oper... |
| mpteq2dfa 44567 | Slightly more general equa... |
| dmmpt1 44568 | The domain of the mapping ... |
| fmptff 44569 | Functionality of the mappi... |
| fvmptelcdmf 44570 | The value of a function at... |
| fmptdff 44571 | A version of ~ fmptd using... |
| fvmpt2df 44572 | Deduction version of ~ fvm... |
| rn1st 44573 | The range of a function wi... |
| rnmptssff 44574 | The range of a function gi... |
| rnmptssdff 44575 | The range of a function gi... |
| fvmpt4d 44576 | Value of a function given ... |
| sub2times 44577 | Subtracting from a number,... |
| nnxrd 44578 | A natural number is an ext... |
| nnxr 44579 | A natural number is an ext... |
| abssubrp 44580 | The distance of two distin... |
| elfzfzo 44581 | Relationship between membe... |
| oddfl 44582 | Odd number representation ... |
| abscosbd 44583 | Bound for the absolute val... |
| mul13d 44584 | Commutative/associative la... |
| negpilt0 44585 | Negative ` _pi ` is negati... |
| dstregt0 44586 | A complex number ` A ` tha... |
| subadd4b 44587 | Rearrangement of 4 terms i... |
| xrlttri5d 44588 | Not equal and not larger i... |
| neglt 44589 | The negative of a positive... |
| zltlesub 44590 | If an integer ` N ` is les... |
| divlt0gt0d 44591 | The ratio of a negative nu... |
| subsub23d 44592 | Swap subtrahend and result... |
| 2timesgt 44593 | Double of a positive real ... |
| reopn 44594 | The reals are open with re... |
| sub31 44595 | Swap the first and third t... |
| nnne1ge2 44596 | A positive integer which i... |
| lefldiveq 44597 | A closed enough, smaller r... |
| negsubdi3d 44598 | Distribution of negative o... |
| ltdiv2dd 44599 | Division of a positive num... |
| abssinbd 44600 | Bound for the absolute val... |
| halffl 44601 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 44602 | Ordering relation for a st... |
| hashssle 44603 | The size of a subset of a ... |
| lttri5d 44604 | Not equal and not larger i... |
| fzisoeu 44605 | A finite ordered set has a... |
| lt3addmuld 44606 | If three real numbers are ... |
| absnpncan2d 44607 | Triangular inequality, com... |
| fperiodmullem 44608 | A function with period ` T... |
| fperiodmul 44609 | A function with period T i... |
| upbdrech 44610 | Choice of an upper bound f... |
| lt4addmuld 44611 | If four real numbers are l... |
| absnpncan3d 44612 | Triangular inequality, com... |
| upbdrech2 44613 | Choice of an upper bound f... |
| ssfiunibd 44614 | A finite union of bounded ... |
| fzdifsuc2 44615 | Remove a successor from th... |
| fzsscn 44616 | A finite sequence of integ... |
| divcan8d 44617 | A cancellation law for div... |
| dmmcand 44618 | Cancellation law for divis... |
| fzssre 44619 | A finite sequence of integ... |
| bccld 44620 | A binomial coefficient, in... |
| leadd12dd 44621 | Addition to both sides of ... |
| fzssnn0 44622 | A finite set of sequential... |
| xreqle 44623 | Equality implies 'less tha... |
| xaddlidd 44624 | ` 0 ` is a left identity f... |
| xadd0ge 44625 | A number is less than or e... |
| elfzolem1 44626 | A member in a half-open in... |
| xrgtned 44627 | 'Greater than' implies not... |
| xrleneltd 44628 | 'Less than or equal to' an... |
| xaddcomd 44629 | The extended real addition... |
| supxrre3 44630 | The supremum of a nonempty... |
| uzfissfz 44631 | For any finite subset of t... |
| xleadd2d 44632 | Addition of extended reals... |
| suprltrp 44633 | The supremum of a nonempty... |
| xleadd1d 44634 | Addition of extended reals... |
| xreqled 44635 | Equality implies 'less tha... |
| xrgepnfd 44636 | An extended real greater t... |
| xrge0nemnfd 44637 | A nonnegative extended rea... |
| supxrgere 44638 | If a real number can be ap... |
| iuneqfzuzlem 44639 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 44640 | If two unions indexed by u... |
| xle2addd 44641 | Adding both side of two in... |
| supxrgelem 44642 | If an extended real number... |
| supxrge 44643 | If an extended real number... |
| suplesup 44644 | If any element of ` A ` ca... |
| infxrglb 44645 | The infimum of a set of ex... |
| xadd0ge2 44646 | A number is less than or e... |
| nepnfltpnf 44647 | An extended real that is n... |
| ltadd12dd 44648 | Addition to both sides of ... |
| nemnftgtmnft 44649 | An extended real that is n... |
| xrgtso 44650 | 'Greater than' is a strict... |
| rpex 44651 | The positive reals form a ... |
| xrge0ge0 44652 | A nonnegative extended rea... |
| xrssre 44653 | A subset of extended reals... |
| ssuzfz 44654 | A finite subset of the upp... |
| absfun 44655 | The absolute value is a fu... |
| infrpge 44656 | The infimum of a nonempty,... |
| xrlexaddrp 44657 | If an extended real number... |
| supsubc 44658 | The supremum function dist... |
| xralrple2 44659 | Show that ` A ` is less th... |
| nnuzdisj 44660 | The first ` N ` elements o... |
| ltdivgt1 44661 | Divsion by a number greate... |
| xrltned 44662 | 'Less than' implies not eq... |
| nnsplit 44663 | Express the set of positiv... |
| divdiv3d 44664 | Division into a fraction. ... |
| abslt2sqd 44665 | Comparison of the square o... |
| qenom 44666 | The set of rational number... |
| qct 44667 | The set of rational number... |
| xrltnled 44668 | 'Less than' in terms of 'l... |
| lenlteq 44669 | 'less than or equal to' bu... |
| xrred 44670 | An extended real that is n... |
| rr2sscn2 44671 | The cartesian square of ` ... |
| infxr 44672 | The infimum of a set of ex... |
| infxrunb2 44673 | The infimum of an unbounde... |
| infxrbnd2 44674 | The infimum of a bounded-b... |
| infleinflem1 44675 | Lemma for ~ infleinf , cas... |
| infleinflem2 44676 | Lemma for ~ infleinf , whe... |
| infleinf 44677 | If any element of ` B ` ca... |
| xralrple4 44678 | Show that ` A ` is less th... |
| xralrple3 44679 | Show that ` A ` is less th... |
| eluzelzd 44680 | A member of an upper set o... |
| suplesup2 44681 | If any element of ` A ` is... |
| recnnltrp 44682 | ` N ` is a natural number ... |
| nnn0 44683 | The set of positive intege... |
| fzct 44684 | A finite set of sequential... |
| rpgtrecnn 44685 | Any positive real number i... |
| fzossuz 44686 | A half-open integer interv... |
| infxrrefi 44687 | The real and extended real... |
| xrralrecnnle 44688 | Show that ` A ` is less th... |
| fzoct 44689 | A finite set of sequential... |
| frexr 44690 | A function taking real val... |
| nnrecrp 44691 | The reciprocal of a positi... |
| reclt0d 44692 | The reciprocal of a negati... |
| lt0neg1dd 44693 | If a number is negative, i... |
| infxrcld 44694 | The infimum of an arbitrar... |
| xrralrecnnge 44695 | Show that ` A ` is less th... |
| reclt0 44696 | The reciprocal of a negati... |
| ltmulneg 44697 | Multiplying by a negative ... |
| allbutfi 44698 | For all but finitely many.... |
| ltdiv23neg 44699 | Swap denominator with othe... |
| xreqnltd 44700 | A consequence of trichotom... |
| mnfnre2 44701 | Minus infinity is not a re... |
| zssxr 44702 | The integers are a subset ... |
| fisupclrnmpt 44703 | A nonempty finite indexed ... |
| supxrunb3 44704 | The supremum of an unbound... |
| elfzod 44705 | Membership in a half-open ... |
| fimaxre4 44706 | A nonempty finite set of r... |
| ren0 44707 | The set of reals is nonemp... |
| eluzelz2 44708 | A member of an upper set o... |
| resabs2d 44709 | Absorption law for restric... |
| uzid2 44710 | Membership of the least me... |
| supxrleubrnmpt 44711 | The supremum of a nonempty... |
| uzssre2 44712 | An upper set of integers i... |
| uzssd 44713 | Subset relationship for tw... |
| eluzd 44714 | Membership in an upper set... |
| infxrlbrnmpt2 44715 | A member of a nonempty ind... |
| xrre4 44716 | An extended real is real i... |
| uz0 44717 | The upper integers functio... |
| eluzelz2d 44718 | A member of an upper set o... |
| infleinf2 44719 | If any element in ` B ` is... |
| unb2ltle 44720 | "Unbounded below" expresse... |
| uzidd2 44721 | Membership of the least me... |
| uzssd2 44722 | Subset relationship for tw... |
| rexabslelem 44723 | An indexed set of absolute... |
| rexabsle 44724 | An indexed set of absolute... |
| allbutfiinf 44725 | Given a "for all but finit... |
| supxrrernmpt 44726 | The real and extended real... |
| suprleubrnmpt 44727 | The supremum of a nonempty... |
| infrnmptle 44728 | An indexed infimum of exte... |
| infxrunb3 44729 | The infimum of an unbounde... |
| uzn0d 44730 | The upper integers are all... |
| uzssd3 44731 | Subset relationship for tw... |
| rexabsle2 44732 | An indexed set of absolute... |
| infxrunb3rnmpt 44733 | The infimum of an unbounde... |
| supxrre3rnmpt 44734 | The indexed supremum of a ... |
| uzublem 44735 | A set of reals, indexed by... |
| uzub 44736 | A set of reals, indexed by... |
| ssrexr 44737 | A subset of the reals is a... |
| supxrmnf2 44738 | Removing minus infinity fr... |
| supxrcli 44739 | The supremum of an arbitra... |
| uzid3 44740 | Membership of the least me... |
| infxrlesupxr 44741 | The supremum of a nonempty... |
| xnegeqd 44742 | Equality of two extended n... |
| xnegrecl 44743 | The extended real negative... |
| xnegnegi 44744 | Extended real version of ~... |
| xnegeqi 44745 | Equality of two extended n... |
| nfxnegd 44746 | Deduction version of ~ nfx... |
| xnegnegd 44747 | Extended real version of ~... |
| uzred 44748 | An upper integer is a real... |
| xnegcli 44749 | Closure of extended real n... |
| supminfrnmpt 44750 | The indexed supremum of a ... |
| infxrpnf 44751 | Adding plus infinity to a ... |
| infxrrnmptcl 44752 | The infimum of an arbitrar... |
| leneg2d 44753 | Negative of one side of 'l... |
| supxrltinfxr 44754 | The supremum of the empty ... |
| max1d 44755 | A number is less than or e... |
| supxrleubrnmptf 44756 | The supremum of a nonempty... |
| nleltd 44757 | 'Not less than or equal to... |
| zxrd 44758 | An integer is an extended ... |
| infxrgelbrnmpt 44759 | The infimum of an indexed ... |
| rphalfltd 44760 | Half of a positive real is... |
| uzssz2 44761 | An upper set of integers i... |
| leneg3d 44762 | Negative of one side of 'l... |
| max2d 44763 | A number is less than or e... |
| uzn0bi 44764 | The upper integers functio... |
| xnegrecl2 44765 | If the extended real negat... |
| nfxneg 44766 | Bound-variable hypothesis ... |
| uzxrd 44767 | An upper integer is an ext... |
| infxrpnf2 44768 | Removing plus infinity fro... |
| supminfxr 44769 | The extended real suprema ... |
| infrpgernmpt 44770 | The infimum of a nonempty,... |
| xnegre 44771 | An extended real is real i... |
| xnegrecl2d 44772 | If the extended real negat... |
| uzxr 44773 | An upper integer is an ext... |
| supminfxr2 44774 | The extended real suprema ... |
| xnegred 44775 | An extended real is real i... |
| supminfxrrnmpt 44776 | The indexed supremum of a ... |
| min1d 44777 | The minimum of two numbers... |
| min2d 44778 | The minimum of two numbers... |
| pnfged 44779 | Plus infinity is an upper ... |
| xrnpnfmnf 44780 | An extended real that is n... |
| uzsscn 44781 | An upper set of integers i... |
| absimnre 44782 | The absolute value of the ... |
| uzsscn2 44783 | An upper set of integers i... |
| xrtgcntopre 44784 | The standard topologies on... |
| absimlere 44785 | The absolute value of the ... |
| rpssxr 44786 | The positive reals are a s... |
| monoordxrv 44787 | Ordering relation for a mo... |
| monoordxr 44788 | Ordering relation for a mo... |
| monoord2xrv 44789 | Ordering relation for a mo... |
| monoord2xr 44790 | Ordering relation for a mo... |
| xrpnf 44791 | An extended real is plus i... |
| xlenegcon1 44792 | Extended real version of ~... |
| xlenegcon2 44793 | Extended real version of ~... |
| pimxrneun 44794 | The preimage of a set of e... |
| caucvgbf 44795 | A function is convergent i... |
| cvgcau 44796 | A convergent function is C... |
| cvgcaule 44797 | A convergent function is C... |
| rexanuz2nf 44798 | A simple counterexample re... |
| gtnelioc 44799 | A real number larger than ... |
| ioossioc 44800 | An open interval is a subs... |
| ioondisj2 44801 | A condition for two open i... |
| ioondisj1 44802 | A condition for two open i... |
| ioogtlb 44803 | An element of a closed int... |
| evthiccabs 44804 | Extreme Value Theorem on y... |
| ltnelicc 44805 | A real number smaller than... |
| eliood 44806 | Membership in an open real... |
| iooabslt 44807 | An upper bound for the dis... |
| gtnelicc 44808 | A real number greater than... |
| iooinlbub 44809 | An open interval has empty... |
| iocgtlb 44810 | An element of a left-open ... |
| iocleub 44811 | An element of a left-open ... |
| eliccd 44812 | Membership in a closed rea... |
| eliccre 44813 | A member of a closed inter... |
| eliooshift 44814 | Element of an open interva... |
| eliocd 44815 | Membership in a left-open ... |
| icoltub 44816 | An element of a left-close... |
| eliocre 44817 | A member of a left-open ri... |
| iooltub 44818 | An element of an open inte... |
| ioontr 44819 | The interior of an interva... |
| snunioo1 44820 | The closure of one end of ... |
| lbioc 44821 | A left-open right-closed i... |
| ioomidp 44822 | The midpoint is an element... |
| iccdifioo 44823 | If the open inverval is re... |
| iccdifprioo 44824 | An open interval is the cl... |
| ioossioobi 44825 | Biconditional form of ~ io... |
| iccshift 44826 | A closed interval shifted ... |
| iccsuble 44827 | An upper bound to the dist... |
| iocopn 44828 | A left-open right-closed i... |
| eliccelioc 44829 | Membership in a closed int... |
| iooshift 44830 | An open interval shifted b... |
| iccintsng 44831 | Intersection of two adiace... |
| icoiccdif 44832 | Left-closed right-open int... |
| icoopn 44833 | A left-closed right-open i... |
| icoub 44834 | A left-closed, right-open ... |
| eliccxrd 44835 | Membership in a closed rea... |
| pnfel0pnf 44836 | ` +oo ` is a nonnegative e... |
| eliccnelico 44837 | An element of a closed int... |
| eliccelicod 44838 | A member of a closed inter... |
| ge0xrre 44839 | A nonnegative extended rea... |
| ge0lere 44840 | A nonnegative extended Rea... |
| elicores 44841 | Membership in a left-close... |
| inficc 44842 | The infimum of a nonempty ... |
| qinioo 44843 | The rational numbers are d... |
| lenelioc 44844 | A real number smaller than... |
| ioonct 44845 | A nonempty open interval i... |
| xrgtnelicc 44846 | A real number greater than... |
| iccdificc 44847 | The difference of two clos... |
| iocnct 44848 | A nonempty left-open, righ... |
| iccnct 44849 | A closed interval, with mo... |
| iooiinicc 44850 | A closed interval expresse... |
| iccgelbd 44851 | An element of a closed int... |
| iooltubd 44852 | An element of an open inte... |
| icoltubd 44853 | An element of a left-close... |
| qelioo 44854 | The rational numbers are d... |
| tgqioo2 44855 | Every open set of reals is... |
| iccleubd 44856 | An element of a closed int... |
| elioored 44857 | A member of an open interv... |
| ioogtlbd 44858 | An element of a closed int... |
| ioofun 44859 | ` (,) ` is a function. (C... |
| icomnfinre 44860 | A left-closed, right-open,... |
| sqrlearg 44861 | The square compared with i... |
| ressiocsup 44862 | If the supremum belongs to... |
| ressioosup 44863 | If the supremum does not b... |
| iooiinioc 44864 | A left-open, right-closed ... |
| ressiooinf 44865 | If the infimum does not be... |
| icogelbd 44866 | An element of a left-close... |
| iocleubd 44867 | An element of a left-open ... |
| uzinico 44868 | An upper interval of integ... |
| preimaiocmnf 44869 | Preimage of a right-closed... |
| uzinico2 44870 | An upper interval of integ... |
| uzinico3 44871 | An upper interval of integ... |
| icossico2 44872 | Condition for a closed-bel... |
| dmico 44873 | The domain of the closed-b... |
| ndmico 44874 | The closed-below, open-abo... |
| uzubioo 44875 | The upper integers are unb... |
| uzubico 44876 | The upper integers are unb... |
| uzubioo2 44877 | The upper integers are unb... |
| uzubico2 44878 | The upper integers are unb... |
| iocgtlbd 44879 | An element of a left-open ... |
| xrtgioo2 44880 | The topology on the extend... |
| tgioo4 44881 | The standard topology on t... |
| fsummulc1f 44882 | Closure of a finite sum of... |
| fsumnncl 44883 | Closure of a nonempty, fin... |
| fsumge0cl 44884 | The finite sum of nonnegat... |
| fsumf1of 44885 | Re-index a finite sum usin... |
| fsumiunss 44886 | Sum over a disjoint indexe... |
| fsumreclf 44887 | Closure of a finite sum of... |
| fsumlessf 44888 | A shorter sum of nonnegati... |
| fsumsupp0 44889 | Finite sum of function val... |
| fsumsermpt 44890 | A finite sum expressed in ... |
| fmul01 44891 | Multiplying a finite numbe... |
| fmulcl 44892 | If ' Y ' is closed under t... |
| fmuldfeqlem1 44893 | induction step for the pro... |
| fmuldfeq 44894 | X and Z are two equivalent... |
| fmul01lt1lem1 44895 | Given a finite multiplicat... |
| fmul01lt1lem2 44896 | Given a finite multiplicat... |
| fmul01lt1 44897 | Given a finite multiplicat... |
| cncfmptss 44898 | A continuous complex funct... |
| rrpsscn 44899 | The positive reals are a s... |
| mulc1cncfg 44900 | A version of ~ mulc1cncf u... |
| infrglb 44901 | The infimum of a nonempty ... |
| expcnfg 44902 | If ` F ` is a complex cont... |
| prodeq2ad 44903 | Equality deduction for pro... |
| fprodsplit1 44904 | Separate out a term in a f... |
| fprodexp 44905 | Positive integer exponenti... |
| fprodabs2 44906 | The absolute value of a fi... |
| fprod0 44907 | A finite product with a ze... |
| mccllem 44908 | * Induction step for ~ mcc... |
| mccl 44909 | A multinomial coefficient,... |
| fprodcnlem 44910 | A finite product of functi... |
| fprodcn 44911 | A finite product of functi... |
| clim1fr1 44912 | A class of sequences of fr... |
| isumneg 44913 | Negation of a converging s... |
| climrec 44914 | Limit of the reciprocal of... |
| climmulf 44915 | A version of ~ climmul usi... |
| climexp 44916 | The limit of natural power... |
| climinf 44917 | A bounded monotonic noninc... |
| climsuselem1 44918 | The subsequence index ` I ... |
| climsuse 44919 | A subsequence ` G ` of a c... |
| climrecf 44920 | A version of ~ climrec usi... |
| climneg 44921 | Complex limit of the negat... |
| climinff 44922 | A version of ~ climinf usi... |
| climdivf 44923 | Limit of the ratio of two ... |
| climreeq 44924 | If ` F ` is a real functio... |
| ellimciota 44925 | An explicit value for the ... |
| climaddf 44926 | A version of ~ climadd usi... |
| mullimc 44927 | Limit of the product of tw... |
| ellimcabssub0 44928 | An equivalent condition fo... |
| limcdm0 44929 | If a function has empty do... |
| islptre 44930 | An equivalence condition f... |
| limccog 44931 | Limit of the composition o... |
| limciccioolb 44932 | The limit of a function at... |
| climf 44933 | Express the predicate: Th... |
| mullimcf 44934 | Limit of the multiplicatio... |
| constlimc 44935 | Limit of constant function... |
| rexlim2d 44936 | Inference removing two res... |
| idlimc 44937 | Limit of the identity func... |
| divcnvg 44938 | The sequence of reciprocal... |
| limcperiod 44939 | If ` F ` is a periodic fun... |
| limcrecl 44940 | If ` F ` is a real-valued ... |
| sumnnodd 44941 | A series indexed by ` NN `... |
| lptioo2 44942 | The upper bound of an open... |
| lptioo1 44943 | The lower bound of an open... |
| elprn1 44944 | A member of an unordered p... |
| elprn2 44945 | A member of an unordered p... |
| limcmptdm 44946 | The domain of a maps-to fu... |
| clim2f 44947 | Express the predicate: Th... |
| limcicciooub 44948 | The limit of a function at... |
| ltmod 44949 | A sufficient condition for... |
| islpcn 44950 | A characterization for a l... |
| lptre2pt 44951 | If a set in the real line ... |
| limsupre 44952 | If a sequence is bounded, ... |
| limcresiooub 44953 | The left limit doesn't cha... |
| limcresioolb 44954 | The right limit doesn't ch... |
| limcleqr 44955 | If the left and the right ... |
| lptioo2cn 44956 | The upper bound of an open... |
| lptioo1cn 44957 | The lower bound of an open... |
| neglimc 44958 | Limit of the negative func... |
| addlimc 44959 | Sum of two limits. (Contr... |
| 0ellimcdiv 44960 | If the numerator converges... |
| clim2cf 44961 | Express the predicate ` F ... |
| limclner 44962 | For a limit point, both fr... |
| sublimc 44963 | Subtraction of two limits.... |
| reclimc 44964 | Limit of the reciprocal of... |
| clim0cf 44965 | Express the predicate ` F ... |
| limclr 44966 | For a limit point, both fr... |
| divlimc 44967 | Limit of the quotient of t... |
| expfac 44968 | Factorial grows faster tha... |
| climconstmpt 44969 | A constant sequence conver... |
| climresmpt 44970 | A function restricted to u... |
| climsubmpt 44971 | Limit of the difference of... |
| climsubc2mpt 44972 | Limit of the difference of... |
| climsubc1mpt 44973 | Limit of the difference of... |
| fnlimfv 44974 | The value of the limit fun... |
| climreclf 44975 | The limit of a convergent ... |
| climeldmeq 44976 | Two functions that are eve... |
| climf2 44977 | Express the predicate: Th... |
| fnlimcnv 44978 | The sequence of function v... |
| climeldmeqmpt 44979 | Two functions that are eve... |
| climfveq 44980 | Two functions that are eve... |
| clim2f2 44981 | Express the predicate: Th... |
| climfveqmpt 44982 | Two functions that are eve... |
| climd 44983 | Express the predicate: Th... |
| clim2d 44984 | The limit of complex numbe... |
| fnlimfvre 44985 | The limit function of real... |
| allbutfifvre 44986 | Given a sequence of real-v... |
| climleltrp 44987 | The limit of complex numbe... |
| fnlimfvre2 44988 | The limit function of real... |
| fnlimf 44989 | The limit function of real... |
| fnlimabslt 44990 | A sequence of function val... |
| climfveqf 44991 | Two functions that are eve... |
| climmptf 44992 | Exhibit a function ` G ` w... |
| climfveqmpt3 44993 | Two functions that are eve... |
| climeldmeqf 44994 | Two functions that are eve... |
| climreclmpt 44995 | The limit of B convergent ... |
| limsupref 44996 | If a sequence is bounded, ... |
| limsupbnd1f 44997 | If a sequence is eventuall... |
| climbddf 44998 | A converging sequence of c... |
| climeqf 44999 | Two functions that are eve... |
| climeldmeqmpt3 45000 | Two functions that are eve... |
| limsupcld 45001 | Closure of the superior li... |
| climfv 45002 | The limit of a convergent ... |
| limsupval3 45003 | The superior limit of an i... |
| climfveqmpt2 45004 | Two functions that are eve... |
| limsup0 45005 | The superior limit of the ... |
| climeldmeqmpt2 45006 | Two functions that are eve... |
| limsupresre 45007 | The supremum limit of a fu... |
| climeqmpt 45008 | Two functions that are eve... |
| climfvd 45009 | The limit of a convergent ... |
| limsuplesup 45010 | An upper bound for the sup... |
| limsupresico 45011 | The superior limit doesn't... |
| limsuppnfdlem 45012 | If the restriction of a fu... |
| limsuppnfd 45013 | If the restriction of a fu... |
| limsupresuz 45014 | If the real part of the do... |
| limsupub 45015 | If the limsup is not ` +oo... |
| limsupres 45016 | The superior limit of a re... |
| climinf2lem 45017 | A convergent, nonincreasin... |
| climinf2 45018 | A convergent, nonincreasin... |
| limsupvaluz 45019 | The superior limit, when t... |
| limsupresuz2 45020 | If the domain of a functio... |
| limsuppnflem 45021 | If the restriction of a fu... |
| limsuppnf 45022 | If the restriction of a fu... |
| limsupubuzlem 45023 | If the limsup is not ` +oo... |
| limsupubuz 45024 | For a real-valued function... |
| climinf2mpt 45025 | A bounded below, monotonic... |
| climinfmpt 45026 | A bounded below, monotonic... |
| climinf3 45027 | A convergent, nonincreasin... |
| limsupvaluzmpt 45028 | The superior limit, when t... |
| limsupequzmpt2 45029 | Two functions that are eve... |
| limsupubuzmpt 45030 | If the limsup is not ` +oo... |
| limsupmnflem 45031 | The superior limit of a fu... |
| limsupmnf 45032 | The superior limit of a fu... |
| limsupequzlem 45033 | Two functions that are eve... |
| limsupequz 45034 | Two functions that are eve... |
| limsupre2lem 45035 | Given a function on the ex... |
| limsupre2 45036 | Given a function on the ex... |
| limsupmnfuzlem 45037 | The superior limit of a fu... |
| limsupmnfuz 45038 | The superior limit of a fu... |
| limsupequzmptlem 45039 | Two functions that are eve... |
| limsupequzmpt 45040 | Two functions that are eve... |
| limsupre2mpt 45041 | Given a function on the ex... |
| limsupequzmptf 45042 | Two functions that are eve... |
| limsupre3lem 45043 | Given a function on the ex... |
| limsupre3 45044 | Given a function on the ex... |
| limsupre3mpt 45045 | Given a function on the ex... |
| limsupre3uzlem 45046 | Given a function on the ex... |
| limsupre3uz 45047 | Given a function on the ex... |
| limsupreuz 45048 | Given a function on the re... |
| limsupvaluz2 45049 | The superior limit, when t... |
| limsupreuzmpt 45050 | Given a function on the re... |
| supcnvlimsup 45051 | If a function on a set of ... |
| supcnvlimsupmpt 45052 | If a function on a set of ... |
| 0cnv 45053 | If ` (/) ` is a complex nu... |
| climuzlem 45054 | Express the predicate: Th... |
| climuz 45055 | Express the predicate: Th... |
| lmbr3v 45056 | Express the binary relatio... |
| climisp 45057 | If a sequence converges to... |
| lmbr3 45058 | Express the binary relatio... |
| climrescn 45059 | A sequence converging w.r.... |
| climxrrelem 45060 | If a sequence ranging over... |
| climxrre 45061 | If a sequence ranging over... |
| limsuplt2 45064 | The defining property of t... |
| liminfgord 45065 | Ordering property of the i... |
| limsupvald 45066 | The superior limit of a se... |
| limsupresicompt 45067 | The superior limit doesn't... |
| limsupcli 45068 | Closure of the superior li... |
| liminfgf 45069 | Closure of the inferior li... |
| liminfval 45070 | The inferior limit of a se... |
| climlimsup 45071 | A sequence of real numbers... |
| limsupge 45072 | The defining property of t... |
| liminfgval 45073 | Value of the inferior limi... |
| liminfcl 45074 | Closure of the inferior li... |
| liminfvald 45075 | The inferior limit of a se... |
| liminfval5 45076 | The inferior limit of an i... |
| limsupresxr 45077 | The superior limit of a fu... |
| liminfresxr 45078 | The inferior limit of a fu... |
| liminfval2 45079 | The superior limit, relati... |
| climlimsupcex 45080 | Counterexample for ~ climl... |
| liminfcld 45081 | Closure of the inferior li... |
| liminfresico 45082 | The inferior limit doesn't... |
| limsup10exlem 45083 | The range of the given fun... |
| limsup10ex 45084 | The superior limit of a fu... |
| liminf10ex 45085 | The inferior limit of a fu... |
| liminflelimsuplem 45086 | The superior limit is grea... |
| liminflelimsup 45087 | The superior limit is grea... |
| limsupgtlem 45088 | For any positive real, the... |
| limsupgt 45089 | Given a sequence of real n... |
| liminfresre 45090 | The inferior limit of a fu... |
| liminfresicompt 45091 | The inferior limit doesn't... |
| liminfltlimsupex 45092 | An example where the ` lim... |
| liminfgelimsup 45093 | The inferior limit is grea... |
| liminfvalxr 45094 | Alternate definition of ` ... |
| liminfresuz 45095 | If the real part of the do... |
| liminflelimsupuz 45096 | The superior limit is grea... |
| liminfvalxrmpt 45097 | Alternate definition of ` ... |
| liminfresuz2 45098 | If the domain of a functio... |
| liminfgelimsupuz 45099 | The inferior limit is grea... |
| liminfval4 45100 | Alternate definition of ` ... |
| liminfval3 45101 | Alternate definition of ` ... |
| liminfequzmpt2 45102 | Two functions that are eve... |
| liminfvaluz 45103 | Alternate definition of ` ... |
| liminf0 45104 | The inferior limit of the ... |
| limsupval4 45105 | Alternate definition of ` ... |
| liminfvaluz2 45106 | Alternate definition of ` ... |
| liminfvaluz3 45107 | Alternate definition of ` ... |
| liminflelimsupcex 45108 | A counterexample for ~ lim... |
| limsupvaluz3 45109 | Alternate definition of ` ... |
| liminfvaluz4 45110 | Alternate definition of ` ... |
| limsupvaluz4 45111 | Alternate definition of ` ... |
| climliminflimsupd 45112 | If a sequence of real numb... |
| liminfreuzlem 45113 | Given a function on the re... |
| liminfreuz 45114 | Given a function on the re... |
| liminfltlem 45115 | Given a sequence of real n... |
| liminflt 45116 | Given a sequence of real n... |
| climliminf 45117 | A sequence of real numbers... |
| liminflimsupclim 45118 | A sequence of real numbers... |
| climliminflimsup 45119 | A sequence of real numbers... |
| climliminflimsup2 45120 | A sequence of real numbers... |
| climliminflimsup3 45121 | A sequence of real numbers... |
| climliminflimsup4 45122 | A sequence of real numbers... |
| limsupub2 45123 | A extended real valued fun... |
| limsupubuz2 45124 | A sequence with values in ... |
| xlimpnfxnegmnf 45125 | A sequence converges to ` ... |
| liminflbuz2 45126 | A sequence with values in ... |
| liminfpnfuz 45127 | The inferior limit of a fu... |
| liminflimsupxrre 45128 | A sequence with values in ... |
| xlimrel 45131 | The limit on extended real... |
| xlimres 45132 | A function converges iff i... |
| xlimcl 45133 | The limit of a sequence of... |
| rexlimddv2 45134 | Restricted existential eli... |
| xlimclim 45135 | Given a sequence of reals,... |
| xlimconst 45136 | A constant sequence conver... |
| climxlim 45137 | A converging sequence in t... |
| xlimbr 45138 | Express the binary relatio... |
| fuzxrpmcn 45139 | A function mapping from an... |
| cnrefiisplem 45140 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 45141 | A non-real, complex number... |
| xlimxrre 45142 | If a sequence ranging over... |
| xlimmnfvlem1 45143 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 45144 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 45145 | A function converges to mi... |
| xlimconst2 45146 | A sequence that eventually... |
| xlimpnfvlem1 45147 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 45148 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 45149 | A function converges to pl... |
| xlimclim2lem 45150 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 45151 | Given a sequence of extend... |
| xlimmnf 45152 | A function converges to mi... |
| xlimpnf 45153 | A function converges to pl... |
| xlimmnfmpt 45154 | A function converges to pl... |
| xlimpnfmpt 45155 | A function converges to pl... |
| climxlim2lem 45156 | In this lemma for ~ climxl... |
| climxlim2 45157 | A sequence of extended rea... |
| dfxlim2v 45158 | An alternative definition ... |
| dfxlim2 45159 | An alternative definition ... |
| climresd 45160 | A function restricted to u... |
| climresdm 45161 | A real function converges ... |
| dmclimxlim 45162 | A real valued sequence tha... |
| xlimmnflimsup2 45163 | A sequence of extended rea... |
| xlimuni 45164 | An infinite sequence conve... |
| xlimclimdm 45165 | A sequence of extended rea... |
| xlimfun 45166 | The convergence relation o... |
| xlimmnflimsup 45167 | If a sequence of extended ... |
| xlimdm 45168 | Two ways to express that a... |
| xlimpnfxnegmnf2 45169 | A sequence converges to ` ... |
| xlimresdm 45170 | A function converges in th... |
| xlimpnfliminf 45171 | If a sequence of extended ... |
| xlimpnfliminf2 45172 | A sequence of extended rea... |
| xlimliminflimsup 45173 | A sequence of extended rea... |
| xlimlimsupleliminf 45174 | A sequence of extended rea... |
| coseq0 45175 | A complex number whose cos... |
| sinmulcos 45176 | Multiplication formula for... |
| coskpi2 45177 | The cosine of an integer m... |
| cosnegpi 45178 | The cosine of negative ` _... |
| sinaover2ne0 45179 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 45180 | The cosine of an integer m... |
| mulcncff 45181 | The multiplication of two ... |
| cncfmptssg 45182 | A continuous complex funct... |
| constcncfg 45183 | A constant function is a c... |
| idcncfg 45184 | The identity function is a... |
| cncfshift 45185 | A periodic continuous func... |
| resincncf 45186 | ` sin ` restricted to real... |
| addccncf2 45187 | Adding a constant is a con... |
| 0cnf 45188 | The empty set is a continu... |
| fsumcncf 45189 | The finite sum of continuo... |
| cncfperiod 45190 | A periodic continuous func... |
| subcncff 45191 | The subtraction of two con... |
| negcncfg 45192 | The opposite of a continuo... |
| cnfdmsn 45193 | A function with a singleto... |
| cncfcompt 45194 | Composition of continuous ... |
| addcncff 45195 | The sum of two continuous ... |
| ioccncflimc 45196 | Limit at the upper bound o... |
| cncfuni 45197 | A complex function on a su... |
| icccncfext 45198 | A continuous function on a... |
| cncficcgt0 45199 | A the absolute value of a ... |
| icocncflimc 45200 | Limit at the lower bound, ... |
| cncfdmsn 45201 | A complex function with a ... |
| divcncff 45202 | The quotient of two contin... |
| cncfshiftioo 45203 | A periodic continuous func... |
| cncfiooicclem1 45204 | A continuous function ` F ... |
| cncfiooicc 45205 | A continuous function ` F ... |
| cncfiooiccre 45206 | A continuous function ` F ... |
| cncfioobdlem 45207 | ` G ` actually extends ` F... |
| cncfioobd 45208 | A continuous function ` F ... |
| jumpncnp 45209 | Jump discontinuity or disc... |
| cxpcncf2 45210 | The complex power function... |
| fprodcncf 45211 | The finite product of cont... |
| add1cncf 45212 | Addition to a constant is ... |
| add2cncf 45213 | Addition to a constant is ... |
| sub1cncfd 45214 | Subtracting a constant is ... |
| sub2cncfd 45215 | Subtraction from a constan... |
| fprodsub2cncf 45216 | ` F ` is continuous. (Con... |
| fprodadd2cncf 45217 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 45218 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 45219 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 45220 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 45221 | The sequence ` S ` of fini... |
| dvsinexp 45222 | The derivative of sin^N . ... |
| dvcosre 45223 | The real derivative of the... |
| dvsinax 45224 | Derivative exercise: the d... |
| dvsubf 45225 | The subtraction rule for e... |
| dvmptconst 45226 | Function-builder for deriv... |
| dvcnre 45227 | From complex differentiati... |
| dvmptidg 45228 | Function-builder for deriv... |
| dvresntr 45229 | Function-builder for deriv... |
| fperdvper 45230 | The derivative of a period... |
| dvasinbx 45231 | Derivative exercise: the d... |
| dvresioo 45232 | Restriction of a derivativ... |
| dvdivf 45233 | The quotient rule for ever... |
| dvdivbd 45234 | A sufficient condition for... |
| dvsubcncf 45235 | A sufficient condition for... |
| dvmulcncf 45236 | A sufficient condition for... |
| dvcosax 45237 | Derivative exercise: the d... |
| dvdivcncf 45238 | A sufficient condition for... |
| dvbdfbdioolem1 45239 | Given a function with boun... |
| dvbdfbdioolem2 45240 | A function on an open inte... |
| dvbdfbdioo 45241 | A function on an open inte... |
| ioodvbdlimc1lem1 45242 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 45243 | Limit at the lower bound o... |
| ioodvbdlimc1 45244 | A real function with bound... |
| ioodvbdlimc2lem 45245 | Limit at the upper bound o... |
| ioodvbdlimc2 45246 | A real function with bound... |
| dvdmsscn 45247 | ` X ` is a subset of ` CC ... |
| dvmptmulf 45248 | Function-builder for deriv... |
| dvnmptdivc 45249 | Function-builder for itera... |
| dvdsn1add 45250 | If ` K ` divides ` N ` but... |
| dvxpaek 45251 | Derivative of the polynomi... |
| dvnmptconst 45252 | The ` N ` -th derivative o... |
| dvnxpaek 45253 | The ` n ` -th derivative o... |
| dvnmul 45254 | Function-builder for the `... |
| dvmptfprodlem 45255 | Induction step for ~ dvmpt... |
| dvmptfprod 45256 | Function-builder for deriv... |
| dvnprodlem1 45257 | ` D ` is bijective. (Cont... |
| dvnprodlem2 45258 | Induction step for ~ dvnpr... |
| dvnprodlem3 45259 | The multinomial formula fo... |
| dvnprod 45260 | The multinomial formula fo... |
| itgsin0pilem1 45261 | Calculation of the integra... |
| ibliccsinexp 45262 | sin^n on a closed interval... |
| itgsin0pi 45263 | Calculation of the integra... |
| iblioosinexp 45264 | sin^n on an open integral ... |
| itgsinexplem1 45265 | Integration by parts is ap... |
| itgsinexp 45266 | A recursive formula for th... |
| iblconstmpt 45267 | A constant function is int... |
| itgeq1d 45268 | Equality theorem for an in... |
| mbfres2cn 45269 | Measurability of a piecewi... |
| vol0 45270 | The measure of the empty s... |
| ditgeqiooicc 45271 | A function ` F ` on an ope... |
| volge0 45272 | The volume of a set is alw... |
| cnbdibl 45273 | A continuous bounded funct... |
| snmbl 45274 | A singleton is measurable.... |
| ditgeq3d 45275 | Equality theorem for the d... |
| iblempty 45276 | The empty function is inte... |
| iblsplit 45277 | The union of two integrabl... |
| volsn 45278 | A singleton has 0 Lebesgue... |
| itgvol0 45279 | If the domani is negligibl... |
| itgcoscmulx 45280 | Exercise: the integral of ... |
| iblsplitf 45281 | A version of ~ iblsplit us... |
| ibliooicc 45282 | If a function is integrabl... |
| volioc 45283 | The measure of a left-open... |
| iblspltprt 45284 | If a function is integrabl... |
| itgsincmulx 45285 | Exercise: the integral of ... |
| itgsubsticclem 45286 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 45287 | Integration by u-substitut... |
| itgioocnicc 45288 | The integral of a piecewis... |
| iblcncfioo 45289 | A continuous function ` F ... |
| itgspltprt 45290 | The ` S. ` integral splits... |
| itgiccshift 45291 | The integral of a function... |
| itgperiod 45292 | The integral of a periodic... |
| itgsbtaddcnst 45293 | Integral substitution, add... |
| volico 45294 | The measure of left-closed... |
| sublevolico 45295 | The Lebesgue measure of a ... |
| dmvolss 45296 | Lebesgue measurable sets a... |
| ismbl3 45297 | The predicate " ` A ` is L... |
| volioof 45298 | The function that assigns ... |
| ovolsplit 45299 | The Lebesgue outer measure... |
| fvvolioof 45300 | The function value of the ... |
| volioore 45301 | The measure of an open int... |
| fvvolicof 45302 | The function value of the ... |
| voliooico 45303 | An open interval and a lef... |
| ismbl4 45304 | The predicate " ` A ` is L... |
| volioofmpt 45305 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 45306 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 45307 | The Lebesgue measure of op... |
| volicofmpt 45308 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 45309 | The Lebesgue measure of a ... |
| voliccico 45310 | A closed interval and a le... |
| mbfdmssre 45311 | The domain of a measurable... |
| stoweidlem1 45312 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 45313 | lemma for ~ stoweid : here... |
| stoweidlem3 45314 | Lemma for ~ stoweid : if `... |
| stoweidlem4 45315 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 45316 | There exists a δ as ... |
| stoweidlem6 45317 | Lemma for ~ stoweid : two ... |
| stoweidlem7 45318 | This lemma is used to prov... |
| stoweidlem8 45319 | Lemma for ~ stoweid : two ... |
| stoweidlem9 45320 | Lemma for ~ stoweid : here... |
| stoweidlem10 45321 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 45322 | This lemma is used to prov... |
| stoweidlem12 45323 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 45324 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 45325 | There exists a ` k ` as in... |
| stoweidlem15 45326 | This lemma is used to prov... |
| stoweidlem16 45327 | Lemma for ~ stoweid . The... |
| stoweidlem17 45328 | This lemma proves that the... |
| stoweidlem18 45329 | This theorem proves Lemma ... |
| stoweidlem19 45330 | If a set of real functions... |
| stoweidlem20 45331 | If a set A of real functio... |
| stoweidlem21 45332 | Once the Stone Weierstrass... |
| stoweidlem22 45333 | If a set of real functions... |
| stoweidlem23 45334 | This lemma is used to prov... |
| stoweidlem24 45335 | This lemma proves that for... |
| stoweidlem25 45336 | This lemma proves that for... |
| stoweidlem26 45337 | This lemma is used to prov... |
| stoweidlem27 45338 | This lemma is used to prov... |
| stoweidlem28 45339 | There exists a δ as ... |
| stoweidlem29 45340 | When the hypothesis for th... |
| stoweidlem30 45341 | This lemma is used to prov... |
| stoweidlem31 45342 | This lemma is used to prov... |
| stoweidlem32 45343 | If a set A of real functio... |
| stoweidlem33 45344 | If a set of real functions... |
| stoweidlem34 45345 | This lemma proves that for... |
| stoweidlem35 45346 | This lemma is used to prov... |
| stoweidlem36 45347 | This lemma is used to prov... |
| stoweidlem37 45348 | This lemma is used to prov... |
| stoweidlem38 45349 | This lemma is used to prov... |
| stoweidlem39 45350 | This lemma is used to prov... |
| stoweidlem40 45351 | This lemma proves that q_n... |
| stoweidlem41 45352 | This lemma is used to prov... |
| stoweidlem42 45353 | This lemma is used to prov... |
| stoweidlem43 45354 | This lemma is used to prov... |
| stoweidlem44 45355 | This lemma is used to prov... |
| stoweidlem45 45356 | This lemma proves that, gi... |
| stoweidlem46 45357 | This lemma proves that set... |
| stoweidlem47 45358 | Subtracting a constant fro... |
| stoweidlem48 45359 | This lemma is used to prov... |
| stoweidlem49 45360 | There exists a function q_... |
| stoweidlem50 45361 | This lemma proves that set... |
| stoweidlem51 45362 | There exists a function x ... |
| stoweidlem52 45363 | There exists a neighborhoo... |
| stoweidlem53 45364 | This lemma is used to prov... |
| stoweidlem54 45365 | There exists a function ` ... |
| stoweidlem55 45366 | This lemma proves the exis... |
| stoweidlem56 45367 | This theorem proves Lemma ... |
| stoweidlem57 45368 | There exists a function x ... |
| stoweidlem58 45369 | This theorem proves Lemma ... |
| stoweidlem59 45370 | This lemma proves that the... |
| stoweidlem60 45371 | This lemma proves that the... |
| stoweidlem61 45372 | This lemma proves that the... |
| stoweidlem62 45373 | This theorem proves the St... |
| stoweid 45374 | This theorem proves the St... |
| stowei 45375 | This theorem proves the St... |
| wallispilem1 45376 | ` I ` is monotone: increas... |
| wallispilem2 45377 | A first set of properties ... |
| wallispilem3 45378 | I maps to real values. (C... |
| wallispilem4 45379 | ` F ` maps to explicit exp... |
| wallispilem5 45380 | The sequence ` H ` converg... |
| wallispi 45381 | Wallis' formula for π :... |
| wallispi2lem1 45382 | An intermediate step betwe... |
| wallispi2lem2 45383 | Two expressions are proven... |
| wallispi2 45384 | An alternative version of ... |
| stirlinglem1 45385 | A simple limit of fraction... |
| stirlinglem2 45386 | ` A ` maps to positive rea... |
| stirlinglem3 45387 | Long but simple algebraic ... |
| stirlinglem4 45388 | Algebraic manipulation of ... |
| stirlinglem5 45389 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 45390 | A series that converges to... |
| stirlinglem7 45391 | Algebraic manipulation of ... |
| stirlinglem8 45392 | If ` A ` converges to ` C ... |
| stirlinglem9 45393 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 45394 | A bound for any B(N)-B(N +... |
| stirlinglem11 45395 | ` B ` is decreasing. (Con... |
| stirlinglem12 45396 | The sequence ` B ` is boun... |
| stirlinglem13 45397 | ` B ` is decreasing and ha... |
| stirlinglem14 45398 | The sequence ` A ` converg... |
| stirlinglem15 45399 | The Stirling's formula is ... |
| stirling 45400 | Stirling's approximation f... |
| stirlingr 45401 | Stirling's approximation f... |
| dirkerval 45402 | The N_th Dirichlet Kernel.... |
| dirker2re 45403 | The Dirichlet Kernel value... |
| dirkerdenne0 45404 | The Dirichlet Kernel denom... |
| dirkerval2 45405 | The N_th Dirichlet Kernel ... |
| dirkerre 45406 | The Dirichlet Kernel at an... |
| dirkerper 45407 | the Dirichlet Kernel has p... |
| dirkerf 45408 | For any natural number ` N... |
| dirkertrigeqlem1 45409 | Sum of an even number of a... |
| dirkertrigeqlem2 45410 | Trigonomic equality lemma ... |
| dirkertrigeqlem3 45411 | Trigonometric equality lem... |
| dirkertrigeq 45412 | Trigonometric equality for... |
| dirkeritg 45413 | The definite integral of t... |
| dirkercncflem1 45414 | If ` Y ` is a multiple of ... |
| dirkercncflem2 45415 | Lemma used to prove that t... |
| dirkercncflem3 45416 | The Dirichlet Kernel is co... |
| dirkercncflem4 45417 | The Dirichlet Kernel is co... |
| dirkercncf 45418 | For any natural number ` N... |
| fourierdlem1 45419 | A partition interval is a ... |
| fourierdlem2 45420 | Membership in a partition.... |
| fourierdlem3 45421 | Membership in a partition.... |
| fourierdlem4 45422 | ` E ` is a function that m... |
| fourierdlem5 45423 | ` S ` is a function. (Con... |
| fourierdlem6 45424 | ` X ` is in the periodic p... |
| fourierdlem7 45425 | The difference between the... |
| fourierdlem8 45426 | A partition interval is a ... |
| fourierdlem9 45427 | ` H ` is a complex functio... |
| fourierdlem10 45428 | Condition on the bounds of... |
| fourierdlem11 45429 | If there is a partition, t... |
| fourierdlem12 45430 | A point of a partition is ... |
| fourierdlem13 45431 | Value of ` V ` in terms of... |
| fourierdlem14 45432 | Given the partition ` V ` ... |
| fourierdlem15 45433 | The range of the partition... |
| fourierdlem16 45434 | The coefficients of the fo... |
| fourierdlem17 45435 | The defined ` L ` is actua... |
| fourierdlem18 45436 | The function ` S ` is cont... |
| fourierdlem19 45437 | If two elements of ` D ` h... |
| fourierdlem20 45438 | Every interval in the part... |
| fourierdlem21 45439 | The coefficients of the fo... |
| fourierdlem22 45440 | The coefficients of the fo... |
| fourierdlem23 45441 | If ` F ` is continuous and... |
| fourierdlem24 45442 | A sufficient condition for... |
| fourierdlem25 45443 | If ` C ` is not in the ran... |
| fourierdlem26 45444 | Periodic image of a point ... |
| fourierdlem27 45445 | A partition open interval ... |
| fourierdlem28 45446 | Derivative of ` ( F `` ( X... |
| fourierdlem29 45447 | Explicit function value fo... |
| fourierdlem30 45448 | Sum of three small pieces ... |
| fourierdlem31 45449 | If ` A ` is finite and for... |
| fourierdlem32 45450 | Limit of a continuous func... |
| fourierdlem33 45451 | Limit of a continuous func... |
| fourierdlem34 45452 | A partition is one to one.... |
| fourierdlem35 45453 | There is a single point in... |
| fourierdlem36 45454 | ` F ` is an isomorphism. ... |
| fourierdlem37 45455 | ` I ` is a function that m... |
| fourierdlem38 45456 | The function ` F ` is cont... |
| fourierdlem39 45457 | Integration by parts of ... |
| fourierdlem40 45458 | ` H ` is a continuous func... |
| fourierdlem41 45459 | Lemma used to prove that e... |
| fourierdlem42 45460 | The set of points in a mov... |
| fourierdlem43 45461 | ` K ` is a real function. ... |
| fourierdlem44 45462 | A condition for having ` (... |
| fourierdlem46 45463 | The function ` F ` has a l... |
| fourierdlem47 45464 | For ` r ` large enough, th... |
| fourierdlem48 45465 | The given periodic functio... |
| fourierdlem49 45466 | The given periodic functio... |
| fourierdlem50 45467 | Continuity of ` O ` and it... |
| fourierdlem51 45468 | ` X ` is in the periodic p... |
| fourierdlem52 45469 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 45470 | The limit of ` F ( s ) ` a... |
| fourierdlem54 45471 | Given a partition ` Q ` an... |
| fourierdlem55 45472 | ` U ` is a real function. ... |
| fourierdlem56 45473 | Derivative of the ` K ` fu... |
| fourierdlem57 45474 | The derivative of ` O ` . ... |
| fourierdlem58 45475 | The derivative of ` K ` is... |
| fourierdlem59 45476 | The derivative of ` H ` is... |
| fourierdlem60 45477 | Given a differentiable fun... |
| fourierdlem61 45478 | Given a differentiable fun... |
| fourierdlem62 45479 | The function ` K ` is cont... |
| fourierdlem63 45480 | The upper bound of interva... |
| fourierdlem64 45481 | The partition ` V ` is fin... |
| fourierdlem65 45482 | The distance of two adjace... |
| fourierdlem66 45483 | Value of the ` G ` functio... |
| fourierdlem67 45484 | ` G ` is a function. (Con... |
| fourierdlem68 45485 | The derivative of ` O ` is... |
| fourierdlem69 45486 | A piecewise continuous fun... |
| fourierdlem70 45487 | A piecewise continuous fun... |
| fourierdlem71 45488 | A periodic piecewise conti... |
| fourierdlem72 45489 | The derivative of ` O ` is... |
| fourierdlem73 45490 | A version of the Riemann L... |
| fourierdlem74 45491 | Given a piecewise smooth f... |
| fourierdlem75 45492 | Given a piecewise smooth f... |
| fourierdlem76 45493 | Continuity of ` O ` and it... |
| fourierdlem77 45494 | If ` H ` is bounded, then ... |
| fourierdlem78 45495 | ` G ` is continuous when r... |
| fourierdlem79 45496 | ` E ` projects every inter... |
| fourierdlem80 45497 | The derivative of ` O ` is... |
| fourierdlem81 45498 | The integral of a piecewis... |
| fourierdlem82 45499 | Integral by substitution, ... |
| fourierdlem83 45500 | The fourier partial sum fo... |
| fourierdlem84 45501 | If ` F ` is piecewise coni... |
| fourierdlem85 45502 | Limit of the function ` G ... |
| fourierdlem86 45503 | Continuity of ` O ` and it... |
| fourierdlem87 45504 | The integral of ` G ` goes... |
| fourierdlem88 45505 | Given a piecewise continuo... |
| fourierdlem89 45506 | Given a piecewise continuo... |
| fourierdlem90 45507 | Given a piecewise continuo... |
| fourierdlem91 45508 | Given a piecewise continuo... |
| fourierdlem92 45509 | The integral of a piecewis... |
| fourierdlem93 45510 | Integral by substitution (... |
| fourierdlem94 45511 | For a piecewise smooth fun... |
| fourierdlem95 45512 | Algebraic manipulation of ... |
| fourierdlem96 45513 | limit for ` F ` at the low... |
| fourierdlem97 45514 | ` F ` is continuous on the... |
| fourierdlem98 45515 | ` F ` is continuous on the... |
| fourierdlem99 45516 | limit for ` F ` at the upp... |
| fourierdlem100 45517 | A piecewise continuous fun... |
| fourierdlem101 45518 | Integral by substitution f... |
| fourierdlem102 45519 | For a piecewise smooth fun... |
| fourierdlem103 45520 | The half lower part of the... |
| fourierdlem104 45521 | The half upper part of the... |
| fourierdlem105 45522 | A piecewise continuous fun... |
| fourierdlem106 45523 | For a piecewise smooth fun... |
| fourierdlem107 45524 | The integral of a piecewis... |
| fourierdlem108 45525 | The integral of a piecewis... |
| fourierdlem109 45526 | The integral of a piecewis... |
| fourierdlem110 45527 | The integral of a piecewis... |
| fourierdlem111 45528 | The fourier partial sum fo... |
| fourierdlem112 45529 | Here abbreviations (local ... |
| fourierdlem113 45530 | Fourier series convergence... |
| fourierdlem114 45531 | Fourier series convergence... |
| fourierdlem115 45532 | Fourier serier convergence... |
| fourierd 45533 | Fourier series convergence... |
| fourierclimd 45534 | Fourier series convergence... |
| fourierclim 45535 | Fourier series convergence... |
| fourier 45536 | Fourier series convergence... |
| fouriercnp 45537 | If ` F ` is continuous at ... |
| fourier2 45538 | Fourier series convergence... |
| sqwvfoura 45539 | Fourier coefficients for t... |
| sqwvfourb 45540 | Fourier series ` B ` coeff... |
| fourierswlem 45541 | The Fourier series for the... |
| fouriersw 45542 | Fourier series convergence... |
| fouriercn 45543 | If the derivative of ` F `... |
| elaa2lem 45544 | Elementhood in the set of ... |
| elaa2 45545 | Elementhood in the set of ... |
| etransclem1 45546 | ` H ` is a function. (Con... |
| etransclem2 45547 | Derivative of ` G ` . (Co... |
| etransclem3 45548 | The given ` if ` term is a... |
| etransclem4 45549 | ` F ` expressed as a finit... |
| etransclem5 45550 | A change of bound variable... |
| etransclem6 45551 | A change of bound variable... |
| etransclem7 45552 | The given product is an in... |
| etransclem8 45553 | ` F ` is a function. (Con... |
| etransclem9 45554 | If ` K ` divides ` N ` but... |
| etransclem10 45555 | The given ` if ` term is a... |
| etransclem11 45556 | A change of bound variable... |
| etransclem12 45557 | ` C ` applied to ` N ` . ... |
| etransclem13 45558 | ` F ` applied to ` Y ` . ... |
| etransclem14 45559 | Value of the term ` T ` , ... |
| etransclem15 45560 | Value of the term ` T ` , ... |
| etransclem16 45561 | Every element in the range... |
| etransclem17 45562 | The ` N ` -th derivative o... |
| etransclem18 45563 | The given function is inte... |
| etransclem19 45564 | The ` N ` -th derivative o... |
| etransclem20 45565 | ` H ` is smooth. (Contrib... |
| etransclem21 45566 | The ` N ` -th derivative o... |
| etransclem22 45567 | The ` N ` -th derivative o... |
| etransclem23 45568 | This is the claim proof in... |
| etransclem24 45569 | ` P ` divides the I -th de... |
| etransclem25 45570 | ` P ` factorial divides th... |
| etransclem26 45571 | Every term in the sum of t... |
| etransclem27 45572 | The ` N ` -th derivative o... |
| etransclem28 45573 | ` ( P - 1 ) ` factorial di... |
| etransclem29 45574 | The ` N ` -th derivative o... |
| etransclem30 45575 | The ` N ` -th derivative o... |
| etransclem31 45576 | The ` N ` -th derivative o... |
| etransclem32 45577 | This is the proof for the ... |
| etransclem33 45578 | ` F ` is smooth. (Contrib... |
| etransclem34 45579 | The ` N ` -th derivative o... |
| etransclem35 45580 | ` P ` does not divide the ... |
| etransclem36 45581 | The ` N ` -th derivative o... |
| etransclem37 45582 | ` ( P - 1 ) ` factorial di... |
| etransclem38 45583 | ` P ` divides the I -th de... |
| etransclem39 45584 | ` G ` is a function. (Con... |
| etransclem40 45585 | The ` N ` -th derivative o... |
| etransclem41 45586 | ` P ` does not divide the ... |
| etransclem42 45587 | The ` N ` -th derivative o... |
| etransclem43 45588 | ` G ` is a continuous func... |
| etransclem44 45589 | The given finite sum is no... |
| etransclem45 45590 | ` K ` is an integer. (Con... |
| etransclem46 45591 | This is the proof for equa... |
| etransclem47 45592 | ` _e ` is transcendental. ... |
| etransclem48 45593 | ` _e ` is transcendental. ... |
| etransc 45594 | ` _e ` is transcendental. ... |
| rrxtopn 45595 | The topology of the genera... |
| rrxngp 45596 | Generalized Euclidean real... |
| rrxtps 45597 | Generalized Euclidean real... |
| rrxtopnfi 45598 | The topology of the n-dime... |
| rrxtopon 45599 | The topology on generalize... |
| rrxtop 45600 | The topology on generalize... |
| rrndistlt 45601 | Given two points in the sp... |
| rrxtoponfi 45602 | The topology on n-dimensio... |
| rrxunitopnfi 45603 | The base set of the standa... |
| rrxtopn0 45604 | The topology of the zero-d... |
| qndenserrnbllem 45605 | n-dimensional rational num... |
| qndenserrnbl 45606 | n-dimensional rational num... |
| rrxtopn0b 45607 | The topology of the zero-d... |
| qndenserrnopnlem 45608 | n-dimensional rational num... |
| qndenserrnopn 45609 | n-dimensional rational num... |
| qndenserrn 45610 | n-dimensional rational num... |
| rrxsnicc 45611 | A multidimensional singlet... |
| rrnprjdstle 45612 | The distance between two p... |
| rrndsmet 45613 | ` D ` is a metric for the ... |
| rrndsxmet 45614 | ` D ` is an extended metri... |
| ioorrnopnlem 45615 | The a point in an indexed ... |
| ioorrnopn 45616 | The indexed product of ope... |
| ioorrnopnxrlem 45617 | Given a point ` F ` that b... |
| ioorrnopnxr 45618 | The indexed product of ope... |
| issal 45625 | Express the predicate " ` ... |
| pwsal 45626 | The power set of a given s... |
| salunicl 45627 | SAlg sigma-algebra is clos... |
| saluncl 45628 | The union of two sets in a... |
| prsal 45629 | The pair of the empty set ... |
| saldifcl 45630 | The complement of an eleme... |
| 0sal 45631 | The empty set belongs to e... |
| salgenval 45632 | The sigma-algebra generate... |
| saliunclf 45633 | SAlg sigma-algebra is clos... |
| saliuncl 45634 | SAlg sigma-algebra is clos... |
| salincl 45635 | The intersection of two se... |
| saluni 45636 | A set is an element of any... |
| saliinclf 45637 | SAlg sigma-algebra is clos... |
| saliincl 45638 | SAlg sigma-algebra is clos... |
| saldifcl2 45639 | The difference of two elem... |
| intsaluni 45640 | The union of an arbitrary ... |
| intsal 45641 | The arbitrary intersection... |
| salgenn0 45642 | The set used in the defini... |
| salgencl 45643 | ` SalGen ` actually genera... |
| issald 45644 | Sufficient condition to pr... |
| salexct 45645 | An example of nontrivial s... |
| sssalgen 45646 | A set is a subset of the s... |
| salgenss 45647 | The sigma-algebra generate... |
| salgenuni 45648 | The base set of the sigma-... |
| issalgend 45649 | One side of ~ dfsalgen2 . ... |
| salexct2 45650 | An example of a subset tha... |
| unisalgen 45651 | The union of a set belongs... |
| dfsalgen2 45652 | Alternate characterization... |
| salexct3 45653 | An example of a sigma-alge... |
| salgencntex 45654 | This counterexample shows ... |
| salgensscntex 45655 | This counterexample shows ... |
| issalnnd 45656 | Sufficient condition to pr... |
| dmvolsal 45657 | Lebesgue measurable sets f... |
| saldifcld 45658 | The complement of an eleme... |
| saluncld 45659 | The union of two sets in a... |
| salgencld 45660 | ` SalGen ` actually genera... |
| 0sald 45661 | The empty set belongs to e... |
| iooborel 45662 | An open interval is a Bore... |
| salincld 45663 | The intersection of two se... |
| salunid 45664 | A set is an element of any... |
| unisalgen2 45665 | The union of a set belongs... |
| bor1sal 45666 | The Borel sigma-algebra on... |
| iocborel 45667 | A left-open, right-closed ... |
| subsaliuncllem 45668 | A subspace sigma-algebra i... |
| subsaliuncl 45669 | A subspace sigma-algebra i... |
| subsalsal 45670 | A subspace sigma-algebra i... |
| subsaluni 45671 | A set belongs to the subsp... |
| salrestss 45672 | A sigma-algebra restricted... |
| sge0rnre 45675 | When ` sum^ ` is applied t... |
| fge0icoicc 45676 | If ` F ` maps to nonnegati... |
| sge0val 45677 | The value of the sum of no... |
| fge0npnf 45678 | If ` F ` maps to nonnegati... |
| sge0rnn0 45679 | The range used in the defi... |
| sge0vald 45680 | The value of the sum of no... |
| fge0iccico 45681 | A range of nonnegative ext... |
| gsumge0cl 45682 | Closure of group sum, for ... |
| sge0reval 45683 | Value of the sum of nonneg... |
| sge0pnfval 45684 | If a term in the sum of no... |
| fge0iccre 45685 | A range of nonnegative ext... |
| sge0z 45686 | Any nonnegative extended s... |
| sge00 45687 | The sum of nonnegative ext... |
| fsumlesge0 45688 | Every finite subsum of non... |
| sge0revalmpt 45689 | Value of the sum of nonneg... |
| sge0sn 45690 | A sum of a nonnegative ext... |
| sge0tsms 45691 | ` sum^ ` applied to a nonn... |
| sge0cl 45692 | The arbitrary sum of nonne... |
| sge0f1o 45693 | Re-index a nonnegative ext... |
| sge0snmpt 45694 | A sum of a nonnegative ext... |
| sge0ge0 45695 | The sum of nonnegative ext... |
| sge0xrcl 45696 | The arbitrary sum of nonne... |
| sge0repnf 45697 | The of nonnegative extende... |
| sge0fsum 45698 | The arbitrary sum of a fin... |
| sge0rern 45699 | If the sum of nonnegative ... |
| sge0supre 45700 | If the arbitrary sum of no... |
| sge0fsummpt 45701 | The arbitrary sum of a fin... |
| sge0sup 45702 | The arbitrary sum of nonne... |
| sge0less 45703 | A shorter sum of nonnegati... |
| sge0rnbnd 45704 | The range used in the defi... |
| sge0pr 45705 | Sum of a pair of nonnegati... |
| sge0gerp 45706 | The arbitrary sum of nonne... |
| sge0pnffigt 45707 | If the sum of nonnegative ... |
| sge0ssre 45708 | If a sum of nonnegative ex... |
| sge0lefi 45709 | A sum of nonnegative exten... |
| sge0lessmpt 45710 | A shorter sum of nonnegati... |
| sge0ltfirp 45711 | If the sum of nonnegative ... |
| sge0prle 45712 | The sum of a pair of nonne... |
| sge0gerpmpt 45713 | The arbitrary sum of nonne... |
| sge0resrnlem 45714 | The sum of nonnegative ext... |
| sge0resrn 45715 | The sum of nonnegative ext... |
| sge0ssrempt 45716 | If a sum of nonnegative ex... |
| sge0resplit 45717 | ` sum^ ` splits into two p... |
| sge0le 45718 | If all of the terms of sum... |
| sge0ltfirpmpt 45719 | If the extended sum of non... |
| sge0split 45720 | Split a sum of nonnegative... |
| sge0lempt 45721 | If all of the terms of sum... |
| sge0splitmpt 45722 | Split a sum of nonnegative... |
| sge0ss 45723 | Change the index set to a ... |
| sge0iunmptlemfi 45724 | Sum of nonnegative extende... |
| sge0p1 45725 | The addition of the next t... |
| sge0iunmptlemre 45726 | Sum of nonnegative extende... |
| sge0fodjrnlem 45727 | Re-index a nonnegative ext... |
| sge0fodjrn 45728 | Re-index a nonnegative ext... |
| sge0iunmpt 45729 | Sum of nonnegative extende... |
| sge0iun 45730 | Sum of nonnegative extende... |
| sge0nemnf 45731 | The generalized sum of non... |
| sge0rpcpnf 45732 | The sum of an infinite num... |
| sge0rernmpt 45733 | If the sum of nonnegative ... |
| sge0lefimpt 45734 | A sum of nonnegative exten... |
| nn0ssge0 45735 | Nonnegative integers are n... |
| sge0clmpt 45736 | The generalized sum of non... |
| sge0ltfirpmpt2 45737 | If the extended sum of non... |
| sge0isum 45738 | If a series of nonnegative... |
| sge0xrclmpt 45739 | The generalized sum of non... |
| sge0xp 45740 | Combine two generalized su... |
| sge0isummpt 45741 | If a series of nonnegative... |
| sge0ad2en 45742 | The value of the infinite ... |
| sge0isummpt2 45743 | If a series of nonnegative... |
| sge0xaddlem1 45744 | The extended addition of t... |
| sge0xaddlem2 45745 | The extended addition of t... |
| sge0xadd 45746 | The extended addition of t... |
| sge0fsummptf 45747 | The generalized sum of a f... |
| sge0snmptf 45748 | A sum of a nonnegative ext... |
| sge0ge0mpt 45749 | The sum of nonnegative ext... |
| sge0repnfmpt 45750 | The of nonnegative extende... |
| sge0pnffigtmpt 45751 | If the generalized sum of ... |
| sge0splitsn 45752 | Separate out a term in a g... |
| sge0pnffsumgt 45753 | If the sum of nonnegative ... |
| sge0gtfsumgt 45754 | If the generalized sum of ... |
| sge0uzfsumgt 45755 | If a real number is smalle... |
| sge0pnfmpt 45756 | If a term in the sum of no... |
| sge0seq 45757 | A series of nonnegative re... |
| sge0reuz 45758 | Value of the generalized s... |
| sge0reuzb 45759 | Value of the generalized s... |
| ismea 45762 | Express the predicate " ` ... |
| dmmeasal 45763 | The domain of a measure is... |
| meaf 45764 | A measure is a function th... |
| mea0 45765 | The measure of the empty s... |
| nnfoctbdjlem 45766 | There exists a mapping fro... |
| nnfoctbdj 45767 | There exists a mapping fro... |
| meadjuni 45768 | The measure of the disjoin... |
| meacl 45769 | The measure of a set is a ... |
| iundjiunlem 45770 | The sets in the sequence `... |
| iundjiun 45771 | Given a sequence ` E ` of ... |
| meaxrcl 45772 | The measure of a set is an... |
| meadjun 45773 | The measure of the union o... |
| meassle 45774 | The measure of a set is gr... |
| meaunle 45775 | The measure of the union o... |
| meadjiunlem 45776 | The sum of nonnegative ext... |
| meadjiun 45777 | The measure of the disjoin... |
| ismeannd 45778 | Sufficient condition to pr... |
| meaiunlelem 45779 | The measure of the union o... |
| meaiunle 45780 | The measure of the union o... |
| psmeasurelem 45781 | ` M ` applied to a disjoin... |
| psmeasure 45782 | Point supported measure, R... |
| voliunsge0lem 45783 | The Lebesgue measure funct... |
| voliunsge0 45784 | The Lebesgue measure funct... |
| volmea 45785 | The Lebesgue measure on th... |
| meage0 45786 | If the measure of a measur... |
| meadjunre 45787 | The measure of the union o... |
| meassre 45788 | If the measure of a measur... |
| meale0eq0 45789 | A measure that is less tha... |
| meadif 45790 | The measure of the differe... |
| meaiuninclem 45791 | Measures are continuous fr... |
| meaiuninc 45792 | Measures are continuous fr... |
| meaiuninc2 45793 | Measures are continuous fr... |
| meaiunincf 45794 | Measures are continuous fr... |
| meaiuninc3v 45795 | Measures are continuous fr... |
| meaiuninc3 45796 | Measures are continuous fr... |
| meaiininclem 45797 | Measures are continuous fr... |
| meaiininc 45798 | Measures are continuous fr... |
| meaiininc2 45799 | Measures are continuous fr... |
| caragenval 45804 | The sigma-algebra generate... |
| isome 45805 | Express the predicate " ` ... |
| caragenel 45806 | Membership in the Caratheo... |
| omef 45807 | An outer measure is a func... |
| ome0 45808 | The outer measure of the e... |
| omessle 45809 | The outer measure of a set... |
| omedm 45810 | The domain of an outer mea... |
| caragensplit 45811 | If ` E ` is in the set gen... |
| caragenelss 45812 | An element of the Caratheo... |
| carageneld 45813 | Membership in the Caratheo... |
| omecl 45814 | The outer measure of a set... |
| caragenss 45815 | The sigma-algebra generate... |
| omeunile 45816 | The outer measure of the u... |
| caragen0 45817 | The empty set belongs to a... |
| omexrcl 45818 | The outer measure of a set... |
| caragenunidm 45819 | The base set of an outer m... |
| caragensspw 45820 | The sigma-algebra generate... |
| omessre 45821 | If the outer measure of a ... |
| caragenuni 45822 | The base set of the sigma-... |
| caragenuncllem 45823 | The Caratheodory's constru... |
| caragenuncl 45824 | The Caratheodory's constru... |
| caragendifcl 45825 | The Caratheodory's constru... |
| caragenfiiuncl 45826 | The Caratheodory's constru... |
| omeunle 45827 | The outer measure of the u... |
| omeiunle 45828 | The outer measure of the i... |
| omelesplit 45829 | The outer measure of a set... |
| omeiunltfirp 45830 | If the outer measure of a ... |
| omeiunlempt 45831 | The outer measure of the i... |
| carageniuncllem1 45832 | The outer measure of ` A i... |
| carageniuncllem2 45833 | The Caratheodory's constru... |
| carageniuncl 45834 | The Caratheodory's constru... |
| caragenunicl 45835 | The Caratheodory's constru... |
| caragensal 45836 | Caratheodory's method gene... |
| caratheodorylem1 45837 | Lemma used to prove that C... |
| caratheodorylem2 45838 | Caratheodory's constructio... |
| caratheodory 45839 | Caratheodory's constructio... |
| 0ome 45840 | The map that assigns 0 to ... |
| isomenndlem 45841 | ` O ` is sub-additive w.r.... |
| isomennd 45842 | Sufficient condition to pr... |
| caragenel2d 45843 | Membership in the Caratheo... |
| omege0 45844 | If the outer measure of a ... |
| omess0 45845 | If the outer measure of a ... |
| caragencmpl 45846 | A measure built with the C... |
| vonval 45851 | Value of the Lebesgue meas... |
| ovnval 45852 | Value of the Lebesgue oute... |
| elhoi 45853 | Membership in a multidimen... |
| icoresmbl 45854 | A closed-below, open-above... |
| hoissre 45855 | The projection of a half-o... |
| ovnval2 45856 | Value of the Lebesgue oute... |
| volicorecl 45857 | The Lebesgue measure of a ... |
| hoiprodcl 45858 | The pre-measure of half-op... |
| hoicvr 45859 | ` I ` is a countable set o... |
| hoissrrn 45860 | A half-open interval is a ... |
| ovn0val 45861 | The Lebesgue outer measure... |
| ovnn0val 45862 | The value of a (multidimen... |
| ovnval2b 45863 | Value of the Lebesgue oute... |
| volicorescl 45864 | The Lebesgue measure of a ... |
| ovnprodcl 45865 | The product used in the de... |
| hoiprodcl2 45866 | The pre-measure of half-op... |
| hoicvrrex 45867 | Any subset of the multidim... |
| ovnsupge0 45868 | The set used in the defini... |
| ovnlecvr 45869 | Given a subset of multidim... |
| ovnpnfelsup 45870 | ` +oo ` is an element of t... |
| ovnsslelem 45871 | The (multidimensional, non... |
| ovnssle 45872 | The (multidimensional) Leb... |
| ovnlerp 45873 | The Lebesgue outer measure... |
| ovnf 45874 | The Lebesgue outer measure... |
| ovncvrrp 45875 | The Lebesgue outer measure... |
| ovn0lem 45876 | For any finite dimension, ... |
| ovn0 45877 | For any finite dimension, ... |
| ovncl 45878 | The Lebesgue outer measure... |
| ovn02 45879 | For the zero-dimensional s... |
| ovnxrcl 45880 | The Lebesgue outer measure... |
| ovnsubaddlem1 45881 | The Lebesgue outer measure... |
| ovnsubaddlem2 45882 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 45883 | ` ( voln* `` X ) ` is suba... |
| ovnome 45884 | ` ( voln* `` X ) ` is an o... |
| vonmea 45885 | ` ( voln `` X ) ` is a mea... |
| volicon0 45886 | The measure of a nonempty ... |
| hsphoif 45887 | ` H ` is a function (that ... |
| hoidmvval 45888 | The dimensional volume of ... |
| hoissrrn2 45889 | A half-open interval is a ... |
| hsphoival 45890 | ` H ` is a function (that ... |
| hoiprodcl3 45891 | The pre-measure of half-op... |
| volicore 45892 | The Lebesgue measure of a ... |
| hoidmvcl 45893 | The dimensional volume of ... |
| hoidmv0val 45894 | The dimensional volume of ... |
| hoidmvn0val 45895 | The dimensional volume of ... |
| hsphoidmvle2 45896 | The dimensional volume of ... |
| hsphoidmvle 45897 | The dimensional volume of ... |
| hoidmvval0 45898 | The dimensional volume of ... |
| hoiprodp1 45899 | The dimensional volume of ... |
| sge0hsphoire 45900 | If the generalized sum of ... |
| hoidmvval0b 45901 | The dimensional volume of ... |
| hoidmv1lelem1 45902 | The supremum of ` U ` belo... |
| hoidmv1lelem2 45903 | This is the contradiction ... |
| hoidmv1lelem3 45904 | The dimensional volume of ... |
| hoidmv1le 45905 | The dimensional volume of ... |
| hoidmvlelem1 45906 | The supremum of ` U ` belo... |
| hoidmvlelem2 45907 | This is the contradiction ... |
| hoidmvlelem3 45908 | This is the contradiction ... |
| hoidmvlelem4 45909 | The dimensional volume of ... |
| hoidmvlelem5 45910 | The dimensional volume of ... |
| hoidmvle 45911 | The dimensional volume of ... |
| ovnhoilem1 45912 | The Lebesgue outer measure... |
| ovnhoilem2 45913 | The Lebesgue outer measure... |
| ovnhoi 45914 | The Lebesgue outer measure... |
| dmovn 45915 | The domain of the Lebesgue... |
| hoicoto2 45916 | The half-open interval exp... |
| dmvon 45917 | Lebesgue measurable n-dime... |
| hoi2toco 45918 | The half-open interval exp... |
| hoidifhspval 45919 | ` D ` is a function that r... |
| hspval 45920 | The value of the half-spac... |
| ovnlecvr2 45921 | Given a subset of multidim... |
| ovncvr2 45922 | ` B ` and ` T ` are the le... |
| dmovnsal 45923 | The domain of the Lebesgue... |
| unidmovn 45924 | Base set of the n-dimensio... |
| rrnmbl 45925 | The set of n-dimensional R... |
| hoidifhspval2 45926 | ` D ` is a function that r... |
| hspdifhsp 45927 | A n-dimensional half-open ... |
| unidmvon 45928 | Base set of the n-dimensio... |
| hoidifhspf 45929 | ` D ` is a function that r... |
| hoidifhspval3 45930 | ` D ` is a function that r... |
| hoidifhspdmvle 45931 | The dimensional volume of ... |
| voncmpl 45932 | The Lebesgue measure is co... |
| hoiqssbllem1 45933 | The center of the n-dimens... |
| hoiqssbllem2 45934 | The center of the n-dimens... |
| hoiqssbllem3 45935 | A n-dimensional ball conta... |
| hoiqssbl 45936 | A n-dimensional ball conta... |
| hspmbllem1 45937 | Any half-space of the n-di... |
| hspmbllem2 45938 | Any half-space of the n-di... |
| hspmbllem3 45939 | Any half-space of the n-di... |
| hspmbl 45940 | Any half-space of the n-di... |
| hoimbllem 45941 | Any n-dimensional half-ope... |
| hoimbl 45942 | Any n-dimensional half-ope... |
| opnvonmbllem1 45943 | The half-open interval exp... |
| opnvonmbllem2 45944 | An open subset of the n-di... |
| opnvonmbl 45945 | An open subset of the n-di... |
| opnssborel 45946 | Open sets of a generalized... |
| borelmbl 45947 | All Borel subsets of the n... |
| volicorege0 45948 | The Lebesgue measure of a ... |
| isvonmbl 45949 | The predicate " ` A ` is m... |
| mblvon 45950 | The n-dimensional Lebesgue... |
| vonmblss 45951 | n-dimensional Lebesgue mea... |
| volico2 45952 | The measure of left-closed... |
| vonmblss2 45953 | n-dimensional Lebesgue mea... |
| ovolval2lem 45954 | The value of the Lebesgue ... |
| ovolval2 45955 | The value of the Lebesgue ... |
| ovnsubadd2lem 45956 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 45957 | ` ( voln* `` X ) ` is suba... |
| ovolval3 45958 | The value of the Lebesgue ... |
| ovnsplit 45959 | The n-dimensional Lebesgue... |
| ovolval4lem1 45960 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 45961 | The value of the Lebesgue ... |
| ovolval4 45962 | The value of the Lebesgue ... |
| ovolval5lem1 45963 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 45964 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 45965 | The value of the Lebesgue ... |
| ovolval5 45966 | The value of the Lebesgue ... |
| ovnovollem1 45967 | if ` F ` is a cover of ` B... |
| ovnovollem2 45968 | if ` I ` is a cover of ` (... |
| ovnovollem3 45969 | The 1-dimensional Lebesgue... |
| ovnovol 45970 | The 1-dimensional Lebesgue... |
| vonvolmbllem 45971 | If a subset ` B ` of real ... |
| vonvolmbl 45972 | A subset of Real numbers i... |
| vonvol 45973 | The 1-dimensional Lebesgue... |
| vonvolmbl2 45974 | A subset ` X ` of the spac... |
| vonvol2 45975 | The 1-dimensional Lebesgue... |
| hoimbl2 45976 | Any n-dimensional half-ope... |
| voncl 45977 | The Lebesgue measure of a ... |
| vonhoi 45978 | The Lebesgue outer measure... |
| vonxrcl 45979 | The Lebesgue measure of a ... |
| ioosshoi 45980 | A n-dimensional open inter... |
| vonn0hoi 45981 | The Lebesgue outer measure... |
| von0val 45982 | The Lebesgue measure (for ... |
| vonhoire 45983 | The Lebesgue measure of a ... |
| iinhoiicclem 45984 | A n-dimensional closed int... |
| iinhoiicc 45985 | A n-dimensional closed int... |
| iunhoiioolem 45986 | A n-dimensional open inter... |
| iunhoiioo 45987 | A n-dimensional open inter... |
| ioovonmbl 45988 | Any n-dimensional open int... |
| iccvonmbllem 45989 | Any n-dimensional closed i... |
| iccvonmbl 45990 | Any n-dimensional closed i... |
| vonioolem1 45991 | The sequence of the measur... |
| vonioolem2 45992 | The n-dimensional Lebesgue... |
| vonioo 45993 | The n-dimensional Lebesgue... |
| vonicclem1 45994 | The sequence of the measur... |
| vonicclem2 45995 | The n-dimensional Lebesgue... |
| vonicc 45996 | The n-dimensional Lebesgue... |
| snvonmbl 45997 | A n-dimensional singleton ... |
| vonn0ioo 45998 | The n-dimensional Lebesgue... |
| vonn0icc 45999 | The n-dimensional Lebesgue... |
| ctvonmbl 46000 | Any n-dimensional countabl... |
| vonn0ioo2 46001 | The n-dimensional Lebesgue... |
| vonsn 46002 | The n-dimensional Lebesgue... |
| vonn0icc2 46003 | The n-dimensional Lebesgue... |
| vonct 46004 | The n-dimensional Lebesgue... |
| vitali2 46005 | There are non-measurable s... |
| pimltmnf2f 46008 | Given a real-valued functi... |
| pimltmnf2 46009 | Given a real-valued functi... |
| preimagelt 46010 | The preimage of a right-op... |
| preimalegt 46011 | The preimage of a left-ope... |
| pimconstlt0 46012 | Given a constant function,... |
| pimconstlt1 46013 | Given a constant function,... |
| pimltpnff 46014 | Given a real-valued functi... |
| pimltpnf 46015 | Given a real-valued functi... |
| pimgtpnf2f 46016 | Given a real-valued functi... |
| pimgtpnf2 46017 | Given a real-valued functi... |
| salpreimagelt 46018 | If all the preimages of le... |
| pimrecltpos 46019 | The preimage of an unbound... |
| salpreimalegt 46020 | If all the preimages of ri... |
| pimiooltgt 46021 | The preimage of an open in... |
| preimaicomnf 46022 | Preimage of an open interv... |
| pimltpnf2f 46023 | Given a real-valued functi... |
| pimltpnf2 46024 | Given a real-valued functi... |
| pimgtmnf2 46025 | Given a real-valued functi... |
| pimdecfgtioc 46026 | Given a nonincreasing func... |
| pimincfltioc 46027 | Given a nondecreasing func... |
| pimdecfgtioo 46028 | Given a nondecreasing func... |
| pimincfltioo 46029 | Given a nondecreasing func... |
| preimaioomnf 46030 | Preimage of an open interv... |
| preimageiingt 46031 | A preimage of a left-close... |
| preimaleiinlt 46032 | A preimage of a left-open,... |
| pimgtmnff 46033 | Given a real-valued functi... |
| pimgtmnf 46034 | Given a real-valued functi... |
| pimrecltneg 46035 | The preimage of an unbound... |
| salpreimagtge 46036 | If all the preimages of le... |
| salpreimaltle 46037 | If all the preimages of ri... |
| issmflem 46038 | The predicate " ` F ` is a... |
| issmf 46039 | The predicate " ` F ` is a... |
| salpreimalelt 46040 | If all the preimages of ri... |
| salpreimagtlt 46041 | If all the preimages of le... |
| smfpreimalt 46042 | Given a function measurabl... |
| smff 46043 | A function measurable w.r.... |
| smfdmss 46044 | The domain of a function m... |
| issmff 46045 | The predicate " ` F ` is a... |
| issmfd 46046 | A sufficient condition for... |
| smfpreimaltf 46047 | Given a function measurabl... |
| issmfdf 46048 | A sufficient condition for... |
| sssmf 46049 | The restriction of a sigma... |
| mbfresmf 46050 | A real-valued measurable f... |
| cnfsmf 46051 | A continuous function is m... |
| incsmflem 46052 | A nondecreasing function i... |
| incsmf 46053 | A real-valued, nondecreasi... |
| smfsssmf 46054 | If a function is measurabl... |
| issmflelem 46055 | The predicate " ` F ` is a... |
| issmfle 46056 | The predicate " ` F ` is a... |
| smfpimltmpt 46057 | Given a function measurabl... |
| smfpimltxr 46058 | Given a function measurabl... |
| issmfdmpt 46059 | A sufficient condition for... |
| smfconst 46060 | Given a sigma-algebra over... |
| sssmfmpt 46061 | The restriction of a sigma... |
| cnfrrnsmf 46062 | A function, continuous fro... |
| smfid 46063 | The identity function is B... |
| bormflebmf 46064 | A Borel measurable functio... |
| smfpreimale 46065 | Given a function measurabl... |
| issmfgtlem 46066 | The predicate " ` F ` is a... |
| issmfgt 46067 | The predicate " ` F ` is a... |
| issmfled 46068 | A sufficient condition for... |
| smfpimltxrmptf 46069 | Given a function measurabl... |
| smfpimltxrmpt 46070 | Given a function measurabl... |
| smfmbfcex 46071 | A constant function, with ... |
| issmfgtd 46072 | A sufficient condition for... |
| smfpreimagt 46073 | Given a function measurabl... |
| smfaddlem1 46074 | Given the sum of two funct... |
| smfaddlem2 46075 | The sum of two sigma-measu... |
| smfadd 46076 | The sum of two sigma-measu... |
| decsmflem 46077 | A nonincreasing function i... |
| decsmf 46078 | A real-valued, nonincreasi... |
| smfpreimagtf 46079 | Given a function measurabl... |
| issmfgelem 46080 | The predicate " ` F ` is a... |
| issmfge 46081 | The predicate " ` F ` is a... |
| smflimlem1 46082 | Lemma for the proof that t... |
| smflimlem2 46083 | Lemma for the proof that t... |
| smflimlem3 46084 | The limit of sigma-measura... |
| smflimlem4 46085 | Lemma for the proof that t... |
| smflimlem5 46086 | Lemma for the proof that t... |
| smflimlem6 46087 | Lemma for the proof that t... |
| smflim 46088 | The limit of sigma-measura... |
| nsssmfmbflem 46089 | The sigma-measurable funct... |
| nsssmfmbf 46090 | The sigma-measurable funct... |
| smfpimgtxr 46091 | Given a function measurabl... |
| smfpimgtmpt 46092 | Given a function measurabl... |
| smfpreimage 46093 | Given a function measurabl... |
| mbfpsssmf 46094 | Real-valued measurable fun... |
| smfpimgtxrmptf 46095 | Given a function measurabl... |
| smfpimgtxrmpt 46096 | Given a function measurabl... |
| smfpimioompt 46097 | Given a function measurabl... |
| smfpimioo 46098 | Given a function measurabl... |
| smfresal 46099 | Given a sigma-measurable f... |
| smfrec 46100 | The reciprocal of a sigma-... |
| smfres 46101 | The restriction of sigma-m... |
| smfmullem1 46102 | The multiplication of two ... |
| smfmullem2 46103 | The multiplication of two ... |
| smfmullem3 46104 | The multiplication of two ... |
| smfmullem4 46105 | The multiplication of two ... |
| smfmul 46106 | The multiplication of two ... |
| smfmulc1 46107 | A sigma-measurable functio... |
| smfdiv 46108 | The fraction of two sigma-... |
| smfpimbor1lem1 46109 | Every open set belongs to ... |
| smfpimbor1lem2 46110 | Given a sigma-measurable f... |
| smfpimbor1 46111 | Given a sigma-measurable f... |
| smf2id 46112 | Twice the identity functio... |
| smfco 46113 | The composition of a Borel... |
| smfneg 46114 | The negative of a sigma-me... |
| smffmptf 46115 | A function measurable w.r.... |
| smffmpt 46116 | A function measurable w.r.... |
| smflim2 46117 | The limit of a sequence of... |
| smfpimcclem 46118 | Lemma for ~ smfpimcc given... |
| smfpimcc 46119 | Given a countable set of s... |
| issmfle2d 46120 | A sufficient condition for... |
| smflimmpt 46121 | The limit of a sequence of... |
| smfsuplem1 46122 | The supremum of a countabl... |
| smfsuplem2 46123 | The supremum of a countabl... |
| smfsuplem3 46124 | The supremum of a countabl... |
| smfsup 46125 | The supremum of a countabl... |
| smfsupmpt 46126 | The supremum of a countabl... |
| smfsupxr 46127 | The supremum of a countabl... |
| smfinflem 46128 | The infimum of a countable... |
| smfinf 46129 | The infimum of a countable... |
| smfinfmpt 46130 | The infimum of a countable... |
| smflimsuplem1 46131 | If ` H ` converges, the ` ... |
| smflimsuplem2 46132 | The superior limit of a se... |
| smflimsuplem3 46133 | The limit of the ` ( H `` ... |
| smflimsuplem4 46134 | If ` H ` converges, the ` ... |
| smflimsuplem5 46135 | ` H ` converges to the sup... |
| smflimsuplem6 46136 | The superior limit of a se... |
| smflimsuplem7 46137 | The superior limit of a se... |
| smflimsuplem8 46138 | The superior limit of a se... |
| smflimsup 46139 | The superior limit of a se... |
| smflimsupmpt 46140 | The superior limit of a se... |
| smfliminflem 46141 | The inferior limit of a co... |
| smfliminf 46142 | The inferior limit of a co... |
| smfliminfmpt 46143 | The inferior limit of a co... |
| adddmmbl 46144 | If two functions have doma... |
| adddmmbl2 46145 | If two functions have doma... |
| muldmmbl 46146 | If two functions have doma... |
| muldmmbl2 46147 | If two functions have doma... |
| smfdmmblpimne 46148 | If a measurable function w... |
| smfdivdmmbl 46149 | If a functions and a sigma... |
| smfpimne 46150 | Given a function measurabl... |
| smfpimne2 46151 | Given a function measurabl... |
| smfdivdmmbl2 46152 | If a functions and a sigma... |
| fsupdm 46153 | The domain of the sup func... |
| fsupdm2 46154 | The domain of the sup func... |
| smfsupdmmbllem 46155 | If a countable set of sigm... |
| smfsupdmmbl 46156 | If a countable set of sigm... |
| finfdm 46157 | The domain of the inf func... |
| finfdm2 46158 | The domain of the inf func... |
| smfinfdmmbllem 46159 | If a countable set of sigm... |
| smfinfdmmbl 46160 | If a countable set of sigm... |
| sigarval 46161 | Define the signed area by ... |
| sigarim 46162 | Signed area takes value in... |
| sigarac 46163 | Signed area is anticommuta... |
| sigaraf 46164 | Signed area is additive by... |
| sigarmf 46165 | Signed area is additive (w... |
| sigaras 46166 | Signed area is additive by... |
| sigarms 46167 | Signed area is additive (w... |
| sigarls 46168 | Signed area is linear by t... |
| sigarid 46169 | Signed area of a flat para... |
| sigarexp 46170 | Expand the signed area for... |
| sigarperm 46171 | Signed area ` ( A - C ) G ... |
| sigardiv 46172 | If signed area between vec... |
| sigarimcd 46173 | Signed area takes value in... |
| sigariz 46174 | If signed area is zero, th... |
| sigarcol 46175 | Given three points ` A ` ,... |
| sharhght 46176 | Let ` A B C ` be a triangl... |
| sigaradd 46177 | Subtracting (double) area ... |
| cevathlem1 46178 | Ceva's theorem first lemma... |
| cevathlem2 46179 | Ceva's theorem second lemm... |
| cevath 46180 | Ceva's theorem. Let ` A B... |
| simpcntrab 46181 | The center of a simple gro... |
| et-ltneverrefl 46182 | Less-than class is never r... |
| et-equeucl 46183 | Alternative proof that equ... |
| et-sqrtnegnre 46184 | The square root of a negat... |
| natlocalincr 46185 | Global monotonicity on hal... |
| natglobalincr 46186 | Local monotonicity on half... |
| upwordnul 46189 | Empty set is an increasing... |
| upwordisword 46190 | Any increasing sequence is... |
| singoutnword 46191 | Singleton with character o... |
| singoutnupword 46192 | Singleton with character o... |
| upwordsing 46193 | Singleton is an increasing... |
| upwordsseti 46194 | Strictly increasing sequen... |
| tworepnotupword 46195 | Concatenation of identical... |
| upwrdfi 46196 | There is a finite number o... |
| hirstL-ax3 46197 | The third axiom of a syste... |
| ax3h 46198 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 46199 | A closed form showing (a i... |
| aibandbiaiaiffb 46200 | A closed form showing (a i... |
| notatnand 46201 | Do not use. Use intnanr i... |
| aistia 46202 | Given a is equivalent to `... |
| aisfina 46203 | Given a is equivalent to `... |
| bothtbothsame 46204 | Given both a, b are equiva... |
| bothfbothsame 46205 | Given both a, b are equiva... |
| aiffbbtat 46206 | Given a is equivalent to b... |
| aisbbisfaisf 46207 | Given a is equivalent to b... |
| axorbtnotaiffb 46208 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 46209 | Given a is equivalent to (... |
| axorbciffatcxorb 46210 | Given a is equivalent to (... |
| aibnbna 46211 | Given a implies b, (not b)... |
| aibnbaif 46212 | Given a implies b, not b, ... |
| aiffbtbat 46213 | Given a is equivalent to b... |
| astbstanbst 46214 | Given a is equivalent to T... |
| aistbistaandb 46215 | Given a is equivalent to T... |
| aisbnaxb 46216 | Given a is equivalent to b... |
| atbiffatnnb 46217 | If a implies b, then a imp... |
| bisaiaisb 46218 | Application of bicom1 with... |
| atbiffatnnbalt 46219 | If a implies b, then a imp... |
| abnotbtaxb 46220 | Assuming a, not b, there e... |
| abnotataxb 46221 | Assuming not a, b, there e... |
| conimpf 46222 | Assuming a, not b, and a i... |
| conimpfalt 46223 | Assuming a, not b, and a i... |
| aistbisfiaxb 46224 | Given a is equivalent to T... |
| aisfbistiaxb 46225 | Given a is equivalent to F... |
| aifftbifffaibif 46226 | Given a is equivalent to T... |
| aifftbifffaibifff 46227 | Given a is equivalent to T... |
| atnaiana 46228 | Given a, it is not the cas... |
| ainaiaandna 46229 | Given a, a implies it is n... |
| abcdta 46230 | Given (((a and b) and c) a... |
| abcdtb 46231 | Given (((a and b) and c) a... |
| abcdtc 46232 | Given (((a and b) and c) a... |
| abcdtd 46233 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 46234 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 46235 | Operands in a biconditiona... |
| nabctnabc 46236 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 46237 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 46238 | From one negated implicati... |
| twonotinotbothi 46239 | From these two negated imp... |
| clifte 46240 | show d is the same as an i... |
| cliftet 46241 | show d is the same as an i... |
| clifteta 46242 | show d is the same as an i... |
| cliftetb 46243 | show d is the same as an i... |
| confun 46244 | Given the hypotheses there... |
| confun2 46245 | Confun simplified to two p... |
| confun3 46246 | Confun's more complex form... |
| confun4 46247 | An attempt at derivative. ... |
| confun5 46248 | An attempt at derivative. ... |
| plcofph 46249 | Given, a,b and a "definiti... |
| pldofph 46250 | Given, a,b c, d, "definiti... |
| plvcofph 46251 | Given, a,b,d, and "definit... |
| plvcofphax 46252 | Given, a,b,d, and "definit... |
| plvofpos 46253 | rh is derivable because ON... |
| mdandyv0 46254 | Given the equivalences set... |
| mdandyv1 46255 | Given the equivalences set... |
| mdandyv2 46256 | Given the equivalences set... |
| mdandyv3 46257 | Given the equivalences set... |
| mdandyv4 46258 | Given the equivalences set... |
| mdandyv5 46259 | Given the equivalences set... |
| mdandyv6 46260 | Given the equivalences set... |
| mdandyv7 46261 | Given the equivalences set... |
| mdandyv8 46262 | Given the equivalences set... |
| mdandyv9 46263 | Given the equivalences set... |
| mdandyv10 46264 | Given the equivalences set... |
| mdandyv11 46265 | Given the equivalences set... |
| mdandyv12 46266 | Given the equivalences set... |
| mdandyv13 46267 | Given the equivalences set... |
| mdandyv14 46268 | Given the equivalences set... |
| mdandyv15 46269 | Given the equivalences set... |
| mdandyvr0 46270 | Given the equivalences set... |
| mdandyvr1 46271 | Given the equivalences set... |
| mdandyvr2 46272 | Given the equivalences set... |
| mdandyvr3 46273 | Given the equivalences set... |
| mdandyvr4 46274 | Given the equivalences set... |
| mdandyvr5 46275 | Given the equivalences set... |
| mdandyvr6 46276 | Given the equivalences set... |
| mdandyvr7 46277 | Given the equivalences set... |
| mdandyvr8 46278 | Given the equivalences set... |
| mdandyvr9 46279 | Given the equivalences set... |
| mdandyvr10 46280 | Given the equivalences set... |
| mdandyvr11 46281 | Given the equivalences set... |
| mdandyvr12 46282 | Given the equivalences set... |
| mdandyvr13 46283 | Given the equivalences set... |
| mdandyvr14 46284 | Given the equivalences set... |
| mdandyvr15 46285 | Given the equivalences set... |
| mdandyvrx0 46286 | Given the exclusivities se... |
| mdandyvrx1 46287 | Given the exclusivities se... |
| mdandyvrx2 46288 | Given the exclusivities se... |
| mdandyvrx3 46289 | Given the exclusivities se... |
| mdandyvrx4 46290 | Given the exclusivities se... |
| mdandyvrx5 46291 | Given the exclusivities se... |
| mdandyvrx6 46292 | Given the exclusivities se... |
| mdandyvrx7 46293 | Given the exclusivities se... |
| mdandyvrx8 46294 | Given the exclusivities se... |
| mdandyvrx9 46295 | Given the exclusivities se... |
| mdandyvrx10 46296 | Given the exclusivities se... |
| mdandyvrx11 46297 | Given the exclusivities se... |
| mdandyvrx12 46298 | Given the exclusivities se... |
| mdandyvrx13 46299 | Given the exclusivities se... |
| mdandyvrx14 46300 | Given the exclusivities se... |
| mdandyvrx15 46301 | Given the exclusivities se... |
| H15NH16TH15IH16 46302 | Given 15 hypotheses and a ... |
| dandysum2p2e4 46303 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 46304 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 46305 | Replacement of a nested an... |
| adh-minim 46306 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 46307 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 46308 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 46309 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 46310 | Fourth lemma for the deriv... |
| adh-minim-ax1 46311 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 46312 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 46313 | Sixth lemma for the deriva... |
| adh-minim-ax2c 46314 | Derivation of a commuted f... |
| adh-minim-ax2 46315 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 46316 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 46317 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 46318 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 46319 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 46320 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 46321 | Third lemma for the deriva... |
| adh-minimp-sylsimp 46322 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 46323 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 46324 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 46325 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 46326 | Fourth lemma for the deriv... |
| adh-minimp-ax2 46327 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 46328 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 46329 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 46330 | If a class with one elemen... |
| eusnsn 46331 | There is a unique element ... |
| absnsb 46332 | If the class abstraction `... |
| euabsneu 46333 | Another way to express exi... |
| elprneb 46334 | An element of a proper uno... |
| oppr 46335 | Equality for ordered pairs... |
| opprb 46336 | Equality for unordered pai... |
| or2expropbilem1 46337 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 46338 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 46339 | If two classes are strictl... |
| eubrv 46340 | If there is a unique set w... |
| eubrdm 46341 | If there is a unique set w... |
| eldmressn 46342 | Element of the domain of a... |
| iota0def 46343 | Example for a defined iota... |
| iota0ndef 46344 | Example for an undefined i... |
| fveqvfvv 46345 | If a function's value at a... |
| fnresfnco 46346 | Composition of two functio... |
| funcoressn 46347 | A composition restricted t... |
| funressnfv 46348 | A restriction to a singlet... |
| funressndmfvrn 46349 | The value of a function ` ... |
| funressnvmo 46350 | A function restricted to a... |
| funressnmo 46351 | A function restricted to a... |
| funressneu 46352 | There is exactly one value... |
| fresfo 46353 | Conditions for a restricti... |
| fsetsniunop 46354 | The class of all functions... |
| fsetabsnop 46355 | The class of all functions... |
| fsetsnf 46356 | The mapping of an element ... |
| fsetsnf1 46357 | The mapping of an element ... |
| fsetsnfo 46358 | The mapping of an element ... |
| fsetsnf1o 46359 | The mapping of an element ... |
| fsetsnprcnex 46360 | The class of all functions... |
| cfsetssfset 46361 | The class of constant func... |
| cfsetsnfsetfv 46362 | The function value of the ... |
| cfsetsnfsetf 46363 | The mapping of the class o... |
| cfsetsnfsetf1 46364 | The mapping of the class o... |
| cfsetsnfsetfo 46365 | The mapping of the class o... |
| cfsetsnfsetf1o 46366 | The mapping of the class o... |
| fsetprcnexALT 46367 | First version of proof for... |
| fcoreslem1 46368 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 46369 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 46370 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 46371 | Lemma 4 for ~ fcores . (C... |
| fcores 46372 | Every composite function `... |
| fcoresf1lem 46373 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 46374 | If a composition is inject... |
| fcoresf1b 46375 | A composition is injective... |
| fcoresfo 46376 | If a composition is surjec... |
| fcoresfob 46377 | A composition is surjectiv... |
| fcoresf1ob 46378 | A composition is bijective... |
| f1cof1blem 46379 | Lemma for ~ f1cof1b and ~ ... |
| f1cof1b 46380 | If the range of ` F ` equa... |
| funfocofob 46381 | If the domain of a functio... |
| fnfocofob 46382 | If the domain of a functio... |
| focofob 46383 | If the domain of a functio... |
| f1ocof1ob 46384 | If the range of ` F ` equa... |
| f1ocof1ob2 46385 | If the range of ` F ` equa... |
| aiotajust 46387 | Soundness justification th... |
| dfaiota2 46389 | Alternate definition of th... |
| reuabaiotaiota 46390 | The iota and the alternate... |
| reuaiotaiota 46391 | The iota and the alternate... |
| aiotaexb 46392 | The alternate iota over a ... |
| aiotavb 46393 | The alternate iota over a ... |
| aiotaint 46394 | This is to ~ df-aiota what... |
| dfaiota3 46395 | Alternate definition of ` ... |
| iotan0aiotaex 46396 | If the iota over a wff ` p... |
| aiotaexaiotaiota 46397 | The alternate iota over a ... |
| aiotaval 46398 | Theorem 8.19 in [Quine] p.... |
| aiota0def 46399 | Example for a defined alte... |
| aiota0ndef 46400 | Example for an undefined a... |
| r19.32 46401 | Theorem 19.32 of [Margaris... |
| rexsb 46402 | An equivalent expression f... |
| rexrsb 46403 | An equivalent expression f... |
| 2rexsb 46404 | An equivalent expression f... |
| 2rexrsb 46405 | An equivalent expression f... |
| cbvral2 46406 | Change bound variables of ... |
| cbvrex2 46407 | Change bound variables of ... |
| ralndv1 46408 | Example for a theorem abou... |
| ralndv2 46409 | Second example for a theor... |
| reuf1odnf 46410 | There is exactly one eleme... |
| reuf1od 46411 | There is exactly one eleme... |
| euoreqb 46412 | There is a set which is eq... |
| 2reu3 46413 | Double restricted existent... |
| 2reu7 46414 | Two equivalent expressions... |
| 2reu8 46415 | Two equivalent expressions... |
| 2reu8i 46416 | Implication of a double re... |
| 2reuimp0 46417 | Implication of a double re... |
| 2reuimp 46418 | Implication of a double re... |
| ralbinrald 46425 | Elemination of a restricte... |
| nvelim 46426 | If a class is the universa... |
| alneu 46427 | If a statement holds for a... |
| eu2ndop1stv 46428 | If there is a unique secon... |
| dfateq12d 46429 | Equality deduction for "de... |
| nfdfat 46430 | Bound-variable hypothesis ... |
| dfdfat2 46431 | Alternate definition of th... |
| fundmdfat 46432 | A function is defined at a... |
| dfatprc 46433 | A function is not defined ... |
| dfatelrn 46434 | The value of a function ` ... |
| dfafv2 46435 | Alternative definition of ... |
| afveq12d 46436 | Equality deduction for fun... |
| afveq1 46437 | Equality theorem for funct... |
| afveq2 46438 | Equality theorem for funct... |
| nfafv 46439 | Bound-variable hypothesis ... |
| csbafv12g 46440 | Move class substitution in... |
| afvfundmfveq 46441 | If a class is a function r... |
| afvnfundmuv 46442 | If a set is not in the dom... |
| ndmafv 46443 | The value of a class outsi... |
| afvvdm 46444 | If the function value of a... |
| nfunsnafv 46445 | If the restriction of a cl... |
| afvvfunressn 46446 | If the function value of a... |
| afvprc 46447 | A function's value at a pr... |
| afvvv 46448 | If a function's value at a... |
| afvpcfv0 46449 | If the value of the altern... |
| afvnufveq 46450 | The value of the alternati... |
| afvvfveq 46451 | The value of the alternati... |
| afv0fv0 46452 | If the value of the altern... |
| afvfvn0fveq 46453 | If the function's value at... |
| afv0nbfvbi 46454 | The function's value at an... |
| afvfv0bi 46455 | The function's value at an... |
| afveu 46456 | The value of a function at... |
| fnbrafvb 46457 | Equivalence of function va... |
| fnopafvb 46458 | Equivalence of function va... |
| funbrafvb 46459 | Equivalence of function va... |
| funopafvb 46460 | Equivalence of function va... |
| funbrafv 46461 | The second argument of a b... |
| funbrafv2b 46462 | Function value in terms of... |
| dfafn5a 46463 | Representation of a functi... |
| dfafn5b 46464 | Representation of a functi... |
| fnrnafv 46465 | The range of a function ex... |
| afvelrnb 46466 | A member of a function's r... |
| afvelrnb0 46467 | A member of a function's r... |
| dfaimafn 46468 | Alternate definition of th... |
| dfaimafn2 46469 | Alternate definition of th... |
| afvelima 46470 | Function value in an image... |
| afvelrn 46471 | A function's value belongs... |
| fnafvelrn 46472 | A function's value belongs... |
| fafvelcdm 46473 | A function's value belongs... |
| ffnafv 46474 | A function maps to a class... |
| afvres 46475 | The value of a restricted ... |
| tz6.12-afv 46476 | Function value. Theorem 6... |
| tz6.12-1-afv 46477 | Function value (Theorem 6.... |
| dmfcoafv 46478 | Domains of a function comp... |
| afvco2 46479 | Value of a function compos... |
| rlimdmafv 46480 | Two ways to express that a... |
| aoveq123d 46481 | Equality deduction for ope... |
| nfaov 46482 | Bound-variable hypothesis ... |
| csbaovg 46483 | Move class substitution in... |
| aovfundmoveq 46484 | If a class is a function r... |
| aovnfundmuv 46485 | If an ordered pair is not ... |
| ndmaov 46486 | The value of an operation ... |
| ndmaovg 46487 | The value of an operation ... |
| aovvdm 46488 | If the operation value of ... |
| nfunsnaov 46489 | If the restriction of a cl... |
| aovvfunressn 46490 | If the operation value of ... |
| aovprc 46491 | The value of an operation ... |
| aovrcl 46492 | Reverse closure for an ope... |
| aovpcov0 46493 | If the alternative value o... |
| aovnuoveq 46494 | The alternative value of t... |
| aovvoveq 46495 | The alternative value of t... |
| aov0ov0 46496 | If the alternative value o... |
| aovovn0oveq 46497 | If the operation's value a... |
| aov0nbovbi 46498 | The operation's value on a... |
| aovov0bi 46499 | The operation's value on a... |
| rspceaov 46500 | A frequently used special ... |
| fnotaovb 46501 | Equivalence of operation v... |
| ffnaov 46502 | An operation maps to a cla... |
| faovcl 46503 | Closure law for an operati... |
| aovmpt4g 46504 | Value of a function given ... |
| aoprssdm 46505 | Domain of closure of an op... |
| ndmaovcl 46506 | The "closure" of an operat... |
| ndmaovrcl 46507 | Reverse closure law, in co... |
| ndmaovcom 46508 | Any operation is commutati... |
| ndmaovass 46509 | Any operation is associati... |
| ndmaovdistr 46510 | Any operation is distribut... |
| dfatafv2iota 46513 | If a function is defined a... |
| ndfatafv2 46514 | The alternate function val... |
| ndfatafv2undef 46515 | The alternate function val... |
| dfatafv2ex 46516 | The alternate function val... |
| afv2ex 46517 | The alternate function val... |
| afv2eq12d 46518 | Equality deduction for fun... |
| afv2eq1 46519 | Equality theorem for funct... |
| afv2eq2 46520 | Equality theorem for funct... |
| nfafv2 46521 | Bound-variable hypothesis ... |
| csbafv212g 46522 | Move class substitution in... |
| fexafv2ex 46523 | The alternate function val... |
| ndfatafv2nrn 46524 | The alternate function val... |
| ndmafv2nrn 46525 | The value of a class outsi... |
| funressndmafv2rn 46526 | The alternate function val... |
| afv2ndefb 46527 | Two ways to say that an al... |
| nfunsnafv2 46528 | If the restriction of a cl... |
| afv2prc 46529 | A function's value at a pr... |
| dfatafv2rnb 46530 | The alternate function val... |
| afv2orxorb 46531 | If a set is in the range o... |
| dmafv2rnb 46532 | The alternate function val... |
| fundmafv2rnb 46533 | The alternate function val... |
| afv2elrn 46534 | An alternate function valu... |
| afv20defat 46535 | If the alternate function ... |
| fnafv2elrn 46536 | An alternate function valu... |
| fafv2elcdm 46537 | An alternate function valu... |
| fafv2elrnb 46538 | An alternate function valu... |
| fcdmvafv2v 46539 | If the codomain of a funct... |
| tz6.12-2-afv2 46540 | Function value when ` F ` ... |
| afv2eu 46541 | The value of a function at... |
| afv2res 46542 | The value of a restricted ... |
| tz6.12-afv2 46543 | Function value (Theorem 6.... |
| tz6.12-1-afv2 46544 | Function value (Theorem 6.... |
| tz6.12c-afv2 46545 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 46546 | Corollary of Theorem 6.12(... |
| funressnbrafv2 46547 | The second argument of a b... |
| dfatbrafv2b 46548 | Equivalence of function va... |
| dfatopafv2b 46549 | Equivalence of function va... |
| funbrafv2 46550 | The second argument of a b... |
| fnbrafv2b 46551 | Equivalence of function va... |
| fnopafv2b 46552 | Equivalence of function va... |
| funbrafv22b 46553 | Equivalence of function va... |
| funopafv2b 46554 | Equivalence of function va... |
| dfatsnafv2 46555 | Singleton of function valu... |
| dfafv23 46556 | A definition of function v... |
| dfatdmfcoafv2 46557 | Domain of a function compo... |
| dfatcolem 46558 | Lemma for ~ dfatco . (Con... |
| dfatco 46559 | The predicate "defined at"... |
| afv2co2 46560 | Value of a function compos... |
| rlimdmafv2 46561 | Two ways to express that a... |
| dfafv22 46562 | Alternate definition of ` ... |
| afv2ndeffv0 46563 | If the alternate function ... |
| dfatafv2eqfv 46564 | If a function is defined a... |
| afv2rnfveq 46565 | If the alternate function ... |
| afv20fv0 46566 | If the alternate function ... |
| afv2fvn0fveq 46567 | If the function's value at... |
| afv2fv0 46568 | If the function's value at... |
| afv2fv0b 46569 | The function's value at an... |
| afv2fv0xorb 46570 | If a set is in the range o... |
| an4com24 46571 | Rearrangement of 4 conjunc... |
| 3an4ancom24 46572 | Commutative law for a conj... |
| 4an21 46573 | Rearrangement of 4 conjunc... |
| dfnelbr2 46576 | Alternate definition of th... |
| nelbr 46577 | The binary relation of a s... |
| nelbrim 46578 | If a set is related to ano... |
| nelbrnel 46579 | A set is related to anothe... |
| nelbrnelim 46580 | If a set is related to ano... |
| ralralimp 46581 | Selecting one of two alter... |
| otiunsndisjX 46582 | The union of singletons co... |
| fvifeq 46583 | Equality of function value... |
| rnfdmpr 46584 | The range of a one-to-one ... |
| imarnf1pr 46585 | The image of the range of ... |
| funop1 46586 | A function is an ordered p... |
| fun2dmnopgexmpl 46587 | A function with a domain c... |
| opabresex0d 46588 | A collection of ordered pa... |
| opabbrfex0d 46589 | A collection of ordered pa... |
| opabresexd 46590 | A collection of ordered pa... |
| opabbrfexd 46591 | A collection of ordered pa... |
| f1oresf1orab 46592 | Build a bijection by restr... |
| f1oresf1o 46593 | Build a bijection by restr... |
| f1oresf1o2 46594 | Build a bijection by restr... |
| fvmptrab 46595 | Value of a function mappin... |
| fvmptrabdm 46596 | Value of a function mappin... |
| cnambpcma 46597 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 46598 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 46599 | The sum of two complex num... |
| leaddsuble 46600 | Addition and subtraction o... |
| 2leaddle2 46601 | If two real numbers are le... |
| ltnltne 46602 | Variant of trichotomy law ... |
| p1lep2 46603 | A real number increasd by ... |
| ltsubsubaddltsub 46604 | If the result of subtracti... |
| zm1nn 46605 | An integer minus 1 is posi... |
| readdcnnred 46606 | The sum of a real number a... |
| resubcnnred 46607 | The difference of a real n... |
| recnmulnred 46608 | The product of a real numb... |
| cndivrenred 46609 | The quotient of an imagina... |
| sqrtnegnre 46610 | The square root of a negat... |
| nn0resubcl 46611 | Closure law for subtractio... |
| zgeltp1eq 46612 | If an integer is between a... |
| 1t10e1p1e11 46613 | 11 is 1 times 10 to the po... |
| deccarry 46614 | Add 1 to a 2 digit number ... |
| eluzge0nn0 46615 | If an integer is greater t... |
| nltle2tri 46616 | Negated extended trichotom... |
| ssfz12 46617 | Subset relationship for fi... |
| elfz2z 46618 | Membership of an integer i... |
| 2elfz3nn0 46619 | If there are two elements ... |
| fz0addcom 46620 | The addition of two member... |
| 2elfz2melfz 46621 | If the sum of two integers... |
| fz0addge0 46622 | The sum of two integers in... |
| elfzlble 46623 | Membership of an integer i... |
| elfzelfzlble 46624 | Membership of an element o... |
| fzopred 46625 | Join a predecessor to the ... |
| fzopredsuc 46626 | Join a predecessor and a s... |
| 1fzopredsuc 46627 | Join 0 and a successor to ... |
| el1fzopredsuc 46628 | An element of an open inte... |
| subsubelfzo0 46629 | Subtracting a difference f... |
| fzoopth 46630 | A half-open integer range ... |
| 2ffzoeq 46631 | Two functions over a half-... |
| m1mod0mod1 46632 | An integer decreased by 1 ... |
| elmod2 46633 | An integer modulo 2 is eit... |
| smonoord 46634 | Ordering relation for a st... |
| fsummsndifre 46635 | A finite sum with one of i... |
| fsumsplitsndif 46636 | Separate out a term in a f... |
| fsummmodsndifre 46637 | A finite sum of summands m... |
| fsummmodsnunz 46638 | A finite sum of summands m... |
| setsidel 46639 | The injected slot is an el... |
| setsnidel 46640 | The injected slot is an el... |
| setsv 46641 | The value of the structure... |
| preimafvsnel 46642 | The preimage of a function... |
| preimafvn0 46643 | The preimage of a function... |
| uniimafveqt 46644 | The union of the image of ... |
| uniimaprimaeqfv 46645 | The union of the image of ... |
| setpreimafvex 46646 | The class ` P ` of all pre... |
| elsetpreimafvb 46647 | The characterization of an... |
| elsetpreimafv 46648 | An element of the class ` ... |
| elsetpreimafvssdm 46649 | An element of the class ` ... |
| fvelsetpreimafv 46650 | There is an element in a p... |
| preimafvelsetpreimafv 46651 | The preimage of a function... |
| preimafvsspwdm 46652 | The class ` P ` of all pre... |
| 0nelsetpreimafv 46653 | The empty set is not an el... |
| elsetpreimafvbi 46654 | An element of the preimage... |
| elsetpreimafveqfv 46655 | The elements of the preima... |
| eqfvelsetpreimafv 46656 | If an element of the domai... |
| elsetpreimafvrab 46657 | An element of the preimage... |
| imaelsetpreimafv 46658 | The image of an element of... |
| uniimaelsetpreimafv 46659 | The union of the image of ... |
| elsetpreimafveq 46660 | If two preimages of functi... |
| fundcmpsurinjlem1 46661 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 46662 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 46663 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 46664 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 46665 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 46666 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 46667 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 46668 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 46669 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 46670 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 46671 | Every function ` F : A -->... |
| fundcmpsurinj 46672 | Every function ` F : A -->... |
| fundcmpsurbijinj 46673 | Every function ` F : A -->... |
| fundcmpsurinjimaid 46674 | Every function ` F : A -->... |
| fundcmpsurinjALT 46675 | Alternate proof of ~ fundc... |
| iccpval 46678 | Partition consisting of a ... |
| iccpart 46679 | A special partition. Corr... |
| iccpartimp 46680 | Implications for a class b... |
| iccpartres 46681 | The restriction of a parti... |
| iccpartxr 46682 | If there is a partition, t... |
| iccpartgtprec 46683 | If there is a partition, t... |
| iccpartipre 46684 | If there is a partition, t... |
| iccpartiltu 46685 | If there is a partition, t... |
| iccpartigtl 46686 | If there is a partition, t... |
| iccpartlt 46687 | If there is a partition, t... |
| iccpartltu 46688 | If there is a partition, t... |
| iccpartgtl 46689 | If there is a partition, t... |
| iccpartgt 46690 | If there is a partition, t... |
| iccpartleu 46691 | If there is a partition, t... |
| iccpartgel 46692 | If there is a partition, t... |
| iccpartrn 46693 | If there is a partition, t... |
| iccpartf 46694 | The range of the partition... |
| iccpartel 46695 | If there is a partition, t... |
| iccelpart 46696 | An element of any partitio... |
| iccpartiun 46697 | A half-open interval of ex... |
| icceuelpartlem 46698 | Lemma for ~ icceuelpart . ... |
| icceuelpart 46699 | An element of a partitione... |
| iccpartdisj 46700 | The segments of a partitio... |
| iccpartnel 46701 | A point of a partition is ... |
| fargshiftfv 46702 | If a class is a function, ... |
| fargshiftf 46703 | If a class is a function, ... |
| fargshiftf1 46704 | If a function is 1-1, then... |
| fargshiftfo 46705 | If a function is onto, the... |
| fargshiftfva 46706 | The values of a shifted fu... |
| lswn0 46707 | The last symbol of a not e... |
| nfich1 46710 | The first interchangeable ... |
| nfich2 46711 | The second interchangeable... |
| ichv 46712 | Setvar variables are inter... |
| ichf 46713 | Setvar variables are inter... |
| ichid 46714 | A setvar variable is alway... |
| icht 46715 | A theorem is interchangeab... |
| ichbidv 46716 | Formula building rule for ... |
| ichcircshi 46717 | The setvar variables are i... |
| ichan 46718 | If two setvar variables ar... |
| ichn 46719 | Negation does not affect i... |
| ichim 46720 | Formula building rule for ... |
| dfich2 46721 | Alternate definition of th... |
| ichcom 46722 | The interchangeability of ... |
| ichbi12i 46723 | Equivalence for interchang... |
| icheqid 46724 | In an equality for the sam... |
| icheq 46725 | In an equality of setvar v... |
| ichnfimlem 46726 | Lemma for ~ ichnfim : A s... |
| ichnfim 46727 | If in an interchangeabilit... |
| ichnfb 46728 | If ` x ` and ` y ` are int... |
| ichal 46729 | Move a universal quantifie... |
| ich2al 46730 | Two setvar variables are a... |
| ich2ex 46731 | Two setvar variables are a... |
| ichexmpl1 46732 | Example for interchangeabl... |
| ichexmpl2 46733 | Example for interchangeabl... |
| ich2exprop 46734 | If the setvar variables ar... |
| ichnreuop 46735 | If the setvar variables ar... |
| ichreuopeq 46736 | If the setvar variables ar... |
| sprid 46737 | Two identical representati... |
| elsprel 46738 | An unordered pair is an el... |
| spr0nelg 46739 | The empty set is not an el... |
| sprval 46742 | The set of all unordered p... |
| sprvalpw 46743 | The set of all unordered p... |
| sprssspr 46744 | The set of all unordered p... |
| spr0el 46745 | The empty set is not an un... |
| sprvalpwn0 46746 | The set of all unordered p... |
| sprel 46747 | An element of the set of a... |
| prssspr 46748 | An element of a subset of ... |
| prelspr 46749 | An unordered pair of eleme... |
| prsprel 46750 | The elements of a pair fro... |
| prsssprel 46751 | The elements of a pair fro... |
| sprvalpwle2 46752 | The set of all unordered p... |
| sprsymrelfvlem 46753 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 46754 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 46755 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 46756 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 46757 | The value of the function ... |
| sprsymrelf 46758 | The mapping ` F ` is a fun... |
| sprsymrelf1 46759 | The mapping ` F ` is a one... |
| sprsymrelfo 46760 | The mapping ` F ` is a fun... |
| sprsymrelf1o 46761 | The mapping ` F ` is a bij... |
| sprbisymrel 46762 | There is a bijection betwe... |
| sprsymrelen 46763 | The class ` P ` of subsets... |
| prpair 46764 | Characterization of a prop... |
| prproropf1olem0 46765 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 46766 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 46767 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 46768 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 46769 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 46770 | There is a bijection betwe... |
| prproropen 46771 | The set of proper pairs an... |
| prproropreud 46772 | There is exactly one order... |
| pairreueq 46773 | Two equivalent representat... |
| paireqne 46774 | Two sets are not equal iff... |
| prprval 46777 | The set of all proper unor... |
| prprvalpw 46778 | The set of all proper unor... |
| prprelb 46779 | An element of the set of a... |
| prprelprb 46780 | A set is an element of the... |
| prprspr2 46781 | The set of all proper unor... |
| prprsprreu 46782 | There is a unique proper u... |
| prprreueq 46783 | There is a unique proper u... |
| sbcpr 46784 | The proper substitution of... |
| reupr 46785 | There is a unique unordere... |
| reuprpr 46786 | There is a unique proper u... |
| poprelb 46787 | Equality for unordered pai... |
| 2exopprim 46788 | The existence of an ordere... |
| reuopreuprim 46789 | There is a unique unordere... |
| fmtno 46792 | The ` N ` th Fermat number... |
| fmtnoge3 46793 | Each Fermat number is grea... |
| fmtnonn 46794 | Each Fermat number is a po... |
| fmtnom1nn 46795 | A Fermat number minus one ... |
| fmtnoodd 46796 | Each Fermat number is odd.... |
| fmtnorn 46797 | A Fermat number is a funct... |
| fmtnof1 46798 | The enumeration of the Fer... |
| fmtnoinf 46799 | The set of Fermat numbers ... |
| fmtnorec1 46800 | The first recurrence relat... |
| sqrtpwpw2p 46801 | The floor of the square ro... |
| fmtnosqrt 46802 | The floor of the square ro... |
| fmtno0 46803 | The ` 0 ` th Fermat number... |
| fmtno1 46804 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 46805 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 46806 | The second recurrence rela... |
| fmtnodvds 46807 | Any Fermat number divides ... |
| goldbachthlem1 46808 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 46809 | Lemma 2 for ~ goldbachth .... |
| goldbachth 46810 | Goldbach's theorem: Two d... |
| fmtnorec3 46811 | The third recurrence relat... |
| fmtnorec4 46812 | The fourth recurrence rela... |
| fmtno2 46813 | The ` 2 ` nd Fermat number... |
| fmtno3 46814 | The ` 3 ` rd Fermat number... |
| fmtno4 46815 | The ` 4 ` th Fermat number... |
| fmtno5lem1 46816 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 46817 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 46818 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 46819 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 46820 | The ` 5 ` th Fermat number... |
| fmtno0prm 46821 | The ` 0 ` th Fermat number... |
| fmtno1prm 46822 | The ` 1 ` st Fermat number... |
| fmtno2prm 46823 | The ` 2 ` nd Fermat number... |
| 257prm 46824 | 257 is a prime number (the... |
| fmtno3prm 46825 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 46826 | Any power of two is coprim... |
| fmtnoprmfac1lem 46827 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 46828 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 46829 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 46830 | Divisor of Fermat number (... |
| fmtnofac2lem 46831 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 46832 | Divisor of Fermat number (... |
| fmtnofac1 46833 | Divisor of Fermat number (... |
| fmtno4sqrt 46834 | The floor of the square ro... |
| fmtno4prmfac 46835 | If P was a (prime) factor ... |
| fmtno4prmfac193 46836 | If P was a (prime) factor ... |
| fmtno4nprmfac193 46837 | 193 is not a (prime) facto... |
| fmtno4prm 46838 | The ` 4 `-th Fermat number... |
| 65537prm 46839 | 65537 is a prime number (t... |
| fmtnofz04prm 46840 | The first five Fermat numb... |
| fmtnole4prm 46841 | The first five Fermat numb... |
| fmtno5faclem1 46842 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 46843 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 46844 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 46845 | The factorisation of the `... |
| fmtno5nprm 46846 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 46847 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 46848 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 46849 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 46850 | The mapping of a Fermat nu... |
| prminf2 46851 | The set of prime numbers i... |
| 2pwp1prm 46852 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 46853 | Every prime number of the ... |
| m2prm 46854 | The second Mersenne number... |
| m3prm 46855 | The third Mersenne number ... |
| flsqrt 46856 | A condition equivalent to ... |
| flsqrt5 46857 | The floor of the square ro... |
| 3ndvds4 46858 | 3 does not divide 4. (Con... |
| 139prmALT 46859 | 139 is a prime number. In... |
| 31prm 46860 | 31 is a prime number. In ... |
| m5prm 46861 | The fifth Mersenne number ... |
| 127prm 46862 | 127 is a prime number. (C... |
| m7prm 46863 | The seventh Mersenne numbe... |
| m11nprm 46864 | The eleventh Mersenne numb... |
| mod42tp1mod8 46865 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 46866 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 46867 | If ` P ` is a Sophie Germa... |
| lighneallem1 46868 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 46869 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 46870 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 46871 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 46872 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 46873 | Lemma 3 for ~ lighneal . ... |
| lighneal 46874 | If a power of a prime ` P ... |
| modexp2m1d 46875 | The square of an integer w... |
| proththdlem 46876 | Lemma for ~ proththd . (C... |
| proththd 46877 | Proth's theorem (1878). I... |
| 5tcu2e40 46878 | 5 times the cube of 2 is 4... |
| 3exp4mod41 46879 | 3 to the fourth power is -... |
| 41prothprmlem1 46880 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 46881 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 46882 | 41 is a _Proth prime_. (C... |
| quad1 46883 | A condition for a quadrati... |
| requad01 46884 | A condition for a quadrati... |
| requad1 46885 | A condition for a quadrati... |
| requad2 46886 | A condition for a quadrati... |
| iseven 46891 | The predicate "is an even ... |
| isodd 46892 | The predicate "is an odd n... |
| evenz 46893 | An even number is an integ... |
| oddz 46894 | An odd number is an intege... |
| evendiv2z 46895 | The result of dividing an ... |
| oddp1div2z 46896 | The result of dividing an ... |
| oddm1div2z 46897 | The result of dividing an ... |
| isodd2 46898 | The predicate "is an odd n... |
| dfodd2 46899 | Alternate definition for o... |
| dfodd6 46900 | Alternate definition for o... |
| dfeven4 46901 | Alternate definition for e... |
| evenm1odd 46902 | The predecessor of an even... |
| evenp1odd 46903 | The successor of an even n... |
| oddp1eveni 46904 | The successor of an odd nu... |
| oddm1eveni 46905 | The predecessor of an odd ... |
| evennodd 46906 | An even number is not an o... |
| oddneven 46907 | An odd number is not an ev... |
| enege 46908 | The negative of an even nu... |
| onego 46909 | The negative of an odd num... |
| m1expevenALTV 46910 | Exponentiation of -1 by an... |
| m1expoddALTV 46911 | Exponentiation of -1 by an... |
| dfeven2 46912 | Alternate definition for e... |
| dfodd3 46913 | Alternate definition for o... |
| iseven2 46914 | The predicate "is an even ... |
| isodd3 46915 | The predicate "is an odd n... |
| 2dvdseven 46916 | 2 divides an even number. ... |
| m2even 46917 | A multiple of 2 is an even... |
| 2ndvdsodd 46918 | 2 does not divide an odd n... |
| 2dvdsoddp1 46919 | 2 divides an odd number in... |
| 2dvdsoddm1 46920 | 2 divides an odd number de... |
| dfeven3 46921 | Alternate definition for e... |
| dfodd4 46922 | Alternate definition for o... |
| dfodd5 46923 | Alternate definition for o... |
| zefldiv2ALTV 46924 | The floor of an even numbe... |
| zofldiv2ALTV 46925 | The floor of an odd numer ... |
| oddflALTV 46926 | Odd number representation ... |
| iseven5 46927 | The predicate "is an even ... |
| isodd7 46928 | The predicate "is an odd n... |
| dfeven5 46929 | Alternate definition for e... |
| dfodd7 46930 | Alternate definition for o... |
| gcd2odd1 46931 | The greatest common diviso... |
| zneoALTV 46932 | No even integer equals an ... |
| zeoALTV 46933 | An integer is even or odd.... |
| zeo2ALTV 46934 | An integer is even or odd ... |
| nneoALTV 46935 | A positive integer is even... |
| nneoiALTV 46936 | A positive integer is even... |
| odd2np1ALTV 46937 | An integer is odd iff it i... |
| oddm1evenALTV 46938 | An integer is odd iff its ... |
| oddp1evenALTV 46939 | An integer is odd iff its ... |
| oexpnegALTV 46940 | The exponential of the neg... |
| oexpnegnz 46941 | The exponential of the neg... |
| bits0ALTV 46942 | Value of the zeroth bit. ... |
| bits0eALTV 46943 | The zeroth bit of an even ... |
| bits0oALTV 46944 | The zeroth bit of an odd n... |
| divgcdoddALTV 46945 | Either ` A / ( A gcd B ) `... |
| opoeALTV 46946 | The sum of two odds is eve... |
| opeoALTV 46947 | The sum of an odd and an e... |
| omoeALTV 46948 | The difference of two odds... |
| omeoALTV 46949 | The difference of an odd a... |
| oddprmALTV 46950 | A prime not equal to ` 2 `... |
| 0evenALTV 46951 | 0 is an even number. (Con... |
| 0noddALTV 46952 | 0 is not an odd number. (... |
| 1oddALTV 46953 | 1 is an odd number. (Cont... |
| 1nevenALTV 46954 | 1 is not an even number. ... |
| 2evenALTV 46955 | 2 is an even number. (Con... |
| 2noddALTV 46956 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 46957 | An odd nonnegative integer... |
| nnoALTV 46958 | An alternate characterizat... |
| nn0oALTV 46959 | An alternate characterizat... |
| nn0e 46960 | An alternate characterizat... |
| nneven 46961 | An alternate characterizat... |
| nn0onn0exALTV 46962 | For each odd nonnegative i... |
| nn0enn0exALTV 46963 | For each even nonnegative ... |
| nnennexALTV 46964 | For each even positive int... |
| nnpw2evenALTV 46965 | 2 to the power of a positi... |
| epoo 46966 | The sum of an even and an ... |
| emoo 46967 | The difference of an even ... |
| epee 46968 | The sum of two even number... |
| emee 46969 | The difference of two even... |
| evensumeven 46970 | If a summand is even, the ... |
| 3odd 46971 | 3 is an odd number. (Cont... |
| 4even 46972 | 4 is an even number. (Con... |
| 5odd 46973 | 5 is an odd number. (Cont... |
| 6even 46974 | 6 is an even number. (Con... |
| 7odd 46975 | 7 is an odd number. (Cont... |
| 8even 46976 | 8 is an even number. (Con... |
| evenprm2 46977 | A prime number is even iff... |
| oddprmne2 46978 | Every prime number not bei... |
| oddprmuzge3 46979 | A prime number which is od... |
| evenltle 46980 | If an even number is great... |
| odd2prm2 46981 | If an odd number is the su... |
| even3prm2 46982 | If an even number is the s... |
| mogoldbblem 46983 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 46984 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 46985 | Lemma for ~ perfectALTV . ... |
| perfectALTV 46986 | The Euclid-Euler theorem, ... |
| fppr 46989 | The set of Fermat pseudopr... |
| fpprmod 46990 | The set of Fermat pseudopr... |
| fpprel 46991 | A Fermat pseudoprime to th... |
| fpprbasnn 46992 | The base of a Fermat pseud... |
| fpprnn 46993 | A Fermat pseudoprime to th... |
| fppr2odd 46994 | A Fermat pseudoprime to th... |
| 11t31e341 46995 | 341 is the product of 11 a... |
| 2exp340mod341 46996 | Eight to the eighth power ... |
| 341fppr2 46997 | 341 is the (smallest) _Pou... |
| 4fppr1 46998 | 4 is the (smallest) Fermat... |
| 8exp8mod9 46999 | Eight to the eighth power ... |
| 9fppr8 47000 | 9 is the (smallest) Fermat... |
| dfwppr 47001 | Alternate definition of a ... |
| fpprwppr 47002 | A Fermat pseudoprime to th... |
| fpprwpprb 47003 | An integer ` X ` which is ... |
| fpprel2 47004 | An alternate definition fo... |
| nfermltl8rev 47005 | Fermat's little theorem wi... |
| nfermltl2rev 47006 | Fermat's little theorem wi... |
| nfermltlrev 47007 | Fermat's little theorem re... |
| isgbe 47014 | The predicate "is an even ... |
| isgbow 47015 | The predicate "is a weak o... |
| isgbo 47016 | The predicate "is an odd G... |
| gbeeven 47017 | An even Goldbach number is... |
| gbowodd 47018 | A weak odd Goldbach number... |
| gbogbow 47019 | A (strong) odd Goldbach nu... |
| gboodd 47020 | An odd Goldbach number is ... |
| gbepos 47021 | Any even Goldbach number i... |
| gbowpos 47022 | Any weak odd Goldbach numb... |
| gbopos 47023 | Any odd Goldbach number is... |
| gbegt5 47024 | Any even Goldbach number i... |
| gbowgt5 47025 | Any weak odd Goldbach numb... |
| gbowge7 47026 | Any weak odd Goldbach numb... |
| gboge9 47027 | Any odd Goldbach number is... |
| gbege6 47028 | Any even Goldbach number i... |
| gbpart6 47029 | The Goldbach partition of ... |
| gbpart7 47030 | The (weak) Goldbach partit... |
| gbpart8 47031 | The Goldbach partition of ... |
| gbpart9 47032 | The (strong) Goldbach part... |
| gbpart11 47033 | The (strong) Goldbach part... |
| 6gbe 47034 | 6 is an even Goldbach numb... |
| 7gbow 47035 | 7 is a weak odd Goldbach n... |
| 8gbe 47036 | 8 is an even Goldbach numb... |
| 9gbo 47037 | 9 is an odd Goldbach numbe... |
| 11gbo 47038 | 11 is an odd Goldbach numb... |
| stgoldbwt 47039 | If the strong ternary Gold... |
| sbgoldbwt 47040 | If the strong binary Goldb... |
| sbgoldbst 47041 | If the strong binary Goldb... |
| sbgoldbaltlem1 47042 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 47043 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 47044 | An alternate (related to t... |
| sbgoldbb 47045 | If the strong binary Goldb... |
| sgoldbeven3prm 47046 | If the binary Goldbach con... |
| sbgoldbm 47047 | If the strong binary Goldb... |
| mogoldbb 47048 | If the modern version of t... |
| sbgoldbmb 47049 | The strong binary Goldbach... |
| sbgoldbo 47050 | If the strong binary Goldb... |
| nnsum3primes4 47051 | 4 is the sum of at most 3 ... |
| nnsum4primes4 47052 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 47053 | Every prime is "the sum of... |
| nnsum4primesprm 47054 | Every prime is "the sum of... |
| nnsum3primesgbe 47055 | Any even Goldbach number i... |
| nnsum4primesgbe 47056 | Any even Goldbach number i... |
| nnsum3primesle9 47057 | Every integer greater than... |
| nnsum4primesle9 47058 | Every integer greater than... |
| nnsum4primesodd 47059 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 47060 | If the (strong) ternary Go... |
| evengpop3 47061 | If the (weak) ternary Gold... |
| evengpoap3 47062 | If the (strong) ternary Go... |
| nnsum4primeseven 47063 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 47064 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 47065 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 47066 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 47067 | If the binary Goldbach con... |
| bgoldbtbndlem1 47068 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 47069 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 47070 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 47071 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 47072 | If the binary Goldbach con... |
| tgoldbachgtALTV 47075 | Variant of Thierry Arnoux'... |
| bgoldbachlt 47076 | The binary Goldbach conjec... |
| tgblthelfgott 47078 | The ternary Goldbach conje... |
| tgoldbachlt 47079 | The ternary Goldbach conje... |
| tgoldbach 47080 | The ternary Goldbach conje... |
| grimfn 47086 | The graph isomorphism func... |
| grimdmrel 47087 | The domain of the graph is... |
| isgrim 47089 | An isomorphism of graphs i... |
| grimprop 47090 | An isomorphism of graphs i... |
| grimf1o 47091 | An isomorphism of graphs i... |
| isuspgrim0lem 47092 | An isomorphism of simple p... |
| isuspgrim0 47093 | An isomorphism of simple p... |
| uspgrimprop 47094 | An isomorphism of simple p... |
| isuspgrimlem 47095 | Lemma for ~ isuspgrim . (... |
| isuspgrim 47096 | A class is an isomorphism ... |
| grimidvtxedg 47097 | The identity relation rest... |
| grimid 47098 | The identity relation rest... |
| grimuhgr 47099 | If there is a graph isomor... |
| grimcnv 47100 | The converse of a graph is... |
| grimco 47101 | The composition of graph i... |
| brgric 47102 | The relation "is isomorphi... |
| brgrici 47103 | Prove that two graphs are ... |
| dfgric2 47104 | Alternate, explicit defini... |
| gricbri 47105 | Implications of two graphs... |
| gricushgr 47106 | The "is isomorphic to" rel... |
| gricuspgr 47107 | The "is isomorphic to" rel... |
| gricrel 47108 | The "is isomorphic to" rel... |
| gricref 47109 | Graph isomorphism is refle... |
| gricsym 47110 | Graph isomorphism is symme... |
| gricsymb 47111 | Graph isomorphism is symme... |
| grictr 47112 | Graph isomorphism is trans... |
| gricer 47113 | Isomorphism is an equivale... |
| gricen 47114 | Isomorphic graphs have equ... |
| opstrgric 47115 | A graph represented as an ... |
| ushggricedg 47116 | A simple hypergraph (with ... |
| 1hegrlfgr 47117 | A graph ` G ` with one hyp... |
| upwlksfval 47120 | The set of simple walks (i... |
| isupwlk 47121 | Properties of a pair of fu... |
| isupwlkg 47122 | Generalization of ~ isupwl... |
| upwlkbprop 47123 | Basic properties of a simp... |
| upwlkwlk 47124 | A simple walk is a walk. ... |
| upgrwlkupwlk 47125 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 47126 | In a pseudograph, the defi... |
| upgrisupwlkALT 47127 | Alternate proof of ~ upgri... |
| upgredgssspr 47128 | The set of edges of a pseu... |
| uspgropssxp 47129 | The set ` G ` of "simple p... |
| uspgrsprfv 47130 | The value of the function ... |
| uspgrsprf 47131 | The mapping ` F ` is a fun... |
| uspgrsprf1 47132 | The mapping ` F ` is a one... |
| uspgrsprfo 47133 | The mapping ` F ` is a fun... |
| uspgrsprf1o 47134 | The mapping ` F ` is a bij... |
| uspgrex 47135 | The class ` G ` of all "si... |
| uspgrbispr 47136 | There is a bijection betwe... |
| uspgrspren 47137 | The set ` G ` of the "simp... |
| uspgrymrelen 47138 | The set ` G ` of the "simp... |
| uspgrbisymrel 47139 | There is a bijection betwe... |
| uspgrbisymrelALT 47140 | Alternate proof of ~ uspgr... |
| ovn0dmfun 47141 | If a class operation value... |
| xpsnopab 47142 | A Cartesian product with a... |
| xpiun 47143 | A Cartesian product expres... |
| ovn0ssdmfun 47144 | If a class' operation valu... |
| fnxpdmdm 47145 | The domain of the domain o... |
| cnfldsrngbas 47146 | The base set of a subring ... |
| cnfldsrngadd 47147 | The group addition operati... |
| cnfldsrngmul 47148 | The ring multiplication op... |
| plusfreseq 47149 | If the empty set is not co... |
| mgmplusfreseq 47150 | If the empty set is not co... |
| 0mgm 47151 | A set with an empty base s... |
| opmpoismgm 47152 | A structure with a group a... |
| copissgrp 47153 | A structure with a constan... |
| copisnmnd 47154 | A structure with a constan... |
| 0nodd 47155 | 0 is not an odd integer. ... |
| 1odd 47156 | 1 is an odd integer. (Con... |
| 2nodd 47157 | 2 is not an odd integer. ... |
| oddibas 47158 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 47159 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 47160 | The structure of all odd i... |
| nnsgrpmgm 47161 | The structure of positive ... |
| nnsgrp 47162 | The structure of positive ... |
| nnsgrpnmnd 47163 | The structure of positive ... |
| nn0mnd 47164 | The set of nonnegative int... |
| gsumsplit2f 47165 | Split a group sum into two... |
| gsumdifsndf 47166 | Extract a summand from a f... |
| gsumfsupp 47167 | A group sum of a family ca... |
| iscllaw 47174 | The predicate "is a closed... |
| iscomlaw 47175 | The predicate "is a commut... |
| clcllaw 47176 | Closure of a closed operat... |
| isasslaw 47177 | The predicate "is an assoc... |
| asslawass 47178 | Associativity of an associ... |
| mgmplusgiopALT 47179 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 47180 | Slot 2 (group operation) o... |
| intopval 47187 | The internal (binary) oper... |
| intop 47188 | An internal (binary) opera... |
| clintopval 47189 | The closed (internal binar... |
| assintopval 47190 | The associative (closed in... |
| assintopmap 47191 | The associative (closed in... |
| isclintop 47192 | The predicate "is a closed... |
| clintop 47193 | A closed (internal binary)... |
| assintop 47194 | An associative (closed int... |
| isassintop 47195 | The predicate "is an assoc... |
| clintopcllaw 47196 | The closure law holds for ... |
| assintopcllaw 47197 | The closure low holds for ... |
| assintopasslaw 47198 | The associative low holds ... |
| assintopass 47199 | An associative (closed int... |
| ismgmALT 47208 | The predicate "is a magma"... |
| iscmgmALT 47209 | The predicate "is a commut... |
| issgrpALT 47210 | The predicate "is a semigr... |
| iscsgrpALT 47211 | The predicate "is a commut... |
| mgm2mgm 47212 | Equivalence of the two def... |
| sgrp2sgrp 47213 | Equivalence of the two def... |
| lmod0rng 47214 | If the scalar ring of a mo... |
| nzrneg1ne0 47215 | The additive inverse of th... |
| lidldomn1 47216 | If a (left) ideal (which i... |
| lidlabl 47217 | A (left) ideal of a ring i... |
| lidlrng 47218 | A (left) ideal of a ring i... |
| zlidlring 47219 | The zero (left) ideal of a... |
| uzlidlring 47220 | Only the zero (left) ideal... |
| lidldomnnring 47221 | A (left) ideal of a domain... |
| 0even 47222 | 0 is an even integer. (Co... |
| 1neven 47223 | 1 is not an even integer. ... |
| 2even 47224 | 2 is an even integer. (Co... |
| 2zlidl 47225 | The even integers are a (l... |
| 2zrng 47226 | The ring of integers restr... |
| 2zrngbas 47227 | The base set of R is the s... |
| 2zrngadd 47228 | The group addition operati... |
| 2zrng0 47229 | The additive identity of R... |
| 2zrngamgm 47230 | R is an (additive) magma. ... |
| 2zrngasgrp 47231 | R is an (additive) semigro... |
| 2zrngamnd 47232 | R is an (additive) monoid.... |
| 2zrngacmnd 47233 | R is a commutative (additi... |
| 2zrngagrp 47234 | R is an (additive) group. ... |
| 2zrngaabl 47235 | R is an (additive) abelian... |
| 2zrngmul 47236 | The ring multiplication op... |
| 2zrngmmgm 47237 | R is a (multiplicative) ma... |
| 2zrngmsgrp 47238 | R is a (multiplicative) se... |
| 2zrngALT 47239 | The ring of integers restr... |
| 2zrngnmlid 47240 | R has no multiplicative (l... |
| 2zrngnmrid 47241 | R has no multiplicative (r... |
| 2zrngnmlid2 47242 | R has no multiplicative (l... |
| 2zrngnring 47243 | R is not a unital ring. (... |
| cznrnglem 47244 | Lemma for ~ cznrng : The ... |
| cznabel 47245 | The ring constructed from ... |
| cznrng 47246 | The ring constructed from ... |
| cznnring 47247 | The ring constructed from ... |
| rngcvalALTV 47250 | Value of the category of n... |
| rngcbasALTV 47251 | Set of objects of the cate... |
| rngchomfvalALTV 47252 | Set of arrows of the categ... |
| rngchomALTV 47253 | Set of arrows of the categ... |
| elrngchomALTV 47254 | A morphism of non-unital r... |
| rngccofvalALTV 47255 | Composition in the categor... |
| rngccoALTV 47256 | Composition in the categor... |
| rngccatidALTV 47257 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 47258 | The category of non-unital... |
| rngcidALTV 47259 | The identity arrow in the ... |
| rngcsectALTV 47260 | A section in the category ... |
| rngcinvALTV 47261 | An inverse in the category... |
| rngcisoALTV 47262 | An isomorphism in the cate... |
| rngchomffvalALTV 47263 | The value of the functiona... |
| rngchomrnghmresALTV 47264 | The value of the functiona... |
| rngcrescrhmALTV 47265 | The category of non-unital... |
| rhmsubcALTVlem1 47266 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 47267 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 47268 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 47269 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 47270 | According to ~ df-subc , t... |
| rhmsubcALTVcat 47271 | The restriction of the cat... |
| ringcvalALTV 47274 | Value of the category of r... |
| funcringcsetcALTV2lem1 47275 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 47276 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 47277 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 47278 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 47279 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 47280 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 47281 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 47282 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 47283 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 47284 | The "natural forgetful fun... |
| ringcbasALTV 47285 | Set of objects of the cate... |
| ringchomfvalALTV 47286 | Set of arrows of the categ... |
| ringchomALTV 47287 | Set of arrows of the categ... |
| elringchomALTV 47288 | A morphism of rings is a f... |
| ringccofvalALTV 47289 | Composition in the categor... |
| ringccoALTV 47290 | Composition in the categor... |
| ringccatidALTV 47291 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 47292 | The category of rings is a... |
| ringcidALTV 47293 | The identity arrow in the ... |
| ringcsectALTV 47294 | A section in the category ... |
| ringcinvALTV 47295 | An inverse in the category... |
| ringcisoALTV 47296 | An isomorphism in the cate... |
| ringcbasbasALTV 47297 | An element of the base set... |
| funcringcsetclem1ALTV 47298 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 47299 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 47300 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 47301 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 47302 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 47303 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 47304 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 47305 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 47306 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 47307 | The "natural forgetful fun... |
| srhmsubcALTVlem1 47308 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 47309 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 47310 | According to ~ df-subc , t... |
| sringcatALTV 47311 | The restriction of the cat... |
| crhmsubcALTV 47312 | According to ~ df-subc , t... |
| cringcatALTV 47313 | The restriction of the cat... |
| drhmsubcALTV 47314 | According to ~ df-subc , t... |
| drngcatALTV 47315 | The restriction of the cat... |
| fldcatALTV 47316 | The restriction of the cat... |
| fldcALTV 47317 | The restriction of the cat... |
| fldhmsubcALTV 47318 | According to ~ df-subc , t... |
| opeliun2xp 47319 | Membership of an ordered p... |
| eliunxp2 47320 | Membership in a union of C... |
| mpomptx2 47321 | Express a two-argument fun... |
| cbvmpox2 47322 | Rule to change the bound v... |
| dmmpossx2 47323 | The domain of a mapping is... |
| mpoexxg2 47324 | Existence of an operation ... |
| ovmpordxf 47325 | Value of an operation give... |
| ovmpordx 47326 | Value of an operation give... |
| ovmpox2 47327 | The value of an operation ... |
| fdmdifeqresdif 47328 | The restriction of a condi... |
| offvalfv 47329 | The function operation exp... |
| ofaddmndmap 47330 | The function operation app... |
| mapsnop 47331 | A singleton of an ordered ... |
| fprmappr 47332 | A function with a domain o... |
| mapprop 47333 | An unordered pair containi... |
| ztprmneprm 47334 | A prime is not an integer ... |
| 2t6m3t4e0 47335 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 47336 | For any finite subset of `... |
| nn0sumltlt 47337 | If the sum of two nonnegat... |
| bcpascm1 47338 | Pascal's rule for the bino... |
| altgsumbc 47339 | The sum of binomial coeffi... |
| altgsumbcALT 47340 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 47341 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 47342 | An element of the (base se... |
| zlmodzxz0 47343 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 47344 | The scalar multiplication ... |
| zlmodzxzadd 47345 | The addition of the ` ZZ `... |
| zlmodzxzsubm 47346 | The subtraction of the ` Z... |
| zlmodzxzsub 47347 | The subtraction of the ` Z... |
| mgpsumunsn 47348 | Extract a summand/factor f... |
| mgpsumz 47349 | If the group sum for the m... |
| mgpsumn 47350 | If the group sum for the m... |
| exple2lt6 47351 | A nonnegative integer to t... |
| pgrple2abl 47352 | Every symmetric group on a... |
| pgrpgt2nabl 47353 | Every symmetric group on a... |
| invginvrid 47354 | Identity for a multiplicat... |
| rmsupp0 47355 | The support of a mapping o... |
| domnmsuppn0 47356 | The support of a mapping o... |
| rmsuppss 47357 | The support of a mapping o... |
| mndpsuppss 47358 | The support of a mapping o... |
| scmsuppss 47359 | The support of a mapping o... |
| rmsuppfi 47360 | The support of a mapping o... |
| rmfsupp 47361 | A mapping of a multiplicat... |
| mndpsuppfi 47362 | The support of a mapping o... |
| mndpfsupp 47363 | A mapping of a scalar mult... |
| scmsuppfi 47364 | The support of a mapping o... |
| scmfsupp 47365 | A mapping of a scalar mult... |
| suppmptcfin 47366 | The support of a mapping w... |
| mptcfsupp 47367 | A mapping with value 0 exc... |
| fsuppmptdmf 47368 | A mapping with a finite do... |
| lmodvsmdi 47369 | Multiple distributive law ... |
| gsumlsscl 47370 | Closure of a group sum in ... |
| assaascl0 47371 | The scalar 0 embedded into... |
| assaascl1 47372 | The scalar 1 embedded into... |
| ply1vr1smo 47373 | The variable in a polynomi... |
| ply1sclrmsm 47374 | The ring multiplication of... |
| coe1id 47375 | Coefficient vector of the ... |
| coe1sclmulval 47376 | The value of the coefficie... |
| ply1mulgsumlem1 47377 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 47378 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 47379 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 47380 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 47381 | The product of two polynom... |
| evl1at0 47382 | Polynomial evaluation for ... |
| evl1at1 47383 | Polynomial evaluation for ... |
| linply1 47384 | A term of the form ` x - C... |
| lineval 47385 | A term of the form ` x - C... |
| linevalexample 47386 | The polynomial ` x - 3 ` o... |
| dmatALTval 47391 | The algebra of ` N ` x ` N... |
| dmatALTbas 47392 | The base set of the algebr... |
| dmatALTbasel 47393 | An element of the base set... |
| dmatbas 47394 | The set of all ` N ` x ` N... |
| lincop 47399 | A linear combination as op... |
| lincval 47400 | The value of a linear comb... |
| dflinc2 47401 | Alternative definition of ... |
| lcoop 47402 | A linear combination as op... |
| lcoval 47403 | The value of a linear comb... |
| lincfsuppcl 47404 | A linear combination of ve... |
| linccl 47405 | A linear combination of ve... |
| lincval0 47406 | The value of an empty line... |
| lincvalsng 47407 | The linear combination ove... |
| lincvalsn 47408 | The linear combination ove... |
| lincvalpr 47409 | The linear combination ove... |
| lincval1 47410 | The linear combination ove... |
| lcosn0 47411 | Properties of a linear com... |
| lincvalsc0 47412 | The linear combination whe... |
| lcoc0 47413 | Properties of a linear com... |
| linc0scn0 47414 | If a set contains the zero... |
| lincdifsn 47415 | A vector is a linear combi... |
| linc1 47416 | A vector is a linear combi... |
| lincellss 47417 | A linear combination of a ... |
| lco0 47418 | The set of empty linear co... |
| lcoel0 47419 | The zero vector is always ... |
| lincsum 47420 | The sum of two linear comb... |
| lincscm 47421 | A linear combinations mult... |
| lincsumcl 47422 | The sum of two linear comb... |
| lincscmcl 47423 | The multiplication of a li... |
| lincsumscmcl 47424 | The sum of a linear combin... |
| lincolss 47425 | According to the statement... |
| ellcoellss 47426 | Every linear combination o... |
| lcoss 47427 | A set of vectors of a modu... |
| lspsslco 47428 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 47429 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 47430 | Equivalence of a _span_ of... |
| rellininds 47434 | The class defining the rel... |
| linindsv 47436 | The classes of the module ... |
| islininds 47437 | The property of being a li... |
| linindsi 47438 | The implications of being ... |
| linindslinci 47439 | The implications of being ... |
| islinindfis 47440 | The property of being a li... |
| islinindfiss 47441 | The property of being a li... |
| linindscl 47442 | A linearly independent set... |
| lindepsnlininds 47443 | A linearly dependent subse... |
| islindeps 47444 | The property of being a li... |
| lincext1 47445 | Property 1 of an extension... |
| lincext2 47446 | Property 2 of an extension... |
| lincext3 47447 | Property 3 of an extension... |
| lindslinindsimp1 47448 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 47449 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 47450 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 47451 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 47452 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 47453 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 47454 | Implication 2 for ~ lindsl... |
| lindslininds 47455 | Equivalence of definitions... |
| linds0 47456 | The empty set is always a ... |
| el0ldep 47457 | A set containing the zero ... |
| el0ldepsnzr 47458 | A set containing the zero ... |
| lindsrng01 47459 | Any subset of a module is ... |
| lindszr 47460 | Any subset of a module ove... |
| snlindsntorlem 47461 | Lemma for ~ snlindsntor . ... |
| snlindsntor 47462 | A singleton is linearly in... |
| ldepsprlem 47463 | Lemma for ~ ldepspr . (Co... |
| ldepspr 47464 | If a vector is a scalar mu... |
| lincresunit3lem3 47465 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 47466 | Lemma 1 for properties of ... |
| lincresunitlem2 47467 | Lemma for properties of a ... |
| lincresunit1 47468 | Property 1 of a specially ... |
| lincresunit2 47469 | Property 2 of a specially ... |
| lincresunit3lem1 47470 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 47471 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 47472 | Property 3 of a specially ... |
| lincreslvec3 47473 | Property 3 of a specially ... |
| islindeps2 47474 | Conditions for being a lin... |
| islininds2 47475 | Implication of being a lin... |
| isldepslvec2 47476 | Alternative definition of ... |
| lindssnlvec 47477 | A singleton not containing... |
| lmod1lem1 47478 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 47479 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 47480 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 47481 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 47482 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 47483 | The (smallest) structure r... |
| lmod1zr 47484 | The (smallest) structure r... |
| lmod1zrnlvec 47485 | There is a (left) module (... |
| lmodn0 47486 | Left modules exist. (Cont... |
| zlmodzxzequa 47487 | Example of an equation wit... |
| zlmodzxznm 47488 | Example of a linearly depe... |
| zlmodzxzldeplem 47489 | A and B are not equal. (C... |
| zlmodzxzequap 47490 | Example of an equation wit... |
| zlmodzxzldeplem1 47491 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 47492 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 47493 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 47494 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 47495 | { A , B } is a linearly de... |
| ldepsnlinclem1 47496 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 47497 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 47498 | The class of all (left) ve... |
| ldepsnlinc 47499 | The reverse implication of... |
| ldepslinc 47500 | For (left) vector spaces, ... |
| suppdm 47501 | If the range of a function... |
| eluz2cnn0n1 47502 | An integer greater than 1 ... |
| divge1b 47503 | The ratio of a real number... |
| divgt1b 47504 | The ratio of a real number... |
| ltsubaddb 47505 | Equivalence for the "less ... |
| ltsubsubb 47506 | Equivalence for the "less ... |
| ltsubadd2b 47507 | Equivalence for the "less ... |
| divsub1dir 47508 | Distribution of division o... |
| expnegico01 47509 | An integer greater than 1 ... |
| elfzolborelfzop1 47510 | An element of a half-open ... |
| pw2m1lepw2m1 47511 | 2 to the power of a positi... |
| zgtp1leeq 47512 | If an integer is between a... |
| flsubz 47513 | An integer can be moved in... |
| fldivmod 47514 | Expressing the floor of a ... |
| mod0mul 47515 | If an integer is 0 modulo ... |
| modn0mul 47516 | If an integer is not 0 mod... |
| m1modmmod 47517 | An integer decreased by 1 ... |
| difmodm1lt 47518 | The difference between an ... |
| nn0onn0ex 47519 | For each odd nonnegative i... |
| nn0enn0ex 47520 | For each even nonnegative ... |
| nnennex 47521 | For each even positive int... |
| nneop 47522 | A positive integer is even... |
| nneom 47523 | A positive integer is even... |
| nn0eo 47524 | A nonnegative integer is e... |
| nnpw2even 47525 | 2 to the power of a positi... |
| zefldiv2 47526 | The floor of an even integ... |
| zofldiv2 47527 | The floor of an odd intege... |
| nn0ofldiv2 47528 | The floor of an odd nonneg... |
| flnn0div2ge 47529 | The floor of a positive in... |
| flnn0ohalf 47530 | The floor of the half of a... |
| logcxp0 47531 | Logarithm of a complex pow... |
| regt1loggt0 47532 | The natural logarithm for ... |
| fdivval 47535 | The quotient of two functi... |
| fdivmpt 47536 | The quotient of two functi... |
| fdivmptf 47537 | The quotient of two functi... |
| refdivmptf 47538 | The quotient of two functi... |
| fdivpm 47539 | The quotient of two functi... |
| refdivpm 47540 | The quotient of two functi... |
| fdivmptfv 47541 | The function value of a qu... |
| refdivmptfv 47542 | The function value of a qu... |
| bigoval 47545 | Set of functions of order ... |
| elbigofrcl 47546 | Reverse closure of the "bi... |
| elbigo 47547 | Properties of a function o... |
| elbigo2 47548 | Properties of a function o... |
| elbigo2r 47549 | Sufficient condition for a... |
| elbigof 47550 | A function of order G(x) i... |
| elbigodm 47551 | The domain of a function o... |
| elbigoimp 47552 | The defining property of a... |
| elbigolo1 47553 | A function (into the posit... |
| rege1logbrege0 47554 | The general logarithm, wit... |
| rege1logbzge0 47555 | The general logarithm, wit... |
| fllogbd 47556 | A real number is between t... |
| relogbmulbexp 47557 | The logarithm of the produ... |
| relogbdivb 47558 | The logarithm of the quoti... |
| logbge0b 47559 | The logarithm of a number ... |
| logblt1b 47560 | The logarithm of a number ... |
| fldivexpfllog2 47561 | The floor of a positive re... |
| nnlog2ge0lt1 47562 | A positive integer is 1 if... |
| logbpw2m1 47563 | The floor of the binary lo... |
| fllog2 47564 | The floor of the binary lo... |
| blenval 47567 | The binary length of an in... |
| blen0 47568 | The binary length of 0. (... |
| blenn0 47569 | The binary length of a "nu... |
| blenre 47570 | The binary length of a pos... |
| blennn 47571 | The binary length of a pos... |
| blennnelnn 47572 | The binary length of a pos... |
| blennn0elnn 47573 | The binary length of a non... |
| blenpw2 47574 | The binary length of a pow... |
| blenpw2m1 47575 | The binary length of a pow... |
| nnpw2blen 47576 | A positive integer is betw... |
| nnpw2blenfzo 47577 | A positive integer is betw... |
| nnpw2blenfzo2 47578 | A positive integer is eith... |
| nnpw2pmod 47579 | Every positive integer can... |
| blen1 47580 | The binary length of 1. (... |
| blen2 47581 | The binary length of 2. (... |
| nnpw2p 47582 | Every positive integer can... |
| nnpw2pb 47583 | A number is a positive int... |
| blen1b 47584 | The binary length of a non... |
| blennnt2 47585 | The binary length of a pos... |
| nnolog2flm1 47586 | The floor of the binary lo... |
| blennn0em1 47587 | The binary length of the h... |
| blennngt2o2 47588 | The binary length of an od... |
| blengt1fldiv2p1 47589 | The binary length of an in... |
| blennn0e2 47590 | The binary length of an ev... |
| digfval 47593 | Operation to obtain the ` ... |
| digval 47594 | The ` K ` th digit of a no... |
| digvalnn0 47595 | The ` K ` th digit of a no... |
| nn0digval 47596 | The ` K ` th digit of a no... |
| dignn0fr 47597 | The digits of the fraction... |
| dignn0ldlem 47598 | Lemma for ~ dignnld . (Co... |
| dignnld 47599 | The leading digits of a po... |
| dig2nn0ld 47600 | The leading digits of a po... |
| dig2nn1st 47601 | The first (relevant) digit... |
| dig0 47602 | All digits of 0 are 0. (C... |
| digexp 47603 | The ` K ` th digit of a po... |
| dig1 47604 | All but one digits of 1 ar... |
| 0dig1 47605 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 47606 | The integers 0 and 1 corre... |
| dig2nn0 47607 | A digit of a nonnegative i... |
| 0dig2nn0e 47608 | The last bit of an even in... |
| 0dig2nn0o 47609 | The last bit of an odd int... |
| dig2bits 47610 | The ` K ` th digit of a no... |
| dignn0flhalflem1 47611 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 47612 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 47613 | The digits of the half of ... |
| dignn0flhalf 47614 | The digits of the rounded ... |
| nn0sumshdiglemA 47615 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 47616 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 47617 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 47618 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 47619 | A nonnegative integer can ... |
| nn0mulfsum 47620 | Trivial algorithm to calcu... |
| nn0mullong 47621 | Standard algorithm (also k... |
| naryfval 47624 | The set of the n-ary (endo... |
| naryfvalixp 47625 | The set of the n-ary (endo... |
| naryfvalel 47626 | An n-ary (endo)function on... |
| naryrcl 47627 | Reverse closure for n-ary ... |
| naryfvalelfv 47628 | The value of an n-ary (end... |
| naryfvalelwrdf 47629 | An n-ary (endo)function on... |
| 0aryfvalel 47630 | A nullary (endo)function o... |
| 0aryfvalelfv 47631 | The value of a nullary (en... |
| 1aryfvalel 47632 | A unary (endo)function on ... |
| fv1arycl 47633 | Closure of a unary (endo)f... |
| 1arympt1 47634 | A unary (endo)function in ... |
| 1arympt1fv 47635 | The value of a unary (endo... |
| 1arymaptfv 47636 | The value of the mapping o... |
| 1arymaptf 47637 | The mapping of unary (endo... |
| 1arymaptf1 47638 | The mapping of unary (endo... |
| 1arymaptfo 47639 | The mapping of unary (endo... |
| 1arymaptf1o 47640 | The mapping of unary (endo... |
| 1aryenef 47641 | The set of unary (endo)fun... |
| 1aryenefmnd 47642 | The set of unary (endo)fun... |
| 2aryfvalel 47643 | A binary (endo)function on... |
| fv2arycl 47644 | Closure of a binary (endo)... |
| 2arympt 47645 | A binary (endo)function in... |
| 2arymptfv 47646 | The value of a binary (end... |
| 2arymaptfv 47647 | The value of the mapping o... |
| 2arymaptf 47648 | The mapping of binary (end... |
| 2arymaptf1 47649 | The mapping of binary (end... |
| 2arymaptfo 47650 | The mapping of binary (end... |
| 2arymaptf1o 47651 | The mapping of binary (end... |
| 2aryenef 47652 | The set of binary (endo)fu... |
| itcoval 47657 | The value of the function ... |
| itcoval0 47658 | A function iterated zero t... |
| itcoval1 47659 | A function iterated once. ... |
| itcoval2 47660 | A function iterated twice.... |
| itcoval3 47661 | A function iterated three ... |
| itcoval0mpt 47662 | A mapping iterated zero ti... |
| itcovalsuc 47663 | The value of the function ... |
| itcovalsucov 47664 | The value of the function ... |
| itcovalendof 47665 | The n-th iterate of an end... |
| itcovalpclem1 47666 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 47667 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 47668 | The value of the function ... |
| itcovalt2lem2lem1 47669 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 47670 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 47671 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 47672 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 47673 | The value of the function ... |
| ackvalsuc1mpt 47674 | The Ackermann function at ... |
| ackvalsuc1 47675 | The Ackermann function at ... |
| ackval0 47676 | The Ackermann function at ... |
| ackval1 47677 | The Ackermann function at ... |
| ackval2 47678 | The Ackermann function at ... |
| ackval3 47679 | The Ackermann function at ... |
| ackendofnn0 47680 | The Ackermann function at ... |
| ackfnnn0 47681 | The Ackermann function at ... |
| ackval0val 47682 | The Ackermann function at ... |
| ackvalsuc0val 47683 | The Ackermann function at ... |
| ackvalsucsucval 47684 | The Ackermann function at ... |
| ackval0012 47685 | The Ackermann function at ... |
| ackval1012 47686 | The Ackermann function at ... |
| ackval2012 47687 | The Ackermann function at ... |
| ackval3012 47688 | The Ackermann function at ... |
| ackval40 47689 | The Ackermann function at ... |
| ackval41a 47690 | The Ackermann function at ... |
| ackval41 47691 | The Ackermann function at ... |
| ackval42 47692 | The Ackermann function at ... |
| ackval42a 47693 | The Ackermann function at ... |
| ackval50 47694 | The Ackermann function at ... |
| fv1prop 47695 | The function value of unor... |
| fv2prop 47696 | The function value of unor... |
| submuladdmuld 47697 | Transformation of a sum of... |
| affinecomb1 47698 | Combination of two real af... |
| affinecomb2 47699 | Combination of two real af... |
| affineid 47700 | Identity of an affine comb... |
| 1subrec1sub 47701 | Subtract the reciprocal of... |
| resum2sqcl 47702 | The sum of two squares of ... |
| resum2sqgt0 47703 | The sum of the square of a... |
| resum2sqrp 47704 | The sum of the square of a... |
| resum2sqorgt0 47705 | The sum of the square of t... |
| reorelicc 47706 | Membership in and outside ... |
| rrx2pxel 47707 | The x-coordinate of a poin... |
| rrx2pyel 47708 | The y-coordinate of a poin... |
| prelrrx2 47709 | An unordered pair of order... |
| prelrrx2b 47710 | An unordered pair of order... |
| rrx2pnecoorneor 47711 | If two different points ` ... |
| rrx2pnedifcoorneor 47712 | If two different points ` ... |
| rrx2pnedifcoorneorr 47713 | If two different points ` ... |
| rrx2xpref1o 47714 | There is a bijection betwe... |
| rrx2xpreen 47715 | The set of points in the t... |
| rrx2plord 47716 | The lexicographical orderi... |
| rrx2plord1 47717 | The lexicographical orderi... |
| rrx2plord2 47718 | The lexicographical orderi... |
| rrx2plordisom 47719 | The set of points in the t... |
| rrx2plordso 47720 | The lexicographical orderi... |
| ehl2eudisval0 47721 | The Euclidean distance of ... |
| ehl2eudis0lt 47722 | An upper bound of the Eucl... |
| lines 47727 | The lines passing through ... |
| line 47728 | The line passing through t... |
| rrxlines 47729 | Definition of lines passin... |
| rrxline 47730 | The line passing through t... |
| rrxlinesc 47731 | Definition of lines passin... |
| rrxlinec 47732 | The line passing through t... |
| eenglngeehlnmlem1 47733 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 47734 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 47735 | The line definition in the... |
| rrx2line 47736 | The line passing through t... |
| rrx2vlinest 47737 | The vertical line passing ... |
| rrx2linest 47738 | The line passing through t... |
| rrx2linesl 47739 | The line passing through t... |
| rrx2linest2 47740 | The line passing through t... |
| elrrx2linest2 47741 | The line passing through t... |
| spheres 47742 | The spheres for given cent... |
| sphere 47743 | A sphere with center ` X `... |
| rrxsphere 47744 | The sphere with center ` M... |
| 2sphere 47745 | The sphere with center ` M... |
| 2sphere0 47746 | The sphere around the orig... |
| line2ylem 47747 | Lemma for ~ line2y . This... |
| line2 47748 | Example for a line ` G ` p... |
| line2xlem 47749 | Lemma for ~ line2x . This... |
| line2x 47750 | Example for a horizontal l... |
| line2y 47751 | Example for a vertical lin... |
| itsclc0lem1 47752 | Lemma for theorems about i... |
| itsclc0lem2 47753 | Lemma for theorems about i... |
| itsclc0lem3 47754 | Lemma for theorems about i... |
| itscnhlc0yqe 47755 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 47756 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 47757 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 47758 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 47759 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 47760 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 47761 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 47762 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 47763 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 47764 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 47765 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 47766 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 47767 | The intersection points of... |
| itsclc0b 47768 | The intersection points of... |
| itsclinecirc0 47769 | The intersection points of... |
| itsclinecirc0b 47770 | The intersection points of... |
| itsclinecirc0in 47771 | The intersection points of... |
| itsclquadb 47772 | Quadratic equation for the... |
| itsclquadeu 47773 | Quadratic equation for the... |
| 2itscplem1 47774 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 47775 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 47776 | Lemma D for ~ 2itscp . (C... |
| 2itscp 47777 | A condition for a quadrati... |
| itscnhlinecirc02plem1 47778 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 47779 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 47780 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 47781 | Intersection of a nonhoriz... |
| inlinecirc02plem 47782 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 47783 | Intersection of a line wit... |
| inlinecirc02preu 47784 | Intersection of a line wit... |
| pm4.71da 47785 | Deduction converting a bic... |
| logic1 47786 | Distribution of implicatio... |
| logic1a 47787 | Variant of ~ logic1 . (Co... |
| logic2 47788 | Variant of ~ logic1 . (Co... |
| pm5.32dav 47789 | Distribution of implicatio... |
| pm5.32dra 47790 | Reverse distribution of im... |
| exp12bd 47791 | The import-export theorem ... |
| mpbiran3d 47792 | Equivalence with a conjunc... |
| mpbiran4d 47793 | Equivalence with a conjunc... |
| dtrucor3 47794 | An example of how ~ ax-5 w... |
| ralbidb 47795 | Formula-building rule for ... |
| ralbidc 47796 | Formula-building rule for ... |
| r19.41dv 47797 | A complex deduction form o... |
| rmotru 47798 | Two ways of expressing "at... |
| reutru 47799 | Two ways of expressing "ex... |
| reutruALT 47800 | Alternate proof for ~ reut... |
| ssdisjd 47801 | Subset preserves disjointn... |
| ssdisjdr 47802 | Subset preserves disjointn... |
| disjdifb 47803 | Relative complement is ant... |
| predisj 47804 | Preimages of disjoint sets... |
| vsn 47805 | The singleton of the unive... |
| mosn 47806 | "At most one" element in a... |
| mo0 47807 | "At most one" element in a... |
| mosssn 47808 | "At most one" element in a... |
| mo0sn 47809 | Two ways of expressing "at... |
| mosssn2 47810 | Two ways of expressing "at... |
| unilbss 47811 | Superclass of the greatest... |
| inpw 47812 | Two ways of expressing a c... |
| mof0 47813 | There is at most one funct... |
| mof02 47814 | A variant of ~ mof0 . (Co... |
| mof0ALT 47815 | Alternate proof for ~ mof0... |
| eufsnlem 47816 | There is exactly one funct... |
| eufsn 47817 | There is exactly one funct... |
| eufsn2 47818 | There is exactly one funct... |
| mofsn 47819 | There is at most one funct... |
| mofsn2 47820 | There is at most one funct... |
| mofsssn 47821 | There is at most one funct... |
| mofmo 47822 | There is at most one funct... |
| mofeu 47823 | The uniqueness of a functi... |
| elfvne0 47824 | If a function value has a ... |
| fdomne0 47825 | A function with non-empty ... |
| f1sn2g 47826 | A function that maps a sin... |
| f102g 47827 | A function that maps the e... |
| f1mo 47828 | A function that maps a set... |
| f002 47829 | A function with an empty c... |
| map0cor 47830 | A function exists iff an e... |
| fvconstr 47831 | Two ways of expressing ` A... |
| fvconstrn0 47832 | Two ways of expressing ` A... |
| fvconstr2 47833 | Two ways of expressing ` A... |
| fvconst0ci 47834 | A constant function's valu... |
| fvconstdomi 47835 | A constant function's valu... |
| f1omo 47836 | There is at most one eleme... |
| f1omoALT 47837 | There is at most one eleme... |
| iccin 47838 | Intersection of two closed... |
| iccdisj2 47839 | If the upper bound of one ... |
| iccdisj 47840 | If the upper bound of one ... |
| mreuniss 47841 | The union of a collection ... |
| clduni 47842 | The union of closed sets i... |
| opncldeqv 47843 | Conditions on open sets ar... |
| opndisj 47844 | Two ways of saying that tw... |
| clddisj 47845 | Two ways of saying that tw... |
| neircl 47846 | Reverse closure of the nei... |
| opnneilem 47847 | Lemma factoring out common... |
| opnneir 47848 | If something is true for a... |
| opnneirv 47849 | A variant of ~ opnneir wit... |
| opnneilv 47850 | The converse of ~ opnneir ... |
| opnneil 47851 | A variant of ~ opnneilv . ... |
| opnneieqv 47852 | The equivalence between ne... |
| opnneieqvv 47853 | The equivalence between ne... |
| restcls2lem 47854 | A closed set in a subspace... |
| restcls2 47855 | A closed set in a subspace... |
| restclsseplem 47856 | Lemma for ~ restclssep . ... |
| restclssep 47857 | Two disjoint closed sets i... |
| cnneiima 47858 | Given a continuous functio... |
| iooii 47859 | Open intervals are open se... |
| icccldii 47860 | Closed intervals are close... |
| i0oii 47861 | ` ( 0 [,) A ) ` is open in... |
| io1ii 47862 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 47863 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 47864 | Open neighborhood and neig... |
| sepnsepo 47865 | Open neighborhood and neig... |
| sepdisj 47866 | Separated sets are disjoin... |
| seposep 47867 | If two sets are separated ... |
| sepcsepo 47868 | If two sets are separated ... |
| sepfsepc 47869 | If two sets are separated ... |
| seppsepf 47870 | If two sets are precisely ... |
| seppcld 47871 | If two sets are precisely ... |
| isnrm4 47872 | A topological space is nor... |
| dfnrm2 47873 | A topological space is nor... |
| dfnrm3 47874 | A topological space is nor... |
| iscnrm3lem1 47875 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 47876 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem3 47877 | Lemma for ~ iscnrm3lem4 . ... |
| iscnrm3lem4 47878 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 47879 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 47880 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 47881 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 47882 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 47883 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 47884 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 47885 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 47886 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 47887 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 47888 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 47889 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 47890 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 47891 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 47892 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 47893 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 47894 | A completely normal topolo... |
| iscnrm3v 47895 | A topology is completely n... |
| iscnrm4 47896 | A completely normal topolo... |
| isprsd 47897 | Property of being a preord... |
| lubeldm2 47898 | Member of the domain of th... |
| glbeldm2 47899 | Member of the domain of th... |
| lubeldm2d 47900 | Member of the domain of th... |
| glbeldm2d 47901 | Member of the domain of th... |
| lubsscl 47902 | If a subset of ` S ` conta... |
| glbsscl 47903 | If a subset of ` S ` conta... |
| lubprlem 47904 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 47905 | The set of two comparable ... |
| lubpr 47906 | The LUB of the set of two ... |
| glbprlem 47907 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 47908 | The set of two comparable ... |
| glbpr 47909 | The GLB of the set of two ... |
| joindm2 47910 | The join of any two elemen... |
| joindm3 47911 | The join of any two elemen... |
| meetdm2 47912 | The meet of any two elemen... |
| meetdm3 47913 | The meet of any two elemen... |
| posjidm 47914 | Poset join is idempotent. ... |
| posmidm 47915 | Poset meet is idempotent. ... |
| toslat 47916 | A toset is a lattice. (Co... |
| isclatd 47917 | The predicate "is a comple... |
| intubeu 47918 | Existential uniqueness of ... |
| unilbeu 47919 | Existential uniqueness of ... |
| ipolublem 47920 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 47921 | The domain of the LUB of t... |
| ipolub 47922 | The LUB of the inclusion p... |
| ipoglblem 47923 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 47924 | The domain of the GLB of t... |
| ipoglb 47925 | The GLB of the inclusion p... |
| ipolub0 47926 | The LUB of the empty set i... |
| ipolub00 47927 | The LUB of the empty set i... |
| ipoglb0 47928 | The GLB of the empty set i... |
| mrelatlubALT 47929 | Least upper bounds in a Mo... |
| mrelatglbALT 47930 | Greatest lower bounds in a... |
| mreclat 47931 | A Moore space is a complet... |
| topclat 47932 | A topology is a complete l... |
| toplatglb0 47933 | The empty intersection in ... |
| toplatlub 47934 | Least upper bounds in a to... |
| toplatglb 47935 | Greatest lower bounds in a... |
| toplatjoin 47936 | Joins in a topology are re... |
| toplatmeet 47937 | Meets in a topology are re... |
| topdlat 47938 | A topology is a distributi... |
| catprslem 47939 | Lemma for ~ catprs . (Con... |
| catprs 47940 | A preorder can be extracte... |
| catprs2 47941 | A category equipped with t... |
| catprsc 47942 | A construction of the preo... |
| catprsc2 47943 | An alternate construction ... |
| endmndlem 47944 | A diagonal hom-set in a ca... |
| idmon 47945 | An identity arrow, or an i... |
| idepi 47946 | An identity arrow, or an i... |
| funcf2lem 47947 | A utility theorem for prov... |
| isthinc 47950 | The predicate "is a thin c... |
| isthinc2 47951 | A thin category is a categ... |
| isthinc3 47952 | A thin category is a categ... |
| thincc 47953 | A thin category is a categ... |
| thinccd 47954 | A thin category is a categ... |
| thincssc 47955 | A thin category is a categ... |
| isthincd2lem1 47956 | Lemma for ~ isthincd2 and ... |
| thincmo2 47957 | Morphisms in the same hom-... |
| thincmo 47958 | There is at most one morph... |
| thincmoALT 47959 | Alternate proof for ~ thin... |
| thincmod 47960 | At most one morphism in ea... |
| thincn0eu 47961 | In a thin category, a hom-... |
| thincid 47962 | In a thin category, a morp... |
| thincmon 47963 | In a thin category, all mo... |
| thincepi 47964 | In a thin category, all mo... |
| isthincd2lem2 47965 | Lemma for ~ isthincd2 . (... |
| isthincd 47966 | The predicate "is a thin c... |
| isthincd2 47967 | The predicate " ` C ` is a... |
| oppcthin 47968 | The opposite category of a... |
| subthinc 47969 | A subcategory of a thin ca... |
| functhinclem1 47970 | Lemma for ~ functhinc . G... |
| functhinclem2 47971 | Lemma for ~ functhinc . (... |
| functhinclem3 47972 | Lemma for ~ functhinc . T... |
| functhinclem4 47973 | Lemma for ~ functhinc . O... |
| functhinc 47974 | A functor to a thin catego... |
| fullthinc 47975 | A functor to a thin catego... |
| fullthinc2 47976 | A full functor to a thin c... |
| thincfth 47977 | A functor from a thin cate... |
| thincciso 47978 | Two thin categories are is... |
| 0thincg 47979 | Any structure with an empt... |
| 0thinc 47980 | The empty category (see ~ ... |
| indthinc 47981 | An indiscrete category in ... |
| indthincALT 47982 | An alternate proof for ~ i... |
| prsthinc 47983 | Preordered sets as categor... |
| setcthin 47984 | A category of sets all of ... |
| setc2othin 47985 | The category ` ( SetCat ``... |
| thincsect 47986 | In a thin category, one mo... |
| thincsect2 47987 | In a thin category, ` F ` ... |
| thincinv 47988 | In a thin category, ` F ` ... |
| thinciso 47989 | In a thin category, ` F : ... |
| thinccic 47990 | In a thin category, two ob... |
| prstcval 47993 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 47994 | Lemma for ~ prstcnid and ~... |
| prstcnid 47995 | Components other than ` Ho... |
| prstcbas 47996 | The base set is unchanged.... |
| prstcleval 47997 | Value of the less-than-or-... |
| prstclevalOLD 47998 | Obsolete proof of ~ prstcl... |
| prstcle 47999 | Value of the less-than-or-... |
| prstcocval 48000 | Orthocomplementation is un... |
| prstcocvalOLD 48001 | Obsolete proof of ~ prstco... |
| prstcoc 48002 | Orthocomplementation is un... |
| prstchomval 48003 | Hom-sets of the constructe... |
| prstcprs 48004 | The category is a preorder... |
| prstcthin 48005 | The preordered set is equi... |
| prstchom 48006 | Hom-sets of the constructe... |
| prstchom2 48007 | Hom-sets of the constructe... |
| prstchom2ALT 48008 | Hom-sets of the constructe... |
| postcpos 48009 | The converted category is ... |
| postcposALT 48010 | Alternate proof for ~ post... |
| postc 48011 | The converted category is ... |
| mndtcval 48014 | Value of the category buil... |
| mndtcbasval 48015 | The base set of the catego... |
| mndtcbas 48016 | The category built from a ... |
| mndtcob 48017 | Lemma for ~ mndtchom and ~... |
| mndtcbas2 48018 | Two objects in a category ... |
| mndtchom 48019 | The only hom-set of the ca... |
| mndtcco 48020 | The composition of the cat... |
| mndtcco2 48021 | The composition of the cat... |
| mndtccatid 48022 | Lemma for ~ mndtccat and ~... |
| mndtccat 48023 | The function value is a ca... |
| mndtcid 48024 | The identity morphism, or ... |
| grptcmon 48025 | All morphisms in a categor... |
| grptcepi 48026 | All morphisms in a categor... |
| nfintd 48027 | Bound-variable hypothesis ... |
| nfiund 48028 | Bound-variable hypothesis ... |
| nfiundg 48029 | Bound-variable hypothesis ... |
| iunord 48030 | The indexed union of a col... |
| iunordi 48031 | The indexed union of a col... |
| spd 48032 | Specialization deduction, ... |
| spcdvw 48033 | A version of ~ spcdv where... |
| tfis2d 48034 | Transfinite Induction Sche... |
| bnd2d 48035 | Deduction form of ~ bnd2 .... |
| dffun3f 48036 | Alternate definition of fu... |
| setrecseq 48039 | Equality theorem for set r... |
| nfsetrecs 48040 | Bound-variable hypothesis ... |
| setrec1lem1 48041 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 48042 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 48043 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 48044 | Lemma for ~ setrec1 . If ... |
| setrec1 48045 | This is the first of two f... |
| setrec2fun 48046 | This is the second of two ... |
| setrec2lem1 48047 | Lemma for ~ setrec2 . The... |
| setrec2lem2 48048 | Lemma for ~ setrec2 . The... |
| setrec2 48049 | This is the second of two ... |
| setrec2v 48050 | Version of ~ setrec2 with ... |
| setrec2mpt 48051 | Version of ~ setrec2 where... |
| setis 48052 | Version of ~ setrec2 expre... |
| elsetrecslem 48053 | Lemma for ~ elsetrecs . A... |
| elsetrecs 48054 | A set ` A ` is an element ... |
| setrecsss 48055 | The ` setrecs ` operator r... |
| setrecsres 48056 | A recursively generated cl... |
| vsetrec 48057 | Construct ` _V ` using set... |
| 0setrec 48058 | If a function sends the em... |
| onsetreclem1 48059 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 48060 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 48061 | Lemma for ~ onsetrec . (C... |
| onsetrec 48062 | Construct ` On ` using set... |
| elpglem1 48065 | Lemma for ~ elpg . (Contr... |
| elpglem2 48066 | Lemma for ~ elpg . (Contr... |
| elpglem3 48067 | Lemma for ~ elpg . (Contr... |
| elpg 48068 | Membership in the class of... |
| pgindlem 48069 | Lemma for ~ pgind . (Cont... |
| pgindnf 48070 | Version of ~ pgind with ex... |
| pgind 48071 | Induction on partizan game... |
| sbidd 48072 | An identity theorem for su... |
| sbidd-misc 48073 | An identity theorem for su... |
| gte-lte 48078 | Simple relationship betwee... |
| gt-lt 48079 | Simple relationship betwee... |
| gte-lteh 48080 | Relationship between ` <_ ... |
| gt-lth 48081 | Relationship between ` < `... |
| ex-gt 48082 | Simple example of ` > ` , ... |
| ex-gte 48083 | Simple example of ` >_ ` ,... |
| sinhval-named 48090 | Value of the named sinh fu... |
| coshval-named 48091 | Value of the named cosh fu... |
| tanhval-named 48092 | Value of the named tanh fu... |
| sinh-conventional 48093 | Conventional definition of... |
| sinhpcosh 48094 | Prove that ` ( sinh `` A )... |
| secval 48101 | Value of the secant functi... |
| cscval 48102 | Value of the cosecant func... |
| cotval 48103 | Value of the cotangent fun... |
| seccl 48104 | The closure of the secant ... |
| csccl 48105 | The closure of the cosecan... |
| cotcl 48106 | The closure of the cotange... |
| reseccl 48107 | The closure of the secant ... |
| recsccl 48108 | The closure of the cosecan... |
| recotcl 48109 | The closure of the cotange... |
| recsec 48110 | The reciprocal of secant i... |
| reccsc 48111 | The reciprocal of cosecant... |
| reccot 48112 | The reciprocal of cotangen... |
| rectan 48113 | The reciprocal of tangent ... |
| sec0 48114 | The value of the secant fu... |
| onetansqsecsq 48115 | Prove the tangent squared ... |
| cotsqcscsq 48116 | Prove the tangent squared ... |
| ifnmfalse 48117 | If A is not a member of B,... |
| logb2aval 48118 | Define the value of the ` ... |
| comraddi 48125 | Commute RHS addition. See... |
| mvlraddi 48126 | Move the right term in a s... |
| mvrladdi 48127 | Move the left term in a su... |
| assraddsubi 48128 | Associate RHS addition-sub... |
| joinlmuladdmuli 48129 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 48130 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 48131 | Join AB-CB into (A-C) on L... |
| mvlrmuld 48132 | Move the right term in a p... |
| mvlrmuli 48133 | Move the right term in a p... |
| i2linesi 48134 | Solve for the intersection... |
| i2linesd 48135 | Solve for the intersection... |
| alimp-surprise 48136 | Demonstrate that when usin... |
| alimp-no-surprise 48137 | There is no "surprise" in ... |
| empty-surprise 48138 | Demonstrate that when usin... |
| empty-surprise2 48139 | "Prove" that false is true... |
| eximp-surprise 48140 | Show what implication insi... |
| eximp-surprise2 48141 | Show that "there exists" w... |
| alsconv 48146 | There is an equivalence be... |
| alsi1d 48147 | Deduction rule: Given "al... |
| alsi2d 48148 | Deduction rule: Given "al... |
| alsc1d 48149 | Deduction rule: Given "al... |
| alsc2d 48150 | Deduction rule: Given "al... |
| alscn0d 48151 | Deduction rule: Given "al... |
| alsi-no-surprise 48152 | Demonstrate that there is ... |
| 5m4e1 48153 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 48154 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 48155 | Resolution rule. This is ... |
| testable 48156 | In classical logic all wff... |
| aacllem 48157 | Lemma for other theorems a... |
| amgmwlem 48158 | Weighted version of ~ amgm... |
| amgmlemALT 48159 | Alternate proof of ~ amgml... |
| amgmw2d 48160 | Weighted arithmetic-geomet... |
| young2d 48161 | Young's inequality for ` n... |
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